Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors
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Polaritons 209
function in 2D and 3D.
Problem 11.3: Compute the polariton transformation coefficients u
q
and
v
q
and the polariton spectrum Ω
q
by evaluating the commutator [p, H] once
using the polariton Hamiltonian, Eq. (11.44), and once using Eq. (11.42)
together with the exciton–photon Hamiltonian.
Problem 11.4: Show that the operators for the lower and upper polariton
branches commute, e.g., [p
1
,p
†
2
]=0.
Problem 11.5: Diagonalize the microcavity exciton–photon Hamiltonian
(11.56) and derive the dispersion relation, Eq. (11.58), by applying the
polariton transformation according to Eqs. (11.42) – (11.47).
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Chapter 12
Semiconductor Bloch Equations
The previous two chapters analyze the interband transitions in semicon-
ductors in the linear regime focusing on the correct treatment of the all
important Coulomb attraction between the conduction-band electron and
the valence-band hole. In this and the following chapters, we will now ex-
tend that treatment into the nonlinear regime where several electron–hole
pairs are excited and the interactions between these pairs have to be con-
sidered. In situations with elevated electron–hole–pair densities, we have
to deal with finite numbers of electrons and holes in the conduction and
valence band which are coupled dynamically to the interband polarization.
In this chapter, we derive the set of coupled differential equations which
governs the coupled dynamics of electrons, holes and the optical polariza-
tion in the spectral vicinity of the semiconductor band gap. We call these
equations the semiconductor Bloch equations since they are the direct gen-
eralization of the equations for the free-carrier transitions, which can be
put into the form of Bloch equations, as has been shown in Chap. 5.
12.1 Hamiltonian Equations
We start from the Hamiltonian
H = H
el
+ H
l
, (12.1)
where
211
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
212 Quantum Theory of the Optical and Electronic Properties of Semiconductors
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
'
(12.2)
is the electron part of the Hamiltonian in two-band approximation. Here,
we include only those Coulomb interaction processes that conserve the par-
ticle number in each band. In other words, interband Coulomb processes
in which, e.g., one electron and one hole are annihilated and two electrons
are created are not taken into account due to the relatively large energy
gap between both bands.
H
I
= −
k
E(t)(a
†
c,k
a
v,k
d
cv
+ h.c.) , (12.3)
describes the interband dipole coupling to the light field.
For the subsequent calculations, it is convenient to transform the total
Hamiltonian into the electron–hole representation (see Chap. 11). Inserting
the definitions (11.21) and (11.22),
β
†
−k
= a
v,k
(12.4)
and
α
†
k
= a
†
c,k
, (12.5)
into the Hamiltonian (12.1) and restoring normal ordering of the electron
and hole operators, we obtain
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Semiconductor Bloch Equations 213
H =
k
&
E
e,k
α
†
k
α
k
+ E
h,k
β
†
−k
β
−k
'
+
1
2
k,k
,q=0
V
q
&
α
†
k+q
α
†
k
−q
α
k
α
k
+β
†
k+q
β
†
k
−q
β
k
β
k
−2α
†
k+q
β
†
k
−q
β
k
α
k
'
−
k
E(t)
&
d
cv
α
†
k
β
†
−k
+h.c.
'
, (12.6)
electron–hole Hamiltonian
where constant terms have been left out. The single particle energies in
(12.6) are
E
e,k
= E
c,k
=
e,k
and
E
h,k
= −E
v,k
+
q=0
V
q
=
h,k
, (12.7)
showing that the kinetic energy of the holes includes the Coulomb exchange
energy −
V
q
n
v,|k−q|
with n
v,|k−q|
=1for the full valence band. The low-
intensity interband transition energy is therefore
∆E
k
= E
c,k
− E
v,k
+
q=0
V
q
. (12.8)
In extension of the analysis in Chap. 10, we now want to derive the coupled
equations of motion for the following elements of the reduced density matrix
α
†
k
α
k
= n
e,k
(t)
β
†
−k
β
−k
= n
h,k
(t)
β
−k
α
k
= P
he
(k,t) ≡ P
k
(t) .
(12.9)
We proceed as in Chap. 10 and compute the Hamiltonian equations of
motion, but this time for all the expectation values (12.9). Straightforward
operator algebra yields
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214 Quantum Theory of the Optical and Electronic Properties of Semiconductors
i
d
dt
− (
e,k
+
h,k
)
P
k
=(n
e,k
+ n
h,k
− 1)d
vc
E(t)
+
k
,q=0
V
q
$
α
†
k
+q
β
−k+q
α
k
α
k
−β
†
k
+q
β
−k+q
β
k
α
k
+ β
−k
α
†
k
−q
α
k
α
k−q
−β
−k
β
†
k
−q
β
k
α
k−q
%
, (12.10)
∂
∂t
n
e,k
= −2Im
d
cv
E(t) P
∗
k
+ i
k
,q=0
V
q
$
α
†
k
α
†
k
−q
α
k−q
α
k
−α
†
k+q
α
†
k
−q
α
k
α
k
+ α
†
k
α
k−q
β
†
k
−q
β
k
−α
†
k+q
α
k
β
†
k
−q
β
k
%
, (12.11)
∂
∂t
n
h,k
= −2Im
d
cv
E(t) P
∗
k
+ i
k
,q=0
V
q
$
β
†
−k
β
†
k
−q
β
−k−q
β
k
−β
†
−k+q
β
†
k
−q
β
−k
β
k
+ α
†
k
+q
α
k
β
†
−k
β
−k+q
−α
†
k
+q
α
k
β
†
−k−q
β
−k
%
. (12.12)
As in our derivation of the polarization equation in Chap. 10, we now
again split the four-operator terms into products of densities and interband
polarizations plus the unfactorized rest. This way we separate the equations
of motion into the Hartree–Fock and scattering parts
∂
∂t
A =
∂
∂t
A
HF
+
∂
∂t
A
scatt
. (12.13)
Here, the scattering terms are simply defined as the differences between
the full and the Hartree–Fock terms so that the decomposition is formally
exact. However, at later stages we will have to make approximations to
obtain manageable expressions for the scattering terms.
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Semiconductor Bloch Equations 215
Using the decomposition (12.13) for all the four-operator terms in
Eq. (12.10) – (12.12), one can write the resulting equations as
∂P
k
∂t
= −i(e
e,k
+ e
h,k
)P
k
(12.14)
−
i
&
n
e,k
+ n
h,k
− 1
'
d
cv
E(t)+
q=k
V
|k−q|
P
q
+
∂P
k
∂t
scatt
,
∂n
e,k
∂t
= −
2
Im
d
cv
E(t)+
q=k
V
|k−q|
P
q
P
∗
k
+
∂n
e,k
∂t
scatt
, (12.15)
∂n
h,k
∂t
= −
2
Im
d
cv
E(t)+
q=k
V
|k−q|
P
q
P
∗
k
+
∂n
h,k
∂t
scatt
, (12.16)
where again the renormalized single-particle energies
e
i,k
=
i,k
+Σ
exc,i
(k)=
i,k
−
q
V
|k−q|
n
i,q
,i= e, h (12.17)
have been introduced.
Approximations for the scattering terms of Eqs. (12.14) – (12.16) will
be discussed in Sec. 12.3 and in later chapters of this book. Here, we note
that in all three equations the combination
1
d
cv
E(t)+
q=k
V
|k−q|
P
q
appears. In generalization of the Rabi frequency (2.49) for an atomic sys-
tem, we therefore introduce the generalized Rabi frequency
ω
R,k
=
1
d
cv
E +
q=k
V
|k−q|
P
q
, (12.18)
generalized Rabi frequency
emphasizing the fact, that the electron–hole system does not react to the
applied field E(t) alone, but to the effective field which is the sum of the
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
216 Quantum Theory of the Optical and Electronic Properties of Semiconductors
applied field and the internal dipole field of all generated electron–hole
excitations.
With the definition (12.18), Eqs. (12.14) – (12.16) can be written in the
very compact form
∂P
k
∂t
= −i(e
e,k
+ e
h,k
)P
k
− i(n
e,k
+ n
h,k
− 1)ω
R,k
+
∂P
k
∂t
scatt
∂n
e,k
∂t
= −2Im(ω
R,k
P
∗
k
)+
∂n
e,k
∂t
scatt
∂n
h,k
∂t
= −2Im(ω
R,k
P
∗
k
)+
∂n
h,k
∂t
scatt
. (12.19)
semiconductor Bloch equations
Since Eqs. (12.19) are the generalization of the optical Bloch equations
(5.30) – (5.32), we call them the semiconductor Bloch equations.
These semiconductor Bloch equations constitute the basis for most of
our understanding of the optical properties of semiconductors and semi-
conductor microstructures. Depending on the strength and time dynamics
of the applied laser light field contained in the generalized Rabi frequency
(12.18), one can distinguish several relevant regimes:
• the low excitation regime in which the exciton resonances — some-
times accompanied by exciton molecule (biexciton) resonances — dom-
inate the optical properties. The interaction with phonons provides the
most important relaxation and dephasing mechanism. As the density
increases gradually, scattering between electron–hole excitations also
becomes important.
• the high excitation regime in which an electron–hole plasma is excited.
Here, the screening of the Coulomb interaction by the optically excited
carriers and the collective plasma oscillations are the relevant physi-
cal phenomena. The main dissipative mechanism is the carrier–carrier
Coulomb scattering.
• the quasi-equilibrium regime which can be realized on relatively long
time scales. Here, the excitations have relaxed into a quasi-equilibrium
and can be described by thermal distributions. The relatively slow
approach to equilibrium can be described by a semi-classical relaxation
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Semiconductor Bloch Equations 217
and dephasing kinetics.
• the ultrafast regime in which quantum coherence and the beginning
dissipation determine the optical response. The process of decoherence,
of the beginning relaxation and the build-up of correlations, e.g., by a
time-dependent screening are governed by the quantum kinetics with
memory structure of the carrier–carrier and carrier–phonon scattering.
The optical light field can consist of two or more light pulses propagating in
equal or different directions with the same or different carrier frequencies,
pulse envelopes, and intensities, respectively. The semiconductor Bloch
equations are excellently suited to analyze such multi-pulse experiments and
the wide variety of physical phenomena, such as relaxation, dephasing, and
build-up of correlations. Among the many different possible configurations
in nonlinear spectroscopy, there are two basic experimental set-ups that we
will analyze in the following chapters of this book:
• As four-wave mixing (FWM) spectroscopy, we denote a typical exper-
imental configuration where two or three pulsed laser beams hit the
sample under different angles of incidence and variable time delays.
The pulses induce polarizations in the crystal. These polarizations in-
terfere, forming a lattice (grating), from which one obtains refracted
signals in various orders. The decay of the FWM signal is often stud-
ied to analyze dephasing processes in detail.
• Time resolved differential transmission spectroscopy (DTS) refers to ex-
periments where two more or less co-linearly propagating pulsed laser
beams with equal or different carrier frequencies hit the sample with
variable time delay. The carriers excited by the first, often stronger
pulse (the pump pulse) modify the absorption of the second, usually
weak pulse (the probe pulse). Such DTS-experiments give direct in-
formation about build-up and decay of nonlinearities induced by the
pump, especially about the relaxation kinetics of the electron–hole ex-
citations.
The analysis of such experiments under various conditions in terms of the
semiconductor Bloch equations will be a major issue in some of the following
chapters of this book.
Generally, the scattering terms in the semiconductor Bloch equations
(12.19) describe all the couplings of the polarizations and populations, i.e.,
of the single-particle density matrix elements to higher-order correlations,
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
218 Quantum Theory of the Optical and Electronic Properties of Semiconductors
such as two-particle and phonon- or photon-assisted density matrices. How-
ever, in many cases one can identify specific physical mechanisms that dom-
inate the scattering terms in some of the excitation regimes listed above.
For example, in the low excitation regime often the coupling of the excited
carriers to phonons determines relaxation and dephasing, whereas at high
carrier densities carrier–carrier scattering dominates. For relatively long
pulses, Markov approximations for the scattering processes are often justi-
fied, and the scattering terms can be described by Boltzmann-like scattering
rates due to carrier–phonon or carrier–carrier scattering, respectively.
For ultra-short pulses, the Markov approximations may break down.
Here, quantum kinetics with its memory structure has to be used to describe
the effects of scattering processes which are often not completed during the
action of a light pulse. In this regime, the quantum coherence of the electron
states influence the scattering kinetics in an important way and give rise to
a mixture of coherent and dissipative effects.
In all situations, the semiconductor Bloch equations are a very suit-
able theoretical framework which, however, has to be supplemented with
an appropriate treatment of the scattering terms in order to describe the
various aspects of the rich physics which one encounters in pulse excited
semiconductors. In general, the semiconductor Bloch equations have to be
treated together with the Maxwell equations for the light field in order to
determine the optical response. This self consistent coupling of Maxwell
and semiconductor Bloch equations (for shortness also called Maxwell–
semiconductor–Bloch equations) is needed as soon as spatially extended
structures are analyzed where light propagation effects become important.
Relevant examples are the polariton effects analyzed in the previous chap-
ter, as well as semiconductor lasers or the phenomenon of optical bistability
discussed later in this book. In optically thin samples, however, where prop-
agation effects are unimportant, the transmitted light field is proportional
to the calculated polarization field. Under these conditions one can treat
the semiconductor Bloch equations separately from Maxwell’s equations to
calculate the optical response.
Superficially, the semiconductor Bloch equations seem to be diagonal
in the momentum index k, but in reality already the Coulomb terms in
the generalized Rabi frequency (12.18) and in the exchange energy (12.17)
and even more the scattering terms lead to strong couplings of all momen-
tum states. As simple limiting cases, the semiconductor Bloch equations
reproduce both the optical Bloch equations for free-carrier transitions (see
Chap. 5) and the Wannier equation for electron–hole pairs.