10. Appendix

626 Appendix B
As an example, we will calculate the probability for the scattering process
in which the electron absorbs a phonon. In this case the electron-phonon matrix
element is given by (under the simplifying assumption that eq is parallel
to q):
〈k, Nq|HeLA|k ′ , Nq 1〉 2
2
(ac) (q · eq) 2
〈k, Nq|C q exp[i(q · r ˆt)]k′
, Nq 1〉
(ac)
q
2
q
2NVρˆ
2NVρˆ
(q) 2 Nq‰(Ek Ek ′ Eq)‰(k k ′ q)
In obtaining the above expression we have used the result that the probability
of absorbing a phonon is proportional to Nq. Substituting this matrix element
into the Fermi Golden Rule we obtain the total scattering rate of the electron
out of the state k via absorption of phonon:
2apple
2
P(k) (ac)
k ′
2NVρˆ
,q
2apple
(ac)
2
2NVρ
q
(q) 2 Nq‰(Ek Ek ′ Eq)‰(k k ′ q)
q
2 Nq‰(Ek Ekq Eq)
ˆ
It should be noted that the LA phonon frequency ˆ vsq where vs is the
LA phonon velocity and therefore cannot be taken outside the summation
over q in the above expression. Notice that P(k) is essentially the same as
(5.41). In the high temperature limit, where kBT ≫ ˆ and Nq ≫ 1, we can
approximate Nq ∼ kBT/(vsq) so that the q 2 term inside the summation sign
is cancelled by the Nq/ˆ term. The final result can be reduced to:
P(k) ∼ ‰(Ek Ekq Eq)
except for a constant of proportionality which depends on material properties,
such as the density, sound velocity, and the deformation potential. The
way to calculate the allowed values of q is shown geometrically in Fig. 5.1(a).
The summation over q can be converted into an integral over q as shown in
(5.42a). The resultant expression is:
P(k)
a 2 c kBT
8apple 2 ρv 2 s
2appleq 2 dq d cos £‰
2 q
2m ∗
(q 2k cos £) v2q
After integration over cos £ and q the final expression becomes:
a
P(k)
2
∗
ckBTm q2
max
2k
4appleρ 3 v 2 s
where qmax represents the maximum value of the wave vector of the phonon
absorbed. We will now make the approximation that qmax ∼ 2k as in 5.2.4.
With this simplification the probability of absorbing a phonon is given by:
a
P(k)
2 √
∗
2
ckBTm 2ackBT(m k
∗ ) 3/2
E 1/2
2appleρ 3 v 2 s
2appleρ 4 v 2 s
2