10. Appendix

Solution to Problem 6.21 645
by a simple constant Í. The valence electron wave functions in an atom are
normalized to the total number of valence electrons/atom over the entire
space. However, the valence band wave functions in a crystal are normalized
over a unit cell by the total number of valence electrons/unit cell. Thus,
the probability of finding a valence electron near the atomic nuclei will be
enhanced in the crystal leading to a stronger S-O coupling. In other words:
¢crystal Í¢atom with Í 1. As an example, ¢atom 0.219 eV for the 4p
electrons in Ge (see, for example, the Atomic Spectra Database at the NIST
website: http://physics.nist.gov/PhysRefData/ASD/index.html). The corresponding
¢0 for the p-like top valence band at k 0 in Ge crystal is 0.296 eV. Thus,
the enhancement factor Í is about 1.4, which is quite substantial.
For a binary compound semiconductor, like GaAs, we expect that that
S-O coupling for the valence electrons will depend on the S-O coupling of
the outermost filled core electrons in both the cation and the anion. A simple
approach would be to assume that ¢crystal Í1¢atom1 Í2¢atom2. If the electron
spends equal time around the cation and the anion one would expect the
ratio Í1/Í2 1. In ionic compounds this is usually not the case. For example,
in GaAs the valence electrons spend more time around the As atoms so one
expects that ÍAs/ÍGa 1. Indeed a ratio of ÍV/ÍIII 1.86 has been suggested
for III-V compounds. An even larger ratio is expected for the II-VI semiconductors.
So far we have been considering the effect of HS-O on the degenerate plike
valence band states at the zone center. When the electron wave vector k
increases from the zone center along a general direction, one would not expect
the above model based on a spherical potential to be valid. Hence one
may not be able to predict the contribution of HS-O to the band splitting. Using
the k · p approach, the orbital part of the Hamiltonian of the electron as
given by (2.35) is:
H H0 [(k) 2 /2m] (/m)k · p
Where H0 is the Hamiltonian for k 0. The k · p term now appears as an
additional contribution to the crystal potential term and as a result HSO becomes
k dependent. Instead of changing the crystal potential in response to
the non-zero value of k we can interpret this effect as the electron now moving
with velocity v k/m. We will choose a moving coordinate system O ′
which is traveling with the electron (the electron rest frame). In this moving
frame O ′ the spin-orbit Hamiltonian is again given by: HSO(v 0)
[(/4c2m2 ][(∇V × p) · Û]. However, the electron momentum p ′ in the stationary
(or laboratory frame) O is now given by p ′ p k. Thus the spin-orbit
Hamiltonian in the frame O is:
HSO(k) [h/4c 2 m 2 ][(∇V × p ′ ) · Û]
[/4c 2 m 2 ][(∇V × p) · Û (∇V × k) · Û]
In this Hamiltonian V and p both refer to the electron rest frame and, therefore,
are calculated just as for the zone center electrons. In general, the second
term is much smaller than the first one. The reasoning is like this: when the