Band Theory, Kittel chapter 7

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Institute of Solid State Physics
Technische Universität Graz
Band Theory, Kittel chapter 7
Calculate the dispersion relation for electrons in a crystal

Kronig-Penney model
I
II
2 2
d ψ
− + Vx () ψ =
2
2mdx
Eψ
Solutions can be found in region I and region II
Match boundary conditions

Kronig-Penney model
Solutions can be found that are simultaneous eigenfunctions of the
Hamiltonian and the translation operator.
Eigenfunctions of the translation operator can be found in terms of any
two linearly independent solutions. A convenient choice is:
dψ
1
dψ
2
ψ1(0) = 1, (0) = 0, ψ2(0) = 0, (0) = 1.
dx
dx

Kronig-Penney model
for 0 < x < b
ψ ( x) = cos( kx), ψ ( x)
=
1 1 2
for b < x < a
sin( kx
1
)
k
1
k
=
2 mE ( −V1
)
1 2
ψ
ψ
x k x b kb
k sin( k ( x−b))sin( kb)
1 2 1
1() = cos(
2( − ))cos(
1
) −
,
k2
cos( k ( x−b))sin( kb) sin( k ( x−b))cos( kb)
2 1 2 1
2() x = +
.
k1 k2
k
=
2 mE ( −V2
)
2 2
Except for the coefficients, these are the same solutions as we found for light
in a layered material.

Kronig-Penney model
at x = a
ψ
ψ
a k a b kb
k sin( k ( a−b))sin( kb)
1 2 1
1() = cos(
2( − ))cos(
1
) −
,
k2
cos( k ( a−b))sin( kb) sin( k ( a−b))cos( kb)
2 1 2 1
2() a = +
.
k1 k2
The translation operator translates the function a distance a.
⎡ψ
( )
( )
1
x + a ⎤ ⎡T11 T12
⎤⎡ψ1
x ⎤
⎢ ⎥=
⎢ ⎥.
ψ ( x + a)
⎢
T T
⎥
⎣ ⎦ ψ ( x)
⎣ 2 ⎦ 21 22 ⎣ 2 ⎦
The elements of the translation operator can be evaluated at x = a.

If |α| > 2, the potential acts like a mirror for electrons
Kronig-Penney model
⎡ dψ
1 ⎤
ψ1
( )
( a) ( a
⎡ψ
)
1
x+
a ⎤ ⎢ dx ⎥⎡ψ1( x)
⎤
⎢ ⎥= ⎢ ⎥⎢ ⎥
⎣ψ2( x+ a)
⎦ ⎢ dψ
2
ψ2
( ) ( )
( x)
ψ2
a a ⎥⎣ ⎦
⎢⎣
dx ⎥⎦
The eigen functions and eigen values are
2 ψ
2( a) 1
ψ
± ( x) = ψ1() x + ψ2(), x λ±= ( α±
δ)
,
dψ
2() a
− ψ
2
1()
a ± δ
dx
δ
2
= α −
dψ
2()
a ⎛k2 k ⎞
1
α = ψ1() a + = 2cos( k2( a−b)
) cos( kb
1 ) − ⎜ + ⎟sin( k2( a−b)
) sin ( kb
1 ).
dx ⎝k1 k2⎠
4

Kronig-Penney model
The eigen values:
1
λ± = ( α±
δ)
2
δ
2
= α −
4
dψ
2()
a ⎛k2 k ⎞
1
α = ψ1() a + = 2cos( k2( a−b)
) cos( kb
1 ) − ⎜ + ⎟sin( k2( a−b)
) sin ( kb
1 ).
dx ⎝k1 k2⎠
If |α| > 2, the eigenvalues are real. The solutions grow or decay exponentially.
If |α| < 2, the eigenvalues are complex and on the unit circle.
ika
λ = e
k
2
1 1 4
tan
⎛ − −α
=± ⎞
a ⎜ α ⎟
⎝ ⎠

Kronig-Penney model
|α| < 2
⎛
tanka =±⎜
⎜
⎝
2
4−α
α
⎞
⎟
⎠

Kronig-Penney model
free electrons
(a) The energy-wave number dispersion relation. The dashed line is the Fermi energy. (b) The
density of states. (c) The internal energy density (solid line) and Helmholtz free energy density
(dashed line). (d) The chemical potential (solid line) and the specific heat (dashed line). All of
the plots were drawn for a square wave potential with the parameters: V = 12.5 eV, a =2×10 -
10
m, b =5×10 -11 m, and an electron density of n = 3 electrons/primitive cell.

Band structure in 1-D

from Ashcroft and Mermin
Extended, reduced, and
repeated zone schemes
Extended
k′ = k + k + G
el el photon
Repeated
E ' = E + ω
el el photon
Reduced