Nonequilibrium semiconductor dynamics

0079-6727(95)00001-l

@ant. Electr. 1995, Vol. 19, No. 415,Pp. 307-462 Copyright @I 1995Blsevier Science Ltd Printd-in -&at Britain. All rights reserved 0079-6727/95

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R. BINDER * and S. W. KOCH + * Optical Sciences Center, University of Arizona, Tucson, AZ 85721, U.S.A. + Fachbereich Physik und Zentrum fiir Materialwissenschaften, Philipps-Universitgt Marburg, Renthof 5, 35032 Marburg, Germany

CONTENTS 1. Introduction 2. Theoretical Framework 2.1. Maxwell Equations 2.2. Hartree-Fock Theory 2.3. Band-Coupling Effects 2.4. Application to Two-Pulse and FWM Cotiguration 2.5. General Remarks on Correlation Effects 2.6. Green’s Function Approach 2.7. Screened-Hartree-Fock Theory 2.7.1. Expansion into eigenfunctions 2.7.1.1. General expansion 2.7.1.2. Spatially homogenous and isotropic systems 2.7.2. Quasi-static approximation 2.7.3. Evaluation of Keldysh matrices 2.7.4. Free-particle Green’s functions 2.7.5. Two approximation schemes 2.7.5.1. The Kadanoff-Baym ansatz 2.7.5.2. The generalized Kadanoff-Baym ansatz 2.7.6. Correlation contributions to the semiconductor Bloch Equations 2.7.6.1. Two-time formalism 2.7.6.2. Kadanoff-Baym formalism. 2.7.6.3. Generalized Kadanoff-Baym formalism 2.7.7. The longitudinal polarization function 2.7.7.1. Two-time formalism 2.7.7.2. Kadanoff-Baym formalism 2.7.7.3. Generalized Kadanoff-Baym formalism 2.7.8. The screened Coulomb potential 2.7.8.1. Two-time formalism 2.7.8.2. Kadanoff-Baym formalism 2.7.8.3. Generalized Kadanoff-Bavm formalism 2.7.9. Selfenergies and generalized dikerential scattering cross-sections 2.7.9.1. Two-time formalism 2.7.9.2. Kadanoff-Bavm formalism 2.7.9.3. Generalized i(adanoff-Baym formalism 2.8. The xt3)-Approach 3. Numerical Results of Selected Phenomena 3.1. Basic Excitation Dynamics 3.2. Coherent Dynamics 3.3. The Optical Stark Effect 3.4. Multi-Wave Mixing and Photon Echo 3.4.1. Results for a two-band semiconductor 3.4.2. Band-mixing effects 3.4.3. Excitation-induced-dephasing effects 3.5. Pulse Propagation 3.5.1. Pulses centered at the exciton resonance 3.5.1.1. Coherent pulse propagation

307

308 310 310 316 327 330 336 338 346 348 348 350 351 353 355 356 356 359 361 361 363 366 367 367 367 369 369 369 370 372 373 373 374 378 381 389 389 389 400 409 409 416 423 424 424 424

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R. Binder and S. W. Koch

3.5.1.2. Four-wave mixing and pulse propagation 3.52. Pulse propagation in amplifiers 3.6. Coherent Electric Field Effects 4. Conclusion 5. Acknowledgements 6. Appendix A: The Functional Derivative Technique 7. Appendix B: The Luttinger Hamiltonian References

428 431 439 444 445 446 :86

1. INTRODUCTION

Electrons and holes in an excited semiconductor structure establish a many-body Fermion system which exhibits characteristic dynamical features that occur on a wide range of different time scales Optical excitation with short pulses of sufficient intensity generates an interband polarization in the semiconductor which is accompanied by nonequilibrium electron and hole distributions Through clever experimental techniques it is possible to follow, with femtosecond time resolution, the dynamic evolution of the system from its initial state all the way into quasi-thermal equilibrium and beyond. The theoretical analysis of the nonequilibrium semiconductor dynamics requires an understanding of the time-dependent many-body interactions in the electronic system which is coupled to the light field. Since the generation of the initial state occurs on the same time scale as the fastest interaction processes, the optical and electronic dynamics have to be treated consistently. This differs considerably from quasi-equilibrium pump-probe situations where the optical probe field acts only as a probe which is used to investigate many-body interactions in excited semiconductors and which does not explicitely enter the theoretical analysis The many-body theory of highly excited semiconductors in quasi-equilibrium was mainly developed in the 197Os,first for electron-hole liquids in indirect-gap semiconductors (reviewed, for example, by Rice(‘)) and then for direct-gap semiconductors Later, a detailed Green’s funo tion theory for quasi-equilibrium electron-hole plasmas in semiconductors and semiconductor lasers has been developed. Comprehensive presentations of these theories and references to the corresponding original publications in this field are given by Haug and Schmitt-Rinkt2) and by Zimmermann.(3) In contrast to the quasi-equilibrium situation, optical pulses in most femtosecond experiments do not simply play the role of a passive external probe. Instead, the light pulses am an integral part of the nonequilibrium system. Prime examples for the dynamic coupling between light field and semiconductor dynamics are the so-called coherent phenomena. In fact, most coherent optical effects known from atomic systems (see, e.g. Ref. (4)) have analogies in semiconductors. In basically all cases, however, the specific semiconductor nonequilibrium many-body effects lead to characteristic differences For example, resonant semiconductor excitation leads to oscillations in the electron-hole density (generalized Rabi oscillations), where the oscillation frequency is significantly increased in comparison to that in atomic systemst5-7) This modification of the Rabi frequency is a direct consequence of the Coulomb interaction between the charge carriers, which results in an induced optical field. The induced field plays an important role in the analysis of most coherent effects in semiconductors The most important examples, which have been investigated recently, include the optical Stark effect,(*-I’) the temporal response of the optically excited electron-hole density,” 8-2o) photon echo and four-wave mixing signals,(21-30) ftee-induction decay,(31) and pulse propagation effects in semiconductors(32-34) An important aspect in the understanding of ultrafast optical nonlinearities of semiconductors, such as GaAs, are effects related to the light-field polarization vectors of the incoming beams as well as the scattered light fields The analysis of these effects requires a detailed

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knowledge of the hand structure (in particular, the valence band structure). Most of the relevant bandstructure features of zincblende semiconductors have been discussed by Luttinger and Kohn.(35,3a) Other important publications regarding banstructure effects close to the Ipoint and investigations of excitonic effects beyond the two-band model include Refs (37-52) as well as Ref. (53). Recent investigations of light-field polarization effects and ultrafast spin dynamics include Refs (27, 54-67). In addition to coherent phenomena which have their counterparts in atomic systems, semiconductors exhibit effects which are characteristic for solids only. An example of current interest are the so-called Bloch oscillations, i.e. the periodic movement of an electronic wavepacket through the Brillouin zone of a crystal with an applied electric field. Even though Bloch oscillations were predicted long ago, (68*6g) they could be observed only recently using the minibandstructure in semiconductor superlattices Generally, one important aspect for the analysis of coherent phenomena is the analysis of all those effects that lead to the decay of coherence, i.e. the dephasing and relaxation processes Among the best studied aspects of these phenomena in semiconductors are the incoherent scattering processes of the elementary excitations Examples of important recent investigations in this area include carrier-phonon, carrier-impurity and carrier-carrier scattering.(70-‘08’ More complete accounts of relaxation processes in semiconductors can be found, for example, in recent proceedings of the conference on “Hot Carriers in Semiconductors” and review papers therein.(‘Og) Several recent books deal with various topics related to linear and nonlinear optical properties of semiconductor structures, for example Refs (1 IO-1 16). In this review we present a detailed outline of the theoretical analysis of coherent optical effects and nonequilibrium phenomena in semiconductors We restrict ourselves to the so-called semiclassical approximation where the light-field is treated classically but the semiconductor medium is analyzed using many-body quantum theory. For later reference throughout this article, we first summarize the basic Maxwell theory of light interacting with a polarizable dielectric medium. We then discuss systematically the nonequilibrium many-body theory of electronic semiconductor excitations Using the standard formulation of quantum field theory we derive the coupled equations of motion for the semiconductor interband polarization and the carrier populations in the different quantum states Closing these equations through the time-dependent Hartree-Fock approximation, we obtain the so-called semiconductor Bloch equations @BE). (I23‘3,‘5,‘6,’‘5,“742” These equations have been used sucessfully to analyze many aspects of the coherent optical semiconductor phenomena mentioned above. In order to deal with correlations of the charge carrier system, we have to extend our analysis beyond the Hartree-Fock level. For this purpose we adopt the technique of nonequilibrium Green’s functions(1’2~‘22-‘27) The theory of Green’s functions for nonequilibrium systems is essentially a generalization of - and in many respects similar to - equilibrium Green’s function techniques(‘2s-‘3’) The nonequilibrium Green’s function method allows us to include screening and memory effects into the general analysis For most problems of practical interest, the resulting coupled integro-differential equations become extremely complicated at this level so that numerical solutions, even with today’s best and fastest supercomputers, become very time consuming, if not impossible. Hence, one has to introduce appropriate approximation schemes, which (hopefully) preserve the most important physical effects As examples we discuss the Kadanoff-Baym approach,(122’ the generalized Kadanoff-Baym ansatz of Lipavsky, Spicka and Velicky,u32’ and a general nonlinear susceptibility scheme of Axt and Stahl, (‘33’which uses the strength of the external field as the only smallness parameter in the problem. After the main theory part of this paper we present a number of applications of the general nonequilibrium theory. First, we discuss the ultrafast optical response for excitation below

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the exciton resonance (optical Stark effect). We then study generalized Babi oscillations for resonant excitation and discuss the temporal behavior of four-wave mixing and photon echo signals for a variety of conditions As a first step, we deal with these problems solving the SBE for the model of a two-band semiconductor. The two-band approximation reduces the complexity of the numerical calculations while it keeps most of the physical insight into the relevant many-body effects However, since all multi-valence-band and valence-band mixing effects are completely neglected, such studies cannot be used to analyze the dependence of the optical response on the polarizations of the exciting light fields We therefore extend the twoband calculations to include the valence-band mixing and the quantum-confinement induced bandstructurc modifications In addition to the analysis of coherent ultrafast phenomena in thin samples, where a spatially homogeneous situation can be assumed, we also analyze the effects of femtosccond pulse propagation in extended semiconductor systems, We study propagation phenomena for resonant and nonresonant excitation in active, i.e. partially inverted, and passive semiconductor systems Furthermore, we investigate the influence of propagation effects on four-wave mixing experiments In a separate section we briefly discuss the coherent effects of semiconductors in the presence of a static or time-varying electric field. The phenomena include the formation of a WannierStark ladder in the absorption spectrum, the generation of THz radiation as a consequence of the electronic Bloch oscillations, as well as quantum beats in four-wave mixing experiments and dynamic localization for a time-dependent external field. All the examples in Section 3 have been chosen mainly to illustrate specific applications of the nonequilibrium many-body theory. We concentrate largely on those phenomena for which we have participated directly in the theoretical analysis Hence, we do not want to claim completeness of the topics selected or even attempt to give a full listing of the many exciting experimental and theoretical contributions to the rapidly growing field of coherent and nonequilibrium phenomena in semiconductors.

2. THEORETICAL

FRAMEWORK

2.1. Maxwell Equations In a semiclassical approach the analysis of optical semiconductor properties breaks down into one part, where one deals with the light field(s) using the material polarization as an input, and the other part where the material polarization is computed for given optical field(s). Under stationary conditions it is often possible to reduce the analysis of the light field to a quasi-linear response treatment, where the knowledge of the material absorption and refrao tive index coefficients (optical susceptibility) are sufficient. In order to discuss ultrafast and coherent phenomena, however, the properties of light pulses traversing through semiconductors become an interesting subject in their own right. Here it is generally not possible to predict the outcome of experiments on the basis of linear response investigations of the pure semiconductor properties Therefore, and for reasons of completeness, we start our discussion by mviewing Maxwell’s equations, concentrating on those approximations which render these equations possible for numerical solutions in conjunction with the relevant semiconductor equations After specifying the wave propagation equation for the optical field amplitude ,!?, we introduce the slowly-varying-envelope approximation. We then focus on two special situations First, we discuss the four-wave-mixing (FWM) scenario where, in the simplest case, two pulses, which propagate at a small angle, excite the semiconductor and create a density grating by means of a phase mask. Here, we will restrict the analysis to optically very thin semiconduc-

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tors Propagation through optically thick materials will be studied only for single pulses and co-propagating pulse trains The corresponding propagation equation turns out to be a special case of the FWM propagation equation. We begin with the macroscopic Maxwell equations for a nonmagnetic material (& = ii) in Gauss units:

where the index A4 indicates macroscopic sources, b = i? +47r&, and 3~ is the dipole density of the semiconductor. We assume that the optical field inside the semiconductor is transverse, and tepresent_all v_ector_fields in-terms of longittdin_al (index ~5) and transverse (index T) components:+E = EL + ET with V x EL = 0 and V . ET = 0. We take ‘V x’ of Eqn. (2) and separate the resulting equation into its longitudinal and transverse part. The longitudinal equation can easily be integrated over time. Taking then ‘V a’ of this equation and using Eqn. (1) we obtain immediately the equation of continuity, &

+ V - &f,L

= 0.

(5)

The transverse part of the equation is the propagation equation

(6) We now assume that all macroscopic sources vanish and deal with a boundary value problem, where the light pulse is known at the surface of the sample (Z = 0). In the case of a spatially homogeneous semiconductor such as a high quality bulk GaAs system, we can proceed assuming that the incident field induces a polarization &, which in turn renormalizes the propagating field inside the sample, until the field reaches the end facet, Z = L, where L is the sample length. If one wants to know, for example, the far field seen by optical detectors, one solves the propagation equation for Z > L, which no longer involves the semiconductor properties, except for the input at Z = L (e.g. Ref. (134)). For the situation realized in spatially inhomogeneous systems, such as multiple-quantumwell structures, the analysis has to be modified. Here, the induced resonant polarization is confined to the regions of the quantum wells, whereas the field propagates through the wells and the barriers._Hence, a c_orrect treatment requires a detailed analysis of the different length scales on which P,u,T and ET vary, including partial reflections at the boundaries between the quantum wells and barriers (13~)In this article, however, we will not analyze these mom complicated situations, but concentrate on situations where spatial homogeneity can be assumed. The optical field has the general form _&r-(2,t) = !??c(ii,t> cos(& = ;$i(,

t) ,#o

’ k - qt) - g8(k, t) sin&, _ * R - mot) + cc

’ ii - wet) (7)

where cc denotes the complex conjugate. Here the slowly varying envelope ri(i2, t) = !2,&

t) + l&k,

t)

(8)

R. Binder and S. W. Koch

312

is complex whenever a sin-contribution is mixed with the cos-contribution. Two cases are of particular importance for this paper: linearly polarized light (!& II !&) and circularly polarized light. Here, we denote by ‘e, and Z,, the unit vectors in X and Y direction, respectively. If !& = !GX and !& = G,, we have circularly polarized light (which we call right-handed circularly polarized), and if & = IE& and !& = -!G$ we have left-handed circularly polarized light. The optical frequency wo is at this point arbitrary, important only in that the envelope varies slowly on a time scale w.-I. This condition can be re-formulated in terms of the Fourier transform * -D &(&co, = dkdt CitK’ R- wt) &-(i&t>

The spectrum of 2 should be centered around w = wg. B is then nonzero only around w = wg and w = -wg. In the case co = cog, we assume that we can neglect $* with respect to 9. This simplifies the propagation equation

Inserting Eqn. (9) and &(k,

0) = +(k

- &, w - 00) + @(-k

- i&, --w - WI))

(11)

into Eqn. (10) and neglecting z* with respect to 9 and @* with respect to L@,we obtain

The back transformation

into the space and time domain yields {-$($-ic.u0)2-(0+i&)2]!E@,t)= -$

(& -iw0)2!?&2,t).

(13)

To separate out the background contribution &, (@b) of the transverse semiconductor polarization i)M,r = &, + ? (!&J = % + i)) we proceed in the usual fashion, lumping into & all contributions that arise as a consequence of optical transitions in the medium which are not in Tsonance with the incident light field. Furthermore, we assume that !& responds linearly to E, and the corresponding response function is a rapidly converging Taylor expansion in frequency space around the point wg. Additionally, we assume the response to be local and without any spatial memory. We can thus write !&,(k, w - WI-J)= x(o)

$(ii, w - WO)

(14)

which follows from &.,r(& t) = if the nonresonant

I

dt’x(t

terms are neglected. With

- t’&(k,

1’)

(15)

Nonequilibrium semiconductor dynamics

x(w) = x(wo) + (w - oo)x’(wo) +

;ku- cL)o)2f(wo)

313

(16)

we obtain

(17)

The index ‘0’ indicates the argument 00. Because we have to take the Tend time derivative of ?+, in Eqn. (13) we would obtain time derivatives of the field envelope T of higher than second order. These higher order derivatives will be neglected. Inserting Eqn. (18) into Eqn. (13) we obtain lengthy prefactors in front of the various time derivatives, containing combinations of cog and ~0. These prefactors can be simplified by defining a background dispersion relation: P(o)

= $

(1 + 4rrx(w)).

(19)

To proceed, we first assume that & = K(wo) which allows us to eliminate the terms in the propagation equation which carry no space or time derivatives As a second step we differentiate Eqn. (19) with respect to o twice and evaluate the first and second derivatives at w = -WC,. Assuming x(w) to be real, i.e. x(-w) = x(w), we find that K& is essentially the pmfactor in front of the first time derivative of !& and k$‘& + &” is the prefactor of the second derivative, respectively. Here, the index ‘0’ at E4; and k$’ indicates the first and second derivative is taken at -wg. With this definition, Eqn. (13) becomes

Here, $ contains the response of all transitions with energy around hwo. Its calculation is the actual objective of the micrsocopic theory. Usually, Eqn. (20) is too complicated to be solved numerically if simultaneously ? contains the full information about the microscopic details of the nonlinear optical semiconductor response. In the remainder of this section we therefore discuss approximations that are valid in specific situations To do this, we assume that all envelopes have typical spatial and temporal extensions denoted by AX, AY, AZ, and At. Then, the derivatives in Eqn. (20) can approximately be replaced by a/M + l/AX etc. Also, we assume that all wavevectors have only two components, & = (K,, 0, KS). For an order of magnitude estimate, we can also replace && - wg /cz and G’& + 6” -* l/s. Assuming wg > l/At, we can neglect the second order time derivative on the lhs of Eqn. (20) with respect to the first order time derivative, i.e. we neglect group velocity dispersion effects On the rhs of Eqn. (20) we keep, by the same argument, only the cot term. IfRcoo = 1.5 eV (the bandgap energy of GaAs), then 00 = 2.3 fs-‘, so that an optical cycle takes about 3 fa In this case our approximation scheme could safely be used to discuss phenomena caused by pulses in the 100 fs range, whereas the analysis of experiments done with 10 fs pulses would stretch the regime of validity. Concerning spatial effects, we have to distinguish light scattering or FWM configurations from waveguide geometries In a light scattering experiment, the field is incident at a small angle (& 4 K.) and has as spot size of about AX. If the absorption length, which typically

R. Binder and S. W. Koch

314

is in the pm regime, is less than the sample thickness, we have to choose the spatial extension in propagation direction, AZ, as the absorption length. Then, as long as the spot size is large in comparison to the absorption length, one can neglect the second transverse derivative with respect to the second derivative in Z. This eliminates diffraction effects, which is a sensible approximation in the thin film limit. (For investigations of the diffraction problem see, for example, Ref. (110). Recent investigations of transverse effects include Refs (136, 137)) Because of the thin film approximation and the almost normal incidence, the term KJAX can be neglected with respect to K,/AZ. More critical is the question, whether the second Z-derivative is small in comparison to K,/AZ. For Kz = nbcuo/c we find, for the above 00 and a background refractive index tzt, = 3.56 a value of 27/.fm-’ for Kz. Since typical gain or absorption lengths are of the order of lOMm,we might keep only the term KJAZ. With these approximations the Maxwell equation is easily accessible for numerical solutions, even though the computation of the nonlinear semiconductor response is usually very involved. The situation in a waveguide is slightly different. Here, Kx = 0 and the transverse derivatives are determined by the transverse mode structure of the waveguide. If we consider only the lowest transverse mode AX would be roughly the diameter of the waveguide, if the boundary condition in X would be zero. This, however, corresponds to an infinite dielectric mismatch between the guiding and the cladding materials In a more realistic situation, the light extends well into the cladding material, and the transverse derivative inside the guiding material is often very small. Hence, we can again neglect the transverse derivative, so that 47r s(k, t) = i 7 We can write Eqn. background group is nb = Jm. the group velocity

wf z

l- ZP(R, t).

(21) in a more common form with the help of the following delinitions The velocity is vg = l/G , & = nbwO/c, where the background refractive index We also use a coordinate frame where the time n is retarded according to vg:

q=*_Z 5 =

z.

vg

(22) (23)

(Note that in Ref. (4) the retarded time is called <.) For computational simplicity we furthermore assume that all functions depend only on one transverse direction (X), rather than on X and Y. Equation (21) then takes the form

Let us first discuss Eq!. (24) for the FWM c+se. The input field at 5 = 0 consists of two waves with wavevectors & = (iKx, 0, K,) and Kl = (-;K,, 0, K,) with a small input angle between the two beams, 9 = K,/K,:

(25) m=O

where we used the definition of Eqn. (7). To determine the angle of the output field, we assume that Kz does not change during the propagation inside the sample. Then we only have to compute the transverse phase factors (in X direction) of the output field. This, of course, can only be done on the basis of the semiconductor equations We-show in Section (2.2.) that the S-dependence of !P stems mainly from the C-dependence of T, since even in the case of

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315

quantum confined systems like multiple quantum wells we assume that the k-scale is large in comparison to the well width, so that the resulting microscopic spatial dependence does not appear in the propagation equation. Furthermore, we will show that a field of the form of Eqn. (25) yields, within the rotating wave approximation, a polarization of the form

(26) For the formal solution of Eqn. (24),

we need to know the $-dependence

of L? at 2 > 0. Iterating Eqn. (27) we find in first order,

=i47-++[X,~(5=o),t71

+ !&X,~=O,rj).

(28)

According to Eqn. (26) the output field contains in this order all scattering orders m f O,+l, +2, * * *. In principle, one could iterate Eqn. (27) further. However, in the next order, P would have to be evaluated in the presence of the source field Eqn. (28), which already contains all scattering orders so that no additional scattering orders are generated. Since the new source field has a <-dependent phase the angle of any given scattering order 0, = mKJK, becomes Kz dependent. To avoid this problem we assume that the sample length L is sufficiently small so that the propagation induced phase error_ can_be neglected. For later reference, we call the m = 0 (i.e. K = 6) direction the pump direction, m = 1 (K = K,) the probepire$ion, m = - 1 (i.e. k = 2& - Kt) the F WM or photon echo direction, and m = 2 (K = 2K,-&) the phase conjugate direction. Except for the pump and probe directions, the fields are solely caused by the nonlinear response of the semiconductor. We will see that the investigation of the various field directions yields detailed information about coherent many-body effects, because the many-body effects influence different directions differently. In the probe direction, it is meaningful to define the differential transmission T(r

o)

I

,

=

Ihh.a2 - I~lK = 09m)12 I%- = O,co)P

-

(29)

Using only the first-order field, Eqn. (28), we have to neglect the terms - e and find

The corresponding absorption coefficient a for the intensity I(C, o) = 12, (c, w) I2 is given by o( = -T, /?& if we assume Z = Zae-“c = Zo(l - ac). Of course, a depends on all pulse parameters, in particular on the delay time fd between the first and second pulses Another numerically feasible way to study propagation dynamics in a FWM scenario is to simply use the transverse expansions for the E-field and the polarization (25) and (26), respectively, in the propagation equation Eqn. (24). Now, however, we use these expansions

R. Binder and S. W. Koch

316

for every point in the Z-direction, not only at 2 = 0. We then have immediately the following propagation equations for the individual modes: (31) Finally, let us go back to Eqn. (24) and discuss the application of this equation to waveguides In the simplest approximation, one can assume that the E-field is homogeneous in the transverse direction. We therefore can simply integrate numerically the g-dependence for arbitrary propagation lengths, starting at < = 0 with the appropriate boundary conditions for the incoming pulse. In a more realistic approach to an index guided waveguide, where the nonlinear medium is confined to a small region in transverse direction, whereas the transverse E-Field mode extends according the guiding efficiency more or less into the outside region, one can average Eqn. (24) with the transverse mode function of the E-Field. This leaves the formal structure of Eqn. (24) essentially unchanged. It only introduces a light field wing fattor in front of the polarization. That describes the fact that only a certain fraction of the field is modified by the semiconductor properties contained in !P. 2.2. Hartree-Fock

Theory

The basic Hamiltonian of the electrons which are moving in an external lattice potential and which are subjected to an external classical electric field is H =

s +;

d&+($t)

&‘t)q(x’t)

s

dx’dx”cCl+(x’t)~+(x”t)Y(i’

- i”)cy(&“t)cy(&‘t).

(32)

Here we define the combination of spatial and spin variables as p = {?, s}, which should not be confused with the component X of the space vector k Correspondingjy, the sum over spatial and spin variables is J & = 2, J d3r. The one-particle Hamiltonian h contains the external light field, described by the potential (A, a), and the potential of the ionic lattice (UL):

where ma and e0 are the mass and charge on an electron in vacuum. At this point we formulate the theory in terms of real-space field operators w(g), because later we will use various models to describe the nonexcited semiconductor and each model suggests its own expansion of W(xt) into a complete set of functions p,,(i). An often convenient choice is to use as expansion set the eigenfuntions of the one-particle Hamiltonian (Eqn. (33)) in the absence of the light field (A, ‘P). The corresponding one-particle Schriidinger equation is

= E#“(iS)

.

(34)

The one-particle quantum numbers n contain the wavevector &, the Bloch-band index v, and in quantum cot&red systems also the subband index 1. Hem we have formally taken into account all Coulomb effects by means of the selfenergy g which describes the nonexcited semiconductor. Of course, all these zero-excitation effects am included in the Hamiltonian, Eqn. (32).

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317

A very pragmatic approach to separate the zero-excitation effects from the changes caused by the optical excitation is discussed in Ref. (126). We adopt this approach here. We first derive the equations of motion describing the optical excitation, and in the end subtract all energy renormalizations which are left in the zero-excitation limit, because they should already be included in the E”. We will come back to this point in Eqn. (75). We also note that the full solution of Eqn. (34) is practically never needed. In most cases, all the information relevant to optical experiments can be reduced to a few parameters, such as the effective masses and optical dipole moments The field operators are given in the Heisenberg picture, and obey the equation of motion

=

J

h(g)q/(&l)+ &‘V(i

-ih-$+(xl)

- i”)cC/+(x’r)~(x’t)cCl(xz)

(35)

= [H, w+(xt)l(36)

where the commutation

relations

r4dxt>, w+(x’tH+ = m-x’,

bP+w,

w+(x't)l+

=0

[W(Xl), W(x’t)l+ = 0

(37)

have been used. As discussed in Section (2.1.), the quantity that gives measurable information in an experiment in which the dipole-approximation is valid is the dipole density P(t) = &C-e,)

G) = -$ (-eO) J&F

(4~~(xt)Np)).

(38)

In Eqn. (38) the expectation value (I&~(xt)@(&)) is the diagonal element of the reduced oneparticle density matrix obtained by using the density operator of the system before the light field is coupled to the semiconductor. Technically, we assume that there is a time to at which the initial conditions of Eqns (35) and (36) are given. In Eqn. (38) we have left out the spatial dependence, because our quantum mechanical treatment assumes spatial homogeneity, which is justified by the large difference between the lattice constant and the optical wavelength. This length difference permits the assumption that, on a length scale where the light field is constant, the semiconductor properties are almost those of an infinite homogeneous sample. In fact, there are good reasons to believe that as little as 10 unit cells are sufficient to justify the concept of effective masses Although on this mesoscopic length scale the light field can be assumed to be constant, the variation of the light field on a scale comparable to the optical wavelenght gives rise to important observable effects, as discussed in Section (2.1.). To incorparate these effects into the quantum theory of the crystal, we will allow a parametric space dependence of the dipole-density (38) via its dependence on the potential (A, @) in the Hamiltonian (33). In this case of a spatially quasihomogeneous field, we replace 2(7,t) -1&J), which makes it possible to choose the gauge

(39)

R. Binder and S. W. Koch

318

Ti(T,t) =o

w

+((i,t)=eoj:~E(k,l).

(41)

The parametric k-dependence of the polarization density is now solely a consequence of the appeareance of i?(k, t) in the equations of motion. In order to derive the equation of motion for 3 within a Hart=-Fock decoupling scheme, we present in this section the basic steps in a way which might seem unnecessarily complicated. In Section (2.6.), however, we will extend the formalism and there it will prove convenient to keep the theory somewhat more flexible. The expression for the polarization density (38) contains the particle density

An equation of motion is readily obtained by differentiating t. Using Eqns (35) and (36) we obtain

(qt GLV~>~(%)) with respect to

x (cl/+(~f)~+(Xjt)(II(_Xjt)((l(X,t))

(43)

where, again, x = 3, s. Because the polarization density (38) contains only matrix elements which are diagonal in x, one could try and evaluate Eqn. (43) at xl = z2 = x. That, however, eliminates the Coulomb contribution and yields, as it should be,(‘22) only the continuity equation: $(il)

= -Gj(?t).

(44)

To obtain this result we have used Eqn. (42),

(i(6)> = -&

=--

ih

2m0

((vJ+GI) [a,m,],- p,ul+(i,)] l/m)), (91

-

$2)

(*+(?22)(Cl(7,))

Ihci,

(45)

and summed Eqn. (43) over the spin index s. Before we discuss the proper treatmeant of the equation of motion for the polarization density, we first indroduce our basic approximation scheme for the case ii + ?z. In this case, Eqn. (43) couples two-operator expectation values to four-operator expectation values which describe higher quantum correlations influencing the one-particle expectation values To approximate the four-operator expectation values in terms of two-operator expectation values, we adopt the decoupling (factorization) scheme

This factorization scheme is formally equivalent to the Hartree-Fock approximation. However, in general the wavefunction will be quite different from the Hartree-Fock ground state wavefunction which is the only wavefunction for which Eqn. (46) would be rigorous (equal

Nonequilibrium semiconductor dynamics

319

sign instead of approximately equal). One should, therefore, regard the approximation (46) merely as the simplest possible approximation yielding a closed set of equations for the twooperator expectation values A quantitative test of the quality of this approximation can only be achieved by going beyond this Hat-tree-Fock like decoupling. In Section (2.7.), we will discuss the simplest extension of the Hat-tree-Fock decorrelation, the Screened-Hartree-Fock (SHF) approximation. Using the Hartree-Fock like decoupling scheme, Eqn. (43) becomes

ih%(rl/+&t)(I/(&,t))

=

{jlH(i,t)-hHG2t)3 (W+(&-t)W(Zlt)) dg3[VG, - h) -

-

v(F2

-

i3)l

I x


w+

(X3W(X,G)

,

(47)

where the Hartree Hamiltonian contains, in addition to the field of the ions, the static electric field of the electrons UH:

jrH(i,t)

= -g 0

+ Us

+ Us

+ coil - ii(kt)

(48)

with uH(;,

t)

=

=

=

ds VG, -

i3)

(W+ (x3tW+

s

(X3t))

(49)

d3r3 V(i, - T3)2 (Wt (x3t)Wt (X3t)) s3

(50)

d3r3 V(31 - T3) n(&t) .

(51)

Generally, the lattice potential U. cancels to a great extend the electronic or Harttee potential, and within the so-called jellium model, &(‘r) = constant, one can expect the cancellation to be exact if the system is spatially homogenous Concerning the optical excitation, this requires homogeneity of the electric field E, which of course is only true in the infinite wavelength limit. Nevertheless there are situtations where the macroscopic field, UL+ UH, is less important than the short-range correlation described by the explicit Coulomb term in Eqn. (47). On the other hand, even a system which is spatially homogenous with respect to measurable quantities like the polarization density will in general exhibit density fluctuations, which predominantly affect the field UH. This will be the basis of our approach to describe Coulomb screening in Section (2.6.). Generally, screening arises as a consequence of carrier correlations due to carrier-carrier scattering, and in this section we assume that on a fs-timescale we can neglect these processes Let us come back to the question of the limit ?i = ‘;2 in the equation of motion (43) or (47). To describe the underlying microscopic aspects of the polarization density we have to note that the x-integration in Eqn. (38) is over the dipole density and not just the density. The factor ‘: gives a weighting to different contributions from the integral. This can be discussed more easily, when the field operators are expanded in terms of the eigenfuction qn defined by Eqn. (34). If the Hat-tree field UH is time independent, it is advantagous to replace UL by UL + UH in Eqn. (34). In the following, we will make this assumption, so that U,q no longer occurs explicitly. Then, using tp(xt)

= ~an(t)v,(x), n

(52)

R. Binder and S. W. Koch

320

Eqn. (47) becomes

Equation (55) defines the electronic dipole matrix elemen_t ii,,,,,. Again, the quantum number n contains the Bloch-band index Y, the momentum vector k and, in quantum confined systems, the subband label 8. We now expand Eqn. (47) with the help of Eqns (52) and (53) and project out the coefficients

Here the matrix element of the Coulomb potential is

It is convenient to introduce at this point the Fourier transform of the Coulomb potential with respect to the direction of free carrier motion. In a bulk semiconductor we have a 3dimensional wave vector i and V(x,y,z)

=

1

e-i;--r 47re2 7.

4 In order to account for dielectric screening of the Coulomb potential in the absence of free charge carriers, we use the effective charge e = eO/ &, where G is the low-frequency dielectric constant of the unexcited sample. For GaAs caculations, we use eO = 12.7. In a quantum well the wave vector is Zdimensional, + = (qx, q,), and V(x,y,z)

=

ewi(qXx + W)

1

27Te2 e-4l4

4

.

4

In a quantum wire the wave vector is l-dimensional,

0 = qx, and the Coulomb potential is

V(x,JJ,z) = c e-@xX 2ezKo(f&Z), 9

(60)

with the modified Bessel function ko. Note, that 6 is in principle not restricted to the first Brillouin zone and that

c=+-ddq (2rr)d 4

(61)

Nonequilibrium semiconductor dynamics

321

without the additional factor Ld in front of the d-dimensional integral (d = 3 in bulk, 2 in quantum wells, and 1 in quantum wires). In spatially homogenoues systems and in thin quantum confined systems where only one subband has to be taken into account, the matrix elements of V, Eqn. (57), can be approximated more easily when the representations (58), (59) or (60) am used. In particular, we have in a 3D-system (the analysis is formally similar in quantum confined systems): Vn,n*njQ= 1

Vq h I .A

=7

In3)

b41

e@.

‘:

In2)

9

(62)

4

where the Bloch-matrix elements (n,l e-i+ - F In2) =

I

dxd,

(LC)e+

* ’ vnz (xl

(63)

have been introduced. Before deriving in Section (2.3.) the SBE for more general cases including band-coupling effects, we will restrict ourselves in the following to the simplest approximation that allows an explicit evaluation of the matrix elements of I/ and p. If * .r; . ; V”(S) = cP&,s) = u,@) eik - F = zf&#,S) d , (64) where u,,.J(~, s) is the lattice periodic part of the Bloch function, it can be shown easily that (nl I e-‘b . i

In2)

= 6,,,

6&+p-&$.

(65)

(A more quantitative statement for the validity of Eqn. (65) is given by Eqn. (524) in Appendix B.) In the following we always assume that the reciprocal lattice vector G is zero. An investigation beyond the G = 0 approximation has been performed by Kuklinski and Mukamel.(‘2’) The resulting dipole-dipole interaction between unit-cells is similar to the corresponding interaction in Frenkel exciton systems(’ 38)or the dipoledipole interaction between quantum-wells in multiple-quantum-well systems (13g~140) The Coulomb matrix element is now given by Vn,n2ngi.q = V&

-

$3)

6&&,&&

&,,,&,,.

(66)

Here, VGj) = V, is defined by Eqn. (58),

Again, one can similarly treat quantum confined structures in the limit of a single subband, if one takes the Fourier transforms according to Eqn. (59) or Eqn. (60), respectively. The approximation (64) can be called “effective mass approximation”, because the momentum matrix element is essentially the carrier momentum, which similar to the case of free particles: (nl I (-ifi@

In2)

= 6gkz

{&,,hl;~

+ jYIY)

(68)

where jyy is defined in Eqn. (488). Since we are more interested in the dipole than in the momentum matrix elements, we now use approximation (64) and (524) for the evaluation of h

I(-e,Pln2)

=

t2M3

ieo L3 a&, ai;z

-

i2)6 WY + %,lcJc,~ -

Here, we have formally replaced 6k,k2 by (2rr)3/L3S(&i - &) (assuming that the momentum states form a continuous set) to stress the similarity of the intraband dipole moment with that

R. Binder and S. W. Koch

322

of free particles and to be cble to replace i by the k-derivative. Of course, this intraband dipole moment for the different k-states does not lead to an nonzero dc-polarizability, because the latter is obtained from the average taken with the corresponding detribution functions If the distribution functions have the inversion symmetry, j”(k) = j”(-k), the corresponding total intraband polarization is zero. For optical purposes more important is the second contribution to Eqn. (69), which couples conduction and valence bands, fiN and PVC.To compute the optical polarization P, we need only consider the cv and vc contributions to Eqn. (55), (70) because only those terms oscillate with the optical frequency [i.e. 0~ exp(+iwor)]. In Eqn. (70) we have introduced an additional (phenomenological) k-dependence of the optical dipole matrix element fro. This is beyond the effective mass approximation (69), which would lead to artificial divergencies in the k-sum of Eqn. (70). Later, we come back to this point [see Eqn. (96)l. Using Eqns (65) and (69) in Eqn. (62) we have

+ (a+ ,,j&av& ) ie3,,(kz,“] -

c

v,

[ (a~,$&&)

(a;,J3+ij%,,#c,+i)

&;4 -

(a~,4+pv3.~3+$

(a~,p,,,i,)

] -

(71)

The initial conditions of Eqn. (71) are (ak,k+,,k,)

= G,,

(72)

a~,,k~G,At)

where j&,&k) is the initial carrier distribution function of band v. Specifically, in a noninverted semiconductor all conduction band distribution functions are initially zero, and all valence band distribution functions are initially one. Because both the transition matrix element and the initial conditions are diagonal in k, the solution of Eqn. (71) is also diagonal in k. Within the two-band model, the Bloch indices are restricted to v = {v, c}. Mom precicely, we have two identical and uncoupled sets of Bloch equations for v = {v t, c t } and v = (v 1, c 11, mspectively, where the two spin functions are denoted by 1t , 11. The dipole matrix element is diagonal in the spin indices Introducing the optical polarization function P(k) = (&z,,k)

(73)

and the distribution functions (74) we obtain from Eqn. (71) with vi = c, vz = v (with the same spin), and &I =g=g ihiP

= (4 - ~1) P(&) - G:(k)

- & {f(k)

- fe(k))

Nonequilibrium semiconductor dynamics

323

and, with vi = c, v2 = c,

i&f(k) =-ji$k) kP*(k) P(k)/&,(i) E* - c b [p”(W(~ + i, - P* (5 + $P(k)] . ’

+

’

4

In Eqn. (76) the f”f” terms have cancelled out. Also, since we restrict ourselves here to the fully coherent approximation, where (77) As discussed afier Eqn. (34), we still have to investigate Eqn. (75) in the low excitation limit, fc(k) --, 0, P(k) * 1, to make sure that all explicit selfenergies vanish in this lim$. We see immediately that in this limit the transition energy (i.e. the prefactor in front of P(k)) is

However, within the parabolic band approximation,

the one-particle energies are (78) (79)

These energies contain already all zero-density correlation effects, and therefore we have to substract the term 1, Vqfrom the right-hand-side of Eqn. (75). Since even in the zero-density limit one-particle states are not eigenstates of the system, we introduce now a phenomenological zero-density dephasing rate y”. It describes incoherent scattering effects in the low density limit, mainly due to carrier-phonon and impurity scattering. Since dephasing results form scattering of valence band or conduction band states, we formally replace 9 = E: - ~1 by Ek - iy’. In order to stress the formal similarity of Eqns (75) and (76) with the atomic Bloch equations, one can introduce the renormalized dipole energyu3) M(k) Then, with f(k)

= fi$(k) * k + 2 V(k - &)P(k’) . k

(80)

= 1 - f*(k), and the excitation induced selfenergy Z$ = -

1 V(k’- i)f”(k’) ,

031)

(a = e, h) ,

k

we obtain the final equations of motion ( Ek + 2; + i$ - iy”) P(k) -[l -P(k)

- f”Wl

hwd

,

(82)

R. Binder and S. W. Koch

324

$f”m

= 2 Im [a* (&P(k)]

.

(83)

The two-band semiconductor Bloch equations (SBE) ate the basis of our investigations of coherent effects in semiconductors These equations yield information about the linear and nonlinear Coulomb interaction effects on optical signals The fundamental differences between the SBE and the atomic Bloch equations are twofold. First, in a semiconductor the spectrum consists of discrete resonances and a continuum part, which has the lower bound E* and exhibits an unstructured, more or less constant absorption on the high-energy side of E,. This general behavior is basically the same in one, two, and three dimensions, although the respective free-particle density of states is proportional to JET, constant, or 1IJEE, respectively. Typical zero-density absorption spectra for the spectral range around the bandgap are shown in Fig. 1. The binding energy of the ground-state exciton increases with decreasing dimension, and the absorption at the bandedge is nonzero and finite, in contrast to the free-particle denstity of states in three- and one-dimensional systems In later sections we investigate the consequences of the mixed spectrum and compare it to atomic cases, where either discrete lines prevail (homogeneous atomic sytems) or a rather structureless continuum of states dominates (inhomogeneously broadened systems). Another major difference between semiconductors and atomic systems are the so-called exchange effects In Eqns (82) and (83) we call all nonlinear terms, i.e. all products of the form fP and P* P exchange terms. Physically, these terms represent field induced band-gap shtinkage (0~ C Vfl and interaction-induced light field renormalization (0~ C VP). The notation to label these terms as exchange terms stems from the Green’s function approach (see Section (2.6.)), where all Hartree contributions are called direct interaction (meaning the direct interaction with the macroscopic mean field), whereas the exchange Hartree-Fock diagrams give rise to all Coulomb contributions in Eqns (82) and (83). With this definition, even the linear interaction is included, which leads to the binding of the electron-hole pairs Note, however, that many publications dealing with exciton theory deviate from our nomenclature since there the linear exciton binding potential and the Hartree terms are denoted as exchange interaction. In this paper, we are not dealing with spatially inhomogeneous systems or spatial polariton effects (effects due to the nonzero wavevector of the E-field). We can therefore unambigously label all nonlinear terms in the SBE as exchange terms The discussion of the consequences of the exchange terms a chief focus of this paper. Their temporal behavior and transient effects on spectral properties give rise to a wealth of coherent effects, which are intimately related to the semiconductor aspect of optical phenomena. An impo_rtant consequence of the excha;ge e@cts is the renormalization of the dipole energy i& - E by the Coulomb term & V(k’ - k)P(K), Eqn. (80). This interaction-induced effective field renormalization is sometimes called local field effect. In our approximation of the separation of macroscopic and mi:roscopic length scale the term ‘local field’ is appropriate, since at each macroscopic point R the Coulomb interaction changes the effective dipole energy according to Eqn. (80). However, the term should not be confused with other ‘local field corrections’, such as the well known local field effects in longitudinal E-fields in dielectric media, described by the Clausius-Mossotti equation, (134)the renormalization of the longitudinal RPA dielectric screening function by exchange and correlation effects as described, for example,‘by the Hubbard correction factor,(t4i) or the &’+ 0 terms in the inverse dielectric function of a crystal.(‘29pi4L142) In Eqns (82) and (83) we left out the contributions from the intraband dipole moments, such as &. The mason for that was that we are mainly interested in the response to external fields which is in or close to the resonance of the fundamental interband tansition. Thus, the induced optical polarization has a central frequency close to wo = E,/h, whereas the distribution functions contain essentially no temporal oscillations The coupling of the optical field via

325

Nonequilibrium semiconductor dynamics

bulk

quantum well

.,&

.,; quantum wire

-20

-16

-12

-8

ENERGY (E-Es)

-4

0

[meV]

Fig. 1. Linear exciton absorption spectra for GaAs systems. The parameters for the two-band model are the background dielectric constant, e,, = 12.7, the electron mass, m, = 0.067~ .(mo = free electron mass), and, in the bulk case, average hole mass mh = 0.193~~10.In the quantum conlined cases the heavy-hole mass is mh = 0.107~. The quantum we.11 thickness is SOA and the effective quantum wire width is 3oA. Et is the bandgap energy of the unexcited system and differs for the three cases shown because of the quantum confinement shift in the quantum well and quantum wire.

intraband dipole interaction would yield, for example, terms like ficckP in the equation for P. However, these terms have central frequencies around 00 + wg. Therefore, they are strongly detuned from the quantity P and can be neglected. An additional consequence of these frequency arguments is the validity of the so-called rotating wave approximation, which will be performed in Section (2.4.). This simple situation of the separability of quantities with different center frequencies has to be modified if multiphoton processes become important. Especially, this is the case if the optical center frequency to optical is much less than the bandgap, )Iwg = Eg/2. For a discussion of the &-contribution process and, more general, two-photon processes see, for example, Refs (143-150). Another situation in which the frequency-separation still works, but which requires the intraband dipole matrix elements to be taken into account, is the case of an applied dc-electric field. In particular, multiple quantum wells subject to dc-fields are of current interest. Since

R. Binder and S. W. Koch

326

the main part of this paper does not deal with d&field effects, we will outline only briefly how they can easily be incorporated into the present formalism. In Section (3.6.) we will discuss examples of electric field effects, whereas in all other parts of Section (3.) no dc-field will be considered. We use the term dc-field to distinguish between optical fields and those, which are either time-independent (i.e. are strictly dofields) or oscillate at low frequencies WI such that WI < WC). In order to amend the SBE with a dc-field contribution, we only have to assume that the field in Eqn. (41) contains, in addition to the applied_optic$ fie!d, a dc-field Edc. This yields immediately the following additional contribution to ki = kz = k contribution of Eqn. (71):

- (a+ Y,kav,.I)

(~v2k%‘rh>mid,].

(84)

Here, we have already used the fact that the intraband dipole matrix element is diagonal in the Bloch band indices and that the contributions proportional to the interband matrix element will yield terms which do not oscillate at the frequency of (a&a,,> . We can now use the explicit form of the matrix element to evaluate Eqn. (84). Since the matrix element is essentially the derivative-of a delta-function, see Eqn. (69), we can easily integrate by parts and evaluate the resulting K-integration. We then i-e-write the two contributions to Eqn. (84) such that they result from the product rule applied to the following differentiation:

1

ihaat (a+y,&,,d dc =

i’JLc’ & (a~,~%~$ )1

(85)

k=k.

In particular, we obtain the familiar drift term in the equation for the distribution function

&f’(6)=2 Im [a*(-k)p(-$11+ eo&jc &fh($)1

(87)

’

,@&

and the corresponding

term in the equation for the optical polarization

*

_

+

. -

-11 - f(k) - f”(k)1 n(k) + &&ic -

-5(k) k=k ak

-

(88) In Eqn. (87) we have used the transformation j”(i) = 1 - fh(-&.Also, the functions depend now on k, not only on k (as in Eqns (82) and (83)), since &c is in general anisotropic. The dc-field contributions show that, in contrast to a spatially homogeneous light field, the spatially homogenoues dc-field couples to expectation values which describe spatial inhomgeneities, (aLai) with 6 + k. However, this induced inhomogeneity is infktitesimally small, since only the derivative with respect to i is needed. Loosely speaking, one could say that the d&field couples only neatest neighbors in momentum space. The reason why the optical field can couple to the semiconductor without coupling various momentum states is that jiN requires the implicit spatial inhomogeneity which stems from the fact that, for example, s and p orbitals are different.

Nonequilibrium semiconductordynamics

327

2.3. Band-Coupling Efects Although Eqns (82) and (83) contain already the most important spectral and nonlinear exchange properties of a semiconductor, in real systems, notably in III-V compounds, further complications arise from band coupling effects within the manifold valence bands A practical approach to include such effects into the theory of optical nonlinearities is based on the Luttinger-Kane theory, which is outlined briefely in Appendix B. There we discuss the Luttinger Hamiltonian 3f- for III-V compounds, Eqn. (498), the equation for the eigenfunctions of 3f, Eqn. (486), and the extension of the Luttinger theory to quantum confined systems by means of the envelope function approach. The central approximation made above for deriving the SBE without band-coupling was the effective mass approximation, Eqns (65) and (64). This approximation yields a diagonal optical dipole-moment matrix, Eqn. (69), and the Bloch-band dependence 6,, ,n 6,,, for the Coulomb matrix element, Eqn. (62). These relations are generalized to include band-coupling in Eqns (522) and (531) for bulk systems and in Eqns (525) and (533) for quantum wells At this point, we could directly insert Eqns (522) and (53 1) into equation Eqn. (56) and extract the equations for the relevant polarization and distribution functions To evaluate the resulting set of equations, we need to know the eigenvectors of the Luttinger Hamiltonian. Although in the bulk case the eigenvectors can indeed be found analytically, and in the quantum well case an analytical derivation of the eigenvectors is possible if only one subband is taken into account, we proceed at this point differently and eliminate the eigenvectors from the equations of motion for the polarization and the distribution functions Instead of the eigenvectors, the Luttinger Hamiltonian itself will appear in the equations This procedure allows a simpler analysis of the symmetry properties of the equations of motion. In the following we outline the elimination of the Luttinger eigenfunction matrix W for the bulk case. For quantum confined systems the steps are formally identical if only one subband is taken into account. The main step is a basis transformation of the one-particle eigenfunction. As is apparent from Eqn. (56), we have chosen one-particle states which diagonalize the oneparticle Hamiltonian. We now define a new basis by means of the unitary transformation (89) (90) We insert Eqns (89) and (90) in Eqn. (56) and operate from the left with

After using the orthormality properties of the unitary matrix W, we obtain from the eigenenergy terms on the rhs contributions of the form (91) However, from the defining Eqn. (486) we have immediately (92) Y

allowing us to eliminate the_eigenvectors of ti in favor of 3f itself. Since the dipole-matrix element (522) is diagonal in k and initially the system is spatially homogeneous, all expectation values remain of the form

R. Binder and S W. Koch

328

b&a~p

) = Q,k

(a;pgp)

(93)

for all times The same holds for the the expectation values of (c+c) . It is now straightforward to derive the following equation of motion:

Here we used the abbreviation (95) Equation (94) is the general Hartree-Fock equation of motion including band coupling within the so-called k - j-theory. We want to stress at this point, that the k-dependence of the optical matrix element within this theory is restricted to wavevectors close to the zone center. Therefore, it is possible that in Eqn. (94) the matrix element is fully k-independent, which is a consequence of the special choice of basis functions It is well known, however, that the k-dependence of ficv is necessary to prevent a divergence in the calculation of the linear refractive index. In writing Eqn. (94) with a k-independent matrix ele_ment, we tacitly assume that in reality, i.e. in calculations beyo_nd second order k . j-theory, a k-dependence of fi, will appear that indeed removes the large-k divergency. Concerning optical signals, especially from polarized pump and probe pulses, the important aspects of the Luttinge_r theory are mainly the symmetry aspects of, e.g. Eqn. (94). These involve indeed only small ks, which are on the order of ,a; ‘. Equation (94) covers these symmetry properties correctly. The large-k cut-off can formally be prevented by replacing -0 CICV- P:(k)

.

(96)

In numerical calculations it is convenient to choose this k-dependence as fig(k)=

fi: for k < k,,, 0 fork>k,,,,

(97)

where typically the maximum k-value, k,,,, is chose? on the order of 20aj’. A_though the Coulomb interaction can, in principle, couple low k-states with arbitrary high k-states, the numerical solutions of the SBE show that it is sufficient to use k,,, such that, for all times, the quantity k;jYk,,,) is small [e.g. 10m4times smaller than the maximum value of the function k2fa(k)]. Clearly, the choice of k,,, depends on both the excitation and the initial conditions If, for example, and initially noninverted semiconductor is excited resonantly at the Is-exciton resonance, km is determined by the shape of the Is-exciton wave function. If the initial conditions are those of an optical amplifier, k,,, is determined by the initial Fermi distribution of the lighter species of charge carriers, The matrix elements fi,$ for the hh and lh transitions are shown in Fig. 2 [the states are discussed in Appendix B]. A considerable simplification arises if the light field vector has only two vector components, e.g. Ex and E,. In this case the inner product j&E does not allow

Nonequilibrium semiconductor dynamics

j=-lL2

329

j=+lt2

Fig. 2. Schematic of the dipole transitions at the Brillioun zone center in cubic semiconductors with large spin-orbit splitting of the valence band. The three basis vectors jI1 (i = x,y, z) of the optical dipole moment fi I fi&(k = 0) are of equal length, which is (sle,xlx) , where (sl is the orbital s-like Bloch wave function of the conduction band minimum p and Ix) is one of the three p-like Bloch wave functions of the valence band maximum p.

a coupling between the two subsystems {s = l/2, j = +3/2, j = -l/2} and {S = -l/2, j = -3 /2, j = + l/2}. For this case we evaluate now Eqn. (94) explicitly. The band indices v become s and j for the conduction and valence band, respectively. We also perform the transformation from the valence band to the hole band description according to K,,(i;) j&,(6

= -H$,

6-i) ,

= 6”,“, - &,

C-k).

(98) (99)

Throughout this paper we keep the original valence-band indices even after the transformation from the conduction-band-valence-band description to the electron-hole picture. In particular, we label the hole distribution functions with the valence-band-state angular momenta j. The corresponding hole angular momentum would be -j (see, for example, Ref. (128)). We defer a more detailed discussion of the electron-hole picture to Section (2.8.). From Eqn. (94) we obtain (see also Ref. (27))

R. Binder and S. W. Koch

330

-

1 V(k - K,&,(K) K

+

1 V(k - i, fi(i)PsJ(k) + ~~J’&P&‘) k

I'

[

h&$?(&) = 2 Im 1,

jij,(k) * k + 1 V(K - &)P$,(k) [

P

* 1

I

p,/(i)

I

,

W)

(101)

and

+

1 V(k - &)[Ps;‘(-KE-,(-i>-

P;‘(-ip&‘~]

k,s -

1 V(K - 5, [j-j,”(C)j-j!‘,’(Ii, - fj,” (k,$‘,’

(k,] .

P,,”

(102) Here, we assumed again that the term & V, is accounted for in the (experimentally measured) hole energies, which enter 3fh, and we explicitly included again the low-density dephasing constant tip. 2.4. Appkation

to Two-Pulse and FWM Configuration

If the external light field consists of more than one light pulse, the nonlinear coupling of the light with the semiconductor might lead to the formation of density and polarization gratings inside the material. These gratings are the origin of the so-called multi-wave mixing phenomena. In this section we evaluate the theory for the case of two light pulses We first consider a two-band semiconductor and at the end of this section we extend the analysis to deal with a quantum-well structure, including band-mixing effects within the Luttinger theory. In a two-pulse experiment with noncopropagating pulses we have (compare Eqn. (7)) (103) We arc labeling the pump pulse by m = 0 and the probe by m = 1, respectively. Inserting Eqn. (103) into Eqns (82) and (83) we see that the parametric dependence of the functions P and f on k is of the general form

where m,m’ are integer numbers and F is either f or P. A significant simplifkation arises, however, if one works in the rotating wave approximation @WA). This means that only resonant contributions of Eqn. (103) in Eqns (82) and (83) are taken into account:

Nonequilibrium semiconductor dynamics

331

(104) The RWA leaves Eqns (82) and (83) formally unchanged if we replace P by its slowly varying amplitude P, p&; A, t) = fp(i;; k, t) $($

* k - wet) ,

(105)

the laser field by

(106) and the transition energy by Ek+Ek--hWg.

(107)

In this case, the only possible spatial Fourier components of f, Eqn. (83), are those with dzfirerencesof wavevectors, leading to a simple density grating structure. This can be seen easily by constructing an iterative solution of Eqns (82) and (83). Because light scattering off these density gratings creates the spatial Fourier components of P, Eqn. (82), the only possible components here are again wavevector differences, centered around either & or & : P(&;k,t)

=

eimtL f WI=--00

- &I

. k pgrn,

I t)

WW

Here, P(O) is the polarization traveling in pump direction, P(1) is traveling in probe direction, ?(--1) is going in the photon echo or FWM direction 2& - &, and P(2) in the conjugate direction 2& - &. Only P(0) and P(1) have source terms in Eqn. (82). The contributions in all other directions are created by the scattering of ‘P(O) or P(1) off the density grating f(+l),f(+2), * * -. If the pump and probe pulses are very different in strength, then fo, which has no spatial dependence and thus does not act as a grating, is dominated by the pump pulse. The important aspect of the spatial Fourier transformation in the thin film limit is the X-dependence of the phase. The discussion in this section, where we write, for clarity, the full wavevector dependence, reduces trivially to the discussion in Section (2.1.) where only the case 2 = 0 was considered. The back-transformation of Eqns (108) and (109) is

(110)

(111) 0

where 4 = (& - &J - k and f(m) = f* C-m). In order not to over-complicate the notation, we set h = 1 and omit now in the spatial Fourier transform of Eqns (82) and (83) the carrier

R. Biidcr and S. W. Koch

332

momentum variable i and its integrations, which remain the same as in Eqns (82) and (83). The dependence on the light scattering order m is i$P(m)= (E-$)P(m)

- 1 C[Vf(m-m')]P(m') a=a,h m’

-Cl(m)+ 1 Cf"(m-m')&

(112)

a=c,h m’

#f(m)

= C Cl* (m’ - m)!P(m’)

- fl(m’

+ m)P*(m’),

(113)

m’ where * O(m) =jim+

+ [W(m)]

(114)

and [Vf ] and [VP] indicates the k-integration according to Eqn. (82) and Eqn. (83), mspectively. Because the creation of higher order polarization components !P(m)(with m + 0,l)is a consequence of the light scattering, the set of equa_tions(112) and (113) converges in general rapidly with increasing m. Specifically,if the pulse Et is very weak, one can restrict Eqns (112) and (113) to m = 0, SC1. The result is similar to a linearization procedure(‘51)of Eqns (82) and (83), except for not very important contributions of the form f (-1)&W) in the P(O) equation. Another case of practical importance is realized when the pulses are of similar intensity. To describe the lowest order light scattering processes in this case, one has to include the m = 0,+l and +2 contributions For completeness, we write out the full set of equations for this configuration (all functions depend, in addition to the m-dependence given, on I and k, and, in principle, on the macroscopic spatial coordinate 2):

-NO) [l-2f(O)] +Z(-1) P(1) + 2f(-1) n(1) + Z(1) !P(-1) + 2f(1) n(-1) + Z(-2) P(2) + 2f(-2)n(2) , @(l)

(115)

= {E - iy* + 2(O)] P(1) - a(l) [l - 2fCO)] +uuno) + 2f(l)nco, + Z(2) P(-1) + 2f(2)n(-1) +X(-l)

+Y-1)

P(2) + 2f(-1) n(2) ,

(116)

= {Q - iy* + Z(O)] P(-1) -U-l) [1-2f(O)] + 2(-l) P(O) + 2f(-1) ha(O) + U-2) !P(l) + 2f(-2)n(l),

(117)

333

Nonequilibrium semiconductor dynamics

= (Q - iyo + Z(O)} !P(2)

@J(2)

- a(2) [l - WO)] + E(2) P’(O) + 2-02) a(o) + Z(l) ‘p(1) + 2.01) n(l)

iif

=

nco,P*(o)

R*(O) !P’(O) -

+ n*(l)

(118)

,

!P(l) - n(l) P*(l)

+0*(--l)

P(-1)

- n(-1)

!P*(-1)

+ a* (2) P(2) - n(2) P* (2) I

iif

=0*(-l)

P(0)

+ n*(o) P(1) + n*(-2)

(119)

- Q(l) P*(o) - .Q(2) P*(l)

!P(-1)

- n(0) F*(-l)

(120)

+ n* (1) P(2) , and Q(2)

= a*(-2)!P(o)

- L2(2)!P* (0)

+ a*(-l)!P(l)

- n(l)P*(-1)

+ n* (O)P(2) .

(121)

Here, we assume the electron and hole distributions to be exactly equal. The more general case of unequal distributions can be obtained simply by replacing 2f(k; m) by f’(k m) + r” (k m). The selfenergy is defined as C(k; m) = - 1 V(k - &, 2f(k’; m) ,

(122)

k

and the generalized Babi frequency is, as before, O(k;m) = ifiN * !Em+ 1 V(k

- k, !P(K;m)

.

(123)

k

As a consequence of the exchange terms, the SBE define a set of nonlinear equations, the solution of which usually requites numerical treatment with supporting analytical considerations for special limiting cases or simplified models The temporal behavior of the exchange terms is, in general, different from the temporal shape of the light pulses To discuss qualitatively the duration of the interaction induced nonlinearities, one has to distinguish the case of resonant absorption, where the pulses have spectral overlap with the linear or nonlinear absorption, from the so-called adiabatic following (nonresonant) regime. Adiabatic following occurs for semiconductor states which have no absorptive overlap with the exciting pulse spectrum. It is characterized by the temporal test& tion of the field-induced polarization and inversion to those times when the pulse is present. In the resonant case, the exchange effects remain present when the pulse is gone. The time scale for the self-energy renormalization is 7’1,which, for our considerations, is practically infinite,

334

R. Binderand S. W. Koch

and the polarization-induced field renormalization decays on a time scale TZ which depends strongly on the details of the system. Of course, the Coulomb interaction affects different scattering orders !?Yk;m) differently, and we will discuss in the following this dependence qualitatively for resonant excitation conditions The discussion is based only on the general structure of Eqns (115)-(121) and does not require knowledge of the exact solutions The only assumption is that the two external pulses are very short pulses with the delay time &j. Let us first consider the case of a weak probe pulse (El) and a strong pump pulse (20). We can neglect all functions with m = +2. If the probe pulse comes after the pump, we assign a positive sign to the delay time &j. In typical FWM experiments, the probe beam comes before the pump, and therefore a positive delay time could be assigned to this case (see, e.g. Ref. (21)). The main source terms for the pump and probe polarization, P(k;O) and ?Yk; l), respectively, are the pump and probe fields The occurence of the Coulomb potential in R(k; 0) and Q(k, 1) affects predominantly the linear (absorption) properties, giving rise to the excitonic resonances and the Sommerfeld enhancement of the continuum absorption. Both, pump and probe polarization am affected by the pump induced band-gap reduction @(k;O)), but the influence on the probe is restricted to positive delay times The pump induced Pat&blocking term [ 1 - 2f(k; O)] affects pump and probe in a similar way. Another important aspect is the scattering of pump light at the density grating f(k; 1). This affects the evolution of the probe signal P(k; 1) and gives rise to the FWM signal !P(k; - 1). The density grating is created only when the pump-probe delay time is not large in comparison with the dephasing time. Then, for Ed> 0, the source contribution in Eqn. (120) is R(k; l)!.P* (k;O) and for rd < 0 it is R* (k; O)!P(k; 1). This argument is not affected by the interaction induced field contribution to R. Although the grating f(k; 1) is generated for positive and negative delay times even without the Coulomb contribution in Eqn. (120), the effect of the grating on both the probe and the FWM signal strongly differs with and without Coloumb interaction. The light scattering into the probe direction is described by the product of the grating f(k; 1) with the Rabi frequency n(k;O) (Eqn. (116)), and into the FWM direction it is given by f(k; -l)fl(k;O) (Eqn. (117)). Since f(k; -1) = S*(k; 1) the behavior of the scattering in both directions is essentially the same. We do not explicitly discuss the corresponding selfenergy terms Z(k; +l)!P(k;O), because the& temporal properties are similar to the corresponding induced field terms f(k; + 1) 1 V(k - b’)P(k’, 0). If the pump pulse 2(O) comes after the probe, the light sc_attering occurs after the pump pulse. If, however, the delay time is positive, the source term T(0) is already gone by the time the grating is created, i.e. by the time when the probe pulse arrives. However, the interaction induced field contribution to R&O) does not vanish after the external pulse is gone, and it can indeed produce scattering which leads ta a FWM signal even for positive delay times The effects of the scattering into the probe direction have been investigated in Ref. (117). For negative delay times they give rise to the so-called coherent oscillations in the differential transmission spectrum (see, for example, Refs (152) and (153)). The corresponding effects on FWM signals have been observed and reported in Ref. (21). Additional interference effects, which am of importance for inhomogeneously broadened transitions as well as the semiconductor continuum absorption, change the scattering behavior significantly. Such inhomogeneous systems exhibit essentially only the regular photon echo behavior, i.e. the FWM signal arises only for negative delay times and is strongly centered around one delay time after the second pulse. If_both pulses are of similar or equal intensity we cannot neglect higher than linear powers of ITi and need to consider the m = &2 terms This gives rise to a signal !P(k,2) into the conjugate direction, which is created by scattering of the probe field at the first-order density grating Cf(k; 1)Nk; 1) in Eqn. (118)) and by scattering of the pump beam at the second order

Nonequilibrium semiconductor dynamics

335

grating (J(k; 2)Nk; 0)). All source terms for the second order grating involve the first order grating. Without Coulomb interaction, the only sources for f(k; 2) which can act as a source for !?Yk;2) (i.e. which is not proportional to !P(k; 2)) is the contribution n(k; l)!P* (k; -1). This term, however, is zero if the pulse !+?Z (0) comes after pulse IE(1). (Again, we assume there is no temporal pulse overlap.) But in this case also the other sounx for P(k; 2), i.e. the term f(k; l)fi(k; l), vanishes As a consequence, for negative delay time any FWM signal in conjugate direction is solely due to the nonlinear Coulomb interactions At the end of this section we wish to specify the four-wave mixing equations for a quantumwell structure. We perform the above analysis for Eqns (100)-(102) and obtain

* +

c fi$ (k)

. 2Em’ f,,th

m - m’)

t-i;

/ -

C V(k k,mJ

- 6) [fi(k;m

+ 1 f$(-P;m i’ + kxd V(k

- m’)‘P,f(E;m’)]

-K> [fl(L;m

+ 1 f&(-K;m 1’

- m’)P&;m’)

1

- m’)!P8,(P;m’)

- m’)Psy(k;m’)]

+

m’) -ii;(k) . +j%P$(k;

+p& Y(i2)

[P$@,m’

,

(124)

1

- mP,i(Km’)

,I

- rP,J(k;m’

+ m)P$@;m’)]

,

(125)

R. Binder and S. W. Koch

336

- P$(-&;m’

-

1

V(k-

+ m)PSj(-k;m’)] P) [jj+,dK;m - m’)f;y(~;m’)

k ,/“,id

These equations will be discussed in Section (3.4.2.) where we also present numerical solutions 2.5. General Remarks on CorrelationEfects In the previous section we ignored all incoherent carrier-carrier scattering processes This is a first approximation for the description of ultrafast optical effects, where the light pulse duration is comparable, or-ideally--even less than the carrier-carrier scattering and polarization dephasing times Generally, these times depend on the details of the excitation conditions, since, e.g. the carrier scattering rates are nonlinear functions of the carrier distributions In the low density regime the dephasing times can be as long as 10 ps, and the main microscopic effect in III-V and II-VI compounds is carrier phonon scattering. Many studies on carrier-carrier scattering in highly excited semiconductors show that, depending on the details of the excitation conditions, these times can be as short as the shortest available light pulses Clearly, in such cases the completely coherent description within the Hartree-Fock approximation becomes questionable. However, even with today’s supercomputers it is a formidable task to attempt the analysis of fs-optical experiments including a fully time-dependent and nonMarkovian description of the carrier dynamics For this reason, there exists a large variety of approximation schemes for the inclusion of carrier-scattering and other incoherent processes in the semiconductor Bloch equations All of these schemes can be viewed as approximations to the full miscroscopic equations, which we are going to derive in this section. Before doing so, we summarize the most common approaches to deal with incoherent phenomena. First, one can include the phenomenological dephasing and carrier relaxation times TZ and TI , respectively. By carrier relaxation we think of all carrier scattering processes, which lead to the decay of population in a quantum state, including both interband and intraband carrier relaxation. However, since most of the effects discussed in this review occur on fs or ps time scales, we arc not concerned with optical recombination, which takes place on a ns time scale. Therefore, our most relevant TI-processes are the intraband scattering processes which drive the sytems toward a quasi-equilibrium state, which is characterized by the electron and hole Fermi functions pF (k) and fj (k). At the simplest level, the inclusion of the intraband scattering terms in the SBE amounts to the addition of

in the equations for !P and f, respectively. Although not apparent, them is an important difference between Eqns (127) a_nd(128) and the cormpsoning atomic Ti and Tz: Eqn. (128) leads to a coupling between all k-states, because it preserves only the total density,

Nonequilibrium semiconductor dynamics

gne =-&d,

337

(129)

with n“ = 2 &p(g). Thus, electrons excited at a certain momentum & will in general scatter into another k-state. Such an electron transition is not necessarily accompanied by a transition of a hole between the same two states The resulting difference in electron and hole popula$ions could not occur if the k-states were really independent two-level systems, where p(k) = - fh 6). The model (127) and (128) constitutes the basic and simplest way to include dephasing processes in the numerical solutions of the SBE. Most numerical results presented in this paper are based on this simple model. With respect to the relative importance of Eqns (127) and (128), one often finds that the dephasing, Eqn. (127), dominates the excitation dynamics since it determines the degree of coherence and, in the region of discrete resonances, the effective absorption of the light pulses Thus, another practical way to approximately treat incoherent phenomena is to take Eqn. (127) into account and leave out Eqn. (128). Later in this article Section (3.5.2.), we discuss a situation where the relaxation rate approximation can be justified quantatively. (154)We then also present numerical calculations for the scattering times Ti and TZ in inverted semiconductors Generally, the relaxation rate approximation is a certain kind of Markov process, since the scattering terms depend only on the time t and not on earlier times t’ < t. This is certainly an approximation since carrier scattering processes as such are not incoherent. One evidence for non-Markovian behavior is already evident in linear absorption spectra, where the exciton resonance generally exhibits a non-Lomntzian lineshape (Urbach tails), whereas Eqn. (127) predicts a Lorentzian lineshape. Corresponding low-density line-shape investigations have been A phenomenological approachus7) performed, for example, by Haug and Liebler et al. (155,156) to improve the relaxation rate aproximation is to incorporate memory effects into the scattering term via the replacement I

yP(i;t)

s

-

dt’y(t

- t’)P(k

t’) ,

(130)

--oo

where y = h/ Tz. Here y(t) is a function which is peaked at t = 0 and whose width determines the memory time. In addition to the time dependence, the dephasing rate will in general depend on the distribution functions of the carriers This creates additional dependences of the relaxation process, e.g. on the wavevector a&;

t1 -

YW(k

t) ,

(131)

or, more .generally YP& t) -

&&k,P(k;t,. k

(132)

If the system is close to a thermal quasi-equilibrium, the dependence of y on the distribution functions might be reduced to a dependence on the density and temperature of the carriers In particular, we investige later in this article the density dependence of y, which can be described phenomenologically by YP& t) + y(n)P(& t) ,

(133)

where n is the carrier density. Also, in a system with more than two bands, y will be different for different transitions, J&6KVVS-c

R. Binder and S. W. Koch

338

ypeyk; 2) + yc~v2Pc,v2(~ 1) ,

(134)

or even contain coupling terms between different optical polarizations In order to approach incoherent phenomena, which are formally beyond the HartreeFock decoupling, Eqn. (46), we will use a nonequilibirium Green’s function technique, which It has been applied extensively to optical is often called the Keldysh technique. (123-125~158) involving nonequilibrium situations, and it contains the limiting cases problems (126~127~15s-161) of quasi-equilibrium electron-hole plasmas Generally, the measurable quantities in optical experiments am given in terms of one-particle expectation values (ai2(t)a,, (I)) , which we call ~%$:!!~(t). In Section (2.2.) we were mainly interested in optical+ polarization fucctions, nl = lc, &) and n2 = {v, It}, and distribution functions, ni = {a, k} and a2 = {a, k} with a = c,v. The polarization function determines the optically allowed transitions, or the possible energy-differences between one-particle states Formally this is a consequence of the fact that both time arguments in a& and a, are the same. To calculate the possible one-particle energies one has to compute the correlation functions f”(k) (see, e.g. Ref. (129)). The physical processes which lead to the most rapid decay of the correlation function are intraband carrier-carrier scattering processes To incorporate these processes in the SBE, one has to account at least for certain retardation effects in Eqn. (43). Technically, the simplest way would be to derive the equation of motion for the four-operator, six-operator, and higher expectation value, factorize it similar to the factorization (46), and integrate the resulting equation with respect to time. Similar approaches with projection operator techniques have been used, for example, in Ref. (117) . Altough very transparent, those iterative methods are not easily extended to an infinite order perturbation series in the Coulomb potential. For this reason, diagram techniques seem preferable, as long as they are not based on thermal equilibrium conditions As an example for non-Green’s function techniques, we will discuss in Section (2.8.) the so-called x(“)- approach, which is based on an expansion of all equations of motion in powers of the external light field. 2.6. Green’sFunction Approach The general way to include carrier correlations starts with the equation of motion for the two-time function

ihG+-(xlhtx$d

= (Wt(z$~)(II(x~~I1) ,

(135)

where we used again x = Ii, $1. Since in the diagrammatic expansion of a nonequilibrium system all possible time-orderings occur, the definition of G has to be extended to include the time-ordered function,

Nonequilibrium semiconductor dynamics ihG++(~,t1,>~2)

339

= U-+w(~1hM+Q2~2)) = Wl -

- 22) kNxl~lw+(~2~2)) w2

-

t1)

(136)

w+b2~2)VI(_X,m

,

(137-I

I

(138)

the anti-time ordered function ihG--Cc,tl,&tz)

= -

(T-W(xlf1)Wt(&2f2))

= WI -

002

t2)

(ccl+ (&212)(II(X,h

-

11) wx,~lM+(&2~2))

1)

and

= (w/(x~~~)(v~(A~~)) .

iM-+(slh,z&

(139)

Equations (135)(139) can formally be unified by assigning the so-called Keldysh indices b = +, - to the time variables Furthermore, one defines the operator T, which is T+ if both times carry a ‘+‘-index, T_ if both times carry a ‘-‘-index, and which orders the operator with ‘-‘index to the left of the operator with a ‘+‘-index in the mixed cases, accompanied by a sign change for each exchange of two Fermion operators This is called contour-ordering, since formally the time axis has been transformed into the contour form - 00 to + 00 (Keldysh index ‘+‘) and back from +co to -w (Keldysh index ‘-‘). The terminology refers specifically to the time-evolution operator in the interaction representation [see, for example, Ref. (125) and Appendix A]. The definition of the Keldysh matrix which contains all the above mentioned components can be chosen to be ihG(x,tlbl,z2f2bd

= b2 (T,~(xlt,bl)cyt(~22t2b2))

= bzWi;Q

(w(~,i;)w+(~&))

(w+(3GM(x,~)) .

W(M)

-

(140)

In order to make the notation more compact we have defined a generalized O-function, for for O(t;lg = for 1 @(tt t2) for i O(Zl 0 1

where we have defined i; = {t, bl

}. This

22)

b, bl bl bl

= = = =

+ + -

b2 = b2 = b2 = b2 =

+ + -

(141)

function fulfils the relation

and its time derivatives are

$Wfiiz) =b2 t&626(t, 1

- z2)

$O(r;&) =- b2 &,,b26(tl

- ~2) .

(142) (143)

2

The equation of motion for the Keldysh matrix G can now be obtained in a way similar to Eqn. (43). We assume that we know the initial value of G, i.e. G(iZ) at tt = tz z to, and that IO is in the remote past before the optical excitation takes places Formally we will take the

R. Binder and S. W. Koch

340

If an electron-hole plasma is optically created at t = 0, say, the scattering and relaxation processes in the plasma are not affected by the limit to - -OQ, i.e. we do not apply Bogolyubovs principle of asymptotic vanishing of correlations (see, for example, Ref. (162)) to the optically created plasma. Of course it is not a trivial matter to obtain the proper initial (or boundary) values of G. If, for example, the system is initially in a thermal equilibrium state, i.e.

limit to + -m.

and (T,~(x,tibi)tP~

(S&M)

= TrIM’Mxllibi)Wt

(h&)1

,

(145)

where po is the statistical operator of the grand-canonical ensemble, one would have to solve the full equilibrium many-body problem in order to obtain the correct initial value. In this paper, however, we are dealing only with two simple initial condition. One initial condition is that of an intrinsic semiconductor without any excited charge carriers (the initial many-body problem determines only the bandstructure parameters and the background dielectric constant E,), and the other is that of an initially inverted semiconductor with an electron-hole plasma in quasi-thermal equilibrium (in this case the initial correlations lead to energy renormalizations and broadenings which will be treated in the same way as the time dependent renormalizations which are the focus of this paper). In principle, knowledge of G(t&,) is not sufficient to determine G(ti 12)since the Coulomb interaction couples G to four-operator expectation values A thorough discussion of initial correlations, including initial values for expectation values with four or more operators, can be found in Ref. (125). In the simple approach presented in this section we will assume the validity of standard diagrammatic methods (i.e. four-operator expectation values can be expressed in terms of two-operator expectation values). There are two differences between the equation of motion for G and Eqn. (43). First, in order to determine G for all tr and t2 (1 to), one has to formulate two equations of motion (one in tl and one in tz). The second difference is the time derivative of the O-functions The g-function obtained from this derivative, together with the commutation relations (37), yields a simple S-function contribution to the time derivative of G. One obtains, with Eqns (35) and (142),

it+(E) 1

= b(i- 2)+h(i)G(i2) - bz

s dW(i- 3)

(T,cy(i)~(3+)~+(3++)~+(2)) ,

f

(146)

and, with Eqns (36) and (143),

-ih$(ii)

= b(i- 2)+ir@G(iZ) - bz &

where the notation i = {3isi zib, } and

I

dV(?

- 3,

(T,~(i)~(3--)ly+(3-)cti+(Z)) ,

(147)

Nonequilibrium semiconductor dynamics

341

is used. The Coulomb potential has been assigned the formal time dependence

The notation I* denotes the time argument tl + 616 (with 6 - +O) . Similarly, i ++ indicates 11+ 2b16 and I-- indicates II - 216. These 6s are introduced so that the time ordering operator T, restores in an obvious way the operator ordering according to Eqn. (35) in Eqn. (146) and according to Eqn. (36) in Eqn. (147), respectively.Since Eqns (35) and (36) contain three operators each with fixed ordering we use two different ‘deltas’ to unambigously recover the correct order. For example, in Eqn. (146) it is obvious that Tc orders the three operators with arguments t; such that the operator with time argument fi + is placed to the right of the operator with the argument fi ++. The operator with the argument fi is placed even further to the right. To recover the equations for the density matrix, where II = t2, we have to set 12= II + 3br6 in Eqn. (146) and tr = t2 - 3b26 in Eqn. (147). One can introduce the four-operator Keldysh Green’s function bzb; G(i925’) = (ih)2 (T,~(i)~(~)cl/+(3’)~+(2))

(149)

to t-e-writeEqn. (146) in the form ih cG(ii)

= s(i - 2) + ir(i)G(iZ)

1

_.. _-

- ih

d3 bJV(i - ?UG(13+23++).

(150)

and Eqn; (147) as aG(iZ)

-ih at2

= 6(i - 2) + h@)G(iZ) - iA d? bjV(2 - ~)G(i?-??) I

.

(151)

Equations (150) and (151) am still a simple generalization of Eqn. (43) to allow for unequal time-arguments of the two operators and the various time-orderings Specifically,it is still ‘local’ in time, since the time-integration over t3 does not really describe retardation effects (due to the d-function in Eqn. (148), or, in other words, due to the local time structure of the interaction Hamiltonian). Such retardation effects are, however, to be expected, if one breaks the operator hierarchy and deals, e.g. only with two-operator expectation values For this, one can write the four-operator Green’s function as a functional derivative of a suitably defined generating functional, which, for simplicity, we call again G(i?>. The dependence of the generating functional on the additional external longitudinal potential U, is implicitly contained in the subsequent equations, but only two quantities related to the generating functional are needed: the generating function evaluated at U, = 0 and its derivative evaluated at U, = 0. In general, U,, is a function of ht and formally depends also on the Keldysh index b. We can uow write (see Appendix A) -_

G(i323’) = -a

ex

+ G(33+)G(iZ)

(152)

to formally eliminate the four operator Green’s function. This is still not an obvious simplification, because instead of having two different Green’s functions we have now two different values of the generating functional, which itself is a much more complicated quantity than

R. Binder and S. W. Koch

342

the Green’s function. However, this re-formulation of the problem leads to a practical way to introduce a diagrammatical perturbation expansion. The equation obtained by substituting Eqn. (152) into Eqn. (150) can-be used as starting point for such a diagrammatic ex_pansion. One possible way is to iterate G( 12) in the functional derivative with respect to U&(3), starting from the noninteracting Green’s function. This procedure is described e.g. in Ref. (122) and, specifically for the Keldysh formalism, in Ref. (125) . This expansion scheme is based on the noninteracting Green’s function. In order to avoid the appearance of noninteracting Green’s functions in the diagrammatic expansion without simultaneously complicating the rules for constructing the diagrams, one has to extend the equations for G(i%). Technically, this extension is based on the repeated change of variables and the consequent application of the chain-rule in the evaluation of the functional derivatives Before discussing these technical details, we would like to introduce the functions which will result from the change of variables As additional functions, one usually generates the selfenergy Z(i2), which contains information on both the renormalization of the one-particle energies and the scattering rates determining the one-particle distribution functions, the longitudinal polarization function II( which describes the possible one-particle transitions as a results of longitudinal electric fields (the latter can either be an external field or the result of charge density fluctuations in the system), the screened Coulomb potential W(12), which differs from the bare Coulomb potential because of the possibility of one-particle transitions as described by II( brought about by charge density fluctuations, and because of the related possibility of collective excitations, the vertex function I’(i23), which serves to complete formally the set of equations We discuss these functions below. Although the expanded set of functions still does not lead to a closed set of equations (an additional function, aZ/aG, occurs), it allows for a perturbative solution by means of iterating Z in the derivative X/aG. The formal structure of these equations will turn out to be essentially Z= WTG l-I=Gl-G w=v+vl-Iw

F=l+EGrG. All details of the equations are intentionally left out to stress the general structure of the equations The equations will be completed as we derive them now in detail. To define the selfenergy, one eliminates first the four-particle Green’s function in Eqn. (150) with the help of Eqn. (152). The second term in (152) gives rise to the induced (or Hartree) potential, which, together with the external potential, is called the effective potential: ueJf(i)

=

v,,(i) -

ih

s

dZbgqi- 2)G(ZZ+).

(153)

It is easily verified that the Hartree potential in (153) is exactly the one given in Eqn. (51). The functional derivative contribution to Eqn. (150) defines the selfenergy via ih

_ _ s-

aG(i2) d3 b3 VU - 3) av,,(3+)

=

-

s

--

--

d3 2(13)G(32)

,

(154)

Nonequilibrium semiconductor dynamics

and the functional derivative contribution via

343

to Eqn. (151) defines the corresponding

selfenergy

__

ih

J

d3 b3 V(? -

3) aF[&

P

Jdj

G(i3)Z(3?)

.

(155)

ex

With the help of Eqns (152), (153) and (154) we obtain from Eqn. (150) the Dyson equation:

{ih+

- P(i)

I

- u,(i)}

G(U)

-

Jd3 z(U)

G(Q) = b(i - 2)

(156) where AH is defined in Eqn. (48) and, again, the limit V,,(i) Similarly, we obtain from Eqn. (151)

U,(i))G(i>)

-hH(ti) -

{-ih: 2

-

+ 0 is implicitely assumed.

Id3 G(i3) I(%?) = s(i - 2) (157)

To solve Eqn. (154) for Z requires the existence of the inverse of G defined by

J

d3 G-‘(ij)

G(32,

=

d? G(i3) G-l(%)

= 6(i - 2, .

(158)

In particular, the time integration ranges from -co to +co, and we assume that the homogeneous contributions to G, which fulfill the relation

J

dg G-%3)

Gh”“‘(3?) = 0

do not have to be considered. Using then G-’ allows us to write a unique solution of the equation of motion of G. As usual, we need only the existence of G-’ to set up the following equations, not the explicit form of it. After all, in the end we have to resort to perturbational methods to solve for G. The basic procedure to generate the functions Il, W and T is to repeatedly change variables and apply the chain rule of the functional derivatives We will assume that the effective field and the external field are connected by a one-to-one correspondence. From Eqn. (154) we find, with Eqn. (158), __

X(12) = ihb, = -ihb, = -ifib, = -ihbl

= -ibl

___ _ J

aG(i4)

d34 I’(1 - 3) av,(g+)

s

G-’ (42)

(159)

d34 V(i - 3) G(iii)

(160)

&;i V(i’

(161)

J J

- 3) G(i;i)

-mm

d345 V(i+ - 3) G(l4)

45 w(si+)

G(i4) r(k?S).

avsf/(S,

av,,(l)

(162) (163)

In going,from Eqn. (160) to Eqn. (161) we have made use of the fact that bl = b3. In Eqn. (163) we have introduced the screened Coulomb potential

R. Binder and S. W. Koch

344

(164) and the vertex function ___

(i?)

aG-’

r(123) = au&)

(165)

*

The screened Coulomb potential or, equivalently, the inverse dielectric function

c-l(E) = av,,w x4& ’

(166)

can be written in terms of the polarization function

n(i2) =

$(iz

-ihb,

(167)

df

in the following way [using Eqn. (153)]:

m?ff(l) av,&

(168) =

6(i - 2)

x(33+) aueffGv v(i - 3) au&u

=

s(i -

2) +

(169)

au&

J

(170)

In this way, we obtain E-I

(iZ) = s(i - 2) +

I

d3da

v(i - 3>I-I(%$)r-’ (siz)

(171)

s

@‘(Ai).

(172)

and, from Eqn. (164),

w(X) = v(i - 2) +

dgd;i V(? - 3)ll(34)

For later reference, we wish to also introduce the generalized density-density function S( iz), which is defined in analogy to the relation s=

correlation

e-1 -1=nw

as

s(i% = fd3 n(i3) ~(32).

(173)

Multiplying Eqn. (172) with II and integrating over the first argument of W we find

s(E) = sO(iZ)+ where we used the definition

I

d3 svi3)

SO%

(174)

Nonequilibrium semiconductor dynamics

sO(iZ) = Jdjn(D)v(Z -

345

3).

(175)

To obtain the screened Coulomb potential, one can either solve the integral equation Eqn. (172), or, equivalently, solve Eqn. (174) and use

w(Z)= v(i - 2) +

Jd3

~(2 - 7) s(3) .

(176)

In order to finally obtain a closed set of equations we proceed now and express the longitudinal polarization Il in terms of the vertex function T: n(E)

J

= it&

djd;l G(i3) I’(%iZ)G(;ii+).

(177)

To obtain this, we have used

a

-

avsf/(4

jd3

G-W)

G(32) = o,

(178)

which allows us to transfer the derivative from G to G-l. Finally, we need an explicit expression for G-i in terms of G. It is given by G-W% = G;‘(E)

- z(iZ)

(179)

with G,-‘(TZ) =

t

a$

I

- h(i) - u,,(i)]

6(i - 2)

(180)

which can be verified using the Dyson equation (156). Although Eqn. (179) is a re-formulation of the problem rather than a solution (G;’ and Z are still unkown functions of G!) it allows us to perform formally the derivative of G-i with respect to &JY. Using

aueffw _ = s(i au4//(2)

- 2)

(181)

we obtain for f -_

___

r(l23) = - s(i - Z)6(i - ?) -

a’(122 aud3)

= -6(i - Z)s(i - 3) - Jdaj s

(182) a~~~?!1 c

(183)

= - s(i - Z)s(i - 3)

(184) = - s(i - Z)s(i - 3)

where we used Eqns (158) and (178) to obtain Eqn. (184) and Eqn. (165) to obtain Eqn. (185). Equations (156), (163), (172), (177) and (185) constitute the starting point for the diagrammatic expansion. The only crucial assumption made so far is that G-i exists and that G is uniquely determined from Dyson’s equation as a solution which is obtained by operating with G-i on the equation.

R. Binder and S W. Koch

346

The more severe assumption to be made now is that iXli3G in Eqn. (185) can be expressed as a function of G. If this is true, one can iterate the equations by first neglecting the aZ/aG term, ___ I’(123) = - s(i - 2) b(i - 3) .

UW

This yields an explicit expression for all other functions in terms of G. In particular, it yields an explicit expression for Z, which can be used in the next iteration step to calculate aZ/aG. The zero-order iteration, Eqn. (186), is called screened-Hartrec-Fock (SHF) approximation or ‘neglect of vertex corrections’. The above equations are all derived from the time derivative with respect to t’ rather than ~2.This is sufficient because the two selfenergies defined respectively in Eqns (154) and (155) arc the same functions To see this we write Eqn. (156) formally as (G;’ - Z) G = 1 which yields with Eqn. (158) Z = G;’ - G-' . The Dyson equation corresponding to Eqn. (151) is G(Gi’ - 2) = 1 resulting in the same expression fo Z. The fact that Gi’ in the left and right handed Dyson equation is the same follows directly from

1

-ih$

2

- P(Z)

*

I

s(i-1)

=

1

ih$--P(i)j&(i--2). 1

In general, the diagrammatic approach to the many-body system requires two basic properties to be fulfilled. Clearly, if we take all diagrams into account, we must recover the true solution of the problem. To investigate this questions, one has to analyze the possibility of a generalized Wick’s decomposition. Some general requirements regarding the form of the initial density matrix po are discussed in Ref. (125). The second problem is whether or not a diagrammatic approximationis consistent with the conservation laws of transport theory, for example Eqn. (44). A discussion of this question can be found in Ref. (122, 125). The SHF approximation fulfills this requirement, and we also assume that the first requirement (possibility of Wick’s decomposition) is fulfilled in our system. We thus evaluate the theory now in the SHF approximation (186). 2.7. Screened-Harlree-Fock Theory The relevant equations are superpositions of Eqns (156) and (157). Subtracting these equations yields 2%{&

+ +-]

G(E) = {it”(i) - P(2)}

G(E)

lEti% G(32, - G(i?) X(%)1 ,

whereas the sum yields G(E) = @R(i) +P(Z))

G(E)

Z(i3) G(%?) + G(i3) Z(32)] + 2 s(i - 2) .

We wish to evaluate all equations in the screened-Hartme-Fock (186)). Within this approximation we obtain the selfenergy

WfO

(SHF) approximation

(Eqn.

Nonequilibrium semiconductor dynamics

347

z(iZ)= ihb,w(Zi+)c(iZ),

(189)

where the screened Coulomb potential

w(X)= v(i - 2)+ s d34

~(2 - 3) n(34)

w(4i)

(190)

is evalutated with the longitudinal polarization function

n(E) = -ihb,c(E) G(Zi+) .

(191)

As we will discuss later in detail, this approximation for fI is equivalent to the so-called randomphase approximation (RPA). For this reason, the terms random-phase approximation and screened-Hartree-Fock approximation are sometimes used to specify the same approximation. The reason for building the di$rence of the two Dyson equations, which yields the sum of the time derivatives (Eqn. (187)), is the intention to generalize Eqn. (47), where the sum is simply a consequence of the product rule for the time derivative. Of course, in a quasithermal equilibrium such time derivatives vanish. In this case, the only interesting information about the many-body correlations comes from the correlations with tl f t2. This can be investigated by forming the sum of the two Dyson equations, which yield the difference of the time derivatives with respect to 21and 22. These two limits suggest the definition of macroscopic (t) and microscopic (t,.) (relative) times,(‘22) t, = t1 - t2 l=-

(192)

t1 +t2 2 ’

(193)

so that

a _E+?

atl-

2 at

a ia -=_-_at2 2 at

at, a at;

(194) (195)

Another possible choice is(163) II = tl - 12

t =t1,

(196) (197)

with

Z

a _~+~ - at

at,

(198) (199)

Clearly, Eqn. (187) describes the macroscopic time dependence, whereas Eqn. (188) describes the microscopic time dependence. Until now, only a few attempts to treat both time dependences numerically exact have been successful. 063) Usually, various approximation schemes arc used to deal with the two-time problem, or, in other words, to deal with nonequilibrium particle correlations This point will be discussed further below. We need to stress that the choice (193) for the induced time can bear problems The exact solution of the SBE, of course, does not depend on the specific form of the transformations in Eqns (192) and (193), but typical approximation schemes do. In particular, the choice of the macroscopic time can affect

348

R. Binderand S W. Koch

certain results within commonly used approximation schemes. We will discuss the question of the separation of time scales in more detail in connection with the Kadanoff-Baym approach and the generalized Kadanoff-Baym approach, Section (2.751.). We also wish to stress that approximate evaluations of the theory (including the numerical solutions) are chosen such that they are best suited to the specific problem under consideration. If one is interested, for example, in the fs-dynamics of an optically exited semiconductor, one usually uses very simple models for the one-particle energy levels In other words, a practical description of the ultrafast carrier kinetics can often be obtained only at the expense of the dynamical description of carrier correlations We use here the terms ‘kinetic’ and ‘dynamical’ properties in the usual sense: kinetics refers to the time evolutions of distribution functions and dynamical properties refers to carrier correlations which determine correlation functions and spectral properties of the system. The dynamical correlations are the central problem of equilibrium theory, as there is no kinetic evolution in an equilibrium state. Clearly, in a nonequilibrium system both aspects have to be treated simultaneously, but, again, the neglection of dynamical correlations and energy renormalizations might be justified in specific problems dealing with ultrafast kinetic processes To discuss Eqns (187)-(191) and common ways to approach these equations numerically, we proceed in the following way: (1) Before discussing the time structure and common approximations with respect to the two-time problem, we will investigate the spatial/spin structure. For this we expand all functions into the set of eigenfunctions P”. As the most relevant limiting case, we also give’ the equations for a homogeneous and isotropic two-band model. (2) The simplest approximations for the complicated time-structure are the Hartme-Fock and the quasi-static screened-Hartree-Fock approximation. We derive the semiconductor Bloch equations in this approximation. (3) In order to approach the dynamical correlation problem, we present some useful formulas and evaluation techniques for the Keldysh matrices (4) For reference, we list the Green’s functions for noninteracting particles (5) Two approximation schemes will be discussed: the Kadanoff-Baym ansatz (along with the Markov approximation in the macroscopic time), and the generalized Kadanoff-Baym ansatz (with the special case of the so-called free generalized Kadanoff-Baym ansatz). (6) We discuss the equations of motion for G with special emphasis on the correlation contributions (7) We discuss the longitudinal polarization function. (8) We discuss the screened Coulomb potential, where the screening is described by the longitudinal polarization function. (9) We discuss specific forms of the selfenergies, the scattering contributions to the generalized Bloch equations and the generalized differential scattering cross-sections 2.7.1. Expansion into eigenjiictions 2.7.1.1. General expansion. In this step we use the representation

and analogous expansions for the dipole moment fi and the selfenergy Z, and we define the screened Coulomb matrix element corresponding to Eqn. (57)

(201)

Nonequilibrim

semiconductor dynamics

349

Hem, the time arguments still contain the Keldysh indices, t; = {ti, b1}. We obtain for Eqns (187) and (188) iR

1; f &]

=

-

c [fin,", E(t,) P3”2(i;fi)

G""'*(f;fi)

T

(E,, + E”*} G"'"(fiG) G”‘“J(~~, p

*3**~(f2)]

"3

+

1

dt;

"3

[ Z"'"3(fii;)

G"3"*(t;l;)

s

+

G”ln3(f;ij) Zn3”*(fifi) ]

+ (1 + 1) SC4 -G)

,

(202)

and for Eqn. (189) P’“*(t;&) = ihb, 1

w,,,,,,,&X+)

G”3m((f;i2).

(203)

n3n*

For later reference, we insert Eqn. (203) into Eqn. (202) and obtain

i+3 -

1

={E,,

G"'"z(fi&)

[8n,,h)

ff3”z(t;G)

-

T

E,,} G""'*(t;h)

Gn’“3(t;&) ri.,,&)]

"3

+ih

1

Jd&

[b,W

",",",,(iji;+)

Gn3”Vi;ij) Gn5”*(fi&)

"3"4"5 +

+

b3 WQ”2”s”3(M’)

G”‘“‘(fifi)

Gn5”’(t;t;)

(1 + 1) b(t; - 4).

1 (204)

The screened Coulomb potential is given by W“,“2”3”.,(&?) = 6,,“2”3”,(I; - iz)

+

c f dkit

K,,,,,

OYd bsnsn,ns Gr4) K,n,n3ns Wi )

“s”‘n*

(205) with II n,n2n3,(i;&) =

-ih

bl G”l”z(fit;) G”3w(f&) .

(206)

If one interprets the Coulomb interaction Vn,n2n3n4 (&t;) as a scattering process in which the one particle changes from state ni to state n3 and the other from n2 to n4, and the scattering of the two particles occurs simultaneously (formally due to the time-delta function) then the screened interaction can be interpreted as a sequence of scattering processes, where at the time,tz the scattering from ni to n3 creates a pair of particles in the respective states n6 and ~27.These propagate from time 22 to ti when they are destroyed simultaneously with the transition of the particle form state n2 to n4. The possibility of the propagation of the intermediate states over a nonzero time interval gives rise to the retardation of the interaction and justifies one to call the screening process dynamical correlation. We will see below how

these possible scattering processes yield effects important for the semiconductor. First, it yields

350

R. Binder and S. W. Koch

the dephasing of the optical polarization (in the equation for Gy_), and secondly it drives the distributions (cc Gc,C_,G”+“_)to the quasi-thermal equilibrium state. Equation (206) also shows that screening is not only caused by the quantum mechanical correlations within the same state, c;“1”1(&fi), but also by ‘off-diagonal’ correlations @I”*(4 fi), ni f n2. Typically the initial conditions together with the external forces create only a limited number of cohemnces In our cay, the electric dip_ole field can only create functions with the same momentum index, ni = {vi, k} and nz = {VZ,k}. If the exte_mal field would have a spatial variation we could even have coherence between two different k-states In systems which are in thermal equilibrium we have quite generally G”L’Q= b,,,, @*“I with ni = {v, k}, and the temporal correlation function G”*“’(1it2) simply describes the decay of the one-particle state ni with increasing time-difference ti - 22. Although in a nonequilibrium state screening is affected by all possible coherence functions (including optical polarization functions), one has to estimate which of the correlation functions dominate the screening if numerical calculations should become feasable. One criterion to reduce the number of terms in the polarization function arises in the theory of incoherent scattering processes, which will be discussed below. The basic argument will be that peaks in the Fourier spectrum (with respect to 12- ti) of W should be around typical intraband energies, rather than interband energies A general decomposition scheme of all equations of motion into even and odd functions (with respect to the main oscillation frequency) is discussed in Ref. (126). 2.7.1.2. Spatially homogenous and isotropic systems. Until now we have classified the oneparticle state by the quantum number nt, which comprises Bloch-band and wavevector indices In a spatially homogenous and isotropic system the translational and rotational-invariance allows for a considerable simplification of the above equations with respect to the k-coupling. In this case, we can replace G”‘“Z* d&i* GVIW(k,) ,

(207)

where v is the Blochband index. Similarly, we can replace

With Eqns (207) and (208) we obtain for Eqn. (202)

+ 1

Y s

dij [ ZyLy(Ic,t;ij) G@“(k;fit;) i Gy’” (k, t; &) lP-

+ (1 + 1) S(t; - &, .

(k; f&) ] (209)

The i-coupling is, of course, the same which one would obtain from a spatial Fourier transform with respect to the difference variables in space, i.e. & ti i?i - 72. Many publications use this result as a starting point and avoid the mom complicated general expansion. For quantum confined systems, however, it is useful to start with the general expansion, because the translational invariance is broken in at least one direction. For the selfenergy, Eqn. (203), we obtain

Nonequilibrium semiconductor dynamics

351

The screened Coulomb potential can be discussed most easily within the effective-mass approximation, Eqn. (66). If we up the_same structure for the screened potential, i.e. replace V by W in Eqn. (66), and replace ki - k3 by 3, we obtain from Eqn. (205)

In the important case that both the unscreened and the screened Coulomb potential do not couple different bands (i.e. W,,,,, oc 5,, 6 vzvr) , the equation for W can be simplified and, in agreement with the usual definition of the intraband polarization function, all remaining summations over band and momentum indices which occured in Eqn. (211) can be transfered into the definition of Il: (212)

In a two:band system without optical coherence (G”“” oc a,,), the Bloch band summation in Eqn. (213) reduces to the usual spin summation. If an optical polarization is present, the intraband polarization contains a contribution 0~ PP. In general, the inclusion of bandcoupling effects is not easy. However, if the Coulomb matrix element contains a factorized formfactor, as in Eqn. (531), and if all interband coherences can be neglected, G”‘lyz K 6y,y, the following approximate treatment is possible. One can use the same formfactor which describes band coupling in the bare Coulomb matrix element for the screened Coulomb potential. The factorization properties allow for a cancellation of parts of the formfactor in the equation for W, and the remaing parts of the formfactor can be transferred into the definition of Tl. One finally obtains the result

zY W&*(W) ?y,,,(id 2.

(214)

2.7.2. Quasi-static approximation Before further discussing the complicated time structure of Eqns (209)--(213), we would like to point out that in the quasi-static screening limit, Eqn. (204) immediately yields the generalized semiconductor Bloch equation, Eqn. (71), the main difference being that the bare Coulomb potential is replaced by the statically screened potential. To see this, let us first mcover the SBE in the Hartree-Fock approximation, i.e. using the unscreened Coulomb potential. In this case, the selfenergy is

F(E) = ihb,

v(Z -

Using the definition for V (Eqn. (148)), the relations

i+) G(iZ).

R. Binder and S. W. Koch

352

(216) (217-l

G++(zI,~I + E) = -G+-01~1) -E) = G+-(t,tl)

G--(~I,zI

,

which follow from the definition Eqns (136) and (138), and ljy6(t1 we obtain the usual Hartme-Fock

c::(12)

(218)

f E - tz) = 601 - tz) ,

selfenergy

= ZHF(12) = -b(l1 - Zz)V(?, - ?2) (~+(xzt,)~(x,t,))) p+

= p + = 0

(219)

(220)

’

We finally recover the Bloch equations Eqn. (71), if we set tl = t2 = t in the ‘+-’ component of Eqn. (187). Within the quasi-static screening approximation one assumes that the screened Coulomb potential has the same time structure as the unscreened Coulomb potential (Eqn. (148)). Physically, screening is a dynamical process which requires time in order to move charges and hence screen the potential. However, one can assume the fi(tl - t&approximation to be good if the time scale for tl and 22 is long compared to typical scattering and screening times The &&-approximation is made because we want to model the original equations of motion as closely as possible, in which two-operator functions are coupled to four-operator functions (Eqn. (43)). In specific evaluations of the screened potential one can also show that W+- = W-+ = 0. (In the plasmon-pole approximation for bulk semiconductor, for example, W+- (tz-tl ) is, after Fourier transformation from tl -tz to w, proportional to 6(w--opt). Since the plasma frequency wpl + 0, the co = O-component vanishes Equivalently, W+_ (22 - tl ) has no b(tl - t&contribution.) Replacing V by W in Eqn. (215) we obtain now immediately the semiconductor Bloch equations with the screened instead of the unscreened Coulomb potential. Note that the subtraction of C, V’ in the final equations is unaffected by the screening, since it is a zero-density renormalization. This divergency was formally treated in Ref. (126), where the selfenergy of the unexcited two-band semiconductor was subtracted from the Hartree-Fock selfenergy Eqn. (189). Generalizing that procedure to the multi-valence band systems we replace

Here, the indices w3 are restricted to those of valence-bands only, i.e. w3 = {v3,&}. If the Coulomb potential has the simple form (66), only the diagonal elements of the valence-band selfenergies Zvvlvlhave to be renormalized. For the important case of a two-band semiconductor we wish to explicitly give the SBE (transformed into the electron-hole picture) in the quasi-static screening model: iRiP

= (Ek + Z’(k) + &k) -(l

-j-Y&

-r’(k))

- iy”) P(k) ha(k)

&PW=2Zm[a*WPWI Rn(k) =

,

,

ficvw E + 1 W(k - k)P(K) , ’

K

(222)

(223) (224)

Nonequilibrium semiconductor dynamics

Wk) = - 1

WC2- i)fw) + 6,”

k

x [W(k - Ii, -

353

V(P - i)] .

(225)

k

The first term in Eqn. (225) is called the statically screened exchange contribution and the second term, which stems from the renormalization procedure of Eqn. (221), is called the Coulomb hole term. This contribution is also discussed in detail in Ref. (130) as a consequence of screening retardation and in Ref. (2) as the limiting case of the time-dependent retarded selfenergy of an e - h plasma. It gives rise to the terminology quasi-static screening, which is more than a mere replacement of V by W in the SBE. Unfortunately, the specific form of W in the static-screening limit is not readily deduced from Eqn. (205). We will therefore continue to evaluate the time dependent equations and in the end take the static limit of the specific time-dependent model (see Eqns (339) and (340) in Section (2.7.8.2.)). A significant disadvantage of the quasistatic screening model is that within this model screening does not act as an additional source of dephasing. The self-energy Eqn. (225) is real and describes only the renormalization of the one-particle energies, not a broadening. Screening itself is, at least in quasi-thermal equilibrium, a consequence of incoherent carriercarrier scattering, which in turn results in a decay of the optical coherence. This effect of ‘dynamical correlations’ of the carrier system has been neglected so far. 2.7.3. Evaluationof Keldysh matrices To treat dynamical correlations in general we have to deal with the full Keldysh matrix. In the following we will extract the relevant Keldysh components from the equations for G, Z, W, and fl. The main task here is to eliminate the linear dependencies between the various Keldysh components, and unfortunately there is no unique way to do that. The best choice depends largely on the approximation scheme for the two-time structure. Because of the need for further approximations, in the end the various choices are not equivalent. We will, therefore, discuss several possibilities used in the literature to eliminate Keldysh components We discuss here, as examples, the approach of Schafer and Treusch(iz6) and the more recent treatment of Hartmann and Schafer. (163)For simplicity, we discuss the linear dependence for the space-dependent functions, not for the matrix elements with respect to Pi. The linear dependence for G follows directly from the definition, which yields, G++(12) + G+_(12) = G--(12) =

~o(t,

-

t2) ([~(Z,h),

+ G-+(12)

*+&2t2)1+)

I G,(12)

(226) (227)

(228)

and G++(12) - G-+(12) = G--(12) =

-j+(t2

I GJ12)

-II)

uw(x,tI),

- G+-(12)

~+(&2t2)1+)

(229) (230) (231)

where 1 = {xi, tl }. Clearly, Eqn. (226) defines the retarded Green’s function G,(12) and Eqn. (229) defines the advanced Green’s function G,( 12). In the limit tl - t2 k 0 we have ihG, + 1 and ipIGa -. - 1, independent of the coordinates xi, x2 and independent of tl . Equations (226) and (229) allow us to eliminate, for example, G ++ and G- - and keep either G+ -, G- +, and Gr or keep G+-, G,, and GO.(lza) From the definition of the retarded Green’s function in terms of nontime-ordered expectation values, Eqn. (227), one obtains immediately JPOE 19:415-o

R. Binder and S W. Koch

354

G,(12) = @(tl - tz) [G-+(12) + G+-(12)]

,

(232)

and from Eqn. (230) G,(12) = -O(tz - 11) [G-+(12) + G+-(12)]

.

(233)

The mininum number of functions to be calculated is therefore two (e.g. G+- and G-+), which reflects the fact that we are dealing with two independent properties of the system: spectral and kinetic properties Concerning the notation, we should point out that the definition in this papers differs by a minus sign for G+- from the corresponding definitions in Ref. (122): G+_( 12) = -G’( 12)

(234)

G-+(12) = G’(12).

(235)

Concerning the selfenergy, one can prove the same linear dependence as stated for G in Eqns (226) and (229) from the general definition, Eqn. (159). We will only outline the proof here. First, one has to prove that G-’ has the same linear dependencies as G. One can show that directly from the defining equation G-’ G = 1. Using finally the fact that aG/XJ,( +) = -aG/XJ,(-) one can show directly that Z has the same linear dependencies as G. There is only one important difference between the relations of G and those for 1. As one can see from Eqn. (215), the selfenergy can have a nonretarded and nonadvanced contribution propotional to b(zl - 22). Whereas this contribution still preserves the linear-dependencies relations, we have to modify the generalization of Eqns (232) and (233)u2’) and explicitly account for the &contribution ZHF: 2,(12) = P-712)

+ @(t, - 22) [X-+(12) + E+_(12)] ,

(236)

&(12) = 2712)

- @(t2 - t1) [2_+(12) + 2+_(12)]

(237)

and

Direct inspection of Eqn. (191) yields the corresponding l&(12) = 00, - t2) [l--+(12)

.

relations for II:

+ II+_(l2)]

,

l&(12) = -@(I2 - 21) [l-l-+(12) + n+_(l2)]

(238) .

(23%

It is instructive to prove the linear dependencies for the polarization Il for the specific example of the screened-Hartrce-Fock approximation. We find with Eqn. (191) l-l++ + n+- = -ih (G++G++ + G+_G_+)

(240)

= -ih ({G, - G+-} {G, - G+_} +G+- (G, - Go - G+_))

(241)

= -ih (G,G, - G,G+_ - G+_G,)

(242)

= +ih (G,G+_ + G+_G,) .

(243)

In going from Eqn. (242) to Eqn. (243) we have used the fact that

G,G, 0~ @(t, - tz)O(tz - 11) = 0 .

Gw

since a similar calculation for IL- + II-+ yields the same result as in Eqn. (243), the linear dependence of Il is again the same as that for G. The same holds for IV, because the dependence on the Keldysh indices of the equation for W, Eqn. (190), is exactly that of Eqn. (191). In conclusion we have, for all functions

Nonequilibrium semiconductor dynamics

355

F = {G,Z, W,l-I} the relations F,, + F+_ = FL + F_+ = F, F;, -F-+

= F-- -F+-

= Fa .

(245) (246)

Concerning the multiplications of Keldysh matrices in Eqns (18~( 19 1) or, equivalently, (204 j (206), we find only two kinds of multiplication. The first kind is Ab,bz

=

c

(247)

Bb,b&bjbz

b3 where {A, B, C} are {G, 1, G} in Eqn. (187) and {W,ll, W} in Eqn. (190) (note that the bare Coulomb potential is diagonal in the Keldysh indices). The matrix elements in this case are A+_ = B,C+- + B+-C,

(248)

A_+ = B,C_+ + B_+C,

(24%

A, = B,C,

(250)

A, = B,C, .

(251)

Ab,bz

(252)

The second case is of the form = h Bbzb, Cb,bz

where {A, B, C} are {Z, W, G} in Eqn. (189) and {fI, G, G} in Eqn. (191). The matrix elements are A+_ = B_+C+_

(25%

A-,

(254)

= -B+_C_+

A, = -B+_C, - BaC+-

(255)

A, = -B+_C, - B,.C+ .

(256)

In principle, we have now to, solve the set of equations for all linearly independent Keldysh elements of all Bloch matrix elements of all functions F = {G, Z, W, ll}, all of which depend on two time arguments tit t2. This is clearly not feasable and the next approximation step will be guided by the properties of the Green’s functions of the noninteracting system. Also, the fact that we are trying to generalize the SBE, which contain kinetic functions for the G+components in nonequilibrium systems, will influence the approximation scheme. The specific approximation would be quite different if our goal was the calculation of spectral properties of equilibrium systems, rather than kinetic properties of nonequilibrium systems 2.7.4. Free-particle Green’sfunctions To obtain a feeling for sensible approximations we recall the solutions for noninteracting particles with no external field present. In addition, some of these functions will be used later to replace the full selfconsistent functions occuring in the equations of motion. The Heisenberg operators for noninteracting particles have the time dependence a,(t) = e -&J/h

a,

a:(t) = eiEntth a,t . The Green’s function are then

(257) (258)

R. Binder and S W. Koch

356 ihGy(l,Z~)

=

e -k,

II /h

~hgff~(I,12)

=

e-k,tl

+ bt2lh

ihlQn2(t,t2)

,

[a,,,

C&m)

/h + W2lh

ihG:‘“2(t1t2) = @(II - 12) e -ib,tllh = e(I,

(a;zan,)

+

_ t2) e-&tl!R

= -@(I2

- I,)

_

(25%

h2t21h

([a;2,an,~+)

+ k2t2/h

e -&,I1

wa

] t

6n,n2 ,

/h + iEn2t21h fin,,,

(261)

.

(262)

lfn1 = n2 the ‘+--’ and ‘- +’ components depend only on the relative time tr = II- 12and the distribution function,

The

jh($:l'(t,)

=

e-jb,blh

fm

ihG?‘,“‘(t,)

=

e --ih,tJft

[1 _ f”‘]

(263)

,

.

w4)

Fourier transforms in this case ate iRGY!‘(o) = 2rr6(w -6,/h)

f”’ ,

iRdll:‘(c0)=27rS(c0--e,,/h)

(265)

[l-f”‘]

,

(266)

with 6 4 +O. For nr + nz the retarded free-particle function vanishes In this approximation the retarded function contains only spectral information and is not related to the particle distribution function. In the free-particle approximation the only nonvanishing cohemcce function (i.e. ni + n2) is the polarization created by the external light field. For nr = {c,k}, n2 = {v,k} we have

ihc_(&, 2112)= =e

e -i+

Ifi + +2IA

-i$tl

/R + k$/h

i)lG?+(kt112) = - e

-iEjtl/h

+ i+lh

(a~kaa) p(i)

,

(268)

p(i) .

(269)

Since the polarization results from a timetranslation-symmetry-breaking field it has a macroscopic time dependence. With the Kadanoff-Baym choice, Eqns (192) and (193), we find

ihc_(&, 2,; I)

= e

--i($ - .$)2/h - i($ + QJ(2h)

P(i;,

9

(270)

and the corresponding Fourier transform with respect to t, reads --i($ - $)2/h ihGy_(&,w;t) = e x2TT6(w - (6 + $)/(2h))

P(k) .

(271)

The oscillation with respect to I in Eqn. (271) is the usual optical oscillation_which can be transformed out by going into the rotating frame Without an external field P(k) is zero, and Eqn. (271) assumes that an optical field has created an optical polarization without inducing a t-dependence except for the trivial phase factor. 2.7.5. Tko approximation schemes 2.7.5.1. The Ka&no&Baym ansatz. The solutions for free particles presented in Section (2.7.4.) are in certain ways utilized in all approaches to the case of interacting particles. In

Nonequilibrium semiconductor dynamics

357

the two-time approach, one splits off ‘rotating’ exponentials according to Eqn. (259). Such a transformation into a k-dependent rotating frame is, of course, only a means to improve numerical stabilitiy, not an approximation. In the Kadanoff-Baym approximation and the generalized Kadanoff-Baym approximation the formal similarity of interacting and noninteracting Green’s functions suggests certain approximations We discuss first the so-called KadanoffBaym ansatz. Then we will discuss the generalized Kadanoff-Baym ansatx of Lipavslj, SpiEka and Velicky .(’32) The Kadanoff-Baym approach, and, more specifically, the Markov approximation in the macroscopic time, is based on the following observation. Under the intluence of an external field all functions which are time independent in the noninteracting, no-external field case become f-dependent, but ideally in a way that the variation in t is slow compared with the variation in t,. (Again, the fast variation in t of the optical polarization function can be removed by going into the rotating frame. The slow t-variation is that of the amplitude in the rotating frame, P(t).) It is assumed that the functions vary slowly in t, and that all correlations die out on a time scale short compared to a typical varation in z. Equilibrium theory shows that the Coulomb interaction leads to a renormalization of the one-particle energies Within the so-called quasi-particle approximation (see, e.g. Ref. (130)) we have En -+ (en n

=E”+Zy”(W=Ey),

(272)

where, again, o denotes the Fourier transform variable with respect to 2,. Denoting the inverse correlation time as Y”,

(273) one sees that the renormalized correlation functions, which can be obtained from Eqns (263) and (270) upon replacing E” by tip, die out on the time scale of r;‘. The same holds for Z, as it is proportional to G. To be specific, let us consider time integrals of the form 11 a(t1t2)

=

dt3

b(tl

t3)

dt3t2)

s -cm

(274)

which appear in Eqns (187) and (190). Transformation to microscopic and macroscopic variables (Eqns (192) and (193)), i.e. a(t1t2)

*

AOr;t)

9

(275)

yields 11

A(l,;I)

=

I

dt’ B(t, - 2’;t + t’/2) C(t’; I + (I’ - 2,)/2) *

(276)

-co The Markov approximation in the macroscopic time variable reads tr

A&; t) =

I

-*

or, after taking the Fourier transform,

dt’ B(t, - t’; t) at’; 2)

(277)

358

R. Binder and S. W. Koch

A(w;t)

= B(w;t)C(w;

(278)

I) .

Note that B can be evaluated at the macroscopic time t because its time argument, t + f/2, is restricted to small values of t’. The mason for that is that in Eqn. (276) C is essentially proportional to exp(-rlt’l), i.e. it vanishes for large t’. Similarly, C can be evaluated at the macroscopic time t because in (276) B contains a factor exp(-rlt, - 1’1) (i.e. it vanishes for large tr - 1’) which restricts the macroscopic time argument of C, t + (t’ - &l/2, essentially to t. If the correlation functions would grow as a function of the microscopic time, the above arguments could not be applied directly. The Fourier transforms of Eqns (189) and (191), which are of the general form -40112) = Bbti)

C(tltz)

(279)

become A(co;t)

=

do’

xB(d;

t) C(w +

a’; 1) .

(280)

As shown by Henneberger, the Taylor expansion with respect to the induced time can in principle be taken to infinite order. (161~164~16~ That makes the Kadanoff-Baym ansatz correct and removes all ambiguities with respect to the choice of the macroscopic time t. The approximation (277) can be called ‘Markov approximation in the induced time’, because within this approximation the Dyson equation for G in the induced time does not couple different induced times anymore. This approximation does not require that the functional dependence of all function with respect to t is strictly linear. The only strict requirement for this approximation is that the functions B and C in Eqn. (276) vanish if their microscopic time argument gets in the same order of magnitude in which the linear expansion of the functions with respect to t breaks down. In other words, quantum mechanical correlations must die out fast on a time scale of the external perturbations Much of the theory of quantum transport in the last thirty years is based on the approximation (277) (or its spatial analog), because it renders most equations analytically solvable or numerically tractable. Of course, in the field of fs-semiconductor optics the approximation (277) might not always be fullfilled, since the timescale of the external field reaches not only the scale of quantum mechanicle correlations, but sometimes even the time scale of optical cycles, i.e. h / (Ed--E,). In those cases the KadanoffBaym ansatz and approximation (277) can only be regarded as a first approach to the problem, and fs optical properties computed in this scheme should always be compared with msults, where all incoherent carrier correlations are neglected (e.g. the HF-factorization). A possible problem with the Markov approximation becomes more apparent if one transforms Eqn. (277) back into the two-time formalism:

dlll2)

=

s

dt3b

-cm

(l,+!?$?,!?$h)

c(!i+2+!+).

Specifically, for tz = tl the comparison of Eqn. (281) with Eqn. (274) exhibits a problem with causality. Whereas in both equations the second argument of b and the first argument of c have values less than tl, we find in Eqn. (281) that the first argument of b and the second argument of c to be larger than tl if t3 < 11. Approximation (277) can be viewed as the opposite to the fully coherent calculation, because the former certainly overestimates carrier correlations whereas the latter neglects them completely. An important consequence of the Markov approximation (277) is the breaking of

Nonequilibrium

semiconductor

dynamics

359

the time-inversion symmetry. Later we will analyze scattering and dephasing processes (within this approximation) which drive the system toward thermal equilibrium. The corresponding scattering rates will justify approximation (277) a posteriori (see Section (2.7.9.2.)). However, apart from showing that, in this respect, this approximation is not intrinsically inconsistent, it seems difficult to evaluate a theory in this approximation scheme in a way which allows the continuous transition from fully coherent carrier dynamics to the completely incoherent carrier scattering dynamics This question can be discussed more easily within the generalized Kadanoff-Baym ansatz. We should also mention another possible problem of approximation (277), the solution of which is discussed in detail in Ref. (122). The neglection of first-order (in the induced variables) contributions in an equation of motion, which is a first-order differential equation with respect to the induced variables, is in principle not correct. A typical problem arising from such an inconsistency is the well-known fact that the carrier-carrier scattering contribution to the Boltzmann equation conserves only the kinetic energy, not kinetic plus interaction energy. Although a solution to this problem is discussed in detail by Kadanoff and Baym,(‘22) the numerical solution is presently only practical if the gradient terms are neglected. For completeness we have to generalize Eqn. (277) to the case where either B or C is an optical polarization function. If for example B is such a function, then we have to transform into the rotating frame, B(l,;t) = e -iwof I?&; I)

(282)

before the Taylor expansion is performed, which yields J(w; 2) = B(w; I) C(w - cue/2; 2) ,

(283)

where ;i is defined analog to Eqn. (282). If C is the optical function we have &w;t)

= B(w + wo/2;t) Z‘(w;r) .

(284)

Within this approximation scheme all equations, Eqns (18%(191), are ‘local’ in the induced time argument t, which allow for a direct numerical integration of the z-dependence. Later we will discuss this in detail. 2.7.5.2. The generalizedKadano$-Baym ansau. In order to avoid the Markov approximation and retain certain features of the two-time formalism, the generalized Kadanoff-Baym ansatz has recently been applied to the problem of carrier-carrier scattering.(‘27~‘6U68) The basic idea, which makes this ansatz, given in Ref. (132), of practical importance, is the assumption that the spectral properties of the systems are generally not as strongly time-dependent as the kinetic functions, and thus might be treated (in lowest order) as time-independent. That means that the retarded and advanced Green’s function, which depend only implicitly on the kinetic properties of the system (i.e. on G+_), can be approximately treated as being independent of the ‘induced’ or ‘marcroscopic’ time. The general ansatz for the kinetic Green’s function is then (in real space coordinates) G+_(12) =

J

d3 [G,(l3)e+-(32)

- e+-(13)&(32)1

,

(285)

with e+_(12) = ihG+-(12) 601 and

22) ,

(286)

R. Binder and S W. Koch

360

G-+(12) =

f

d3 [G,(13)e-+(32)

- e-+t13)Gat32)]

(287)

,

with e-+(12)

= &G-+(12) s(tt - 12) .

(288)

As is easily verified, this ansatz retains the general linear dependencies

among the various Keldysh components of G, Eqns (226) and (229). Expanding into one-particle eigenfunctions we obtain

(289) and

@‘?%ir,)G;3”2(t,t2)]

(290)

.

For a spatially homogeneous system we have

(291) and

G?,'1(k;t,t2) = 1 [q’YJ(k;t,f2) [6,,

-f@‘Yk;tz)]

-

Yl

NWJ-f”“(k;td]

GyYktt12)]

*

(292)

Here, we used again the definition j-“‘“(k;f)

= ihG:?(k;fz)

.

(293)

For VI = v2 we obtain for f the regular distribution function, whereas for vi = c and vz = v f denotes now the optical polarization function P. To make this ansatz practical, one still needs a simple yet appropriate model for the retarded and advanced Green’s functions Especially desirable is a model in which Grla depends only on time-differences, i.e. G,la = G,,,,(ti - t2). The simplest possible choice for that is the free particle Green’s functions, Eqns (261) and (262). In Ref. (168) this is called free generalized Kadanoff-Baym ansatz (FGKB). Within the generalized Kadanoff-Baym ansatz the functions G+_ (ti 22) and G_+ (ti 12) am no longer independent functions, since both am completely determined by the single time expectation values which, in turn, are connected by the commutation relations for the operators CZ+and a (in Eqn. (292) we have used this fact explicitly). This can be understood intuitively because now essentially only kinetic properties are being computed, whereas spectral properties am assumed to be known. This can be seen explicitly in the FGKB. In comparison to the full two-time approach, the generalized Kadanoff-Baym ansatz has thus two advantages: it reduces the kinetic Green’s functions (i.e. the ‘+-’ and ‘-+’ components) to single time expecation values and it reduces the number of functions to be calculated from two to one (for every pair of one-particle quantum numbers {ni, 122)).

Nonequilibrim

semiconductor dynamics

361

The importance of the generalized Kadanoff-Baym ansatz exceeds by far the applications and effects discussed in this paper. Further discussions and improvements of the GKB are presently being developed (see, for example, the proceedings of the NOEKS 4 conference(16p)). 2.7.6. Correlation contributions to the semiconductor Bloch Equations In order to discuss correlation and incoherent scattering effects 2.7.6.1. Two-timeformalism. in the semiconductor Bloch equations, we first divide the equations for the ‘+ -’ and ‘- +’ components of kinetic functions Gy’e (k; tI 12) into a coherent contribution, which was discussed already in Section (2.2.), and a scattering contribution, to be discussed in the following.

The term scattering contribution is used here because in certain limits these contributions describe the incoherent carrier-carrier scattering. On a fs-timescale, however, it is not always possible to distinguish between the coherent evolution of the charge carrier systems and the incoherent scattering processes, for both processes are manifestations of the same basic chargecarrier coupling due to the Coulomb interaction. To be specific, we restrict ourselves from now on to spatially homogeneous and isotropic systems and assume that the screened and unscreened Coulomb potential matrix elements are of the form Vy,y2Wy40~ 6,,W,6yV,. Also, we will assume that there are no coherences between different conduction bands or different valence bands, respectively, GQQ oc &,crGclcl and Pl”z a 6v,vzG"lvl. Furthermore, the optical field should couple each conduction band with only one valence band. Following the discussion in Section (2.7.1. l.), one can easily extend the following analysis to inhomogeneous systems, systems with more general coherence functions, or systems with more general Coulomb matrix elements The coherent contribution to Eqn. (294) is simply obtained from Eqn. (209) by using the Hartree-Fock selfenergy Eqn. (215) (together with Eqn. (221)):

-ih

1 V(k-k) v,P -G??

[G??(k’;ttt,)

G??(k;t,t~)

(k; t, 12) G?? (k’; tztz)]

To obtain the corresponding equation for G-+ one has to exchange in Eqn. (295) ‘+-’ with ‘-+‘. From Eqn. (295) one obtains-after multiplication with i)l, setting tl = t2 = t, and transformation into the electron-hole picture-exactly the Hartree-Fock semiconductor Bloch equations, Eqns (75), (76), and (77). For the incoherent contribution we obtain

(296)

R. Bin&r and S. W. Koch

362

where we have introduced the scattering contributions

St, and Sow,

and the correlation contribution

{%%t,l3)

[G??(k;t3t2)

-G%(k;r1t3)

The corresponding

[Z:?(k;t3t2)

+ G??(k;t3t2) + E??(~;I~I~)

] ]} .

(299)

‘- +‘-equations are

(300) with 12

!R?l,‘L(k;t,t2) = {E’?(k;r,z3) -G?,‘3(k;tlt3)

[G??(k;t3t2) [E:?(k;t3t2)

+ G?,y(k;t312)] + E??(k;t3t~)]}

.

(301)

We will see shortly that for the distribution function f = ihG”” the ‘in’ and ‘out’ scattering contributions St,, and St,, reduce, in the limit tt = t2, to the familiar form Tln(l - f) and foUfS, respectively. Obviously, the correlation contribution !R.vanishes in this limit. Also, in this limit the scattering time derivative of G+ _ and G_+ differ only by a minus sign, showing that in any approximation scheme that relies only on the single-time functions it is sufficient to compute only one of the two Green’s functions To obtain the above Eqns (296) and (300) from Eqn. (209) we can, for example, use Eqns (248) and (249) and eliminate the retarded and advanced functions with the help of Eqns (232) and (233) and Eqns (236) and (237). Equations (296) and (300) are the general two-time formalism equations for G. For the case of electron-phonon interaction similar equations were solved by Hartmann and Schiifer.(*63) Due to the complex structure of the selfenergy in the case of electron-electron interaction, which will be discussed in more detail in Section (2.7.9.1.), these equations have not been solved yet. We will therefore proceed and discuss these equations for the two approximation schemes Kadanoff-Baym ansatz combined with the Markov approximation in the macroscopic time (KB) and the free generalized Kadanoff-Baym ansatz (FGKB).

Nonequilibrium semiconductor dynamics

363

2.7.6.2. Ka&no&Baym formalism. Although the Kadanoff-Baym formalism might be easier to analyze if one keeps the retarded and advanced functions, we will use in the following Eqn. (296) and evaluate it using the steps outlined in Section (2.7.5.1.). This allows for an easier comparison of the various approaches (two-time, KB and FGKB). We introduce first the times given in Eqns (192) and (193) and, to extract the kinetic equations, set f, = 0. Clearly, in this limit !R = 0 [Eqn. (299)]. We also perfom the Markov approximation in the macroscopic time, Eqn. (277). The simplest terms are the intraband scattering terms such as Sff and SF& The corresponding intravalence band scattering rates have the same appearence as Sfj and S$$ with only c and v exchanged. We obtain 0

J [~:‘_(k,-t’;r)Ge_c+~~,~‘;~) +

S;;(k,t, = 0;t) = -

df’

G”C+(k,-t’;z)ZC,C_(k,z’;t)

+Z~_(k,--t’;t)

G?+(k,z’;~) + G~+‘,k,-t’;f)~V,C_(k,t’;t)]

.

The intraband contribution can easily be simplified by choosing the representation (assuming now that f depends on I). For the cv-components we use the symmetry [XP(-r,)]*

= -AN(&) ,

(302) (264)

(303)

where A can be G or 2, and transform into the rotating frame according to Eqn. (282). We obtain Sf;(k,t, = 0;t) = $Y:-(k,w

= $/h;t)

[l -f’(k;t)] (304)

Similarly; we obtain for the out-scattering term S;c,,(k,z, = 0;t) = ;F+(k,w + jdl’2Re{%T!‘+(k,i;t)

= &h;t)

[@(k,r’;f)]*}

f”(k;t) .

(305)

0

As is well kown, the intraband scattering rates are related to the Fourier transforms of the selfenergies with respect to the difference time t,, evaluated at the particle energy: Tf,,(k; t) = j+_

(k, EL/h; t)

(306)

and T&&t)

= ~~~+(k,&R;~)

.

(307)

In addition to these scattering contributions we have in Eqns (304) and (305) the interband terms, which are similiar to the field renormalization terms C Y(k - k’)P(k’)*P(k) in the Hartme-Fock approximation. To evaluate these terms we need the solution (or at least a reasonable ansatz) for the dependence of Gcv on t,. One possible ansatz (not a solution!) is a factorization of the form

R. Binder and S W. KD&

364

@_(k,t,;t) = g+-(k,t,;t)@-ws)

(308)

with the obvious constraint g+-(k,O;d

= 1.

(309)

There is a corresponding ansatz for the ‘-+‘-component with ‘-’ and ‘+’ exchanged in Eqn. (308). With this ansatx we obtain [l -

Sfi(k,f, = 0;t) = +!_(k,&/R;t)

f”(k;t)l

- jdt’ZReI$~~_(k,r’;t)gr+(k,r’;r)

P*(k,l)j

,

(310)

0

where we used

iA[iF+(k,O;r)l* = rp*tk;o,

(311)

and, similarly, SF,(k,z, = 0;1) =

+

i

dt’2Re

0

;ZEC+(k&R;I) [l -f”(k;t)]

~ea(k,I’;r)g:_(k,t’;t)

i

P*(k,l)j

.

(312)

Common to all possible gs is that the oscillation frequency of g with respect to 6 is roughly of the order of half the bandgap. One can see this by examining the free-particle Gcy(Eqn. (270)), which suggests the choice g+_ (k, t,; t) = e--i(E; + ~~)~,Icm

.

(313)

An even simpler choice would result from the neglection of the energy dispersion in Eqn. (3 13), i.e. g+_(/&t,;r)

= +w(2*)

.

(314)

The most common ansatz is the so-called Shindo approximation,tl’@ g+-(k,t,;t)

=

f(k)

1 -P(k)

--i(E;/h +

f”(k) e

g-+

(k, t,; t) =

f(k)

f-C(k)

e-i(Q’R - y,t,

,

(315)

!

f(k)

--i(E;/h + q,,

[l - f”(k)1 e

_

- 11 -_P(k)l

e

-i(Lqh

- yb,

.

I

(316) To obtain Eqns (315) and (3 16) one has to compute GNpofor noninteracting particles (V = 0) from Eqn. (295), evaluated in the rotating wave approximation with a stationary light field.

365

Nonequilibrium semiconductor dynamics

Assuming that, due to the time-independence of !E’,the time-derivative on the left-hand side of Eqn. (295) vanishes, the solution for Gmeois obtained immediately. The factor g is then defined as g(k ~1 =

Gcv** (k, 2,)

(317)

w~*(k,O) ’

The generalization of this ansatz, which contains the correct limiting behavior, is discussed in Ref. (126) . We wish to emphasize that all the above choices for the t,-dependence of Gcv can lead to severe problems in certain parameter regimes Obviously, Eqns (3 15) and (316) are not defined if ,fY(k) = f’(k). Also, the derivation of these equations assumes explicitly stationary solutions, and it is not obvious that the generalization to explicitly time-dependent distribution functions is always meaningful. The ansatz might not be valid if effects nonlinear in the light field amplitude are important. The problems with Eqn. (313) are less obvious., but we want to mention that in a system with optical pktsmons the resulting expression for the selfenergy might diverge. This, however, is not the case in a two-dimensional system.(171) Having expressed these warnings, we would like to proceed and evaluate the scattering terms within the Shindo approximation. Using the identity (310 (with S d +0), we obtain Sf;(k, f, = O;t) = +_tk,

G/h; t) [1 - fc(k; ?)I - fqk) 1 - f”(k, tl Aw-E;-F--i6 1 - f’(kA - fiw_EF,+yQ._i6

Similarly, we obtain for the out-scattering contribution

x

! p(k)

1 I’ p*(k;l)

k 1P*(k;t) I.

f’(k, I) AUbEC+~-ib2

The corresponding interband terms are $g(k,O;tj

=

1 P(k, 11 - j-W 0

I

(319)

(320)

dw 2rr

1 -jYkd

1 - fc(W

hw-E~-hwo-id-hw-&-i6

1 -f”(k;r) 1 - fYW) nw-&~+~5-hw-Efk+)1Wo+ib t

Zy_(k, w; t) 1 XXk,

,:,,]

R. Binder and S. W. Koch

366

I- f’(k;l) 1 -f&t) hw-EVk-_hwo/2-i6-hco-E~+hwo/2+iS

xi%:(k, w; t>

(321)

and iz(k,O;z)

=

f”(k,l)

1 -f”(k;t)

dw [S 2TT

f’(kt)

p(kt) hw-E~-_hwo-i6-hw-E;-ii6

f”(k;t) P(k;t) hw-$+i6_hw-&+hcug+ib

Z’_‘+(k, w; t) jY+tk,w;t)] x!P(k; t)

f”(k; t) PO? t) ho-E~-_hwo/2-ib-hw-E;+hwo/2+i6

xi%?“+ (k, w; t).

(322)

We will continue the discussion of these relations in Section (2.7.9.2.). 2.7.6.3. GeneralizedKaabnoff-Baymformalism. Whereas in the previous subsection the scattering rates were evaluated at the same (macroscopic) time as the time derivative of the kinetic functions f and !P, the generalized Kadanoff-Baym formalism (GKB) of Lipavskjr, SpiZka, and Velicky(‘32) offers a practical possibility to treat retardation and memory effects in the kinetic equation. Generally, the time derivatives of f and P depend on the values of those functions at earlier times Also, kinetic retardation effects should account for the dynamical processes described by the w-dependencies which occur in the KB formalism. To keep the analysis simple, we will employ in the following the ‘free generalized Kadanoff-Baym formalism’ (FGKB).(“j7) Using Eqn. (292) together with the retarded and advanced functions for noninteracting particles, Eqns (261) and (262), we immediately obtain

iZEY(k;tlt3) [*.

+

i.IT?(k;t3tl)

e-iEp(t3-t1) esipF

[~5,,-fY3~(k;t~)]

01- t3) [S,,

-F""(k;t3)]

1

-

(323)

A comparison with the KB result (e.g. Eqn. (3 19)) shows that in the GKB the memory effects of the kinetic quantities and the dynamical correlation effects are not seperable any more. Similar to Eqn. (323) we have the out-scattering term

Nonequilibrium

361

semiconductor dynamics

2.1.7. The longitudinalpolarizationfunction 2.7.7.1. Two-timeformalism. Within the present theoretical model, the correlation effects and kinetic processes can be described in terms of the longitudinal polarization function II. It is determined by the response of the charge density to an effective longitudinal (electro-static) potential, which can be caused either by an external potential or the charge density fluctuations within the system. The retarded component, lI,, determines the effective (retarded) screened Coulomb potential. According to Eqns (238) and (239) only the ‘+-’ and ‘-+’ components of ll need to be known for a knowledge of the retarded and advanced components Within the exact two-time formalism and the SHF, no further transformations have to be performed, and we obtain directly from Eqn. (213) n,(q; 2122)= ihO(tl - td 1 1 [G?? (k; 1~22)G? VW ,$ - G’:?(k;t,t2) lI,(q;fltd

= -ihEW

G!%&

G;121,)] ,

- tl) 11 [G??(k;f112) YW k

- G!??((k;t,t2) G!?:(j;-

(&- p; t2tl)

i;tM]

(325) G??(i-

.

G;zztl) (324)

The physical contents of these relations might become more clear from a comparison with the corresponding Kadanoff-Baym relations 2.7.7.2. Ka&no$-Baymformalism. In order to evaluate the intraband polarization (325) and (326) within the Kadanoff-Baym formalism, we use again Eqn. (264) to represent the Green’s functions which are diagonal in the Bloch band indices, and, to be specific, we use the Shindo approximation, Eqns (3 15) and (316), for the off-diagonal (optical) Green’s functions Within the Kadanoff-Baym formalism and the Markov approximation in t, we use the representation (263) and (264) with the parametric time-argument I and obtain for Eqns (325) and (326), after Fourier transforming from fr to CL),

R. Binder and !S.W. Koch

368

f(k;t)

-f”&+~;z,

-hw+E~-$+9+Rwo+i6 .f(k;l) -f”(&+‘i;t, hw + & - of+*+ i6

I

(327)

where the small 6 stems from the O-functions (lim6 + +O). If no optical polarization is present, but the occupation functions am nonzero (which is the case for times longer than the dephasing time but shorter than the recombination time), only the first line of Eqn. (327) contributes to the screening. This is the well-known Lindhard (or RPA) formula. In this case we obtain from the imaginary part of &lo the well-known intraband pair excitation continua: Iml-I,(q, co;t) = -7r c

[j-Y&)

- j-Y& + Zj;t)] G(hco + $ - $+J

.

v,t

(328) This contains the energy-conserving &function.The frequency argument (u is the energy difference of two one-particle states and the distribution functions assure that a transition is only possible from an occupied to unoccupied (or vice versa) state. The sum over v = {c t , c 1 , v t , v 1} indicates that, contrary to quantum-mechanical exchange effects, all charge carriers contribute to screening. If the optical polarization is present, it too can contribute to intraband transitions (second term of Eqn. (327)). Of course, !P enters quadratically since one P (one ‘photon’) can only cause an interband transition. Other than that the general structure of the optically induced screening terms is very similar to the usual intraband screening terms They too contain the information about the Pauli blocking restrictions as well as the energy conservation. The occurence of these terms is a consequence of the fact that, within the Shindo approximation, the retarded (advanced) interband Green’s function is not zero. Any approximation in which the retarded (advanced) interband Green’s function is zero (like the noninteracting particle approximation, Eqn. (3 13), and the free generalized Kadanoff-Baym ansatz), does not include optically.induced intraband screening. This can be seen quite generally from the fact that Il, = -G+-G, - G,G+- [see Eqn. (255)]. However, even in this approximation, the functions lI+- and II-+, which enter the differential cross-section (see Section (2.7.9.)) contain terms proportional to !P!P*. The Shindo approximation shows in a simple way how the external field can lead to v-cband coupling (and hence to screening contributions):

@(k, w;t)

=

P”(k; t) f(k;l) -_fYkl)

1 x i hw-&+hwo/2+ib-

1 hw-El-hw0/2+i6

1 ’ (329)

Contrary to the free-particle approximation, within the Shindo approximation the external field induces poles roughly in the middle of the energy gap. The dispersion follows exactly that of the valence and conduction band. The intraband transitions between these induced states give rise to the optically induced screening effects (see also Ref. (172)). The ocCurence of the &function in Eqn. (328), or, mom generally, in the imaginary part of Eqn. (327), is a consequence of the separation of time-scales In general such an energy conservation law is not valid. Clearly, even in a self-consistent Kadanoff-Baym approximation,

Nonequilibrium semiconductor dynamics

369

in which the one-particle energies have to be mnomalixed with the complex self-energies, such an energy conservation will be ‘smeared out’, even if the separation of time-scales is valid. For the application of femtosecond dynamics, the perhaps more important aspect is the nonseparability of time scalea Clearly, Eqns (325) and (326) do not seem to contain a oneparticle energy conservation law, and this can be discussed in mom detail within the generalized Kadanoff-Baym ansatz. If, for example, the energy difference AE = E:- $+p is of the order of the exciton binding energy in GaAs (4.2 mev), then, in order for AEI /h >> 1, we need a timescale such that I >> 0.15 ps 2.7.7.3. GeneralizedKadanofl-Baymformalism. Using, as before, the representation for G+_ and G-+ discussed in Section (2.7.5.2.), we obtain after a short calculation iRII,(q;t,rz)

= @(t, - fz) C

eWitEkY - Ei-#)(tl - tz)‘h

V.k ij;rz) -fYStz)]

[I’&

iMI,(q; t1z2)= -0(t2 - tl) C

e

,

--i(ei - e&)01

(330) -

t2)lh

V.E

-fV(kt*)]

[fd-&I,

-

(331)

The main difIerence between the KB and the GKB polarization is that in the GKB case the density change at time 11, caused by an effective potential at time t2, is evaluated with the distribution function at the (earlier) time of the potential perturbation, whereas in the KB or, more precisely, in the t-Markov approximation, all functions are evaluated at the same time t. In that respect, the KB and the GKB represent opposite exptremes of minimum (KB) and maximum (GKB) memory effects The exact two-time formalism can be expected to yield intermediate results what the memory effects concerned. Until now, however, this last statement has not been proven rigorously. The second difference is the absence of optically induced screening effects in Eqns (330) and (331), which is a consequence of Gy= = 0 in the free generalized Kadanoff-Baym ansatz. This point has already been discussed in Section (2.7.7.2.). 2.7.8. The screened Coulomb potential 2.7.8.1. Two-timeformalism. Again, within the two-time formalism the general equations of the SHF approximation have to be solved directly, without any prior simplification. For the retarded Coulomb potential one has to solve

s

(332)

dt3 $,,(q; t,t9 S&q; t3t2) ,

(333)

W/a(q;f1fd = f’,&t~ - 12) + V, dh Hr/o(q;tlf3)Wr/a(q;t3t2)t or, equivalently,

GAq; flfz) = $,a(q;t,td

+

s

r2

with $,,(q;

t122) = Jf&,&;

w2)

,

(334)

370

R. Binder and S W. Koch

where fIrlo is given by Eqns (325) and (326). As above, the easiest way to obtain insight into the physi.cal meaning is to look at the equations in the Kadanoff-Baym approximation. Because of its specific time-structure, the formal advan2.7.8.2. Kaaiano$-Baym formalism. tage of the KB ansatz lies in the simplicity with which the Fourier transform with respect to the relative time zr of essentially all equations can be taken. Transforming all quantities in Eqns (332) and (333) to relative and ‘center-of-mass’ times t and t,, respectively, performing the Markov approximation in t, i.e. approximate, for example, in ll+,(q; tl td -= t1 + t3

I1 + 12- 02 - ts) ~ 11+t2 _ t

2

2

2 (335)

(because (t2+) is the relative time argument of W,,,(q; t&)), and taking the Fourier transform with respect to t,, we obtain immediately &Aq,

w; I) = vq + vq l-b/&, w;t) Wdq,

w;t) ,

(336)

and

Sdq, w; t) =

$Qq, w t) + $,,(q,w; t) %kA

0;

t) -

(337)

The popularity of the Kadanoff-Baym ansatz (beyond the regime of its strict validity) becomes clear from the simplicity of Eqn. (336) (or (333)). The equations are simple algebraic equations They correspond to the infmite summation of bubble diagrams (the TVA-form for n). We obtain for the screened potential the well-known form

w t) =

W/&,

v,

1 - vql-Ir/cr(qlwit) -

The quasi-static approximation, discussed in Section (2.7.2.), is now readily obtained by taking the LO= O-component of Eqn. (338). With W,(q, 0; t) I W(q; t) we have

w(q;t, = 1 -

v,

(339)

vqn(q;t)

where n(q;t)

=

1

v,k

fwif)

-fQ+ii;t) F; - F;+p

*

(340)

To consider optically induced screening in the same way, one can easily amend Eqn. (340) by adding the !P!P*-contribution of Eqn. (327) with w + iS = 0. Numerous simplified results for ll(q; t) are known for the limit q + 0, such as the result for quasi-thermal equilibrium l/l$l(q;

t) = -Ld ;

E

)

(341)

where pv is the chemical potential of the particles in band v (note that we do not have a spin summation in the density ny = LWd&f”(k), since the band index v labels explicitly the individual spin components of the bands). Away from quasi-thermal equilibrium, the following forms of ll(q; t) are easy to use in numerical calculations for three dimensions (d = 3),

Nonequilibrium semiconductor dynamics

371

and in two dimensions (d = 2),

1

my f”(k

= 0;t).

Y

The two-dimensional static screening has several strange aspects, such as the independence of density at zero temperature (a discussion of this feature can be found, e.g. in Ref. (173)), and the fact that them is no static screening contribution in a nonequilibrium plasma in which the k = O-component off is not occupied (see, e.g. Ref. (101)). The co # 0 components to the screening function contain, apart from the intraband pair excitations, the collective plasmon modes Therefore one needs to look at the full densitydensity correlation function S,(q, w; t) in order to discuss screening processes With

Sk?,Wl) =

$(q, w;z) 1

_

cqq,

w;t)

(344)

it is clear that the dominant contribution stems from the zeros of (1 - So). More precisely, the back-Fourier transform from o to t, is dominated by the poles of S,(q, w; t) with the smallest imaginary part. Such a damping (Landau damping) exists even if the imaginary parts of the one-particle states are neglected. The dispersion relation of the corresponding plasma excitations (plasmons) follows from the condition 1 - S”(q, w) = 0. More details of plasma dynamics can be found in many plasma text books (for example Refs (174) and (175)). Recent discussions of plasma excitations in nonequilibrium quasi-one-dimensional plasma are given in Refs (176) and (177). For small q, we have (345) In 3D, the result for S is particularly simple: limS,(q, w; 2) =

P-0

w;,(t) (w + ibJ2 - w$(t)

with (347) The phenomenological

extension to nonzero q is (see, for example, Refs (178) and (179))

s(q’w”) =

w;,(I) (w

+

ifi)

_

o;(t)

(348)

with the effective plasmon dispersion (for d = 2,3)

u$(t) = u$[(f) [ 1 + $+]+:[&+&12. Here, the screening vector

K

is defined by

(349)

R. Binder and S. W. Koch

372 Kd -

V,rI(q;t)

1 = lim,-’

(3 50)

.

Simihar to Eqn. (328), the imaginary part of S is determined by the possible longitudinal excitations of the system, Ims,(q,o;f)

=

-77

$$

(S(w-

Wq(f)) - S(f.u +

w,(t))) .

(351)

4

For the dispersion fitting parameter C we choose (unless otherwise stated) in all numerical calculations presented in this paper C = 4. The reason for this choice is that a stationary excitor& probe spectrum of a quasi-thermal system, calculated within the quasi-static approximation, shows a significant red-shift of the exciton resonance for C < 4. Since such a ted shift is not observed experimentially, the choice C = 4 seems to be appropriate. 2.7.8.3. Generalized Kaabno$-Baum formalism. Formally, the equations for the retarded and advanced components of the screened Coulomb potential are exactly the same in the generalized Kadanoff-Baym formalism and the two-time formalism. The only difference is the specific form of the intraband polarization. The generalization of the plasmon-pole approximation in the FGKB ansatz was performed by El Sayed et a1.(“j7) The equation for the density-density correlation function (333) can be investigated if one uses the GKB results for the retarded (advanced) polarizability, Eqns (330) and (331). In the limit q - 0, we have hn#q;t,r2)

= -WI

- t2) [II - 221402)

*

(352)

Therefore, in this limit it is straightforward to take the second time-dervative of the retarded version of Eqn. (333) with respect to tr . One obtains the generalization of Eqn. (346): (353) The initial conditions follow directly from Eqns (333) and (352): mwz)It,=,~

= !ie

(354)

$(w2)

=o,

(355) (356)

=

-c$&)

.

(357)

El Sayed et al. (16n find an approximate solution to this initial value problem and extend the solution to nonzero q-values in a way corresponding to going from Eqn. (346) to Eqn. (348): ml; ttt2) = -@(t, - 22)

402) $qiyjqg

sin

Id23 i

LO&)

.

(358)

1

Apparently, for a description of transient screening both the time-dependence of the plasmafrequency and the ‘energy uncertainty’ am important.

373

Nonequilibrium semiconductor dynamics

2.7.9. Selfenergies and generalized dl@erential scattering cross-sections Since the selfenergy components Z+_ and Z_+ can be in2.7.9. I. no-time formalism. tepreted as scattering rates, which, in turn, are obtained from W+_ and W_+ by a wave-vector summation, = ihx

Z??((k;tltd

Z!y(k;t,t2)

W_+(k -&;tztl)

k

= -iR 1

GT?(k’;t,tz),

W+-(k - ktztl)

G!??(k’;tltZ) ,

k

(359)

(360)

one might call the quantities iW-+ and iW+- generalized differential scattering cross-sections These ‘+ -’ and ‘- +’ components of the screened potential can be expressed quite generally (i.e. not restricted to the assumptions of spatial homogeneity and isotropy) as

W+-(ml) = W-+ Ml) =

I J

dtdt4 Kttzt3)~+-(t3td Wa(t4tl) ,

(361)

dbdt4 K(tzt,VI-+

(362)

(t&l

WaO4tI 1 .

Here, the surpressed variables and summations are those of Eqn. (205) or Eqn. (21 l), respeu tively. The proof of Eqns (361) and (362)(‘*O)is analogous to the corresponding relationship for G+-.(‘26) Evaluating Eqn. (190) for the ‘+-’ compoment, we obtain I

d5 ~1(25)W+-(51) =

d45 Y(2-4)l-I+_(45)W,(51),

(363)

where we have defined(130) ~,(12) = 6(1 - 2) -

d3 V(1 - 3)l-I,(32) .

(364)

With W,(21) =

I

d3 ~;‘(23)V(l

- 3),

(365)

which follows from the retarded component of Eqn. (190), we obtain from Eqn. (363), upon operating with E-’ from the left, the relation (361). A similar proof holds for Eqn. (362). For spatially homogeneous systems Eqns (361) and (362) take the simple form W+-(q; tztl) = W-+ (4; lztl) =

s I

hit4

K(q; tztdn+-

dhdt4 WXq; htdn-+

(q; ml

W,tq; tat11 ,

(q; t&d Wq;

t.dl) ,

(366) (367)

where the intraband polarizations follow simply from Eqn. (206) or (213), respectively:

n+-(q;t&) = -ih 1

G!‘?(k;tlt2)

G?F(i-ij;t2tl),

(368)

ll-+(q;tlt2)

G!!?(k;tltz)

G??(&-G;t2tl).

(36%

= +iR 2 v2~

This concludes the derivation of the equations describing the electronic properties of the coupled semiconductor-light system within the screened Hartme-Fock approximation in the

R. Binder and S. W. Koch

314

full two-time formalism, which takes into acount all retardation and timedependent correlation effects To be solved in a selfconsistent fashion are Eqns (294), (295), (296), (300), (332), (325), (326), (359), (360), (366), (367), (368) and (369). For the near future, this does not seem to be possible. To obtain more insight into the physical meaning, we will start again with the discussion of the Kadanoff-Baym approach.

2.7.9.2. Ka&no&Baym formalism. Within the Kadanoff-Baym formalism the generalized scattering rates iI+_ and iL+ can be obtained readily from Eqns (359) and (360) along the lines discussed above (i.e. employing the Markov approximation with respect to t and, for example, the Shindo approximation): ZT(k,w;t)

zy+((k,w;I)

= 1 K+(k k =

-&,Ej; - ho;t)

1 w+-
-Rw;t)

jYk’;t)

,

[l -_Y@‘;z)l

(370)

I

(371)

P and

[

W-+ (k - #c,EL + hwd2 - hw; t) jYk’;t)

- W-+ p-i&

[

w+- (k-K&

-hwo/2-hw;f)

(372)

[l-f(k’;t)]

+Rw0/2-Rw;z)

- w+_ (k-8&

fc(k’;r)] ,

-Rwo/2-hw;t)

[l-f”(k’;z)l]

.

(373) In order to perform the Markov approximation with respect to t = (21 + 22)/2 in Eqns (366) and (367), we note that the macroscopic time argument of W,, for example, should be approximated by I, 12 + t3 -=

2

t2+t1

-

2

+

1 -(t3

2

- t4 + t4

-

21) =

t2 + t1 -

2

(374)

since (23- f4) and (t.4-21 1 are the microscopic time arguments of II and W,, respectively. Similar considerations hold for the other macroscopic times in Eqns (366) and (367), and as a result all macroscopic times arguments are t, which allows us to perform the Fourier transform with regard to the microscopic time arguments For the contributions of II containing products of interband function (such as GcvGyc), we perform the transformation into the rotating frame on both interband functions and then perform the Markov approximation. To be specific, let us use the Shindo approximation, Eqns (315) and (316), to evaluate the internal w-integrations After a short calculation similar to that discussed in Section (2.7.7.2.), we 6nd

Nonequilibrium semiconductor dynamics

375

iW-+(q, w;t) = --27l wrtq, w; 2) watq, w; 2)

iw+-(q,w;t)

= +27~ W,(q,o;t)WJq,w;t)

+

[F(X + zj;tpm*(kt) + cc] B(&,&w;t,

1 cvk

I )

(376)

with

x (11 - P(k; t)] pc& + +;I) b(hw + E; - E;+@) -[l

- f’(k;I)]

j-Y& + G;t) 6(hW + E; - E;+a - hwo)

-[l

-f&t)]

f”(i; + P;t) S(hw + Eyk- E;+4 + hwo)

+[1-f(k;f)]f”(~+~;t)6(hw+~~-ECk+g)].

(377)

To obtain B from D one has to replace all f’s by [ 1 - fl. Just as in the perturbative approach to the theory of elastic collisions,(‘*‘) the differential cross-section contains the square of the interaction matrix element, W,(q, w) W,(q, co) = I W,(q, wj I2 (valid for real w; the analytical properties of W, and W, are explicitly accounted for by the i&), and the energy conserving b-function. In addition, we have the fermionic Pauli blocking factors f[ 1 - f], the screening of the interaction, which is brought about by the same scattering processes as described by the differential cross-section, the overall time-dependence on the macroscopic time t, and the contributions from the optical polarization. In the absence of the optical polarization, the scattering terms ST and rU [Eqns (319) and (320)] yield the well known Boltzrnann scattering expressions

.syqk;rr) = SV,~ r:(k;f)

[l -F’(k;l)l

(378)

with

(379) and (380)

R. Binder and S. W. Koch

316

with

(381) If the system is close to quasi-thermal equilibrium, one can evaluate the scattering rates using Fermi functions fF(k). With the help of the identity fF@

+

hw)[l -

fF(dl

= gE(hm)[fF(d

-

fF(E

+ hw)] ,

(382)

where gr, is the Bose function with zero-chemical potential, and the relation IW(q,o;t)121dUq,w;t)

=

ImW(q,wt),

(383)

which follows from the definition of W, one obtains the familiar result

and

r;u(kf)=-21

~{w~~;+~-E;;I)}

0

gE($+p- E;) r1 - _I# + gir)l .

(385)

Hem, the only time-dependent quantities are the temperature and the density of the individual plasma components We wish to stress that the dephasing contributions to the optical polarization equation, Eqns (321) and (322), contain in general not only the intraband scattering terms Tm + r,,,,, but also polarization scattering terms The latter are those y-contributions to

(386) which are off-diagonal in &. The simplest and most intuitive relation which connects the optical dephasing rate wiih{he sum of the intraband scattering rates, and which neglects the off-diagonal elements of y(k, ti), can be obtained from the quasi-particle approximation. This approximation is best justified if the optical polarization is small (like in the linear optical regime), and if electron-hole correlaticns can be neglected (which, to a certain degree, is the case in the optical amplifiers where the plasma density is signi6cantly higher than the Mott density). As discussed above, in this situation one can essentially fix the frequency argument of the selfenergy, Zya(w) 2: ZTU(9’). Of course, the scattering rates and the imaginary part of the selfenergy are related by Zy(o) + Z”+(o) = 2i In&(w), where Z, is the ‘dephasing rate’ (more precicely, the coherence-loss rate) for each band. In order to stay close to our present formalism, we would like to show how the quasi-particle approximation for the optical transitions can formally be introduced into Sg and sf. In contrast to the Shindo approximation, one has to use

Nonequilibrium semiconductor dynamics

g(r) = e-e

377

- wo/2)r

for the P-contribution and g(t) = emi(&; + WO/2)z for the Xwcontribution. The Y-contribution is to be neglected in this case. This way one obtains the familiar result

(387) which yields the optical dephasing rate

(388) ye would like to note in passing that a very simple model for off-diagonal dephasing r(k, K) can be obtained if one uses the ansatz Eqn. (314) instead of the Shindo approximation. Assuming, again, that one can evaluate the scattering rates in quasi-thermal equilibrium, we obtain

x [ f;(k”;l)

+ g&;r

- G)]

-Im{W(k-L,&-Eg;r))

-Im{W(Q

-Z,~i;t){

[fi(k;r)

+g&)]

.

(38%

Since, in this model, we essentially approximate the intraband tran_sition energies by fixing one frequency argument to the band minimun or maximum, P’(R = 0), this model can only be regarded as a very crude analytical approximiation for off-diagonal dephasing. Also, numerical investigations show that in the low density regime the dephasing processes seem to be significantly overestimated by Eqn. (389). Finally, we wish to mention that the complex energy renormalization of the one-particle energies, which is formally given by CT& E;; t), leads, in principle, to a self-consistency requirement. In each ansatz for Gvlfi (k, W; t) one has to replace the energies E; by the shifted and broadened one-particle energies (see Eqn. (272)). Since Z+- and Z-+ determine &la, one has to compute these quantities selfconsistently for each macroscopic time t. In contrast to the KB ansatz, the two-time formalism does not contain such an explicit self-consistency requirement. There the solution is inherently self-consistent, and the de-correlation of twopoint functions with two dilTerent time arguments should be a natural consequence of the various time-scales involved in the temporal structure of the equations of motion. The set of equations to be solved in the Kadanoff-Baym ansatz, evaluated within the Markov-approximation in the macroscopic time and the Shindo approximation, is (in addition

378

R. Binder and S W. Koch

to the semiconductor optical Bloch equation) Eqns (319), (320), (321), (322), (327), (338), (370), (371), (372), (373), (375) and (376). 2.7.9.3. GeneralizedKaaiznoff-Baym formalism. As mentioned above, the screening functions (or, equivalently, the differential cross-sections) do not contain interband polarizations if the retarded and advanced Green’s functions am approximated by noninteracting particle Green’s functions Thus, within the free generalized Kadanoff-Baym formalism we obtain ;w-+(gW

= 1

K(q;t4t1)

~$$%xf$lzll)

WYZ tW3 [

- 24) e

-

$9

(13 -

6 ww - f vly (&; 14)] f tiV’ (h -

+ Wt4 [

-i(EZ

5 YY

-i(EF

t3) e

- f%;t,,]

-

$7

14)/h

R 14)

03 -

fT&-

f4)lh

ii;t3)}

(390)

and j!++-(q;2112)

= -

v~I-+3$4w(q;12131

NW3

-

24) e

fVT&;t4)

-iC$

[6,,

+ 6904 -

- EQ(t3

-

- f”“(L -i(Ek”

t3) e

fY’Ykl3)

W,(q;f4tl)

[&,,

24)/h

G;t4)]

- ‘&‘(I3

- t4)/h

-pwLg;r,$.

(391)

The generalized scattering rates now become +Yk;1112)

k+(q;121,)

= 1 0 X

WI

-

t2) e

-i.&

01 -

22)/h

7%

+ i;12)

i + Wt2 -

21) e

-iE&

01 -

12)/h

f’%+

ij;t1)

(392) and

$‘,n(k;l,12)= - 1 Lw+-(q;12tl) 0 X

i

WI - 22) e

+ WI2

-i$+,

- TV> e

01 -

12)/h [

-ieF+$ (21 -

t2)lh

6 VI+2 -fTZ+~;tz,]

6 E WY2 -pQ+&l,)]

. I

(393)

Nonequilibrium semiconductor dynamics

379

Equations (392) and (393), together with the optical Bloch equations and Eqns (323), (324), (330), (331), (332), (390) and (391) are the basic equations of the FGKB. In order to obtain more insight into the temporal structure of the scattering rates, one can evaluate St, and S,, explicitly within the static screening approximation, w,/l&;

t2t11

=

601

-

12)

w(q;h)

(394)

,

which yields

1 1

S2(k;tlf,) = ;

YV4VS

E6 v,vs

i

-

fVTC

W(q;t,)

W(q;td

_m

~;t,)l fmvw;13)[Gqv4

+

i(--E;:8 +‘E;

pp

I’ dt

J-f

-E&

-

p"(k

- ij;t3)]f"n(k;13)f

-

f”“(K

- k;t3)1 PYW3))

+ EF)(Z, --13)/h

e

+

116

vlw - pyi+ -i(-ET+9

i.j;t3)] pYk’;t3)[6Y3v4 +E;

- EZ_$ + q)(z,

- t3)lh

e

(395)

and

I

1

LP”(k e

+ kt3)l

i(--E;+q

+ E;

[S,,

- E&@

-fwv4w;t3)1[pv3(k + EF)(Z,

-

ij;t3m6v5q

-j-v5Yk;13)1)

- 23)/h

+ i IF2

cs + i; f3)l -i(-Ez+g

e

[&3v4

+ E; -E&

-fmv4w;13)l[pY3(k +$)(I1

- $i~t3)1[&,y

-fwvvkt3H)

- fj)/h

(396)

The general structure of Eqns (395) and (396) is, of course, similar to those in the KadanoffBaym approach. The internal sum over v3 and v4 generates terms like [ 1 - f]f and !P*P (these are the intraband screening terms, including the optical polarization induced screeing). It also contains the appropriate spin summations The scattering terms See in the equation for the particle distribution functions contain contributions of the form [ 1 - f”]p and P!P* (generated from the sum over VS), and optical selfenergy contributions SN cont_ai, terms like Py and f’!?‘. The latter terms define a very general dephasing function r(S K; ti t2), containing both ‘P-P-scattering and memory effects The temporal structure of the coupling to the Coulomb potentials (i.e. one potential contains the time variable tl and the other the retarded time variable t3) can easily be understood. In a non-Green’s function approach, the time evolution of a 2-point function is proportional to the Coulomb potential and a 4-point function. The time evolution of the 4-point function is again related to the Coulomb potential and 6-point functions As is well known, factorizing the 6-point functions and formally integrating the differential equation for the 4-point functions yields essentially the Boltzmann equation. Within the model of a phenomenological timedependence of the (screened) Coulomb potential, such an approach yields immediately the

R. Binder and S. W. Koch

380

temporal structure of Eqns (395) and (396). Of course, the advantage of the GKB is that it yields the equations for the screened potential as well. Equations (395) and (396) show clearly why the GKB could also be called ‘maximum memory’ approach. To determine the time derivative of a polarization or distribution function at time 11 the scattering integrals are evaluated in a way such that all polarizations and distributions are evaluated at the retarded time 13. We need to mention, however, that the memory effects will be reduced if one does not use the free generalized Kadanoff-Baym ansatz. Instead of using noninteracting particle energies, one should use renormalized energies (see the discussion at the end of Section (2.7.9.2.)). Just like in the Kadanoff-Baym approach, this leads to a self-consistency requirement, since the renormalization is determined by the scattering rates and vice versa. A simple model for such self-consistent calculations was discussed by Tran Thoai and Haug.(‘@ The imaginary parts of the energy renormalization limits the effective range of integration to times within the dephasing time. If the distribution functions and screened Coulomb potentials are essentially timeindependent SF (k, tl ti ) and Szt (k, ti tl ) reduce to the usual Boltzmann scattering terms in the limit of large times tl . More precisely, if the carrier distributions are created at time to (we can let to -, -co), and the condition (4,

k+P

+

E; - $_, + $)(t, - 20)/h x=-1

(397)

is fulfilled for all relevant energies E;, the time integral reduce to the energy conserving Sfunction: 1(-E&, + E; - E;+ + E[)(l, - t3)/h

+ 2rr 6(-E;+@ + E; - E&

+cc

+ El) .

The energy differences are typically of the order of a few meV If E = 4.2 meV and t is given in ps we have ~z/h = 6.38. This shows that on a 10 fs time-scale Et/h = 0.0638 < 1. El Sayed el al. suggest to set the exponential time factors altogether to unity if one is dealing with few fs-dynamics(‘68) It is also pointed out in Ref. (168) that in this case one even can take an unscreened Coulomb potential because the sum of I$ over i does not diverge. In the long-time limit only plasma-screening can remove the divergence of the scattering, since plasma-screening prevents the Coulomb interaction to diverge as q - 0. We would like to point out here that the use of the screened Hartree-Fock approximation bears an’intrinsic problem when dealing with the optical excitation of semiconductor. The justification of the SHF approximation is based on the fact that in a plasma the screened Coulomb potential can be regarded, in a certain way, as a small quantity, allowing for a perturbative approach with W as the smallness parameter. (This statement refers only to the fact, that Z = WG, in general the neglection of vertex corrections amounts to the neglect of certain kinds of diagrams, and, hence, is not strictly equivalent to a pertubation expansion in W. Also, since G is a function of W, not even Z is strictly proportional to W.) In a high density plasma the SHF approximation can be very good. However, to describe optical excitation, one has to start from a zero-density electron-hole plasma. In this case the initial kinetics might not be well described within an approach, where the essentially unscreened Coulomb potential is the smallness parameter. Clearly, the possible divergence of scattering terms in certain evaluations of the SHF approximation indicates the nonapplicability of the SHE On the other hand, in order to obtain numerical results, one is often limited to the use of the SHE As shown above, even the free generalized Kadanoff-Baym ansatz has, until now, not even been approached without further approximations like the static-screening approximation. From a

Nonequilibrium semiconductor dynamics

381

practical point of view, one has therefore to regard the SHF as the currently most feasable approach to a detailed microscopic theory of electron-hole plasma dynamics in semiconductors Also, one should keep in mind that in a fs-excitation process the validity of the SHF improves on a fs time-scale, and in pie-excited or electrically pumped semiconductors (optical amplifiers and lasers) the SHF can savely be regarded as fully sufficient microscopic description. We will discuss in the following section a completely different approach, the X(“)-approach, which does not have the shortcomings of a dynamically changing criterion of applicability. Its shortcoming is the extremely complicated structure of the equations of motion even in the lowest order expansion (,J$~)). 2.8. The x(3)-Approach Whereas in Sections (2.6.) and (2.7.) we discussed a diagrammatic approach to correlation effects in a system of charge carriers, we turn now to a recently developed approach which combines the classical nonlinear suszeptibility ($“)) approach of nonlinear optics(182-‘84)with the problem of the infinite hierarchy of equations of motions due to the Coulomb interaction. Since in many-body problems no exact solutions seem to be possible, one is usually confronted with the question what the best choice for a smallness parameter is The ‘small’ quantity then serves as an expansion parameter in a perturbative approach. The approach in Sections (2.6.) and (2.7.) was valid for a plasma, in which the screened Coulomb interaction is the small quantity. The unscreened Coulomb potential is not a small quantity and thus-in general-is not suitable for an expansion (note, however, that certain equations like the semiconductor Bloch equations, do not yield unreasonable results within the (unscreened) Hartree-Fock (HF) approximation, only the corresponding carrier-carrier scattering rates diverge in the HF approximation). In nonlinear optics the external light field is the small quantity. Since the light field is an externally controllable parameter, the ~(“1-expansion is in a way more satisfactory than the Green’s function techniques (at least for the description of ultrafast nonequilibrium processes). Recently Axt and Stahl extended the nonlinear optics approach to optically excited semiconductors, (‘33*185) In that approach, no assumption about the smallness of the Coulomb interaction has to be made. For any given order of the external field, the number of equations for the various correlation functions is finite. The disadvantage of this approach is that even in lowest order, in which nonlinear optical processes are described (i.e. the x(3)-approximation), the equations arc so difficult that without further simplifying assumptions they are almost completely inaccessible to numerical solutions Also, to the best of our knowledge, no general proof has yet been given that shows that the individual terms in the xc”)-expansion are convergent for arbitrary orders of n. Since the expansion coefficients are functionals of the unscreened Coulomb potential, it is not a priori clear that no divergences similar to the one mentioned above occur in the x(“)-analysis Before we present a more detailed discussion of the xfnf -approach (along the lines of Ref. (186)), we would like to briefely recall some basic features of nonlinear optics, in particular the exact form of the smallness parameter. The simplest case of a x(“) -expansion is that of linear optics, where all xt3)-terms are being neglected in comparison to x (‘)-terms. We would like to discuss this approximation for the case of the semiconductor Bloch equations, Eqns (82) and (83). The procedure is similar for any given order, but is illustrated most easily in lowest order. At this point, we can even neglect the Coulomb terms in Eqns (82) and (83), since they do not affect the general cut-off criteria of the x(*)-expansion. In the central part of this section, we will then discuss the Coulomb contributions to the ~(“1-equations in detail. To keep the following argument simple, we also make the rotating wave approximation and assume that f’(k; I 1 = fh (k; I) = f(k; 2). All these assumptions are made for reasons of simplicity, but they could easily be dropped. If

R. Binder and S. W. Koch

382

formally integrate Eqns (82) and (83) , and insert the result for f(k; t) into the result for P(k; t), we obtain the following integral equation for the envelope of the optical polarization:

we

P(k, 2) = P”‘(Lz., t) + 1’

I 4R

I -m

dt’

s --m

dt” ~“‘(k;

t - t’)n(t’)

Im[ R* (t”) P(k; I”)]

(399) with t

@“(k-f),

c

dt’x”‘(k

= -

, t - t’) M-W’)

,

VW

-co

X”)(k.t) , =

1 ih

e

-q, h

-

iyO)t I

(401)

and (402)

Iteration of Eqn. (399) yields the nonlinear polarizations !P(I), Pt3), !Pc5)etc, which ultimately allow us to compute the correct !P = C !P(“). Specifically, in 3rd order we have

!F’(k;t)

= -4h2 jdl’

jdl”

Im[ n’(P)

Z(P)

f df”’ x”‘(k. , I - t’) CUt’)

xc),,

t” - f”‘)] .

(403)

To justify the linear approximation for the description of a short pulse excitation, the condition lW(~t)I ,

<< ]P”‘(/Ct)] ,

(404)

must be fulfilled for all times t and momenta k. One could also employ weaker conditions by comparing appropriate time and/or momentum averages of PC31and !P(‘). It is instructive to evaluate condition (404) for the example of a very short pulse n(t) = A 6(t) ,

(405)

where the dimensionless quantity A represents the area (time integral) of the pulse. Inserting Eqn. (405) in Eqns (400) and (403) we find

(406) Similiar considerations apply to higher order expansions As a result, in the coherent regime a small pulse area is sufficient to justify a x(“) -expansion. This statement differs from the smallness criterion usually used in nonlinear optics. If one considers, for example, a stationary light field,

Nonequilibrium semiconductor dynamics

n(t)

383

WV

=

the above criterion is not applicable (in fact, the ratio P(3)/!Po) would diverge, since the area of a stationary light field in infinite). In such a case, one has to consider the effects of spontaneous emission, by adding a term -f&k; t) to the density equation (83). The equilibrium between constant excitation and recombination can make a ~(“1-expansion meaningful. For the Fourier components of Ip(“) one obtains, with the field (407),

P(3)(k; 0)

I

w)(k;

W)

I

=

1 Im (x(i)& W))( 772rsp

h lnolz

,

(4433)

which shows that the use of nonlinear susceptilities is best justified for off-resonant excitation (i.e. small x(i) (k w)). Hence, we & that we can define various smallness parameters according to the specific physical situation. However, all the possible expansions have in common that the results are classified in orders of the external light field. In the case of the Bloch equations, the lowest order (with respect to the light field) of the optical polarization is one (i.e. !P oc T’ in lowest order), and for the distribution (or carrier density) it is two cf CC!E2 in lowest order). Let us now turn to the problem of the interaction-induced hierachial structure of the equations of motion. In Eqn. (43) the Coulomb interaction couples a two-operator expectation value to a four-operator expectation value. Since the mininum order with respect to the E-field is connected to the question how many carriers the E-field generates, it is advangageous for the following argument to discuss specific expectation values for creation and annihilations operators of electrons and holes rather than the general space dependent field operators q(g) or conduction and valence band operators First we expand the w(xt)-operators in the Hamiltonian (32) according to Eqns (52) and (53), perform the basis transformation (89) and (90) so that fro appears in the Hamiltonian and the Coulomb contribution reduces to the simple form (532). We then transform into the electron-hole picture by means of cb + ag if v is a conduction band index,

cbt - agt if v is a conduction band index, c&-b+-

-kl

CL -. b-i,

if v is a valence band index, if v is a valence band index.

(409)

Owing to the transformation (89) and (90), the operators at and a am not the same as the original operators defined in Eqns (52) and (53). We use here again the conduction and valence band angular momentum labels s and i for the electron and hole operators, respectively, but at this point we do not use any specific properties of the angular momentum eigenstates Whether or not the one-particle states am eigenstates of the angular momentum is immaterial throughout this section, because we will not make any specific assumptions about the form of the dipole matrix element fi$. The resulting electron-hole Hamiltonian has the familiar form

384 (410) Since we are about to perfo_rma perturbation expansion, which leads to very long equations even if only low orders of S(l) am considered, we will compress the notation to an almost

extreme degree. The first step in this compression is the introduction of the antisymmetrized Coulomb potential Vac,(lE6) 1 v,& = *$V’k,-k$~,,~&vj

-

~k,-QI~v,,~b\l.~,}6k,tkz,k~+kt

(411)

where the plus sign is used for the intraband interaction, i.e. all four band indices denote either electron or hole bands, and the minus sign indicates the interband (electron-hole) interaction, i.e. two band indices are electron indices and two are hole indices This form of the Coulomb potential has the properties J/‘u _ I/m’* 1234 -

Vrn 1234

(412)

4321

= - by34

=

- by43

(413)

*

Although in our case Vos is real, we allow V” in Eqn. (412) to be a mom general (hermitian) interaction potential. The antisymmetrized dipole energy reads E,Y = ‘6&&,~

(M * B(t)

(414)

where the plus sign is used if VI is a conduction band _ index and the minus is used if _VI is a valence band index, i.e. Ef& = -6,-p&W . E(t) and Ezvkc = +6,_pi&k) . E(t). We can now use electron (VI = S) and hole (VI = j) creation and annihilations operators, (x/ (= o&J and al (E ctk,,,,), respectively, and rewrite the Hamiltonian (410) as

(415) The one-particle energies ~12are the electron energies 6,,,6~I~z$ if both band indices am electron indices, zero if one index is an electron and the other a hole index, and 6&fjh if both indices refer to holes (This specific form of ~12is acutally not important for the following discussion; the only important aspect is that the kinetic energy is a bilinear form.) The equations of motion for the Heisenberg operators are (with k = I throughout this section‘\ (416) and

a t-- -

i-q at

c

CE~~CX~ 2

2,3,4

v2?4,

azt &4

-

E;* U2 .

1 2

Finally we introduce the expectation value of a normally ordered product of electron and hole operators, N”,,,,, with n creation operators tit and m annihilation operators a. For example, we have

Nonequilibrim semiconductor dynamics

N1,3(1234) =

385

(418)

(&w3~4)

and %-q(123456)

= (&;af&x5a6)

.

(419)

We want to stress that all operators are Heisenberg operators with the same time argument I (which we suppressed to keep the notation compact). Within the x(“)-approach no correlation functions with different times are needed. Instead, we have the full hierarchial structure of the equations of motion. The expectation value in N “,,,, is taken with respect to the zerotemperature ground state of the semiconductor, ac(to)lO) = 0 (with to - -00). Whereas the x(“)-expansion can be performed for any other initial condition, this method is best suited if the initial state contains no electrons and no holes In this case the xc”)-theory yields a finite number of equations, whereas the number would be inihnite if initially charge carriers were present. If the system is initially in a thermal state, it depends on the thermal excitation [K exp(kT/E,)] whether or not the assumption of an initial zero charge carrier density is justified. (In pure GaAs, thermal excitation at room temperature can be neglected. In reality, the main contribution of initial charge carries are ionized impurities) The semiconductor optical Bloch equations (SBE) are the equations of motion for iwo?pera_tor expectation values In the pynt notation, the ‘cv’ component of No,3 (12) (with k, = k2 = k) is the optical polariz$ion P(k) and the corresponding ‘cc’ component of Ni,i(12) is the distribution function f’(k). Using (416) and (417) we find, after bringing the expectations values into normal order, the following equations of motion for the two-operator expectation values (generalized SBE):

+&(12)=1

[Ed’hd32) + Q3%2(13)1

3

+ 1

[v,?4,5%3(3425)

-

V3~45~3(3145)1

345 + 1 [~%f’4,(32) 3

+%%o

itNl,(l2)=1

-EgJVj,(31)],

(420)

[-$3%1(32) + 93%1(13)1

3 -

1 [V&n/,2(3452) - v&%(1345)1 345

- 1 [Epj*%2(32) 3

- E@%o(l%l

.

(421)

We introduced in Eqn. (420) the N&3 = 1 term, which yields the only contribution proportional to E’. It acts as a sort of ‘seed’ for all carrier distribution and correlation functions, given the initial condition of zero-density. In general, higher order correlation functions, i.e. expectation values with more operators, are of higher order in the external field. This nontrivial observation(133) is due to the fact that the Coulomb potential couples an expectation value N,, only to the same sort of expecation value, or to one with more operators, such as Yv-“+I,,,,+I. For example, in Eqn. (420) No3 is Coulomb-coupled to No3 and Ni3. However, the equation of motion for Ni3, JPOE 19:4/5-F

R. Binder and S. W. Koch

386

i$f,,(1234)

= 1 [ -&N,3(5234) :&,,(1254) + 1 [ V37556N

+ E&v-13(1534) + E&V-13(1235)]

3 (1564)

+ V&W

3(

1256)

~?‘&N,U356H - 2 [V&%&(567234) - V&,%&( 156734) 561 +v3&3\/24( 152674) - v&-7v24( 15236711 + EgWl(l4)

+ ggv-l1(12)

- ~$gvl1(13)

- 2 [Ep3*%&(5234) - EgN22(1534) :&v&(1524)

- Eg%&(l523)]

,

(422)

shows that its source terms are proportional to ENi 1. Since, on the other hand, the source terms of NII are proportional to EJV&, and hence of minimum order two, the minimum order of JV7v;3 is three. Let us now turn to the %-term , which occurs in the ‘density’-equation (421). Its equation of motion is

i$‘&(l234) =c [-@i&(5234)

- E&N22(1534)

:E&&2(1254) +E&h&(1235)] - c V’&J%2(5634) 56

- V4~~&2(1256)1

+ c [I’&, JV33(562734) - I’&2N33 (156734) 567 +V3?&7_Tv33(125674) - V4~67N33(123567)] - Eg*.7voz(34) + E,nSJ+J20(12) + 1 [Eg*N,3(2534) - E$*N,3(1534) 56 +E3y9V31(1254) - E4yN3, (1253)] .

(423

If we restrict ourselves to the xt3) -regime, in which we can neglect all terms of order E4 and higher, we can drop all terms proportional to EN13 (because JVi3 is already of order three) and -TV33(which is of order E4, as can be seen from its equation of motion). We therefore have

+%(1234)

= 2 [--E&%&(5234) - E&%-33(1534) :E3&(1254)

+ E&&(1235)]

- 2 V&,X22(5634) 56

- v&$%(1267)]

- Eg* J% (34) + E4yJ%o ( 12) + O(E4) .

(424)

However, in third order, the factorized form of the function 3vz2, i.e. the function y22 = N20%12, obeys exactly the same equation as JV&. Thus, within the x(3)-approximation, the

Nonequilibrium semiconductor dynamics

387

equation for the particle distribution is formally identical with the f-equation obtained in Section (2.2.) from the Hartme-Fock decoupling, Eqn. (83). The factorization for n/22 is a special case of the more general factorization result discussed in Ref. (186) . If both the first and second index of N are even, we have %,,2,,,

= N~,,,cJV&,

in lowest order in E .

(425)

A similar but slightly more complicated (thus less practicable) factorization holds if the first and second index of N are both odd. We want to stress that, in spite of the formal similarity of the f-equation obtained from the Hat-tree-Fock factorization and the xC3)-equation for Nil, the factorization (425) is different from the usual HartreeFock factorization. As can be seen from the example of n/22, the correlation functions factorize in minimum order into ‘polarization like’ functions In other words, JV& factorizes into N203v02 if the two creation operators create an electron-hole pair and the two annihilation operators destroy an electronhole pair. Otherwise the factorized functions JV& and n/02 would clearly be zero (note that the present formalism does not contain anomalous propagators like , because they do not have an optical source term). If, for example, all four operators in N22 would be conduction band operators, the expectation value would not be of minimum order. Such an intraband two-particle correlation function is of minimum order four, because it is similar to a product of two distribution functions A quantitative comparison with the usual Hartree-Fock decoupling could be achieved by defining the correlation contribution 69&,,, as difference between the Hartme-Fock factorized form of -TV;, and the full L&,. This would allow one to introduce the Hat-tree Hamiltonian in the equation of motion of N,, and also to test quantitatively the validity of the HartreeFock factorization by comparing the correlation contribution SN,, with the full solution Nm. The factorization rule (425) can be applied to the six-operator function N& in Eqn. (422). In the third order, we have LN& = N203v04. Since the other six operator function, n/33, which occurs in Eqn. (423), is of minimum order four, the only missing equation for a full xC3)analysis it that for N&J: iiNo4(1234)

= 2 [~i59&(5234)

+ .&N&(1534)

3

+&%t(1254)

+ %%4(1235)]

+ 1 CJ’2%5N14(5634)+ V3~56%4(1564) 56 --v;s6%4(2564) + v7$63v04( 1256) -v4?&&(1356)

+ I$563v04(2356)3

+ 2 [V&9Jl5(567234)

- V-$&n/,5(516734)

561

+v$$,%5(512674)

- v4y6,%-,5(512367)]

+ E2ai’9-02(34)+ E$M12(14) - E3asN12(24) + GPM12)

- Eg%2(13)

+ c [EE%,(5234) 56

+ Eg%2(23)

- E,“N,3(5134)

+Eg%3(5124) - E;;n/,,(5123)]

.

(426)

It is clear that the minimum order here is three (the term EN&). Again, the higher order contributions Ni5 and ENi3 in Eqn. (426) can be neglected within the x(3)-approximation.

R. Bin&r and S. W. Koch

388

NI

6 3 5P

0123456

-M

Fig. 3. Sketch of the coupling between normal ordered expectation values NN,M. The numbers in the circles are for the minimum order of the corresponding function in the external field amplitude. The dotted lines denote the coupling due to the dipole interaction between the light field and the semiconductor. The solid lines denote the coupling due to the Coulomb interaction. (From Ref. (186))

Similar to the equation for JV&, in which the kinetic energy terms and Coulomb contributions are equivalent to the quantum-mechanical one-atom (or hydrogen) problem, the kinetic energy and Coulomb contributions in Eqn. (426) are those of the quantum-mechanical two-atom problem. Hence, in analogy to calling P = JV&‘excitonic function’, the functions JV& can be called ‘biexcitonic’. Again, if the indices in JV& are not pair-wise ‘cv’, but all, for example, ‘c’, N22 cannot describe the interaction and possible binding of two excitons (biexciton), but only scattering states We have already discussed the scattering problem in connection with the Boltzmann equation in Section (2.7.9.). The biexcitonic problem based on the x(3)-approximation is discussed in detail in Ref. (185) . The complete set of equations within the x13)-approximation is now: Eqn. (420), Eqn. (421) with the factorized terms JV& = N&V& (we do not need to solve Eqn. (423)), Eqn. (422) with the’factorized terms N24 = N2oNo4 and N22 = N&N& and Eqn. (426) without the Nis and BNr3 contributions Functions like L&O can be obtained from No2 by talking the complex conjugate and reversing the list of arguments A graphical representation of the functions needed in a xc3)-analysis is shown in Fig. 3. Finally, we would like to list the general coupling mechanisms and minimum orders, as, for example, discussed in Ref. (186). In general the kinetic energy couples only N”,, to JV&,, , the Coulomb potential couples N,,, and Nn+l,,,,+i to N,,,,, and the optical E-field couples -7Vn-l,m+l,J%+I,~-I, X-Z,,,,, and J&+2 to N,,,. The minimum order in the external field for initial states with zero-density am -7V2n,2m = OW+“‘)

n/Zn+1,2m+1= 0(En+m+2) .

(427) (428)

The above analysis shows that, if the x(“) -expansion is valid, it is possible to test the validity of diagrammatic approaches by comparing the results obtain from Green’s function calculations with the x(*) -results To evaluate, for example, the validity of the screened-Hartree-Fock

Nonequilibrium semiconductor dynamics

389

approximation, one would expand the screened Coulomb potential in terms of the intraband polarization lI, i.e. W = Y(1 + Vfl + VlIVfl . . s), and observe that II is proportional to the induced density and thus of order E2. This allows for an expansion of the SHF-semiconductorBloch equations in orders of the external field. Moreover, this procedure allows us, in principle, to identify the various correlation terms in the xcn)-approach as screening contributions Of course, in any finite order in the E-field we can not obtain the familiar screened Coulomb potential W = V/(1 - VII), as the latter is obviously of infinite order in E. As another practical advantage of the xfn) -approach, one can discuss optically-induced charge carrier correlation effects (or excitation-induced-dephasing effects) on four-wave-mixing signals In Section (3.4.3.) we will address this question in a partly phenomenological approach, but a mote rigorous analysis based on the x(“)-expansionu86) confirms essentially the results presented in Section (3.4.3.).

3. NUMERICAL

RESULTS OF SELECTED PHENOMENA

3.1. Basic Excitation Dynamics 3.2. Coherent Dynamics We begin the discussion of the coherent bandedge excitation dynamics with the fully coherent case. We neglect all incoherent processes by setting all dephasing and relaxation rates to zero. That allows us to study the limiting case of the fastest possible excitation processes in semiconductors Although in this section we will use GaAs as an example, which, due to its small exciton binding energy, is probably not the substance where this limiting case can ever be reached experimentally, the following analysis is of importance for two reasona First, the inclusion of incoherent processes will often only quantitatively modify the coherent results, and an understanding of the coherent dynamics is a pmrequisit for the understanding of the more realistic results, which are discussed in the following section. Secondly, substances can be engineered (for example by means of quantum confinement), such that the exciton binding energy can become very large. In materials with large exciton binding energy one can use very short pulses for the investigation of the exciton dynamics, because in this case it is possible that the spectrally broad pulse centered at the exciton resonance still has no spectral overlap with the higher lying excitons and the continuum absorption. If it becomes possible in the future to realize very good and thin quantum wires, it might be possible even in the case of GaAs to separate the timescale of the coherent exciton excitation from the incoherent dephasing processes All numerical results in this and the following section are obtained from a numerical solution of Eqns (82) and (83). The initial conditions for the polarization and distribution functions are zero. The time integration is performed with a 4th-order Runge-Kutta method, and the step size is chosen such that the oscillation with the frequency ~(k*)/h can be integrated with an accuracy of a few percent. Here, k,,, is the largest k-value in the set of discrete k-values used for the Gaussion quadrature solution of all k-integrals The Gaussian quadrature contains subintervals so that it is possible to accumulate k-points in those regions which determine the result most sensitively. For example, if the excitation process is such that only low lying excitons are excited, it is important to select the k-points in a way that they resolve the exciton wave functions The cut-off wavenumber k,,, is chosen such that all polarizations and distribution functions have dropped of significantly at k,,, (at least 0.1%). The investigation of the coherent semiconductor dynamics can be divided into two main regimes The first is the resonant excitation. In this case the lowest bound exciton msonance

390

R. Binder and S. W. Koch

0.0 ‘ITME[PSI

Fig. 4. Excitation density vs. time for a pulse centered at the exciton resonance of bulk GaAs (the exciton Bohr radius is 0~ = 135A). All dephasing and relaxation rates are set to zero. The pulse amplitude has as sech shape and the intensity has a duration of 0.1~s (FWHM). The peak dipole energy is &!&J = 34.75 meV, corresponding to pulse area (= time integral of the dipole energy divided by A) of 37~. The thick lines show the results within the rotating wave approximation, and the fast oscillating thin line around the thick lines shows the results obtained without the rotating wave approximation (the gap energy was chosen as Eg = 1.52eV).

(or the interband continuum) defines the spectral region of interest (see Fig. 1). The other case is that of the large detuning limit, which means that the pulse is centered far below the exciton resonance. We will also study intermediate cases As a first example we show in Fig. 4 the temporal behavior of the excitation density if the pulse is centered at the exciton resonance of bulk GaAs The pulse duration is 100 fs (FWHM of intensity), and the intensity is chosen such that the area of the pulse amplitude is 37r. In this case an atomic transition would yield an inversion that exhibits 1.5 Rabi flops One Rabi flop is the evolution of the inversion from zero, before the arrival of the pulse, up to maximum inversion and back to zero. Hence, after 1.5 Rabi flops the system ends up in the excited state after the 37-r pulse has passed through. Figure 4 shows that a different temporal evolution occurs in the case of a semiconductor. For the parameters chosen, the full solution exhibits roughly two Rabi cycles (corresponding more or less to a 47r pulse in an atomic system). When all nonlinear interaction terms (the so-called exchange contributions) are switched off in the calculation, the inversion resembles much more the behavior of an isolated atomic transition, altough even in this,case the Rabi oscillation seems to be blurred out. We will show below that this is a consequence of the spectral overlap of the pulse with higher lying exciton resonances Figure 4 also serves to justify the rotating-wave approximation. Not making this approximation, we find almost the exact same inversion dynamics, only that now the time-dependence is modified by a small fast oscillation around its mean value. The fast oscillation has roughly the frequency 2EE,lh.

Nonequilibrium semiconductor dynamics

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-0.14

0.0

0.14

391

0.28

TIME [ps] Fig. 5. (a) Corresponding to the results with exchange effects in Fig. 4 the real part (dashdotted line) and imaginary part (dotted line) of the induced polarization. The solid line is the density (same as in Fig. 4). (b) Same as in (a) but without exchange effects.

The induced polarization for this case is shown in Fig. 5a. The structure in the time dependence of the imaginary part of the polarization reflects that of the inversion, which is similar to the Bloch vector rotation in the atomic case. Note, however, that the conservation law of the SBE, Eqns (82) and (83), is

Km12= _ml

-f&,1

(429

which applies to every k-state, not to the total polarization and carrier density. As a striking feature of Fig. 5a we see that the real part of the polarization dominates over the imaginary part. In a resonantly excited 2-level system the real part would be exactly zero since a real part of P can only be excited for a finite detuning. However, in the semiconductor the exchange contributions can also do that. Leaving out the exchange effects we find that the real part is still much bigger than the imaginary part of P, Fig. 5b, suggesting that the signal contains a large amount of detuned resonances If a 100 fs pulse is centered in the Is-exciton resonance of bulk GaAs, it excites a significant amount of higher lying excitons as well as continuum transitions This affects in particular the possibility of self-induced transparency (SIT). In an inhomogeneously broadened two-level system, the se&shaped 27r pulse produces exactly one Rabi flop for each state, independentof the detuning. In the semiconductor this mechanism is modified by the exchange effects In Fig.’ 6 we show results for longer pulses with narrower spectra and less overlap with high

R. Binder and S. W. Koch

392

0.24

-2.8

-1.4

0.0

1.4

2.8

7

14

‘mm [PSI

-7

0 IME

[PSI

Fig. 6. (a) Excited density vs time for resonant exciton excitation. The pulse duration is Ips (FWHM) and the area is 3rr. ‘Ihe solid line shows the results of the full calculation including the exchange effects, which are left out in the dotted line. Incoherent effects are neglected. (b) Same as (a) but for a 5 ps pulse. The area is again 37~.

lying exciton resonances In particular, the Sps-pulse has practically no overlap with other than the Is-exciton resonance, as can be seen from the result without exchange effects, which is almost exactly that of an atomic transition subject to a 3rr-pulse. The number of additional oscillations caused by the exchange interaction increases strongly if the pulse duration increases (for constant pulse area). The corresponding time-resolved polarization dynamics shown in Fig. 7 show the expected result that the real part of P, which is dominating if the exchange effects are taken into account (Fig. 7a) almost vanishes if exchange contributions are neglected (Fig. 7b). In order to 6nd a 100 fs pulse which minimizes the final density, we vary in Fig. 8 the area around 17~. We do not choose OTT(as in the atomic case, since the exchange contributions increase the effective Rabi frequency, and for a 100 fs-pulse this increase amounts roughly to a factor of two. Fig. 8 shows clearly that it is impossible to excite the initially noninverted system with a 100 fs pulse and bring the system back to its initial state. Figure 9 shows the final density for an initially unexcited semiconductor as a function of

Nonequilibrim

semiconductor dynamics

393

&:l :;:

” ._..__......

‘.

-14

-7

.,’

’

0

7

14

TIME [PSI

Fig. 7. (a) Corresponding to the results with exchange effects in Fig. 6b the real part (dashdotted line) and imaginary part (dotted line) of the induced polarization. The solid line is the density (same as in Fig. 6b). (b) Same as in (a) but without exchange effects.

the pulse area for the 100 fs pulse discussed earlier and for a 5 ps pulse. Note, however, that the approximations made in this section, i.e. the total neglect of damping and dephasing mechanisms, is very unrealistic for a pulse duration of 5 ps Hence, the comparison of the two pulses should only serve as an illustration of the mathematical properties of the equations We see in Fig. 9 that the 10Qfs pulse yields for no area the possibility of complete Babi flops However, the 5 ps-pulse, which has essentially no spectral overlap with higher lying excitons, allows one Rabi flop in the density for an intensity such that fi,!& = 0.0562 meV. For this case we show in Fig. 10 the full polarization and inversion dynamics We see that after the pulse is gone the system has almost exactly recovered its inital state, so that this pulse acts as an ‘effective 27-r’pulse. The large real part of the polarization indicates that the dynamics is not that of an isolated transition, although, in order to achieve such an 2rr behavior, we had to exclude the spectral overlap of the pulse with other than the Is-exciton resonances Another difference to Rabi flops of isolated transitions is apparent in Fig. 9. An isolated transition will always be completely inverted if resonantly excited by a TTpulse. In Fig, 9, however, the maximum inversion roughly scales with the inverse pulse duration. The 5 ps ‘effective r-r-pulse’ with the area of approximately 0.12~~ creates fifty times less density compared to the 0.1 ps pulse with the area 0.6~. In order to analyze the above results, we first transform the SBE, Eqns (82) and (83), into an exciton basis For this purpose we use the complete set of hydrogen-like wavefunctions

R. Binder and S W. Koch

394

0.0 1 -0.26

-0.13

0.00

0.13

0.26

‘MIME [PSI Fig. 8. (a) Excited density vs time for resonant exciton excitation. The pulse duration is 0.1~s (FWHM) and the pulse area is indicated. ‘Ihe pulse shape is again sech-like. Incoherent effects are neglected. p,,(k), where the quantum number n labels both the discrete and the continuous part of the spectrum. We expand

P(k) = 1 P,,p,,(k)

(430)

f(k) = &MkI n

(431)

and obtain, neglecting all incoherent effects, for the components of the polarization equation, Eqn. (821, ihgPn = E,P~ + 1 &f!Pn~ n’ 1, -- ,w~bP~cr=0,

+ 2fn3.

(432)

Here, E” is the detuning of the exciton with quantum number n with respect to the center frequency of the field envelope,

and the nondiagonal selfenergy is

Cnl = 12 II” w

P; (k)

Nonequilibrium semiconductor dynamics

0.0

0.5

1.0

1.5

395

2.0

AREA [unitsof x] Fig. 9. Final density vs pulse area for excitation of the exciton resonance with a XC/Ipulse. The pulse duration (FWHM of intensity) is 5 ps (solid line) and 0.1 ps (dotted line). Incoherent effects are neglected.

We have already used the fact that in the fully coherent case the distribution functions for electrons and holes are equal, which allows us to replace fe + fj, by 2s. For the components of Eqn. (83) we obtain (435) where the nondiagonal Coulomb contribution

to the induced field is

In Eqn. (435) we have assumed, for simplicity, that all functions p”(i) are real. Equations (432) and (435) are most easily analyzed if only one state (usually the Is-exciton) is excited by the field. According to Eqn. (432) this happens if the Fourier spectrum of the external field envelope is centered at the 1s exciton, EI# = 0, if its width is small in comparison to the ls-2s separation, and if the spectral broadening due to the nonlinear terms is negligible. Also, the shift of the exciton resonance associated with Z must be small. If these conditions are fulfilled, we can write pn = &,iPlS.

(437)

The first immediate consequence of Eqn. (437) can be drawn for the case where the exchange contributions are neglected, Znng = 0 and dnnr = 0. In this case only fis can be create+ irrespective of the pulse intensity. Consequently the distribution function as function of k always follows the shape of the Is-exciton function:

R. Biider and S. W. Koch

396

Fig. 10. Corresponding to the point at 0.243~ of the 5 ps-pulse in Fig. 9 the real part (dash-dotted line) and imaginary part (dotted line) of the induced polarization vs time. ‘Ihe solid line is the density. Exchange effects are included, incoherent effects are neglected.

f(k) = fls 94,(k) -

(438)

Of course, this approximation is not meaningful if f (k= 0) 1 1. This can easily happen, because the conservation law (429), which ensures 0 I f(k) I 1 for all k, does not hold without exchange terms The full equations for P,, and fn,including the exchange contributions, can k analyzed easily only in the low intensity limit. In this limit we tid from Eqn. (429) with f(k)
f(k)= 12W12.

(439)

From the inverse of Eqn. (431) we find, with the approximations (439) and (437),

which shows that, in contrast to the case without exchange effects, fn+ &,lsfls. Instead, we obtain from the inverse of Eqn. (440) the expected result

f(k) = lv,,W12 l%12.

ww

The total density is now given by n = Wd2 9 and the selfenergy (434) by

(442)

Nonequilibrium semiconductor dynamics

f%

397

k

8 ,****--.-.*- ......._._.._.* _.. --a 3 6 WAVE NUMBER [unitsof $1

0

0

1 WAVENUMBER[unitsof as-‘]

2

Fig. 11. Distribution functions vs wave amber at time t = 0 (i.e. at the pulse maximum). The pulse duration is 5 ps and the area is 0.0277. Results shown include exchange effects (solid line), and exclude exchange effects (dotted line). The triangles show the squak of the 1~ wavefunction, normaliid to coincided at k = 0 with the nwerically obtained distribution, and the crosses show the l+wwefunction, normalized to coincided at k = 0 with the corresponding numerical results without exchange effects. Incoherent effects are neglected. Inset: The distribution function multiplied by k2, the bulk volume element in &ace.

The form (443) has been evaluated analytically by Keldysh and Kozlovus7) for bulk systems: Lls

= T

l%12u;&.

WV

Here, UB is the bulk exciton Bohr radius and ER is the exciton Rydberg energy (binding energy). Eqn. (443) has been discussed in detail by Schmitt-Bink et al(‘**) In particular, the analogy of Eqn. (443) with the Heitler-London exchange integral has been noticed there. The Is-contribution to the distribution function is given by El1 et al.:(16) f-h =

~P,,(wmA~v

=- 72 d-&ms12

1%12 -

(445) w3

The numerical solution of the full SBE yields for a very low intensity 5 ps-pulse the distribution functions shown in Fig. 11. The density as a function of time is practically the same with and without exchange contributions It shows the expected aggreement with the Is-wavefunction

Increasing the intensity of the 5 ps-pulse to 0.0562 meV, which is the ‘effective 27r-pulse’ of Fig. 10, we 6nd for the distribution function the results shown in Fig. 12. At the time of

398

R. Binder and !S.W. Koch

WAVE NUMBER [units of aBe’] Fig. 12. Distribution functions vs wave number at time 2 = 0 (solid line) and after the

pulse (dotted line). The pulse duration is 5 ps and the area is 0.0211. The crosses and the circles show the square of the Is-wavefunction, normalized to coincided at k = 0 with the numerically obtained distribution for t = 0 and t = Q), respectively.

the pulse maximum, the distribution diffiers significantly from I~,,(k)12, showing that the exchange effects have ‘mixed in’ other exciton states Unlike a-level systems, where resonant excitation yield unity inversion at the time of the pulse maximum, no k-state is populated more than 0.5. After the pulse is gone, the distribution follows again exactly the square of the exciton function. In this case the pulse intensity was chosen to yield a very small final density. The maximum inversion of about 0.5 at k = 0 is not a strict upper limit. Figure 13 shows the distribution for a pulse with amplitude 23.17 meV (corresponding to a 2rr area). In this case, however, the Pauli blocking causes a more pronounced deviation of the distribution from lq+,(k)12, ‘smearing out’ the distribution in k-space considerably. The corresponding calculation without exchange effects yields unphysical distributions with values greater than one, but, of course, the shape is that of the exciton wave function, as discussed above. As mentioned above, since we omitted the damping and dephasing terms of the SBE in this section, the results for the 100 fs-pulse are certainly more realistic than those of the 5 ps-pulse. As shown in Fig. 9, the density generated after a 100 fs-pulse is minimized for an area of 1.4~~.The corresponding distribution functions with and without exchange effects are shown in Fig. 14. The distribution extends to very high wavenumbers The comparison of the result with and without exchange effects shows that the correct Pauli blocking (included only in the calculations with exchange effects) causes a considerable reduction of the excitation of lowmomentum carriers The occupation of high-momentum carriers, as well as the total density, is less significantly affected by the exchange effects We have seen in Fig. 4 that if the pulse spectrum extends high into the continuum region the exchange effects are less drastic than in the case where only bound exciton states determine the optical response. In order to spectrally separate the bound exciton states beyond the bulk semiconductor regime, one often uses the effects of quantum confinement. As shown in Fig. 1, the lowest exciton resonance becomes more dominating as the dimension is reduced form 3D to 1D. Of course, in the pure lD-limit (the limit of zero quantum-wire thickness) the exciton binding energy diverges and no continuum resonances remain. In other words, an ideal quantum wire allows an arbitarily large separation of the lowest exciton resonance from all other resonances

Nonequilibrium semiconductor dynamics

399

WAVE NUMBER lunia of %-‘I

I

2

1

0

WAVENUMBER[unitsof ag’]

Fig. 13. Same as Fig. 11, but for a 5 ps-pulse with area 27~.

z

1.8 ...--..

g s 1.2

i3

WAVENUMEIER

5 m 0.6

[Units 0fS”l

5 g 0.0

0

1

2

3

4

5

WAVENUMBER[unitsof a~-‘]

Fig. 14. game as Fig. 11, but for a 0.1 ps-pulse with area 1.41~.

The main question for the coherent optical repsonse is the effect of the increased spectral separation of the exciton in quantum wells and quantum wirea Intuitively one might expect that a well separated exciton behaves more like an ideal two-level system. On the other hand, the spectral separation is caused by the (linear) Coulomb interaction. Increasing the Coulomb interaction in the linear regime might be associated with an increased influence of the nonlinear exchange effecta The exchange contributions are also in!Juenced by phase-space filling effects

400

R. Binder and S. W. Koch

These are generally increased as the dimension is decreased, because it is easier to fill up a onedimensional energy-band structure than, for example, a 3D-band structure, where for every energy the full solid angle of momentum states exists In order to quantitatively investigate the dimensional dependence of the nonlinear manybody effects, we have solved the SBE for the bulk, quantum well, and quantum wire cases simultaneously. The well width is chosen as 5OA. This thickness is thin enough so that the neglect of the light-hole subband is justified. On the other hand it is thick enough that the infinite barrier approximation, which ensures that the subband functions am completely inside the well and not spread out into the barrier, is a good approximation. To analyze the dimensional variation of the density for pure exciton excitation we show in Fig. 15 the results for a spectrally narrow 3rr-pulse. Whereas the results for the bulk and quantum well case am very similar, the quantum wire shows a less significant influence of the exchange effects Choosing an effective wire thickness of 5OA we find even less enhancement for the Babi frequency (not shown). This might be surprising, since the linear Coulomb interaction causes the exciton binding energy to be larger than in the quantum well and bulk cases for both effective wire thicknesses (&, = 17.4 meV for the 3OA wire and 11.9 meV for the 5OA wire). It is clear, that a large exciton binding energy or the exciton oscillator strength does not simultaneoulsy mean that the nonlinear exchange effects are large. To further investigate the influence of the dimension, we show in Fig. 16 the final density which is left in the semiconductor after a 0.1 ps-pulse, centered at the Is-exciton resonance, has passed through. Again, bulk and quantum well systems show very similar behavior. An almost ‘effective 2rr-pulse’ can be achieved with an area of about 1.4rr. In the quantum wire case the ‘effective 2rr-pulse’ has an ama of about 1.6rr-1.8rr for effective wire thicknesses between 30 and 5OA. For the case of a 1OA wire, where the binding energy is 36.5meV, the value reduces to 1.4rr. For these minima we show in Fig. 17 the corresponding time-resolved density in units of a$, d = 1,2,3. Th e contrast between the maximum density (around t = 0) and the final density is comparable in all three casea However, the distribution functions at t = 0, Fig. 18, show a strong dependence on the dimensionality at small momenta. Finally, we want to stress that all the coherent effects discussed above are generally weakened if incoherent effects are being considered in addition to the purely coherent processes As an example, we show in Fig. 19 how the Babi oscillations shown as solid line in Fig. 6b are being ‘smeared out’ by additional incoherent effects A nonzero dephasing rate increases the spectral overlap between the pulse and the exciton resonance and, hence, leads to an absorptive component in the density which roughly follows the time-integral of the intensity (dotted line in Fig. 19). If we also include relaxation terms in the j-equation (dashed line), the saturation effects become weaker because the carriers scatter out of the states where they are excited. This leads to an additional increase of the absorption. Finally, the inclusion of the static screening reduces the effective field and hence decreases the effective Babi frequency. It also adds an additional contribution to the bandgap renormalization (Coulomb-hole term) which in turn increases the absorption of the field (dash-dotted line). 3.3. The OpticalStark Efect In this section we solve the semiconductor Bloch equations to discuss some aspects of the excitonic optical Stark effect. (is*) The optical Stark effect is well known from atomic systems,(4) where for resonant excitation a light-induced splitting of the atomic levels occurs, which can be understood in the dressed state picture as transition from the pure atomic states to the eigenstates of the coupled light atom system. For large detunings of the exciting laser beam from the optical transition energy the optical Stark effect leads to a shift of the resonance which is proportional to the intensity of the laser beam.

Nonequilibrium semiconductor dynamics

-7

0

401

7

14

TIME [PSI Fig. 15. Excitation density vs time for a pulse centered at the exciton resonance of GaAs systems (the exciton Bohr radius is ug = 135 A). All &phasing and relaxation processes are neglected. The pulse. amplitude has as sech shape and the intensity has a duration of 5 ps (FWHM). The peak dipole energy is 34.75 meV, corresponding to a 3rr-pulse. The quantum-well thickness is 50 A and the effective quanhrm wire thickenss are 30 A. The solid lines (dotted lines) show results with (without) exchange interaction.

To ob&xve the optical Stark effect in semiconductors for transitions in the spectral vicinity of the fundamental absorption edge, one has to use short optical pulses because of the rapid dephasing processes in real systems Using nonresonant excitation with femtosecond pulses, the optical Stark effect has been observed as a transient blue shift of the exciton resonanm@. 14,189,190) J?SI9,4/5-9

R. Binder and S. W. Koch

402

0.8

H

(4

/

0.6

/

/

.

/

\

\

I . ..’ .*.. ::

0.4

/

\

Y

c %

\

. . ,- . . . .

/

\

\

.. ..

-2 3

.’ *.

I

.,.-

\

*.

.*

\

a. *.

.*

\

,.**

iz 2

;_

0.2

8

H

.

/

\

1.2 1.o 0.8 0.6 0.4 0.2

0.0

0.5

1.0 AREA

1.5

2.0

[unitsof ~1

Fig. 16. Final density vs pulse area for excitation of the exciton resonance with a se&pulse. The pulse duration (FWHM of intensity) is 0.1 ps. Incoherent effects are neglected. (a) Solid line: Bulk GaAs (in units of aj3, where a~=135 A is the exciton Bohr radius), whith an exciton binding energy of E&.2 meV. Dotted line: GaAs quantum well with thickness 50 A (in units of ai’), &=10.3 meV. (b) GaAs quantum wire with effective thickness 50 A (solid line), J&=11.7 meV, 30 A (dotted line), &=17.4 meV, and 10 A (dash-dotted line), Ebz36.5 meV. The dashed line in (a) and (b) shows, for comparison, the final inversion of a two-level system in arbitray units.

To theoretically study this phenomenon in its pure form we numerically solve the semiconductor Bloch equations, Eqns (82) and (83), without screening (IV = V) and without relaxation terrns(i5’) We compute the absorption spectrum of a probe pulse for different time delays td from the pump pulse. It is assumed that pump and probe pulses propagate into slightly different directions, such that they overlap in the sample but are seperately detectable after the sample. This geometry is schematically shown in Fig. 20. This makes it possible to investigate e.g. the probe transmission with and without the simultaneous presence of the pump pulse. As an example of the computed probe absorption spectra we show in Fig. 21 spectra for several td for the case of bulk GaAs excited by a Gaussian pump pulse with an amplitude FWHM of 120 fs whose central frequency is detuned 10 & below the bandgap energy E,. The peak height of the pump pulse is such that j&,&o/2 = 2 ER at t = 0. The probe pulse has the same shape and duration as the pump pulse, but is centered energetically at the exciton resonance to monitor the transient exciton shift. The peak height of the probe pulse

Nonequilibrium

semiconductor dynamics

403

1.6

;jt; IL i’!

’

\

/ i . e*-.

/ /

i -t *

1, ;:,!

i,/

--,, \,

i/ *:

:*. \

I:

j:

‘.

.---._._._.

.._........._......_...

./’ * -0.14

0.14

0.0

0.28

‘ITME [PSI Fig. 17. Time-resolved excitation density for three points of Fig. 16. The examples shown correspond to the minima in Fig. 16, which are at an area of 1.4rr in the bulk case (shown here as solid line), at 1.3~~in the quantum well case (dotted line), and at 1.647~in the case of the 30A quantum wire (dash-dotted line).

z

1.0

0 6 0.8 ‘06 3’ 5 0.4 g 0.2 Ei 0.0

0

2

4

6

WAVENUMBER [unitsof age’]

Fig. 18. Electron distribution functions at time I = 0 (peak intensity) for the three cases shown in Fig. 17.

404

R. Binder and S W. Koch

0.08 r

1

.s E

**.:-. ..........._._.__..........

-

0.04

g z z 8

0.02 ’

0.00 -14

-7

0

7

14

TIME [ps] Fig. 19. Excitation density vs time for a pulse centered at the excitoo resonance of bulk GaAs The pulse duration (FWHM of intensity) is 5 ps The solid line is again the density shown as solid line in Fig. 6b, i.e. with exchange contributions but without dephasing, relaxation, and screening. The dotted line shows the result with a dephasing time of 5 ps (no relaxation, no screening), the dashed line shows the result with dephasing and relaxation in .the relaxation rate approximation modeling carrier-carrier scattering (both times are chosen to be 5 ps). The result shown as dash-dotted line includes dephasiog, relaxation, and quasistatic screening of the Coulomb potential.

2K1

-

Ko

Fig. 20. Sketch of the opt&al excitation geometry. IO addition to the directions of and probe beam, $ and gl, re_spectively,the photon echo or FWM direction 2& the conjugate direction 2% - & are indicated. (For the discussion of polarization rules in quantum wells in Section (3.4.2.) we assmne that the angle between Kl very small.)

the_pump - 4, and selection and K2 is

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-0.2-2 ENERGY (E-E,)/ER Fig. 21. Probe spectra for various delay times f,j in comparison to the linear exciton spectrum (solid line); Id = -600 fs (dash-double-dotted lie), rd = -60 fs Oong dashed fine), rd = -0 fs (dash-dotted line), rd = +60 fs (short-dashed line). The excitation pulse is centered 10 ER below the bandgap energy Et of the unexcited GoAs semiconductor, where Ea is the bulk exciton binding energy (4.2 mev). The pulse duration is 120 fs (FWHM of the field amplitude). (From Ref. (151).)

is ,& .?if?, 12 = 0.01E~ at I = fd. We assumed a constant phenomenological dephasing time of 0.87 ps, which corresponds to an exciton lineshape width before the excitation of y = 0.2E~. For a convenient comparison of the results we normalize in this section all absorption spectra such that the Is-exciton peak has a height of 1 before the excitation. Differential transmission spectra (DTS) are obtained for a given pump-probe delay time as the difference between the probe absorption in the absence and presence of the pump pulse. The resulting DTS corresponding to Fig. 21 are shown in Fig. 22. Since the pulse width is much shorter than the dephasing time, the resulting exciton blue shift is substantially smaller than for stationary (cw-) excitation”@ with the same peak intensity. This phenomenon is a consequence of the fact that for pulsed excitation one has different pump fields at different times.ti5’) Other important effects in ultrafast pump-probe experiments ate the coherent scattering of the pump-pulse into the probe direction as well as the energy redistribution within the probe pulse. As analyzed in detail in Refs (117,120,152), the coherent scattering of the pump into the probe direction leads to oscillations which are observable in the DTS at negative delay times The period of these oscillations is essentially given by f,f. For nonresonant excitation these coherent oscillations are accompanied by the exciton blue-shift characteristic for the optical Stark effect. However, similar coherent oscillations are also present for other pump and probe tunings, such as for spectral hole burning experiments in the interband continuum or for exciton bleaching studies.(i2’) Figures 21 and 22 show that the exciton resonance is substantially bleached for fd - 0 , i.e. for situations where pump-probe overlap occurs This bleaching is caused by the transiently excited carriers which give rise to significant phase-space tiling effects The temporal behavior of the excited density is displayed in Fig. 23. It clearly exhibits ultrafast adiabatic following

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‘;; 0.6 ‘2 3

/ -\ /

’

’ -0.61 -2

ENERGY -,E-Eg),E, Fig. 22. Differential transmission spectra corresponding to Fig. 21. (From Ref. (Ul).)

behavior,(4s 19’)i.e. the temporal carrier density follows the pump pulse envelope, as is expected for the assumed configuration where we have negligible spectral overlap between the pump pulse and the exciton resonance. As a consequence of the significant decrease of the carrier density after the pulse, the bleaching and shifting of the exciton almost vanishes at positive delay times (Fig. 21). The residual small bleaching is caused by the nonzero amount of remaining carrier density which is mainly a consequence of the many-body Coulomb interaction (see the detailed discussion in the previous section). In addition to the bleaching for td + 0, Fig. 21 shows that the low energy tail of the exciton resonance exhibits also a pronounced region of increased transmission. As analyzed in Ref. (151) this transmission increase is not true optical gain but merely caused by energy redistribution within the pulse (see also Ref. (15)). In the results shown so far in this section, the main effect of the non-Lotentzian exciton lineshape was simulated by using different dephasing rates in the spectral region of the pumppulse and at the exciton peak. (Is) For the results shown in the figures we chose fl = O.Ol& at hwo. The microscopic justification for such an approach lies in the calculation of the dephasing as result of carrier-phonon and carrier-trier scattering. If one uses a non-Markov approximation in such a dynamic calculation, one finds a decreasing dephasing rate for increasing detuning below the exciton resonance. In order to avoid the step-like frequency dependence of p, we use the dynamical model for y given in Eqn. (130), which can be justified using a microscopic model for LO-phonon scattering. In the simplest approximation, r(t) can be written as product of phase factor and a peaked function with a half width corresponding to a typical scattering time which determines the range of the memory effects Hem we assume a Gaussian memory with

where all constants were chosen to approximately fit the half width, the maximum, and the energetic position of a Lorentzian Is-exciton lineshape. As illustrated in Fig. 24 for T,,, =

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0.6

A : E 0.4 0 5 0

-0.16

-0.08

0.00

TIME

0.08

0. 1

(ps)

Fig. 23. Excited carrier density (solid line) and light-field intensity (dashed line) corresponding to Fig. 21. (From Ref. (151).)

0.09 ps, only the low energy tail of the spectrum is strongly affected by the time-dependent dephasing, in agreement with the assumption of a dephasing rate that decreases with increasing detuning. Clearly, Fig. 24 demonstrates that the resulting modifications for&f = 0 are relatively insignificant. Hence, the main features of the excitonic optical Stark effect are not affected by the details of the exciton lineshape, as long as the negligible spectral overlap between exciton pulse and the exciton resonance is ensured. In order to analyze the influence of the Coulomb interaction on the exciton bleaching and shifting in greater detail, we calculate the spectrum at rd = 0 without the exchange contributions Figure 25 indicates that for the intensity chosen these contributions slightly increase the blue-shift, the bleaching and the gain at the low-energy tail. However they do not change the results qualitatively. In Ref. (5) we have shown that the modifications in the computed transmission spectra caused by the exchange terms become larger with increasing carrier excitation, i.e. for higher intensities or smaller spectral detuning of the pump from the exciton resonance. Drastic modifications of the results are caused by the density-grating terms in our equations which are responsible for the pump-probe mixing effects The neglect of these terms leads to a spectrum exhibiting only resonance saturation, i.e. bleaching of the exciton due to phase-space filling effects As can be seen in Fig. 25, both the blue shift and the low-energy gain are completely gone. To further examine the phase-space filling effects, we show in Fig. 26 the distribution function ye(k) for t = 0. Obviously, the distribution reaches far into the band, with and without inclusion of the exchange contributions For comparison, we also plot the result obtained when we neglect all Coulomb terms, as well as the square of the Is-exciton wavefunction. In the case without Coulomb-interaction the distribution is almost constant over a large k-range, indicating that the differences in detuning from the pump pulse are negligible for the displayed k-values On the other hand, if the excitation pulse would excite only the Is-exciton, the carrier distribution should roughly follow the square of the Is-wavefunction p,,(k). Figure 26 shows large discrepancies between the Is-wavefunction and the full results These discrepancies am

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-0.2_* ENERGY (E-&)/ER Fig. 24. Linear spectnrm and fd = 0 spectrum of Fig. 21 (solid line and dash-dotted line), together with the corresponding spectra using time-dependent &phasing (long-dashed line and short-dashed line, respecively). (From Ref. (151).)

-0.2-* ENERGY (E-&)/ER ‘Fig. 25. Gkulations without exchange terms (long-dashed line) and without density-grating terms (short-dashed line) in comparison to the corresponding td = 0 sepctrum of Fig. 21. The solid curve is the linear spectrum. (From Ref. (151).)

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0.0 Wave

vector

kae

Fig. 26. Carrier distribution function j,(k) for t = 0 (solid line) corresponding to Fig. 21. For comparison, f&) is also computed without exchange terms (long-dashed line) and without Coulombinteractions(dash-dottedline). The short-dashedline showsthe square of the Is-excitonwavefunction,normalizedat k = 0 to f&k). (From Ref. (151).)

not simply a consequence of saturation effects To stress this fact, we plot the results in Fig. 26 such that the value of q&(k) coincides with f(k = 0). For the correct normalization of 91,(k), its square would have values larger than 1 for small k, drastically increasing the differences that are already displayed in Fig. 26. The occupation of larger k-states in the full results indicates that in the assumed large detuning limit the pump pulse causes similar modifications of the lsexciton and of the higher exciton and continuum states, so that the applicability of analytical estimations neglecting these states is questionable. 3.4. Multi- Wave Mixing and Photon Echo 3.4.1. Results for a two-band semiconductor In this and the following section we apply the semiconductor Bloch equations to study twopulse four-wave mixing (FWM) in semiconductors First, in this section, we assume a twoband model for the semiconductor, following the work of Lindberg et al.(24) and in the next section we then include a more realistic valence-bandstructure and the resulting band-mixing effects(27) Unless noted otherwise, we restrict ourselves to the simplest experimental geometry, shown schematically in Fig. 20. One of the transient phenomena observable in two-pulse FWM is the effect of a photon echo, i.e. the emission of an echo pulse at the time &j after the system has been excited by two separate pulses at the times t = -td and t = 0, respectively. Such photon echoes have been observed, and analyzed in atomic gases with inhomogeneous broadening, i.e. a distribution of the relevant transition energy. The theoretical description of photon echoes in atomic gasesc4) is mainly based on a model which treats the individual atoms as independent oscillators In a semiconductor the concept of independent oscillators is lost because of the continuum of excited states and the strong Coulomb coupling between them. Hence, in general the free motion after an initial excitation

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Fig. 27. Time resolved signal in photon echo direction for excitation at the exciton resonance. The material parameters are chosen for CdSe. The dephasing rate is set to zero (no dephasing and no population relaxation), the center frequency of both pubes is at the exciton resonance bwe = Rwt = Els, the pulse width is 100 fs (FWHM of the amplitude), the delay time is ld = -300 fs (puke $1 comes 300 fs before $2, and the field strengths are j& . $1 = 0.02& and ficv . C&= 0.2&. (From Ref. (24).)

is very complicated due to the nonlinearity of the system. In order to concentrate on the many-body Coulomb effects in semiconductor FWM experiments we start our discussion for the ideal case, where we assume a simple two-band structure and ignore all dissipative terms in the SBE. In Fig. 27 we show the result of numerical investigations(24) where we assumed a pulse delay time rd = -300 fs and the field strengths _ P, - 9, /2 = 0.01E~ and ficv - &o/2 = 0.1E~. The computed time dependent signal is plotted for the situation where the exciting fields are in exact resonance with the Is-exciton. The most important and somewhat surprising feature is that the signal seems to rise without limit. The rise is absent in atomic systems and is completely a consequence of the many-body Coulomb effects, or, more specifically, of the exchange contributions in the SBE. To verify this fact, we show in Fig. 28 that if the exchange contributions are neglected the signal exhibits only a constant level signal superimposed on relatively small amplitude oscillations due to the quantum beats between the 1s and 2s-excitor-is A comparison of the signal magnitudes in Fig. 27 and Fig. 28 indicates that in the full calculations of Fig. 27 the small quantum beat signatures are completely masked by the much stronger rising signal. Clearly, in a real system the signal would not rise forever, since higher nonlinearities and the omitted dissipative contributions lead to an eventual decay at later times To study the effects of the intrinsic inhomogeneous broadening of the semiconductor continuum states we show in Fig. 29 results which are obtained for excitation of the serniconductor at the band edge. All other parameters are kept the same as in Fig. 27. Now the signal shows a clear echo which has its maximum about 300fs after the second pulse. The exchange contributions in this case do not significantly change the overall shape of the signal; however, omitting these terms, the signal strength decreases by roughly one order of magnitude. Clearly, in real semiconductors the polarization dephasing and the scattering of the popu-

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Fig. 28. same as Fig. 27, but exchange terms are neglected. (From Ref. (24).)

TIME (fs) Fig. 29. Same as Fig. 27, but for excitation at the bandedge @IWO= hwl = Es). (From Ref. (24).)

lation play an important role. In fact, photon echo experiments are often used to measure the dephasing rates In what follows we therefore include the dephasing and the scattering simply as phenomenological decay terms As an example of the results we plot in Fig. 30 the temporal signal evolution for various delay times from -400 to - 150 fs for the case when the center frequency of the excitation is at the Is-exciton resonance and the intensity of the probe pulse is 1% of the pump pulse intensity. A constant dephasing time of 200 fs has been assumed. We

412

R. Binder and S. W. Koch

TIME

(fs)

Fig. 30. Time resolved signal in photon echo direction. The material parametersare chosen for CdSe (m, = 0.125m0, mh = 0.431m0, ER = 16 meV, ~0 = 9), and the dephasing rate is (Tz =

200 fs). The center frequency of both pulses is at the exciton resonance (hwo =_hwl = E~J, the pulse width is 100 fs (FWHM of the amplitude), the field strengths are fiN . TI = 0.02& and jiN - &I = 0.2&, and the delay times fd are -150, -175, -200, -300 and -400 fs, corresponding to decreasing signal maximum, respectively. Negative delay times indicate that pulse $1 comes before $2. (From Ref. (24).)

see that the signal starts to rise immediately after the second pulse hits the sample. In the case of independent two-level systems such an instantaneous signal indicates the dominance of a single resonance, rather than a broadband continuum. In the semiconductor, this resonance is the exciton. The fact that the underlying one-particle states are continuously distributed is not significantly reflected in the signal. The time at which the total signal peaks is not related to the delay time between the two pulses It is determined by the original rising part of the signal and the subsequent decay because of the dephasing. However, in a measurement situation this kind of signal could be confused with a real photon echo if, by coincidence, the delay time &jand the signal peak time are not too different or if only time-integrated measurements are performed. The decrease of the signal magnitude with increasing delay time is due to the dephasing during the time interval between the pulses To investigate the influence of the excitation strength on the two-pulse FWM results, we plot in Fig. 3 1 a series of curves where we vary the strength of the first pulse. For these calculations we keep the center frequency of the pulses at the Is-exciton resonance,_fix the delay time at td = -400 fs, and choose the fixed strength of the second pulse as iiev . EoEo/2= 0.1E~. Figure 3 1 shows that for low intensities the signal has a single peak near zero time, whereas for large intensities the peak of the signal is at 400 fs, which is the correct photon echo position in this case. The curves for intermediate intensities show the transition from signals which are characteristic for ‘free induction decay’ to those characteristic for a ‘photon echo’. A double peaked structure is seen to appear between the limiting cases The unusual dynamic behavior of the emitted signal for excitation at the exciton resonance can be understood by analyzing the dynamic resonance conditions For this purpose we plot in Fig. 32 the time dependence of the band edge corresponding to Fig. 30 (top curve) and

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8A

‘;

0

6?m z < 5 v,

4-

2-

700 TIME (fs) Fig. 31. Time-resolved signal in the photon echo direction for increasing strength of the first pulse 21. Excitation occurs at the exciton resonance, the dephasing time is Zj = 200 fs, the delay time is t,j = -400 fs, the pulse duration is for both pulses 100 fs (FWHM of the amplitude), and the peak amplitude of the second pulse is fifl . & = 0.2&. With increasing dash length, the curves are for j& . $1 = 0.02&, 0.06E~, 0.1E~, 0.14&, and O.l& (solid line). (From Ref. (24).)

to Fig. 31. First we analyze the situation corresponding to Fig. 30. From the top curve in Fig. 32 we see that the first pulse, which is very weak, leads to a shift of only about 0.3E~, whereas the second pulse causes a shift of the order of PER. In the case of such a large shift, a pulse which is originally centered at the 1s exciton resonance causes direct excitation of the continuum states after the bandgap has shifted by ER. Nevertheless, this situation does not lead to a photon echo in Fig. 30, since the phase shifts necessary for the echo am produced in the time interval between the pulses, and during this time interval the band-edge shift is still too small to allow significant continuum excitation by the first pulse. However, as can be seen from the series of curves in Fig. 32, by increasing the intensity of the first pulse, we shift and excite immediately part of the continuum so that the free propagation before the second pulse is sufficient to cause the needed phase shifts for the photon echo. With increasing intensity of the first pulse the gap shift before the arrival of the second pulse increases and exceeds ER. Consequently the band states are shifted in resonance with the field and the photon echo signal develops The second peak in Fig. 31 develops as soon as the band states shift into resonance. The weighting between the first and second peaks is determined by the relative duration of exciton excitation and band-state excitation. Finally, we want to investigate in this section the dependence of the FWM signal on the sample dimension. For this comparison we choose GaAs paramenters and a center frequency of the two pulses at the respective Is-exciton energy of the bulk, qunatum well, and quantum wire. Figure 33 shows the results obtained with a weak first pulse centered at t = -600 fs and strong second pulse at t = 0. In order to allow for a realistic comparison, the same dephasing constant is chosen for all systems, TX= 800 fs Screening and relaxation processes are neglected. In all three cases the signal (solid lines) starts at the t = 0 and is peaked around t=l ps The bulk signal exhibits a small step around t = 500 fs This can be understood by

414

R. Binder and S. W. Koch

10 TIME (fs) Fig. 32. Renormalized band edge as a function of time. _@ is the unrenormalized band gap. The delay time is td = -400 fs and the dl%erent curves are for jiev . $1 = 0.02&, 0.06& O.Ll&,0.14&, and O.lE,, from top to bottom, respectivley. All other parameters are the same as in Fig. 30. (From Ref. (24).)

comparing the full signal with the signal obtained without any exchange terms (dotted lines). We see double peaked structures where the first peak is due to the free-polarization decay (which stems from the Is-exciton), and the second peak is the regular photon echo. The pulse duration (100 fs FWHM) is so short that the pulse spectrum has considerable overlap with the continuum absorption. The step in the full signal which occurs roughly at the positive delay time indicates the underlying regular photon echo contribution. Of course, the full signal is domintated by the interaction of the Is-exciton with all other transitions, and, as in Fig. 30, the signal is peaked at a time which is a complicated function of the interaction induced rise time (Fig. 27) and the dephasing time. To further clarify this point, we show in Fig. 34a results for various delay times Clearly, the first peak (which only leads to the step in the signal for the smallest delay time) occurs roughly at the delay time. The dominating free induction decay contribution prevents the temporal shape of this echo-related peak from reflecting the pulse’s duration. To prove that the interaction of the continuum states does not alter the signal duration in such a drastic manner, we plot in Fig. 34b the signal for the delay time of 600 fs, where now the center frequency is well inside the continuum (hwo = E,+20 mev). We see that the duration of the signal is the same, independent of the interaction. Only the signal size increases strongly if the linear Coulomb interaction is switched on, and even more if the nonlinear exchange effects are taken into account. A step-like feature, as in the bulk case, is not observed in Fig. 33 in the signals of the quantum confined structures As a consequence of the increased binding energy in these structures, the overlap with the continuum of the 100 fs-pulse centered at the Is-exciton resonance of the quantum well and quantum wire, respectively, is strongly reduced. The interaction free signal is a pure free-polarization decay. The interaction leads, similar to the bulk case, to an increase of the signal magnitude and a delay of the signal peak. We note, however, that the system with the highest exciton binding energy (the quantum wire), exhibits the smallest average signal

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TIME [PSI Fig. 33. Four-wave mixing signals for a bulk, quanta well, and quantum wire system. The material is GaAs with the same parameter of Fig. 1, except for the dephasing time which is T2 = 800 fs in these calculations. Excitation occurs at the exciton resonance of the respective system. The delay time is rd = -600 fs, the pulse duration is for both pulses 100 fs (intensity FWHM), and the peak amplitudes are ficv . !& = O.OlJ& for the first pulse and O.lEa for the second pulse (the pulse sequence is shown at the top of the figure). The solid lines show the full results, and the dotted lines are results obtained without exchange contributions.

R. Binder and S. W. Koch

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-1

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3

‘ITMEIpsl Fig. 34. (a) Four-wave mixing signals for a bulk for various delay times: td = 600 fs (solid line), 800 fs (&shed line), and 1.2 ps (dotted line). Ah other parameters are the same as in Fig. 33 (the solid line is the same as in Fig. 33). (b) Four-wave mixing signals four continuum excitation: the pulse center frequency of both pulses is 20 meV above the bandgap. The dotted line shows the full result, the dashed line shows results including the linear Coulomb interaction but no exchange contribution, and the solid line shows results without Coulomb interaction.

increase due to the exciton-exciton interaction. Again, as in the case of the Rabi oscillations, a strong linear interaction does not necessarily indicate a strong nonlinear interaction, if one compares systems with different dimensions We finally note that the beating observed in the quantum well and quantum wire signals is the Is-2s-polarization beating. This assignment can easily be verified by comparing the energetic Is-2s-exciton separation in Fig. 1. It is interesting to note that this beating does not occur in the interaction-free signal (dotted lines). That means that the spectral overlap of the external pulse with the 2s-exciton is indeed negligible. However, the renormalized field has obviously a significant overlap with the 2s-resonance. 3.4.2. Bqnd-mixing eflects In this section we expand the discussion of transient four-wave mixing experiments to analyze the influence of the polarization state of the exciting light pulses For this purpose we have to include the relevant aspects of the realistic valence-band structure in III-V compounds

Nonequilibrium semiconductor dynamics

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(-l/2,2) (-1/2J)

417

(+1/N

(l/2,0)

(l/2,-2)

Fig. 35. Electron-hole-pair states resulting from the Luttinger Hamiltonian. The quantum numbers in brackets are the z-component, j, of the electron or hole angular momentum, respectively, and the envelope orbital momentum index 8. The arrows indicate the strong dipole-allowed transitions_ (From Ref. (27).)

as done by Hu et al. (27) The technical details of the band-structure analysis are summarized in Appendix B, we mainly concentrate on the numerical analysis of the resulting multi-band semiconductor Bloch equations (124)-( 126). To analyze these equations we us_ethe symmetry principle that in a quantum well all functions of the in-plane momentum vector k remain unchanged if&is rotated by 27~(see also Appendix Bl). As a consequence, one can introduce the representationC4*)

!Psj(k;m, t) =

,f

C?‘,i(k,E;m, t) emi’@

(449)

I=-00 for P and corresponding expansions for all other functions Using this representation in Eqns (124) to (126), we obtain coupled equations for s-exciton polarization, !?‘$j(k, 4’ = 0; m, t), and d-exciton polarization, Psj(k,J = 2;m, t). This is a consequence of the +-dependence of the Luttinger matrix elements, (503) and (505), which allows only A8 = 0 and A& = 2 coupling. The matrix elements of 3f,jl (k, 4 - 4’) vanish unless I - 4?’ = 0 and j - j’ = 0, or 8 - Y = k2 and j - j’ = ~2. Therefore, an exciton with envelope angular momentum lz = d?and valenceband angular momentum j is only coupled to an exciton with I’ = I + 2 and j’ = j T 2. For example, as shown in Fig. 35, an s-state heavy-hole exciton is coupled to a d-state lighthole exciton and an s-state Ih-exciton is coupled to a d-state hh-exciton. In Fig. 35 we have assigned the exciton orbital momentum Z to the holes and the orbital momentum 8 = 0 to the electrons Note, that we use the conventional definition of heavy-hole (j = +3/2) and light-hole (j = f l/2) although the angular momentum is not a good quantum number for one-particle states with finite momentum. The electron-hole-pair states axe two-fold degenerate. Hence, the complicated multiband structure can effectively be reduced to a two-fold degenerate system, which can be labeled according to the electron spin in the respective conduction band. The s = l/2 conduction .WX I0:W-H

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R. Binder and S. W. Koch

band is optically coupled to the s - hh band with j = -3/2 by a &- E-field, i.e. a circularly polarized field with polarization vector G_ = (& - i&,)/fl. In such an excitation process the d-lh band is optically excited only because of the hh-lh coupling. Since the transition between the uncoupled d-lh band and the s = l/2 conduction band is dipole forbidden, the d-lh component in this excitation process is considerably smaller than its hh counterpart. Hence, in comparison to the dominating s-hh contribution the d-lh contribution is negligible. To illustrate the field polarization effects, we numerically solve Eqns (124) to (126) for m = 0, k 1, choosing the incident fields as circularly polarized (K &*) and assuming that the propagation directions are nearly perpendicular to the plane of the quantum-well layers Before we present our numerical results, we first discuss the situation of oppositely circularly polarized pump and probe pulses, choosing c+ (&_.) polarized pump (probe). For this purpose, we analyze how these pulses interact with the subset of s = - l/2 conduction band and valence bands j = 312 and j = -l/2 (see Fig. 35). The possible echo signals are due to P-1,2,&k; -1) and !P_ I/Z,_ 1,2(/c;- 1), where again the argument - 1 in P indicates the FWM direction. We see_from Eqn. (124) that !P_1~2,&k; -1) stems from the scattering terms fi-i,2;3,2 . 20 f&&k; - 1) and !P_i,~;~,~(k, - 1)f$2,3,2(k; - 1). However, the population pulsation f$,2,3,2(i; - 1) has to be created by &l/2:3,2 * $00, (fi-1,2;3,2' gi,,*, which is zero since the pump and probe fields are oppositeJy polarized. The source terms in_Eqn. (124) for-the signal P-1 12,~I&; - 1) are -1). Inspection of Eqn. (126) Li12;-112 . 20 f&-1,2(k; -l), and P- 1/2;-1,2(k, -l)f&,,_,,,(k; shows, however, that f$2,_1,2(i; -1) = 0 in this case. It is straightforward to extend these arguments to other cases Thus, within the Hartree-Fock approximation evaluated in a timedependent xc3)-analysis, the echo signals are exactly zero if pump and probe fields have opposite circular polarizations Hence, our analysis indicates that for oppositely polarized pump and probe beams any experimentally observed photon echo signal has to be attributed to deviations of the system from a perfectly symmetric thin quantum well, excited in the low intensity limit. For example, random fluctuations of the quantum-well interfaces introduce a sample-dependent effective coupling between the hh s- and d-states As a consequence of this disorder-induced coupling, (61) four-wave-mixing signals become dipole allowed even if the pump and probe fields are oppositely circular polarized. Besides well-width fluctuations, other mechanisms, such as long-range exchange interactions(60) as well as scattering by acoustic phonons,@@ can also cause deviations from the ideal results To distinguish between the different effects it is essential to perform detailed time-resolved four-wave-mixing experiments in quantum wells under systematically varied sample and pump-probe conditions In addition to real spin transfer processes, there are other incoherent scattering processes, which do not need a spin-flip but still would affect, e.g. the &+ exciton even if dipole transitions involve only the valence and conduction bands for the &_ exciton. Before we discuss these scattering processes, we continue the study of the ideal coherent system for the case that the pump and probe fields have the same circular polarization. In this case the SBE for the sixband system (only s-state, no d-state excitons) reduce to the well-studied two-band Bloch equations discussed in the previous section. For the subsequently discussed numerical results(27) we choose a quantum-well width of 100 A with GaAs material parameters To reduce the substantial numerical complexity we assume perfect quantum confinement conditions and a purely two-dimensional Coulomb-potential. Furthermore, we use Gaussian light pulses with a 100 fs intensity full width at half maximum (FWHM). The maxima of the pulse amplitudes are 0.01 meV for the pump and 0.001 meV for the probe, and the dipole matrix elements of the hh transition are a factor J5 larger than those of the Ih transitions We use a delay time between the pulses of td = -300 fs, i.e. the probe pulse comes 300 fs earlier than the pump pulse which is centered at t = 0 fs We evaluate Eqns (124)-(126) for various pump and probe conditions for circularly co-

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0.8

TIME (PSI Fig. 36. Computed four-wave-mixing signal in the direction 2$ - & for a 100 A GaAs quantum well (yt = 6.85, y2 = 2.1, y3 = 2.9, m, = O.O67rnr,, e0 = 12.7). The depbasing time is 2” = 200 fs. The light pulses have amplitudes ji-1,2;3,2 - 6 = 0.02 meV and jj-1/2;3/2 . $1 = 0.002 meV. The delay time is td = 300 fs and the central frequency (same for both pubes) is at the heavy-hole exciton resonance. The arrows indicate the center time of the pump 0 fs and probe -300 fs, respectively. The inset shows the corresponding linear absorption spectrum. The arrow in the inset marks the probe and pump center frequency. The results shown as the dashed curve neglect the off-diagonal elements of the Luttinger Hamiltonian while the solid curve is computed using the full Luttinger Hamiltonian, Rqn. (520). (From Ref. (27).)

polarized light. In Fig. 36 we show the computed signal in the FWM direction for pump and probe pulse frequencies centered at the hh exciton resonance. The FWM signal has the same polarization as that of the incident pulses The solid curve in Fig. 36 shows the results of the effective mass approximation (i.e. hole states with angular momentum 8 > 0 are neglected). The dashed curve in Fig. 36 is obtained by numerically integrating the full equations including the s- and d-exciton coupling. The inset shows the computed linear absorption for the same material and pulse parameters As discussed before, the hh-lh coupling quite generally is expected to be weak. Figure 36 gives a quantitative estimate of this, clearly showing the validity of the effective-mass approximation in this situation. The signal in Fig. 36 is attributed to free-induction decay. We see from this calculation that the lh bands do not significantly contribute to the signal if the pump is in resonance with the Is-hh exciton. The reasons are the large frequency detuning of the pulse with respect to the lh exciton and the l/d smaller matrix element of the Ih transition. In Fig. 37 we show the results for the case where the center frequencies of pump and probe pulses are at the l&h exciton with otherwise unchanged parameters The signal again exhibits free-induction-decay-like behavior. However, the contribution from the hh bands (centered around 0.2 ps) is almost one order of magnitude smaller than that from the lh band. In this situation the signal is mostly due to the lh exciton states, the hh continuum contribution is very weak even though it is resonantly excited. Energetically nondegenerate four-wave-mixing results are shown in Fig. 38. Hem the probe pulse is tuned to the Is-/h exciton resonance and the pump is at the Is-hh exciton. Similar to the results in Fig. 36, the Zh bands are not excited in this case. The signal is still the freeinduction decay from the hh exciton. In contrast to the rather simple signal shape shown in

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R. Binder and S. W. Koch

I355845 835 825 WAVELENGTH lnm)

-0.8 -0.4 0.0 0.4 TIME(ps)

0.8

1.2

Fig. 37. Same as Fig. 36 but both pump and probe pulses are tuned at the light-hole exciton resonance. The top two curves are the computed four-wave-mixing signals using the Luttinger Hamiltonian (solid line) and the simplified six-band model (dashed line). The two curves in middle are the light-hole contributions (the higher one is obtained from the sixband equations). The lowest curve centered around 0.2~s (dash-dotted line) is the heavy-hole contribution which is almost identical in both calculations. (From Ref. (27).)

Fig. 38a, however, pronounced oscillations appear in the time-resolved signal if the Ih exciton energy is chosen as the center frequency of the pump pulse and the probe is tuned into the hh exciton resonance (Fig. 38b). These quantum-beat oscillations are due to the interference between the Is-hh exciton and its excited states In Fig. 39 we see the results obtained when both pulses am tuned slightly above the lh bandgap. The hh continuum contribution (dashed line) yields a simple echo behavior, whereas the lh contribution contains oscillatory features These oscillations are the quantum beats between the ground and excited states of the Ih exciton. The two maxima from the lh signal are separated by approximately 220 fs while the ideal quantum beat period is 200 fa Similar quantum beat phenomena in the photon echo measurement of semiconductor quantum dots have already been predicted theoretically.(28) Quantum beats between hh and lh excitons occur if the pulse spectra are broad enough to cover both exciton resonancea To simulate this situation we leave all pulse parameters unchanged, but choose a slightly thicker quantum-well of 130 A, where the hh-lh splitting is considerably smaller than in the previous examples Figure 40 shows the computed signal for this situation. The inset shows that the pulse spectra for the parameters chosen cover both exciton resonances in the linear absorption spectrum, making the hh and Ih quantum beats possible. .The more detailed analysis shows that the signal from the lh band is relatively simple whereas the hh band signal has oscillating features which can be attributed to the quantum beats between the exciton ground state and excited states The additional oscillations in the total signal am the quantum beats between the hh and lh states The results discussed so far are based on a theory, in which exciton-exciton interactions are taken into acccount only to the degree of fermionic Hartree-Fock exchange contributions In other words, we have neglected the possibility of bound exciton states, i.e. biexcitons The main reason for that is that the biexciton binding energy (i.e. the difference between the biexciton energy and two exciton energies) is rather small in GaAa However, it recently became appeamnt that biexcitonic effects in thin GaAs quantum wells can be significant, and that the biexciton

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(a) .

WAVELENGTH (r,ml

0.06

’ -0.8

-0.4

0.0 0.4 TIME(ps)

0.8

1.2

Fig. 38. (a) Same as Fig. 36, but the probe pulse center frequency is tuned into the light-hole exciton resonance. (b) Same as (a) but with exchanged pump and probe center frequencies. (From Ref. (27).)

Table 1. Polarization selection rules for the two-beam case. The polarization directions are denoted by X, Y (linear) and u+, (T_(circularly polarized light). dbd means determined by dynamics (from Ref. (67)) Beam 1

Beam 2

2Kr - Kz

2K2 - KI

u-

no signal X Y WA

no signal X X dbd

X X X

binding energy in those systems (which can be as large a 2 mev) is not always negligible.(63) In order to derive selection rules for FWM-signal without any simplifying assumption other than the validity of the Luttinger Hamiltonian, Lindbergog2) suggested a scaling analysis in which all polarization and distribution functions are given as products of certain combinations of the external light field amplitudes 2 and functions, which depend only on l$12.(a7) The only crucial assumptions in that investigation are that all pulses are essentially colin:ar and that the rotating wave approximation is applicable. The precise form of the explicitly T-dependent prefactors was explained as a consequence of the underlying rotational symmetry of the

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R. Binder and !J. W. Koch

Fig. 39. Same as Fig. 36, but both pump and probe have center frequencies at the band continuum (Rwe = Rwt = 1.518eV) (From Ref. (27).)

-0.8 -0.4 0.0 0.4 TIME(ps)

0.8

1.2

Fig. 40. Computed echo signal for a 130-A GaAs quantum well. The parameters are the same as in Fig. 36, except pump and probe center frequencies are changed (hwe = hwt = 1.44 ev). The solid curve is the net echo signal, the short-dashed and long-dashed curve correspond to the heavy-hole and light-hole band contributions, respectively. The solid curve in the inset is the computed linear absorption of the same sample and the spectrum of probe and pump pulses is given as the dashed line. (From Ref. (27).)

Hamiltonian, which also explains why the above analysis is based on the selection rules for the quantum numbers connected to the two-dimensional rotation symmetry of the quantum well, j, s and 8. The discussion in Ref. (67) is, however, not restricted to quantum wells A summary of the resulting selection rules is given in Table 1. Further results containing three-

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pulse selection rules are given in Ref. (67). 3.4.3. Excitation-induced-dephasingeffects In the previous section we saw that, in a thin quantum well oppositely circularly polarized, light pulses do not yield a four-wave mixing (FWM) signal, as long as there is no disorder or not enough time for real spin-flip processes of the holes to occur. This, however, does not imply that in general G+-excitons and &--excitons are uncoupled. To investigate such a coupling without invoking spin-flip phenomena we drop in this section the earlier assumption that the pump pulse is so weak that the pump-induced excitation density does not contribute to additional excitation induced dephasing (RID) processes (see Wang et al.(64)).To include such processes into Eqns (124)-(126) we have to replace

which corresponds to the general model (133). If the effect of the pump pulse is such that the change of the dephasing rate is small, then we can expand y about the initial value. We allow in the following for a generally nonzero initial density, which, for example, may be created by an optical pie-excitation pulse. Denoting the pre-excited density no, we have

r&z) = yij + An &(n = no) , with rij = Ysj(n = no) and

i 1

xg(k,t)

+ xf,@,t)

- [nG+ni]

(451)

.

(452)

In general, the computation of ys,(n) is very difficult. If, however, the system is in a quasiequilibrium, one can compute the dephasing rate within the Screened-Hat-tree-Fock approximation (see, e.g. Ref. (2)) +gB(P, -i=$)] ImIV(i-rl-‘,P,-P,). r,(kn) = 1 [J-$(k)

(453)

k

In the case of valence-band coupling, 4 would be the dispersion according to the diagonalized Luttinger Hamiltonian. In Eqn. (453) the dephasing rate depends on the wave-vector i, but the off-diagonal elements of y (see Eqn. (132)) have, for simplicity, been neglected. If we insert Eqn. (451) into Eqn. (124) to (126) we can perform the spatial Fourier analysis easily. We obtain the following replacement for the simple dephasing Eqn. (124):

+

1 _f~jGC;m- m’) i

1

Psj(ii;m’).

(454)

For clarity of the following discussion, we neglect in the following the light-hole bands, so that the relevant transitions are only the ii, -transition involving the s = - l/2 electron and the &--transition involving the s = + l/2 electron (Figs 2 and 35). An effective coupling of the excitons corresponding to these two transitions can be achieved in the following way. Assume,

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Koch

for example, the pump beam creates &+-excitons These excitons screen the Coulomb potential by means of intraband scattering. The screening of the Coulomb potential affects then also the a_-excitons, because, quite generally, the screening of the Coulomb potential between any two carriers depends on the total exciton density, n = &na (na is the exciton density of e-excitons). The change of screening affects the binding energy of the exciton, the bandgap renormalization, and the dephasing rate. Our numerical studies have shown that the change in the dephasing rate is the dominant effect that couples &+ and &--excitons on a sub-ps timescale without spin-flip processes To investigate this effect in a FWM experiment, one has to use linearly polarized pump and probe pulses, because of the absence of a FWM signal for oppositely circularly polarized pulses(64) Using linearly polarized beams, one always obtains a FWM signal, and the signal depends on the relative polarization of the beams If the beams are parallel polarized, we denote the signal by 41, and if the beam polarizations are perpendicular, we denote it 4. Additional information can be obtained if the system is pre-excited with a certain exciton density. We restrict ourselves in this section to a third-order expansion in the pump and probe field amplitudes of Eqns (124) to (126). In this limit, the expansion of the dephasing up to linear order in th_edensity (Eqn. (451)) becomes exact and the y’-term multiplying the density grating 0: Cjf;j(k;m - m’ = rt1) in Eqn. (124) yields a G--i;, coupling. An analytic perturbation analysis shows that the two gratings (j = +3/2) are out of phase if the pulses are polarized perpendicularly. In that case the dephasing induced contributions to the signal ZL are zero. For parallel polarized pulses, the gratings are in phase yielding a nonzero contribution to 41. Consequently, the ratio 41/Z, is a sensitive function of y’. In a pi-e-excited system, y’, which is evaluated at the pre-excitation density no, may vary as a function of no if the dephasing processes described by r(n) undergo saturation effects with increasing no. A typical variation of the dephasing rate (at T = 20 K) is shown in the inset of Fig. 41. The system chosen in Fig. 41 is bulk GaAs, where the hh-Zh degeneracy is lifted, for example by external strain, so that, similar to thin quantum wells, the hh- and Z/r-resonances are practically decoupled. Figure 41 shows that, similar to the perturbational analysis, ZL depends less on no than 41 because there is essentially no EID-contribution in IL. In the high density limit shown in Fig. 41 the EID contribution vanishes for both I1 and III, because y’ + 0. Hence, the ratio 41/Z, approaches unity. We also show in Fig. 41 results without exchange terms in the SBE. As discussed before (see also Ref. (24)), the exchange effects dominate the nonlinear optical signal for a two-band semiconductor. In our current model the contributions from both subsystems, &+ and &-, experience this effect. Only 41, however, is additionally affected by EID since EID itself is influenced by the exchange effects, although only indirectly via the density grating. Removal of the exchange terms means that both Z, and 41, are reduced. Because 41 depends on the EID and the exchange effects, and the EID is not fully removed along with the exchange terms, and since ZLis almost solely determined by the exchange terms, the ratio 41/IL is enhanced without exchange terms Investigation of EID therefore yields previously unaccessible information on ultrafast many-body effects in semiconductors with zincblende structure. As we have mentioned at the end of Section (2.8.), similar results have been obtained on the basis of a x(3)-calculation.(‘86) 3.5. Pulse Propagation 3.5.1. Pulses centered at the exciton resonance 3.5.1.1. Coherent pulse propagation. We begin our discussion of pulse propagation effects with the work of Knorr et al.(26133)To investigate the influence of the many-body effects on the coherent propagation of short pulses and to compare these results to the phenomenon

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0.01 1.0

’

425

’ ’ -y--,-_ 3.0 5.0 7,o 9.0 DENSITY ( 1015cm-3)

Fig. 41. Ratio of co- and cross-linearly polarized FWM signals in strained bulk GaAs as function of the preexcited exciton density (solid line). The &shed line shows results (divided by 10) without exchange interactions. Inset : density dependence of the underlying dephasing model. (From Ref. (64))

of self-induced transparency (SIT) in atomic systems, we neglect in this section all relaxation processes and screening effects, We use the coupled Maxwell semiconductor Bloch equations, i.e. Eqns (82) and (83) together with (455) Equation (455) is the same as Eqn. (24) except that there is no dependence on the transverse coordinate x, and that the vector properties are suppressed. Of course, within the two-band approximation the description of vectorial properties of the light field are not possible anyway. From this set of equations one can derive the relation(33)

between the energy of the pulse and the density after the pulse-excitation. Here, we define the pulse energy density (energy/beam area) as (457) According to Eqn. (457) the possibility for lossless propagation in an initially uninverted semiconductor is given for pulses which return the density to zero after the pulse. For pulse propagation in two-level systems this condition is satisfied if the pulse area is an integer multiple of 27r. These optical fields conserve their area while propagating and show pulse break-up for areas in excess of 2rr. For example, a 47r-pulse breaks into two 27r-pulses, which travel without distortion through the medium after the break-up.(49 lg3plg4) We begin the anylsis with the propagation of a 27r-pulse centered at the exciton resonance of bulk CdSe (the material parameters are the same as in Section 3.4.1.). We choose here a

426

R. Binder and S. W. Koch

-200

-100

0

100

TIME (fs)

200

300

Fig. 42. Intensity (solid line) and density (dashed line) profles for an initial Zrr-area se&shaped light pulse at different positions in the semiconductor sample for the case for coherent propagation. 10 = (10-4~R2f~)/(8n&). (From Ref. (33).)

short pulse (100 fs intensity FWHM), which has a certain spectral overlap with higher exciton states Figure 42 shows the computed pulse intensity (solid lines) and generated carrier density (dashed lines) at different positions in the sample. It can be seen from Fig. 42 that, in contrast to the two-level dynamics, the initial 2rr-pulse develops a shoulder and becomes shorter during the first part of the propagation. As discussed in Section 3.2., in this parameter regime the number of Rabi flops is nearly doubled compared to the same excitation conditions for a twolevel system (Fig. 42, dashed line for z = 0). However, due to the remaining density in the system after the pulse, the energy of the electromagnetic field is slowly absorbed during its propagation. This can be seen from Fig. 42 by comparing the initial intensity profile and the profile after the propagation. Due to this absorption the effective Rabi frequency is gradually reduced so that the second Rabi flop cannot be maintained during the propagation through the sample. In the case of a two-level system, a 2rr-pulse would create one Rabi flop following the shape of the pulse. The approximate doubling of Rabi flops in the semiconductor, which causes an oscillating temporal structure of the source terms in Eqn. (459, leads to peak amplification and temporal compression of the field. Knorr et al. have shown that SIT exactly analogous to that in a two-level system cannot occur. (33)However, complete oscillations of the induced density am possible in the limit of the weak external fields and vanishing dephasing.(‘g5) To investigate the phenomenon of resonant pulse break-up in semiconductors we solve the coupled Maxwell-SBE for an input pulse of 4rr. We show in Fig. 43 the temporal profile of the pulse intensity at different positions in the material. As can be seen from this figure, the propagating 4rr-pulse eventually breaks into three quite well separated smaller pulses Two of the three pulses have originally almost the same amplitude, whereas the third one is substantially smaller. During propagation, the amplitude of the first pulse increases at the expense of the two other onea We note two main differences in comparison to the propagation of a 4rr-pulse in a two-level system. First of all, a 4rr-pulse in a semiconductor induces approximately four Rabi flops (effective 87r-pulse). 6) In a two-level system these four Rabi

Nonequilibrium semiconductor dynamics

421

, -200

0

400

200 TIME (fs) -

Fig. 43. Temporal intensity protiles for an initially 4x-area and se&shaped pulse at different positions in the semiconductor for the case of coherent propagation. Pulse break-up into three pulses is observed. This specific pulse would show total break-up into two Zrr-pulses when traveling in a two-level system. (From Ref. (26).)

flops would lead to a breakup into four separated pulses, whereas we observe only three pulses in the semiconductor case. Additionally, small portions of the pulse energy are absorbed during the propagation. It was found by McCall and Hahn (1g3*‘g4) that the area of the optical field is the appropriate measure of the coherent interaction strength between light and matter. The famous area theorem shows that for the propagation of nonfrequency-modulated pulses in two-level systems, pulse areas which are integer multiples of 27r are stable steady state solutions Since the refractive index for a homogeneously broadened two-level resonance is perfectly antisymmetric around the resonance energy, no net frequency modulation occurs for resonantly propagating pulses as long as pulse shape and excitation density are symmetric. The resulting temporal changes in the refractive index of the medium vanish. In semiconductors, on the other hand, the refractive index change is not symmetric and its zero crossing is blue shifted in comparison to the exciton resonance. Hence, all propagating pulses which resonantly excite the exciton accumulate a phase modulation. To take this into consideration, we introduce as the measure of the coherent interaction strength the modulus of the area A,,

&03

=

l/2

do Rep,EE(r, r7) * + 1

[I

drl

1 2

b.@E(C,

r7)

.

(458)

For vanishing imaginary part, as well as for vanishing phase modulation of the pulse, the area defined in Eqn. (458) reduces to the usual area definition. In Fig. 44 we plot the computed pulse area as a function of the propagation distance for various input pulses It can be seen that the area of a rr-pulse is more or less conserved, whereas the area of larger pulses decreases to rr in an oscillatory fashion. As a reminder, we mention that in a two-level-system area conservation occurs for all integer multiples of 27r-pulses, leading to several stable branches

R. Binder and S. W. Koch

428

0

200

400

600

PROPAGATION DISTANCE (orb units) -

800

Fig. 44. Effective area, defined in Eqn. (458). of different real input pulses vs propagation distance. (From Ref. (26).)

in a plot like Fig. 44 (see, for example, Ref. (4)). Obviously, the semiconductor scenario is substantially different. The conservation of a rr-pulse instead of the 2rr-pulse in atomic systems is not too surprising, since this is simply the consequence of the fact that a rr-pulse in a semiconductor already generates one Rabi flop. However, even including pulses with an initial area of 4rr, we did not find a higher branch in Fig. 44 with stable areas We believe that this again is a consequence of the finite density remaining in the sample after each single Rabi flop. Due to the same reasoning, the rr-branch in Fig. 44 is also only a quasi-stable solution. The pulse is initially successfully stabilizing its area but it cannot avoid the absorption losses which eventually lead to decay for longer propagation distances Similar behavior has been found for pulse propagation under the simultaneous influence of two-level and Kerr nonlinearitiest’g6)

3.5.1.2. Four-wave mixing and pulse propagation In this section we investigate the modifications of pulse propagation effects on multi-wave mixing experiments in spatially extended semiconductorst30p ‘g7-200) To illustrate the main features we concentrate on the configuration where the exciton is resonantly excited and the spectral overlap of the pulse with all energetically higher excitons or with the continuum is minimized. The FWM propagation equation (31) has been evaluated numerically by Schulze et aI.(201) considering only resonant Is-exciton excitation and ignoring all other contributions in the SBE. As an example of the results we show in Fig. 45 the time-resolved FWM signal for different time delays after propagation through a 2pm sample. For these calculations we assumed Gaussian input pulses with intensity FWHM of 300 fs and same center frequency (i.e. w, = wg in Eqn. (31)).

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429

T/P8 0.6 0.3

-0.6 0

2

4

6

TIME (ps) Fig. 45. The computed time resolved four-wave-mixing signal is plotted for a sample of length 2~ for different time delays T between the input pubes. The dashed lines show the results obtained without the exchange contributions in the semiconductor Bloch equations. The input pulses are assumed to have a Gaussian shape and an intensity FWHM of 300 fs, the other material parameters are chosen to be representative for bulk GaAs (From Ref. (2011.1

Figure 45 shows that the propagation effects induce an oscillatory structure in the FWM signal for positive and negative ti.me delays (solid line). Such propagation induced oscillations occur already for single pulses(201) In order to separate the propagation induced features from the modifications caused by the Coulomb effects, we also show in Fig. 45 the solution of the equations without exchange terms (dashed line). We clearly recognize that the propagation leads to a signal for negative time delays and to time-delayed components, i.e. temporal signal break up. An overview of the relative importance of the propagation effects on the time-resolved FWM signals is presented in Fig. 46, where the sample length is varied between 0 - 6~ for a fixed delay time td = 0.3 ps The figure clearly shows that the single peaked FWM signal for small sample length breaks-up into several peaks for increasing propagation length. We see that the magnitude of the signal shows a pronounced peak for sample lengths in the range of 1 - 2pm. This peak results from the growth of the signal for short propagation lengths, due to the quadratic increase of the signal with length, and the onset of destructive interference when propagation becomes important for longer samples Figure 46 demonstrates that for longer samples not only the peak value decays, but also the temporal decay of the signal depends strongly on the sample length. For samples longer than 2pm we see a growth of subsequent peaks in the time domain. The modulation is strongly length dependent and for longer samples, in excess of approximately lpm, the propagation effects dominate the FWM signal.(20’) In Fig. 47 we plot the time-integrated signal for different sample lengths First of all we note a propagation dependent modification of the signal decay. A transition occurs from simple decay to oscillatory behavior with increasing propagation length. The modulation frequency of the signal increases with increasing propagation length.

R. Binder and S. W. Koch

430

Fig. 46. The computed time resolved four-wave-mixing signal is plotted as function of time and propagation length. The time delay between the input pulses is tied at T = 0.3 pa (From Ref. (201).)

-2

0 DELAY

2 TIME

4

6

(ps)

Fig. 47. The time integrated four-wave mixing signal is plotted as function of the delay time T between the pulses for different sample lengths. Only for the very short sample length of 0.1~ the signal decay time constant agrees with the input dephasing time of 2 pa For longer samples propagation effects modify the decay behavior. (From Ref. (201).)

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As a rough estimate from these numerical results we conclude that for samples longer than lprn the FWM signals are significantly influenced by propagation effects making the simple extraction, e.g. of dephasing times out of FWM experiments, problematic. Experimentally, the relevance of propagation effects should therefore be checked carefully, e.g. by measuring the temporal distortion of a single pulse. If pulse reshaping is observed, propagation is certainly also important in analyzing the FWM results 3.5.2. Pulse propagation in amplifiers In contrast to pulses propagating in absorbing media, one expects for amplifying media the growth of the input pulse at the expense of the medium inversion. However, Knorr et al. (202) found that, besides pulse amplification, also lossless propagation and even pulse absorption can occur. For long propagation distances we find a propagation induced adiabatic following scenario leading to drastic pulse reshaping and shortening (see Indik et aI.(34)). In inverted semiconductors, which exhibit gain and therefore a high excitation density prior to the arrival of the pulse, dephasing of the interband polarization and screening of the Coulomb potential are of importance even if the intensity of the propagating pulse is low. Thus, inclusion of dephasing and scattering terms in the semiconductor Bloch equations is important. In Ref. (154) the time evolution of an initially disturbed Fermi distribution for electrons and holes is computed as an initial value problem with the relaxation rates (379) and (381). The distribution functions of the electrons and holes are shown in Fig. 48, and the corresponding relaxation rates r = Tin + rour are shown in Fig. 49. The relaxation times are of the order of 50 fs, not only in the k-region which corresponds the gain region (below 6aj’), but even high in the absorption region. As a consequence, concerning the propagation of high intensity pulses, like ultrafast 2rrpulses, incoherent phenomena are likely to dominate over the coherent inversion dynamics related to Rabi oscillations Nevertheless, before we study this more realistic case of large relaxation and dephasing rates, we briefely discuss the problem of Rabi oscillations in optical amplifiers under idealized conditions, i.e. we neglect all dephasing and relaxation processes completely. First, we show in Fig. 50 the density response of an amplifier with initial carrier density n = 2.5 x 10’8cm-3 at T= 300K. The center frequency of the external pulse is close to the gain peak of the linear absorption spectrum. The pulse is se&shaped and the pulse duration is 150 fs (intensity FWHM). We see from Fig. 50 how, with increasing peak intensity (i.e. increasing pulse area), the density response develops oscillations The oscillations are, however, not sin-shaped, and in the case of the highest intensity they appear as variation of an envelope-like structure. It is important to note that the final density, i.e. the density which remains when the pulse has passed through, is always less than the initial density. As a consequence, the pulse would gain energy according to the net density loss caused by the pulse, which is, of course, nothing but the regular stimulated emission (see Eqn. (456)). We will see later that strong incoherent processes can change this result. In order to study the coherent density response to a 27~-pulse, which, in an inhomogeneously broadenened system of two-level systems would not yield any effective density change, we study in Fig. 51 the effects of a 2m-se&-pulse. The result without Coulomb interaction shows the regular two-level dynamics The final density is the same as the initial density. This is not true if we include the coherent Coulomb exchange interaction. Of course, we have to choose different center frequencies for the case with and without Coulomb intercation, since the bandedge renormalization alters the spectral position of the peak gain. In Fig. 51 all results are shown for the case where the pulse is centered in frequency roughly at the respective peak gain. We see from Fig. 51 that the Couloumb effects slightly increase the effective Rabi

R. Binder and S. W. Koch

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3

3

6

WAVE NUMBER kaB Fig. 48. Relaxation of initially disturbed Fermi distribution functions for density a, = nh = 3 x 10’8cm-3 and temperature T = 300K, based on the relaxation rates (379) and (381). Shown are the distribution functions of electron (a) and holes (b) in units of the inverse exction Bohr radius in GaAs, aj’, for the times f = 0 (dotted lines), t=21 fs (short-dashed line), t=75 fs (long-dashed line) and t=796 fs(solid line). (After Ref. (154).)

frequency, which is in agreement with the results for noninverted semiconductors, see Section 3.2.. The increase of the Babi frequency in amplifiers is, however, siginificantly smaller than in noninverted systems In addition, the quasi-static screening of the Coulomb potential reduces this increase even more. Thus, the coherent Coulomb interaction processes constitute only a Flatively small perturbation on the density response with respect to the model of independent k-states A much more significant effect of the Coulomb interaction in amplifiers is due to incoherent scattering processes, which we discuss in the following. We now include the effects of dephasing and carrier relaxation due to carrier-carrier scattering in ‘the study of short optical pulse propagation in GaAs semiconductor amplifiers We solve the SBE including quasi-static screening, a dephasing rate y = h / 60 fs and the corresponding scattering terms within the relaxation rate approximation, with Te = Th = y. Contrary to most studies of amplifier nonlinearities which concentrate on the centerfrequency dependence of the pulse, we focus in the following on the intensity dependence. The center frequency will be 6xed throughout this section. The intensity Z is chosen such that the se&pulse with a FWHM of 150 fs is fully engulfed in the gain region of the linear spectrum of the amplifier. The initial carrier density is 2.5 x 10’8cm-3 at T = 300 K. The sample length is taken as one linear gain length, & = g;’ , where go is the linear gain, so that the transmitted energy is’amplified by a factor of e1 for a small input pulse with the same frequency spectrum as the strong pulse. Figure 52 shows examples of our numerical results for various peak input intensities The input (solid lines) and transmitted (dashed lines) pulse profiles for low (Z = Zs/4), intermediate (Z = ZO= 1.8GW/cm2), and high (Z = 410) peak input intensities are shown in Fig. 52a . The corresponding variation of the pulse energy, normalized to the input pulse energy, with scaled

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semiconductor dynamics

433

WAVE NUMBER kaB

Fig. 49. Carrier-carrier scattering rates of electrons (a) and of holes (b) extracted from Fig. 48. Shown are only the rates for I = 0 (dotted lines) and ~796 fs (solid line). (After Ref. (154))

T

CT 7.0

“0 3

.”

c

z

6.5

6.0

-0.4

-0.2

0.0

0.2

0.4

TIME[ps] Fig. 50. Temporal variation of the carrier density at the front face of a GaAs semiconductor amplitjer. The initial density is n = 2.5 x lO’*~m-~ at T= 3OOK. The pulse, with center frequency at the gain peak, has a duration of 150 fs (intensity FWHM), and the peak dipole energy is 10 meV (short-dashed line), 20 meV (dotted line), 40 meV (dash-dotted line) and 80 meV (long dashed line). All dephasing and relaxation contributions are neglected.

R. Binder and S. W. Koch

434

6.2 ,.._..._.......... T-

6.1,

5.7 -0.28

-0.14

0.0

0.14

0.28

TM! [PSI Fig. 5 1. Temporal variation of the carrier density at the front face of a GaAs semiconductor amplifier. The initial conditions are the same as in Fig. 50. The se&shaped pulse has an area of 27r. Results without Coulomb interaction are shown as dotted line. In this case the center frequency is Rwe=50 meV above the unrenormalized band edge. This corresponds to the gain peak. In the case with Coulomb interaction, shown as solid line (including screening) and dashed line (without screening) the center frequency is 20 meV above the unrenormalized band edge, which is again rougly at the peak gain. Again, all dephasing and relaxation contributions are neglected.

propagation distance go<, is shown in Fig. 52b. For the low input-intensity pulse (dashed line in Fig. 52b) amplification of the pulse energy is clearly seen. In contrast, for the intermediate intensity case (dotted line), the pulse energy propagates essentially unchanged, and for the high intensity case (dash-dotted line) the transmitted pulse suffers even a net absorption. The intermediate intensity (Z = lo) case shown in Fig. 52 represents a special situation. Here the amplifying states and the nonresonant absorbing states have equal weight and the density returns to its initial value after the pulse. This means that there is no net absorption or amplification, which is reflected by the fact that the transmitted pulse energy is almost identical to that of the input pulse. Under these conditions the pulse therefore undergoes lossless propagation. Figure 53a shows the temporal variation of the carrier density at the front face of the amplifier. Whereas the low intensity leads to a net decrease of the density (corresponding to an energy gain of the pulse), the high intensity result shows an effective increase of the density. Even though the optimized pulse intensity ZOleads to density variations during the pulse, at the end the density returns back to its original value before the arrival of the pulse. To understand these phenomena, we recall that them is a continuum of electronic states in an inverted semiconductor. Prior to the pulse these states are partially populated by the electron-hole plasma according to Fermi-Dirac distributions The one-particle states are only inverted in the spectral region between the renormalized bandgap and the chemical potential, which is determined by the total carrier density. Hence, one always has the coexistence of a spectral gain region and a region of optical absorption above the chemical potential. Our analysis shows (*O*)that the results in Fig. 52 am caused by the interplay of the states in the gain and absorption regions To appreciate this feature, one has to realize that, even when

Nonequilibrium semiconductor dynamics

435

/’

,I’ ‘\,

(a)

// /

// */ ;

\

‘,x4 \ \\ \.

\

-0.1

-0.2

0.0

0.1

0.2

TIME [PSI

/_/=-I&I

(b) C-

c_

cc

b

C.~~............................................................

----A_._._, -.-.__

-*-.-._.

4 XI, 0.5

0 z

[units of

1 &)-‘I

Fig. 52. (a) Input (solid line) and output (dashed line) pulse intensity profiles of propagation over one linear amplification length g;’ m a GaAs amplifier and three different peak input intensities: Is/4 (top), Is = ~.~GW/CPP?(middle), and T = 10 (bottom) (for clarity the respective base lines for the different intensities are shied). (b) The corresponding plots of pulse energy transmission (pulse energy divided by the input pulse energy) vs go< for Is/4 (dashed line), Is = 1.8GW/c& (dotted line), and Z = 41s (dash-dotted line). An input pulse duration of 150 fs FWHM was used in all calculations_ (From Ref. (202))

exciting into the gain region, the optical pulse interacts also with absorbing states These states have a nonzero linewidth and therefore give rise to absorption of the pulse. For high-intensity pulses, when gain saturation becomes appreciable, this absorption can overcompensate the amplification since the weighting of the absorbing states is high (it increases a k2 due to the semiconductor density of states). For the high intensity (I = 410) we show in Fig. 53b the distribtuion functions at the front face of the amplifier before, during, and after the pulse. The transient spectral hole at t = 0 around a certain k-value, which corresponds to the center frequency of the pulse, is accompanied by a broad decrease of carrier population of low k-states On the other hand, the high k-state

436

R. Binder and S. W. Koch

I

IO/4 d

-0.2

-0.1

0.0

0.1

0.2

0.3

TIME[ps] 8 1.0

& 5 L

ioo~

5 0.5

0

9 6 3 WAVENUMBER[unitsof ae-7

12

Fig. 53. (a) Temporal variation of the carrier density at the front face of the sample under the same conditions as for Fig. 52 for I = Is/4 (dash-dotted line), I = Is (dotted line), and I = 41s (dash-dotted line). (b) Electron distributions fe(k) (thin lines) and hole distributions fh(k) (thick lines) at the front face of the waveguide. The distributions are shown for times before the pulse (solid lines), at the peak of the pulse, t = 0 (dotted lines), and after the pulse when a thermal quasi-equilibrium due to carrier-carrierscattering is reached (dash-dotted lines). The peak intensity is 410.

population is increased. Due to the fast carrier-carrier scattering the distributions teaches a quasi-thermal equilibrium shortly after the pulse has passed through. This quasi-thermal equilibrium is characterized by a slightly increased carrier density and a greatly increased carrier temperature (T >6OOK). Thus, the absorption of carriers above the Fermi edge is an additional carrier heating mechanism. Other heating mechanisms, such as two-photon absorption and free carrier absorption, are not included in the model discussed here. As another interesting example we study the propagation of ultrashort pulses over long distances in semiconductor amplifiers (34)For this purpose we solve again the semiconductor Bloch equations and the Maxwell equation for propagation in one dimension, but this time for a very weak se&-input pulse with an intensity of 0.0045GW/cm2 and a full width at half maximum of 150 fa The carrier frequency of this pulse is chosen to coincide with the gain maximum and the pulse spectrum is again fully inside the semiconductor gain region.

Nonequilibrium semiconductor dynamics

437

0

_

_

3091

4091

C>

s .? a, --2

2191

-0.5

0.0

0.0

t (PS)

t (PSI

0.5

Fig. 54. The intensity profile of the propagating pulse is shown after propagation over the amplification lengths (in units of the linear amplification length gl) indicated in the figures (a)-(d). The calculations are done for the very low initial peak intensity of O.O04SGW/c~, which appears as an almost flat line in (a). Note that (b) and (d) are scaled by factors of l/2 and 2/3, respectively. (From Ref. (34).)

Our numerical results for the temporal pulse intensity Z = (j&.,,m$I2 at different positions in the amplifier are shown in Fig. 54. Starting with a very low peak intensity, the pulse is propagated numerically over a total distance of 35 gain lengths As can be seen from Fig. 54a (l-10 gain lengths) the pulse is amplified without temporal distortion over about 10 gain lengths After this distance a shoulder builds up in the trailing part of the pulse (Fig. 54b, lO20 gain lengths). This shoulder gradually develops into a separate pulse which splits off from the main part of the pulse (Fig. 54c, 20-35 gain lengths). During this time the first part of the pulse is continuously amplified and considerably shortened. Beginning at a propagation distance of about 30 gain lengths the sharp pulse remains basically nearly unchanged during the subsequent propagation, Fig. 54c. The corresponding temporal behavior of the total electron-hole density at different positons in the amplifier is shown in Fig. 55. Figure 55a demonstrates that the density decreases as the function of time, as expected for linear pulse amplification. This behavior begins to change

438

R. Binder and S. W. Koch

5.9

-

6.4 6.3

C> i

/

0.0

0.0

t (PS)

t (PSI

Fig. 55. The carrier density profiles are plotted corresponding to the intensity profiles in Fig. 54. (From Ref. (34).)

after a propagation distance of about 7 gain lengths when the temporal density evolution starts to exhibit an increase after the initial decrease. Surprisingly, after propagation for about 25 gain lengths the density increases as a function of time, immediately after the pulse onset. This behavior is unexpected in an amplifier medium where stimulated emission, at least at the beginning of the pulse, should be expected. After 30-35 gain lengths the density increase adiabatically follows the light intensity (Fig. 54c, Fig. 55~). These results can be explained as follows. After its original amplification and after accumulation of enough energy the pulse saturates the gain causing maximum carrier depletion and hence a minimum in the decreasing carrier density. However, as discussed above, due to the spectral overlap of the light pulse with absorbing states energetically slightly above the chemical potential the density starts to increase after reaching the gain saturation energy. For longer propagation distances the density causes a shoulder in the pulse profile due to the decrease and increase of the electron-hole density. The temporal shoulder in the carrier density is caused by the interplay of amplification for one part and absorption for a different part of the pulse. Since the shoulder has a slower

Nonequilibrium semiconductor dynamics

439

time dependence than the amplified part of the pulse it induces a slower temporal component in the density developement. Consequently, the temporal rise of the density after the gain saturation is devided into two components: a first interval with fast variations corresponding to the fast part of the pulse, and a second interval with a slow component corresponding to the slowly varying part at the end of the pulse (Fig. 55b). The different density behavior in both intervals yields different absorption rates for the trailing part of the pulse because the temporal derivative of the density drives the spatial change of the pulse intensity: &-IIE12 0~ &,N. Therefore, pulse break-up occurs at the transition of the described intervals At this point another mechanism becomes important, since the density starts to increase at the time the pulse is switched on (Fig. 55c, about 25 gain lengths), showing that the interaction of the light pulse with the absorbing part of the semiconductor continuum is increased. However, the states close to the pulse spectrum which cause the absorption in Fig. 55b cannot be responsible for this behavior. These states which overlap with the pulse spectrum by their absorption lineshape am stronger saturated for increasing field intensity. Therefore, an enhanced interaction must take place with the states where the overlap of the noninverted states is due to the refractive index and not due to the absorption. Similar to excitation in the large detuning limit discussed above, this adiabatic following(4*‘9’,203)occurs for off-resonant states well above the chemical potential if the pulse is strong enough. The corresponding electron-hole density follows the pulse intensity, hence no real absorption is noticed after the pulse is switched off. Through virtual excitation by a high peak intensity the far offresonant states transiently contribute strongly to the total electron-hole density because they are weighted with a monotonically increasing density of states for increasing detuning. Under suitable conditions this adiabatic following overcompensates the amplification by the gain. We have seen a similiar behaviour in Fig. 50, where the long-dashed line (highest intensity) exhibits clear adiabating following features (apart from the small Babi-oscillation-like modulations). The above results can be modified significantly if additional incoherent effects become important. For example, carrier cooling by means of carrier-phonon interactions can reduce the heating mechanism discussed above if the corresponding relaxation rate is comparable to the pulse duration. In addition, memory effects leading to non-Lorentzian lineshapes reduce the absorption and heating processes 3.6. Coherent Electric Field EfJects In this section we briefly discuss coherent effects, which can occur if a homogeneous quasistationary (dc) electric field is applied to a photoexcited semiconductor, along the lines of the work of Meier et al.(204-206) To describe these phenomena, the SBE that include the applied electric field have been derived in Section 2.2.. As before, the solution of these extended SBE, Eqns (88), (86) and (87), yields coherent linear and nonlinear optical effects in semiconductors, including the effects of the additional homogeneous static or low-frequency time-dependent electric field. The phenomena described by the extended SBE include the so-called Bloch oscillations, i.e. the periodic motion of an electronic wavepacket through the Brillouin zone, and their counterpart in the frequency domain, the Wannier-Stark-ladder (WSL) in the linear absorption spectrum. The periodicity interval for Bloch oscillations is TB = h/ (e&&r), where Edc is the electric field amplitude and al the lattice constant. Although both phenomena have been postulated a long time ago, (68p69) they have only recently been observed in semiconductor superlattices(207-209) So far Bloch oscillations could not be observed in bulk semiconductors because TB is longer than typical particle scattering times so that the electronic wavepacket has spread out before an oscillation cycle is completed. Hovever, in semiconductor superlattices the periodicity length al is not the relatively small atomic lattice constant but the total

440

R. Binder and S. W. Koch

thickness of the quantum well plus barrier in the growth direction of the superlattice This total thickness is a design parameter which can be modified by changing the sample growth conditions Hence, the periodicity length can be increased so that smaller Bloch oscillation times can be obtained. In currently available semiconductor superlattice structures the Bloch oscillation times could be shortened sufficiently so that for experimentally realizable electric fields TB can become shorter than the coherence destroying carrier scattering times To theoretically analyze the basic effects we use a very simple model for a semiconductor superlattice, employing a one-dimensional tight-binding model. (209-212) For the Coulomb interaction we assume an on-site interaction, which retains the most important excitonic features, namely a single bound excitonic state and an ionization continuum. Within the simple model the linear absorption spectra and the nonlinear response, i.e. fourwave mixing (FWM) and terahertz (THz) emission signals have been computed in Refs (204, 206). The FWM signal is dominated by the interband polarization dynamics with the strong influence of the Coulomb exchange interaction, as in the field free case, whereas the THz dynamics is mainly governed by the external field. For the case of an applied oscillating electric field the effect of dynamic localization,(21i~213) i.e. the localization of electrons induced by an oscillating electric field, is discussed and it is shown that this phenomenon should lead to charateristic signatures which can be observed using THz emission spectroscopy. We base our discussion on the semiconductor Bloch_equations, Eqns (88), (86) and (87), which contain the dofield terms proportional to the k-derivatives of the polarization and distribution functions The? contributions can formally be eliminated by introducing a moving coordinate frame through k&) = k + co/h I’ &,(t’)dr’. In this coordinate frame the partial derivatives with respect to the quasi-momentum disappear and the extended SBE reduce to the field-free equations However, as an additional condition, the quasi-momentum obeys the so-called acceleration theorem@“)

These equations have been solved numerically in Refs (204,206) to study the influence of the electric field on the linear and nonlinear response of a semiconductor superlatice. For these calculations a simple model for the semiconductor superlattice has been employed. The carrier movement was assumed to be one-dimensional (perpendicular to the layer), i.e. k 4 k,, & (L/27r) j dk,. Th e miniband structure has been modeled using one-dimensional tight-binding dispersions for electrons and heavy-holes, EL = .ro - %os(k,a& 2

(460)

EV = $cos(k,n,) k;

(461)

where al is the combined well and barrier width (periodicity unit) in the superlattice, EOis the gap between the centers of the minibands, Ae and AV are the miniband widths for the fieldfree case, simple exponential relaxation and dephasing rates have been used, and an on-site Coulomb-interaction has been assumed, i.e. V(K - k) E V”. This form of the Coulombinteraction leads to substantial numerical simplifications, yet it retains the most important excitonic features, i.e. a bound excitonic state and an ionization continuum.(214) Effects of the full anisotropic coulomb interaction in superlattice structures are discussed in Ref. (205). For the one-dimensional tight-binding model of the semiconductor superlattice the firstorder susceptibility can be calculated analytically. The top parts of Figs 56 and 57 show the maxima of the resulting absorption spectra as a function of the applied field amplitude. The zero of the energy scale has been chosen to correspond to an optical transition to the center

441

Nonequilibrium semiconductor dynamics

30 z

20

& '0 0 G L Q) -10 I=

-20 -30 0

10 (meV)

5 eFd

15

20

0 k ._ m N z

.

1

0 real I

I

I

I

I

2

4

6

I

I

time I

0 delay

2 time

(ps)

4 (ps)

6

Fig. 56. (Top) For a relatively wide miniband (combined bandwith 40 meV) in a semiconductor superlattice the Wannier-Stark ladder, i.e. the energetic position of the peaks in the linear absorption spectrum, is plotted as function of the applied dc-field e,,!!$a~, which is denoted here as eFd. For the field value e,h$al = 7.5 meV the computed THz signal (middle) and the four-wave-mixing signal (bottom) are plotted as functions of time. (After Ref. (209.)

of the combined miniband. For vanishing field one sees only the exciton resonance, whereas for finite field values the WSL develops In the range around e,Ed,al = 3.5meV one clearly sees anti-crossing behavior. Such anti-crossings show up whenever the excitonic Is-state comes into resonance with a WSL transition of the excitonic ionization continuum. The spectral positions of the resonances and their oscillator strengths am qualitatively in good agreement with calculations by Dignam and Sipe, (2’S)who have obtained excitonic WSL using a threedimensional approach with variational wavefunctions The middle parts of Figs 56 and 57 show the computed THz emission signals as function of time for two different values of the applied field. Generally, in a THz emission experiment the emitted electric field caused by the coherent motion of charges in the sample is measured.(216-219)After the excitation with a short laser pulse the signal is detected using an optically gated m-dipole antenna. This experimental technique has also successfully been used for the observation of the Bloch oscillations in the time domain.(208)

442

R. Binder and S W. Koch

0

2

4 eFd (meV)

0 real

0 delay

2 time

2 time

6

6

4

6

4

6

(ps)

(ps)

Fig. 57. (Top) For a relatively narrow miniband (combined bandwith 10 mev) in a semiconductor superlattice the Wannier-Stark ladder, i.e. the energetic position of the peaks in the linear absorption spectrum, is plotted as function of the applied dc-field eO&a,, denoted here as eFd. For the field value e,,#&r~ = 3.5 meV the computed THz signal (middle) and the four-wave-mixing signal (bottom) are plotted as functions of time. (After Ref. (206).)

In a semiclassical approximation derivative of the current(220)

the THz signal in a superlattice can be expressed as the

The computed THz signals (in second order in &) shown in Figs 56 and 57 clearly exhibit periodic oscillations which match very well the respective Bloch oscillation frequency. In the bottom parts of Figs 56 and 57 the computed time integrated transient FWM signals am plotted for the same applied fields for which the THz signals ate shown in the corresponding middle parts of the respective figures The calculated FWM signals exhibit oscillations, the frequency of which is not necessarily the same as that of the THz signals, since the FWM beats represent quantum beats induced by the Bloch oscillations of the optically excited carriers The oscillation frequency is given by the energy difference of the dominant transitions in the

Nonequilibrium semiconductor dynamics

443

linear spectra. In the presence of excitonic effects this energy difference is generally different from the nominal WSL spacing. Consequently, the comparison of the THz and FWM signals shows differences in the temporal evolution. These differences are most pronounced for electric fields corresponding to the region where excitonic modifications are preeminent in the linear absorption spectra. Only for relatively strong fields, where the absorption spectra show clear WSL characteristics, the oscillation periods of FWM and THz signals become similar. It has been discussed in the literature (211)that, for time-dependent external fields, one may observe the effect of dynamic localization in semiconductor superlattices Dynamic localization in this context can be identified by monitoring, e.g. the temporal evolution of the expectation value of the square of the position operator of an initially localized electron. If, with increasing time that expectation value increases without bounds, the initially localized electron delocalizes This delocalization is the normal situation without applied field. However, if one applies an oscillating electric field, such as &(t) = E&cos(w!~) one can reach a situation where the expectation value of the square of the position operator remains bounded, indicating that the electron remains localized. This is the phenomenon of dynamic localization. The conditions for dynamic localization are that the amplitude and the frequency of the electric field are in a certain relation to each other, i.e. the ratio

eJ&a

r=hwl

(463

has to correspond to a root of the Bessel function Jo. It has been shown(2’3) that the oscillating electric field leads to a change of the effective bandwidth AF as

AF

=

IJo(dI(Ac+ A,) .

(464)

Equation (464) shows that if Jo(r) is equal zero the band is flat and therefore the electrons have an infinite effective mass and arc localized. To illustrate these phenomena, we show in Fig. 5X calculated absorption spectra of the model superlattice as a function of r for hwr = 20 meV.(206)We see that the spectral position of the exciton oscillates as function of r. These oscillations are a consequence of the oscillations of the bandwidth according to Eqn. (464), which can also be seen in Fig. 58. For very high r, the Bessel-function in Eqn. (464) is zero indicating that the miniband bandwidth vanishes Consequently, the exciton resonance tends to a spectral position which is situated at 10 meV, i.e. the Coulomb interaction energy VO, below the center of the band. Figure 59b shows calculated THz emission signals for some values of the ratio r, and Fig. 59a shows the corresponding linear spectra in the vicinity of the 1s exciton resonance. For the upper three curves the ratio is equal to 0.5,O.S and 1.1 and the signal exhibits oscillations with a time period which is given by the oscillation time of the oscillating electric field. The fourth curve in Fig. 59b shows the signal for r=2.4, which appoximately corresponds to the first root of JO. As a consequence of the dynamic localization the THz signal is strongly reduced. The lower curve, for which the ratio has been increased to 4, exhibits again oscillations with a higher frequency. The frequency increase is a consequence of the fact that the electric field, which shifts the k-vectors of the electrons across the Brillouin zone, is so strong that it leads to more than one complete transversal of the electron wavepacket through the Brillouin zone for each cycle of the electric field.

R. Binder and S. W. Koch

444

5

IO eFd/h

15

20

Fig. 58. The superlattice absorption spectrum is shown as contour lines for different strengths of the applied time-dependent field e,J&a, I eFd. The oscillations in the energetic position of the exciton peak and the collapse and reappearance of the continuum absorption for increasing field strength is clearly visible. (From Ref. (206).)

4. CONCLUSION

In this article we concentrate heavily on the formal development of the nonequilibrium many-body theory for semiconductors and the interaction of the material excitations with ultrashort light pulses Furthermore, we analyze a number of physical examples for which we numerically evaluate the resulting coupled equations Since it was our main emphasis to use these examples as illustrations of the theory we do not discuss the corresponding experiments in detail. However, in this final chapter we want to mention some experimental results, for which the theory can provide qualitative, in some cases even quantitative analysis Sucessful comparisons of the current theory with experiments include studies of coherent oscillations,‘g’ optical Stark effect and adiabatic following, (221)time resolved and time integrated FWM,(25,64,222,223) Rabi flapping,(7) pulse propagation in waveguides(224) and quantum beats.(225) Clearly, this list of references is not complete, it should only serve as an illustration for publications dealing with direct theory-experiment comparisons Theory predictions which we have discussed in this review, but which have not (yet) been tested experimentally, include the pulse propagation studies in amplifiers as well as most of the theoretical results dealing with self-induced transparency in semiconductors. Furthermore, there are also very detailed experiments, for which the present theory has to be extended. One example are biexcitonic effects in polarization dependent four-wave mixing,

Nonequilibrium semiconductor dynamics

-5

-10

-15

Energy

(meV)

0

1

2

Time

3

4

5

6

(ps)

Fig. 59. (a) Examples of the absorption spectra from Fig. 58 and (b) the corresponding THz signals. For the different curves the ratio I = e,J$=al/Rwl is equal to 0.5, 0.8, 1.1, 2.4 and 4 (from top to bottom). (From Ref. (206).)

see e.g. Refs (63, 197,226), which currently can only be analyzed using the simplified, partially phenomenological approaches discussed in these articles Besides these biexcitonic effects, the current theory has not yet been extended to correctly include phenomena such as two-photon absorption or other higher-order effects such as phonon-assisted excited state transitions (socalled free carrier absorption), to name only two examples Additional open challenges include the better treatment of Coulomb correlations such as excitonic effects in the carrier scattering processes, memory effects, time and space resolved problems including disorder, as well as basically all phenomena related to the nonclassical nature of the light field. Hence, one can conclude that, even though the microscopic theory of ultrafast optical processes in semiconductors is fairly well developed and experimentally tested at the screened Hartme-Fock level, there are still many open problems to attack.

5. ACKNOWLEDGEMENTS

We are indebted to the following researchers for sharing some of their knowledge with us: W. Schafer (especially in the field of Green’s function theory), M. Lindberg (semiconductor Bloch equations, four-wave-mixing, x(“) -analysis), Y. Z. Hu (band mixing effects, excitation-induceddephasing effects), A. Knorr (Maxwell-semiconductor-Bloch equations, excitonic pulse propagation effects, pulse propagation in semiconductor amplifiers), T. Meier (static E-field effects), R. Indik (pulse propagation in semiconductor amplifiers), A. Schulze (propagation effects in four-wave-mixing). We are also grateful for many stimulating discussions with the following researchers and their co-workers: M. Bonitz, F. Jahnke, H. Gibbs, E. Gobel, H. Haug, K. Henneberger, J. Moloney, N. Peyghambarian, D. C. Scott, D. Steel, P. Thomas and E. Wright. The original manuscript was typed in LaTeX. For her help in typing the manuscript we thank M. Binder. For his work as system manager of our local computers we am indebted to DC. Scott. We gratefully acknowledge financial support from NSF, OCC (Optical Circuitry Cooperative, University of Arizona), ARO/AFOSR (JSOP), NATO and the DFG (Germany), partially through the Sonderforschungsbereich 383. We also acknowledge grants for CPU time at

R. Binder and S. W. Koch

446

the Pittsburgh Supercomputer Center (PSC) and the Center for Computing and Information Technology (CCIT’) at the University of Arizona.

6. APPENDIX A: THE FUNCTIONAL DERIVATIVE TECHNIQUE

Since the main focus of Sections (2.6.) and (2.7.) is the application of the nonequilibrium Green’s function technique rather than its theoretical foundation, we will summarize in this Appendix only briefly the main steps that allow for the elimination of the two-particle Green’s functions (four-operator expectation values) in Eqns (150) and (151). We will employ the Schwinger functional derivative method. (i5*) More rigorous and comprehensive discussions have been presented, for example, by Kadanoff and Baym,(122) Korenman,(15g) Dubois,(124) Hedin and Lundquist,(‘301 Sto1z(‘2g) and Danielewicz.(‘25) We define the generating functional G as(227) ihG++ (12) = &

(s:[T+S+ww+m

ihS+_(12)

= -,;‘, c

([T-st~+(2)1[~+S+~(l)l) t

(466)

ihG-+(12)

= - l (SJ

([T-st~,(l)I[T+S+~+(2)1) ,

(467)

ih&(12)

= - &

or, equivalently, the corresponding

c

,

(m3f.wW~+t~)lS+) I

Keldysh matrix

ihcj(iZ) =b2 & uxdcy+(Z)) c where 1 = {F1sltl} , i = {lb*},

(465)

,

and the time evolution operators are given by -- i S+ = T+ e

h

I

dt’H: (t’)

--OO

9

(470)

a,

dt’H: 0’)

(471) and SC =

sts+.

(472)

The subscripts ‘+’ and ‘-’ are not only a reminder that the time evolution operator S contains the regular positive time ordering operator and its adjoint operator contains the reverse ordering provided by T- (see also Section (2.6.)), but in the following we will assume that Hi can indeed be distinguished from HI. Equations (465)-(468) are essentially the generalized interaction picture representation of Eqns (136)-(138), provided that H is replaced by H + H’, the interaction picture representation is defined with respect to H’, the time to at which the evolution according to H + H’ starts is at -DO, and appropriate unity S-operators have been inserted in order to allow the upper time-integration limit to be +m. The occumnce of products of

Nonequilibrium semiconductor dynamics

447

time-ordering operators (T+, for example, occurs in Eqn. (465) explicitly as well as in the definition of S+) has to be intepreted in such a way that the cl/-operators have to be brought to the correct time inside the time-integration of S and all exponentials have to be time ordered. For example, T+S++(l) = [T+exp(-(i/A)I,~dt’H~(t’)l~(l)[T+exp(-(i/h)~~,dt’H~(t’)l. Earlier, we used the words ‘generalized interaction picture’ because there is an important difference between the regular interaction picture and the previous definition. Any physical Hamiltonian is the same whether it occurs in S or St. Within the Keldysh formalism one needs to have an H’ which is formally different if it occurs in a positively or negatively timeordered product. In order to obtain the ‘physical limit’ (DuBois (124)),one can either regard the situation where Hi = HL or, as we do in the following, H’ -. 0.“25’ In either case, the physical limit results in (S,) + 1. One reason for the occurence of l/ (SC) in the definition of Cj is the analogy with the regular interaction picture, where the negatively time-orderd S-operator can be ‘factored out’ and formally be written as a denominator (which yields the so-called discomlected diagrams). Of course, in the Keldysh approach to nonequilibrium systems one cannot factor out the St operator. As we will see shortly, the inclusion of l/ (SC) yields the Hartree-contribution as the lowest order representation of the two-particle Green’s function (see also Eqn. (152)). In order to generate four-operator expectation values by means of a functional derivative of G, we use the following form for H’:

I

H;(f) = dx V,x(tb) w+(xt)wW

(473)

for b = {+, -1. The equations of motion for I,U+and w are still given by Eqns (36) and (35). The equations of mo$on for the matrix G are those for G, Eqns (146) and (147), except for the replacement h - h + U,. (129)Hence, in the limit of vanishing U,,, the generating functional G reduces to the Green’s function G. The functional derivative is nonzero only if the derivative of S+ with respect to U&Z+) and the derivative of S! with respect to U&-1 is taken. Specifically, we have

as+ au&t+) &St

au&t-)

=

-~T,S+l+(xt)w ,

(474)

=+LT_Stq+(,)q(g). h

We can now interchange the order of the two operators in Eqns (474) and (475), which yields a minus-sign, and write

We obtain from Eqn. (476)

a

I

icixai-=$-p(ii+), c

which yields, as mentioned above, the Hat-tree-contribution (ih)$(jjZj’)

= (s b2b; c

(477)

to

(T,s,~(i)(y(3)~+(3’)~+(2)) .

(478)

Finally, we can take the functional derivative of Eqn. (469) and obtain with the help of Eqns (477) and (476) the representation of a four-operator expectation value in terms of the functional derivative of a two-operator expectation value,

R. Binder and S. W. Koch

448

aG(iZ) _ a&,(3)

---= +(1323+)

+ G(jj+)G(i>)

.

(479)

The terms two- and four-operator expectation values refer still to the original Heisenberg operators defined by H; the two-operator expectation value (469) contains clearly an infinte number of operators due to the appeamnce of S,. In going from Eqn. (479) to Eqn. (152) we have already, for the purpose of a simplified notation, assumed that G is essentially the same function as G( U,, = 0),and we have also denoted the functional derivative of G, evaluated at v,, = 0, by G. Further applications of the functional derivative method, such as the generation of the one-particle Green’s function from an external potential, which is coupled to only one Fermi operator, can be found, for example, in Refs (125,228).

7. APPENDIX B: THE LUITINGER

HAMILTONIAN

One of the approaches most often used to solve the one-particle Schriidinger equation in 2 semi-empirical fashion is the so-called Luttinger-Kane theory,(35) which is based on the k - j-perturbation theory. The application of that theory to excitons has been developed by Dresselhaus(37) and Baldareschi and Lipari. (38*3g)For reviews see, for example, Refs (38, 39). In this Appendix we summarize only those elements of the Luttinger theory which are relevant in the context of nonlinear semiconductor optics The most important aspect of the Luttinger theory is the correct treatment of the symmetry properties, in particular those of the more involved valence bands Also, the Luttinger theory allows for a straightforward inclusion of the modulation potential to describe quantum confinement effects Formally, the one-particle Schriidinger equation can be written as

1-g+ 0

u-L(h) + v,(i)

1

@(is) = Ee(is) ,

where we assume that all correlations have been incorporated in the lattice potential UL and the confinement potential UC accounts for the variation of one-particle energies due to compositional variations in the system. In a homogeneous semiconductor (UC z 0), the quantum numbers of the one-particle eigenstates are the 3-dimensional wavevector k and the Blochband index v. In general, the Bloch function has the form

with the lattice periodic part u,k(i,s). Within the b - j theory, the lattice periodic part for g f 0 is expanded in the set of k = O-functions, which are assumed to be known. (Instead of k = 0 one can, of course, use any other &). Calling the expansion coefficients JV, we have

With UC+ 0, the wavevector is not a good quantum number anymore. The eigenstates of Eqn. (480) are then of the general form

where we assumed that the external potential does not vary within a unit cell and thus does not couple different Bloch bands In quantum wells, the confinement potential depends only

449

Nonequilibrium semiconductor dynamics

on one spatial coordinate, z, so kx and ky remain good quantum numbers In that case, the eigenfunctions am

From Eqn. (480) one derives easily the equation for the effective expansion coefficients J+& (& = C,,J%/

(i) ,

(485)

which can be written as

In quantum wells we have U,(k) = b-k,,oUc(kz), in which case Eqn. (486) becomes c~~/(~,,,k,)iis,~(~,,,k,) v” = +,1;,, *,+

+ 1 U,(kz - k;)b,db&,,,k;) v”K,

&,I, kz) .

Quantum wires are treated correspondingly with U,(i) = bk,,oUc(ky,k,). In Eqn. (486) the band labels v still comprise all bands and the Hamiltonian terms of eigenenergies E,,J=,, and the momentum matrix elements are

(487)

J-f- is given in

(488) where LO is the linear extension of the crystal unit cell. Furthermore,

Within a modified degenerate perturbation approach’**” one can group various bands (e.g. the close lying valence bands Iv}) and treat the coupling to all other groups perturbatively. One obtains for the group {v, v’} = {v, v’} (490) where the sum over r indicates all remote bands (other than the valence band group under consideration). In practical applications, one uses only the symmetry properties of H,,,t and rewrites Eqn. (490) in the most general form which preserves its symmetry. The general form usually has fewer parameters than Eqn. (490). In this article we deal spec@ally with the valence band structure of cubic III-V compounds Since valence band states at k = 0 are mainly built from the p-orbitals of the isolated atoms, it is a good approximation to label the Bloch function at k = 0 with the angular momentum quantum numbers of the p-states Of course, the states do not really transform according to the p-representation of the 3-dimensional rotation group, but according to the corresponding irreducible representation of the cubic point group. According to Eqn. (482) these states determine the basis of 3-&. To treat the conduction band one can use an equation similar to Eqn. (490). The idea of performing two seperate Luttinger perturbation approaches for the valence and conduction

R. Binder and S W. Koch

450

band has been discussed by Dmsselhaus c3’) Since in III-V compounds the conduction band is only two-fold (spin) degenerate, the corresponding effective conduction band Hamiltonian 3&ti is diagonal in the band index and yields immediately an expression for the effective electron mass tensor, (l/m,)ij, {i,j1 = {x,y,z). To combine the valence and conduction band theories, one can formally introduce

and correspondingly

Since

we find

where we have defined Ev =

(496)

EC&C+ Evbw .

In the following we now focus on the modeling of the valence bands Including the spin of the valence band electrons, the states am labeled by a total angular momentum (in units of h) of J = 3/2 and J = l/2, respectively. A detailed investigation of the GaAs bandstructure shows that the l/Zstate is split from the 3/2-state due to a strong spin-orbit interaction. In the following, we will therefore restrict ourselves to the 3/2-state, which spans the basis

lj) = {13/2), I - l/2), I -3/z), where j is the z-component

II/3 I ,

of J. In this basis, the valence band Luttinger Hamiltonian

(497) reads (498)

with the unit matrix I and

(499)

The matrix elements of f4 are functions of & and the so-called Luttinger parameters yi , ~2, and y3: Q=~2((k2,++2e),

(500)

S = 2&y3(kx - ik,) kz ,

(501)

R= -&[y&

- A$) - Ziy,k,k,]

.

(502)

451

Nonequilibrium semiconductor dynamics

In Eqn. (499) we already neglected anisotropy effects These effects, however, have been found to be very small.(3*) Diagonal&ion of Eqn. (499) shows that the energies E”&depend on the direction of 6, which is called warping. However, in a system like GaAs the experimentally deduced Luttinger parametes are such, that y2 = y3. Within the so-called spherical approximation one sets y2 = y3 and uses the numerical value (y2 + ;y3)12 = ~23 in Eqn. (502). This eliminates the warping effects The matrix elements become Q=r,

(+2k3

=-

16rr 5

s = 2dy3k,,

Y2 k2

kz

y12’m&

9

(503)

Ci4 (504)

=-

J y

y23

k2

Y!;‘(6),

4) .

(505)

Here Yi2) (0,4) are the spherical harmonics in k-space. The description of the conduction band is simplified by the fact that the effective conduction band Hamiltonian is diagonal. In the isotropic limit it determines simply the effective electron mass mc.In this (so-called effective mass approximation) the matrix We, C$)is the unit matrix 6,d and the lattice periodic part of the Bloch function is independent of 5. At this point, the Luttinger matrix still describes the valence-band structure of the bulk material as well as possible quantum-confined structures The eigenenergies for the general system follow from the solution of Eqn. (486) or, specifically for quantum wells, from Eqn. (487). To get a feeling for the relative importance of the terms in Eqn. (499) within the spherical approximation, it is instructive to consider the symmetry properties of the matrix elements More specifically, we consider the contribution of the various matrix elements to a low-density s-exciton state. The electron wave functions are s-like, and therefore the dominant contribution to an s-exciton comes from those valence band matrix elements which are also s-like. Which matrix elements have the highest symmetry depends, however, on the dimensionality of the system. In 3D, we see immediately from Eqns (503)-(505) that all elements of fd are d-like. As was shown in(39) for the calculation of s-excitons, ‘& can be treated as perturbation to the ;yl contribution in Eqn. (498). Thus the hole mass of the four-fold degenerate unperturbed s-excitons is yi lmoand the correction due to the d-like terms is small. A full diagonalization of Eqn. (498) yields two doubly degenerate eigenvalues with effective masses mo/(y,f 2~~~). These masses do not agree with experimentally observed bulk s-exciton masses, since the diagonalization yields a mixture of s and d hole states that do not contribute to the optically excited s exciton with the same strength. Since the discrete energy spectrum of the unperturbed (i.e. s) excitons exhibits an exact Rydberg law (oc l/n2), the full exciton spectrum in bulk is still roughly Rydberg-like. In quantum-wells k, is not a good quantum number, and s-excitons are those which transform according to the m = 0 representation of the two-dimensional rotation group. As seen from Eqn. (503), Q is now part of the (high-symmetry) unperturbed Hamiltonian. The pref-

R.

452

actor 4 of the unperturbed are easily seen to be

Binderand S. W.Koch

Hamiltonian

yields the zero-order effective hole masses, which

mhhll = Yl mlhll

=

YI

m0 + y23

-

y23

These masses determine the in-plane density-of-states (a nomenclature, that becomes apparent below): mhh,z =

mlh,z =

’

m0 ’

The prefactor of kz gives the z-masses

m0 YI

-

b’23

m0 Y1 + b’23

’

which determine the hh-lh splitting, as will be discussed below. The foregoing argument shows that it is the symmetry reduction which yields two exciton peaks, even if the quantum well potential would not shift the hh and Ih exciton differently (which is, of course, not realistic). Since in the quantum-well case the s-contribution to the Hamiltonian is diagonal in the above bas_is,it is reasonable to label the excitons according to the quantum number j, which is good at k = 0. In addition, the optical selection rules will be dominated by the transitions between the conduction band states and the states (497). In quantum wires the s-symmetry reduces to a reflection symmetry, and neither kz nor k, are good quantum numbers In addition to Q we now have to consider R as part of the unperturbed Hamiltonian, because a rotation by rr leaves R unchanged. As a consequence, the unperturbed quantum wire s exciton couples the j = 312 with the j = -l/2 states, and the j = -3/2 with the j = l/2 states This hh-lh coupling in quantum-wires is discussed by Bockehnann and Bastard in Ref. (51). They show, however, that in spite of the coupling the masses in GaAs are still similar to the quantum-well masses We discuss now some technical details of the envelope equations for the case of quantum wells A practical simplification of Eqn. (487) results from the Fourier transformation with respect to kz:

1 {J-L+ (h, -iam + ucwvf~) 9’ = +&,, W$ G,, , z) .

%$fyl (i,,,z) (510)

Here we assumed that the eigenfunctions % are constant over each unit cell. Therefore “l? are called envelope functions The confinement potential gives rise to an additional quantum number 8, which labels the quantum-confinement subbands The normalization of the envelope functions is (511) For 2 practical calculation of the valence band properties, one can p_roceed in the spirit of the k -3 theory and expand the subband eigenfunctions at nonzero kll in eigenfunctions obtained for kll = 0:

Jq# CK,, ,z) = 1 c$y(i;,,)I?$ (0, z) . P

(512)

453

Nonequilibrium semiconductor dynamics

Note that the Luttinger Hamiltonian is diagonal at &II= 0; therefore, the matrix of subband wavefunctions %’ is diagonal and obeys the simple equation {MJO, -ia/az)

+ UC(Z)}%$O,z)

= E,&#(O,Z)

.

(513)

The expansion coefficients C are determined from the equation

=

Q&$k,,),

where & denotes that part of the Hamiltonian The matrix elements

involve matrix elements of (-ia/ ~zY” with element @$

= 5

(514)

which does not vanish in the limit $11- 0.

m = 0, 1,2. In an infinite well the overlap matrix

dz J?; (0, z) * %$ (0, z)

(516)

L

does not depend on v, v’ and is Sot. The matrix element of the wavenumber in z-direction,

U&f/ = dz*:(O,z)*c-i;)%$(O,z) f L

(517)

is zero if 8 = 4?’ and the z-dependent function have definite parity with respect to reflection about z = 0. In particular, this is true for the ground state wavenumber (kz) 5. Thisleads to a significant simplification of the model structure that contains only one (i.e. the lowest) subband. In an infinite potential well the energy matrix elements involve (518) In the following, we restrict ourselves to the limit of very thin quantum wells where it is sufficient to take only the lowest subband into account. With regard to the matrix elements of 7& this means that the element S (Eqn. (504)) vanishes Therefore, the 4 x 4 matrix Ta breaks up into two 2 x 2 matrices, (519) with (520) The foregoing argument can also be applied to the simple conduction band structure. For that, one has to take Jf in Eqn. (510) as being diagonal. The matrix %$,, (&II) becomes the diagonal matrix of the conduction-band envelope functions %!&I), which are normalized according to

454

R. Binder and S. W. Koch

(521) Within the framework of k .&theory, the dipole matrix element between a conduction band and a valence band state for the bulk system is obtained with Eqn. (482):

where 1

-0

&v =

utO(c

-e0j Lo

s) F &,o(i,

s) .

Ix Los

(523)

The momentum conserving delta in the Eqn. (522) function follows from the approximation

if I~l,lk I -=KL,‘. In the quantum well case the dipole matrix element is (525) Until now we have concentrated on one-particle properties We now discuss the evaluation of the two-particle Coulomb Hamiltonian (526) where the matrix elements are defined in Eqn. (57). In 3D, the matrix elements of V are determined by the Bloch matrix elements of exJ(-@i) (see Eqn. (62)). Using the representation (482) we obtain (n,

1 ,-i@

ln2)

=

A

1

s

,--&I + 4 - i3)k

d3rz

4

esih

u~(is)u,(is)

(528)

S

where & and Nt denote the positions and number of ions, Li is the unit cell, and 6 is a reciprocal lattice vector, respectively. Using again the approximation

we obtain

455

Nonequilibrium semiconductor dynamics (n, I e-‘V

In*) = %* +p,k,1 Jc& Y

(k %,(ktz)

9

(530)

and for the Coulomb matrix elements, 1

Vnmn3n*

=

_5 L

47rez %,-k&-h (&-id2

where, again, the quantum numbers ni are {vi, 61. Because of Eqn. (492) we see that the Coulomb matrix element is unequal to zero only if VI and vs are both either conduction or valence bands, and the same has to be true for vz and ~4. Technically speaking, both approximation (491) and (529) have to be fullfled to elimintate so-called Coulomb vertices, i.e. the possibility that, for example VI is a conduction band index and vs a valence band index. We note that Eqn. (5_31)red!ces t,o the one used within the effective mass approximation if we set in the ws kr = k3 and k2 = k4 and use the orthonormality properties of the eigenvector matrix IV. We also note, that the Coulomb Hamiltonian in Eqn. (526) reduces to the form used in Ref. (27), if the transformation (89) and (90) to the nondiagonal Luttinger Hamiltonian is performed. We immediately obtain from Eqn. (526) 1 p = z .,n;3,

1 E?

4=*

(&4 - i2)2 X&,v3&p,

6 i3 _6,h-i3 c,t,c~c&t3

*

(532)

For quantum wells, we proceed similarly and obtain, in analogy to the bulk case Eqn. (531), V4mn3m

= -

1

L2

-

2rre2

sg_i,,J&*

lb-kzl

c

‘I?$;,,,

(i, ,z) *

%‘$,,

(i3, z)

VP6

(533) where it is understood that all i = &IIand the index nf contains the additional subband index & Equation (533) reduces to an equation which is formally identical to Eqn. (531) (with the replacement of 4me2/L3$ with 2m?/L2q) in the limit of thin quantum wells The exponential exp( -qlzl ) becomes unity in this limit and the application of Eqn. (512) with one subband and the normalization Eqn. (511) immediately yields the structure of Eqn. (531). Another frequently used approximation of Eqn. (533) is obtained if only the lowest subband is taken into account but exp(-qlzl) is not set to unity. If, in addition, the barrier height is taken to be infinite, the envelope functions are independent of the Blochband indices and depend only on the wavefunctions ~sin(mz/L), giving rise to a so-called form factor. The simplest method to treat effects due to finite well thickness arises in the additional approximation of decoupled bands In this case the eigenfunction matrices are diagonal, and the Coulomb matrix element becomes

456

R. Bin&r and S. W. Koch

V“,“~tI~n4 = -1 -2=f L* lk4-kzl

%,-&--k2 a,,

&2v,_F(E

-

,

(534)

1*

(535)

Q,

where the form factor F is given by

F(q) =

*

(qL)* + 47r*

&g + m* 8

z

(1 - e-@)47+ - (qL)2 [ (qL)2 + 47r*]

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