Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy

PHYSICAL REVIEW B 15 DECEMBER 2000-IVOLUME 62, NUMBER 23

Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy

Matthew C. Beard, Gordon M. Turner, and Charles A. SchmuttenmaerChemistry Department, Yale University, 225 Prospect Street, P.O. Box 208107, New Haven, Connecticut 06520-8107

~Received 6 December 1999; revised manuscript received 17 March 2000!

The onset and decay of photoconductivity in bulk GaAs has been measured with 200-fs temporal resolutionusing time-resolved THz spectroscopy. A low carrier density (,231016 cm23) with less than 100-meVkinetic energy was generated via photoexcitation. The conductivity was monitored in a noncontact fashionthrough absorption of THz~far-infrared! pulses of several hundred femtosecond duration. The complex-valuedconductivity rises nonmonotonically, and displays nearly Drude-like behavior within 3 ps. The electron mo-bilities obtained from fitting the data to a modified Drude model (6540 cm2 V21 s21 at room temperature withN51.631016 cm23, and 13600 cm2 V21 s21 at 70 K with N51.531016 cm23) are in good agreement withliterature values. There are, however, deviations from Drude-like behavior at the shortest delay times. It isshown that a scalar value for the conductivity will not suffice, and that it is necessary to determine thetime-resolved, frequency-dependent conductivity. From 0 to 3 ps a shift to higher mobilities is observed as theelectrons relax in theG valley due to LO-phonon-assisted intravalley absorption. At long delay times~5–900ps!, the carrier density decreases due to bulk and surface recombination. The time constant for the bulkrecombination is 2.1 ns, and the surface recombination velocity is 8.53105cm/s.

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I. INTRODUCTION

Ultrafast carrier dynamics in semiconductors have beegreat interest for the last few decades, and GaAs is the hspeed semiconductor of choice.1,2 Experimental techniqueof time-resolved optical transmission,3–9 reflection, electro-optic sampling,10,11 luminescence,12–15 four-wavemixing,16,17 and photon echoes,18,19 among others, have acontributed to a more complete understanding of carriernamics in semiconductors. Theoretical models of carriernamics have ranged from coupled rate equations20 to MonteCarlo simulations,21–24 to semiconductor Bloch equations25

and Boltzmann transport theory.26,27 This paper reports onthe time-resolved, frequency-dependent photoconductivitGaAs based on measurement of the intraband absorptio

Time-resolved THz spectroscopy~TRTS! is a noncontactelectrical probe capable of determining photoconductivwith a temporal resolution of better than 200 fs. Methosuch as transient absorption or time-resolved luminescewhich are sensitive to either the sum or the product ofelectron and hole distribution functions, respectively, domeasure the actual conductivity. Unlike luminescence,technique is also applicable to a study of carrier dynamicindirect-gap semiconductors. Even methods which are setive to the diffusion of carriers, such as transient gratingfour-wave mixing, are not able to determine the frequendependent conductivity. The high sensitivity of TRTS makit possible to carry out measurements with photoexcitedrier column densities as low as 531010 cm22 in GaAs~equivalent to a carrier density of 531014 cm23, with a skindepth of 1 mm).

The ability to characterize electrical properties in a nocontact fashion with subpicosecond temporal resolutionnecessary in the field of nanoscale electronics and oelectronics, where it is difficult, if not impossible, to usconventional probes. Knowledge of a material’s frequendependent conductivity is important for its use in electro

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devices. Therefore, one of the motivating factors for twork was to benchmark the TRTS technique on a westudied system before proceeding to study systems thanot as well characterized.

This study was also motivated in part by a series of papby Vengurlekar and Jha that theoretically predicted a dmatic variation of the frequency-dependent mobility of phtoexcited electrons as a function of their initial kinetenergy.26–30 If the initial electron kinetic energy is an integral multiple of the LO-phonon energy, then tremendovariations and oscillations in the frequency-dependent cductivity are predicted during the first couple of picoseconafter photoexcitation. On the other hand, if the initial kineenergy is a half-integral multiple of the LO-phonon energthe onset of photoconductivity is predicted to smoothly aproach Drude-like behavior. We have carried out a full timresolved study of the relaxation dynamics that has produresults directly comparable to these theoretical predictioWhile our results agree with some of the features of Vegurlekar and Jha’s predictions, we do not observe differenbased on whether the initial kinetic energy is integer or hainteger multiples of the LO-phonon energy.

The time-dependent conductivity is given bys(t)5en(t)m(t), wheren(t) is the time-dependent carrier density, m(t) is the time-dependent mobility, ande is the elec-tron charge. The carrier density will change on time scale;5 ps due to surface recombination, and;100 ps due tobulk recombination and diffusion. Therefore, changess(t) will reflect changes inm(t) for times interval,5 ps.The mobility is related to the dynamics of the semiconducthrough both the effective massm* and the carrier scatteringrateg by m(t)5e/@g(t)m* (t)#. The effective mass is timedependent because the photoexcited electron distribuf (E,t) is itself time dependent:

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FIG. 1. Experimental apparatus used to collect TRTS spectraA reference scan is collected witthe chopper in position 1, andphotoexcited difference scan icollected with the chopper in position 2. The THz amplitude ismonitored by changing the relative delay of table 1 or table 2 depending on the type of experimenbeing performed ~see text formore detail!. Delay table 3 is usedto change the relative delay of thpump beam. Plot~a! shows a typi-cal time-domain THz transient oabout 400 fs duration~FWHM!and a signal-to-noise ratio oroughly 1000:1. Plot~b! shows thepower spectrum of the THz pulse

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This has been observed using visible pump/THz probeperiments where the photoexcited electrons have sufficenergy to access theX andL valleys, both of which have lowcurvature (d2E/dk2) and thus low mobility.31–33 The THzresponse is then dominated by the electron population stering back to theG valley rather than a time-dependent sctering rate. To probe the time-dependent scattering rate inpendent of intervalley scattering, photoexcitation into theGvalley with a small excess energy is required.

This work significantly extends some of the most closrelated studies. Nuss and co-workers performed timresolved electro-optic sampling measurements of GaAs,used 2.0-eV photon energies, which allowed electronsscatter to the low mobility satellite valleys.31,34 A time-resolved THz transmission study of GaAs that used simhigh photon energies was reported by Greenet al.35 A studyof GaAs and InP, which also included frequency-resolvspectra 7 ps after photoexcitation, was reported by Set al.36 Groeneveldet al. measured carrier and exciton far-Ispectra at several pump-probe delay timesGaAs-AlxGa12xAs quantum wells.37 Ralph and co-workersinvestigated the overall change in carrier absorption afunction of time, but were only able to determine thfrequency-dependent conductivity on a much longer tiscale because they used bolometric detection ratherelectrooptic sampling.33,38 Flanderset al. reported the aver-age change in THz transmission after photoexcitation,well as frequency resolved absorbance spectra for five puprobe delay times when using excitation energies lenough to prevent intervalley scattering.39 Finally, Schalland Jepsen recently investigated the interface effectspropagation of THz pulses through photoexcited GaA40

While all of these pioneering studies provided a glimpse iphotoexcited carrier dynamics, none fully characterizedtime-resolved, frequency-dependent photoconductivity.

II. EXPERIMENTAL METHODS

The significant developments required to extend TRexperiments to the subpicosecond time scale are only br

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discussed in this section. Additional description of thesevelopments can be found in Ref. 41. It is very importantstress that the datamustbe worked up correctly if meaningful results are to be obtained.

A. Experimental apparatus

The time-resolved THz spectrometer is shown schemcally in Fig. 1. A Spectra Physics regenerative amplifier stem ~Millennia-Tsunami-Merlin-Spitfire! produces a 1-kHzpulse train of 1-mJ, 800-nm pulses of 100 fs duration@fullwidth at half maximum~FWHM!#. This beam is split intothree portions, each of which is responsible for eitherTHz transmitter, THz receiver, or visible photoexcitatioarms of the spectrometer~labeled 1, 2, and 3 respectively!.For reduced bandwidth experiments described below,output of the laser is modified. Metal beam blocks are plain the compressor of the amplifier to block the low- ahigh-frequency regions of the bandwidth of the uncopressed pulse. After recompression, the resultant pulsesa duration of approximately 280 fs~FWHM!, and a reducedbandwidth of about 6 nm.

Terahertz generation occurs through nonresonant oprectification42,43 of the visible pulse in a 1-mm-thick110&ZnTe crystal~II-IV Inc. !. The THz radiation is detected bfree-space electro-optic sampling43–47 in another 0.5-mm-thick ^110& ZnTe crystal. The signal is collected withlock-in amplifier~Stanford Research Systems SR830! phase-locked to an optical chopper which modulates either the Tgeneration arm or the pump beam. We usually collect dwith the chopper in the pump beam; thus, to determinephotoexcited scan, the difference scan must be addedreference scan collected with the chopper in the probe be

A collimated 2-mm-diameter visible beam o230 mJ/pulse is used for THz generation, and a focus200-mm-diameter visible beam of 4.5 nJ/pulse is usedTHz detection. For GaAs experiments, the pump beamergy is attenuated to less than 1mJ/pulse to keep the totanumber of photoexcited carriers below 231016 cm23.

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15 766 PRB 62BEARD, TURNER, AND SCHMUTTENMAER

The THz beam path from the transmitter to the receiveenclosed and purged with dry nitrogen. The sample is plain a variable temperature cryostat~Janis model ST-100, withLakeShore model 321 temperature controller! fitted withsingle-crystalz-cut quartz windows rotated to minimize THgeneration. The sample consists of a 0.46-mm-thick Gwafer in the^100& orientation to ensure that it does not geerate THz radiation upon photoexcitation.48

We find that when the spot size~the diameter at which theintensity of a Gaussian beam falls to 1/e2 of its value at thebeam center! of the pump is the same size as or smaller ththe probe, the extracted spectrum is skewed to higherquencies~see Fig. 2!. Therefore, the spot size of the visib~pump! was at least two times larger than that of the T~probe! beam, so that the probe beam sampled a uniforegion of photoexcitation. The spot sizes were determinedcomparing the beam intensities measured through a sebrass pinholes of known diameter. The visible spot sizeameter was between 9 and 11 mm, and the THz spot sizetypically between 3 and 5 mm. The pinholes were also uto overlap the THz and visible beams at the sample. The Tand visible beams were copropagated to minimize tempsmearing.

B. Data acquisition and analysis

Complete TRTS experiments require collecting a Tdifference scan at a series of pump-probe delay times.refer to this full data set as a two-dimensional~2D! scan. It isalso possible to scan one delay line while holding the otfixed. This is referred to as a 1D scan, and can be colleas such, or obtained numerically from the 2D data. Aprobescan is obtained when the pump delay line~3! is fixedand the THz receiver delay line~2! is scanned. A 1D probescan is only meaningful when obtaining data at least 3after time zero (t50). A 1D pumpscan is obtained when thTHz receiver delay line is fixed at the peak of the THz puand the pump delay line is scanned. A 1D pump scan pvides useful information about the average THz absorpby the sample. For example, 1D pump scans of GaAs~Fig.

FIG. 2. Change in optical density spectra of photoexcited Gaas a function of visible~pump! spot size at a 20-ps pump-probdelay time. The visible spot sizes are 6.0 mm~solid line!, 1.7 mm~dashed line!, and 1.1 mm~dot dashed line!, whereas the THz sposize is 2.3 mm. The spectrum becomes skewed toward higherquencies as the visible spot size becomes smaller than the THzsize.

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3! show that the carriers take longer to return to the higmobility G valley ~Fig. 4! as a function of increasing photoexcitation energy. This type of 1D TRTS is similar to othtime-resolved pump-probe techniques where the changtransmission (DT/T) is measured.

For GaAs the total measuredDT/T is proportional to theabsorption coefficienta, which is related to the mobility ofthe photoexcited electrons. The rise and fall times ofobserved signal can be used to obtain dynamical informa

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FIG. 4. GaAs band structure showing different possible exction schemes. In excitation scheme 1, the optical photon exccarriers high enough in theG valley to allow scattering to theL orX valleys. In this case the onset of photoconductivity will be goerned by the return of the electrons from theL or X into theG valleywhere the mobility is highest. Excitation scheme 2 excites the etron directly into theG valley, and the dynamics will be due trelaxation solely in theG valley.

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about relaxation and scattering rates. However, a 1D puscan is not adequate when the photoresponse includchange in index of refraction, or when the transient absotion has a frequency dependence. In these cases, an obschange inDT/T could be due to a transient distortion of thTHz pulse or a change in index of refraction, and not relato the average THz absorption by free carriers. Only a coplete 2D data set will obtain all the dynamical informatioavailable in a TRTS experiment.

The analysis of the 2D data set is more difficult thtreating each scan as a conventional static time-domain-~TD-THz! difference scan. In such static THz-TDS experi-ments the power spectrum and phase of the reference ispared with that of a photoexcited scan to extract the chain optical density (DOD) and change in phase (Df). Thesevalues are then used to calculate the absorption coefficieaand index of refractionn.49

Two related issues occur in a TRTS experiment that coplicate this simple analysis. First, at early pump-probe detimes the visible excitation pulse arrives at the sample asTHz pulse is propagating through it. The visible pulse affeonly the trailing part of the THz pulse; therefore, the THpulse experiences a different set of propagation parametethe trailing end of the pulse than it did at the beginninSecond, the response of the material to photoexcitationdescribed by its response function, is itself time dependand will evolve during the THz propagation.

Figure 5 illustrates these effects. The figure is a contplot of the THz difference scans@DETHz(t,t9) is equal topump on minus pump off# as a function of pump-probe delay, t9. A line is drawn at 45° to thet9 axis and everythingbelow and to the left of the line is a result of the THz arrivinbefore the visible pulse, while everything to the right aabove the line is obtained when the THz arrives aftervisible pulse. Several THz transients are shown in F5~a!–5~d!. The arrows represent the arrival of the visibpulse, and correspond to the arrows on the 2D grid. Iapparent that the THz difference scans of Figs. 5~a!–5~c! cannot be thought of as static THz-TDS experiments.

The change in response function in GaAs is proportioto the change in conductivity,Ds(t,t9). The measured dif-ference signalDE will be a convolution of the change inresponse function and the input THz pulse. These compltions were formally discussed in the context of photoexction in liquids, and the results can be applied to transiphotoconductivity with only minor modifications.50 Therein,Kindt and Schmuttenmaer showed that scanning dela~THz transmitter! in reverse, with delays 2~THz receiver!and 3 ~pump delay! fixed is equivalent to synchronouslscanning delays 2 and 3 together with delay 1 fixed. Eitscenario results in every portion of measured THz transexperiencing the same delay from the visible pulse.

Collecting the data synchronously is equivalent to americal projection of the 2D data set shown in Fig. 5 alothe constantu5t2t9 axis, and the result is shown iFig. 6~a!. The change in response function can now bescribed byDs(t,u). Sinceu is constant,Ds only dependsparametrically on pump-probe delay time,Ds(t;t9), andtherefore represents the photoexcited optical propertieeacht9. We collect the data conventionally and project oDE(t,u) numerically, because we must also deconvolve

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detector response from the measured wave form as iscussed in Ref. 41. Once the data have been collecteddeconvolved from the detector response as shown in6~b!, each pulse in the grid can be treated as an individTHz scan.

To extract the photoexcited optical constants~absorptioncoefficienta, and index of refractionn) we must considerthe photoexcited path lengthd, and the reflectivity loses athe two interfaces~air/photoexcited GaAs and photoexciteGaAs/bulk GaAs!. We obtaind by employing a model whichincludes diffusion, surface recombination, and bulk recomnation effects, as discussed in Sec. III A. The reflectivlosses at the two surfaces are not knowna priori. In addition,the spatial extent of the THz pulse is much larger thanpath length traversed. Both the reflectivity changes and

FIG. 5. Contour plot of two-dimensional~2D! raw THz differ-ence scans collected by fixing the pump delay line while scannthe probe delay line. Solid lines correspond to positive values, wdashed lines correspond to negative values. The arrows labeled~a!–~d! represent different arrival times of the pump beam, andcorresponding difference scans are shown below the figure.~a! cor-responds to the pump arriving after the main part of the THzstill affecting the trailing part. As the pump arrival time approachthe main feature of the THz pulse@~b! and ~c!# a larger differencescan is measured, until pump delay time~d!, which represents thepump coming well before the probe. The line at 45° representsconstantu5t2t9 axis.

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FIG. 6. Two-dimensional~2D! contour plot ofDE(t,t2t9). Solid lines correspond to positive values, while dashed lines corresponnegative values. Plot~a! is the raw collected data after projection onto the constantu5t2t9 axis. Plot~b! is obtained after the detectoresponse has been deconvolved, and the data projected onto the constantu5t2t9 axis. Plot~a! shows features appearing prior to the pumpulse arrival due to the finite detector response. Plot~b! shows that these features have essentially been removed.

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small path length lead to an effect where the THz pulsepears to propagate with superluminal velocities in photocited GaAs. Schall and Jepsen accounted for this behavioconsidering the small path length and front surface interfonly.40 However, to be most quantitative we have found thinclusion ofboth interfaces is needed. To do so we have usthin-film transmission formulas derivable in the frequendomain from Maxwell’s equations.51 We also find that the‘‘superluminal’’ propagation is entirely accounted for wheemploying the thin-film formulas~and also in our numericapropagation described in Sec. III C!.

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III. RESULTS AND DISCUSSION

To fully characterize the photoconductivity in GaAs,wide range of experiments were performed. The conditioused in these experiments are listed in Table I. In each cthe photon energy was kept to within 100 meV of the baedge to avoid intervalley scattering. Therefore, the datanot affected by velocity overshoot because the carriers cnot access the satellite valleys. Consequently, we probechange in mobility solely in theG valley ~Fig. 4!. We have

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TABLE I. Experimental parameters. Columns A—M list the experimental conditions for the different 2D pump-probe experimenl isthe wavelength of the photoexcitation pulse in nm, KEe,HH and KEe,LH are the kinetic energies in meV of electrons excited from the heaand light-hole bands, respectively, andnHH andnLH are the number of\vLO above the conduction-band minimum for electrons excited frthe heavy- and light-hole bands, respectively. The bandwidth of the visible excitation pulse is given in meV, and the pulse wfemtoseconds is the FWHM of the excitation pulse. The optical skin depth of the absorption in GaAs is given inmm, ~Ref. 54!, and thenascent photoexcited carrier density isN. Data with parameters from columnF are shown as representative data sets in Figs. 10–12.

Temperature 75 Ka Temperature 300 Kb

A B C D E F G H I J K L M

l ~nm! 808 809 803.5 799 793 789 783 778 774 803.6 799 795 786KEe,HH ~meV! 28.3 26.9 36.6 45.4 55.4 64.3 73.5 83.5 91 109 117 124 140KEe,LH ~meV! 14.8 13.9 18.9 23.4 28.3 32.3 37.7 42.5 46.5 55.7 59.6 63.2 71nHH 0.78 0.75 1.01 1.26 1.54 1.78 2.04 2.3 2.5 3.0 3.25 3.44 3nLH 0.13 0.39 0.53 0.65 0.79 0.90 1.0 1.2 1.3 1.5 1.66 1.76 1Bandwidth~meV! 10.4 11.7 11.6 11.8 13.7 13.0 13.0 12.0 11.5 17 20 19 18Pulse width~fs! 280 100 280 280 280 280 280 280 280 100 100 100 100Skin depth (mm) 1.1 0.99 0.94 0.90 0.86 0.82 0.79 0.75 0.72 0.68 0.66 0.64 0N (31016 cm23) 1.8 0.1 0.4 2.2 2.3 2.1 2.4 2.5 2.6 2.0 8.6 5.0 6.1

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maintained the highest temporal resolution by deconvolvthe detector response and numerically tracking the pumpprobe beam delay lines together. This eliminates any artiffrom monitoring a mobility that is changing on a time scawhich is faster than the THz pulse duration.

Our results are presented in four sections. Section Idiscusses the 1D pump results~scanning the pump with fixedprobe delay time!. We determine the bulk and surface recobination dynamics from these studies. Section III B presethe 1D probe results~scanning the probe with fixed pumdelay times!. We show the accuracy of the TRTS methodcomparing the measured data to that predicted by the Dmodel. In Secs. III C and III D we discuss the early timwhere simple 1D probe and 1D pump scans are insufficand 2D scanning is required. Section III C introducesfinite-difference time-domain~FDTD! method which is usedto model the data, and Sec. III D compares the measuredto the FDTD simulations.

A. Bulk and surface recombination dynamics„1D pump…

The carrier recombination dynamics of photoexcitGaAs can be monitored with 1D TRTS experiments in whthe change in THz transmission is measured as a functiothe pump-probe delay time~see Sec. II B!. While 1D pumpscans are of limited use at short pump-probe delay tim(,3 ps), when a change in index of refraction accompana change in absorption, this is not an issue at longer puprobe delay times (.3 ps). The inset of Fig. 7 displays a800-ps duration, room-temperature, 1D pump scan witphotoexcitation arclength of 790 nm. This scan illustrathat the recombination dynamics cannot be described bsingle exponential, as would be expected from a single brecombination process. Instead, the data are fit to a calated carrier density as a function of time after photoexction, which is extracted from the solution of a diffusion eqution with terms for bulk recombination and the opticpenetration depth of the visible excitation pulse.52

FIG. 7. The distribution of carriers as a function of distance inthe sample, calculated from a diffusion model~Ref. 52! which in-cludes terms for both bulk and surface recombination. Solid curare profiles of the distribution at 50-ps intervals, starting 1 ps aphotoexcitation. Att50, the distribution is exponential. The insis a room-temperature 1D pump scan of 800-ps duration~dots!,which is fit to the calculated carrier density as a function of timafter photoexcitation.

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Figure 7 shows the calculated carrier distributionn(x) at50-ps intervals, starting 1 ps after photoexcitation. The vues for tb and s used to calculate the carrier distributionshown are taken from the best fit to the data in the inseFig. 7. Literature values for the ambipolar mobility53 (mab5388 cm2 V21 s21) and absorption coefficient54 (a51620 cm21) were used to obtain values fortb ands of 2.1ns and 8.53105 cm/s, respectively. Since the initial distrbution of carriers att50 is an exponential distribution of thform exp(2ax), the majority of the carriers are near the suface at early times, and surface recombination is the donant relaxation process. It can be seen that at long dtimes, as the carriers simultaneously diffuse into the bulkthe sample and recombine at the surface, bulk recombinabecomes the dominant relaxation process. This accountthe observed double-exponential behavior, and allows uextract rates for both bulk and surface recombination pcesses.

The distributions calculated from this model are usedobtain d, which is used in the calculation ofa and n asdescribed elsewhere.41 For a simple exponential distributiothe path length is the 1/e point, or, equivalently, the totaintegral of the distribution. For more general distributioninvolving the effects of surface recombination, one cantain the path length in an analogous fashion by scalingmaximum of each distribution to unity and integratingobtaind.

B. Mobility at long delay times „1D probe…

We have performed 1D probe scans at pump-probe dtimes of 5, 10, 50, 100, 250, and 500 ps at both 300 andK, where for each delay time the carriers have thermaliand relaxed within the centralG valley. At such long timesafter photoexcitation, the Drude model is expected toscribe the photoexcited optical properties. We have fitdata to a variety of models described below, and find thatresults are best represented by a generalized form ofDrude model. Electron mobilities extracted from this trement of the data are in good agreement with literature valu

The simplest model of electrical conductivity is the Drumodel, which treats conduction electrons as free to munder the influence of an applied electric field but subjeca collisional damping force.55 The frequency-dependencomplex conductivity,s(v), is related to the dielectric susceptibility throughs(v)52 ivx(v). The inverse Fouriertransform ofx(v) yields the time domain response functiox(t), and describes the evolution of the system after apping an electric field.56 Drude theory assumes an exponentevolution with a relaxation time oft. In the frequency do-main the equation for the Drude conductivity is equivalentthe permittivity of a liquid based on Debye theory.57 TheDrude model can be modified in an analagous fashion as‘‘Cole-Cole’’ 58 ~CC! or ‘‘Cole-Davidson’’59 ~CD! modelsfor liquids. The CC and CD~Ref. 60! theories both describe

l-

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response functions with nonexponential relaxations, forample, the CD response function has the form61

x~ t !51

tG~b! S t

t D b21

e2t/t, ~9!

where G(b) is the g function with argumentb, and t isknown as the critical relaxation time since there is actualldistribution of relaxation times. The CC response functihas a similar form in that there is ane2t/t term, but it can notbe written as a closed expression.61 These nonexponentiaresponses lead to a continuous distribution of relaxattimes, and because of the large range of relaxation timinvolved, it is easier to work with a logarithmic distributionIn the case of the Drude model, the continuous distributsimply reduces to ad function, G(ln t8)5d(t2t8). TheCC model has a symmetric logarithmic distribution functiocentered at lnt, whose width increases witha, and becomesa d function whena50.61 The CD model has an asymmeric ~actually one-sided! logarithmic distribution function thatpeaks att, and is zero for all times greater thant. It falls offfor values less thant, falling off more slowly with smallervalues ofb, and reduces to ad function att whenb51.61

Both the CC and CD forms describe a response whichinitially faster than the Drude form~with the CD model be-ing faster than the CC model!. Symmetric and asymmetricdistributions of relaxation times can be included simulneously to form~GD! generalized Drude model.

One of the main assumptions of the Drude model isfree-electron approximation~FEA!. The FEA assumes thathe electrons do not interact with the scatterers. Thusscattering event is independent of the electron energy.FEA accounts for most of the failure of the Drude model55

Inclusion of interactions with the scatterers leads to‘‘memory effect,’’ wherein the momentum and energy of thscattered electron depend on the incoming momentumenergy. The distribution of relaxation times described byCC, CD, and GD models implies memory effect.56,62 Photo-conduction in GaAs does not strictly adhere to the FEA, athe response is better described by including a distributionrelaxation processes in the form of the GD model. Thishardly surprising, since we expect many different scatterprocesses in GaAs. For example, scattering occurs fromphonons, heavy holes, impurities, and accoustic phononname a few. Hill and Dissado62 showed that even a simplcase of two different Debye-type relaxation processes resin a nonexponential behavior of the response function,x(t).

Jeon and Grischkowdky observed a deviation fromDrude model, and modified it to include energy-dependrelaxation from lattice impurity and acoustic phonon scatting events with t21(E)5t i

21(E/kBT0)23/2

1t l21(E/kBT0)1/2, where t21(E) is the overall scattering

rate, t i21 is the impurity scattering rate,t l

21 is the lattice~acoustic phonon! scattering rate, andT0 is an additionalfitting parameter with units of temperature.63 Those scatter-ing events did not account for their measured data, whereCD model did. We too have employed this scattering modand find that it is not as good as the GD model. Invoking

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GD model to fit our data assumes that the true relaxaprocess is the average of a distribution of relaxation timwhich are energy dependent.

The frequency-dependent complex conductivity is dscribed by the GD model as

s~v!5«0vp

2t

~12~ ivt!12a!b, ~10!

where«s is the free-space permittivity,t is the carrier col-lision time, andvp is the plasmon frequency. The plasmofrequency is defined as

vp25Ne2/~«0m* !, ~11!

whereN is the carrier density,e is the electron charge, anm* is the effective mass of the electron in GaA(0.067me).

53 Electron mobilities can be calculated accordito m5e/(m* g), where the carrier damping rateg51/t.Equation~10! reduces to the CC model whenb51, the CDmodel whena50, and the Drude model whena50 andb51.

Fits obtained using CC, CD, and GD models indicate tthe CC model fits well to the real conductivity at frequencbelow 1.2 THz, and that it fits well to the imaginary condutivity at frequencies above 0.9 THz. The CD model coverges to the Drude model for these data (b51), and thusneed not be considered further. The Drude model fits wethe real conductivity at frequencies above 1.2 THz, but ovestimates the imaginary conductivity for frequencies bel0.9 THz. Either model~CC or Drude! individually deviatesfrom the data by as much as 20%, but when combined inGD model reproduce the data to within1/- 3% for all fre-quencies less than 1.6 THz. Other THz spectroscopic stuon doped silicon found that the frequency-dependent condtivity fit well to the CD model,57,63 and studies on dopeGaAs found that the conductivity fit to the Drude model49

We performed pump-probe studies on silicon, and foundthe conductivity fit better to the CC model, which indicatthat the GD behavior we observe for GaAs may be due todifference between photoexcited and doped conduction etrons.

Several additional modifications to the Drude model winvestigated. The Drude conductivity was modified to allothe carriers to have two different damping rates, whichlows for there to be two types of scattering processes.model was then modified to allow for the existence of sface and bulk electrons, each with their own characteriplasmon frequency and damping rate. Finally, the modelmodified to include both electrons and holes as charge cers. None of these models performed nearly as well asGD model.

Based on these extensive comparisons, the GD modelchosen as the best theory to represent the measured dlong pump-probe delay times (.3 ps). Figure 8 displaysthe real and imaginary components of the frequendependent conductivity 100 ps after photoexcitation atand 300 K. The GD fit reproduces the data well, and pduces reasonable values for the electron mobility and cadensity. The 300-K electron mobility obtained from the fitme56540 cm2 V21 s21 with a carrier density, obtained

ns,

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from Eq. ~11!, of N51.631016 cm23. This is slightly lessthan the literature value53 of 8600 cm2 V21 s21. The data at77 K leads to a much higher mobility ofme513600 cm2V21s21, with a carrier density ofN51.531016 cm23. An increase in mobility at lower temperatureis expected due to the decreased number of scattering evIn addition, the carrier densities obtained from the fits asimilar to those calculated based on the spot sizes and insities of the pump beam (N57.131015 cm23 at 300 K andN51.431016 cm23 at 77 K!.

The CC parametera obtained from fits to the data atvariety of pump-probe delay times was found not to depesignificantly on carrier concentration or pump-probe detime. The CD parameter,b, was always greater than 0.8 anfound to converge to 1 in the limit of low carrier concentrtions. These trends are different than those observed byet al.63 They found that the CD term converged to 1 at hicarrier concentrations. Surface effects are an unlikely souof this discrepancy, becausea was found not to depend othe amount of time after photoexcitation~at long times thecarriers are far away from the surface!. The most likely ex-planation for this is that our sample is photoexcited, ndoped. In a doped sample carriers are continuously trapand released from the ionic cores, whereas in a photoexcsample, once an electron recombines with a hole it will nbe released.

The calculated mobilities at 77 K as a function of carrdensity are shown in Fig. 9. The mobility increases ascarrier concentration decreases, and is shown to fit welthe empirical Caughey-Thomas relation.63,64

C. TRTS finite-difference time-domain „FDTD… simulations

In an effort to better understand our results, we havemerically propagated THz pulses through a 0.7-mm-thick

FIG. 8. Real~black circles! and imaginary~white circles! con-ductivity at 300 K~a! and 77 K~b!, measured 100 ps after photoexcitation in GaAs in the100& orientation. The fit~solid line! is thegeneralized Drude~GD! model.

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layer of photoexcited GaAs, bounded on one side by air,the other by nonphotoexcited GaAs. The properties wchanged from insulating to conducting as a function of ti~using the Drude model with a time-dependent scatterrate! by simulating a Gaussian photoexcitation pulse. Wfind that the simulations reproduce the general trends seethe data as will be discussed in Sec. III D.

The FDTD method was used with absorbing boundconditions.65–67 In this method, Maxwell’s equations arsolved numerically. We need only solve a 1D propagatproblem since the radial dimension of the THz spot~2 mm!is much larger than the optical absorption depth of 0.7mm.The magnetic and electric fields are obtained using65

H~s,t,t9!5H~s,t2dt,t9!2Dt

m0Ds@E~s1ds/2,t2dt/2,t9!

2E~s2ds/2,t2dt/2,t9!# ~12!

and

D~s,t,t9!5D~s,t2dt,t9!2Dt

Ds@H~s1ds/2,t2dt/2,t9!

2H~s2ds/2,t2dt/2,t9!#, ~13!

where

D~s,t,t9!5«0«E~s,t,t9!1«0E0

t

E~ t2t!x~t,t9!dt,

~14!

B is the magnetic field andB5m0H, m0 is the permeabilityof free space,E is the electric field,D is the electric displacement, s is the spatial position, andDs is its step size. Thepropagation time ist, with step sizeDt, and the pump-probedelay time ist9. The dielectric constant is«5«0«` , andx(t8,t9) is the dielectric response of the material. The aboFDTD formalism is correct to second order. We have foua spatial step size of 0.025mm is needed to adequatesimulate a photoexcitation depth of 0.7mm, and the tempo-ral step size is 0.05 fs, much smaller than the Courant staity criterion.68

FIG. 9. Mobility at 77 K as a function of carrier concentrationNas determined from the generalized Drude~GD! model. The mobil-ity follows the empirical Caughey-Thomas curve~line!. At lowerconcentrations the mobility is expected to level off again.

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The time-domain dielectric response for photoexcited criers in the Drude model is given by the solution of thDrude equation of motion,

s~ t !52g s~ t !1e

m*E~ t !, ~15!

where g is the damping constant andm* is the effectivemass of the carrier. The solution is66

x~ t8,t9!5vp

2~ t9!

g~ t9!$12exp@2g~ t9!t8#%, ~16!

where vp2(t9)5e2N(t9)/«0m* is the plasmon frequency

which depends on the time since the arrival of the pupulse through the number of photoexcited electronsN that itgenerates. The damping constantg(t9) can also depend ontime to simulate a time-dependent mobility. The aboveponential form of the dielectric response leads to a simpcation in the descritization of the FDTD formulas, and whave followed the procedure of Ref. 66.

We have compared an exponential distribution of carrito a slab distribution using the FDTD method. We find thfor distributions with skin depths less than 3mm, the errorof assuming a slab distribution is less than 5% for carrconcentrations of,1017 cm23. For skin depths larger than3 mm, the error grows linearly with skin depth. Howevewe find that an exact solution can be obtained by dividingdistribution into small slices and expressing the transmisscoefficient in terms of multiple layer formulas.69 Further-more, we investigated the effects of carrier profiles, suchthose described in Sec. III A, where the surface recombtion has depleted the carriers near the surface resultincarrier distributions which are neither exponential nor slWe find that these distributions behave more like the sthan the exponential. In fact, assuming a slab distributintroduces no additional error in obtaining the correct resueven for skin depths greater than 3mm. As a result, weassume a slab distribution with path lengths determineddiscussed in Sec. III A.

D. Behavior at early times „2D pump probe…

One of the motivations for these studies was the desirobserve the dramatic changes predicted in the time-resolfrequency-dependent conductivity for initial electron kineenergies that are integer multiples of the LO-phonon fquency (\vLO).30 The excitation bandwidth was reduced6 nm ~11 meV! so that the spread in kinetic energies wsmaller than the LO-phonon energy~36 meV in GaAs!. Pho-ton energies were varied from less than 13\vLO to almost43\vLO to investigate any trends in the data. Table I tablates the different photon energies used, the relationshipthe electron KE to\vLO , the bandwidth of the pump laseand the optical skin depth at each wavelength. We fouthat, for the most part, any differences as a function of eltron kinetic energy were less than the reproducibility btween data sets. Therefore, it is most productive to idencommon characteristics among the data sets rather thanferences.

Figures 10~a! and 10~c! and 11~a! and 11~c! show repre-sentative 2D plots of the real and imaginary parts of

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FIG. 10. Representative data set for the repart of the time-resolved, frequency-dependeconductivity. The data shown correspond to texperimental parameters given in Table I, columF. ~a! is the measured data,~c! is a contour plot of~a!, ~b! is the FDTD simulated data, and~d! is acontour plot of~b!. The simulation reproduces thdata fairly well.

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experimentally determined conductivity for GaAs. Figur10~b! and 10~d! and 11~b! and 11~d! are the 2D plots of thesimulated real and imaginary parts of the conductivity. Tsimilarity between the actual data and the results of the silation is quite good. To further quantify and understanddata, difference plots are shown in Figs. 12~a!–12~d!. Thedifference plots were generated by subtracting a scaled lotime ~3 ps! scan from entire data set. The long time scan wscaled with a sigmoid that increased from zero to one at tzero with a width equaling the width of the pump pulse. T

eu-e

g-se

difference plots show where the data deviates from its equbrated values, and the main features of the deviation arehighly dependent on the width of the sigmoid.

The data can be divided into three regimes dependingthe pump-probe delay time, and they are initial (20.5–1 ps!,intermediate~1–3 ps!, and steady state (.3 ps). Severalvariations are observed as the conductivity evolves tosteady state~3 ps!. The simulations do not completely reproduce the variations observed in the initial regime; howevthey qualitatively reproduce the trends seen in all three

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FIG. 11. Representative data set for thimaginary part of the time-resolved, frequencdependent conductivity. The data shown corrspond to the experimental parameters givenTable I, column F.~a! is the measured data,~c! isa contour plot of~a!, ~b! is the FDTD simulateddata, and~d! is a contour plot of~b!. Deviationsof the data from the simulations are observedlow frequencies.

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FIG. 12. Difference plots generated by sutracting a scaled long time~3 ps! 1D probe dataset from the data. Difference plots are shown fthe real ~Re@s#! ~a! and imaginary~Im@s#! ~c!parts of measured conductivity. The differencplots for the FDTD simulated Re@s# and Im@s#are shown in~b! and~d!, respectively. Deviationsare most pronounced in Im@s# at low frequencies,but both the Re@s# and the Im@s# contain devia-tions from 0 – 2.5 ps.

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gimes, with the last two being reproduced best. The mobinitially increases rapidly, then decreases, and finallycreases gradually to the steady-state value.

Initially, the conductivity is seen to rise simultaneousfor all frequencies. Then the lower frequencies continuerise while the higher frequencies decrease. A peak atfrequencies is observed in the imaginary part of the condtivity ~Im@s#! that is not reproduced in the simulations~seeFigs. 10–12!. This peak also causes the low frequenciesthe Im(s) to grow in faster than the high frequencies. Tfact that the low frequencies grow in faster than the higfrequencies suggest this is not a detector limitation. TRe@s# is well reproduced by the simulations. These obsertions are partially explained by ballistic transport of eletrons.

Ballistic transport occurs when no scattering centerspresent and thereforeg50 in Eq. ~15!. The conductivityresponse in the frequency domain would be purely imanary, and scale asv21. This might account for the rise in thimaginary part of the conductivity at low frequencies. Tfurther investigate ballistic transport, we allowedg to betime dependent. Specifically, we allowedg to start near zeroand then rise to its limiting steady-state value. We compathis to simulations whereg was held constant. We find thaincluding ballistic electrons enhances the Im@s# at low fre-quencies, but not to the extent observed in the data.conclude that ballistic transport cannot fully account for tbehavior observed in the data.

An instantaneous acceleration of the electrons fromapplied electric field of the THz probe pulse can take plawhen the photoexcitation occurs at the same time thatTHz field is in the sample. At long times, when the pumcomes well before the probe, the electrons will experienc

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gradual increase in the field and will only be gently accelated. At the earliest times, however, the pump creates cduction electrons while the THz field is present. Thus telectrons experience an electric field upon being created.acceleration of the electrons by the field radiates an ational field, and this may account for the observed responThis effect, however, is present in the simulations, and thefore cannot explain the observed deviations. We have vfied that this nonlinear interaction is linearly dependentpump intensity and THz amplitude.70

The second regime of the data encompasses 1–3 ps.data show a peak in the Im@s# at about 0.8 THz, that shifts toabout 0.5 THz over the next 1.5 ps. A time-dependent stering rate was included in the FDTD simulations to invesgate this regime. Based on work of Vengurlekar and Jh26

the scattering rate was large initially, and decreased exnentially to its steady-state value. These simulations quatively reproduce the trends seen in the data. In particular,peak in the Im@s# shifts rapidly from low frequencies~in theinitial regime! to high frequencies, then gradually backlow frequencies. The peak of Im@s# is related to the scattering rateg which in turn is inversely related to the mobilitby m5e/(m* g). A plot of the peak of Im@s# for the averageof five data sets is shown in Fig. 13. These observationsin the spirit with the predictions of Vengurlekar and Jha, wpredicted a time-dependent mobility due to the dynamicsthe photoconducting electrons with the hot phonons.

Furthermore, the mobility increases more than theserved shift in the peak of Im@s# to lower frequencies because the electron effective mass is decreasing as theirtribution evolves from the initial photoexcited distributiona thermalized one. The time-dependent mobility is related

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the effective mass and the scattering rate bym(t)5e/@g(t)m* (t)#. TheG valley of GaAs is slightly nonparabolic, and conduction electrons will have a different effectimassm* depending on the distribution function of the phtoexcited electrons,f (E,t). The effective mass is obtaineusing Eq.~1!. The effective mass is largest immediately folowing photoexcitation, and smallest at the band edge.mobility increases as the distribution relaxes to the bottomthe G valley due to the decrease in both effective massscattering rate.

Two other potential explanations for the behavior btween 1 and 3 ps, velocity overshoot and Coulombic screing, have been considered and ruled out. We have ruledthe possibility of velocity overshoot because of the low eltron kinetic energies and low applied field. Even at the higest photon energies employed, the initial electron kineticergy is 99 meV at 70 K and 117 meV at 300 K. As a wocase scenario, consider a Boltzmann distribution withsame kinetic energy as the photoexcited electrons, whwould correspond to the carriers fully thermalizing amothemselves prior to equilibration with the lattice. With thassumption, only about 5% of the carriers have kineticergy above theL-valley minimum~300 meV!. Therefore, theonly way to transfer significant population into theL valleywould be to apply a large electric field~the Gunn effect!.However, the THz field~measured to be 110 V/cm at thsample! is much smaller than the required fields of aboukV/cm. Coulombic screening is also ruled out, becausedoes not become important until higher excitation densiand applied fields~5 kV/cm! are reached.71 Therefore, weconclude that the observed dynamics are a result of theteractions between the photoexcited electrons and phon

FIG. 13. Plot of the frequency at the peak of the imaginary pof the conductivity as a function of time after photoexcitation. Thvalue corresponds to the scattering rate,g. Variations as a functionof time are attributed to phonon-assisted intravalley relaxation.set ~a! shows far infrared~FIR! absorption assisted by LO-phonoabsorptionand emission. Inset~b! shows that after the electrodistribution has relaxed below\vLO , FIR absorption is only as-sisted by LO-phonon absorption.

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Intraband free-carrier absorption is a phonon-mediaprocess. That is, to conserve both energy and momentumphoton absorption must be accompanied by absorptionemission of a phonon~see the inset of Fig. 13!. Thus thetransient conductivity can depend on the phonon dynamThe observed features exist over the range of times thatcharacteristic of electron-phonon coupling. For example,cent work has shown that as a high density of hot electrcool, a large population of phonons can be created, whcan then assist intraband absorption, and can also be rsorbed by the electrons, thereby slowing the cooling rate72

Since IR photon energies are much greater than thephonon energy of 36 meV, intraband absorption typicaoccurs with either the absorptionor emission of a phononOur situation is somewhat different because the FIR phoenergy is less than the phonon energy. Specifically, as sin the insets of Fig. 13, at early times phonons can be eiabsorbed or emitted to assist the photoabsorption, wherelong delay times, after the carriers have relaxed to the botof the conduction band, the photon energy of absorptionnot great enough to allow emission of a phonon to accopany photoabsorption. This effect is responsible for thecrease ing at intermediate times.

In a series of papers, Vengurlekar and Jha detailed aoretical prediction based on the Boltzmann transport eqtions for photoconducting electrons in GaAs.26–30 Vengurle-kar and Jha predicted a time-dependent mobility duedynamics in theG valley as the electrons cool to the banedge by their interactions with the LO phonons throughFrohlich interaction.73 The momentum relaxation timetm

will depend on electron-phonon dynamics becausetm21

5tLO211t1

21 , wheretLO21 is the LO emission rate andt1

21 isthe quasielastic scattering rate. At early times the photocited electrons can emit phonons, andtm'tLO'150 fs. Af-ter the electrons have cooled to below\vLO , they can nolonger emit LO phonons andtm't1'1 –100 ps. The quasielastic scattering rate describes scattering with the heholes, impurities, and acoustic phonons. Vengurlekar andpredicted that these dynamics occur on a time scale ofps.26 Our data verify these predictions. While the data reva time-dependent scattering rate, we do not observe amatic dependence on whether the photoexcited electronscreated with integer or half-integer multiples of\vLO , aspredicted.30 Vengurlekar and Jha’s predictions provide a dscription of the carrier mobility and scattering rates, but tscope of their work did not include calculation of intrabaTHz absorption. To do so would require explicit inclusionthe coupling of phonons in the intraband absorption proce

The last regime found in the data is the static, long-timlimit. After 3 ps, the carriers have relaxed to the bottomthe conduction band. The mobility is seen to increase olong time scale (.100 ps) as carriers disperse and relathereby lowering the overall carrier density.

IV. CONCLUSIONS

There are some important experimental considerationbe aware of when carrying out TRTS studies. In particularis necessary to project the data onto the constantu5t2t9axis ~either numerically or physically! before further analy-sis. It is also necessary to deconvolve the detector resp

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function if the sample response is faster than the deteresponse. The need for deconvolution becomes evident wthere are features in the data that appear to arrive at negtime delays.

This paper demonstrates that TRTS can accurately msure the carrier dynamics and mobility in a photoexcisample. We chose bulk GaAs as a prototypical benchmsystem, and found good agreement with literature valuelong pump-probe delay times, after the carriers have suciently thermalized.

Finally, this work uncovers a contribution to thfrequency-dependent conductivity on time scales from 0.23 ps that has not been previously observed, to our knoedge. These results are in the spirit of the predictions frVengurlekar and Jha30 We attribute these features to thinfluence of phonon dynamics on intraband free-carriersorption.

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Future work will investigate the Boltzmann transpoequations derived by Vengurlekar and Jha in order to undstand the differences between the observed data and theoretical predictions. In addition, we will extend this typestudy to materials that are difficult to characterize with coventional methods, such as semiconductor quantum dotsdye-sensitized colloidal TiO2.

ACKNOWLEDGMENTS

We would like to thank J. T. Kindt for providing earlyversions of the computer programs used for numerical pupropagation. We would like to acknowledge financial suport from the National Science Foundation CAREER Awaprogram, and the Camille and Henry Dreyfus New FacuAward Program.

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Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy

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Transcript of Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy