Page 251 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
2
—Ξ
+
+
1
ˆ
[
V kk' q,, = iqe -------------------- b + b ]c c ×
q 2MNω q – q k' k
q
(7.34)
(
∫ e i k + q – k')r u∗ k' r)u kr) V
(
,
(
,
d
Lattice
The integration in (7.34) has to be performed over the whole crystal lat-
tice. Since the u kr,( ) are defined in a single lattice cell this integral may
be split into a integration over a such a cell summed over all lattice cell
sites to give
(
∫ e i k + q – k')r u∗ k' r)u kr) V
,
(
(
,
d
Lattice
(7.35)
(
(
,
,
(
d
= ∑ ∫ e i k + q – k')r u∗ k' r)u kr) V
sites cell
With the substitution r = R + r' we obtain
i
(
(
=
δ k + q – k')F kk' ∑ e i k + q – k')R i ×
,
i
(7.36)
(
∫ e i k + q – k')r' u∗ k' r')u kr') V'
,
(
,
(
d
cell
where we set
,
(
,
(
d
F kk' = ∫ u∗ k' r)u kr) V (7.37)
,
cell
Thus we have for the interaction operator
⁄
2
(
ˆ iqe —Ξ ⁄ ( 2MNω )) 12 [ + +
V kq, = F b + b ]c c (7.38)
,
q 1 q k + q k q – q k + q k
We observe two terms in (7.38) and interpret them according to the cre-
ation and destruction operator definitions previously made in Chapter 3:
+
k
b c c destroys an electron with wavevector creates one k + q and
q k + q k
248 Semiconductors for Micro and Nanosystem Technology