We present a spatially resolved semiclassical model for the simulation of semiconductor quantum-dot lasers including a multi-species description for the carriers along the optical active region. The model links microscopic determined quantities like scattering rates and dephasing times, that essentially depend via Coulomb interaction on the carrier densities, with macroscopic transport equations and equations for the optical field

Bandelow, Uwe

Knorr, Andreas

Koprucki, Thomas

Wilms, Alexander

English

We present a spatially resolved semiclassical model for the simulation
of semiconductor quantum-dot lasers including a multi-species description for
the carriers along the optical active region. The model links microscopic
determined quantities like scattering rates and dephasing times, that
essentially depend via Coulomb interaction on the carrier densities, with
macroscopic transport equations and equations for the optical field.78A60
68U2078A60 68U2

Repositorium für Naturwissenschaften und Technik

Weierstraß-Institutfür Angewandte Analysis und Stochastikim Forschungsverbund Berlin e. V.Preprint ISSN 0946 – 8633Modeling of quantum dot lasers with microscopictreatment of Coulomb effectsThomas Koprucki1, Alexander Wilms1,Andreas Knorr2 , Uwe Bandelow1submitted: September 28, 20101 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyE-Mail: Thomas.Koprucki@wias-berlin.deE-Mail: Alexander.Wilms@wias-berlin.deE-Mail: Uwe.Bandelow@wias-berlin.de2 Institute for Theoretical Physics, TU Berlin, Hardenbergstr. 36, 10623 Berlin, GermanyE-Mail: Andreas.Knorr@physik.tu-berlin.deNo. 1543Berlin 20102010 Mathematics Subject Classification. 78A60, 68U20.2008 Physics and Astronomy Classification Scheme. 42.55.Px, 85.35.Be, 73.21.-b, 73.21.Fg,73.21.La.Key words and phrases. quantum dots, optoelectronics, transport, coulomb scattering.Edited byWeierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Mohrenstraße 3910117 BerlinGermanyFax: +49 30 2044975E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/ABSTRACT. We present a spatially resolved semiclassical model for the simulation ofsemiconductor quantum-dot lasers including a multi-species description for the carriersalong the optical active region. The model links microscopic determined quantities likescattering rates and dephasing times, that essentially depend via Coulomb interaction onthe carrier densities, with macroscopic transport equations and equations for the opticalfield.1. INTRODUCTIONDue to many advantages, such as low threshold currents and improved robustness ofoperation, semiconductor quantum dot (QD) lasers have become increasingly interestingfor telecom applications. The need for a detailed understanding of the physics involvedand for optimization of such devices makes them subject for theoretical investigations,comprising modeling and numerical simulation. A suitable model used for the simulationof such devices constitutes a complex multiscale problem and has to cover many importantphysical effects.On the coarsest scale, classical carrier transport through the bulk part of the devicecan be described by a macroscopic model, which are the drift-diffusion equations in ourcase. Additionally, a semiclassical description of the optical field by Maxwell equationsis required. Usually, by selfconsistently coupling of both the electronic and the opticalmodel via the optical gain on one hand, and via the rate of spontaneous and stimulatedrecombination on the other hand, a self-consistent picture for the spatial distributions ofthe carriers and the field intensity (Bandelow et al, 2003) is achieved.However, due to the quantum confinement of carriers on the scale of a few nanometersin QD active regions, c.f. Fig. 1, processes like optical gain, scattering and recombinationof carriers have to be described by a microscopic model. The carrier-carrier scattering viaCoulomb interaction, that dominates at high carrier densities (Nielsen et al, 2004; Nilssonet al, 2005; Lorke et al, 2006) and which we will exclusively take into account within thispaper, causes nonradiative, Auger-like transitions between QDs and carrier reservoirs suchas bulk and wetting layer (WL) states. Thereby, the dynamics of the laser can be stronglyinfluenced by density dependent scattering rates (Viktorov et al, 2006). Also the opticaltransitions between the QD electron and hole states, that result in the optical gain, will bemodified by the Coulomb interaction. This is reflected by a density dependend dephasingtime (Lorke et al, 2006; Dachner et al, 2010), which modifies the optical gain. The activeregion of our considered device comprises optically active QDs grown on a 2D WL asshown in Fig 1a. We will restrict to two-level QDs possessing only a single electron andonly a single hole state, see Fig. 1c.The paper is organized as follows: We present in Section 2 a spatially resolved multi-species electronic model for QD lasers. In Section 3 we address the microscopic calcula-tion of the Coulomb-assisted carrier-carrier scattering and the dephasing processes occur-ring in the model. We demonstrate this approach by calculations of scattering rates anddephasing times for a two-level QD system, including the calculation of the optical gain.2. MULTI SPECIES MODELIn our model we differ between bulk carriers, reservoir carriers, here taken to be con-fined in a two-dimensional WL, and carriers that are localized in the quantum dots asschematically shown in Figs. 1a and 1c. The bulk carriers are freely roaming through the1jnjpn-contactp-contactxywe(x), wh(x)ne(x), nh(x)wetting layer (WL)quantumdotsn(x, y)p(x, y)(a) (b)bulkbulkwettinglayerquantumdotsSine SouteSouthSinhnenhwhwenpCcapnRsp RstimNsCcappwettinglayer(c)FIGURE 1. (a) Scheme of transverse cross-section of an edge-emittingQD laser with a QD-WL active region that contains lens-shaped QDs(b). (c) Scheme of scattering and recombination processes in the quan-tum dot active region described by the multi-species model. Carriers areinjected from the bulk into the wetting layers and enter the quantum dotsby Coulomb scattering. There they contribute to the stimulated recom-bination.layers in devices as depicted in Fig. 1c and are described by classical drift-diffusion equa-tions. The two-dimensional transport of the carriers confined in the WL is described byin-plane drift-diffusion equations, whereas the occupation of the QD-states is governed byrate equations for localized states. All these equations are coupled to each other by corre-sponding carrier-exchange rates and by the electrostatic potential via the Poisson equation.The injection of the bulk carriers into the wetting layers, as depicted in Fig. 1, is describedby phenomenological capture-escape models (Grupen and Hess, 1998; Steiger et al, 2008).For the capture-escape process from the wetting layer into the quantum dots we use scat-tering rates derived from a microscopic model for the dominating Auger-like processes athigh carrier densities, which have their physical origin in Coulomb-scattering. The con-ceptual approach to describe the carriers in low-dimensional nanostructures by a partitionof the carrier density into different species has been proposed by Steiger et al (2008) forthe simulation quantum-well and quantum-wire based optoelectronic nanostructures.For edge-emitters we can restrict our considerations to a transverse cross section Ω⊂R2of the device, as schematically shown in Fig.1a. In order to simplify the presentation ofour model we consider in the following the case of QD-lasers containing only a single QDoptical active layer. We denote the Nabla operator with respect to rt = (x,y)∈Ω by ∇, andits in-plane part with respect to the in-plane coordinate r‖ = (x,0) along the QD layer by∇‖. The bulk carrier densities of the electrons and holes are denoted by n and p and aredefined on the whole cross-section Ω of the device, see Fig. 1a. The carrier densities of theelectrons and holes confined in the WL and in the QDs are denoted by we, wh and ne, nh,respectively. The WL densities we and wh and QD occupation densities ne and nh are sheetdensities depending only on the in-plane coordinate r‖, see Fig. 1a. The evolution of the2carrier densities is governed by the following system of equations:−∇ · (ε0εs∇ϕ) = q(C−ntot + ptot),(1)∂n∂ t− 1q∇ · jn =−R−C3,capn ,(2)∂ p∂ t+1q∇ · jp =−R−C3,capp ,(3)∂we∂ t− 1q∇‖ · jwe =−R̃nr +C2,capn −[Sine (NQD−ne)−Soute ne)] NtotNQD−RWLsp ,(4)∂wh∂ t+1q∇‖ · jwh =−R̃nr +C2,capp −[Sinh (NQD−nh)−South nh)] NtotNQD−RWLsp ,(5)∂ne∂ t= Sine (NQD−ne)−Soute ne−Rstim−Rsp,(6)∂nh∂ t= Sinh (NQD−nh)−South nh−Rstim−Rsp.(7)The net charge density, given by the sum over the the local distributions of the respectiveelectron and hole species, together with the doping profile C, enters the Poisson equa-tion (1) for the electrostatic potential ϕ , where q is the elementrary charge and εs is thestatic permittivity. The bulk electron and hole current densities jn and jp, as well as thosefor the WL, jwe and jwh, are defined in the usual way by gradients of corresponding quasiFermi-potentials, i.e. jn =−qµnn∇ϕn and jwe =−qµwewe∇‖ϕwe, c.f. (Steiger et al, 2008).C3,cap describes the injection from the bulk into the WL by phenomenological capture-escape rates according to Grupen and Hess (1998). C2,cap are the corresponding charge-conserving counterparts in the WL equations. Sine/h and Soute/h are scattering rates describingthe capture and escape processes from the WL into the QDs. These scattering rates dependnonlinearly on we and wh, see Section 3 and Fig. 2a. Ntot is twice the total density ofQDs grown on the WL and NQD is twice the sheet density of resonant QDs. The rates ofspontaneous recombination in the WL and QDs are denoted by RWLsp and Rsp, respectively.The rate of stimulated recombination in the QDs is denoted by Rstim and defined by(8) Rstim =vgLg(ω)|Ξ0|2Ns,where L is the length of the laser, vg is the group velocity and Ξ0(rt) refers to the nor-malized transverse main mode profile. In the case of stable transverse waveguiding andlongitudinal single-mode operation we work with the balance equation(9) Ṅs = vg(2ℑmβ −α0)Ns + rsp.for the numbers of photons Ns, where ℑm[β ] is the modal gain given by the imaginarypart of the corresponding eigenvalue β of the transverse waveguiding problem, α0 areall longitudinal losses by scattering into other modes (radiation modes, e.g.) includingoutcoupling losses. The spontaneous emission rate into the mode is denoted by rsp. Forfurther details we refer to (Bandelow et al, 2003).The material gain caused by the interband polarization from the QDs has the form:g(ω) =− ωn(ω)c|dvc|2L0(NQD−nh−ne)∫Γ2(ε,we,wh)(ε− h̄ω)2 +Γ2(ε,we,wh)2G(ε) dε,(10)where Γ2 = h̄/T2 (11) is the homogeneous broadening of the gain defined by the dephasingtime T2, and G(ε) is the (Gaussian-shaped) distribution of the QD transition energy, whichcauses inhomogeneous broadening. In Eq. (10) n(ω) is the refractive index, c is the speedof light, dvc the dipole matrix element of the QD transition, L0 is a normalization length.If one neglects bulk/WL carrier transport and assumes spatially homogenous WL andQD densities, Eq.s (1)–(9) reduce to a set of rate equations. Such type of rate-equation3models have been applied for the analysis of the electron and hole dynamics of edge-emitting QD lasers (Malic et al, 2006; Lüdge et al, 2008; Lüdge and Schöll, 2009).3. COULOMB SCATTERING BETWEEN QUANTUM DOT AND WETTING LAYER STATESThe calculation of the scattering rates Sin and Sout as found in Eqs. (4)-(7), as well asof the dephasing time entering Eq. (10) of the resonant QD transition, has been treated bydensity matrix theory in the limit of the second order Born approximation. The descriptionresults in a nonlinear dependence of these scattering rates and of the dephasing time on theWL carrier density (Nielsen et al, 2004; Nilsson et al, 2005; Lorke et al, 2006; Malic et al,2007). For precise calculations several scattering channels between the QD and WL haveto be included, as in particular pure dephasing processes for the calculation of T2 (Lorkeet al, 2006). The dephasing time determines the homogeneous line width of the single QDgain, additionally broadened by an inhomogeneous distribution of QDs, see Eq. (10).To describe Sin/out we use a Hamiltonian according to Nielsen et al (2004) containingfree carrier energies of both quantum dot and wetting layer states and carrier-carrier inter-action via the Coulomb matrix element Vabcd =∫d3r∫d3r′φ ∗a (r)φ ∗b (r)V (r−r′)φc(r′)φd(r).The latter is calculated with QD and WL-OPW-wavefunctions as in Nielsen et al (2004),where a, b, c and d are compound indices denoting all possible QD and WL states. Thisincludes the highest WL valence subband, the lowest WL conduction subband and boththe QD valence and conduction ground states, all doubly degenerate due to spin. In thefollowing we will use the indices i and j to denote states restricted to the QDs.To derive the scattering contribution, we consider the temporal evolution of densitymatrices ρab, since they determine all dynamical quantities in Eqs. (4)-(7). For consistencywith our semiclassical description of the WL used in the multi-species model we assumequasi-equilibrium for the WL states by using Fermi-distributions in the following. Theparameters used for the numerical calculations of scattering rates and dephasing times arenear to Lorke (2008) for shallow QDs. Additionally, we used the approximation that thewidth of the WL equals to the height of the QDs and to L0 in Eq. (10), which applicable tothe considered lens-shaped QDs embedded in the WL.The diagonal elements ρa := ρaa of the density matrix are the occupation probabilitiesof the states a. Our EOM for the occupation probabilities of QD states has the same form asin Malic et al (2006). The determination of Sin and of Sout are next to Nielsen et al (2004),but we included screening in the static limit of the Lindhard-formula (Haug and Koch,2004). The resulting scattering rates are depicted in Fig. 2a (for room temperature) andshow a nonlinear dependence on the WL carrier density. The dephasing time T2 = h̄/Γ2 ofthe QD polarization ρi j, i 6= j, has in the markovian limit the formΓ2 =π ∑a,b,c{Wabic(Wicab−Wicba)[ρc(1−ρb)(1−ρa)+(1−ρc)ρbρa]δ (εa + εb− εc− εi)+Wjabc(Wbc ja−Wcb ja) [ρbρc(1−ρa)+(1−ρb)(1−ρc)ρa]δ (ε j + εa− εb− εc)−2(WaiicWjca j−WaiicWjc ja−WiaicWjca j)[(1−ρc)ρa]δ (εa− εc)−(Wabi jWjiba)[ρ j(1−ρb)(1−ρa)+(1−ρ j)ρbρa]δ (εa + εb− ε j− εi)−(WjibcWcbi j)[ρbρc(1−ρi)+(1−ρb)(1−ρc)ρi]δ (ε j + εi− εb− εc)}.(11)Here Wabcd denotes the screened Coulomb matrix elements. In previous publications (Kimet al, 2010; Dachner et al, 2010) an approximation of this expression by contributionspossessing a similar formal structure as the in- and out-scattering terms (first two lines)has been used, which leads to smaller dephasing times. For the numerical evaluation ofΓ2 in this paper all contributions in (11) have been considered. Results for the calcuationof the dephasing time are depicted in Fig. 2b. The gain spectra for our example QD-WL4 0 0.2 0.4 0.6 0.8 1 0  5  10  15  20  25  30Dephasing time T2 [ps]WL carrier density [1011cm-2] 0 1 2 3 4 5 6 7 0  5  10  15  20  25  30Scattering Rate in [ps-1]WL carrier density in [1011cm-2](a) (b) 0 50 100 150 200 250 300 350 400 450 500 0.85  0.9  0.95  1  1.05Material Gain [1/cm]Energy of the test beam [eV]4102030(c)10x10xSinSoutelectronsholesFIGURE 2. Calculated (a) Coulomb scattering rates for electrons (solid)and holes (dotted), (b) dephasing time T2, in dependence on the WL car-rier density. (c) gain spectra for WL densities 4,10,20,30 · 1011cm−2.For the calculation of the scattering rates Auger-like processes accordingto Malic et al (2007) have been considered, together with an inhomoge-noues broadening FWHM of 60 meV. we = wh always assumed.structure are depicted in Fig. 2c. As a result of the Coulomb interaction, they exhibit asaturation behaviour with increasing carrier density.4. CONCLUSIONA semiclassical, spatially resolved, multi-species model for the simulation of semicon-ductor QD lasers has been presented. The electronic model comprises the carriers in thebulk, in the wetting layer, and in the QDs, governed by suited transport equations. Thecoupling between the equations for the wetting layer densities and the equations for theQD densities are determined in particular by microscopically calculated scattering ratesbetween these species. This electronic model is selfconsistently coupled to equations forthe optical field, where in addition a microscopically determined dephasing time enters viathe optical gain. In effect these microscopically calculated quantities become dependent onthe carrier densities in a nonlinear way. This nonlinear behavior is expected to influence thedynamic properties of such devices, as e.g. their modulation response (Lüdge et al, 2008).The presented model goes beyond the existing approaches with respect to the followingtwo aspects: First, in pure rate-equation models the influence of a spatial inhomogenouscurrent injection, of the electrostatic interaction via the Poisson equation, and of the spatialinhomogenous stimulated recombination is neglected. Second, compared to other exist-ing models with spatial resolution, we cover the impact of important QD-WL Coulombinteraction on the scattering rates and on the optical gain by microscopic calculations.ACKNOWLEDGEMENTSThe work of T. Koprucki and A. Wilms has been supported by the Deutsche Forschungs-gemeinschaft (DFG) within the collaborative research center 787 ”Semiconductor Nanopho-tonics”. We acknowledge fruitful discussions with E. Malic, J.-E. Kim, M. Lorke, K.Gärtner, J. Griepentrog and P. Racec.REFERENCESBandelow U, Hünlich R, Koprucki T (2003) Simulation of Static and Dynamic Propertiesof Edge-Emitting Multi Quantum Well Lasers. IEEE J Sel Topics Quantum Electron9(3):798–8065Dachner MR, Malic E, Richter M, Carmele A, Kabuss J, Wilms A, Kim JE, HartmannG, Wolters J, Bandelow U, Knorr A (2010) Theory of carrier and photon dynamics inquantum dot light emitters. Phys Status Solidi B 247(4):809–828Grupen M, Hess K (1998) Simulation of carrier transport and nonlinearities in quantum-well laser diodes. IEEE J Quantum Electron 34(1):120–140Haug H, Koch SW (2004) Quantum Theory of the Optical and Electronic Properties ofSemiconductors. World Scientific, SingaporeKim JE, Malic E, Richter M, Wilms A, Knorr A (2010) Maxwell-bloch equation approachfor describing the microscopic dynamics of QD surface-emitting structures. IEEE JQuantum Electron 46(7):1115–1126Lorke M (2008) Optical gain and laser properties of semiconductor QD systems. PhDthesis, Uni BremenLorke M, Nielsen T, Seebeck J, Gartner P, Jahnke F (2006) Influence of carrier-carrier andcarrier-phonon correlations on optical absorption and gain in quantum-dot systems. PhysRev B 73(8):085,324Lüdge K, Schöll E (2009) Quantum-dot lasers - desynchronized nonlinear dynamics ofelectrons and holes. IEEE J Quantum Electron 45(11):1396–1403Lüdge K, Bormann MJP, Malić E, Hövel P, Kuntz M, Bimberg D, Knorr A, Schöll E (2008)Turn-on dynamics and modulation response in semiconductor quantum dot lasers. PhysRev B 78(3):035,316Malic E, Ahn KJ, Bormann MJP, Hovel P, Scholl E, Knorr A, Kuntz M, Bimberg D (2006)Theory of relaxation oscillations in semiconductor quantum dot lasers. Appl Phys Lett89(10):101,107Malic E, Bormann MJ, Hövel P, Kuntz M, Bimberg D, Knorr A, Schöll E (2007) Coulombdamped relaxation oscillations in semiconductor quantum dot lasers. IEEE J Sel TopicsQuantum Electron 13(5):1242–1248Nielsen T, Gartner P, Jahnke F (2004) Many-body theory of carrier capture and relaxationin semiconductor quantum-dot lasers. Phys Rev B 69(23):235,314Nilsson HH, Zhang JZ, Galbraith I (2005) Homogeneous broadening in quantum dots dueto auger scattering with wetting layer carriers. Phys Rev B 72(20):205,331Steiger S, Veprek RG, Witzigmann B (2008) Unified simulation of transport and lumines-cence in optoelectronic nanostructure. Journal of Computational Electronics 7(4):509–520Viktorov E, Mandel P, Vladimirov A, Bandelow U (2006) Model for mode-locking inquantum dot lasers. Appl Phys Lett 88(20):201,1026

Modeling of quantum dot lasers with microscopic treatment of Coulomb effects

We present a spatially resolved semiclassical model for the simulation of semiconductor quantum-dot lasers including a multi-species description for the carriers along the optical active region. The model links microscopically determined quantities like scattering rates and dephasing times, that essentially depend via Coulomb interaction on the carrier densities, with macroscopic transport equations and equations for the optical field

Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics

Weierstraß-Institutfür Angewandte Analysis und Stochastikim Forschungsverbund Berlin e. V.Preprint ISSN 0946 – 8633Modeling of quantum dot lasers with microscopictreatment of Coulomb effectsThomas Koprucki1, Alexander Wilms1,Andreas Knorr2 , Uwe Bandelow1submitted: September 28, 20101 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyE-Mail: Thomas.Koprucki@wias-berlin.deE-Mail: Alexander.Wilms@wias-berlin.deE-Mail: Uwe.Bandelow@wias-berlin.de2 Institute for Theoretical Physics, TU Berlin, Hardenbergstr. 36, 10623 Berlin, GermanyE-Mail: Andreas.Knorr@physik.tu-berlin.deNo. 1543Berlin 20102010 Mathematics Subject Classification. 78A60, 68U20.2008 Physics and Astronomy Classification Scheme. 42.55.Px, 85.35.Be, 73.21.-b, 73.21.Fg,73.21.La.Key words and phrases. quantum dots, optoelectronics, transport, coulomb scattering.Edited byWeierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Mohrenstraße 3910117 BerlinGermanyFax: +49 30 2044975E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/ABSTRACT. We present a spatially resolved semiclassical model for the simulation ofsemiconductor quantum-dot lasers including a multi-species description for the carriersalong the optical active region. The model links microscopic determined quantities likescattering rates and dephasing times, that essentially depend via Coulomb interaction onthe carrier densities, with macroscopic transport equations and equations for the opticalfield.1. INTRODUCTIONDue to many advantages, such as low threshold currents and improved robustness ofoperation, semiconductor quantum dot (QD) lasers have become increasingly interestingfor telecom applications. The need for a detailed understanding of the physics involvedand for optimization of such devices makes them subject for theoretical investigations,comprising modeling and numerical simulation. A suitable model used for the simulationof such devices constitutes a complex multiscale problem and has to cover many importantphysical effects.On the coarsest scale, classical carrier transport through the bulk part of the devicecan be described by a macroscopic model, which are the drift-diffusion equations in ourcase. Additionally, a semiclassical description of the optical field by Maxwell equationsis required. Usually, by selfconsistently coupling of both the electronic and the opticalmodel via the optical gain on one hand, and via the rate of spontaneous and stimulatedrecombination on the other hand, a self-consistent picture for the spatial distributions ofthe carriers and the field intensity (Bandelow et al, 2003) is achieved.However, due to the quantum confinement of carriers on the scale of a few nanometersin QD active regions, c.f. Fig. 1, processes like optical gain, scattering and recombinationof carriers have to be described by a microscopic model. The carrier-carrier scattering viaCoulomb interaction, that dominates at high carrier densities (Nielsen et al, 2004; Nilssonet al, 2005; Lorke et al, 2006) and which we will exclusively take into account within thispaper, causes nonradiative, Auger-like transitions between QDs and carrier reservoirs suchas bulk and wetting layer (WL) states. Thereby, the dynamics of the laser can be stronglyinfluenced by density dependent scattering rates (Viktorov et al, 2006). Also the opticaltransitions between the QD electron and hole states, that result in the optical gain, will bemodified by the Coulomb interaction. This is reflected by a density dependend dephasingtime (Lorke et al, 2006; Dachner et al, 2010), which modifies the optical gain. The activeregion of our considered device comprises optically active QDs grown on a 2D WL asshown in Fig 1a. We will restrict to two-level QDs possessing only a single electron andonly a single hole state, see Fig. 1c.The paper is organized as follows: We present in Section 2 a spatially resolved multi-species electronic model for QD lasers. In Section 3 we address the microscopic calcula-tion of the Coulomb-assisted carrier-carrier scattering and the dephasing processes occur-ring in the model. We demonstrate this approach by calculations of scattering rates anddephasing times for a two-level QD system, including the calculation of the optical gain.2. MULTI SPECIES MODELIn our model we differ between bulk carriers, reservoir carriers, here taken to be con-fined in a two-dimensional WL, and carriers that are localized in the quantum dots asschematically shown in Figs. 1a and 1c. The bulk carriers are freely roaming through the1jnjpn-contactp-contactxywe(x),wh(x)ne(x), nh(x)wetting layer (WL)quantumdotsn(x, y)p(x, y)(a) (b)bulkbulkwettinglayerquantumdotsSine SouteSouthSinhnenhwhwenpCcapnRsp RstimNsCcappwettinglayer(c)FIGURE 1. (a) Scheme of transverse cross-section of an edge-emittingQD laser with a QD-WL active region that contains lens-shaped QDs(b). (c) Scheme of scattering and recombination processes in the quan-tum dot active region described by the multi-species model. Carriers areinjected from the bulk into the wetting layers and enter the quantum dotsby Coulomb scattering. There they contribute to the stimulated recom-bination.layers in devices as depicted in Fig. 1c and are described by classical drift-diffusion equa-tions. The two-dimensional transport of the carriers confined in the WL is described byin-plane drift-diffusion equations, whereas the occupation of the QD-states is governed byrate equations for localized states. All these equations are coupled to each other by corre-sponding carrier-exchange rates and by the electrostatic potential via the Poisson equation.The injection of the bulk carriers into the wetting layers, as depicted in Fig. 1, is describedby phenomenological capture-escape models (Grupen and Hess, 1998; Steiger et al, 2008).For the capture-escape process from the wetting layer into the quantum dots we use scat-tering rates derived from a microscopic model for the dominating Auger-like processes athigh carrier densities, which have their physical origin in Coulomb-scattering. The con-ceptual approach to describe the carriers in low-dimensional nanostructures by a partitionof the carrier density into different species has been proposed by Steiger et al (2008) forthe simulation quantum-well and quantum-wire based optoelectronic nanostructures.For edge-emitters we can restrict our considerations to a transverse cross sectionΩ⊂R2of the device, as schematically shown in Fig.1a. In order to simplify the presentation ofour model we consider in the following the case of QD-lasers containing only a single QDoptical active layer. We denote the Nabla operator with respect to rt = (x,y)∈Ω by ∇, andits in-plane part with respect to the in-plane coordinate r‖ = (x,0) along the QD layer by∇‖. The bulk carrier densities of the electrons and holes are denoted by n and p and aredefined on the whole cross-section Ω of the device, see Fig. 1a. The carrier densities of theelectrons and holes confined in the WL and in the QDs are denoted by we, wh and ne, nh,respectively. The WL densities we and wh and QD occupation densities ne and nh are sheetdensities depending only on the in-plane coordinate r‖, see Fig. 1a. The evolution of the2carrier densities is governed by the following system of equations:−∇ · (ε0εs∇ϕ) = q(C−ntot + ptot),(1)∂n∂ t− 1q∇ · jn =−R−C3,capn ,(2)∂ p∂ t+1q∇ · jp =−R−C3,capp ,(3)∂we∂ t− 1q∇‖ · jwe =−R˜nr+C2,capn −[Sine (NQD−ne)−Soute ne)] NtotNQD−RWLsp ,(4)∂wh∂ t+1q∇‖ · jwh =−R˜nr+C2,capp −[Sinh (NQD−nh)−South nh)] NtotNQD−RWLsp ,(5)∂ne∂ t= Sine (NQD−ne)−Soute ne−Rstim−Rsp,(6)∂nh∂ t= Sinh (NQD−nh)−South nh−Rstim−Rsp.(7)The net charge density, given by the sum over the the local distributions of the respectiveelectron and hole species, together with the doping profile C, enters the Poisson equa-tion (1) for the electrostatic potential ϕ , where q is the elementrary charge and εs is thestatic permittivity. The bulk electron and hole current densities jn and jp, as well as thosefor the WL, jwe and jwh, are defined in the usual way by gradients of corresponding quasiFermi-potentials, i.e. jn =−qµnn∇ϕn and jwe =−qµwewe∇‖ϕwe, c.f. (Steiger et al, 2008).C3,cap describes the injection from the bulk into the WL by phenomenological capture-escape rates according to Grupen and Hess (1998). C2,cap are the corresponding charge-conserving counterparts in the WL equations. Sine/h and Soute/h are scattering rates describingthe capture and escape processes from the WL into the QDs. These scattering rates dependnonlinearly on we and wh, see Section 3 and Fig. 2a. Ntot is twice the total density ofQDs grown on the WL and NQD is twice the sheet density of resonant QDs. The rates ofspontaneous recombination in the WL and QDs are denoted by RWLsp and Rsp, respectively.The rate of stimulated recombination in the QDs is denoted by Rstim and defined by(8) Rstim =vgLg(ω)|Ξ0|2Ns,where L is the length of the laser, vg is the group velocity and Ξ0(rt) refers to the nor-malized transverse main mode profile. In the case of stable transverse waveguiding andlongitudinal single-mode operation we work with the balance equation(9) N˙s = vg(2ℑmβ −α0)Ns+ rsp.for the numbers of photons Ns, where ℑm[β ] is the modal gain given by the imaginarypart of the corresponding eigenvalue β of the transverse waveguiding problem, α0 areall longitudinal losses by scattering into other modes (radiation modes, e.g.) includingoutcoupling losses. The spontaneous emission rate into the mode is denoted by rsp. Forfurther details we refer to (Bandelow et al, 2003).The material gain caused by the interband polarization from the QDs has the form:g(ω) =− ωn(ω)c|dvc|2L0(NQD−nh−ne)∫ Γ2(ε,we,wh)(ε− h¯ω)2+Γ2(ε,we,wh)2 G(ε) dε,(10)where Γ2 = h¯/T2 (11) is the homogeneous broadening of the gain defined by the dephasingtime T2, and G(ε) is the (Gaussian-shaped) distribution of the QD transition energy, whichcauses inhomogeneous broadening. In Eq. (10) n(ω) is the refractive index, c is the speedof light, dvc the dipole matrix element of the QD transition, L0 is a normalization length.If one neglects bulk/WL carrier transport and assumes spatially homogenous WL andQD densities, Eq.s (1)–(9) reduce to a set of rate equations. Such type of rate-equation3models have been applied for the analysis of the electron and hole dynamics of edge-emitting QD lasers (Malic et al, 2006; Lu¨dge et al, 2008; Lu¨dge and Scho¨ll, 2009).3. COULOMB SCATTERING BETWEEN QUANTUM DOT AND WETTING LAYER STATESThe calculation of the scattering rates Sin and Sout as found in Eqs. (4)-(7), as well asof the dephasing time entering Eq. (10) of the resonant QD transition, has been treated bydensity matrix theory in the limit of the second order Born approximation. The descriptionresults in a nonlinear dependence of these scattering rates and of the dephasing time on theWL carrier density (Nielsen et al, 2004; Nilsson et al, 2005; Lorke et al, 2006; Malic et al,2007). For precise calculations several scattering channels between the QD and WL haveto be included, as in particular pure dephasing processes for the calculation of T2 (Lorkeet al, 2006). The dephasing time determines the homogeneous line width of the single QDgain, additionally broadened by an inhomogeneous distribution of QDs, see Eq. (10).To describe Sin/out we use a Hamiltonian according to Nielsen et al (2004) containingfree carrier energies of both quantum dot and wetting layer states and carrier-carrier inter-action via the Coulomb matrix elementVabcd =∫d3r∫d3r′φ ∗a (r)φ ∗b (r)V (r−r′)φc(r′)φd(r).The latter is calculated with QD and WL-OPW-wavefunctions as in Nielsen et al (2004),where a, b, c and d are compound indices denoting all possible QD and WL states. Thisincludes the highest WL valence subband, the lowest WL conduction subband and boththe QD valence and conduction ground states, all doubly degenerate due to spin. In thefollowing we will use the indices i and j to denote states restricted to the QDs.To derive the scattering contribution, we consider the temporal evolution of densitymatrices ρab, since they determine all dynamical quantities in Eqs. (4)-(7). For consistencywith our semiclassical description of the WL used in the multi-species model we assumequasi-equilibrium for the WL states by using Fermi-distributions in the following. Theparameters used for the numerical calculations of scattering rates and dephasing times arenear to Lorke (2008) for shallow QDs. Additionally, we used the approximation that thewidth of the WL equals to the height of the QDs and to L0 in Eq. (10), which applicable tothe considered lens-shaped QDs embedded in the WL.The diagonal elements ρa := ρaa of the density matrix are the occupation probabilitiesof the states a. Our EOM for the occupation probabilities of QD states has the same form asin Malic et al (2006). The determination of Sin and of Sout are next to Nielsen et al (2004),but we included screening in the static limit of the Lindhard-formula (Haug and Koch,2004). The resulting scattering rates are depicted in Fig. 2a (for room temperature) andshow a nonlinear dependence on the WL carrier density. The dephasing time T2 = h¯/Γ2 ofthe QD polarization ρi j, i 6= j, has in the markovian limit the formΓ2 =pi ∑a,b,c{Wabic(Wicab−Wicba)[ρc(1−ρb)(1−ρa)+(1−ρc)ρbρa]δ (εa+ εb− εc− εi)+Wjabc(Wbc ja−Wcb ja) [ρbρc(1−ρa)+(1−ρb)(1−ρc)ρa]δ (ε j+ εa− εb− εc)−2(WaiicWjca j−WaiicWjc ja−WiaicWjca j)[(1−ρc)ρa]δ (εa− εc)−(Wabi jWjiba)[ρ j(1−ρb)(1−ρa)+(1−ρ j)ρbρa]δ (εa+ εb− ε j− εi)−(WjibcWcbi j)[ρbρc(1−ρi)+(1−ρb)(1−ρc)ρi]δ (ε j+ εi− εb− εc)}.(11)HereWabcd denotes the screened Coulomb matrix elements. In previous publications (Kimet al, 2010; Dachner et al, 2010) an approximation of this expression by contributionspossessing a similar formal structure as the in- and out-scattering terms (first two lines)has been used, which leads to smaller dephasing times. For the numerical evaluation ofΓ2 in this paper all contributions in (11) have been considered. Results for the calcuationof the dephasing time are depicted in Fig. 2b. The gain spectra for our example QD-WL4 0 0.2 0.4 0.6 0.8 1 0  5  10  15  20  25  30Dephasing time T 2 [ps]WL carrier density [1011cm-2] 0 1 2 3 4 5 6 7 0  5  10  15  20  25  30Scattering Rate in [ps-1]WL carrier density in [1011cm-2](a) (b) 0 50 100 150 200 250 300 350 400 450 500 0.85  0.9  0.95  1  1.05Material Gain [1/cm]Energy of the test beam [eV]4102030(c)10x10xSinSoutelectronsholesFIGURE 2. Calculated (a) Coulomb scattering rates for electrons (solid)and holes (dotted), (b) dephasing time T2, in dependence on the WL car-rier density. (c) gain spectra for WL densities 4,10,20,30 · 1011cm−2.For the calculation of the scattering rates Auger-like processes accordingto Malic et al (2007) have been considered, together with an inhomoge-noues broadening FWHM of 60 meV. we = wh always assumed.structure are depicted in Fig. 2c. As a result of the Coulomb interaction, they exhibit asaturation behaviour with increasing carrier density.4. CONCLUSIONA semiclassical, spatially resolved, multi-species model for the simulation of semicon-ductor QD lasers has been presented. The electronic model comprises the carriers in thebulk, in the wetting layer, and in the QDs, governed by suited transport equations. Thecoupling between the equations for the wetting layer densities and the equations for theQD densities are determined in particular by microscopically calculated scattering ratesbetween these species. This electronic model is selfconsistently coupled to equations forthe optical field, where in addition a microscopically determined dephasing time enters viathe optical gain. In effect these microscopically calculated quantities become dependent onthe carrier densities in a nonlinear way. This nonlinear behavior is expected to influence thedynamic properties of such devices, as e.g. their modulation response (Lu¨dge et al, 2008).The presented model goes beyond the existing approaches with respect to the followingtwo aspects: First, in pure rate-equation models the influence of a spatial inhomogenouscurrent injection, of the electrostatic interaction via the Poisson equation, and of the spatialinhomogenous stimulated recombination is neglected. Second, compared to other exist-ing models with spatial resolution, we cover the impact of important QD-WL Coulombinteraction on the scattering rates and on the optical gain by microscopic calculations.ACKNOWLEDGEMENTSThe work of T. Koprucki and A. Wilms has been supported by the Deutsche Forschungs-gemeinschaft (DFG) within the collaborative research center 787 ”Semiconductor Nanopho-tonics”. We acknowledge fruitful discussions with E. Malic, J.-E. Kim, M. Lorke, K.Ga¨rtner, J. Griepentrog and P. Racec.REFERENCESBandelow U, Hu¨nlich R, Koprucki T (2003) Simulation of Static and Dynamic Propertiesof Edge-Emitting Multi Quantum Well Lasers. IEEE J Sel Topics Quantum Electron9(3):798–8065Dachner MR, Malic E, Richter M, Carmele A, Kabuss J, Wilms A, Kim JE, HartmannG, Wolters J, Bandelow U, Knorr A (2010) Theory of carrier and photon dynamics inquantum dot light emitters. Phys Status Solidi B 247(4):809–828Grupen M, Hess K (1998) Simulation of carrier transport and nonlinearities in quantum-well laser diodes. IEEE J Quantum Electron 34(1):120–140Haug H, Koch SW (2004) Quantum Theory of the Optical and Electronic Properties ofSemiconductors. World Scientific, SingaporeKim JE, Malic E, Richter M, Wilms A, Knorr A (2010) Maxwell-bloch equation approachfor describing the microscopic dynamics of QD surface-emitting structures. IEEE JQuantum Electron 46(7):1115–1126Lorke M (2008) Optical gain and laser properties of semiconductor QD systems. PhDthesis, Uni BremenLorke M, Nielsen T, Seebeck J, Gartner P, Jahnke F (2006) Influence of carrier-carrier andcarrier-phonon correlations on optical absorption and gain in quantum-dot systems. PhysRev B 73(8):085,324Lu¨dge K, Scho¨ll E (2009) Quantum-dot lasers - desynchronized nonlinear dynamics ofelectrons and holes. IEEE J Quantum Electron 45(11):1396–1403Lu¨dge K, Bormann MJP, Malic´ E, Ho¨vel P, Kuntz M, Bimberg D, Knorr A, Scho¨ll E (2008)Turn-on dynamics and modulation response in semiconductor quantum dot lasers. PhysRev B 78(3):035,316Malic E, Ahn KJ, Bormann MJP, Hovel P, Scholl E, Knorr A, Kuntz M, Bimberg D (2006)Theory of relaxation oscillations in semiconductor quantum dot lasers. Appl Phys Lett89(10):101,107Malic E, Bormann MJ, Ho¨vel P, Kuntz M, Bimberg D, Knorr A, Scho¨ll E (2007) Coulombdamped relaxation oscillations in semiconductor quantum dot lasers. IEEE J Sel TopicsQuantum Electron 13(5):1242–1248Nielsen T, Gartner P, Jahnke F (2004) Many-body theory of carrier capture and relaxationin semiconductor quantum-dot lasers. Phys Rev B 69(23):235,314Nilsson HH, Zhang JZ, Galbraith I (2005) Homogeneous broadening in quantum dots dueto auger scattering with wetting layer carriers. Phys Rev B 72(20):205,331Steiger S, Veprek RG, Witzigmann B (2008) Unified simulation of transport and lumines-cence in optoelectronic nanostructure. Journal of Computational Electronics 7(4):509–520Viktorov E, Mandel P, Vladimirov A, Bandelow U (2006) Model for mode-locking inquantum dot lasers. Appl Phys Lett 88(20):201,1026

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Modeling of quantum dot lasers with microscopic treatment of Coulomb effects

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