Banyai, Ladislaus

Haug, Hartmut

Heinrich, M.F.

Huegel, Werner

Vu, Q.T.

Wegener, Martin

Heinrich, M. F.

Vu, Q. T.

KITopen

VOLUME 83, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1999First published in:      Photon Echoes from Semiconductor Band-to-Band Continuum Transitionsin the Regime of Coulomb Quantum KineticsW. A. Hügel, M. F. Heinrich, and M. WegenerInstitut für Angewandte Physik, Universität Karlsruhe (TH), Kaiserstraße 12, D-76128 Karlsruhe, GermanyQ. T. Vu, L. Bányai, and H. HaugInstitut für Theoretische Physik, J. W. Goethe-Universität Frankfurt, Robert-Mayer-Strasse 8,D-60054 Frankfurt am Main, Germany(Received 29 January 1999)We show that the coherent response of semiconductor band-to-band continuum transitions at roomtemperature, excited by 11 fs pulses at elevated carrier densities, deviates substantially from thatexpected for an ideal photon echo. A microscopic non-Markovian quantum kinetic theory of bothelectron-electron and electron-LO-phonon scattering reproduces these unusual experimental observationsqualitatively and, furthermore, the famous scaling of the decay time t with carrier density neh accordingto t21  t210 1 cn13eh quantitatively. t, however, cannot be associated with a single unique phaserelaxation time T2.PACS numbers: 78.47.+p, 42.50.Md, 42.65.Re, 63.20.KrConsider a dense gas of electrons and holes generatedat random places in pairs with opposite momentum incoherent quantum states in a semiconductor. It is clearthat the particles will move in such a way that theyscreen their mutual Coulomb interaction. How long doesit take that screening is built up and that scattering hasdestroyed the initially quantum mechanically coherentstate? About one decade ago, Shank et al. addressedthis fundamental question when they studied the variationof the Coulomb scattering in a dense gas of opticallyexcited crystal electrons and holes [1,2] with density nehusing 10 fs pulses and a photon echo technique. Theyreported that the decay time t of the photon echo signalat room temperature scales approximately according tot ~ n21Deh , where D is the dimension of either bulk GaAsD  3 or GaAs quantum wells D  2. At that time,it was tempting and appealing to interpret this seeminglysimple result in terms of the nearest-neighbor scattering ofD-dimensional hard spheres in which case it is simple toshow that the average time between different collisionsscales ~ n21Deh indeed.Today, a number of important questions are still unan-swered. (i) Is it a photon echo? In analogy with a spinecho this would mean that a first short pulse excites thesystem, a second pulse arrives after time delay t21, andthe coherent collective response of the system, the photonecho, peaks at yet another time delay t21 later. Withoutreal-time resolution [1,2], however, one only measures theenergy of the echo signal versus t21 and its decay time t.For a phenomenological dephasing time T2 it is known thatT2  4t holds for a four-wave mixing (FWM) geometryunder these conditions. Not a single experiment, however,has shown directly that the coherent real-time response ofsemiconductor band-to-band continuum transitions underthese conditions is a photon echo indeed. Under quite dif-0031-90079983(16)3313(4)$15.00EVA-STAR (Elektronisches Volltextahttp://digbib.ubka.uni-karlsruhe.de/voferent conditions (T  4 K, 100 fs pulses, and much lowercarrier densities), Ref. [3] showed that the total response isnot a photon echo. Only after spectral filtering (behind thesample) of those components of the FWM-signal resonantwith the band-to-band continuum, were they able to isolatea contribution which corresponded to an echo. The posi-tion of this signal shifted with time delay as expected [4],its width was constant (150 fs) and given by the time reso-lution of their setup for all conditions. The picosecondresponse of inhomogeneously broadened excitonic tran-sitions [5] as well as the response of modulation-dopedquantum wells [6] followed the expectation for an idealphoton echo. Other real-time resolved work has addressedthe coherent response of the homogeneously broadened 1sexciton [7,8]. (ii) Can the absolute values of the measureddecay time t and their scaling with carrier density neh beexplained quantitatively by theory? So far the answer hasbeen no. Such a theory of electron-electron scattering hasto account for the fact that the excited carriers initiallybehave as coherent quantum mechanical waves. In otherwords: This theory has to account for the finite durationof scattering processes. Also, such a theory has to includethe kinetics of the buildup of screening. This means thatthe range of the screened Coulomb potential at time t doesnot only depend on the carrier distribution functions at thatsame time t but also on their history. The recently devel-oped quantum kinetic theory of electron-electron scattering[9] accounts for these aspects in analogy to our previouswork on electron-phonon scattering [10,11]. (iii) Is thefrequently used concept of a dephasing time T2 meaning-ful under these conditions at all? If the memory effects ofscreening and scattering discussed in (ii) are really impor-tant, the T2 concept breaks down.In this Letter, we first reproduce the basic observationsof Shank et al. concerning the time-integrated FWM on© 1999 The American Physical Society 3313rchiv – Scientific Articles Repository) lltexte/65499 VOLUME 83, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1999bulk GaAs. By additional real-time resolved experimentswe address the above questions (i)–(iii). This shedsnew light upon the interpretation of the famoust21  t210 1 cn13eh law.Description of experiment.—Two colinearly polarizedpulses, 1 and 2, arriving at times t  t1 and t  t2 andpropagating with wave vectors q1 and q2, respectively,interact in the sample and lead to a diffracted FWMsignal with wave vector 2 q2 2 q1 which is monitoredas a function of time delay t21  t2 2 t1 and the real-time t. For clarity, we use the convention t2  0. Thus,for a photon echo situation, we expect a signal maximumat time t  t21. On the other hand, a free polarizationdecay would not shift with time delay t21 but rather peakat a fixed point in time t. Using an up-conversion [6]technique with nearly transform-limited 11 fs, sech2t-shaped optical pulses, we measure a time resolution of5 fs (both, for the t21 and for the t domain). The errorin the determination of the absolute positions t  0 andt21  0 via scattered laser light is 610 fs. The electron-hole pair densities neh which are quoted below arealways defined as the sum of the two individual one-pulseFIG. 1. Time-integrated FWM signal versus time delay forthree different optically excited carrier densities neh  0.5, 1,and 2 3 1017 cm23 (from bottom to top). The full trianglesare the result of the quantum kinetic theory for neh  0.57 31017 cm23. The open circles are the time-integrated data ofFig. 2, shifted by 8 fs. The decay time constants t correspondto the dashed straight lines. The left (right) inset shows thevertically displaced and normalized FWM spectra (linear scale)for t21  0 t21  100 fs and the laser spectrum (dashed).Bulk GaAs, T  300 K, excitation with 11 fs optical pulsescentered 50 meV above the band edge.3314carrier densities, i.e., as neh  n1eh 1 n2eh with n2ehn1eh 5. The sample is a high quality GaAsAl0.3Ga0.7Asdouble heterostructure. It contains a 0.6 mm thin layerof GaAs and is glued to a sapphire disk on one sideand is antireflection coated on the other side. From themeasured exciton linewidth of this GaAs sample [10,11],we can safely rule out that any undesired inhomogeneousbroadening is relevant for the present experiments underroom temperature conditions.Experimental results.—First, we relate our experimentto that of Ref. [1]. Figure 1 shows the time-integratedFWM signal versus time delay for three different carrierdensities on the same logarithmic scale. Beside minorstructures, the FWM traces decay with exponential timeconstants t as indicated by the dashed straight lines inFig. 1. (t vs neh is shown in Fig. 5 and will be discussedlater.) Selected spectra of the FWM signal at t21  0 andt21  100 fs, respectively, are shown as insets in Fig. 1.The dashed curve is the laser spectrum. Note that theFWM spectrum at t21  0 becomes broader with increas-ing neh, which indicates that different distributions of op-tical continuum transitions contribute to the FWM signal.Hence, for different neh, t stems from different averages.As an example of the measured real-time responses,Fig. 2 shows the data corresponding to the conditions inFIG. 2. Real-time t resolved FWM signal (linear scale) forthe same conditions as in Fig. 1 neh  2 3 1017 cm23. Thecurves are vertically displaced, the corresponding time delayt21 can be seen on the right-hand side. The scaling factorsof the various curves are different as indicated on the left-hand side for each curve. The full straight lines correspondto the temporal position of the two incident pulses, arriving attimes t  t1 and t  t2  0, respectively; the dashed straightline is the expected position for an ideal photon echo, i.e.,t  t21  t2 2 t1  2t1.VOLUME 83, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1999Fig. 1 neh  2 3 1017 cm23. The real-time responseis a single-side exponential with rise-time constant ,5 fsfor t21 , 0. For t21 . 0, the signal acquires a moresymmetric shape. The position versus time delay roughlyfollows the ideal photon echo up to t21  50 fs. Forlarger time delays, the position of the echo is roughlyconstant again, hence, the actual echo comes earlier thanexpected for an ideal echo. The position and the width(full width at half maximum, FWHM) of the echo can beseen more clearly in Fig. 3, where the data of Fig. 2 andthat of two other carrier densities are analyzed. Note themonotonous rise of the echo width which amounts to morethan a factor of 2. This rise is consistent with the fact thatthe FWM spectra in Fig. 1 become narrower for largertime delays t21 as compared to t21  0. A single uniqueT2 cannot explain these observations [4], neither can awave vector-dependent T2 [6]. This already suggeststhat the deviations from an ideal photon echo responseobserved here are due to many-body correlations, and, inparticular, that they are specific for the regime of Coulombquantum kinetics. We now show that the experimentalevidence can be reproduced by a quantum kinetic theoryof Coulombic electron-electron scattering indeed.Description of theory.—In the framework of Keldyshnonequilibrium Green functions (GF) [12], the scatteringself-energies are taken in the so-called GW approximation,i.e., Skt1, t2  iPq G k2 qt1, t2W qt1, t2. The time-FIG. 3. The position and the width (full width at halfmaximum) of the photon echo signal versus time delay t21 asextracted from the data shown in Fig. 2 and others. The dashedstraight lines correspond to the expected temporal position t foran ideal photon echo, i.e., to t  t21. The width of the echowould be constant for an ideal echo.dependent screened Coulomb potential W q is determinedself-consistently as the solution of the integral equa-tion W qt1, t2  V qdt1 2 t2 1 V qL qt1, t3W qt3, t2.Here a matrix notation is used for the contour timearguments. L qt1, t2 is the generalized random-phase-approximation (RPA) polarization function, whichincludes also off-diagonal elements of the GF. These de-scribe band mixing and excitonic effects in the coherentlyexcited crystal. In this approach the potential W q evolvesfrom the originally unscreened potential V q to a dynami-cally screened potential after times longer than the inverseplasma frequency of the excited carriers. Recently, wesucceeded in solving the semiconductor Bloch equationstogether with the delayed quantum kinetics scatteringintegrals on conventional workstations [9]. Under thepresent experimental conditions, electron-electron scat-tering dominates over electron-phonon scattering. Forcompleteness, however, the quantum kinetics of electron-LO-phonon scattering at T  300 K is also accountedfor [10,13]. In order not to overburden the numeri-cal calculation, we describe the corresponding spectralfunctions, which determine the memory kernels of thescattering integrals, by a free-particle Wigner-Weisskopfapproximation [14]. We use the two-band effective massapproximation with GaAs parameters, i.e., an electron/hole mass ratio of memh  16.7. 11 fs sech2t pulseswith 50 meV excess energy above the band gap excite aFIG. 4. Coulomb quantum kinetic theory: Real-time tresolved FWM signal (linear scale), convoluted with an 11 fssech2t pulse to model the up-conversion process. Thesecurves can be compared with the experimental data shown inFig. 2. neh  2.2 3 1017 cm23, bulk GaAs parameters, T 300 K, excitation with 11 fs optical pulses centered 50 meVabove the band edge.3315VOLUME 83, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1999FIG. 5. Decay time t of the time-integrated FWM signalversus carrier density neh. The experimental data (open circles)and the result of the quantum kinetic theory (full triangles) arecompared with a simple fit according to t21  t210 1 cn13ehwith t0  100 fs. The inset shows the computed FWM spectra(linear scale, normalized and vertically displaced) at t21  0for carrier densities neh: 0.86, 1.34, 2.2, 4.2, 8.66, 13.9, and18.4 3 1017 cm23 (from bottom to top). The dashed curve isthe laser spectrum.total carrier density which is defined as in the experiment,i.e., according to neh  n1eh 1 n2eh with n2ehn1eh  5.The FWM signal propagating with wave vector 2 q2 2 q1is extracted by a projection procedure described inRef. [10].Theoretical results.—Figure 4 shows the FWM signalcomputed this way for a total carrier density of neh 2.2 3 1017 cm23. Figure 4 exhibits the same qualitativedeviations from an ideal photon echo as the experiment(Fig. 2). The echo width increases from 26 fs t21 12.5 fs to 50 fs t21  87.5 fs, which agrees qualitativelywell with the experiment (Fig. 3). The echo maximumoccurs earlier than the ideal echo for large t21. This effect,however, is less pronounced than in the experiment. Atyet larger neh, which are not accessible experimentallyhere, the deviations from the ideal echo become alsomore pronounced in theory. An example of the time-integrated FWM signal is shown by the full triangles inFig. 1. Beside small structures, it decays according to~ e2t21t— in sharp contrast to previous theoretical workwith a bare Coulomb potential, see, e.g., Refs. [15,16],which predicted a ~ e2at321 decay for the early-time andlow-density regime. t vs neh is depicted in Fig. 5 togetherwith the experimental data. In agreement with the earlywork of Shank et al. [1], we obtain a good fit with apower law according to t21  t210 1 cn13eh . t0 describesthe density-independent scattering with LO phonons. Inagreement with the experiment (insets in Fig. 1), thecomputed FWM spectra (inset in Fig. 5) indicate that, for3316different neh, t stems from different averages of opticalband-to-band continuum transitions.A bare Coulomb potential is not adequate for the de-scription of the present FWM experiments, which be-comes clear from a comparison with Ref. [15]. This showsthat screening has already partially developed after sev-eral tens of femtoseconds under conditions where the in-verse plasma frequency, v21pl , is longer than 50 fs neh 2.2 3 1017 cm23. In contrast to this, previous pump/probe experiments [17,18] on bulk GaAs seemed to beless sensitive with respect to screening, so that these ex-periments could be described well with a bare Coulombpotential [18].In conclusion we have observed qualitative deviationsfrom an ideal photon echo response when exciting theband edge of GaAs with 11 fs pulses. These deviationsare reproduced qualitatively by a quantum kinetic theoryof carrier-carrier Coulomb scattering. Theory and experi-ment agree quantitatively in the famous power law ofdecay time versus carrier density t21  t210 1 cn13eh .The deviations of the coherent response from an idealecho, however, show that t cannot simply be related toa single unique dephasing time T2.This work has been supported by the DFG-SPP Quan-tenkohärenz in Halbleitern and by the Krupp-Stiftung.One of us (Q. T. V.) gratefully acknowledges support bythe KAAD. We thank W. Stolz for the high qualitysample and M. U. Wehner who helped to build the setup.[1] P. C. Becker et al., Phys. Rev. Lett. 61, 1647 (1988).[2] J. Y. Bigot et al., Phys. Rev. Lett. 67, 636 (1991).[3] A. Lohner et al., Phys. Rev. Lett. 71, 77 (1993).[4] M. Lindberg et al., Phys. Rev. A 45, 1865 (1992).[5] G. Noll et al., Phys. Rev. Lett. 64, 792 (1990).[6] D. S. Kim et al., Phys. Rev. Lett. 68, 2838 (1992).[7] D. S. Kim et al., Phys. Rev. Lett. 69, 2725 (1992).[8] J.-Y. Bigot et al., Phys. Rev. Lett. 70, 3307 (1993).[9] L. Bányai et al., Phys. Rev. Lett. 81, 882 (1998).[10] L. Bányai et al., Phys. Rev. Lett. 75, 2188 (1995).[11] M. U. Wehner et al., Phys. Rev. Lett. 80, 1992 (1998).[12] H. Haug and J.-P. Jauho, Quantum Kinetics in Transportand Optics of Semiconductors, Springer Series in Solid-State Sciences Vol. 123 (Springer-Verlag, Berlin, 1996).[13] Q. T. Vu et al., Phys. Rev. B 59, 2760 (1999).[14] Using self-consistently computed retarded Greens func-tions for both, Coulomb and LO-phonon scattering, and acarrier density of neh  2 3 1016 cm23, the theory alsoreproduces the non-Markovian phonon oscillations dis-cussed in Ref. [10].[15] Q. T. Vu et al., Europhys. Lett. 40, 323 (1997).[16] V. L. Gurevich et al., Europhys. Lett. 12, 4 (1990).[17] S. Bar-Ad et al., Phys. Rev. Lett. 77, 3177 (1996).[18] F. X. Camescasse et al., Phys. Rev. Lett. 77, 5429 (1996).

Photon echoes from semiconductor band-to-band continuum transitions in the regime of coulomb quantum kinetics

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Photon echoes from semiconductor band-to-band continuum transitions in the regime of coulomb quantum kinetics

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