Subpicosecond study of carrier trapping dynamics in wide-band-gap crystals

PHYSICAL REVIEW B 1 MARCH 1997-IVOLUME 55, NUMBER 9

Subpicosecond study of carrier trapping dynamics in wide-band-gap crystals

P. Martin, S. Guizard, Ph. Daguzan, and G. PetiteService de Recherche sur les Surfaces et l’Irradiation de la Matie`re, Commissariat a` l’Energie Atomique, DSM/DRECAM,

CEN Saclay, 91191 Gif sur Yvette, France

P. D’Oliveira, P. Meynadier, and M. PerdrixService des Photons, Atomes et Mole´cules, Commissariat a´ l’Energie Atomique, DSM/DRECAM, CEN Saclay,

91191 Gif sur Yvette, France~Received 3 October 1996!

Using a very sensitive time-resolved interferometric technique, we study the laser induced carrier trappingdynamics in wide band-gap crystals with 100 fs temporal resolution. The fast trapping of electrons in theband-gap is associated with the formation of self-trapped excitons~STE’s!. The STE’s formation kinetics doesnot depend on the pump laser intensity in SiO2, while the trapping rate increases in NaCl with the excitationdensity. We interpret this result as a direct evidence of exciton trapping in the first case, and an electronictrapping following a hole trapping in the second. This result is explained in terms of electron trajectoriescalculated with a simple Monte Carlo simulation: the electrons can explore a large volume before beingtrapped in NaCl, not in SiO2. A temperature influence on the initial trapping process is observed in KBr, notin NaCl and SiO2. Finally, we find no evidence of STE formation in diamond. This result is in agreement withgeneral consideration about the STE’s formation in terms of lattice elasticity and deformation potentials.@S0163-1829~97!01110-7#

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I. INTRODUCTION

When a lattice is not stable upon creation of electron-hpairs, an electron, a hole, or a bound electron-hole pair~ex-citon! can be trapped in its own lattice deformation. Thself-trapping process, occurring in materials where the cpling between carriers and phonons is strong, has recefor a long time a considerable interest especially in wband gap solids because it is supposed to be the initial stedefect creation mechanisms. The development of femtoond laser pulses is very helpful to understand these phenena because it is now possible to separate the different sof the trapping process and then to perform studies of defformation dynamics. Experiments are usually based on stroscopic measurements: a first pulse~pump! createselectron-hole pairs whereas a second pulse~probe! is used totest the absorption coefficient of the perturbed material aa controlled time delay. Several time-resolved spectroscomeasurements have been performed in alkali halides sucNaCl,1 KBr,2 and in oxides such as SiO2.

3,4 Recently, time-resolved experiments based on the concept of interferomin the frequency domain have been performed.5 This methodis very powerful because it allows a direct observation ofmodification of the dielectric function of the material duethe pump pulse, and thus, to measure the lifetime of phexcited carriers. The evolution of the electronic excitatidensity has been studied in this way in three oxides~SiO2,MgO, Al 2O3).

6 While the lifetime of electrons in the conduction band is respectively 50 ps and 100 ps in MgO aAl 2O3, a mean electron trapping time of 150 fs has bemeasured in quartz. This trapping time coincides withrise time of the self-trapped-exciton~STE! absorption band,leading to the conclusion that the electronic trapping is asciated with the formation of STE’s.4 In order to investigatethe role of the sample ionicity in these phenomena, this

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terferometric technique has been applied to several otherterials going from pure ionic solids~NaCl, KBr! to a purecovalent one~diamond! v ia a material presenting a mixionocovalent character~SiO2). The influence of the pumplaser pulse intensity and sample temperature has beenied for two probe wavelengths: 790 and 618 nm.

The experimental results presented in this work canunderstood with the help of some basic pictures proposeexplain the STE’s formation some time ago, and constitan experimental evidence for these models that are stillsettled out of hand:7 in quartz, STE’s are formed from thelectron and the hole of the same electron-hole pair, whein NaCl, their formation can be roughly interpreted as a tsteps mechanism: first, a hole self-trapping occurs, andan electron is captured by this hole, leading to an STE. Ththe ultrafast carrier trapping is correlated to the STE formtion. We will show that in KBr, contrary to NaCl, the samptemperature plays a role even in the initial trapping proceOur results are consistent with models which predict tSTE’s can exist in a given material, only if the lattice elaticity is large enough.8

In Sec. II, we describe the experimental setup and pciple. In Sec. III, a phenomenological description of the ecited material dielectric function is developed. The expemental data in SiO2, NaCl, KBr, and diamond are presentein Sec. IV, as well as a kinetic model based on rate equatused to emphasize the difference of the STE’s creationnetics between SiO2 and NaCl. The results of a Monte Carlsimulation of electron trajectories in NaCl and SiO2, under-lying the crucial role of the electron-phonon collisions,presented in Sec. V. Finally, the elastic constants of theferent materials studied here will be compared, and a colation between elastic properties and radiation hardnessbe pointed out.

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II. EXPERIMENTAL PRINCIPLE AND SETUP

The technique of interferometry in the frequency domis a powerful and sensitive method to measure the phaseoccurring in the electric field of a probe laser pulse in tsample under study. This phase shift is obtained by anaing the pattern resulting from interferences between a prand a reference beam, at the output of a spectrometer.method was originally used to study the temporal transfunction of optical fibers.9 The fiber was placed in one armof a Mach-Zehnder-type interferometer, and the spectrwas obtained by scanning the wavelength of a tunable laThe technique has been adapted for short laser pulses tosure the self-phase modulation occurring during the propation in optical fibers.10 In this case the spectrum is broad ano tunable source is needed. More recently, time resoexperiments using a pair of identical probe pulses separby a fixed time delay have been performed. In this case,sample is modified by a high intensity pump pulse, the rerence and probe pulses impinging the sample respectbefore and after the pump pulse.11 Interferences between thtwo probe pulses can be observed if the grating of the sptrometer induces at the exit slit a temporal broadening larthan the delay between them. The advantage of this setuto shake off the problem of the path difference betweentwo arms of a small Michelson interferometer and thusremove the problem of the mechanic stability. Furthermosince the optical paths are the same, the noise in the msured phase arising for instance from the roughness ofsample surface can be suppressed by comparing meaments with and without the pump pulse. This method wsuccessfully used to study the dynamics of laser induplasma at metal surfaces,12 the refractive index modificationof dielectric materials induced by a high intensity laser5,6 andthe oscillations of plasma density in the wake of an inteultrashort laser pulse.13

In the present experiments, we used a mode-locTi:sapphire laser operating at a wavelength of 790 nm. Toutput of the oscillator is amplified in a regenerative ampfier, yielding 2 mJ, 120 fs pulses at a 20-Hz repetition rateblock diagram of the experimental setup is drawn in Fig.The first beam splitter reflects 90% of the main beam for

FIG. 1. Schematic representation of the experimental setup.beam splitter, BBO: frequency doubling crystal, DL: delay line, MMichelson interferometer, DM: dichroic mirror, FL: focusing lenS: sample, IL: imaging lens, VC: video camera, CCD: chacoupled device camera.

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pump beam, whose frequency is doubled in a 50mm-thick frequency doubling crystal~BBO!. The fundamen-tal and frequency doubled beams are separated bychroic mirrors~DM!. The coarse setting of the pump beaintensity is achieved by placing a circular aperture before300-mm focal lens that focuses the pump beam onsample, and the fine adjustment obtained by changingpolarization of the beam at 790 nm with a half-wave plalocated before the BBO crystal~not shown on Fig. 1!. Weused pulse energies at 395 nm in the range 2 to 22mJ. Thediameter~at 1/e) of the pump beam is 34, 44, or 58mm,yielding intensities between 231011 and 431012 W/cm2.We used two probe wavelengths, 790 nm~the remaining10% transmitted by the first beam splitter! or 618 nm. Thiswavelength is obtained by white light continuum generatin a water cell on which a part of the main Ti-Sapphire lasbeam at 790 nm is focused. The white light pulse is amfied at 618 nm in three Rhodamine 6-G dye cells pumpedthe second harmonic of a yttrium aluminum garnet~YAG!laser. The probe beam passes through a delay line~DL! anda small pinhole to improve the wave-front quality and icrease the focal spot size. The two identical probe pulsesgenerated in a Michelson interferometer~MI !; the delay be-tween them is adjusted by translating one of the mirrorswas usually set at 12 or 18 ps. The interaction regionimaged on video cameras~VC! with lenses~IL ! to ensure thealignment of pump and probe beams, and at the entranceof the spectrometer. A typical image observed at the outof the spectrometer and digitalized with a charge coupdevice~CCD! camera is shown in Fig. 2: the horizontal axis the wavelength, and the perpendicular axis is parallethe slit of the spectrometer and therefore represents thetical axis in real space. The spatial resolution is of the orof 1 mm. The interference pattern is clearly observed in tpicture. Indeed, if one considers the electric field of the rerence pulseE1(t)5E0(t)e

iv0t and of the probe pulse, separated by a delayDt: E2(t)5E0(t2Dt)eiv0(t2Dt), the powerspectrum of this sequence of two pulses is given

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e FIG. 2. Images of the two probe pulses spectrum obtained aoutput of the spectrometer. Top: without pump pulse, bottompump pulse has crossed the Al2O3 sample 3 ps before the seconprobe pulse. The vertical axis represents the distance from theter of the pump beam, and the horizontal axis the wavelength.

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55 5801SUBPICOSECOND STUDY OF CARRIER TRAPPING . . .

I (v)52I 0(v)@11cos(vDt)#, leading to fringes in the spectrum separated by 2p/Dt. In Fig. 2~b!, a high intensity pumppulse crossed the sample~Al 2O3 in this case! 3 ps before thepassage of the second probe pulse. As a consequencerefractive index experienced by the reference and prpulses are slightly different. This induces a phase shift anpartial absorption, which readily manifest themselves in F2 ~bottom! by a fringe pattern distortion and by a fringcontrast decrease, respectively. LetDF be the phaseshift and T the sample transmittance, the electric fieldthe second probe pulse is nowE2(t)5ATE0(t2Dt)ei [v0(t2Dt)1DF] and the spectrum of the interfering firand second probe pulses is given byI (v)5I 0(v)3@11T12ATcos(vDt1DF)#. Since the perturbation depends on the pump laser intensity which has a Gaussianfile, the distortion of the fringes~the phase shift! and thefringe contrast~proportional to the transmittance! vary ac-cordingly. If we suppose that the refractive index doesvary significantly during the probe pulse, a simple lineline Fourier transform of the image allows us to extractdesired informations: the phase shift and the transmitta~or absorption!, as a function of a radial coordinate that reresents the distance from the center of the pump beamrepeating this operation for different time delays betweenpump and the second probe pulse, we get the temporallution of these two quantities. As already emphasized, amatic signal to noise ratio improvement of the measuphase shift is obtained by comparing the phase obtainedand without pump pulse. The results are also averagedthree laser shots to limit the influence of shot to shot puintensity fluctuations. Furthermore, the sample is displaafter each pump laser shot to work on a fresh area and acumulative effects. We used in this experiment commerciaavailable high purity~impurity content&1 ppm!, 500-mmthick and optically polished samples. The experiment setivity for the phase shift is 231022 rad and the absorptioncoefficient is measured with a confidence of65%.

In Fig. 3, the space-dependent phase shift for three tdelays observed in NaCl at room temperature are presenThe pump intensity is 1,6 TW/cm2 and the probe wavelengtis 618 nm. Let us anticipate the discussion in Sec. III aqualitatively explain these three curves. The phase shiproportional to the modification of the refractive indeDF(r ,t)52pL/l3Dn(r ,t), where r is the distance fromthe center of the pump beam,t the time elapsed since thcrossing of the pump pulse,L is the length over which theprobe and pump beams are overlapped in the sam

FIG. 3. Phase shift as a function of the distance from the ceof the pump beam observed in NaCl for three different time delbetween the pump and the second probe pulses.

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('300 mm in the present experiments! and l the probewavelength. The zero time delay is defined as the coindence of the maxima of the pump and second probe pulA delay -60 fs means that the refractive index of the mateis probed 60 fsbefore the maximum of the pump pulse. Ithat situation, the pump and the second probe pulse areready temporally overlapped and we observe a posiphase shift which is due to the Kerr effect and is therefproportional to the pump laser intensity. Each curve haGaussian profile as associated fits~full lines! show, and thefull width at 1/e, 56mm, is thus in this case the size of thfocused pump beam. For the delay1220 fs, probe and pumppulses no longer overlap and the signal is negative. Tnegative phase shift is proportional to the density of eltrons in the conduction band. Finally, for longer time delathe phase shift is positive and significantly narrower~32mm) than the initial positive one. In this case the positiphase shift is proportional to the density of trapped electroValence electrons must absorb three pump phot(\vp53.14 eV! to bridge the 8.7 eV band-gap in NaCl. Thdensity of trapped electron is thus proportional to the thpower of the pump laser intensity. This explains the narroing of the spatial distribution of trapped electron comparedthe beam intensity profile. The spatial width ratio is almoexactly A3, the ratio expected for a three photon proceFinally the spatial resolution is very useful in two taskmeasuring the pump laser intensityin situ, which is inverselyproportional to the pump beam surface, and checkingorder of the multiphoton process at work in the excitatifrom the valence band to the conduction band.

III. REFRACTIVE INDEX MODEL

To interpret our experimental results, we need a mowhich describes the refractive index ‘‘seen’’ by the firprobe pulse and the second one after the pump pulse. Infollowing equations, only the electronic contribution to thdielectric function is taken into account. We consider thatfirst pulse interacts with the wide band-gap insulator infundamental state and we represent the medium with a polation of two level systems. In that case, the dielectric funtion reads

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refractive indexn0, and we will use this value for the firs~reference! probe pulse in all cases.

We consider now that the second probe pulse ‘‘seethree different electronic populations: a densityNCB of elec-trons in the CB, a densityNtr of electrons trapped somewhein the band-gap, and, of course,N02Nbc2Ntr electrons stillin the ground state. Furthermore, while the pump pulsepresent in the medium, it contributes to the dielectric funtion by inducing a nonlinear polarization known as Keeffect.14 In this context, the dielectric function can be writteas

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m* is the electron effective mass in the conduction bav tr is the energy difference between the fundamental andfirst excited state of the induced defect, 1/t tr the width of thistransition, andf tr is its corresponding oscillator strengtfCB is the oscillator strength standing for the transitionscurring in the CB and 1/te2p simulates the electron-phonocollisions in the CB.xeff

3 is an effective third order nonlineasusceptibility andEp the electric field associated with thpump laser pulse. Note that holes in the valence bandtrapped in the band gap should contribute to the dielecfunction in a similar way. The corresponding terms habeen neglected in Eq.~2! because the effective massesholes are much larger than that of electrons in the mateunder study, except in diamond. It is important to underlthat this hypothesis is not valid if the probe frequenmatches a trapped hole transition since such a resonanhavior can offset the mass ratio. However, this is notsituation prevailing in the materials studied here: in Naand KBr, the optical absorption bands for holes~STE or forH center! lie in the range 3–4 eV, while those for the eletron component are in the range of the probe wavelenused in the present experiments.15 In the case of SiO2, bothelectron and hole absorption bands are peaked above 4far from our probe photon energies.16,17 This situation rein-force the argument of neglecting holes. Finally, let us ephasize that this simplification has in all cases little influenin our conclusion, except may be in the estimation ofexcitation density.

In this model, the optical absorption spectra associawith electrons trapped in the band gap are representedsingle absorption lines. In other words,f tr must also be con-sidered as an adjustable effective parameter because, atin principle, several transitions with different probabilitieare possible. Practically, we will consider in each case othe dominant term, i.e., the one which is the closest to renant transitions for our probe wavelengths.

The phase shift and the induced absorption coefficienteasily be obtained from the above discussed definitions:

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L is the length over which the pump and probe beamssuperposed in the sample, andl52pc/v is the probe wave-length.

In the fitting procedure that will be presented in the nesection, Eqs.~2!–~4! have been usedin extenso. But for thesake of clarity, let us assume that the density of excited etrons is small compared to the density of electrons infundamental state (N0@NCB,Ntr) and that the dampingterms can be neglected. The latter hypothesis is rough,the former is valid since, as we shall see later, the excitadensity in our experiments never exceeds 1019 cm23. Withthese approximations, a first order expansion of Eq.~2! re-placed in Eq.~3! gives

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Although a crude estimate, this expression gives a gorder of magnitude of the observed phase shifts5,6 and ismore convenient to identify the contribution of each term

The first term represents the Kerr effect. It is proportionto the intensity of the pump laserI p and contributesposi-tively to the phase shift because at our probe wavelengthsnonlinear index (n2) is positive. It is observed in all materials and will last as long as the pump and the probe puoverlap in the sample. The second term, which is proptional to the density of electrons that have been excitedthe pump pulse in the conduction band, is alwaysnegative.The last term stands for the trapping of the electrons subquent to a defect formation. Its sign is determined byrelative values ofv tr and v. For example, in the case oshallow traps (v tr,v), its contribution isnegative. Thismeans that the phase shift measurement alone is insufficto distinguish the electrons in the conduction band andelectrons in shallow traps. On the contrary, if the absorptbands associated with the trapped state correspond to wlengths shorter than the probe wavelength (v tr.v), the trap-ping of electrons is revealed by the observation of apositivephase shift.

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This probe photon absorption cross section includescontribution of electrons in the conduction band~first part!and trapped in the band gap~second part!. The first termarises from electron-photon-phonon collisions. It is respsible for laser heating of the electrons in the conductband18,19 and increases with the probe wavelength.

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IV. EXPERIMENTAL RESULTSAND KINETICS MODELIZATION

We now present and quantitatively discuss in detailresults obtained for different materials: NaCl, SiO2, KBr,and C~diamond!. Each point in the curves showing the temporal evolution of the measured phase shifts~and absorption!is obtained by spatially averaging the original data as thpresented in Fig. 3. In the fitting procedure presented belwe consider the real experimental geometry: a Gaussianfile for the pump beam crossed by the probe beam atangle of 10°. This means that for each time delay, Eqs.~3!and ~4! are integrated along the path of the probe beameach value ofr ~Fig. 3!. This allows to take into account thspatial distribution specific for the Kerr effect and for thelectron in the conduction band or trapped in the band g

Equations~3! and ~4! are time-dependent because telectron populationsNCB andNtr are time dependent. Thkinetics of these populations can be simulated by using aof rate equations. The basic idea is to postulate a timependence for the trapping process, to solve numericthe set of rate equations and then to check if the resof Eqs. ~3! and ~4! fit the experimental data. The resulof such simulations are presented in this section for Nand SiO2 because their comparison shows a strikingly dferent behavior.

FIG. 4. Phase shift as a function of time measured in NaClthree pump laser intensities. The probe wavelength is 618 nmthe sample is at room temperature.

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A. The NaCl case

We show in Figs. 4~a!–4~c! the space integrated phasshift for NaCl for various pump intensities ranging fro0.5 to 1.5 T W/cm2. The probe frequency is 618 nm and thsample is at room temperature. As already mentioned,observe in all cases a positive phase shift for delays closzero due to the Kerr effect. This gives us the temporalsponse of the pump-probe system, which is about 150 fsthe end of the pump pulse, the phase shift becomes abrunegative, due to the presence of electrons photoexcited inconduction band. The most remarkable feature in NaCthat the subsequent behavior of the signal depends onpump laser intensity. In the low intensity limit@Fig. 4~a!#, thesignal goes back slowly to zero and becomes slightly potive for a delay equal to 5 ps. For a medium [email protected]~b!#, the phase shift becomes positive for 2 ps delay. Inhighest intensity case@Fig. 4~c!#, the phase shift reachespositive value greater than in the above case but the tdelay is now only 1.3 ps. Keeping in mind that the negatphase shift is proportional to the density of electrons inconduction band and that the onset of a positive phase shthe signature of a trapping process, we conclude from thobservations that thetrapping kinetic is intensity dependen.In other words,the higher the excitation density, the fastthe trapping process.

Absorption measurements are very helpful to identify tnature of the trap. We show in Figs. 5~a! and 5~b!, the ab-sorption as a function of time obtained for comparable puintensities~1.47 and 0.95 T W/cm2) but for two differentprobe wavelengths: 618 and 790 nm. In both cases, a stabsorption peak is present at short times but for longer tdelays, a constant absorption close to 70% is reached in618-nm case whereas this coefficient falls down to less t10% at 790 nm. These experiments have been performedtwo sample temperatures: 300 and 10 K. We did not remany significant temperature dependence in the behaviothe phase shift and of the absorption coefficient for bprobe wavelengths. This indicates that at the time scaleour experiment, i.e., up to 10 ps, the trapping center issame at 10 K and at room temperature. Transient absorpspectra following electron pulse or picosecond laser exction in NaCl at low temperature have been pu

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FIG. 5. Absorption as a function of time measured in NaCltwo pump laser wavelengths: 618 nm~a! and 790 nm~b!.

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lished.15,20 These spectra are attributed to the presenceSTE’s and exhibit a strong absorption band around 2.1('600 nm). We can therefore conclude that the trappprocess at work in our experiment is the formation of STE

We now have to write rate equations describing the teporal evolution of the electron density in the conductibandNCB and in trap levelsNtr . The observed trapping ratincreases with the excitation density. This suggestsbimolecular-type kinetics, an hypothesis already formulain alkali halides.21,22 This suggests also that STE’s aformed in a two-steps process: a hole trapping and thenelectron trapping on a self-trapped hole~STH! site. In thecorresponding rate equations, this two steps trapping prois described as follows: the hole trapping probability is siply an intrinsic trapping rateth , a quantity which is believedto be less than a few ps~Ref. 8! but has not been yet precisely measured, while the trapping probability for an eletron is proportional to the density of trapped holes (Nth)which are not yet occupied by an electron (Nth2Ntr).

These equations contain the populations of freetrapped holes, notedNfh andNth , respectively, and a sourcterm that describes the creation of electron-hole pairs.already pointed out, in NaCl this is a three photon excitatprocess. Finally the rate equations can be written as

~d/dt!NCB5s~3!Fp3N02scapturevNCB~Nth2Ntr!,

~d/dt!Ntr5scapturevNCB~Nth2Ntr!,

~d/dt!Nfh5s~3!Fp3N02Nfh /th ,

~d/dt!Nth5Nfh /th . ~7!

s (3) is a generalized cross section for a three photoncitation process andFp is the photon flux of the pump pulseWe take forN0 the density of Cl atoms. The electron traping rate is proportional to the productscapturev. The meanelectron velocity in the CBv is chosen so that the electrokinetic energy corresponds to the energy of a LO phonwhich is roughly the electron energy after the relaxationthe CB.15,23 scapture is the cross section for electron captuby a self-trapped hole. This quantity has been previouestimated to lie in the range of a few 10214 cm2.21,24

The electron-phonon coupling in NaCl is known to be leefficient than in SiO2 as we will discuss in the next sectionTo perform the simulation, we tookte2p510215 s. The po-sition of the maximum of the absorption band (v tr) and itsassociated width (t tr) are taken from the STE’s absorptiospectrum in NaCl15. The corresponding values are respetively 2.1 and 0.35 eV. In Table I, we present a summarythe set of parameters that fits satisfactorily the measuphase shifts for three pump laser intensities and the abstion for the two probe wavelengths~full lines in Figs. 4 and5!.

Given the relatively important number of adjustable prameters (n2, s (3), te2p , f tr , thole, scapture), we do notclaim that the set of parameters used in the simulation hunique character. A satisfying fit of the five curves cannhowever, be obtained with significantly different values. Othe other hand, it is evident from the data that the trappkinetics cannot be described by a simple exponential lBut an important point to notice is that it is neither possibto fit the phase shift evolution for the three intensities wi

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out the assumption of a preliminary hole trapping. If wassume an instantaneous hole trapping, the electron traprate, initially proportional to the square of the excitation desity, should indefinitely increase with increasing pump laintensity. However, we observe experimentally that aboexcitation densities of the order of 1019 cm23, the delay forwhich the phase shift becomes positive is always the sa(.1.3 ps!. This saturation of the trapping rate unambigously demonstrates that the hole must first be self-trappevery high excitation densities, the electron trapping immeately follows the hole trapping, and the hole trapping ra(thole50.5 ps! is readily measured from the high intensidata.

We can also deduce from this simulation~from the maxi-mum value of the negative phase shift! the electronic densityachieved in the CB. ForI p51,5 T W/cm2 we obtainNCB'831018 cm23 and for I p50,5 T W/cm2,NCB'431017 cm23.

B. The SiO2 case

In Figs. 6~a! and 6~b! the phase shifts measured at 618 nat 300 K in a sample of Quartz (a-SiO2) for two pumpintensities, respectively 3 T W/cm2 and 4 T W/cm2 areshown. The order of the nonlinear process responsible forinjection of valence electrons in the CB has been measuwith the method described above. We found a four phoabsorption process in agreement with the fact that the bagap in Quartz is equal to 10 eV (433.14 eV'12.5 eV!. Asa consequence, the initial excitation density is five timlarger at 4 T W/cm2 than at 3 T W/cm2. As in the abovecase, we observe a positive phase shift due to the Kerr efimmediately followed by a negative phase shift, whichproportional to the density of photoexcited electrons. AsNaCl, the evolution of the phase shift toward a positive vaindicates a trapping of the electrons. However, contrarythe case of NaCl, the trapping kinetics is independent onpump laser intensity. Although the initial excitation densitidiffer by a factor of 5,the time necessary to get the positiphase shift (550 fs) is the same.This behavior is, as in NaClindependent of the sample temperature and of the pr

TABLE I. Set of parameters used in the simulation for NaCl

Nonlinear refractive index (cm2/W) n2 7310216

initial valence electron density (cm23) N0 2.2431022

order of the multiphoton process n 3Multiphoton cross section (cm6 s22) s (3) 2310282

Oscillator strength for the CB fCB 1Electron-phonon scattering rate (s21) 1/te2p 631014

Electron effective mass in theCB ~kg!

m* 0.539.1310231

Hole trapping time~ps! thole 0.5Electron-hole capturecross section (cm2)

scapture 8310214

mean electron velocity in theCB (cm s21)

n 1.453107

STE absorption band maximum~eV! v tr 2.1STE absorption band width~eV! 1/t tr 0.35Oscillator strength for the trap level f tr 0.25

e

atllseThamtueas-these.s

f aattioce

II.-

ec-

ho-d.hiftrrtherac-beeentver,fi-onsngusing,

toousing

he

chin-ongre-tion

not

t


wavelength. The absorption measured at 618 nm in theperimental conditions of Fig. 6~b! is shown in Fig. 7. Weobserve a decrease of the absorption while electronstrapped in the band gap. This initial absorption is dueelectron-phonon collisions. For longer time delays, it fadown to less than 10%, indicating that the probe photonergy is far from the resonances of the trapping center.behavior of the phase shift and of the absorption is the sat 790 nm, and does not depend on the sample tempera

It is now well known that ionizing radiations producSTE’s in SiO2.

25,26 The STE transient absorption band hmaxima at 5.2 and 4.2 eV,16,17 and it has recently been established from time-resolved absorption measurementsthe rise time of the absorption peak at 5.2 eV coincidwithin experimental errors, with the electron trapping tim6

There are therefore no doubt that the trapping procesSiO2 is due to the formation of STE’s.

An intensity independent kinetics is the signature ononsequential or exponential trapping. Accordingly, the requations governing the density of trapped and conducelectrons simply include a multiphoton excitation sourterm and an exponential decay,

~d/dt!NCB5N0s~4!Fp

42NCB/t,

~d/dt!Ntr5NCB/t. ~8!

FIG. 6. Phase shift as a function of time measured in SiO2 fortwo pump laser intensities. The probe wavelength is 618 nm andsample temperature is 300 K.

FIG. 7. Absorption as a function of time measured in SiO2. Thepump intensity is 1.5 T W/cm2. The probe wavelength is 618 nmand the sample temperature is 300 K.

x-

reo

n-eere.

at,

in

en

The parameters for which Eqs.~3!, ~4!, and~8! give the bestfit of the experimental results are summarized in TableThe parametersv tr andt tr are taken from the published transient absorption spectra.16

The electron-phonon coupling is responsible for the eltronic energy loss in the CB and is taken as 1.531015 s21.As already pointed out, this term is also responsible for pton absorption while electrons are in the conduction ban

For small positive delays, say 0 to 200 fs, the phase sdramatically varies from a positive value due to the Keeffect, to a negative value while electrons are excited inconduction band. Therefore the assumption that the reftive index does not change significantly during the propulse is not valid in this situation. This is the origin of thslight disagreement in this temporal region. The agreembetween experimental data and the curve fit is, howequite good for longer time delays, leading to a strong condence in exponential law assumed in the kinetic equatiand in the decay time of 150 fs obtained from the fittiprocedure. Finally the main conclusion is the simultaneotrapping of the hole and the electron: in quartz the trappreally concerns theexciton. Indeed, contrary to alkali halidesno hole trapping is observed ina-SiO2 ~Ref. 27! and accord-ing toab initioHartree-Fock calculation in SiO2 clusters, thestabilization of the STE is due to the electron rather thanthe hole.28 Hole trapping has been suggested in amorphsilica (a-SiO2),

29 but we have measured the same trappkinetics5 and the same absorption rise time4 in a-SiO2 thanin a-SiO2, indicating that the exciton trapping process is tsame in both cases.

C. The KBr case

We present in Figs. 8 the phase shifts~top curves! andabsorption~bottom curves! obtained in KBr with two probewavelengths: 618 nm@Figs. 8~a! and 8~b!# and 790 [email protected]~c! and 8~d!#, at two sample temperatures: 300 K@Figs. 8~a!and 8~c!# and 10 K@Figs. 8~b! and 8~d!#. The pump intensityis the same in all cases and is of the order of 1 T W/cm2. Therange of intensities that can be explored in KBr is mumore limited than in other materials because, for highertensities and especially at 618 nm, the absorption is so strthat it rapidly reaches 100%, making interference measuments impossible. In all cases, we observe a rapid evoluof the signal~phase shift and absorption! during the first 4ps. In the limited accessible intensity range, we could

he

TABLE II. Set of parameters used in the simulation for SiO2.

Nonlinear refractive index (cm2/W! n2 2310216

initial valence electron density N0 2.231022

Order of the multiphoton process n 4Multiphoton cross section~cm8s23) s (4) 2.33102114

Oscillator strength for the CB f cb 1Electron effective mass in the CB~kg! m* 0.539.1310231

Electron-phonon scattering rate~s21) 1/te2p 1.531015

Electron trapping time~fs! t 150Oscillator strength for the trap level f tr 0.4 and 0.15Trap level energy~eV! v tr 5.2and4.2Width of the trap level~eV! 1/t tr 1.5 and 1

m

f


FIG. 8. Phase shift~top curves! and absorp-tion ~bottom curves! measured in KBr at 300 K@left column: ~a!,~c!# and 10 K @right column:~b!,~d!# and at two probe wavelengths: 618 n@top: ~a!,~b!# and 790 nm@bottom: ~c!,~d!#. Thepump intensity is of the order o731011 W/cm2.

seone20ing

ioththebye

t,

ece3

ionde-iththent ofenl,thegTE-

anti-tionande

nd antingi-ient

observe any excitation density influence on the kinetics aNaCl. This last point is not consistent with previous freelectron lifetime measurements in KBr based on photocductivity experiments.22 It has been shown that when thexcitation density is low, the electron lifetime is about 10–ns and the recombination process is due to extrinsic trappFor increasing excitation density, intrinsic trapping followinbimolecular kinetics is observed, but above 1017/cm3, theexperimental resolution did not allow a precise determinatof the lifetime, which becomes less than 10 ps. Frompresent high resolution results we could conclude thattrapping is not sequential but as we noticed in NaCl, abovgiven excitation density the trapping kinetics is limitedthe hole trapping time, and we cannot exclude that the lowexcitation density (;1018/cm3) explored in KBr is alreadyabove this threshold.

The striking feature concerning the phase shift is thabecomes positive after 4;5 ps in all cases except at 300 K618 nm @Fig. 8~a!, top curve#. According to our refractiveindex model, the conclusion of this observation is that eltronic state associated with the trap in the band gap presabsorption bands peaked at energies lower than 2 eV at

in--

g.

neea

st

it

-nts00

K, and higher than 2 eV at 10 K. Of course the real situatcan be more complicated than what our simple modelscribes. In particular it is possible that different states wdifferent absorption bands are populated simultaneously:phase shift in such cases gives an averaged measuremethese different populations. Indeed Alkali halides have beclassified in two classes.30 In type-I materials such as NaCthe STE’s recombine radiatively at low temperature andformation ofF-H pairs occurs via thermal activation alonthe lowest state of the STE. In this case the yield of Sluminescence and stableF-H pairs formation are anticorrelated. In KBr, which belongs to type-II alkali halides,F-Hpairs are created even at low temperature and no suchcorrelation is observed. Recent time resolved absorpmeasurements have shown that the populations of STE’sF centers following electronic excitation evolve in two timscales:2,7 a rapid evolution (t'2 ps! which we readily ob-serve in our phase shift and absorption measurements, aslower (t'40 ps!, that we did not investigate in the preseset of experiments. The ultrafast dynamical process leadto the formation ofF centers is so efficient that the assocated absorption band around 2 eV dominates the trans

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n

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isthlo

shlsthe.

th

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intheo-pret

aanionry

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cele istheistiontheion,

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ob

618


absorption spectrum after only 3 ps.2,7 The behavior of thephase shift after the first 3 ps is therefore controlled bypresence ofF centers. This hypothesis is confirmed by oabsorption data. The absorption at 618 nm mainly due tcenters, increases while the electrons are trapped. At 790the initial absorption is due to the electrons in the conductband and decreases to a value lower than at 618 nm.remaining absorption is due to the STE’s. Moreover, it hbeen established a long time ago31 that the absorption spectrum of F centers in KBr is shifted towards low energy anbroadened when the temperature increases: it is peake600 nm~2.1 eV! at 73 K and at 630 nm~1.95 eV! at 300 K.In that case, it is not surprising that the phase shift is alwpositive at 790 nm and changes its sign at 618 nm whentemperature changes from 10 to 300 K.

D. The diamond case

In Fig. 9 the phase shift obtained for a pure diamosample at 300 K~the pump intensity is 331011 W/cm2) isshown for a probe wavelength of 618 nm. We observe tthe phase shift is negative and remains constant. This beior is identical at both probe wavelengths, whateverpump intensity and the sample temperature. Even for insities as high as 1.5 T W/cm2 we were not able to see anmodification in this behavior.

According to the experimental results, no trapping is vible in diamond. We conclude that, at such time scales, ifelectrons are trapped, they are trapped into levels very cto the bottom of the conduction band.

Because no clear evidence of change in the phaseappears during the time that separates the two probe puwe performed measurements in the relative mode: we lettwo probe pulses cross the sample after the pump pulswe suppose an exponential decay~lifetime t) of the electrondensity in the CB, the phase shift and absorption are incase given by:DF5DF0(e

[2t/t]2e@2(t2Dt)t#) and A5A0(e

(2t/t)1e@2(t2Dt)t#), whereDF0 andA0 are the phaseshift and the absorption just after the pump pulse. Tmethod is less sensitive than the absolute one but mlonger time delays can be investigated. The result of relameasurements performed at room temperature in diamare shown in Figs. 10~a! and 10~b!. The pump wavelengthand intensity are respectively 618 nm and 0.7 T W/cm2. Thefits give 140 and 190 ps as time decay for the phase shiftabsorption, respectively. The latter value is certainly mreliable, due to the larger scattering in the absorption da

FIG. 9. Phase shift measured in diamond at 300 K. The prwavelength is 618 nm and the pump intensity is 331011 W/cm2.

erFmnhes

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se

d

atv-en-

-ese

iftes,eIf

is

schend

nde.

V. ELECTRON TRANSPORT IN NaCl AND SiO 2: MONTECARLO SIMULATION

The experimental results have revealed an importantference between SiO2 and NaCl: the electron trapping timdoes not depend on the conduction electron densitySiO2, while it is strongly dependent on carrier densityNaCl. This observation led us to describe the evolution ofelectron density in the conduction band in NaCl with bimlecular kinetics. The same model has been used to intertime resolved absorption experiments in alkali halides.21 InNaCl, STE’s consist of a hole, relaxed to the form ofCl 2

2 molecular ion and an electron. After the creation ofelectron-hole pair by the incident radiation, the molecularformation results from the self-trapping of a hole in a veshort time, estimated to be less than 1 ps;7 then an electron iscaptured by a hole and a STE is formed. The underlyhypothesis in this two steps process is that electronsexplore a volume large enough to contain many STH’s.the case of SiO2, the STE’s creation rate is governed by aexponential law independent on the excitation density. Eelectron-hole pair remains in interaction until the STEformed. So the difference of behavior of the electron traping time versus the pump intensity can be explained ifelectrons can run away from their hole in NaCl and notSiO2.

Knowing exactly the minimum electron-hole distannecessary to prevent the trapping of an electron by a hoa difficult task, but as a first approximation, one can usesimple following arguments. When an electron-hole paircreated by the pump pulse, there is a coulomb attracbetween them until they get far apart. In order to estimatedistance necessary for the carriers to avoid recombinatone can take the distance (r c) at which the Coulomb energybecomes equal to the thermal energy~3/2 kT!. One findsr c(SiO2)5 97 Å andr c(NaCl)5 63 Å for T5300 K. Thus,if the electron reachesr c with an energy greater than 3/2 kTone can consider very roughly that electron’s capture by

e

FIG. 10. Relative phase shift@top: ~a!# and absorption@bottom:~b!# measured in diamond at 300 K. The probe wavelength isnm and the pump intensity is 831011 W/cm2.

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hole becomes impossible.32 To evaluate the time dependenof the electron energy and the evolution of the electron-hdistance, we have performed a simple Monte Carlo simution, based on the scheme developed by Fischettiet al.33 Theonly interactions considered here are the LO and acouselectron-phonon interactions, respectively treated withFrohlich Hamiltonian and a deformation potential approaThe hole is supposed not to move and there is no interacbetween the electron and the hole. Electron transporSiO2 has been the subject of numerous study, and mphysical data such as electron-phonon coupling constantavailable.33,23,34In NaCl, these quantities seem to be knowwith a smaller precision, and especially electron-acoustphonon coupling. According to several authors,35,36 this in-teraction is smaller than in SiO2. Electron-acoustical phonocollisions are isotropic, so they efficiently randomize telectron direction. Note that in order to take into account tlack of knowledge, we have tried different deformation ptential values for NaCl, ranging from the value usedSiO2 to this value divided by ten without noting significachanges in the results of the simulation. Electrons arejected in the CB with a kinetic energy (nhy2Eg) of 2 eV inSiO2 and 1 eV in NaCl. The result of the Monte Carlo simlation is drawn in Fig. 11. We observe that electrons lotheir energy much more quicker in SiO2 than in NaCl,mainly due to the LO-phonon energy, larger in SiO2 ~150meV! than in NaCl ~30 meV!. When the electron energbecomes of the order of thermal energy, thee-h distance isabout 60 Å @,r c ~SiO2)] in SiO2, whereas, this distance iabout 150 Å @.r c~NaCl!# in the case of NaCl. Consequently, we understand on the basis of this simple modelin NaCl an electron has a much greater chance to esfrom its parent hole and meet another hole, whereasSiO2 an electron always remains in the vicinity of its hoand thus the STE formation involves the electron andhole of the same initial pair. Let us underline that the intaction between electrons and holes is not taken into accin this model. However, the screening of the Coulomb ptential is less efficient in quartz than in NaCl becausestatic dielectric constant is higher in the last case. Therefthe inclusion of such an interaction should still enhancerelative pathways discussed above.

FIG. 11. Monte Carlo simulation of electron trajectoriesSiO2 ~full lines! and NaCl~dashed lines!. The light lines show thekinetic energy~left scale! and the heavy lines the electron-hodistance~right scale!.

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al

s-r

-

e

atpein

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VI. CONCLUSIONS

The technique of interferometry in the frequency domaproved to be a very efficient and sensitive tool for the timresolved study of carriers excitation and relaxation dynamin wide band gap insulators. Compared to for instance tiresolved absorption spectroscopy, that brings fruitful infmation about the absorption bands associated with pernent (F andH centers! and transient states~STE, STH!, theexperiments reported in this paper provide complemeninformation. In particular, they offer the possibility to observe electrons in the conduction band and to measure wgood accuracy the excitation density. From this pointview, the absorption process due to electron-photon-phocollision which occurs while electrons are in the conductiband is noteworthy. Its importance is evident before trappin SiO2 ~Fig. 7! and increases for longer probe wavelengtfor example in NaCl at 790 nm@Fig. 5~b!#, and also in amaterial where no trapping is observed@diamond, Fig.10~b!#.

Our experimental results demonstrate that the ultratrapping of carriers deep in the band gap observed in NaKBr, and SiO2, is in all cases associated with the formatioof STE’s. Furthermore, a detailed study of intensity depdence kinetics reveals two different pictures of STE’s formtion: hole trapping followed by electron capture in NaCl aexciton trapping in SiO2. With the help of a simple MonteCarlo simulation, these different behaviors can be interprein terms of electrons trajectories: the electrons lose theirergy quickly in SiO2, while they have the possibility tomove away from their own holes before being trappedNaCl. The fitting procedure of the trapping kinetics allowsto extract for the first time the intrinsic hole trapping timeNaCl. Bimolecular kinetics are supposed to be a generalin alkali halides, but are not observed in KBr, for excitatiodensities ranging over one decade from to 231017 to231018 cm23. Electron capture is rather as fast as hole traping or occurs simultaneously. Indeed it has been pointedthat different relaxation channels exist for electron hole pain KBr, depending on the time when the electron interawith the hole under relaxation.2

The case of diamond is apparently very similar to the cof Al 2O3 and MgO.6 In these three materials, we do nobserve evidence for STE’s formation, but a slow decreof the electron density in the conduction band with lifetimlying in the range of 100 ps. It has been recently predicthat in diamond, it should be possible to observe setrapping in the case of valence biexcitons.37 The maximumdensity reached in the above experiment is of the orde1019 cm23 ~it is difficult to increase this excitation densitwithout making serious damages to the sample! and we didnot observe this related effect. This can be due to thethat the excitation density in still not high enough.Al 2O3, the transient volume change induced by high-eneelectrons has been ascribed to the formation of STE’38

However, intrinsic recombination luminescence has beenserved in Al2O3 at 7.3 eV,39 with an excitation spectrumsharply peaked at the band edge~exciton resonance!.40 Thismeans that the stoke shift associated with eventual Swould be of the order of 1 eV or less. As already pointed oour method could possibly not distinguish electron in t

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areheent

-


conduction band or trapped close to the bottom of the cduction band. The same type of electron-hole recombinaluminescence has been observed in MgO, indicating esmaller stoke shifts.41

Among the properties that allow to predict the trendexciton trapping in a given material, the ionicity certainplays a role but is not a sufficient criteria to explain oresults, since for instance STE’s would be observed in Mmuch more ionic than SiO2. As already mentioned, a STEassociated with a local deformation of the lattice arounbound electron-hole pair. This can be understood on thesis of simple and well-known scaling arguments. It is knothat the consequence of the addition of an extra chargeperfect lattice is to induce a relaxation of the lattice. Tcorresponding relaxation energy of the system is inversproportional to the number of atomic bounds which suppthis extra charge (Nb) and then is minimum ifNb51. Inother words, the tendency of the system is to form acalled ‘‘small polaron.’’42 This energy which stabilizes thsystem is the sum of two terms coming from the electrphonon interaction~acoustical and optical!. On the otherhand, this localization process has a cost. The latter isportional to 1/Nb

2/3 and the localization results from the competition between these two contributions. Toyozawa aco-workers43 have located electrons, holes, and excitonsvarious solids into a phase diagram. One of the pertin

-

nd

ier

eite

y

tti,

-nn

r

,

aa-

a

lyrt

-

-

o-

dnnt

parameters to decide whether or not an exciton is strapped or free is the ratioC/j, whereC is the deformationpotential andj the elastic constant: the largest this ratio, thighest the self-trapping probability. So, self-trapping isvored in materials with high deformation potential and smelastic constant. In Table III the elastic constants relativethe materials which have been studied in this workshown. It is clear that the elasticity plays a crucial role in tself-trapping process and that our data are in full agreemwith the simple arguments discussed above.

ACKNOWLEDGMENTS

Financial support from CEA-DAM is gratefully acknowledged.

TABLE III. Elastic constants in the~11! direction.

Material C11 ~GPa! STE

C 1076 NoAl 2O3 497 No?MgO 294 NoSiO2 87 YesNaCl 49 YesKBr 34 Yes

os

lstion

v. B

ev.

.P.

ens.

R.

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v.

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Documents

Subpicosecond study of carrier trapping dynamics in wide-band-gap crystals