10. Appendix

The Early Stages of Band-Structures Physics and Struggles 561
ing to learn over the next year or so that most of my inferences were wrong.
How did I go astray?
My first step, safe enough, was to classify the possible types of band-edge
points: those at wavevector k 0, and those at k 0 (multi-valley); for each
of these the states could be degenerate (two or more states of the same energy
and k) or nondegenerate. In surveying the experimental and theoretical
evidence bearing on the choices among these numerous alternatives, I began
by trying to limit the possible choices to those that could occur for band structures
qualitatively similar to that newly calculated by Herman [1] for diamond,
which seemed more reliable than any others that had been made for any material
with this crystal structure. Using the “k · p method” for qualitative estimations
of the energy-band curvatures on moving away from k 0, this
meant that I neglected perturbations of the p-like k 0 states °25 ′, °15 by the
anti-bonding s-like level °2 ′, which is quite high in diamond but, contrary to
my assumption, much lower in silicon and germanium. This neglect turned out
to make me omit the possibility of conduction-band edges on the [111] axes in
k-space for n-germanium, and to retain the possibility of valence-band edges
on the [100] axes for p-silicon.
From this flawed start I tried to narrow the possibilities further by appealing
to experimental evidence, and especially to magnetoresistance. The nearvanishing
of longitudinal magnetoresistance in [100]-type directions was obviously
consistent with multi-valley band-edge regions centered on the [100]type
axes in k-space, and this proved to be the correct identification for n-type
silicon. But, lacking explicit calculations, I assumed that the energy surfaces of
a degenerate hole band at k 0 would be so strongly warped as to preclude
the near-zero [100] longitudinal magnetoresistance observed for p-silicon. So
my predictions were all wrong here. Finally, I had the tedious task of calculating
the complete anisotropy of magnetoresistance for multi-valley models,
which a few months later were shown to give strong evidence for [111]-type
valleys for n-germanium.
What all this illustrates is that to achieve an acceptable understanding of
band structures, each of three types of information sources had to reach a
certain minimum level of sophistication. Band calculations from first principles
had to be made with accuracy and self-consistency in an adequately large
function space. Experimental measurements of properties sensitive to band
structure had to be made under well-controlled conditions. And theoretical
predictions of these properties for different band structure models had to be
available. There were gaps in all three of these sources up to the end of 1953;
it is thus not surprising that Shockley, in writing what was intended as a basic
text for the coming semiconductor age [2], stated, in spite of his awareness of
the diversity of possible band structures, that the theoretical reasoning in the
book would all be based on the simple model with an isotropic effective mass.
Remarkably, in a year or so starting in 1954, each of the three sources filled
itself in sufficiently so that they could pull together (e. g., better theoretical
bands [3], cyclotron resonance [4], magnetoresistance theory [5]) and bandstructure
physics became a solid and accepted component of basic knowledge.