138
6 Band Structure
6.2.3.1 General Wave Equation
In this section, we will discuss the solution of a general wave equation for electrons in a periodic
potential. The solution is investigated particularly at the zone boundary. The potential U is periodic
with the lattice (6.1). It can be represented as a Fourier series with the reciprocal lattice vectors (lattice
vector expansion, cf. (3.19)):
U (r) =
G
U G exp (i G r) .
(6.9)
Since U is a real function, U −G = U
∗
G . The deeper reason for the success of such an approach is that
for typical crystal potentials, the Fourier coefficients decrease rapidly with increasing G, e.g. for the
unscreened Coulomb potential U G ∝ 1/G
2 . The wavefunction is expressed as a Fourier series (or
integral) over all allowed (Bloch) wavevectors K,
(r) =
K
C K exp (i K r) .
(6.10)
The kinetic and potential energy terms in the Schrödinger equation (6.6) are
∇
2
= −
K
K
2 C K exp (i K r)
(6.11a)
U =
G
K
U G C K exp (i (G + K) r) .
(6.11b)
With K
= K + G, (6.11b) can be rewritten as
U =
G
K
U G C K −G exp
i K
r
.
(6.12)
Now, the Schrödinger equation can be written as an (infinite) system of algebraic equations:
(λ K − E) C K +
G
U G C K−G = 0 ,
(6.13)
with λ K =
2 K
2
/(2m).
6.2.3.2 Solution for One Fourier Coefficient
The simplest (non-trivial) potential energy has only one important Fourier coefficient −U (U > 0)
for the shortest reciprocal lattice vector G. Also, we have U −G = U G . Thus, the (one-dimensional)
potential has the form U (x) = −2 U cos(Gx). Then the equation system (6.13) has only two equations
for C K and C K−G , leading to the condition
λ K − E
−U
−U λ K−G − E
= 0 .
(6.14)
We find two solutions
E ± =
λ K + λ K−G
2
±
λ K − λ K−G
2
2
+ U 2 .
(6.15)
6 Band Structure
6.2.3.1 General Wave Equation
In this section, we will discuss the solution of a general wave equation for electrons in a periodic
potential. The solution is investigated particularly at the zone boundary. The potential U is periodic
with the lattice (6.1). It can be represented as a Fourier series with the reciprocal lattice vectors (lattice
vector expansion, cf. (3.19)):
U (r) =
G
U G exp (i G r) .
(6.9)
Since U is a real function, U −G = U
∗
G . The deeper reason for the success of such an approach is that
for typical crystal potentials, the Fourier coefficients decrease rapidly with increasing G, e.g. for the
unscreened Coulomb potential U G ∝ 1/G
2 . The wavefunction is expressed as a Fourier series (or
integral) over all allowed (Bloch) wavevectors K,
(r) =
K
C K exp (i K r) .
(6.10)
The kinetic and potential energy terms in the Schrödinger equation (6.6) are
∇
2
= −
K
K
2 C K exp (i K r)
(6.11a)
U =
G
K
U G C K exp (i (G + K) r) .
(6.11b)
With K
= K + G, (6.11b) can be rewritten as
U =
G
K
U G C K −G exp
i K
r
.
(6.12)
Now, the Schrödinger equation can be written as an (infinite) system of algebraic equations:
(λ K − E) C K +
G
U G C K−G = 0 ,
(6.13)
with λ K =
2 K
2
/(2m).
6.2.3.2 Solution for One Fourier Coefficient
The simplest (non-trivial) potential energy has only one important Fourier coefficient −U (U > 0)
for the shortest reciprocal lattice vector G. Also, we have U −G = U G . Thus, the (one-dimensional)
potential has the form U (x) = −2 U cos(Gx). Then the equation system (6.13) has only two equations
for C K and C K−G , leading to the condition
λ K − E
−U
−U λ K−G − E
= 0 .
(6.14)
We find two solutions
E ± =
λ K + λ K−G
2
±
λ K − λ K−G
2
2
+ U 2 .
(6.15)