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PHYSICS CHEMISTRY
Solid state band Molecular orbital
Valence band, VB HOMO
Conduction band, CB LUMO
Band gap, Eg HOMO-LUMO gap
Fermi energy, EF Chemical potential
n-doping Reduction, pH scale base
p-doping Oxidation, pH scale acid
Direct band gap Dipole allowed
Indirect band gap Dipole forbidden
Phonon or lattice vibration Vibrational mode
Peierls distortion Jahn-Teller effect
Introduction to Band Theory
Electronic Structure of Solids
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Valence electrons from the atoms spread throughout the entire
structure
Molecular orbitals are extended over all the constituent atoms
A large number of overlapping atomic orbitals lead to molecular
orbitals with very similar energies = continuous band
The bands are separated by band gaps (energy values where
there are no available levels)
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Formation of Bands
3d
4s
4p
1 atom NA atoms
Energies of electrons are quantized = can possess only allowed
energies, can occupy only allowed levels, cannot enter forbidden band
gaps
Formation of Electronic Bands
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Molecular orbitals Electronic bands
N atomic orbitals combine to form bonding and antibonding
molecular orbitals = N energy levels
Electronic Bands
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s - orbital
p - orbital
As atoms get closer:
• Bands widen
• Bonding/Antibonding orbitals get
lower/higher in energy
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Bands arise from many MO’s of slightly different energies - different
degree of bonding
The bottom of the band – the lowest energy MO, all bonding character
The top – the highest energy MO with all antibonding character
The rest of the band is formed from all the MO’s with intermediate
bonding character between the two extremes
Electronic Structure of Solids
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Crystal Orbitals
N atoms in the chain = N energy levels and N
electronic states (MO)
The wavefunction for each electronic state:
Ψk = Σ eikna χn
Where:
a = the lattice constant (spacing between
atoms)
n identifies the individual atoms within the
chain
χn represents the atomic orbitals
k = a quantum number that identifies the
wavefunction and the phase of the orbitals
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Bonding
Antibonding
Bloch functions, crystal orbitals
simple example:
infinite one-dimensional array of
s-orbitals
k = wavevector gives the phase
of the AOs as well as the
wavelength of the electron
wavefunction (crystal momentum)
a = lattice constant
n = orbital counter
Large number of discreet levels =
band
Band Theory
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Filling Bands by Electrons
N atoms - 1 electron on each
N levels in a band
Occupied by pairs of electrons
N/2 levels filled
N/2 levels empty
3s
3p
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Bandwidth or Band Dispersion
Bandwidth = energy difference between the highest and lowest level
Bandwidth increases with better orbital overlap
• shorter interatomic distance
• closer energy match
• topology
• density, oxides more diffuse than halides, wider bands
• localization of electrons = narrow bands
Bandwidth arising from sigma > pi > delta overlap
Core orbitals – narrow bands (0.1 eV), 4f in lanthanides
Valence orbitals, s, p – wide bands (10 eV)
Wide bands = Large intermolecular overlap = delocalized eNarrow
bands = Weak intermolecular overlap = localized e-
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Bandwidth or Band Dispersion
Bandwidth = energy
difference between the
highest and lowest level
Bandwidth increases
with better orbital overlap
= shorter interatomic
distance
Bands
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Different types of orbitals (symmetry)
form separate bands
s, p, d bands
Sigma, pi, delta
Distinct bands with a band gap
Overlaping bands
Depends on the separation of the
orbitals and strength of the interaction
between the atoms
Strong interaction = wide bands and a
greater overlap
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Two Dimensional Lattice
Band structure
of a square lattice
of H atoms
(dHH = 2.0 Å)
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Density of States -DOS
a) MO diagram with translational symmetry
b) Density of states (DOS, N(E) dE)
Number of levels available for electrons at different energies per unit
volume of the solid
DOS is zero in the band gap
a) b)
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Rutile TiO2 Band Structure
Ti eg
Ti t2g
O 2p
O 2s
Band structure – spaghetti (a) and DOS (b)
Fermi level
Empty Ti4+ d0
Filled O2
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Contributions to
the total DOS of rutile
(a) Ti and O
(b) Ti d-orbitals, t2g and eg
Rutile TiO2 Band
Structure
Empty Ti4+
Filled O2
Classification of Solids
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Molecular solids – N2, C6H6, I2, …
Van der Waals forces, little change from the gas phase, electronic bands
correspond to empty and filled MOs of the individual molecules
Ionic solids – NaCl, CaF2, NiO, …
Charge transfer from cations to anions, energy bands made up from the atomic
orbitals of cations and anions
NaCl: 3p of Cl is the top filled band, 3s of Na is the lowest empty band
Covalent solids – diamond, Si, …..
Overlap of orbitals and electron sharing between adjacent atoms, filled bands
are made up from bonding MOs, empty bands are made up from antibonding
MOs
Metallic solids – Cu, Mg, W, TiO, ….
Na - very strong overlap of atomic orbitals on adjacent atoms, arising bands
are very broad, 3s, 3p, and 3d merge into a single wide band, electrons move
freely, little attraction to the atomic cores
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The distinction between metallic and non-metallic solids - the orbitals filling
Metallic behavior – a partially filled band, no gap between the top filled level
(Fermi level) and the lowest empty one
Non-metallic behavior – a completely filled level (the valence band) and an
empty one (the conduction band) separated by a band gap
Metallic and Non-metallic Solids
Metallic Non-metallic
Conduction band
Valence band
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Example NaCl, Eg = 9 eV
Ionic Solids
i = ions in the gas phase
ii = ions in the lattice,
Madelung potential,
filled levels stabilized by
positive potential of cations,
empty levels destabilized
iii = polarization energy
iv = electronic bands
Conduction band
Valence band
Fermi Level
20The Fermi level cuts a band in a metal
EF = the thermodynamic chemical potential for electrons in the solid
Metals – boundary between filled and unfilled levels
Nonmetals – situated in the band gap
The Fermi-Dirac distribution function:
P(E) = 1/[1 + exp{(E – EF)/kT}]
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Fermi Level
EF occupation probability ½
Levels at T = 0 K
E < EF occupied - probability = 1
E > EF empty - probability = 0
E = EF probability = ½
The Fermi-Dirac distribution function
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Fermi Level and Conduction
In the filled band every electron is matched by another
- no overall net motion of electric charge
For conduction to occur electrons have to be excited up to the
conduction band by overcoming an activation energy and hence,
the conduction of these compounds increases with temperature
Semiconductors and Insulators
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Band gap = the minimum photon energy required to excite an
electron up to the conduction band from the valence band
The band gap size determines a semiconductor or an insulator
Insulators - a completely filled valence band separated from the
next empty energy band by a large, forbidden gap
Diamond = insulator, a very large band gap of 6 eV
very few electrons have sufficient energy to be promoted and the
conductivity is negligibly small
Conductivity of nonmetallic solids increases with temperature
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Band Gap Energies, kJ mol1
NaCl 840
Diamond 480
ZnO 315
CdO 210
B 170
Si 125
Ge 85
Te, InAs 40
PbTe, InSb 20
α-Sn (grey) 8
Mg, Al, Cu, β-Sn (white)…… 0
1 eV = 1.60210 10−19 J 1 eV (molecule)−1 = 1 eV  NA = 96 485 J mol−1
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Band Gap Energies
1 eV = 1.60210 10−19 J 1 eV (molecule)−1 = 1 eV  NA = 96 485 J mol−1
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Electrical Conductivity
Conductivity of insulators and
semiconductors
increases with temperature
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Bands in Graphite
Graphite is a conductor
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Bands in Diamond
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Semiconductors - a similar band structure to insulators but the
band gap is small, some electrons have sufficient thermal energy
to be promoted up to the empty conduction band
Two types of conduction mechanism in semiconductors:
- Electrons promoted into the conduction band = negative
charge carriers, move towards a positive electrode under an
applied potential
- The holes these electrons leave behind = positive holes
Holes move when an electron enters them - a new positive hole
is created, the positive holes move in an opposite direction to the
electrons
Semiconductors
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Semiconductors
A direct band gap
(InP, InAs, GaAs)
the band edges aligned in k, so
that an electron can transit from
the valence band to the
conduction band, with the
emission of a photon, without
changing considerably the
momentum
An indirect band gap
(Si, Ge, AlSb)
the band edges are not aligned
so the electron does not transit
directly to the conduction band, in
this process both a photon and a
phonon are involved
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INTRINSIC
Intrinsic semiconductors are pure materials with the band
structure, the number of electrons in the conduction band is
determined only by the size of the band gap and the temperature
(more electrons with small band gap and high temperature)
EXTRINSIC
Extrinsic semiconductors are materials where the conductivity is
controlled by adding dopants with different numbers of valenece
electrons to that of the original material
Semiconductors
Semiconductors
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Two fundamental differences between extrinsic and intrinsic
semiconductors:
1) At standard temperatures extrinsic semiconductors tend
to have significantly greater conductivities than comparable
intrinsic ones
2) The conductivity of an extrinsic semiconductor can
easily and accurately be controlled by controlling the
amount of dopant
Materials can be manufactured to exact specifications of
conductivity
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Doping of semiconductors - introducing atoms with more or less
electrons than the parent element
Doping is substitutional, the dopant atoms directly replace the
original atoms
Very low levels of dopant are required, only 1 atom in 109 of the
parent atoms
Extrinsic Semiconductors
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Extrinsic Semiconductors
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Silicon - phosphorous atoms introduce extra electrons (one extra
valence electron for each dopant atom introduced as P)
The dopant atoms form a set of energy levels that lie in the band
gap between the valence and conduction bands, but close to the
conduction band
The electrons in the dopant levels cannot move directly - there is not
enough of them to form a continuous band
The levels act as donor levels because the electrons have enough
thermal energy to get up into the conduction band where they can
move freely
n-type semiconductors = the negative charge carriers or electrons
Extrinsic Semiconductors n-type
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Silicon doping with an element with one less valence electron, such
as Ga, for every dopant atom - an electron missing
Ga forms a narrow, empty band consisting of acceptor levels which
lie just above the valence band, discrete levels if the concentration of
gallium atoms is small
Electrons from the valence band have enough thermal energy to be
promoted into the acceptor levels,
electrons in the acceptor levels cannot contribute to the conductivity
of the material
The positive holes in the valence band left behind by the promoted
electrons are able to move - p-type semiconductors, the positive
holes
Extrinsic Semiconductors p-type
Electronic Bands in Nanoparticles
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Photocatalysis
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Conduction band
Valence band
TiO2 band gaps: 3.0 eV Rutile
3.15 eV Anatase
1972 Fujishima and Honda Photochemical splitting of water
H2O  ½ O2 (g) + H2 (g) ΔG = +237 kJ/mol
• A photon-driven reaction
• TiO2 adsorbs photons with an
energy higher than or equal to
its bandgap (Eg)
• Electrons in the filled VB are
excited to the vacant CB,
leaving holes in the VB
• The generation of the electron–
hole pair
• Electron–hole pair separation
• The separated electrons or
holes migrate to the surface
• Redox reactions at the surface