10. Appendix

564 Appendix A
irradiation of n- or p-type material. By modulating the optical excitation the
detection of the absorption signal was made highly sensitive [3]. In addition,
if there is any doubt about the sign of the carriers, circularly polarized microwaves
can be (and were) used to distinguish the sense of rotation of the
carriers in the magnetic field.
The most surprising result of the original experiments was the observation
of two effective masses (m ∗ ) for the Ge holes: m ∗ /m0 0.04 and 0.3, both approximately
isotropic. Frank Herman and Joseph Callaway had calculated that
the top of the valence band in Ge occurs at the center of thr Brillouin zone
and is threefold degenerate (sixfold with spin), corresponding to p bonding orbitals
on the Ge atoms. This would have given rise to three hole masses. We
suggested [4,5] that the spin–orbit (s.o.) interaction splits the p orbitals into
fourfold degenerate (related to p3/2 orbitals) and twofold degenerate (related
to p1/2 orbitals) bands at the zone-center. We found that the most general form
of the energy of the upper valence bands in the diamond structure to second
order in wavevector k is (2.62)
E(k) Ak 2 ± [B 2 k 4 C 2 (k 2 x k2 y k2 y k2 z k2 z k2 x )]1/2 .
This was perhaps the first application of the spin–orbit interaction in semiconductors.
The “s.o. split-off” or lower band in Ge is 0.30 eV below the top of the
valence band edge. This s.o. splitting and the lower band itself are explored
best by optical absorption. The analysis by Kahn [6] of the available experiments
was an important confirmation of our model developed from cyclotron
resonance.
One of the early applications of the results of cyclotron resonance experiments
in Si and Ge was to the theory of the ionization energies of the shallow
donor and acceptor states in these materials. The approximate ionization energies
are 0.04 eV for electrons and 0.05 eV for holes in Si, and 0.01 eV for
both electrons and holes in Ge. The near equality of the ionization energies
for both electrons and holes was astonishing, at the time, because their band
edge structures were known to be completely different (thanks to cyclotron
resonance). The problem was discussed in the summer of 1954 with visitors
to Berkeley, notably Freeman Dyson and Joaquin Luttinger. The near equality
turns out to be merely a matter of coincidence after the electron and hole
ionization energies are calculated separately.
The donor ionization energy was calculated first at Berkeley [7]. We used
the hamiltonian for an ellipsoidal energy surface at any of the degenerate
band edges for electrons and the dielectric constant of the bulk crystal. The
calculated energies are in good agreement with experiment, at least for donors
with atomic numbers close to that of the host crystal. For heavier donors, central
cell corrections must be made. The acceptor problem is more difficult because
of the fourfold degeneracy of the valence band edges at the zone center,
and is reviewed by Walter Kohn [8], with satisfying results.