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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 8.1 W. Schottky: Naturwissenschaften 26, 843 (1938) 8.4 W. Schottky: Phys. Zeitschr. 41, 570 (1940) 8.2 F. Braun: Pogg. Ann. Physik Chemie 153, 556 (1874) 8.5 J. Bardeen: Phys. Rev. 71, 717 (1947) 8.3 N. F. Mott: Proc. Camb. Philos. Soc. 34, 568 (1938) 8.6 V. Heine: Phys. Rev. 138, A1689 (1965) 160 Part A Fundamental Properties 8.7 C. Tejedor, F. Flores: J. Phys. C 11, L19 (1978) 8.35 A. Qteish, V. Heine, R. J. Needs: Phys. Rev. B 45, 6534 8.8 L. N. Pauling: The Nature of the Chemical Bond (Cor- (1992) nell Univ. Press, Ithaca 1939) 8.36 P. Käckell, B. Wenzien, F. Bechstedt: Phys. Rev. B 8.9 W. Mönch: On the Present Understanding of Schot- 50, 10761 (1994) tky Contacts. In: Festkörperprobleme, Vol. 26, ed. by 8.37 S. Ke, K. Zhang, X. Xie: J. Phys. Condens. Mat. 8, P. Grosse (Vieweg, Braunschweig 1986) p. 67 10209 (1996) 8.10 W. Mönch: Phys. Rev. B 37, 7129 (1988) 8.38 J. A. Majewski, P. Vogl: MRS Internet J. Nitride Semi- 8.11 R. Schmitsdorf, T. U. Kampen, W. Mönch: Surf. Sci. cond. Res. 3, 21 (1998) 324, 249 (1995) 8.39 S.-H. Wei, S. B. Zhang: Phys. Rev. B 62, 6944 (2000) 8.12 J. L. Freeouf, T. N. Jackson, S. E. Laux, J. M. Woodall: 8.40 W. A. Harrison, E. A. Kraut, J. R. Waldrop, R. W. Grant: AppI. Phys. Lett. 40, 634 (1982) Phys. Rev. B 18, 4402 (1978) 8.13 W. Schottky: Phys. Zeitschr. 15, 872 (1914) 8.41 J. S. Nelson, A. F. Wright, C. Y. Fong: Phys. Rev. B 43, 8.14 W. Mönch: Electronic Properties of Semiconductor 4908 (1991) Interfaces (Springer, Berlin, Heidelberg 2004) 8.42 S. B. Zhang, M. L. Cohen, S. G. Louie, D. Tománek, 8.15 H.A. Bethe: MIT Radiation Lab. Rep. 43-12 (1942) M. S. Hybertsen: Phys. Rev. B 41, 10058 (1990) 8.16 K. Takayanagi, Y. Tanishiro, M. Takahashi, S. Taka- 8.43 M. S. Hybertsen, S. G. Louie: Phys. Rev. B 34, 5390 hashi: Surf. Sci. 164, 367 (1985) (1986) 8.17 H.-J. Im, Y. Ding, J. P. Pelz, W. J. Choyke: Phys. Rev. 8.44 R. W. Godby, M. Schlüter, L. J. Sham: Phys. Rev. B 37, B 64, 075310 (2001) 10159 (1988) 8.18 W. J. Kaiser, L. D. Bell: Phys. Rev. Lett. 60, 1406 (1988) 8.45 M. Lazzarino, G. Scarel, S. Rubini, G. Bratina, 8.19 L. D. Bell, W. J. Kaiser: Phys. Rev. Lett. 61, 2368 L. Sorba, A. Franciosi, C. Berthod, N. Binggeli, (1988) A. Baldereschi: Phys. Rev. B 57, R9431 (1998) 8.20 R. Turan, B. Aslan, O. Nur, M. Y. A. Yousif, M. Willan- 8.46 S. Rubini, E. Pellucchi, M. Lazzarino, D. Kumar, der: Appl. Phys. A 72, 587 (2001) A. Franciosi, C. Berthod, N. Binggeli, A. Baldereschi: 8.21 J. Cohen, J. Vilms, R.J. Archer: Hewlett-Packard R&D Phys. Rev. B 63, 235307 (2001) Report AFCRL-69-0287 (1969) 8.47 J. Bardi, N. Binggeli, A. Baldereschi: Phys. Rev. B 54, 8.22 R. W. Grant, J. R. Waldrop, E. A. Kraut: Phys. Rev. R11102 (1996) Lett. 40, 656 (1978) 8.48 S. Picozzi, A. Continenza, G. Satta, S. Massidda, 8.23 S. B. Zhang, M. L. Cohen, S. G. Louie: Phys. Rev. B 34, A. J. Freeman: Phys. Rev. B 61, 16736 (2000) 768 (1986) 8.49 S. Picozzi, G. Profeta, A. Continenza, S. Massidda, 8.24 J. Tersoff: J. Vac. Sci. Technol. B 4, 1066 (1986) A. J. Freeman: Phys. Rev. B 65, 165316 (2002) 8.25 W. Mönch: Appl. Phys. Lett. 86, 162101 (2005) 8.50 J. Hoekstra, M. Kohyama: Phys. Rev. B 57, 2334 (1998) 8.26 W. Mönch: Appl. Phys. Lett. 86, 122101 (2005) 8.51 M. Kohyama, J. Hoekstra: Phys. Rev. B 61, 2672 (2000) 8.27 J. Tersoff: Phys. Rev. Lett. 52, 465 (1984) 8.52 M. Städele, A. J. Majewski, P. Vogl: Phys. Rev. B 56, 8.28 W. Mönch: J. Appl. Phys. 80, 5076 (1996) 6911 (1997) 8.29 A. Baldereschi: Phys. Rev. B 7, 5212 (1973) 8.53 J. A. Majewski, M. Städele, P. Vogl: Mater. Res. Soc. 8.30 D. R. Penn: Phys. Rev. 128, 2093 (1962) Symp. Proc. 449, 917 (1997) 8.31 A. M. Cowley, S. M. Sze: J. Appl. Phys. 36, 3212 (1965) 8.54 N. Binggeli, P. Ferrara, A. Baldereschi: Phys. Rev. B 8.32 W. Mönch: Appl. Surf. Sci. 92, 367 (1996) 63, 245306 (2001) 8.33 A. R. Miedema, F. R. de Boer, P.F. de Châtel: J. Phys. 8.55 B. K. Agrawal, S. Agrawal, R. Srivastava, P. Srivas- F 3, 1558 (1973) tava: Physica E 11, 27 (2001) 8.34 A. R. Miedema, P. F. de Châtel, F. R. de Boer: Physica 8.56 M. R. Laridjani, P. Masri, J. A. Majewski: Mater. Res. 100B, 1 (1980) Soc. Symp. Proc. 639, G11.34 (2001) 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. References 9.1 A. Bunde, K. Funke, M. D. Ingram: Solid State Ionics 9.14 B. I. Shklovskii, A. L. Efros: Electronic Properties of 105, 1 (1998) Doped Semiconductors (Springer, Berlin, Heidelberg 9.2 S. D. Baranovskii, H. Cordes: J. Chem. Phys. 111, 7546 1984) (1999) 9.15 I. P. Zvyagin: Kinetic Phenomena in Disordered 9.3 C. Brabec, V. Dyakonov, J. Parisi, N. S. Sariciftci: Semiconductors (Moscow University Press, Moscow Organic Photovoltaics: Concepts and Realization 1984) (in Russian) (Springer, Berlin, Heidelberg 2003) 9.16 H. Böttger, V. V. Bryksin: Hopping Conduction in 9.4 M. H. Brodsky: Amorphous Semiconductors (Springer, Solids (Wiley, New York 1985) Berlin, Heidelberg 1979) 9.17 H. Overhof, P. Thomas: Electronic Transport in Hy- 9.5 G. Hadziioannou, P. F. van Hutten: Semiconducting drogenated Amorphous Semiconductors (Springer, Polymers (Wiley, New York 2000) Berlin, Heidelberg 1989) 9.6 J. D. Joannopoulos, G. Locowsky: The Physics of Hy- 9.18 H. Bässler: Phys. Status Solidi B 175, 15 (1993) drogenated Amorphous Silicon I (Springer, Berlin, 9.19 P. W. Anderson: Phys. Rev. 109, 1492 (1958) Heidelberg 1984) 9.20 A. L. Efros, M. E. Raikh: Effects of Composition Disor- 9.7 J. D. Joannopoulos, G. Locowsky: The Physics of Hy- der on the Electronic Properties of Semiconducting drogenated Amorphous Silicon II (Springer, Berlin, Mixed Crystals. In: Optical Properties of Mixed Crys- Heidelberg 1984) tals, ed. by R. J. Elliott, I. P. Ipatova (Elsevier, New 9.8 A. Madan, M. P. Shaw: The Physics and Applications York 1988) of Amorphous Semiconductors (Academic, New York 9.21 D. Chattopadhyay, B. R. Nag: Phys. Rev. B 12, 5676 1988) (1975) 9.9 M. Pope, C. E. Swenberg: Electronic Processes in Or- 9.22 J. W. Harrison, J. R. Hauser: Phys. Rev. B 13, 5347 ganic Crystals and Polymers (Oxford Univ. Press, (1976) Oxford 1999) 9.23 I. S. Shlimak, A. L. Efros, I. V. Yanchev: Sov. Phys. 9.10 J. Singh, K. Shimakawa: Advances in Amorphous Semicond. 11, 149 (1977) Semiconductors (Gordon and Breach/Taylor & Fran- 9.24 S. D. Baranovskii, A. L. Efros: Sov. Phys. Semicond. cis, London 2003) 12, 1328 (1978) 9.11 R. A. Street: Hydrogenated Amorphous Silicon, Cam- 9.25 P. K. Basu, K. Bhattacharyya: J. Appl. Phys. 59, 992 bridge Solid State Science Series (Cambridge Univ. (1986) Press, Cambridge 1991) 9.26 S. Fahy, E. P. O’Reily: Appl. Phys. Lett. 83, 3731 (2003) 9.12 K. Tanaka, E. Maruyama, T. Shimada, H. Okamoto: 9.27 V. Venkataraman, C. W. Liu, J. C. Sturm: Appl. Phys. Amorphous Silicon (Wiley, New York 1999) Lett. 63, 2795 (1993) 9.13 J. S. Dugdale: The Electrical Properties of Disordered 9.28 C. Michel, P. J. Klar, S. D. Baranovskii, P. Thomas: Metals, Cambridge Solid State Science Series (Cam- Phys. Rev. B 69, 165211–1 (2004) bridge Univ. Press, Cambridge 1995) 9.29 T. Holstein: Philos. Mag. B 37, 49 (1978) 186 Part A Fundamental Properties 9.30 H. Scher, T. Holstein: Philos. Mag. 44, 343 (1981) 9.58 M. Grünewald, B. Movaghar: J. Phys. Condens. Mat. Part A 9 9.31 A. Miller, E. Abrahams: Phys. Rev. 120, 745 (1960) 1, 2521 (1989) 9.32 N. F. Mott, E. A. Davis: Electronic Processes in Non- 9.59 S. D. Baranovskii, B. Cleve, R. Hess, P. Thomas: J. Crystalline Materials (Clarendon, Oxford 1971) Non-Cryst. Solids 164-166, 437 (1993) 9.33 A. L. Efros, B. I. Shklovskii: J. Phys. C 8, L49 (1975) 9.60 S. Marianer, B. I. Shklovskii: Phys. Rev. B 46, 13100 9.34 M. Pollak: Disc. Faraday Soc. 50, 13 (1970) (1992) 9.35 S. D. Baranovskii, A. L. Efros, B. L. Gelmont, 9.61 B. Cleve, B. Hartenstein, S. D. Baranovskii, M. Schei- B. I. Shklovskii: J. Phys. C 12, 1023 (1979) dler, P. Thomas, H. Baessler: Phys. Rev. B 51, 16705 9.36 I. Shlimak, M. Kaveh, R. Ussyshkin, V. Ginodman, (1995) S. D. Baranovskii, H. Vaupel, P. Thomas, R. W. van 9.62 M. Abkowitz, M. Stolka, M. Morgan: J. Appl. Phys. der Heijden: Phys. Rev. Lett. 75, 4764 (1995) 52, 3453 (1981) 9.37 S. D. Baranovskii, P. Thomas, G. J. Adriaenssens: J. 9.63 W. D. Gill: J. Appl. Phys. 43, 5033 (1972) Non-Cryst. Solids 190, 283 (1995) 9.64 S. J. Santos Lemus, J. Hirsch: Philos. Mag. B 53, 25 9.38 J. Orenstein, M. A. Kastner: Solid State Commun. 40, (1986) 85 (1981) 9.65 H. Bässler: Advances in Disordered Semiconductors. 9.39 M. Grünewald, P. Thomas: Phys. Status Solidi B 94, In: Hopping and Related Phenomena, Vol. 2, ed. by 125 (1979) M. Pollak, H. Fritzsche (World Scientiﬁc, Singapore 9.40 F. R. Shapiro, D. Adler: J. Non-Cryst. Solids 74, 189 1990) p. 491 (1985) 9.66 G. Schönherr, H. Bässler, M. Silver: Philos. Mag. B 9.41 D. Monroe: Phys. Rev. Lett. 54, 146 (1985) 44, 369 (1981) 9.42 B. I. Shklovskii, E. I. Levin, H. Fritzsche, S. D. Bara- 9.67 P. M. Borsenberger, H. Bässler: J. Chem. Phys. 95, novskii: Hopping photoconductivity in amorphous 5327 (1991) semiconductors: dependence on temperature, elec- 9.68 P. M. Borsenberger, W. T. Gruenbaum, E. H. Magin, tric ﬁeld and frequency. In: Advances in Disordered S. A. Visser: Phys. Status Solidi A 166, 835 (1998) Semiconductors, Vol. 3, ed. by H. Fritzsche (World 9.69 P. M. Borsenberger, W. T. Gruenbaum, E. H. Magin, Scientiﬁc, Singapore 1990) p. 3161 S. A. Visser, D. E. Schildkraut: J. Polym. Sci. Polym. 9.43 S. D. Baranovskii, F. Hensel, K. Ruckes, P. Thomas, Phys. 37, 349 (1999) G. J. Adriaenssens: J. Non-Cryst. Solids 190, 117 (1995) 9.70 A. Nemeth-Buhin, C. Juhasz: Hole transport in 1,1- 9.44 M. Hoheisel, R. Carius, W. Fuhs: J. Non-Cryst. Solids bis(4-diethylaminophenyl)-4,4-diphenyl-1,3-buta- 63, 313 (1984) diene. In: Hopping and Related Phenomena, ed. by 9.45 P. Stradins, H. Fritzsche: Philos. Mag. 69, 121 (1994) O. Millo, Z. Ovadyahu (Racah Institute of Physics, 9.46 J.-H. Zhou, S. D. Baranovskii, S. Yamasaki, K. Ikuta, The Hebrew University Jerusalem, Jerusalem 1995) K. Tanaka, M. Kondo, A. Matsuda, P. Thomas: Phys. pp. 410–415 Status Solidi B 205, 147 (1998) 9.71 A. Ochse, A. Kettner, J. Kopitzke, J.-H. Wendorff, 9.47 B. I. Shklovskii, H. Fritzsche, S. D. Baranovskii: Phys. H. Bässler: Chem. Phys. 1, 1757 (1999) Rev. Lett. 62, 2989 (1989) 9.72 J. Veres, C. Juhasz: Philos. Mag. B 75, 377 (1997) 9.48 S. D. Baranovskii, T. Faber, F. Hensel, P. Thomas, 9.73 U. Wolf, H. Bässler, P. M. Borsenberger, W. T. Gruen- G. J. Adriaenssense: J. Non-Cryst. Solids 198-200, 214 baum: Chem. Phys. 222, 259 (1997) (1996) 9.74 M. Grünewald, B. Pohlmann, B. Movaghar, D. Würtz: 9.49 R. Stachowitz, W. Fuhs, K. Jahn: Philos. Mag. B 62, Philos. Mag. B 49, 341 (1984) 5 (1990) 9.75 B. Movaghar, M. Grünewald, B. Ries, H. Bässler, 9.50 S. D. Baranovskii, T. Faber, F. Hensel, P. Thomas: D. Würtz: Phys. Rev. B 33, 5545 (1986) Phys. Status Solidi B 205, 87 (1998) 9.76 S. D. Baranovskii, T. Faber, F. Hensel, P. Thomas: J. 9.51 S. D. Baranovskii, T. Faber, F. Hensel, P. Thomas: J. Phys. C 9, 2699 (1997) Non-Cryst. Solids 227-230, 158 (1998) 9.77 S. D. Baranovskii, H. Cordes, F. Hensel, G. Leising: 9.52 A. Nagy, M. Hundhausen, L. Ley, G. Brunst, Phys. Rev. B 62, 7934 (2000) E. Holzenkämpfer: J. Non-Cryst. Solids 164-166, 529 9.78 J. M. Marshall: Rep. Prog. Phys. 46, 1235 (1983) (1993) 9.79 W. D. Gill: J. Appl. Phys. 43, 5033 (1972) 9.53 C. E. Nebel, R. A. Street, N. M. Johanson, C. C. Tsai: 9.80 O. Rubel, S. D. Baranovskii, P. Thomas, S. Yamasaki: Phys. Rev. B 46, 6803 (1992) Phys. Rev. B 69, 014206–1 (2004) 9.54 H. Antoniadis, E. A. Schiff: Phys. Rev. B 43, 13957 9.81 W. D. Gill: Electron mobilities in disordred and crys- (1991) talline tritroﬂuorenone. In: Proc. Fifth Int. Conf. 9.55 K. Murayama, H. Oheda, S. Yamasaki, A. Matsuda: of Amorphous and Liquid Semiconductors, ed. by Solid State Commun. 81, 887 (1992) J. Stuke, W. Brenig (Taylor and Francis, London 1974) 9.56 C. E. Nebel, R. A. Street, N. M. Johanson, J. Kocka: p. 901 Phys. Rev. B 46, 6789 (1992) 9.82 S. D. Baranovskii, I. P. Zvyagin, H. Cordes, S. Ya- 9.57 B. I. Shklovskii: Sov. Phys. Semicond. 6, 1964 masaki, P. Thomas: Phys. Status Solidi B 230, 281 (1973) (2002)

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