Structure of Solids

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					     Section 4
     (M&T Chapter 7)

Structure and Energetics of
 Metallic and Ionic Solids
               Bonding in Solids
► We have discussed bonding in molecules with three
    Lewis
    Valence Bond
    MO Theory
► The above models aren’t suitable for describing bonding in
  solids (metals, ionic compounds)
► The structures of many solids (e.g. NaCl(s), Fe(s)) are best
  described by a lattice model, in which atoms (ions) of the
  lattice are placed in highly ordered arrangements (crystal
► Arrangement yields maximum net attractive force with
  other ions/atoms in the lattice
► Treat    the atoms of a metal (or ions of an
  ionic substance) as spheres (e.g. marbles).
► If fill a tray until its surface is covered with
  a layer of marbles, we get a picture that
  looks something like the figure
► In this model, all spheres are touching the surface
  of six other spheres (except those on the
  edges/corners) – a hexagonal arrangement
► This arrangement is one layer of what is called a
  “close-packed” arrangement
► See that there are six spaces around each sphere
► If we were to pile another layer on top of this, it
  would appear as follows…

Around each sphere, there
 are a total of six spaces
    When the next close-packed level is added, only three
    of the spaces surrounding each sphere can be occupied

►   So far, the levels are different (the spheres are not located
    in the same space directly above the first level), so we have
    an “AB-type” arrangement for the two layers
►   Note the spaces (hollows) that are now present – two
     “A” – located above a sphere
     “C” – located above a space
If we were to add another close-packed level on top, we could do so in two ways:

                                                            ABC arrangement

                                                            ABA arrangement
► In the top figure (ABC-type), the top layer was
  constructed by placing spheres in the spaces
  labeled “C” of the second layer
► This results in a level that is different than either
  of the first two levels (an “ABC…” arrangement)
► In the bottom figure (ABA-type) the top level was
  created by putting spheres into the spaces labeled
  “A” of the second layer
► The top level here is thus the same as the first
  (bottom) level (the spheres in this level are
  located directly above those of the first level) – an
  “ABAB…” arrangement
    Close-Packed Arrangements
►   These two arrangements are shown below, are
    represented as are ball-and-stick arrangements, and
    not meant to imply that spheres are not touching
►   Which is which?
                         The coordination number (CN)
                         in each of these lattices is 12.

                                                Top layer different
                                                   than bottom
    Close-Packed Arrangements
►    The unit cells are called:
    a) hexagonal close-packed (hcp) –
       describing an “ABABAB…” arrangement
    b) cubic close-packed (ccp) – this describes
       an “ABCABC…” ordering

                                           (this is also known
                                           as a “face-centered
                                           cubic” arranegment)

             hcp                 ccp
   Close-Packed Arrangements
    unit cells are shown below as spherical
► The
   Cubic close-packed (ccp) (face-centered cubic)
   Hexagonal close-packed (hcp)
          Interstitial Holes
► The  spaces between spheres in the
  close-packing arrangements are called
  interstitial holes/interstitial spaces
► Two types:
   Tetrahedral (four point cavities)
   Octahedral (six point cavities - larger)
Non-Close Packed Arrangements
  ► In many solids, the percentage of occupied
    space is less than 74% - these solids assume
    non-close packed arrangements
  ► Arrangements below depict the simplest
    model that incorporates all the information of
    the lattice (a unit cell)
                  52% of space used   68% of space used
What are the
numbers in
these lattices?

                    Simple cubic      Body-centered cubic
circle = bcc
diamond = hcp
+ = ccp (fcc)   Data quoted for T = 298 K
► The  lattice structure adopted by an element
  may change as the temperature and/or the
  pressure is changed
► A substance that exists in more than one
  crystalline form is said to be polymorphic
► Look at figure on next slide – a phase diagram
  for Fe
► Phase diagram – lines (phase boundaries)
  separate different phases of a substance
► On a line, have both phases present in
             Phase Diagram for Iron

Non close-packed
        Phase Diagram for Iron
► At ambient temperature (~ 200 K) and pressure (1
  atm), iron adopts a body-centered cubic (bcc)
  ordering (a-Fe)
   look at Fe at 1000 K, 1 atm (a-Fe). Increase the
    pressure. What happens? (g-Fe)
   Fe at 200K, 1 atm – increase pressure to 100 atm.
    What happens?
► Sometimes,  we can heat an element, changing its
  packing, and rapidly cool it to retain the higher
  temperature structure (quenching) – this allows
  higher temperature structures to be studied at
  ambient temperatures
           Alloys and Intermetallic
► Alloys are intimate mixtures (solid solutions) of
  metals/elements, or a distinct compound
  consisting of these elements, with physical
  properties that differ from those of the elements
  that make it up
► The aim of alloying is to endow the mixture with
  desirable properties that are inherent to the
  metals of the mixture (e.g. hardness, ductility)
► Everyday examples of alloys: (solute(s), solvent)
   Solder (Sn, Pb); low melting point
   Stainless steel (Cr, C, Fe); anti-corrosion, strength,
   Sterling silver (Cu, Ag); doesn’t tarnish
► Substitutional  alloys: created from metals having
  similar rMet (radii), similar coordination number
► Metals are mixed in the molten state (high
  temperature) and allowed to cool gradually
► The solute atoms occupy sites in the lattice that
  would normally be occupied by solvent atoms
► Sterling silver is created from 92.5% Ag, 7.5% Cu
  – both adopt ccp lattices
   rMet(Ag) = 140 pm
   rMet(Cu) = 124 pm   rMet = metallic radius: half the distance between
                        nearest neighbor atoms in a solid-state metallic
►   Interstitial alloys: solute atoms occupy the interstitial
    spaces (cavities) of a host lattice (e.g. carbon steel)
     Low C steel: 0.03 – 0.25% C (steel sheeting)
     Medium C steel: 0.25 – 0.70% (bolts, screws, etc.)
     High C steel: 0.8 – 1.5% (cutting, drilling tools)

►   Tetrahedral holes can contain atoms up to 0.225r,
    where r is the radius of the spheres that make up the
    close-packed lattice
►   Octahedral holes can accommodate larger atoms (e.g.
    C in carbon-steel) – up to 0.41r
      Intermetallic Compounds
► Some melts (combinations of metals in the
 molten state) will solidify in arrays that are
 different than either of the components that
 make up the mixture – these compounds
  are called intermetallic compounds
► Brass (Cu, Zn) is an intermetallic compound
► g-brass has a formula of Cu5Zn8
                          Unit Cells
► The   smallest collection of spheres that describes a
  lattice (when it is repeated) is the unit cell
► Some of the atoms in a unit cell are shared with
  other cells, and so they don’t completely belong to
  one cell            Corner site

Example, the atoms at the corners of the
cell shown on the right are shared with
seven other cells
Each contributes 1/8 of an atom to the
cell shown, and is called a “corner site”

                             Face-centered site
                                             A body-centered cubic unit cell
             Unit Cell Contributions
►   Corner sites each contribute 1/8
    of a sphere to a cell
►   Edge sites contribute ¼ sphere
    to the cell
►   Face sites contribute ½ sphere
    to the cell
►   Sites contained in the cell (e.g.
    the center site of a bcc cell)
    contribute 1 each

    How many spheres (atoms/ions) occupy   Face-centered site
           - a simple cubic unit cell?
           - a body-centered cubic cell?
           - a face-centered cubic cell?
      Dimensions of Cubic Lattices
►   For cubic unit cells, it is possible to determine cell lengths
    (edge) and dimensions of lattice spaces
►   For example, a simple cubic cell will have an edge length
    of 2r, where r is the radius of a sphere      In this diagram, let a = 2r

    It can be shown that the corner-to-corner length 3 a
    indicated in the picture will be 31/2a
Each sphere has a radius of r
Each edge length (length of the edge of the
unit cell) is then 2r (spheres contact at surface)
Can also be shown that                                                         a
           - a sphere of radius up to 0.73r can fit
           in the center
           -the atoms occupy 52% of the available
           space of the unit cell
                                                      Simple cubic unit cell

►   In diamond, C-atoms take
    on a fcc-type arrangement
►   Each C-atom is four-
    coordinate (tetrahedral)
    and so possesses a
    complete octet of electrons
    through covalent bonds
►   Also adopted by Si, Ge, Sn
    and Pb (other C-group
► The  most thermodynamically
  stable form of carbon (diamond
  is metastable), graphite consists
  of layers of carbon sheets that
  are built from fused, 6-
  membered carbon rings
► The bonding that exists within a
  layer is covalent (and
  delocalized), but between
  planes, dispersion forces (non-
  covalent) hold the rings together
  (weak intermolecular force)
                              Binary Lattices

             Rock Salt Lattice (NaCl)
►   Interpenetrating fcc lattices of Na+ ions and Cl- ions (rem: fcc = ccp)
►   Cl- ions are much larger then the Na+ ions (Cl-: 181 pm; Na+: 102 pm).
►   Na+ ions are occupying octahedral holes in the unit cell shown
►   Each Na+, Cl- ion is octahedral (six coordinate)
►   NaCl-type lattice structures exist for many ionic compounds (NaF, NaBr,
    NaI, NaH, LiX, KX, RbX (X = halide), AgF, AgCl, AgBr, MgO, CaO, SrO,
    BaO, MnO, CoO, NiO, MgS, CaS, SrS, BaS)
                     CsCl Lattice
►   Eight coordinate ions (cations and
    anions) – body centered cubic
►   Interpenetrating simple cubic-type
►   Adopted by CsBr, CsI, TlCl, TlBr
            CaF2 (Fluorite) Lattice
►   Eight-coordinate cations (Ca2+, grey spheres)
►   Four coordinate anions (F-, blue spheres)
►   Six cations are face-positioned, shared between adjacent
►   This lattice type adopted for group II metal fluorides,
    BaCl2, and f-block metal dioxides
►   Exchanging the cations and anions in this structure would
    yield an antifluorite lattice – M2X stoichiometry. Adopted
    by some group I metal oxides and sulfides (e.g. Na2S).
 Zinc blende, ZnS (diamond type) lattice

► Similar to the fluorite lattice, with removal of half
  of the anions (so MX2 to MX stoichiometry).
► Looks something like the structure shown for
  diamond – each atom is in a tetrahedral

                                           How many Zn, S
                                           atoms exist in this

  Zn: grey
  S: blue
         Wurtzite (ZnS) lattice
► Wurtzite  formed by
  high temperature
  transition from zinc
► Hexagonal prism unit
  cell with all ions
  tetrahedrally sited
► How many Zn2+, S2-
  ions exist in this
              β-cristobalite (SiO2) lattice

      ►   Again, a lot like a diamond
          structure, but with oxygen
          ions between the
          tetrahedral Si ions.
      ►   Si-O-Si bond angle in
          figure is 180o, while in
          practice, it is found to be
          147o (bonding in SiO2 is
          not purely electrostatic).

In a pure ionic model, electrostatic attraction would be the only
factor that would be expected to hold an ionic lattice together
         Rutile (TiO2) structure
► Oxygen   ions (white)
  are trigonal planar
  while titanium centers
  (black) are octahedral.
► Four oxygen ions are
  face-oriented, while
  two are contained in
  the cell
        Perovskite (CaTiO3) lattice
►   A double oxide (oxygen atoms are coordinated to both
    Ca2+ and Ti4+
►   Ca2+ ion is at center of cube unit cell
►   Ti4+ ions at corners of the cube (eight of these)
►   O2- ions at each edge of the cube (twelve of these)
                 CdCl2, CdI2 Lattices
►   Common for MX2 structures to
    crystallize in this structure
►   Can observe the layers as ABAB
    (layered lattices)
►   I- ions (grey) are arranged in a
    hcp format with the Cd2+ ions
    (blue) occupying octahedral
►   In CdCl2, the arrangement is ccp
►   Attractive forces that exist
    between these planes is weak
    (dispersion forces), and so
    fracture of a crystal of this kind
    usually produces cleavage
    planes                               CdI2 lattice
              Lattice Energies
► We   have already looked at bond dissociation
  enthalpies (energy required to break bonds in
  homonuclear and heteronuclear diatomics) when
  we looked at Pauling electronegativities
► Energy is also required to break apart ionic
  lattices, due to the large amount of electrostatic
  forces that exist between the ions in the lattice
► Coulombic forces (attractions, repulsions)
► Born forces (electron-electron, nucleus-nucleus)
    Energy Between Two Point
►   Consider what happens if we bring two point charges
    from an infinite separation to form an ion pair:
                  Mz+(g) + Xz-(g)  MX(g)
►   We can calculate the change in internal energy (DU)
                             z z e 2 
                      DU  
                             4e d 
                                 o     
►   z is the magnitude of the charge of an ion; |z+/-| is the
    absolute value of this quantity (the modulus)
►   e is the charge of an electron (1.602 x 10-19 C)
►   eo is the permittivity of a vacuum (8.854 x 10-12 C2/J.m)
►   d is internuclear separation
► Because  oppositely charged ions are attracted to
  one another, energy is released in this process
► Consider the attractions and repulsions that exist
  in a rock salt lattice (between oppositely charged
  and like charged ions)

                         z z e 2 
                  DU  
                         4e d 
                             o     
     A Summary of Attractions and
► The  attractions experienced by the Na+ ion are
  summarized as follows:
► 6 Xz- ions, each at a distance d (the ions at the
  face sites)
► 12 Mz+ ions, each at a distance (21/2)d (the ions at
  the edge sites)
► 8 Xz- ions each at a distance (31/2)d (the ions at
  the ions at the corner sites)
► 6 Mz+ ions each at a distance of (41/2)d (imagine
  the next set of blue spheres in figure beyond the
  face gray spheres)
               Madelung Constants
► We must factor these attractions and
 repulsions into the expression:

                              12   2  8         6   2 
                6 z z   
 DU                              z    z z    z ...
        4e o d               2       3         4     

► Convergent  series, which yields a number
 for each lattice type that is called the
 Madelung constant, A
           Madelung Constants
► Ifwe sum the interactions (attractive and
  repulsive) between the ions of this lattice, we get a
  convergent term (for this lattice, the value
  converges to a value of ~1.7476). This value is
  obtained regardless of the actual charges on the
  ions.                           Table 7-2, M. & T.
► Madelung constants (A) are unique for each
  coordination environment (i.e., for each type of
  crystal lattice).
                          A z z e 2 
                   DU  
                          4e d 
                               o      
  Lattice Energy (almost there…)
► The internal energy change for the formation of
  one mole of an ionic lattice in this arrangement is
  then calculated as:
                      LA z  z e 2 
               DU  
                      4e d 
                             o      

►L  = Avogadro’s number (6.022 x 1023 mol-1)
► but what about Born forces? (nuclear-nuclear,
  electron-electron forces)
                          Lattice Energies
   ► If  we consider electrostatic and Born forces,
       we arrive at the Born-Mayer equation
       (evaluated at equilibrium internuclear
       separation, do)                        Correction for
    This equation                                                           Born forces
   will enable us
to predict lattice                 LA | z || z | e 2    r
 energies (called    DU (0 K )                       1  
                                                        d 
   the calculated                      4e o d o           o 
   lattice energy

   ►r    corrects for repulsions at short distances.
       Typically, a value of 30 pm is used for r.
•Lattice energy can also be defined as the energy required to pull apart an ionic lattice
 into its gas-phase ions, as defined in M&T, in which case, it is a positive energy
             Lattice Energies
► Lattice  energy can be defined as the internal
  energy change associated with the formation of
  one mole of the solid from its constituent gas
  phase ions at 0 (zero) Kelvin. Thus, at 0K, the
  lattice energy corresponds to the process:
                Mn+(g) + nX-(g)  MXn(s)

Lattice energies may be estimated by assuming an
  electrostatic model (ions are point charges) – a
  good approximation in some cases. In others, not
  so good.
       Sample Calculation: Lattice
► Calculate        the lattice energy for NaCl (rNa-Cl =
  283 pm)
                                  LA | z || z | e 2    r
                    DU (0 K )                       1  
                                    4 o d o  d o      

              (6.022x1023 )(1.7476) | 1 || 1 | (1.602 x1019 C ) 2     30 pm 
DU (0 K )                                                         283 pm 
                                                                   1        
                4 (8.854 x10 C J m )(283x10 m) 
                              12 2 . 1. 1             12
 766871J / mol
 767kJ / mol
                                                                   rNa+ = 116 pm
                                               Appendix B
                                                                   rCl- = 167 pm
                 Sizes of Ions
► Ions of ionic lattices are treated as hard spheres in
  these models
► Think of the ions (of opposite charge) as just
  touching in this model, thus internuclear
  separation (do), determined experimentally, yields

                  d = rcation + ranion

► This pure, electrostatic model is a (sometimes
  poor) approximation of structure. It assumes zero
  electron density between ion spheres (at odds
  with MO theory)
  Lattice Energy and Born-Haber
► The  Born-Mayer equation calculates an
  estimate of the lattice energy by considering
  various interactions in an electrostatic
► We can also determine the lattice energy by
  employing experimental values (e.g.
  enthalpies of formation, bond dissociation
  enthalpies, etc.). This procedure is called
  the Born-Haber cycle
Relevant Reactions For Sodium Chloride, NaCl

► Atomization
                    Na(s)  Na(g)
► Bond Breaking     Cl2(g)  2Cl(g)       Appendix
► Ionization        Na(g)  Na+(g) + e-      B-2

► Electron Affinity Cl(g) + e  Cl (g)
                             -     -         B-3

► Lattice Energy
                    Na+(g) + Cl-(g)  NaCl(s)

                               Na(s) + ½Cl2(g)  NaCl(s)
► Formation

   Bond enthalpy, DCl2 corresponds to DH for Cl2(g)  2Cl(g) process
                Born-Haber Cycle
 DHof(MXn,s) = DaHo(M,s) + n/2D(X2,g) +
      ΣIE(M,g) + nDEAH(X,g) +
                                   DHolattice(MXn) = DU(0K)
►   Rearranging, we get

    DHolattice(MXn) ≈ DHof(MXn,s) - DaHo(M,s) - n/2D(X2,g)-
                 ΣIE(M,g) - nDEAH(X,g)

Born-Haber cycles are often called “experimental” lattice
  energies, since they are derived from thermochemical
                 Sample Calculation
► Use the Born-Haber cycle to calculate the lattice
  energy of NaCl, given DHof(NaCl,s) = -407.6

► DaHo(Na,s)  (sublimation of Na) = 108 kJ/mol
► D(Cl2,g) (dissociation of Cl2) = 242 kJ/mol*
► ΣIE(Na,g) (appendix B-2) = 495.8 kJ/mol
► DEAH(Cl,g) (appendix B-3) = -349 kJ/mol*

* Defined for Cl2(g)  2Cl(g)
* Defined for Cl(g) + e-  Cl-(g)
             Sample Calculation
► Thus,   the lattice energy is:

   DU(0K) ≈ DHof(MXn,s) - DaHo(M,s) - n/2D(X2,g)-
            ΣIE(M,g) - nDEAH(X,g)

   DU(0K) ≈ (-407.6 kJ/mol) – (108 kJ/mol) - ½( 242
            kJ/mol) – (495.8 kJ/mol) – (-349 kJ/mol)

                  = -783.4 kJ/mol
           Lattice Energies
► Wesee there is only a minor discrepancy
 between the value obtained with the Born-
 Mayer equation (-767 kJ/mol) and the Born-
 Haber cycle (-783 kJ/mol)

    NaCl, there’s only a ~2% difference
► For
 between the calculated and experimental
 energies (an ionic model provides a good
 approximation of NaCl)
                 Lattice Energies
► Since the calculated values agree so well (2% error), we see the
  electrostatic model is a reasonably good assumption for the type
  of bonding which exists in a NaCl(s) lattice
► Not true for layered structures like CdI2(s) – recall the forces
  that exist in this structure
► We also see that for silver halides, the calculated and
  experimental energies differ greatly, in the order
  AgF<AgCl<AgBr<AgI. The bonding with larger halide ions has
  more covalent character, and thus an ionic approximation does
  not hold

                       F   Cl   Br       I
                       Radius Ratios
►A   rough guide for predicting structures
 of salts (cations and anions). Use
 rcation/ ranion, or r+/r-)
      Value of r+/r-     Predicted      Predicted
                         Coordination   Coordination
                         Number of      Geometry of
                         Cation         Cation
      <0.15              2              Linear

      0.15-0.22          3              Trigonal Planar

      0.22-0.41          4              Tetrahedral

      0.41-0.73          6              Octahedral

      >0.73              8              Cubic
                        Radius Ratios
►   These guidelines often yield incorrect predictions –
    example: LiBr (r+/r- = 0.38; tetrahedral)
►   Predict only one coordination geometry for a given
    combination of ions (not helpful for polymorphic samples)
►   Examples:

      What is the coordination number of Ti in rutile? (rTi3+ =
       75 pm; rO2- = 124 pm)

      What is C.N. of Ca in fluorite? (rCa2+ = 126 pm; rF- =
       117 pm)

      What is C.N. of Zn in zinc blende? (rZn2+ = 77pm; rS2- =
       170 pm)
 Electrical Conductivity in Metals,
 Semiconductors, and Insulators
           conductivity is a property displayed by
► Electrical
  metals and some inorganic and organic materials
► Loosely defined as the ability of a substance to
  permit movement of electrons throughout its
► On a molecular level, electrons can be passed
  around (atom-to-atom) by being promoted into
  empty orbitals on other atoms
► Band theory is used to explain conductivity
     Electrical Conductivity and Resistivity

► Resistivity(r) measures a substance’s electrical
  resistance for a wire of uniform cross-section

                    resistivity / Wm length / m 
           a        cross  sectional _ area / m 

► Resistance is measured in ohms (W)
► Conductivity is 1/resistivity. Units of conductivity
  are W-1m-1 or S/m (S = Siemens; S = W-1)
Electrical Conductivity and Resistivity

        “electrical resistivity”

► The resistivity of
 a metal increases
 decreases) with
     Electrical Conductivity and Resistivity

► The   resistivity of a
    semiconductor decreases
    (conductivity increases)
    with increasing temperature

Resistivity with temperature for a semiconductor
   MO Theory Approach to Band Theory
   ►   Consider a line of H-atoms that interact through their valence s-orbitals.
       As more H-atoms interact, more MO’s are created

In the infinite structure, there is a continuum of energy states - a “band” (non-quantized)
Valence orbitals overlap to create a “valence band”
              Band Theory of Metals
In order for electrons to be able to move through a material, they must jump
from an occupied molecular orbital to an unoccupied orbital. In a metal, this
should be easily accomplished (metals are highly conductive).

Lithium (and other alkali metals) have a half-filled valence s-orbital (occupied).
In the infinite solid, there will be a half-full band. Electrons can move into an
unoccupied MO with minimal energy cost (small applied potential)

                                         density of
                                          states is
                                           in the
                                         middle of
                                         the band

                                    Band that is created from occupied
                                    orbitals is called “valence band”
          Band Theory of Metals
►   For the metal, Be (2s2), how does
    electron movement occur? (valence
    band is full)
►   The energy separation of 2s and 2p
    orbitals in Be is small enough that the
    conduction band (band that derives
    from unoccupied orbitals) overlaps the
    valence band

                                          electronic movement can
                                          occur by an electron jumping
                                          from the valence band into an
                                          energy-matched unoccupied
                                          conduction band orbital
        Semiconductors and Band Gaps
►   In many materials (e.g. diamond), there is an energy
    gap (band gap, Eg) between the valence band and
    the conduction band
►   For C, the energy separation that exists between the
    2s and 2p valence orbitals is not small enough for
    valence band-conduction band overlap to occur in the
    bulk material (e.g. diamond)

     Metals have either partially filled valence bands or
      overlapping valence and conduction bands (e.g. Na)
     Semiconductors have fully occupied bands that are
      separated from the valence band by a small energy gap
      (e.g. Si)
     Insulators have fully occupied valence bands that are
      separated from the conduction band by a significant energy
      gap (e.g. diamond)
                  metal with
                 valence band
   insulator                metal with
(e.g. diamond)              overlapping
                           valence and
                         conduction bands

► For semiconductors, thermal energy will
  enable electrons to move into the empty
  conduction band, creating “holes” in the
  valence band (orbitals that were occupied
  by electrons). The mobile electrons (and
  holes) give rise to electrical conductivity.
► Pure materials that are electrically
  conductive called intrinsic semiconductors
         Extrinsic Semiconductors
►   Certain semiconductors exhibit
    enhanced electrical conductivity when
    small quantities of another element
    are present in the semiconductor
    lattice (called doping).
►   The band structure of silicon involves
    a band gap of approximately 106
►   Introduction of either gallium or
    arsenic in very small quantities
    creates an extrinsic semiconductor,
    with a band gap of only about 10         Si adopts a diamond lattice
Extrinsic Semiconductors
               n-Type Semiconductors
►   Arsenic-doped silicon contains
    atoms having an additional
    valence electron (As is gr. 5A, Si
    Gr. 4A). As in a Si lattice use
    four of its electrons in bonding
    (one left over)
►   Even a small number of As
    atoms in the Si matrix creates a
    “donor band” with an energy
    just below the energy of the Si
    conduction band (~10 kJ/mol
►   Thermal energy can excite
    electrons from the donor band
    into the Si conduction band,
    yielding mobile charge carriers

Si is n-doped by introducing P or As atoms
The charge carriers in the Si band structure are electrons: “n-type” semiconductor
             p-Type Semiconductors
►   Ga has one less valence
    electron than Si
►   Introduction of a small number
    of Ga atoms (Gr. 3A) into the Si
    lattice structure creates a hole
    (nothing for silicon’s fourth
    electron to bond to)
►   The energy of these holes
    creates a band just above the
    valence band energy (by ~ 10
    kJ/mol), an “acceptor band”
►   Electrons can occupy this new
    band, leaving holes in the Si
    valence band

The mobile holes in Si band structure yield conductivity – a “p-type” semiconductor
► Semiconductors  can be inorganic or organic
► Inorganic semiconductors consist of main group
  elements (Si, Ge, Ga, As, In,…)
► Organic semiconductors consist of conjugated
  carbon structures, typically oligomers or polymers

                                   S             N
    *                          *           * *       n
                                                         *   *           *
         n   *   *     n   *           n                             n
              Semiconductor Devices

►   Thermal population of unoccupied states makes these
    materials conductive; at 0K, electrons occupy lowest
    possible energies
                           The Fermi energy is energy at which an electron is
                           equally likely to be in occupied and unoccupied bands

p-type semiconductor at 0K (left)                 n-type semiconductor at 0K (left)
     and at T = 298K (right)                           and at T = 298K (right)
           Semiconductor Devices
►   The Fermi level serves to set the relative energies of the p-
    type and n-type interfaces, and is the energy at which an
    e- is equally likely to be in the valence or conduction band.
►   In an intrinsic semiconductor, this lies in the middle of the
    bandgap of the host semiconductor
►   Doping
     Raises the energy of the Fermi level to between the donor band
      and the bottom edge of the host’s conduction band
     Lowers the energy of the Fermi level to the region between the
      acceptor band and the top of the host’s valence band

            intrinsic            p-type              n-type
            Semiconductor Devices p,n-junction
►   When n- and p-type
    semiconductors contact, mobile
    electrons in the n-type layer
    near the interface can migrate
    into the p-type layer, resulting
    in recombination
►   The extra electrons in the p-
    type layer raise its energy and
    new holes in the n-type layer
    lower its energy. Charge
    movement ceases nearly
    immediately, as the p-type layer
    accumulates negative charge
    and the n-type layer positive
►   Recombination results in the
    formation of a depletion zone at
    the junction. This depletion
    zone is accompanied by its own
                                    +   power    -
►    When this kind of device is
     connected to a DC power
     source, a connection can be
     made in two ways:
    1. Negative terminal to n-
        type layer, positive
        terminal to p-type layer
        (forward bias): as the
        potential at the negative
        terminal is made more and
        more negative (and the
        positive terminal more and
        more positive), the
        potential difference of the
        depletion zone can be
        overcome and a current
        flows (e-’s repelled by –
        terminal; holes repelled by
        + terminal)
                                 -   power    +
2.   Negative terminal to p-
     type layer; positive
     terminal to n-type layer
     (reverse bias): electrons
     and holes are pulled away
     from the depletion zone.
     Charge cannot move
     across the junction and
     there is essentially zero
►   Diodes are semiconductor
    devices which permit
    current to flow in one
    direction but not the other
    (current rectifying), made
    through combinations of p-
    type and n-type
►   Such devices are useful in
    electronics (e.g. in
    common circuitry)
                 Photovoltaic Cells
►   In the absence of an
    applied potential, electrons
    can be made to jump from
    valence to conduction
    bands by absorbing
    radiation (e.g. sunlight)
    and through external
    connections, can be used
    to power electronics (solar
    calculators, solar panels)
►   Under reverse bias
    conditions, if the bandgap is
    small enough, visible
    radiation may be sufficient
    to promote electrons from
    the valence band to the
    conduction band (thus
    current flows in the
    presence of light – a
►   Used in sensors
    (photodetectors for UV,
    visible, infrared radiation),
    automatic lights, etc.
    Light Emitting Diodes (LEDs)
► Under forward bias conditions, electrons move from the n-
  type layer (conduction band) into the p-type layer
  (valence band)
► This movement results in recombination (releases energy).
  When electrons fall into the holes of the p-doped layer, if
  the energy change is of the right magnitude, visible light
  will be emitted (luminescence)
► The color of the radiation emitted will depend on the
  bandgap (Eg), so by varying the bandgap (by controlling
  the composition of the semiconductor material), different
  colors can be produced