Gibbs, H. M.

Haug, H.

Khitrova, G.

Kärtner, F. X.

Morgner, U.

Mücke, O. D.

Tritschler, T.

Vu, Q. T.

Wegener, M.

English

KITopen

P H Y S I C A L R E V I E W L E T T E R S week ending28 MAY 2004VOLUME 92, NUMBER 21First published in: Light-Induced Gaps in Semiconductor Band-to-Band TransitionsQ. T. Vu and H. HaugInstitut für Theoretische Physik, Universität Frankfurt, 60054 Frankfurt am Main, GermanyO. D. Mücke, T. Tritschler, and M. WegenerInstitut für Angewandte Physik, Universität Karlsruhe (TH), Wolfgang-Gaede-Straße 1, 76131 Karlsruhe, GermanyG. Khitrova and H. M. GibbsOptical Sciences Center, University of Arizona, Tucson, Arizona 85721, USA(Received 4 December 2003; published 27 May 2004)217403-1We observe a triplet around the third harmonic of the semiconductor band gap when exciting50–100 nm thin GaAs films with 5 fs pulses at 3  1012 W=cm2. The comparison with solutions of thesemiconductor Bloch equations allows us to interpret the observed peak structure as being due to a two-band Mollow triplet. This triplet in the optical spectrum is a result of light-induced gaps in the bandstructure, which arise from coherent band mixing. The theory is formulated for full tight-binding bandsand uses no rotating-wave approximation.DOI: 10.1103/PhysRevLett.92.217403 PACS numbers: 78.20.Bh, 42.50.Md, 42.65.Re, 78.47.+pFIG. 1. (a) Scheme of the two-level system Mollow triplet forresonant excitation, i.e.,   !0. (b) Light-induced gaps (grayareas) are the analog for excitation within the bands of asemiconductor. The square modulus of the Hopfield coefficients[1] of the four resulting bands is superimposed onto them by agray scale: Black corresponds to 1, white to 0. GaAs parametersare used and the Rabi energy hR, the product of dipole matrixelement and peak laser electric field, is chosen as hR=Eg 0:5, with the band gap energy Eg  1:42 eV, h!0=Eg  1:1.The dashed curves show the original conduction and valenceband, respectively. Both (a) and (b) assume monochromaticexcitation as well as negligible damping and are based on theperiod has to be shorter than this value. This brings one rotating-wave approximation.When an intense light field at frequency !0 excitesband-to-band transitions in a semiconductor, additionalenergy gaps can evolve within the original bands[Fig. 1(b)]. These so-called light-induced gaps [1–5] arethe analog of the induced doublets, split by the Rabifrequency R, in a resonantly excited two-level system[Fig. 1(a)]. Here, two out of the four possible opticaltransitions are energetically degenerate. Hence, the opti-cal spectrum consists of three peaks — the famousMollow triplet for atoms [6,7] or excitons [8,9]. In thesemiconductor band case, the light-induced gap in, e.g.,the conduction band arises because the original conduc-tion band and the one-photon sideband of the valenceband lead to an avoided crossing. The correspondingHopfield coefficients [1] determine the amount of con-duction band admixture. It is crucial to note that thisadmixture can be finite even far away from the fictitiouscrossing point. This statement becomes particularly im-portant for Rabi energies hR approaching the photonenergy h!0, while it can be neglected for small Rabienergies. The latter case is tacitly assumed in previousgraphical representations of calculated light-inducedgaps. Importantly, as a result of this finite admixture,optical transitions between the induced bands are possiblethroughout an appreciable fraction of momentum space.Consequently, the transitions acquire considerable spec-tral weight. Again, two out of the four sets of possibleoptical transitions are energetically degenerate and atriplet results, which we will refer to as the two-bandMollow triplet in what follows.Light-induced gaps in semiconductor bands have notunambiguously been observed yet. In order to observe thesplitting, it clearly has to be larger than the damping. Astypical electron dephasing times under relevant condi-tions are on the order of 10 fs or less, the Rabi oscillation0031-9007=04=92(21)=217403(4)$22.50 EVA-STAR (Elektronisches Volltextarchttp://digbib.ubka.uni-karlsruhe.de/volltclose to Rabi frequencies approaching the frequency oflight —a regime which has previously been referred to ascarrier-wave Rabi flopping [10]. In this regime, a splittingin the third-harmonic spectra has been observed [11].However, as the measurements have been comparedonly with calculations on the basis of two-level systems[11], skepticism has been expressed whether this interpre-tation was correct indeed. Furthermore, later theoreticalwork [12] showed that the original experimental data2004 The American Physical Society 217403-1hiv – Scientific Articles Repository) exte/1000010909 FIG. 2. Experiment (solid curves): Spectra of light emittedinto the forward direction by a l  100 nm thin GaAs film onsapphire substrate excited by 5 fs pulses. Lower curve: I 0:24  1012 W=cm2; upper curve: I  2:8  1012 W=cm2 (samescale). We estimate that the latter translates into a peak electricfield inside the GaAs film of ~E0  2:1  109 V=m. Theory(dashed curves): Corresponding result from the semiconductorBloch equations for high ( ~E0  1:65  109 V=m) and low( ~E0  0:23  109 V=m) excitation, respectively.P H Y S I C A L R E V I E W L E T T E R S week ending28 MAY 2004VOLUME 92, NUMBER 21were to some extent obscured by propagation effectswithin the sample.In this Letter we present fully dynamical solutions ofthe semiconductor Bloch equations for tight-bindingbands with semiphenomenological density- and energy-dependent dephasing and relaxation and without employ-ing the rotating-wave approximation. Using parameters ofthe model semiconductor GaAs, we find that the light-induced gaps in the original bands lead to a background-free replica of the two-band Mollow triplet around thethird harmonic of the fundamental band gap. The com-puted emission spectra are in almost quantitative agree-ment with corresponding thin-film measurements. Thisdirect comparison of experiment and theory delivers thefirst unambiguous observation of light-induced gaps insemiconductor band-to-band transitions.In our experiments, we employ 5 fs linearly polarizedoptical pulses from a Ti:sapphire mode-locked laser os-cillator [13]. The pulses are tightly focused onto the GaAsfilm, held at room temperature, by means of a highnumerical aperture (NA  0:5) reflective microscope ob-jective. This leads to a spot radius of about 1 m. Weestimate that an average laser power of 36:5 mW in frontof the sample at fr  81 MHz repetition frequency cor-responds to a peak intensity of I  2:8  1012 W=cm2.With dielectric constant 	  10:9 for GaAs, we furtherestimate a peak electric field of ~E0  2:1  109 V=m in-side the sample. To get to these large fields in the experi-ment without damaging the sample, we have alreadychopped the laser intensity at about 100 Hz frequencyand with about 3% duty cycle. The light emitted by thesample into the forward direction is collected by a secondreflective microscope objective (NA  0:5), spectrallyprefiltered by a sequence of four fused-silica Brewster-angle prisms to suppress the overwhelmingly prominenttransmitted laser spectrum, and is sent into a gratingspectrometer connected to a liquid-nitrogen cooled,back-illuminated, UV-enhanced charge-coupled devicecamera. The GaAs layers of thickness l (withoutAlGaAs barriers) are grown directly on sapphire sub-strate in a molecular-beam epitaxy machine. The growthof the thin GaAs layer on the 0.43 mm thick, 51 mmdiameter epiready sapphire substrate (University Wafer)mounted in an In-free molyblock was performed thesame as for a GaAs wafer with a growth rate of0:25 nm=s. We have investigated four samples withGaAs film thicknesses of l  25 nm (two samples), l 50 nm, and l  100 nm. Only the latter two samples turnout to have a damage threshold sufficiently large to ac-tually perform the experiments at large Rabi energies.Two representative spectra for low and high excitation,respectively, are shown in Fig. 2. The spectra consist ofthe third harmonic and surface second-harmonic genera-tion (SHG). This interpretation has been confirmed byadditional experiments (not shown) in which we havevaried the sample thickness l. At low excitation (lowersolid curve), the third-harmonic spectrum consists of a217403-2single peak around the third harmonic of the GaAs bandgap (3Eg  4:26 eV). At high excitation (upper solidcurve), this peak splits into three, one of which mergeswith the surface SHG. The following theoretical discus-sion will allow us to assign these three peaks to the two-band Mollow triplet arising from the light-induced gapsin the original semiconductor bands.For the theoretical analysis we use the semiconductorBloch equations. It is known [3,4] that they are of theform of the Gross-Pitaevskii equation for the order pa-rameter of a weakly nonideal Bose condensate. Becauseof the large spectral width of the pulses we do not makethe usual effective mass approximation, but rather ac-count for full bands. For simplicity, the (heavy-hole) va-lence (v) and conduction (c) bands are described by cubictight-binding cosine bands. The coefficients are deter-mined by fitting the resulting band structure to theGaAs band gap of Eg  1:42 eV and its effective masses.The light-hole valence band leads to a smaller transitionstrength and is neglected. Furthermore, the Hartree-Fock(HF) Coulomb exchange interaction is taken into account,Hcoul  Xn;mc;v~k; ~qV ~qnm; ~k ~q  n;vm;v an; ~kam; ~k; (1)with the bare Coulomb potential V~q ve2=4V	bP~R0ei ~q ~R=j ~Rj, where 	b is the backgrounddielectric constant, v is the volume of a unit cell, V is thecrystal volume, and ~R is a lattice vector. Note that thiscrystal sum is generally different from the Fourier trans-form of Vr. The density matrix  is defined as nm; ~k ham; ~kan; ~ki. In (1), we have subtracted the exchange energy217403-2P H Y S I C A L R E V I E W L E T T E R S week ending28 MAY 2004VOLUME 92, NUMBER 21for the filled valence band, because that has already beenaccounted for in the band structure calculation. The cou-pling to the light field is treated in the usual dipoleapproximation [14]. We split the time derivative of thedensity matrix into a coherent part and a dissipative partdue to various types of collisions,@@tnm; ~kt @@tnm; ~ktcoh@@tnm; ~ktcoll: (2)The equation of motion of the coherent time evolution is@@tnm; ~ktcohihen; ~kt  em; ~ktnm; ~kt iXn1Rnn1; ~ktn1m; ~kt nn1; ~ktRn1m; ~kt : (3)The HF renormalized energies areen; ~kt  	n; ~k X~k0V~k ~k0 nn; ~k0 t  n;v; (4)with the single-particle energy bands 	n; ~k and the HFrenormalized Rabi energies arehRnm; ~kt d ~kEt X~qV ~qnm; ~k ~qtxnm; (5)where d ~k is the momentum-dependent dipole matrix ele-ment, Et the electric field of the laser pulse, andx 0 11 0is a Pauli spin matrix. Note that the rotating-wave ap-proximation has not been made. As we do not intend totreat the effects of different polarizations of light andmomentum anisotropies, we introduce an isotropic den-sity matrix which depends only on energy by a directionalaveraging.In the regime of a few femtoseconds, the dissipativekinetics is governed by LO-phonon and by Coulombscattering. The latter dominates at carrier densities of1020=cm3 in the present experiments. A full quantumkinetic treatment along the lines of Ref. [15] is numeri-cally too demanding under these conditions. Hence, weemploy a phenomenological description—with parame-ters based on earlier findings [15,16]. For the dephasing ofthe off-diagonal density matrix elements we take intoaccount that the damping is the sum of a constant con-tribution due to LO-phonon scattering and a term propor-tional to the third root of the carrier density n [16]. Toaccount for the additional energy dependence, we use thefollowing dephasing for the various interband polariza-tion (p) components (n  m),@@tnm	; tcoll "p	; tnm	; t ; (6)where 	 is the excess energy with respect to the unrenor-217403-3malized band gap and the density- and energy-dependentdamping rate is given by"p	; t  "0p  cpn1=3t 1 	$0  a	0	 3!: (7)The damping (7) increases with excess energy 	 be-cause of the increasing number of decay channels.Parameters are as follows: "0p  1=300 fs, cp  3:4 108 cm3 fs1, $0  25 meV, a  1 meV, and 	0 1400 meV. For the longitudinal relaxation, however, anenergy-dependent rate would not be a conserving approxi-mation. Hence, we use an energy-independent rate "etfor the longitudinal (e) relaxation@@tnn	; tcoll "etnn	; t  f0	; nt; (8)where f0	; nt is the thermal (T  300 K) equilibriumFermi function belonging to the density at time t and"et  "0e  cen1=3t : (9)Parameters are "0e  1=160 fs and ce  7:8 108 cm3 fs1. Furthermore, we use an optical pulseEt  ~Etcos!0t% with Gaussian envelope~Et  ~E0exp2 ln2t2=t2, carrier photon energyh!0  1:38 eV, carrier-envelope phase %  0 [17] anda full width at half maximum of the temporal inten-sity profile of t  7 fs. Finally, the third-harmonicspectra are proportional to jP!  3!0j2, where P!is the Fourier transform of the total polarizationPt P~kd ~k cv; ~kt  c:c.Theoretical results.—Two calculated third-harmonicspectra are shown in Fig. 2. For the lower field of ~E0 0:23  109 V=m, the computed third-harmonic spectrum(lower dashed curve) consists of a single maximum,roughly centered around the third harmonic of theGaAs band edge, in agreement with experiment (lowersolid curve). The theoretical peak is somewhat broader,which is likely due to the approximations made (pulseshape and/or scattering processes). For the higher field of~E0  1:65  109 V=m, the theoretical spectrum (upperdashed curve) exhibits three maxima around 4.3, 3.8,and 3.5 eV, again in agreement with the experiment (uppersolid curve). Because of their energetic separation weinterpret these three peaks as manifestations of thetwo-band Mollow triplet. The splitting has to be calcu-lated from the appropriately averaged renormalized Rabifrequency. The triplet is not symmetrical around 3Eg butis rather redshifted due to renormalization effects. Alsonote that the absolute value of the estimated ‘‘high’’ peakelectric field of ~E0  2:1  109 V=m inside the GaAs filmis merely 21% larger than the corresponding value of~E0  1:65  109 V=m used in the calculations. This dif-ference is well within the errors of the experimentalestimate and the accuracy with which the theory is ableto determine the correct dipole matrix element. Noticethat the chosen pulse duration of 7 fs is somewhat largerthan the experimental duration of 5 fs (see above). This217403-3FIG. 3. Calculated inversion for various excess energies 	versus time t for a peak electric field of ~E0  1:65 109 V=m. Parameters are identical to Fig. 2. For t  20 fs thecarrier density equals 1:1  1020= cm3. The lower part showsthe laser electric field Et versus time.P H Y S I C A L R E V I E W L E T T E R S week ending28 MAY 2004VOLUME 92, NUMBER 21effectively accounts for remaining pulse distortions dueto propagation effects in the GaAs film [12] and/or forsome residual chirp on the incident pulses. For t  5 fs,the third-harmonic spectra tend to exhibit a shift to-wards higher energies in comparison with the experi-ment. The detailed shape of the third-harmonic spectraalso depends on the shape of the pulse envelope ~Et.For the investigation of this aspect we have used aslightly widened ‘‘experimental pulse shape’’ and found(not shown) that one can further improve the agree-ment with experiment, at the cost of additional fittingparameters.To further support our interpretation in terms of thetwo-band Mollow triplet in the frequency domain corre-sponding to Rabi flopping of band-to-band transitions inthe time domain, we depict the inversion components,cc	; t  vv	; t, in Fig. 3 for various excess energies	. Parameters are identical to high excitation in Fig. 2.The structures with twice the frequency of light due tothe carrier-wave oscillation lead to third-harmonic gen-eration, their modulation to the triplet in Fig. 2. At smallexcess energies (close to the band edge), the inversionnearly performs a complete Rabi flop during the first halfof the pulse (t < 0). A second Rabi flop roughly betweent  1 fs and t  6 fs is already very heavily damped. Atlater times, dissipative rather than coherent kineticsdominates and the low-energy states are filled fromabove. At large excess energies, where the dephasing isstronger, only an irreversible increase of the inversion isleft after the pulse has passed. Based on our experi-ence with quantum kinetics [15,16], we expect that thewindow of coherent dynamics would be widened to acertain extent by the use of a non-Markovian retardeddescription of electron-electron scattering. We havechecked that replacing the above electron-phonon scatter-217403-4ing by electron-phonon quantum kinetics leads to nosignificant changes.In conclusion, we have observed a triplet peak struc-ture in our experiments on thin films of the model semi-conductor GaAs. Fully dynamical solutions of thesemiconductor Bloch equations are in almost quantitativeagreement with the experiment. This comparison allowsus to assign the peak structure to the two-band Mollowtriplet, which is an immediate consequence of the light-induced gaps in the original conduction and valence band,respectively. A key parameter is the Rabi energy. Only ifit becomes comparable to the photon energy, optical tran-sitions between the induced bands have sufficient spectralweight throughout an appreciable fraction of momentumspace. Finally, we note that we have also found the modu-lation due to the two-band Mollow triplet in calculations[18] in a pump/probe geometry for slightly negative delaytimes and for otherwise similar excitation conditions.This shows that light-induced gaps in semiconductorbands can possibly be observed also by other experimen-tal techniques.We thank L. Bányai and P. Gartner for their contribu-tions to the formulation of semiconductor Bloch equationswith full bands. We also thank U. Morgner and F. X.Kärtner for help with setting up the laser as well as forstimulating discussions. The group of H. H. acknowledgesthe support by the DFG (Ha 1197/15-1) and the use ofcomputing facilities at the Frankfurt Center for ScientificComputing. The work of M.W. is supported by ProjectsNo. DFG-We1497/11-1 and No. DFG-We 1497/9-1.H. M. G. and G. K. thank NSF AMOP for support.[1] H. Haug, in Optical Nonlinearities and Instabilities,edited by H. Haug (Academic, New York, 1988), p. 53.[2] V. F. Elesin, Sov. Phys. JETP 32, 328 (1971).[3] C. Comte and G. Mahler, Phys. Rev. B 34, 7164 (1986).[4] S. Schmitt-Rink et al., Phys. Rev. B 37, 941 (1988).[5] F. Jahnke and K. Henneberger, Phys. Rev. B 45, 4077(1992).[6] F.Y. Wu et al., Phys. Rev. Lett. 38, 1077 (1977).[7] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,Atom-Photon Interactions (Wiley, New York, 1992).[8] J.-P. Likforman et al., Phys. Rev. Lett. 79, 3716 (1997).[9] F. Quochi et al., Phys. Rev. Lett. 80, 4733 (1998).[10] S. Hughes, Phys. Rev. Lett. 81, 3363 (1998).[11] O. D. Mücke et al., Phys. Rev. Lett. 87, 057401 (2001).[12] O. D. Mücke et al., Phys. Rev. Lett. 89, 127401 (2002).[13] U. Morgner et al., Opt. Lett. 24, 411 (1999).[14] The ~p  ~A coupling delivers qualitatively similar results.[15] Q.T. Vu et al., Phys. Rev. Lett. 85, 3508 (2000).[16] W. A. Hügel et al., Phys. Rev. Lett. 83, 3313 (1999).[17] In the theory, the third-harmonic spectrum for %  0 ispractically identical to the %-averaged spectrum, whichis measured in the experiment.[18] Q.T. Vu and H. Haug (unpublished).217403-4

Light-induced gaps in semiconductor band-to-band transitions

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Light-induced gaps in semiconductor band-to-band transitions

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