140
6 Band Structure
Fig. 6.4 Complex band
structure q (() according to
(6.19) for two different
values of λ/U
1
0.5
0
0.5
1
0
0.05
0.1
0.15
0.2
0.25
Complex wavevector q'
/ U
=10U
=5U
function. Such solutions occur at surfaces or interfaces. For larger band gaps, the localization length
is smaller (larger q) (Fig. 6.4).
6.2.3.5 Solution in the Vicinity of the Zone Boundary
For K in the vicinity of the zone boundary the solutions (6.15) can be developed. Therefore, we use
the (small) distance from the zone boundary
K = K − G/2. With λ = (
2
/2m) (G
2
/4) we rewrite
still exactly (6.15):
E ±
K
=
2
2m
1
4
G
2
+
K
2
±
4λ
2
K
2
2m
+ U
2
.
(6.20)
For small
K with
2 G
K
2m
|U |, the energy is then approximately given by
E ±
K
∼ = λ ± U +
2
K
2
2m
1 ±
2 λ
U
.
(6.21)
Thus the energy dispersion in the vicinity of the zone boundary is parabolic. The lower state has a
negative curvature, the upper state a positive curvature. The curvature is
m
∗
= m
1
1 ± 2λ/U
≈ ± m
U
2λ
,
(6.22)
and will be later related to the effective mass. The approximation in (6.22) is valid for |U | | 2λ. We
note that in our simple model m
∗ increases linearly with increasing band gap 2U (see Fig. 6.34 for
experimental data).
6.2.4 Kramer’s Degeneracy
E n (k) is the dispersion in a band. The time-reversal symmetry (Kramer’s degeneracy) implies
E n↑ (k) = E n↓ (−k) ,
(6.23)