6.2 Electrons in a Periodic Potential
139
Fig. 6.3 Periodic potential
U (one-dimensional cosine,
black) and the squares of
the wavefunctions −
(red) and + (blue) for the
wavevector at the zone
boundary,
K = G/2 = π/a
6.2.3.3 Solution at the Zone Boundary
We consider the solution at the zone boundary, i.e. at K = G/2. The kinetic energy is then the same
for K = ±G/2, i.e. λ K = λ K−G = (
2
/2m) (G
2
/4) = λ. The determinant (6.14) reads then
(λ − E)
2
− U
2
= 0 .
(6.16)
Thus the energy values at the zone boundary are
E ± = λ ± U =
2
2m
G
2
4
± U .
(6.17)
At the zone boundary, a splitting of the size E + − E − = 2U occurs. The center of the energy gap is given
by the energy λ K of the free-electron dispersion. The ratio of the coefficients is C G/2 /C −G/2 = ∓1.
The ‘−’ solution of (6.17) (lower energy) is a standing cosine wave ( − ), the ‘+’ solution ( + ) is
a standing sine wave as visualized in Fig. 6.3. For the lower-energy (binding) state the electrons are
localized at the potential minima, i.e. at the atoms, for the upper state (antibinding) the electrons are
localized between the atoms. Both wavefunctions have the same periodicity since they belong to the
same wavevector K = G/2. We note that the periodicity of is 2a, while the periodicity of
2 is
equal to the lattice constant a.
6.2.3.4 Gap States
For energies within the gap, solutions with a complex wavevector K = G/2 + i q exist. Solving (6.16)
results (in terms of q
= (
2
/2m) q
2 ) to
E ± = λ − q
2 ±
−4 λ q 2 + U 2 .
(6.18)
For energies E = λ + with −U ≤ ≤ U , the complex part of the wavevector is given by
q
= −(( + 2λ) +
4λ (( + λ) + U 2 .
(6.19)
The maximum value of q is in the center of the band gap ( = 0); for |U | | 2 λ, it is q
2
max ≈ U
2
/(4λ).
At the band edges ( = ±U ), q = 0. q is the characteristic length of an exponentially decaying wave