10. Appendix

Solution to Problem 2.20 597
to the higher energy plane wave states. However, for the purpose of demonstrating
the symmetry of the L wave functions the present approximation is
adequate.
Based on the above assumption, the only pseudopotential form factor
which couples „1 and „2 is: Vg V s 3
cos(g · s) from (2.33) where the vec-
tors g (2apple/a)(1, 1, 1) and s (a/8)(1, 1, 1). Notice that in arriving at (2.33)
we have chosen the origin to be at the mid-point between the two atoms in the
diamond lattice. We note also that the symmetric pseudopotential form factor
Vs 3 is negative for all diamond and zincblende-type semiconductors (see Table
2.21) because of the attractive nature of the potential seen by the electrons.
Thus Vg Vs 3 cos(g · s) Vs 3 cos(3apple/4) (1/√2)Vs 3 is 0.
To calculate the eigenvalues and eigenfunctions we will set up the usual
determinant:
2k2 E Vg
2m
V∗
g
2k2
and diagonalize it.
E
2m
The resultant energies are: E± [2k2 /2m] ±|Vg| where [2k2 /2m] ± Vg.
The corresponding wave functions are given by: (1/2) 1/2 [„1 „2].
For the valence band we have to take the lower energy solution: „v
C1„1 C2„2. Substituting back EV E into the linear equation for C1 and
C2 we get:
{[2k2 /2m] E}C1 VgC2 0orVgC1 VgC2 0.
Since Vg is 0, we obtain: C1 C2. Thus „V (1/2) 1/2 [„1 „2] or the valence
band has odd parity under inversion operation I and hence its symmetry
is L2 ′.
(b) To calculate the bonding and anti-bonding s-orbitals within the tightbinding
approximation we will further simplify the results contained in 2.7.2.
First, we neglect the overlap between the s- and p-orbitals. This has the effect
of reducing the 8 × 8 matrix in Table 2.25 into a 2 × 2 matrix for the s-orbitals
and a 6×6 matrix for the p-orbitals. The effect of this simplification will affect
the accuracy of the bonding and anti-bonding orbital energies but should not
change their parity. Within this approximation there are only 2 orbitals: |S1〉
and |S2〉 corresponding to the s-orbitals of the two atoms inside the primitive
unit cell. The secular equations for their eigenvalues (as obtained from Table
2.25) are now:
Es
Ek Vssg1
0
Vssg ∗ 1
Es Ek
According to (2.82a) the factor
g1 (1/4) exp[i(d1 · k)] exp[i(d2 · k)]
exp[i(d3 · k)] exp[i(d4 · k)]
cos(k1apple/2) cos(k2apple/2) cos(k3apple/2) i sin(k1apple/2) sin(k2apple/2) sin(k3apple/2) ,