Exciton saturation in GaAs multiple quantum wells at room temperatureA. Miller, P. Riblet, M. Mazilu, S. White, T. M. Holden, A. R. Cameron, and P. Perozzo Citation: Journal of Applied Physics 86, 3734 (1999); doi: 10.1063/1.371244 View online: http://dx.doi.org/10.1063/1.371244 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/86/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Excitonic electroluminescence at room temperature in an (In,Ga)As multiple-quantum-well diode Appl. Phys. Lett. 97, 031103 (2010); 10.1063/1.3464559 Time-resolved photoluminescence study of InGaAs/GaAs quantum wells on (111)B GaAs substrates undermagnetic fields J. Appl. Phys. 89, 7875 (2001); 10.1063/1.1376401 Large optical nonlinearity and fast response time in low-temperature grown GaAs/AlAs multiple quantum wells Appl. Phys. Lett. 77, 58 (2000); 10.1063/1.126876 Ultrafast excitonic saturable absorption in ion-implanted InGaAs/InAlAs multiple quantum wells Appl. Phys. Lett. 72, 759 (1998); 10.1063/1.120885 Exciton saturation in room temperature GaAs/AlGaAs multiple quantum wells Appl. Phys. Lett. 71, 936 (1997); 10.1063/1.119694

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Exciton saturation in GaAs multiple quantum wells at room temperatureA. Miller,a) P. Riblet, M. Mazilu, S. White, T. M. Holden, A. R. Cameron, and P. PerozzoSchool of Physics and Astronomy, University of St Andrews, Fife, KY16 9SS, Scotland, United Kingdom

~Received 16 March 1999; accepted for publication 30 June 1999!

Pump-probe experiments in GaAs/AlGaAs multiple quantum well samples are described using bothpicosecond and femtosecond laser pulses in different polarization configurations. The excitonic andfree carrier components of exciton absorption saturation were analyzed as a function of well width.The relative importance of phase space filling, Coulomb screening and broadening contributionswas determined from polarization and laser bandwidth dependencies via spin-dependent andspin-independent components. A five-level model was used to fit the data and nonlinear coefficientsfor the individual contributions were determined for each well width. The Coulomb contributionsarising from screening and broadening of the excitons was found to dominate the absorptionbleaching using picosecond pulses, whereas phase space filling was largest in the femtosecondregime. Exciton phase space filling was found to be four times larger than free carrier phase spacefilling at narrower well widths. A sublinear dependence of exciton broadening with carrier densitywas observed. ©1999 American Institute of Physics.@S0021-8979~99!06919-4#

I. INTRODUCTION

Excitons in multiple quantum well~MQW! semiconduc-tors exhibit large resonant optical nonlinearities1,2 which of-fer a number of useful functions for optoelectronic devices.3

These include laser mode-locking elements,4 saturable ele-ments for controlling the propagation of optical solitons infiber transmission systems,5 all-optical bistable etalons,6 all-optical directional coupler switches,7 and self-electro-opticdevices8 for high speed communications, signal processingand computing.9 The operation and optimization of these de-vices rely on an understanding of the mechanisms that con-tribute to bleaching of the exciton absorption feature at roomtemperature. Previous work has shown that the relative con-tributions of the various mechanisms which cause bleachingcan be distinguished by the use of circularly polarizedlight10,11 and different laser pulse widths12 in pump-probemeasurements. This article describes pump-probe measure-ments in GaAs/AlGaAs MQWs using both picosecond andfemtosecond duration pulses, as a function of wavelength,polarization, bandwidth, carrier density, and well width inorder to quantify the individual contributions to excitonicoptical nonlinearities. Three MQW samples with differentwell widths have been compared in these studies.

II. BACKGROUND

Excitonic absorption features are clearly resolved atroom temperature in quantum wells compared to bulk semi-conductors because confinement leads to larger binding en-ergies and enhanced oscillator strengths. The loss of degen-eracy of light hole~lh! and heavy hole~hh! valence bandsleads to an exciton doublet in MQWs. Bleaching of the ex-citonic features is observed in GaAs/AlGaAs MQWs at be-low 1 mW average power, for laser spot sizes on the order of

10–100mm. This saturation occurs due to the optical gen-eration of excitons or free carriers at two-dimensional~2D!densities in excess of 1010cm22.

Resonant excitation initially creates bound electron-holepairs, i.e., excitons. When the density of excitons approachesthat required to fill space (;1017cm23 based on an excitondiameter of 30 nm for bulk GaAs!, then the number of addi-tional excitons which can be created is reduced by Pauliexclusion, thus decreasing the level of absorption. Knoxet al.13 measured a room-temperature ionization time for ex-citons into free electron-hole pairs of 300 fs. Therefore, ontime scales less than 300 fs, bleaching results primarily fromthe excess exciton density, while on picosecond and longertime scales, it can be assumed that the exciton saturationresults from free carriers.

In order to describe the different nonlinearities respon-sible for exciton saturation, the generalized Wannier equa-tion can be used. This is deduced from the screened Hamil-tonian for a two-band, quasi-two-dimensionalsemiconductor:11,14,15

@\v2Ee,k2Eh,k1 i ~G01G!1Sk,s#xk,s

5~12 f e,s,k2 f h,s,k!Fdk1(k8

Vs~k2k8!xk8,sG ~1!

for the polarizabilityx(\v)5(kdkxk,s(\v) whose imagi-

nary part is proportional to the absorption coefficient.Ee,k ,Eh,k correspond to the single-particle energies of elec-tron and hole, respectively.Vs(k) is the Fourier transformedquasi-two-dimensional screened Coulomb potential,dk thedipole moment~proportional to the modulus of the interbandmomentum matrix element! and f i ,s,k ( i 5$e,h%) are theFermi functions describing the distribution of electrons~holes! in their energy bands with spins. Sk,s , the real partof the self-energy is given bya!Electronic mail: a.miller@st-and.ac.uk

JOURNAL OF APPLIED PHYSICS VOLUME 86, NUMBER 7 1 OCTOBER 1999

37340021-8979/99/86(7)/3734/11/$15.00 © 1999 American Institute of Physics

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Sk,s5(k8

~ f e,s,k1 f h,s,k!Vs~k2k8!

1(k8

Vs~k8!2V~k!, ~2!

whereV(k) is the Fourier transformed unscreened potential.The first term of these summations is the ‘‘exchange

hole’’ energy and results from the Pauli exclusion principlewhich implies that each fermion is surrounded by a regionwhere the probability of the existence of another identicalfermion is very small. This repulsive energy occurs for par-ticles with equal spin and charge.

The second term is the ‘‘Coulomb hole’’ energy whichinvolves equally charged fermions avoiding each other be-cause of Coulomb repulsion. This term is independent of thespin of the particles. A finite homogeneous linewidthG0 , isintroduced to account for dephasing collisions~electron-phonon interactions! and is accompanied with the termGwhich corresponds to the imaginary part of the self-energyrepresenting the carrier-carrier scattering rate.

Finally, the term (12 f e,s,k2 f h,s,k) is the filling factorrepresenting a blockade of transitions by Pauli exclusion,where a state occupied by a fermion is no longer available asa final state in an optical absorption process.

We can therefore distinguish three different mechanismsinducing exciton saturation:

Phase space filling~PSF! is the mechanism resulting di-rectly from the Pauli exclusion principle and will reduce theexcitonic oscillator strength through the filling factor and theexchange hole energy.

Screening is the mechanism resulting from the Coulombinteraction and will reduce the excitonic oscillator strengththrough the screened potentialVs and the Coulomb hole en-ergy.

Broadening is the mechanism resulting from carrier-carrier scattering and will increase the homogeneous line-width of the exciton with the same overall oscillator strengththrough the imaginary part of the self-energy.

PSF depends on the spin of the electron since the fillingfactor and the exchange hole energy are both spin-dependentquantities, in contrast to screening and broadening~in thelow density regime!. This difference has already been ex-ploited in pump-probe experiments in MQWs at roomtemperature11,12where it was possible to separate experimen-tally the effect of PSF from the combined effects of screen-ing and broadening. This is made possible only in MQWsemiconductors because of the lifting of the degeneracy be-tween the light and heavy hole bands due to confinementallowing access to polarization sensitive optical selectionrules, as shown in Fig. 1.

Circularly polarized light resonant with the heavy holeexciton generates 100% spin-polarized electron-hole pairs.The initial spin orientation of the optically generated exci-tons is maintained by the electrons for tens of picosecondsafter exciton ionization.16 Spin relaxation is not a coherentphenomenon. The physics of electron spin relaxation hasbeen analyzed theoretically in bulk17 and MQW18 semicon-ductors at room temperature and is most likely due to the

D’Yakonov and Perel~DP! process.16,17 The DP spin relax-ation mechanism results from spin-orbit splitting of the con-duction band. Spin splitting is equivalent to the existence ofa magnetic field acting on the electron spins. Between colli-sions the spin precesses about the direction of the pseud-ofield, defined by the momentum direction. During colli-sions, changes in momentum cause a rotation of theprecession axis, allowing the spins to flip.

Hole spin relaxation is much faster~subpicosecond timescale at room temperature! because of valence band mixingand the mixed spin character of the light hole states.19,20

Therefore, because PSF is spin dependent, it is possible toseparate out electron PSF and to deduce the spin relaxationtime for free electrons by studying the change in transmis-sion of the probe as a function of time delay under differentpolarization combinations of pump and probe. In addition, astudy of the change in transmission of the probe as a functionof carrier density and excitation wavelength will show therelative importance of broadening as compared to screening.The density dependence of the broadening is complex andnot well established theoretically.

III. EXPERIMENT

Pump-probe experiments were carried out using a self-mode-locked Ti:sapphire laser~Spectra-Physics, Tsunami!producing either 100 fs or 1 ps pulses at 82 MHz. The beamwas split to form pump and probe pulses, and the time ofarrival of the probe pulses at the sample could be varied witha variable optical delay. The pump beam was chopped forsubsequent phase sensitive detection. Each of these beamswas passed through a linear polarizer followed by a quarter-wave plate or a half-wave plate to produce either left or righthanded circularly polarized or linearly polarized light. A setof three samples, S51, KLB, and FK141 were studied.

Sample S51 consists of 60 periods of 4.4 nm GaAsquantum wells surrounded by 17.5 nm Al0.33Ga0.66As barri-ers with no doping background. KLB contains 120 periods of6.5 nm GaAs quantum wells surrounded by 21.2 nmAl0.4Ga0.6As barrier with an intentionalp-type doping back-ground of;1016cm23. FK141 incorporates 15 periods of 9nm GaAs quantum wells surrounded by 10 nm Al0.2Ga0.8Asbarrier with no doping background.

FIG. 1. Selection rules for the transitions from the heavy hole and light holevalence bands to the conduction band in MQWs. The (j , mj ) refer to thequantum numbers for angular momentum and its component along one di-rection. Thes1 and s2 refer to the transitions excited by each sense ofcircularly polarized light and correspond toDmj561, respectively, wherethe propagation direction is used to definemj .

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The GaAs substrate was etched off to allow transmissionmeasurements and an antireflection coating applied to eachsurface for all three samples. Figure 2 shows the hh-lh exci-ton doublet in the linear absorption spectrum for sample S51.

A pump power of 1 mW resonant with a hh excitonabsorption peak creates an estimated carrier density of;531016cm23 for each sample~within a factor 2, taking intoaccount the different absorption coefficients, thicknesses,and well widths!. This corresponds to a two-dimensionaldensity of;1010cm22. ~Although there will be some non-uniformity of carrier density across the wells, particularly inS51 and KLB, within the low density~linear! regime, thisdoes not affect the results.! We measure the change in trans-mission DT of the probe beam and deduce the change inabsorptionDa from the relation:

Dad52 lnS dT

T011D'2

DT

T0, ~3!

whered is the effective thickness of the sample andT0 itslinear transmission. For sufficiently small change in trans-mission, the two quantities are proportional.

IV. WAVELENGTH DEPENDENCE

Bleaching of exciton absorption produces both absorp-tive and refractive optical nonlinearities at moderate opticalpowers.21 There is no observed shift in wavelength of thepeak of the exciton during saturation. Any reduction of thebinding energy caused by Coulomb screening to yield a blue-shift of the exciton is balanced by a redshift resulting fromband-gap renormalization.22 This may not be entirely coinci-dental, since band-gap renormalization arises from a combi-nation of exchange and correlation effects between carrierswhich is a Coulomb interaction, plus the fact that the excitonis a neutral particle.23 Above about 3 mW, the absorptionbecomes progressively more difficult to saturate because

band-gap renormalization causes a redshift of the fundamen-tal band-gap energy and thus a progressively larger densityof states needs to be blocked. At high optical powers, andthus very high carrier densities, band filling will cause theremaining absorption to saturate, but this is normally accom-panied by a redshift of the band gap due to local heating ofthe lattice.

Figure 3 compares differential transmission spectra forthe three GaAs/AlGaAs MQW structures with different wellwidths under conditions of moderate excitation level. Thesespectra were recorded in a pump-probe configuration usinglinearly polarized 1 ps duration pulses. There was a fixeddelay of a few picoseconds between the pump and probepulses. The negative signals are caused by broadening of theexciton features. It may be noted that the magnitude of thenegative signal is largest in the sample with the narrowestquantum wells and clearly occurs on both sides of the hhexciton feature in this case.

V. POLARIZATION DEPENDENCE

Figure 4 shows the results of time-resolved saturationmeasurements carried out at the peak of the hh exciton ab-

FIG. 2. Optical density,ad, as a function of wavelength for sample S51 atroom temperature (hh5heavy hole, lh5light hole)

FIG. 3. Change in transmission as a function of the laser wavelength usinglinear polarization for the three samples FK141, KLB, and S51.

3736 J. Appl. Phys., Vol. 86, No. 7, 1 October 1999 Miller et al.

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sorption in sample KLB at room temperature using 1 pspulses with the three polarization configurations of pump andprobe beams,~a! opposite linear polarization~OLP!, ~b!same circular polarization~SCP!, and ~c! opposite circularpolarization~OCP!. One picosecond pulses have a spectralbandwidth much less than the hh exciton spectral linewidth.The transmission change is due to the combined effects ofPSF, screening and broadening on exciton saturation whichall reduce the absorption at the center of the line. The OLPconfiguration shows a rapid rise in transmission of the probeat zero delay followed by a recovery due to carrier recombi-nation on nanosecond time scales. The signal is thereforeobserved as an almost flat plateau on the picosecond timescales recorded here.~Note that there is no difference usingthe same and opposite linear polarizations.!

When the pump is circularly polarized, all of the carriersgenerated will initially have the same spin orientation. Theprobe can be chosen to have either the same sense of circularpolarization as the pump, SCP, or the opposite circular po-larization, OCP. The SCP result in Fig. 4 shows an initiallyenhanced saturation, compared to the OLP case, and a recov-ery to the plateau level within a time on the order of 100 ps.This enhancement arises from spin-dependent PSF. Theprobe interacts with electron spin states that are twice as fullas those in the OLP case~for the same total number of ex-cited carriers!. For the OCP condition, the initial bleaching isless than for OLP. In this case, immediately after excitationall of the electrons are in the opposite spin state to that beingprobed, so there is no PSF. This results in less bleachinginitially, but as the electron spins randomize, the OCP signalrises to the OLP level. The initial level in the OCP signalclose to zero delay is due to the combined spin-independentCoulomb effects of screening and broadening. The recoverytime of the enhanced~SCP! and reduced~OCP! bleaching

gives a characteristic electron spin relaxation timetspin of 50ps.

For the OCP configuration, the combined influence ofPFS, screening, and broadening sets the transmission changeat the plateau level, whereas the initial change in transmis-sion before any spin relaxation occurs, is due to screeningand broadening alone. Again for the SCP configuration, thedifference between the plateau level and the initial signal issolely due to PSF. We note that the influence of PSF in theSCP configuration, is roughly double that in the OLP con-figuration. This is because there is only one electron avail-able per state~all of the electrons have the same spin in theSCP configuration! compared to the OLP configuration inwhich both spin states are excited. As a result, for the samegenerated electron population, there will be twice as manyoccupied states in the probed spin band in the SCP case ascompared to the OLP case, doubling the influence of PSF.Although the general equation@Eq. ~1!# relates the differentmechanisms in a complex manner, calculations have beenperformed,11 that well describe the results obtained for thethree different polarization configurations in pump-probemeasurements in a low density regime (,1010cm22).

We have been able to separate PSF due to electrons inthis experiment only because the hole spin randomizeswithin 1 ps at room temperature. It is possible that PSF dueto the holes occurs even in the OCP configuration. However,the average in-plane effective mass for holes is much largerthan for electrons, which means that PSF should be domi-nated by the electron contribution. Furthermore, calculationsshow only a small difference when PSF induced by the holesis taken into account.

VI. SPECTRAL BANDWIDTH DEPENDENCE

The broad bandwidth of ultrashort pulses can be used toidentify the broadening contribution to the exciton absorp-tion nonlinearity as described by Holdenet al.12 Figures 5~a!and 5~b! compare results of pump-probe measurements~asdescribed in Sec. V! using 1 ps and 100 fs pulses for sampleS51. One picosecond pulses have a spectral bandwidth of;2 meV, so that the observed transmission change centered onthe peak of the hh exciton of spectral linewidth 6.4 meV~seeFig 2!, is due to the combined effects of PSF, screening andbroadening. On the other hand, for 100 fs pulses, the mea-sured bandwidth is; 10 nm~20 meV!, which is greater thanthe hh exciton absorption linewidth. Consequently, 100 fsduration probe pulses effectively measure the integrated ab-sorption change rather than the change at the line center. Thestriking difference between the two sets of results in Figs.5~a! and 5~b! follows from the absence of any sensitivity tobroadening of the exciton when using the shorter durationpulses. Collisional broadening lowers the absorption at linecenter while it increases it in the wings leaving the integratedabsorption unchanged. A comparison of the initial step sizesfor the OCP configuration for picosecond results, Fig. 5~a!,and femtosecond results, Fig. 5~b!, demonstrates that broad-ening is the dominant contribution to the peak exciton ab-sorption quenching in this sample for picosecond or longerduration pulses.

FIG. 4. Dependence of probe transmission change on optical time delay inpump-probe measurements for three polarization configurations,OLP5opposite linear polarization, SCP5same circular polarization,OCP5opposite circular polarization for sample KLB.

3737J. Appl. Phys., Vol. 86, No. 7, 1 October 1999 Miller et al.

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VII. CARRIER DENSITY DEPENDENCE

So far, we have identified broadening to be the mostsignificant contribution to exciton bleaching in the samplewith the narrowest wells~S51! using longer duration pulsesfrom the wavelength dependence and from the laser band-width dependence in polarization sensitive pump-probe stud-ies. Broadening can also be identified in measurements car-ried out as a function of pump power when the polarizationselection rules are again used to extract the magnitude of thespin-independent contribution to exciton saturation.

Results such as those shown in Fig. 4 using 1 ps pulseswere used to quantify the relative magnitudes of the electronPSF and Coulomb contributions to exciton saturation at lowexcitation from the initial changes in transmission in thethree polarization configurations. We now extend this tostudy the variation as a function of generated excess carrierdensity. Figure 6 plots the spin-dependent~PSF! and spin-independent~Coulomb! components of the initial transmis-sion change of 1 ps pulses as a function of the average pump

power from results similar to those shown in Fig. 4 for thethree MQW samples.

First, we note that the Coulomb contribution is alwayslarger than the PSF contribution for all three well widths.Second, the power dependence is observed to be linear in allcases except for the Coulomb contribution in sample S51.

In all three samples, the spin-independent component,DTPSF, increases linearly with the carrier density. This im-plies that the excitonic oscillator strength decreases linearlywith carrier density~up to 1010cm22) as expected for a third-order nonlinearity. The experiment remains in a low densityregime because the excitonic absorption is not completelysaturated.

For samples KLB and FK141, the spin-independentcomponent,DTOCP, follows the same linear dependence asPSF. This similar behavior is expected for the screeningmechanism and has been calculated previously.12 This com-ponent is significantly larger than PSF and is found to bemore dominant in FK141 than in KLB in comparison with

FIG. 5. Probe transmission changes as a function of delay time for the threepolarizations, SCP, LP, and OCP using~a! 1 ps and~b! 100 fs pulses forsample S51.

FIG. 6. Coulomb~screening and/or broadening! and PFS contributions tothe transmission changes as a function of the pump power at the peak of theheavy hole exciton in the three samples FK141, KLB, and S51.

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PSF. One possible explanation would be the influence ofconfinement. In fact, the quantum wells in sample KLB havea well width of 6.5 nm with an electron confinement energyof 63 meV, whereas the wells in FK141 are 9 nm wide withan electron confinement energy of 37 meV. Previous theo-retical studies12 predicted that PSF should increase its influ-ence with confinement and become the dominant mecha-nism. This evolution is consistent with our measurements,with the exception that the spin-independent component re-mains larger than PSF although its influence decreases withconfinement.

The carrier density dependence of the spin-independentCoulomb contribution is strongly sublinear in the narrowquantum well sample, S51. We propose that the dominantcontribution is broadening whereas screening dominates inthe other two samples. This is supported by the negativechanges in transmission as a function of wavelength asshown in Fig. 2 in which the broadening mechanism clearlyappears in sample S51 at the hh exciton but is not as impor-tant in KLB or FK141. This observation indicates that forsamples KLB and FK141 the two principal mechanisms forsaturation using picosecond pulses are screening and PSF,whereas broadening plays a more important role for S51.

For sample S51,DTOCP shows a clear nonlinear behav-ior. The change in transmission varies linearly up to a pumppower of; 0.2 mW which corresponds to a carrier densityof ;33109 cm22 and is followed by a saturation effect. Ifwe assume an exciton resonance centered at the wavelengthlmax to have a Gaussian shape, the absorption will be

ad5a0d expS 2~l2lmax!

2

G82 D , ~4!

whereG8 is the linewidth of the resonance. If we considerbroadening to be the only mechanism, we have the relation atl5lmax:

G85G0a0d

a0d2 ln~DT/T011!, ~5!

where G0 and a0 are the initial linewidth and absorptioncoefficient before the pump excitation. For relatively smallchanges in transmission and with a coefficienta0d close tounity, relation Eq.~5! becomes

G

G05

G82G0

G0'

DT

a0dT0. ~6!

As a result,DTOCP gives a measure of the change of theexciton linewidth.G corresponds to the same collision rate asdefined in Eq.~1!. In general, scattering of the exciton iscaused by exciton-exciton and exciton-free carrier interac-tions. A direct measurement of collisional broadening of ex-citons in GaAs quantum wells has been reported at lowtemperatures,24 showing separately the scattering induced byfree carriers and excitons. The free carrier-exciton scattering

rate has been observed to increase linearly up to a carrierdensity of 53109 cm22 as predicted by many-body theory atlow density.25 At room temperature, excitons ionize within300 fs which implies that the dominant scattering mechanismcontributing to the increased exciton linewidth will beexciton-free carrier collisions. Our measurements are consis-tent with the low temperature measurements of Ref. 24, butas the carrier density increases above;53109 cm22, thecollision rate begins to saturate. Many-body theory is notwell established for a correct description of the scatteringrates and phenomenological dependencies are usually intro-duced if this effect has to be taken into account.11 We can,however, understand this behavior qualitatively. As the den-sity of carriers becomes more significant, the number of scat-tering events per unit volume increases in proportion to thecarrier density; but simultaneously the carrier-carrier interac-tion, which is nothing but the Coulomb interaction, isscreened, thereby reducing the impact of any collision abovea critical density. From this point of view, the screeningmechanism, as it has been defined in this paper, and thebroadening mechanism have the same physical origin, i.e.,the Coulomb interaction. Screening corresponds to the effectof Coulomb interaction on the oscillator strength, whereasbroadening corresponds to the effect of Coulomb interactionon the resonance linewidth.

In this analysis for sample S51, we have assumed thatthe only mechanism that contributes toDTOCP is broadening.However, the screening mechanism should also be includedas indicated by the spectral bandwidth results discussed inSec. VI. From Fig. 5~b!, the contribution due to screening isabout two thirds of the PSF contribution, so that the broad-ening can be estimated from Fig. 5~a! at around five timesthat of screening at low excitation levels. The quantum wellsin sample S51 are 4.4 nm wide, with an electron confinementenergy of 93 meV. From the analysis of KLB and FK141, weexpect the screening mechanism to be linearly dependent oncarrier density and to have a maximum value of the sameorder of magnitude as PSF. However, such a correction dueto screening would have the effect of reinforcing the satura-tion characteristic of the curve. Besides, we have measuredin single beam experiments that the maximum change intransmission for S51 reaches 15% compared with only 5%for KLB, indicating that the effect of screening onDT is notas strong as the effect of broadening in S51.

As illustrated in Fig. 2, S51 is of high optical qualitywith a sharp well-resolved heavy hole exciton at room tem-perature (G056.4 meV). This is partly due to good materialquality and partly because the well width of 4.4 nm givesclose to the maximum exciton binding energy. Therefore, asmall change of the linewidth can induce a rather largechange in transmission at the maximum of the resonance. Onthe other hand, the excitonic absorption features in KLB andFK141 are not as well resolved which make the broadeningmechanism less sensitive for these two samples. Moreover,the broadening effect may become more important withmore confinement but there is no theory available for such aprediction.

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VIII. TIME DEPENDENCE

Higher time resolution pump-probe measurements using100 fs pulses allow a comparison of exciton absorption satu-ration before and after exciton ionization.12,13 In this section,we will describe an analysis of pump-probe results, whichgives the relative magnitudes of exciton and free carrier con-tributions to spin-dependent and spin-independent hh excitonbleaching. Using 100 fs pulses, broadening is averaged outand the contributions from free carrier phase space filling~spin dependent! and free carrier Coloumb screening~spinindependent! can be isolated and compared. Circular polar-izations were again employed in order to create and probethese spin-dependent and spin-independent mechanisms.

An example of a set of higher time resolution pump-probe scans using 100 fs pulses and recorded within 2 ps ofzero delay are shown in Fig. 7. In his case, a probe beamsteering mirror was vibrated to eliminate any coherence ar-tifacts. Resonant excitation with a 100 fs pulse initially pro-duces a population of cold (k50) bound electron-hole pairs~excitons!. Further creation of excitons is inhibited by exci-tonic PSF and thus the absorption becomes bleached. Atroom temperature, excitons strongly ionize via collisionswith LO phonons within about 300 fs.13 A fast transient fea-ture is seen in SCP and LP configurations in Fig. 7, as theexcitons are rapidly generated and then ionize, creating hotfree carriers. Within the first 1 ps after excitation, excitoncontributions have passed and free carrier contributions toexciton saturation dominate. The enhanced transmissionchange at small delays results from the fact that the presenceof excitons is more efficient in saturating the hh exciton ab-sorption feature than the presence of the same density of freecarriers. This transmission enhancement is observed to belarger for the SCP configuration than LP, confirming that the

spin-dependent PSF component is larger for excitons thanfor free carriers.

On inspection of the OCP trace in Fig. 7, the initial riseshows a step followed by a slow increase in transmissiontowards the LP curve at the spin reorientation rate. Perhapssurprisingly, there is no signature of exciton ionization. AsPSF does not contribute to the initial OCP condition, thisobservation may indicate that screening due to excitons andfree carriers is equal.12 However, an alternative interpretationis that the exciton contribution is minimal because the cor-related opposite charge carriers cancel each other out.26 Inthis latter case, the initial rise in the OCP transmission is dueonly to the free carrier population build up after exciton ion-ization. The theoretical fits for the OCP data described laterare highly sensitive to the pulse duration and the zero delayposition. Further experiments are needed to determine theprecise relative magnitude of the exciton screening.

The schematic in Fig. 8 shows a five-level model con-taining two coupled three-level systems, one for spin-up andone for spin-down carriers with a common ground state. ThecoefficientsG1 andG2 correspond to the recombination ratesof the free carriers and excitons andG3 gives the rate ofexciton ionization. The inverses of these coefficients give thecharacteristic ionization or relaxation times. Another factorchanging the carrier populations is the pump excitation. Thisexcitation is represented by intensity profiles of the pulsedecomposed into the two opposite circular polarizationG1(t) and G2(t). The selection rules, Fig. 1, imply thatG1(t) andG2(t) correspond to the rate of excitation of thecarriers from the ground level into the spin-up and spin-down exciton levels, respectively. Further, we take into ac-count the spin relaxation ratesGs1 and Gs2 , which imply ahomogenization of the spin after the selective excitation ofonly one spin state. Finally, the probe pulse is representedhere to be resonant with the spin-up exciton level.

The evolution of the population of such a five-level sys-tem is given by a system of four coupled first-order differ-ential equations

FIG. 7. Probe transmission changes at the hh exciton absorption peak as afunction of delay using 100 fs pulses for SCP~same circular polarizations!,LP ~same linear polarizations!, and OCP~opposite circular polarizations! asa function of time for sample S51.

FIG. 8. Diagram of the five-level system. The arrows represent the directionof flow of population.

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dnex1

dt52~G21G31Gs1!nex

11Gs1nex21G1~ t !,

dnex2

dt51Gs1nex

12~G21G31Gs1!nex21G2~ t !,

~7!dnFC

1

dt51G3nex

12~G11Gs2!nFC1 1Gs1nFC

2 ,

dnFC2

dt51G3nex

21Gs2nFC1 2~G11Gs2!nFC

2 ,

wherenex1 , nex

2 , nFC1 , andnFC

2 represent the exciton and freecarrier populations in the spin-up and spin-down states.

In order to find a simple analytic solution to this systemof equations, we assume that the intensity profile of the ex-citation is Gaussian. We can distinguish three differentmodes of excitation, which can be summarized into the fol-lowing expressions:

G1~ t !5g01 expS 2

t2

Dt02D ,

~8!

G2~ t !5g02 expS 2

t2

Dt02D ,

whereg06 andDt0 corresponds, respectively, to the intensity

coefficient and duration of the pump pulses. In the case oflinear polarized light, we have

g025g0

151/2. ~9!

For circular polarized light, either the spin-up or spin-downexciton level is excited. For the spin-up excitation, we have

g0151, g0

250, ~10!

and for the spin-down case,

g0150, g0

251. ~11!

To solve Eq.~7! we transform it into two uncoupledsystems each corresponding to a three-level system. Intro-ducing the sum and difference of populations

sex5nex11nex

2 , sFC5nFC1 1nFC

2 ,~12!~12!

dex5nex12nex

2 , dFC5nFC1 2nFC

2 ,

the evolution equations of the two three-level systems read

d

dtsex52~G21G3!sex1G1~ t !1G2~ t !,

~13!d

dtsFC51G3sex2G1sFC,

and

d

dtdex52~G21G312Gs1!dex1G1~ t !2G2~ t !,

~14!d

dtdFC51G3dex2~G112Gs2!dFC.

By analyzing these two systems of equations, we see thatthey correspond indeed to simple three-level systems.

In the s system, Eq.~13!, the excitation is given by thesum of G1(t) and G2(t) and is further characterized byexciton recombination and ionization rates,G2 andG3 . Thefree carrier recombination rate isG1 . We remark that allthree excitation configurations@Eqs.~9!–~11!# have the sameeffect on thes system and thus this system is invariant withrespect to the polarization. Indeed, this is to be expectedbecause of the symmetry of switching between spin-up andspin-down states.

Similarly, in the d system, Eq.~14!, the exciton term,dex, has recombination and ionization rates given by (G2

12Gs1) and G3 . The ‘‘free carrier’’ recombination rate is(G112Gs2). Its excitation corresponds to the difference be-tweenG1(t) andG2(t). In this case, the system is not ex-cited at all by the linear polarization excitation, Eq.~9!,whereas the circular polarization excitation corresponds ei-ther to a ‘‘positive,’’ Eq.~10!, or ‘‘negative,’’ Eq. ~11!, ex-citation.

To determine the solution of Eqs.~13! and~14!, we needto assume the initial conditions, i.e., the state of the popula-tions before the excitation pulse. We take them to be zero att52`. The exact solutions read

sex~ t !5~g011g0

2!KG21G3~ t !,

sFC~ t !5~g011g0

2!@KG1~ t !2KG21G3

~ t !#,

~15!dex5~g0

12g02!KG21G312Gs1

~ t !,

dFC5~g012g0

2!@KG112Gs2~ t !2KG21G312Gs1

~ t !#,

where

KG~ t !5Ap

2Dt0 expS 2Gt1

G2Dt02

4 D3F11erfS 2t2GDt0

2

2Dt0D G . ~16!

Using the population evolution of Eqs.~15! and~12!, wecan define the induced transmission change of the probebeam due to the saturation effects. Indeed the density ofexcitons, and subsequently free carrier pairs, created by thepump pulse under the conditions described in earlier sectionswas estimated to be 1010cm22. At this density, Boltzmannstatistics apply and we can assume that the strength of thesaturation and transmission change is proportional to the in-stantaneous particle density. Thus, using the variable delaytbetween the pump and probe pulses, one can scan the popu-lation evolution after an excitation. The transmission changeof the probe pulse is given by the integration of the satura-tion effects over the duration of the pulse and is proportionalto

DT~t!}E2`

1`

G1~ t2t!@pexnex1~ t !1~sFC1pFC!nFC

1 ~ t !

1sFCnFC2 ~ t !#dt, ~17!

where the coefficients in front of the populations give theamount of interaction with the probe pulse. Further, in thecase of free carriers, we take two contributions into account.

3741J. Appl. Phys., Vol. 86, No. 7, 1 October 1999 Miller et al.

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One is due to the screening effect,sFC, and is present forboth spin-up and spin-down populations of free carriers. Theother part corresponds to the magnitude of the PSF,pFC, andthus is only present on the spin-up side if the circularly po-larized probe pulses are probing this transition. In the case ofexciton populations, we include only a PSF contribution,pex, because the screening effects are negligible.

Another benefit of using Gaussian intensity profiles forthe excitation pulse is that it is possible to evaluate the inte-gral given by Eq.~17! exactly. To simplify the solution weconsidered the recombination ratesG1 and G2 and the spinrelaxation ratesGs1 andGs2 to be negligible in comparisonwith the ionization rateG3 . This has no effect on the calcu-lated transmission change because of the large difference be-tween these rates.

After simplification, the transmission change is propor-tional to

DT~t!}pexg01KG3

l ~t!1sFC~g011g0

2!@KG1

l ~t!2KG3

l ~t!#

1pFC@ 12~g0

11g02!KG1

l ~t!2g01KG3

l ~t!

1~g012g0

2!K2Gs2

l ~t!# ~18!

with

KGl ~t!5exp~2Gt!F11erfS t2GDt0

2

&Dt0D G . ~19!

Both short scan pump-probe measurements with a probedelay of a few picoseconds, Fig. 9, and long scans of up to120 ps, Fig. 10, were carried out for the three samples using100 fs pulses. In this way, both the exciton ionization tran-sient and spin relaxation were time resolved. The experimen-tal data from scans on both time scales were then fitted si-multaneously using the three pump configurations for each.This greatly improved the parameter confidence, as the spinrelaxation times derived from the opposite and same circularpolarization must be the same for each sample. Further, thesimultaneous fit of the three polarization configurations madepossible the determination of the relative contributions ofscreening and PSF to exciton saturation. In Fig. 9, whichshows the short time scale scans for sample S51, we haveplotted the individual contributions from spin-up and spin-down excitons and free carriers, as well as the resulting to-tals. Figure 9~a! and 9~b! show that it takes almost 1 ps forthe free carrier concentration to become established. Fortimes less than 1 ps, both exciton and free carrier densitiesmust be taken into account. An exciton ionization time ofaround 250 fs was determined from the decays in the initialpeaks for SCP and LP results. The decays in the longer timescale results in Fig. 10 are determined by spin relaxation, andclearly show the well width dependence of the spin reorien-tation time by a comparison of the three samples.

We summarize the results of the fits for the threesamples using 100 fs pulses in Table I. The magnitudes ofthe contributions have been normalized to the free carrierscreening component in order to illustrate trends in the satu-ration components as a function of well width. We stress thatit is important to carry out this fitting procedure rather than

estimate the step sizes in the pump-probe data, in order toaccurately determine the relative importance of the differentcomponents. Note that the parameters,sFC, pFC, and pex

were defined above@Eq. ~17!# as nonlinear coefficients perexcess electron-hole pair. This definition gives values ofpFC

andpex which are twice as large as the nonlinear coefficientsdefined with respect to the polarization dependence in Fig. 4~see Holdenet al.12!.

We first note that both the exciton and free carrier PSFcontributions are always larger than the spin-independentfree carrier Coulomb contribution. This is contrary to theresults shown in Fig. 6 obtained with picosecond pulses. Inthis case, broadening has been eliminated using 100 fspulses. These results therefore imply that broadening mustprovide some contribution to the Coulomb component forFK141 and KLB in Figs. 6~a! and 6~b!, but the linear density

FIG. 9. Change of probe transmission as a function of delay time using 100fs pulses for sample S51 using the same three polarization configurations asFig. 7~a! SCP,~b! LP, and~c! OCP. The solid lines are fits to the experi-mental data~dots!. Also shown are the components of the contributions tothe transmission change from spin-up and spin-down free carriers and exci-tons deduced from the fits assuming the probe is coupled to the spin-upstates~Fig. 8!.

3742 J. Appl. Phys., Vol. 86, No. 7, 1 October 1999 Miller et al.

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dependencies point to the dominance of the screening com-ponent. From Table I, it can be observed that when broaden-ing is eliminated, the ratio of free carrier PSF to screening isgreater than unity for all three samples, ranging from 1.5 to3.6 as the confinement increases.

Second, the results in Table I show that, consistent withtheory,26 both pFC andpex increase with confinement. Exci-ton PSF is larger than free carrier PSF~for equal excitationdensities! for all three well widths, but the ratio of thesecontributions decreases with increasing confinement. A pos-sible explanation may be found by referring again to Eq.~1!.We stated in Sec. II that PSF comprises both a filling factorand an exchange term. We may expect the free carrier con-tribution to be primarily via the filling factor whereas bothterms will contribute in the presence of excess excitons. Thisfollows from the fact that cold excitons will have a higherrepulsive potential because they are more highly localizationthan free carriers. Additionally, whereas optically generatedexcitons are created close tok50, the free carriers createdby thermal ionization of excitons will occupy a range ofkstates thus reducing the filling factor. Greater quantum con-finement will serve to increase the localization of the freecarriers thus making the exchange part relatively more im-portant.

Finally, we observe that the well width dependence ofthe spin relaxation time is consistent with the DP model asreported by Brittonet al.16

IX. CONCLUSION

In these studies, the three main mechanisms responsiblefor the heavy hole exciton saturation in MQWs at room tem-perature, i.e., PSF, screening, and broadening, have beenstudied in samples with different well widths using thepump-probe technique. Both picosecond and femtosecondpulses were employed in experiments carried out as a func-tion of wavelength, polarization, and excitation level. Wehave shown that the broadening mechanism plays a domi-nant role for picosecond and longer pulses when the hh ex-citon absorption resonance is narrow, and we have demon-strated for the first time a saturation of the carrier-carriercollision rate at low density (;53109 cm22). This is ofimportance for any optoelectronic device using room tem-perature excitonic nonlinearities in quantum wells. In addi-tion, measurements such as four-wave mixing which involveresonant excitonic optical nonlinearities must be interpretedappropriately in the light of this observation. In sampleswhere broadening is less important, we have shown thatscreening remains an important contribution although PSFbecomes increasingly dominant in narrower wells. ExcitonicPSF is approximately four times larger than the free carriercomponent in narrower wells.

ACKNOWLEDGMENTS

The authors would like to thank Ian Galbraith for usefuldiscussions and Karl Woodbridge, Geoff Duggan, and DonMcDaniels for the MQW samples used in these studies.EPSRC is acknowledged for financial support.

1Optical Nonlinearities and Instabilities in Semiconductors, edited by H.Haug ~Academic, San Diego, 1988!.

2S. Schmitt-Rink, D. S. Chemla, and D. A. B. Miller, Adv. Phys.38, 89~1989!.

FIG. 10. Change of probe transmission using 100 fs pulses at long timedelays for samples FK141, KLB, and S51 together with numerical fits~solidlines! for the three polarization configurations as defined in Fig. 7.

TABLE I. Results of fitting pump-probe data using 100 fs pulses.pFC andpex are the relative magnitudes of the PSF contributions to exciton bleachingper electron-hole pair due to free carriers and excitons, respectively, nor-malized to the free carrier screening contribution per electron-hole pair,sFC .tspin is the fitted spin relaxation.

SampleWell width

~nm! sFC pFC pex pex /pFC

tspin

~ps!

FK141 9.0 160.05 1.560.1 11.160.5 7.4 180615KLB 6.5 160.05 3.260.1 13.260.3 4.1 90610S51 4.4 160.03 3.660.1 14.060.2 3.9 5065

3743J. Appl. Phys., Vol. 86, No. 7, 1 October 1999 Miller et al.

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and K. Woodbridge, J. Opt. Soc. Am. B6, 567 ~1989!.22C. Weber, C. Klingshirn, D. S. Chemla, D. A. B. Miller, J. E. Cunning-

ham, and C. Ell, Phys. Rev. B38, 12748~1988!.23S. Schmitt-Rink, D. S. Chemla, and D. A. B. Miller, Adv. Phys.38, 89

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