- Email: [email protected]
Theory of coherent effects in semiconductors
Journal of Luminescence 83}84 (1999) 1}6
Theory of coherent e!ects in semiconductors S.W. Koch!,*, C. Sieh!, T. Meier!, F. Jahnke!, A. Knorr!, P. Brick", M. HuK bner", C. Ell", J. Prineas", G. Khitrova", H.M. Gibbs" !Department of Physics and Material Sciences Center, Philipps University, Renthof 5, D-35032 Marburg, Germany "Optical Sciences Center, University of Arizona, Tucson, AZ 85721, USA
Abstract A microscopic theory is applied to discuss coherent excitation e!ects in semiconductors. The theory is evaluated to analyze ultrafast absorption changes around the exciton resonance in semiconductor quantum-well structures where the relative strength of the di!erent many-body contributions can be manipulated by proper selection of pump and probe polarizations. For two-band systems and oppositely circular polarization it is shown that Coulombic correlation dynamics dominates the optical response. As a consequence, the optical Stark e!ect in this con"guration corresponds to a red shift of the exciton resonance in striking contrast to the usual atomic-like blue shift. The predictions are con"rmed by experiments using high-quality InGaAs quantum-well systems. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Coherent spectroscopy; Optical Stark e!ect; Electronic correlations
1. Introduction The excitonic optical Stark e!ect was the "rst major coherent e!ect observed in semiconductor systems [1}3]. Since the original discovery, substantial progress has been made in the understanding of such ultrafast nonlinear optical e!ects but the Stark e!ect problem is still good for surprising new results [4]. A big challenge in the theoretical description of nonlinear experiments like four-wave mixing and pump}probe is the proper inclusion of Coulombinduced many-body carrier correlations. Already in the low-density s(3)-limit (third order in the optical
* Corresponding author. Fax: #49-6421-2827076. E-mail address: [email protected] (S.W. Koch)
"eld) correlations contribute strongly to the nonlinear optical response [4}14]. Di!erent schemes have been developed for the theoretical analysis: If plasma excitation dominates bound exciton complexes are of less importance and correlation e!ects can be described well using the second Born approximation in the Markov limit [11,12]. However, for relatively weak, resonant excitation conditions biexcitonic and memory e!ects need to be considered explicitly [4,13,14] which can be done, e.g. using the dynamics controlled truncation scheme [7}9]. In this paper we summarize the results of our analysis of absorption changes at the 1s-exciton resonance which are induced by a coherent pump pulse [4,13]. After brie#y reviewing our model and the theoretical approach in Section 2, we compare theoretical results for various detunings and polarizations of the pump and probe pulses with
0022-2313/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 0 6 5 - 4
2
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
experiments on high-quality InGaAs quantum wells with spectrally very narrow exciton line width. Since the InGaAs quantum wells used in the experiment have a rather large splitting between the heavy- and light-hole exciton of about 15 meV, it is su$cient to consider only the heavy holes for the experimental con"guration discussed in Section 3. In Section 4 we extend our theoretical model to include also light holes. It is shown that the observability of the red shift crucially depends on the splitting between the heavy- and light-hole excitons and also on the detuning of the pump pulse. Our main results are brie#y summarized in the concluding Section 5.
2. Brief description of approach and model To theoretically model correlation e!ects we compute the optical response in the coherent s(3)limit without further approximations [7}9]. This is done by numerically solving the relevant equations of motion for the single-exciton amplitude p and the correlated part of the two-exciton amplitude B [4,11,13]. Omitting the various sums and indices the basic structure of these equations is given by L p"i[u p#kE(1!2pHp)#
ed dipole matrix elements describing the optical transitions between these bands [6,10]. The monopole}monopole Coulomb interactions < between ij particles at sites i and j are included. The spatial variation is given by a regularized potential: < " ij ; d/(Di!jDd#a ), where d is the distance between 0 0 the sites and ; and a are parameters characteriz0 0 ing the strength of the interaction and the spatial variation. In our numerical study we use the electron and heavy-hole couplings of J "8 meV % and J "0.8 meV between neighboring sites, )) ; "8 meV and a /d"0.5.1 0 0 The theory allows us to identify three types of optical nonlinearities: the total di!erential absorption signal is the sum of a Pauli-blocking term (JkEpHp) and Coulomb-induced many-body nonlinearities, which can further be separated into a "rst-order term (J
(1)
3. Numerical and experimental results
(2)
Numerical and experimental results on absorption changes following resonant excitation of the 1s exciton resonance have been presented in Ref. [4]. For both co-circularly (both r`) and oppositely circularly (r`r~) polarized pump and probe pulse there is bleaching at the exciton and excited state absorption due to exciton two-exciton transitions.
Here, u and u are energies of a single- and a p B two-exciton state, respectively, k is the transition dipole, E the external laser "eld, and < the Coulomb interaction. In Ref. [4] we have numerically solved Eqs. (1) and (2), both for a fully twodimensional model of the quantum-well structure [12] and for a simpli"ed, quasi-one-dimensional system [13]. As shown in Ref. [4] the spectra obtained for both models are in good qualitative agreement. Therefore, in the following we only present numerical results for the simpler one-dimensional tight-binding model. The two spindegenerate electron and heavy-hole bands are considered, and we use the usual circularly polariz-
1 The parameters chosen in the model give an exciton binding energy of 8 meV and a biexciton binding energy of 1.4 meV. Homogeneous broadening is introduced phenomenologically through decay rates of c "(3 ps)~1 for single-excitons and 1 c "(1.5 ps)~1 for two excitons. The pump pulse has a Gaussian B envelope with 1.18 ps (full-width at half-maximum) duration for the pulse intensity. These values are close to the experimental situation.
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
Fig. 1. Calculated di!erential absorption spectra for excitation 4.5 meV below the 1s exciton resonance and time delay q"0 ps. (a) Co-circularly polarized pump and probe pulses (r`r`), the lower panel shows the three contributions to the signal (solid: Pauli-blocking, dashed: "rst-order Coulomb, and dotted: Coulomb correlation). (b) and (c) same for cross-linear (xy) and co-linear (xx) polarizations, respectively. (d) Opposite-circularly polarized pump and probe pulses (r`r~). The dotted line represents the result of a calculation without biexcitons, as discussed in the text.
For co-circular excitation the excited state absorption is solely induced by unbound two-exciton states and appears energetically above the exciton resonance, whereas for oppositely circular excitation also signatures of a bound biexciton show up energetically below the exciton resonance [4,13]. Numerical and experimental results on resonant absorption changes are in good qualitative agreement [4].
3
We now investigate the situation where the pump pulse is tuned 4.5 meV below the exciton resonance. Fig. 1(a)}(d) displays the theoretical results for r`r`, xy, xx, and r`r~ excitation. For the "rst three cases the di!erential absorption corresponds to a blue shift, whereas for r`r~ a clear red shift appears. Note that for r`r~ excitation both the Pauli-blocking and the "rst-order Coulomb-induced nonlinearity, i.e., the Hartree}Fock contribution, vanish identically [4,13]. Thus in this case the total signal is purely induced by Coulomb correlations. The physical origin of the red shift can be analyzed in more detail by looking at the individual contributions to the signal. For r`r`, xy, and xx polarization and 4.5 meV detuning below the exciton the Pauli-blocking and the "rst-order Coulomb term always induce a blue shift, whereas the Coulomb correlations always correspond to a red shift, see Fig. 1(a)}(c). The fact that also for r`r` excitation, where no bound biexcitons are excited, the correlation term alone still corresponds to a red shift clearly demonstrates that this shift is not directly related to the existence of a bound biexciton. To substantiate this claim we performed additional calculations of the di!erential absorption spectra in r`r~ con"guration. To eliminate the bound biexciton contribution also for r`r~ excitation we arti"cially dropped the six terms containing the attractive and repulsive Coulomb terms for two electrons and two holes in the homogeneous part of the equation of motion for the two-exciton amplitude B; see Refs. [7}9,13]. In this case, also for r`r~ excitation no bound biexcitons exist. The resulting spectrum displayed in Fig. 1(d) shows that the signal amplitude is somewhat reduced, however, the red shift clearly persists. Thus, the correlation dynamics determines the red shift, but the exact structure of the two-exciton states seems to be unimportant. The linear absorption spectra of the 8.5 nm thin In Ga As quantum-well sample used in the 0.04 0.96 experiments are dominated by a heavy-hole exciton resonance with a spectrally very narrow line width of 0.56 meV, energetically separated well below all other exciton resonances. Hence, the pump}probe experiments are in#uenced only by heavy-hole exciton states and their higher-order correlations
4
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
(this is crucial, see discussion in Section 4). Both, pump and probe pulses were generated by an actively mode-locked Ti : sapphire laser. The spectrally broad 100 fs short pulses were used for probing the hh-exciton resonance while pumping simultaneously with a pulse spectrally narrowed by an external pulse shaper. The spectral width of the pump pulses was reduced to about 2.3 meV resulting in a temporal width of 1.5 ps. The spectrally resolved pump}probe experiments were performed in transmission geometry at helium temperatures. The experimental results displayed in Fig. 2(a)}(d) clearly show that in agreement with the theory a blue shift is observed for r`r`, xx, and xy polarized pump- and probe-pulses. For r`r~, however, a clear red shift is seen in Fig. 2(d).
4. Numerical results including heavy- and light-holes Since there have been numerous experimental investigations on the optical Stark e!ect in GaAs/ AlGaAs-type quantum wells, one might wonder why the red shift as reported here was not observed before. In this section we address this point and show that the observability of the red shift depends crucially on the energetic separation between heavy- and light-hole excitons as well as on the detuning. We therefore extend the approach presented in Ref. [13] to include both heavy- and light-holes using the typical selection rules and the three to one ratio of the oscillator strengths [10]. This extension is straightforward, since in Ref. [13] the approach was already formulated in a multiband formalism. Here the heavy- and light-hole transitions are coupled by two mechanisms: (i) they share a common electronic state [10] and (ii) via the Coulomb coupling. The in-plane dispersion of the valence band structure in quantum wells is modeled by considering heavy- and light-hole masses according to the Luttinger}Hamiltonian. For GaAs parameters we get m "m /(c #c )" )) 0 1 2 0.112m and m "m /(c !c )"0.211m . Further 0 -) 0 1 2 0 band-mixing e!ects are neglected here for simplicity. For the conduction band electrons we use m "0.0665m and take the same coupling of % 0 J "8 meV as used above. The valence band %
Fig. 2. Experimental di!erential absorption spectra at time delay q"0 ps and o!-resonant excitation 4.5 meV below the 1s hh-exciton resonance. (a) co-circularly, (b) cross-linearly, (c) colinearly, and (d) opposite-circularly polarized pump and probe pulses, after Ref. [4].
masses then enter into the model by taking the nearest-neighbor coupling to be inversely proportional to the mass, which yields J "J m /m " )) % % )) 4.75 meV and J "J m /m "2.52 meV. The site -) % % -) energies for the light-holes are chosen relative to the heavy holes to have an adjustable splitting between the heavy- and light-hole excitons. Fig. 3 shows calculated di!erential absorption spectra for co-circular and oppositely circular excitation considering both heavy- and light-holes for various splittings between the heavy- and lightholes excitons (a) and (b), as well as for various detunings (c) and (d). Considering a detuning of 4.5 meV below the 1s heavy-hole exciton and a splitting of 15 meV between the heavy- and light-hole excitons (which is the splitting present in the InGaAs quantum-well sample investigated above, see Fig. 2) we reproduce the blue shift for r`r` and the red shift for r`r~ excitation, see Fig. 3(a)
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
Fig. 3. Calculated di!erential absorption spectra including heavy- and light-holes. (a) Co-circularly and (b) opposite-circularly polarized pump and probe pulses, respectively, for a "xed detuning of 4.5 meV below the 1s hh-exciton resonance and various splittings between the heavy- and light-hole excitons, solid: 15 meV, dashed: 5 meV, dotted: 2.5 meV, dash-dot: 1 meV, and dash}dot}dot: 0 meV. (c) Co-circularly and (d) oppositecircularly polarized pump and probe pulses, respectively, for a "xed splitting between the heavy- and light-hole excitons of 15 meV for various detunings, solid: 4.5 meV, dashed: 6.75 meV, dotted: 9 meV, and dash-dot: 18 meV.
and (b). For reduced heavy-light splitting the blue shift for r`r` excitation remains unchanged. However, for r`r~ excitation the amplitude of the red shift strongly decreases with decreasing splitting. For a splitting of less than 1 meV the red shift has been changed into a blue shift. We thus conclude that the red shift should be well pronounced only in samples with a signi"cant heavy-light splitting. For small splittings the coupling between heavy- and light-hole excitons induces a blue shift which overcompensates the red shift induced by the heavy-hole exciton alone.
5
For a "xed heavy-light splitting taken as 15 meV in Fig. 3(c) and (d) the di!erential absorption spectra strongly depend on the detuning. For cocircular excitation the blue shift present for a detuning of 4.5 meV survives also for larger detunings and is only reduced in amplitude with increased detuning. For opposite circular excitation, however, besides a strong reduction in amplitude also the direction of the shift changes with increasing detuning. For very large detuning we again obtain a switch over from red to blue shift. The reason for this behavior is that for very large detuning, heavyand light-hole excitons are both o!-resonant and we enter the regime of complete adiabatic following where the frequency-dependent optical response decays only weakly with increasing resonance frequency [15]. Thus with increasing detuning the relative weight of the light-hole exciton increases, which then due to heavy-hole}light-hole coupling induces the blue shift at the heavy-hole exciton. The results presented in Fig. 3 nicely display that the Coulomb-memory-induced red shift for oppositely circular excitation is very sensitive to both the heavy-hole}light-hole splitting and the pump detuning. Thus it should only be observable in samples with su$ciently large exciton splittings and only in a certain detuning range. The exact detuning and splitting ranges that allow to observe the red shift depend on the heavy- and light-hole masses and on the ratio of their oscillator strengths.
5. Summary A microscopic theory capable of describing coherent excitation e!ects in semiconductors including many-body Coulomb-induced carrier correlations is reviewed. The theory is applied to analyze polarization-dependent ultrafast absorption changes at the 1s exciton resonance. For twoband semiconductors it is shown that the optical Stark e!ect changes from a blue shift for co-circular polarized pump and probe pulses to a red shift for oppositely circular conditions. The numerical results are in good qualitative agreement with experiments performed on high-quality InGaAs quantum wells. A theoretical analysis including heavy- and light-holes shows that the occurrence of the red
6
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
shift for oppositely circular polarized excitation depends sensitively on both the splitting between the heavy- and light-hole excitons and the detuning. In Ref. [13] the approach has been generalized to treat also e!ects of energetic disorder. Furthermore, the theory has been applied to excitons in microcavities [13], yielding numerical results on re#ection changes that are in good qualitative agreement with recent experiments [16]. In Ref. [14] the theory is applied to an incoherent situation. Whereas one could expect that correlations might not be important in this situation, strong correlation e!ects on the di!erential absorption induced by incoherent thermalized excitons and free electron}hole pairs are found [14]. Another point demonstrated in Ref. [14] is that at low temperatures populated free electron}hole pairs induce stronger bleaching at the exciton resonance than populated excitons of the same density. The theoretical approach has further been extended to include e!ects of higher intensities (coherent s(5)limit) where it yields results on the excitonic di!erential absorption in good agreement with experimental results [17]. In addition, work is in progress dealing with the analysis of disorder-induced dephasing in four-wave mixing spectroscopy, which can only be understood properly if both disorder and correlations are treated microscopically [18]. Acknowledgements The Marburg part of this work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Sonderforschungsbereich 383, the QuantenkohaK renz Schwerpunkt, the Heisenberg program (FJ) and the Leibniz prize (SWK), and by the John von Neumann Institut fuK r Computing, Forschungszentrum JuK lich, Germany, through
grants for extended CPU time on their supercomputer systems. The Tucson part acknowledges the support by NSF (AMOP and LWT), AFOSR/ DARPA, JSOP (AFOSR and ARO), and COEDIP. C.E. thanks the DFG for partial support. References [1] D. FroK hlich, A. NoK the, K. Reimann, Phys. Rev. Lett. 55 (1985) 1335. [2] A. Mysyrowicz, D. Hulin, A. Antonetti, A. Migus, W.T. Masselink, H. Morkoc, Phys. Rev. Lett. 56 (1986) 2748. [3] A. Von Lehmen, D.S. Chemla, J.E. Zucker, J.P. Heritage, Opt. Lett. 11 (1986) 609. [4] C. Sieh, T. Meier, F. Jahnke, A. Knorr, S.W. Koch, P. Brick, M. HuK bner, C. Ell, J. Prineas, G. Khitrova, H.M. Gibbs, Phys. Rev. Lett. 82 (1999) 3112. [5] H. Wang, K.B. Ferrio, D.G. Steel, Y.Z. Hu, R. Binder, S.W. Koch, Phys. Rev. Lett. 71 (1993) 1261. [6] K. Bott, O. Heller, D. Bennhardt, S.T. Cundi!, P. Thomas, E.J. Mayer, G.O. Smith, R. Eccleston, J. Kuhl, K. Ploog, Phys. Rev. B 48 (1993) 17 418. [7] V.M. Axt, A. Stahl, Z. Phys. B 93 (1994) 195. [8] V.M. Axt, A. Stahl, Z. Phys. B (1994) 205. [9] M. Lindberg, Y.Z. Hu, R. Binder, S.W. Koch, Phys. Rev. B 50 (1994) 18 060. [10] K. Bott, E.J. Mayer, G.O. Smith, V. Heuckeroth, M. HuK bner, J. Kuhl, T. Meier, A. Schulze, M. Lindberg, S.W. Koch, P. Thomas, J. Opt. Soc. Am. B 13 (1996) 1026. [11] W. SchaK fer, D.S. Kim, J. Shah, T.C. Damen, J.E. Cunningham, K.W. Goosen, L.N. Pfei!er, K. KoK hler, Phys. Rev. B 53 (1996) 16 429. [12] F. Jahnke, M. Kira, S.W. Koch, G. Khitrova, E.K. Lindmark, T.R. Nelson Jr., D.V. Wick, J.D. Berger, O. Lyngnes, H.M. Gibbs, K. Tai, Phys. Rev. Lett. 77 (1996) 5257. [13] C. Sieh, T. Meier, A. Knorr, F. Jahnke, P. Thomas, S.W. Koch, Europ. Phys. J. B, in press. [14] T. Meier, S.W. Koch, Phys. Rev. B 59 (1999) 13 202. [15] R. Binder, S.W. Koch, M. Lindberg, N. Peyghambarian, W. SchaK fer, Phys. Rev. Lett. 65 (1990) 899. [16] X. Fan, H. Wang, H.Q. Hou, B.E. Hammons, Phys. Rev. B 57 (1998) 9451. [17] P. Brick, T. Meier, S.W. Koch et al., unpublished. [18] S. Weiser, T. Meier, P. Thomas, S.W. Koch et al., unpublished.
Theory of coherent e!ects in semiconductors S.W. Koch!,*, C. Sieh!, T. Meier!, F. Jahnke!, A. Knorr!, P. Brick", M. HuK bner", C. Ell", J. Prineas", G. Khitrova", H.M. Gibbs" !Department of Physics and Material Sciences Center, Philipps University, Renthof 5, D-35032 Marburg, Germany "Optical Sciences Center, University of Arizona, Tucson, AZ 85721, USA
Abstract A microscopic theory is applied to discuss coherent excitation e!ects in semiconductors. The theory is evaluated to analyze ultrafast absorption changes around the exciton resonance in semiconductor quantum-well structures where the relative strength of the di!erent many-body contributions can be manipulated by proper selection of pump and probe polarizations. For two-band systems and oppositely circular polarization it is shown that Coulombic correlation dynamics dominates the optical response. As a consequence, the optical Stark e!ect in this con"guration corresponds to a red shift of the exciton resonance in striking contrast to the usual atomic-like blue shift. The predictions are con"rmed by experiments using high-quality InGaAs quantum-well systems. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Coherent spectroscopy; Optical Stark e!ect; Electronic correlations
1. Introduction The excitonic optical Stark e!ect was the "rst major coherent e!ect observed in semiconductor systems [1}3]. Since the original discovery, substantial progress has been made in the understanding of such ultrafast nonlinear optical e!ects but the Stark e!ect problem is still good for surprising new results [4]. A big challenge in the theoretical description of nonlinear experiments like four-wave mixing and pump}probe is the proper inclusion of Coulombinduced many-body carrier correlations. Already in the low-density s(3)-limit (third order in the optical
* Corresponding author. Fax: #49-6421-2827076. E-mail address: [email protected] (S.W. Koch)
"eld) correlations contribute strongly to the nonlinear optical response [4}14]. Di!erent schemes have been developed for the theoretical analysis: If plasma excitation dominates bound exciton complexes are of less importance and correlation e!ects can be described well using the second Born approximation in the Markov limit [11,12]. However, for relatively weak, resonant excitation conditions biexcitonic and memory e!ects need to be considered explicitly [4,13,14] which can be done, e.g. using the dynamics controlled truncation scheme [7}9]. In this paper we summarize the results of our analysis of absorption changes at the 1s-exciton resonance which are induced by a coherent pump pulse [4,13]. After brie#y reviewing our model and the theoretical approach in Section 2, we compare theoretical results for various detunings and polarizations of the pump and probe pulses with
0022-2313/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 0 6 5 - 4
2
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
experiments on high-quality InGaAs quantum wells with spectrally very narrow exciton line width. Since the InGaAs quantum wells used in the experiment have a rather large splitting between the heavy- and light-hole exciton of about 15 meV, it is su$cient to consider only the heavy holes for the experimental con"guration discussed in Section 3. In Section 4 we extend our theoretical model to include also light holes. It is shown that the observability of the red shift crucially depends on the splitting between the heavy- and light-hole excitons and also on the detuning of the pump pulse. Our main results are brie#y summarized in the concluding Section 5.
2. Brief description of approach and model To theoretically model correlation e!ects we compute the optical response in the coherent s(3)limit without further approximations [7}9]. This is done by numerically solving the relevant equations of motion for the single-exciton amplitude p and the correlated part of the two-exciton amplitude B [4,11,13]. Omitting the various sums and indices the basic structure of these equations is given by L p"i[u p#kE(1!2pHp)#
ed dipole matrix elements describing the optical transitions between these bands [6,10]. The monopole}monopole Coulomb interactions < between ij particles at sites i and j are included. The spatial variation is given by a regularized potential: < " ij ; d/(Di!jDd#a ), where d is the distance between 0 0 the sites and ; and a are parameters characteriz0 0 ing the strength of the interaction and the spatial variation. In our numerical study we use the electron and heavy-hole couplings of J "8 meV % and J "0.8 meV between neighboring sites, )) ; "8 meV and a /d"0.5.1 0 0 The theory allows us to identify three types of optical nonlinearities: the total di!erential absorption signal is the sum of a Pauli-blocking term (JkEpHp) and Coulomb-induced many-body nonlinearities, which can further be separated into a "rst-order term (J
(1)
3. Numerical and experimental results
(2)
Numerical and experimental results on absorption changes following resonant excitation of the 1s exciton resonance have been presented in Ref. [4]. For both co-circularly (both r`) and oppositely circularly (r`r~) polarized pump and probe pulse there is bleaching at the exciton and excited state absorption due to exciton two-exciton transitions.
Here, u and u are energies of a single- and a p B two-exciton state, respectively, k is the transition dipole, E the external laser "eld, and < the Coulomb interaction. In Ref. [4] we have numerically solved Eqs. (1) and (2), both for a fully twodimensional model of the quantum-well structure [12] and for a simpli"ed, quasi-one-dimensional system [13]. As shown in Ref. [4] the spectra obtained for both models are in good qualitative agreement. Therefore, in the following we only present numerical results for the simpler one-dimensional tight-binding model. The two spindegenerate electron and heavy-hole bands are considered, and we use the usual circularly polariz-
1 The parameters chosen in the model give an exciton binding energy of 8 meV and a biexciton binding energy of 1.4 meV. Homogeneous broadening is introduced phenomenologically through decay rates of c "(3 ps)~1 for single-excitons and 1 c "(1.5 ps)~1 for two excitons. The pump pulse has a Gaussian B envelope with 1.18 ps (full-width at half-maximum) duration for the pulse intensity. These values are close to the experimental situation.
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
Fig. 1. Calculated di!erential absorption spectra for excitation 4.5 meV below the 1s exciton resonance and time delay q"0 ps. (a) Co-circularly polarized pump and probe pulses (r`r`), the lower panel shows the three contributions to the signal (solid: Pauli-blocking, dashed: "rst-order Coulomb, and dotted: Coulomb correlation). (b) and (c) same for cross-linear (xy) and co-linear (xx) polarizations, respectively. (d) Opposite-circularly polarized pump and probe pulses (r`r~). The dotted line represents the result of a calculation without biexcitons, as discussed in the text.
For co-circular excitation the excited state absorption is solely induced by unbound two-exciton states and appears energetically above the exciton resonance, whereas for oppositely circular excitation also signatures of a bound biexciton show up energetically below the exciton resonance [4,13]. Numerical and experimental results on resonant absorption changes are in good qualitative agreement [4].
3
We now investigate the situation where the pump pulse is tuned 4.5 meV below the exciton resonance. Fig. 1(a)}(d) displays the theoretical results for r`r`, xy, xx, and r`r~ excitation. For the "rst three cases the di!erential absorption corresponds to a blue shift, whereas for r`r~ a clear red shift appears. Note that for r`r~ excitation both the Pauli-blocking and the "rst-order Coulomb-induced nonlinearity, i.e., the Hartree}Fock contribution, vanish identically [4,13]. Thus in this case the total signal is purely induced by Coulomb correlations. The physical origin of the red shift can be analyzed in more detail by looking at the individual contributions to the signal. For r`r`, xy, and xx polarization and 4.5 meV detuning below the exciton the Pauli-blocking and the "rst-order Coulomb term always induce a blue shift, whereas the Coulomb correlations always correspond to a red shift, see Fig. 1(a)}(c). The fact that also for r`r` excitation, where no bound biexcitons are excited, the correlation term alone still corresponds to a red shift clearly demonstrates that this shift is not directly related to the existence of a bound biexciton. To substantiate this claim we performed additional calculations of the di!erential absorption spectra in r`r~ con"guration. To eliminate the bound biexciton contribution also for r`r~ excitation we arti"cially dropped the six terms containing the attractive and repulsive Coulomb terms for two electrons and two holes in the homogeneous part of the equation of motion for the two-exciton amplitude B; see Refs. [7}9,13]. In this case, also for r`r~ excitation no bound biexcitons exist. The resulting spectrum displayed in Fig. 1(d) shows that the signal amplitude is somewhat reduced, however, the red shift clearly persists. Thus, the correlation dynamics determines the red shift, but the exact structure of the two-exciton states seems to be unimportant. The linear absorption spectra of the 8.5 nm thin In Ga As quantum-well sample used in the 0.04 0.96 experiments are dominated by a heavy-hole exciton resonance with a spectrally very narrow line width of 0.56 meV, energetically separated well below all other exciton resonances. Hence, the pump}probe experiments are in#uenced only by heavy-hole exciton states and their higher-order correlations
4
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
(this is crucial, see discussion in Section 4). Both, pump and probe pulses were generated by an actively mode-locked Ti : sapphire laser. The spectrally broad 100 fs short pulses were used for probing the hh-exciton resonance while pumping simultaneously with a pulse spectrally narrowed by an external pulse shaper. The spectral width of the pump pulses was reduced to about 2.3 meV resulting in a temporal width of 1.5 ps. The spectrally resolved pump}probe experiments were performed in transmission geometry at helium temperatures. The experimental results displayed in Fig. 2(a)}(d) clearly show that in agreement with the theory a blue shift is observed for r`r`, xx, and xy polarized pump- and probe-pulses. For r`r~, however, a clear red shift is seen in Fig. 2(d).
4. Numerical results including heavy- and light-holes Since there have been numerous experimental investigations on the optical Stark e!ect in GaAs/ AlGaAs-type quantum wells, one might wonder why the red shift as reported here was not observed before. In this section we address this point and show that the observability of the red shift depends crucially on the energetic separation between heavy- and light-hole excitons as well as on the detuning. We therefore extend the approach presented in Ref. [13] to include both heavy- and light-holes using the typical selection rules and the three to one ratio of the oscillator strengths [10]. This extension is straightforward, since in Ref. [13] the approach was already formulated in a multiband formalism. Here the heavy- and light-hole transitions are coupled by two mechanisms: (i) they share a common electronic state [10] and (ii) via the Coulomb coupling. The in-plane dispersion of the valence band structure in quantum wells is modeled by considering heavy- and light-hole masses according to the Luttinger}Hamiltonian. For GaAs parameters we get m "m /(c #c )" )) 0 1 2 0.112m and m "m /(c !c )"0.211m . Further 0 -) 0 1 2 0 band-mixing e!ects are neglected here for simplicity. For the conduction band electrons we use m "0.0665m and take the same coupling of % 0 J "8 meV as used above. The valence band %
Fig. 2. Experimental di!erential absorption spectra at time delay q"0 ps and o!-resonant excitation 4.5 meV below the 1s hh-exciton resonance. (a) co-circularly, (b) cross-linearly, (c) colinearly, and (d) opposite-circularly polarized pump and probe pulses, after Ref. [4].
masses then enter into the model by taking the nearest-neighbor coupling to be inversely proportional to the mass, which yields J "J m /m " )) % % )) 4.75 meV and J "J m /m "2.52 meV. The site -) % % -) energies for the light-holes are chosen relative to the heavy holes to have an adjustable splitting between the heavy- and light-hole excitons. Fig. 3 shows calculated di!erential absorption spectra for co-circular and oppositely circular excitation considering both heavy- and light-holes for various splittings between the heavy- and lightholes excitons (a) and (b), as well as for various detunings (c) and (d). Considering a detuning of 4.5 meV below the 1s heavy-hole exciton and a splitting of 15 meV between the heavy- and light-hole excitons (which is the splitting present in the InGaAs quantum-well sample investigated above, see Fig. 2) we reproduce the blue shift for r`r` and the red shift for r`r~ excitation, see Fig. 3(a)
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
Fig. 3. Calculated di!erential absorption spectra including heavy- and light-holes. (a) Co-circularly and (b) opposite-circularly polarized pump and probe pulses, respectively, for a "xed detuning of 4.5 meV below the 1s hh-exciton resonance and various splittings between the heavy- and light-hole excitons, solid: 15 meV, dashed: 5 meV, dotted: 2.5 meV, dash-dot: 1 meV, and dash}dot}dot: 0 meV. (c) Co-circularly and (d) oppositecircularly polarized pump and probe pulses, respectively, for a "xed splitting between the heavy- and light-hole excitons of 15 meV for various detunings, solid: 4.5 meV, dashed: 6.75 meV, dotted: 9 meV, and dash-dot: 18 meV.
and (b). For reduced heavy-light splitting the blue shift for r`r` excitation remains unchanged. However, for r`r~ excitation the amplitude of the red shift strongly decreases with decreasing splitting. For a splitting of less than 1 meV the red shift has been changed into a blue shift. We thus conclude that the red shift should be well pronounced only in samples with a signi"cant heavy-light splitting. For small splittings the coupling between heavy- and light-hole excitons induces a blue shift which overcompensates the red shift induced by the heavy-hole exciton alone.
5
For a "xed heavy-light splitting taken as 15 meV in Fig. 3(c) and (d) the di!erential absorption spectra strongly depend on the detuning. For cocircular excitation the blue shift present for a detuning of 4.5 meV survives also for larger detunings and is only reduced in amplitude with increased detuning. For opposite circular excitation, however, besides a strong reduction in amplitude also the direction of the shift changes with increasing detuning. For very large detuning we again obtain a switch over from red to blue shift. The reason for this behavior is that for very large detuning, heavyand light-hole excitons are both o!-resonant and we enter the regime of complete adiabatic following where the frequency-dependent optical response decays only weakly with increasing resonance frequency [15]. Thus with increasing detuning the relative weight of the light-hole exciton increases, which then due to heavy-hole}light-hole coupling induces the blue shift at the heavy-hole exciton. The results presented in Fig. 3 nicely display that the Coulomb-memory-induced red shift for oppositely circular excitation is very sensitive to both the heavy-hole}light-hole splitting and the pump detuning. Thus it should only be observable in samples with su$ciently large exciton splittings and only in a certain detuning range. The exact detuning and splitting ranges that allow to observe the red shift depend on the heavy- and light-hole masses and on the ratio of their oscillator strengths.
5. Summary A microscopic theory capable of describing coherent excitation e!ects in semiconductors including many-body Coulomb-induced carrier correlations is reviewed. The theory is applied to analyze polarization-dependent ultrafast absorption changes at the 1s exciton resonance. For twoband semiconductors it is shown that the optical Stark e!ect changes from a blue shift for co-circular polarized pump and probe pulses to a red shift for oppositely circular conditions. The numerical results are in good qualitative agreement with experiments performed on high-quality InGaAs quantum wells. A theoretical analysis including heavy- and light-holes shows that the occurrence of the red
6
S.W. Koch et al. / Journal of Luminescence 83}84 (1999) 1}6
shift for oppositely circular polarized excitation depends sensitively on both the splitting between the heavy- and light-hole excitons and the detuning. In Ref. [13] the approach has been generalized to treat also e!ects of energetic disorder. Furthermore, the theory has been applied to excitons in microcavities [13], yielding numerical results on re#ection changes that are in good qualitative agreement with recent experiments [16]. In Ref. [14] the theory is applied to an incoherent situation. Whereas one could expect that correlations might not be important in this situation, strong correlation e!ects on the di!erential absorption induced by incoherent thermalized excitons and free electron}hole pairs are found [14]. Another point demonstrated in Ref. [14] is that at low temperatures populated free electron}hole pairs induce stronger bleaching at the exciton resonance than populated excitons of the same density. The theoretical approach has further been extended to include e!ects of higher intensities (coherent s(5)limit) where it yields results on the excitonic di!erential absorption in good agreement with experimental results [17]. In addition, work is in progress dealing with the analysis of disorder-induced dephasing in four-wave mixing spectroscopy, which can only be understood properly if both disorder and correlations are treated microscopically [18]. Acknowledgements The Marburg part of this work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Sonderforschungsbereich 383, the QuantenkohaK renz Schwerpunkt, the Heisenberg program (FJ) and the Leibniz prize (SWK), and by the John von Neumann Institut fuK r Computing, Forschungszentrum JuK lich, Germany, through
grants for extended CPU time on their supercomputer systems. The Tucson part acknowledges the support by NSF (AMOP and LWT), AFOSR/ DARPA, JSOP (AFOSR and ARO), and COEDIP. C.E. thanks the DFG for partial support. References [1] D. FroK hlich, A. NoK the, K. Reimann, Phys. Rev. Lett. 55 (1985) 1335. [2] A. Mysyrowicz, D. Hulin, A. Antonetti, A. Migus, W.T. Masselink, H. Morkoc, Phys. Rev. Lett. 56 (1986) 2748. [3] A. Von Lehmen, D.S. Chemla, J.E. Zucker, J.P. Heritage, Opt. Lett. 11 (1986) 609. [4] C. Sieh, T. Meier, F. Jahnke, A. Knorr, S.W. Koch, P. Brick, M. HuK bner, C. Ell, J. Prineas, G. Khitrova, H.M. Gibbs, Phys. Rev. Lett. 82 (1999) 3112. [5] H. Wang, K.B. Ferrio, D.G. Steel, Y.Z. Hu, R. Binder, S.W. Koch, Phys. Rev. Lett. 71 (1993) 1261. [6] K. Bott, O. Heller, D. Bennhardt, S.T. Cundi!, P. Thomas, E.J. Mayer, G.O. Smith, R. Eccleston, J. Kuhl, K. Ploog, Phys. Rev. B 48 (1993) 17 418. [7] V.M. Axt, A. Stahl, Z. Phys. B 93 (1994) 195. [8] V.M. Axt, A. Stahl, Z. Phys. B (1994) 205. [9] M. Lindberg, Y.Z. Hu, R. Binder, S.W. Koch, Phys. Rev. B 50 (1994) 18 060. [10] K. Bott, E.J. Mayer, G.O. Smith, V. Heuckeroth, M. HuK bner, J. Kuhl, T. Meier, A. Schulze, M. Lindberg, S.W. Koch, P. Thomas, J. Opt. Soc. Am. B 13 (1996) 1026. [11] W. SchaK fer, D.S. Kim, J. Shah, T.C. Damen, J.E. Cunningham, K.W. Goosen, L.N. Pfei!er, K. KoK hler, Phys. Rev. B 53 (1996) 16 429. [12] F. Jahnke, M. Kira, S.W. Koch, G. Khitrova, E.K. Lindmark, T.R. Nelson Jr., D.V. Wick, J.D. Berger, O. Lyngnes, H.M. Gibbs, K. Tai, Phys. Rev. Lett. 77 (1996) 5257. [13] C. Sieh, T. Meier, A. Knorr, F. Jahnke, P. Thomas, S.W. Koch, Europ. Phys. J. B, in press. [14] T. Meier, S.W. Koch, Phys. Rev. B 59 (1999) 13 202. [15] R. Binder, S.W. Koch, M. Lindberg, N. Peyghambarian, W. SchaK fer, Phys. Rev. Lett. 65 (1990) 899. [16] X. Fan, H. Wang, H.Q. Hou, B.E. Hammons, Phys. Rev. B 57 (1998) 9451. [17] P. Brick, T. Meier, S.W. Koch et al., unpublished. [18] S. Weiser, T. Meier, P. Thomas, S.W. Koch et al., unpublished.