10. Appendix

662 Appendix B
the latter k the reciprocal lattice vector (2apple/a0)(111) we find that the energy at
the point (apple/a0)(1 ‰)(111) should be the same as that of (apple/a0)(1 ‰)(111).
Hence the slope of the energy vs. ‰ vanishes. There are no other symmetry
points (U, W, orK) for which the slope perpendicular to the (111) faces vanishes.
In the case of germanium the group of the k vector at L is isomorphic to
D3d (see p. 55). This group has even higher symmetry than C3v. Hence, the
slope under consideration also vanishes. In the case of the X-point matters
are more complicated because of the non-symmorphic nature of the groups
as discussed in 2.4.2. The relevant k-vector representations are all two-fold
degenerate (X1, X2, X3, X4 in Table 2.19). With the exception of X4 these representations
split along ¢ becoming non-degenerate. The representations that
split have equal and opposite slopes along ¢ (so that the average slope remains
zero). Hence for them no van Hove singularities are obtained. All the
others reach the X point with zero slope.
Solution to Problem 8.10
Let us first consider the case of the d orbital states (l 2) under a tetrahedral
field (point group symmetry Td, which is equivalent to the ° point of a
zincblende crystal). The splitting pattern is the same for a cubic field of point
group symmetry Oh (similar to the ° point of the diamond crystal) although
the corresponding representations may be labeled differently, according to Tables
2.3 (Td) or2.5 (Oh). The d-functions are even upon inversion, hence reflections
and C2 rotations must have the same characters, given by (8.24) when
considering the full rotation group. These characters are, for the five operations
relevant to the Td group:
E C2 S4 Û C3
5 1 1 1 1
(8.25)
In order to decompose the fivefold degenerate orbital d states into those belonging
to irreducible representations of the Td point group we use the orthogonality
relations (2.11) and the Table 2.3 of characters of the Td group.
Performing the appropriate multiplications and sums, the right hand side
of (2.11) becomes: h (the number of Td group operations 24) for the °3(E)
and °4(T2) representations and zero for all others. Hence, the fivefold d orbital
states split into a triplet °4(T2) and a doublet °3(E) in a field of either
tetrahedral or cubic symmetry. In the case of electronic band structures this
will happen for the d states at k 0.
Let us consider d-orbital states with a spin of 1
2
attached to them (one-
electron states). Multiplication of angular momentum l 2 (orbit) with
s 1/2 (spin) gives rise to two sets of states, a sextuplet with J 5/2 and
a quadruplet with J 3/2. In order to investigate the effect of a field of either
Td (the case we are considering here) or Oh symmetry we must use the