10. Appendix

640 Appendix B
with the diamond- and zincblende-type crystal structure. The main features of
this curve are that: Eg is almost independent of T for T 100 K and then decreases
linearly with T at higher temperatures. There are a few notable exceptions
to this behavior. For example, the direct band gap of IV–VI chalcogenide
semiconductors like PbS, PbSe and PbTe increases with T (see Fig. 6.14(b)).
The band gaps of the chalcopyrite semiconductors AgGaS2 and AgGaSe2 exhibit
a small blue-shift with increase in T at T 100 K before decreasing with
T (see, for example, P.W. Yu, W.J. Anderson and Y.S. Park: Anomalous temperature
dependence of the energy gap of AgGaS2, Solid State Commun. 13,
1883 (1973)). Finally, the exciton energy (whose temperature dependence is
similar to the band gap at low T) in cuprous iodide shows a shallow minimum
as a function of T at low temperatures (see: J. Serrano, Ch. Schweitzer, C.T.
Lin, K. Reimann, M. Cardona, and D. Fröhlich: Electron-phonon renormalization
of the absorption edge of the cuprous halides. Phys Rev B65125110
(2002)).
Because the temperature dependence of Eg shown in Fig. 6.44 is highly
nonlinear, especially around the “knee” at 100 K, it is not possible to derive
(6.161) by simply expanding Eg(T) as a Taylor series in T. Instead, one has
to consider what are the effects of T on Eg. In general, T can change Eg via
one of these two effects. The first effect is associated with thermal expansion,
an effect that results from the anharmonicity of the lattice. In other words, it
involves phonon-phonon interaction (see, for example, C. Kittel: Introduction
to Solid State Physics (Wiley, New York, 1995)). Typically, at T ∼ 300 K this
effect is small since the coefficient of linear expansion is ∼ 5 × 106 K1 .For
a semiconductor with these typical parameters: bulk modulus ∼100 GPa and
pressure coefficient dEg/dP ∼ 100 meV/GPa (see Table of Physical Parameters
of Tetrahedral Semiconductors in the inside cover), the contribution of
this effect to the temperature coefficient is ∼0.15 meV/K. This is about a factor
of 3 smaller than the typically observed value of dEg/dT ∼ 0.5 meV/K. The
second effect is present even if the size of the unit cell does not change with T
and arises directly from the electron-phonon interaction (whose Hamiltonian
Hep is discussed in Chapter 3). Of the two effects, usually the second effect
has a larger magnitude although they may have different sign. This happens in
the case of the indirect band gap of Si where the gap increases with thermal
expansion. One approach to estimate the electron-phonon effect is to assume
that Hep is weak enough for the change in Eg to be calculated by second-order
perturbation theory (see, for example, Eq. (2.38)). Using perturbation theory
one obtains:
Eg(T) Eg(0)
〈g,0|Hep|i, ∓q〉〈i, ∓q|Hep|g,0〉
n(ˆk,q) 1
1
±
2 2
i,k,q
Eiq Eg ± ˆk,q
In this expression |g,0〉 represent the electronic initial state of the system
where an electron is in the conduction band and a hole is in the valence
band at zone-center. |i, q〉 represents an intermediate electronic state where
the electron-hole pair has been scattered to the state i by either emitting ()
or absorbing () a phonon with wave vector q and belonging to the branch