10. Appendix

Ùm
∞
NLOq
0
2 apple
dq
0
∞
Solution to Problem 5.5 629
where P(k, k ′ ) is the probability per unit time for the electron to be scattered
from the initial state k to the final state k ′ . The summation over k ′ can be
converted into an integration over q by invoking wave vector conservation. In
the case of scattering by LO phonons the expression for PLO is then given by
(5.48). The corresponding expression for (k/Ùm) then becomes:
k
q
∝
d cos £
2
(q2 q2 0 )
q cos £‰(Ek ′ (Ek ELO))
·
0
apple
0
(NLO 1)q 2 dq
q
d cos £
2
(q2 q2 0 )
(q cos £)‰(Ek ′ (Ek ELO))
To eliminate the integration over cos £ one notes that the ‰-functions can be
written as:
and
‰[Ek ′ (Ek ELO)] (2m ∗ / 2 )‰[(2kq cos £ q 2 ) (2m ∗ / 2 )ELO]
‰[Ek ′ (Ek ELO)] (2m ∗ / 2 )‰[(2kq cos £ q 2 ) (2m ∗ / 2 )ELO]
After integration over cos £, one eliminates the delta functions and obtains:
k
Ùm
∝
q1max
q1min
NLO dq q 2m∗ELO 2
q
q2max
q2min
(NLO 1) dq q 2m∗ELO 2
q
Using the results from (a) and (b) for the limits of integration, we can perform
the integration over q. The above results have been obtained under the
assumption that the screening wave vector q0 can be set to zero. This assumption
is valid when q1min and q2min are both larger than q0 or if the concentration
of electrons is low. [Note that in case of scattering by acoustic phonons
via the piezoelectric interaction it is impossible to neglect q0 since the minimum
value of q is zero]. For LO phonons we have shown that both q1min and
q2min can never be zero (except in the limit Ek ⇒∞) so this problem will
not arise. To obtain (5.51) we have to perform the integration and put in the
appropriate limits in the above expression. The result is:
1
∝ NLO
Ùm
q 2 1max q2 1min
(NLO 1)
k 2
ELO
Ek
ln
q1max
q1min
q2 2max q2 2min
k2 ELO
q2max
ln
Ek q2min