10. Appendix

638 Appendix B
the surface plasmon are completely decoupled since the photon frequency
approaches infinity when the wave vector becomes infinite while the surface
plasmon frequency remains finite. Hence, retardation of the surface plasmon
is completely negligible. When k || is finite there is coupling between the surface
plasmon and the photon and therefore the surface plasmon is known as
a surface-polariton in analogy to the phonon-polariton discussed in 6.4.1.
(c) The dispersion of the surface phonon-polariton can be obtained simply by
substituting ÂA by 1 and ÂB by the expression for the dielectric function of
the optical phonon (6.110b). To obtain the surface phonon frequency without
retardation we again set ÂB 1. Since ÂB approaches ∞ for ˆ slightly
larger than ˆT and ÂB 0asˆ approaches ˆL, ÂB must be equal to 1 at
some ˆ lying between ˆT and ˆL.
Solution to Problem 6.13
(a) In Figure 3.1 we found that the symmetry of the zone center optical
phonon of Si is °25 ′ (this is true for any material with the diamond crystal
structure, including Ge and gray tin). From the character table for the ° point
in the diamond structure (Table 2.16) we find that the wave function of the
°25 ′ optical phonon is triply degenerate. The crystal symmetry is lowered when
the crystal is subjected to a uniaxial strain causing the triply degenerate optical
phonon to split.
For a [100] applied strain the crystal symmetry is lowered from cubic to
tetragonal. Before the crystal is strained the optical phonons whose vibrations
are polarized along the [100], [010] and [001] directions are degenerate. After
the crystal is strained, the vibrations polarized along the [100] axis are expected
to have a different frequency than those polarized perpendicular to the
strain axis. Since the crystal remains invariant under S4 symmetry, operations
of the strained crystal (provided the four-fold axis of rotation is parallel to
the [100] axis) we expect the optical phonons polarized along the [010] and
[001] axes to remain degenerate. Thus we conclude that the optical phonon
in Ge will split into a doublet (polarized perpendicular to the strain axis) and
a singlet (with polarization parallel to the strain axis). The effect of an [100]
uniaxial strain on the symmetry of the q 0 optical phonons is similar “in a
sense” to making the phonon wave vector q non-zero and directed along the
[100] axis. In both cases the triple degeneracy of the phonon is split. As shown
in Fig. 3.1, when the optical phonons propagate along the ¢ direction their
frequencies split into two, corresponding to symmetries ¢2 ′ and ¢5. The character
table of the group of ¢ (Table 2.20) shows that ¢2 ′ is a singlet while ¢5
is a doublet. Using similar arguments we can show that a tensile stress along
the [111] direction will split the optical phonon in Ge into a doublet (§3) and
a singlet (§1). Whether the singlet or triplet phonon state will have a lower
frequency cannot be determined by symmetry alone. This and also the magnitude
of the splitting between the singlet and triplet states can be determined