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Band Theory• This is a quantum-mechanical treatment of bonding
in solids, especially metals and semiconductors.• The spacing between energy levels is so minute in
metals that the levels essentially merge into a band.• When the band is occupied by valence electrons, it is
called a valence band.• A partially filled or low lying empty band of energy
levels, which is required for electrical conductivity, is a conduction band.
• Band theory provides a good explanation of metallic luster and metallic colors.
<Ref> 1. “The Electronic Structure and Chemistry of Solids” by P.A. Cox
2. “Chemical Bonding in Solids” by J.K. Burdett
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From Molecular Orbitals to Band Theory
H2
Bond order = ½ ( # of bonding electrons - # of anti-bonding electrons )
Electron configuration of H2 : (σ1s)2
B.O. of H2 = ½ (2 - 0) = 1
4
M.O. from Linear Combinations of Atomic Orbitals (LCAO)
∑=Ψn
nn xcx )()( χ
χn(x) : atomic orbital of atom nCn : coefficient
For H2 molecule, Ψbonding = c1ϕ1s(1) + c2ϕ1s(2) = 1/√2(1+S) [ϕ1s(1) + ϕ1s(2) ]
~ 1/√2 [ϕ1s(1) + ϕ1s(2) ]
Ψantibonding = c1ϕ1s(1) - c2ϕ1s(2) = 1/√2(1-S) [ϕ1s(1) - ϕ1s(2) ]
~ 1/√2 [ϕ1s(1) - ϕ1s(2) ]
where, S = ∫ϕ1s(1)* ϕ1s(2) > 0 Overlap integral
5
+ +
Constructive Interference for bonding orbital
The electron density is given byρ(x) = Ψ*(x) Ψ(x) =|Ψ(x)|2
7
Energies of the States
∫∫=
kk
kkk
HE
ψψ
ψψ*
* ˆ
βαβα+≈
++
=S
Ebonding 1if S~0 (neglecting overlap)
βαβα−≈
−−
=S
E gantibondin 1
α+β
-β∫ <= 0)1(ˆ)1( 1
*1 ss Hψψα
Coulomb integral
∫ <= 0)2(ˆ)1( 1*1 ss Hψψβ
Exchange integral
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Species Electron configuration
B.O. Bond energy (kJ/mol)
Bond length (pm)
H2 (σ1s)2 1 435 74
H2+ (σ1s)1 ½ 269 106
H2- (σ1s)2(σ1s*)1 ½ 238 108
He2 (σ1s)2(σ1s*)2 0 - -
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2nd Period Homo-nuclear Diatomic Molecules
Electron configuration of Li2 : KK(σ1s)2
B.O. of Li2 = ½ (2 - 0) = 1
17
Density of state= dn/dE
n = number of states
(a) (b)
Density of states in (a) metal, (b) semimetal (e.g. graphite).
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Fermi level- the highest occupied orbital at T= 0
(a) (b)
Fermi distribution (a) at T= 0, and (b) at T> 0.
The population decays exponentially at energies well above the Fermi level.
Population,1
1/)( +
= − kTEeP µ where, µ = chemical potential
When E= µ, P= 1/2
20
(a) population (b)
Fermi distribution and the band gap at T> 0 for (a) Intrinsic semiconductor, (b) Insulator
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empty
filled
Cyclic system with n ≥ 4 atoms, jth levelEj = α + 2βcos2jπ/n , j = 0, 1, 2, 3 ….
Cyclic ring
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If Ψ(x) is the wave function along the chain
Periodic boundary condition:The wavefunction repeats after N lattice spacingsOr, Ψ(x+ Na) = Ψ(x) (1)
The electron density is given byρ(x) = Ψ*(x) Ψ(x) (2)
The periodicity of electron density ⇒ ρ(x +a) = ρ(x) (3)
34
ρ(x +a) = ρ(x) (3)
This can be achieved only if Ψ(x+ a) = µ Ψ(x) (4)µ is a complex number µ* µ = 1 (5)
Through n number of lattice space Ψ(x+ na) = µn Ψ(x) (6)Through N number of lattice space Ψ(x+ Na) = µN Ψ(x) (7)
Since Ψ(x+ Na) = Ψ(x), µN = 1 (8)⇒ µ = exp(2πip/ N) = cos(2πp/ N) + i sin(2πp/ N) (9)Where, i = √-1, and p is an integer or quantum number
Define another quantum number k (Wave number or Wave vector)k = 2πp/(N a) (10)⇒ µ = exp(ika) (11)
considering wave function repeats after N lattice spacings (N a) ~ λ
Although p = 0, ±1, ±2, …. , If N is very large in a real solid ⇒ k is like a continuous variable
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Since Ψ(x+ a) = µ Ψ(x) (4)Ψ(x+ a) = µ Ψ(x) = exp(ika) Ψ(x) (12)
Free electron wave like Ψ(x)= exp(ikx) = cos(kx) + i sin(kx) (13)can satisfy above requirement
A more general form of wave functionBloch function Ψ(x) = exp(ikx) µ(x) (14)
and, µ(x+a) = µ(x) a periodic function, unaltered by moving from one atom to anothere.g. atomic orbitals
⇒The periodic arrangement of atoms forces the wave functions of e- to satisfy the Bloch function equation.
36
real
imaginary
λ= ∞
λ= 2π/k
λ= 2a
wavelength
Ψ(x) =µ(x)= ϕ1s
Ψ(x) = exp(ikx) ϕ1s
Free e-
Real part of
restricted e-
37
λ= ∞
λ= 2a
Wave vector (Wave number) k = 2π/λ1. Determining the wavelength of a crystal orbital2. In a free electron theory,
k α momentum of e- ↔ conductivity3. -π/a ≤ k ≤ +π/ a often called the First Brillouin Zone
Anti-bonding between all nearby atoms
node
node
E
38
Crystal Orbitals from Linear Combinations of Atomic Orbitals (LCAO)
∑=Ψn
nn xcx )()( χ
χn(x) : atomic orbital of atom nCn : coefficient Cn = exp(ikx) = exp(ikna)
∑=Ψn
n xiknax )()exp()( χ Bloch sums of atomic orbitals
(15)
(16)
(17)
From eq (10), k = 2πp/(N a) for quantum number p of repeatingunit N
Consider a value k’, corresponding to a number of p + Nk’ = 2π(p + N)/(N a) = k + 2π/a
Cn’ = exp{i(k + 2π/a )na}= exp(ikna)∙exp(i2πn) = Cn
⇒ A range of 2π/a contains N allowed values of kHowever, Since k can be negative, usually let -π/a ≤ k ≤ +π/ a
⇒ First Brillouin Zone
39
1-D Periodic
X0 X1 X2 X3 X4 X5 X6
a
Bloch function Ψk = Σn e-ikna Xnwhere Xn atomic wavefunction
k value
Index of translation between 0 – π/aor, 0 – 0.5 a* (a* = 2π/a)
Reciprocal lattice
40
σ-bond Xn = ϕ1s orbitalΨ(0) = Σn e0 Xn = Σn Xn
= X0 + X1 + X2 + X3 + X4 + X5 + X6 + …k = 0
λ = ∞X0 X1 X2 X3 X4 X5 X6
k = π/a= 0.5 a*
Ψ (π/a) = Σn e-inπ Xn = Σn (-1)n Xn= X0 - X1 + X2 - X3 + X4 - X5 + X6 - …
X0 X1 X2 X3 X4 X5 X6 λ = 2a
k = π/2a= 0.25 a*
Ψ (π/2a) = Σn e-inπ/2 Xn = Σn (-1)n/2 Xn= X0 + iX1 - X2 - iX3 + X4 + iX5 - X6 + …
X0 X1 X2 X3 X4 X5 X6 λ = 4a
node
41
Energies of the States
∫∫=
kk
kkkE
ψψ
ψψ*
*H
Express Ψk and Ψk* as Bloch sums
[ ]∑ ∫∑∫= = ⎭
⎬⎫
⎩⎨⎧
Η−=ΗN
nnm
N
mkk xxkmni
1
*
1
* ˆ)(expˆ ψψ
[ ]∑ ∫∑∫= = ⎭
⎬⎫
⎩⎨⎧
−=N
nnm
N
mkk xxkmni
1
*
1
* )(expψψ
Ek = α + 2βcos(ka)
∫= nn χχα H*
∫ Η= nm χχβ ˆ* If m and n are neighbors
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Xn = ϕ2p orbitalΨ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …
Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …node
48
k = 0zp0ϕ
zpa*5.0
ϕ
zpa*25.0
ϕ
zpa6
*ϕ
2
0z
dϕ2
*5.0z
d
aϕ
3
0z
fϕ
3*5.0
zf
aϕ
0.5 a*
0.25 a*
1/6 a*
bonding
antibonding
bonding
antibonding
a
π/a
π/2a
π/3a
λ
2a
∞
4a
6a
∞
∞
2a
2a
k
50
π- bondEj = α + 2βcos jπ/(n+1)
j = 1, 2, 3, ……, n
( ) ( )1sin
12
centerr of orbital 1
+⎥⎦⎤
⎢⎣⎡
+=
Φ
Φ=∑=
nrj
nC
C
jr
r
n
rrjrj
ππ
ψ
The evolution of the π-orbital picture for conjugated linear polyenes.
52
Binary Chain
Bloch function
[ ]∑=
+=ΨN
nnknkb BbAaiknak
1)()()exp()( χχ
[ ]∑=
−=ΨN
nnknka BaAbiknak
1)()()exp()( χχ
Where, χ(A)n and χ(B)n are atomic orbitals at position n
53
χ(A) = s- orbital, χ(B) = σ p- orbital
nknkn BbAaX )()( χχ +=
Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …
No effective overlap between orbitals ⇒ non-bonding
Effective overlap between orbitals ⇒ bonding
Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …
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χ(A) = s- orbital, χ(B) = σ p- orbital
nknkn BaAbX )()( χχ −=
Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …
Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …
Antibonding between neighbor orbitals
No effective overlap between orbitals ⇒ non-bonding
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Nearly-free electron model
Ψ = exp(ikx)= cos(kx) + isin(kx)
E = ½ mv2 + V= 2p2/m + V
de Broglie’s formulaMomentum p = h/λwhere h: Planck constant
λ= 2π/kp = hk/2π ⇒ p α k
64
Effect of Distortion
A comparison of the change in the energy levels and energy bandsassociated with (a) the Jahn-Teller distortion of cyclobutadieneand (b) the Peierls distortion of polyacetylene.
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