January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
158 Quantum Theory of the Optical and Electronic Properties of Semiconductors
Coulomb correlation effects not included in the Hartree–Fock theory. The
corresponding term which appears in addition to the screened exchange
self-energy, is called the Coulomb hole self-energy,
CH
. The origin of
this name is due to the strong reduction of the pair correlation function
R
ss
(|r − r
| =0)also for electrons with different spin, s = s
, when the
proper Coulomb correlations are included in the many-electron wave func-
tion. In analogy to the “exchange hole” discussed in Chap. 7, this correlation
effect is referred to as “Coulomb hole”. Hence, the Coulomb-hole self-energy
describes the reduction of the total energy due to the fact that the elec-
trons avoid each other because of their mutual Coulomb repulsion. The
renormalized single-particle energy is instead of Eq. (9.24)
e
k
=
k
+Σ
SX
(k)+Σ
CH
. (9.24a)
The derivation of the Coulomb-hole contribution to the self-energy is
presented in later chapters of this book. There, we also discuss the corre-
sponding change of the effective semiconductor band gap. Here, we only
want to mention that this contribution can be calculated as the change of
the self-interaction of a particle with and without the presence of a plasma,
i.e.,
Σ
CH
=
1
2
lim
r→0
[V
s
(r) − V (r)] . (9.33)
Ignoring for the moment the term ∝ q
4
in the effective plasmon frequency ω
q
of the single-plasmon-pole approximation, one obtains for V
s
(r) the Yukawa
potential, Eq. (8.61), both in 2d and 3d. Using the Yukawa potential in
Eq. (9.33) we obtain the Coulomb-hole self-energy as
Σ
CH
= −
e
2
2
0
κ, (9.34)
where we have included static screening through
0
and κ is the screening
wave number in 2d or 3d, respectively.
For the three-dimensional system, it is possible to improve the result
(9.34) by analytically evaluating V
s
(r) using the full static plasmon–pole
approximation,
V
s
(r)=
L
3
(2π)
3
d
3
qV
s
(q)e
−iq·r
, (9.35)