10. Appendix

Solution to Problem 6.19(a) 641
k. ˆk,q and n(ˆk,q) are, respectively, the energy and occupancy of the this
phonon mode. The summation in the above expression is over all the intermediate
electron states and phonon branches. Notice that energy does not have
to be conserved in the transition to the intermediate state (this is called a virtual
transition) while the wave vector has to be conserved: this has already
been taken into account in the summation. We have purposely put a minus
sign in front of the second term on the right hand side of the above equation
to emphasize that this term is always positive. The reason is that Eg is
the lowest possible energy state of an electron-hole pair in the semiconductor
(when excitonic effects are neglected). Most of the possible intermediate
states will have higher energies than Eg so the energy denominator will usually
be positive (one exception is when the electron or hole scatters back into
the same initial state via the emission of a zone-energy optical phonon). The
above equation can be simplified by considering only electron-hole pair states
near the band gap since the energy denominator will make contributions from
the high energy states negligible. In addition, one can limit the summation to
only intraband scattering processes (ie |i〉 involves the same bands as those
forming the gap). This approach is sometimes referred to as the Fan mechanism
after H.Y. Fan (H.Y. Fan: Temperature Dependence of the Energy Gap
in Semiconductors. Phys. Rev. 82, 900 (1951)). Another simplification is to assume
that one phonon branch dominates the scattering. For example, the longitudinal
optical phonon in zincblende-type semiconductors tends to have the
strongest interaction with the electron-hole pairs near the band gap via the
Fröhlich interaction. Using this approximation, one can limit the summation
over phonon modes to essentially over one “average phonon mode” with energy
ˆ. The net result of all the above simplifications is that:
Eg(T) Eg(0) A[n(ˆ) 1] An(ˆ) Eg(0) A[2n(ˆ) 1]
where the phonon occupancy n(ˆ) is given by:
1
n(ˆ)
exp(ˆ/kBT) 1
kB being the Boltzmann constant and A is a negative parameter. The term
[2n(ˆ) 1] results from the probabilities for phonon emission (proportional
to n 1) and absorption (proportional to n). Usually this expression gives a
fairly good fit to the experimental result provided ˆ is adjustable.
One should note that in some special cases the interband scattering contributions
to the electron-phonon scattering may become as important as the
intraband scattering. When this happens it becomes difficult to estimate even
just the sign of the electron-phonon effect because the interband scattering
terms can be either positive or negative. In principle, scattering to the lower
energy band tends to increase the gap while scattering to the higher energy
bands will decrease the gap. Which process will win out may depend on the
detailed band structure and the electron-phonon interaction. In other words, it
is difficult to calculate the temperature dependence of the band gap from first
principles.