Springer Handbook of Electronic and Photonic Materials162-211

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					                                                                                                                                 137




                    Photoconduct
                         7. Photoconductivity in Materials Research


                                                            7.1    Steady State Photoconductivity
 Photoconductivity is the incremental change in
                                                                   Methods .............................................   138
 the electrical conductivity of a substance upon
                                                                   7.1.1  The Basic Single-Beam
 illumination. Photoconductivity is especially
                                                                          Experiment ..............................        138
 apparent for semiconductors and insulators,                       7.1.2  The Constant Photocurrent
 which have low conductivity in the dark.                                 Method (CPM) ...........................         141
 Significant information can be derived on the                      7.1.3  Dual-Beam Photoconductivity
 distribution of electronic states in the material and                    (DBP) .......................................    141
 on carrier generation and recombination processes                 7.1.4  Modulated Photoconductivity
 from the dependence of the photoconductivity                             (MPC) .......................................    141
 on factors such as the exciting photon energy,
 the intensity of the illumination or the ambient           7.2    Transient Photoconductivity
 temperature. These results can in turn be used                    Experiments........................................     142
 to investigate optical absorption coefficients or                  7.2.1  Current Relaxation
                                                                          from the Steady State ................           143
 concentrations and distributions of defects in the
                                                                   7.2.2 Transient Photoconductivity
 material. Methods involving either steady state
                                                                          (TPC) ........................................   143
 currents under constant illumination or transient
                                                                   7.2.3 Time-of-Flight Measurements
 methods involving pulsed excitation can be used                          (TOF) ........................................   144
 to study the electronic density of states as well as              7.2.4 Interrupted Field Time-of-Flight
 the recombination. The transient time-of-flight                           (IFTOF)......................................    145
 technique also allows carrier drift mobilities to be
 determined.                                                References .................................................. 146




Photoconductivity has traditionally played a significant     nation over time, will offer insights into the structure and
role in materials research, and most notably so in the      electronic properties of the material under investigation.
study of covalently bonded semiconductors and insu-         However, given the fact that three separate processes
lators. Indeed, since it is the incremental conductivity    are involved in the production of a specific photocur-
generated by the absorption of (optical) photons, photo-    rent, it follows that any analysis of experimental data
conductivity can be most clearly resolved in situations     in terms of system parameters will require a sufficiently
where the intrinsic dark conductivity of the material is    comprehensive data set that will allow for differentiation
low. This conductivity in the dark, leading to “dark cur-   between alternative interpretations. For instance, a low

                                                                                                                                 Part A 7
rent”, is due to the thermal equilibrium density of free    photocurrent may be the result of a low optical absorp-
carriers in the material and must be subtracted from        tion coefficient at the given photon energy, but it may
any measured current in order to obtain the actual pho-     also be due to significant geminate recombination of the
tocurrent. The basic processes that govern the magnitude    photogenerated electron–hole pairs, or it may reflect the
of the photocurrent are the generation of free electrons    formation of excitons. The combined use of different
and holes through the absorption of incident photons,       types of photoconductivity experiments is therefore of-
the transport of those free carriers through the material   ten advisable, as is the combination of photoconductivity
under the influence of an electric field, and the recombi-    with related experiments such as photoluminescence or
nation of the photoexcited electrons and holes. The study   charge collection.
of any of those aspects as a function of the characteris-       A wide variety of experimental techniques based on
tics of the current-inducing illumination, as well as the   photoconductivity have come into general use over the
study of their development upon changes in that illumi-     years. They can be divided into two main groups, one
  138        Part A   Fundamental Properties



                      involving steady state photoconductivity (SSPC), where             measurements. Recombination can be studied via TPC,
                      the focus is on stationary photocurrent levels, and a sec-         but the temperature dependence of SSPC can also be
                      ond one involving transient effects (TPC) where the                used to identify different recombination mechanisms,
                      time evolution of the photocurrent is studied. We will             while details of the electronic density of states (DOS)
                      use this division in our survey of the various methods,            in the band gap of a semiconductor can be inferred ei-
                      but we should point out that SSPC can also be meas-                ther from the spectral response of the SSPC or from
                      ured through ac excitation. The information that can be            a proper analysis of TPC. Detailed discussions of the
                      obtained about the material under investigation is gen-            general principles of photoconductivity may be found in
                      erally not specific to either the SSPC or TPC method                the standard monographs by Bube [7.1, 2], Ryvkin [7.3]
                      that is used, but will depend on the wider context of the          and Rose [7.4].



                      7.1 Steady State Photoconductivity Methods
                      7.1.1 The Basic Single-Beam Experiment                             a significant number of materials with widespread prac-
                                                                                         tical applications, one of either the product μn Δn or the
                      The simplest photoconductivity experiment uses a con-              product μp Δ p turns out to be much larger than the other
                      stant monochromatic light source to generate equal                 because of strongly unequal carrier mobilities. For in-
                      excess densities of free electrons and holes, Δn = Δ p,            stance, the electron term dominates in intrinsic silicon,
                      that lead to a change in the conductivity by
                          Δσ = σph = e(μn Δn + μp Δ p) ,                        (7.1)      a)                                            Light

                      where e is the electronic charge and μn and μp are
                      the electron and hole mobilities, respectively. The ba-                                                                    A
                                                                                                                     n = n0 + Δn
                      sic experimental arrangement is illustrated in Fig. 7.1a,                                      p = p0 + Δn
                      where L and A are the length and the cross-sectional
                      area of the sample and the photocurrent Iph corresponds                                             L
                      to σph AF, where F = V/L is the electric field applied.                                Ld + Iph     +        –
                      The end surfaces of the sample are covered by a metallic
                      electrode. However, since materials of current interest                                                 V
                      are often used in thin film rather than bulk form, in-                b)                                              Light
                      terdigitated electrodes of the type shown in Fig. 7.1b                    Thin film
                      are frequently used in actual measurement geometries.
                      In general, a fraction of the photogenerated carriers be-
                      comes immobilized by getting trapped at various defects                       V                                        Substrate
                      such that not every part of Δn and Δ p contributes                                               Ld + Iph
                      equally to the photoconductivity in (7.1). The effect
                      of such trapping on the photoconductivity is reflected                c)
Part A 7.1




                      in the use of values for the mobilities μn and μp that               G
                      are lower – and not necessarily symmetrically lower –
                      than the theoretical free-carrier mobility μ0 . In fact, for                      Illumination

                      Fig. 7.1 (a) Basic arrangement for photoconductivity mea-
                      surements, with V the applied voltage, L the sample length
                                                                                           Δn
                      and A the cross-sectional area. Id , n 0 and p0 are the current
                      and the carrier densities in the dark, and Iph , Δn, Δ p are the
                      incremental values caused by the illumination. (b) Example                                τG
                      of interdigitated electrode configuration for a thin film sam-
                      ple. (c) Schematic time development of the excess carrier                 0                                 toff               Time t
                      concentration Δn in response to a period of illumination
                                          Photoconductivity in Materials Research     7.1 Steady State Photoconductivity Methods   139



while the photocurrent in chalcogenide glasses is carried     simplicity, the frequently encountered case of photo-
by holes. In those instances, (7.1) effectively reduces to    conductivity dominated by one type of carrier (known
a one-carrier equation.                                       as the majority carrier), and assuming electrons to be
    In the μn Δn or μp Δ p products, the mobility μi          the majority carrier, the recombination rate can be writ-
is a material parameter that, in general, will depend on              −1
                                                              ten as τn = b( p0 + Δ p), where b is a recombination
temperature and sample characteristics, while the excess      constant, and p0 and Δ p are the equilibrium and ex-
carrier density Δn = Δ p is determined by a combination       cess minority carrier densities. It then follows that the
of material and external parameters. Phenomenolog-            photoconductivity
ically, the excess density Δn can be written as the
product Gτi , where G is the rate of generation of free           σph ∝ Δn = Gτn = G/b( p0 + Δ p)
electrons and holes per unit volume, and τi is the av-                     = G/b( p0 + Δn) .                          (7.5)
erage lifetime of the excess carrier. Introducing these
quantities into (7.1) leads to the form                       Equation (7.5) indicates that a linear relationship
                                                              σph ∝ G holds for Δn         p0 (a low excess carrier den-
   σph = eG(μn τn + μp τp ) ,                         (7.2)   sity), while high excitation levels with Δn        p0 lead to
which explicitly displays the mobility–lifetime products      σph ∝ G 1/2 . These linear and quadratic recombination
that are frequently used to characterize photoconductors.     regimes are also referred to as mono- and bimolecular
The relationship between the steady state values of Δn        recombination. For a given light source and tempera-
and G is illustrated in Fig. 7.1c, where the build-up and     ture, variations in G correspond to variations in the light
                                                                                                     γ
decay of Δn when the illumination is turned on and            intensity I0 , and therefore σph ∝ I0 with 1/2 ≤ γ ≤ 1.
turned off are also shown. Those time-dependent aspects       The value of γ itself will of course depend on the light
of photoconductivity will be addressed in a later section.    intensity I0 . However, I0 is not the only factor that deter-
    The generation rate G is defined by                        mines the value of γ : intermediate γ values may indicate
                                                              a Δn ≈ p0 condition, but they may equally be caused
    G = η(I0 /hν)(1 − R)[1 − exp(−αd)]/d ,            (7.3)   by a distribution of recombination centers, as outlined
                                                              below [7.4].
where η is the quantum efficiency of the generation pro-            From a materials characterization point of view,
cess, I0 is the incident illumination intensity (energy per   SSPC offers the possibility of using the above equations
unit time and unit area), hν is the photon energy, R is the   to determine the absorption coefficient as a function
reflection coefficient of the sample, α is the optical ab-      of the energy of the incoming photons, and thus ex-
sorption coefficient of the material, and d is the sample      plore the electronic density of states around the band
thickness. A quantum efficiency η < 1 signifies that, due       gap of a semiconductor. When single-crystalline sam-
to geminate recombination of the carriers or of exciton       ples of materials with sufficiently well-defined energy
formation, not every absorbed photon generates a free         levels are studied, maxima corresponding to specific op-
electron and hole that will contribute to the photocur-       tical transitions may be seen in the photoconductivity
rent. The values of the parameters η, R and α depend,         spectra. A recent example, involving the split valence
in general, on the wavelength of the illuminating light.      band of a p-CdIn2 Te4 crystal, may be found in You
Consequently, monochromatic illumination from a tun-          et al. [7.5]. Another example is given in Fig. 7.2, where
able light source can be used to obtain energy-resolved       the spectral distribution of the photocurrent is shown

                                                                                                                                   Part A 7.1
information about the sample, while illumination with         for optical-quality diamond films prepared by chem-
white light will only offer a global average. Under many      ical vapor deposition [7.6]. The rise in photocurrent
experimental circumstances, the condition αd 1 will           around 5.5 eV corresponds to the optical gap of dia-
hold over a significant energy range (when the sample          mond, while the shoulders at ≈ 1.5 eV and ≈ 3.5 eV
thickness is small with respect to the optical absorp-        signal the presence of defect distributions in the gap.
tion depth of the material). Equation (7.3) can then be       The data in Fig. 7.2 were obtained under ac conditions
simplified to                                                  using chopped light and a lock-in amplifier. The changes
    G ∼ η(I0 /hν)(1 − R)α .
      =                                               (7.4)   in the observed phase shift can then also be used to locate
                                                              the energies at which transitions to specific features of
The free-carrier lifetimes of the excess electrons and        the density of states (DOS) become of importance. The
holes, τn and τp , in (7.2) are governed by recombi-          use of ac excitation and lock-in detection has the added
nation with carriers of opposite sign. Assuming, for          advantage of strongly reducing uncorrelated noise, but
  140        Part A      Fundamental Properties



                                                                                                            Fig. 7.2 Room temperature ac photocurrent spectra, meas-
                                      Phase shift (deg)                                                     ured at 7 Hz, after various treatments of CVD diamond
                               120
                                                                  S2-Oxidized                               layers deposited at 920 ◦ C (S2) and 820 ◦ C (S3) (after [7.6])
                                90
                                                                                                            cess, depend on the temperature through the ap-
                                60
                                                                  S2-Hydrogenated                           proximate Fermi–Dirac occupation probability func-
                                30                                                                          tion exp[(E − E F )/kT ], thus making recombination
                                                                  S2-As grown                               a temperature-dependent process. In photoconductors,
                                  0                                                                         recombination is mediated by carrier traps in the
                                      0          1          2           3           4        5       6
                                                                                    Photon energy (eV)
                                                                                                            bandgap. The presence of discrete trapping levels leads
                                      Photocurrent (A)                                                      to thermally activated photocurrents, with the activa-
                             10– 4
                                                                                                            tion energy indicating the energetic positions of the
                                                                                                            traps. Main and Owen [7.8] and Simmons and Tay-
                                                       S2-As grown
                             10– 6                                                                          lor [7.9] showed that the positive photocurrent activation
                                                                              S2-Hydrogenated
                                                                                                            energy in the monomolecular recombination regime cor-
                                –8
                                                                                                            responds to the distance above the Fermi level of a donor-
                             10                                                                             like center, while a negative activation energy value
                                                                              S2-Oxidized
                                                                                                            in the bimolecular region refers to the energy position
                             10–10                                                   S3-Oxidized            above the valence band edge of an acceptor-like cen-
                                                                                                            ter. Figure 7.3 illustrates this photocurrent behavior for
                                                                                                            amorphous As2 Se3 [7.7]. The above pattern is character-
                             10– 12                                                                         istic of chalcogenide glasses, where the intrinsic charged
                                                                                                            defects with negative effective correlation energy act as
                                                                                                            recombination centers [7.10]. SSPC measurements can
                             10– 14
                                      0          1          2           3           4        5       6      thus determine the recombination levels of those defects.
                                                                                    Photon energy (eV)           In highly photosensitive materials, such as selenium
                                                                                                            or hydrogenated amorphous silicon (a-Si:H), measure-
                         care must be taken to ensure that the ac frequency re-                             ments in the monomolecular region are hindered by the
                         mains lower than the response rate of the investigated                             problem of satisfying the Δn        p0 condition. In addi-
                         system over the spectral range of interest.                                        tion, the SSPC temperature dependence in a-Si:H does
                             The equilibrium free-carrier densities n 0 and                                 not exhibit a definite activation energy due to the pres-
                         p0 , which play a role in the recombination pro-                                   ence of a more distributed and complex set of traps that
                                                                                                            even induce regions of superlinear dependence on light
                                                                                                            intensity [7.11]. This illustrates that SSPC analysis is
               a)                                                b)
                       I (A)                                          I (A)                                 not necessarily straightforward.
                                                                                                                 Whenever the electronic density of states in the band
              10– 8
                                                                                                            gap of a photoconductor consists of a distribution of traps
                                                                                                            (as is the case in amorphous materials), a quasi-Fermi
                                                                                                            level E qF = E F − kT ln(1 + Δn/n), linked to the excess
Part A 7.1




                                                                                                            carrier density, can be defined. This quasi-Fermi level
                                                                                      ΔEb = 0.30 eV         will – to a first approximation – correspond to the de-
              10– 9

                                                                                                            Fig. 7.3a,b Temperature dependence of the steady state
                                                                                                            dark and photocurrents in an a-As2 Se3 bulk sample, il-
                                                                                                            luminated at 1.55 eV with intensities of 0.84, 3.5, 9.8,
              10– 10                                                                                        38 and 120 × 1012 photons/cm2 s (a), and illuminated
                       ΔEm = 0.17 eV                   Id
                                                                                                            at 1.85 eV with intensities of 0.56, 1.7, 4.6, 27 and
                                                Eo = 0.88 eV                                                77 × 1012 photons/cm2 s (b). ΔE m and ΔE b represent the
                                                                                                            photocurrent activation energies in the monomolecular and
                       2.6      2.8       3.0     3.2 3.4         2.6         2.8    3.0    3.2 3.4
                                                                                                            bimolecular recombination regimes respectively, and E σ is
                                                  103 /T (K–1)                              103 / T (K–1)
                                                                                                            the activation energy of the dark current Id (after [7.7])
                                          Photoconductivity in Materials Research     7.1 Steady State Photoconductivity Methods        141



marcation level that divides the DOS into a shallower
                                                                                     Mono             Beam       Sample
part where carriers will be trapped and subsequently                                 chromator        splitter
                                                                   Lamp
re-emitted and a deeper part where traps have become                                                                         D2
recombination centers. In other words, varying the light
intensity influences both carrier generation and recom-
bination rates. When several trapping centers with quite                  Chopper
different characteristics are present in the photoconduc-
tor, shifts in the positions of the quasi-Fermi levels can                                       D1
                                                        γ
then produce unexpected results. Instances of σph ∝ I0 ,                                                                  Electronic
with γ > 1 (as referred to above) will be observed                                                                        circuits
for some materials, while combinations that actually
produce negative photoconductivity, σph < 0, have also        Fig. 7.4 Schematic diagram of an ‘absolute’ CPM set-up. Photode-
been encountered [7.4].                                       tector D1 is used to regulate the intensity of the lamp, while detector
                                                              D2 measures the transmitted light (after [7.12])
7.1.2 The Constant Photocurrent Method
      (CPM)                                                   diode. Main et al. [7.14] showed that, in the dc mode,
                                                              transitions involving initially unoccupied DOS levels
The constant photocurrent method (CPM) has been used          raise the absorption above the value that is seen with the
        eˇ
by Vanˇ cek and coworkers [7.12, 13] to determine the         ac technique. Systematic comparison of dc and ac results
optical absorption coefficient as a function of photon         allows us, therefore, to distinguish between occupied
energy, α(E), via (7.2–7.4). In CPM, the photocurrent         states below the operative Fermi level and unoccupied
is kept constant by continually adjusting the light in-       ones above it. In cases where the quantum efficiency of
tensity I0 while the photon energy is scanned across          carrier generation η can be taken as unity, CPM gives
the spectrum. The constant photocurrent implies that the      α(E) directly as 1/I0 , and this method is widely used, for
quasi-Fermi levels have immobile positions and thus that      example for hydrogenated amorphous silicon. However,
the free-carrier lifetime is a constant, τ. It then follows   for materials such as chalcogenide glasses or organic
that in                                                       semiconductors where η itself is energy-dependent, it is
   σph = eμτ(I0 /hν)(1 − R)ηα                         (7.6)
                                                              only the product ηα that is readily obtained.

the product (I0 /hν)α will remain constant, and that α        7.1.3 Dual-Beam Photoconductivity (DBP)
can be determined from it, provided that any energy
dependencies for the parameters μ, R and η of (7.6) are       Like the CPM discussed above, the dual-beam pho-
negligible. The value at which the photocurrent is fixed       toconductivity (DBP) technique is used to determine
can be chosen freely, but will in practice be dictated by     the sub-bandgap optical absorption in a photoconduc-
the low-absorption region of the sample. However, since       tor. A constant, uniformly absorbed illumination I0 is
even low-level photocurrents can still be measured with       used to establish a constant excess carrier density in
high precision, the method is especially useful at low        the material, and hence a constant free-carrier lifetime
values of optical absorption where standard transmission      τ. The chopped signal I (E) of a low-intensity, tunable
measurements lose their accuracy.                             light source is added to this background to generate

                                                                                                                                        Part A 7.1
    In ‘absolute’ CPM, the optical transmission through       variations in photoconductivity δσph (E). Synchronous
the film is measured at the same time as the photocurrent,     lock-in detection of the small ac signal then provides
and the data from the two measurements are combined in        the information needed to deduce α(E). By carrying out
order to remove optical interference fringes from the data    measurements at different values of the background il-
and to fix the value of the proportionality constant [7.12].   lumination intensity, DBP allows the photoconductor
The experimental arrangement used in such absolute            absorption to be tested for changing quasi-Fermi level
CPM measurements is shown schematically in Fig. 7.4.          positions. Changes in the resolved α(E) curves can then
The CPM experiment can be operated with either dc             be used to obtain information on the DOS distribution
or ac illumination, but the absorption spectra retrieved      in the sample. An example of this use of DBP can be
will not be identical. AC illumination can be obtained        found in Günes et al. [7.15], where differences in absorp-
using a mechanical chopper (as suggested in Fig. 7.4),        tion between annealed and light-soaked hydrogenated
but also, for instance, from an ac-driven light-emitting      amorphous silicon samples are studied.
  142        Part A   Fundamental Properties



                      7.1.4 Modulated Photoconductivity (MPC)                         where Φ and Iac are the phase and intensity of the
                                                                                      ac photocurrent, kB is the Boltzmann constant, T the
                      The experimental technique that has become known as             temperature, ν0 the attempt-to-escape frequency and
                      modulated photoconductivity (MPC) is used to deter-             ω the modulation frequency. At the low-frequency
                      mine the energetic distribution of states in the bandgap of     end, recombination and trapping in deep states deter-
                      a photoconductor by analyzing the phase shift between           mine the phase shifts and the DOS varies according
                      ac photoexcitation and the ensuing ac photocurrent              to tan(Φ)/ω. The transition between the two re-
                      as a function of the modulation frequency of the                gions is tied to the position of the quasi-Fermi
                      light [7.17,18]. Figure 7.5 shows the essential parts of an     levels and can, therefore, be shifted by changing the
                      MPC set-up, and illustrates the phase difference between        illumination intensity. MPC works best with pho-
                      the illumination and the photocurrent. Two modulation           toconductors where one carrier type dominates the
                      frequency ranges with distinct characteristics are identi-      current, and therefore only one side of the bandgap
                      fied. In the high-frequency region, from a few Hz up to          needs be taken into account in the analysis. Exam-
                      the kHz range, the signal is dominated by carrier release       ples of MPC-determined DOS profiles are given in
                      from traps, with a release rate that matches the modu-          Fig. 7.6 [7.16]. The figure shows a profile for the con-
                      lation frequency. The usual assumption, that the release        duction band side of the bandgap of an as-deposited
                      probability decreases exponentially with the trap depth         polymorphous silicon sample, as well as those for the
                      according to r ∝ exp(−E/kT ), gives the link between            sample following light soaking and after subsequent
                      the measured phase shift and the DOS of the material.           annealing.
                      The relationship between the two is expressed by
                            g(E) ∝ sin(Φ)/Iac , E = kB T ln(ν0 /ω) ,          (7.7)         19
                                                                                                 N (E) (cm–3 eV–1)
                                                                                       10


               Sinusoidal             Photoconductor
               driver                                                                  1018                                            Annealed
                            Emitter                             Iph                                                                    66 h (420 K)
                                                                                                                                       + 139 h (460 K)
                                                                                       1017
                                                                V

                                                                                       1016           As depos.
               Light intensity                         Photocurrent
                                                                                                      Light soaked
                                                                                                      Ann. 66 h (420 K)
                                                                                       1015
                                                                                          0.1         0.2    0.3     0.4   0.5   0.6    0.7      0.8   0.9
                                                                                                                                              Ec – E(eV)
               0                              t        0                       t
                                                                                      Fig. 7.6 DOS below the CB edge of a polymorphous sil-
             Fig. 7.5 Schematic diagram of an MPC set-up (upper frames), and          icon sample deposited at 423 K and measured (by MPC)
             of the phase relationship between the exciting light intensity and the   as-deposited, after light soaking, and after two stages of
             resulting photocurrent (lower frames)                                    subsequent annealing (after [7.16])
Part A 7.2




                      7.2 Transient Photoconductivity Experiments
                      The study of transient aspects of photoconductivity can         easier to analyze. Nevertheless, a simple exponential
                      relate to either the build-up or the relaxation of steady       decay of the photocurrent, as sketched in Fig. 7.1c, will
                      state photocurrents, or to a material’s response to pulsed      only be observed when a unique recombination path is
                      excitation. While the SSPC turn-on transient reflects the        followed, a situation that is the exception rather than
                      interplay between generation and recombination of car-          the rule. Transient photoconductivity (TPC) caused by
                      riers (an interplay that often leads to a current overshoot     pulsed excitation is generally simpler to analyze. Indeed,
                      at high excitation levels), the SSPC relaxation upon            whereas a quasi-equilibrium distribution of trapped pho-
                      turn-off only involves recombination and is therefore           togenerated carriers will build up or be present in the
                                         Photoconductivity in Materials Research     7.2 Transient Photoconductivity Experiments   143



photoconductor’s bandgap under SSPC, the TPC ex-              they will have been immobilized a number of times by
periments can be analyzed against the background of           various traps that are present in the material. Since the
the thermal equilibrium distribution of carriers in the       carrier distributions are in thermal equilibrium at the
material.                                                     start of the experiment, both the trapping sites for elec-
                                                              trons above the Fermi level and the hole trapping sites
7.2.1 Current Relaxation                                      below EF are empty, such that the newly created carriers
      from the Steady State                                   are not excluded from any of those trapping sites. Given
                                                              that carrier release from a trap is a thermally activated
Upon termination of steady state illumination, the gen-       process with the trap depth being the activation energy,
eration term drops out of the rate equation that describes    deeper traps immobilize carriers for longer times and
the nonequilibrium carrier distribution, but the carrier      lead to lower values for the transient current. As shal-
density itself and the operative recombination process        lower states release trapped carriers sooner, retrapping
are not altered. Consequently, the initial photocurrent de-   of those carriers will lead to increased occupation of the
cay will be governed by whatever recombination mode           deeper states and further reduction of the current level.
existed under SSPC conditions. Spectroscopic analysis         To allow this thermalization of the excited carriers to
of the relaxation current in terms of the distribution of     run its full course until recombination sets in, the ex-
states in the bandgap can be readily achieved in the case     periments are traditionally carried out in the so-called
of monomolecular recombination [7.19], with the prod-         secondary photocurrent mode, whereby the sample is
uct of photocurrent and time being proportional to the        supplied with ohmic electrical contacts and carrier loss
DOS:                                                          is by recombination only. Coplanar electrode geome-
    Iph (t)t ∝ g(E) ,   E = kB T ln(ν0 t) .           (7.8)   tries (gap cells) are mostly used. Expressions that link
                                                              the transient current to the distribution of localized states
In (7.8), kB T is the Boltzmann energy and ν0 is the          can be derived [7.20], but they are difficult to invert in
attempt-to-escape frequency. When, on the other hand,         the general case. Nevertheless, as long as recombination
bimolecular recombination dominates, the link between         can be neglected, the relationship g(E) ∝ [I(t)t]−1 can
the current and the distribution of recombination centers     be used as a first-order estimate.
is much less direct and spectroscopic analysis is difficult.       For the special case of an exponential DOS, the
Unfortunately, bimolecular recombination is dominant          solution is straightforward: a g(E) ∝ exp(−E/E 0 ) dis-
in good photoconductors.                                      tribution of trapping levels leads to a power law for
    In spite of the above, relaxation of the steady state     the transient current I (t) ∝ t −(1−α) with α = kB T/E 0 .
current has often been used to obtain a first-order esti-      In other words, the width of the exponential distribu-
mate for free-carrier lifetimes, even when this had to be     tion E 0 can be deduced from the slope of the power
done on a purely phenomenological basis due to a lack         law decay of the current. Essentially exponential distri-
of sufficient information on the recombination mecha-          butions were found to dominate the valence band tail
nisms involved. An exponential fit to the initial part of      of equilibrated amorphous As2 Se3 samples over a wide
the decay is then often used to make the estimate. In         energy range [7.7], but no other examples have emerged.
cases where more than one – sometimes vastly different            An elegant way to circumvent the difficulties posed
– recombination mechanisms are operative, this initial        by a time domain analysis of the transient current is to
decay does not necessarily represent the most significant      transpose the current decay into the frequency domain

                                                                                                                                   Part A 7.2
proportion of carriers. This is certainly the case when-      by a Fourier transform [7.21]. Since the TPC current
ever so-called persistent photoconductivity is observed;      decay is the photoconductor’s response to an impulse
one of the relaxation times involved is then longer than      excitation, its Fourier transform gives the frequency re-
the observation time.                                         sponse I (ω) of that photoconductor. In fact, this I (ω)
                                                              corresponds to the photocurrent intensity Iac as used in
7.2.2 Transient Photoconductivity (TPC)                       the MPC method, and the same procedures can thus be
                                                              used to extract the information on density and energy
In the standard transient photoconductivity (TPC) exper-      distribution of localized states in the band gap. Not just
iment, free carriers are excited into the transport band at   Fourier transform but also Laplace transform techniques
time t = 0 by a short light pulse. They are then moved        have been applied to the conversion of TPC signals into
along by the electric field until their eventual disappear-    DOS information. A comparison and discussion of the
ance through recombination, but before this happens           results may be found in [7.22]. Examples of Fourier
  144        Part A   Fundamental Properties



                              DOS (relative)                                                                                Pulsed laser
                       1020

                       1019

                       1018                                                                                +++++++
                                               FT
                                                                                                                           L Sample
                       1017                                                                     Va
                                               HFT                                                                               v (t)
                       1016                                                                  Pulsed                                                       tT
                                                                                             bias         I (t)
                       1015                                                                                            R
                          0.2          0.3     0.4      0.5       0.6         0.7
                                                                    Ec – E (eV)
                      Fig. 7.7 DOS below the conduction band edge in a-Si:H,
                      obtained through Fourier transforms of the transient pho-     Fig. 7.8 TOF measurement set-up shown for the case of
                      tocurrent; HFT: the high-resolution analysis of [7.22], FT:   holes being drifted through the length L of the sample
                      the earlier analysis according to [7.21]                      by a positive applied voltage. Choosing the resistance R
                                                                                    that generates an output voltage to be low minimizes RC
                      transform TPC analysis, as originally proposed and as         distortion of the signal at short times; choosing it to be high
                      developed since, are shown in Fig. 7.7 for an a-Si:H          enhances the detectability of weak signals at the expense of
                      sample. Whereas the energy range that can be probed is        time resolution
                      limited in MPC by the frequency range of the lock-in
                      amplifier, it is the smallest resolution time of the detec-    tT (the time needed for the charge sheet to cross the sam-
                      tion system that limits the range in the case of TPC, the     ple), the drift mobility μd can be calculated according
                      latter one being generally more advantageous.                 to μd = L/tT F, where L is the sample length and F the
                                                                                    applied electric field. The essential elements of a TOF
                      7.2.3 Time-of-Flight Measurements (TOF)
                                                                                           Photocurrent (arb. units)
                      The time-of-flight (TOF) experiment, originally de-             2.0
                      signed to determine the drift mobility of free carriers                                          Me R R'               Me R R'
                      in high-mobility materials, has been highly successfully
                      adaptated to low-mobility materials such as organic or
                                                                                                                                                 n
                      amorphous semiconductors [7.24], where it has been             1.5                                    R' Me R                  R'
                      used for drift mobility measurements but also as an al-                                                    R:              n-decyl
                      ternative TPC technique to study the energy distribution
                                                                                                                                 R': n-hexyl
                      of localized states. While majority carriers will dominate
                      photocurrents in traditional TPC, TOF allows indepen-          1.0
                      dent measurements with majority and minority carriers,
                      and thus independent examinations of the valence band
                      side and conduction band side of the band gap.
Part A 7.2




                          For TOF measurements, the sample consists of               0.5
                      a layer of the photoconducting material sandwiched be-
                      tween two electrodes that are blocking carrier injection
                      into the sample. At least one of the electrodes must be
                                                                                     0.0
                      semitransparent to permit the photoexcitation of free                    0                  1                      2                           3
                      carriers in the material just beyond the illuminated con-                                                                                t / ttr
                      tact by a strongly absorbed light flash. Depending on the      Fig. 7.9 Time-of-flight transients measured at 243 K
                      polarity of the electric field applied across the sample,      in methyl-substituted ladder-type poly(para)phenylene
                      either electrons or holes will then be drifted through the    (MeLPPP) with 60 kV/cm (line) and 300 kV/cm (•) ap-
                      sample. At their arrival at the back contact, the current     plied, and normalized to a transit time set to 90% of the
                      will drop since the blocking contact ensures that only the    pre-transit current. The inset shows the chemical structure
                      primary photocurrent is measured. From the transit time       of MeLPPP (after [7.23])
                                             Photoconductivity in Materials Research    7.2 Transient Photoconductivity Experiments   145



                                                                 and the information about the distribution of gap states
         I (A)
                                                                 g(E) that is contained in the current transient can
 10– 4                                                 10 V      be extracted in the same ways. Both pre-transit cur-
                                                                 rent transients and measured drift mobility values have
                                                                 been employed in the past to estimate the DOS in the
 10– 5                                                           band tails of disordered semiconductors. In the lat-
                                                                 ter case, specific g(E) functions are explored through
 10– 6                                                           trap-controlled transport modeling to reproduce the ex-
                                                                 perimental dependence of μd on the temperature and
            Temperature (°C)                                     the electric field. This technique has since been replaced
   –7                48
 10
                     24
                                                                 by the more direct procedures described in preceding
                      5                                          sections.
                   – 13                                              At times longer than the TOF transit time, a steeper
 10– 8             – 30                                          current decay testifies to the fact that carriers are leaving
         10– 7     10– 6       10–5   10–4      10–3             the sample. The post-transit current that is then ob-
                                                         t (s)
                                                                 served is increasingly due to the emission of carriers
Fig. 7.10 Example of TOF hole transients measured at sev-        that were trapped in states deep in the bandgap. Pro-
eral temperatures, as indicated, with 10 V applied across a      vided that the conditions are such that the probability of
5.6 μm-thick a-Si:H sample grown in an expanding ther-           subsequent deep retrapping of the same carriers is negli-
mal plasma at 0.85 nm/s and 250 ◦ C substrate temperature,       gible, a proper analysis of these post-transit TOF current
and sandwiched between Mo contacts (after [7.25])                transients permits the elucidation of the distribution of
                                                                 localized states deeper in the gap [7.27] with, as in (7.8),
measuring circuit are displayed in Fig. 7.8. The transit         g(E) ∝ I(t)t expressing the correspondence.
time can be measured directly on the current trace, in
which case it is variously defined as the time at which           7.2.4 Interrupted Field Time-of-Flight
the current has dropped by values ranging from 10% to                  (IFTOF)
50% (the latter one being most commonly used), or it
can be obtained by integrating the current and using the         The interrupted field time-of-flight (IFTOF) experiment
time at which the collected charge saturates. Obtaining          differs from the time-of-flight experiment described
a true value of μd requires that the field F be uniform
and constant during the carrier transit, which means that
                                                                   Photocurrent (100 μA / Div)
F should only be applied a short time before the optical
excitation and that the transit time should be short with           a)
respect to the dielectric relaxation time in the material.
Figure 7.9 shows TOF transients in a conjugated poly-
mer whereby a 10% drop is used to define the transit
time.
    In materials with a wide distribution of localized
gap states, as is generally the case in disordered pho-                           j1

                                                                                                                                      Part A 7.2
toconductors, the drifting charge package spreads out               b)
along the length of the sample, and a representative tran-                                                 T1'
sit time can only be discerned as a change of slope                                              j2
in a double-logarithmic plot of current versus time.
The curves in Fig. 7.10 (from [7.25]) illustrate such                                      ti
behavior. Measurements at different temperatures and
applied fields are then needed to ascertain that the ob-                  O   T1                       T2
served feature marks an actual carrier transit rather
than deep trapping of the photogenerated charge. In                                                         Time (5 μS / Div)
the materials that exhibit this anomalously dispersive           Fig. 7.11a,b Comparison of current traces in TOF (a) and
transport, the pre-transit current will have the charac-         IFTOF (b) experiments. The applied electric field is turned
teristics of the TPC described in the previous section,          off in case (b) for a length of time ti (after [7.26])
 146       Part A   Fundamental Properties



                    in the previous section in that the applied field that                the sample, recombination parameters can be studied
                    drives the photogenerated carrier packet through the                 too [7.29].
                    sample is turned off for some period of time before                      Another interesting method for studying the recom-
                    the carriers have completed their transit. As illustrated            bination process is – just like IFTOF – based on a simple
                    in Fig. 7.11, a lower current intensity is measured                  modification of the TOF experiment: after generating
                    when the field is turned on again, signalling that                    free carriers through one contact and drifting the slower
                    some of the drifting carriers have become immobi-                    type of carrier into the sample, a second light pulse
                    lized in deep traps [7.28]. By studying the drop in                  through the other contact sends a sheet of oppositely
                    current as a function of the interruption time ti , the              charged carriers towards the first one. The two carrier
                    deep-trapping lifetime of the carriers can be evaluated.             packages will cross and some electrons and holes will
                    Recombination can be routinely neglected in TOF ex-                  recombine during that crossing, thereby affecting the
                    periments since only one type of carrier drifts through              observed current levels and providing a way to study the
                    the sample, but by charging a sample with carriers                   recombination process. An elegant example of the ap-
                    of one polarity before performing an IFTOF experi-                   plication of this technique to amorphous selenium can
                    ment that drifts carriers of the opposite polarity through           be found in Haugen and Kasap [7.30].


                    References

                    7.1    R. H. Bube: Photoconductivity of Solids (Wiley, New           7.15   M. Günes, C. Wronski, T. J. McMahon: J. Appl. Phys.
                           York 1960)                                                           76, 2260 (1994)
                    7.2    R. H. Bube: Photoelectronic Properties of Semicon-            7.16   C. Longeaud, D. Roy, O. Saadane: Phys. Rev. B 65,
                           ductors (Cambridge Univ. Press, Cambridge 1992)                      85206 (2002)
                    7.3    S. M. Ryvkin: Photoelectric Effects in Semiconductors         7.17   H. Oheda: J. Appl. Phys. 52, 6693 (1981)
                           (Consultants Bureau, New York 1964)                           7.18   R. Brüggemann, C. Main, J. Berkin, S. Reynolds:
                    7.4    A. Rose: Concepts in Photoconductivity and Allied                    Philos. Mag. B 62, 29 (1990)
                           Problems (Krieger, Huntington 1978)                           7.19   M. S. Iovu, I. A. Vasiliev, E. P. Colomeico, E. V. Emelia-
                    7.5    S. H. You, K. J. Hong, T. S. Jeong, C. J. Youn, J. S. Park,          nova, V. I. Arkhipov, G. J. Adriaenssens: J. Phys.
                           D. C. Shin, J. D. Moon: J. Appl. Phys. 95, 4042 (2004)               Condens. Mat. 16, 2949 (2004)
                    7.6    M. Nesládek, L. M. Stals, A. Stesmans, K. Iak-                7.20   A. I. Rudenko, V. I. Arkhipov: Philos. Mag. B 45, 209
                           oubovskii, G. J. Adriaenssens, J. Rosa, M. Vanˇˇek:   ec             (1982)
                           Appl. Phys. Lett. 72, 3306 (1998)                             7.21   C. Main, R. Brüggemann, D. P. Webb, S. Reynolds:
                    7.7    G. J. Adriaenssens: Philos. Mag. B 62, 79 (1990) and                 Solid State Commun. 83, 401 (1992)
                           references therein                                            7.22   C. Main: J. Non-Cryst. Solids 299, 525 (2002)
                    7.8    C. Main, A. E. Owen: In: Electronic and Structural            7.23   D. Hertel, A. Ochse, V. I. Arkhipov, H. Bässler: J. Imag.
                           Properties of Amorphous Semiconductors, ed. by                       Sci. Technol. 43, 220 (1999)
                           P. G. Le Comber, J. Mort (Academic, London 1973)              7.24   W. E. Spear: J. Non-Cryst. Solids 1, 197 (1969)
                           p. 527                                                        7.25   M. Brinza, E. V. Emelianova, G. J. Adriaenssens: Phys.
                    7.9    J. G. Simmons, G. W. Taylor: J. Phys. C 7, 3051 (1974)               Rev. B 71, 115209 (2005)
                    7.10   G. J. Adriaenssens, N. Qamhieh: J. Mater. Sci. Mater.         7.26   S. Kasap, B. Polishuk, D. Dodds, S. Yannacopoulos:
                           El. 14, 605 (2003)                                                   J. Non-Cryst. Solids 114, 106 (1989)
Part A 7




                    7.11   H. Fritzsche, B.-G. Yoon, D.-Z. Chi, M. Q. Tran: J.           7.27   G. F. Seynhaeve, R. P. Barclay, G. J. Adriaenssens,
                           Non-Cryst. Solids 141, 123 (1992)                                    J. M. Marshall: Phys. Rev. B 39, 10196 (1989)
                    7.12   M. Vanˇˇek, J. Koˇka, A. Poruba, A. Fejfar: J. Appl.
                                   ec           c                                        7.28   S. Kasap, B. Polishuk, D. Dodds: Rev. Sci. Instrum.
                           Phys. 78, 6203 (1995)                                                61, 2080 (1990)
                    7.13   M. Vanˇˇek, J. Koˇka, J. Stuchlík, A. Tˇíska: Solid State
                                   ec         c                     r                    7.29   S. Kasap, B. Fogal, M. Z. Kabir, R. E. Johanson,
                           Commun. 39, 1199 (1981)                                              S. K. O’Leary: Appl. Phys. Lett. 84, 1991 (2004)
                    7.14   C. Main, S. Reynolds, I. Zrinˇˇak, A. Merazga: Mater.
                                                          sc                             7.30   C. Haugen, S. O. Kasap: Philos. Mag. B 71, 91
                           Res. Soc. Symp. Proc. 808, 103 (2004)                                (1995)
                                                                                                                               147




   8. Electronic Properties of Semiconductor Interfaces
                    Electronic Pro
 In this chapter we investigate the electronic
                                                           8.1    Experimental Database ........................ 149
                                                                  8.1.1  Barrier Heights of Laterally
 properties of semiconductor interfaces. Semi-
                                                                         Homogeneous Schottky Contacts . 149
 conductor devices contain metal–semiconductor,
                                                                  8.1.2 Band Offsets of Semiconductor
 insulator–semiconductor, insulator–metal and/or                         Heterostructures ....................... 152
 semiconductor–semiconductor interfaces. The
 electronic properties of these interfaces de-             8.2    IFIGS-and-Electronegativity Theory ....... 153
 termine the characteristics of the device. The
 band structure lineup at all these interfaces is          8.3    Comparison of Experiment and Theory ..               155
                                                                  8.3.1 Barrier Heights
 determined by one unifying concept, the con-
                                                                        of Schottky Contacts ..................        155
 tinuum of interface-induced gap states (IFIGS).
                                                                  8.3.2 Band Offsets of Semiconductor
 These intrinsic interface states are the wave-
                                                                        Heterostructures .......................       156
 function tails of electron states that overlap                   8.3.3 Band-Structure Lineup
 the fundamental band gap of a semiconduc-                              at Insulator Interfaces ...............        158
 tor at the interface; in other words they are
 caused by the quantum-mechanical tunnel-                  8.4    Final Remarks ..................................... 159
 ing effect. IFIGS theory quantitatively explains
                                                           References .................................................. 159
 the experimental barrier heights of well-
 characterized metal–semiconductor or Schottky
 contacts as well as the valence-band offsets of            semiconductor heterostructures. Insulators are
 semiconductor–semiconductor interfaces or                  viewed as semiconductors with wide band gaps.




In his pioneering article entitled Semiconductor Theory    the very simple and therefore attractive Schottky–Mott
of the Blocking Layer, Schottky [8.1] finally explained     rule, Bardeen [8.5] proposed that electronic interface
the rectifying properties of metal–semiconductor con-      states in the semiconductor band gap play an essen-
tacts, which had first been described by Braun [8.2],       tial role in the charge balance at metal–semiconductor
as being due to a depletion of the majority carri-         interfaces.
ers on the semiconductor side of the interface. This            Heine [8.6] considered the quantum-mechanical tun-
new depletion-layer concept immediately triggered          neling effect at metal–semiconductor interfaces and
a search for a physical explanation of the barrier         noted that for energies in the semiconductor band gap,
heights observed in metal–semiconductor interfaces,        the volume states of the metal have tails in the semi-
or Schottky contacts as they are also called in order      conductor. Tejedor and Flores [8.7] applied this same
to honor Schottky’s many basic contributions to this       idea to semiconductor heterostructures where, for ener-
field.                                                      gies in the band-edge discontinuities, the volume states
    The early Schottky–Mott rule [8.3, 4] proposed that    of one semiconductor tunnel into the other. The continua
n-type (p-type) barrier heights were equal to the dif-     of interface-induced gap states (IFIGS), as these evanes-
ference between the work function of the metal and         cent states were later called, are an intrinsic property of
                                                                                                                               Part A 8




the electron affinity (ionization energy) of the semi-      semiconductors and they are the fundamental physical
conductor. A plot of the experimental barrier heights      mechanism that determines the band-structure lineup at
of various metal–selenium rectifiers versus the work        both metal–semiconductor contacts and semiconductor
functions of the corresponding metals did indeed re-       heterostructures: in other words, at all semiconductor
veal a linear correlation, but the slope parameter was     interfaces. Insulator interfaces are also included in this,
much smaller than unity [8.4]. To resolve the failure of   since insulators may be described as wide-gap semi-
 148       Part A   Fundamental Properties



                                                                                   the difference in the electronegativities of the atoms in-
                                      W
                                                                                   volved in the interfacial bonds also describes the charge
                                                                                   transfer at semiconductor interfaces. Combining the
                                             Wci
                                                                                   physical IFIGS and the chemical electronegativity con-
                                ΦBn                                                cept, the electric-dipole contributions of Schottky barrier
                        WF                                              Wcb        heights as well as those of heterostructure band offsets
                                                                                   vary proportional to the difference in the electronega-
                                                                                   tivities of the metal and the semiconductor and of the
                                                                        Wvb
                                                                                   two semiconductors, respectively. The electronegativi-
                                Metal / n – semiconductor                      z   ties of the Group IV elemental and the IV–IV, III–V, and
                                                W                                  II–VI compound semiconductors are almost equal, since
                                                                                   the elements that constitute these semiconductors are all
                                                                                   placed in the middle of the Periodic Table. Hence, the
                                                                                   IFIGS dipole terms of the respective semiconductor het-
                      W lci                                                        erostructures will be small and may be neglected [8.9].
                                      ΔWc                                          The valence-band offsets of nonpolar, of lattice-matched
                                                                       W rci
                                                                       W rvi       and of metamorphic heterostructures should thus equal
                                      ΔWv                                          the difference between the branch-point energies of the
                      W lvi
                                                                                   semiconductors in contact.
                                 Semiconductor – semiconductor                 z
                                                                                        The theoreticians appreciated Heine’s IFIGS con-
                    Fig. 8.1 Schematic energy-band diagrams of metal–              cept at once. The initial reluctance of most experi-
                    semiconductor contacts and semiconductor heterostruc-          mentalists was motivated by the observation that the
                    tures. WF : Fermi level; ΦBn : barrier height; Wv and Wc :     predictions of the IFIGS theory only marked upper lim-
                    valence-band maximum and conduction-band minimum,              its for the barrier heights observed with real Schottky
                    respectively; ΔWv and ΔWc : valence- and conduction-           contacts [8.10]. Schmitsdorf et al. [8.11] finally re-
                    band offset, respectively; i and b: values at the interface    solved this dilemma. They found a linear decrease in
                    and in the bulk, respectively; r and l: right and left side,   the effective barrier height with increasing ideality fac-
                    respectively                                                   tors for their Ag/n-Si(111) diodes. Such behavior has
                                                                                   been observed for all of the Schottky contacts investi-
                    conductors. Figure 8.1 shows schematic band diagrams           gated so far. Schmitsdorf et al. attributed this correlation
                    of an n-type Schottky contact and a semiconductor              to patches of decreased barrier heights and lateral di-
                    heterostructure.                                               mensions smaller than the depletion layer width [8.12].
                        The IFIGS continua derive from both the valence-           Consequently, they extrapolated their plots of effective
                    and the conduction-band states of the semiconduc-              barrier height versus ideality factor to the ideality factor
                    tor. The energy at which their predominant character           determined by the image-force or Schottky effect [8.13]
                    changes from valence-band-like to conduction-band-             alone; in this way, they obtained the barrier heights of the
                    like is called their branch point. The position of the         laterally homogeneous contacts. The barrier heights of
                    Fermi level relative to this branch point then deter-          laterally uniform contacts can also be determined from
                    mines the sign and the amount of the net charge in the         capacitance–voltage measurements (C/V ) and by ap-
                    IFIGS. Hence, the IFIGS give rise to intrinsic interface       plying ballistic-electron-emission microscopy (BEEM)
                    dipoles. Both the barrier heights of Schottky contacts         and internal photoemission yield spectroscopy (IPEYS).
                    and the band offsets of heterostructures thus divide up        The I/V , C/V, BEEM, and IPEYS data agree within the
                    into a zero-charge-transfer term and an electric-dipole        margins of experimental error.
                    contribution.                                                       Mönch [8.14] found that the barrier heights of lat-
                        From a more chemical point of view, these interface        erally homogeneous Schottky contacts as well as the
Part A 8




                    dipoles may be attributed to the partial ionic character of    experimentally observed valence band offsets of semi-
                    the covalent bonds between atoms right at the interface.       conductor heterostructures agree excellently with the
                    Generalizing Pauling’s [8.8] electronegativity concept,        predictions of the IFIGS-and-electronegativity theory.
                                                  Electronic Properties of Semiconductor Interfaces      8.1 Experimental Database   149



8.1 Experimental Database
8.1.1 Barrier Heights of Laterally                            of vacuum. The ideality factor n describes the voltage
      Homogeneous Schottky Contacts                           dependence of the barrier height and is defined by

     I/V Characteristics                                          1 − 1/n = ∂ΦBn /∂e0 Vc .
                                                                              eff
                                                                                                                          (8.4)
The current transport in real Schottky contacts occurs via
                                                              For real diodes, the ideality factors n are generally found
thermionic emission over the barrier provided the dop-
                                                              to be larger than the ideality factor
ing level of the semiconductor is not too high [8.15]. For
doping levels larger than approximately 1018 per cm3 ,                                          −1
                                                                                    δΦif
                                                                                      0
the depletion layer becomes so narrow that tunnel or              n if = 1 −                         ,                    (8.5)
field emission through the depletion layer prevails.                               4e0 |Vi0 |
The current–voltage characteristics then become ohmic         which is determined by the image-force effect only.
rather than rectifying.                                           The effective barrier heights and the ideality factors
    For thermionic emission over the barrier, the             of real Schottky diodes fabricated under experimentally
current–voltage characteristics may be written as (see,       identical conditions vary from one specimen to the next.
for example, [8.14])                                          However, the variations of both quantities are correlated,
                                                              and the ΦBn values become smaller as the ideality factors
                                                                         eff
    Ite = A A∗ T 2 exp −ΦBn /kB T exp(e0 Vc /nkB T )
                         eff
             R                                                increase. As an example, Fig. 8.2 displays ΦBn versus
                                                                                                                 eff

          × [1 − exp(−e0 Vc /kB T )] ,                (8.1)   n data for Ag/n-Si(111) contacts with (1 × 1)   i - unrecon-

                                                              structed and (7 × 7)i -reconstructed interfaces [8.11]. The
where A is the diode area, A∗ is the effective Richard-
                             R                                dashed and dash-dotted lines are the linear least-squares
son constant of the semiconductor, and kB , T, and e0 are     fits to the data points. The linear dependence of the effec-
Boltzmann’s constant, the temperature, and the elec-          tive barrier height on the ideality factor may be written
tronic charge, respectively. The effective Richardson         as
constant is defined as
                                                                  ΦBn = ΦBn − ϕp (n − n if ) ,
                                                                   eff   nif
                                                                                                                          (8.6)
           4πe0 kB m ∗     m∗
    A∗
     R   =           n
                       = AR n ,                       (8.2)
              h3           m0                                 where ΦBn is the barrier height at the ideality factor
                                                                         nif

                                                              n if . Several conclusions may be drawn from this rela-
where AR = 120 A cm−2 K−2 is the Richardson con-              tion. First, the ΦBn − n correlation shows that more than
                                                                                eff
stant for thermionic emission of nearly free electrons
into vacuum, h is Planck’s constant, and m 0 and m ∗ are
                                                      n
the vacuum and the effective conduction-band mass of                  Effective barrier height (eV)
                                                               0.75
electrons, respectively. The externally applied bias Va di-
vides up into a voltage drop Vc across the depletion layer
of the Schottky contact and an IR drop at the series resis-    0.72
tance Rs of the diode, so that Vc = Va − IRs . For ideal
(intimate, abrupt, defect-free, and, above all, laterally                                                      (1 × 1)i
homogeneous) Schottky contacts, the effective zero-bias        0.69
barrier height ΦBn equals the difference ΦBn − δΦif be-
                 eff                        hom       0

tween the homogeneous barrier height and the zero-bias         0.66        Ag / n-Si(111)
image-force lowering (see [8.14])                                          Nd = 1 × 1015 cm–3                  (7 × 7)i
                                                   1/4                     T = 293 Κ
                    2e2 Nd
                      0
                                                               0.63
   δΦif = e0
     0
                                 e0 Vi0 − kB T           ,         1.0                     1.1               1.2
                                                                                                                                     Part A 8.1




                (4π)2 ε2 εb ε3
                       ∞     0
                                                                                                            Ideality factor
                                                      (8.3)   Fig. 8.2 Effective barrier heights versus ideality factors de-
                                                              termined from I/V characteristics of Ag/n-Si(111)-(7 × 7)i
where Nd is the donor density,    e0 |Vi0 |is the zero-bias   and -(1 × 1)i contacts at room temperature. The dashed and
band bending, ε∞ and εb are the optical and the bulk di-      dash-dotted lines are the linear least-squares fits to the data.
electric constant, respectively, and ε0 is the permittivity   After [8.11]
  150        Part A   Fundamental Properties



                      one physical mechanism determines the barrier heights
                                                                                           V / V0i
                      of real Schottky contacts. Second, the extrapolation of        1.0
                      ΦBn versus n curves to n if removes all mechanisms
                         eff

                      that cause a larger bias dependence of the barrier height
                      than the image-force effect itself from consideration.
                      Third, the extrapolated barrier heights ΦBn are equal to
                                                                  nif

                      the zero-bias barrier height ΦBnhom − δΦ 0 of the laterally
                                                               if                    0.5
                      homogeneous contact.
                           The laterally homogeneous barrier heights obtained
                      from ΦBn versus n curves to n if are not necessarily
                                eff

                      characteristic of the corresponding ideal contacts. This
                      is illustrated by the two data sets displayed in Fig. 8.2,                                                                    1.0
                      which differ in the interface structures of the respective       0
                                                                                                                                   0.5
                                                                                            – 0.2
                      diodes. Quite generally, structural rearrangements such                        x / zdep 0
                                                                                                                                         z / zdep
                                                                                                                  0.2
                      as the (7 × 7)i reconstruction are connected with a redis-                                         0
                      tribution of the valence charge. The bonds in perfectly       Fig. 8.3 Calculated potential distribution underneath and
                      ordered bulk silicon, the example considered here, are        around a patch of reduced interface potential embedded
                      purely covalent, and so reconstructions are accompanied       in a region of larger interface band-bending. The lateral
                      by electric Si+Δq −Si−Δq dipoles. The Si(111)-(7 × 7)         dimension and the interface potential reduction of the patch
                      reconstruction is characterized by a stacking fault in        are set to two tenths of the depletion layer width z dep and
                      one half of its unit mesh [8.16]. Schmitsdorf et al. [8.11]   one half of the interface potential of the surrounding region
                      quantitatively explained the experimentally observed re-
                      duction in the laterally homogeneous barrier height of        ory gives the variation in the depletion layer capacitance
                      the (7 × 7)i with regard to the (1 × 1)i diodes by the        per unit area as (see [8.14])
                      electric dipole associated with the stacking fault of the                                                               1/2
                                                                                        Cdep = e2 εb ε0 Nd /2 e0 Vi0 − Vc − kB T
                                                                                                0                                                    .
                      Si(111)-7 × 7 reconstruction.
                           Patches of reduced barrier height with lateral di-                                                                       (8.7)
                      mensions smaller than the depletion layer width that          The current through a Schottky diode biased in the
                      are embedded in large areas of laterally homogeneous          reverse direction is small, so the IR drop due to the
                      barrier height is the only known model that explains          series resistance of the diode may be neglected. Con-
                      a lowering of effective barrier heights with increasing       sequently, the extrapolated intercepts on the abscissa of
                      ideality factors. In their phenomenological studies of        1/Cdep versus Va plots give the band bending e0 |Vi0 |
                                                                                         2

                      such patchy Schottky contacts, Freeouf et al. [8.12]          at the interface, and together with the energy dis-
                      found that the potential distribution exhibits a saddle       tance Wn = WF − Wcb from the Fermi level to the
                      point in front of such nanometer-size patches of reduced      conduction band minimum in the bulk, one obtains
                      barrier height. Figure 8.4 explains this behavior. The        the flat-band barrier height ΦBn ≡ ΦBn = e0 |Vi0 | + Wn
                                                                                                                   fb      hom

                      saddle-point barrier height strongly depends on the volt-     which equals the laterally homogeneous barrier height
                      age drop Vc across the depletion layer. Freeouf et al.        of the contact.
                      simulated the current transport in such patchy Schot-             As an example, Fig. 8.4 displays the flat-band barrier
                      tky contacts and found a reduction in the effective           heights of the same Ag/n-Si(111) diodes that are dis-
                      barrier height and a correlated increase in the ideal-        cussed in Fig. 8.2. The dashed and dash-dotted lines are
                      ity factor as they reduced the lateral dimensions of the      the Gaussian least-squares fits to the data from the diodes
                      patches. However, they overlooked the fact that the bar-      with (1 × 1)i and (7 × 7)i interface structures, respec-
                      rier heights of the laterally homogeneous contacts may        tively. Within the margins of experimental error the peak
                      be obtained from ΦBn versus n plots, by extrapolating
                                            eff                                     C/V values agree with the laterally homogeneous bar-
Part A 8.1




                      to n if .                                                     rier heights obtained from the extrapolations of the I/V
                                                                                    data shown in Fig. 8.2. These data clearly demonstrate
                           C/V Characteristics                                      that barrier heights characteristic of laterally homoge-
                      Both the space charge and the width of the depletion lay-     neous Schottky contacts can be only obtained from I/V
                      ers at metal–semiconductor contacts vary as a function        or C/V data from many diodes fabricated under identical
                      of the externally applied voltage. The space-charge the-      conditions rather than from a single diode. However, the
                                                          Electronic Properties of Semiconductor Interfaces    8.1 Experimental Database   151



      Number of diodes                                                      Probability (%)
 20                                                                    15
                              Ag / n-Si(111)                                   Pd / n-6H-SiC
                                               (1 × 1)i                        T = 293 K
 15
                   (7 × 7)i                                            10

 10

                                                                        5
  5


  0                                                                     0
      0.65         0.70            0.75         0.80           0.85              1.1           1.2        1.3           1.4
                                    Flat-band barrier height (eV)                                       BEEM barrier height (eV)
Fig. 8.4 Histograms of flat-band barrier heights deter-                Fig. 8.5 Histograms of local BEEM barrier heights of two
mined from C/V characteristics of Ag/n-Si(111)-(7 × 7)i               Pd/n-6H-SiC(0001) diodes with ideality factors of 1.06
and -(1 × 1)i contacts at room temperature. The data were             (gray solid bars) and 1.49 (empty bars). The data were ob-
obtained with the same diodes discussed in Fig. 8.2. The              tained by fitting the square law (8.8) to 800 BEEM Icoll /Vtip
dashed and dash-dotted lines are the Gaussian least-squares           spectra each. Data from Im et al. [8.17]
fits to the data. After [8.11]
                                                                      The local barrier heights are determined by fitting rela-
effective barrier heights and the ideality factors vary as            tion (8.8) to measured Icoll /Vtip characteristics recorded
a function of the diode temperature. Hence, effective bar-            at successive tip positions along lateral line scans. Fig-
rier heights and ideality factors evaluated from the I/V              ure 8.5 displays histograms of the local BEEM barrier
characteristics for one and the same diode recorded at                heights of two Pd/n-6H-SiC(0001) diodes [8.17]. The
different temperatures are also suitable for determining              diodes differ in their ideality factors, 1.06 and 1.49,
the corresponding laterally homogeneous barrier height                which are close to and much larger, respectively, than
(see [8.14]).                                                         the value n if = 1.01 determined solely by the image-
                                                                      force effect. Obviously, the nanometer-scale BEEM
    Ballistic-Electron-Emission Microscopy                            histograms of the two diodes are identical although
In ballistic-electron-emission microscopy (BEEM)                      their macroscopic ideality factors and therefore their
[8.18], a tip injects almost monoenergetic electrons into             patchinesses differ. Two important conclusions were
the metal film of a Schottky diode. These tunnel-injected              drawn from these findings. First, these data suggest
electrons reach the semiconductor as ballistic electrons              the existence of two different types of patches, intrin-
provided that they lose no energy on their way through                sic and extrinsic ones. The intrinsic patches might be
the metal. Hence, the collector current Icoll is expected             correlated with the random distributions of the ion-
to set in when the ballistic electrons surpass the metal–             ized donors and acceptors which cause nanometer-scale
semiconductor barrier; in other words, if the voltage Vtip            lateral fluctuations in the interface potential. A few
applied between tip and metal film exceeds the local po-               gross interface defects of extrinsic origin, which es-
tential barrier ΦBn (z)/e0 . Bell and Kaiser [8.19] derived
                  loc                                                 cape BEEM observations, are then responsible for
the square law                                                        the variations in the ideality factors. Second, Gaus-
                                                  2
                                                                      sian least-squares fits to the histograms of the local
      Icoll (z) = R∗ Itip e0 Vtip − ΦBn (z)
                                     loc
                                                              (8.8)   BEEM barrier heights yield peak barrier heights of
                                                                      1.27 ± 0.03 eV. Within the margins of experimental er-
for the BEEM Icoll /Vtip characteristics, where Itip is               ror, this value agrees with the laterally homogeneous
                                                                                                                                           Part A 8.1




the injected tunnel current. BEEM measures local bar-                 value of 1.24 ± 0.09 eV which was obtained by extrap-
rier heights; most specifically, the saddle-point barrier              olation of the linear least-squares fit to a ΦBn versus n
                                                                                                                      eff

heights in front of nanometer-sized patches rather than               plot to n if . The nanometer-scale BEEM histograms and
their lower barrier heights right at the interface.                   the macroscopic I/V characteristics thus provide iden-
    BEEM is the experimental tool for measuring spatial               tical barrier heights of laterally homogeneous Schottky
variations in the barrier height on the nanometer-scale.              contacts.
  152        Part A   Fundamental Properties



                          Internal Photoemission Yield Spectroscopy
                                                                                           [Y(hω) × hω]1/2
                      Metal-semiconductor contacts show a photoelectric re-         0.03
                      sponse to optical radiation with photon energies smaller
                      than the width of the bulk band gap. This effect is caused
                      by photoexcitation of electrons from the metal over the
                                                                                    0.02
                      interfacial barrier into the conduction band of the semi-
                      conductor. Experimentally, the internal photoemission
                      yield, which is defined as the ratio of the photoinjected
                      electron flux across the barrier into the semiconductor        0.01
                                                                                                                                           Pt / p-Si (001)
                      to the flux of the electrons excited in the metal, is meas-                                                           Na = 8 × 1015 cm–3
                      ured as a function of the energy of the incident photons.                                                            T = 50 K
                      Consequently, this technique is called internal photoe-       0.00
                      mission yield spectroscopy (IPEYS). Cohen et al. [8.21]          0.25                    0.30                         0.35             0.40
                      derived that the internal photoemission yield varies as                                                                Photon energy (eV)
                      a function of the photon energy ω as                         Fig. 8.6 Spectral dependence of the internal photoemis-
                                                                                               √
                                                     2
                                                                                   sion yield Y ( ω) · ω of a Pt/p-Si(001) diode versus the
                         Y ( ω) ∝          IPEYS
                                      ω − ΦBn              ω.              (8.9)   photon energy of the exciting light. The dashed line is the
                                                                                   linear least-squares fit to the data for photon energies larger
                      Patches only cover a small portion of the metal–             than 0.3 eV. Data from Turan et al. [8.20]
                      semiconductor interface, so the threshold energy
                        IPEYS will equal the barrier height Φ hom of the later-
                      ΦBn                                                          The valence-band discontinuity is then given by (see
                                                               Bn
                      ally homogeneous part of the contact minus the zero-bias     Fig. 8.7)
                      image-force lowering δΦif .0
                                                                                       ΔWv = Wvir − Wvil = Wi (nr lr ) − Wi (nlll )
                          In Fig. 8.6, experimental [Y ( ω) · ω]1/2 data for
                      a Pt/p-Si(001) diode [8.20] are plotted versus the en-                 + [Wvbr − Wb (nr lr )] − [Wvbl − Wb (nlll )] ,
                      ergy of the exciting photons. The dashed line is the                                                                                    (8.10)
                      linear least-squares fit to the data. The deviation of the    where nr lr and nlll denote the core levels of the semi-
                      experimental [Y ( ω) · ω]1/2 data towards larger values      conductors on the right (r) and the left (l) side of the
                      slightly below and above the threshold is caused by the      interface, respectively. The subscripts i and b charac-
                      shape of the Fermi–Dirac distribution function at finite      terize interface and bulk properties, respectively. The
                      temperatures and by the existence of patches with barrier
                      heights smaller and larger than ΦBn .
                                                         hom
                                                                                                                        W
                      8.1.2 Band Offsets
                                                                                                                                       Wvil
                            of Semiconductor Heterostructures                                                                Δ Wv
                                                                                                                    Wvir
                                                                                           Wvbl – Wb(n l ll)
                      Semiconductors generally grow layer-by-layer, at
                      least initially. Hence, core-level photoemission spec-
                      troscopy (PES) is a very reliable tool and the                                                        Wi (n l l l)      Wvbr – Wb(n r lr)
                      one most widely used to determine the band-
                      structure lineup at semiconductor heterostructures.                                      Wi (nr lr)
                      The valence-band offset may be obtained from the
                      energy positions of core-level lines in X-ray pho-                   Bulkleft                 Interface                     Bulkright       z
                      toelectron spectra recorded with bulk samples of             Fig. 8.7 Schematic energy band diagram at semiconductor
                      the semiconductors in contact and with the inter-            heterostructures. Wvb and Wvi are the valence-band maxima
Part A 8.1




                      face itself [8.22]. Since the escape depths of the           and Wb (nl) and Wi (nl) are the core levels in the bulk and at
                      photoelectrons are on the order of just 2 nm, one            the interface, respectively. The subscripts l and r denote the
                      of the two semiconductors must be sufficiently thin.          semiconductors on the right and the on the left side of the
                      This condition is easily met when heterostructures           interface. ΔWv is the valence-band offset. The thin dashed
                      are grown by molecular beam epitaxy (MBE) and                lines account for possible band-bending from space-charge
                      PE spectra are recorded during growth interrupts.            layers
                                      Electronic Properties of Semiconductor Interfaces     8.2 IFIGS-and-Electronegativity Theory   153



energy difference Wi (nr lr ) − Wi (nlll ) between the core   in the bulk of the two semiconductors are evaluated
levels of the two semiconductors at the interface is deter-   separately.
mined from energy distribution curves of photoelectrons           Another widely used technique for determining band
recorded during MBE growth of the heterostructure. The        offsets in heterostructures is internal photoemission
energy positions Wvbr − Wb (nrlr ) and Wvbl − Wb (nlll )      yield spectroscopy. The procedure for evaluating the
of the core levels relative to the valence-band maxima        IPEYS signals is the same as described in Sect. 8.1.1.


8.2 IFIGS-and-Electronegativity Theory
Because of the quantum-mechanical tunneling effect,               The IFIGS are made up of valence-band and
the wavefunctions of bulk electrons decay exponentially       conduction-band states of the semiconductor. Their net
into vacuum at surfaces or, more generally speaking, at       charge depends on the energy position of the Fermi
solid–vacuum interfaces. A similar behavior occurs at         level relative to their branch point, where their character
interfaces between two solids [8.6, 7]. In energy regions     changes from predominantly donor- or valence band-
of Schottky contacts and semiconductor heterostructures       like to mostly acceptor- or conduction band-like. The
where occupied band states overlap a band gap, the            band-structure lineup at semiconductor interfaces is thus
wavefunctions of these electrons will tail across the in-     described by a zero-charge-transfer term and an electric
terface. The only difference to solid–vacuum interfaces       dipole contribution.
is that the wavefunction tails oscillate at solid–solid in-       In a more chemical approach, the charge trans-
terfaces. Figure 8.8 schematically explains the tailing       fer at semiconductor interfaces may be related to the
effects at surfaces and semiconductor interfaces. For the     partly ionic character of the covalent bonds at inter-
band-structure lineup at semiconductor interfaces, only       faces. Pauling [8.8] described the ionicity of single
the tailing states within the gap between the top va-         bonds in diatomic molecules by the difference between
lence and the lowest conduction band are of any real          the electronegativities of the atoms involved. The bind-
importance since the energy position of the Fermi level       ing energies of core-level electrons are known to depend
determines their charging state. These wavefunction tails     on the chemical environment of the atoms or, in other
or interface-induced gap states (IFIGS) derive from the       words, on the ionicity of their chemical bonds. Figure 8.9
continuum of the virtual gap states (ViGS) of the com-        displays experimentally observed chemical shifts for
plex semiconductor band structure. Hence, the IFIGS           Si(2p) and Ge(3d) core levels induced by metal adatoms
are an intrinsic property of the semiconductor.
                                                                    Adatom-induced core-level shift (eV)
 a)                          ψ ψ*                                           Si(111)
                                                                            Si(001)
                                                                 0.5
                                                                            Ge(111)
                                                                            Ge(001)                         Au

                                                                                                   Ag
                                                                 0.0                          In
      Metal, Semiconductor          Vacuum             z                                                   Sn

 b)                          ψ ψ*
                                                               – 0.5   Cs


                                                                              –1                     0                      1
                                                                                          Electronegativity difference Xm – Xs
                                                                                                                                     Part A 8.2




      Metal, Semiconductor          Semiconductor      z
                                                              Fig. 8.9 Chemical shifts of Si(2p) and Ge(3d) core levels
                                                              induced by metal adatoms on silicon and germanium sur-
Fig. 8.8a,b Wavefunctions at clean surfaces (a) and at        faces, respectively, as a function of the difference X m − X s
metal–semiconductor and semiconductor–semiconductor           in the metal and the semiconductor electronegativities in
interfaces (b) (schematically)                                Pauling units. After [8.14]
  154        Part A   Fundamental Properties



                      on silicon and germanium surfaces as a function of the        Table 8.1 Optical dielectric constants, widths of the di-
                      difference X m − X s between the Pauling atomic elec-         electric band gap, and branch-point energies of diamond-,
                      tronegativity of the metal and that of the semiconductor      zincblende- and chalcopyrite-structure semiconductors and
                      atoms. The covalent bonds between metal and substrate         of some insulators
                      atoms still persist at metal–semiconductor interfaces, as      Semiconductor           ε∞                   Wdg (eV)
                                                                                                                                              p
                                                                                                                                             Φbp (eV)
                      ab-initio calculations [8.23] have demonstrated for the        C                        5.70            14.40           1.77
                      example of Al/GaAs(110) contacts. The pronounced               Si                      11.90             5.04           0.36a
                      linear correlation of the data displayed in Fig. 8.9 thus
                                                                                     Ge                      16.20             4.02           0.18a
                      justifies the application of Pauling’s electronegativity
                                                                                     3C-SiC                   6.38             9.84           1.44
                      concept to semiconductor interfaces.
                                                                                     3C-AlN                   4.84            11.92           2.97
                          The combination of the physical IFIGS and the
                                                                                     AlP                      7.54             6.45           1.13
                      chemical electronegativity concept yields the barrier
                      heights of ideal p-type Schottky contacts and the              AlAs                     8.16             5.81           0.92
                      valence-band offsets of ideal semiconductor heterostruc-       AlSb                    10.24             4.51           0.53
                      tures as                                                       3C-GaN                   5.80            10.80           2.37
                                     p                                               GaP                      9.11             5.81           0.83
                            ΦBp = Φbp − S X (X m − X s )                  (8.11)
                                                                                     GaAs                    10.90             4.97           0.52
                      and                                                            GaSb                    14.44             3.8            0.16
                                         p    p                                      3C-InN                  –                 6.48           1.51
                            ΔWv = Φbpr − Φbpl + D X (X sr − X sl ) ,      (8.12)
                                                                                     InP                      9.61             5.04           0.86
                                                  p
                      respectively, where Φbp = Wbp − Wv (Γ ) is the energy          InAs                    12.25             4.20           0.50
                      distance from the valence-band maximum to the branch           InSb                    15.68             3.33           0.22
                      point of the IFIGS or the p-type branch-point energy.          2H-ZnO                   3.72            12.94           3.04b
                      It has the physical meaning of a zero-charge-transfer          ZnS                      5.14             8.12           2.05
                      barrier height. The slope parameters S X and D X are           ZnSe                     5.70             7.06           1.48
                      explained at the end of this section.                          ZnTe                     7.28             5.55           1.00
                          The IFIGS derive from the virtual gap states               CdS                      5.27             7.06           1.93
                      of the complex band structure of the semiconduc-
                                                                                     CdSe                     6.10             6.16           1.53
                      tor. Their branch point is an average property of the
                                                                                     CdTe                     7.21             5.11           1.12
                      semiconductor. Tersoff [8.24, 27] calculated the branch-
                                        p                                            CuGaS2                   6.15             7.46           1.43
                      point energies Φbp of Si, Ge, and 13 of the III–V
                                                                                     CuInS2                   6.3*             7.02           1.47
                      and II–VI compound semiconductors. He used a lin-
                      earized augmented plane-wave method and the local              CuAlSe2                  6.3*             6.85           1.25
                      density approximation. Such extensive computations             CuGaSe2                  7.3*             6.29           0.93
                      may be avoided. Mönch [8.28] applied Baldereschi’s             CuInSe2                  9.00             5.34           0.75
                      concept [8.29] of mean-value k-points to calculate             CuGaTe2                  8.0*             5.39           0.61
                      the branch-point energies of zincblende-structure com-         CuInTe2                  9.20             4.78           0.55
                      pound semiconductors. He first demonstrated that the            AgGaSe2                  6.80             5.96           1.09
                      quasi-particle band gaps of diamond, silicon, germa-           AgInSe2                  7.20             5.60           1.11
                      nium, 3C-SiC, GaAs and CdS at the mean-value k-point           SiO2                     2.10                            3.99c
                      equal their average or dielectric band gaps [8.30]             Si3 N4                   3.80                            1.93c
                                                                                     Al2 O3                   3.13                            3.23c
                            Wdg = ωp / ε∞ − 1 ,                           (8.13)
                                                                                     ZrO2                     4.84                           ≈ 3.2c
                       where ωp is the plasmon energy of the bulk valence            HfO2                     4.00                            2.62c
                                                               p
                      electrons. Mönch then used Tersoff’s Φbp values, calcu-        ∗ε    = n 2 , a [8.24], b [8.25], c [8.26]
Part A 8.2




                                                                                       ∞
                      lated the energy dispersion Wv (Γ ) − Wv (kmv ) of the top-
                      most valence band in the empirical tight-binding approx-
                      imation (ETB), and plotted the resulting branch-point         fit to the data of the zincblende-structure compound
                                                   p
                      energies Wbp−Wv (kmv ) = Φbp+[Wv (Γ )−Wv (kmv )]ETB           semiconductors [8.28]
                      at the mean-value k-point kmv versus the widths of
                                                                                           p
                      the dielectric band gaps Wdg . The linear least-squares          Φbp = 0.449 · Wdg−[Wv (Γ )−Wv (kmv )]ETB , (8.14)
                                  Electronic Properties of Semiconductor Interfaces    8.3 Comparison of Experiment and Theory   155



indicates that the branch points of these semiconductors     states (MIGS), as the IFIGS in Schottky contacts are
lie 5% below the middle of the energy gap at the mean-       traditionally called, and plotted the (e2 /ε0 )Dgs /2qgs
                                                                                                        0
                                                                                                               mi    mi

value k-point. Table 8.1 displays the p-type branch-point    values versus the optical susceptibility ε∞ − 1. The
energies of the Group IV elemental semiconductors, of        linear least-squares fit to the data points yielded [8.32]
SiC, and of III–V and II–VI compound semiconductors,
as well as of some insulators.                                   A X /S X − 1 = 0.1 · (ε∞ − 1)2 ,                 (8.16)
    A simple phenomenological model of Schottky con-         where the reasonable assumption εi ≈ 3 was made.
tacts with a continuum of interface states and a constant        To a first approximation, the slope parameter D X
density of states Dis across the semiconductor band gap      of heterostructure band offsets may be equated with
yields the slope parameter [8.31, 32]                        the slope parameter S X of Schottky contacts, since the
                                                             IFIGS determine the intrinsic electric-dipole contribu-
    S X = A X / 1 + e2 /εi ε0 Dis δis ,
                     0                              (8.15)
                                                             tions to both the valence-band offsets and the barrier
where εi is an interface dielectric constant. The pa-        heights. Furthermore, the Group IV semiconductors and
rameter A X depends on the electronegativity scale           the elements constituting the III–V and II–VI compound
chosen and amounts to 0.86 eV/Miedema-unit and               semiconductors are all placed in the center columns of
1.79 eV/Pauling-unit. For Dis → 0, relation (8.15)           the Periodic Table and their electronegativities thus only
yields S X → 1 or, in other words, if no interface-induced   differ by up to 10%. Consequently, the electric-dipole
gap states were present at the metal–semiconductor in-       term D X · (X sr − X sl ) may be neglected [8.9], so that
terfaces one would obtain the Schottky–Mott rule. The        (8.12) reduces to
extension δis of the interface states may be approxi-
                                                                 ΔWv ∼ Φbpr − Φbpl
                                                                        p      p
                                                                     =                                            (8.17)
mated by their charge decay length 1/2qis . Mönch [8.32]
                   mi       mi
used theoretical Dgs and qgs data for metal-induced gap      for practical purposes.



8.3 Comparison of Experiment and Theory
8.3.1 Barrier Heights of Schottky Contacts                   experimental techniques, I/V , BEEM, IPEYS and PES,
                                                             yield barrier heights of laterally homogeneous Schottky
Experimental barrier heights of intimate, abrupt, clean      contacts which agree within the margins of experimental
and (above all) laterally homogeneous Schottky con-          error.
tacts on n-Si and n-GaAs as well as n-GaN, and the               Second, all experimental data are quantitatively ex-
three SiC polytypes 3C, 6H and 4H are plotted in             plained by the branch-point energies (8.14) and the slope
Figs. 8.10 and 8.11, respectively, versus the differ-        parameters (8.16) of the IFIGS-and-electronegativity
ence in the Miedema electronegativities of the metals        theory. As was already mentioned in Sect. 8.1.1, the
and the semiconductors. Miedema’s electronegativi-           stacking fault, which is part of the interfacial Si(111)-
ties [8.33, 34] are preferred since they were derived        (7 × 7)i reconstruction, causes an extrinsic electric dipole
from properties of metal alloys and intermetallic com-       in addition to the intrinsic IFIGS electric dipole. The lat-
pounds, while Pauling [8.8] considered covalent bonds        ter one is present irrespective of whether the interface
in small molecules. The p- and n-type branch-point           structure is reconstructed or (1 × 1)i -unreconstructed.
            p
energies, Φbp = Wbp − Wv (Γ ) and Φbp = Wc − Wbp , re-
                                    n                        The extrinsic stacking fault-induced electric dipole
spectively, add up to the fundamental band-gap energy        quantitatively explains the experimentally observed bar-
Wg = Wc − Wv (Γ ). Hence, the barrier heights of n-type      rier height lowering of 76 ± 2 meV.
Schottky contacts are                                            Third, the IFIGS lines in Figs. 8.11a and 8.11b were
                                                             drawn using the branch-point energies calculated for
   ΦBn = Φbp + S X (X m − X s ) .
    hom   n
                                                    (8.18)
                                                                                                                                 Part A 8.3




                                                             cubic 3C-GaN and 3C-SiC, respectively, since relation
The electronegativity of a compound is taken as the ge-      (8.12) was derived for zincblende-structure compounds
ometric mean of the electronegativities of its constituent   only. However, the Schottky contacts were prepared
atoms.                                                       on wurtzite-structure 2H-GaN and not just on cu-
   First off all, the experimental data plotted in           bic 3C-SiC but also on its hexagonal polytypes 4H
Figs. 8.10 and 8.11 clearly demonstrate that the different   and 6H. The good agreement between the experimen-
  156        Part A   Fundamental Properties



                       a) Schottky                                           Electronegativity      a) Barrier height Φn = 1
                                                                                                                         Bn                              Electronegativity
                             barrier height (eV)                                   (Miedema)               (eV)                                                (Miedema)
                               2              3                4                5                            2              3               4                5
                       1.0                                                                                                                                     Au
                                n-Si                                                     Ir   Pt             n-GaN(0001)                                           Pd
                                                                                    Au               1.2
                                                           Ag-(1 × 1)   i                                                                           Ag
                                                                                                                                                                      Pt
                       0.8                   Al-(1× 1)i
                                             Pb-(1× 1)i                                              0.8                                     Pb                Ni
                                                                               Ag-(7 × 7)i
                                  Na-(3× 1)   i                             Al-(7× 7)i
                       0.6                                                  Pb-(7× 7)i                                                          IFIGS theory
                                                                                                     0.4
                                             Na-(7× 7)i
                                                                            IFIGS theory
                                Cs-(7× 7)i                                                                   Cs
                       0.4                                                                           0.0
                          –3             –2             –1            0            1                              –3             –2           –1               0
                                                     Electronegativity difference Xm– Xs                                           Electronegativity difference Xm– Xs
                       b) Schottky                                           Electronegativity
                             barrier height (eV)                                   (Miedema)        b) Schottky                                          Electronegativity
                                    2          3               4             5          6               barrier height (eV)                                    (Miedema)
                               n-GaAs                                                Pt             2.0         2           3                   4             5
                                                                              Au                             n-SiC                                               Ni
                       1.0                                                                                                                               Mo
                                                                MnSb                                                            IFIGS theory
                                                               Al                    Pd             1.5                                  Ti
                                                          Ti
                                                                                                                                                                Au
                                                                               Ni                                               4H                              Ni        Pt
                       0.8                                                                          1.0                                                              Pd
                                                                                                                                6H                   Cu
                                                                                                                  Cs
                                                     IFIGS theory                                                                                              Au
                                                                                                    0.5
                                  Cs                                                                                            3C                  IFIGS theory
                       0.6
                         –3            –2           –1          0           1         2
                                                     Electronegativity difference Xm– Xs            0.0
                                                                                                       –4              –3             –2          –1             0
                                                                                                                                     Electronegativity difference Xm– Xs
                      Fig. 8.10a,b Barrier heights of laterally homogeneous
                      n-type silicon (a) and GaAs Schottky contacts (b) ver-                       Fig. 8.11a,b Barrier heights of laterally homogeneous
                      sus the difference in the Miedema electronegativities                        n-type GaN(0001) (a) and 3C-, 4H-, and 6H-SiC Schottky
                      of the metals and the semiconductors. The               and                  contacts (b) versus the difference in the Miedema elec-
                        , , , and          symbols differentiate the data from                     tronegativities of the metals and the semiconductors. (a):
                      I/V , BEEM, IPEYS, and PES measurements, respec-                             The      , , and symbols differentiate the data from I /V ,
                      tively. The dashed and the dash-dotted lines are the                         BEEM, IPEYS, and PES measurements, respectively. The
                      linear least-squares fits to the data from diodes with                        solid IFIGS line is drawn with S X = 0.29 eV/Miedema-unit
                                                                                                          p
                      (1 × 1)i -unreconstructed and (7 × 7)i -reconstructed inter-                 and Φbp = 2.37 eV. (b): The , , and symbols differ-
                      faces, respectively. The solid IFIGS lines are drawn                         entiate data of 4H-, 6H- and 3C-SiC Schottky contacts,
                                                                  p
                      with S X = 0.101 eV/Miedema-unit and Φbp = 0.36 eV for                       respectively. The solid IFIGS lines are drawn with the band
                                                                                                                                  p
                      silicon (a) and with S X = 0.08 eV/Miedema-unit and                          gaps of the polytypes minus Φbp = 1.44 eV of cubic 3C-SiC
                         p
                      Φbp = 0.5 eV for GaAs (b). After [8.14]                                      and S X = 0.24 eV/Miedema-unit. After [8.14]

                      tal data and the IFIGS lines indicates that the p-type                       8.3.2 Band Offsets
                      branch-point energies are rather insensitive to the spe-                           of Semiconductor Heterostructures
                      cific bulk lattice structure of the semiconductor. This
Part A 8.3




                      conclusion is further justified by the band-edge discon-                      In the bulk, and at interfaces of sp3 -coordinated semi-
                      tinuities of the semiconductor heterostructures, which                       conductors, the chemical bonds are covalent. The
                      were experimentally observed and are discussed in                            simplest semiconductor–semiconductor interfaces are
                      Sect. 8.3.2, and by the band-edge offsets of 3C/2H                           lattice-matched heterostructures. However, if the bond
                      homostructures that were calculated for various semi-                        lengths of the two semiconductors differ then the inter-
                      conductors [8.35–39].                                                        face will respond with tetragonal lattice distortions. Such
                                      Electronic Properties of Semiconductor Interfaces      8.3 Comparison of Experiment and Theory   157



                                                                 the other hand, causes extrinsic electric dipoles. Their
 a) Heterostructures: non-polar                                  components normal to the interface will add an ex-
    Valence-band offset (eV)
                                                                 trinsic electric-dipole contribution to the valence-band
 2.0                                                             offset. In the following, only nonpolar, lattice-matched
                                                Ge / CdS
                               Ge / ZnSe
                                                                 isovalent, and metamorphic heterostructures will be dis-
 1.5                                                             cussed.
                    ZnSe / GaAs                  CdSe / GaSb         The valence-band offsets at nonpolar, in other words
 1.0    Ge / GaAs                                                (110)-oriented, heterostructures of compound semicon-
       AlP/GaP                                                   ductors should equal the difference in the branch-point
 0.5                        AlAs / GaAs
                                                                 energies of the two semiconductors in contact pro-
                        InAs / GaSb
                                                                 vided the intrinsic IFIGS electric-dipole contribution
 0.0            GaAs / InAs
                                                                 can be neglected, see relation (8.17). Figure 8.12a dis-
        0.0         0.5           1.0         1.5          2.0   plays respective experimental results for diamond- and
                    Difference of branch-point energies (eV)     zincblende-structure semiconductors as a function of
                                                                 the difference in the branch-point energies given in Ta-
 b) Heterostructures: metamorphic
                                                                 ble 8.1. The dashed line clearly demonstrates that the
    Valence-band offset (eV)
                                                                 experimental data are execellently explained by the the-
 2.0                                                ZnS /Ge
                                           2H-GaN / GaAs         oretical branch-point energies or, in other words, by the
                                                                 IFIGS theory.
 1.5
                                                                     As an example of lattice-matched and isovalent het-
                                                                 erostructures, Fig. 8.13 shows valence-band offsets for
 1.0    2H-GaN / 2H-AlN              2H-GaN / 6H-SiC             Al1−x Gax As/GaAs heterostructures as a function of
          AlSb / ZnTe                 2H-GaN / 3C-SiC            the alloy composition x. The IFIGS branch-point en-
 0.5
       Si / Ge                                                   ergies of the alloys were calculated assuming virtual
 0.0                                                             Al1−x Gax cations [8.28], and were found to vary lin-
              ZnTe / CdTe
                                                                 early as a function of composition between the values
        0.0         0.5           1.0         1.5          2.0   of AlAs and GaAs. More refined first-principles calcu-
                    Difference of branch-point energies (eV)
                                                                 lations yielded identical results [8.41, 42]. Figure 8.13
Fig. 8.12a,b Valence band offsets at nonpolar (110)-             reveals that the theoretical IFIGS valence-band offsets
oriented (a) and metamorphic semiconductor heterostruc-          fit the experimental data excellently.
tures (b) versus the difference between the p-type                   Figure 8.12b displays valence-band offsets for meta-
branch-point energies of the semiconductors in contact.          morphic heterostructures versus the difference in the
After [8.14]                                                     branch-point energies of the two semiconductors. The

pseudomorphic interfaces are under tensile or compres-
                                                                        Valence-band offset (eV)
sive stress. If the strain energy becomes too large then          0.6
it is energetically more favorable to release the stress by                                              Al1–x Gax As / GaAs
the formation of misfit dislocations. Such metamorphic
interfaces are almost relaxed.
                                                                  0.4
     In contrast to isovalent heterostructures, the chem-
ical bonds at heterovalent interfaces require special
attention, since interfacial donor- and acceptor-type
bonds may cause interfacial electric dipoles [8.40]. No           0.2
                                                                           IFIGS theory
such extrinsic electric dipoles will exist normal to non-
polar (110) interfaces. However, polar (001) interfaces
                                                                                                                                       Part A 8.3




behave quite differently. Acceptor bonds or donor bonds           0.0
normal to the interface would exist at abrupt heterostruc-           0.0                           0.5                   1.0
tures. But, for reasons of charge neutrality, they have to                                                    Composition x
be compensated by a corresponding density of donor               Fig. 8.13 Valence band offsets of lattice-matched and iso-
bonds and acceptor bonds, respectively. This may be              valent Al1−x Gax As/GaAs heterostructures as a function of
achieved by an intermixing at the interface which, on            alloy composition x. After [8.14]
  158        Part A   Fundamental Properties



                      dashed line indicates that the experimental results are
                                                                                    a) Valence-band offset (eV)
                      again excellently described by the theoretical IFIGS
                      data. This is true not only for heterostructures between
                                                                                    4                       CdTe
                      cubic zincblende- and hexagonal wurtzite-structure
                      compounds but also for wurtzite-structure Group III                   Ge                         SiC        ZnS
                      nitrides grown on both cubic 3C- and hexagonal 6H-
                                                                                    2                                                     GaN
                      SiC substrates. These observations suggest the following
                      conclusions. First, all of the heterostructures considered                                                 Al0.3Ga0.7N       SiO2
                                                                                                 Si
                      in Fig. 8.12b are only slightly (if at all) strained, al-
                                                                                    0                                                              Al2O3
                      though their lattice parameters differ by up to 19.8%.
                      Second, the calculations of the IFIGS branch-point                                                                           HfO2
                      energies assumed zincblende-structure semiconductors.                                                             GaN        Si3N4
                      These values, on the other hand, reproduce the experi-            0                   1                    2               3
                      mental valence band offsets irrespective of whether the                                                    Branch-point energy (eV)
                                                                                    b)
                      semiconductors have zincblende, wurtzite or, as in the            Barrier height (eV)                    Electronegativity (Miedema)
                      case of 6H-SiC, another hexagonal-polytype structure.              3              4                      5             6
                      These findings again support the conclusion drawn from
                                                                                    5                            IFIGS theory
                      the GaN and SiC Schottky barrier heights in the previous
                      section, that the IFIGS branch-point energies are rather
                                                                                                                  Ag            Au
                      insensitive to the specific semiconductor bulk lattice                                                                Pt
                                                                                    4                                  Cu
                      structure.                                                                                                      Pd

                      8.3.3 Band-Structure Lineup                                                                          W
                                                                                                                                 Ni
                                                                                                      Hf
                            at Insulator Interfaces                                 3                       Al
                                                                                                                      Cr

                      The continuing miniaturization of complementary                                  Mg                                       Metal / SiO2
                      metal–oxide–semiconductor (CMOS) devices requires             2
                                                                                                 –3              –2                  –1              0
                      gate insulators where the dielectric constants (κ) are                                                                         Xm – XSiO2
                      larger than the value of the silicon dioxide conven-
                                                                                   Fig. 8.14 (a) Valence band offsets of SiO2 , Si3 N4 , Al2 O3
                      tionally used. At present, the high-κ insulators Al2 O3 ,
                                                                                   and HfO2 heterostructures versus the p-type branch-point
                      ZrO2 , and HfO2 are being intensively studied. Insula-
                                                                                   energies of the respective semiconductors. (b) n-type barrier
                      tors may be considered to be wide-gap semiconductors.
                                                                                   heights of SiO2 Schottky contacts versus the difference be-
                      Hence, relations (8.11) and (8.12) also apply to insulator
                                                                                   tween electronegativities of the metal and SiO2 . The dashed
                      Schottky contacts and heterostructures. Unfortunately,
                                                                                   line is the linear least-squares fit to the data points. The
                      the branch-point energies of these insulators cannot
                                                                                   solid IFIGS line is drawn with Φbp = 5 eV (Wg = 9 eV) and
                                                                                                                     n
                      be obtained from relation (8.14) since it is valid for
                                                                                   S X = 0.77 eV/Miedema-unit (ε∞ = 2.1). After [8.25]
                      zincblende-structure compound semiconductors only.
                      However, the experimental band offsets reported for          can also be adopted for the Al2 O3 , HfO2 and Si3 N4
                      SiO2 , Si3 N4 , Al2 O3 , and HfO2 heterostructures may       heterostructures, where less experimental results are
                      be plotted as a function of the branch-point energies of     available. Hence, the data displayed in Fig. 8.14a pro-
                      the respective semiconductors [8.26]. Figure 8.14a re-       vide a means of determining the branch-point energies
                      veals that the valence-band offsets become smaller with        p
                                                                                   Φbp (ins) of SiO2 , Si3 N4 , and the high-κ oxides Al2 O3
                      increasing branch-point energy of the semiconductors.        and HfO2 . The dashed lines in Fig. 8.14a are the lin-
                      Moreover, the data points reported for the many different    ear least-squares fits to the respective data points.
                      SiO2 heterostructures studied indicate a linear depen-       The experimental slope parameters ϕvbo range from
Part A 8.3




                      dence for the valence-band offsets on the branch-point       1.16 to 1.23 for HfO2 and SiO2 heterostructures,
                      energy of the semiconductors, which may be written as        respectively, while relation (8.12) predicts ϕvbo = 1
                                          p           p
                         ΔWv = ϕvbo Φbp (ins) − Φbp (sem) ,               (8.19)   provided that the electric dipole term D X · (X sr − X sl )
                                                                                   vanishes. However, as well-established as this simpli-
                      since the valence band offsets of insulator homostruc-       fying assumption is for the classical semiconductor
                      tures will definitely vanish. Such a linear relationship      heterostructures discussed in Sect. 8.3.2, it has ques-
                                                                   Electronic Properties of Semiconductor Interfaces   References   159



tionable validity for the insulators considered here         only available for SiO2 . Figure 8.14b displays the bar-
since they are much more ionic. Hence, the difference        rier heights of SiO2 Schottky contacts as a function of
ϕvbo − 1 may be attributed to intrinsic electric-dipole      the electronegativity difference X m − X SiO2 , where the
layers at these insulator–semiconductor interfaces. The      electronegativity of SiO2 is estimated as 6.42 Miedema-
                                 p
p-type branch-point energies Φbp of the insulators ob-       units. The linear least-squares fit
tained from the linear least-squares fits are displayed in          ΦBn = (4.95 ± 0.19) + (0.77 ± 0.10)
Table 8.1.
    The reliability of these branch-point energies may                   × (X m − X SiO2 )[eV]                     (8.20)
be checked by, for example, analyzing barrier heights        to the experimental data agrees excellently with the
of respective insulator Schottky contacts. Such data are     prediction from the IFIGS-and-electronegativity theory.


8.4 Final Remarks
The local density approximation to density functional        contacts [8.47]. However, ab-initio LDA-DFT barrier
theory (LDA-DFT) is the most powerful and widely used        heights of Al-, Ag-, and Au/p-GaN contacts [8.48,49], as
tool in theoretical studies of the ground-state properties   well as of Al- and Ti/3C-SiC(001) interfaces [8.50,51],
of solids. However, excitation energies such as the width    strongly deviate from the experimental results.
of the energy gaps between the valence and conduction            As already mentioned, ab-initio LDF-DFT va-
bands of semiconductors cannot be correctly obtained         lenc band offsets of Al1−x Gax As/GaAs heterostruc-
from such calculations. The fundamental band gaps of         tures [8.41, 42] reproduce the experimental results
the elemental semiconductors C, Si and Ge as well as of      well. The same holds for mean values of LDF-DFT
the III–V and II–VI compounds are notoriously under-         valence-band offsets computed for different interface
estimated by 25 to 50%. However, it became possible to       configurations of GaN- and AlN/SiC heterostruc-
compute quasi-particle energies and band gaps of semi-       tures [8.52–56].
conductors from first principles using the so-called GW           The main difficulty which the otherwise extremely
approximation for the electron self-energy [8.43, 44].       successful ab-initio LDF-DFT calculations encounter
The resulting band gap energies agree to within 0.1 to       when describing semiconductor interfaces is not the
0.3 eV with experimental values.                             precise exchange-correlation potential, which may be
    For some specific metal–semiconductor contacts, the       estimated in the GW approximation, but their remark-
band-structure lineup was also studied by state-of-the-art   able sensitivity to the geometrical and compositional
ab-initio LDA-DFT calculations. The resulting LDA-           structure right at the interface. This aspect is more
DFT barrier heights were then subjected to a-posteriori      serious at metal–semiconductor interfaces than at het-
corrections which consider quasi-particle effects and, if    erostructures between two sp3 -bonded semiconductors.
necessary, spin-orbit interactions and semicore-orbital      The more conceptual IFIGS-and-electronegativity the-
effects. However, comparison of the theoretical results      ory, on the other hand, quantitatively explains not only
with experimental data gives an inconsistent picture.        the barrier heights of ideal Schottky contacts but also
The mean values of the barrier heights of Al- and            the valence-band offsets of semiconductor heterostruc-
Zn/p-ZnSe contacts, which were calculated for differ-        tures. Here again, the Schottky contacts are the more
ent interface configurations using ab-initio LDA-DF           important case, since their zero-charge-transfer barrier
theory and a-posteriori spin-orbit and quasi-particle cor-   heights equal the branch-point energies of the semicon-
rections [8.45, 46], agree with the experimental data to     ductors, while the valence-band offsets are determined
within the margins of experimental error. The same           by the differences in the branch-point energies of the
conclusion was reached for Al/Al1−x Gax As Schottky          semiconductors in contact.
                                                                                                                                    Part A 8




References

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Part A 8
                                                                                                                                  161




                     Charge Transp
                       9. Charge Transport in Disordered Materials




                                                                                                                                  Part A 9
                                                              9.1    General Remarks on Charge Transport
 This chapter surveys general theoretical concepts
                                                                     in Disordered Materials ........................ 163
 developed to qualitatively understand and to
 quantitatively describe the electrical conduction
 properties of disordered organic and inorganic               9.2    Charge Transport in Disordered Materials
                                                                     via Extended States ............................. 167
 materials. In particular, these concepts are applied
 to describe charge transport in amorphous and
 microcrystalline semiconductors and in conjugated            9.3    Hopping Charge Transport in Disordered
                                                                     Materials via Localized States ...............       169
 and molecularly doped polymers. Electrical
                                                                     9.3.1 Nearest-Neighbor Hopping.........              170
 conduction in such systems is achieved through
                                                                     9.3.2 Variable-Range Hopping ............            172
 incoherent transitions of charge carriers between
                                                                     9.3.3 Description of Charge-Carrier
 spatially localized states. Basic theoretical ideas                        Energy Relaxation and Hopping
 developed to describe this type of electrical                              Conduction in Inorganic
 conduction are considered in detail. Particular                            Noncrystalline Materials.............         173
 attention is given to the way the kinetic coefficients               9.3.4 Description of Charge Carrier
 depend on temperature, the concentration of                                Energy Relaxation and Hopping
 localized states, the strength of the applied                              Conduction in Organic
 electric field, and the charge carrier localization                         Noncrystalline Materials.............         180
 length. Charge transport via delocalized states in
 disordered systems and the relationships between             9.4    Concluding Remarks ............................ 184
 kinetic coefficients under the nonequilibrium
 conditions are also briefly reviewed.                         References .................................................. 185




Many characteristics of charge transport in disordered        of a single modifier with the same total concentration of
materials differ markedly from those in perfect crys-         ions. A comprehensive description of these effects can
talline systems. The term “disordered materials” usually      be found in the review article of Bunde et al. [9.1]. Al-
refers to noncrystalline solid materials without perfect      though these effects sometimes appear puzzling, they
order in the spatial arrangement of atoms. One should         can be naturally and rather trivially explained using rou-
distinguish between disordered materials with ionic           tine classical percolation theory [9.2]. The description of
conduction and those with electronic conduction. Dis-         ionic conduction in glasses is much simplified by the in-
ordered materials with ionic conduction include various       ability of ions to tunnel over large distances in the glass
glasses consisting of a “network-formers” such as SiO2 ,      matrix in single transitions. Every transition occurs over
B2 O3 and Al2 O3 , and of “network-modifiers” such as          a rather small interatomic distance, and it is relatively
Na2 O, K2 O and Li2 O. When an external voltage is ap-        easy to describe such electrical conductivity theoreti-
plied, ions can drift by hopping over potential barriers in   cally [9.2]. On the other hand, disordered systems with
the glass matrix, contributing to the electrical conduction   electronic conduction have a much more complicated
of the material. Several fascinating effects have been ob-    theoretical description. Transition probabilities of elec-
served for this kind of electrical conduction. One is the     trons between spatially different regions in the material
extremely nonlinear dependence of the conductivity on         significantly depend not only on the energy parameters
the concentration of ions in the material. Another beau-      (as in the case of ions), but also on spatial factors such as
tiful phenomenon is the so-called “mixed alkali effect”:      the tunnelling distance, which can be rather large. The
mixing two different modifiers in one glass leads to an        interplay between the energy and spatial factors in the
enormous drop in the conductivity in comparison to that       transition probabilities of electrons makes the develop-
  162      Part A   Fundamental Properties



                    ment of a theory of electronic conduction in disordered       dered materials. It was (and still is) a highly challenging
Part A 9




                    systems challenging. Since the description of electronic      task to develop a consistent theory of charge transport
                    conduction is less clear than that of ionic conduction, and   in such systems. On the other hand, the explosion in
                    since disordered electronic materials are widely used for     research into charge transport in disordered materials
                    various device applications, in this chapter we concen-       is related to the various current and potential device
                    trate on disordered materials with the electronic type of     applications of such systems. These include the appli-
                    electrical conduction.                                        cation of disordered inorganic and organic materials in
                         Semiconductor glasses form one class of such mater-      photovoltaics (the functioning material in solar cells),
                    ials. This class includes amorphous selenium, a-Se and        in electrophotography, in large-area displays (they are
                    other chalcogenide glasses, such as a-As2 Se3 . These         used in thin film transistors), in electrical switching
                    materials are usually obtained by quenching from the          threshold and memory devices, in light-emitting diodes,
                    melt. Another broad class of disordered materials, inor-      in linear image sensors, and in optical recording de-
                    ganic amorphous semiconductors, includes amorphous            vices. Readers interested in the device applications of
                    silicon a-Si, amorphous germanium a-Ge, and their al-         disordered materials should be aware that there are nu-
                    loys. These materials are usually prepared as thin films       merous monographs on this topic: the literature on this
                    by the deposition of atomic or molecular species. Hy-         field is very rich. Several books are recommended (see
                    drogenated amorphous silicon, a-Si:H, has attracted           [9.3–12]), as are numerous review articles referred to in
                    much attention from researchers, since incorporation          these books.
                    of hydrogen significantly improves conduction, mak-                In this chapter we focus on disordered semiconduc-
                    ing it favorable for use in amorphous semiconductor           tor materials, ignoring the broad class of disordered
                    devices. Many other disordered materials, such as hy-         metals. In order to describe electronic transport in
                    drogenated amorphous carbon (a-C:H) and its alloys,           disordered metals, one can more or less successfully
                    polycrystalline and microcrystalline silicon are similar      apply extended and modified conventional theoretical
                    to a-Si:H in terms of their charge transport properties.      concepts developed for electron transport in ordered
                    Some crystalline materials can also be considered to          crystalline materials, such as the Boltzmann kinetic
                    be disordered systems. This is the case for doped crys-       equation. Therefore, we do not describe electronic
                    tals if transport phenomena within them are determined        transport in disordered metals here. We can recom-
                    by randomly distributed impurities, and for mixed crys-       mend a comprehensive monograph to interested readers
                    tals with disordered arrangements of various types of         (see [9.13]), in which modern concepts about conduction
                    atoms in the crystalline lattice. In recent years much re-    in disordered metals are presented beautifully.
                    search has also been devoted to the study of organic              Several nice monographs on charge transport in dis-
                    disordered materials, such as conjugated and molecu-          ordered semiconductors are also available. Although
                    larly doped polymers and organic glasses, since these         many of them were published several years ago (some
                    systems has been shown to possess electronic proper-          even decades ago), we can recommend them to the in-
                    ties similar to those of inorganic disordered materials,      terested reader as a source of information on important
                    while they are easier to manufacture than the latter          experimental results. These results have permitted re-
                    systems.                                                      searchers the present level of understanding of transport
                         There are two reasons for the great interest of re-      phenomena in disordered inorganic and organic mater-
                    searchers in the conducting properties of disordered          ials. A comprehensive collection of experimental data
                    materials. On the one hand, disordered systems repre-         for noncrystalline materials from the books specified
                    sent a challenging field in a purely academic sense. For       above would allow one to obtain a picture of the modern
                    many years the theory of how semiconductors perform           state of experimental research in the field.
                    charge transport was mostly confined to crystalline sys-           We will focus in this chapter on the theoretical de-
                    tems where the constituent atoms are in regular arrays.       scription of charge transport in disordered materials,
                    The discovery of how to make solid amorphous mater-           introducing some basic concepts developed to describe
                    ials and alloys led to an explosion in measurements of        electrical conduction. Several excellent books already
                    the electronic properties of these new materials. How-        exist in which a theoretical description of charge trans-
                    ever, the concepts often used in textbooks to describe        port in disordered materials is the main topic. Among
                    charge carrier transport in crystalline semiconductors        others we can recommend the books of Shklovskii and
                    are based on an assumption of long-range order, and so        Efros [9.14], Zvyagin [9.15], Böttger and Bryksin [9.16],
                    they cannot be applied to electronic transport in disor-      and Overhof and Thomas [9.17]. There appears to be
                    Charge Transport in Disordered Materials    9.1 General Remarks on Charge Transport in Disordered Materials   163



a time gap in which comprehensive monographs on the            some new and rather powerful theoretical concepts were




                                                                                                                                  Part A 9.1
theoretical description of electrical conduction in disor-     developed. We present these concepts below, along with
dered materials were not published. During this period         some more traditional ones.



9.1 General Remarks on Charge Transport in Disordered Materials
Although the literature on transport phenomena in dis-         on the energy spectrum in the vicinity and below the
ordered materials is enormously rich, there are still          mobility edge (in the band tails). Unfortunately this en-
many open questions in this field due to various prob-          ergy spectrum is not known for almost all disordered
lems specific to such materials. In contrast to ordered         materials. A whole variety of optical and electrical in-
crystalline semiconductors with well-defined electronic         vestigation techniques have proven unable to determine
energy structures consisting of energy bands and en-           this spectrum. Since the experimental information on
ergy gaps, the electronic energy spectra of disordered         this spectrum is rather vague, it is difficult to develop
materials can be treated as quasi-continuous. Instead of       a consistent theoretical description for charge transport
bands and gaps, one can distinguish between extended           ab initio. The absence of reliable information on the
and localized states in disordered materials. In an ex-        energy spectrum and on the structures of the wavefunc-
tended state, the charge carrier wavefunction is spread        tions in the vicinity and below the mobility edges can
over the whole volume of a sample, while the wave-             be considered to be the main problem for researchers at-
function of a charge carrier is localized in a spatially       tempting to quantitatively describe the charge transport
restricted region in a localized state, and a charge car-      properties of disordered materials.
rier present in such a state cannot spread out in a plane          An overview of the energy spectrum in a disordered
wave as in ordered materials. Actually, localized elec-        semiconductor is shown in Fig. 9.1. The energy levels
tron states are known in ordered systems too. Electrons        εv and εc denote the mobility edges for the valence and
and holes can be spatially localized when they occupy          conduction bands, respectively. Electron states in the
donors or acceptors or some other impurity states or           mobility gap between these energies are spatially local-
structural defects in ordered crystalline materials. How-      ized. The states below εv and above εc can be occupied
ever, the localized states usually appear as δ-like discrete   by delocalized holes and electrons. Some peaks in the
energy levels in the energy spectra of such materials.         DOS are shown in the mobility gap, which can be created
In disordered semiconductors, on the other hand, en-           by some defects with particularly high concentrations.
ergy levels related to spatially localized states usually      Although there is a consensus between researchers on
fill the energy spectrum continuously. The energy that          the general view of the DOS in disordered materials,
separates the extended states from the localized ones in
disordered materials is called the mobility edge. To be
                                                                       g(ε) [cm–3eV–1]
precise, we will mostly consider the energy states for
                                                                1021
electrons in the following. In this case, the states above
the mobility edge are extended and the states below the
edge are localized. The localized states lie energetically      1020
above the extended states for holes. The energy region
between the mobility edges for holes and electrons is
                                                                1019
called the mobility gap. The latter is analogous to the
band gap in ordered systems, although the mobility gap
contains energy states, namely the spatially localized          1018
states. Since the density of states (DOS), defined as the
number of states per unit energy per unit volume, usu-
                                                                1017
ally decreases when the energy moves from the mobility                          εv                          εc
edges toward the center of the mobility gap, the energy                                                          Energy
regions of localized states in the vicinity of the mobility    Fig. 9.1 Density of states of a noncrystalline semiconductor
edges are called band tails. We would like to emphasize        (schematic); εv and εc correspond to mobility edges in the
that the charge transport properties depend significantly       conduction band and the valence band, respectively
  164        Part A   Fundamental Properties



                      the particular structure of the energy spectrum is not            Another important characteristic of the electrical
Part A 9.1




                      known for most disordered systems. From a theoretical         properties of a disordered material is its alternating
                      point of view, it is enormously difficult to calculate this    current (AC) conductivity measured when an external
                      spectrum.                                                     alternating electric field with some frequency ω is ap-
                          There are several additional problems that make the       plied. It has been established in numerous experimental
                      study of charge transport in disordered materials more        studies that the real part of the AC conductivity in most
                      difficult than in ordered crystalline semiconductors. The      disordered semiconductors depends on the frequency
                      particular spatial arrangements of atoms and molecules        according to the power law
                      in different samples with the same chemical composition
                                                                                        Re σ(ω) = Cωs ,                                     (9.2)
                      can differ from each other depending on the preparation
                      conditions. Hence, when discussing electrical conduc-         where C is constant and the power s is usually smaller
                      tion in disordered materials one often should specify         than unity. This power law has been observed in nu-
                      the preparation conditions. Another problem is related        merous materials at different temperatures over a wide
                      to the long-time relaxation processes in disordered sys-      frequency range. This frequency dependence differs
                      tems. Usually these systems are not in thermodynamic          drastically from that predicted by the standard kinetic
                      equilibrium and the slow relaxation of the atoms toward       theory developed for quasi-free charge carriers in crys-
                      the equilibrium arrangement can lead to some changes          talline systems. In the latter case, the real part of the AC
                      in electrical conduction properties. In some disordered       conductivity has the frequency dependence
                      materials a long-time electronic relaxation can affect the
                                                                                                      ne2     τ
                      charge transport properties too, particularly at low tem-         Re σ(ω) =                   ,                       (9.3)
                      peratures, when electronic spatial rearrangements can be                         m 1 + ω2 τ 2
                      very slow. At low temperatures, when tunneling electron       where n is the concentration of charge carriers, e is
                      transitions between localized states dominate electri-        the elementary charge, m is the effective mass and τ
                      cal conduction, this long-time electron relaxation can        is the momentum relaxation time. Since the band elec-
                      significantly affect the charge transport properties.          trons in crystalline semiconductors usually have rather
                          It is fortunate that, despite these problems, some        short momentum relaxation times, τ ≈ 10−14 s, the con-
                      general transport properties of disordered semiconduc-        tribution of charge carriers in delocalized states to the
                      tors have been established. Particular attention is usually   AC conductivity usually does not depend on frequency
                      paid to the temperature dependence of the electrical          at ω τ −1 . Therefore, the observed frequency depen-
                      conductivity, since this dependence can indicate the un-      dence described by (9.2) should be ascribed to the
                      derlying transport mechanism. Over a broad temperature        contribution of charge carriers in localized states.
                      range, the direct current (DC) conductivity in disordered          One of the most powerful tools used to study the
                      materials takes the form                                      concentrations of charge carriers and their mobilities
                                                                                    in crystalline semiconductors is the provided by mea-
                                                    β
                                            Δ(T )                                   surements of the Hall constant, RH . Such measurements
                         σ = σ0 exp −                   ,                   (9.1)
                                            kB T                                    also provide direct and reliable information about the
                                                                                    sign of the charge carriers in crystalline materials. Un-
                      where the pre-exponential factor σ0 depends on the un-        fortunately, this is not the case for disordered materials.
                      derlying system and the power exponent β depends on           Moreover, several anomalies have been established for
                      the material and also sometimes on the temperature            Hall measurements in the latter systems. For example,
                      range over which the conductivity is studied; Δ(T ) is        the sign of the Hall constant in disordered materials
                      the activation energy. In many disordered materials, like     sometimes differs from that of the thermoelectric power,
                      vitreous and amorphous semiconductors, σ0 is of the or-       α. This anomaly has not been observed in crystalline
                      der of 102 –104 Ω−1 cm−1 . In such materials the power        materials. The anomaly has been observed in liquid and
                      exponent β is close to unity at temperatures close to         solid noncrystalline semiconductors. Also, in some ma-
                      and higher than the room temperature, while at lower          terials, like amorphous arsenic, a-As, RH > 0, α < 0,
                      temperatures β can be significantly smaller than unity.        while in many other materials other combinations with
                      In organic disordered materials, values of β that are         different signs of RH and α have been experimentally
                      larger than unity also have been reported. For such           established.
                      systems the value β ≈ 2 is usually considered to be                In order to develop a theoretical picture of the trans-
                      appropriate [9.18].                                           port properties of any material, the first issues to clarify
                     Charge Transport in Disordered Materials    9.1 General Remarks on Charge Transport in Disordered Materials       165



are the spectrum of the energy states for charge carri-         wells. The eigenstates in such a model are delocal-




                                                                                                                                       Part A 9.1
ers and the spatial structure of such states. Since these       ized with wavefunctions of the Bloch type. This is
two central issues are yet to be answered properly for          trivial. The problem is to find the solution for a fi-
noncrystalline materials, the theory of charge transport        nite degree of disorder (W = 0). The result from the
in disordered systems should be considered to be still in       Anderson model for such a case is described as fol-
its embryonic stage.                                            lows. At some particular value for the ratio W/(z I ),
     The problem of deducing electron properties in a ran-      where z is the coordination number of the lattice, all
dom field is very complicated, and the solutions obtained        electron states of the system are spatially localized.
so far only apply to some very simple models. One               At smaller values of W/(z I ) some states in the outer
of them is the famous Anderson model that illustrates           regions of the DOS are localized and other states in
the localization phenomenon caused by random disor-             the middle of the DOS energy distribution are spa-
der [9.19]. In this model, one considers a regular system       tially extended, as shown schematically in Fig. 9.3.
of rectangular potential wells with randomly varying            This is one of the most famous results in the trans-
depths, as shown schematically in Fig. 9.2. The ground          port theory of disordered systems. When considering
state energies of the wells are assumed to be randomly          this result, one should note the following points. (i)
distributed over the range with a width of W. First,            It was obtained using a single-electron picture without
one considers the ordered version of the model, with            taking into account long-range many-particle interac-
W equal to zero. According to conventional band the-            tions. However, in disordered systems with localized
ory, a narrow band arises in the ordered system where           electrons such interactions can lead to the localization
the energy width depends on the overlap integral I              of charge carriers and they often drastically influence
between the electron wavefunctions in the adjusting             the energy spectrum [9.14]. Therefore the applicability
                                                                of the single-electron Anderson result to real systems
                                                                is questionable. (ii) Furthermore, the energy structure
 Energy
                                                                of the Anderson model shown in Fig. 9.3 strongly con-
                                                                tradicts that observed in real disordered materials. In
                                                                real systems, the mobility gap is located between the
                                                                mobility edges, as shown in Fig. 9.1, while in the An-
                                                                derson model the energy region between the mobility
                                                                edges is filled with delocalized states. Moreover, in
                                                                one-dimensional and in some two-dimensional systems,
                                                                the Anderson model predicts that all states are local-
                                                                ized at any amount of disorder. These results are of
                                                      Space     little help when attempting to interpret the DOS scheme
Fig. 9.2 Anderson model of disorder potential                   in Fig. 9.1.
                                                                     A different approach to the localization problem is
 ε
                                                                to try to impose a random potential V (x) onto the band
                                                                structure obtained in the frame of a traditional band
                                                                theory. Assuming a classical smoothly varying (in space)


                                                                 a)                     b)                     c)

                                                        g (ε)




                                                                Fig. 9.4a–c Disorder potential landscape experienced by a charge
                                                                carrier (a). Regions with energies below some given energy level
                                                                E c are colored black. In frame (b) this level is very low and there
Fig. 9.3 Density of states in the Anderson model. Hatched       is no connected path through the system via black regions. In frame
regions in the tails correspond to spatially localized states   (c) the level E c corresponds to the classical percolation level
 166         Part A   Fundamental Properties



                      potential V (x) with a Gaussian distribution function           where
Part A 9.1




                                                                                                          1
                                   1     V2                                               f (ε) =                    .                      (9.7)
                          F(V ) = √ exp − 2                 ,                 (9.4)                 1 + exp   ε−εF
                                 ε0 2π   2ε0                                                                  kB T

                      one can solve the localization problem using the classi-        Here T is the temperature and kB is the Boltzmann
                      cal percolation theory illustrated in Fig. 9.4. In Fig. 9.4a,   constant.
                      an example of a disorder potential experienced by elec-             The Fermi level in almost all known disordered
                      trons is shown schematically. In Fig. 9.4b and Fig. 9.4c        semiconductors under real conditions is situated in the
                      the regions below a given energy level E c are colored          mobility gap – in the energy range which corresponds
                      black. In Fig. 9.4b this level is positioned very low, so       to spatially localized electron states. The charge carrier
                      that regions with energies below E c do not provide a con-      mobility μ(ε) in the localized states below the mobil-
                      nected path through the system. In Fig. 9.4c an infinite         ity edge is much less than that in the extended states
                      percolation cluster consisting only of black regions ex-        above the mobility edge. Therefore, at high tempera-
                      ists. The E c that corresponds to the first appearance of        tures, when a considerable fraction of electrons can be
                      such a connected path is called the classical percolation       found in the delocalized states above the mobility edge,
                      level [9.14]. Mathematically soluving the percolation           these states dominate the electrical conductivity of the
                      problem shows that the mobility edge identified with the         system. The corresponding transport mechanism under
                      classical percolation level in the potential V (x) is shifted   such conditions is similar to that in ordered crystalline
                      with respect to the band edge of the ordered system by          semiconductors. Electrons in the states within the en-
                      an amount ξε0 , where ξ ≈ 0.96 towards the center of the        ergy range of the width, of the order kB T above the
                      bandgap [9.15]. A similar result, though with a different       mobility edge, dominate the conductivity. In such a case
                      constant ξ, can be obtained via a quantum-mechanical            the conductivity can be estimated as
                      treatment of a short-range potential V (x) of white-noise          σ ≈ eμc n(εc )kB T ,                               (9.8)
                      type [9.20]. As the amplitude ε0 of the random potential
                      increases the band gap narrows, while the conduction            where μc is the electron mobility in the states above
                      and valence bands become broader. Although this result          the mobility edge εc , and n(εc )kB T is their concen-
                      is provided by both limiting models – by the classical          tration. This equation is valid under the assumption
                      one with a long-range smoothly varying potential V (x)          that the typical energy scale of the DOS function g(ε)
                      and by the quantum-mechanical one with a short-range            above the mobility edge is larger than kB T . The posi-
                      white-noise potential V (x) – none of the existing theories     tion of the Fermi level in disordered materials usually
                      can reliably describe the energy spectrum of a disordered       depends on temperature only slightly. Combining (9.6)–
                      material and the properties of the charge carrier wave-         (9.8), one obtains the temperature dependence of the DC
                      functions in the vicinity of the mobility edges, in other       conductivity in the form
                      words in the energy range which is most important for                                  Δ
                      charge transport.                                                  σ = σ0 exp −              ,                        (9.9)
                                                                                                            kB T
                           The DC conductivity can generally be represented
                                                                                      described by (9.1) with β = 1 and constant activation
                      in the form
                                                                                      energy, which is observed in most disordered semicon-
                          σ =e      μ(ε)n(ε) dε ,                             (9.5)   ductors at high temperatures.
                                                                                           In order to obtain the numerical value of the conduc-
                      where e is the elementary charge, n(ε) dε is the con-           tivity in this high-temperature regime, one needs to know
                      centration of electrons in the energy range between ε           the density of states in the vicinity of the mobility edge
                      and ε + dε and μ(ε) is the mobility of these electrons.         g(εc ), and also the magnitude of the electron mobility μc
                      The integration is carried out over all energies ε. Under       in the delocalized states above εc . While the magnitude
                      equilibrium conditions, the concentration of electrons          of g(εc ) is usually believed to be close to the DOS value
                      n(ε) dε is determined by the density of states g(ε) and         in the vicinity of the band edge in crystalline semicon-
                      the Fermi function f (ε), which depends on the position         ductors, there is no consensus among researchers on the
                      of the Fermi energy εF (or a quasi-Fermi energy in the          magnitude of μc . In amorphous semiconductors μc is
                      case of the stationary excitation of electrons):                usually estimated to be in the range of 1 cm2 /V s to
                                                                                      10 cm2 /V s. Unfortunately, there are no reliable theo-
                          n(ε) = g(ε) f (ε) ,                                 (9.6)   retical calculations of this quantity for most disordered
                    Charge Transport in Disordered Materials     9.2 Charge Transport in Disordered Materials via Extended States   167



materials. The only exception is provided by so-called         such systems. This method can be extended to other dis-




                                                                                                                                    Part A 9.2
mixed crystals, which are also sometimes called crys-          ordered materials, provided the statistical properties of
talline solid solutions. In the next section we describe the   the disorder potential, essential for electron scattering,
theoretical method which allows one to estimate μc in          are known.



9.2 Charge Transport in Disordered Materials via Extended States
The states with energies below εv and above εc in disor-       no consensus between the theorists on how to describe
dered materials are believed to possess similar properties     charge carrier transport in the latter states. Moreover, it
to those of extended states in crystals. Various exper-        is not clear whether the energy at which the carrier mo-
imental data suggest that these states in disordered           bility drops coincides with the mobility edge or whether
materials are delocalized states. However, traditional         it is located above the edge in the extended states. Nu-
band theory is largely dependent upon the system having        merous discussions of this question, mostly based on
translational symmetry. It is the periodic atomic struc-       the scaling theory of localization, can be found in spe-
ture of crystals that allows one to describe electrons and     cial review papers. For the rest of this section, we skip
holes within such a theory as quasi-particles that exhibit     this rather complicated subject and instead we focus on
behavior similar to that of free particles in vacuum, al-      the description of charge carrier transport in a semicon-
beit with a renormalized mass (the so-called “effective        ductor with a short-range random disorder potential of
mass”). The energy states of such quasi-particles can          white-noise type. This seems to be the only disordered
be described by their momentum values. The wavefunc-           system where a reliable theory exists for charge carrier
tions of electrons in these states (the so-called Bloch        mobility via extended states above the mobility edge.
functions) are delocalized. This means that the proba-         Semiconductor solid solutions provide an example of a
bility of finding an electron with a given momentum             system with this kind of random disorder [9.20–25].
is equal at corresponding points of all elementary cells            Semiconductor solid solutions Ax B1−x (mixed crys-
of the crystal, independent on the distance between the        tals) are crystalline semiconductors in which the sites
cells.                                                         of the crystalline sublattice can be occupied by atoms
     Strictly speaking, the traditional band theory fails in   of two different types, A and B. Each site can be oc-
the absence of translational symmetry – for disordered         cupied by either an A or a B atom with some given
systems. Nevertheless, one still assumes that the charge       probability x between zero and unity. The value x is of-
carriers present in delocalized states in disordered ma-       ten called the composition of the material. Due to the
terials can be approximately described by wavefunctions        random spatial distributions of the A and B atoms, lo-
with a spatially homogeneous probability of finding             cal statistical fluctuations in the composition inside the
a charge carrier with a given quasi-momentum. As for           sample are unavoidable, meaning that mixed crystals
crystals, one starts from the quasi-free particle picture      are disordered systems. Since the position of the band
and considers the scattering effects in a perturbation         edge depends on the composition x, these fluctuations
approach following the Boltzmann kinetic description.          in local x values lead to the disorder potential for elec-
This description is valid if the de Broglie wavelength of      trons and holes within the crystal. To be precise, we will
the charge carrier λ = / p is much less than the mean          consider the influence of the random potential on a con-
free path l = vτ, where τ is the momentum relaxation           duction band electron. Let E c (x) be the conduction band
time and p and v are the characteristic values of the          minimum for a crystal with composition x. In Fig. 9.5
momentum and velocity, respectively. This validity con-        a possible schematic dependence E c (x) is shown. If the
dition for the description based on the kinetic Boltzmann      average composition for the whole sample is x0 , the lo-
equation can also be expressed as /τ          ε, where ε is    cal positions of the band edge E c (x) fluctuate around
the characteristic kinetic energy of the charge carriers,      the average value E c (x0 ) according to the fluctuations
which is equal to kB T for a nondegenerate electron gas        of the composition x around x0 . For small deviations in
and to the Fermi energy in the degenerate case. While          composition Δx from the average value, one can use the
this description seems valid for delocalized states far        linear relation
from the mobility edges, it fails for energy states in the
vicinity of the mobility edges. So far, there has been             E c (x0 + Δx) = E c (x0 ) + αΔx ,               (9.10)
 168         Part A   Fundamental Properties



                                                                                     Charge carriers in mixed crystals are scattered by com-
Part A 9.2




                             Band edge Ec (eV)
                       3.0                                                           positional fluctuations. As is usual in kinetic descriptions
                                                                                     of free electrons, the fluctuations on the spatial scale of
                       2.5                                                           the order of the electron wavelength are most efficient.
                       2.0                                                           Following Shlimak et al. [9.23], consider an isotropic
                                                                                     quadratic energy spectrum
                       1.5
                                                                                               p2
                       1.0                                                              εp =       ,                                      (9.16)
                                                                                               2m
                       0.5                                                           where p and m are the quasi-momentum and the effec-
                       0.0                                                           tive mass of an electron, respectively. The scattering rate
                                                     x0   x0 + x                     for such an electron is
                                                                   Composition x
                                                                                               2π            2
                      Fig. 9.5 Schematic dependence of the conduction band               νp =            Vq      1 − cos ϑq δ ε p − ε p−q ,
                      edge εc on composition x in a mixed crystal Ax B1−x                               q
                                                                                                                                         (9.17)
                      where
                                                                                     where ϑq is the scattering angle and
                            dE c (x)
                         α=                      .                          (9.11)              2       1
                              dx          x=x0                                             Vq       =       d3r exp (iqr) V (r)V (0) .   (9.18)
                                                                                                        Ω
                      If the deviation of the concentration of A atoms from its
                      mean value in some region of a sample is ξ(r) and the          The quantity Ω in this formula is the normalization vol-
                      total concentration of (sub)lattice sites is N, the devia-     ume. Using the correlation function (9.14), one obtains
                      tion of the composition in this region is Δx = ξ(r)/N,         the relation
                      and the potential energy of an electron at the bottom of                  2     α2 x(1 − x)
                      the conduction band is                                               Vq       =             ,                       (9.19)
                                                                                                          ΩN
                                     ξ(r)                                            which shows that the scattering by compositional
                           V (r) = α      .                                (9.12)
                                      N                                              fluctuations is equivalent to that by a short-range po-
                      Although one calls the disorder in such systems a “short-      tential [9.23]. Substituting (9.19) into (9.17) one obtains
                      range” disorder, it should be noted that the consideration     the following expression for the scattering rate [9.20]
                      is valid only for fluctuations that are much larger than the             α2 x(1 − x)m p
                      lattice constant of the material. The term “short-range”          νp =                 .                       (9.20)
                      is due to the assumption that the statistical properties                    π 4N
                      of the disorder are absolutely uncorrelated. This means        This formula leads to an electron mobility of the fol-
                      that potential amplitudes in the adjusting spatial points      lowing form in the framework of the standard Drude
                      are completely uncorrelated to each other. Indeed, it          approach [9.20, 23]
                      is usually assumed that the correlation function of the
                                                                                            π 3/2        e 4N
                      disorder in mixed crystals can be approximated by a               μC = √ 2                           .             (9.21)
                      white-noise correlation function of the form                          2 2 α x(1 − x)m 5/2 (kB T )1/2

                             ξ(r)ξ(r ) = x(1 − x)Nδ(r − r ) .               (9.13)
                                                                                     Very similar formulae can be found in many recent pub-
                                                                                     lications (see for example Fahy and O’Reily [9.26]). It
                      The random potential caused by such compositional              has also been modified and applied to two-dimensional
                      fluctuations is then described by the correlation func-         systems [9.27] and to disordered diluted magnetic semi-
                      tion [9.20]                                                    conductors [9.28].
                                                                                         It would not be difficult to apply this theoretical
                             V (r)V (r ) = γδ(r − r )                       (9.14)   description to other disordered systems, provided the
                      with                                                           correlation function of the disorder potential takes the
                                                                                     form of (9.14) with known amplitude γ . However, it
                                 α2                                                  is worth emphasizing that the short-range disorder of
                         γ=         x(1 − x) .                              (9.15)
                                 N                                                   white-noise type considered here is a rather simple
           Charge Transport in Disordered Materials     9.3 Hopping Charge Transport in Disordered Materials via Localized States   169



model that cannot be applied to most disordered ma-                  In the following section we present the general con-




                                                                                                                                    Part A 9.3
terials. Therefore, we can conclude that the problem of         cepts developed to describe electrical conduction in
theoretically describing charge carrier mobility via de-        disordered solids at temperatures where tunneling tran-
localized states in disordered materials is still waiting to    sitions of electrons between localized states significantly
be solved.                                                      contribute to charge transport.



9.3 Hopping Charge Transport in Disordered Materials
    via Localized States
Electron transport via delocalized states above the
mobility edge dominates the electrical conduction of dis-
ordered materials only at temperatures high enough to
cause a significant fraction of the charge carriers fill these
states. As the temperature decreases, the concentration
                                                                                                                   εj
of the electrons described by (9.9) decreases exponen-                                     rij
tially and so their contribution to electrical conductivity
diminishes. Under these circumstances, tunneling tran-                  εi
sitions of electrons between localized states in the band
                                                                                  α
tails dominate the charge transport in disordered semi-
conductors. This transport regime is called hopping             Fig. 9.6 Hopping transition between two localized states i
conduction, since the incoherent sequence of tunneling          and j with energies of εi and ε j , respectively. The solid and
transitions of charge carriers resembles a series of their      dashed lines depict the carrier wavefunctions at sites i and
hops between randomly distributed sites. Each site in this      j, respectively; α is the localization radius
picture provides a spatially localized electron state with
some energy ε. In the following we will assume that the         frequency (≈ 1013 s−1 ), although a more rigorous ap-
localized states for electrons (concentration N0 ) are ran-     proach is in fact necessary to determine ν0 . This should
domly distributed in space and their energy distribution        take into account the particular structure of the electron
is described by the DOS function g(ε):                          localized states and also the details of the interaction
             N0     ε                                           mechanism [9.29, 30].
    g(ε) =      G         ,                           (9.22)        When an electron transits from a localized state i to
             ε0     ε0
                                                                a localized state j that is higher in energy, the transi-
where ε0 is the energy scale of the DOS distribution.           tion rate depends on the energy difference between the
    The tunneling transition probability of an electron         states. This difference is compensated for by absorbing
from a localized state i to a localized state j that is lower   a phonon with the corresponding energy [9.31]:
in energy depends on the spatial separation rij between
the sites i and j as                                                                                2rij
                                                                    νij (r, εi , ε j ) = ν0 exp −
                         2rij                                                                        a
    νij (r) = ν0 exp −          ,                     (9.23)
                          α                                                                      ε j − εi + ε j − εi
                                                                                      × exp −                           .
where α is the localization length, which we assume                                                     2kB T
to be equal for sites i and j. This length determines the                                                               (9.24)
exponential decay of the electron wavefunction in the lo-
calized states, as shown in Fig. 9.6. The pre-exponential       Equations (9.23) and (9.24) were written for the case
factor ν0 in (9.23) depends on the electron interaction         in which the electron occupies site i whereas site j is
mechanism that causes the transition. Usually it is as-         empty. If the system is in thermal equilibrium, the occu-
sumed that electron transitions contributing to charge          pation probabilities of sites with different energies are
transport in disordered materials are caused by interac-        determined by Fermi statistics. This effect can be taken
tions of electrons with phonons. Often the coefficient           into account by modifying (9.24) and adding terms that
ν0 is simply assumed to be of the order of the phonon           account for the relative energy positions of sites i and
  170        Part A   Fundamental Properties



                       j with respect to the Fermi energy εF . Taking into ac-        uations the energy-dependent terms in (9.24) and (9.25)
Part A 9.3




                      count these occupation probabilities, one can write the         do not play any significant role and the hopping rates
                      transition rate between sites i and j in the form [9.31]        are determined solely by the spatial terms. The rate of
                                                                                      transition of an electron between two sites i and j is de-
                                           2rij
                         νij = ν0 exp −                                               scribed in this case by (9.23). The average transition rate
                                            a                                         is usually obtained by weighting this expression with the
                                          |εi − εF | + ε j − εF + ε j − εi            probability of finding the nearest neighbor at some par-
                               × exp −                                          .     ticular distance rij , and by integrating over all possible
                                                        2kB T
                                                                                      distances:
                                                                             (9.25)
                                                                                                  ∞
                      Using these formulae, the theoretical description of
                                                                                           ν =        dr ν0
                      hopping conduction is easily formulated. One has to
                      calculate the conductivity provided by transition events                   0
                      (the rates of which are described by (9.25)) in the man-                            2r                      4π 3
                                                                                                 × exp −        4πr 2 N0 exp −       r N0
                      ifold of localized states (where the DOS is described by                            α                        3
                      (9.22)).                                                                ≈ πν0 N0 α .
                                                                                                        3
                                                                                                                                            (9.26)

                      9.3.1 Nearest-Neighbor Hopping                                  Assuming that this average hopping rate describes the
                                                                                      mobility, diffusivity and conductivity of charge carriers,
                      Before presenting the correct solution to the hopping           one apparently comes to the conclusion that these quan-
                      problem we would like to emphasize the following. The           tities are linearly proportional to the density of localized
                      style of the theory for electron transport in disordered        states N0 . However, experiments evidence an exponen-
                      materials via localized states significantly differs from        tial dependence of the transport coefficients on N0 .
                      that used for theories of electron transport in ordered              Let us look therefore at the correct solution to the
                      crystalline materials. While the description is usually         problem. This solution is provided in the case considered
                      based on various averaging procedures in crystalline sys-       here, N0 α3 1, by percolation theory (see, for instance,
                      tems, in disordered systems these averaging procedures          Shklovskii and Efros [9.14]). In order to find the trans-
                      can lead to extremely erroneous results. We believe that        port path, one connects each pair of sites if the relative
                      it is instructive to analyze some of these approaches           separation between the sites is smaller than some given
                      in order to illustrate the differences between the de-          distance R, and checks whether there is a continuous
                      scriptions of charge transport in ordered and disordered        path through the system via such sites. If such a path
                      materials. To treat the scattering rates of electrons in        is absent, the magnitude of R is increased and the pro-
                      ordered crystalline materials, one usually proceeds by          cedure is repeated. At some particular value R = Rc ,
                      averaging the scattering rates over the ensemble of scat-       a continuous path through the infinite system via sites
                      tering events. A similar procedure is often attempted           with relative separations R < Rc arises. Various math-
                      for disordered systems too, although various textbooks          ematical considerations give the following relation for
                      (see, for instance, Shklovskii and Efros [9.14]) illustrate     Rc [9.14]:
                      how erroneous such an approach can be in the case of
                                                                                          4π
                      disordered materials.                                                  N0 Rc = Bc ,
                                                                                                 3
                                                                                                                                            (9.27)
                           Let us consider the simplest example of hopping                 3
                      processes, namely the hopping of an electron through            where Bc = 2.7 ± 0.1 is the average number of neighbor-
                      a system of isoenergetic sites randomly distributed in          ing sites available within a distance of less than Rc . The
                      space with some concentration N0 . It will be always as-        corresponding value of Rc should be inserted into (9.23)
                      sumed in this chapter that electron states are strongly         in order to determine kinetic coefficients such as the mo-
                      localized and the strong inequality N0 α3 1 is ful-             bility, diffusivity and conductivity. The idea behind this
                      filled. In such a case the electrons prefer to hop between       procedure is as follows. Due to the exponential depen-
                      the spatially nearest sites and therefore this transport        dence of the transition rates on the distances between
                      regime is often called nearest-neighbor hopping (NNH).          the sites, the rates for electron transitions over distances
                      This type of hopping transport takes place in many real         r < Rc are much larger than those over distances Rc .
                      systems at temperatures where the thermal energy kB T           Such fast transitions do not play any significant role as
                      is larger than the energy scale of the DOS. In such sit-        a limiting factor in electron transport and so they can
           Charge Transport in Disordered Materials     9.3 Hopping Charge Transport in Disordered Materials via Localized States   171



                                                                lief of many researchers in the validity of the procedure




                                                                                                                                    Part A 9.3
                                   Rc                           based on the averaging of hopping rates is so strong that
                                                                the agreement between (9.28) and experimental data is
                                                                often called occasional. We would like to emphasize
                                                                once more that the ensemble averaging of hopping rates
                                                                leads to erroneous results. The magnitude of the aver-
                                                                age rate in (9.26) is dominated by rare configurations
                                                                of very close pairs of sites with separations of the order
                                                                of the localization length α. Of course, such pairs allow
                                                                very fast electron transitions, but electrons cannot move
                                                                over considerable distances using only these close pairs.
                                                                Therefore the magnitude of the average transition rate is
                                                                irrelevant for calculations of the hopping conductivity.
                                                                The correct concentration dependence of the conductiv-
                                                                ity is given by (9.28). This result was obtained under
                                                                the assumption that only spatial factors determine tran-
                                                                sition rates of electrons via localized states. This regime
                                                                is valid at reasonably high temperatures.
                                                                     If the temperature is not as high and the ther-
Fig. 9.7 A typical transport path with the lowest resistance.   mal energy kB T is smaller than the energy spread
Circles depict localized states. The arrow points out the       of the localized states involved in the charge trans-
most “difficult” transition, with length Rc                      port process, the problem of calculating the hopping
                                                                conductivity becomes much more complicated. In this
be neglected in calculations of the resistivity of the sys-     case, the interplay between the energy-dependent and
tem. Transitions over distances Rc are the slowest among        the distance-dependent terms in (9.24) and (9.25) de-
those that are necessary for DC transport and hence such        termines the conductivity. The lower the temperature,
transitions determine the conductivity. The structure of        the more important the energy-dependent terms in the
the percolation cluster responsible for charge transport        expressions for transition probabilities of electrons in
is shown schematically in Fig. 9.7. The transport path          (9.24) and (9.25) become. If the spatially nearest-
consists of quasi-one-dimensional segments, each con-           neighboring sites have very different energies, as shown
taining a “difficult” transition over the distance ≈ Rc .        in Fig. 9.8, the probability of an upward electron transi-
Using (9.23) and (9.27), one obtains the dependence             tion between these sites can be so low that it would be
of the conductivity on the concentration of localization        more favorable for this electron to hop to a more dis-
sites in the form                                               tant site at a closer energy. Hence the typical lengths of
                       γ
    σ = σ0 exp −        1/3
                              ,                    (9.28)
                    αN0                                           Energy
where σ0 is the concentration-independent pre-
exponential factor and γ = 1.73 ± 0.03. Such arguments
do not allow one to determine the exponent in the kinetic                                   1
                                                                                                        2
coefficients with an accuracy better than a number of the                                                               εF
order of unity [9.14]. One should note that the quantity
in the exponent in (9.28) is much larger than unity for
a system with strongly localized states when the inequal-
ity N0 α3 1 is valid. This inequality justifies the above
derivation. The dependence described by (9.28) has been                                                 Spatial coordinate
confirmed in numerous experimental studies of the hop-           Fig. 9.8 Two alternative hopping transitions between oc-
ping conductivity via randomly placed impurity atoms            cupied states (filled circles) and unoccupied states. The
in doped crystalline semiconductors [9.14]. The drastic         dashed line depicts the position of the Fermi level. Tran-
difference between this correct result and the erroneous        sitions (1) and (2) correspond to nearest-neighbor hopping
one based on (9.26) is apparent. Unfortunately, the be-         and variable-range hopping regimes, respectively
   172       Part A   Fundamental Properties



                      electron transitions increase with decreasing tempera-         the typical hopping distance from (9.29) as a function of
Part A 9.3




                      ture. This transport regime was termed “variable-range         the energy width Δε in the form
                      hopping”. Next we describe several useful concepts
                      developed to describe this transport regime.                       r(Δε) ≈ [g(εF )Δε]−1/3 ,                          (9.30)

                                                                                     and substitute it into (9.24) in order to express the typical
                      9.3.2 Variable-Range Hopping                                   hopping rate
                      The concept of variable-range hopping (VRH) was put                                2[g(εF )Δε]−1/3    Δε
                      forward by Mott (see Mott and Davis [9.32]) who con-               ν = ν0 exp −                    −            .    (9.31)
                                                                                                                 α         kB T
                      sidered electron transport via a system of randomly
                      distributed localized states at low temperatures. We start     The optimal energy width Δε that provides the maxi-
                      by presenting Mott’s arguments. At low temperatures,           mum hopping rate can be determined from the condition
                      electron transitions between states with energies in the       dν/ dΔε = 0. The result reads
                      vicinity of the Fermi level are most efficient for transport                                3/4
                                                                                                    2kB T
                      since filled and empty states with close energies can only          Δε =                          .                   (9.32)
                      be found in this energy range. Consider the hopping con-                    3g1/3 (εF )
                      ductivity resulting from energy levels within a narrow         After substitution of (9.32) into (9.31) one obtains Mott’s
                      energy strip with width 2Δε symmetric to the Fermi             famous formula for temperature-dependent conductivity
                      level shown in Fig. 9.9. The energy width of the strip         in the VRH regime
                      useful for electron transport can be determined from the
                                                                                                                   1/4
                      relation                                                                              T0
                                                                                         σ = σ0 exp −                      ,               (9.33)
                          g(εF ) · Δε · r 3 (Δε) ≈ 1 .                     (9.29)
                                                                                                            T

                      This criterion is similar to that used in (9.27), although     where T0 is the characteristic temperature:
                      we do not care about numerical coefficients here. Here                          β
                      we have to consider the percolation problem in four-               T0 =               .                              (9.34)
                                                                                                kB g(εF )α3
                      dimensional space since in addition to the spatial terms
                      considered in Sect. 9.3.1 we now have to consider the          Mott gave only a semi-quantitative derivation of (9.33),
                      energy too. The corresponding percolation problem for          from which the exact value of the numerical constant β
                      the transition rates described by (9.25) has not yet been      cannot be determined. Various theoretical studies in 3-D
                      solved precisely. In (9.29) it is assumed that the energy      systems suggest values for β in the range β = 10.0 to
                      width 2Δε is rather small and that the DOS function g(ε)       β = 37.8. According to our computer simulations, the
                      is almost constant in the range εF ± Δε. One can obtain        appropriate value is close to β = 17.6.
                                                                                         Mott’s law implies that the density of states in
                                                                                     the vicinity of the Fermi level is energy-independent.
                               ε                                                     However, it is known that long-range electron–electron
                                                                                     interactions in a system of localized electrons cause
                                                                                     a gap (the so-called Coulomb gap) in the DOS in the
                                                                                     vicinity of the Fermi energy [9.33,34]. The gap is shown
                                                                                     schematically in Fig. 9.10. Using simple semiquantita-
                                                                                     tive arguments, Efros and Shklovskii [9.33] suggested
                       2Δ ε   εF                                                     a parabolic shape for the DOS function
                                                                                                 η κ3
                                                                                         g(ε) =       (ε − εF )2 ,                     (9.35)
                                                                                                  e6
                                                                                     where κ is the dielectric constant, e is the elementary
                                                                                     charge and η is a numerical coefficient. This result was
                                                                                     later confirmed by numerous computer simulations (see,
                                                                           g(ε )     for example, Baranovskii et al. [9.35]). At low temper-
                      Fig. 9.9 Effective region in the vicinity of the Fermi level   atures, the density of states near the Fermi level has
                      where charge transport takes place at low temperatures         a parabolic shape, and it vanishes exactly at the Fermi
            Charge Transport in Disordered Materials    9.3 Hopping Charge Transport in Disordered Materials via Localized States   173



energy. As the temperature rises, the gap disappears (see,     nential, while in organic materials it is usually assumed




                                                                                                                                    Part A 9.3
for example, Shlimak et al. [9.36]).                           to be Gaussian. In these cases, new concepts are needed
     As we have seen above, localized states in the vicin-     in order to describe the hopping conduction. In the next
ity of the Fermi energy are the most useful for transport      section we present these new concepts and calculate the
at low temperatures. Therefore the Coulomb gap es-             way the conductivity depends on temperature and on the
sentially modifies the temperature dependence of the            concentration of localized states in various significantly
hopping conductivity in the VRH regime at low temper-          noncrystalline materials.
atures. The formal analysis of the T -dependence of the
conductivity in the presence of the Coulomb gap is sim-        9.3.3 Description of Charge-Carrier Energy
ilar to that for the Mott’s law discussed above. Using the           Relaxation and Hopping Conduction
parabolic energy dependence of the DOS function, one                 in Inorganic Noncrystalline Materials
arrives at the result
                  ⎡               ⎤                            In most inorganic noncrystalline materials, such as vitre-
                              1/2
                        ˜
                        T0                                     ous, amorphous and polycrystalline semiconductors, the
     σ = σ0 exp ⎣−                ⎦                 (9.36)
                        T                                      localized states for electrons are distributed over a rather
                                                               broad energy range with a width of the order of an elec-
      ˜ ˜                       ˜
with T0 =βe2/(καkB ), where β is a numerical coefficient.       tronvolt. The DOS function that describes this energy
    Equations (9.33) and (9.36) belong to the most fa-         distribution in such systems is believed to have a purely
mous theoretical results in the field of variable-range         exponential shape
hopping conduction. However these formulae are usu-                         N0       ε
ally of little help to researchers working with essentially        g(ε) =      exp −         ,                      (9.37)
                                                                            ε0       ε0
noncrystalline materials, such as amorphous, vitreous
or organic semiconductors. The reason is as follows.           where the energy ε is counted positive from the mobility
The above formulae were derived for the cases of either        edge towards the center of the mobility gap, N0 is the to-
constant DOS (9.33) or a parabolic DOS (9.36) in the en-       tal concentration of localized states in the band tail, and
ergy range associated with hopping conduction. These           ε0 determines the energy scale of the tail. To be precise,
conditions can usually be met in the impurity band of          we consider that electrons are the charge carriers here.
a lightly doped crystalline semiconductor. In the most         The result for holes can be obtained in an analogous way.
disordered materials, however, the energy distribution of      Values of ε0 in inorganic noncrystalline materials are be-
the localized states is described by a DOS function that       lieved to vary between 0.025 eV and 0.05 eV, depending
is very strongly energy-dependent. In amorphous, vit-          on the system under consideration.
reous and microcrystalline semiconductors, the energy               It is worth noting that arguments in favor of a purely
dependence of the DOS function is believed to be expo-         exponential shape for the DOS in the band tails of
                                                               inorganic noncrystalline materials described by (9.37)
            ε
                                                               cannot be considered to be well justified. They are
                                                               usually based on a rather ambiguous interpretation of
                                                               experimental data. One of the strongest arguments in
                                                               favor of (9.37) is the experimental observation of the
                                                               exponential decay of the light absorption coefficient for
                                                               photons with an energy deficit ε with respect to the en-
 2Δ ε ε F                                                      ergy width of the mobility gap (see, for example, Mott
                                                               and Davis [9.32]). One should mention that this argu-
                                                               ment is valid only under the assumption that the energy
                                                               dependence of the absorption coefficient is determined
                                                               solely by the energy dependence of the DOS. However,
                                                               in many cases the matrix element for electron excitation
                                                               by a photon in noncrystalline materials also strongly
                                                               depends on energy [9.14, 37]. Hence any argument for
                                                       g(ε )   the shape of the DOS based on the energy dependence
Fig. 9.10 Schematic view of the Coulomb gap. The insert        of the light absorption coefficient should be taken very
shows the parabolic shape of the DOS near the Fermi level      cautiously. Another argument in favor of (9.37) comes
   174       Part A   Fundamental Properties



                      from the measurements of dispersive transport in time-       level discovered by Grünewald and Thomas [9.39] and
Part A 9.3




                      of-flight experiments. In order to interpret the observed     by Shapiro and Adler [9.40] for equilibrium hopping
                      time dependence of the mobility of charge carriers, one      transport.
                      usually assumes that the DOS for the band tail takes             Shklovskii et al. [9.42] have shown that the same
                      the form of (9.37) (see, for example, Orenstein and          energy level εt also determines the recombination and
                      Kastner [9.38]). One of the main reasons for such an         transport of electrons in the nonequilibrium steady state
                      assumption is probably the ability to solve the problem      under continuous photogeneration in a system with the
                      analytically without elaborate computer work.                DOS described by (9.37).
                          In the following we start our consideration of the           It is clear, then, that the TE determines both equi-
                      problem by also assuming that the DOS in a band tail of      librium and nonequilibrium and both transient and
                      a noncrystalline material has an energy dependence that      steady-state transport phenomena. The question then
                      is described by (9.37). This simple function will allow      arises as to why this energy level is so universal that
                      us to introduce some valuable concepts that have been        electron hopping in its vicinity dominates all transport
                      developed to describe dynamic effects in noncrystalline      phenomena. Below we derive the TE by considering
                      materials in the most transparent analytical form. We        a single hopping event for an electron localized deep in
                      first present the concept of the so-called transport en-      the band tail. It is the transport energy that maximizes
                      ergy, which, in our view, provides the most transparent      the hopping rate as a final electron energy in the hop,
                      description of the charge transport and energy relaxation    independent of its initial energy [9.43]. All derivations
                      of electrons in noncrystalline materials.                    below are carried out for the case kB T < ε0 .
                                                                                       Consider an electron in a tail state with energy εi .
                           The Concept of the Transport Energy                     According to (9.24), the typical rate of downward hop-
                      The crucial role of a particular energy level in the hop-    ping of such an electron to a neighboring localized state
                      ping transport of electrons via localized band-tail states   deeper in the tail with energy ε j ≥ εi is
                      with the DOS described by (9.37) was first recognized
                                                                                                             2r(εi )
                      by Grünewald and Thomas [9.39] in their numerical               ν↓ (εi ) = ν0 exp −              ,               (9.38)
                      analysis of equilibrium variable-range hopping conduc-                                   α
                      tivity. This problem was later considered by Shapiro         where
                      and Adler [9.40], who came to the same conclusion as                    ⎡                 ⎤−1/3
                                                                                                     ∞
                      Grünewald and Thomas, namely that the vicinity of one                    4π
                      particular energy level dominates the hopping transport         r(ε) ≈ ⎣           g(x) dx ⎦         .           (9.39)
                                                                                                3
                      of electrons in the band tails. In addition, they achieved                    εi
                      an analytical formula for this level and showed that its
                      position does not depend on the Fermi energy.                The typical rate of upward hopping for such an electron
                           Independently, the rather different problem of          to a state less deep in the tail with energy ε j ≤ εi is
                      nonequilibrium energy relaxation of electrons by hop-                                    2r(εi − δ)    δ
                      ping through the band tail with the DOS described by            ν↑ (εi , δ) = ν0 exp −              −        ,   (9.40)
                                                                                                                   α        kB T
                      (9.37) was solved at the same time by Monroe [9.41]. He
                      showed that, starting from the mobility edge, an elec-       where δ = εi − ε j ≥ 0. This expression is not exact. The
                      tron most likely makes a series of hops downward in          average nearest-neighbor distance, r(εi − δ), is based on
                      energy. The manner of the relaxation process changes         all states deeper than εi − δ. For the exponential tail,
                      at some particular energy εt , which Monroe called the       this is equivalent to considering a slice of energy with a
                      transport energy (TE). The hopping process near and be-      width of the order ε0 . This works for a DOS that varies
                      low TE resembles a multiple-trapping type of relaxation,     slowly compared with kB T , but not in general. It is
                      with the TE playing a role similar to the mobility edge.     also assumed for simplicity that the localization length,
                      In the multiple-trapping relaxation process [9.38], only     α, does not depend on energy. The latter assumption
                      electron transitions between delocalized states above the    can be easily jettisoned at the cost of somewhat more
                      mobility edge and the localized band-tail states are al-     complicated forms of the following equations.
                      lowed, while hopping transitions between the localized           We will analyze these hopping rates at a given tem-
                      tail states are neglected. Hence, every second transi-       perature T , and try to find the energy difference δ that
                      tion brings the electron to the mobility edge. The TE        provides the fastest typical hopping rate for an electron
                      of Monroe [9.41] coincides exactly with the energy           placed initially at energy εi . The corresponding energy
           Charge Transport in Disordered Materials       9.3 Hopping Charge Transport in Disordered Materials via Localized States         175



difference, δ, is determined by the condition




                                                                                                                                            Part A 9.3
                                                                                       g(ε )
     dν↑ (εi , δ)
                  =0.                              (9.41)
         dδ
Using (9.37), (9.39) and (9.40), we find that the hopping
rate in (9.40) has its maximum at
                   3ε0 (4π/3)1/3 N0 α
                                        1/3                         εt
    δ = εi − 3ε0 ln                     .           (9.42)
                           2kB T
The second term in the right-hand side of (9.42) is called         δ
the transport energy εt after Monroe [9.41]:
                                  1/3                               εi
                  3ε0 (4π/3)1/3 N0 α
      εt = 3ε0 ln                         .              (9.43)
                         2kB T
                                                                       ε
We see from (9.42) that the fastest hop occurs to the
state in the vicinity of the TE, independent of the ini-          Fig. 9.11 Hopping path via the transport energy. In the left frame, the
tial energy εi , provided that εi is deeper in the tail than      exponential DOS is shown schematically. The right frame depicts the
εt ; in other words, if δ ≥ 0. This result coincides with         transport path constructed from upward and downward hops. The
that of Monroe [9.41]. At low temperatures, the TE εt             upward transitions bring the charge carrier to sites with energies
is situated deep in the band tail, and as the tempera-            close to the transport energy εt
ture rises it moves upward towards the mobility edge.
At some temperature Tc , the TE merges with the mo-               for instance, Shklovskii et al. [9.42]). We will consider
bility edge. At higher temperatures, T > Tc , the hopping         only one phenomenon here for illustration, namely the
exchange of electrons between localized band tail states          hopping energy relaxation of electrons in a system with
becomes inefficient and the dynamic behavior of elec-              the DOS described by (9.37). This problem was studied
trons is described by the well-known multiple-trapping            initially by Monroe [9.41].
model (see, for instance, Orenstein and Kastner [9.38]).               Consider an electron in some localized shallow en-
At low temperatures, T < Tc , the TE replaces the mo-             ergy state close to the mobility edge. Let the temperature
bility edge in the multiple-trapping process [9.41], as           be low, T < Tc , so that the TE, εt , lies well below the
shown in Fig. 9.11. The width, W, of the maximum of the           mobility edge, which has been chosen here as a refer-
hopping rate is determined by the requirement that near           ence energy, ε = 0. The aim is to find the typical energy,
εt the hopping rate, ν↑ (εi , δ), differs by less than a factor   εd (t), of our electron as a function of time, t. At early
of e from the value ν↑ (εi , εi − εt ). One finds [9.42]           times, as long as εd (t) < εt , the relaxation is governed
                                                                  by (9.38) and (9.39). The depth εd (t) of an electron in
    W=      6ε0 kB T .                                  (9.44)
                                                                  the band tail is determined by the condition
For shallow states with εi ≤ εt , the fastest hop (on aver-
                                                                       ν↓ [εd (t)] t ≈ 1 .                               (9.45)
age) is a downward hop to the nearest spatially localized
state in the band tail, with the rate determined by (9.38)        This leads to the double logarithmical dependence
and (9.39). We recall that the energies of electron states        εd (t) ∝ ε0 ln[ln(ν0 t)] + C, where constant C depends on
are counted positive downward from the mobility edge              ε0 , N0 , α in line with (9.38) and (9.39). Indeed, (9.38)
towards the center of the mobility gap. This means that           and (9.45) prescribe the logarithmic form of the time
electrons in the shallow states with εi ≤ εt normally hop         dependence of the hopping distance, r(t), and (9.37)
into deeper states with ε > εi , whereas electrons in the         and (9.39) then lead to another logarithmic dependence
deep states with εi > εt usually hop upward in energy             εd [r(t)] [9.41]. At the time
into states near εt in the energy interval W, determined
                                                                             −1          3ε0
by (9.44).                                                             tC ≈ ν0 exp                                       (9.46)
    This shows that εt must play a crucial role in those                                 kB T
phenomena, which are determined by electron hopping               the typical electron energy, εd (t), approaches the TE εt ,
in the band tails. This is indeed the case, as shown in           and the style of the relaxation process changes. At t > tc ,
numerous review articles where comprehensive theo-                every second hop brings the electron into states in the
ries based on the concept of the TE can be found (see,            vicinity of the TE εt from where it falls downward in
  176        Part A   Fundamental Properties



                      energy to the nearest (in space) localization site. In the      charge carriers over energy. Hence it is invalid for de-
Part A 9.3




                      latter relaxation process, the typical electron energy is       scribing the energy relaxation in the exponential tails, in
                      determined by the condition [9.41]                              which electron can move over the full energy width of
                                                                                      the DOS (from a very deep energy state toward the TE)
                          ν↑ [εd (t), εt ] t ≈ 1 ,                           (9.47)   in a single hopping event.
                                                                                          In the equilibrium conditions, when electrons in the
                      where ν↑ [εd (t), εt ] is the typical rate of electron hop-     band tail states are provided by thermal excitation from
                      ping upward in energy toward the TE [9.41]. This                the Fermi energy, a description of the electrical con-
                      condition leads to a typical energy position of the re-         ductivity can easily be derived using (9.5)–(9.7) [9.39].
                      laxing electron at time t of                                    The maximal contribution to the integral in (9.5) comes
                                                                                      from the electrons with energies in the vicinity of the
                                                                                      TE εt , in an energy range with a width, W, described
                          εd (t) ≈ 3ε0 ln [ln (ν0 t)] − ε0 8/ N0 α3      .   (9.48)   by (9.44). Neglecting the temperature dependence of the
                                                                                      pre-exponential factor, σ0 , one arrives at the temperature
                      This is a very important result, which shows that in            dependence of the conductivity:
                      a system where the DOS has a pure exponential energy
                      dependence, described by (9.37), the typical energy of                                2r(εt )       εF − εt
                      a set of independently relaxing electrons would drop                σ ≈ σ0 exp −      −1/3
                                                                                                                      −             ,       (9.49)
                      deeper and deeper into the mobility gap with time. This                              Bc α            kB T
                      result is valid as long as the electrons do not interact with
                      each other, meaning that the occupation probabilities of        where coefficient Bc ≈ 2.7 is inserted in order to take
                      the electron energy levels are not taken into account.          into account the need for a charge carrier to move over
                      This condition is usually met in experimental studies           macroscopic percolation distances in order to provide
                      of transient processes, in which electrons are excited by       low-frequency charge transport.
                      short (in time) pulses, which are typical of time-of-flight           A very similar theory is valid for charge transport in
                      studies of the electron mobility in various disordered ma-      noncrystalline materials under stationary excitation of
                      terials. In this case, only a small number of electrons are     electrons (for example by light) [9.42]. In such a case,
                      present in the band tail states. Taking into account the        one first needs to develop a theory for the steady state
                      huge number of localized band tail states in most disor-        of the system under stationary excitation. This the-
                      dered materials, one can assume that most of the states         ory takes into account various recombination processes
                      are empty and so the above formulae for the hopping             for charge carriers and provides their stationary con-
                      rates and electron energies can be used. In this case the       centration along with the position of the quasi-Fermi
                      electron mobility is a time-dependent quantity [9.41].          energy. After solving this recombination problem, one
                      A transport regime in which mobility of charge carriers         can follow the track of the theory of charge transport
                      is time-dependent is usually called dispersive transport        in quasi-thermal equilibrium [9.39] and obtain the con-
                      (see, for example, Mott and Davis [9.32], Orenstein and         ductivity in a form similar to (9.49), where εF is the
                      Kastner [9.38], Monroe [9.41]). Hence we have to con-           position of the quasi-Fermi level. We skip the corre-
                      clude that the transient electron mobility in inorganic         sponding (rather sophisticated) formulae here. Interested
                      noncrystalline materials with the DOS in the band tails         readers can find a comprehensive description of this sort
                      as described by (9.37) is a time-dependent quantity and         of theory for electrical conductivity in the literature (see,
                      the transient electrical conductivity has dispersive char-      for instance, Shklovskii et al. [9.42]).
                      acter. This is due to the nonequilibrium behavior of the             Instead, in the next section we will consider a very
                      charge carriers. They continuously drop in energy during        interesting problem related to the nonequilibrium en-
                      the course of the relaxation process.                           ergy relaxation of charge carriers in the band tail states.
                           In some theoretical studies based on the Fokker–           It is well known that at low temperatures, T ≤ 50 K, the
                      Planck equation it has been claimed that the maximum            photoconductivities of various inorganic noncrystalline
                      of the energy distribution of electrons coincides with          materials, such as amorphous and microcrystalline semi-
                      the TE εt and hence it is independent of time. This state-      conductors, do not depend on temperature [9.44–46].
                      ment contradicts the above result where the maximum             At low temperatures, the TE εt lies very deep in the
                      of the distribution is at energy εd (t), given by (9.48).       band tail and most electrons hop downward in energy,
                      The Fokker–Planck approach presumes the diffusion of            as described by (9.38) and (9.39). In such a regime, the
           Charge Transport in Disordered Materials     9.3 Hopping Charge Transport in Disordered Materials via Localized States            177



photoconductivity is a temperature-independent quan-            where




                                                                                                                                             Part A 9.3
tity determined by the loss of energy during the hopping
                                                                                ∞
of electrons via the band-tail states [9.47]. During this                                                                  ε
hopping relaxation, neither the diffusion coefficient D              N(ε) =              g(ε) dε = N0 exp −                      .   (9.52)
                                                                                                                           ε0
nor the mobility of the carriers μ depend on tempera-                           ε
ture, and the conventional form of Einstein’s relationship
μ = eD/kB T cannot be valid. The question then arises           It was assumed in the derivation of (9.51) that eFx ε0 .
as to what the relation between μ and D is for hopping              Due to the exponential dependence of the hopping
relaxation. We answer this question in the following            rate on the hopping length r, the electron predominantly
section.                                                        hops to the nearest tail state among the available states
                                                                if r α, which we assume to be valid. Let us calculate
    Einstein’s Relationship for Hopping Electrons               the average projection x on the field direction of the
Let us start by considering a system of nonequilib-             vector r from the initial states at energy ε to the near-
rium electrons in the band tail states at T = 0. The            est available neighbor among sites with a concentration
only process that can happen with an electron is its hop        N(ε, x) determined by (9.51). Introducing spherical co-
downward in energy (upward hops are not possible at             ordinates with the angle θ between r and the x-axis, we
T = 0) to the nearest localized state in the tail. Such         obtain [9.48]
a process is described by (9.37)–(9.39). If the spatial
distribution of localized tail states is isotropic, the prob-               2π                π
ability of finding the nearest neighbor is also isotropic             x =            dφ            dθ sin θ
in the absence of the external electric field. In this case,                 0             0
the process of the hopping relaxation of electrons resem-
bles diffusion in space. However, the median length of                              ∞
a hop (the distance r to the nearest available neighbor),                   ×           [ dr · r 3 cos(θ) · N(ε, r cos θ)]
                                       −1
as well as the median time, τ = ν↓ (r), of a hop [see
                                                                                0
(9.38)] increases during the course of relaxation, since
the hopping process brings electrons deeper into the tail.                               ⎡         2π            π
Nevertheless, one can ascribe a diffusion coefficient to                     × exp ⎣−                    dφ           dθ sin θ
such a process [9.42]:
                                                                                                  0          0
          1                                                                                                              ⎤
    D(r) = ν↓ (r)r 2 .                                (9.50)                        r
          6
                                                                                                       (ε, r cos θ)⎦ .
                                                                                                  2N
                                                                            ×            dr r                                       (9.53)
Here ν↓ (r)r 2 replaces the product of the “mean free
path” r and the “velocity” r · ν↓ (r), and the coefficient                       0
1/6 accounts for the spatial symmetry of the problem.
                                                                Substituting (9.51) for N(ε, r cos θ), calculating the in-
According to (9.37)–(9.39) and (9.50), this diffusion
                                                                tegrals in (9.53) and omitting the second-order terms
coefficient decreases exponentially with increasing r and
hence with the number of successive electron hops in the                                          2
relaxation process.                                                     eN −1/3 (ε)F
                                                                                                         1,                         (9.54)
    In order to calculate the mobility of electrons during                  ε0
hopping relaxation under the influence of the electric
field, one should take into account the spatial asymmetry        we obtain
of the hopping process due to the field [9.47, 48]. Let us
consider an electron in a localized state at energy ε. If an                eFN −2/3 (ε) Γ (5/3)
                                                                     x =                          ,                                 (9.55)
external electric field with a strength F is applied along                      3ε0      (4π/3)2/3
direction x, the concentration of tail states available to
this hopping electron at T = 0 (in other words those that       where Γ is the gamma-function and N(ε) is determined
have energies deeper in the tail than ε) is [9.47]              by (9.52). Equation (9.55) gives the average displace-
                                                                ment in the field direction of an electron that hops
                           eFx                                  downward from a state at energy ε to the nearest avail-
    N(ε, x) = N(ε) 1 +             ,                  (9.51)
                            ε0                                  able neighbor in the band tail. The average length r of
  178        Part A   Fundamental Properties



                      such a hop is                                                  drops deeper into the tail. However, for any F, there
Part A 9.3




                                  ∞
                                                                                     is always a boundary energy in the tail below which
                                                               4π                    the condition eFx ε0 cannot be fulfilled and where
                           r =         dr4πr 3 N(ε) exp −         N(ε)r 3            nonlinear effects play the decisive role in the hopping
                                                                3
                                  0                                                  conduction of charge carriers. In the next section we
                                                −1/3                                 show how one can describe these nonlinear effects with
                                      4π N(ε)              4
                              =                        Γ        .           (9.56)   respect to the applied electric field.
                                         3                 3
                      One can ascribe to the hopping process a mobility                   Nonlinear Effects in Hopping Conduction
                                                                                     Transport phenomena in inorganic noncrystalline ma-
                              v     x ν( r )
                         μ=     =                                                    terials, such as amorphous semiconductors, under the
                              F         F                                            influence of high electric fields are the foci for intensive
                              eN −2/3 (ε) ν ( r ) Γ (5/3)                            experimental and theoretical study. This is due to obser-
                            =                                               (9.57)
                                    3ε0          (4π/3)2/3                           vations of strong nonlinearities in the dependencies of
                      and a diffusion coefficient                                     the dark conductivity [9.11,52,53], the photoconductiv-
                                                                                     ity [9.49] and the charge carrier drift mobility [9.54–56]
                             1 2                                                     on the field for high electric fields. These effects are most
                          D=   r ν( r )
                             6                                                       pronounced at low temperatures, when charge transport
                             1                    Γ 2 (4/3)                          is determined by electron hopping via localized band tail
                            = N −2/3 (ε) ν ( r )            .               (9.58)
                                                                                     states (Fig. 9.12).
                             6                   (4π/3)2/3
                                                                                          Whereas the field-dependent hopping conductivity
                      Expressions (9.57) and (9.58) lead to a relationship           at low temperatures has always been a challenge to
                      between μ and D of the form                                    describe theoretically, theories for the temperature de-
                               2Γ (5/3) e          e                                 pendence of the hopping conductivity in low electric
                         μ=      2 (4/3) ε
                                            D ≈ 2.3 D .                     (9.59)   fields have been successfully developed for all of the
                               Γ          0        ε0
                                                                                     transport regimes discussed: for the dark conductiv-
                      This formula replaces the Einstein’s relationship              ity [9.39], for the drift mobility [9.41], and for the
                      μ = eD/kB T for electron hopping relaxation in the             photoconductivity [9.42]. In all of these theories, hop-
                      exponential band tail. Several points should be noted          ping transitions of electrons between localized states
                      about this result. First of all, one should clearly real-      in the exponential band tails play a decisive role, as
                      ize that (9.59) is valid for nonequilibrium energy-loss        described above in (9.37)–(9.59).
                      relaxation in which only downward (in energy) transi-
                      tions between localized states can occur. This regime
                      is valid only at low temperatures when the TE εt is                     Photoconductance (Ω–1)
                      very deep in the band tail. As the temperature increases,
                      the upward hops become more and more efficient                    10–9
                      for electron relaxation. Under these circumstances, the                  T(K)
                      relation between μ and D evolves gradually with ris-                     100
                      ing temperature from its temperature-independent form           10–10
                      at T = 0 to the conventional Einstein’s relationship,
                                                                                                80
                      μ = eD/kB T [9.50, 51]. Secondly, one should realize              –11
                                                                                      10
                      that (9.59) was derived in the linear regime with respect
                                                                                                60
                      to the applied field under the assumption that eFx ε0 .
                      According to (9.55), the quantity x is proportional to          10–12
                                      −2/3
                      N −2/3 (ε) = N0      exp [2ε/(3ε0 )], in other words it in-               50
                      creases exponentially during the course of the relaxation                 40
                                                                                                30
                      toward larger localization energies ε. This means that          10–13     20
                      for deep localized states in the band tail, the condition                       1                10                  100
                      eFx ε0 breaks down. The boundary energy for appli-                                                    Electric field F (kV / cm)
                      cation of the linear theory depends on the strength of the     Fig. 9.12 Dependence of the photoconductivity in a-Si:H
                      electric field, F. As F decreases, this boundary energy         on the electric field at different temperatures [9.49]
            Charge Transport in Disordered Materials    9.3 Hopping Charge Transport in Disordered Materials via Localized States   179



     Shklovskii [9.57] was the first to recognize that a




                                                                                                                                    Part A 9.3
                                                                                  F           g(ε )
strong electric field plays a similar role to that of tem-
                                                                 0
perature in hopping conduction. In order to obtain the
field dependence of the conductivity σ(F ) at high fields,                                                            g(ε )
Shklovskii [9.57] replaced the temperature T in the well-                                 0
known dependence σ(T ) for low fields by a function
Teff (F ) of the form                                                         x

            eFα                                                                               eFx
   Teff =       ,                                      (9.60)
            2kB
                                                                 ε
where e is the elementary charge, kB is the Boltzmann
constant, and α is the localization length of electrons
                                                                                          ε
in the band tail states. A very similar result was ob-
tained later by Grünewald and Movaghar [9.58] in                Fig. 9.13 Tunneling transition of a charge carrier in the
their study of the hopping energy relaxation of elec-           band tail that is affected by a strong electric field. Upon
trons through band tails at very low temperatures and           traveling the distance x, the carrier acquires the energy
high electric fields. The same idea was also used by             eFx, where F is the strength of the electric field, and e is
Shklovskii et al. [9.42], who suggested that, at T = 0,         the elementary charge
one can calculate the field dependence of the stationary
photoconductivity in amorphous semiconductors by re-            T = 20 K in Fig. 9.12, with the low-field photocon-
placing the laboratory temperature T in the formulae of         ductivity at T = Teff = eFα as measured by Hoheisel
                                                                                          2kB
the low-field finite-temperature theory by an effective           et al. [9.44] and by Stradins and Fritzsche [9.45],
temperature Teff (F) given by (9.60).                           we come to the conclusion that the data agree quan-
    It is easy to understand why the electric field plays        titatively if one assumes that the localization length
a role similar to that of temperature in the energy relax-      α = 1.05 nm [9.42], which is very close to the value
ation of electrons. Indeed, in the presence of the field,        α ≈ 1.0 nm found for a-Si:H from independent esti-
the number of sites available at T = 0 is significantly          mates [9.11]. This comparison shows that the concept
enhanced in the field direction, as shown in Fig. 9.13.          of the effective temperature based on (9.60) provides
Hence electrons can relax faster at higher fields. From          a powerful tool for estimating transport coefficient non-
the figure it is apparent that an electron can increase its      linearity with respect to the electric field using the
energy with respect to the mobility edge by an amount           low-field results for the temperature dependencies of
ε = eFx in a hopping event over a distance x in the di-         such coefficients.
rection prescribed by the electric field. The process is              However, experiments are usually carried out not at
reminiscent of thermal activation. The analogy becomes          T = 0 but at finite temperatures, and so the question of
tighter when we express the transition rate for this hop        how to describe transport phenomena in the presence of
as                                                              both factors, finite T and high F, arises. By studying the
                                                                steady state energy distribution of electrons in numer-
                    2x                  2ε
   ν = ν0 exp −           = ν0 exp −                            ical calculations and computer simulations [9.59, 60],
                    α                  eFα                      as well as straightforward computer simulations of the
                         ε                                      steady-state hopping conductivity and the transient en-
     = ν0 exp −                   ,                    (9.61)
                    kB Teff (F)                                 ergy relaxation of electrons [9.61], the following result
                                                                was found. The whole set of transport coefficients can
where Teff (F) is provided by (9.60).                           be represented by a function with a single parameter
    This electric field-induced activation at T = 0 pro-         Teff (F, T )
duces a Boltzmann tail to the energy distribution
                                                                                                      β 1/β
function of electrons in localized states as shown by                                 β   eFα
numerical calculations [9.59, 60]. In Fig. 9.12, the field-           Teff (F, T ) = T + γ                     ,     (9.62)
                                                                                           kB
dependent photoconductivity in a-Si:H is shown for
several temperatures [9.49]. If we compare the pho-             where β ≈ 2 and γ is between 0.5 and 0.9 depending
toconductivity at the lowest measured temperature,              on which transport coefficient is considered [9.61]. We
 180         Part A   Fundamental Properties



                      are aware of no analytical theory that can support this       the hopping transport mechanism. Examples include
Part A 9.3




                      numerical result.                                             polyvinylcarbazole (PVK) or bis-polycarbonate (Lexan)
                           To wrap up this section we would like to make the        doped with either strong electron acceptors such as trini-
                      following remark. It is commonly claimed in the scien-        trofluorenone acting as an electron transporting agent, or
                      tific literature that transport coefficients in the hopping     strong electron donors such as derivatives of trypheny-
                      regime should have a purely exponential dependence            lamine of triphenylmethane for hole transport [9.62,63].
                      on the applied electric field. The idea behind such state-     To avoid the need to specify whether transport is carried
                      ments seems rather transparent. Electric field diminishes      by electrons or holes each time, we will use a general
                      potential barriers between localized states by an amount      notation of “charge carrier” below. The results are valid
                      Δε = eFx, where x is the projection of the hopping ra-        for both types of carrier – electrons or holes. Charge car-
                      dius on the field direction. The field should therefore         riers in disordered organic materials are believed to be
                      diminish the activation energies in (9.24) and (9.25) by      strongly localized [9.18,62–64]. The localization centers
                      this amount, leading to the term exp(eFx/kB T ) in the ex-    are molecules or molecular subunits, henceforth called
                      pressions for the charge carrier mobility, diffusivity and    sites. These sites are located in statistically different en-
                      conductivity. One should, however, take into account          vironments. As a consequence, the site energies, which
                      that hopping transport in all real materials is essentially   are to great extent determined by electronic polarization,
                      described by the variable-range hopping process. In such      fluctuate from site to site. The fluctuations are typically
                      a process, as discussed above, the interplay between spa-     on the order of 0.1 eV [9.65]. This is about one order
                      tial and energy-dependent terms in the exponents of the       of magnitude larger than the corresponding transfer in-
                      transition probabilities determine the conduction path.       tegrals [9.65]. Therefore carrier wavefunctions can be
                      Therefore it is not enough to solely take into account        considered to be strongly localized [9.65].
                      the influence of the strong electric field on the activation        As discussed above, the crucial problem when de-
                      energies of single hopping transitions. One should con-       veloping a theoretical picture for hopping transport is
                      sider the modification of the whole transport path due to      the structure of the energy spectrum of localized states,
                      the effect of the strong field. It is this VRH nature of the   DOS. It is believed that, unlike inorganic noncrystalline
                      hopping process that leads to a more complicated field         materials where the DOS is believed exponential, the en-
                      dependence for the transport coefficients expressed by         ergy dependence of the DOS in organic disordered solids
                      (9.60)–(9.62).                                                is Gaussian (see Bässler [9.18] and references therein),
                           We have now completed our description of elec-
                      tron transport in inorganic disordered materials with                        N0     ε2
                      exponential DOS in the band tails. In the next section            g(ε) =     √ exp − 2             ,                (9.63)
                                                                                                 ε0 2π    2ε0
                      we tackle the problem of charge transport in organic
                      disordered materials.
                                                                                    where N0 is the total concentration of states and ε0 is
                      9.3.4 Description of Charge Carrier Energy                    the energy scale of the DOS. The strongest evidence in
                            Relaxation and Hopping Conduction                       favor of such an energy spectrum in disordered organic
                            in Organic Noncrystalline Materials                     materials is the ability to reproduce the observed experi-
                                                                                    mentally temperature dependence of the carrier mobility
                      Electron transport and energy relaxation in disordered        and that of hopping conductivity assuming the Gaussian
                      organic solids, such as molecularly doped polymers,           DOS in computer simulations [9.18, 66]. It has been
                      conjugated polymers and organic glasses, has been the         observed in numerous experimental studies [9.67–73]
                      subject of intensive experimental and theoretical study       that the temperature dependence of the drift mobility
                      for more than 20 years. Although there is a wide ar-          of charge carriers in disordered organic solids takes the
                      ray of different disordered organic solids, the charge        form
                      transport process is similar in most of these materials.                               2
                                                                                                        T0
                      Even at the beginning of the 1980s it was well under-             μ ∝ exp −                                         (9.64)
                      stood that the main transport mechanism in disordered                             T
                      organic media is the hopping of charge carriers via
                      spatially randomly distributed localized states. Binary       with a characteristic temperature T0 , as shown in
                      systems like doped polymeric matrices provide canoni-         Fig. 9.14a. Computer simulations and theoretical cal-
                      cal examples of disordered organic materials that exhibit     culations [9.65, 66, 74, 75] with the Gaussian DOS
                   Charge Transport in Disordered Materials             9.3 Hopping Charge Transport in Disordered Materials via Localized States   181



described by (9.63) give a dependence of the form                              systems with a purely exponential DOS (9.37). The an-




                                                                                                                                                    Part A 9.3
                                                                               swer to this question is yes. The reason becomes clear
                                           2
                                ε0                                             if one considers the behavior of a single charge carrier
      μ ∝ exp − C                               ,                     (9.65)
                               kB T                                            in an empty system. In an empty system with an expo-
                                                                               nential DOS, a charge carrier always (on average) falls
where C is a numerical coefficient. Computer simula-                            downward in energy if kB T < ε0 [see (9.45)–(9.48)], and
tions [9.65,66] give a value C ≈ 0.69 for this coefficient,                     its mobility continuously decreases with time; however,
and analytical calculations [9.74, 75] predict a similar                       in a system with a Gaussian DOS, a particular energy
value of C ≈ 0.64. Equation (9.65) is often used to de-                        level ε∞ determines the equilibrium energy position of
termine the parameter ε0 of the DOS from experimental                          a charge carrier. When it is located at some site with high
measurements of the ln(μ) versus (1/T )2 dependences                           energy in the Gaussian DOS, the charge carrier first hops
(see, for example, Ochse et al. [9.71]).                                       via localized states so that its average energy εd (t) de-
    One may wonder whether the theoretical description                         creases until it achieves the energy level ε∞ after some
of hopping conduction and carrier energy relaxation in                         typical time period τrel . At times t < τrel the behavior
a system with a Gaussian DOS (9.63) should differ sig-                         of the carrier qualitatively resembles that seen for the
nificantly from the theory described above for disordered                       purely exponential DOS. The downward hops are then
                                                                               replaced by relaxation hops that send the carrier up-
                                                                               ward to the transport energy, and the carrier mobility at
 a) Mobility (cm2 / Vs)                                                        t < τrel decreases with time. However, in contrast with
  10– 3                                                                        the case for the exponential DOS, in a Gaussian DOS
  10– 4                                                                        the carrier mobility becomes time-independent after a
  10– 5                                                                        time τrel , when the average carrier energy reaches the
                                                    (1)
  10–6                                                                         level ε∞ . At t > τrel , the dispersive transport regime with
  10– 7                                                                        time-dependent carrier mobility is replaced by a quasi-
  10– 8                                                                        equilibrium so-called “Gaussian transport” regime, in
  10– 9                                                                        which the spatial spreading of the carrier packet with
 10– 10                                   (3)
                                                                               time can be described by the traditional diffusion picture
 10– 11
 10– 12
                                     (4)                                       with a time-independent diffusion coefficient.
 10– 13                                                                            The peculiarity of the hopping energy relaxation
                                                                       (2)
 10– 14                                                                        of charge carriers in a system with a Gaussian DOS
          5           10       15          20        25    30       35         described above makes it easier to describe charge trans-
                                                          (1000 / T)2 [K–2]
                                                                               port at times t > τrel than in the case of the exponential
 b)       In [μe– 1 k BTr– 2 (ε t)v0–1]                                        DOS. In the latter case, only the presence of a sig-
 – 10            50                                                            nificant number of carriers in a quasi Fermi level can
                 70                                                            make kinetic coefficients such as mobility, diffusivity
 – 15           100
                                                                               and conductivity time-independent and hence conven-
 – 20            150                                                           tionally measurable and discussible quantities. In the
              ε0 (meV)                                                         case of the Gaussian DOS, these kinetic coefficients are
 – 25
                                                                               not time-dependent at times t > τrel . Moreover, in diluted
 – 30                                                                          systems one can calculate these coefficients by consid-
                                                                               ering the behavior of a single charge carrier. This makes
 – 35                                                                          theoretical considerations of electrical conductivity in
               6           8    10         12       14    16      18     20    organic disordered solids with a Gaussian DOS much
                                                          (1000 / T)2 [K–2]    easier than when considering inorganic noncrystalline
Fig. 9.14a,b Temperature dependence of the zero-field                           materials with an exponential DOS. Let us now calcu-
mobility in organic semiconductors. Experimental data                          late ε∞ , τrel and μ in disordered organic solids with a
(a): (1) di-p-tolylphenylamine containing (DEASP)-                             Gaussian DOS.
traps [9.69]; (2) (BD)-doped polycarbonate [9.70];                                 Computer simulations [9.66] and analytical calcu-
(3) (NTDI)-doped poly(styrene) [9.68]; (4) (BD)-doped                          lations [9.74, 75] show that the mean energy of the
TTA/polycarbonate [9.72]. Theoretical results (b) were                         independently hopping carriers, initially distributed ran-
obtained via (9.73)                                                            domly over all states in the Gaussian DOS, decreases
  182        Part A   Fundamental Properties



                      with time until it approaches the thermal equilibrium                 above the TE to the spatially nearest sites with rates
Part A 9.3




                      value                                                                 determined by (9.38) and (9.39).
                                      ∞                                                          Now that we have clarified the relaxation kinetics
                                                    ε
                                           ε exp − kB T g(ε) dε                             of charge carriers in the Gaussian DOS, it is easy to
                                   −∞                                          ε2           calculate the relaxation time τrel and the drift mobility
                          ε∞ =                                           =−     0
                                                                                   .
                                       ∞                                      kB T          μ. We consider the case ε∞ < εt < 0, which corre-
                                                  ε
                                           exp − kB T g(ε) dε                               sponds to all reasonable values of material parameters
                                      −∞                                                    N0 α3 and kB T/ε0 [9.76]. The energy relaxation of most
                                                                                   (9.66)   carriers with energies ε in the interval ε∞ < ε < εt
                      The time τrel required to reach this equilibrium is of key            occurs via a multiple trapping-like process, well de-
                      importance in the analysis of experimental data [9.65],               scribed in the literature (see, for example, Orenstein
                      since at t < τrel the carrier mobility decreases with                 and Kastner [9.38] or Marschall [9.78]). Below εt the
                      time (dispersive transport) until it reaches its equilib-             average energy of the carriers ε(t) moves logarithmi-
                      rium, time-independent value at t ≈ τrel . It has been                cally downward with time t. States above ε(t) achieve
                      established by computer simulations that τrel strongly                thermal equilibrium with states at εt at time t, while
                      depends on temperature [9.18]:                                        states below ε(t) have no chance at time t to exchange
                                                                                            carriers with states in the vicinity of εt . Hence the oc-
                                                           2
                                                   ε0                                       cupation of those deep states does not correspond to
                          τrel ∝ exp          B                                    (9.67)   the equilibrium one, being determined solely by the
                                                  kB T
                                                                                            DOS of the deep states. The system reaches thermal
                      with B ≈ 1.07. Given that the same hopping processes                  equilibrium when the time-dependent average energy
                      determine both μ and τrel , researchers were puzzled for              ε(t) achieves the equilibrium level ε∞ , determined by
                      many years by the fact that they had different coefficients            (9.66). This happens at t = τrel . Since the relaxation
                      B and C (in other words they have different temperature               of carriers occurs via thermal activation to the level
                      dependencies) [9.65]. Below we show how to calculate                  εt , the relaxation time τrel is determined by the time
                      both quantities – μ and τrel – easily, and we explain their           required for activated transitions from the equilibrium
                      temperature dependencies (obtained experimentally and                 level ε∞ to the transport energy εt . Hence, accord-
                      by computer simulations as expressed by (9.64), (9.65)                ing to (9.40) and (9.47), τrel is determined by the
                      and (9.67)).                                                          expression
                          Our theoretical approach is based on the concept of
                      transport energy (TE), introduced in Sect. 9.3.3, where it                       −1       2r(εt ) εt − ε∞
                                                                                               τrel = ν0 exp           +             .          (9.70)
                      was calculated for the exponential DOS given by (9.37).                                     α       kB T
                      Literally repeating these calculations with the Gaussian
                      DOS, given by (9.63), we obtain the equation [9.76, 77]               From (9.68)–(9.70) it is obvious that the activation en-
                                      ⎡ x               ⎤4/3                                ergy of the relaxation time depends on the parameters
                                              √
                                               2                                            N0 α3 and kB T/ε0 . Hence, generally speaking, this de-
                                 x2       ⎢                      ⎥
                          exp             ⎢        exp(−t 2 ) dt ⎥                          pendence cannot be represented by (9.67) and, if at
                                  2       ⎣                      ⎦
                                                                                            all, the coefficient B should depend on the magnitude
                                           −∞
                                                                                            of the parameter N0 α3 . However, numerically solving
                                                         −1/3   kB T                        (9.68)–(9.70) using the value N0 α3 = 0.001, which was
                          = 9(2π)1/2 N0 α3                           .             (9.68)
                                                                 ε0                         also used in computer simulations by Bässler [9.18,65],
                                                                                            confirms the validity of (9.67) with B ≈ 1.0. This re-
                      If we denote the solution of (9.68) as X t (N0 α3 , kB T/ε0 ),        sult is in agreement with the value B ≈ 1.07 obtained
                      then the transport energy in the Gaussian DOS is equal                from computer simulations [9.18, 65]. A way to de-
                      to                                                                    scribe the temperature dependence of the relaxation
                          εt = ε0 · X t N0 α3 , kB T/ε0 .                          (9.69)   time τrel by (9.67) is provided by the strong temper-
                                                                                            ature dependence of ε∞ in the exponent in (9.70),
                      Charge carriers perform thermally activated transitions               while the temperature dependencies of the quantities εt
                      from states with energies below the TE, εt , to the states            and r(εt ) in (9.70) are weaker and they almost cancel
                      with energies close to that of the TE [9.76]. Charge car-             each other out. However, if N0 α3 = 0.02, the relax-
                      riers hop downward in energy from states with energies                ation time is described by (9.67) with B ≈ 0.9. This
           Charge Transport in Disordered Materials                        9.3 Hopping Charge Transport in Disordered Materials via Localized States                 183



shows that (9.67) can only be considered to be a good




                                                                                                                                                                     Part A 9.3
approximation.                                                                                                                                         μn (1 / r2)
                                                                                    In [μe–1 k BTr–2 (ε t) v0–1]       N0–1/ 3 (nm)                   [V sec]–1
    Now we turn to the calculation of the carrier drift
                                                                                           0.6         0.8             1.0            1.2          1.4
mobility μ. We assume that the transition time ttr nec-                              –3
essary for a carrier to travel through a sample is longer                                                                    Theory (kT / σ = 0.3)
than τrel , and hence the charge transport takes place                               –6                                      Experiment
                                                                                                                                                               109
under equilibrium conditions. As described above, ev-
ery second jump brings the carrier upward in energy to                               –9
the vicinity of εt , and is then followed by a jump to
the spatially nearest site with deeper energy, determined                                                                                                      108
                                                                                    – 12
solely by the DOS. Therefore, in order to calculate the
drift mobility μ, we must average the hopping transi-                               – 15
tion times over energy states below εt , since only these                                                                                                      107
states are essential to charge transport in thermal equi-                           – 18
librium [9.77, 80]. Hops downward in energy from the
level εt occur exponentially faster than upward hops to-
                                                                                    – 21                                                                       106
wards εt . This means that one can neglect the former in
the calculation of the average time t . The carrier drift
                                                                                    – 24
mobility can be evaluated as
                                                                                                                         slope 1.73                            105
          e     r 2 (εt )                                                           – 27
   μ≈                       ,                                         (9.71)
         kB T       t
                                                                                           0     1    2      3     4     5     6       7    8     9 10
where r(εt ) is determined via (9.39), (9.63), (9.68) and                                                                                       (N0 α3)–1/ 3
(9.69). The average hopping time takes the form [9.80]
                                                                                   Fig. 9.15 Concentration dependence of the drift mobility
          ⎡ ε          ⎤−1     ε                                                   evaluated from (9.73) (solid line), and the depen-
                t                               t

     t =⎣           g(ε) dε⎦           ×             −1
                                                    ν0 g(ε)                        dence observed experimentally (circles) for TNF/PE and
                                                                                   TNF/PVK [9.79]
             −∞                            −∞
                                1/3
                        2r(εt )Bc               εt − ε                                 In Fig. 9.14b, the dependence of the drift mobil-
          × exp                            +               dε ,           (9.72)
                                a                kB T                              ity on the temperature at N0 α3 = 0.01 is depicted for
                                                                                   several values of ε0 . The sensitivity of the mobility
where Bc ≈ 2.7 is the percolation parameter. This nu-                              to temperature is clear from this picture. Comparison
merical coefficient is introduced into (9.72) in order to                           of these dependencies with experimental measurements
warrant the existence of an infinite percolation path over                          of ln(μ) versus (1/T )2 [some are shown in Fig. 9.14a]
the states with energies below εt . Using (9.63), (9.68),                          provides information on the energy scale, ε0 , of the
(9.69), (9.71) and (9.72), one obtains the following re-                           DOS (see, for example, Bässler [9.18] and Ochse
lation for the exponential terms in the expression for the                         et al. [9.71]).
carrier drift mobility:                                                                In Fig. 9.15, the dependence of the drift mobility
                                                                                   on N0 α3 is shown for kB T/ε0 = 0.3. Experimental data
            er 2 (εt )ν0                                                           from Gill [9.81] are also shown in the figure. It is clear
    ln μ/
               kB T                                                                that the slope of the mobility exponent as a function of
            ⎡                             √                       ⎤−1/3            (N0 α3 )−1/3 given by the theory described above agrees
                 √                     Xt/ π
            ⎢4 π                                                  ⎥                with the experimental data. At a very low concentration
       = −2 ⎣          N 0 α3                       exp(−t 2 ) dt ⎦
                3Bc                                                                of localized states, N0 , when the probability of carrier
                                       −∞                                          tunneling in space dominates the transition rate in (9.24),
                                            2                                      charge carriers hop preferentially to the nearest spatial
             X t ε0 1                ε0
         −         −                                .                 (9.73)       sites. In this regime of nearest-neighbor hopping, the
             kB T    2              kB T
                                                                                   concentration dependence of the drift mobility is de-
It is (9.73) that determines the dependence of the carrier                         scribed by (9.28), as illustrated by the dashed line in
drift mobility on the parameters N0 α3 and kB T/ε0 .                               Fig. 9.15.
  184        Part A   Fundamental Properties



                          So far we have discussed the drift mobility of charge     where n is the concentration of charge carriers in the
Part A 9.4




                      carriers under the assumption that the concentration of       material and μ is their drift mobility. If, however,
                      charge carriers is much less than that of the localized       the concentration n is so large that the Fermi energy
                      states in the energy range relevant to hopping transport.     at thermal equilibrium or the quasi-Fermi energy at
                      In such a case one can assume that the carriers perform       stationary excitation is located significantly higher (en-
                      independent hopping motion and so the conductivity can        ergetically) than the equilibrium energy ε∝ , a more
                      be calculated as the product                                  sophisticated theory based on the percolation approach
                                                                                    is required [9.82]. The result obtained is similar to that
                         σ = enμ ,                                        (9.74)    given by (9.49).


                      9.4 Concluding Remarks
                      Beautiful effects have been observed experimentally by        surd results if applied to hopping transport in disordered
                      studying the charge transport in disordered organic and       materials. One can use ideas from percolation theory in-
                      inorganic materials. Among these, the transport coeffi-        stead to adequately describe charge transport. One of the
                      cients in the hopping regime show enormously strong           most important ideas in this field is so-called variable-
                      dependencies on material parameters. The dependence           range hopping (VRH) conduction. Although the rate of
                      of the charge carrier mobility on the concentration of lo-    transitions between two localized states is a product of
                      calized states N0 (Fig. 9.15) spreads over many orders        exponential terms that are separately dependent on the
                      of magnitude, as does its dependence on the tempera-          concentration of localized states N0 , the temperature of
                      ture T (Fig. 9.14) and on the (high) electric field strength   the system T , and also on the field strength F (for high
                      F (Fig. 9.12). Such strong variations in physical quan-       field strengths), it is generally wrong to assume that
                      tities are typical, say, in astrophysics, but they are not    the carrier drift mobility, diffusivity or conductivity can
                      usual in solid state physics. This makes the study of         also be represented as the product of three functions that
                      the charge transport in disordered materials absolutely       are separately dependent on N0 , T and F. Instead one
                      fascinating. The strong dependencies of kinetic coeffi-        should search for a percolation path that takes into ac-
                      cients (like drift mobility, diffusivity and conductivity)    count the exponential dependences of the hopping rates
                      in disordered materials on various material parameters        on all of these parameters simultaneously. Such a pro-
                      makes these systems very attractive for various device        cedure, based on strong interplay between the important
                      applications. Since they are relatively inexpensive to        parameters in the exponents of the transition rates, leads
                      manufacture too, it is then easy to understand why dis-       to very interesting and (in some cases) unexpected re-
                      ordered organic and inorganic materials are of enormous       sults, some of which were described in this chapter.
                      interest for various technical applications.                  For example, it was shown that the effect of a strong
                           These materials also provide a purely academic           electric field on transport coefficients can be accounted
                      challenge with respect to their transport phenomena.          for by renormalizing the temperature. Most of the ideas
                      While traditional kinetic theories developed for crys-        discussed in this chapter were discussed in the early
                      talline materials are largely dependent on the systems        works of Mott and his coauthors (see, for example, Mott
                      having translational symmetry, there is no such symme-        and Davis [9.32]). Unfortunately, these ideas are not yet
                      try in disordered materials. However, we have shown           known to the majority of researchers working in the field
                      in this chapter that it is still possible to develop a re-    of disordered materials. Moreover, it is often believed
                      liable theoretical approach to transport phenomena in         that transport phenomena in different disordered ma-
                      disordered materials. Particularly interesting is the hop-    terials need to be described using different ideas. Mott
                      ping transport regime. In this regime, charge carriers        based his ideas, in particular the VRH, mostly on in-
                      perform incoherent tunneling jumps between localized          organic glassy semiconductors. Most of the researchers
                      states distributed in space and energy. The enormously        that are studying amorphous inorganic semiconductors
                      strong (exponential) dependence of the transition rates       (like a-Si:H) are aware of these ideas. However, new re-
                      on the distances between the sites and their energies call    searchers that are working on more modern disordered
                      for a completely new set of ideas compared to those           materials, such as organic disordered solids and dye-
                      for crystalline solids. Conventional transport theories       sensitized materials, are often not aware of these very
                      based on the averaging of transition rates lead to ab-        useful and powerful ideas developed by Mott and his
                                                                             Charge Transport in Disordered Materials        References   185



followers that can be used to describe charge transport         space. No correlations between the spatial positions of




                                                                                                                                          Part A 9
in inorganic disordered systems. In this chapter we have        the sites and the energies of the electronic states at these
shown that the most pronounced charge transport effects         sites were considered here. Some theoretical attempts to
in inorganic and organic disordered materials can be            account for such correlations can be found in the litera-
successfully described in a general manner using these          ture, although the correlations have not been calculated
ideas.                                                          ab initio: instead they are inserted into a framework
     Although we have presented some useful ideas for           of model assumptions. This shows how far the field of
describing charge transport in disordered systems above,        charge transport in disordered materials is from a desir-
it is clear that the theoretical side of this field is still     able state. Since these materials are already widely used
embyonic. There are still no reliable theories for charge       in various technical applications, such as field transistor
transport via extended states in disordered materials. Nor      manufacture, light-emitting diodes and solar cells, and
are there any reliable theoretical descriptions for the spa-    since the sphere of such applications is increasing, the
tial structure of the localized states (DOS) in organic and     authors are optimistic about the future of research in this
inorganic noncrystalline materials. All of the theoretical      field. The study of fundamental charge transport prop-
concepts presented in this chapter were developed us-           erties in disordered materials should develop, leading
ing very simple models of localization centers with a           us to a better understanding of the fundamental charge
given energy spectrum that are randomly distributed in          transport mechanisms in such systems.


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