We present a theory of scanning local optical spectroscopy in quantum structures taking into account structural disorder. The calculated spatially resolved spectra show the individual spectral lines from the exciton states localized by the disordered potential as well as the quasicontinua spectra at positions close to the potential barriers in agreement with the experimental findings

Elsaesser, T.

Emiliani, V.

Intonti, F.

Lienau, Ch.

Nötzel, R.

Ploog, K.H.

Low-dimensional semiconductor structures grown by molecular beam epitaxy on a patterned (311)A GaAs substrate are investigated by near-field spectroscopy at a temperature of 10 K. In particular, the two-dimensional potential profiles of quantum wire and coupled wire-dot structures are determined from photoluminescence (PL) measurements with a spatial resolution of 150 nm. Also presented is an optical method for investigating carrier transport in low-dimensional systems involving performing spatially resolved PL excitation measurements on the wire-dot structure

Pure OAI Repository

Near-field optical imaging and spectroscopy of single GaAs quantum wires

A novel semi-classical theory of exciton-polaritons in quantum wells (QWs) is presented. Time-resolved coherent optical spectra of QWs are described by the light coupling with ensembles of localised quantum dot-like exciton states. The ratio between reflection and resonant Rayleigh scattering (RRS) intensity gives a direct measure of the density of these excitonic dots. Good agreement with experimental data on time-resolved angle-dependent reflection and RRS from a GaAs single QW is achieved

Kavokin, A.V.

Malpuech, G.

Langbein, W.

Southampton (e-Prints Soton)

Theory of propagation and scattering of exciton-polaritons in quantum wells

Di Stefano, Omar

Savasta, Salvatore

Martino, Giovanna

Girlanda, Raffaello

Messanae Universitas Studiorum

Looking at Localized Excitons in Quantum Structures:A Theoretical DescriptionO. Di Stefano1), S. Savasta, G. Martino, and R. GirlandaDipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate,Universitá di Messina, Salita Sperone 31, 98166 Messina, ItalyWe present a theory of scanning local optical spectroscopy in quantum structures taking into ac-count structural disorder. The calculated spatially resolved spectra show the individual spectrallines from the exciton states localized by the disordered potential as well as the quasicontinuaspectra at positions close to the potential barriers in agreement with the experimental findings.Static structural disorder on mesoscopic length scale is an unavoidable feature of low-dimensional semiconductor structures such as quantum wells (QWs) and quantum wires(QWRs). These structures exhibit, in addition to the impurities, lattice imperfections,etc., which are also present in homogeneous bulk crystals, interface imperfections whichcan involve different components with different length scales. Disorder in these quan-tum structures results in an effective 2D (QWs) or 1D (QWRs) potential with spatialcorrelation which tends to localize the center of mass motion of excitons. The opticalspectra of semiconductor nanostructures, which near the absorption edges are domi-nated by excitons, yield valuable information on the quality of the interfaces and thegrowth process [1, 2]. In order to investigate in detail the optical properties and theeffects of disorder on the individual electronic quantum states, optical probes combin-ing very high spatial and spectral resolutions are required [3–6].Theoretical approaches modeling the interaction of quantum structures with highly inho-mogeneous light fields have been recently presented [7–13]. Here we present theoreticalcalculations of local optical spectroscopy in QWs and in QWRs taking into account struc-tural disorder. Excitons, produced by a low-optical excitation behave as an effective quan-tum particle of mass M whose wave-function runs at mesoscopic dimensions over manyatomic units. These calculations show how local optical probes can partially map these me-soscopic wave-functions. The theoretical calculations consist of two steps in close analogywith real experiments. The first step is the growth of the sample. Each sample correspondsto a single realization of disorder with prescribed statistical properties. Once obtained thesamples, we calculate the spectra, each spectrum obtained with the probe tip fixed at apoint on the QWplane or along the wire free axis is a single complete calculation.In reasonable good quality structures, the amplitudes of the confinement energy fluc-tuations are typically one order of magnitude smaller than the binding energy of the 1Sexciton [14]. In this limit the relative exciton motion described by the effective wave-function f1SðqehÞ may be assumed undistorted by disorder. So disorder affects signifi-1) Corresponding author; Tel.: +39 090 6765393; Fax: +39 090 391382;e-mail: distef@ortica.unime.itcantly only the centre of mass (COM) motion through an effective potential VðrÞ. Inthe following we describe the highly inhomogeneous light field by assuming a Gaussianprofile gRðrÞ centered around the beam position R. The total absorption under localillumination is proportional to [10–12]agðR;wÞ / Im hgRj½w id Ĥ1j gRi ; ð1Þwhere the Hamiltonian Ĥ describes the exciton COM motion and, in the coordinaterepresentation, is given by Ĥr ¼ ðh2=2MÞr2 þ VðrÞ þ hw0, where hw0 is the 1S exci-ton energy level in the ideal (disorder free) structure, r is the coordinate along the QWplane or the wire free axis, and VðrÞ is the effective disorder potential felt by excitons.The degree of local information on the excitonic wave functions provided by agðR;wÞis determined by the comparison between the scale of the spatial variations of thewavefunction and the spatial extension of the beam [10]. For example, for a spatiallyhomogeneous light field, agðR;wÞ is proportional to the square modulus of the spatialintegral of the excitonic wave function. For a infinitely narrow probe beam agðR;wÞ isproportional to the square modulus of the wavefunction calculated at R. Although notdirect, significative information on absorption under local illumination can also be ob-tained by subwavelength excitation spectroscopy [6, 15].We model the random potential VðrÞ felt by the excitons as a zero mean, Gaussdistributed and spatially correlated process [16] defined by the propertyVðrÞVðr0Þh i ¼ v20 e rr0j j2=2x2 ; where . . .h i denotes ensemble average over random config-urations, v0 is the width of the energy distribution, and x is the correlation length char-acterizing the potential fluctuations. This effective potential is a result of the convolu-tion of the actual disorder potential felt by electrons and holes with the 1S excitonwave function fðqehÞ. In our simulation, after a single realization of disorder, thegrowth of the sample, calculations are carried out in real space mapping, on a fine meshof points, Hr, which is then tridiagonalized by using the Lanczos algorithm startingfrom the normalized initial state gRj i. The relevant spectrum comes from the inversionof the resolvent matrix ðwþ idHÞ1 which is very fast in the tridiagonal form of theHamiltonian H [11]. For both the structures taken in examination, we adopt an excitonkinetic mass of m ¼ 0:25m0 typical for AlAs/GaAs quantum wells.For calculations on QWRs we have considered a system size of L ¼ 6 mm which hasbeen divided into n ¼ 3000 steps, d ¼ 80 meV.Figure 1 (1D) displays the specific potential reali-zation of the sample (in the investigated region)which has been obtained using the specific correla-tion length x ¼ 30 nm and a strength v0 ¼ 1:73 meV.The calculations yield a two-dimensional matrix ofdata agðZ;wÞ.Figure 2 displays spatially resolved spectraagðZ;wÞ, obtained by using beams with differentspatial extension, as a function of photon energy w(1-D)(2-D)xEnergy(meV)100 nmyz-55Fig. 1. Realizations of disorder potential with correlationlength x ¼ 30 nm for a QWR (1D) and a QW (2D)and beam position Z along the free z-axis of the wire. At high spatial resolution (a, b) it ispossible to observe sharp and spatially narrow structures from the individual eigenstatesof the quantum system that provide detailed information on the spatial extension of theoptically active quantum states. In agreement with experiments [17], we can observe atlow energy structures fully localized inside only one potential well and, increasing theenergy, extended structures arising from tunneling states between wells. Lowering spatialresolution the structures broaden, information on the potential profile is partially lostand, owing to the nonlocal character of light-matter interaction in semiconductors, cancel-lation effects can be observed due to destructive spatial interference of the correspondingquantum state [10, 12]. However we notice that, even when using a beam extensions ¼ 300 nm well beyond the spatial correlation of the disorder potential, a number ofindividual structures remain visible in agreement with experimental findings [5].The local spectra in QWs have been calculated by considering a square region of4 mm2 which has been reproduced with a 500 500 mesh. Periodic boundary conditionshave been adopted. For all the calculated spectra we used a homogeneous broadeningd ¼ 60 meV. The calculations yield a three-dimensional matrix of data agðx; y;wÞ. Fig-ure 1 (2D) displays a 2D gray scale image of the region of the specific realized samplewhere local spectra have been calculated. The disordered potential has been obtainedby using a correlation length x ¼ 30 nm and a strength v0 ¼ 1:5 meV. Figure 3i displaysa spectrum calculated with a beam with s ¼ 250 nm and centered in the middle of thewindows in Fig. 1 (2D). Figures 3a–h show five 2D images generated by sectioning thedata agðx; y;wÞ in planes of constant energy w indicated by letters in Fig. 3i. Theimages, calculated using a beam with s ¼ 40 nm, show the different behaviour of two-dimensional excitons when varying the excitation energy and give information on thespatial extension of the quantum states which determine the sharp peaks in Fig. 3i. Atlow energy the images (Figs. 3a–e) display the lowest states of the exciton centre ofmass motion located at the individual potential minima. Increasing the excitation en-ergy, tunneling starts and we find an increasing number of structures which origin fromexcited states shared by different potential minima. An analogous behaviour has beenobserved in experimental spectroscopic images [3, 15].In summary we have presented a theoretical formulation of scanning local opticalspectroscopy in disordered quantum structures. We have performed calculations for aQWR and for a QW with interfacial roughness. We have reported calculations obtainedfor the specific disorder realization. We point out that the method presented here isTip position along the wire free axisEnergy(a) (b) (c)(d) (e) (f)6-4100 nmFig. 2. Local absorption agðZ;wÞ as a func-tion of beam position Z and photon energyalong the free z-axis of the wire, obtainedfor the sample in Fig. 1 (1D)) and by usingdifferent spatial resolutions: a) s ¼ 20 nm, b)40, c) 80, d) 120, e) 210, f) 300 nm. The linein the bottom of Fig. 3e measures 100 nm.The energy range on the vertical axis is be-tween 4 and 6 meV for all the panelsgeneral and can be applied for arbi-trary disorder configurations. Theimages presented here show that ourtheoretical simulations can give a use-ful contribution to the interpretationof experimental investigations andcharacterizations. Furthermore the di-rect comparison of spectroscopicimages with the realized disorder po-tential shown here, puts forward thecorrelation between structural disorder and spectroscopic near field images in QWs andQWRs. Moreover, these results have been obtained assuming that disorder does notperturb the e–h relative motion. This is not true in narrow quantum structures wherethis approximation becomes questionable. Work is in progress in order to relax thisapproximation and in order to include more realistic disorder potentials.Acknowledgement We greatfully acknowledge useful discussions with V. Savona.References[1] C. Weisbuch, R. Dingle, A. C. Grossard, and W. Wiegmann, Solid State Commun. 37, 219 (1981).[2] M. A. Herman, D. Bimberg, and J. Christen, J. Appl. Phys. 70, R1 (1991).[3] H. F. Hess, E. Betzig, T. D. Harris, L. N. Pfeiffer, and K. W. West, Science 264, 1740 (1994).[4] A. Zrenner, L. V. Butov, M. Hagn, G. Abstreiter, G. Böhm, and G. Weinmann, Phys. Rev.Lett. 72, 3382 (1994).[5] J. Hasen, L. N. Pfeiffer, A. Pinczuk, S. He, K. W. West, and B. S. Dennis, Nature (London)390, 54 (1997).[6] D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, Science 273, 87 (1996);Phys. Rev. Lett. 76, 3005 (1996).[7] E. Runge and R. Zimmermann, phys. stat. sol. (b) 206, 167 (1998).[8] B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, Phys. Rev. B 55, 13715 (1997).[9] G. W. Bryant, Appl. Phys. Lett. 72, 768 (1998).[10] O. Mauritz, G. Goldoni, F. Rossi, and E. Molinari, Phys. Rev. Lett. 82, 847 (1999);Phys. Rev. B 62, 8204 (2000).[11] S. Savasta, G. Martino, and R. Girlanda, Phys. Rev. B 61, 13852 (2000).[12] O. Di Stefano, S. Savasta, G. Martino, and R. Girlanda, Phys. Rev. B 62, 11071 (2000).[13] O. Di Stefano, S. Savasta, G. Martino, and R. Girlanda, Appl. Phys. Lett. 77, 2804 (2000).[14] R. F. Schnabel, R. Zimmermann, D. Bimberg, H. Nickel, R. Lösch, and W. Schlapp, Phys. Rev.B 46, 9873 (1992).[15] Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).[16] S. Glutsch and F. Bechstedt, Phys. Rev. B 50, 7733 (1994).[17] F. Intonti, V. Emiliani, C. Lienau, T. Elsaesser R. Nötzel and K. H. Ploog, Phys. Rev. B 63,075313 (2001).100 nm(a) (b)(c) (d) (e)(f) (g) (h)(i)-4 -2 0 2 4 6Energy (meV)AbsorptionhgfedcbaFig. 3. a)–h) 2D images generated by sec-tioning the data SIgðx; y;wÞ in planes ofconstant energy indicated by letters inFig. 3i. i) Spectrum obtained using a Gaus-sian beam with s ¼ 250 nm centered inthe middle of the image in Fig. 1 (2D)

Looking at Localized Excitons in Quantum Structures: A Theoretical Description

A novel semi-classical theory of exciton–polaritons in quantum wells (QWs) is presented. Time-resolved coherent optical spectra of QWs are described by the light coupling with ensembles of localised quantum dot-like exciton states. The ratio between reflection and resonant Rayleigh scattering (RRS) intensity gives a direct measure of the density of these excitonic dots. Good agreement with experimental data on time-resolved angle-dependent reflection and RRS from a GaAs single QW is achieved

Langbein, Wolfgang Werner

Online Research @ Cardiff

8 páginas, 2 figuras, 1 tabla.-- PACS numbers  73.21.Hb; 78.55.Cr; 78.67.Lt; S7.11; S7.12.Temperature dependent photoluminescence studies have been carried out on several samples containing self-assembled GaxIn1–xAs/InP quantum wires. A red-shift of the emission wavelength is observed when increasing the Ga content in the alloy, but the overall optical quality decreases. In the case of x = 0.15, the photoluminescence is not sensible to temperature and the emission band is conserved until room temperature is reached, which could be explained if the nanostructures are considered to be almost amorphous.This work has been supported by the spanish CICYT under the
project No. TIC99-1035-C02.Peer reviewe

Alén, Benito

Biswas, Dipankar

Martínez Pastor, Juan Pascual

García Martínez, Jorge Manuel

González Sotos, Luisa

Digital.CSIC

Optical Properties of SelfAssembled GaxIn xAsInPQuantum WiresB Alen  a D Biswas   a J MartnezPastory a J M Garca  b and L Gonzalez  b a Instituto de Ciencia de Materiales Universidad de ValenciaP O Box  	 Valencia Spain b Instituto de Microelectronica de Madrid  CNM CSICIsaac Newton   Tres Cantos  Madrid Spain Dated Sep  AbstractTemperature dependent photoluminescence studies have been carried out on several samplescontaining self	assembled GaxInxAsInP quantum wires A red	shift of the emission wavelengthis observed when increasing the Ga content in the alloy but the overall optical quality decreasesIn the case of x   the photoluminescence is not sensible to temperature and the emissionband is conserved until room temperature which can only be explained if the nanostructres areconsidered almost amorphousPACS numbers Hb 	Cr 	Lt S  On leave from Institute of Radiophysics and Electronics  APC Road Calcutta  Indiay Corresponding author Phone   	  Fax   	  email martinepuvesI INTRODUCTIONSelfassembled quantum nanostructures are especially suitable to be incorporated in theactive region of diode lasers with lower threshold currents and higher eciencies than conventional quantum well lasers InAsInP selfassembled quantum wires  QWRs are particularly promising candidates for light emitting devices in the 	  m wavelength regionOnce the growth conditions are xed the best way to tune the emission wavelength is theuse of a ternary alloy instead of the binary compounds in the structure either InAs  QWRmaterial or InP  barrier material In this work we have used GaxInxAsInP instead ofInAs as QWR materialThe temperature stability of the lasing wavelength is a very important parameter to becontrolled for appropriate use in bre optic based devices Due to the temperature dependence of the band gap energy of the active region in semiconductor lasers the emission wavelength varies considerably during operation This change can be compensated externally bythermal stabilisers like articially introduced gratings Some other more intrinsic strategiescan be put in practice Nearly temperature independent emission bands have been reportedon GaInAs QWRs created by the straininduced laterallayer ordering  SILO process andon InAsGaAs quantum dots with an overgrowth layer of GaInAs In this work we willshow how GaInAs alloys with moderate Ga content give rise to ecient selfassembled QWRnanostructures whose emission wavelength is practically constant in the whole examinedtemperature range up to room temperatureII SAMPLES AND EXPERIMENTSeveral GaxInxAs based QWR samples have been grown by MBE on InP  substrates under similar conditions A  nm thick InP buer layer was grown in all samplesin the     surface reconstruction at TS C and a beam equivalent pressure  BEP P    Torr The GaxInxAs layers  x   for samples  and  x   forsample 	 and x   for sample  were deposited at a BEP  As   Torr agrowth rate of  MLs and TS C An annealing at C under arsenic pressurewas followed during  s After annealing the samples were either immediately cooleddown and taken out of the growth chamber for morphological studies by atomic force miSample x dGaInAs ML h  nm w  nm   nm and                TABLE I The deposited thickness  dGaInAs height  h width  w and period   of GaxInxAsQWRs are showncroscopy  AFM or covered with a nmthick InP cap layer for optical characterizationThe gallium concentration  x  and deposited thickness of GaxInxAs  dGaInAs are givenin Table I for these samples The average values for height  h width  w and pitch period  of the QWR array in each sample are also given in Table I as obtained by the statisticalanalysis of AFM images on similar uncapped samples The deposited amount of GaInAs isa nominal value Due to the AsP exchange at the InP surface exposed to As pressure alarger amount of InAs could be involved in the InAs QWR formation More details aboutgrowth and structural properties can be found on Refs Photoluminescence studies as a function of temperature and excitation density have beencarried out using the  nm line of an Ar ion laser as excitation source The light wasdispersed by a  m double monochromator and synchronously detected by a pin InGaAsextended photodiode working at room temperatureIII RESULTS AND DISCUSSIONFigure  shows the photoluminescence spectra of the four samples at  K It is worthnoting the strong redshift of the emission more than  meV from Gafree samples tosample    Ga The observed redshift is explained by the reduction of the quantumconnement due to the larger size of the wires formed in samples 	 and  as listed in TableI The lower carrier connement in bigger QWRs is more important than the band gapincrease in the GaInAs alloys Moreover given that the strain relaxation is proportional tothe nanostructure volume an additional redshift can be expected for these samples This istrue independently of the anisotropic nature of the strain which is thought to be triaxial inour QWRs		The characteristic emission of samples  and  is not a single broad gaussian band butinstead several gaussian components can be well resolved as shown in Figs  a and  bTwo consecutive gaussian components are reasonably explained under the hypothesis ofwires whose height diers in one monolayer This hypothesis is compatible with the factthat most of the components are observed under a large range of excitation densities  Wcm and temperatures  		 K Calculated bandtoband optical transitions arerepresented by the vertical dashed lines in Figs  a and  b for QWR heights from  to	 ML reproducing quite well the experimental peak energies of the dierent componentsThis situation does not apply for samples 	 and  as shown in gures  c and  dTheir corresponding PL spectra consist in broad emission bands whose asymmetry can bereasonably reproduced by two gaussians On the one hand the ground states for dierentQWR height families are closer because the quantum connement is reduced with increasing the wire dimensions On the other hand AFM images obtained on similar samples showthat size and shape uctuations are more important both in number and amplitude withincreasing gallium content Therefore larger inhomogeneous broadening and exciton localization eects are expected in these samples Hence the gaussian components depicted inFigs  c and  d are associated with excitonic recombination in wires with a broad sizedistribution and they can not be related directly to a given unique height In sample 	 stillis possible to nd a correlation between calculated values and the experimental two gaussianpeak energies if the wire heights are in the range  to  ML in agreement with the averagesize given in Table I However in sample  the model fails overestimating the transitionenergy in a reasonable range of wire heights In this case the lack of homogeneity aectsthe nature of the emission itself The emission band of sample  is comparatively narrowerthan that of sample 	 possibly due to the dominant contribution of localized excitons inwire defects  islands observed in the AFM images of sample  typically bigger than theaverage wire dimensions This hypothesis is consequent with the PL dependence on theincident excitation power The low energy side of the PL band saturates with respect to itshigh energy side when increasing the excitation density  in the range  to  kWcmgiving rise to a net blueshift of the emission band by more than  meV This blueshift isnot observed in the other samples Only the existence of an important density of localizedstates as compared to near free or mobile excitons can explain reasonably the nature ofthe emission in this sampleFIG  PL spectra at  K of the four studied samplesMore dierences arise from the photoluminescence dependence on temperature In gure a we observe how the high energy components of the PL spectra for samples  and quench faster than the low energy ones This can be explained by the thermal activatedescape of carriers from the smallest wires towards the InP barrier as the temperature isincreased which makes PL spectra to be dominated by bigger wires at higher temperaturesThe situation changes with increasing Ga content in the QWR layer In gure  b thetemperature evolution of the PL band in sample 	 can be followed by its gaussian componentscentered at  and  eV at low temperatures The high energy component fastlydisappear at around  K  the PL linewidth becomes now comparable to that found insample  may be due to thermal escape of carriers out to the InP barriers as occurs forsamples  and  These carriers can recombine non radiatively in the barrier or be retrappedin larger wires and exciton localization defects From  K up to  K the integratedintensity of the PL band decreases by about a factor  with any clear activated energyFIG  PL spectra of the four samples at dierent temperaturespossibly because we are dealing mostly with strongly localized excitons On the contrarythe most important phenomenology observed in sample  is the absence of any change inits PL spectum with temperature either the linewidth and peak energy as observed in Fig bIn samples  and 	 the PL spectra shifts to red at a rate around 	 meVK and meVK respectively slightly slower than the InAs bulk value This slight dierencecan be attributed to the particular role played by localized and mobile excitons in therecombination dynamics at each temperature andor the possible inuence of a less ecientexcitonphonon interaction in onedimensional nanostructures Either localized or mobileexcitons are subject to the interaction with optical phonons and a redshift of the emission isexpected at lower or higher temperatures Therefore additional reasons should be given forsample  where the redshift rate is almost one order of magnitude lower  	 meVKThe situation in sample  can be more similar to that found in Refs  In the threesystems even if dierent we have an apparent common issue the GaInAs alloy Howeverthe key point most than the alloy itself is the highly inhomogeneous strain elds producedby size or compositional uctuations in the alloy Triaxial strain elds are evident in selfassembled nanostructures  wires and dots	 If we take into account the reduced thicknessof the InGaAs layers the density of uctuations and its range of variation and the presenceof a ternary compound with possible compositional deviations the result can be a fullydisordered or amorphous nanostructure The elastic domains are extended anisotropicallyover a length scale slightly larger than the lattice parameter and hence the vibrationalproperties of the InGaAs layer will be seriously aected In covalent compounds the phonondispersion relation of the crystal disappears in the amorphous phase of the solid due tothe large distribution of oscillation frequencies associated with the disorder Furthermoremost of the optical properties of amorphous semiconductors are explained by means ofstrong localization eects which is also the case for sample  As pointed out in Ref the complicated evolution of the triaxial strain elds following the thermal expansionof the distorted lattice is not enough to compensate the natural redshift found in bulkTherefore an additional mechanism of suppression of the excitonphonon interaction due tothe amorphous nature of the GaInAs nanostructures is needed to explain our results andthose in Refs IV CONCLUSIONSIn summary we have investigated several samples containing selfassembledGaxInxAsInP QWRs The size and compositional uctuations have been pointed outto be responsible for the dierences found in the optical properties with increasing x Atemperature variation lower than  nmK is observed for the emission associated to excitons strongly localized in the case of the highest gallium content sampleThis work has been supported by the spanish CICYT under the project no TIC	CC Walther W Hoerstel H Niehus J Erxmeyer and WT Masselink J CrystGrowth    L Gonzalez J M Garca R Garca J MartnezPastor C Ballesteros and FBriones Appl Phys Lett    J Braoult M Gendry O Marty M Pitaval J Olivares G Grenet and GHollinger Appl Surf Sci 	   J Wu Y P Zeng Z Z Sun F Lin B Xu and ZG Zeng J Cryst Growth   DE Wohlert ST Chou AC Chen KY Cheng and KC Hsieh Appl Phys Lett   K Mukai and M Sugawara Appl Phys Lett    JM Garca L Gonzalez MU Gonzalez JP Silveira Y Gonzalez and FBriones J Cryst Growth    	M Notomi J Hammersberg J Zeman H Weman M Potemski H Sugiura and TTamamura Phys Rev Lett    B Alen J MartnezPastor A GarcaCristobal L Gonzalez and J M GarcaAppl Phys Lett    

phys stat sol (a)

Optical Properties of Self-Assembled GaxIn1–xAs/InP Quantum Wires

The radiative coupling between heavy-hole excitons in ZnSe double quantum wells is investigated experimentally and theoretically. The double quantum wells have a distance of half or quarter the wavelength (λ/2, λ/4) of the light resonant to the excitonic transition. The exciton energy splitting is tuned via a quantum well thickness gradient to control the radiative coupling. While in the λ/2 case a superradiant mode is observed, in the λ/4 case an anti-splitting is found. The results are compared with theoretical predictions using transfer matrix calculations. The residual disorder-induced inhomogeneous broadening, which is smaller than the radiative broadening of a single quantum well, significantly reduces the radiative coupling effects

Mann, Ch.

Woggon, U.

Klude, M.

Hommel, D.

Radiative coupling of excitons in ZnSe double quantum wells

We show theoretically that the introduction of a cold electron gas into quantum microcavities induces efficient electron-polariton scattering. This process allows the condensation of the polaritons accumulated at the bottleneck to the final emitting state with a transition time of a few picoseconds, opening the way to a new generation of low-threshold light-emitting devices

Kavokin, A.

Di Carlo, A.

Baumberg, J.J.

Polariton Lasing Due to the Exciton–Electron Scatteringin Semiconductor MicrocavitiesG. Malpuech1) (a), A. Kavokin (b), A. Di Carlo (a), J. J. Baumberg (c),F. Compagnone (a), P. Lugli (a), and M. Zamfirescu (b)(a) INFM – Dept. Elect. Eng., University of Rome ‘‘Tor Vergata“,Via di Tor Vergata 110, I-00133 Roma, Italy(b) LASMEA, CNRS – Universite´ Clermont-II ‘‘Blaise Pascal“, 24 Avenue des Landais,F-63177 Aubiere Cedex, France(c) Department of Physics and Astronomy, University of Southampton, SO17 1BJ, UK(Received September 26, 2001; accepted October 31, 2001)Subject classification: 42.55.Sa; 71.36.þc; 78.47.þp; S7.12The issue of polariton lasing in n-doped microcavities is addressed. We show theoretically that theintroduction of a cold electron gas into quantum microcavities induces efficient electron–polaritonscattering. This process allows the condensation of the polaritons accumulated at the bottleneck tothe final emitting state with a transition time of a few picoseconds, opening the way to a newgeneration of low-threshold light emitting devices.Achievement of polariton lasing at room temperature has become a major challengefor the semiconductor physicists of today. It requires new materials and new physicalprinciples. GaN, showing stable excitons at room temperature, is an excellent candidatefor being a basic material of a new generation of semiconductor lasers [1]. The funda-mental problem to overcome is the bottleneck effect, i.e. kinetic blocking of the polar-iton relaxation down to the ground state that prevents the Bose condensation of polar-itons necessary for polariton lasing. Here we propose a new polariton relaxationmechanism in a microcavity that is based on the scattering of polaritons with free elec-trons. This allows polaritons to efficiently relax from the bottleneck [2–4] region to thelowest energy state once free electrons are introduced into the active region either viadoping or by photo-excitation. This opens the way to realization of low-threshold laser-like devices based on cavity polaritons.The basic principle of a polariton laser is illustrated in Fig. 1a using the dispersioncurve of the lower branch exciton-polaritons in a typical microcavity. The strong cou-pling regime creates a trap containing a small number of polariton states at energiesbelow all other states in the semiconductor. This polariton trap is sharp with a depthequal to nearly half the splitting W between the two polariton modes. Polaritons in thetrap are half photon and half exciton and have properties suitable for the Bose conden-sation of exciton-polaritons once sufficiently populated. Recombination from this statein the Bose condensation regime is coherent and monochromatic, characteristic of laseremission. The relaxation of polaritons into the k ¼ 0 state is found to be stimulated ifthe population of the final state is larger than unity [5].The major problem to be resolved on the way towards realization of polariton laserscomes from a relaxation bottleneck [2, 3] in the dispersion relation. For small in-plane1) Corresponding author; e-mail: malpuech@lasmea.univ-bpclermont.fr# WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0031-8965/02/19003-0181 $ 17.50þ.50/0phys. stat. sol. (a) 190, No. 1, 181–186 (2002)wave vectors, acoustic phonons cannot assist in the polariton relaxation, and polaritonsaccumulate in a broad bottleneck region. Nevertheless, in recent years new experimen-tal results have appeared giving hope for realization of polariton lasers [7, 8]. As wewill show, this strong luminescence originates from the trap states thanks to polariton–polariton scattering processes (Fig. 1a). Under resonant excitation, the high efficiency ofpair scattering processes between polaritons has been shown under pulsed excitation[5, 9], cw excitation [10], and theoretically [11]. These results clearly demonstrate thatcavity polaritons behave as bosons rather than fermions and that they can condense inthe trap if a resonant scattering process is available. However, polariton–polariton scat-tering is only efficient for resonant optical pumping of the cavity. The exciton–excitonscattering is too weak to allow the polariton trap population to exceed unity [3, 4].Although polariton–polariton scattering is promising for realising optical devices suchas parametric oscillators, it is unsuitable for polariton lasers. Here we show that if aconcentration of free electrons is introduced into the structure, polaritons can be effi-ciently scattered from the bottleneck region to the ground state (Fig. 1b). Through thisprocess, a polariton passes its excess energy and wave vector to a free electron, asdescribed byeðqÞ þ PðkÞ ! eðqþ k k0Þ þ Pðk0Þ ; ð1ÞEðk0Þ þ h2ðqþ k k0Þ22me h2q22me¼ EðkÞ ; ð2Þwhere e denotes an electron, E is the polariton energy, and me is the electron effec-tive mass. The polaritons cannot be re-ionized because the excited electrons rapidlythermalise with the electron distribution due to the electron–electron scattering that weexpect to be extremely efficient, providing a ‘thermal lock’ for the trapped polaritons.The advantage of this scattering mechanism with respect to previously discussed ones isthat (i) the matrix element of electron–exciton scattering is quite large [13], and (ii) theelectron has a much lighter mass than an exciton, thus the energy relaxation of a polar-iton from the bottleneck region to the ground state requires fewer scattering events or182 G. Malpuech et al.: Polariton Lasing Due to Exciton–Electron ScatteringFig. 1. a) Dispersion relations ofupper and lower polaritons, show-ing the dominant pair scatteringof polaritons feeding energy intothe trap at k ¼ 0. b) Similar polar-iton–electron scattering process.The excited carriers quickly relax,preventing re-ionizationa lower carrier density than for polariton–polariton or polariton–phonon scattering.Note also that we neglect the trions in the present model. In case of trion formation,collisions with free electrons may cause the resonant dissociation of trions, that mightbe stimulated. This process will be considered in detail in our further publications.We simulate the relaxation of cavity polaritons excited non-resonantly in a steady-state regime. A Boltzmann equation is solved numerically in two-dimensional reciprocalspace,dnkdt¼ Pk  Gknk  nkPk0Wk! k0 ðnk0 þ 1Þ þ ðnk þ 1ÞPk0Wk0 ! knk0 ; ð3Þwhere nk is the polariton distribution function in the state having a wave vector k, Pk isthe generation term describing optical pumping, Gk is the recombination rate composedof both radiative and non-radiative components, and Wk!k0 is the total scattering ratebetween the states k and k0. This scattering rate is comprised of three terms in ourmodel, namely, polariton–acoustic phonon scattering (rate Wa), polariton–polaritonscattering (rate Wx), and polariton–electron scattering (rate We). Only the intra-branchscattering within the lower polariton branch is considered. The cavity photon lifetime tcis taken to be 8 ps, whereas the radiative life time of the QW excitons is taken to be20 ps (typical for InGaAs QWs). The polariton non-radiative lifetime is chosen constantover the whole reciprocal space (1 ns). The polariton–acoustic phonon and the polari-ton–polariton scattering rates are calculated as in [2]. The polariton–electron scatteringrate is calculated using the Fermi golden rule asWek!k0 ¼2phPqM2Nqð1Nqþk0kÞ d Eðk0Þ  EðkÞ  h22meðq2  jqþ k k0j2Þ" #; ð4Þwhere q is the wave vector of an electron required for the resonant scattering and Nq isthe Fermi-Dirac electron distribution function. The electron distribution is not stronglyperturbed by the exciton–electron scattering in the low-pumping regime we consider.Only a small part of electrons is accelerated by scattering with excitons. Thus we assumethe equilibrium Fermi-distribution for the electron gas. Initially, an electron concentra-tion ne ¼ 1010 cm2 and a temperature of 10 K are assumed. The exciton screening isintrinsically taken into account in the calculated polariton dispersion curve EðkÞ: M isthe matrix element of interaction between an electron and a polariton. In the Bornapproximation, M ¼ Mdir Mexc, corresponding to the triplet configuration (þ, parallelelectron spins) and the singlet configuration (, antiparallel electron spins). Further, weshall only include the stronger electron–polariton scattering of the triplet configuration.Mdir ¼ 2pe2Aexkxk0jk k0j ½ð1þ z2hÞ32  ð1þ z2eÞ32 ; ð5ÞMexc ¼ 8pe2Aexkxk0ða2 þ jqbek0j2Þ1=2½ð1þ z2cÞ32  ð1þ 4z2hÞ32 ; ð6Þwhere e is the dielectric susceptibility of the QW, xk is the exciton fraction of the polar-iton state having a wave vector k, ze; h ¼ ð1=2Þbe; hjk k0j a, zc ¼ bekþ jk k0j  qð Þ a,phys. stat. sol. (a) 190, No. 1 (2002) 183be; h ¼ me;h=ðme þmhÞ, where me and mh are electron and hole effective masses, re-spectively, a is the in-plane Bohr radius of the QW exciton, and A is a normalisationarea.Figure 2 shows the equilibrium distribution function under continuous pumping for acavity having a normal-mode splitting of 5 meV containing a single QW, and for zerodetuning of cavity and exciton modes. For all the curves the pump power absorbed bythe single QW is set to be 4.2 W/cm2 between k ¼ 3 106 and 5 106 cm1, roughlyequivalent to an excess of energy of 20 meV. Taking into account only the acousticphonon scattering (curve a in Fig. 2), a thermalised population is seen only beyondk ¼ 2 104 cm1 (the bottleneck region) where the polaritons accumulate. Equilibriumis reached after 10 ns leaving a total polariton population of 2.5 1010 cm2. Includingboth polariton–polariton and polariton–acoustic phonon scattering processes (curve bin Fig. 2) shows partial relaxation of the bottleneck and a flat polariton distribution.However, the equilibrium polariton density in the cavity remains the same, close to thesaturation density for excitons ( 5 1010 cm2). The population of the ground state isclose to unity. This result is in excellent agreement with experimental results obtainedup to now [4], which shows that the threshold amplification for the population of thetrap state is reached when the strong coupling regime is likely to be destroyed.When all three scattering processes listed above are taken into account (curve c inFig. 2), a huge occupation number of the lowest energy state of more than 104 isachieved. This system thus acts as a polariton laser in that the scattering of polaritonsinjected at high k by optical or electrical pumping is stimulated by the population ofthe low k-states. In this situation the light power emitted in a cone of 1 is 3.3 W/cm2and the efficiency of the energy transfer from pump to emitted light is about 80%. Thelight emitted by the cavity is much more directional and comes from a smaller number184 G. Malpuech et al.: Polariton Lasing Due to Exciton–Electron ScatteringFig. 2. Distribution function of polaritons at 10 K when non-resonantly pumped with a power of4.2 W/cm2. Results are shown for (a) polariton–acoustic phonon scattering (dotted), (b) as (a) pluspolariton–polariton scattering (dashed), and (c) as (b) plus polariton–electron scattering (solid).The thin dotted line shows the equilibrium Bose distribution function with zero chemical potentialof states that in the case of curve b. The equilibrium polariton density in the cavity isnow 1:25 109 cm2, i.e. 20 times lower than in the cases of curves a and b. The thindotted line in Fig. 2 shows the equilibrium polariton density from a Bose distributionfunction plotted for zero chemical potential. The good agreement with curve (c) clearlydemonstrates that a thermodynamic equilibrium is achieved for this value of the chemi-cal potential, which is a signature of Bose condensation of polaritons.This is confirmed by tracking the temporal evolution of the population of the groundstate for different excitation powers (Fig. 3a). For the strongest pumping, the populationis strongly enhanced at very short times as soon as it exceeds unity, demonstrating theBose amplification of the final state population. The dashed curve corresponds to thethreshold condition for Bose amplification, producing an equilibrium population of thek ¼ 0 state slightly exceeding unity, but without significant amplification occurring. Foran excitation area of diameter 100 mm this excitation density corresponds to an extraor-dinary low threshold power of 15 mW.Figure 3b shows the temporal evolution of the population of the ground state fordifferent doping. The equilibrium density is strongly enhanced and exceeds the lasingthreshold for electron densities above 1010 cm2.Though the calculations shown here were done for conventional GaAs-based micro-cavities that are presently available, they can be easily extended to future GaN-basedmicrocavities. Once the strong coupling in GaN-cavities is achieved, they will becomemost suitable for studies of the polariton lasing for two basic reasons: first, experimentsat room temperature should become possible, thanks to the stability of excitons in ni-trides, and second, much higher levels of doping will be possible for the same reason.Acknowledgements This work has been supported by the European Commission inthe framework of the Improving of Human Potential contract CLERMONT HPRN-CT-phys. stat. sol. (a) 190, No. 1 (2002) 185Fig. 3. a) Polariton occupation of the k ¼ 0 state versus time for excitation powers of 0.42 W/cm2(dashed), 4.2 W/cm2 (dotted), and 168 W/cm2 (solid). The pump is turned on at t ¼ 0. The corre-sponding total polariton equilibrium densities are 7:0 108 cm2, 1:3 109 cm2, and1:3 1010 cm2, respectively. b) Same as a) for a pump power of 0.42 W/cm2 and electron dopingof 2:5 109 cm2 (dashed), 1010 cm2 (dotted), and 4 1010 cm2 (solid)1999-00132, HEFCE JR98SOBA, FMRX-CT970134, and EPSRC GR/N18598. One ofus (A.D.C.) acknowledges the ONR support.References[1] Y. Yamamoto, Nature 405, 629 (2000).[2] F. Tassone et al., Phys. Rev. B 56, 7554 (1997).F. Tassone et al., Phys. Rev. B 59, 10830 (1999).[3] A.I. Tartakovskii et al., Phys. Rev. B 62, R2283 (2000).[4] P. Sellenart et al., Phys. Rev. B 62, R16263 (2000).[5] P. Savvidis et al., Phys. Rev. Lett. 84, 1547 (2000).[6] T. E. Sole, Vertical Surface Emitting Lasers, Wiley, New York 1995.[7] Le Si Dang et al., Phys. Rev. Lett. 81, 3920 (1998).[8] P. Senellart et al., Phys. Rev. Lett. 82, 1233 (1999).[9] R. Huang et al., Phys. Rev. B 61, R7854 (2000).[10] R.M. Stevenson et al., Phys. Rev. Lett. 85, 3680 (2000).[11] C. Ciuti et al., Phys. Rev. B 62, R4825 (2000).[12] C. Benoit a` la Guillaume et al., Phys. Rev. 177, 567 (1969).H. Haug et al., phys. stat. sol. (b), 82, 531 (1977).S.W. Koch et al., phys. stat. sol. (b) 89, 43 (1978).[13] I. Galbraith et al., J. Cryst. Growth 159, 667 (1996).186 G. Malpuech et al.: Polariton Lasing Due to Exciton–Electron Scattering

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Exciton-electron scattering in semiconductor microcavities: Tool for polariton lasing

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