Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors

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Quantum Dots 399

Using for f(R) simply a Gaussian distribution around

R = 20 Å, we obtain

for CdS quantum dots the results shown in Fig. 20.5. One clearly sees

the energetically lowest one-pair resonances, which merge to a continuous

structure with increasing width of the size distribution. Absorption spectra

similar to those in Fig. 20.5 are experimentally observed in many quantum

dot systems.

In the nonlinear regime, one has to solve the full set of Bloch equations

(20.56). Assuming a pump–probe conﬁguration, we obtain increasing levels

of one- and two-pair-state populations with increasing intensities of the

pump beam. For such a situation, Hu et al. (1996) computed the series of

probe absorption spectra, shown in Fig. 20.6. We see a gradual bleaching

of the energetically lowest transitions until, at suﬃciently high excitation

level, negative absorption, i.e., optical gain occurs. This gain can be used

to produce semiconductor lasers with quantum dots as active material.

For these lasers, one typically uses arrays of quantum dots based on III-V

materials. More details can be found, e.g. in Bimberg et al. (1999).

REFERENCES

The ”classical” papers introducing the theory of semiconductor quantum

dots are:

L.E.Brus,J.Chem.Phys.80, 4403 (1984)

Al. L. Efros and A.L. Efros, Sov. Phys. Semicond., 16, 772(1982)

Reference books on semiconductor quantum dots include:

L. Banyai and S.W. Koch, Semiconductor Quantum Dots, World Scientiﬁc

Publ., Singapore (1993)

U. Woggon, Optical Properties of Semiconductor Quantum Dots, Springer

Tracts in Modern Physics 136, Springer Verlag, Berlin (1997)

D. Bimberg, M. Grundmann, and N.N. Ledentsov, Quantum Dot Het-

erostructures, Wiley and Sons, New York (1999)

The work presented in this chapter is largely based on:

Y.Z.Hu,M.Lindberg,andS.W.Koch,Phys.Rev.B42, 1713 (1990)

see also:

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400 Quantum Theory of the Optical and Electronic Properties of Semiconductors

Y.Z. Hu, H. Giessen, N. Peyghambarian, and S.W. Koch, Phys. Rev. B53,

4814 (1996)

PROBLEMS

Problem 20.1: Compute the eﬀective Coulomb interaction potential be-

tween two point charges in a dielectric sphere of radius R and background

dielectric constant 

which is embedded in a medium with background

dielectric constant 

Problem 20.2: Solve Eq. (20.21) for a quantum-box. Discuss the single-

particle energy eigenvalues as function of box-length.

Problem 20.3: Evaluate the Liouville equation (20.51) for the density

matrix given in Eq. (20.55). Show, that the density-matrix elements obey

the multilevel Bloch equations (20.56).

Problem 20.4: Expand the multilevel Bloch equations (20.56) in third

order in the pump ﬁeld and derive the corresponding nonlinear optical

susceptibility.

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Chapter 21

Coulomb Quantum Kinetics

In this ﬁnal chapter, we come back to the discussion of the scattering terms

in the semiconductor Bloch equations (12.19). So far, we have treated

only scattering of carriers with LO-phonons (see chapter 12). However,

for the proper analysis of conﬁgurations with ﬁnite carrier densities, we

have to include carrier–carrier scattering and the screening of the Coulomb

potential.

Screening by optically generated carriers in a semiconductor can be

treated quite naturally in the framework of nonequilibrium Green’s func-

tions, where the interaction is described by a dynamical two-time-dependent

propagator. This allows us to investigate, e.g., the dynamical build-up of

screening after ultrafast carrier excitation. This dynamics reﬂects the time

it takes the generated carriers to rearrange themselves in order to optimally

screen their mutual interaction potential.

Because nonequilibrium, or Keldysh Green’s functions may be less fa-

miliar to the reader, we present a relatively simple introduction into this

technique in Appendix B. For a more complete coverage, we refer to the

books quoted at the end of the chapter.

We note at this point, that the theory of reduced density matrices also

allows us to treat screening. Here, one has to include the equations of

motion of all two-particle (or four-point) density matrices, as has been

known in plasma physics for quite some time. Because this treatment is less

explored and quite involved in a two-band semiconductor, we refer again to

the literature quoted at the end of this chapter. Generally, it turns out that

both methods have their own advantages and are particularly well suited

for speciﬁc approximations, as will be shown below.

After presenting the general formulation of the problem, we proceed in

this chapter by ﬁrst treating the interaction as a given statically screened

401

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402 Quantum Theory of the Optical and Electronic Properties of Semiconductors

object. This is a good assumption for time scales that are long compared

to the inverse plasma frequency. We evaluate the scattering integral by

including the direct and exchange contributions. This approximation is

called the second Born approximation.

The quantum kinetics of the femtosecond build-up of screening is then

treated in the last section of this chapter. The screening develops roughly

within the period of a plasma oscillation. In a next step, we then treat the

combined dynamical interactions by the exchange of longitudinal plasmons

and optical phonons. In order to reduce the complexity, the exchange

interaction is neglected in this discussion.

21.1 General Formulation

As derived in Appendix B, the scattering integral for the single-time density

matrix ρ(t) in a spatially homogeneous situation is given by

∂ρ

(t)

∂t



scatt

= −



−∞





(t, t



,t) − Σ

(t, t



,t)

− G

(t, t



)Σ



,t)+G

(t, t



)Σ



,t)



. (21.1)

quantum kinetic collision integral

Here, the density matrix, the particle propagators G

and the scat-

tering self-energies Σ

, Σ

are all matrices in the band indices. Eq. (21.1)

shows that the dynamics of the single-time density matrix is coupled to

two-time propagators, so that one does not have a closed equation for the

density matrix. Without scattering, i.e., for free particles or particles which

experience only a mean ﬁeld interaction, one can show that the two-time

propagators can be expressed in terms of single-time density matrices and

a retarded or advanced Green’s function which describes the correlation

between the times t and t



(t, t



)=ia

†



)a(t) = −G

(t, t



)ρ(t



)+ρ(t)G

(t, t



) . (21.2)

generalized Kadanoﬀ–Baym ansatz

This equation is often called generalized Kadanoﬀ–Baym ansatz. It holds

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Coulomb Quantum Kinetics 403

in systems where the interaction is weak. Once the scattering self-energies

are also expressed in terms of the particle propagators, the generalized

Kadanoﬀ–Baym ansatz allows one to close the equation. Naturally, the

retarded and advanced Green’s functions have still to be evaluated self-

consistently in the mean-ﬁeld approximation. By self-consistently we mean

that both quantities have to be computed jointly within the numerical

integration of the coupled equations for ρ and the spectral functions. In this

way, these spectral functions can still describe nontrivial renormalizations

such as the Hartree–Fock band-edge renormalizations, excitonic eﬀects, or

the optical Stark eﬀect.

Let us now turn to the evaluation of the scattering self-energy. Instead

of formulating the perturbation theory in terms of unperturbed Green’s

functions and bare interaction potentials, we apply a self-consistent scheme

in which the self-energies are expressed by the dressed particle propagators

and screened interaction potentials. In Fig. 21.1, the Feynman diagrams

are given for those terms that are included in this treatment. The ﬁrst

diagram, called the GW approximation, would actually be the exact one, if

the full vertex correction was included. The second diagram of Fig. 21.1 is

the ﬁrst vertex correction or the exchange self-energy, as can be seen from

the crossing particle lines.

t’tt

k’’−k’

k’−k

t’ t

k’’

k’ k’

k’ − k

k+k’’−k’

Fig. 21.1 Feynman diagrams for the scattering self-energy with the GW self-energy and

the exchange self-energy, also called ﬁrst vertex correction.

To show that the second diagram in Fig. 21.1 is indeed a vertex correc-

tion, we redraw it in the way shown in Fig. 21.2. This way, we obtain the

same type of diagram as the ﬁrst one in Fig. 21.1, however, the left vertex

(or interaction point) is renormalized. Naturally, an inﬁnite series of higher

and higher vertex corrections should be included to make the self-energy

exact. Unfortunately, such a procedure is not possible.

Not even the summation of a selected class of vertex corrections, called

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404 Quantum Theory of the Optical and Electronic Properties of Semiconductors

k+k’’−k’k−k’

’

k’’−k’

k’

Fig. 21.2 The scattering exchange self-energy drawn as vertex correction of the GW

self-energy.

the ladder diagrams shown in Fig. 21.3, have so far been evaluated in the

nonequilibrium many-body theory, because of its complexity.

It is important to remember that all nonequilibrium Feynman diagrams

have to be evaluated for times which are ordered on the Keldysh contour

which runs from minus inﬁnity — where the system was in equilibrium —

through the two times of the Green’s function back to minus inﬁnity (see

Appendix B). The ﬁrst part of this contour is called the positive branch,

the part running backwards the negative branch. In order to be sure that

in a “lesser” function the ﬁrst time argument is always earlier (less) than

the second one, the ﬁrst argument has to be on the positive and the second

one on the negative branch of the Keldysh contour. The GW self-energy is

thus

<GW

(t, t



)=i





k−k



(t, t







,t) . (21.3)

Here, the propagators always start at the vertex at the right time argument,

and run to the vertex at the left time argument. Therefore, the time argu-

ments of the interaction W are interchanged. Because t>t



on the contour

= +

Fig. 21.3 The ladder approximation for the Coulomb vertex.

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Coulomb Quantum Kinetics 405

W is a “greater” function W



,t). In a next step, we want to express

this “greater” potential in terms of the retarded and advanced interaction

potential, W

(t, t



) and W

(t, t



), which determine the dielectric response.

WVV W

Fig. 21.4 Integral equation for the screened potential W .

The screened potential W obeys in contour time space an integral equa-

tion shown symbolically in Fig. 21.4. Similarly, the retarded potential obeys

a simple Dyson equation in the form

(t, t



)=V

δ(t, t



)+V

(t, t



) . (21.4)

Here, a matrix notation for the real time arguments is used and it is implied

that one has to integrate over repeated time arguments. V

is the bare

Coulomb potential, and L

(t, t



) is the retarded irreducible polarization

function.

If we write Eq. (21.4) for the lesser part of the potential, the ﬁrst term

δ(t, t



) does not contribute, because the two time arguments are on diﬀer-

ent branches of the contour and therefore cannot be equal. In the random

phase approximation (RPA), L

(t, t



) is simply a bubble of Green’s func-

tions describing intraband scattering. The lesser part of the potential is

then

(t, t



)=V

(t, t



)]

, (21.5)

where all times are considered still on the contour. With the Langreth

theorem described in Appendix B, the product (LW )

can be evaluated

and one ﬁnds

(t, t



)=V

(t, t



)+V

(t, t



) , (21.6)

where now all times are real times, i.e., no longer contour times.

If we now multiply Eq. (21.4) with V

−1

, we ﬁnd (V

−1

− L)W

=1,so

that W

r,−1

= V −L. Similarly, Eq. (21.6) gives (V

−1

−L)W

= L

r,−1

. Multiplying the last equation with W

yields the result

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406 Quantum Theory of the Optical and Electronic Properties of Semiconductors

>,<

(t, t



)=W

(t, t

>,<



) . (21.7)

optical theorem for the scattering potential

This relation is called optical theorem. Here, we have implied that the

same relation can be derived for the greater function if the polarization L

is also taken as a greater function (see also Problem 21.3). The relation

(21.7) shows that the GW scattering self-energy is actually quadratic in the

spectral functions W

and W

tt’

Fig. 21.5 Polarization bubble.

If we evaluate the polarization diagram of Fig. 21.5 in RPA we ﬁnd

(t, t



)=−i



k+q

(t, t



,t) . (21.8)

The GW scattering self-energy becomes

<GW

(t, t



)=











(t, t







)

× G





,t) . (21.9)

The three propagators G

<,>

yield in the scattering rate the initial and ﬁnal

state occupation factors, if one expresses the functions G

<,>

by the density

matrices via the generalized Kadanoﬀ–Baym ansatz.

The evaluation of the exchange scattering self-energy is considerably

more involved. Still in the contour time formalism, we get from the diagram

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Coulomb Quantum Kinetics 407

of Fig. 21.2

(t, t



)=−









(t, t



−k

,t)

× G



−k





k−k







) . (21.10)

In order to write this diagram in real time, the most eﬃcient way is to

sum for each contour time integral over its branch index α,whichis+

for the positive branch and − for the negative branch. In this notation,

(t, t



)=Σ

+−

(t, t



), showing that the time t is on the earlier positive

branch while t



is on the later, negative branch. Symbolically written, the

exchange self-energy yields four terms with the following combination of

Keldysh indices:



,α

+α

−α

−



+α

−+

−

+ G

+−

−α

−−

−

)

= G

−+

+−

+ G

+−

−+

−−

+−

+ G

−+

+−

−+

−−

+ G

+−

−+

−−

. (21.11)

The second term in the ﬁnal result contains three G

<,>

propagators and

the ++ and −− components of the potential. With the relations

= W

− W

+−

−−

= W

+ W

+−

, (21.12)

we can transform this term into a scattering rate with two spectral func-

tions of the potential and three particle propagators, which has again the

structure of the GW term. This term is thus the exchange Coulomb scat-

tering rate which belongs to the direct Coulomb scattering rate of the GW

scattering self-energy. One sees, however, that generally the vertex correc-

tion contains also contributions of third and fourth order in the interaction

potential, if one expresses the potential via the optical theorem by the re-

tarded and advanced components of the potential. We will not consider

these higher order terms any further.

The second-order vertex contribution to the scattering self-energy is

<ex

(t, t



)=−











(t, t



−k

,t)

× G



−k





k−k







) . (21.13)

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408 Quantum Theory of the Optical and Electronic Properties of Semiconductors

So far, in the literature, this exchange contribution has been considered

mainly in the long-time regime, where a quasi-equilibrium screened poten-

tial can be assumed.

21.2 Second Born Approximation

We now explicitly derive the scattering rates as determined by the GW

and exchange scattering self-energies in the long-time limit. The quantum

kinetic study of the time-dependent build-up of screening will show that

this approximation is reasonable on time scales that are large compared to

the inverse electron–hole plasma frequency.

Under these, with respect to the screening dynamics, quasi-stationary

conditions, the two-time retarded and advanced statically screened poten-

tial becomes singular in the time diﬀerence

r,a

)=W

)δ(t

− t

) , (21.14)

where W

(t) is the quasi-equilibrium statically screened Coulomb poten-

tial, which still depends parametrically on the carrier densities. Note that

the form (21.14) holds only for a statically screened potential, a frequency-

dependent screening would result in a potential which depends on the dif-

ference time coordinate. With (21.14), all time integrals in the scattering

self-energy vanish, and only the time-integral in the scattering rate (21.1)

remains. Consistently with the above assumptions, we will make a Markov

approximation for this last integration. Naturally, all particle propaga-

tors are still matrices in the band index. After applying the generalized

Kadanoﬀ–Baym ansatz (21.2) to all particle propagators in the self-energies

and the scattering rates, only spectral, i.e., retarded and advanced, Green’s

functions and single-time density matrices remain.

Since we are in the long-time limit, we assume that the oﬀ-diagonal

spectral functions have already decayed. As a further simplifying approx-

imation, we take the diagonal elements of the spectral functions in the

eﬀective-particle approximation. With G

(t, t



)=[G



,t)]

∗

we get

ii,k

(t, t



)=−



Θ(t − t



−



i,k

−iγ

(t−t



)

ii,k

(t, t





Θ(t



− t)e

−



i,k

−iγ

(t−t



)

. (21.15)

Here, ε

i,k

are the mean-ﬁeld renormalized single-particle energies and γ

is a quasi-particle broadening. In the Markov approximation, the slowly