Thirdorder nonlinearities and coherent transient grating effects of narrowgapsemiconductors in the midinfraredS. Hughes, C. M. Ciesla, B. N. Murdin, C. R. Pidgeon, D. A. Jaroszynski, and R. Prazeres Citation: Journal of Applied Physics 78, 3371 (1995); doi: 10.1063/1.359964 View online: http://dx.doi.org/10.1063/1.359964 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/78/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Wavelength-dependent resonant homodyne and heterodyne transient grating spectroscopy with adiffractive optics method: Solvent effect on the third-order signal J. Chem. Phys. 116, 9333 (2002); 10.1063/1.1473653 Thirdorder nonlinearity of semiconductor doped glasses at frequencies below band gap Appl. Phys. Lett. 67, 13 (1995); 10.1063/1.115473 Auger generation suppression in narrowgap semiconductors using the magnetoconcentration effect J. Appl. Phys. 71, 5706 (1992); 10.1063/1.350505 Nonlinear optical coefficients of narrowgap semiconductors J. Vac. Sci. Technol. A 8, 1215 (1990); 10.1116/1.576948 Tunneling through narrowgap semiconductor barriers Appl. Phys. Lett. 48, 644 (1986); 10.1063/1.96731

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Third-order nonlinearities and coherent transient grating effects of narrow-gap semiconductors in the midinfrared

S. Hughes and C. M. Ciesla Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

B. N. Murdin FOM Institute for Plasma Physics, Rijnhuizen P.0. Box 1207, 3430 BE Nieuwegein, The Netherlands

C. R. Pidgeona) Department of Physics, Heriot- Watt University, Edinburgh EH14 4AS, United Kingdom

D. A. Jaroszynski and R. Prazeres LURE, Ba”timent 209d. Universite’ de Paris-Sad, 91405 Orsay Cedex, Paris, France

(Received 29 March 1995; accepted for publication 11 May 1995)

Picosecond excitation-probe measurements using a far-infrared free-electron laser (CLIO) have revealed large nonlinearities for indium antimonide at 4.7 ,um. A theoretical model is described to determine the cw third-order nonlinear susceptibility and the interband relaxation time of the semiconductor which were found to be -8.6X lo-” m2 VP2 and 0.3 ns, respectively. Furthermore, the observation of the associated coherent transient grating effects allows us to obtain a coherence time of the laser system (2.5 ps) and the x (3) of the transient grating which was found to be -7 3X lo-l3 m2 Ve2. The measurements were performed at room temperature on undoped bulk Ins; Q 1995 American Institute of Physics.

I. INTRODUCTION

The recent evolution of high-power, “tunable” free- electron lasers [FELs) is currently enabling scientists to study directly the dynamic properties of a range of nonlinear materials, at longer wavelengths than were hitherto acces- sible with conventional picosecond laser sources. For ex- ample, the two-photon absorption coefficients’ and recombi- nation lifetimes” have now been determined for narrow-gap semiconductors such as InSb over a wide range of wave- lengths and excitation conditions, and the far-IR (FIR) inter- subband lifetimes have been recorded in GaAs quantum wells3 at 70 ,um. Such parameters are important for the de- velopment, understanding, and indeed the exploitation of practical semiconductor devices. Third-harmonic generation experiments with the cw Santa Barbara FEL have demon- strated enhanced nonresonant third-order nonlinearities.” Here we utilize the time resolution (ps) of CL10 (Collabora- tion pour Laser Infrarouge i Orsay) to determine the satura- tion and transient grating values of xi3’ and the associated relaxation times in narrow-gap semiconductors. Previous measurements were confined to cw or longer-pulse fixed- frequency laser determinations.s-7 These materials have a host of applications, including ultrafast switching, high- fluence power limiting (induced absorption), and infrared detection/emission.

Experimentally, carrier dynamics such as Auger recom- bination can be accurately studied by employing dual-beam, time-resolved excitation-probe measurements, where the probe energy transmittance can be measured as a function of delay between the two pulses. The signal shows a rise on a scale of the pulse duration, followed by an exponential decay with a time constant equal to the inverse relaxation rate of

SElectronic mail: phycrpl@clust.hw.ac.uk

the excited carriers. In addition, when the two beams overlap at delay times less than the coherence time of the laser exci- tation, coherent transient grating effects may be observed. 1~ this situation the standing waves created by interference of the excite and the probe beams gives rise to a carrier con- centration modulation throughout the sample, which results in a phase grating. This manifests in, for example, the so- called “coherent artifact,“8’9 where a portion of the pump light is diffracted into the probe direction giving an enhance- ment of the sample transmittance around zero delay. Broadly speaking the height of this feature is determined bq’ the third- order nonlinear susceptibility tensor element J$&,, and the width is determined by the coherence length of the laser; the e and p subscripts respectively refer to the excitation and probe beam polarizations. Only if the laser is ‘not Fourier transform limited, i.e., with some phase modulation, will the additional coherence “spike” be resolvable on top of the consequently slower rise of the “background” excitation- probe signal.

We present here a macroscopic theory of the dynamics which includes such effects as laser frequency chirp and pulse shape. This is essential for the correct interpretation of excite-probe experiments around zero delay, in particular in- terband recombination studies in semiconductors and future studies of coherent transient material effects. The model is utilized to extract values of xc3’ and the relaxation times in narrow-gap semiconductors whose band gaps lie in the mid-IR (MIR), from such measurements using CLIO.

II. THEORY

For our theoretical approach, we seek to obtain the third- order nonlinear polarization contributions, which are intro- duced here from a macroscopic point of view. Microscopic expressions may also be derived, for example, from pertur-

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bative analysis of the allowed electronic transitions in the medium in the electric-dipole approximation.‘.!‘- In the pres- ence of one-photon resonance, the induced third-order non- linear polarization can be written

P(3)(mu9 = ~03~ Kfjkfim - 0,~)

tribution to describe diffraction of the excite beam into the probe direction due to a coherent interaction between the two pulses may be written similarly,

x;&b, - w,o) P(3)(~,f,r)spike= fo

7c

Xexp[i(kj-kk+kJ.r]&iEj(w,t) Xexp[i(k,-k,+k,).r]i,E,(w,t)

l-t

I I

X E;(o,t’)El(o,t’)A(t-t’)dt’+c.c., --ca

(1) where Ej,k,l(~,t) represent the time profiles of the input fields oscillating at frequency w, and Gi are the different polarization contributions. The dephasing rate of the optical transition is typically much larger than any optical pumping terms and, hence, appears outside the time integration in the Kijkl factor, which is related to the third-order susceptibility elements and nonlinear relaxations as discussed below. This would not be the case, for example, when dealing with co- herent transient oscillations of the material and formally one would use a polarization of the form

@3)(f) = J-f-f

qS(t’,t”,t”‘)E(t’)E(t”)E(t”‘).

The response function inside the above integration is taken to be

A(t-t’)=exp - -

X J

E:(w’)E,(d) --m

Xexp[i$(t’)]+c.c., (4)

if a decay lifetime rC is assumed; physically this lifetime characterizes the decay of the coherent transient grating via

I t E~(w,t’)EJw,t’)exp[i~(t’)]dt’. --m (5)

This contribution only becomes finite during a period when the two input fields are “coherent” with one another, and differs very much from the “incoherent” background term, which is simply the impulse response convolved with the intensity correlation function. For the situation where the co- herence time might be less than the pulse duration (which is the case for the experimental observations presented later), one has to establish a mechanism for dephasing. It is known” that there can be some frequency chirp in the FEL and we therefore introduce a linear frequency chirp of the form o--mfa,t. The task is now to evaluate integraIs of the form

where several important assumptions have been made. First, we have ignored reorientational effects, such that anisotropic

/ID exp( -$)exp( -w) +

state filling (induced dicroism) is sufficiently fast to wipe out any orientational grating effects. In this sense, our analysis applies to parallel polarized excite and probe beams, where a purely excitation-concentration grating will be formed inside the material, and the symmetry of the nonlinearity will sim- ply be the product of two linear susceptibilities.8 Second, the generated carriers, created by the intense excite pulse, will relax exponentially with the electron-hole recombination tune rot, which has been introduced phenomenologically.

-. ” ,

xexp[iia(2t’rd-d)]dt’,

s g 1.0

‘S= c6 6i 0.8

25

.s 0.6 3 b J? 0.4 c 0 $.j 0.2 z

iz 0.0 22

With a mind to excitation probe measurements, first we introduce the background polarization, which describes satu- ration and changes in the effective optical constant of the sample due to the excitation beam alone,

P(3)(w,rhack= ~0 x$?‘,,,,(‘0~ - w-w)

701

Xexp[i(k,-k,+k,).r]SiE,(w,t)

J I

X E~iw’&?iw’) 0 5 10 15 20 -cv Time delay (ps)

(3) FIG. 1. Complex integration [Fq. (6)] t o investigate the role of frequency chirp constant a, in determining the coherence time. The determined T= are

where xf2eecw is the long-lived third-order nonlinear suscep- the HW e-* maximum values of the peak signal, from which an empirical

tibility tensor element. The appropriate transient grating con- relationship is determined. As a, was varied from lO23 to 10” s-*, the coherence time increased from 1 to 13.5 ps.

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where 7. corresponds to the pulse duration [half-width (IIW) e-l maximum], and Td is the time delay between the two pulses, excite and probe.

Examples of this contribution (which can be rewritten as a complex error function) are depicted in Fig. 1. A range of simulations is shown for different a, chirp parameters and a pulse duration of 15 ps. Defining a coherence time T,, as the HW at e-t maximum for these curves we find that rC = (T&)-‘.F or on 1 g er coherence times, the integral is sim- ply the convolution of the pulses; as one expects, i.e., for coherence times longer than the pulse duration, it is impos- sible to determine the coherence time from excite-probe measurements.

Assuming a probe beam with field EP traveling in the y

I .

direction, the “slowly varying” wave equation may be writ- ten

(7)

where cue is the linear absorption coefficient. The differential transmittance change can then be expressed as

&J7= T = Vnl- Tiin) T v , I L lin

where T,t and Z’rin are the nonlinear and linear transmittance of the probe beam, respectively. In the small-signal limit, the solution subsequently becomes

AY= c E~(w,t’,O)E,(o,t’,O)A(t-t’)dt’ dt+ J --m

E;(u,t,O)E&,t,O)

f t

x3-$-Wt)l E,*(o,t’,O)E,(o,t’,O)exp[i+(t’)]dt’ dt J--m

where

(b(t) = ; @h-d- 4). c

c= Wx~Lw~ k 4; 701 G (w;+w,2)v2roJ;;’ (11)

‘m(xgL,) k w: = SC 2ao (w;+w~)v2roJ;;’ (12)

and wP and w,, respectively, refer to the probe and excite beam spot sizes (HW e-s maximum irradiance). Note Eqs. (11) and (12) are equal because we assume parallel polarized excite and probe beams and consequently the two tensor el- ements have the same symmetry properties (see, for ex- ample, Ref. 8).

Since the probe pulse is merely a delayed version of the excite pulse, the contributions to differential transmittance can be decomposed into the following two expressions:

Ad%? Im (E,(t-7,)(‘1f~lE,(t’)12A(t-t~)dtf dt -93

-I- C.C., (13)

A.$&o: Co I E:tt- ~d)Eett) -m

XeXp[ -id(t)]f~~ET(t')E,(t'- rd)

Xexp[iqS(t’)]dt’ dt+c.c.,

J. Appl. Phys., Vol. 78, No. 5, 1 September 1995

(14)

I

where the first integral represents the third-order background contribution and the second corresponds to the coherent tran- sient grating-spike response. i’

The slowly varying background response (13) is merely the convolution of the impulse response of the material and the irradiance-time correlation of the pulses and the induced nonlinearity relaxes with the recombination time rot. This positive change in transmittance is associated with band fill- ing which causes a strong dynamic blue shift of the IR- absorption edge, i.e., the dynamic Moss-Burstein shift (DMS), and, hence, saturation occurs. In addition excite- probe transmittance measurements may produce a spike at around zero delay, described by Eq. (14); this contribution is also positive since it is caused by constructive interference between the original probing wave and the coherently dif- fracted portion of the excite wave. Note that this signal may be negative, for example, when induced absorption over- comes the effect of saturation and destructive interference is~ exhibited.”

The effect on the differential transmitted probe energy can be obtained from Eq. (9), an example of which is shown in Fig. 2 for a coherence time, ~,=8 ps and a 15 ps pulse duration; also depicted are the two separate contributions implied by Eqs. (13) and (14) above. For the calculations the recombination time was taken to be much longer than the pulse duration.

III. EXPERIMENT The experimental setup used has been reported

previously” for MIR, picosecond, pump-probe studies of Au- ger recombination in samples of undoped InSb. In the ex- periments described here the F+EL operated with pulses sepa- rated by 32 ns at 4.7 q. The sample was an undoted bulk film of I&b, intrinsic (n-lot6 cmm3) at room temperature.

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Bo cl 5; Time delay (ps)

FIG. 2. Theoretical simulation of the slowly varying third-order background response (chain line) and a third-order coherence spike contribution (dashed line). Also shown is the sum of the nonlinear processes (solid line). The coherence time and pulse duration were assumed to be 8 and 15 ps, respectively.

The sample was mounted on a 2 mm pinhole at 300 K. The excite beam was normally incident, focused, and had a HW e -* maximum spot size o, of 0.25 mm. A small part of the beam was diverted with a ZnSe beam splitter into a mov- able retroreilector which acted as a delay line. The probe was then directed (unfocused, w,=4.2 mrn) onto the sample at * 10” to the excite beam. The effective excite and probe energy fluences per micropulse were estimated to be 50 and 5 $cmp2, respectively, inchrding losses to beam splitters and windows; the micropulses have 8 ps pulse duration. The transmittance of the probe was measured as a function of optical delay rd between excite and probe.

The excite-probe data-i.e., probe transmittance versus probe delay after the excite pulse-are shown in Fig. 3. Band

0.8

0 50 100

Time delay (ps)

FIG. 3. Time-resolved excitation-probe measurements for pulses polarized parallel to each other, for au excite input fluence of 50 CCJ cm-*. Also shown is the theoretical simulation which allows the determination of the recom- bination lifetime from the first excited state, the coherence time of the laser, and the third-order nonlinear susceptibility elements, Im(cw) and w%$3) ) **x .

3374 J. Appl. Phys., Vol. 78, No. 5, 1 September 1995 Hughes et al.

filling as the excite radiation is absorbed causes a dynamic blue shift in the lR-absorption edge (DMS) and leads to pro- nounced absorption bleaching. By fitting the third-order non- linear background response, we obtain first the recombina- tion time of 0.3 (20.1) ns (assumed constant) and a steady- state Im(&,) of -8.6 (*0.5)X10-” m2V-*. This is equivalent to w - 6 X low3 esu, which is approximately the same cw magnitude estimated by ps four-wave-mixing experiments13 in the near infrared (-1.5 ,um) for GaInAsP, and agrees well with the cw determinations for InAs.”

Also seen in the data is the coherent artifact at zero delay, from which we determine a coherence time of 2.5 (2 1) ps and an ImGs&) of -7.2 (+0.5)X lo-l3 m* V-s. Future experiments have been planned which will employ the four-wave-mixing geometry to determine the real contri- bution of this susceptibility, in connection with other tech- niques for determining the pulse characteristics of length and coherence time such as second-harmonic-generated autocorrelation.”

The theory applied neglects many-body effects, resulting from fermionic exchange and Coulomb repulsion. However, for semiconductors with narrow gaps, the Coulomb forces are strongly screened, and consequently many-body effects are not so pronounced.

IV. CONCLUSIONS

In conclusion a theoretical model was applied to analyze the response of InSb using time-resolved excite-probe mea- surements, obtained for the free-electron laser CLIO. The phenomena of saturation of the excited-state and coherent transient grating effects were taken into account. By fitting to experiment, an excited-state recombination time of 0.3 ns was extracted, and values for the imaginary contributions to the third-order nonlinear susceptibilities, ,Y&.,~ and ,&ix, were obtained. Further, by modeling the resulting coherence spike, the characteristic coherence time of CL10 was deter- mined to be 2.5 ps. This lifetime becomes important when one is dealing with polarizations in the probe direction of the form EJE,*E, , for example, where the e and p subscripts, respectively, refer to the excitation and probe beams.

ACKNOWLEDGMENTS

The authors would like to acknowledge Professor Brian Wherrett for useful discussions. This work was performed as part of the research program of the “Stichting ‘voor Funda- mental der Materie” (FOM) with financial support from the “Nederlandse Organisatie voor Wetenschappelijk Onder- zoek” (NWO). One of the authors (C.M.C.) is grateful to the DRA, Malvern, for a CASE studentship.

‘ B. N. Murdin, A. K. Kar, C. R. Pidgeon, D. A. Jaroszynski, J. M. Ortega, R. Prazeres, F. Glotin, and D. C. Hutchings, Opt. Mater. 2, 89 (1993).

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