10. Appendix

616 Appendix B
(a) The mode Grüneisen parameter Áˆ of a phonon mode is defined as:
Áˆ d(ln ˆ)/d(ln v) where v is the volume of the crystal. In the following
table we list the mode Grüneisen parameter for various phonon modes in
semiconductors.
Table 1 of Problem 3.17. Mode Grüneisen parameter of phonons in selected diamondand
zincblende-type semiconductors (TO transverse optical phonon, LO longitudinal
optical phonon, TA transverse acoustic phonon and LA longitudinal acoustic
phonon).
Semiconductors Zone Center
Phonons
Zone Edge Phonons
TO LO L X
Si 0.98 0.9(LA) 1.3(TA) 0.9(LA) 1.4(TA)
Ge 1.12 1.53(TA)
GaAs 1.39 1.23 1.5(TO) 1.7(TA) 1.73(TO) 1.62(TA)
GaP 1.09 0.9 1.5(TO) 0.81(TA) 1.0(LA) 0.72(TA)
InP 1.44 1.24 1.4(TO) 2.0(TA) 1.4(TO) 2.1(TA)
ZnS 0.95 1.0(TO) 1.5(TA) 1.0(TO) 1.2(TA)
ZnTe 1.7 1.2 1.7(LO) 1.0(TA) 1.8(LO) 1.55(TA)
The experimental values contained in this table are obtained from:
B.A. Weinstein and R. Zallen: Pressure-Raman Effects in Covalent and Molecular Solids
in Light Scattering in Solids IV, edited by Cardona and G. Güntherodt (Springer-Verlag,
Berlin, 1984) p. 463–527. Citations to the original publications for these values can be
found in this article’s references.
This table shows that for most diamond- and zincblende-type semiconductors
the value of Áˆ for the optical phonons is around 1, although it can be as large
as 1.7 in ZnTe. However, the value of Áˆ is usually negative for the zoneedge
TA phonons. This is related to the fact that the diamond and zincblende
lattices are unstable against shear distortion except for the restoring forces due
to the bond charges (see Section 3.2.4).
(b) The linear thermal expansion coefficient · of a solid is defined usually as:
· 1
L
where L is the length of the solid, T the temperature and P
L T P
the pressure. It is related to the volume thermal coefficient of expansion ‚ by
the expression:
‚ 1
v
3·
v T P
where v is the volume of the solid. To relate ‚ to the mode Grüneisen parameter
of the solid we start with the thermodynamic relation between the
pressure P and the Helmholtz free energy F:
F
P .
v T
For a semiconductor, where there are no free electrons, the free energy contains
mainly two contributions: one, which we shall denote as º, is due to the