Gartner, P.

Jahnke, F.

Lorke, M.

Nielsen, T. R.

Seebeck, J.

Juelich Shared Electronic Resources

John von Neumann Institute for ComputingMany-Body Effects in Semiconductor QuantumDotsMichael Lorke, Jan Seebeck, Torben R. Nielsen,Paul Gartner, and Frank Jahnkepublished inNIC Symposium 2006 ,G. Mu¨nster, D. Wolf, M. Kremer (Editors),John von Neumann Institute for Computing, Ju¨lich,NIC Series, Vol. 32, ISBN 3-00-017351-X, pp. 167-172, 2006.c© 2006 by John von Neumann Institute for ComputingPermission to make digital or hard copies of portions of this work forpersonal or classroom use is granted provided that the copies are notmade or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwiserequires prior specific permission by the publisher mentioned above.http://www.fz-juelich.de/nic-series/volume32Many-Body Effects in Semiconductor Quantum DotsMichael Lorke, Jan Seebeck, Torben R. Nielsen, Paul Gartner, and Frank JahnkeInstitute for Theoretical PhysicsUniversity of Bremen28334 Bremen, GermanyE-mail: {mlorke, jseebeck, tnielsen, frank.jahnke}@itp.uni-bremen.de,gartner@physik.uni-bremen.de1 IntroductionIn recent years, semiconductor quantum dots (QDs) have been studied extensively due topossible applications in optoelectronic devices like LEDs, lasers, or amplifiers1, 2. In therapidly emerging field of quantum information technology, QDs have been successfullyused to demonstrate the generation of single photons or correlated photon pairs3–5. Fur-thermore, the strong coupling regime for QD emitters in optical microcavities has beendemonstrated6, 7. A common aspect in fundamental studies and practical applications ofQDs is the critical role of correlation and scattering processes of carriers which are studiedwithin this project.Optical studies of QDs have been recently focused on self-assembled systems whichare typically grown in the Stranski-Krastanoff mode. The resulting QDs are randomlydistributed on a two-dimensional wetting layer (WL). The energy spectrum of this systemconsists of discrete states which correspond to a three-dimensional carrier localization inthe QDs and a quasi-continuum of states at higher energies in connection with the two-dimensional motion of carriers within the WL. Scattering processes of carriers betweenthe localized QD states as well as between localized and delocalized states are possibledue to the Coulomb interaction in addition to carrier-phonon interaction.2 Quantum Kinetics of Carrier-Phonon InteractionAt low carrier densities and elevated temperatures the interaction of carriers with LO-phonons provides the dominant contribution to scattering channels for redistributing car-riers in QDs as well as to the dephasing of optical excitations. Due to the discrete natureof the QD energy spectra, a phonon bottleneck has been predicted by energy conservationarguments based on Fermi’s golden rule: if the energy spacing between QD states doesnot exactly match the LO-phonon energy ωLO, carrier scattering would not be possiblein this picture. The phonon-bottleneck problem is still a debated topic because there isexperimental evidence for8, 9 as well as against it10, 11. Recently it has been discussed, thata perturbational treatment of the carrier-phonon interaction based on Fermi’s golden ruleis not applicable to discrete QD systems.Because of the strong interaction between carriers and phonons in QDs, a theoreticaldescription has to be based on polarons12, 13. The physical picture of a polaron as a carrierin a crystal interacting with a surrounding cloud of lattice distortions is shown in Fig. 1a.The energy spectrum of polarons differs significantly from that of free carriers as shown in167− −−−+ +++− +−− −+ +−+−+−+ −+−+−10 10 10 10 100(a) WL (k=0)10 10 10 10 Density of states (b)	0 2Energy  (h_ω - EG) / h_ωLO10 10 10 10 (c) 	Figure 1. Electron interacting with surrounding cloud of lattice distortions (left) and density of states (DOS) fora polaron at the bandedge a), in the p-shell b) and in the s-shell c) (right). Vertical lines indicate the δ-like DOSof free electrons.Fig. 1. There and in the following examples we consider QDs with two confined shells forelectrons and holes. For the assumed cylindrical symmetry they are called s- and p-shellwhere the latter is two-fold degenerate in addition to the spin degeneracy14. In the polaronpicture, energy renormalizations that incorporate shifts and broadening of the electronicstates as well as phonon satellites and hybridization effects immediately invalidate thesimple arguments for the phonon bottleneck.To study the experimentally observed fast carrier relaxation in QDs on the level of amicroscopic theory, we solve numerically the coupled quantum-kinetic equations for thecarrier and polarization dynamics. These equations incorporate high-dimensional scatter-ing integrals for the carrier-phonon interaction as well as memory integrals over the historyof the system14, 15.The scattering rate between polaron states is proportional to the overlap of the density-of-states (DOS) of the involved states, shifted by one phonon energy against each other.From Fig. 1 one immediately finds that in the free particle picture scattering between QDstates only is possible if the level-spacing exactly matches the LO-phonon energy due to theδ-like DOS. This corresponds to the prediction of the phonon bottleneck. In the polaronpicture even for large detunings between the level spacing and the LO-phonon energy,sufficient overlap is present which gives rise to fast carrier scattering.168a) b)s−shellp−shells−shellp−shellrelaxationphotonelectronsholes-1 0 1 2 3 4Time in ps00.050.10.15Populationp-shells-shellelectronsFigure 2. Resonant optical excitation condition for the semiconductor QD system a) and corresponding temporalevolution of the population probability for the QD states. Calculations are done for InGaAs/GaAs QDs at 300Kand a detuning of 10% between level spacing and phonon energy.In the following example we consider the situation where a resonantly tuned laser pulseexcites carriers only in the p-shell and the subsequent relaxation of carriers into the s-shellis studied, see Fig. 2a. The corresponding evolution of the carrier population is shownin Fig. 2b. First the p-shell is populated due to pulse excitation which is followed by afast relaxation into the s-shell. Rabi-oscillations due to memory effects indicate the strongcoupling regime for the carrier-phonon interaction. At later times a decrease of the QDpopulation is observed due to escape of carriers into the WL, which is also included in thecalculation.Our results show that even for materials with weak polar coupling, like InGaAs, QDpolarons lead to fast carrier scattering on a sub-picosecond timescale which is influencedonly weakly by the level spacing14.3 Quantum Kinetics of Carrier-Carrier InteractionFor optoelectronic applications of semiconductor nano-structures the optical gain spectraof the active material are of central importance. Under the assumption that the carriersystem is in a thermodynamic quasi-equilibrium, we evaluate the temperature and carrier-density dependence of the optical gain spectra theoretically. To achieve this, the knowledgeof the many-body effects is of central importance, because the line-broadening as well asthe lineshape of the optical spectra is governed by dephasing and correlation processes16.At room temperatures and elevated carrier densities the important dephasing mechanismsare the carrier-carrier Coulomb interaction as well as the interaction of carriers with LOphonons.For the case of a high excitation density, the screening of the Coulomb interaction in thecoupled QD-WL system justifies a treatment of the carrier-carrier scattering in the second-order Born approximation while the carrier-phonon interaction is included in the randomphase approximation (RPA). It turns out that in the coupled QD-WL system the Markov169holeselectronswetting layer(WL)s−shell p−shellFigure 3. Shell structure of the combined QD-WL system showing localized QD states for electrons (holes)which are energetically below a quasi-continuum of delocalized WL states corresponding to the free motion inthe WL. The red lines show the allowed optical transitions.approximation cannot be applied and renormalized quasi-particle properties (which in thecase of low carrier densities are the polarons) need to be included for a proper descrip-tion of the optical spectra17. In Fig. 3 the electronic structure of the discussed system isschematically shown. We consider two confined shells for electrons as well as for holesso that from the allowed transitions (red lines in Fig. 3) we expect three resonances inthe optical spectra, namely the ground state resonance, the excited state resonance and theexcitonic resonance of the WL.-150 -100 -50 0E - EG  [meV]00.511.5Im χ   [a.u.]nSYS = 1 1010cm-2nSYS = 5 1010cm-2nSYS = 1 1011cm-2nSYS = 2 1011cm-2nSYS = 5 1011cm-2nSYS = 1 1012cm-2-150 -100 -50-0.0300.030.06Figure 4. Optical absorption spectra for the combined QD-WL system including interaction-induced dephasingand line shifts due to Coulomb interaction and carrier-phonon interaction for various total carrier densities. Theinset shows a scale up of the QD resonances.Calculations are done for InGaAs/GaAs QDs at 300K.170-150 -140 -130E - EG  [meV]00.010.020.030.040.05Im χ   [a.u.]e-phCoule-ph + Coul-90 -80 -70E - EG  [meV]00.10.20.3-30 -20 -10 0E - EG  [meV]0123s-shell p-shellnSYS = 5 1010cm-2WLFigure 5. Influence of carrier-carrier and carrier-phonon interaction on the optical absorption spectra. Given arethe energetic region of the ground state resonance, excited state resonance, and excitonic resonance of the WL.The red (blue) line is a calculation with carrier-phonon (carrier-carrier) interaction only, while the black lineshows the full calculation.Calculations are done for InGaAs/GaAs QDs at 300K.In Fig. 4 the results of the calculations are presented. Given is the absorption vs. en-ergy (relative to the band-gap energy EG). We obtain the mentioned three resonances ataround -130meV,-85meV, and -15meV, respectively. With increasing carrier density theresonances are bleached out, reach transparency, and switch to negative absorption (op-tical gain). Additionally observed is a distinct, correlation-induced red-shift of the QDresonances with increasing carrier-density, which has also been found in recent photolumi-nescence experiments18.In Fig. 5 we compare the role of carrier-carrier and carrier-phonon interaction for afixed carrier density. We find that both mechanisms are equally important to properlydescribe the broadening of all three resonances even at intermediate to high excitationconditions.4 SummaryA quantum-kinetic treatment for carrier-phonon and carrier-carrier interaction in semicon-ductor quantum dots has been used to analyze the efficiency of scattering processes andtheir influence on the carrier-density dependent optical spectra. The absence of a phononbottleneck is explained in terms of polaron renormalizations. The combined influence ofthe scattering processes is responsible for the observed dephasing and line-shift effects inthe optical gain spectra.171AcknowledgmentsThis work was supported by the Deutsche Forschungsgemeinschaft. We acknowledge agrant for CPU time at the NIC, Forschungszentrum Ju¨lich.References1. Y. Masumoto and T. Takagahara, eds., Semiconductor Quantum Dots (Springer-Verlag, Berlin, 2002), 1st ed.2. P. Michler, ed., Single Quantum Dots (Springer-Verlag, Berlin, 2003), 1st ed.3. P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto,Nature 406, 968 (2000).4. E. Moreau, I. Robert, L. Manin, V. Thierry-Mieg, J. M. Gerad, and I. Abram, Phys.Rev. Lett. 87, 183601 (2001).5. M. Pelton, C. Santori, J. Vuckovic, B. Zhang, G. S. Solomon, J. Plant, and Y. Ya-mamoto, Phys. Rev. Lett. 89, 233602 (2002).6. J. P. Reithmaier, G. Sek, A. Lo¨ffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V.Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).7. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell,O. B. Shchekin, and D. G. Deppe, Nature 432, 200 (2004).8. J. Urayama, T.B. Norris, J. Singh, and P. Bhattacharya. Observation of phonon bot-tleneck in quantum dot electronic relaxation. Phys. Rev. Lett., 86:4930, 2001.9. W.W.E. Minnaert, A.Yu. Silov, W. van der Vleuten, J.E.M. Haverkort, and J.H. Wolter.Fro¨hlich interaction in InAs/GaAs self-assembled quantum dots. Phys. Rev. B,63:75303, 2001.10. E. Tsitsishvili, R.v. Baltz, and H. Kalt. Temperature dependence of polarizationrelaxation in semiconductor quantum dots. Phys. Rev. B, 66:R161405, 2002.11. E. Pe´ronne, F. Fossard, F.H. Julien, J. Brault, M. Gendry, B. Salem, G. Bremond, andA. Alexandrou. Dynamic saturation of an intersublevel transition in self-organizedInAs/InxAl1−xAs quantum dots. Phys. Rev. B, 67:205329, 2003.12. T. Inoshita and H. Sakaki. Density of states and phonon-induced relaxation of elec-trons in semiconductor quantum dots. Phys. Rev. B, 56:R4355, 1997.13. O. Verzelen, R. Ferreira, G. Bastard, T. Inoshita, and H. Sakaki. Polaron effects inquantum dots. phys. stat. sol. (a), 190:213, 2002.14. J. Seebeck, T.R. Nielsen, P. Gartner, and F. Jahnke. Polarons in semiconductor quan-tum dots and their role in the quantum kinetics of carrier relaxation. Phys. Rev. B,71:125327, 2005.15. J. Seebeck, T.R. Nielsen, P. Gartner, and F. Jahnke. Quantum kinetic the-ory of phonon-assisted carrier transitions in nitride-based quantum-dot systems.arXiv:cond-mat/cond-mat/0509692, 2005.16. H. Haug and S.W. Koch. Quantum Theory of the Optical and Electronic Propertiesof Semiconductors, World Scientifiv Publ., Singapore, 4. edition, 2004.17. M. Lorke, T. R. Nielsen, J. Seebeck, P. Gartner, and F. Jahnke , Influence of carrier-carrier and electron-phonon correlations on optical absorption and gain in quantum-dot systems, arXiv.org:cond-mat/0509543, 2005.18. K. Matsuda, K. Ikeda, T. Saiki, H. Saito, K. Nishi Carrier-carrier interaction in singleIn0.5Ga0.5As quantum dots at room temperature investigated by near-field scanningoptical microscope, Appl. Phys. Lett., 83, 2250 (2003).172

Many-Body Effects in Semiconductor Quantum Dots

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Many-Body Effects in Semiconductor Quantum Dots

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