10. Appendix

Solution to Problem 6.19(a) 639
if the phonon deformation potentials are known. See Problem 6.23 (new for
the 4 th edition) for a discussion of the strain Hamiltonian for optical phonons.
(b) Along high symmetry directions in a zincblende-type crystal, such as
GaAs, the “nearly zone-center” optical phonons are split by the Coulomb interaction
between the transverse effective charges e∗ of the ions into TO and
LO phonons. The polarization of the LO phonon is along the direction of
propagation of the phonon while the TO phonon is polarized perpendicular to
the direction of propagation. When the crystal is subjected to a uniaxial strain,
it is necessary to specify the direction of the uniaxial strain relative to that of
the phonon propagation direction. Since, in many experiments, the strain direction
is perpendicular to that of the phonon propagation (the exceptional
case being a forward scattering experiment), let us consider the case of an
uniaxial strain along the [100] direction while the phonon wave vector q is
along the [010] direction. Without strain the TO phonon polarized along the
[100] and [001] directions are degenerate. After the application of the strain
along the [100] direction this degeneracy is removed. The LO phonon which
is polarized along [010] is non-degenerate and, therefore, cannot exhibit any
strain-induced splitting. The spring constant of the LO phonon involves two
contributions: a “short-range mechanical” restoring force which is equal to
that of the TO phonon and a long range Coulomb force which depends on e∗ .
If the uniaxial strain does not affect the Coulomb force we expect the straininduced
shift to be similar to that of the TO phonon along the [001] axis. In
case the strain changes also e∗ then we will find the strain-induced shift of the
LO phonon to be different from that of the [001] polarized TO phonon. Thus
the difference between the stressed-induced shifts of the LO phonon and TO
phonons in zincblende-type semiconductors can be used to study the effect of
strain on e∗ . See the following references for further details:
(1) F. Cerdeira, C.J. Buchenauer, F.H. Pollak and M. Cardona: Stress-induced
Shifts of First-Order Raman Frequencies of diamond- and Zinc-Blende-Type
Semiconductors. Phys. Rev. B5, 580 (1972).
(2) E. Anastassakis and M. Cardona in Semiconductors and Semimetals Vol.
55 (1998).
One should note that it is possible to separate the optical phonons into TO
and LO modes only for q along high symmetry directions. For the zincblendetype
semiconductors the only other direction (in addition to the [100] and
[111] directions) for which this is possible is the [110] direction. How a uniaxial
strain along the [110] direction will affect the optical phonons is left as
an exercise.
Solution to Problem 6.19(a)
Figure 6.44 shows the temperature (T) dependence of the direct band gap
(Eg) ofGefrom0Kto600K.Thiscurve is representative of the temperature
dependence of the fundamental band gap of most direct gap semiconductors