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A Self-Consistent Static Model of the Double-Heterostructure Laser

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A Self-Consistent Static Model of the Double-Heterostructure Laser IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-17, NO. 9, SEPTEMBER 1981 1941 A Self-Consistent Static Model of the Double-Heterostructure Laser Abstract-A new static model of the double-heterostructure laser is even as an approximation, neglects twovery important effects: presentedwhich treats the p-njunction in the laser in a consistent first,the effect on the electricad characteristics oflateral manner.The .soiution makes use of thefinite-element method to treat complex diodegeometries. The model is valid above lasing thresh- carrier drift and diffusion and, second, the saturation of junc- old cdshows both the saturationin the diode junction voltageat tionvoltage (and carrier populations) associated with lasing thresholdas well as lateral mode shifts associated with spatial hole threshold. A more reasonable condition to apply to the diode burning.Several geometries have been analyzed and somespecific junction in the double-heterostructure laser is to assume the results are presented as illustration. continuity of the carrier quasi-Fermi levels across the hetero- junction interfaces. Thisassumption leads naturally to the I.INTRODUCTION saturationof the diode voltage at lasing threshold,and is OUBLE-HETEROSTRUCTURE injection lasershave consistent with semiconductor physics. However, the use of Drecently become objects of intense interest as compact, this model of the diode junction requires the use of a different highly efficient sources of coherent light. With this in mind, solution method from that ofprevious models. laser diode modeling is potentially a tool of great value, both Another model specifically designed to treat the behavior of to understand the effects seen in real laser diodes as well as to a narrow planar stripe laser treats the diode junction in this predict and possibly optimize the behavior of as yet unfabri- manner using a highly simplified geometry [7] . The simplifi- cated devices. cations involved in this model, however, make it impossible to A large number of authors have constructed highly simpli- generalize. fied and idealized modelsof thedouble-heterostructure in- In this paper, a model of the double-heterostructure laser jection laser to illustrate qualitatively the effects of material diode is presented which treats the diode junction in the man- andstructural parameters on device behavior [l] . These ner described above. Fundamental relationships that describe models are quiteuseful to correlate observedlaser current the device electrical and optical characteristics are derived and thresholdswith device parameters,but are oflittle use in simultaneously solved in aself-consistent manner to yield both understandingthe device behavior above threshold. This, the electrical and optical behavior of the device. The model however,,is one of the most important aspects of laser diode is designed for use both above and below lasing threshold. To performance. give as much freedom as possible in the treatment of device There are at present several generalmodels of the laser geometry, the finite-element method is adopted as a solution diode above lasing threshold [2] -[6] . However, these models technique. A numberof interesting geometries have been make assumptions about the electrical characteristics of the examined and somespecific results will be presented. diode that are incorrect. Specifically, in eachmodel the To begin with, some simplifying assumptions will be made. diode p-n junction is assumed to behave accordingto It should be stressed that these are not fundamental limita- tions of the model, but rather good approximations that can 4l-P j = jo exp- be applied to a large fraction of the device geometries in use. nkT First, since longitudinal effects are minor in most devices, where j represents the injectedelectron and hole current onlya lateral, two-dimensionalmodel willbe used. All densities (which are assumed to be equal), jo and n are material longitudinal variations willbe av'eraged over. Second,the parameters, q is the electronic charge,cp is the junctionvoltage, active layer in the device will be assumed to be thin compared k is Boltzmann's constant, and T is the absolute temperature. tothe carrier diffusion lengths, so thatelectron and hole This is not a fundamental relationship. It can be derived for a densities can be assumed to be constant across the active layer one-dimensionalp-n junction frommore fundamental rela- thickness. Third, cladding layer Ijandgaps will be assumed to tionships. The use of this relationship in laser diode modeling, be large enough so thatminority carrier ieakage fromthe active layer can be neglected compared to the majority carrier Manuscript received January 27, 1981; revised April 20, 1981. This densities. Thisleads tothe simplification thatoutside the work was supported by the Office of Naval Research and the National active layer we need only solve an ohmic conduction problem. Science Foundation underthe Optical CommunicationProgram. Fourth, the diode waveguide is assumed to be treatable by the D. P. Wilt was with the California Institute of Technology, Pasadena, effective permittivity method. CA 91 125. He is now with Bell Laboratories,Murray Hill, NJ 07974. A. Yarivis with the California Institute ofTechnology, Pasadena, We now break up the model into two coupled subproblems, CA 91125. the electrical model and the optical model. 0018-9197/81/0900-1941$00.75 0 1981 IEEE 1942 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-17, NO. 9, SEPTEMBER 1981 H O.lpm typlcal Fig. 1. Lateral cross section of thetypical double-heterostructure Fig. 2. Representative band structure diagram for the diode junction in laser. a double-heterostructure laser under high forward bias. 11. THE ELECTRICALMODEL andjunction voltageis both implicit andnonlocal, making The typical geometry of the device modeled is presented in the solution much more difficult. Fig. 1. It consists oftwo ohmic conduction regions, onea Referring to Fig. 2, we have drawna representative band p-typesemiconductor, the other an n-typesemiconductor, diagramof the p-n heterojunction interface underforward and a thin active layer that is partially surrounded by isotype bias. The detailed spike structure at the interfaces is assumed cladding, in this case n-type, and partially sandwiched in be- to be washed out by interfacial mixing, as occurs in liquid tweenp-type and n-type cladding. The only cases excluded phase epitaxial material. Inthis diagram, the carrier quasi- at this time are those where injection occurs from a remote Fermi levels appear as straight lines due to the assumption that junctionor across a homojunction in the active layer. As the active layer is thin compared to the diffusion length. In stated,the problemwith regard to the electrical character- the case of the active layer surrounded by isotype cladding, istics of the device breaks into four coupled problems: two again the continuity of the quasi-Fermilevels is assumed. ohmicconduction problems in theisotype cladding layers, With this assumption and Poisson's equation for the electro- and two continuity relationships in the active layer. static potential in the active layer Inthe two isotypecladding layers we solveLaplace's equation: v2V = -P (6) =0 v (uVpp) = 0 (2) where cp is the electrostatic potential, p is the charge density, V . (UVpn) = 0 (3) e is the relative permittivity, and eo is the permittivity of free where u is the conductivity and pp and p, are the electrostatic space; we can relate the electron and hole densities in the ac- potentials in either region. These equations are subject to the tive layer to thepotential difference across the p-n hetero- boundary conditions pp = ppo on SIPand p, = pno on Sin, junction. Noting thatthe typical Debye length for these the equipotential ohmic contacts of the device; 08 * Bpp = 0 devicesis onthe orderof 100 8,we will assume quasi- on Szp and uii Vp, = 0 on Szn, the open surfaces where no neutrality and write normal current flows; and pp = pp(y) on SJP and p, = p,(y) p+Nd+=ntN; (7) on SBn, the heterojunction interfaces where the potential will be assumed to be a function of the lateral coordinate to the interface.The outward pointing normal to thesurfaces is represented by 8. Thesolution to thisproblem yields theinjected current densities into the active layer: 1 . .=-og.~ P, In1 .Vp on s3p (4) j,, = -uX Vpn on SBn (5) and theelectrostatic potential inside eachof the regions, n and p are the electron and hole densities, Ni and Ni are whichfor self-consistency must be related tothe potential the ionized donor and acceptor densities, N, and Nu are the distribution along theactive layer. effective densities of states in theconduction and valence This relationship is provided in the model presented here by bands, 4, and I,!J~are electronand hole quasi-Fermi levels, the boundary condition on the heterojunction interfaces and E, and E, are conduction and valence band edges, E, is the the semiconductor continuity relations. This is in contrast to bandgap of the active layer, and pn and pp are the electrostatic [2] -[6 J where (1) is used for this purpose. In comparison, potentialson either side ofthe p-n heterojunction. Fll2 is the resulting relationship used here between injected current the Fermi function: WILT AND YARIV: DOUBLE-HETEROSTRUCTURE LASER 1943 The drift and diffusion term that appears in these equations requiresmore elaboration. Using thedegenerate Einstein relations, we have Theseequations completely define the electron and hole -t Jp = PPpVrLp (22) densities as a function of the potential difference across the --r p-n heterojunction. Jn = npn OrLn (23) To relate the injected current density now to the potential where pn and pp are the electron and hole mobilities. An ad- alongthe active layer, we mustlook at thecontinuity ditional and important complication thatwe wish to include is relationships: the case where the active layer thickness may vary. Since we have already separated off the injected current densities from --dp-Gp-Up--O.Jp=O 1- drift and diffusion currents, we must be careful to force the dt e drift and diffusion current to flow parallel to the heterojunc- dn 1-t tion interfaces, or equivalently, to conserve carriers.
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