VOLUME 77, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 1996

ermany

Ultrafast Electron Redistribution through Coulomb Scattering in Undoped GaAs:Experiment and Theory

F. X. Camescasse, A. Alexandrou, and D. HulinLaboratoire d’Optique Appliquée, Ecole Nationale Supérieure de Techniques Avancées-Ecole Polytechnique,

Unité de Recherche Associée au Centre National de la Recherche Scientifique 1406, Centre de l’Yvette,F-91761 Palaiseau Cedex, France

L. Bányai, D. B. Tran Thoai,* and H. HaugInstitut für Theoretische Physik, J. W. Goethe Universität Frankfurt, Robert-Mayer-Strasse 8, D-60054 Frankfurt a. M., G

(Received 9 July 1996)

We report the observation of spectral hole burning exclusively due to the nonequilibrium electronpopulation in a nondegenerate pump-test configuration. The rapid redistribution of electrons aswell as the other features of the differential absorption spectra are well described by a theoryusing quantum-kinetic bare Coulomb collisions in the framework of the semiconductor Blochequations. [S0031-9007(96)01740-1]

PACS numbers: 78.47.+p, 42.65.Re, 71.10.–w, 78.20.Bh

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The redistribution of nonequilibrium carrier populationin semiconductors has attracted considerable interesthe last two decades. The tremendous progress of ftosecond lasers in terms of pulse duration and stabhas rendered possible the observation of the initial staof carrier relaxation [1–6] and the study of very low carier densities [6]. However, studying the contributionsdifferent scattering mechanisms such as LO-phononcarrier-carrier scattering remains a difficult task, becamost experiments measure a combination of electronhole dynamics and the signals in ultrashort-pulse expments contain coherence effects [7] and are not sopopulation dependent. Indeed, standard pump-test expments [1,3,5,6] measure the absorption saturation duthe Pauli exclusion principle and are sensitive to the sof the electron and hole distribution functions (fe andfh,respectively) while time-resolved luminescence expements [2] measure the productfefh. A selective inves-tigation of the hole dynamics has been used in Ref.to measure the heavy-hole thermalization time. Howevthis method cannot measure the complete hole distribuand the initially injected hole population.

Here we have used a modified pump-test scheme inder to isolate the electron dynamics [4]: the pump puexcites electrons from the heavy-hole (HH) and light-ho(LH) valence bands while the test pulse probes thesorption saturation of the interband transition from tsplit-off (SO) valence band to the conduction band C (sinset of Fig. 1). Because of the large spin-orbit splting in GaAs (340 meV), no holes are present in tSO band and the differential absorption signal2Da ­awithout2pump 2 awith2pump depends on the electron distribution only. This method has a further important avantage: pump and test are at different wavelengths whallows the observation of spectral hole burning due toinitially injected electron population without any contribution from the induced-grating coherence effect [7,9] whi

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considerably complicates the interpretation of standpump-test experiments. Moreover, due to the isotromatrix element of the SO-C transition, the measured snal is equally sensitive to the presence of electrons wall possible wave vector directions.

We report the first observation of hole burning whiccan be attributed exclusively to the electron populatioWhile in previous experiments hole burning was ndiscernible [4], recent ameliorations of the experimensetup have permitted the observation of hole-burnsignals for carrier densities ranging from a few1015 toa few 1018 cm23 and for excess photon energies rangifrom 50 to 110 meV. The ensemble of these results wbe discussed elsewhere. In this Letter, we concenton the very short pump-test delay times at moderdensities, the rapid redistribution of electrons causing

FIG. 1. Differential absorption spectra for the followinpump-test delay times:280, 240, 0, 40, and80 fs. The insetshows the pump-test configuration.

© 1996 The American Physical Society 5429

VOLUME 77, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 1996

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disappearance of hole burning, and the comparison ofearly-time behavior with theory.

We used a Ti:sapphire mode-locked oscillator (Cohent Mira) and regenerative amplifier system (CoherRegA) both pumped by an argon-ion laser. Part ofoutput generates a spectral continuum and is used atest pulse. After chirp compensation of the continuwith a combination of prisms and gratings we obtanearly Fourier-transform-limited pulses with a duration30 and 130 fs for the test and pump pulses, respectivThe pulses were focused down to50 and150 mm for thetest and pump, respectively, on the sample which waintrinsic GaAs layer of thicknessd ­ 0.65 mm antireflec-tion coated on both sides and held at 15 K. In orderminimize the noise, a shutter is used in the optical pof the pump at an 8-Hz rate. In addition, a referenbeam is simultaneously detected on a different trackthe CCD detector and is used to normalize the transmtest beam.

The differential absorption spectra for a pump width15 meV and a pump energy of 1.589 eV (excess eneof 70 meV with respect to the band gap) and for variopump-test delay times are shown in Fig. 1. The cardensity was estimated to be6 3 1016 cm23. The zerodelay is defined as the coincidence of the pump andmaxima and is taken at the middle of the integratsignal rise time. The spectra show two broad peakabout 1.913 and 1.950 eV due to spectral hole burnassociated with the electron populations photoexcfrom the LH and HH bands, respectively. Note thattwo peaks disappear already before the end of the ppulse. While it is clear that the signal in the spectregion from 1.88 to 1.96 eV is dominated by the inductransmission due to the electron population, the induabsorption above 1.97 eV and the oscillatory structaround 1.86 eV cannot be easily explained. Furthermeven in the hole-burning region the differential absorptspectra do not directly reflectfestd due to energy-timeuncertainty and excitonic Coulomb effects. Therefoa theoretical analysis in terms of a quantum-kineapproach is necessary, since the commonly used thebased on the golden-rule long-time limit are not applicaat such ultrashort times.

Quantum kinetics is a generic name for the theoryscribing kinetics with memory on very short time sca(see Ref. [10] for a review). The Markovian rate eqution which has been so successful in the description ocosecond and nanosecond phenomena should be regas a limiting case of the quantum kinetics. In the expmental results described in this Letter, many scattemechanisms are involved. It is most interesting to loat the limited short-time regime where Coulomb sctering dominates because, although the quantum kiics of the electron-LO-phonon interaction (at low carrdensities) has already received attention in the pastyears [11–15] and some observed quantum-kinetic effhave been explained [16], no treatment of the quant

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kinetic Coulomb scattering for real experiments has beattempted yet.

Coulomb scattering presents peculiar features whrequire imperatively a quantum-kinetic formulation.is well known already from the equilibrium theory oscreening that the screened Coulomb potential has a1

q2

singularity asq ! 0 at any finite frequencyv. The sin-gularity is absent only atv ­ 0. However, a vanishingfrequency implies an infinite time. Therefore, the sigularity is always present and plays an important roat short time scales. This singularity which corresponto that of the bare Coulomb potential is fatal for thBoltzmann equation since the argument of the enerconservingd function also vanishes atq ­ 0 and thecollision integral diverges. The energy-time uncertainwhich is taken into account by quantum kinetics automacally eliminates the divergence [17].

For times less than a typical plasma period, screenis negligible [17] and thus the relatively complicated thory of time-dependent screening [18–20] can be avoidWe may also simplify the theoretical task by restrictinour calculations to times less than or comparable witheffective interband polarization decay time. In our cofiguration, where there is no interference betweenpump and test polarizations, one can use a simple pnomenological description of the polarization collisioterm and concentrate only on the quantum-kinetic cosion terms of the electron and hole populations excitedthe pump.

The semiconductor Bloch equations [21] for the poplations and polarizations in the case of the pump field a

≠fe,$kstd≠t

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. (3)

Here a ­ HH, LH and the renormalized energiese andRabi frequenciesV are given by

ei,$kstd ­ e0i,$k

2X$k0

V$k2$k0fi,$k0std, i ­ e, HH, LH ,

(4)

h̄VPa,$k

std ­ da,$kEPstd 1 2X$k0

V$k2$k0pa,$k0std . (5)

In the above equations,EPstd is the envelope of the pumpfield with frequencyvP, da,$k are the respective interbandipole matrix elements, andV $q is the Fourier transform ofthe Coulomb potential. Since we consider an isotromodel with dipole matrix elements independent of t

VOLUME 77, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 1996

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field polarization and$k, we take them to be equal foheavy and light holes.

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The quantum-kinetic collision terms for the populatioare (i, i0 ­ e, HH, LH):

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In the Markovian limit one gets from this equation thusual golden-rule rate equation.

The phenomenological collision term of the polariztion is

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pa,$kstd , a ­ HH, LH . (7)

In the case of the test pulse, one may retain onlyelectron population created by the pump and, therefwe have to consider only the test polarization equation∑

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where the unrenormalized energy of the SO holese0SO,$k

and the renormalized SO Rabi frequency were introduc

h̄VTSO,$k

std ­ dSO,$kET std 1 2X$k0

V$k2$k0pSO,$k0std . (9)

HereET is the envelope of the test field having the carrfrequencyvT and dSO,$k the SO dipole matrix elementTo obtain the absorption spectrum, one has to perforFourier transform of the test polarization summed oall $k.

The electron population excited by the pump acts fias a final-state blocking factor on the right hand sof Eq. (8) and second as a band shift through the Fenergy of the renormalized electron energies [Eq. (These effects are all mixed up, vary in time, and gFourier transformed and therefore it is very difficultdiscuss them separately. In addition, specific Coulospectral effects of the Wannier operator in the polarizatequation (exciton and Coulomb enhancement) impedsimple additive interpretation.

Using a 130-fs pump pulse we performed calculatioof the excited populations up to 300 fs which corresponroughly to the plasma period at our pair density of6 3

1016 cm23. We tookT2 ­ 130 fs. The effective mass ratios were taken to be integer (mHHyme ­ 6, mLHyme ­1, mSOyme ­ 2) for convenience of the numerical algorithm. The numerical calculation on a discrete latticek-space points neglects low-momentum-transfer contritions which in the Coulomb case are important. Neertheless, we take into account low-momentum-trancollisions within a Landau approximation through a Tayexpansion around$q ­ 0. The complete quantum-kineticalculation gives rise to an electron population as show

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Fig. 2. The electron-population peaks are rapidly smeaout and already at about 300 fs after the pump maximthe distribution is very close to a nondegenerate Fermitribution. The calculated differential absorption specwith a 30-fs test pulse are shown in Fig. 3. We did nconsider delay times longer than 80 fs since their callation involves information on the electron population ftimes above 300 fs due to the Fourier transform.

The agreement with the experiment is surprisingly goalthough in many details quantitatively rough. The moremarkable achievement is the prediction that at ab80 fs after the pump maximum the induced hole burnis smeared out. This is related to the fact that the etron population is almost in equilibrium already at abo300 fs after the pump maximum. The only fit paramter was the phenomenological polarization relaxation tiT2. However, if T2 is taken comparable or larger thathe pump duration, it affects only the negative partsthe spectra slightly. In the comparison of theory andperiment, one has to take into account that the exactergy positions of the various features are affected byroughness of the electron energy discretization of ab5 meV, by the slightly modified effective masses as was by the inaccuracy of the numerical Fourier transfoThe more pronounced valley above the band thresholdabout 15 meV) as compared to the experiment may beto the neglect of LO-phonon emission, which providescooling of the electron system.

Both the experimental and the theoretical differential asorptions show a final-state occupation effect (hole buing) due to our narrow-band excitation and an oscillat

FIG. 2. Quantum-kinetic evolution of the electron populatio

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VOLUME 77, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 1996

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FIG. 3. Calculated differential absorption spectra for the sadelay times as in Fig. 1.

that looks like an energy shift in the excitonic region, whthe negative signal on the high-energy side of the exction is mainly due to the nonlinearity introduced by thproduct of the occupation factor with the Coulomb forterm

ih̄

X$k0

V$k2$k0 pSO,$k0 std f1 2 fe,$kstdg , (10)

which stems from the replacement of the Rabi frequeby the renormalized one in the presence of the Coulointeraction [see Eq. (9)]. Actually, this excitonic enhancment term plays an important role also in the oscillatostructure on the low-energy side in addition to the trCoulomb band shift of Eq. (4).

In conclusion, we have observed hole burning inpump-test configuration free from coherence effectssucceeded in giving a satisfactory description of the dferential absorption spectra for ultrashort delay times wthe semiconductor Bloch equations using the quantukinetic bare Coulomb collision term for the populationWe stress that the use of quantum kinetics is mandadue to the Coulomb singularity. The main success oftheory is determined by the structure of the semiconduBloch equations but the numerical prediction of the effetive intraband relaxation time (smearing out of the hoburning) is due entirely to quantum kinetics. Our theretical approach was highly simplified due to the specexperimental configuration implying a differential signdetermined only by the electron population.

An improved version of the theory should include tquantum-kinetic polarization collision term, the buildupscreening, LO-phonon collisions, as well as the transitto the Markovian behavior in order to extend its applicbility to higher densities and longer times. To incorporasuch improvements in the theory for the test beam wstill be an insurmountable task due to the Fourier traform, which requires an enormous time interval.

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We are grateful to R. Planel and V. Thierry-Mie(Laboratoire de Microstructures et de MicroélectroniquBagneux, France) for providing us with the GaAs samand to M. Joffre and J.-P. Likforman for helpful discussions. We acknowledge partial support by the Eropean Training and Mobility for Researchers prograunder Grant No. ERBFMGECT950019. The theoreticwork has been supported by the Deutsche Forschungmeinschaft.

*Permanent address: Institute of Physics, Mac Dinh ChiHo Chi Minh City, Vietnam.

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[9] ”Moving” gratings can be formed by pump and testdifferent wavelengths but in this case the pump diffractiappears at the pump wavelength and at the positof a sharp absorption resonance such as the excitoabsorption line. See H. J. Eichler, Opt. Acta24, 631(1977).

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K. Henneberger, Phys. Rev. B49, 7337 (1994).[20] D. B. Tran Thoai and H. Haug, Z. Phys. B91, 199 (1993).[21] H. Haug and S. W. Koch,Quantum Theory of the Optica

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