Theory of Ultrafast Phenonmna in Photo Excited Semiconductor

REVIEWS OF MODERN PHYSICS, VOLUME 74, JULY 2002

Theory of ultrafast phenomena in photoexcited semiconductors

Fausto Rossi

Istituto Nazionale per la Fisica della Materia (INFM) and Dipartimento di Fisica,Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Tilmann Kuhn

Institut fur Festkorpertheorie, Westfalische Wilhelms-Universitat, Wilhelm-Klemm-Str. 10,D-48149 Munster, Germany

(Published 30 August 2002)

The authors review the physics of ultrafast dynamics in semiconductors and their heterostructures,including both the observed experimental phenomena and the theoretical description of the processes.These are probed by ultrafast optical excitation, generating nonequilibrium states that can bemonitored by time-resolved spectroscopy. Light pulses create coherent superpositions of states, andthe dynamics of the associated phase relationships can be directly investigated by means ofmany-pulse experiments. The commonly used experimental techniques are briefly reviewed. A varietyof different phenomena can be described within a common theoretical framework based on thedensity-matrix formalism. The important interactions of the carriers included in the theoreticaldescription are the phonon interactions, the interactions with classical and quantum light fields, andthe Coulomb interaction among the carriers themselves. These interactions give rise to a stronginterplay between phase coherence and relaxation, which strongly affects the nonequilibriumdynamics. Based on the general theory, the authors review the physical phenomena in varioussemiconductor structures including superlattices, quantum wells, quantum wires, and bulk media.Particular results which have played a central role in understanding the microscopic origins of therelaxation processes are discussed in detail.

CONTENTS

I. Introduction 895A. Nonequlibrium carrier dynamics in photoexcited

semiconductors 896B. Experimental techniques 899C. Aim and outline of the paper 900

II. Theoretical Background 901A. Physical system 901B. Kinetic description 903C. Single-particle dynamics 905

1. Semiclassical generation rate 9052. Homogeneous system, homogeneous

excitation 9063. Homogeneous system, inhomogeneous

excitation 907D. Carrier-phonon interaction 908

1. First order: Coherent phonons 9082. Second order: Scattering and dephasing 9093. Homogeneous system 9104. Third order: Collisional broadening 911

E. Carrier-carrier interaction 9121. First order: Excitons and renormalization 9122. Second order: Scattering and dephasing 9133. Homogeneous system 9134. Second order: Screening 9145. Third order: Collisional broadening 9166. Coulomb interaction in doped

semiconductors 916F. Carrier-photon interaction 917

1. First order: Coherent electromagnetic fields 9172. Second order: Absorption and luminescence 9183. Homogeneous system 9184. Transitions between band and impurity

states 918

0034-6861/2002/74(3)/895(56)/$35.00 895

5. Influence of other interaction mechanisms 919G. Theoretical modeling of typical experiments 919

III. Selected Experimental and Theoretical Results 921A. Line shape of luminescence spectra 921

1. Coherent carrier photogeneration 9212. Luminescence line shape 9233. Band-to-acceptor luminescence spectra 924

B. Coherent features in pump-probe experiments 924C. Temporal and spectral shape of four-wave-

mixing signals 926D. Coherent control phenomena 929E. Charge oscillations in double quantum wells 930F. Bloch oscillations and Wannier-Stark localization

in superlattices 9321. Two equivalent pictures 9332. Bloch oscillation analysis 934

G. Carrier-phonon quantum kinetics 9361. Memory effects and energy-time

uncertainty 9362. Nonequilibrium phonons and energy

conservation 9373. Phonon quantum beats 9384. Carrier-phonon quantum kinetics in

inhomogeneous systems 940H. Carrier-carrier quantum kinetics 942

IV. Summary and Conclusions 944Acknowledgments 945References 945

I. INTRODUCTION

The magic word faster has always been one of themajor challenges in the development of semiconductormicroelectronics and optoelectronics (Capasso, 1990;Shah, 1992). For many years this has basically been a

©2002 The American Physical Society

896 F. Rossi and T. Kuhn: Ultrafast phenomena in photoexcited semiconductors

concern for device and chip designers; however, todaysemiconductor technology has reached a level where thecharacteristic time scales of the underlying physical pro-cesses may determine the speed limits. Investigation ofthese ultrafast dynamics has thus become a strategicfield both in basic research and from a technologicalpoint of view. Recent developments in ultrafast laserphysics and technology now allow us to study the veryinitial interaction processes of nonequilibrium carriers ina semiconductor (Phillips, 1994; Shah, 1999), which aredirectly related to the microscopic details of the cou-pling mechanisms. Therefore time-resolved laser spec-troscopy has become an essential tool in modern semi-conductor physics.

Linear optical spectroscopy of semiconductors hasprovided invaluable information on electronic bandstructures, phonons, plasmons, single-particle spectra,and defects. These are impressive contributions, but in-formation on the details of interaction processes amongthe elementary excitations is often much more difficultto obtain. In many cases it enters only in a strongly av-eraged way, e.g., in terms of momentum or phase relax-ation times determining the spectral linewidth. Here, ul-trafast optical spectroscopy can do much more. Indeed,an optical excitation has the ability to generate nonequi-librium carrier and exciton distributions, and time-resolved spectroscopy provides the best means for deter-mining the temporal evolution of such distributionfunctions. Furthermore, by means of ultrafast pulses, co-herent superpositions of states can be generated and thedynamics of such phase-related quantities can be ana-lyzed. When these unique strengths are combined withspatial imaging techniques and/or specific low-dimensional structures, optical spectroscopy becomes apowerful tool for investigating a wide variety of phe-nomena related to relaxation and transport dynamics insemiconductors (Shah, 1999). It is this wide range thatmakes ultrafast optical spectroscopy a preferred tech-nique for obtaining fundamental new information aboutthe nonequilibrium, nonlinear, and transport propertiesof semiconductors.

Generally speaking, the optical excitation of a semi-conductor creates both interband excitations, i.e., a co-herent interband polarization, and intraband excitations,i.e., electron and hole distributions as well as intrabandpolarizations. The time evolution of these quantities isgoverned by a nontrivial interplay between phase coher-ence and energy relaxation. Indeed, scattering processestend to destroy the coherence, leading to a dephasing ofinterband and intraband polarizations. Furthermore,they lead to a relaxation of the distribution functionstowards the respective equilibrium distributions. Typicaltime scales for scattering processes in semiconductorsare in the range of picoseconds or femtoseconds and theresulting dynamics are generally termed ultrafast. It isthe aim of an ultrafast optical experiment to provideinformation on the details of this temporal evolution,which, in turn, gives insight into the fundamental pro-cesses governing microscopic carrier dynamics. Before

Rev. Mod. Phys., Vol. 74, No. 3, July 2002

going into detail let us start with a brief historical over-view and a description of typical experimental tech-niques.

A. Nonequilibrium carrier dynamics in photoexcitedsemiconductors

The investigation of nonequilibrium carrier dynamicsin optically excited semiconductors started in the late1960s with the analysis of the energy relaxation process(Shah and Leite, 1969) by using cw excitation. The mea-surement of the carrier temperature as a function of thecw laser intensity—obtained from the luminescencespectrum—gave insight into the power loss from the car-riers to the lattice, i.e., in carrier-phonon scattering pro-cesses. In subsequent years, these investigations wereextended to different materials and excitation condi-tions. While band-to-band luminescence spectra gaveonly a combination of electron and hole temperatures,direct information on the electron distribution functionwas obtained by studying band-to-acceptor lumines-cence spectra of doped semiconductors (Ulbrich, 1977).

However, as stressed before, only by using a pulsedexcitation can one directly investigate dynamical pro-cesses. Here, the pulse duration limits the temporalresolution and therefore restricts the phenomena thatcan be studied. The typical time scales for most of theprocesses discussed in the present review range from afew femtoseconds to a few picoseconds. Therefore theapplication of time-resolved nonlinear optical spectros-copy to the study of dynamical processes in semiconduc-tors is closely related to the ability to produce laserpulses on these time scales. Such laser sources becameavailable for semiconductor studies in the late 1970s(Shank et al., 1979). Since then, a great number of phe-nomena have been studied, first mainly focusing on in-coherent dynamics, i.e., the nonequilibrium dynamics ofdistribution functions, and subsequently analyzing moreand more coherent phenomena, i.e., the dynamics of op-tically created interband and intraband polarizations.

A typical scenario for the dynamics of distributionfunctions is plotted schematically in Fig. 1: The laserpulse with a given photon energy and a certain spectralwidth determined by its duration creates electron-holepairs in a more or less localized region in k space. Thisinitial distribution then relaxes due to the presence ofscattering processes. In polar semiconductors on ul-trafast time scales, there are typically two mechanisms ofparticular importance: Due to the polar coupling to lon-gitudinal optical (LO) phonons, the carriers may losetheir initial kinetic energy to the lattice. Since opticalphonons in the relevant region close to the center of theBrillouin zone have a negligible dispersion, this leads tothe buildup of replicas of the initial distribution shifteddownwards by multiples of the phonon energy [Fig.1(c)]. The scattering among the electrons themselvesdue to the Coulomb interaction, on the other hand, con-serves the total kinetic energy; however, it leads to aspreading in k space [Fig. 1(d)] and eventually, to aFermi-Dirac distribution in which the temperature is de-

897F. Rossi and T. Kuhn: Ultrafast phenomena in photoexcited semiconductors

termined by the initial excess energy. If, as is always thecase in a real semiconductor, both mechanisms arepresent, both energy relaxation and thermalization to-wards a Fermi-Dirac distribution occur simultaneously[Fig. 1(e)]; the respective time scales, however, arestrongly dependent on the excitation conditions, in par-ticular on the carrier density.

These thermalization and relaxation processes havebeen studied in great detail over the past two decades,both experimentally and theoretically, in bulk semicon-ductor materials as well as in a variety of heterostruc-tures. The most commonly used experimental tech-niques for these studies have been luminescence, inwhich the photons created by the radiative recombina-tion of electrons and holes are detected, and pump-probe measurements, in which the change in the absorp-tion (or reflection) of a probe beam caused by the priorexcitation of electron-hole pairs by the pump beam isobserved. Since it is impossible to cite all the work, wemention only some of the phenomena that have turnedout to be important under certain excitation conditions.In the case of sufficiently high excitation densities, it hasbeen found that the distribution function of LO phononsis driven substantially out of equilibrium and that this‘‘hot-phonon effect’’ may drastically reduce the coolingprocess (van Driel, 1979; Potz and Kocevar, 1983; Koce-var, 1985; Lugli et al., 1989). The dynamics of the non-equilibrium phonons have been studied directly by Ra-man measurements (von der Linde, Kuhl, andKlingenberg, 1980; Ryan and Tatham, 1992). If the exci-tation energy is above the threshold for transitions tosatellite valleys in the conduction band, intervalley tran-sitions due to carrier-phonon interaction are a very ef-fective scattering process mainly because of the highdensity of states in these valleys (Shah et al., 1987;Oberli, Shah, and Damen, 1989).

FIG. 1. The dynamics of distribution functions: (a) excitationby a short laser pulse with a certain excess energy above theband gap; (b) the resulting distribution of electrons and holesas well as the subsequent relaxation of the electron distribu-tion due to (c) electron-phonon scattering, (d) electron-electron scattering, and (e) both types of scattering processes.


Starting in the mid-1980s, the field of coherent excita-tions in semiconductors became an increasingly activeresearch area. Even if essentially carrier relaxation pro-cesses were monitored in the measurements mentionedabove, it turned out that features related to coherencewere present in these signals, as will be discussed inmore detail in Sec. III. Under certain conditions, how-ever, coherent aspects may be dominant. In the case ofpump-probe spectra, this holds most prominently if thepump pulse is nonresonant with optical transitions, i.e.,if it is tuned into the band-gap region where it does notcreate real populations. Here it gives rise to shifts andsplittings of the exciton line, which is known as the op-tical or ac Stark effect. This effect has been extensivelyinvestigated since the mid-1980s (Mysyrowicz et al.,1986; Schmitt-Rink and Chemla, 1986; Balslev and Stahl,1988; Schmitt-Rink, Chemla, and Haug, 1988; Combes-cot and Combescot, 1989; Joffre et al., 1989).

Besides pump-probe and luminescence measure-ments, there are other techniques that rely completelyon the phase coherence in the carrier system, thus pro-viding direct information on the dynamics of coherentinterband and intraband polarizations. The most promi-nent of these techniques are four-wave-mixing (FWM)experiments and the detection of coherently emitted ra-diation in the terahertz range.

Many physical systems exhibit inhomogeneous broad-ening. In the case of a semiconductor in the excitonicregion, this is typically due to some disorder in thesample, while in the band-to-band continuum region thek dependence of transition energies may also be inter-preted as such broadening. Then, coherent polarizationrapidly decays due to destructive interference of the dif-ferent frequency components, and thus it is difficult toextract information on the true loss of phase coherencedue to dephasing processes. In the case of magneticresonance, the spin-echo technique was introduced in1950 (Hahn, 1950), to eliminate the decay due to inho-mogeneous broadening and thus to make possible themeasurement of dephasing times (so-called T2 times). Inthe 1960s, due to the availability of laser sources, echoexperiments were brought into the optical regime andphoton echoes were first observed in ruby (Kurnit,Abella, and Hartmann, 1964; Abella, Kurnit, and Hart-mann, 1966). Since dephasing times are much shorter insemiconductors, very short pulses are required for suchtechniques. In 1985 photon echoes from delocalized ex-citons in semiconductors were observed by using 7-pspulses (Schultheis, Sturge, and Hegarty, 1985), and a fewyears later photon echoes from band-to-band transitionswere measured with 6-fs pulses (Becker et al., 1988).These photon echoes were typically studied by means ofdegenerate four-wave-mixing experiments, which will bedescribed in the next section. In such experiments anexciton phase coherence time of 7 ps was obtained inGaAs at low temperatures (Schultheis et al., 1986); inaddition, the dephasing time T2 of an electron-holeplasma was shown to depend on the carrier density naccording to T2;n20.3 (Becker et al., 1988). Not only dosuch experiments provide information on the decay of


the coherence, they also are useful in the study of othercoherent phenomena, e.g., quantum beats due toquantum-mechanical superpositions of states, which ex-hibit a splitting caused by a variety of physicalphenomena.1

If such superpositions are excited between states withdifferent spatial localizations, they are the source ofelectromagnetic radiation with a frequency given by theenergy splitting. Often this frequency is in the terahertzrange; in this regime the electric-field strength may bedirectly measured, in contrast to the optical regime,where typically only intensities can be measured. Such aterahertz emission was first observed from asymmetricdouble-quantum-well structures (Roskos et al., 1992),which opened up the field of terahertz spectroscopy insemiconductors (Planken et al., 1992; Waschke et al.,1993; Nuss et al., 1994; Dekorsy, Auer, et al., 1995;Leitenstorfer et al., 1999).

Another direct approach to coherent phenomena isthe technique of coherent control by two temporallyseparated, phase-locked pulses (Planken et al., 1993; He-berle, Baumberg, and Kohler, 1995). If the optical polar-ization created by the first pulse is still present in thesample, this polarization can constructively or destruc-tively interfere with the second pulse, leading to dynam-ics in the system that strongly depend on the relativephases of the two pulses. These dynamics can then beprobed by the reflection or transmission change, i.e., apump-probe technique, or by the four-wave-mixing sig-nal induced by a third pulse. It should be noted thatthere is a second type of coherent-control experiment inwhich superposition of a one-photon and a two- orthree-photon excitation by two simultaneous pulses isused to control the final state in the case of degeneracy(Dupont et al., 1995; Atanasov et al., 1996). An overviewof different applications of coherent control can befound in Potz and Schroeder (1999). Generally, coherentcontrol makes use of the full time dependence of theelectric-field vector of the light pulse, including intensity,phase, and polarization. Pulses of arbitrary shape withina wide range of parameters can be created by using aliquid-crystal-display spatial light modulator (Weineret al., 1990). Then the inverse problem can be formu-lated: Which pulse shape produces the desired dynamicsin the sample? This question of constructing an optimalinteraction Hamiltonian has been addressed mainly inthe context of coherent control of chemical reactions(Shapiro and Brumer, 1986; Warren, Rabitz, andDahleh, 1993; Shapiro and Brumer, 1997); it has beenshown that efficiencies (for example, of multiphotonionization processes) can indeed be substantially in-creased by using an evolutionary algorithm (Assionet al., 1996; Baumert et al., 1997).

Coherences do not only exist in the electronic sub-system of the semiconductor. In spatially inhomoge-

1See, for example, Gobel et al. (1990); Leo, Damen, et al.(1990); Schoenlein et al. (1993); Banyai et al. (1995); Mayeret al. (1995); Joschko et al. (1997).


neous systems or in systems with sufficiently low symme-try, optical excitation may also generate coherentphonons, i.e., phonons with a nonvanishing expectationvalue of the lattice displacement, in contrast to incoher-ent phonons, for which only the mean-square displace-ment is nonzero. The excitation of coherent phonons insemiconductors was first observed by optically excitingthe surface field of n-doped GaAs (Cho, Kutt, and Kurz,1990). Here the differential reflectivity change exhibitedclear modulations with the phonon frequency. If the gen-erated phonons are infrared active, they will also di-rectly emit electromagnetic radiation with a frequency inthe terahertz range, which can be detected in the waydescribed above (Dekorsy, Auer, et al., 1995). Coherentphonons have been observed in many materials and indifferent types of heterostructures; a recent review onthis topic is that of Dekorsy, Cho, and Kurz (2000). Par-ticularly interesting are situations in which the phononscouple to other types of elementary excitations withwhich they are nearly resonant. In addition to the gen-eration of coherent phonons, the coupling of suchphonons to plasmons (Cho et al., 1996), to Bloch oscil-lations in superlattices (Dekorsy, Kim, Cho, Kohler, andKurz, 1996), or to intersubband plasmon modes in quan-tum wells (Dekorsy, Kim, Cho, Kurz, et al., 1996) hasbeen studied.

The enormous progress in the experimental study ofultrafast phenomena has been paralleled by an increas-ingly refined theoretical understanding. In fact, thisprogress has often been possible only because of thestrong collaboration between theory and experiment.While the first studies on energy relaxation were mod-eled by simple rate equations for the mean carrier en-ergy, detailed understanding of carrier relaxation andthermalization processes required a modeling of the dis-tribution functions of the carriers involved. Their tem-poral evolution is governed by the Boltzmann transportequation, which, in general, can only be solved numeri-cally. Monte Carlo simulations have proven to be a tech-nique well suited for this purpose (Jacoboni and Reggi-ani, 1983; Jacoboni and Lugli, 1989). Additionally,modeling coherent phenomena requires taking into ac-count the interband (or intraband/intersubband) polar-ization. On the mean-field level the dynamics are de-scribed by the semiconductor Bloch equations (Huhnand Stahl, 1984; Lindberg and Koch, 1988a), a generali-zation of the well-known optical Bloch equations (Allenand Eberly, 1987). Scattering processes that give rise torelaxation and dephasing can be introduced in aBoltzmann-like (semiclassical) way (Binder et al., 1992;Kuhn and Rossi, 1992b). Such a density-matrix approachhas recently been generalized to describe quantum sys-tems with open boundaries (Rossi, di Carlo, and Lugli,1998).

On very short time scales even the description of scat-tering processes in terms of rates obtained from Fermi’sgolden rule is no longer sufficient. Quantum-kinetictheories that overcome this limitation have been devel-oped based on different approaches, in particular non-equilibrium Green’s functions (Haug and Jauho, 1996;


Haug, 2001) and density matrices (Bonitz, 1998; Kuhn,1998). These theories have been shown to describe thedynamics on a femtosecond time scale with very highaccuracy. A more detailed discussion of theoretical ap-proaches will be given in Sec. II.

B. Experimental techniques

As already mentioned above, essentially two differentclasses of experiments have been used for the study ofcarrier relaxation processes: luminescence and pump-probe measurements. In both cases a pump pulse is usedto generate electron-hole pairs and bring the semicon-ductor into a state far from thermal equilibrium. In aluminescence experiment the radiation emitted in a di-rection different from that of the incident pulse due torecombination processes is analyzed spectrally and/ortemporally. This is shown schematically in Fig. 2(a). De-pending on the temporal resolution, different techniqueshave to be used: Temporal resolution in the range of 10ps can be achieved by direct techniques by using eitherfast photodiodes or a streak camera that provides spec-tral and temporal information simultaneously. Highertime resolution is obtained by gating the luminescencesignal with a second delayed laser pulse. Both the signal

FIG. 2. Schematic representation of typical experimental set-ups for the study of ultrafast phenomena in semiconductors:(a) single-pulse excitation in which the secondary emission(resonant Rayleigh scattering or luminescence) is detected in adirection different from that of incidence; (b) pump-probe ex-periments or four-wave mixing in the two-pulse self-diffractiongeometry; (c) three-pulse four-wave-mixing experiments.


and the delayed pulse are focused on a nonlinear crystal,which creates a sum-frequency signal only in the pres-ence of such a gating laser pulse. In this upconversiontechnique the temporal resolution is limited only by thelaser-pulse duration; thus resolution in the 10-fs range ispossible. In an intrinsic semiconductor the luminescenceis due to the recombination of an electron in the con-duction band with a hole in the valence band. In a fullyincoherent picture, according to Fermi’s golden rule, thesignal is essentially proportional to the product of thedistribution functions of electrons and holes. This com-plicates the interpretation of experimental results. Analternative is the use of doped semiconductors, e.g.,p-doped samples, in which the band-to-acceptor lumi-nescence directly monitors the distribution function ofelectrons.

In a pump-probe experiment the semiconductor is ex-cited by a pump pulse traveling in a direction q1 [pulse 1in Fig. 2(b)], and the dynamics of the carriers induced bythis excitation are studied by looking at some propertyrelated to a delayed probe pulse in a direction q2 . Themost commonly used technique is transmission or reflec-tion spectroscopy, in which the change in the transmis-sion or reflection of the probe pulse—induced by thepump—is measured as a function of the time delay be-tween the two pulses.2 By using a broadband probepulse, one therefore obtains differential transmission/reflection spectra. In a purely incoherent free-carrierpicture the absorption is changed due to phase-spacefilling, and these signals provide information on the sumof the electron and hole populations in the opticallycoupled states. Again, the interpretation of the results isfacilitated if the spectra are determined by a single dis-tribution function. This can be achieved by exploitingoptical transitions for the probe in a different spectralrange, e.g., by pumping the heavy and light hole-to-conduction-band transitions and probing the splitoff tothe conduction-band transition. A variation of thepump-probe technique is electro-optic sampling, inwhich the difference between two polarization compo-nents of the transmitted/reflected signal is analyzed, pro-viding information, for example, on the birefringence in-duced by the optically excited dynamics. Instead of thechange in the transmitted/reflected signal, the change inthe Raman-scattering signal generated by the probepulse can also be measured. The dynamics of photoex-cited phonons as well as electronic excitations can bestudied using this technique.

It is clear that the interpretation of luminescence andpump-probe experiments in terms of a fully incoherentfree-carrier picture is valid only under limited condi-tions. In intrinsic semiconductors at sufficiently low car-rier densities, absorption and luminescence spectra inthe region close to the band gap are strongly dominatedby excitonic effects. Even high up in the band, pump-induced changes in the Coulomb enhancement may sig-nificantly influence pump-probe spectra. Furthermore,

2In Fig. 2 only the transmission case is plotted.


on time scales comparable to or shorter than the char-acteristic dephasing times, the signals may be consider-ably modified by coherence effects. Therefore a detailedanalysis of luminescence and pump-probe spectra in theultrafast regime also provides information on coherentphenomena in the semiconductor. In Sec. III we shalldiscuss several examples in which such phase-related ef-fects play a dominant role.

The most popular technique that provides direct infor-mation on coherence in semiconductors is four-wave-mixing spectroscopy. It can be performed in both a two-pulse and a three-pulse configuration, as shownschematically in Figs. 2(b) and (c).3 For clarity let usstart with the three-pulse configuration by assuming thatthe time delay T12 between the two laser pulses 1 and 2with wave vectors q1 and q2 is zero. In this case thesepulses create an interference pattern with wave vector6(q22q1) on the sample, which translates into a densitygrating when the light is absorbed. This density gratingresults in a refractive index grating, which may diffract athird pulse with incident wave vector q3 into various dif-fraction orders q31n(q22q1), where n is an integer. Infour-wave-mixing the first diffracted order (n51) ismeasured; here, three interacting incident waves inter-act, giving rise to a fourth emitted wave, which explainsthe name of the technique. Access to the coherent po-larization in the sample is now obtained if pulses 1 and 2are temporally separated. In this case there is no longera direct interference pattern on the sample. However, aslong as the microscopic interband polarization createdby pulse 1 is still at least partly present when pulse 2arrives, the interaction of a pulse with wave vector q2with the interband polarization in the direction of q1again results in a transient grating which can diffractpulse 3. Thus, by varying T12 , one obtains informationon the dynamics and the lifetime of the polarization, i.e.,on the dephasing time. It should be noted that a micro-scopic interband polarization may still be present evenif, in the case of a continuous spectrum due to destruc-tive interference of different microscopic components,there is no longer macroscopic polarization in thesample. This is exactly why four-wave-mixing spectros-copy can distinguish between homogeneous and inho-mogeneous broadening. In the more often used two-pulse setup, pulse 2 simultaneously creates the gratingand is diffracted by this grating; hence the name self-diffraction geometry. Besides analyzing the signal in atime-integrated way, one can also spectrally disperse it ina monochromator or temporally resolve it by means ofan upconversion technique, as discussed in the case ofluminescence, which provides additional information onthe dynamics of the interband polarization.

Both pump-probe and four-wave-mixing experimentscan be used to study coherent-control phenomena. Inthis case pulse 1 is replaced by a pair of phase-lockedpulses with variable delay traveling in the same direction

3Four-wave-mixing experiments can also be performed in areflection geometry.


q1 . The carrier dynamics induced by these pulses thendepend on the relative phase between these two pulsesand can be analyzed by the second pulse traveling indirection q2 . In general, measuring the transmitted sig-nal in direction q2 (or the corresponding reflected direc-tion) and the four-wave-mixing signal in direction 2q22q1 yields complementary information on the dynamicsof distribution functions and polarizations.

If the optically excited interband polarizations coupleelectronic states with different spatial localizations, thedynamics are associated with a time-dependent dipolemoment, which, according to classical electrodynamics,acts as a source of electromagnetic radiation. Often thisradiation is in the terahertz spectral range. In a typicalterahertz-emission experiment a short laser pulse excitesthe system in a superposition of states, thus creating anintraband polarization. The intraband polarization oscil-lates according to the energy splitting of the correspond-ing states and emits a pulse of electromagnetic radiationwith the corresponding frequency. This radiation is thencollimated and transmitted to an optically gated photo-conductive antenna, which measures the electric field ofthe radiation. Thus the general setup is the same as Fig.2(a), except that the emitted signal in direction q2 is inthe terahertz range. For a more detailed discussion ofexperimental techniques, we refer the reader to thebook by Shah (1999).

C. Aim and outline of the paper

Ultrafast spectroscopy of semiconductors has been anextremely active field of research and has led to manynew insights into phenomena of fundamental impor-tance in semiconductor physics and technology. Suchrapid development has been accompanied by a growingtheoretical understanding of the basic processes govern-ing the ultrafast nonequilibrium dynamics of photoex-cited carriers. However, theoretical activity in the fieldhas often been focused on specific problems within dif-ferent perspectives.

The aim of this paper is to provide a cohesive discus-sion of ultrafast phenomena in semiconductors. Morespecifically, our primary goal is to show how a variety ofapparently different phenomena can be described withinthe same theoretical framework based on the density-matrix formalism. Within this approach one can describethe strong interplay between coherent and incoherent(i.e., phase-breaking) phenomena that characterizes theultrafast electro-optical response of semiconductor bulkand heterostructures.

In terms of this general theory, we shall review a va-riety of physical phenomena in different semiconductorstructures such as bulk systems, superlattices, quantumwells, and quantum wires. The field is so active and ex-tensive that an exhaustive treatment of all the researchwould be impossible. Thus we have to limit ourselves toa discussion of selected theoretical and experimental re-sults, including recent developments, which have led tofundamental new insights into many diverse aspects ofsemiconductor physics. We shall concentrate on phe-


nomena in which carrier dynamics play the central role.Therefore phenomena like resonant Rayleigh scatteringwill be briefly mentioned but not discussed in detail.Furthermore, we shall restrict ourselves to the case inwhich the exciting light field can be treated as an exter-nal field initializing the carrier dynamics, which excludespropagation effects as well as phenomena related to mi-crocavities. Finally, all the semiconductor structuresmentioned above have in common a continuous spec-trum. The increasingly active field of ultrafast dynamicsin quantum dot structures has its own set of theoreticalconsiderations, which will not be covered by this review.We shall briefly come back to this point in our conclud-ing remarks.

The paper is organized as follows: In Sec. II we dis-cuss the fundamentals of the density-matrix formalismapplied to the analysis of the electro-optical response ofsemiconductor bulk and heterostructures. The contribu-tions to the equations of motion due to various interac-tion mechanisms are derived and their physical meaningis explained. In the last part of the section we discusshow typical experiments are modeled within the frame-work presented earlier. A selection of fundamental re-sults, both experimental and theoretical, is presented inSec. III. Finally, in Sec. IV we summarize and draw someconclusions.

II. THEORETICAL BACKGROUND

A. Physical system

In order to study the optical and transport propertiesof semiconductor bulk and heterostructures, let us con-sider a gas of carriers in a crystal under the action of anapplied electromagnetic field. The carriers will experi-ence mutual interaction as well as interaction with thephonon modes of the crystal. This physical system canbe described by the following Hamiltonian:

H5Hc1Hp1Hcc1Hcp1Hpp . (1)

The first term describes the noninteracting-carrier sys-tem in the presence of the external electromagneticfield, while the second one refers to the free-phononsystem. The last three terms describe many-body contri-butions: carrier-carrier, carrier-phonon, and phonon-phonon interactions, respectively.

In order to discuss their explicit form, let us introducethe usual second-quantization field operators C†(r) andC(r). They describe, respectively, the creation and thedestruction of an electron in r. In terms of the abovefield operators, the carrier Hamiltonian Hc can be writ-ten as

Hc5E drC†~r!

3F F2i\¹r1e

cA~r,t !G2

2m02ew~r,t !1Vc~r!GC~r!.

(2)


Here, Vc(r) denotes an effective single-particle poten-tial due to the perfect crystal plus the valence electrons,m0 is the free-electron mass, and 2e is the charge of theelectron. A(r,t) and w(r,t) denote, respectively, the vec-tor and scalar potentials corresponding to the externalelectromagnetic field:

E~r,t !521c

]

]tA~r,t !2¹rw~r,t !,

B~r,t !5¹r3A~r,t !. (3)

The above equation reflects the well-known gauge free-dom: there is an infinite number of possible combina-tions of A and w which give rise to the same electromag-netic fields E and B. This gauge invariance will becrucial in understanding the relationship between Blochoscillations and Wannier-Stark localization, and will bediscussed in more detail in Sec. III.F.1.

Since we are interested in the electro-optical proper-ties as well as in the ultrafast dynamics of photoexcitedcarriers, the electromagnetic field acting on the crystal—and the corresponding electromagnetic potentials—willbe regarded as the sum of two different contributions:one part which will be treated dynamically in the equa-tions of motion (term 1) and an additional static (electricand/or magnetic) field (term 2) which will be included inthe evaluation of single-particle basis states. Term 1 con-tains the time-dependent laser field responsible for theintraband as well as interband electronic excitations, butit may also contain additional static or dynamic electricfields not included in the definition of the basis states.However, we assume that all these contributions to term1 are spatially sufficiently slowly varying (on the atomicscale) that they are well described by the scalar potential

w1~r,t !52E1~r,t !•r. (4)

With this particular choice of the electromagnetic po-tentials, the Hamiltonian in Eq. (2) can be rewritten asHc5Hc

01Hcf , where

Hc05E drC†~r!

3F F2i\¹r1e

cA2~r,t !G2

2m02ew2~r,t !1Vc~r!GC~r!

(5)

describes the carrier system in the crystal under the ac-tion of the static field 2 only,4 while

Hcf52eE drC†~r!w1~r,t !C~r! (6)

describes the carrier-field interaction.In analogy to the carrier system, when bq

† and bq de-note the creation and destruction operators for a pho-non with wave vector q, the free-phonon Hamiltoniantakes the form

4It should be noted that, depending on the gauge, the poten-tials may be time dependent even for static fields.


Hp5(q

\vqbq†bq , (7)

where vq is the phonon dispersion. For simplicity, herewe restrict ourselves to a single branch of bulk phonons;the generalization to several branches as well as to othertypes of modes, e.g., confined phonons, is obvious.

Let us now discuss the explicit form of the many-bodycontributions. The carrier-carrier interaction is de-scribed by the two-body Hamiltonian

Hcc512 E drE dr8C†~r!C†~r8!Vcc~r2r8!C~r8!C~r!,

(8)

where Vcc denotes the Coulomb potential screened bythe valence electrons as well as by those phonon degreesof freedom that are not taken into account dynamically.The coupling between carriers and phonons is describedby the Hamiltonian

Hcp5E drC†~r!Vcp~r!C~r!, (9)

where

Vcp5(q

@ gqbqeiq•r1 gq* bq†e2iq•r# (10)

is the potential induced by the lattice vibrations, linear-ized in the displacements of the nuclei. Here, the explicitform of the coupling function gq depends on the particu-lar phonon branch (acoustic, optical, etc.) as well as onthe coupling mechanism considered (deformation poten-tial, polar coupling, etc.).

Finally, let us briefly comment on the phonon-phononcontribution Hpp : The free-phonon Hamiltonian Hp inEq. (7), which describes a system of noninteractingphonons, by definition accounts only for the harmonicpart of the lattice potential. However, nonharmonic con-tributions of the interatomic potential can play an im-portant role in determining the lattice dynamics inhighly excited systems (Kash and Tsang, 1989), sincethey are responsible for the decay of optical phononsinto phonons of lower frequency. In our second-quantization picture, these nonharmonic contributionscan be described in terms of a phonon-phonon interac-tion which, in general, induces transitions between free-phonon states. Here, we shall not discuss the explicitform of the phonon-phonon Hamiltonian Hpp respon-sible for such a decay. For the results discussed in thisreview, it is not important either because the phononsystem remains sufficiently close to thermal equilibriumor because the characteristic time scale of this decay(about 7 ps for the decay of LO phonons in bulk GaAs;von der Linde et al., 1980; Kash et al., 1985; Shah, 1992)is considerably longer than the femtosecond time scalestypically studied in ultrafast experiments.

It is well known that the coordinate representationused so far is not the most convenient one for describingthe electron dynamics within a crystalline semiconduc-tor. In general, it is more convenient to employ the rep-resentation given by the eigenstates of a suitably chosen


noninteracting-carrier Hamiltonian, since it automati-cally accounts for some of the symmetries of the system.

In this spirit, let us denote with $fn(r,t)% the set ofeigenfunctions of the noninteracting-carrier Hamil-tonian in Eq. (5) and with en(t) the corresponding en-ergy levels. Since the Hamiltonian is, in general, a func-tion of time, the basis functions fn and the energies enmay be time dependent. Here, the label n denotes, ingeneral, a set of discrete and/or continuous quantumnumbers. In the absence of the static field 2, the abovewave functions will correspond to the well-known Blochstates of the crystal, and the index n will reduce to thewave vector k plus the band (or subband) index n. In thepresence of a homogeneous magnetic field, the eigen-functions fn , after performing an effective mass ap-proximation, may instead correspond to Landau states.Finally, in a constant and homogeneous electric field,depending on the gauge chosen, there exist two equiva-lent representations: the accelerated Bloch states andthe Wannier-Stark picture. We shall come back to thispoint in Sec. III.F.1 when discussing the relationship be-tween Bloch oscillations and Wannier-Stark localization.

Let us now reconsider the system Hamiltonian intro-duced so far in terms of fn . As a starting point, we shallexpand the second-quantization field operators in termsof the new wave functions:

C~r!5(n

fn~r,t !an , C†~r!5(n

fn* ~r,t !an† . (11)

The above expansion defines the new set of second-quantization operators an

† and an ; they describe, respec-tively, the creation and destruction of an electron instate n .

In the case of a semiconductor structure (the only oneconsidered here), the energy spectrum en of thenoninteracting-carrier Hamiltonian (5) is always charac-terized by two well-separated energy regions: the va-lence and the conduction band. Also, in the presence ofan applied electromagnetic field, the effective lattice po-tential Vc gives rise to a large energy gap. Therefore weare dealing with two energetically well-separated re-gions, which suggests the introduction of the usualelectron-hole picture. This corresponds to a separationof the set of states $fn% into conduction states $f i

e% andvalence states $f j

h%. Thus the creation operator an† intro-

duced in Eq. (11) will also be divided into electron cre-ation and hole destruction operators ci

† and dj† , while

the destruction operators an will be divided into electrondestruction and hole creation operators ci and dj

† . Interms of the new electron-hole picture, the expansion(11) is given by

C~r!5(i

f ie~r,t !ci1(

jf j

h* ~r,t !dj† ,

C†~r!5(i

f ie* ~r,t !ci

†1(j

f jh~r,t !dj . (12)

If we now insert the above expansion into Eq. (5), thenoninteracting-carrier Hamiltonian takes the form


Hc0~ t !5(

ie i

e~ t !ci†ci1(

je j

h~ t !dj†dj

5He0~ t !1Hh

0~ t !. (13)

As already pointed out, the above Hamiltonian is ingeneral time dependent. We shall discuss this feature inthe following section, where we shall derive our set ofkinetic equations.

Let us now write the carrier-field interaction Hamil-tonian [Eq. (6)] in terms of our electron-hole represen-tation:

Hcf5(i ,i8

Eii8e(cf)ci

†ci81(j ,j8

Ejj8h(cf)dj

†dj8

1(i ,j

@U ij(cf)ci

†dj†1U ij

(cf)* djci# . (14)

Here, Ell8e ,h(cf) denote the electron and hole intraband

and U ij(cf) the interband matrix elements of the scalar

potential w1(r,t):

Ell8e ,h(cf)

56eE drf le ,h* ~r,t !E1~r,t !•rf l8

e ,h~r,t !, (15a)

U ij(cf)5eE drf i

e* ~r,t !E1~r,t !•rf jh* ~r,t ! (15b)

where the upper (positive) sign refers to electrons andthe lower (negative) sign to holes. Equations (7), (13),and (14) define the single-particle Hamiltonian Hsp

5Hp1Hc01Hcf .

Similarly, the carrier-carrier Hamiltonian (8) can berewritten as

Hcc512 (

i1i2i3i4

Vi1i2i3i4

ee ci1

† ci2

† ci3ci4

112 (

j1j2j3j4

Vj1j2j3j4

hh dj1

† dj2

† dj3dj4

2 (i1i2j1j2

Vi1j1j2i2

eh ci1

† dj1

† dj2ci2

, (16)

where Vl1l2l3l4

ee/hh/eh are the Coulomb matrix elements of thetwo-body Coulomb potential in our f representation.The first two terms describe the repulsive electron-electron and hole-hole interaction, while the third de-scribes the attractive electron-hole interaction. The ef-fective single-particle contributions that appear due tothe reordering of the operators are assumed to be in-cluded in Hc

0(t). Here, we restrict ourselves to themonopole-monopole contributions, which means thatwe neglect terms that do not conserve the number ofelectron-hole pairs, i.e., impact-ionization and Auger re-combination processes (Quade et al., 1994), as well asthe interband exchange interaction. The former assump-tion is typically well satisfied in the range of energiesand densities considered here; the latter is justified if thelongitudinal-transverse splitting of the exciton is muchsmaller than its binding energy (Egri, 1985).


Finally, we rewrite the carrier-phonon interactionHamiltonian introduced in Eq. (9):

Hcp5 (ii8,q

@gqii8ci

†bqci81gqii8* ci8

† bq†ci#

2 (jj8,q

@gqjj8dj

†bqdj81gqjj8* dj8

† bq†dj# , (17)

where

gqll85 gqE drf l* ~r,t !eiq•rf l8~r,t !. (18)

In Eq. (17) we can clearly recognize four different con-tributions corresponding to phonon absorption andemission by electrons and holes. The interband termshave been neglected because of their strong off-resonance nature.

B. Kinetic description

Many experimentally observable quantities like cur-rent densities, phonon populations, photon numbers, orelectronic polarizations which act as sources for emittedelectromagnetic radiation are single-particle quantities.They can be expressed in terms of single-particle densitymatrices like the intraband electron and hole densitymatrices f ii8

e5^ci

†ci8& and f jj8h

5^dj†dj8& , the correspond-

ing interband density matrix pji5^djci&, or the phononoccupation number nq5^bq

†bq&. The diagonal elementsof these density matrices describe the occupation prob-abilities of the respective states, while the off-diagonalelements determine the degree of quantum-mechanicalsuperposition of the two states involved. The expecta-tion value is taken with respect to the initial state of thesystem, which in most cases discussed in this review isthe vacuum of electron-hole pairs and a thermal phonondistribution.

The primary goal of a kinetic theory of ultrafast pro-cesses is to calculate the temporal evolution of the quan-tities introduced above, which constitute the kineticvariables of the system. However, due to the many-bodynature of the problem, an exact solution is generally notpossible, except for some simple model systems (Zim-mermann and Wauer, 1994; Meden et al., 1995; Axt,Herbst, and Kuhn, 1999; Castella and Zimmermann,1999). Such exact calculations provide valuable informa-tion on specific features of certain experiments, but for afull understanding realistic semiconductor models haveto be considered, which then can only be treated ap-proximately.

Different techniques for the theoretical treatment ofthe dynamics of many-body systems have been devel-oped in the past. Among the most commonly used meth-ods are the nonequilibrium Green’s-function techniqueand the density-matrix formalism. The former, intro-duced in the 1960s by Kadanoff and Baym (1962) andKeldysh (1965), is an extension of the well-known equi-librium or zero-temperature Green’s-function theory tononequilibrium systems. The basic ingredient is a


contour-ordered Green’s function for which, as in theequilibrium case, a Dyson equation can be formulated.This contour-ordered function can be separated intofour types of two-time Green’s functions. The centralapproximation in this formalism is the identification ofthe self-energy by selecting a specific subset of the infi-nitely many diagrams, to define the interaction processestaken into account. Since in most cases the quantitiesthat are directly related to experimental observables de-pend on one time only, the question arises whether thetheory can be reduced to single-time variables. Kadanoffand Baym (1962) showed that the Boltzmann equationcan be recovered if the two-time Green’s function G, iswritten as a product of its time-diagonal part—i.e., thesingle-particle density matrix—and a spectral function ofnoninteracting carriers. This ansatz was extended byLipavsky, Spicka, and Velicky (1986) to the so-calledgeneralized Kadanoff-Baym ansatz, which correctlytreats causality. Under nonequilibrium conditions, how-ever, this is only an approximation. An introduction tothe theory of nonequilibrium Green’s functions with ap-plications to many problems in transport and optics ofsemiconductors can be found in the book by Haug andJauho (1996). A recent review is that of Haug (2001).

In the density-matrix formalism, one starts directlywith the equations of motion for the single-particle den-sity matrices. Due to the many-body nature of the prob-lem, the resulting set of equations of motion is notclosed; instead, it constitutes the starting point of an in-finite hierarchy of higher-order density matrices. Asidefrom differences related to the quantum statistics of thequasiparticles involved, this is equivalent to theBogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hi-erarchy in classical gas dynamics (Bogoliubov, 1967; Mc-Quarrie, 1976). The central approximation to obtain atractable problem in this formalism is the truncation ofthe hierarchy. This can be based on different physicalpictures. In this review we shall use the argument thatcorrelations involving an increasing number of particleswill become less and less important. On the classicallevel this corresponds to the derivation of the Vlasovequation (mean field) and the Boltzmann equation (two-particle correlations) from the BBGKY hierarchy. Analternative scheme was introduced by Axt and Stahl(1994a): the so-called ‘‘dynamics-controlled truncation.’’Here, the basic idea is an expansion in powers of theexciting laser field, as is done in nonlinear optics whennonlinear susceptibilities are introduced. It turns outthat for the usual case of a pair-conserving many-bodyHamiltonian and a system in which the initial state isgiven by the vacuum of electron-hole pairs, the hierar-chy can be rigorously truncated to any order in the driv-ing field. This approach is particularly useful whenhigher-order Coulomb correlations like biexcitons andcorrelated two-exciton states are important. Further-more, it allows a unified treatment of semiconductorsand molecular structures. For a detailed discussion ofthis formalism, the reader is referred to the review ar-ticle by Axt and Mukamel (1998).


The standard procedure for deriving the set of kineticequations, i.e., the equations of motion for the relevantkinetic variables, is to derive the equations of motion forthe electron and hole operators introduced in Eq. (12):

ci5E drf ie* ~r,t !C~r!,

dj5E drf jh* ~r,t !C†~r!. (19)

By applying the Heisenberg equation of motion for thefield operator C, one can easily obtain the followingequations of motion:

d

dtci5

1i\

@ci ,H#11i\ (

i8Zii8

e ci8 , (20a)

d

dtdj5

1i\

@dj ,H#11i\ (

j8Zjj8

h dj8 , (20b)

where

Zii8e

5i\E drS ]

]tf i

e* ~r,t ! Df i8e

~r,t !,

Zjj8h

5i\E drS ]

]tf j

h~r,t ! Df j8h* ~r,t !. (21)

Here we neglect minor contributions due to couplingbetween valence and conduction bands. Compared tothe more conventional Heisenberg equations of motion,they contain an extra term, the last one. It accounts forthe possible time dependence of our f representation,which will induce transitions between different states ac-cording to the matrix elements Ze ,h.

By combining the above equations of motion with thedefinitions of the kinetic variables, we can schematicallywrite the resulting set of kinetic equations as

d

dtF5

d

dtFuH1

d

dtFuf, (22)

where F denotes the generic kinetic variable. They ex-hibit the same structure as the equations of motion [Eq.(20)] for the electron and hole creation and destructionoperators: a first term induced by the system Hamil-tonian H (which does not account for the time variationof the basis states) and a second term induced by thetime dependence of the basis functions f.

The explicit form of this second term is given by

d

dtfi1i2

e uf51i\ (

i3

@Zi2i3

e f i1i3

e 2Zi3i1

e f i3i2

e # , (23a)

d

dtfj1j2

h uf51i\ (

j3

@Zj2j3

h fj1j3

h 2Zj3j1

h fj3j2

h # , (23b)

d

dtpj1i1

uf51i\ F(

j2

Zj1j2

h pj2i11(

i2

Zi1i2

e pj1i2G . (23c)

As we shall see, these contributions play a central role inthe description of Zener tunneling in superlatticeswithin the vector-potential representation.


The first term—which is the only one present if atime-independent basis is considered—is the sum of dif-ferent contributions corresponding to the various partsof the Hamiltonian. In the remainder of this section weshall discuss the time evolution induced by the differentcontributions to the total Hamiltonian.

C. Single-particle dynamics

The time evolution induced by the single-particleHamiltonian is easily obtained from the Heisenbergequations of motion in Eq. (20), resulting in a closed setof kinetic equations for the single-particle density matri-ces. The intraband contributions have the same struc-ture as those in Eq. (23); therefore the time dependenceof the basis states can be directly included, which isagain related to the gauge invariance of our formulationas discussed in Rossi (1998). The resulting equations ofmotion are

d

dtfi1i2

e u(sp)51i\ (

i3

@Ei2i3

e(sp)f i1i3

e 2Ei3i1

e(sp)f i3i2

e #

11i\ (

j1

@Ui2j1

(sp)pj1i1* 2Ui1j1

(sp)* pj1i2# , (24a)

d

dtpj1i1

u(sp)51i\ F(

j2

Ej1j2

h(sp)pj2i11(

i2

Ei1i2

e(sp)pj1i2G1

1i\ FUi1j1

(sp)2(i2

Ui2j1

(sp)f i2i1

e 2(j2

Ui1j2

(sp)f j2j1

h G ,

(24b)

where the effective electron and hole single-particle en-ergies (intraband energies) are Ell8

e ,h(sp)5e l

e ,hd ll81Ell8e ,h(cf)

1Zll8e ,h and the effective field (interband energy) is

U ij(sp)5U ij

(cf) . The equations for the hole density matrixf jj8

h typically have the same structure as those for theelectron density matrix. Therefore, in general, we shallgive explicitly only the equations for fe and p .

It should be noted that Eqs. (24) are nothing morethan a multilevel generalization of the optical Blochequations. Indeed, by restricting ourselves to the case ofa single electron state i and a single hole state j , and byidentifying the elements of the density matrix r of a two-level system with our variables according to r225f ii

e ,r11512f jj

h , and r125pji , we recover the well-knownoptical Bloch equations for a two-level system (Allenand Eberly, 1987).

For the derivation of the equations of motion, no spe-cific time dependence of the electric field has been as-sumed. The importance of the different contributions,however, is strongly determined by the frequency of theelectromagnetic excitation. For a static or low-frequencyfield the intraband terms—describing carrier transportphenomena—are usually the most important ones. In-deed, in this case the interband terms are strongly off-resonant; they give rise to Zener tunneling, which re-quires very high fields to become relevant. If, in


contrast, the frequency of the field is of the order of theband gap, the creation of a coherent interbandpolarization—and the resulting generation of electron-hole pairs—is the dominant process.

1. Semiclassical generation rate

Many transport and relaxation phenomena in opti-cally excited semiconductors, in particular if the relevanttime scales are not too short, can be described quite wellon a purely incoherent or semiclassical level, where dis-tribution functions are the only dynamical variables.Therefore one of the purposes of a quantum-kinetictheory is to derive the semiclassical theory as a limitingcase and to study the approximations involved in such aderivation. The general procedure employed to obtainthe semiclassical limit is the same for all types of inter-actions. As we shall see, it consists of an adiabatic elimi-nation of variables involving quantum-mechanical corre-lations by means of a Markov approximation, assumingthat the system was initially uncorrelated. In this sectionwe discuss this approach for the case of interaction dy-namics induced by an external field.

In the semiclassical limit the system is completely de-termined by the distribution functions of electrons andholes. Thus all off-diagonal elements have to be elimi-nated. As discussed above, the dominant terms in Eq.(24) are determined by the frequency of the field. In thefollowing we shall consider the case of a time-independent single-particle basis fn and an optical (in-terband) excitation. By keeping only the nearly resonantparts in the equations of motion, i.e., performing a‘‘rotating-wave’’ approximation (Haug and Koch, 1993),we get

d

dtfii

e 5(j

gji~ t !,d

dtfjj

h 5(i

gji~ t !, (25a)

d

dtpji5

1i\

~e jh1e i

e!pji11i\

U ij0* ~ t !e2ivLt@12f ii

e 2f jjh # ,

(25b)

where the generation rate is

gji~ t !51i\

@U ij0 ~ t !e2ivLtpji* 2U ij

0 ~ t !eivLtpji# , (26)

the effective field in the rotating-wave approximation isUij5U ij

0 (t)e2ivLt, and U ij0 (t) denotes the slowly varying

part of U. In this case the above polarization equationcan be formally integrated:

pji~ t !51i\

e2ivLtE0

`

dte2i(v ji2vL)tUij0 ~ t2t!

3@12f iie ~ t2t!2f jj

h ~ t2t!# , (27)

where \v ji5e jh1e i

e . In the case of a continuous spec-trum, the summation over the final states eventuallyleads to a finite memory depth due to destructive inter-ference of the different frequency contributions. Thesemiclassical limit is then obtained with two assump-tions: First, within the Markov approximation one as-


sumes that the dominant time dependence is given bythe exponential in Eq. (27) and therefore that the carrierdistribution and field amplitude are sufficiently slowlyvarying that their value at time t can be taken out of theintegral. Second, in order to have a well-defined initialcondition, the field is adiabatically switched on accord-ing to U ij

0 (t)5limh→0U ij0 (t)eht. Then, the polarization is

an instantaneous function of the carrier distribution andfield according to

pji~ t !52ip

\U ij

0 ~ t !e2ivLt@12f iie ~ t !2f jj

h ~ t !#D~v ji2vL!,

(28)

where we have introduced the function

D~x !52i

plim

h→0

1x2ih

5d~x !2i

p

Px

. (29)

Here P denotes the principal value. This leads to thesemiclassical generation rate

gji~ t !52p

\2 uU ij0 ~ t !u2@12f ii

e ~ t !2f jjh ~ t !#d~v ji2vL!,

(30)

i.e., Fermi’s golden rule. The general procedure for ob-taining a semiclassical rate, which has been performedhere for the case of light-matter interaction, is the samefor all coupling mechanisms: The interaction process in-troduces a new variable describing the correlation asso-ciated with this interaction. In the present case this is theelectron-hole correlation due to the light field describedby pji . This new variable is then adiabatically eliminatedon the basis of a Markov approximation and the as-sumption of an initially uncorrelated system.

Completely neglecting the time dependence of thefield amplitude has led us to a monochromatic genera-tion rate. Any pulse with a finite duration, however, ischaracterized by a finite spectral width. Often thisbroadening is introduced by multiplying the generationrate by the spectral intensity of the pulse and integratingover the light frequency. More rigorously, this broaden-ing can be derived from the time-integrated generationobtained from Eqs. (26) and (27) after a change of theintegration variables according to

Gji5E2`

`

dtgji~ t !

51\2 E

2`

`

dtE2`

`

dt8F12f iie S t2

12

t8D2f jjh S t2

12

t8D G3U ij

0* S t112

t8DU ij0 S t2

12

t8D e2i(v ji2vL)t8. (31)

If we now perform the Markov approximation for thedistribution functions only and identify the time of gen-eration with the central time t , we arrive at a semiclas-sical generation rate of the form


gji~ t !51\2 @12f ii

e ~ t !2f jjh ~ t !#

3E2`

`

dt8U ij0* S t1

12

t8D3U ij

0 S t212

t8D e2i(v ji2vL)t8, (32)

i.e., a rate determined by the time-dependent spectrumof the pulse. In the absence of other types of interactionsand when phase-space filling effects can be neglected,this generation rate leads to the correct final distributionof electrons and holes. However, it is not strictly causal:The generation rate at time t also depends on the am-plitude of the light field at later times. This is a conse-quence of the definition of an instantaneous spectrumwhich violates energy-time uncertainty. In the case of aGaussian pulse U ij

0 (t)5Uijexp@2t2/tL2 #, Eq. (32) leads to

gji~ t !5A2p

\2 @12f iie ~ t !2f jj

h ~ t !#uUiju2tL

3expF22t2

tL2 GexpF2

12

tL2 ~v ji2vL!2G , (33)

i.e., to a generation rate according to the product of thetemporal and spectral shape of the pulse. In general,however, such a decomposition is not possible.

2. Homogeneous system, homogeneous excitation

The set of kinetic equations derived so far is valid inany single-particle basis and for any type of semiconduc-tor structure. In many experimentally relevant systems,however, symmetries occur that may reduce the com-plexity of the problem. The most important is the homo-geneous system. This can be a bulk semiconductor thatis homogeneous in all three dimensions, but it also ap-plies to low-dimensional structures like quantum wellsand wires in the so-called strong-confinement limit, i.e.,if only one carrier subband contributes to the dynamics.These structures are therefore homogeneous in d di-mensions, where d is two or one. In such homogeneoussystems Bloch functions with a k vector in the d dimen-sions can be chosen as basis states. Then, all interactionmechanisms in the Hamiltonian satisfy momentum con-servation. If, in addition, the k dependence of thelattice-periodic part of the Bloch functions for electrons@ue(r)# and holes @uh(r)# is neglected—which corre-sponds to an effective-mass approximation—the Cou-lomb and carrier-phonon matrix elements Vq and gq de-pend on the momentum transfer q only; moreover, theinterband dipole matrix element

M52eEVc

drue* ~r!ruh* ~r! (34)

is k independent.If such a homogeneous system is excited by a spatially

homogeneous electric field E(t), the effective energiesand fields are given by


E kk8e ,h

5Feke ,h6ieEi•

]

]kGdkk8 , (35a)

Ukk852M•Edk,2k8 , (35b)

where Ei is the component of the electric field in thed-dimensional homogeneous subspace. Due to the di-agonality of these matrices, only spatially homogeneousdynamical variables—i.e., electron and hole distributionfunctions fk

e5^ck†ck& and fk

h5^dk†dk&—and diagonal inter-

band polarizations pk5^d2kck& will be generated. Theysatisfy the following equations of motion:

d

dtfk

eu(sp)5e

\Ei•

]

]kfk

e21i\

E•~Mpk* 2M* pk!, (36a)

d

dtpku(sp)5

1i\

~eke1e2k

h !pk1e

\Ei•

]

]kpk

21i\

M•E~12fke2f2k

h !. (36b)

The intraband terms involving Ei describe accelerationof the carriers due to the electric field. It is the same fordiagonal and off-diagonal parts and it agrees with theBoltzmann drift term. The interband terms couple dis-tribution functions and polarization. Again, as discussedabove, the relevance of the various contributions de-pends on the frequency of the electromagnetic field. Foroptical fields the drift terms can usually be neglected,while for low-frequency fields the interband terms are ofminor importance. If a static or low-frequency field andan optical field are applied simultaneously, the drift termin Eq. (36b) gives rise to the static or dynamical Franz-Keldysh effect in the optical absorption (Franz, 1958;Keldysh, 1958; Jauho and Johnsen, 1996).

3. Homogeneous system, inhomogeneous excitation

If the spatially homogeneous system is excited by aspatially inhomogeneous electric field, the dynamicalvariables become inhomogeneous, and off-diagonal den-sity matrices have to be included. Due to the structure ofthe crystal Hamiltonian, a momentum representation isoften still useful, but a real-space representation can alsobe chosen. In both cases the general equations of motion(24) with the corresponding intraband and interband en-ergies have to be applied. An alternative approach is amixed momentum and real-space representation, the so-called Wigner representation (Wigner, 1932). This is par-ticularly useful for three purposes: (i) The intrabanddensity matrix in this representation has the closest simi-larity to the classical distribution function; therefore it isbest suited for a comparison to semiclassical kinetics de-scribed by the Boltzmann equation. (ii) For sufficientlyslowly varying spatial inhomogeneities a gradient expan-sion can be performed. (iii) Boundary conditions thatoccur in systems with open boundaries—for example,due to the coupling to contacts—can be incorporated(Frensley, 1990).


The single-particle density matrices in the Wigner rep-resentation are obtained by performing a Fourier trans-formation with respect to the relative momentum ac-cording to

fke~r!5(

qeiqr^ck2~1/2!q

† ck1~1/2!q&, (37a)

fkh~r!5(

qeiqr^dk2~1/2!q

† dk1~1/2!q&, (37b)

pk~r!5(q

eiqr^d2k1~1/2!qck1~1/2!q&. (37c)

Here, r is again a vector within the d-dimensional homo-geneous subspace. A lengthy but straightforward calcu-lation allows us to derive the new set of kinetic equa-tions in the Wigner picture; for the case of fe thederivation can be found in Hess and Kuhn (1996). Acharacteristic feature is that these kinetic equations arenonlocal in space. By performing a Taylor expansion, wecan formally transform the resulting equations into par-tial differential equations of infinite order according to

d

dtfk

e~r!51i\ (

n ,m50

` in1m

2n1mn!m! S ]

]k8•

]

]rDnS ]

]r8•

]

]kD m

3$@~21 !n2~21 !m#Ek8e

~r8!fke~r!

1~21 !nUk8~r8!pk* ~r!

2~21 !mUk8* ~r8!pk~r!%u

r85rk85k, (38a)

d

dtpk~r!5

1i\

Uk~r!11i\

3 (n ,m50

` in1m

2n1mn!m! S ]

]k8•

]

]rDnS ]

]r8•

]

]kD m

3$@~21 !nEk8e

~r8!1~21 !mE2k8h

~r8!#pk~r!

2~21 !nUk8~r8!f2kh ~r!

2~21 !mUk8~r8!fke~r!%u

r85rk85k, (38b)

where the space-dependent energies and effective fieldsare

Eke ,h~r!5(

qeiqrEk1~1/2!q,k2~1/2!q

e ,h 5eke ,h6eEi~r,t !•r,

(39a)

Uk~r!5(q

eiqrUk1~1/2!q,k2~1/2!q52M•E~r,t !. (39b)

In each order (n ,m) there are spatial derivatives of theorder n1m . If the length scales of the inhomogeneitiesare sufficiently large, then it can be expected that withincreasing order the contributions will be of decreasingimportance. Let us briefly discuss the structure of thelowest-order contributions.

The zeroth order (n5m50) is given by

d

dtfk

e~r,t !5d

dtf2k

h ~r,t !5gk~r,t !, (40a)


d

dtpk~r,t !5

1i\

@Eke~r,t !1E2k

h ~r,t !#pk~r,t !11i\

Uk~r,t !

3@12fke~r,t !2f2k

h ~r,t !# , (40b)

where the generation rate is

gk~r,t !51i\

@Uk~r,t !pk* ~r,t !2Uk* ~r,t !pk~r,t !# . (41)

Here, the spatial coordinate enters only as a parameter;locally, the dynamics coincide with those of the homoge-neous case and there are no transport effects. As weshall see in Sec. III, in many cases this lowest-order pic-ture is sufficient to describe pump-probe as well as four-wave-mixing experiments.

The first-order contributions (n51, m50 and n50,m51) are given by

d

dtfk

e(1)~r!51\ H 2

]Eke~r!

]k•

]fke~r!

]r1

]Eke~r!

]r•

]fke~r!

]k

2]Uk~r!

]k•

]pk* ~r!

]r1

]Uk~r!

]r•

]pk* ~r!

]k

2]Uk* ~r!

]k•

]pk~r!

]r1

]Uk* ~r!

]r•

]pk~r!

]k J .

(42)

The first two terms on the right-hand side (rhs) corre-spond to the Boltzmann drift terms in phase space. Theother terms can be interpreted as a local generation ratedue to the flow of polarization into or out of the phase-space element. Thus this first-order approximation levelcan be regarded as a two-band generalization of the con-ventional Boltzmann equation.

Higher-order terms then give rise to such typicallyquantum-mechanical features as tunneling.

D. Carrier-phonon interaction

Having considered the equations of motion derivedfrom the single-particle Hamiltonian, we now come tothe many-body contributions. Here, the equations ofmotion for the kinetic variables are no longer closed;instead they give rise to an infinite hierarchy of equa-tions.

Let us start with the case of carrier-phonon interac-tion. The corresponding Hamiltonian introduced in Eq.(17) leads to the following contributions to the equa-tions of motion:

d

dtfi1i2

e u(cp)51i\ (

i3 ,q@gq

i2i3sqe ,i1i31gq

* i3i2sqe ,i3i1*

2gqi3i1sq

e ,i3i22gq* i1i3sq

e ,i2i3* # , (43a)

d

dtpj1i1

u(cp)51i\ (

i2 ,q@gq

i1i2tq(1),j1i21gq

* i2i1tq(2),j1i2#

21i\ (

j2 ,q@gq

j1j2tq(1),j2i11gq

* j2j1tq(2),j2i1# .

(43b)


The equation of motion for the phonon distributionfunction has the same structure:

d

dtnqu(cp)52

1i\ (

i1 ,i2

@gqi1i2sq

e ,i1i22gq* i1i2sq

e ,i1i2* #

11i\ (

j1 ,j2

@gqj1j2sq

h ,j1j22gq* j1j2sq

h ,j1j2* # . (44)

Here, new variables, the so-called phonon-assisted den-sity matrices, have been introduced (Zimmermann,1990):

sqe ,i1i25^ci1

† bqci2&, sq

h ,j1j25^dj1

† bqdj2&, (45a)

tq(1),j1i15^dj1

bqci1&, tq

(2),j1i15^dj1bq

†ci1&. (45b)

These variables describe correlations between carriersand phonons. The quantity sq

e ,i1i2 , for example, relatesan initial state with one electron in the electronic one-particle state i2 and a phonon with wave vector q to afinal state with only an electron in i1 . Thus its temporalevolution contains information on an electronic transi-tion from i2 to i1 by phonon absorption as well as thereverse process by phonon emission. The equations ofmotion for the above phonon-assisted density matricesinvolve expectation values of four operators, and there-fore an infinite hierarchy of equations shows up. To ob-tain a solution, this hierarchy has to be truncated atsome level. As has been discussed in Sec. II.B, trunca-tion schemes based on different ideas have been pro-posed in the literature. Here we shall use a correlationexpansion based on the assumption that correlations in-volving an increasing number of carriers or phonons areof decreasing importance.

1. First order: Coherent phonons

The lowest order in the hierarchy is obtained by ne-glecting all correlations between carriers and phonons.This corresponds to a factorization according to

sqe ,i1i2'^ci1

† ci2&^bq&5f i1i2

e Bq , (46)

where we have introduced the coherent-phonon ampli-tude Bq5^bq&. It is easy to see that a nonvanishingcoherent-phonon amplitude is equivalent to a nonvan-ishing Fourier component of the lattice polarization andthus to a displacement of the ions (Scholz, Pfeifer, andKurz, 1993; Kuznetsov and Stanton, 1994). This is incontrast to the usual phonon occupation number, whichis determined by the ionic mean-square displacementonly.

On this approximation level the contributions ofcarrier-phonon interaction to the equations of motion ofthe single-particle density matrices can be expressed interms of nondiagonal energy renormalizations accordingto

d

dtfi1i2

e u(cp ,1)51i\ (

i3

@Ei2i3

e(cp ,1)f i1i3

e 2Ei3i1

e(cp ,1)f i3i2

e # , (47a)


d

dtpj1i1

u(cp ,1)51i\ F(

j2

Ej1j2

h(cp ,1)pj2i11(

i2

Ei1i2

e(cp ,1)pj1i2G ,

(47b)

where the self-energy matrices are

Ei1i2

e(cp ,1)5(q

@gqi1i2Bq1gq

i2i1* Bq* # , (48a)

Ej1j2

h(cp ,1)52(q

@gqj1j2Bq1gq

j2j1* Bq* # , (48b)

while the equation of motion for the phonon amplitudeis given by

d

dtBq52ivqBq

11i\ F (

i1 ,i2

gq* i2i1f i1i2

e 2 (j1 ,j2

gq* j2j1f j1j2

h G . (49)

2. Second order: Scattering and dephasing

The next step in the hierarchy is obtained by takinginto account deviations of the phonon-assisted densitymatrices from the lowest-order factorization previouslyintroduced, e.g.,

dsqe ,i1i25sq

e ,i1i22f i1i2

e Bq . (50)

In order to determine the equation of motion for thisphonon-assisted correlation, we first derive the equationof motion for the corresponding phonon-assisted densitymatrix. Assuming a time-independent basis and neglect-ing the carrier-field part of the Hamiltonian, we obtain

d

dtsq

e ,i1i2521i\

~e i1

e 2e i2

e 2\vq!sqe ,i1i2

21i\ (

i3 ,q8@g

q8

i3i1^ci3

† ci2bq8bq&

1gq8

i1i3* ^ci3

† ci2bq8

† bq&2gq8

i2i3^ci1

† ci3bqbq8&

2gq8

i3i2* ^ci1

† ci3bqbq8

† &#

11i\ (

i3 ,i4

gqi3i4* ^ci1

† ci4

† ci3ci2

&

21i\ (

j3 ,j4

gqj3j4* ^ci1

† dj4

† dj3ci2

&. (51)

Thus the equation of motion involves expectation valuesof four operators: electron-phonon and electron-electron two-particle density matrices. In the spirit ofthe correlation expansion previously discussed, thesequantities have to be decomposed into all possiblelower-order factorizations, the remaining part describingtwo-particle correlations. Such a decomposition is givenby

^ci3

† ci2bq8

† bq&5f i3i2

e Bq8* Bq1ds

q8

e ,i2i3* Bq1dsqe ,i3i2Bq8

*

1f i3i2

e nqdq,q81d^ci3

† ci2bq8

† bq&. (52)


Here, as in the following, nq only refers to the distribu-tion of incoherent phonons defined as nq5^bq

t bq&2 uBqu2 and we have assumed that this distribution isspace independent, i.e., diagonal in q. A generalizationto the nondiagonal case is straightforward. The equationof motion for the phonon-assisted correlation is ob-tained by inserting these decompositions into Eq. (51)and by subtracting the equations of motion for the sec-ond term on the rhs of Eq. (50). Then, the first andsecond terms on the rhs of Eq. (52) cancel, the thirdleads to a renormalization of the single-particle energiesby the coherent-phonon contributions, the fourth leadsto the scattering part, and the last describes the influ-ence of two-particle correlations. Again, the hierarchycan be truncated by neglecting these higher-order corre-lations, which results in the following equation of mo-tion:

d

dtdsq

e ,i1i2521i\ (

i3 ,i4

~Ei3i1

e d i2i4

2Ei2i4

e d i1i32\vqd i1i3

d i2i4!dsq

e ,i3i4

11i\ (

i3 ,i4

gqi4i3* @~nq11 !f i1i4

e ~d i3i22f i3i2

e !

2nqf i3i2

e ~d i1i42f i1i4

e !#

21i\ (

j1 ,j2

gqj2j1* pj1i1

* pj2i2, (53)

where the renormalized energies are given by

Ei1i2

e 5e i1

e d i1i21E i1i2

e(cp ,1) . (54)

If the time dependence is calculated with the full single-particle Hamiltonian, the energy e i1

e is replaced by thefull single-particle energy matrix, as given in Eq. (24).Furthermore, the effective field leads to a coupling ofdifferent types of phonon-assisted correlations (Schilp,Kuhn, and Mahler, 1994a). Such equations for the fourtypes of phonon-assisted correlations, together with theequations for the single-particle density matrices, consti-tute the basis for the analysis of electron-phonon quan-tum kinetics. Results based on this theoretical approachwill be reviewed and discussed in Sec. III.G.

Again, the semiclassical limit is obtained by adiabaticelimination of the phonon-assisted correlations. This canbe performed following the same procedure described inSec. II.C for the carrier-light interaction. We shall againneglect all off-diagonal energy renormalizations. This isalso in the spirit of the Boltzmann theory, in which scat-tering processes occur between free-carrier states. Westress that when performing the Markov approximation,one must properly take into account the fast oscillationsof interband and intraband polarizations induced by H0 .Within this approximation scheme, the phonon-assistedcorrelation is given by


dsqe ,i1i252ip (

i3 ,i4

D~2e i4

e 1e i3

e 1\vq!gqi4i3*

3@~nq11 !f i1i4

e ~d i3i22f i3i2

e !

2nqf i3i2

e ~d i1i42f i1i4

e !#1ip

3 (j1 ,j2

D~e j1

h 2e j2

h 1\vq!gqj2j1* pj1i1

* pj2i2. (55)

If this phonon-assisted correlation is now inserted intothe equation of motion for the single-particle densitymatrix, it becomes evident that the principal-value partof D is associated with energy renormalizations—describing the polaron corrections to the bandstructure—while the d-function part is associated withirreversible scattering and dephasing processes. Typi-cally, the dominant polaronic features are a rigid shift ofthe bands and a slight modification of the effective mass.In this case, these effects can be included in H0 , sincethey are always present in any experiment determiningthe band structure. Therefore the principal-value contri-butions will be neglected hereafter. However, this deri-vation shows that in a quantum-kinetic treatment, thepolaron shift is always included. This must be taken intoaccount when comparing quantum-kinetic to semiclassi-cal results.

The second-order carrier-phonon contributions in theMarkov limit, for example, for the electron density ma-trix, can be written in the general form

d

dtfi1i2

e u(cp ,2)5(i3

@2G i2i3

e ,out(cp ,2)f i1i3

e 2G i1i3

e ,out(cp ,2)* f i2i3

e*

1G i2i3

e ,in(cp ,2)~d i1i32f i1i3

e !

1G i1i3

e ,in(cp ,2)* ~d i2i32f i2i3

e* !#

11i\ (

j1

@U i2j1

e(cp ,2)pj1i1* 2U i1j1

e(cp ,2)* pj1i2# .

(56)

The explicit form of the various matrices G(cp ,2) andU (cp ,2) due to the second-order carrier-phonon interac-tion appearing on the rhs of Eq. (56) can be found inKuhn (1998). In the next section the results for the spe-cial case of a homogeneous system in momentum repre-sentation will be given explicitly. In the fully semiclassi-cal limit, where all off-diagonal elements of theintraband density matrices and all interband polariza-tions are neglected, the well-known Boltzmann-like scat-tering contributions due to phonon emission and absorp-tion are recovered.

3. Homogeneous system

As in the case of single-particle dynamics, let us dis-cuss the special case of a homogeneous system that iseither homogeneously or inhomogeneously excited. Thefirst-order contributions in the momentum representa-tion are given by the self-energy matrices


Ekk8e ,h(cp ,2)

5gk2k8e ,h Bk2k81gk82k

e ,h* Bk82k* , (57)

and the generation of coherent phonons is described by

d

dtBq52ivqBq1

1i\ (

k@gq

e* fk,k1qe 2gq

h* fk,k1qh # .

(58)

When the system is homogeneously excited, onlycoherent-phonon amplitudes with q50 can be gener-ated. According to Eq. (58), however, this requires dif-ferent coupling matrix elements gq for electrons andholes. Thus no coherent phonons are generated for thecase of Frohlich interaction, in which (Madelung, 1978)

gq5gqe5gq

h5iF2pe2\vLO

V S 1«`

21«s

D G1/2 1q

, (59)

where «` is the optical dielectric constant, «s is the staticdielectric constant, and vLO is the LO phonon fre-quency. This is usually the most important type ofcarrier-phonon interaction for ultrafast dynamics in po-lar semiconductors. The carrier-phonon self-energy inWigner representation depends on r only. In the case ofFrohlich interaction, it can be written as E e ,h

56eF(r,t), where the electrostatic potential is deter-mined by the Poisson equation DF54p¹•P(r,t), andthe lattice polarization is given by

Plat5F\vLO

8pV S 1«`

21«s

D G1/2

(q

qq

eiqr~Bq2B2q* !. (60)

It is interesting to analyze the long-wavelength limit ofEq. (58): The matrix element diverges as q21. Under thecondition

(k

fkke 5(

kfkk

h , (61)

i.e., the condition of charge neutrality, this divergencecancels in Eq. (49), leading to a finite value of B0 whichis equivalent to a phononic dipole moment of the struc-ture. According to classical electrodynamics, such a di-pole moment, when oscillating, acts as a source of elec-tromagnetic radiation typically in the terahertz range.

The second-order contributions, in the case of homo-geneous excitation, can be written in a slightly differentway, which can be more easily interpreted on physicalgrounds. By introducing proper transition matricesWe ,h(cp) and W e ,h(cp) as well as effective fields U e ,h(cp),we obtain

d

dtfk

eu(cp ,2)52(q

@Wk2q,ke(cp) fk

e~12fk2qe !

2Wk,k2qe(cp) fk2q

e ~12fke !#

11i\

@Uke(cp ,2)pk* 2Uk

e(cp ,2)* pk# , (62a)


d

dtpku(cp ,2)52(

qF ~W k2q,k

e(cp) 1W 2(k2q),2kh(cp) !pk

2S gqh

gqe W k,k2q

e(cp) 1gq

e

gqh W 2k,2(k2q)

h(cp) D pk2qG ,

(62b)

where the transition matrices are

Wk2q,ke ,h(cp)5

2p

\ (6

ugqe ,hu2d~ek2q

e ,h 2eke ,h6\vq!

3S nq112

612 D , (63)

W k2q,ke ,h(cp)5

p

\ (6

ugqe ,hu2D~ek2q

e ,h 2eke ,h6\vq!

3F S nq112

712 D fk2q

e ,h

1S nq112

612 D ~12fk2q

e ,h !G , (64)

and the effective field is

U ke ,h(cp)5ip(

qgq

egqh* (

6~6pk2q!D~ek

h ,e2ek2qh ,e 6\vq!.

(65)

Here, we have assumed gqe/gq

h5gqe* /gq

h* , which holdsboth for deformation-potential and polar-couplingmechanisms. In the equations of motion for the distribu-tion functions, we thus recover the usual Boltzmannscattering term consisting of in- and out-scattering con-tributions due to phonon emission and absorption. Ifthere is an interband polarization in the system, there isan additional contribution, which is quadratic in pk . Theprefactor shows that this term is due to the simultaneousinteraction of electrons and holes with the same phononmode. This type of process is sometimes called polariza-tion scattering (Kuznetsov, 1991). On the other hand, ac-cording to its structure, it can also be regarded as a co-herent generation-recombination process, since it hasthe same structure as the generation rate in Eq. (26).However, (i) the effective fields Uk

e ,h(cp) are different forelectrons and holes, and (ii) it conserves the number ofelectrons and holes.

In the equation for polarization, we also find two dif-ferent types of contributions. We have a loss term pro-portional to pk , due to processes involving eitherelectron-phonon or hole-phonon interaction. Thereal part of the corresponding matrices is related to theBoltzmann scattering matrices by

Re W k2q,ke ,h(cp)5

12

@Wk2q,ke ,h(cp)~12fk2q

e ,h !1Wk,k2qe ,h(cp)fk2q

e ,h # .

(66)

Thus a scattering process of a carrier both into and outof the state k leads to a loss of interband coherence.However, there is an additional contribution that mayreduce this loss of coherence, which is again related to


the simultaneous interaction of electrons and holes withthe same phonon. If, as in the case of Frohlich interac-tion, both matrix elements coincide, we obtain

d

dt (kpku(cp ,2)50. (67)

In this case dephasing is the net result of two oppositecontributions, whose relative magnitude is dictated bythe k dependence of both the single-particle energiesand the scattering matrix elements. In particular, it de-pends strongly on the energy transfer in the scatteringprocess.

When the system is inhomogeneously excited and aWigner representation is chosen, a gradient expansioncan also be performed for the second-order contribu-tions. Here, however, only the zeroth order is usuallytaken, which means that all variables in Eq. (62) dependparametrically on the space coordinate r. This is in thespirit of Boltzmann’s Stoßzahlansatz, in which scatteringprocesses are treated as pointlike in space and time. Asfor the Markov approximation, in which energy-time un-certainty is neglected, here the position-momentum un-certainty is neglected, and a scattering process betweenwell-defined momentum states occurs at some well-defined position.

4. Third order: Collisional broadening

The correlation expansion can be continued by takinginto account two-particle correlations. In this section webriefly discuss the structure of these contributions. How-ever, to simplify the notation we shall limit ourselves tothe case of a homogeneous single-band system. Its gen-eralization to the multiband case with arbitrary basisfunctions is straightforward.

The equations of motion for two-particle correlationsinvolve expectation values of five operators, the two-particle phonon-assisted density matrices. Like thephonon-assisted density matrices, these expectation val-ues contain information not only on two-phonon emis-sion or absorption processes, but also on virtual pro-cesses related to the emission and reabsorption ofphonons. The hierarchy is truncated by a factorizationinto single-particle and phonon-assisted density matri-ces. The resulting equation of motion, for example, forthe correlation appearing in Eq. (52), is of the form

d

dtd^ck1q2q8

† ckbq8† bq&

521i\

~ek1q2q8e

2eke !d^ck1q2q8

† ckbq8† bq&

21i\

gq8~12fk1q2q8e

1nq8!dsk1q,q,ke

11i\

gq8~fke1nq!dsk1q2q8,q,k2q8

e

21i\

gq* ~12fke1nq!dsk1q,q8,k1q2q8

e*

11i\

gq* ~fk1q2q8e

1nq8!dsk,q8,k2q8e* . (68)


For a further simplification of this contribution, we keepin mind that Eq. (51) involves a summation over q8.Since the phonon-assisted correlation is a complex quan-tity, in a first approximation we can assume that all con-tributions involving a summation of this quantity aresmall due to random phases at different momenta. Thisallows us to neglect the last three terms in Eq. (68),which can now be formally solved according to

d^ck1q2q8† ckbq8

† bq&

521i\

gq8E0

`

dtei(ek1q2q8e

2eke)t/\

3@12fk1q2q8e

~ t2t!1nq8~ t2t!#dsk1q,q,ke ~ t2t!.

(69)

After inserting this result in Eq. (51), we obtain a closedequation of motion for the phonon-assisted density ma-trix, which, however, contains a memory term. This canbe eliminated by again performing a Markov approxi-mation. As a result, the third-order contributions giverise to second-order self-energy corrections in the equa-tions of motion for ds according to

d

dtdsk1q,q,k

e 521i\

~E k1qe(cp ,2)2E k

e(cp ,2)* !dsk1q,q,ke ,

(70)

where the complex second-order carrier-phonon self-energy is

E ke(cp ,2)52ip (

q8,6ugq8u

2D~eke2ek1q8

e6\vq!

3@~nq8112 6 1

2 !fk1q8e

~nq8112 7 1

2 !~12fk1q8e

!# .

(71)

Here, the real part of the self-energy describes the factthat the scattering processes occur between renormal-ized polaronic states, while the imaginary part describesa collisional broadening. In the Green’s-function ap-proach this collisional broadening appears in the re-tarded Green’s function if the two-time functions are re-duced to one-time functions by means of the generalizedKadanoff-Baym ansatz (Lipavsky et al., 1986) and if aMarkov approximation for the retarded Green’s func-tion is performed (Haug, 1992; Tran Thoai and Haug,1993). However, it turns out that this approximationleads to a strong overestimation of the role played bythe two-particle correlations. In particular, the imaginarypart leads to a violation of energy conservation (Schilpet al., 1995) as well as to an overestimation of the broad-ening. It has been shown that a non-Markovian decay ofthe retarded Green’s function—corresponding to thenon-Markovian dynamics of the phonon-assisted corre-lation [Eq. (69)]—improves the result (Haug and Ban-yai, 1996). However, as has been shown for carrier-carrier scattering, it does not completely restore theconservation of the total energy (Bonitz, Semkat, andHaug, 1999). In contrast, if all third-order terms aretaken into account, it can be shown analytically that thetotal energy is conserved.


E. Carrier-carrier interaction

Many optical properties of semiconductors arestrongly influenced by Coulomb interaction. Thereforelet us now discuss the contributions to the system dy-namics due to the carrier-carrier interaction Hamil-tonian Hcc [Eq. (16)].

The corresponding term for the electron density ma-trix, is

d

dtfi1i2

e u(cc)51i\ (

i3 ,i4 ,i5

@Vi2i3i4i5

ee Ki1 ,i3 ,i4 ,i5

2Vi5i4i3i1

ee Ki5 ,i4 ,i3 ,i2#

21i\ (

i3 ,j1 ,j2

@Vi2j1j2i3

eh Ni1 ,j1 ,j2 ,i3

2Vi3j2j1i1

eh Ni3 ,i2 ,j1 ,i2# . (72)

It involves two-particle density matrices like

Ki1 ,i2 ,i3 ,i45^ci1

† ci2

† ci3ci4

& ,

Ni1 ,j1 ,j2 ,i25^ci1

† dj1

† dj2ci2

& . (73)

Functions involving other combinations of four fermionoperators appear in the equations for the hole densitymatrix and the interband polarization. The variableKi1 ,i2 ,i3 ,i4

, for example, is related to a transition of twoelectrons from the initial states i3 and i4 to the finalstates i2 and i1 , i.e., an electron-electron scattering pro-cess, but it also contains information on the joint occu-pation probabilities of two states, e.g., i15i4 and i25i3 . Again, Eq. (72) constitutes the starting point of aninfinite hierarchy of equations of motion for density ma-trices with an increasing number of carriers. As antici-pated, this is the quantum-mechanical analog of the clas-sical BBGKY hierarchy, in which the equation of motionfor an N-particle distribution function involves the (N11)-particle distribution functions (Bogoliubov, 1967;Carruthers and Zachariasen, 1983).

1. First order: Excitons and renormalization

The lowest-order contribution due to carrier-carrierinteraction is obtained by factorizing the two-particledensity matrices into single-particle ones,

Ki1 ,i2 ,i3 ,i45f i1i4

e f i2i3

e 2f i1i3

e f i2i4

e . (74)

This corresponds to the Hartree-Fock or mean-fieldlevel, in which all correlations between the carriers areneglected. The corresponding equations of motion aregiven by


d

dtfi1i2

e u(cc ,1)51i\ (

i3

@Ei2i3

e(cc ,1)f i1i3

e 2Ei3i1

e(cc ,1)f i3i2

e #

11i\ (

j1

@U i2j1

(cc ,1)pj1i1* 2U i1j1

(cc ,1)* pj1i2# ,

(75a)

d

dtpj1i1

u(cc ,1)51i\ F(

j2

Ej1j2

h(cc ,1)pj2i11(

i2

Ei1i2

e(cc ,1)pj1i2G1

1i\ FU i1j1

(cc ,1)2(i2

U i2j1

(cc ,1)f i2i1

e

2(j2

U i1j2

(cc ,1)f j2j1

h G , (75b)

where the self-energy matrices due to Hartree and Fockcontributions are

Ei1i2

e(cc ,1)52 (i3 ,i4

Vi1i3i2i4

ee f i3i4

e 1 (i3 ,i4

Vi1i3i4i2

ee f i3i4

e

2 (j3 ,j4

Vi1j3j4i2

eh fj3j4

h , (76a)

Ej1j2

h(cc ,1)52 (j3 ,j4

Vj1j3j2j4

hh fj3j4

h 2 (i3 ,i4

Vi3j1j2i4

eh fi3i4

e

1 (j3 ,j4

Vj1j3j4j2

hh fj3j4

h , (76b)

and the internal-field matrix due to the Fock contribu-tions of the electron-hole interaction is

Ui1j1

(cc ,1)52 (i2 ,j2

Vi1j1j2i2

eh pj2i2. (77)

Thus we again obtain the same structure of equations asin the single-particle case, but with renormalized ener-gies and fields. Here, the self-energies describe band-gap-renormalization effects as well as induced poten-tials, while the internal field gives rise to excitoniceffects and Coulomb enhancement. The single-particleequations including first-order carrier-carrier contribu-tions are usually called semiconductor Bloch equations.They were derived in real-space representation by Huhnand Stahl (1984) and in momentum representation bySchmitt-Rink and Chemla (1986) and Lindberg andKoch (1988a).

2. Second order: Scattering and dephasing

As in the case of carrier-phonon interaction, the nextstep in the correlation hierarchy is obtained by includingtwo-particle correlations like

dKi1 ,i2 ,i3 ,i45Ki1 ,i2 ,i3 ,i4

2f i1i4

e f i2i3

e 1f i1i3

e f i2i4

e , (78)

which describe deviations from the corresponding fac-torizations. The equations of motion for these quantitiesinvolve three-particle density matrices. The hierarchycan be truncated by factorizing the three-particle density


matrices into products of three single-particle densitymatrices, resulting in equations like

i\d

dtdKi1 ,i2 ,i3 ,i4

5~e i4

e 1e i3

e 2e i2

e 2e i1

e !

3dKi1 ,i2 ,i3 ,i41See1Seh, (79)

where the source terms See and Seh involve only single-particle density matrices. The intraband term

See5 (i5 ,i6 ,i7 ,i8

~Vi5i6i7i8

ee 2Vi6i5i7i8

ee !

3@f i1i8

e f i2i7

e ~d i6i32f i6i3

e !~d i5i42f i5i4

e !

2~d i1i82f i1i8

e !~d i2i72f i2i7

e !f i6i3

e f i5i4

e # (80a)

exhibits a structure similar to carrier-carrier scatteringterms in the Boltzmann equation, but in an off-diagonalgeneralization. The interband term

Seh5 (i5 ,i6 ,j1 ,j2

Vi5j1j2i6

eh @pj1i1* pj2i4

~d i2i6f i5i3

e 2f i2i6

e d i5i3!

1pj1i2* pj2i3

~d i1i6f i5i4

e 2f i1i6

e d i5i4!

2pj1i1* pj2i3

~d i2i6f i5i4

e 2f i2i6

e d i5i4!

2pj1i2* pj2i4

~d i1i6f i5i3

e 2f i1i6

e d i5i3!# (80b)

gives rise to polarization scattering; it modifies the scat-tering processes as long as there are coherent interbandpolarizations present. The set of equations for two-particle correlations constitutes the starting point for thestudy of carrier-carrier quantum kinetics in the density-matrix formalism. Some recent results on this topic willbe reviewed in Sec. III.H.

The semiclassical limit is then obtained by adiabaticelimination of the two-particle correlations, exactly as inthe case of the phonon-assisted correlations discussed inSec. II.D. The result can be cast into the same form as inEq. (56), but with different functions G(cc ,2) and U (cc ,2)

replacing G(cp ,2) and U (cp ,2); its explicit form can againbe found in Kuhn (1998). In the next section we shallexplicitly discuss the carrier-carrier contributions in mo-mentum representation.


In a homogeneous material the Coulomb matrix ele-ments depend on the momentum transfer q only. Herewe shall also assume that the three matrix elements cor-responding to electron-electron, hole-hole, and electron-hole interactions are equal. This is true in bulk semicon-ductors, where

Vq54pe2

V«rq2 , (81)

but it also holds in low-dimensional structures if the con-finement wave functions for electrons and holes are thesame, i.e., in the limit of infinitely high barriers. For fi-nite barriers the matrix elements become significantly


different for q values larger than the inverse confine-ment length. However, since the small-wavelength be-havior dominates in most of the features discussed here,the assumption of equal matrix elements is often a goodapproximation, even in this case. The value of «r de-pends on the treatment of the phonon dynamics: ifphonon-induced correlations due to the Frohlich inter-action are taken into account dynamically, it coincideswith the optical dielectric constant; otherwise, the staticvalue has to be taken.

The first-order contribution in momentum representa-tion is given by the intraband and interband energies,

E kk8e ,h(cc ,1)

52(q

@Vqfk81q,k1qe ,h

7Vk2k8

3~fq,q2k81ke

2fq,q2k81kh

!# , (82a)

U kk8(cc ,1)

52(q

Vqp2k82q,k1q , (82b)

with Fock (;Vq) and Hartree (;Vk2k8) terms. Herethe upper sign refers to electrons and the lower sign toholes. If the system is homogeneously excited, the Har-tree terms cancel due to charge neutrality. The intrabandFock terms describe band-gap renormalizations, whilethe interband terms describe Coulomb correlations be-tween electrons and holes. In particular, they give rise toexcitonic features in optical spectra.

In a Wigner representation the first-order terms canbe written as

E ke ,h(cc ,1)~r!52(

k8Vk2k8fk8

e ,h~r!7eF~r!, (83a)

Uk~r!52(k8

Vk2k8pk8~r!, (83b)

where the induced potential satisfies the Poisson equa-tion

¹2F~r!524p

«rr~r!5

4pe

«rV (k

@fke~r!2fk

h~r!# , (84)

r(r) being the charge density.The second-order contributions in the case of homo-

geneous excitation are given by

d

dtfk

eu(cc ,2)52(q

@Wk2q,ke(cc ,2)fk

e~12fk2qe !

2Wk,k2qe(cc ,2)fk2q

e ~12fke !#

11i\

@U ke(cc ,2)pk* 2U k

e(cc ,2)* pk# , (85a)

d

dtpku(cc ,2)52(

q@~W k2q,k

e(cc ,2)1W 2(k2q),2kh(cc ,2) !pk

2~W k,k2qe(cc ,2)1W 2k,2(k2q)

h(cc ,2) !pk2q# , (85b)

where the transition matrices are


Wk2q,ke ,h(cc ,2)5

p

\uVqu2 (

n85e ,h(k8

D~ek2qe ,h 1ek81q

n8

2ek8n82ek

e ,h!@fk8n8~12fk81q

n8 !2pk81q* pk8#

1c.c., (86)

W k2q,ke ,h(cc ,2)5

p

\uVqu2 (

n85e ,h(k8

D~ek2qe ,h 1ek81q

n8

2ek8n82ek

e ,h!@2pk81q* pk8

1fk8n8~12fk81q

n8 !~12fk2qe ,h !

1fk2qe ,h fk81q

n8 ~12fk8n8!# , (87)

and the effective field is

U ke ,h(cc ,2)5ip(

k8,q(

n85e ,huVqu2D~ek

h ,e1ek8n8

2ek81qn8 2ek2q

h ,e !@fk81qn8 2fk8

n8#pk2q . (88)

The structure of the equations is the same as in the caseof carrier-phonon interaction, and therefore everythingthat has been stated above also holds in this case. Themain difference is that now the transition matrices [Eq.(86)] no longer coincide with the Boltzmann transitionmatrices. Instead, we have additional contributions dueto the interband polarization, which, in general, removethe positive-definiteness and consequently the possibilityof interpreting such terms as transition rates. Only whenthe polarization has decayed are the Boltzmann ratesrecovered. For the polarization we again obtain contri-butions with the structure of out-scattering (;pk) andin-scattering (;pk2q) terms, which, due to equal Cou-lomb matrix elements, satisfy the sum rule

d

dt (kpku(cc ,2)50. (89)

This compensation between in- and out-scattering termsis essential in order to reproduce a physically reasonabledensity dependence of carrier-carrier scattering-induceddephasing (Rossi, Haas, and Kuhn, 1994; Haas, Rossi,and Kuhn, 1996).

4. Second order: Screening

In the previous section the carrier-carrier scatteringterms were derived in the second Born approximation.While this gives well-defined, finite results on aquantum-kinetic level (El Sayed, Banyai, and Haug,1994), it is well known that in the semiclassical limit thetotal scattering rate for a bare Coulomb potential di-verges due to the long-range nature of this interaction.Usually, this divergence is removed by taking a screenedCoulomb potential whose screening is described, for ex-ample, by the Lindhard formula. This dielectric functionis obtained by studying the response of the carrier sys-tem to an external potential within the random-phaseapproximation (Haug and Koch, 1993), i.e., in the


present case, from the first-order carrier-carrier contri-butions. However, as shown by several authors in thefield of plasma physics in the early 1960s (Lenard, 1960;Balescu, 1961; Guernsey, 1962; Wyld and Fried, 1963),screening appears self-consistently within the second-order density-matrix approach. Some recent derivationsapplied to condensed-matter physics can be found in thework of Hohenester and Potz (1997) and Bonitz (1998).Here we shall briefly review the derivation of the dy-namically screened scattering rates by essentially follow-ing Wyld and Fried (1963) and restricting the discussionto the homogeneous single-band case.

The equation of motion for the distribution functionin a one-band model is given by

i\d

dtfk5(

k8,qVq@dKk,k8,k81q,k2q2dKk2q,k81q,k8,k# .

(90)

The equation of motion for the two-particle correlationsis obtained in the same way as in the previous section,leading to three-particle density matrices. While therethe three-particle correlations were factorized intosingle-particle density matrices only, now all factoriza-tions into lower-order correlations are included. Thusadditional contributions appear due to a factorization ofa three-particle density matrix into a distribution func-tion times a two-particle correlation. Neglecting thethree-particle correlations, we obtain the resulting equa-tion of motion:

i\d

dtdKk,k8,k81q,k2q5~Ek2q1Ek81q2Ek82Ek!

3dKk,k8,k81q,k2q1(i51

5

Si , (91)

where the renormalized energies are Ek5ek2(qVqfk2q . A series of five terms that describe differ-ent physical phenomena results from the factorization(Wyld and Fried, 1963).

S1 leads to the scattering in Born approximation asdiscussed in the previous section:

S15~Vq2Vk2k82q!@fkfk8~12fk81q!~12fk2q!

2fk2qfk81q~12fk8!~12fk!# ; (92a)

S2 gives rise to the screening of the Coulomb potentialin the random-phase approximation (RPA);

S25VqF ~fk2fk2q!(k9

dKk9,k8,k81q,k92q

1~fk82fk81q!(k9

dKk,k9,k91q,k2qG ; (92b)

S3 contains exchange corrections to the screening, whichare necessary to satisfy the correct antisymmetry of dKupon exchange of two particles:


S352Vk2k82qF ~fk82fk2q!(k9

dKk,k9,k91k2k82q,k81q

1~fk2fk81q!(k9

dKk9,k8,k2q,k92k1k81qG ; (92c)

S4 describes the repeated scattering of two particlesfrom each other and leads to the exact T matrix:

S45~12fk2q2fk81q!(q8

Vq8dKk,k8,k81q1q8,k2q2q8

2~12fk2fk8!(q8

Vq8dKk2q8,k81q8,k81q,k2q ;

(92d)

and, finally, S5 contains terms that in the Green’s-function language would be called vertex corrections tothe screening terms S2 and S3 :

S552~fk2fk2q!(q8

Vq8dKk2q8,k8,k81q,k2q2q8

2~fk82fk81q!(q8

Vq8dKk,k81q8,k81q1q8,k2q

2~fk82fk2q!(q8

Vq8dKk,k81q8,k81q,k2q1q8

2~fk2fk81q!(q8

Vq8dKk2q8,k8,k81q2q8,k2q .

(92e)

Due to the divergence of Vq for small q, the directterm in S1 and the term S2 are expected to dominate,since they involve no summation over the Coulomb ma-trix element. Keeping only these contributions is equiva-lent to the random-phase approximation. The solutionof Wyld and Fried is based on the observation that, inthis case, Eq. (91) can be factorized in the followingsense: If the operator Fk,q5ck2q/2

† ck1q/2 satisfies the op-erator analog of the linearized Vlasov equation, i.e.,

d

dtFk,q5

1i\

~Ek1q/22Ek2q/2!Fk,q

21i\

Vq~fk1q/22fk2q/2!(k8

Fk8,q , (93)

then Eq. (91) with only the direct part of S1 and S2 is adirect consequence of Eq. (93). Such a factorizationproperty is also the basis for the derivation of screeningin Hohenester and Potz (1997). By solving Eq. (93) inthe Markov approximation and assuming an initially un-correlated system, we can calculate the two-particle cor-relation. With this result Eq. (90) reads

d

dtfk52

2p

\ (k8,q

U Vq

«@q,~Ek2q2Ek!/\#U2

3d~Ek2q1Ek81q2Ek82Ek!@fkfk8~12fk81q!

3~12fk2q!2fk2qfk81q~12fk8!~12fk!# . (94)


This corresponds to the Boltzmann equation with theCoulomb potential dynamically screened by theLindhard dielectric function,

«~q,v!512Vq(k

fk2fk2q

Ek2Ek2q2\v. (95)

Without the Markov approximation, Eq. (91) with thecontributions S1 and S2 serves as a starting point for aquantum-kinetic investigation of carrier-carrier scatter-ing, including the buildup of screening at early times.

5. Third order: Collisional broadening

The next order in the hierarchy is obtained in thesame way as in the case of carrier-phonon interaction:The equations of motion for three-particle correlationshave to be set up and the resulting four-particle densitymatrices have to be factorized into all kinds of lower-order terms. Among the many possible factorizationsthere is one class of terms having the structure of self-energy corrections to the second-order equation, thusresulting in second-order energy renormalizations andcollisional broadening terms in the equation for the two-particle correlation. Here, however, due to the strongdominance of small-angle scattering, particularly at lowdensities, this approximation overestimates the broaden-ing in a much more dramatic way than in carrier-phononscattering. Such a strong overestimation of the colli-sional broadening due to carrier-carrier scattering whenin-scattering terms are neglected has been studied byRossi et al. (1994) and Haas et al. (1996) for the case ofcarrier photogeneration. In particular, it has been shownhow the inclusion of additional terms with the structureof in-scattering contributions leads to the correct physi-cal behavior. Within the Green’s-function approach astrong violation of energy conservation and unphysicallylarge broadening resulting from an exponentially decay-ing memory function—corresponding to a Lorentzianspectral function—have been demonstrated (Bonitzet al., 1999). In the same paper the authors show thatagain the replacement of the exponential by a hyper-bolic secans improves the results but does not restoreenergy conservation. On the other hand, neglecting thedecay of the retarded Green’s function—which isequivalent to taking the second-order density-matrixtheory—is in surprisingly good agreement with the re-sults of a two-time Green’s-function calculation.

6. Coulomb interaction in doped semiconductors

In the previous sections we have derived equations ofmotion for the case of an intrinsic (or undoped) semi-conductor. If, in contrast, the semiconductor is doped,the doping impurities can provide or accept electrons orholes, thereby changing their charge state. In this casethe Coulomb interaction with the carriers bound at im-purities has to be taken into account as well. In general,this interaction contributes at each order to our equa-tions of motion. The first-order or Hartree term de-scribes the electrostatic potential produced by thecharged impurities. It is necessary to maintain charge


neutrality. In a homogeneous system it is a divergentterm at q50 which exactly cancels the divergence of theother Coulomb contributions. In inhomogeneous sys-tems it is responsible for built-in fields. Examples of thiscase are modulation-doped quantum-well structures inwhich impurities provide conduction electrons to thequantum wells. The second-order terms give rise tocarrier-impurity scattering processes that contribute tomomentum relaxation, and they may also lead to Auger-or impact-ionization-like transitions between free andbound carrier states. In this section we shall address aspecific case which is important if optical transitions be-tween free and bound states are involved. Experimentsbased on such transitions, in particular the study ofband-to-acceptor luminescence, have provided valuableinformation on ultrafast carrier dynamics in optically ex-cited semiconductors, as will be discussed in more detailin Sec. III.A.3.

Let us consider a homogeneous semiconductor that isslightly p-doped with shallow acceptors. At zero tem-perature the Fermi level is in the middle between thetop of the valence band and the acceptor level. Thus atsufficiently low temperatures the acceptors are unoccu-pied, and it is convenient to treat the acceptor occupa-tion in an electron picture. We thus denote by si

† and sithe creation and destruction of an electron at the accep-tor position ri . Within the hydrogenic impurity modelthe wave function of the acceptor state is given by

c~r!5uV~r!f~r2ri!, (96)

where uV denotes the periodic part of the Bloch func-tion at the top of the valence band and f is the 1s hy-drogen wave function. If we again neglect Coulombterms leading to transitions between impurity levels andfree carrier states, which for the purpose studied hereare not important, we have three additional contribu-tions to the Coulomb interaction Hamiltonian, due toelectron-acceptor, hole-acceptor, and acceptor-acceptorinteractions. For sufficiently low doping concentrationand negligible occupation of the acceptor levels, theacceptor-acceptor interaction is not important, and theHamiltonian is given by

Hcca 5Hcc

ea1Hccha

5 (k,q,i

Vqi ck2q

† si†sick2 (

k,q,iVq

i dk2q† si

†sidk , (97)

where the Coulomb matrix elements are

Vqi 5

e2

V«rE drdr8uf~r82ri!u2

eiqr

ur2r8u, (98)

which in the case of the 1s wave function reduces to

Vqi 5Vqeiqribq5Vqeiqri@11~ 1

2 qaB!2#22. (99)

Here, Vq is the Coulomb matrix element of the homo-geneous semiconductor [Eq. (81)] and aB is the acceptorBohr radius.

The relevant dynamical variable to describe opticaltransitions between the conduction band and the impu-rity level is the band-to-acceptor polarization,


pka5(

ie ikri^si

†ck&5(i

e ikripki . (100)

The first-order contribution to its equation of motiondue to the complete (band and impurity) Coulomb in-teraction is given by

d

dtpk

au(cc ,1)521i\ (

qVq~fk1q

e pka2bqfk

epk1qa !. (101)

Here, the first term is due to the renormalization of theconduction band by electron-electron interaction, whilethe second term gives rise to Coulomb enhancement andexcitonic effects in the transition (we have again ne-glected terms involving acceptor occupations). Bothterms are proportional to the electron distribution func-tion and therefore negligible at low densities.

The second-order contribution is obtained in the sameway as the interband polarization. Here we shall giveonly the result in the semiclassical limit, which can bewritten in a form similar to Eq. (85b):

d

dtpk

au(cc ,2)52(q

@W k2q,ke(cc ,2)pk

a2bqW k,k2qe(cc ,2)pk2q

a #

2(q,i

@bq2W k2q,k

a(cc ,2)ei(k2q)ripki

2bqW k,k2qa(cc ,2)eikripk2q

i # , (102)

where the matrix W k,k2qe(cc ,2) is given in Eq. (87) and the

matrix W k,k2qa(cc ,2) is

W k,k2qa(cc ,2)5

p

\uVqu2 (

n5e ,h(k8

d~ek81qn

2ek8n

!

3fk8n

~12fk81qn

!. (103)

If the impurity positions are randomly distributed, thislast part vanishes. In contrast to interband polarization,here we do not have an exact symmetry between in- andout-scattering terms because the matrix elements forelectron-electron and electron-impurity interaction aredifferent. However, for q values smaller than the inverseBohr radius of the impurity we have bq'1 and the sym-metry is approximately recovered, leading again to astrong compensation between the two terms.

F. Carrier-photon interaction

Typically, the laser pulses used to excite semiconduc-tors in the ultrafast regime have a very high degree ofcoherence and are well described by a classical lightfield, as was done in the previous sections. However,there are certain features that can only be described interms of a quantum-mechanical treatment of the electro-magnetic radiation. Among these are all kinds of phe-nomena that affect the photon statistics, like the genera-tion of squeezed light by nonlinear optical techniques(Fox et al., 1995; Dabbicco et al., 1996) or the opticalStark effect with nonclassical light (Altevogt, Puff, andZimmermann, 1997). The most important process, how-


ever, which requires a quantum description of light, isspontaneous emission. Often the characteristic timescales for spontaneous emission are much longer thanthose for other interaction processes like carrier-carrieror carrier-phonon scattering. Therefore, for the model-ing of the ultrafast carrier dynamics, it can be neglected.However, in luminescence experiments the spontane-ously emitted photons are the quantities that are de-tected, and a theory is required that relates the proper-ties of the emitted radiation to the carrier dynamics. Inthis section we shall derive equations of motion describ-ing the rate of spontaneously emitted photons.

The quantized light field is described by creation anddestruction operators aqn

† and aqn for a photon withwave vector q and polarization component n. The free-photon Hamiltonian is then given by

Hl5(q,n

\vqaqn† aqn , (104)

where the dispersion relation is vq5cq , c being the ve-locity of light.

The exact form of the interaction Hamiltonian, as wellas the interpretation of the photon operators, dependson the choice of gauge. In the Coulomb gauge, by ne-glecting terms quadratic in the field, we obtain thecarrier-light Hamiltonian

Hcl5 (ij ,q,n

@mq,nij c i

†aqndj†1mq,n

ij* djaqn† ci# , (105)

where we have considered interband transitions only inthe rotating-wave approximation. The multipole formof the interaction can be obtained by performingthe Power-Zienau-Woolley transformation (Cohen-Tannoudji, Dupont-Roc, and Grynberg, 1989; Kira et al.,1999; Savasta and Girlanda, 1999), which results in thesame form of the interaction Hamiltonian but with dif-ferent matrix elements, and an additional field-independent term. This latter term—the dipole self-energy—has the same operator structure as the electron-hole interaction. For the present case (in which we areonly interested in the properties of the spontaneouslyemitted photons and neglect the feedback on the carrierdynamics) the choice of the gauge plays a minor role;the properties are essentially determined by the struc-ture of the carrier-light Hamiltonian [Eq. (105)]. A re-cent review of the quantum theory of carrier-light inter-action in semiconductors is that of Kira et al. (1999).

1. First order: Coherent electromagnetic fields

The structure of the above light-matter Hamiltonian issimilar to that of carrier-phonon interaction. Thereforethe structure of the resulting equations of motion is alsosimilar, the main difference being that carrier-phononinteraction is an intraband process, while carrier-photoninteraction is an interband process. To first order in thecorrelation expansion, coherent photon amplitudes areexcited according to


d

dtâqn&5ivqâqn&1

1i\ (

ijmq,n

ij* pji . (106)

These coherent amplitudes correspond to a classical co-herent electromagnetic field, and a description on thislevel coincides with the classical treatment of the lightfield in the previous sections. In particular, Eq. (106)shows that the interband polarization acts as a source fora classical light field.

2. Second order: Absorption and luminescence

Even in the absence of a coherent amplitude, photonscan be emitted. These incoherent photons are describedby the average photon occupation number,

Nqninc5âqn

† aqn&2uâqn&u2. (107)

In the absence of a coherent contribution, the rate ofchange of this variable determines the photon flux in thedirection q. This photon flux is obtained from the equa-tion of motion

d

dtNqn

inc521i\ (

ij@mqn

ij rqnji* 2mqn

ij* rqnji # , (108)

where we have introduced the incoherent photon-assisted density matrices

rqnji 5Š~aqn

† 2âqn* &!djci‹. (109)

Thus, in complete analogy with carrier-phonon interac-tion, these photon-assisted density matrices appear asnew dynamical variables. Their temporal evolution isdetermined by the equation of motion

d

dtrqn

j1i151i\

@e j1

h 1e i1

e 2\vq#rqnj1i1

21i\ (

i2 ,j2

mqni2j2~^ci2

† dj2

† dj1ci1

&2pj2i2* pj1i1

!

21i\ (

i2 ,q8,n8m

q8n8

i2j1Š~aqn

† 2âqn* &!ci2

† aq8n8ci1‹

21i\ (

j2 ,q8,n8m

q8n8

i1j2Š~aqn

† 2âqn* &!dj1aq8n8dj2

†‹.

(110)

Here, only the single-particle Hamiltonian in a time-independent basis has been used for the evolution of thecarrier operators. We shall address the role of interac-tion mechanisms in Sec. II.F.5. From Eq. (110) it can beseen that the source term for the emission of incoherentphotons (first term on the rhs) is a two-particle densitymatrix, the function N introduced in Eq. (73), where theinterband coherent part has been subtracted. However,it may still include intraband coherences. If, in accor-dance with our correlation-expansion scheme, all two-particle correlations are neglected, the terms on the rhscan be factorized into products of single-particle densitymatrices according to


d

dtrqn

j1i1u(cl)521i\ (

i2 ,j2

mqni2j2f i2i1

e f j2j1

h

11i\

~Nqninc1uâqn&u2!

3S 12(i2

mqni2j1f i2i1

e 2(j2

mqni1j2f j2j1

h D .

(111)

The first source term gives rise to spontaneous emission,while the second one gives rise to absorption and stimu-lated emission. In the following we shall neglect this sec-ond term, since we assume that the photons immediatelyleave the system, making their occupation number neg-ligible.

In the semiclassical limit, when the photon-assisteddensity matrices are adiabatically eliminated and onlydiagonal density matrices are taken into account, we getthe following rate of emitted photons:

d

dtNqn

inc52p

\ (ij

umqnij u2d~e j

h1e ie2\vq!f ii

e f jjh , (112)

i.e., the well-known Fermi’s golden rule result for whichthe luminescence intensity is proportional to the productof electron and hole distribution functions of the opti-cally coupled states.


The carrier-photon interaction Hamiltonian in mo-mentum representation is given by

Hcl5 (k,q,n

@mq,nck†aqnd2k1q

† 1mq,n* d2k1qaqn† ck# . (113)

Then, the rate of emitted photons in the semiclassicallimit reads

d

dtNqn

inc52p

\ (k

umqnu2d~e2k1qh 1ek

e2\vq!fkef2k1q

h .

(114)

In many cases the momentum of the photon is negligiblecompared to other characteristic momenta in the carrierdynamics, and replacing k2q by k on the rhs of Eq.(114) is a very good approximation.

4. Transitions between band and impurity states

The theory developed so far for interband transitionsis easily translated to transitions between band and im-purity states. The band-to-acceptor interaction Hamil-tonian reads

Hcl5 (i ,k,q,n

@mq,ngkck†aqnsi1mq,n* gksi

†aqn† ck# , (115)

where gk58(paB3 /V)1/2@11(kaB)2#22 is the Fourier

transform of the acceptor wave function. After factor-ization and under the assumption of a negligible accep-tor occupation, this yields for the spectrum


d

dtNqn

inc52p

\ (k

umqnu2ugku2d~eke2ea2\vq!fk

e , (116)

i.e., the same result as for the band-to-band case, inwhich the hole distribution function is replaced by thetime-independent quantity ugku2. Here, ea is the energyof the acceptor level in the electron picture. This showsthat band-to-acceptor luminescence spectra provide di-rect information on the electron distribution function.

5. Influence of other interaction mechanisms

In the previous sections only the contributions of thesingle-particle Hamiltonian to the equation of motionfor the photon-assisted density matrix have been takeninto account. Of course, like its semiclassical counter-part, the interband polarization, the photon-assisteddensity matrix will also be influenced by other interac-tion mechanisms. We can easily obtain the contributionsdue to these interaction Hamiltonians by noticing thatwe can write the equation of motion as

d

dtrqn

j1i1u(cc ,cp)5 K ~aqn† 2âqn* &!

d

dt~djci!u(cc ,cp)L . (117)

Thus we obtain exactly the same operator combinationsas in the case of interband polarization, which are onlymultiplied by a photon creation operator. After factor-ization, if this polarization is replaced by the photon-assisted density matrix, all terms linear in the interbandpolarization that have been derived in the previous sec-tions remain the same. Terms without an interband po-larization do not contribute, since we have neglectedphoton-induced intraband transitions in the Hamil-tonian. Terms with higher powers of the interband po-larization give additional contributions where one of thepolarizations is replaced by the photon-assisted densitymatrix. In many cases, however, particularly if lumines-cence from band states instead of exciton states is stud-ied, these terms are of minor importance.

The results obtained in the previous sections for theinterband polarization can now be directly applied tothe photon-assisted density matrix. The first-order con-tributions due to carrier-phonon and carrier-carrier in-teraction are obtained by simply adding the respectiveself-energies [Eqs. (48) and (76)] to the single-particleenergies in Eq. (110). In particular, in the case of a ho-mogeneous system we obtain (Kuhn and Rossi, 1992b)

d

dtrqn

k 51i\ (

k8akk8~vq!rqn

k821i\

mkfkef2k

h , (118)

where

akk8~vq!5F eke1e2k

h 2\v2(k9

Vk2k9~fk9e

1f2k9h

!G3dkk82~12fk

e2f2kh !Vk2k8 . (119)

This equation, supplemented by the respective stimu-lated terms as well as by a phenomenological dephasing


rate and combined with the respective equations forelectron, hole, and photon distribution functions, hasbeen called the semiconductor luminescence equation(Kira, Jahnke, and Koch, 1998).

The semiclassical luminescence spectrum is given by

d

dtNqn

inc51i\ (

k,k8$mkmk8

* @a21~vq!#kk8

2mk* mk8@a* 21~vq!#kk8%fk8e f2k8

h . (120)

With the bare Coulomb potential and in the absence ofband-gap renormalization, the inverse of the matrixa—including an infinitesimal imaginary part—corresponds to the exciton propagator, which can be cal-culated analytically. Taking into account screening,band-gap renormalization, and a finite dephasing rate,we can calculate the inverse numerically by using tech-niques that have been developed for the calculation ofquasiequilibrium absorption spectra (Schmitt-Rink,Lowenau, and Haug, 1982; Haug, 1988).

The second-order contributions due to carrier-phononand carrier-carrier interaction are given by

d

dtrqn

k u(2)52(q

~Wk2q,krqnk 2Wk,k2qrqn

k2q!, (121)

where the transition matrices are the same as in Eqs.(64) and (87). Thus the same compensation effects be-tween in- and out-scattering terms apply here as well.This will be discussed in more detail in Sec. III.A.2.

Including these terms, the equation of motion for thephoton-assisted density matrix again has the same formas in Eq. (118), but with the matrix a given by

akk8~vq!5F eke1e2k

h 2\vq2(k9

Vk2k9~fk9e

1f2k9h

!

2i\(k9

Wk9,kGdkk82~12fke2f2k

h !Vk2k8

1i\Wk,k8 . (122)

For sufficiently slowly varying distribution functions, thesemiclassical spectrum is again obtained by inverting thismatrix, which now includes broadening due to carrier-phonon and carrier-carrier scattering processes. A nu-merical solution of the equation of motion for thephoton-assisted density matrix yields the quantum-kinetic spectrum.

Band-to-acceptor spectra, including many-body anddephasing processes, can be obtained in the same way,by replacing the self-energies and transition matriceswith those derived in Sec. II.E.6 for the band-to-acceptor polarization.

G. Theoretical modeling of typical experiments

Virtually all experiments in the ultrafast time domainare carried out in the optical regime, which means thatthe semiconductor is optically excited by a short laser


pulse and the detected signal is again electromagneticradiation. Therefore the modeling of such experimentshas to take into account three basic features: the cre-ation of electronic excitations by the exciting light field,the subsequent carrier dynamics in the semiconductor,and the generation of the emitted electromagnetic radia-tion. The excitation of the semiconductor by a classicalelectromagnetic field has been treated in Sec. II.C, whilethe generation of coherent and incoherent radiation hasbeen described in Sec. II.F. For the modeling of the car-rier dynamics we have to distinguish between two cases:excitation by a single pulse, as is typical of luminescenceor terahertz-emission experiments, and excitation bytwo pulses traveling in different directions, as is the casein pump-probe and four-wave-mixing experiments.

If the semiconductor is excited by a single pulse with asufficiently large spatial extension, the equations of mo-tion as derived in Sec. II for the homogeneous case canbe directly applied. In this case the direction of the inci-dent pulse is the only preferred direction, and the coher-ent emission will also take place in this direction. Due todisorder, light scattering in other directions, the so-called Rayleigh scattering, is also possible. An incoherentemission of photons, on the other hand, may occur inany direction; this is usually called luminescence. Sincein many cases it is difficult to distinguish between thesetwo contributions, the more general name secondaryemission, including both phenomena, has become com-mon usage. Here we shall concentrate on some aspectsrelated to luminescence. Recent results on Rayleighscattering and on other aspects of the secondary emis-sion have been presented by Wang et al. (1995), Haackeet al. (1997), Birkedal and Shah (1998), Woerner andShah (1998), Garro et al. (1999), and Haacke et al.(2000).

Many luminescence experiments have been inter-preted on a purely incoherent basis. In this case the car-rier dynamics are completely described in terms of dis-tribution functions, and their temporal evolution isdictated by Boltzmann equations. The spectral and tem-poral shapes of the corresponding generation and emis-sion rates are obtained from Fermi’s golden rule. Withinthis approach the luminescence spectrum due to band-to-band transitions is a direct probe of the product ofelectron and hole distribution functions at the corre-sponding transition energy, while the band-to-acceptorspectrum directly monitors the electron distributionfunction. Because the Boltzmann equation is a rateequation, it is well suited to stochastic simulations basedon the well-known Monte Carlo method. Sophisticatedprograms involving a large variety of scattering mecha-nisms as well as details of the band structure have beendeveloped and applied to many experimental investiga-tions (Osman and Ferry, 1987; Goodnick and Lugli,1988; Stanton, Bailey, and Hess, 1988; Lugli et al., 1989;Rieger et al., 1989; Hohenester et al., 1993; Rota et al.,1993, 1995; Supancic et al., 1996). If coherence phenom-ena do not play an important role—as is typically thecase on not-too-short time scales—a good agreement be-tween theory and experiment has been achieved and


much has been learned about the characteristic times ofthe various scattering processes. To overcome the prob-lem of screening, which is particularly important forcarrier-carrier scattering processes, molecular-dynamicsschemes were developed that directly simulate in phasespace the dynamics of an ensemble of interacting elec-trons, while still treating the interactions with the latticewithin a Monte Carlo scheme. Compared to MonteCarlo simulations using a time-dependent quasistaticscreening model, the molecular-dynamics results showeda faster initial broadening at high carrier densities, whichis in better agreement with experiments (Hohenesteret al., 1993; Rota et al., 1993). However, as will be dis-cussed in more detail in Sec. III.A, it turns out that forthis initial broadening the inclusion of coherence phe-nomena may be essential.

Coherence in the photogeneration process is takeninto account by treating the interband polarization as anindependent dynamical variable. If the scattering pro-cesses are treated on a semiclassical level, it is still pos-sible to use a generalized Monte Carlo approach to treatthe scattering terms in the equations for the distributionfunctions (Kuhn and Rossi, 1992a, 1992b) or in theequations for the polarization (Rossi et al., 1994; Haaset al., 1996), but direct-integration methods have alsobeen widely used (Rappen, Peter, and Wegener, 1994;Jahnke and Koch, 1995; Jahnke et al., 1996; Potz, 1996b;Joschko et al., 1997).

Coherence and correlation phenomena in the photo-emission process are included by taking into account thephoton-assisted density matrices introduced in Sec. II.F.As discussed there, on this level excitonic effects, Cou-lomb enhancement, and renormalization, as well asbroadening due to scattering processes, can be included.If memory effects (‘‘electron-photon quantum kinetics’’)are neglected, the photon-assisted density matrices canbe adiabatically eliminated and the spectrum is obtainedby performing a matrix inversion in which, however, thematrix is a function of time.

Typical terahertz signals constitute coherently emittedelectromagnetic radiation. Therefore a quantized treat-ment of the field is not necessary in this case. Instead,they are calculated directly from the intraband polariza-tion or from the oscillating current density induced bythe exciting light field, which act as sources for a classicalfield.

If the semiconductor is excited by two or more pulsestraveling in different directions or by a strongly localizedpulse, the excitation corresponds to a spatially inhomo-geneous electric-field distribution and thus the fulltheory, including nondiagonal single-particle density ma-trices, has to be used. If the typical length scales intro-duced by the excitation are sufficiently large, however,the Wigner representation in combination with the gra-dient expansion—as introduced in Sec. II.C.3—can beconveniently employed. The characteristic length scalein a two-pulse pump-probe or four-wave-mixing experi-ment is the period of the transient grating created in thesample. If l is the wavelength of the exciting pulses anda the angle between the two incident directions, this pe-


riod is given by l/@2 sin(a/2)# , which is typically at leastin the micrometer range. Since in the absence of strongelectric fields transport processes over this distance canusually be neglected on a femtosecond time scale, thedynamics are well described by the lowest-order contri-bution in the gradient expansion, i.e., by the equationsof motion for the homogeneous case in which all thequantities now depend parametrically on the spatial co-ordinate r. For an exciting electric field of the form

E~r,t !5E1~ t !eiq1r1E2~ t !eiq2r1c.c.

5eiq1r@E1~ t !1E2~ t !ei(q22q1)r#1c.c., (123)

all dynamical variables can be expanded in a Fourierseries (Lindberg, Binder, and Koch, 1992) as5

fk~r!5(n

fk(n)ein(q22q1)r,

pk~r!5eiq1r(n

pk(n)ein(q22q1)r. (124)

Here, fk(0) is the spatially averaged distribution function,

fk(1) and fk

(21) describe transient grating terms, andhigher orders of fk describe a nonsinusoidal shape of thisgrating. pk

(0) and pk(1) describe polarizations traveling in

the directions of the pulses; thus, if pulse 2 is the probepulse, pump-probe signals are calculated from pk

(1) . Inparticular, assuming an optically thin sample by neglect-ing any propagation effects, as is always done in thisreview, the differential transmission spectrum DT(v) isobtained from the total polarization in the probe direc-tion P(1)5(kMpk

(1) by Fourier transformation: accord-ing to

DT~v!;Im@E2* ~v!dP(1)~v!#

uE2~vu2 , (125)

where dP(1) is the difference between the polarizationcalculated with and without the pump pulse (Lindbergand Koch, 1988b; Balslev, Zimmermann, and Stahl,1989).

The diffracted polarizations pk(2) and pk

(21) give rise tofour-wave-mixing signals. Again, in the absence ofpropagation effects, the emitted fields are essentiallyproportional to the polarizations P(n)5(kMpk

(n) , wheren52 and 21; therefore both temporally and spectrallyresolved signals are directly given by uP(t)u2 anduP(v)u2, respectively. Since the feedback from higher tolower orders in the Fourier expansion is usually not verystrong, the series is truncated either at the desired orderor at most one order above. With this technique the cal-

5It should be noted that this Fourier expansion only holds inthe case of a rotating-wave approximation and in the absenceof Coulomb terms that do not conserve the number ofelectron-hole pairs. Otherwise, the clear separation betweendistribution functions and polarizations is lost and a differentexpansion involving additional Fourier components has to beperformed.


culation of pump-probe and four-wave-mixing signalsrequires increasing the number of variables by a factorof about 4 with respect to the single-pulse calculation.An alternative method that avoids the increase in thenumber of variables has been proposed by Banyai et al.(1995; Haug and Jauho, 1996). It is based on the obser-vation that if the phase difference f between the pulsesis treated as a continuous variable equal to (q22q1)rand the dynamics are calculated for all phases f, thepolarization components in the various directions can beobtained by a Fourier transform,

pk(n)5

12p E

0

2p

pk~f!einfdf . (126)

This method is particularly useful in quantum-kineticcalculations, in which the number of variables in the ho-mogeneous case is often already very large and a furtherincrease is not possible due to memory limitations.

III. SELECTED EXPERIMENTAL AND THEORETICALRESULTS

A. Line shape of luminescence spectra

As mentioned above, measurements of the lumines-cence spectrum were among the first optical experimentsto yield information on nonequilibrium carrier dynamicsin semiconductors, and they are still widely applied to avariety of semiconductor materials and structures. For along time they were rather successfully interpreted interms of a fully incoherent picture based on the Boltz-mann equation and transition rates obtained from Fer-mi’s golden rule. However, it turned out that several fea-tures in the spectra—particularly in spectral regionsdirectly related to the carrier photogeneration process—were hard to explain. In this section we shall discusseffects due to a coherent description of carrier dynamics.For this purpose we shall first compare the carrier pho-togeneration process treated in an incoherent picturewith the corresponding coherent description based onthe semiconductor Bloch equations. Then we shall ana-lyze the photoemission process on the same level. Fi-nally, we shall show that this coherent treatment indeedprovides a much better explanation of experimentallyobserved band-to-acceptor luminescence spectra.

1. Coherent carrier photogeneration

According to Fermi’s golden rule, the generation rateof electrons and holes is given by a product of temporaland spectral shapes of the exciting laser pulse, as ob-tained in the case of a Gaussian pulse in Eq. (33). In acoherent picture, on the other hand, carrier generationis a two-step process according to Eq. (26): First, thelight field creates an interband polarization and then thepolarization itself interacts again with the field, creatingelectron and hole populations. The photogenerationrates obtained in these two pictures are plotted at fourdifferent times in Fig. 3 for the case of a 150-fs pulsecentered at 1.68 eV and material parameters corre-sponding to bulk GaAs. In the case of Fermi’s golden


rule, the spectral shape is time independent, except forthe phase-space-filling terms, which are responsible forthe slight difference between the curves at 2100 fs and100 fs. In contrast, the generation rate obtained from theBloch equations in the absence of carrier-phonon andcarrier-carrier scattering exhibits a pronounced time de-pendence of the spectral shape. The generation startsout very broad; with increasing time the shape becomesnarrower, and negative values appear in the wings. Froma physical point of view this can be easily understood:As in the semiclassical case, the spectral width is deter-mined by energy-time uncertainty. Due to causality,however, only the time from the onset of the laser pulseup to the observation time determines the broadening,leading to a spectrally very broad rate at early times.With increasing time, the energy uncertainty decreases,and since there is still complete phase coherence be-tween the carriers and the laser field, those carriers gen-erated off-resonance perform a stimulated recombina-tion leading to negative wings. As a result, the time-integrated generation rate agrees with the semiclassicalcase as long as phase-space filling is not important, andone should expect that for measurements performed af-ter the exciting pulse has gone, there should be no bigdifference.

The situation changes if scattering processes are takeninto account. The semiclassical rate is not affected bythese processes, since on a Boltzmann level all interac-tion processes are treated independently. In the Blochcase, on the other hand, the generation rate is still givenby Eq. (26); the polarization dynamics, however, arestrongly influenced by carrier-phonon and carrier-carrierinteractions. In the present case of a homogeneous sys-tem and an excitation high up in the band, the mostimportant contributions are given by the second-orderterms, which lead to a dephasing of the interband polar-ization.

Figure 4 shows generation rates for different pulse in-

FIG. 3. Generation rate of a 150-fs laser pulse as a function ofwave vector at different times: (a) semiclassical and (b) coher-ent picture. No dephasing processes have been taken into ac-count. After Kuhn and Rossi, 1992b.


tensities (specified by the final carrier density) obtainedfrom the semiconductor Bloch equations, taking into ac-count first-order carrier-carrier contributions (Hartree-Fock) as well as second-order carrier-carrier and carrier-phonon contributions on a Markovian level according toEqs. (62) and (85). Now the shape of the generation ratebecomes strongly density dependent. Due to dephasingby scattering processes, coherence is lost and, as a con-sequence, the stimulated recombination processes in thewings are inhibited, thereby reducing the narrowing ofthe spectral shape with increasing time. The dominantprocess for this density dependence is carrier-carrierscattering. At lower densities dephasing is essentiallydue to carrier-phonon scattering only. The correspond-ing dephasing rate is of the order of 200 fs; therefore, onthe time scale of the pulse, it is not yet very efficient. Weobserve only a slight reduction of the negative parts.With increasing density, however, the loss of coherenceincreases and the negative parts are more and more re-duced. The generation rate remains broad during thepulse, leading to a much broader carrier distribution af-ter the pulse, compared to the semiclassical case. In Sec.III.A.3 we shall show how this broadening of the carriergeneration process influences band-to-acceptor spectra.

It should be noted that a physically reasonable densitydependence is obtained only if both in- and out-scattering-type second-order contributions as given inEq. (85) are included, particularly in the case of carrier-carrier scattering. Sometimes it has been argued that,due to random phases, in-scattering contributions can beneglected. Then, the scattering terms reduce to ak-dependent dephasing rate, given by the total out-

FIG. 4. Coherent generation rate of a 150-fs pulse at threedifferent densities, including dephasing due to carrier-carrierand carrier-phonon scattering. After Leitenstorfer et al., 1996.


scattering rate. As discussed in detail by Rossi et al.(1994) and Haas et al. (1996), this results in an essen-tially density-independent, very strong broadening. Thephysical reason for this strong overestimation of thedephasing at low densities is the assumption inherent inthis approximation that each scattering process com-pletely destroys the coherence. However, at low densi-ties most of the two-body scattering processes are char-acterized by a very small momentum transfer q, which,in turn, leads to a very small energy exchange and thusto very small dephasing.

In addition to the scattering-induced broadening ofthe generation rate, there are, of course, other cases inwhich the semiclassical rate [Eq. (33)] no longer holds.Close to the semiconductor band gap, carrier-light inter-action is strongly influenced by excitonic effects. In thesemiconductor Bloch equations such effects are de-scribed by the first-order carrier-carrier contributions, inparticular by the internal field. Here, a semiclassical gen-eration rate can again be obtained by adiabatic elimina-tion of the polarization, which, however, has to be per-formed in an exciton basis. If the laser pulse intensity isvery high, the assumption of slowly varying distributionfunctions is no longer satisfied. Here, the semiconductorBloch equations give rise to Rabi-type oscillations, wellknown from the physics of two-level systems. Since acontinuum of energies is involved in band-to-band exci-tations, no complete Rabi oscillations are possible; nev-ertheless, the total generation rate still exhibits non-monotonic behavior (Kuhn and Rossi, 1992b; Furst,Leitenstorfer, Nutsch, et al., 1997). Even for excitonicexcitation, which is closer to a two-level model, Cou-lomb effects and the presence of the continuum stronglymodify the Rabi oscillations, but in this case multipleoscillations with a frequency depending linearly on thefield amplitude have been observed (Schulzgen et al.,1999).

2. Luminescence line shape

Luminescence is the inverse process to the generationof carriers by light absorption. Therefore features simi-lar to those discussed in the previous section should oc-cur in the emission process as well, leading again tomodifications in the luminescence spectra when com-pared to the Fermi’s golden rule result. The theoreticaldescription is somewhat more complicated, since, as dis-cussed in Sec. II.F, it requires a quantum-mechanicaltreatment of the light field in which the photon-assisteddensity matrix, instead of the interband polarization, isthe relevant variable introduced by the interaction. Nev-ertheless, the many-body contributions appear in thesame way as in the case of the interband polarization,since neither carrier-phonon nor carrier-carrier interac-tions couple directly to the photons. Therefore, the samescattering matrices responsible for the broadening of thegeneration process give rise to the broadening of theband-to-band luminescence spectrum.

In order to focus on the broadening of the lumines-cence spectrum—and to avoid the broadening of the dis-


tribution function due to the scattering of the excitedcarriers—here, we show luminescence spectra obtainedfor a stationary distribution function of electrons andholes, generated by a laser pulse according to the mod-els in the previous section (Kuhn, Rossi, et al., 1996;Leitenstorfer et al., 1996). Since the distribution func-tions after the pulse are constant, the spectra are ob-tained by inverting the matrix in Eq. (122). In Figs.5(a)–(c) luminescence spectra, calculated according tofour different models, are plotted for three different car-rier densities. The dot-dashed lines show the fully semi-classical result obtained by calculating both generationand luminescence according to Fermi’s golden rule. Inthis case the spectrum is simply given by the square ofthe exciting laser spectrum multiplied by the density ofstates. The dashed lines show the luminescence spectraobtained by solving the semiconductor Bloch equations,taking into account the broadening of the generationprocess, while the luminescence process is described bythe semiclassical formula, thus introducing no additionalbroadening. The dotted lines show the opposite case:semiclassical generation and luminescence includingbroadening. These curves essentially agree with thedashed ones, which again demonstrates the fact that it isthe same physics that broadens the transitions. Finally,the solid lines display the spectra in which broadeninghas been included for both the generation and emissionprocesses. From these results it is clear that, with risingdensity, the semiclassical model increasingly underesti-mates the width of the luminescence spectra; the broad-ening of both generation and emission processes has tobe taken into account.

FIG. 5. Spectral profile of band-to-band [(a)–(c)] and band-to-acceptor [(d)–(f)] luminescence at different densities calcu-lated for a carrier distribution generated by a 150-fs pulse:dot-dashed line, semiclassical generation and recombination;dashed line, broadened generation and semiclassical recombi-nation; dotted line, semiclassical generation and broadened re-combination; solid line, broadened generation and recombina-tion. After Leitenstorfer et al., 1996.


Instead of measuring band-to-band luminescence, inp-doped semiconductors one can measure band-to-acceptor luminescence, which, as has been mentionedabove, has the advantage that it does not involve freeholes. Therefore it is usually easier to interpret. Let usnow look at the broadening in these experiments. Parts(d)–(f) of Fig. 5 show band-to-acceptor spectra calcu-lated under the same conditions as in parts (a)–(c) forband-to-band spectra. Here, we no longer find an agree-ment between the dashed and dotted lines; instead, thedashed lines essentially agree with the solid ones. Thereason for this behavior can be understood as follows:Since the generation is a band-to-band transition, whilethe emission is a band-to-acceptor transition, there is nolonger symmetry between the two processes and there isno reason why the broadening should be the same. Infact, the broadening of the generation process is muchgreater, which can be traced back to the fact thatdephasing of band-to-acceptor transitions is due to scat-tering processes of the electrons only, while the dephas-ing of band-to-band transitions is due to scattering pro-cesses of both electrons and holes, the latter usuallybeing much more efficient because of the much higherdensity of states. Therefore we can conclude that forband-to-acceptor spectra the broadening of the emissionprocess is negligible compared to that of the generationprocess.

3. Band-to-acceptor luminescence spectra

In the previous section we have clearly seen that aproper inclusion of the interband polarizationdynamics—in particular their density-dependentdephasing—should noticeably modify luminescencespectra. Therefore the question arises whether such cal-culations can improve the agreement with experimen-tally observed spectra (Leitenstorfer, Lohner, Elsaesser,et al., 1994; Leitenstorfer et al., 1996). Figure 6 demon-strates that this is indeed the case. Here, time-integratedband-to-acceptor spectra calculated on a semiclassical(Boltzmann) as well as on a coherent (semiconductorBloch) level for three different densities are comparedto experimental results. In agreement with the findingsof the previous section, here the broadening of the emis-sion process has been neglected in the calculations. Atlow densities the spectra exhibit pronounced phononreplicas due to the emission of optical phonons, whilecarrier-carrier scattering has essentially no effect at thelowest density. In contrast, with increasing density thereplicas are more and more washed out due to increas-ing carrier-carrier scattering. The overall behavior issimilar in the Boltzmann and Bloch cases; however,there are remarkable differences, particularly in the re-gion of the peak at the highest energy (marked by heavylines), which are due to those carriers’ not yet havingemitted a phonon. In the Boltzmann case this peak isvisible up to the highest density because in this picturethe carriers are always generated with a narrow distribu-tion, which is subsequently broadened by carrier-carrierscattering. In contrast, in the Bloch case the generation


process is broadened as discussed above, which is clearlyin much better agreement with the experimental results.

B. Coherent features in pump-probe experiments

Much like luminescence experiments, pump-probe ex-periments in the band-to-band region have been per-formed for many years to obtain information on the dy-namics of carrier distribution functions. Measurementsof the transmission change of a probe pulse due to aprevious pump pulse were among the first time-resolvedstudies of the nonequilibrium dynamics in semiconduc-tors (Shank et al., 1979), and the transmission changehas been completely interpreted in terms of Pauli block-ing of the optical transitions. However, the direct sourcefor a differential transmission or reflection signal is athird-order polarization created in the sample by thepump and probe pulses. Therefore, on a time scale ofthe order of the dephasing time, the coherent dynamicsof this polarization will influence the signal. Besides theusual phase-space-filling term present when the pumppulse precedes the probe pulse, there are two additionalcontributions which are also well known from two-levelsystems (Chachisvilis, Fidder, and Sundstrom, 1995):First, if the probe pulse precedes the pump pulse by atime delay of the order of or shorter than the dephasingtime, the pump pulse perturbs the decay of the probepolarization by suddenly increasing the dephasing rate.This gives rise to the perturbed free polarization decay.Second, if the pulses overlap temporally, they induce agrating in the sample which can diffract the pump pulse

FIG. 6. Band-to-acceptor luminescence spectra at three differ-ent excitation densities: left and center panels, theoreticallycalculated; right panel, experimentally observed. The unre-laxed peak (heavy lines) reflects the broadening of the genera-tion processes as included in the coherent model. After Leiten-storfer, Lohner, Elsaesser et al., 1994.


into the probe direction. If the probe pulse precedes thepump pulse, these phenomena give rise to spectral oscil-lations in the differential transmission with a frequencydetermined by the time delay between both pulses(Fluegel et al., 1987; Lindberg and Koch, 1988c; Likfor-man et al., 1995). This shows that, on time scales compa-rable to or shorter than dephasing times, pump-probespectra are quite complicated to interpret. Grating ef-fects can be even more complicated if, for example, inspatially resolved pump-probe experiments, a nearly col-linear excitation has to be used. In this case different,diffracted orders can no longer be distinguished, leadingto a superposition of pump-probe, four-wave-mixing,and possibly higher-order contributions (Otremba et al.,1999). Besides these features general to pump-probe ex-periments, there are additional modifications in a semi-conductor due to many-body effects. Electron and holedistributions give rise to band-gap renormalization and,therefore, to a spectral shift. Furthermore, they modifythe Coulomb enhancement, which again may lead to aninduced absorption in some spectral regions (Furst, Le-itenstorfer, Laubereau, and Zimmermann, 1997). Cou-lomb correlation effects are even more pronounced inthe excitonic region of the spectrum. Here it has beenshown that Coulomb sources may even strongly domi-nate the spectra (Bartels et al., 1997). In particular, inthe case of pumping and probing with two countercircu-larly polarized laser pulses, Pauli blocking—as well asother mean-field contributions—is completely absentand only correlations give rise to a signal (Smith et al.,1994; Axt, Victor, and Stahl, 1996; Sieh et al., 1999). In

FIG. 7. Differential transmission signals after band-to-bandexcitation of bulk GaAs with 20-fs pulses: (a) experimentallyobserved signals at four different energies; (b) calculated sig-nal at 1.56 eV: solid line, the full model; dashed line, withoutinter-valence-band (IV) polarization; dotted line, without scat-tering. After Joschko et al., 1997, 1998.


the following we shall address in more detail a specificcoherent feature in band-to-band spectra.

The Pauli blocking terms in a density-matrix theoryinvolve not only distribution functions—i.e., diagonaldensity-matrix elements—but, if the optical excitationoverlaps spectrally with several transitions, also off-diagonal elements describing phase coherence within agiven band or between different conduction or valencebands, respectively. Since such off-diagonal elements os-cillate in time with a frequency given by the correspond-ing energy splitting, they give rise to quantum beats inpump-probe signals when plotted as a function of thedelay time between pump and probe pulse. In the exci-tonic region such quantum beats have been observed formany years in different systems, where the interferingstates were given by excitons in coupled quantum wells(Leo, Gobel, et al., 1991), spin states in a magnetic field(Bar-Ad and Bar-Joseph, 1991) as well as heavy- andlight-hole excitons (Schmitt-Rink et al., 1992). In theband-to-band regime, on the other hand, there is inmany cases a continuous variation of splitting energies,for example, in the case of heavy- and light-hole bands.Therefore, even if a superposition is excited, it is notobvious whether quantum beats can be observed inpump-probe signals.

Figure 7(a) shows experimentally observed pump-probe signals at different detection energies obtainedfrom bulk GaAs excited by 20-fs pump and probe pulsesat 1.61 eV (Joschko et al., 1997). The pulses overlap witha broad manifold of transitions from the heavy- and the

FIG. 8. Differential transmission spectra calculated (a) withand (b) without inter-valence-band polarizations.


light-hole band to the conduction band, but they do notoverlap with the exciton at about 1.5 eV and thereforeclearly probe band-to-band transitions. While undersuch conditions a spectrally integrated signal exhibits nooscillations, here oscillations are clearly visible on thelow-energy side, in particular at 1.56 eV. The variation ofthe oscillation frequency with the excess energy gives aclear hint that these are indeed heavy-hole/light-holequantum beats. This interpretation is confirmed by the-oretical results obtained from the semiconductor Blochequations. In Fig. 7(b) the calculated signal at 1.56 eV(solid line) is compared with the results of calculationsin which the inter-valence-band polarizations (dashedline) and the carrier-carrier and carrier-phonon scatter-ing (dashed line) have been switched off. Figures 8(a)and (b) show the complete spectra as functions of thephoton energy and the delay time with and withoutinter-valence-band polarization, respectively. If the inter-valence-band polarizations (off-diagonal elements of thehole density matrix) are switched off, these oscillationsare absent, unambiguously demonstrating their origin asheavy-hole/light-hole quantum beats in the continuum.From an analysis of the various contributions in the the-oretical model, the spectra can be understood in detail(Joschko et al., 1998): The strong asymmetry with re-spect to the spectral center of the pulses is due to theCoulomb enhancement, which strongly increases thelow-energy part; the negative feature around zero timedelay is mainly due to the grating effect discussed above;and the damping of the oscillations is due to a combina-tion of dephasing by scattering processes and of the in-homogeneous broadening of the transition frequencies.

The occurrence of quantum beats in spectrally re-solved signals can be easily understood from a three-band model of noninteracting carriers. Assuming thatthe pump pulse at time 2TD creates electron and heavy-and light-hole distributions fk

e , fkh , and fk

l , as well asan inter-valence-band polarization fk

lh5 f klh exp@(ivk

lh

2G lh)(t1TD)# , where \vklh is the heavy-hole/light-hole

splitting energy at a given k and G lh the correspondingdephasing rate, the linear susceptibility due to a probepulse at time t50 is given by

x~v!5(k

uMhu2~fke1fk

h!

\v2eke2ek

h1i\Gh

1(k

uMlu2~fke1fk

l !

\v2eke2ek

l 1i\G l

1(k

Ml* Mhf klh

\v2eke2ek

h1i\Gh

E~v1vklh!

E~v!

3e(2ivklh

2G lh)TD1(k

Mh* Mlf klh

*\v2ek

e2ekl 1i\G l

3E~v2vk

lh!

E~v!e(ivk

lh2G lh)TD. (127)

Here, Gh and G l are interband dephasing rates ofthe heavy-hole-to-conduction-band and light-hole-to-


conduction-band transitions, respectively, Mh and Ml

are the corresponding dipole matrix elements, and E(v)is the Fourier transform of the electric field of the probepulse. We have assumed that the distribution functionsas well as the inter-valence-band polarization envelopef k

lh are slowly varying during the probe pulse. This resultclearly shows that there are oscillating contributions inthe spectrum (third and fourth terms). They are dampednot only by G lh, but also by inhomogeneous broadening,since at a given frequency v, a range of k states (thecorresponding width being determined by the interbanddephasing rate) around the resonant transition fre-quency contributes to the susceptibility. Furthermore,these terms involving inter-valence-band polarizationsdepend on the spectral shape of the probe pulse, andthey are shifted towards lower (third term) and higher(fourth term) frequency compared to the diagonal (firstand second) term. The question remains why oscillationsare seen only on the low-energy side in both the experi-mental and the theoretical spectra. This behavior is alsoconfirmed by Eq. (127) when the k summation is per-formed. It turns out that when we calculate the imagi-nary part of the susceptibility, the two terms resultingfrom the decomposition exp(6ivk

lhTD)5cos vklhTD

6i sin vklhTD in the third term add constructively, while

in the fourth term they oscillate out of phase, thus can-celing each other.

Recently, heavy-hole/light-hole quantum beats in theband-to-band continuum—in particular, their dephasingdynamics—have also been analyzed on the level of aCoulomb quantum-kinetic approach (Mieck and Haug,1999). It has been found that they should persist inpump-probe spectra even in the density range between1017 and 1018 cm23. In the excitonic part of the spectrumthe heavy-hole/light-hole systems more closely resemblea three-level system, and one might expect a simplerinterpretation. However, it turns out that here Coulombeffects are even more important. In particular,Coulomb-induced correlations between excitons mayeven change the sign of the signal or modify the phase ofthe beats (Bartels et al., 1997), and they efficientlycouple heavy-hole and light-hole excitons (Meier et al.,2000).

C. Temporal and spectral shape of four-wave-mixingsignals

In a four-wave-mixing experiment the sample is ex-cited by two pulses traveling in directions q1 and q2 , andthe signal is measured in the diffracted direction 2q22q1 . If the sample is optically thin, it is usually a goodassumption to model the emitted electric field directlyby the optical polarization in this direction. To lowestorder this is a third-order polarization, but, of course,with increasing power of the pulses, higher-order polar-izations also contribute to the signal. For the case of atwo-level system and pulses arriving at times t52TDand t50, respectively, with pulse durations shorter thanthe dephasing time, it can be shown that the third-order


polarization in the diffracted direction is directly propor-tional to the value of the phase-conjugated polarizationat t50 created by the first pulse at t52TD . Since thetime-integrated signal decays with a time constant T2/2,this technique provides direct information on thedephasing time T2 . As a function of time, the polariza-tion decays with the dephasing time; it exhibits a freepolarization decay. If instead of a single two-level systeman inhomogeneously broadened ensemble of such sys-tems is considered, the signal is not emitted immediatelyafter the second pulse due to interference of the differ-ent frequency contributions. Instead, its maximum oc-curs at time t5TD : It exhibits photon-echo behavior.The time-integrated signal in this case decays with atime constant of T2/4. Such an echo signal was first ob-served in magnetic resonance (Hahn, 1950), here calledthe spin echo, with dephasing times of the order of 10ms. In the visible range the first photon echoes wereobserved in ruby (Kurnit et al., 1964; Abella et al., 1966)with dephasing times of the order of 100 ns.

In a semiconductor the spectrum consists of both adiscrete excitonic and a continuous band-to-band part.Furthermore, depending on the sample and the experi-mental conditions, the exciton line may be either homo-geneously or inhomogeneously broadened. Thereforeone might expect a more complicated temporal behaviorof the four-wave-mixing signal, depending on a varietyof parameters. For inhomogeneously broadened two-dimensional excitons in quantum wells excited by12.6-ps pulses, it was shown during the early stages ofcoherent spectroscopy in semiconductors that this sys-tem exhibits a photon echo (Schultheis et al., 1985).Similar results were obtained for excitons in mixed crys-tals (Noll et al., 1990). For weak disorder, however, thesignals can only be understood in detail if the Coulombinteraction is also included in the model (Jahnke et al.,1994).

For homogeneously broadened excitons it was foundthat the expected free polarization decay is modified bymany-body effects, which, particularly at higher excita-tion power, strongly dominate the temporal shape of thesignal (Leo, Wegener, et al., 1990; Kim, Shah, Damen,et al., 1992; Mycek et al., 1992; Weiss et al., 1992). Thisbehavior is qualitatively well reproduced by calculationsbased on the semiconductor Bloch equations (Wegeneret al., 1990; Lindberg et al., 1992). The most prominentdeviation from the two-level case, which also manifestsitself in the time-integrated signal, is the appearance of acontribution at negative delay times. Qualitatively, thiscan be understood on the mean-field level on the basisof a two-level system, which, in addition to the externalfield, is subject to a local field proportional to the polar-ization. Here, the polarization created by the first pulsein the direction q2 , which is still present at the arrival ofthe second pulse, can be diffracted into the observeddirection 2q22q1 . This contribution, however, is re-moved by sufficiently strong inhomogeneous broaden-ing. At positive delay times the Coulomb interactiongives rise to a delayed contribution in the signal, thedelay being determined by the dephasing time. Besides


these mean-field effects of the Coulomb interaction,there are also modifications to the noninteracting carriercase which are related to electronic correlations inducedby the Coulomb interaction. As has been shown in detailin Sec. II.E.2, such correlations give rise to nonlinearscattering and relaxation terms. This type of dephasinghas been introduced in a simplified way as excitation-induced dephasing (Wang et al., 1993; Hu et al., 1994),i.e., a dephasing rate that increases linearly with the car-rier density. However, these correlations lead to morethan just dephasing. Variables like Bj1 ,i1 ,j2 ,i2

5^dj1ci1

dj2ci2

&, which were not discussed in Sec. II be-cause in a correlation expansion they enter at a higherorder, include effects related to biexcitons as well asexciton-exciton correlations (Axt and Stahl, 1994b).They again modify four-wave-mixing signals; in particu-lar, they give rise to biexcitonic quantum beats with afrequency determined by the biexciton binding energy(Pantke et al., 1993; Mayer et al., 1994; Bartels et al.,1995). Even if bound biexcitons are excluded due to ex-citation with two pulses of the same circular polariza-tion, the signals are strongly dominated by such Cou-lomb correlations, particularly at negative delay times(Kner et al., 1998). We shall come back to the features ofbiexcitons and exciton-exciton correlations in Sec. III.Hwhen discussing Coulomb quantum kinetics. Quantita-tively, it has been found that correlation effects are typi-cally more important than local field corrections for thedeviations of four-wave-mixing signals in the excitonicregime from the limiting case of noninteracting two-level systems described by the optical Bloch equations.

In the band-to-band continuum case, which, at least inthe limit of noninteracting carriers, is equivalent to aninhomogeneously broadened ensemble of two-level sys-tems, a photon echo behavior is expected. The dephas-ing in this case was studied as a function of carrier den-sity by using 6-fs pulses (Becker et al., 1988), and thesignal was indeed attributed to a photon echo. However,in that work the signal was not time resolved; thereforea clear proof was not possible. A clear photon echo be-havior, as found in calculations based on the semicon-ductor Bloch equations for the case of excitation in theband-to-band continuum (Lindberg et al., 1992; Glutsch,Siegner, and Chemla, 1995), was observed by Lohneret al. (1993) after spectral filtering of the continuum con-tribution, and recently by Hugel et al. (1999) in room-temperature measurements with 11-fs pulses. In the fol-lowing we shall discuss in more detail the resultsobtained by Lohner et al.

In that experiment a bulk GaAs sample was excitedby 100-fs pulses slightly above the band edge but stilloverlapping with the exciton. The diffracted signal wasthen spectrally as well as temporally analyzed. The spec-trally resolved signal for zero time delay at four differentexcitation densities is shown in Fig. 9(a). The spectrumof the laser pulse is included in Fig. 9(b) as a dashedline. Even if the pulse maximum is in the band, at lowdensities the spectrum is completely determined by theexciton. Such a strong enhancement of the exciton con-tribution has been found in quantum wells by scanning


the excitation frequency across the exciton line (Kim,Shah, Cunningham, et al., 1992). With increasing densitythe exciton is screened and an additional free-carriercontribution appears which is essentially given by thepulse spectrum. This behavior is qualitatively well repro-duced by calculations based on the semiconductor Blochequations, as shown in Fig. 9(b). The slight differences,in particular the slight redshift of the exciton and thestronger free-carrier contribution at higher densities, canbe attributed to limitations of the quasistatic screeningmodel used in these calculations and to possible uncer-tainties in the exact determination of the density. Theinteresting point here is that the exciton is still clearlyvisible at densities more than one order of magnitudehigher than the Mott density at T510 K (Ulbrich, 1988),which demonstrates that the nonequilibrium distributionis much less effective in screening the electron-hole in-teraction. At a given density the spectral shape of thesignal also strongly depends on the delay (Leitenstorfer,Lohner, Rick, et al., 1994): With increasing negative de-lay, the excitonic contribution increases, since here thecontinuum contribution vanishes due to inhomogeneousbroadening. For positive delay times the excitonic con-tribution is decreasing faster than the continuum contri-bution, which can be attributed to destructive interfer-ence between the polarizations of bound and unboundstates. A similar two-component behavior of the spec-trally resolved four-wave-mixing signal was found at thedirect gap of germanium (Rappen et al., 1993).

As one might expect from the discussion above, thetemporal shape of the signal in this case will be quitecomplicated. However, by performing a spectral filteringat either the exciton or excitation frequency, one canextract the two characteristic signal types. This is shown

FIG. 9. Spectrally resolved four-wave-mixing (FWM) signalsof bulk GaAs excited by two 100-fs pulses with zero delay, 4meV above the band gap: (a) experimental and (b) theoreticalresults at different excitation densities. After Lohner et al.,1993.


in Figs. 10(a) and (b) for the experimental data and10(c) and (d) for the theoretical results. The continuumcontribution [Figs. 10(a) and (c)] exhibits clear photonecho behavior, while the excitonic contribution [Figs.10(b) and (d)] exhibits essentially a free polarization de-cay modified by many-body effects as discussed above.This demonstrates that in a semiconductor these typesof signals, which were initially shown to be characteristicof different atomic or magnetic systems, occur simulta-neously, and furthermore they can be clearly separated.

If the semiconductor is excited with a shorter pulsehigher up in the band, the excitonic contribution is es-sentially absent and the time-resolved four-wave-mixingsignal exhibits photon echo behavior. This is clearly seenin Figs. 11(a) and (b), where experimental and theoret-ical results, respectively, are shown corresponding to anexcitation with 11-fs pulses centered at 50 meV abovethe band gap (Hugel et al., 1999). The calculations

FIG. 10. Measured [(a), (b)] and calculated [(c), (d)] tempo-rally resolved four-wave-mixing (FWM) signals for differentdelay times: (a), (c) after filtering at the excitation frequency;(b), (d) after filtering at the exciton frequency. After Lohneret al., 1993.

FIG. 11. Temporally resolved four-wave-mixing (FWM) sig-nals for different delay times for the case of excitation by 11-fspulses centered 50 meV above the band edge: (a) experiment;(b) theory. After Hugel et al., 1999.


shown here are based on a quantum-kinetic treatment ofCoulomb and carrier-phonon scattering processes interms of nonequilibrium Green’s functions, which takesinto account the dynamical buildup of screening.

D. Coherent control phenomena

The existence of a four-wave-mixing signal in the caseof temporally nonoverlapping laser pulses in differentdirections is clear proof of the coherence in the material.Coherence means that the system is characterized notonly by amplitudes but also by phases. This phase sensi-tivity is demonstrated even more clearly by coherent-control experiments, in which the response of the systemis measured after excitation with two temporally non-overlapping phase-locked laser pulses in the same direc-tion. In a semiclassical picture the first laser pulse cre-ates a certain density of electron-hole pairs. If thesystem is far from inversion, then a second pulse simplyadds another density, which, in the case of equal pulseintensities, is essentially the same as the first pulse. In acoherent picture, as has been discussed in detail above,first a polarization is created and then the interaction ofthe polarization with the light field leads to the genera-tion of a carrier density. Thus the second pulse may in-terfere constructively or destructively with the polariza-tion left over from the first pulse. In the absence ofdephasing, constructive interference results in a finalcarrier density that is four times the density of the firstpulse alone, while in the case of destructive interference,all carriers generated by the first pulse are removed,leaving an unexcited sample. Therefore, as a function ofthe delay time between the pulses, the final carrier den-sity exhibits an oscillatory behavior. In the presence ofdephasing, these oscillations are damped with increasingdelay time until finally, for times much longer than thedephasing time, the semiclassically expected result—i.e.,twice the density generated by the first pulse—is recov-ered.

This was shown by Heberle et al. (1995) for excitonsin quantum wells. Experimental and theoretical resultsfor the case of two pulses with different intensities areplotted in Fig. 12 at different delay times (Heberle et al.,1996). The sensitivity with respect to the delay time(here given in units of Thh52p/Ehh , where Ehh is theenergy of the heavy-hole exciton), as well as the de-creasing splitting between the constructive and destruc-tive curves with increasing delay time, is clearly visible.In the experiment the density was extracted from thedifferential reflectivity change, measured with a thirdpulse. The deviations from a constant value after thepulses are due to the fact that this quantity is not exactlyproportional to the density; instead, as discussed in Sec.III.B.1, it also depends on intraband coherences and thetime-dependent momentum distribution of the carriers.

Coherent-control techniques using two temporallyseparated phase-locked pulses have been applied to avariety of systems. If the pulses are perpendicularly po-larized, they do not interfere directly and therefore theirrelative phase does not influence the total exciton den-


sity. However, they may control the spin density in aquantum well sample (Heberle et al., 1996). This can bebest understood by decomposing the pulses into theircircular polarization components. The first pulse createsa superposition of excitons with spin sz511 and sz521. If the delay time is t125(n61/4)Thh , one of thecircular components of the second pulse interferes con-structively with the corresponding polarization compo-nent created by the first pulse, while the other compo-nent interferes destructively. As a result, one of theangular momentum components is removed and theother is enhanced, leaving a net angular momentum inthe sample. At integer or half-integer multiples of Thh ,the interference effectively results in a linear polariza-tion, thus creating no net angular momentum.

All kinds of quantum beats can be coherently con-trolled. They arise due to the excitation of a superposi-tion of two energetically separated levels by a short laserpulse with a spectrum overlapping both transitions. Ingeneral, the delay times corresponding to destructive in-terference of the two transitions are different. Thereforeit is possible to select delay times when one componentis selectively switched off, removing the quantum beatsfrom the signal (Kuhn et al., 1999). This has been shownfor heavy-hole/light-hole beats in quantum wells (He-berle et al., 1995), quantum beats due to charge oscilla-

FIG. 12. Coherent-control signals for excitation of a GaAsquantum well structure with two phase-locked pulses ofslightly different intensities centered at the heavy-hole excitonat delay times of n times the heavy-hole exciton oscillationperiod: (a)–(c), measured differential reflection of a probepulse; and (d)–(f), calculated exciton densities. In (a) and (d)only the signals corresponding to an excitation by pulse one ortwo have been included. After Heberle et al., 1996.


tions in coupled quantum wells (Luo et al., 1993; Plan-ken et al., 1993), and phonon quantum beats (Wehneret al., 1998; Steinbach et al., 1999). We shall come backto the case of phonon quantum beats in Sec. III.G whenwe discuss quantum-kinetic phenomena.

In the case of degenerate final states of an opticaltransition, coherent-control techniques can be used forselective excitations. Here, the quantum interference be-tween a one-photon and a two- or three-photon transi-tion is typically used to control the final states. Based onsuch an approach, the outcome of chemical reactions(Brumer and Shapiro, 1995) or the photoionization ofatoms (Chen, Yin, and Elliot, 1990) can be controlled. Ina semiconductor this technique allows one to create aphotocurrent in the absence of an applied voltage be-cause the superposition of the light pulses removes thesymmetry between the generation of carriers with oppo-site momenta (Dupont et al., 1995; Atanasov et al., 1996;Hache et al., 1997), which is always present if a singlepulse is used. Again, another type of phase-dependent,i.e., coherently controlled, carrier dynamics is obtainedin semiconductor nanostructures if interband and inter-subband transitions are simultaneously excited byphase-locked fields in their respective frequency ranges(Potz, 1997a, 1997b, 1998).

E. Charge oscillations in double quantum wells

For all the phenomena discussed in the previous sec-tions, spatial inhomogeneities were irrelevant for thecarrier dynamics. In pump-probe and four-wave-mixingexperiments inhomogeneities introduced by the differ-ent pulse directions, particularly the induced transientgrating, were necessary for selecting the desired opticalsignal; however, all space dependences were treatedparametrically and thus all spatial transport phenomenawere neglected. This was justified by the large lengthscales in the micrometer range. The situation changes ifinhomogeneities occur on a nanometric scale. In this andthe following section, we shall discuss some phenomenarelated to the spatial dynamics of optical carrier excita-tion in the growth direction of multiple quantum wellstructures.

In an asymmetric double quantum well with a suffi-ciently thin barrier, the states in the two wells arecoupled due to tunneling. Typically, in the flat-band casethis coupling is not very strong because of the differentenergies of bound states in the two wells. By applying anelectric field, however, one can bring states in the wideand narrow wells into resonance (see the inset in Fig.13). The resulting delocalized states are then energeti-cally separated by the tunnel splitting and, if the spectralwidth of the exciting laser pulse is larger than this split-ting, a superposition of the two states is excited, leadingto quantum beats, as discussed above for the case ofheavy and light holes. Such beats have been observedboth in the differential transmission and in the four-wave-mixing signal (Leo, Shah, et al., 1991, 1992). Thedifference with respect to the heavy-hole/light-hole case,however, is the fact that here the superposition leads to


a spatial oscillation of the electronic wave packet. Sincethe hole states are not in resonance, the holes essentiallyremain in the wide well. Thus an oscillating dipole mo-ment is created. Neglecting the interband cotributions,which oscillate much faster, we obtain the dipole mo-ment from the density matrices according to

P5(i1i2

(k

Mi1i2

e f i1i2 ,ke 2(

j1j2(

kMj1j2

h f j2j1 ,kh , (128)

where Mi1i2

e and Mj1j2

h are the electron and hole dipolematrix elements between subbands i1 and i2 or j1 and j2 ,respectively, and k is the in-plane momentum. In Fig.13(a) the dipole moment excited by a 160-fs laser pulseis plotted as a function of time. This result is obtainedfrom a solution of the semiconductor Bloch equationsincluding two electron and one hole subband (Binder,Kuhn, and Mahler, 1994; Kuhn, Binder, et al., 1994).Here, the dashed and dotted lines show the diagonal andoff-diagonal contributions, respectively. The off-diagonal part describes the coherent superposition; it os-cillates and finally decays due to dephasing processes.6

The diagonal part approaches a finite value due to thedifferent localizations of electrons and holes. Accordingto classical electrodynamics, such a time-dependent di-pole moment emits electromagnetic radiation propor-tional to its second derivative, which is plotted in Fig.

6Here dephasing has been treated by phenomenological in-terband and intersubband rates. Results in which the dephas-ing due to carrier-carrier scattering has been treated on thesemiclassical (Boltzmann-Bloch) level can be found in the ar-ticles of Potz, Ziger, and Kocevar (1995), Potz (1996a), andBinder (1997).

FIG. 13. Oscillating wave packet in a GaAs/AlGaAs asymmet-ric double quantum well structure: (a) dipole moment (solidline, total moment; dashed line, diagonal contribution; dottedlines, off-diagonal contribution) and; (b) emitted electromag-netic radiation. The inset shows the band edges and the elec-tronic states. After Kuhn et al., 1994.


13(b).7 Such radiation in the terahertz range has indeedbeen observed experimentally (Roskos et al., 1992; Nusset al., 1994). Figure 14 shows electromagnetic transientsemitted from an asymmetric double quantum well struc-ture with the same parameters as above at different val-ues of the bias field (Roskos et al., 1992). A coherentemission of terahertz radiation has also been observedin superlattices, as will be discussed in the next section.

Spatial inhomogeneities give rise to two additionalfeatures included in the general theory discussed abovewhich are absent in the homogeneous case: coherentphonons and Hartree contributions. An oscillating elec-tronic wave packet polarizes the crystal lattice. This lat-tice polarization is described by coherent opticalphonons according to Eq. (60). Figure 15 shows the elec-tron charge density as well as the lattice polarization asfunctions of space and time. It can clearly be seen thatwhenever the electronic wave packet is localized in thenarrow well, the lattice polarization is maximal, while itessentially vanishes if the electrons are in the wide well,where their charge density is compensated by the holes.The coherent-phonon amplitudes increase linearly withthe density of excited carriers. Therefore they give adensity-dependent contribution to the electron and hole

7It should be noted that the oscillation frequencies are deter-mined by the exciton energies. The additional appearance ofsingle-particle energies in the spectrum under certain condi-tions is an artifact of the truncation of the hierarchy as dis-cussed by Axt, Bartels, and Stahl (1996) and Haring Bolivaret al. (1997).

FIG. 14. Measured coherent electromagnetic transients emit-ted from an asymmetric double-quantum well structure at dif-ferent bias fields for a photon energy of 1.54 eV. After Roskoset al., 1992.


self-energies [Eq. (48)]. As shown by the squares in Fig.16, they tend to reduce the frequency of the electronicoscillation. However, the Hartree terms also increaselinearly with carrier density and give rise to density-dependent self-energies [Eq. (76)]. It turns out that theytend to increase the oscillation frequency (triangles inFig. 16). If both contributions are taken into account,the latter dominates, as shown by the diamonds. Physi-cally, this density dependence can be well understoodfrom classical electrodynamics: The Hartree terms de-scribe the electrostatic forces between electrons andholes. They are attractive, thus increasing the oscillationfrequency. These forces, however, are screened by thelattice, which reduces this increase. These calculations

FIG. 15. Oscillating wave packet in the asymmetric doublequantum well structure of Fig. 13: (a) electron charge density;(b) induced lattice polarization as functions of position andtime.

FIG. 16. Oscillation frequency of the dipole moment in theasymmetric double quantum well structure of Fig. 13 as a func-tion of the excited carrier density for free carriers, includingspace-charge effects (Hartree terms), and/or coherentphonons. After Binder, Preisser, and Kuhn, 1997.


therefore include the lattice screening on a fully dynami-cal and microscopic basis. Frequency changes due to in-duced electric fields have been used to obtain direct in-formation on the spatial dynamics of charge oscillationsin superlattices (Lyssenko et al., 1997).

F. Bloch oscillations and Wannier-Stark localization insuperlattices

Ever since the initial applications of quantum me-chanics to the dynamics of electrons in solids, the analy-sis of Bloch electrons moving in a homogeneous electricfield has been of central importance. By employingsemiclassical arguments, Bloch (1928) demonstratedthat a wave packet, given by a superposition of single-band states peaked about some quasimomentum \k,moves with a group velocity given by the gradient of theenergy-band function with respect to the quasimomen-tum and that the rate of change of the quasimomentumis proportional to the applied field F. This is often re-ferred to as the ‘‘acceleration theorem’’:

\k5eF. (129)

Thus, in the absence of interband tunneling and scatter-ing processes, the quasimomentum of a Bloch electronin a homogeneous and static electric field will be uni-formly accelerated into the next Brillouin zone in arepeated-zone scheme (or equivalently undergoes anumklapp process back into the first zone). The corre-sponding motion of the Bloch electron through the pe-riodic energy-band structure, shown in Fig. 17(a), iscalled Bloch oscillation; it is characterized by an oscilla-tion period tB5h/(eFd), where d denotes the latticeperiodicity in the field direction.

There are two mechanisms impeding a fully periodicmotion: interband tunneling and scattering processes.Interband tunneling is an intricate problem and still isthe subject of a continuing debate. Early calculations ofthe tunneling probability into other bands in which theelectric field is represented by a time-independent scalar

FIG. 17. Bloch oscillations and Wannier-Stark localization insuperlattices: (a) the field-induced coherent motion of an elec-tronic wave packet initially created at the bottom of a mini-band in the Bloch (miniband) picture; (b) transitions from thevalence band (VB) to the conduction band (CB) of a superlat-tice in the Wannier-Stark picture. Here, the width of the mini-band exceeds the LO phonon energy ELO , so that LO phononscattering is possible. After (a) von Plessen et al., 1994 and (b)Waschke et al., 1993.


potential were made by Zener (1934), who used aWentzel-Kramers-Brillouin generalization of Blochfunctions; by Houston (1940), who used acceleratedBloch states (Houston states); and subsequently byKane (1959) and Argyres (1962), who employed thecrystal momentum representation. Their calculations ledto the conclusion that the tunneling rate per Bloch pe-riod is much less than unity for electric fields up to106 V/cm for typical band parameters corresponding toelemental or compound semiconductors.

Despite the apparent agreement among these calcula-tions, the validity of employing the crystal momentumrepresentation or Houston functions to describe elec-trons moving in a nonperiodic (crystal plus externalfield) potential has been disputed. The starting point ofthe controversy was the original paper by Wannier(1960). He pointed out that, due to the translationalsymmetry of the crystal potential, if f(r) is an eigen-function of the scalar-potential Hamiltonian (corre-sponding to the perfect crystal plus the external field)with eigenvalue e, then any f(r1nd) is also an eigen-function with eigenvalue e1nDe , where De5eFd is theso-called Wannier-Stark splitting and d is the primitivelattice vector along the field direction. He concludedthat the translational symmetry of the crystal gives riseto a discrete energy spectrum, the so-called Wannier-Stark ladder. The states corresponding to these equidis-tantly spaced levels are localized, as schematicallyshown in Fig. 17(b) for the case of a semiconductor su-perlattice.

The existence of such energy quantization was dis-puted by Zak (1968), who pointed out that in an infinitecrystal the scalar potential 2F•r is not bounded, whichimplies a continuous energy spectrum. Thus the mainpoint of the controversy was related to the existence (orabsence) of Wannier-Stark ladders. More precisely, thepoint was to decide whether interband tunneling [ne-glected in the original calculation by Wannier (1960)] isstrong enough to destroy the Wannier-Stark energyquantization (and the corresponding Bloch oscillations).

It was only during the last decade that this contro-versy came to an end. From a theoretical point of view,most of the formal problems related to the nonperiodicnature of the scalar potential (superimposed on the pe-riodic crystal potential) were finally removed by using avector potential representation of the applied field (Kit-tel, 1963; Krieger and Iafrate, 1986). Within such a vec-tor potential picture, upper boundaries for the interbandtunneling probability were established at a rigorouslevel, showing that an electron may execute a number ofBloch oscillations before tunneling out of the band(Krieger and Iafrate, 1986; Nenciu, 1991), in qualita-tively good agreement with the earlier predictions of Ze-ner (1934) and Kane (1959).

The second mechanism impeding a fully periodic mo-tion is scattering by phonons, impurities, etc. [see Fig.17(a)]. This results in lifetimes shorter than the Blochperiod tB for all reasonable values of the electric field,so that Bloch oscillations should not be observable inconventional solids.


In superlattices, however, the situation is much morefavorable because of the smaller Bloch period tB result-ing from the small width of the mini-Brillouin zone inthe field direction (Bastard, 1989).

Indeed, the existence of Wannier-Stark ladders as wellas Bloch oscillations in superlattices has been confirmedby a number of recent experiments (Shah, 1999). Thephotoluminescence and photocurrent measurements ofthe biased GaAs/GaAlAs superlattices performed byMendez and co-workers (1988), together with the elec-troluminescence experiments by Voisin and co-workers(1988), provided the earliest evidence of field-inducedWannier-Stark ladders in superlattices. A few yearslater, Feldmann and co-workers (1992) were able tomeasure Bloch oscillations in the time domain through afour-wave-mixing experiment originally suggested byvon Plessen and Thomas (1992). A detailed analysis ofthe Bloch oscillations in the four-wave-mixing signal(which reflects the interband dynamics) was also per-formed by Leo and co-workers (Leo, Haring Bolivar,et al., 1992; Leisching et al., 1994).

In addition to the above interband polarization analy-sis, Bloch oscillations have been detected by monitoringthe intraband polarization, which, in turn, is reflected byanisotropic changes in the refractive index (Shah, 1999).Measurements based on transmittive electro-optic sam-pling were performed by Dekorsy and co-workers (1994;Dekorsy, Ott, et al., 1995). Finally, Bloch oscillationswere measured through a direct detection of terahertzradiation in semiconductor superlattices (Waschke et al.,1993; Roskos et al., 1994).

1. Two equivalent pictures

Let us now apply the theoretical approach presentedin Sec. II to the case of a semiconductor superlattice inthe presence of a uniform (space-independent) electricfield. The noninteracting carriers within the superlatticecrystal will then be described by the Hamiltonian Hc

0 inEq. (5), where now the electrodynamic potentials A2and w2 (in the following simply denoted as A and w)correspond to a homogeneous electric field E2(r,t)5F(t).

As pointed out in Sec. II.A, the natural quantum-mechanical representation is given by the eigenstates ofthis Hamiltonian:

F F2i\¹r1e

cA~r,t !G2

2m02ew~r,t !1Vl~r!Gfn~r!

5enfn~r!. (130)

However, due to the gauge freedom discussed in Sec. II,there is an infinite number of possible combinations ofA and w—and therefore of possible Hamiltonians—thatdescribe the same homogeneous electric field F(t). Inparticular, one can identify two independent choices: thevector-potential gauge


A~r,t !52cEt0

tF~ t8!dt8, w~r,t !50 (131)

and the scalar-potential gauge

A~r,t !50, w~r,t !52F~ t !•r. (132)

As shown by Rossi (1998), the two independent choicescorrespond to the well-known Bloch oscillation andWannier-Stark pictures, respectively. They simply reflecttwo equivalent quantum-mechanical representationsand therefore, any physical phenomenon can be de-scribed in both pictures.

More specifically, within the vector potential picture[Eq. (131)], the eigenfunctions fn in Eq. (130) are theso-called accelerated Bloch states (or Houston states;Houston, 1940; Kittel, 1963; Krieger and Iafrate, 1986).As discussed by Rossi (1998), this time-dependent rep-resentation constitutes a natural basis for the descriptionof Bloch oscillations, i.e., it provides a rigorousquantum-mechanical derivation of the accelerationtheorem [Eq. (129)], thus showing that this is not asimple semiclassical result.8 Within this representation,Bloch oscillations are fully described by the diagonalterms of the intraband density matrix (semiclassical dis-tribution functions). Therefore nondiagonal elementsdescribing phase coherence between different Blochstates do not contribute to the intraminiband dynamics.However, they are of crucial importance for the descrip-tion of interminiband dynamics, i.e., field-induced Zenertunneling, which in this Bloch state representation origi-nates from the time variation of our basis states [see Eq.(23)].

In contrast, within the scalar potential picture [Eq.(132)], the eigenfunctions fn in Eq. (130) are the well-known Wannier-Stark states (Wannier, 1960). Contraryto the previous Bloch picture, within this representationthe intraminiband Bloch dynamics originate from aquantum interference between different Wannier-Starkstates, thus involving nondiagonal elements of the intra-band density matrix.

In the remainder of this section we shall review a fewsimulated experiments on ultrafast carrier dynamics insemiconductor superlattices (Je et al., 1995; Koch et al.,1995; Meier et al., 1995; Rossi et al., 1995; Rossi, Gulia,et al., 1996; Rossi, Meier, et al., 1996). In this case, theBloch representation discussed in Sec. III.F.1 was em-ployed, limiting the set of interband density-matrix ele-ments to the diagonal ones, i.e., i5j . In addition, inco-herent scattering processes were treated within the usualMarkov limit discussed in Sec. II.D. Due to the rela-tively low electric fields considered, the ‘‘intracollisional

8The acceleration theorem [Eq. (129)] and the correspondingBloch oscillation dynamics are often regarded as a semiclassi-cal result compared to the Wannier-Stark picture. On the con-trary, they correspond to two different fully quantum-mechanical pictures.


field effect’’ (Brunetti, Jacoboni, and Rossi, 1989)—i.e.,the action of the field during the scattering process—wasneglected.9

In the simulated experiments reviewed here, the fol-lowing superlattice model was employed: The energydispersion and the corresponding wave functions alongthe growth direction (k i) were computed within thewell-known Kronig-Penney model (Bastard, 1989),while for the in-plane direction (k') an effective-massmodel was used. Only coupling to GaAs bulk phononswas considered. This, of course, is a simplifying approxi-mation which neglects any superlattice effect on thephonon dispersion, such as confinement of opticalmodes in the wells and barriers and the presence of in-terface modes (Rucker, Molinari, and Lugli, 1992; Moli-nari, 1994). However, while these modifications have im-portant consequences for phonon spectroscopies (likeRaman scattering), they are far less decisive for trans-port phenomena.10

2. Bloch oscillation analysis

We shall start by discussing the scattering-induceddamping of Bloch oscillations. In particular, we shallshow that in the low-density limit this damping is mainlydetermined by optical-phonon scattering (Rossi, Meier,et al., 1995, 1996), while at high densities the mainmechanism responsible for the suppression of Bloch os-cillations is found to be carrier-carrier scattering (Rossi,Gulia, et al., 1996).

All of the simulated experiments presented in this sec-tion refer to the superlattice structure considered byMeier et al. (1995): 111-Å GaAs wells and 17-ÅAl0.3Ga0.7As barriers. For such a structure there hasbeen experimental evidence for terahertz emission fromBloch oscillations (Roskos et al., 1994).

In the first set of simulated experiments, an initial dis-tribution of photoexcited carriers (electron-hole pairs) isgenerated by a 100-fs Gaussian laser pulse in resonancewith the first miniband exciton (\vL'1540 meV). Thestrength of the applied electric field is assumed to be 4kV/cm, which corresponds to a Bloch period (tB5h/eFd) of about 800 fs.

In the low-density limit (corresponding to a weak la-ser excitation), incoherent scattering processes do notalter the Bloch oscillation dynamics. This is due to thefollowing reasons: In agreement with recent experimen-tal (Roskos et al., 1994; von Plessen et al., 1994) and the-oretical (Meier et al., 1995; Rossi, Meier, et al., 1995,1996) investigations, at low temperature, scattering withLO phonons is not permitted and scattering with acous-tic phonons is unimportant for superlattices character-

9A detailed analysis of the intracollisional field effect in su-perlattices in the high-field regime can be found in Hader et al.(1997), where numerical solutions based on Wannier-Stark andplane-wave bases are compared.

10Indeed, it is now well known (Molinari, 1994) that the totalscattering rates are sufficiently well reproduced if the phononspectrum is assumed to be bulklike.


ized by a miniband width smaller than the LO phononenergy—as for the structure considered here—and forlaser excitations close to the band gap. Moreover, in thislow-density regime carrier-carrier scattering plays norole: Due to the quasielastic nature of Coulomb colli-sions, most of the scattering processes in the low-densitylimit are characterized by a very small momentum trans-fer. As a consequence, the momentum relaxation alongthe growth direction is negligible. As a result, on thispicosecond time scale the carrier system exhibits coher-ent Bloch oscillation dynamics, i.e., negligible scattering-induced dephasing. This can be clearly seen from thetime evolution of the carrier distribution as a function ofk i (i.e., averaged over k') shown in Fig. 18. During thelaser photoexcitation (t50) the carriers are generatedaround k i50, where the transitions are close to reso-nance with the laser excitation. According to the accel-eration theorem, the electrons are then shifted in kspace. When the carriers reach the border of the firstBrillouin zone, they are Bragg reflected. After about 800fs, corresponding to the Bloch period tB , the carriershave completed one oscillation in k space. As expected,the carriers execute Bloch oscillations without losing thesynchronism of their motion by scattering. This is againshown in Figs. 18(b)–(d), where we have plotted (b) themean kinetic energy, (c) the current, and (d) its timederivative, which is proportional to the emitted far field,i.e., the terahertz radiation. All three quantities exhibitoscillations characterized by the same Bloch period tB .Due to the finite width of the carrier distribution in kspace [see Fig. 18(a)], the amplitude of the oscillationsof the kinetic energy is somewhat smaller than the mini-band width. Since the scattering-induced dephasing isnegligible for this excitation condition, the oscillations ofthe current are symmetric around zero, which implies

FIG. 18. Full Bloch oscillation dynamics corresponding to alaser photoexcitation resonant with the first-miniband exciton:(a) time evolution of the electron distribution as a function ofk i ; (b) average kinetic energy; (c) current; and (d) terahertzsignal corresponding to the Bloch oscillations in (a).


that the time average of the current is equal to zero, i.e.,there is no dissipation.

As already pointed out, this ideal Bloch oscillationregime is typical of laser excitation close to the gap inthe low-density limit. Let us now discuss the case of laserphotoexcitation high in the band, still at low densities.Figure 19(a) shows the terahertz signal as obtained froma set of simulated experiments corresponding to differ-ent laser excitations (Meier et al., 1995). The differenttraces correspond to the emitted terahertz signal for in-creasing excitation energies. We can clearly see the pres-ence of Bloch oscillations in all cases. However, the os-cillation amplitude and decay (effective damping) isexcitation dependent.

In the case of laser excitation resonant with the firstminiband exciton considered above (see Fig. 18), wehave a strong terahertz signal. The amplitude of the sig-

FIG. 19. Bloch oscillations corresponding to laser photoexci-tation high in the band: (a) total terahertz signals for eightdifferent spectral positions of the exciting laser pulse (1540,1560, . . . , 1680 meV, from bottom to top); (b) individual tera-hertz signals of the electrons and holes in the different bandsfor a central spectral position of the laser pulse of 1640 meV;(c) experimentally observed terahertz transients for differentexcitation energies extending from just below the fundamentalband gap up to well into the second miniband. After [(a), (b)]Koch et al., 1995 and (c) Roskos et al., 1994.


nal decreases when the excitation energy is increased.Additionally, there are also some small changes in thephase of the oscillations, which are induced by theelectron-LO phonon scattering.

When the laser energy comes into resonance with thetransitions between the second electron and hole mini-bands (\vL'1625 meV), the amplitude of the tera-hertz signal increases again. The corresponding tera-hertz transients show an initial part, which is stronglydamped, and some oscillations for longer times that aremuch less damped. For a better understanding of theseresults, we show in Fig. 19(b) the individual terahertzsignals, originating from the two electron and twoheavy-hole minibands for the excitation with \v51640 meV. The Bloch oscillations performed by theelectrons within the second miniband are stronglydamped due to intra- and interminiband LO phononscattering processes (Meier et al., 1995; Rossi et al.,1995). Since the width of this second miniband (45 meV)is somewhat larger than the LO phonon energy, intra-miniband scattering is also possible whenever the elec-trons are accelerated into the high-energy region of theminiband. The terahertz signal originating from elec-trons within the first miniband shows an oscillatory be-havior with a small amplitude and a phase that is deter-mined by the time the electrons need to relax down tothe bottom of the band.

At the same time, the holes in both minibands exhibitundamped Bloch oscillations, since the minibands are soclose in energy that no LO phonon emission can occurunder these excitation conditions. The analysis showsthat at early times the tetrahertz signal is mainly deter-mined by the electrons within the second miniband. Atlater times the observed signal is due to heavy holes andelectrons within the first miniband.

The above theoretical analysis closely resembles ex-perimental observations obtained for a superlatticestructure very similar to the one modeled here as shownin Fig. 19(c) (Roskos et al., 1994). In these experimentsterahertz emission from Bloch oscillations was found.For some excitation conditions the oscillations were as-sociated with resonant excitation of the second mini-band. The general behavior of the magnitude of the sig-nals, the oscillations, and the damping are close to theresults shown in Figs. 19(a) and (b). In superlattices witha miniband width larger than the LO phonon energy, ithas been found that the terahertz radiation may even beenhanced by phonon emission because this process givesrise to a narrowing of the electron distribution (Wolteret al., 1997).

Finally, in order to study the density dependence ofthe Bloch oscillation damping, let us go back to the caseof laser excitations close to the gap. Figure 20(a) showsthe total (electrons plus holes) terahertz radiation as afunction of time for three different carrier densities.With increasing carrier density, carrier-carrier scatteringbecomes more and more important: Due to Coulombscreening, the momentum transfer in carrier-carrier scat-tering increases (its typical value being comparable tothe screening wave vector). This can be seen in Fig.


20(a), where for increasing carrier densities we realizean increasing damping of the terahertz signal. However,for the highest carrier density considered here, we alsodeal with a damping time of the order of 700 fs, which ismuch longer than the typical dephasing time, i.e., thedecay time of the interband polarization, associated withcarrier-carrier scattering. The dephasing time is typicallyinvestigated by means of four-wave-mixing measure-ments, and such multipulse experiments can be simu-lated as well (Lohner et al., 1993; Leitenstorfer, Lohner,Rick, et al., 1994). From a theoretical point of view, aqualitative estimate of the dephasing time is given by thedecay time of the incoherently summed polarization(Kuhn and Rossi, 1992b). Figure 20(b) shows this inco-herently summed polarization as a function of time forthe same three carrier densities as Fig. 20(a). As ex-pected, the decay times are always much shorter thanthe corresponding damping times of the terahertz signals[note the different time scales in Figs. 20(a) and (b)].This difference, discussed in more detail by Rossi, Gulia,et al., (1996) and Rossi (1998), can be understood as fol-lows: The fast decay times of Fig. 20(b) reflect the inter-band dephasing, i.e., the sum of the electron and holescattering rates. In particular, for the Coulomb interac-tion this means the sum of electron-electron, electron-hole, and hole-hole scattering. As in the case of bulkGaAs discussed in Sec. III.A, this last contribution isknown to dominate and determines the dephasing timescale. On the other hand, the total terahertz radiation inFig. 20(a) is the sum of the electron and hole contribu-tions. However, due to the small value of the hole mini-band width compared to that of the electron, the elec-tron contribution dominates. This means that theterahertz damping in Fig. 20(a) mainly reflects thedamping of the electron contribution. This decay, inturn, reflects the intraband dephasing of electrons, whichis due to electron-electron and electron-hole scatteringonly, i.e., there are no hole-hole contributions.

From the above analysis we can conclude that the de-cay time of the terahertz radiation due to carrier-carrierscattering differs considerably from the correspondingdephasing times obtained from a four-wave-mixing ex-periment: The first is a measurement of the intraband

FIG. 20. Bloch oscillations corresponding to laser photoexci-tation close to the gap: (a) total terahertz radiation as a func-tion of time; (b) incoherently summed polarization as a func-tion of time. After Rossi, Gulia, et al., 1996.


dephasing, while the second reflects the interbanddephasing.11

G. Carrier-phonon quantum kinetics

Most of the theoretical results discussed so far havebeen obtained by treating energy relaxation and dephas-ing processes on a semiclassical level, i.e., in terms ofscattering rates. The Markov approximation leading tothese rates, however, always assumes a separation be-tween the time scales relevant for the interaction-induced correlations and the dynamics of distributionfunctions or the envelope of polarizations. On a femto-second time scale this separation is no longer satisfiedand quantum-kinetic phenomena are of increasing im-portance either because they quantitatively modify thesemiclassical results or because they introduce com-pletely new features not present in a semiclassical pic-ture. In the following sections we shall review some phe-nomena in which quantum kinetics play an essentialrole. We shall concentrate on carrier-phonon quantumkinetics and then discuss a few results of the currentlyvery active field of Coulomb quantum kinetics.

1. Memory effects and energy-time uncertainty

As discussed in the theory part of this review, in thedensity-matrix formalism each interaction mechanismintroduces new types of dynamical variables. For the in-teraction with a classical light field, these correspond tothe various interband polarizations, while for the carrier-phonon coupling they are given by phonon-assisted den-sity matrices. Semiclassical transition rates are obtainedif these new variables are adiabatically eliminated bymeans of the Markov approximation. In the previoussections we have extensively discussed phenomena thatshowed the failure of this approximation in the case ofthe interband polarization and that could only be ex-plained by treating the interband polarization as an in-dependent dynamical variable. In particular, in Sec.III.A.1 we discussed the time-dependent broadening ofcarrier photogeneration associated with these dynamics.

Similarly, the carrier-phonon interaction is treated onthe quantum-kinetic level if the phonon-assisted densitymatrices are obtained from the solution of equations ofmotion like Eq. (53). The full set of equations for thecase of a homogeneous semiconductor can be found inSchilp et al. (1994a).

Broadening phenomena related to electron-phononquantum kinetics can be observed most clearly in a one-band model in which this is the only type of interaction.

11We stress that this difference between intraband and inter-band dephasing in superlattices is the same as was discussed inSec. III.A.3 for the case of bulk semiconductors, where thebroadening of the photoexcited carrier distribution is mainlydetermined by the decay of the interband polarization (inter-band dephasing), while the subsequent energy broadening ofthe electron distribution is due to electron scattering only (in-traband dephasing; see Fig. 6).


In Figs. 21(a) and (b) we compare the electron distribu-tions as functions of energy and time obtained from asemiclassical (Boltzmann) and a quantum-kinetic calcu-lation; here we consider the energy relaxation of an ini-tial Gaussian distribution within a one-band electronmodel interacting with LO phonons. In the semiclassicalcase replicas due to the emission of one, two, or threephonons appear on the low-energy side. Due to energyconservation in each scattering process, they exhibit thesame spectral shape of the initial distribution, in com-plete analogy with the time-independent shape of thesemiclassical generation rate in Fig. 3. The quantum-kinetic result, on the other hand, exhibits a strong time-dependent broadening. The replicas are initially verybroad; with increasing time they become narrower andapproach the semiclassical shape. This is again a conse-quence of energy-time uncertainty: At early times thesingle-particle energy is not yet a well-defined quantity.Figure 21(c) shows the distribution functions for themore realistic case of a two-band semiconductor (Schilpet al., 1994a, 1994b) in which the carriers are generatedby a 100-fs laser pulse. Now there is a time-dependentbroadening due to both the light absorption and thephonon emission process. The former is responsible forthe broadening of the highest energy peak, while thelatter broadens the subsequent replicas. It can be clearly

FIG. 21. Electron distribution functions in bulk GaAs: (a)semiclassical calculation; (b) and (c) quantum-kinetic calcula-tions. (a) and (b) show the relaxation of a given initial distri-bution in a one-band model, while (c) is the result of a two-band calculation for the case of an excitation by a 100-fs laserpulse. After Schilp, Kuhn, and Mahler, 1994a, 1994c.


seen that there is initially no minimum between two rep-licas; these minima build up with increasing time due todestructive interference at the semiclassically forbiddentransitions. Such time-dependent broadening due tophonon scattering were observed in two-color pump-probe experiments (Furst, Leitenstorfer, Laubereau, andZimmermann, 1997) in which electron-hole pairs weregenerated by a 120-fs laser pulse and the transmissionchange of a weak, spectrally broad 25-fs pulse was mea-sured. The corresponding differential transmission spec-tra for various delay times between pump and probepulse are shown in Fig. 22. The same features were alsoobtained in calculations based on the Green’s-functionformalism (Banyai et al., 1992; Tran Thoai and Haug,1993; Schmenkel, Banyai, and Haug, 1998) as well as inexactly solvable models of electron-phonon interaction(Meden et al., 1996; Schonhammer and Wohler, 1997;Schonhammer 1998).

2. Nonequilibrium phonons and energy conservation

In the previous section we have seen that energy-timeuncertainty is a characteristic feature of quantum kinet-ics. However, in the absence of an external light field, wehave a closed electron-phonon system, and the energy ofsuch a closed system should be constant without anyuncertainty. Here we want to address the question ofenergy conservation in greater detail. In the semiclassi-cal case the system is completely determined by the dis-tribution functions of electrons and holes. In the relax-ation process the electrons lose energy, which is takenup by the phonons. This is shown in Fig. 23(a), where

FIG. 22. Spectrally resolved transmission changes in GaAsmeasured for different time delays at a carrier density of 831014 cm23 (the dashed line denotes the excitation spectrum).After Furst, Leitenstorfer, Laubereau, and Zimmermann,1997.


the mean energies of the electrons and phonons areplotted as functions of time for the case of relaxationfrom a given initial distribution. As is clear from thesemiclassical scattering rates, the sum of the two ener-gies is a constant. Figure 23(b) shows the same energiesobtained from the quantum-kinetic calculation. Now thesum of electron and phonon energy is no longer con-stant; however, if the interaction energy given by theexpectation value of the interaction Hamiltonian is in-cluded, a constant total energy is recovered (Schilpet al., 1995). The initial increase of the electron and pho-non energies is balanced by the buildup of a negativeinteraction energy due to electron-phonon correlations.Thus we directly observe the buildup of polarons froman initially uncorrelated electron-phonon system.

In Sec. II.D.4 we discussed how the correlation expan-sion could be continued to take into account higher-order correlations. As shown there, neglecting the off-diagonal part and treating the diagonal part in Markovapproximation results in a complex self-energy describ-ing a damping of electron-phonon correlations. How-ever, it has been shown both analytically and numeri-cally that this approximation violates the conservation ofthe total energy (Schilp et al., 1995). In addition, itstrongly overestimates the broadening of the distribu-tion functions, in clear contrast to results known fromexactly solvable models and from experiments. How-ever, if all terms of the next order are taken into ac-count, it can be shown analytically that energy conserva-tion is again satisfied. Numerically, this ‘‘fourth Bornapproximation’’ has been studied for a one-dimensionalmodel (Zimmermann et al., 1998). It turns out that inmany cases it is a better approximation to completelyneglect third-order terms than to use the Markovianself-energy approximation.

FIG. 23. One-band model including nonequilibrium phononsin bulk GaAs: (a) semiclassical and (b) quantum-kinetic calcu-lation of total energy (solid line), as well as contributions dueto electrons (dashed lines), phonons (dotted line), and carrier-phonon interaction (dot-dashed line) as functions of time.


The interaction energy in Fig. 23(b) has a contributionthat oscillates with the phonon frequency. These oscilla-tions can be traced back analytically to the divergence atq50 in the Frohlich coupling matrix element, since thisgives rise to a divergence in the frequency spectrum ofthe phonon-assisted density matrices (Binder, Schilp,and Kuhn, 1998). Also, in the case of the phonon energythe oscillations are due to phonons with very small wavevectors mainly in a range that is semiclassically not al-lowed. In the semiclassically allowed region the non-equilibrium phonon distribution in a bulk semiconduc-tor is quite close to its semiclassical value. In a one-dimensional system, however, this is different. Here thesemiclassical model yields very sharp peaks in the pho-non distribution because, for an electron with a givenmomentum, the emission of phonons with only two dis-tinct wave vectors is compatible with energy and mo-mentum conservation. In contrast, in the quantum-kinetic case energy-time uncertainty leads to a smoothdistribution function (Binder et al., 1998).

3. Phonon quantum beats

Phonon quantum kinetics modify the carrier distribu-tion functions that can be measured, for example, inpump-probe experiments. However, experimentally ob-servable signals are in general changed only quantita-

FIG. 24. Phonon quantum beats: (a) incoherently summed po-larization as a function of time in bulk GaAs after excitationby a 50-fs pulse: dotted line, semiclassical calculation; dashedline, quantum-kinetic (qk) calculation with thermal phonons;solid line, quantum kinetic calculation with nonequilibriumphonons. (b) Four-wave-mixing signals at different densitiesexhibiting phonon quantum beats: solid line, experiment; dot-ted line, theory. After (a) Schilp et al., 1995 and (b) Banyaiet al. (1995).


tively. For a clear proof of quantum kinetics, it would bedesirable to have a phenomenon that is present only in aquantum-kinetic treatment. The situation is similar tothe case of the coherent interband polarization discussedabove: An explicit treatment of that variable quantita-tively changes the carrier photogeneration process; phe-nomena like four-wave-mixing or coherent control in thecase of temporally nonoverlapping pulses are simply notpresent if the polarization is adiabatically eliminated.Such a phenomenon, which is present only due tocarrier-phonon quantum kinetics, has indeed beenfound: If the semiconductor is excited by a sufficientlyshort laser pulse, the interband polarization exhibits anoscillatory decay, the oscillation frequency being of theorder of the phonon frequency (Tran Thoai and Haug,1993; Schilp et al., 1994a, 1995; Banyai, Vu, and Haug,1998). This is shown in Fig. 24(a) where the incoherentlysummed polarization is plotted as a function of time.While in the semiclassical case it is smoothly decaying,in quantum-kinetic cases with both thermal and non-equilibrium phonons, oscillations are present. These os-cillations are phonon quantum beats, and they arise dueto the simultaneous excitation of a direct optical transi-tion and a phonon-assisted one; therefore they rely onan electron-phonon correlation. The faster decay of thepolarization in the presence of nonequilibrium phononsis due to enhanced phonon absorption. Phonon quan-tum beats have been experimentally observed in time-integrated four-wave-mixing experiments. Figure 24(b)shows the diffracted signal at different densities afterexcitation of bulk GaAs with two 14.2-fs pulses (Banyaiet al., 1995). The dots are results of quantum-kinetic cal-culations. Phonon quantum beats have also been ob-tained in the case of quantum wells (Wehner et al.,1998).

Like other quantum beat phenomena, electron-phonon quantum beats can be controlled by a secondphase-locked pulse (Wehner et al., 1998; Steinbach et al.,1999). If the delay time corresponds to destructive inter-ference on the phonon-assisted transition, the beats areeliminated from the signal, as can be clearly seen in Fig.25(a), where the incoherently summed polarization isplotted as a function of time for the case of excitationwith two phase-locked 15-fs pulses whose delay timevaries from 42.8 fs (top) to 48.2 fs (bottom) in steps of0.3 fs (Axt et al., 1999; Kuhn et al., 1999). Figure 26shows corresponding experimentally observed four-wave-mixing signals (Wehner et al., 1998) for similar ex-citation conditions. Qualitatively, the appearance ofphonon quantum beats and their coherent control canbe understood quite well in terms of a simple modelbased on a two-level system coupled to a single phononmode. This model has the advantage of being exactlysolvable and thus the linear as well as the nonlinear re-sponse can be given exactly (Axt et al., 1999; Castellaand Zimmermann, 1999). Here the linear spectrum con-sists of a series of lines separated by the phonon energycorresponding to the zero-phonon line and sidebandsdue to phonon-assisted transitions (Mahan, 1990); thusthe excitation of phonon quantum beats by a pulse that


spectrally overlaps with at least two lines becomes obvi-ous. Since for this model the exact solution is known, itcan also be applied to systems with a stronger electron-phonon coupling in which the quantum-kinetic approachbased on the correlation expansion breaks down. In-deed, it has been shown that the coherent control ofphonon quantum beats in ZnSe—where clear signaturesof multiphonon transitions are observed—can be wellexplained (Steinbach et al., 1999). The drawback of thissimple model is, however, that it does not provide anydephasing, which has to be put in by hand. Therefore itcannot reproduce another interesting feature seen in ex-periment, namely, the fact that the decay of the four-wave-mixing signals also depends on the phase differ-ence between the two exciting pulses. This behavior iswell reproduced by the quantum-kinetic semiconductormodel, as is shown in Fig. 25(b), where the inverse of thedecay time extracted from the curves in Fig. 25(a) isplotted as a function of the delay time between thepulses. This clearly demonstrates that scattering pro-cesses can be influenced as long as the correlation be-tween initial and final states has not yet died out.

FIG. 25. Incoherently summed polarization: (a) excitation witha pair of phase-locked 15-fs pulses with delay times rangingfrom 42.8 fs (top) to 48.2 fs (bottom) in steps of 0.3 fs; (b)extracted decay constant as a function of the delay time. Pa-rameters refer to bulk GaAs. After Axt et al., 1999.


4. Carrier-phonon quantum kinetics in inhomogeneoussystems

In the results obtained from a quantum-kinetic treat-ment of carrier-phonon interaction discussed so far, wehave always assumed a spatially homogeneous excita-tion. Technically, this means that the single-particle den-sity matrices depend on only one wave vector k, and thephonon-assisted density matrices depend on two wavevectors. In the case of an inhomogeneous excitation, thiscorresponds to the zeroth order in a gradient expansion,as discussed in Sec. II.C.3. This is a good approximationif all inhomogeneities—introduced, for example, by theexciting laser pulses—occur on sufficiently large lengthscales, such that transport phenomena are not relevanton the ultrafast time scale considered in this review.However, much like the time scales the length scales inoptical experiments are also continuously reduced;therefore the interest in optically induced transport phe-nomena on ultrafast time scales is rapidly increasing.

Depending on the required spatial resolution, differ-ent experimental techniques have been developed toperform a spatially resolved optical excitation and/or de-tection. Spot sizes of the order of 1 mm have beenachieved by using lenses or microscope objectives(Yoon, Wake, and Wolfe, 1992; Otremba et al., 1999).The theoretical limitation of this technique, which isgiven by the diffraction limit of about l/2, where l is thewavelength of the light, can be overcome by using asolid immersion lens. A spot size of 355 nm correspond-ing to 0.41l was demonstrated with this method (Voll-mer et al., 1999). Alternatively, the light can be transmit-ted through holes in metal masks with diameters in themicron and submicron range (Hillmer et al., 1988; Gam-mon et al., 1996; Sonnichsen et al., 2000). The highestspatial resolution, of the order of 100 nm and less, can

FIG. 26. Four-wave-mixing (FWM) signals in GaAs measuredfor the case of excitation with a pair of phase-locked 15-fspulses with different delay times. After Wehner et al., 1998.


be achieved by using scanning near-field optical micro-scopes (Betzig et al., 1991; Hess et al., 1994; Richteret al., 1997; Guenther et al., 1999). In this case transportprocesses occur on a femtosecond time scale.

The semiclassical description of transport phenomenais again based on the Boltzmann picture in which scat-tering events occur locally in space and time betweenstates with well-defined energy and momentum. The dy-namical variable in the semiclassical theory is the single-particle distribution function. Its closest analog in quan-tum mechanics is the Wigner function defined in Eq.(37). Formally, the scattering terms are obtained fromthe equation of motion for the Wigner function as thezeroth order in the gradient expansion discussed in Sec.II.C.3, together with the Markov approximation, whichis equivalent to a zeroth-order gradient expansion intime. In the previous sections we have clearly seen thaton ultrafast time scales the conservation of the single-particle energies, which appears as a result of the Mar-kov approximation, is no longer satisfied. Instead,energy-time uncertainty strongly affects the dynamics.Similarly, on very short length scales the assumption ofslowly varying dynamical variables, which is the basis forthe gradient expansion, is no longer fulfilled. The uncer-tainty relation between position and momentum makesscattering process nonlocal in space. Both kinds of un-certainty relations are treated correctly if, in the case ofcarrier-phonon interaction, the single-particle and thephonon-assisted density matrices are taken as indepen-dent dynamical variables. Here the quantum-kinetictreatment has the additional advantage of being inde-pendent of the choice of the single-particle basis. This isin clear contrast to the semiclassical Boltzmann case, inwhich the Markov approximation requires selecting theinitial and final states of a scattering process, which, inturn, depends on this basis.

In this section we shall discuss a few phenomena re-lated to the spatiotemporal dynamics of locally gener-ated carriers. By using a short-pulse excitation through anear-field microscope, a spatially localized wave packetof electrons and holes is created. The subsequent dy-namics of such a wave packet strongly depend on theexcitation condition (Steininger et al., 1996; Hanewinkelet al., 1999): An excitation resonant with the 1s excitoncreates electron-hole pairs that are strongly bound, andthe resulting wave packet exhibits spatial broadening.An excitation high up in the band-to-band continuumproduces electron-hole pairs, which, in the absence ofphonon interaction, propagate like pulses through thesample. Electron-hole correlation effects are of minorimportance in this case. Phonon emission leads to en-ergy loss and thus to carrier group-velocity relaxation; asa consequence, it induces spatial broadening of thepulses with a subsequent transition from a ballistic to adiffusive transport regime (Steininger et al., 1997; Knorret al., 1998). This behavior, which can be well under-stood on a semiclassical level of description, is againmodified on ultrashort time scales by quantum-kineticfeatures. In Fig. 27 the Wigner function fk

e(r) for aGaussian wave packet prepared at time t50 at z50


with an excess energy of 120 meV is plotted at time t5100 fs. It is obtained from a semiclassical [Fig. 27(a)]and a quantum-kinetic [Fig. 27(b)] treatment of carrier-phonon interaction. These calculations have been per-formed for a GaAs cylindrical quantum wire model witha single subband. The broadening of the phonon replicasdue to energy-time uncertainty in the quantum-kineticresults—which was discussed above for the case of abulk semiconductor—is again clearly visible. Here thisbroadening also affects the spatial dynamics, due to oc-cupation of momentum states inaccessible by semiclassi-cally allowed processes. This phenomenon is analyzed inmore detail in Fig. 28, where the electron density ne(r)5V 21(kfk

e(r) [Figs. 28(a) and (b)] and the mean kineticenergy Ê(r)&5@ne(r)V#21(k (\2k2/2m) fk

e(r) [Figs.28(c) and (d)] are plotted along the wire axis after 100 fsand after 200 fs (Herbst, Axt, and Kuhn, 2000). Notethat the energies are only plotted at those positionswhere this quantity is well defined, i.e., where the den-sity is noticeably different from zero. For clarity the ini-tial values have also been included. The results corre-spond to three levels of the theory: (i) a calculationneglecting phonons (dot-dashed lines), (ii) a calculationincluding phonon scattering by using the semiclassicalBoltzmann approach (solid lines), and (iii) a fullquantum-kinetic treatment of the electron-phonon inter-action (dotted lines). Without phonons we find the ex-pected ballistic transport of electrons. The distributionremains Gaussian for all times and moves outward alongthe wire axis. Initially, the energy distribution is, by con-struction, position independent. Since highly energeticparticles cover a longer distance (compared to low-

FIG. 27. Wigner function at t5100 fs for a Gaussian wavepacket in a GaAs quantum wire prepared at t50, z50, andE5120 meV: (a) semiclassical and (b) quantum-kinetic calcu-lation.


energy ones), at later times the mean kinetic energy in-creases monotonically with the distance from the origin,as can be clearly seen in Figs. 28(c) and (d). In the low-temperature limit considered here, phonons can only beemitted. Therefore, in the Boltzmann case, electrons canonly lose energy, leading to a slowing down, as con-firmed by the corresponding electron density in Figs.28(a) and (b); we now find a noticeable electron densitybehind the ballistic wave front, but no density is built upahead of the wave front. Correspondingly, Ê(r)& is al-ways below the respective value obtained in the ballisticcase. Only at the front edge of the wave—formed bycarriers that have not yet emitted a phonon—is the bal-listic value reached.

In the quantum-kinetic case, also, phonons can onlybe emitted. Nevertheless, as a consequence of theenergy-time uncertainty, it is possible for some of theparticles to increase their kinetic energy as long as thecollision is not yet complete. The effect of this type ofprocess is clearly visible in Figs. 28(a) and (b), where thequantum-kinetic calculation predicts a small but finiteelectron occupation ahead of the ballistic wave front,i.e., we find electron densities in space regions that areout of reach from a semiclassical point of view. Asshown in Figs. 28(c) and (d), the particles ahead of theballistic wave front indeed have mean kinetic energiesdrastically increased compared to the highest values ob-tained in the ballistic description or in the Boltzmanncase. The occurrence of high kinetic energies at earlytimes is known from the quantum-kinetic treatment ofthe spatially homogeneous case (Schilp et al., 1994a).The new aspect here is that in the spatially inhomoge-neous case this effect is accompanied by a spatial sepa-ration of the highly energetic particles from the classicalwave front. For space regions that could be reachedwithin a semiclassical description, the Boltzmann result

FIG. 28. GaAs quantum wire along the wire axis at differenttimes for an initial excess energy of 120 meV: (a) and (b) elec-tron densities; (c) and (d) spatially resolved mean kinetic en-ergies obtained from a semiclassical calculation of a Bloch os-cillation (solid lines), a quantum-kinetic (QK; dotted lines),and a noninteracting-carrier (dot-dashed lines) calculation.The thin lines represent the initial values at time t50. AfterHerbst et al., 2000.


agrees quite well with the predictions of the quantum-kinetic theory.

Although initially the wire is locally neutral, the mo-tion of the electrons leads to a charge separation induc-ing an electrical field. According to Eq. (58), the inducedpotential acts as a source for the generation of coherentphonons. Figure 29 displays the z component of the lat-tice polarization Plat [Eq. (60)] along the wire axis after100 and 400 fs, together with the corresponding electrondensities. In addition to the curves obtained from a com-plete treatment including coherent and incoherentphonons (solid lines), we also plot results where the in-coherent phonons have been switched off (dotted lines).Here, spatial oscillations of the phonon polarization,whose amplitudes decrease with increasing distancefrom the origin, are clearly visible. The spatial extent ofthe lattice vibrations equals the region between the bal-listic wave fronts. The electrons passing a given spaceposition excite lattice vibrations that oscillate with theLO frequency at that site. Because these oscillationsstart in different sites at different times, the vibrationsare translated into spatial modulations. The decrease inthe phonon amplitudes is due to the fact that chargesaccumulated in a small region of a 1d wire induce anelectric field that decreases with increasing distance,thus reducing the forces responsible for the ion displace-ments. The inclusion of incoherent phonons leads to aneffective damping of the phonon oscillations, which be-comes stronger at later times. It should be noted thatEq. (58) is complete within the model considered, i.e.,there are no terms left out due to the truncation proce-dure. It follows that in the absence of anharmonicforces, the only way the incoherent phonons may influ-ence the coherent amplitudes is by their impact on thedistribution functions. The damping of the phonon am-plitudes is thus due to the spatial broadening of thewave packet and to the corresponding reduction of theinduced potentials that generate the coherent phonons.Coherent phonons can be experimentally detected byvarious techniques either in the optical range or directly

FIG. 29. GaAs quantum wire along the wire axis for an initialexcess energy of 80 meV: (a) and (b) electron densities; (c) and(d) z component of the lattice polarization including only co-herent phonons (dotted lines) and both coherent and incoher-ent phonons (solid lines). After Herbst et al., 2000.


in the terahertz frequency range of the phonon oscilla-tions. If coherent phonons are excited close to a surface,they modulate the reflectivity of the crystal and thus giverise to an oscillatory contribution in differential reflec-tion signals. Figure 30 shows such reflectivity changesdue to coherent phonons in bulk GaAs that were gener-ated by the charge separation of carriers excited in astrong surface electric field due to Fermi level pinning atthe surface (Cho et al., 1990). A recent review on coher-ent phonon phenomena in condensed matter is that ofDekorsy, Cho, and Kurz (2000).

H. Carrier-carrier quantum kinetics

As in the case of carrier-phonon coupling, in a carrier-carrier interaction the Boltzmann picture of instanta-neous scattering events occurring between states withwell-defined energies loses its validity over ultrashorttime scales. The temporal evolution of the single-particle density matrices is no longer completely deter-mined by specifying their values at a given time. In con-trast, it also depends on the values at previous times; thedynamics become non-Markovian. In the density-matrixapproach discussed in this review, this corresponds totreating the two-particle density matrices as indepen-dent variables and solving the corresponding equationsof motion like Eq. (79), in which higher-order correla-tion terms have been factorized into single-particle den-sity matrices. Besides scattering, the Coulomb interac-tion in a charged many-particle system, however, givesrise to an additional important feature: The interactionbetween two carriers is screened by the presence of allthe other carriers. It is intuitively clear that the carriersystem requires some time to react to a perturbation andthus to build up the screening. The characteristic time isessentially given by the inverse of the plasmon fre-quency (El Sayed, Schuster, et al., 1994) which, for mod-erate densities in GaAs, is of the order of 100 fs. There-fore this buildup of screening occurs on the same timescale relevant for quantum-kinetic phenomena. Here,

FIG. 30. Time-resolved reflectivity changes of (100)-orientedintrinsic GaAs for the case of excitation with a 50-fs pulse at 2eV exhibiting oscillations due to the coherent phonons. AfterCho et al., 1990.


we shall briefly review some of the results that havebeen achieved in this currently very active field of Cou-lomb quantum kinetics.

As discussed above, it takes about 100 fs for thescreening to build up. On the other hand, this meansthat on very short times of the order of 10 fs, screeningdoes not play an important role, and the calculations canbe performed with an unscreened potential. It is inter-esting to note that, while in the Boltzmann case Cou-lomb scattering rates with an unscreened potential di-verge, in a quantum-kinetic treatment all contributionsremain finite. On this level it has been found that theincoherently summed interband polarization, as well astime-integrated four-wave-mixing signals, exhibit a non-exponential decay with a density dependence of thecharacteristic decay time according to n21/3 (El Sayed,Banyai, and Haug, 1994; Vu et al., 1997), thus confirmingthe measured density dependence (Becker et al., 1988).Furthermore, the ultrafast redistribution of electrons asmeasured in pump-probe experiments was explainedqualitatively (Camescasse et al., 1996). Comparingquantum-kinetic and semiclassical calculations based onthe same static screening model, it turned out that thenon-Markovian relaxation is slightly delayed with re-spect to the Markovian one (Schafer, 1996).

For times approaching 100 fs the buildup of screeninghas to be taken into account. In the density-matrix ap-proach this requires including additional terms in theequations of motion of the two-particle density matrices,as discussed in Sec. II.E.4. In a Green’s-function ap-proach the screened potential is treated as a dynamicalvariable and a corresponding Dyson equation has to besolved. This leads to a considerable increase in theamount of computation. Nevertheless, such calculationsare possible today, and a series of interesting resultshave been obtained in the past few years. Indeed, thedecay of the interband polarization turns out to be inbetween the results obtained with a bare and with astatically screened Coulomb potential (Banyai, Vu,Mieck, and Haug, 1998). On this level, a good quantita-tive agreement between theory and experiment has beenachieved for both four-wave-mixing (Banyai, Vu, andHaug, 1998; Hugel et al., 1999) and pump-probe signals(Vu et al., 1999). For systems excited by an inhomoge-neous external potential, the generation and damping ofplasmons has also been studied (Kwong and Bonitz,2000).

Besides scattering and screening, there are other phe-nomena in a two- (or multi-) band semiconductor thatare related to density matrices involving four operators.Among these are transitions involving two electron-holepairs, which have attracted considerable interest in re-cent years. Typically, such two-pair states consist in abound state, the biexciton, and an exciton-exciton scat-tering continuum. While the bound biexciton has oftenbeen treated in terms of a few-level model, the con-tinuum is more complicated and is directly related toquantum kinetics. In the density-matrix approach thecentral quantity for these phenomena is the two-pairtransition Bj1 ,i1 ,j2 ,i2

introduced in Sec. III.C. However, it


turns out that for the kind of experimental conditions inwhich these effects are most important, the correlationexpansion method described in this review is not themost useful one for truncating the hierarchy, mainly be-cause it treats all two-particle density matrices on thesame level. Instead, virtually all calculations in this fieldhave been based on the dynamics-controlled truncationscheme (Axt and Stahl, 1994a, 1994b; Axt and Muka-mel, 1998), in which the variables are classified accord-ing to their order in the driving electric field. By for-mally solving the equation of motion for the two-pairtransition, one can derive a memory term in theequation for the interband polarization, which againshows the quantum-kinetic nature of this contribution(Ostreich, Schonhammer, and Sham, 1995; Axt, Victor,and Kuhn, 1998). Some aspects related to suchCoulomb-induced correlations in pump-probe and four-wave-mixing signals have already been discussed above.These correlations have been found to be particularlyimportant for describing the polarization dependence ofpump-probe and four-wave-mixing experiments (Axtet al., 1995; Mayer et al., 1995; Schafer et al., 1996). Onereason for their importance is the fact that the corre-lated exciton-exciton continuum strongly compensatessome mean field contributions and, therefore, it is essen-tial to obtain the correct line shape (Haase et al., 2000)as well as the transient polarization (Bartels et al., 1998)of four-wave-mixing experiments in which both excitonand biexciton contributions are present. Furthermore,such correlations may effectively couple different transi-tions and thus remove the clear distinction betweenquantum beats that arise from an interference in thesemiconductor material and polarization interference,where the signals interfere in the detector (Phillips andWang, 1999; Smirl et al., 1999). In higher-order dif-fracted signals—like in a six-wave-mixing experiment—even density matrices involving six operators have beenfound to be important for a correct description of thespectra because there are again strong compensation ef-fects (Bolton et al., 2000).

All the investigations discussed so far have been re-stricted to those contributions of the Coulomb interac-tion that conserve the number of electron-hole pairs, anapproximation that has also been applied in the theorypart of this review. In the case of high carrier densitiesand strong external fields, it is well known that othercontributions also become important, leading, for ex-ample, to Auger recombination and impact ionization.In particular, the latter process is an interesting candi-date for a quantum-kinetic treatment because it involvesthreshold behavior: Semiclassically, an electron mustreach at least an energy corresponding to one band gapabove the minimum of the band before it can create anadditional electron-hole pair. Since in a quantum-kinetictreatment the scattering process is not instantaneous,the electron can still gain energy from the electric field.Therefore with increasing field the threshold for impactionization is shifted to lower fields. This has been calcu-lated both for idealized parabolic bands (Quade et al.,1994) and for realistic materials based on full band struc-


tures (Redmer et al., 2000). Formally similar processesoccur in doped semiconductor nanostructures, where theCoulomb interaction leads to transitions between differ-ent subbands. Here again the threshold behavior of thesemiclassical theory is smoothed on short time scalesdue both to the energy-time uncertainty of the intersub-band transition and to an initial fast broadening causedby intrasubband scattering (Prengel, Scholl, and Kuhn,1997; Prengel and Scholl, 1999a, 1999b).

IV. SUMMARY AND CONCLUSIONS

The aim of the present paper was to provide a reviewof ultrafast phenomena in photoexcited semiconductors.The primary goal was to present a cohesive discussion ofboth coherent—i.e., phase-related—and incoherent—i.e., phase-breaking—processes in semiconductor bulkand heterostructures, as well as of their mutual interplayon a subpicosecond time scale. After a brief historicaloverview and a description of typical experimental tech-niques, we have shown how different phenomena can bedescribed and explained within the same theoreticalframework, based on the density-matrix formalism. Byapplying a correlation expansion we have derived thecontributions to the equations of motion for the relevantkinetic variables corresponding to various interactionmechanisms and we have discussed their physical mean-ing. Based on this theoretical approach, we have re-viewed a number of experimental and theoretical resultscrucial to understanding the microscopic origin of manyultrafast phenomena in semiconductors.

The main conclusion of the analysis presented in thepaper is twofold: On the one hand, purely macroscopicor phenomenological models, commonly used to de-scribe carrier relaxation and dephasing in the early daysof ultrafast optics, are no longer adequate for describingthe nonequilibrium dynamics of interacting carrier-phonon systems on a subpicosecond time scale. In con-trast, a kinetic description—based, for example, on thedensity-matrix formalism but of course also on other ap-proaches like the Green’s-function approach—is re-quired. On the other hand, the study of photoexcitedcarrier and phonon degrees of freedom on extremelyshort (femtosecond) time scales shows a variety of phe-nomena that cannot be explained in terms of the usualsemiclassical Boltzmann theory. This is mainly ascribedto a failure of the conventional Markov approximationover ultrashort time scales as well as a failure of theindependent treatment of different interaction mecha-nisms. One is then forced to employ fully quantum-kinetic formulations, which implies extending the set ofkinetic variables to higher-order correlation functionslike phonon-assisted and specific many-particle densitymatrices.

Generally speaking, the continuous improvement oftime-resolved optical spectroscopy, intimately related tothe ability to generate laser-pulse sequences on shorterand shorter time scales, has been accompanied by a pro-gressive refinement of their theoretical description, i.e.,from macroscopic to kinetic or quantum-kinetic models.


At this point a few comments are in order. As alreadymentioned, the field of ultrafast dynamics in semicon-ductors is so vast that it is not possible to treat all phe-nomena in a single review. For example, for most of thephenomena discussed in this review, we could treat theexciting light field as an external field and neglect thefeedback of the carrier system to this field. This waspossible because the samples were assumed to be opti-cally thin. In optically thick samples propagation effectsbecome important, leading, for example, to the forma-tion of polaritons (Frohlich et al., 1991) and self-inducedtransmission (Giessen et al., 1998). In this case interac-tion mechanisms may also give rise to new, renormalizedstates, for example, between polaritons and phonons,which have been called ‘‘phonoritons’’ (Hanke et al.,1999). Another example is the case of optically coupledsubsystems like multiple quantum well structures inwhich each quantum well ‘‘sees’’ the light field that iscoherently emitted by the other wells. Depending on theinterwell distance, these fields may add up constructivelyor destructively, thereby modifying the carrier dynamicsin the wells (Hubner et al., 1996). Recent progress in thefabrication and characterization of semiconductor mi-crocavities (Khitrova et al., 1999) offers the possibility ofstudying ultrafast carrier dynamics in the presence ofstrong light-matter coupling. Such systems constitute aunique laboratory for the study of basic phenomena re-lated to coupled exciton-light dynamics.

It clearly follows that a proper description of such ef-fects requires treating electromagnetic degrees of free-dom explicitly, which implies further extending our setof dynamical variables. If the light field is treated quan-tum mechanically, photon populations and variousphoton-assisted density matrices have to be includedalong with average fields, as described in Sec. II.F. Inthis respect, there is a natural parallelism between pho-tons and phonons: for both, the strong coupling withcarrier degrees of freedom introduces significant corre-lations in terms of photon- and phonon-assisted densitymatrices, which results in new renormalized states, i.e.,polaritons and polarons, respectively.

All the experiments discussed in this paper have beeninterpreted in terms of the correlation expansion of thedensity-matrix approach. The assumption that correla-tions involving an increasing number of carriers are ofdecreasing relevance has allowed us to limit the descrip-tion to a few higher-order density matrices. In the semi-classical case they could even be adiabatically elimi-nated. Typically, this assumption is fulfilled in thepresence of short-range interactions. The Coulomb po-tential, however, is a long-range interaction and it is onlythe presence of the many-body system that, due toscreening, reduces the length scale of this interaction.Indeed, as already mentioned in Sec. III.H, at suffi-ciently low densities there are experiments that requiredensity matrices involving at least six operators for aproper description (Bolton et al., 2000).

The assumption is even less fulfilled in quasi-zero-dimensional nanostructures, the so-called quantum dotsor semiconductor macroatoms (Jacak, Hawrylak, and


Wojs, 1998). Due to their discrete or atomiclike energyspectrum, they are intrinsically few-electron systems.Such macroatoms typically exhibit few-carrier effectsboth in transport (Schmidt et al., 1995; Tarucha et al.,1996; Rontani et al., 1998) and in optics (Dekel et al.,1998; Motohisa et al., 1998; Hohenester, Rossi, and Mo-linari, 1999), due to the limited number of confined elec-trons and/or holes in the nanostructure: the energyneeded to add an extra electron or electron-hole pair tothe system depends on the current number of electronsand/or holes. In order to describe such few-carrier ef-fects, correlations have to be properly taken into ac-count; in particular, the mean-field or Hartree-Fock de-scription, commonly employed as a first step in systemsof higher dimensionality, cannot be used to describesingle semiconductor macroatoms, for which the de-tailed knowledge of the microscopic few-electron wavefunction is required. Therefore the introduction ofquasi-zero-dimensional systems requires on the theoret-ical side a careful consideration of the relevant correla-tions between the particles.

We are therefore led to conclude that ultrafast opticalspectroscopy of semiconductors in the last three decadeshas allowed us to improve our understanding of non-equilibrium carrier dynamics significantly, from macro-scopic or phenomenologic models to kinetic andquantum-kinetic treatments, to partially microscopic ap-proaches.

We stress that a partially microscopic treatment of thecarrier dynamics in few-electron systems, apart frompractical difficulties, on the one hand leads to a reformu-lation of the concept of dephasing, and on the otherhand raises once again the measurement problem. In-deed, semiconductor macroatoms are currently consid-ered as potential candidates for quantum computation/information12 processing devices. To this end, a recentstudy (Zanardi and Rossi, 1998) has shown that a propertailoring of few-electron states may lead to a strong sup-pression of phonon-induced decoherence processes inquantum dot arrays. The key point is that in a quantum-correlated few-electron system, each electron does notinteract individually with environmental degrees of free-dom, in clear contrast to any single-particle kinetic for-mulation.

Finally, a crucial point for a proper modeling of ul-trafast experiments on few-electron systems is the de-scription of the measurement process. Since quantumcorrelation, i.e., entanglement effects, may play a signifi-cant role in such systems, it is vital to describe at thesame microscopic level (i) the initial-state preparation,(ii) its quantum-mechanical evolution, and (iii) the mea-surement process. On the theoretical side this is the onlyway to account properly for the increasingly sophisti-

12See, for example, Molotkov (1996), Loss and DiVincenzo(1998), Zanardi and Rossi (1998), and Biolatti et al. (2000); fora review on quantum computation, see, Steane (1998) and Di-Vincenzo and Bennet (2000).


cated experiments in ultrafast optics, for which the sepa-ration between ‘‘measured system’’ and ‘‘detector’’ be-comes ill defined.

ACKNOWLEDGMENTS

We wish to thank V. M. Axt for many fruitful discus-sions and for carefully reading the manuscript. We alsothank the many co-workers from our own groups as wellas from several other places with whom we had the plea-sure of collaborating over the past few years on thestudy of ultrafast dynamics in semiconductors. Theirwork as well as many discussions with other colleaguesactive in the field have strongly contributed to this re-view. This work was supported by the Commission ofthe European Union in the framework of the Trainingand Mobility of Researchers (TMR) network UltrafastQuantum Optoelectronics and by the Deutsche Fors-chungsgemeinschaft within the SchwerpunktprogrammQuantenkoharenz in Halbleitern.

REFERENCES

Abella, I. D., N. A. Kurnit, and S. R. Hartmann, 1966, Phys.Rev. 141, 391.

Allen, L., and J. H. Eberly, 1987, Optical Resonance and Two-Level Atoms (Dover, New York).

Altevogt, T., H. Puff, and R. Zimmermann, 1997, Phys. Rev. A56, 1592.

Argyres, P. N., 1962, Phys. Rev. 126, 1386.Assion, A., T. Baumert, J. Helbing, V. Seyfried, and G. Gerber,

1996, Chem. Phys. Lett. 259, 488.Atanasov, R., A. Hache, J. L. P. Hughes, H. M. van Driel, and

J. E. Sipe, 1996, Phys. Rev. Lett. 76, 1703.Axt, V. M., G. Bartels, and A. Stahl, 1996, Phys. Rev. Lett. 76,

2543.Axt, V. M., M. Herbst, and T. Kuhn, 1999, Superlattices Micro-

struct. 26, 117.Axt, V. M., and S. Mukamel, 1998, Rev. Mod. Phys. 70, 145.Axt, V. M., and A. Stahl, 1994a, Z. Phys. B: Condens. Matter

93, 195.Axt, V. M., and A. Stahl, 1994b, Z. Phys. B: Condens. Matter

93, 205.Axt, V. M., A. Stahl, E. J. Mayer, P. Haring Bolivar, S. Nusse,

K. Ploog, and K. Kohler, 1995, Phys. Status Solidi B 188, 447.Axt, V. M., K. Victor, and T. Kuhn, 1998, Phys. Status Solidi B

206, 189.Axt, V. M., K. Victor, and A. Stahl, 1996, Phys. Rev. B 53, 7244.Balescu, R., 1961, Phys. Fluids 4, 94.Balslev, I., and A. Stahl, 1988, Solid State Commun. 67, 85.Balslev, I., R. Zimmermann, and A. Stahl, 1989, Phys. Rev. B

40, 4095.Banyai, L., D. B. Tran Thoai, E. Reitsamer, H. Haug, D. Stein-

bach, M. U. Wehner, M. Wegener, T. Marschner, and W. Stolz,1995, Phys. Rev. Lett. 75, 2188.

Banyai, L., D. B. Tran Thoai, C. Remling, and H. Haug, 1992,Phys. Status Solidi B 173, 149.

Banyai, L., Q. T. Vu, and H. Haug, 1998, Phys. Rev. B 58,R13 341.

Banyai, L., Q. T. Vu, M. Mieck, and H. Haug, 1998, Phys. Rev.Lett. 81, 882.

Bar-Ad, S., and I. Bar-Joseph, 1991, Phys. Rev. Lett. 66, 2491.


Bartels, G., V. M. Axt, K. Victor, A. Stahl, P. Leisching, and K.Kohler, 1995, Phys. Rev. B 51, 11 217.

Bartels, G., G. C. Cho, T. Dekorsy, H. Kurz, A. Stahl, and K.Kohler, 1997, Phys. Rev. B 55, 16 404.

Bartels, G., A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J.Gutowski, 1998, Phys. Rev. Lett. 81, 5880.

Bastard, G., 1989, Wave Mechanics Applied to SemiconductorHeterostructures (Les Editions de Physique, Les Ulis).

Baumert, T., T. Brixner, V. Seyfried, M. Strehle, and G. Gerber,1997, Appl. Phys. B: Lasers Opt. B65, 779.

Becker, P. C., H. L. Fragnito, C. H. Brito Cruz, R. L. Fork, J.E. Cunningham, J. E. Henry, and C. V. Shank, 1988, Phys.Rev. Lett. 61, 1647.

Betzig, E., J. K. Trautmann, T. D. Harris, J. S. Weiner, and R.L. Kostelak, 1991, Science 251, 1468.

Binder, E., 1997, Koharenzzerfall in optisch angeregten Halble-itern (Neue Wissenschaft, Frankfurt am Main).

Binder, E., T. Kuhn, and G. Mahler, 1994, Phys. Rev. B 50,18 319.

Binder, E., D. Preisser, and T. Kuhn, 1997, Phys. Status SolidiB 204, 87.

Binder, E., J. Schilp, and T. Kuhn, 1998, Phys. Status Solidi B206, 227.

Binder, R., D. Scott, A. E. Paul, M. Lindberg, K. Henneberger,and S. W. Koch, 1992, Phys. Rev. B 45, 1107.

Biolatti, E., R. C. Iotti, P. Zanardi, and F. Rossi, 2000, Phys.Rev. Lett. 85, 5647.

Birkedal, D., and J. Shah, 1998, Phys. Rev. Lett. 81, 2372.Bloch, F., 1928, Z. Phys. 52, 555.Bogoliubov, N. N., 1967, Lectures on Quantum Statistics (Gor-

don and Breach, New York), Vol. 1.Bolton, S. R., U. Neukirch, L. J. Sham, D. S. Chemla, and V.

M. Axt, 2000, Phys. Rev. Lett. 85, 2002.Bonitz, M., 1998, Quantum Kinetic Theory (Teubner, Stutt-

gart).Bonitz, M., D. Semkat, and H. Haug, 1999, Eur. Phys. J. B 9,

309.Brumer, P., and M. Shapiro, 1995, Sci. Am. (Int. Ed.) 272 (3),

34.Brunetti, R., C. Jacoboni, and F. Rossi, 1989, Phys. Rev. B 39,

10 781.Camescasse, F. X., A. Alexandrou, D. Hulin, L. Banyai, D. B.

Tran Thoai, and H. Haug, 1996, Phys. Rev. Lett. 77, 5429.Capasso, F., 1990, Physics of Quantum Electron Devices

(Springer, Berlin).Carruthers, P., and F. Zachariasen, 1983, Rev. Mod. Phys. 55,

245.Castella, H., and R. Zimmermann, 1999, Phys. Rev. B 59,

R7801.Chachisvilis, M., H. Fidder, and V. Sundstrom, 1995, Chem.

Phys. Lett. 234, 141.Chen, C., Y.-Y. Yin, and D. S. Elliot, 1990, Phys. Rev. Lett. 64,

507.Cho, G. C., T. Dekorsy, H. J. Bakker, R. Hvel, and H. Kurz,

1996, Phys. Rev. Lett. 77, 4062.Cho, G. C., W. Kutt, and H. Kurz, 1990, Phys. Rev. Lett. 65,

764.Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, 1989,

Photons and Atoms (Wiley, New York).Combescot, M., and R. Combescot, 1989, Phys. Rev. B 40,

3788.Dabbicco, M., A. M. Fox, G. von Plessen, and J. F. Ryan, 1996,

Phys. Rev. B 53, 4479.


Dekel, E., D. Gershoni, E. Ehrenfreund, D. Spektor, J. M.Garcia, and P. M. Petroff, 1998, Phys. Rev. Lett. 80, 4991.

Dekorsy, T., H. Auer, C. Waschke, H. J. Bakker, H. G. Roskos,H. Kurz, V. Wagner, and P. Grosse, 1995, Phys. Rev. Lett. 74,738.

Dekorsy, T., G. C. Cho, and H. Kurz, 2000, in Light Scatteringin Solids VIII: Fullerenes, Semiconductor Surfaces, CoherentPhonons, edited by M. Cardona and G. Guntherodt, Topicsin Applied Physics Vol. 76 (Springer, Berlin), p. 169.

Dekorsy, T., A. M. T. Kim, G. C. Cho, K. Kohler, and H. Kurz,1996, in Ultrafast Phenomena X, edited by P. F. Barbara, J. G.Fujimoto, W. H. Knox, and W. Zinth, Springer Series inChemical Physics Vol. 62 (Springer, Berlin), p. 382.

Dekorsy, T., A. M. T. Kim, G. C. Cho, H. Kurz, A. V. Kuz-netsov, and A. Forster, 1996, Phys. Rev. B 53, 1531.

Dekorsy, T., P. Leisching, K. Kohler, and H. Kurz, 1994, Phys.Rev. B 50, 8106.

Dekorsy, T., R. Ott, H. Kurz, and K. Kohler, 1995, Phys. Rev. B51, 17 275.

DiVincenzo, D. P., and C. Bennet, 2000, Nature (London) 404,247.

Dupont, E., P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R.Wasilewski, 1995, Phys. Rev. Lett. 74, 3596.

Egri, I., 1985, Phys. Rep. 119, 363.El Sayed, K., L. Banyai, and H. Haug, 1994, Phys. Rev. B 50,

1541.El Sayed, K., S. Schuster, H. Haug, F. Herzel, and K. Hen-

neberger, 1994, Phys. Rev. B 49, 7337.Feldmann, J., K. Leo, J. Shah, D. A. B. Miller, J. E. Cunning-

ham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S.Schmitt-Rink, 1992, Phys. Rev. B 46, 7252.

Fluegel, B., N. Peyghambarian, G. Olbright, M. Lindberg, S.W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti,1987, Phys. Rev. Lett. 59, 2588.

Fox, A. M., J. J. Baumberg, M. Dabbicco, B. Huttner, and J. F.Ryan, 1995, Phys. Rev. Lett. 74, 1728.

Franz, W., 1958, Z. Naturforsch. A 13A, 484.Frensley, W. R., 1990, Rev. Mod. Phys. 62, 745.Frohlich, D., A. Kulik, B. Uebbing, A. Mysyrowicz, V. Langer,

H. Stolz, and W. von der Osten, 1991, Phys. Rev. Lett. 67,2343.

Furst, C., A. Leitenstorfer, A. Laubereau, and R. Zimmer-mann, 1997, Phys. Rev. Lett. 78, 3733.

Furst, C., A. Leitenstorfer, A. Nutsch, G. Trankle, and A.Zrenner, 1997, Phys. Status Solidi B 204, 20.

Gammon, D., E. S. Snoke, B. V. Shanabrook, D. S. Katzer, andD. Park, 1996, Phys. Rev. Lett. 76, 3005.

Garro, N., M. J. Snelling, S. P. Kennedy, R. T. Phillips, and K.H. Ploog, 1999, Phys. Rev. B 60, 4497.

Giessen, H., A. Knorr, S. Haas, S. W. Koch, S. Linden, J. Kuhl,M. Hetterich, M. Grun, and C. Klingshirn, 1998, Phys. Rev.Lett. 81, 4260.

Glutsch, S., U. Siegner, and D. S. Chemla, 1995, Phys. Rev. B52, 4941.

Gobel, E. O., K. Leo, T. C. Damen, J. Shah, S. Schmitt-Rink,W. Schafer, J. F. Muller, and K. Kohler, 1990, Phys. Rev. Lett.64, 1801.

Goodnick, S. M., and P. Lugli, 1988, Phys. Rev. B 38, 10 135.Guenther, T., V. Emiliani, F. Intonti, C. Lienau, and T. El-

saesser, 1999, Appl. Phys. Lett. 75, 3500.Guernsey, R., 1962, Phys. Rev. 127, 1446.Haacke, S., S. Schaer, B. Deveaud, and V. Savona, 2000, Phys.

Rev. B 61, R5109.


Haacke, S., R. A. Taylor, R. Zimmermann, I. Bar-Joseph, andB. Deveaud, 1997, Phys. Rev. Lett. 78, 2228.

Haas, S., F. Rossi, and T. Kuhn, 1996, Phys. Rev. B 53, 12 855.Haase, B., U. Neukirch, J. Gutowski, J. Nurnberger, W. Fasch-

inger, M. Behringer, D. Hommel, V. M. Axt, G. Bartels, andA. Stahl, 2000, J. Cryst. Growth 214, 856.

Hache, A., Y. Kostoulas, R. Atanasov, J. L. P. Hughes, J. E.Sipe, and H. M. van Driel, 1997, Phys. Rev. Lett. 78, 306.

Hader, J., T. Meier, S. Koch, F. Rossi, and N. Linder, 1997,Phys. Rev. B 55, 13 799.

Hahn, E. L., 1950, Phys. Rev. 80, 580.Hanewinkel, B., A. Knorr, P. Thomas, and S. Koch, 1999, Phys.

Rev. B 60, 8975.Hanke, L., D. Frohlich, A. L. Ivanov, P. B. Littlewood, and H.

Stolz, 1999, Phys. Rev. Lett. 83, 4365.Haring Bolivar, P., F. Wolter, A. Muller, H. Roskos, H. Kurz,

and K. Kohler, 1997, Phys. Rev. Lett. 78, 2232.Haug, H., 1988, in Optical Nonlinearities and Instabilities in

Semiconductors, edited by H. Haug (Academic, San Diego),p. 53.

Haug, H., 1992, Phys. Status Solidi B 173, 139.Haug, H., 2001, in Ultrafast Physical Processes in Semiconduc-

tors, edited by K. T. Tsen, Semiconductors and SemimetalsNo. 67 (Academic, San Diego), p. 205.

Haug, H., and L. Banyai, 1996, Solid State Commun. 100, 303.Haug, H., and A.-P. Jauho, 1996, Quantum Kinetics in Trans-

port and Optics of Semiconductors (Springer, Berlin).Haug, H., and S. W. Koch, 1993, Quantum Theory of the Op-

tical and Electronic Properties of Semiconductors (World Sci-entific, Singapore).

Heberle, A. P., J. J. Baumberg, E. Binder, T. Kuhn, K. Kohler,and K. H. Ploog, 1996, IEEE J. Sel. Top. Quantum Electron.2, 769.

Heberle, A. P., J. J. Baumberg, and K. Kohler, 1995, Phys. Rev.Lett. 75, 2598.

Herbst, M., V. M. Axt, and T. Kuhn, 2000, Phys. Status Solidi B221, 419.

Hess, H. F., E. Betzig, T. D. Harris, L. N. Pfeiffer, and K. W.West, 1994, Science 264, 1740.

Hess, O., and T. Kuhn, 1996, Phys. Rev. A 54, 3347.Hillmer, H., S. Hansmann, A. Forchel, M. Morohashi, E. Lo-

pez, H. P. Meier, and K. Ploog, 1988, Appl. Phys. Lett. 53,1937.

Hohenester, U., and W. Potz, 1997, Phys. Rev. B 56, 13 177.Hohenester, U., F. Rossi, and E. Molinari, 1999, Solid State

Commun. 111, 187.Hohenester, U., P. Supancic, P. Kocevar, X. Q. Zhou, W. Kutt,

and H. Kurz, 1993, Phys. Rev. B 47, 13 233.Houston, W. V., 1940, Phys. Rev. 57, 184.Hu, Y., R. Binder, S. Koch, S. Cundiff, H. Wang, and D. Steel,

1994, Phys. Rev. B 49, 14 382.Hubner, M., J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R.

Hey, and K. Ploog, 1996, Phys. Rev. Lett. 76, 4199.Hugel, W. A., M. F. Heinrich, M. Wegener, Q. T. Vu, L. Banyai,

and H. Haug, 1999, Phys. Rev. Lett. 83, 3313.Huhn, W., and A. Stahl, 1984, Phys. Status Solidi B 124, 167.Jack, L., P. Hawrylak, and A. Wojs, 1998, Quantum Dots

(Springer, Berlin).Jacoboni, C., and P. Lugli, 1989, The Monte Carlo Method for

Semiconductor Device Simulations (Springer, Vienna).Jacoboni, C., and L. Reggiani, 1983, Rev. Mod. Phys. 55, 645.Jahnke, F., and S. W. Koch, 1995, Appl. Phys. Lett. 67, 2278.


Jahnke, F., M. Koch, T. Meier, J. Feldmann, W. Schafer, P.Thomas, S. W. Koch, E. O. Gobel, and H. Nickel, 1994, Phys.Rev. B 50, 8114.

Jahnke, F., et al., 1996, Phys. Rev. Lett. 77, 5257.Jauho, A. P., and K. Johnsen, 1996, Phys. Rev. Lett. 76, 4576.Je, K.-C., T. Meier, F. Rossi, and S. Koch, 1995, Appl. Phys.

Lett. 67, 2978.Joffre, M., D. Hulin, A. Migus, and M. Combescot, 1989, Phys.

Rev. Lett. 62, 74.Joschko, M., M. Woerner, T. Elsaesser, E. Binder, T. Kuhn, R.

Hey, H. Kostial, and K. Ploog, 1997, Phys. Rev. Lett. 78, 737.Joschko, M., M. Woerner, T. Elsaesser, E. Binder, T. Kuhn, R.

Hey, H. Kostial, and K. Ploog, 1998, OSA Trends Opt. Pho-tonics Ser. 18, 80.

Kadanoff, L. P., and G. Baym, 1962, Quantum Statistical Me-chanics (Benjamin, New York).

Kane, E. O., 1959, J. Phys. Chem. Solids 12, 184.Kash, J. A., and J. C. Tsang, 1989, in Light Scattering in Solids

IV: Electron Scattering, Spin Effects, SERS and Morphic Ef-fects, edited by M. Cardona and G. Guntherodt, Topics inApplied Physics Vol. 54 (Springer, Berlin), p. 423.

Kash, J. A., J. C. Tsang, and J. M. Hvam, 1985, Phys. Rev. Lett.54, 2151.

Keldysh, L. V., 1958, Sov. Phys. JETP 34, 788.Keldysh, L. V., 1965, Sov. Phys. JETP 20, 1018.Khitrova, G., H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch,

1999, Rev. Mod. Phys. 71, 1591.Kim, D.-S., J. Shah, J. E. Cunningham, T. C. Damen, W. Scha-

fer, M. Hartmann, and S. Schmitt-Rink, 1992, Phys. Rev. Lett.68, 1006.

Kim, D.-S., J. Shah, T. C. Damen, W. Schafer, F. Jahnke, S.Schmitt-Rink, and K. Kohler, 1992, Phys. Rev. Lett. 69, 2725.

Kira, M., F. Jahnke, W. Hoyer, and S. W. Koch, 1999, Prog.Quantum Electron. 23, 189.

Kira, M., F. Jahnke, and S. Koch, 1998, Phys. Rev. Lett. 81,3263.

Kittel, C., 1963, Quantum Theory of Solids (Wiley, New York).Kner, P., W. Schafer, R. Lovenich, and D. S. Chemla, 1998,

Phys. Rev. Lett. 81, 5386.Knorr, A., F. Steininger, B. Hanewinkel, S. Kuckenburg, P.

Thomas, and S. W. Koch, 1998, Phys. Status Solidi B 206, 139.Kocevar, P., 1985, Physica B & C 134B, 155.Koch, S., T. Meier, T. Stroucken, A. Knorr, J. Hader, F. Rossi,

and P. Thomas, 1995, in Microscopic Theory of Semiconduc-tors: Quantum Kinetics, Confinement and Lasers, edited by S.W. Koch (World Scientific, Singapore), p. 81.

Krieger, J. B., and G. J. Iafrate, 1986, Phys. Rev. B 33, 5494.Kuhn, T., 1998, in Theory of Transport Properties of Semicon-

ductor Nanostructures, edited by E. Scholl (Chapman andHall, London), p. 173.

Kuhn, T., V. M. Axt, M. Herbst, and E. Binder, 1999, in Coher-ent Control in Atoms, Molecules, and Semiconductors, editedby W. Potz and W. A. Schroeder (Kluwer Academic, Dor-drecht), p. 113.

Kuhn, T., E. Binder, F. Rossi, A. Lohner, K. Rick, P. Leisching,A. Leitenstorfer, T. Elsaesser, and W. Stolz, 1994, in CoherentOptical Interactions in Semiconductors, Vol. 330 of NATOAdvanced Study Institutes Series B: Physics, edited by R. T.Phillips (Plenum, New York), p. 33.

Kuhn, T., and F. Rossi, 1992a, Phys. Rev. Lett. 69, 977.Kuhn, T., and F. Rossi, 1992b, Phys. Rev. B 46, 7496.Kuhn, T., F. Rossi, A. Leitenstorfer, A. Lohner, T. Elsaesser, W.

Klein, G. Boehm, G. Traenkle, and G. Weimann, 1996, in Hot


Carriers in Semiconductors, edited by K. Hess, J.-P. Leburton,and U. Ravaioli (Plenum, New York), p. 199.

Kurnit, N. A., I. D. Abella, and S. R. Hartmann, 1964, Phys.Rev. Lett. 13, 567.

Kuznetsov, A. V., 1991, Phys. Rev. B 44, 13 381.Kuznetsov, A. V., and C. J. Stanton, 1994, Phys. Rev. Lett. 73,

3243.Kwong, N.-H., and M. Bonitz, 2000, Phys. Rev. Lett. 84, 1768.Leisching, P., P. Haring Bolivar, W. Beck, Y. Dhaibi, F. Brugge-

mann, R. Schwedler, H. Kurz, K. Leo, and K. Kohler, 1994,Phys. Rev. B 50, 14 389.

Leitenstorfer, A., T. Elsaesser, F. Rossi, T. Kuhn, W. Klein, G.Boehm, G. Traenkle, and G. Weimann, 1996, Phys. Rev. B 53,9876.

Leitenstorfer, A., S. Hunsche, J. Shah, M. Nuss, and W. Knox,1999, Phys. Rev. Lett. 82, 5140.

Leitenstorfer, A., A. Lohner, T. Elsaesser, S. Haas, F. Rossi, T.Kuhn, W. Klein, G. Boehm, G. Traenkle, and G. Weimann,1994, Phys. Rev. Lett. 73, 1687.

Leitenstorfer, A., A. Lohner, K. Rick, P. Leisching, T. El-saesser, T. Kuhn, F. Rossi, W. Stolz, and K. Ploog, 1994, Phys.Rev. B 49, 16 372.

Lenard, A., 1960, Ann. Phys. (N.Y.) 10, 390.Leo, K., T. C. Damen, J. Shah, E. O. Gobel, and K. Kohler,

1990, Appl. Phys. Lett. 57, 19.Leo, K., E. O. Gobel, T. C. Damen, J. Shah, S. Schmitt-Rink,

W. Schafer, J. F. Muller, K. Kohler, and P. Ganser, 1991, Phys.Rev. B 44, 5726.

Leo, K., P. Haring Bolivar, F. Bruggemann, and R. Schwedler,1992, Solid State Commun. 84, 943.

Leo, K., J. Shah, T. C. Damen, A. Schulze, T. Meier, S.Schmitt-Rink, P. Thomas, E. Gobel, S. Chuang, M. S. C. Luo,W. Schafer, K. Kohler, and P. Ganser, 1992, IEEE J. QuantumElectron. 28, 2498.

Leo, K., J. Shah, E. O. Gobel, T. C. Damen, S. Schmitt-Rink,W. Schafer, and K. Kohler, 1991, Phys. Rev. Lett. 66, 201.

Leo, K., M. Wegener, J. Shah, D. S. Chemla, E. O. Gobel, T. C.Damen, S. Schmitt-Rink, and W. Schafer, 1990, Phys. Rev.Lett. 65, 1340.

Likforman, J.-P., M. Joffre, G. Cheriaux, and D. Hulin, 1995,Opt. Lett. 20, 2006.

Lindberg, M., R. Binder, and S. W. Koch, 1992, Phys. Rev. A45, 1865.

Lindberg, M., and S. W. Koch, 1988a, Phys. Rev. B 38, 3342.Lindberg, M., and S. W. Koch, 1988b, J. Opt. Soc. Am. B 5,

139.Lindberg, M., and S. W. Koch, 1988c, Phys. Rev. B 38, 7607.Lipavsky, P., V. Spicka, and B. Velicky, 1986, Phys. Rev. B 34,

6933.Lohner, A., K. Rick, P. Leisching, A. Leitenstorfer, T. El-

saesser, T. Kuhn, F. Rossi, and W. Stolz, 1993, Phys. Rev. Lett.71, 77.

Loss, D., and D. P. DiVincenzo, 1998, Phys. Rev. A 57, 120.Lugli, P., P. Bordone, L. Reggiani, M. Rieger, P. Kocevar, and

S. M. Goodnick, 1989, Phys. Rev. B 39, 7852.Luo, M. S. C., S. L. Chuang, P. C. M. Planken, I. Brener, and

M. C. Nuss, 1993, Phys. Rev. B 48, 11 043.Lyssenko, V. G., G. Valusis, F. Loser, K. Leo, M. M. Dignam,

and K. Kohler, 1997, Phys. Rev. Lett. 79, 301.Madelung, O., 1978, Introduction to Solid-State Theory

(Springer, Berlin).Mahan, G., 1990, Many-Particle Physics (Plenum, New York).Mayer, E. J., et al., 1994, Phys. Rev. B 50, 14 730.


Mayer, E. J., et al., 1995, Phys. Rev. B 51, 10 909.McQuarrie, D. A., 1976, Statistical Mechanics (Harper and

Row, New York).Meden, V., J. Fricke, C. Wohler, and K. Schonhammer, 1996,

Z. Phys. B: Condens. Matter 99, 357.Meden, V., C. Wohler, J. Fricke, and K. Schonhammer, 1995,

Phys. Rev. B 52, 5624.Meier, T., S. W. Koch, M. Phillips, and H. Wang, 2000, Phys.

Rev. B 62, 12 605.Meier, T., F. Rossi, P. Thomas, and S. Koch, 1995, Phys. Rev.

Lett. 75, 2558.Mendez, E., F. Agullo-Rueda, and J. Hong, 1988, Phys. Rev.

Lett. 60, 2426.Mieck, B., and H. Haug, 1999, Phys. Status Solidi B 213, 397.Molinari, E., 1994, in Confined Electrons and Photons: New

Physics and Applications, edited by E. Burstein and C. Weis-buch (Plenum, New York).

Molotkov, S. N., 1996, JETP Lett. 64, 237.Motohisa, J., J. J. Baumberg, A. P. Heberle, and J. Allam,

1998, Solid-State Electron. 42, 1335.Mycek, M.-A., S. Weiss, J.-Y. Bigot, S. Schmitt-Rink, D. S.

Chemla, and W. Schaefer, 1992, Appl. Phys. Lett. 60, 2666.Mysyrowicz, A., D. Hulin, A. Antonetti, A. Migus, W. T. Mas-

selink, and H. Morkoc, 1986, Phys. Rev. Lett. 56, 2748.Nenciu, G., 1991, Rev. Mod. Phys. 63, 91.Noll, G., U. Siegner, S. G. Shevel, and E. O. Gobel, 1990, Phys.

Rev. Lett. 64, 792.Nuss, M. C., P. C. M. Planken, I. Brener, H. G. Roskos, M. S.

C. Luo, and S. L. Chuang, 1994, Appl. Phys. B: Lasers Opt.B58, 249.

Oberli, D. Y., J. Shah, and T. C. Damen, 1989, Phys. Rev. B 40,1323.

Osman, M. A., and D. K. Ferry, 1987, Phys. Rev. B 36, 6018.Ostreich, T., K. Schonhammer, and L. J. Sham, 1995, Phys.

Rev. Lett. 74, 4698.Otremba, R., S. Grosse, M. Koch, J. Feldmann, V. M. Axt, T.

Kuhn, and W. Stolz, 1999, Solid State Commun. 109, 317.Pantke, K.-H., D. Oberhauser, V. G. Lyssenko, J. M. Hvam,

and G. Weimann, 1993, Phys. Rev. B 47, 2413.Phillips, M., and H. Wang, 1999, Solid State Commun. 111,

317.Phillips, R. T., 1994, Ed., Coherent Optical Processes in Semi-

conductors (Plenum, New York).Planken, P. C. M., I. Brener, M. C. Nuss, M. S. C. Luo, and S.

L. Chuang, 1993, Phys. Rev. B 48, 4903.Planken, P. C. M., M. C. Nuss, I. Brener, K. W. Goossen, M. S.

C. Luo, S. L. Chuang, and L. Pfeiffer, 1992, Phys. Rev. Lett.69, 3800.

Potz, W., 1996a, Appl. Phys. Lett. 68, 2553.Potz, W., 1996b, Phys. Rev. B 54, 5647.Potz, W., 1997a, Phys. Rev. Lett. 79, 3262.Potz, W., 1997b, Appl. Phys. Lett. 71, 395.Potz, W., 1998, Appl. Phys. Lett. 72, 3002.Potz, W., and P. Kocevar, 1983, Phys. Rev. B 28, 7040.Potz, W., and W. A. Schroeder, 1999, Eds., Coherent Control in

Atoms, Molecules, and Semiconductors (Kluwer, Dordrecht).Potz, W., M. Ziger, and P. Kocevar, 1995, Phys. Rev. B 52, 1959.Prengel, F., and E. Scholl, 1999a, Phys. Rev. B 59, 5806.Prengel, F., and E. Scholl, 1999b, Semicond. Sci. Technol. 14,

379.Prengel, F., E. Scholl, and T. Kuhn, 1997, Phys. Status Solidi B

204, 322.


Quade, W., E. Scholl, F. Rossi, and C. Jacoboni, 1994, Phys.Rev. B 50, 7398.

Rappen, T., U.-G. Peter, and M. Wegener, 1994, Phys. Rev. B49, 10 774.

Rappen, T., U.-G. Peter, M. Wegener, and W. Schafer, 1993,Phys. Rev. B 48, 4879.

Redmer, R., J. R. Madureira, N. Fitzer, S. M. Goodnick, W.Schattke, and E. Scholl, 2000, J. Appl. Phys. 87, 781.

Richter, A., G. Behme, M. Supitz, C. Lienau, T. Elsaesser, M.Ramsteiner, R. Notzel, and K. H. Ploog, 1997, Phys. Rev.Lett. 79, 2145.

Rieger, M., P. Kocevar, P. Lugli, P. Bordone, L. Reggiani, andS. M. Goodnick, 1989, Phys. Rev. B 39, 7866.

Rontani, M., F. Rossi, F. Manghi, and E. Molinari, 1998, Appl.Phys. Lett. 72, 957.

Roskos, H. G., M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A.M. Fox, S. Schmitt-Rink, and K. Kohler, 1992, Phys. Rev.Lett. 68, 2216.

Roskos, H., C. Waschke, R. Schwedler, P. Leisching, Y. Dhaibi,H. Kurz, and K. Kohler, 1994, Superlattices Microstruct. 15,281.

Rossi, F., 1998, in Theory of Transport Properties of Semicon-ductor Nanostructures, edited by E. Scholl (Chapman andHall, London), p. 283.

Rossi, F., A. di Carlo, and P. Lugli, 1998, Phys. Rev. Lett. 80,3348.

Rossi, F., M. Gulia, P. Selbmann, E. Molinari, T. Meier, P. Tho-mas, and S. Koch, 1996, in The Physics of Semiconductors,edited by M. Scheffler and R. Zimmermann (World Scientific,Singapore), p. 1775.

Rossi, F., S. Haas, and T. Kuhn, 1994, Phys. Rev. Lett. 72, 152.Rossi, F., T. Meier, P. Thomas, S. W. Koch, P. E. Selbmann, and

E. Molinari, 1995, Phys. Rev. B 51, 16 943.Rossi, F., T. Meier, P. Thomas, S. W. Koch, P. E. Selbmann, and

E. Molinari, 1996, in Hot Carriers in Semiconductors, editedby K. Hess, J.-P. Leburton, and U. Ravaioli (Plenum, NewYork), p. 157.

Rota, L., P. Lugli, T. Elsaesser, and J. Shah, 1993, Phys. Rev. B47, 4226.

Rota, L., F. Rossi, P. Lugli, and E. Molinari, 1995, Phys. Rev. B52, 5183.

Rucker, H., E. Molinari, and P. Lugli, 1992, Phys. Rev. B 45,6747.

Ryan, J. F., and M. C. Tatham, 1992, in Hot Carriers in Semi-conductor Nanostructures: Physics and Applications, editedby J. Shah (Academic, Boston), p. 345.

Savasta, S., and R. Girlanda, 1999, Phys. Rev. B 59, 15 409.Schafer, W., 1996, J. Opt. Soc. Am. B 13, 1291.Schafer, W., D. S. Kim, J. Shah, T. C. Damen, J. E. Cunning-

ham, K. W. Goosen, L. N. Pfeiffer, and K. Kohler, 1996, Phys.Rev. B 53, 16 429.

Schilp, J., T. Kuhn, and G. Mahler, 1994a, Phys. Rev. B 50,5435.

Schilp, J., T. Kuhn, and G. Mahler, 1994b, Semicond. Sci. Tech-nol. 9, 439.

Schilp, J., T. Kuhn, and G. Mahler, 1994c, in Proceedings of the22nd International Conference on the Physics of Semiconduc-tors, Vancouver, edited by D. J. Lockwood (World Scientific,Singapore), p. 341.

Schilp, J., T. Kuhn, and G. Mahler, 1995, Phys. Status Solidi B188, 417.

Schmenkel, A., L. Banyai, and H. Haug, 1998, J. Lumin. 76-77,134.


Schmidt, T., M. Tewordt, R. H. Blick, R. J. Haug, D. Pfann-kuche, K. von Klitzing, A. Forster, and H. Luth, 1995, Phys.Rev. B 51, 5570.

Schmitt-Rink, S., and D. S. Chemla, 1986, Phys. Rev. Lett. 57,2752.

Schmitt-Rink, S., D. S. Chemla, and H. Haug, 1988, Phys. Rev.B 37, 941.

Schmitt-Rink, S., J. Lowenau, and H. Haug, 1982, Z. Phys. B:Condens. Matter 47, 13.

Schmitt-Rink, S., et al., 1992, Phys. Rev. B 46, 10 460.Schoenlein, R. W., D. M. Mittelman, J. J. Shiang, A. P. Alivi-

satos, and C. V. Shank, 1993, Phys. Rev. Lett. 70, 1014.Scholz, R., T. Pfeifer, and H. Kurz, 1993, Phys. Rev. B 47,

16 229.Schonhammer, K., 1998, Phys. Rev. B 58, 3518.Schonhammer, K., and C. Wohler, 1997, Phys. Rev. B 55,

13 564.Schultheis, L., J. Kuhl, A. Honold, and C. W. Tu, 1986, Phys.

Rev. Lett. 57, 1797.Schultheis, L., M. D. Sturge, and J. Hegarty, 1985, Appl. Phys.

Lett. 47, 995.Schulzgen, A., R. Binder, M. E. Donovan, M. Lindberg, K.

Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian,1999, Phys. Rev. B 82, 2346.

Shah, J., 1992, Ed., Hot Carriers in Semiconductor Nanostruc-tures: Physics and Applications (Academic, Boston).

Shah, J., 1999, Ultrafast Spectroscopy of Semiconductors andSemiconductor Nanostructures (Springer, Berlin).

Shah, J., B. Deveaud, T. C. Damen, W. T. Tsang, A. C. Gos-sard, and P. Lugli, 1987, Phys. Rev. Lett. 59, 2222.

Shah, J., and R. C. C. Leite, 1969, Phys. Rev. Lett. 22, 1304.Shank, C. V., R. L. Fork, R. F. Leheny, and J. Shah, 1979, Phys.

Rev. Lett. 42, 112.Shapiro, M., and P. Brumer, 1986, J. Chem. Phys. 84, 4103.Shapiro, M., and P. Brumer, 1997, J. Chem. Soc., Faraday

Trans. 93, 1263.Sieh, C., T. Meier, F. Jahnke, A. Knorr, and S. Koch, 1999,

Phys. Rev. Lett. 82, 3112.Smirl, A. L., M. J. Stevens, X. Chen, and O. Buccafusca, 1999,

Phys. Rev. B 60, 8267.Smith, G. O., E. J. Mayer, J. Kuhl, and K. Ploog, 1994, Solid

State Commun. 92, 325.Sonnichsen, C., A. C. Duch, G. Steininger, M. Koch, G. von

Plessen, and J. Feldmann, 2000, Appl. Phys. Lett. 76, 140.Stanton, C. J., D. W. Bailey, and K. Hess, 1988, IEEE J. Quan-

tum Electron. 24, 1614.Steane, A., 1998, Rep. Prog. Phys. 61, 117.Steinbach, D., G. Kocherscheidt, M. U. Wehner, H. Kalt, M.

Wegener, K. Ohkawa, D. Hommel, and V. M. Axt, 1999,Phys. Rev. B 60, 12 079.

Steininger, F., A. Knorr, T. Stroucken, P. Thomas, and S. W.Koch, 1996, Phys. Rev. Lett. 77, 550.

Steininger, F., A. Knorr, P. Thomas, and S. W. Koch, 1997, Z.Phys. B: Condens. Matter 103, 45.

Supancic, P., U. Hohenester, P. Kocevar, D. Snoke, R. M. Han-nak, and W. W. Ruhle, 1996, Phys. Rev. B 53, 7785.

Tarucha, S., D. G. Austing, T. Honda, R. J. van der Hage, andL. P. Kouwenhoven, 1996, Phys. Rev. Lett. 77, 3613.

Tran Thoai, D. B., and H. Haug, 1993, Phys. Rev. B 47, 3574.Ulbrich, R., 1977, Solid-State Electron. 21, 51.Ulbrich, R. G., 1988, in Optical Nonlinearities and Instabilities

in Semiconductors, edited by H. Haug (Academic, San Di-ego), p. 121.


van Driel, H. M., 1979, Phys. Rev. B 19, 5928.Voisin, P., J. Bleuse, C. Bouche, S. Gaillard, C. Alibert, and A.

Regreny, 1988, Phys. Rev. Lett. 61, 1639.Vollmer, M., H. Giessen, W. Stolz, W. W. Ruhle, L. Ghislain,

and V. Elings, 1999, Appl. Phys. Lett. 74, 1791.von der Linde, D., J. Kuhl, and H. Klingenberg, 1980, Phys.

Rev. Lett. 44, 1505.von Plessen, G., T. Meier, J. Feldmann, E. O. Gobel, P. Tho-

mas, K. W. Goossen, J. M. Kuo, and R. F. Kopf, 1994, Phys.Rev. B 49, 14 058.

von Plessen, G., and P. Thomas, 1992, Phys. Rev. B 45, 9185.Vu, Q., L. Banyai, H. Haug, F. Camescasse, J.-P. Likforman,

and A. Alexandrou, 1999, Phys. Rev. B 59, 2760.Vu, Q., L. Banyai, P. Tamborenea, and H. Haug, 1997, Euro-

phys. Lett. 40, 323.Wang, H., K. Ferrio, D. Steel, Y. Z. Hu, R. Binder, and S. W.

Koch, 1993, Phys. Rev. Lett. 71, 1261.Wang, H., J. Shah, T. C. Damen, and L. N. Pfeiffer, 1995, Phys.

Rev. Lett. 74, 3065.Wannier, G. H., 1960, Phys. Rev. 117, 432.Warren, W. S., H. Rabitz, and M. Dahleh, 1993, Science 259,

1581.Waschke, C., H. G. Roskos, R. Schwedler, K. Leo, H. Kurz,

and K. Kohler, 1993, Phys. Rev. Lett. 70, 3319.


Wegener, M., D. S. Chemla, S. Schmitt-Rink, and W. Schafer,1990, Phys. Rev. A 42, 5675.

Wehner, M. U., M. H. Ulm, D. S. Chemla, and M. Wegener,1998, Phys. Rev. Lett. 80, 1992.

Weiner, A., D. Leaird, J. Patel, and J. Wullert, 1990, Opt. Lett.15, 326.

Weiss, S., M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, and D. S.Chemla, 1992, Phys. Rev. Lett. 69, 2685.

Wigner, E., 1932, Phys. Rev. 40, 749.Woerner, M., and J. Shah, 1998, Phys. Rev. Lett. 81, 4208.Wolter, F., H. G. Roskos, P. Haring Bolivar, G. Bartels, H.

Kurz, H. T. Grahn, and R. Hey, 1997, Phys. Status Solidi B204, 83.

Wyld, H. W., and B. D. Fried, 1963, Ann. Phys. (N.Y.) 23, 374.Yoon, H. W., D. R. Wake, and J. P. Wolfe, 1992, Phys. Rev. B

92, 13 461.Zak, J., 1968, Phys. Rev. Lett. 20, 1477.Zanardi, P., and F. Rossi, 1998, Phys. Rev. Lett. 81, 4752.Zener, C., 1934, Proc. R. Soc. London, Ser. A 145, 523.Zimmermann, R., 1990, Phys. Status Solidi B 159, 317.Zimmermann, R., and J. Wauer, 1994, J. Lumin. 58, 271.Zimmermann, R., J. Wauer, A. Leitenstorfer, and C. Furst,

1998, J. Lumin. 76-78, 34.

Theory of Ultrafast Phenonmna in Photo Excited Semiconductor

Documents