Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors
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Retarded Green’s Function for Electrons 159
where
V
s
(q)=
4πe
2
0
q
2
1+
1
1+
q
2
κ
2
+
$
ν
q
ω
pl
%
2
. (9.36)
The resulting expression is
V
s
(r)=
e
2
0
r
e
−ra
+
cos(ra
−
)+(4/u
2
− 1)
−1/2
sin(ra
−
)
for u ≤ 2
=
e
2
2
0
r
b_e
−rc
+
+ b
+
e
−rc
−
for u>2 , (9.37)
where
u =
ω
pl
κ
2
C
1/2
,a
±
=
κu
2
2
u
± 1 ,b
±
=1±
1 −
4
u
2
−1/2
,
and
c
±
=
κu
√
2
1 ±
1 −
4
u
2
1/2
1/2
.
The screened Coulomb interaction potential according to Eq. (9.37) is plot-
ted in Fig. 9.1 for low and high plasma density and compared to the Yukawa
potential, Eq. (8.61), for the same parameter values. A prominent feature
that we note is the more effective screening with increasing plasma density
causing the strong reduction of the effective Coulomb potential. Whereas
the Yukawa potential is always negative, indicating electron–electron re-
pulsion, the plasmon–pole approximation yields an effective Coulomb po-
tential, which becomes attractive at large distances and sufficiently high
plasma densities.
Evaluating the Coulomb hole self-energy, Eq. (9.33), for the potential
given by Eq. (9.37) we obtain
Σ
CH
= −a
0
κE
0
$
2
1−4/u
2
%
1/2
(b
+
− b_) for u ≥ 2
= −a
0
κE
0
u
(1 +
2
u
)
1/2
− (1 +
2
u
)
−1/2
for u<2
(9.38)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
160 Quantum Theory of the Optical and Electronic Properties of Semiconductors
-2
-1
0
1
0.5
1 1.5 2
SPP n1
SPP n2
YUK n1
YUK n2
r/a
0
V/E
sR
Fig. 9.1 Comparison of the screened Coulomb potential V
s
(r) using the single-plasmon-
pole approximation (SPP), Eq. (9.37) and the Yukawa potential (YUK), Eq. (8.61), for
the plasma densities n
1
=10
−3
a
−3
0
and n
2
= a
−3
0
, respectively. Here, a
0
is the 3d Bohr
radius, and E
R
= E
0
is given by Eq. (9.39).
where
E
0
=
e
2
2a
0
0
= E
R
. (9.39)
An evaluation of Eq. (9.38) is shown in Fig. 9.2 for parameter values which
are typical for bulk GaAs. We see that Σ
CH
varies almost linearly with
1/r
s
over a wide range of density values.
At the end of this chapter, we want to mention that the retarded Green’s
function is by no means the only Green’s function which obeys a differential
equation with a singular inhomogeneity. There exists, e.g., a time-ordered
Green’s function which is uniquely related to the retarded Green’s function
in equilibrium systems, but for which one has strict diagrammatic rules
for developing perturbative approximations. As already mentioned, the
situation is different in nonequilibrium systems, such as optically excited
semiconductors. Here, a general nonequilibrium Green’s function theory
exists, which allows us to calculate not only the spectral properties of the
system (contained in the retarded Green’s function) but also the kinetic
evolution of the distribution of the excitations in the system described by
the particle propagator. Furthermore, this general nonequilibrium Green’s
function theory provides also strict diagrammatic rules for the retarded
Green’s function. An introduction to the theory of nonequilibrium Green’s
functions is given in Chap. 21 and Appendix B of this book.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Retarded Green’s Function for Electrons 161
-6
-4
-2
0 1 2 3
S
CH R
/E
1/r
s
Fig. 9.2 Coulomb hole self-energy, Eq. (9.38), as function of particles density expressed
in terms of 1/r
s
, Eq. (7.62). The used parameter values are a
0
=1.096 · 10
−6
cm,
T =10K, E
0
=5.32meV, m
e
/m =1.12, m
h
/m =9.1,wherem is the reduced electron–
hole mass.
REFERENCES
S. Doniach and E.H. Sondheimer, Green’s Functions for Solid State Physi-
cists, Benjamin, Reading, Mass. (1974)
A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Physics,
McGraw–Hill, New York (1971)
G.D. Mahan, Many Particle Physics, Plenum Press, New York (1981)
D. Pines and P. Nozieres, The Theory of Quantum Liquids,Benjamin,
Reading, Mass. (1966)
PROBLEMS
Problem 9.1: Follow the steps discussed in this chapter to prove
Eq. (9.18).
Problem 9.2: Evaluate Eq. (9.18) for a quasi-2d system. Use the approx-
imations discussed for the 3d case to obtain an estimate for the exchange
self-energy.
Problem 9.3: Perform the Fourier transform in (9.10) by closing the fre-
quency integration in the upper or lower complex frequency plane depend-
ing on the sequence of t and t
.
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Chapter 10
Excitons
In the preceding chapters, we discussed the quantum statistics and the
many-body Coulomb effects in an electron plasma with positive background
(jellium). The results obtained are clearly relevant for a metal since it has a
partially filled conduction band. Moreover, these results are also important
for a dielectric medium such as a semiconductor, since they describe the
carrier–carrier interactions within the same band, i.e., the intraband inter-
actions. However, as already discussed in Chap. 5, the optical properties
of semiconductors are mostly related to interband transitions between the
valence and conduction bands.
As it turns out, one can separate the many-body treatment of the
electron–hole system in excited semiconductors quite naturally into the
determination of spectral properties and kinetic properties. As spectral, we
denote energy shifts and the broadening due to interactions, i.e., the renor-
malizations of the states due to the many-body interactions. The kinetics,
on the other hand, deals with the development of the particle distributions
in the renormalized states. The optical properties are mainly linked with
the interband kinetics, whereas the transport properties are connected with
the intraband kinetics of electrons and/or holes, depending on the kind of
free carriers in the semiconductor.
The analysis of the semiconductor interband transitions for variable
excitation conditions is discussed in the following chapter. In the present
chapter, we consider only the low-excitation regime, where an extremely
small density of electrons and holes exist. We therefore concentrate on the
analysis of one-electron-hole pair effects.
163
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
164 Quantum Theory of the Optical and Electronic Properties of Semiconductors
10.1 The Interband Polarization
In Chap. 5, we discussed the optical susceptibility due to free-carrier tran-
sitions in semiconductors. In the present chapter, we now generalize this
treatment to include also the Coulomb effects. Again, we first compute the
optical susceptibility from which we then get the absorption coefficient and
the refractive index.
We start by analyzing the macroscopic interband polarization induced
by the coherent monochromatic classical light field E(r,t).Thepolarization
P(t) is defined as the expectation value of the electric dipole er as
P(t)=
s
d
3
r
ˆ
ψ
†
s
(r,t) er
ˆ
ψ
s
(r,t)
=
s
d
3
rtr[ρ
0
ˆ
ψ
†
s
(r,t) er
ˆ
ψ
s
(r,t)] . (10.1)
In Eq. (10.1), the average is taken with the equilibrium statistical operator
ρ
0
describing the system at the initial time before the field was switched
on. If one introduces the reduced one-particle density matrix ρ(r, r
,t) as
the single-time correlation function
ρ
ss
(r, r
,t)=
ˆ
ψ
†
s
(r,t)
ˆ
ψ
s
(r
,t) , (10.2)
one can rewrite Eq. (10.1) as
P(t)=
s
d
3
rρ
ss
(r, r
,t)|
r=r
,s=s
er . (10.3)
These general definitions now have to be adapted to a semiconductor by
expanding the electron field operators into an appropriate basis.
In the case of a spatially homogeneous system, one is dealing with com-
pletely delocalized electrons. Then the Bloch functions ψ
λ
(k, r) are the
appropriate set for expanding the field operators
ˆ
ψ
s
(r,t)=
λ,k
a
λ,k,s
(t)ψ
λ
(k, r) . (10.4)
Inserting this expansion into Eq. (10.1) yields
P(t)=
s,λ,λ
,k,k
a
†
λ,k,s
a
λ
,k
,s
d
3
rψ
∗
λ,k
(r) er ψ
λ
,k
(r) . (10.5)
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Excitons 165
The integral has been evaluated in Chap. 5, Eqs. (5.11) – (5.20). We use
the result (5.20) in the form
d
3
rψ
∗
λ,k
(r) er ψ
λ
,k
(r) δ
k,k
d
λλ
, (10.6)
where λ = λ
. Inserting Eq. (10.6) into Eq. (10.5), we obtain the polariza-
tion in a spatially homogeneous system as
P(t)=
k,s,λ,λ
a
†
λ,k,s
a
λ
,k,s
(t)d
λλ
=
k,s,λ,λ
P
λλ
,k,s
(t)d
λλ
, (10.7)
where we introduced the pair function
P
λλ
,k,s
(t)=a
†
λ,k,s
a
λ
,k,s
(t) . (10.8)
As in Chap. 5, we restrict the treatment to the optical transitions be-
tween the valence and conduction bands of the semiconductor (two-band
approximation). Furthermore, we suppress the spin index s from now on,
assuming that it is included in k. The following calculations will not de-
pend on s, only the spin summation in the definition of the polarization
(10.1) will lead to an extra prefactor of 2 in the final result. Choosing λ = v
and λ
= c, the pair function, Eq. (10.8) becomes
P
vc,k
(t)=a
†
v,k
a
c,k
(t) . (10.9)
This quantity is a representation of the off-diagonal elements of the reduced
density matrix. In an equilibrium system without permanent dipole mo-
ment, these off-diagonal matrix elements vanish. However, the presence
of the light induces optical transitions between the bands. Therefore, the
interband polarization P
vc
in such an externally driven system is finite.
In second quantization, we write the interaction Hamiltonian, Eq. (2.4),
between the electric field and the semiconductor electrons as
H
I
=
d
3
r
ˆ
ψ
†
(r)(−er) ·E(r,t)
ˆ
ψ(r) . (10.10)
Assuming that the electric field has a simple exponential space dependence,
E(r,t)=E(t)
1
2
&
e
iq·r
+c.c.
'
, (10.11)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
166 Quantum Theory of the Optical and Electronic Properties of Semiconductors
and using the expansion (10.4) for spatially homogeneous systems, we ob-
tain
H
I
−
k
E(t)(a
†
c,k
a
v,k
d
cv
+ h.c.) , (10.12)
where we took the limit q → 0 (dipole approximation) and defined d
cv
as
the projection of the dipole d
cv
in the direction of the field E.Theinter-
action Hamiltonian, (10.12), shows how the applied field causes transitions
of electrons between valence and conduction band.
Besides the interaction with the external field, we also have to consider
the kinetic and Coulomb contributions from the electrons. These effects are
described by the Hamiltonian (7.29). For our present purposes, we have to
extend the treatment of Chap. 7 by including also the band index λ:
H
el
=
λ,k
E
λ,k
a
†
λ,k
a
λ,k
+
1
2
k,k
q=0
λ,λ
V
q
a
†
λ,k+q
a
†
λ
,k
−q
a
λ
,k
a
λ,k
. (10.13)
This Hamiltonian is obtained from Eq. (7.29) by including the summa-
tion over the band indices λ, λ
. Furthermore, we have omitted all those
Coulomb terms which do not conserve the number of electrons in each band.
Such terms are neglected because they would describe interband scattering,
i.e., promotion of an electron from valence to conduction band or vice versa
due to the Coulomb interaction, which is energetically very unfavorable.
For the two-band model, we restrict the band summation to λ, λ
= c, v
and obtain
H
el
=
k
(E
c,k
a
†
c,k
a
c,k
+ E
v,k
a
†
v,k
a
v,k
'
+
1
2
k,k
,q=0
V
q
$
a
†
c,k+q
a
†
c,k
−q
a
c,k
a
c,k
+ a
†
v,k+q
a
†
v,k
−q
a
v,k
a
v,k
+2a
†
c,k+q
a
†
v,k
−q
a
v,k
a
c,k
%
. (10.14)
For simplicity, we use the single particle energies in effective mass approx-
imation and write the conduction and valence band energies as
E
c,k
=
c,k
= E
g
+
2
k
2
/2m
c
, (10.15)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Excitons 167
and
E
v,k
=
v,k
=
2
k
2
/2m
v
. (10.16)
The full Hamiltonian of the electrons in the valence and conduction
bands interacting with the light field is now given by
H = H
I
+ H
el
. (10.17)
To derive the equation of motion of the polarization, we use the Heisenberg
equation for the individual operators. An elementary but lengthy calcula-
tion for the isotropic, homogeneous case yields
i
d
dt
− (
c,k
−
v,k
)
P
vc,k
(t)=
n
c,k
(t) − n
v,k
(t)
d
cv
E(t)
+
k
,q=0
V
q
$
a
†
c,k
+q
a
†
v,k−q
a
c,k
a
c,k
+ a
†
v,k
+q
a
†
v,k−q
a
v,k
a
c,k
+ a
†
v,k
a
†
c,k
−q
a
c,k
a
c,k−q
+ a
†
v,k
a
†
v,k
−q
a
v,k
a
c,k−q
%
, (10.18)
where, as usual,
n
λ,k
= a
†
λ,k
a
λ,k
. (10.19)
The first line of Eq. (10.18) is basically the same as Eq. (5.30) of the free-
carrier theory. The four-operator terms appear as a consequence of the
Coulomb part of the electron Hamiltonian (10.14).
To proceed with our calculations, we again make a random phase ap-
proximation to split the four-operator terms in Eq. (10.18) into products
of densities and interband polarizations. For example, we approximate
a
†
c,k
+q
a
†
v,k−q
a
ck
a
c,k
P
vc,k
n
c,k
δ
k−q,k
, (10.20)
because the term ∝ δ
q,0
does not contribute. As result, we obtain
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
168 Quantum Theory of the Optical and Electronic Properties of Semiconductors
i
d
dt
− (e
c,k
− e
v,k
)
P
vc,k
(t)
=
n
c,k
(t) − n
v,k
(t)
d
cv
E(t)+
q=k
V
|k−q|
P
vc,q
. (10.21)
dynamics of interband polarization (pair) function
In Eq. (10.21), we introduced the renormalized frequencies
e
λ,k
= e
λ,k
+Σ
exc,λ
(k) (10.22)
with the exchange self-energy
Σ
exc,λ
(k)=−
q=k
V
|k−q|
n
λ,q
, (10.23)
compare Eq. (9.21).
As in the case of free carriers, Eq. (5.30), we see that Eq. (10.21) cou-
ples the dynamics of the interband polarization to the evolution of the
carrier distribution functions. Consequently, we need additional equations
for ∂n
c,k
(t)/∂t and ∂n
v,k
(t)/∂t. The resulting coupled equations would
then be the optical Bloch equations (5.30) – (5.32) with the additional
Coulomb effects included. We will discuss these semiconductor Bloch equa-
tions in Chap. 12. For the purposes of the present chapter, we eliminate
the population dynamics by making the quasi-equilibrium assumption dis-
cussed in Chap. 5. In quasi equilibrium, we assume that the time scales
of interest are sufficiently long so that the rapid scattering processes have
already driven the carrier distributions to equilibrium in the form of quasi-
stationary Fermi–Dirac distributions. We can then make the replacements
n
c,k
(t) → f
c,k
and n
v,k
(t) → f
v,k
,
and Eq. (10.21) simplifies to