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
1
Magnesium metal
2
From Molecular Orbitals to Band Theory
H2
Bond order = ½ ( # of bonding electrons - # of antibonding electrons )
Electron configuration of H2 : (σ1s)2
B.O. of H2 = ½ (2 - 0) = 1
3
M.O. from Linear Combinations of Atomic Orbitals (LCAO)
Ψ ( x ) = ∑ cn χ n ( x )
n
χn(x) : atomic orbital of atom n
Cn : 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
4
Constructive Interference for bonding orbital
+
+
The electron density is given by
ρ(x) = Ψ*(x) Ψ(x) =|Ψ(x)|2
5
Destructive Interference for antibonding orbital
+
-
6
Energies of the States
Ek =
* ˆ
ψ
∫ k Hψ k
*
ψ
∫ kψ k
α +β
Ebonding =
Eantibonding
≈α +β
1+ S
α −β
=
≈α −β
1− S
if S~0 (neglecting overlap)
α = ∫ψ 1*s (1) Hˆ ψ 1s (1) < 0
Coulomb integral
β = ∫ψ (1) Hˆ ψ 1s (2) < 0
*
1s
α
-β
+β
Exchange integral
7
(He)2 molecule is not present!
8
Species
Electron
configuration
B.O.
Bond
energy
(kJ/mol)
435
Bond
length (pm)
H2
(σ1s)2
1
H2+
(σ1s)1
½
269
106
H2-
(σ1s)2(σ1s*)1
½
238
108
He2
(σ1s)2(σ1s*)2
0
-
-
74
9
10
2nd Period Homo-nuclear Diatomic
Molecules
Electron configuration of Li2 : KK(σ1s)2
B.O. of Li2 = ½ (2 - 0) = 1
11
Hetero-nuclear Diatomic Molecule
Lewis Structure
12
Chemical bond from molecules to solids
1 D array of atoms
orbitals
empty
filled
13
The 2s Band in Lithium Metal
Anti-bonding
Conduction band
e-
e-
Valence band
Bonding
14
Band Overlap in Magnesium
Conduction band
Valence band
15
Band Structure of Insulators
and Semiconductors
16
Density of state
= dn/dE
n = number of states
(a)
(b)
Density of states in (a) metal, (b) semimetal
(e.g. graphite).
17
Conductivity of Graphite
insulator
e- -conductor
18
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.
1
Population, P = ( E − µ ) / kT
where, µ = chemical potential
e
+1
When E= µ, P= 1/2
19
(a)
population
(b)
Fermi distribution and the band gap at T> 0 for
(a) Intrinsic semiconductor, (b) Insulator
20
Extrinsic semiconductor: (a) n-type, e.g. P doped Si
(b) p-type, e.g. Ga doped Si.
21
p-type
n-type
p-n junction
22
23
LUMO
LUMO
HOMO
HOMO
24
One-dimensional chain with n π-orbitals, jth level
Ej = α + 2βcosjπ/(n+1) , j =1, 2, 3 ….
25
Linear Conjugated Double Bonds
Antibonding
E
Bonding
π-M.O.
26
Cyclic ring
empty
filled
Cyclic system with n ≥ 4 atoms, jth level
Ej = α + 2βcos2jπ/n , j = 0, 1, 2, 3 ….
27
The π-Molecular Orbitals of Benzene
E
28
29
30
31
Elementary Band Theory
32
If Ψ(x) is the wave function along the chain
Periodic boundary condition:
The wavefunction repeats after N lattice spacings
Or, Ψ(x+ Na) = Ψ(x)
(1)
The electron density is given by
ρ(x) = Ψ*(x) Ψ(x)
(2)
The periodicity of electron density ⇒ ρ(x +a) = ρ(x) (3)
33
ρ(x +a) = ρ(x)
(3)
This can be achieved only if Ψ(x+ a) = µ Ψ(x)
µ is a complex number µ* µ = 1
(4)
(5)
Through n number of lattice space Ψ(x+ na) = µn Ψ(x)
Through N number of lattice space Ψ(x+ Na) = µN Ψ(x)
(6)
(7)
Since Ψ(x+ Na) = Ψ(x), µN = 1
⇒ µ = exp(2πip/ N) = cos(2πp/ N) + i sin(2πp/ N)
Where, i = √-1, and p is an integer or quantum number
(8)
(9)
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
34
Since Ψ(x+ a) = µ Ψ(x)
Ψ(x+ a) = µ Ψ(x) = exp(ika) Ψ(x)
(4)
(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 function
Bloch function Ψ(x) = exp(ikx) µ(x)
and, µ(x+a) = µ(x)
(14)
a periodic function, unaltered by
moving from one atom to another
e.g. atomic orbitals
⇒The periodic arrangement of atoms forces the wave functions of
e- to satisfy the Bloch function equation.
35
real
Free ewavelength
imaginary
λ= ∞
Ψ(x) =µ(x)= ϕ1s
Real part
of
restricted e-
λ= 2π/k
λ= 2a
Ψ(x) = exp(ikx) ϕ1s
36
λ= 2a
node
Anti-bonding between
all nearby atoms
E
node
λ= ∞
Wave vector (Wave number) k = 2π/λ
1. Determining the wavelength of a crystal orbital
2. In a free electron theory,
k α momentum of e- ↔ conductivity
3. -π/a ≤ k ≤ +π/ a often called the First Brillouin Zone
37
Crystal Orbitals from Linear Combinations of Atomic Orbitals
(LCAO)
Ψ ( x ) = cn χ n ( x )
(15)
∑
n
χn(x) : atomic orbital of atom n
Cn : coefficient
Cn = exp(ikx) = exp(ikna)
Ψ ( x) = ∑
n
(16)
exp(ikna ) χ n ( x) Bloch sums of atomic orbitals
(17)
From eq (10), k = 2πp/(N a) for quantum number p of repeating
unit N
Consider a value k’, corresponding to a number of p + N
k’ = 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 k
However, Since k can be negative, usually let -π/a ≤ k ≤ +π/ a
38
⇒ First Brillouin Zone
1-D Periodic
a
X0
X1
X2
Bloch function
X3
X4
X5
X6
Ψk = Σn e-ikna Xn
where Xn atomic wavefunction
k value
Index of translation between 0 – π/a
or, 0 – 0.5 a*
(a* = 2π/a)
Reciprocal lattice
39
σ-bond
Xn = ϕ1s orbital
k=0
Ψ(0) = Σn e0 Xn
= Σn Xn
= X0 + X1 + X2 + X3 + X4 + X5 + X6 + …
X0
X1
X2
X3
X4
X5
X6
λ=∞
Ψ (π/a) = Σn e-inπ Xn
= Σn (-1)n Xn
= X0 - X1 + X2 - X3 + X4 - X5 + X6 - …
k = π/a
= 0.5 a*
X0
X1
X2
X3
X4
X5
λ = 2a
X6
Ψ (π/2a) = Σn e-inπ/2 Xn
= Σn (-1)n/2 Xn
= X0 + iX1 - X2 - iX3 + X4 + iX5 - X6 + …
k = π/2a
= 0.25 a*
X0
X1
X2
X3
X4
X5
λ = 4a
X6
node
40
Energies of the States
Ek =
*
ψ
∫ k Ĥψ k
*
ψ
∫ kψ k
Express Ψk and Ψk* as Bloch sums
N
⎧
⎫
* ˆ
ˆ
⎨∑ exp[i (n − m)k ]∫ xm Η xn ⎬
∫ψ Η ψ k = ∑
n =1 ⎩ m =1
⎭
N
⎫
⎧N
*
*
⎨∑ exp[i (n − m)k ]∫ xm xn ⎬
∫ψ kψ k = ∑
n =1 ⎩ m =1
⎭
*
k
N
Ek = α + 2βcos(ka)
α = ∫ χ n*Ĥχ n
β = ∫ χ m* Ηˆ χ n
If m and n are neighbors
41
Ek = α + 2βcos(ka)
and β < 0
E
Energy as a function of k for s-band
42
Xn = ϕ2p orbital
Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …
node
Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …
43
σ-bond
1st Brillouin zone
44
DOS(E)dE
= # of levels between E and E + dE
45
46
s-band as a function of a
k = 2π/ λ
k = 0 → 0.5a*
λ= ∞ → 2a
47
ϕ 0p
k
a
k=0
z
ϕ0p.5a
z
*
ϕ0p.25a
z
*
2a
0.5 a*
ϕ0
d
z2
ϕ0.5a
z2
f
ϕ0
f
*
z3
ϕ0.5a
z3
*
∞
π/2a 4a
1/6 a*
π/3a 6a
z
*
d
π/a
0.25 a*
ϕ ap
6
λ
bonding
∞
antibonding
2a
antibonding
2a
bonding
∞
48
σ bond
49
π- bond
Ej = α + 2βcos jπ/(n+1)
j = 1, 2, 3, ……, n
n
ψ j = ∑ C jr Φ r
r =1
Φ r π orbital of r center
⎤ sin rjπ
C jr = ⎡ 2
(n + 1)
⎢⎣ (n + 1)⎥⎦
The evolution of the π-orbital picture for conjugated linear
polyenes.
50
The evolution of the π energy levels of an infinite onedimensional chain (-CH-)n.
51
Binary Chain
Bloch function
N
Ψb (k ) = ∑ exp(ikna )[ak χ ( A) n + bk χ ( B ) n ]
n =1
N
Ψa (k ) = ∑ exp(ikna )[bk χ ( A) n − ak χ ( B ) n ]
n =1
Where, χ(A)n and χ(B)n are atomic orbitals at position n
52
χ(A) = s- orbital, χ(B) = σ p- orbital
X n = ak χ ( A) n + bk χ ( B ) n
Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …
No effective overlap between orbitals ⇒ non-bonding
Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …
Effective overlap between orbitals ⇒ bonding
53
E
Non-bonding
B band
54
χ(A) = s- orbital, χ(B) = σ p- orbital
X n = bk χ ( A) n − ak χ ( B ) n
Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …
No effective overlap between orbitals ⇒ non-bonding
Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …
Antibonding between neighbor orbitals
55
antibonding
non-bonding
non-bonding
α2
α1
bonding
56
57
Nearly-free electron model
Ψ = exp(ikx)
= cos(kx) + isin(kx)
E = ½ mv2 + V
= 2p2/m + V
de Broglie’s formula
Momentum p = h/λ
where h: Planck constant
λ= 2π/k
p = hk/2π ⇒ p α k
58
1st Brillouin zone
Energy gap is produced due to
periodic potential
59
60
61
Schematic showing the method of generating the
band structure of the solid.
62
chain
63
Effect of Distortion
A comparison of the change in the energy levels and energy bands
associated with (a) the Jahn-Teller distortion of cyclobutadiene
and (b) the Peierls distortion of polyacetylene.
64
65