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 boththe saturationin thediode 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 heterojunction 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 makeassumptions 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, orequivalently, to conserve carriers. We can -_ -Gn-Unt-V-Jn=O. dt e assume thatthe magnitudeof thecurrent flow is constant across the active layer, but the changing of the active layer G, and G, are electron and hole generation rat9, U, a2d Up thickness gives an additional termwihen we take the divergence are electron and hole recombination rates, and J, and Jp are in (1 3) and (14). With the condition that active layer thick- electron and hole drift and diffusion currents. ness varies slowly with respect toy, we have for these terms Injected current can most easily be included in these equations as a generation term. Thermal generation is neglectable in the laser diode, which operates under high forwardbias: -V.J1 -+ =-11dt -.-+- d np - e e (t dy dy) ( ':;)' Theseterms are easilyseen to ble conservative, as desired. The derivatives of the quasi-Fermi levels that appear in these terms must, of course, .be treated self-consistently with the solution to theohmic conduction problem. The identifica- where tis the active layer thickness. tion is providedby the assumption of continuity of quasi- The recombination terms consist of both nonradiative (trap, Fermi levels acrossthe interfaces, as before. Neglecting the surface recombination) and radiative (spontaneous and stimu- contribution of carriers that leak over the confining hetero- lated) terms. The forms used for these are barriers, this allows us to identifywith the Fermi level in the p-cladding and rLn with the Fermi level in the n-cladding along thep-n junction region. In the case where the active layer is surrounded by isotype cladding, we do this for the majority carrier; for the minority carrier we instead demand that the injected minority carrier current density be zero. With these relationships, the electrical behavior of the device is completely defined. It is interesting to note that at no point in the analysis was theassumption of equal injected current densities orthe assumption of ambipolar diffusion required. These are not necessarily bad approximations, but 7, and T~ are effective nonradiative minority carrier lifetimes they cannot bederived from the relationships above.The andmay include the effects of leakageover theconfining difficulty liesin the fact that the electron and hole popula- heterojunction barriers. S is a surface recombination velocity, tions are essentially in equilibrium with their isotype cladding ys being the location of the surface interface. B is a material layers. An interesting facet of this is that symmetric devices constant, P is the optical power density, g is the local optical withp- and n-type layers interchanged butwith identical gain ofthe medium, and liw is thephoton energy.In this conductivities do not behave identically. model, the gain term is assumed to be of the form Fromthe standpoint of solving the electrical behaviorof themodel, the problem is to find an electrostaticpotential g =go +glpP+glnn+gzpnPn. g=go (21) distribution and quasi-Fermi levels in the active layer that are These relationships are simplified forms of more general rela- consistent with all of the relationships set down above. tionships, making use of the fact that the laser diode operates in the high forward bias regime. Of course, to be consistent 111. THEOPTICAL MODEL with the assumption that the active layer is thin and that the Theoptical model presented here is quite similar tothat electron and hole densities are uniform across it, the relation- presented elsewhere [2] -[6] . In brief, effective permittivity shipfor the stimulated emission recombination rate (20) formalism is used to find the TE modes of a perturbed slab must be averaged over the direction normal to the active layer. waveguide.TE modes are treated because they are experi- 1944 JOURNALIEEE OF QUANTUMELECTRONICS, VOL. QE-17, NO. 9, SEPTEMBER 1981 mentally known to dominate the behavior of the semiconduc- variation in theperpendicular (x) direction. Todo this, we tor laser. Modalgains are either found directly fromthe substitute into thevariational equation the trial form solution of the waveguide eigenmode equation or from perturbation theory if the mode profile is only slightly perturbed. Here double-heterostructure lasers split into two equivalence Since the function we will allow to vary, Y,is a function only classes, thosewhere carriers contribute significantly tothe of the lateral variable y, we can integrate out the variable x in waveguide problem and those where carriers may be treated the variational principle to get an effective variational princi- as aperturbation. Roughly speaking, these two classes cor- ple involving only Y and y: respond to devices with geometric structures that define the waveguide modes (e.g., buried heterostructure lasers [8] and channeled substrate lasers [9] , [lo] ) and devices that have no built-in geometric waveguide structure(eg., beryllium implanted lasers [ll] and oxide-stripe lasers [12]). Ths is a rough divisionbecause there are important laser structures that have geometric waveguiding and still use the carrier popu- where lations to define the optical modes [13]. The optical model presentedhere, while forconvenience limited to TE modes and effective indexformalism, is capableof treating both classes of semiconductor lasers. With the assumption of a TE mode, the eigen equation for The normalization condition (29) and the field equation for the waveguide modes of the laser simplifies to X,(28), have been used to simplify this expression. The Euler equation for thisvariational expression is then

(-$ + k2 eeff- p2 Y = 0 (33) where u is the (scalar) TE electric field and /3 is themode 1 propagationconstant. The magnitudeof the wave vector is where the effective permittivity is k and E representsthe complex relative permittivityof the medium. For convenience, we will take the x coordinate to be normal to the active layer and the y coordinate parallel to Thesecond term in this expression, yi, is usually quite the active layer. This eigen equation canbe presented in small and is neglected here. This leaves us with the expression variational form as

forthe effective permittivity.The field Y willbe assumed normalized accordingto

J_,J_, dx dy u2 (27) To apply the effective permittivity formalism to this equa- so that the normalization on the field u is tion, a variational form willbe assumedand the variational principle (27) will beused to derive anEuler equation for the lateral modal field. (37) The procedure applied to the problem is to first solve the one-dimensional waveguide problem for the lowest modeX: Theadvantage of approaching the effective permittivity problemfrom the standpoint of the variational principle, aside from the inclusion of a term which we have neglected, is that it assures in a sense that the best approximation to the (an effective variation in the normal (x) direction to the active true modal field is found. If first-order perturbation theory is layer). The lateral coordinate y is consideredhere to be a applied to the modal profiles found (assuming the extra term parameter. Consistent with the use of complex permittivities, is not neglected), the lowest nonzero corrections to the modal the normalization condition on-this field will be taken as field involve overlap integrals of the field X with higher order modes in the x direction, or equivalently, corrections involving overlap integrals of X with itself are not present. dxX2 = 1. 1: 1: The modal gain is related to thepropagation constant p We would now like to find then the best possible approxima- gmode = 2 Im &ode (38) tion to the true modal field using this field X to represent the which, if the proper permittivity is used, is exact. If one wishes WILT AND WILT YARIV: DOUBLE-HETEROSTRUCTURE LASER 1945 to use a modal profile and propagation constant determined Evaluation of this expressionyields (43). with a different permittivity, lowest order perturbation theory In this model, the distributed loss term is assumed to be a to findthe propagation constant (and thusmodal gain) is constant, although its dependence upon p and n can easily be appropriate: included in a manner similar to thegain expression (21). Note, however, that this distributed loss is not equivalent to a gain term. The difference be.tween the two is that the d(p2)= Jmdx dy u2k2de (3 9) 1--m -m gain term also appears in the stimulated recombination rate [see (20)] while the loss term does not. This loss term repre- which simplifies to sents nonretrievable loss mechanisms suchas scattering. Thismodel assumes all opticalmodes to have the same do = ”p dx dy u2 de. facet reflectivity. This is probablya good approximation as 2P -m lI we have taken them all to have the :same mode profile X. The variation in facet reflectivity between modes can, of course, To treat a laser, one must, of course, include the effects of be included in the calculation with minor complication. the longitudinal cavity. Inthe simplest form, these are the Depending upon which equivalence class the device under roundtrip phase condition(which is hereneglected) to give consideration is judged to fall into, geometrically guided or theFabry-Perot modes (often called longitudinalmodes) carrier guided, the lateral mode profiles can be found once and and the roundtrip gain condition that the optical gain in the onlyperturbation theory canbe applied to findthe modal cavity balances the optical loss in the cavity plus the radiation gain and loss, orthe lateral mode profiles foundfor every losses. This model neglects the contribution of spontaneous value of the carrier populations, while the solutionto the elec- emission to the optical power flow in the cavity, but, if de- tricalproblem is being iteratively sought. If the deviceis sired, thiscontribution is easily included.This relationship carrier guided,of course, the dependence of both real and can be stated as imaginaryparts of the permittivity onelectron and hole density must be included in the modal calculation. In principle and practice, either type of device can be treated. How- ever, for the carrier guided device, the solution of the eigen- where &ode is the modal gain, amode is the modal loss, L is mode equation at each iteration can. be quite time consuming. the device length, and Rmodeis the mode mirrorreflectivity. Hybridtechniques involving bothexact and perturbation The optical power density in the device can be represented methods are usually more reasonablefor this type ofdevice. as N.SOLUTION TECHNIQUE As a first step,the functional relationship betweenthe junction voltage and carrier population densities is solved [see (7)-(1 l)] . This is done using a nonlinear root-finding technique. Since this is only material dependent, it need be done where Pi is thetotal powerflowing inthe cavity (average only once for a given material and doping density. over length of backward and forward traveling waves). This The two ohmic conduction problems [(2)-(5)] are treated optical power is related to the actual power emitted from both using the finite-element method with triangular elements and mirrors by linear interpolationfunctions. Since this problem is linear, the solution can be stated in the forrn of an equivalent Green’s (43) functionfor each region that relates the injectedcurrent density to thepotential distribution along the junction where Po is the total power output from the device. This can boundary: be shown in the following manner. The actual power distribution in the laser diode is (47) J i

J i where thepotentials on the conta.cts are qpo and qno,as before,and ppj cnd qnj are thenodal potentials along the junction interface. The potential along the junction interface where the diodemirrors are locatedat kL/2. It is easily is assumed to vary linearly between the junction nodes. The verified that the average over the length of the diode of (44) fi are linear interpolation functions along the interface. yields (42). The total power emitted from the laser facets is Theproblem then reduces to satisfying thecontinuity given by relationships in the active layer [see (13) and (14)] subject to Po = Pi(l - R) (Z+ Z- (47) and (48). Simultaneously, of course, the optical modes E)+ (- $-)) ofthe structure and their stimulatedemission (if they are 1946 IEEE JOURNAL OF QUANTUMELECTRONICS, VOL. QE-17, NO. 9, SEPTEMBER 1981

-Zn diffusion TABLE I ,,-n GaAs MATERIALPARAMETERS USEDIN THE MODEL I I J

p Gao.65A~O.3, As

p Gao.8oA'0.20 As -k Ga0.95 "0.05

Gao.8oAlo.2oAs

G00.65A10.35AS

&----.I 2.5pm + n GaAs(substrate) Fig. 3. Lateral cross section of the example device treated. This type of device is characterized as a nonplanar large optical cavity laser.

above threshold)must be included. If a mode isreceiving stimulated emission, itsgain is held constant accordingto (41). This problem, in order to be consistent with the solution of theohmic conduction problem, is formulated also inthe finite-elementfashion. The electrical modelhere is one- dimensional and linear interpolation functions are used. The finite-element equations are derived using Galerkin's method. The optical mode problem is treated as both a slab waveguide problem [(28) for the mode profile X] and a finite-element problem [(33)for the modeprofile Y]where a one-dimensional grid and first-order Hermite interpolation functions are used. Again this is done to achieve compatibilitybetween the subproblems. Up to fourlateral modes are includedin the STRUCTURALPARAMETERS FOR THE DEVICEANALYZED calculation. conductivity 1 refractive index 1 layer In this formulation, the problem reduces to solving a non- ohm-' cm-' , (E = n2) I linear system ofequations for the nodal values of the two n+GaAs substrate 1000 3.64 -0,05281 quasi-Fermi levels GP and $, and the optical power outputs n Gao.65 AIo.35 As 200 3.39 nGao.eoA10,20As 200 3.50 in each of the modes. The only free parameter in the model _- - then is the voltage difference between the equipotential con- ___p Ga0.95A10.05As 3.64 + dn,,r,,er, tacts, a global boundary condition. In practice this is allowed P Ga0.80A10.2GAs 8 3. 50 to vary and, instead, the total current through the deviceis specified. An iterativetechnique ofthe modifiedNewton form is used to find the appropriate solution to the nonlinear simultaneous equations. chosen to be compatible with both direct experimental mea- surementsand measured broad-area lasing currentthreshold V. RESULTS densities [I] . The n-GaAs top layer in the structure is used Several device structures have been analyzed, including both only as a blocking layer, which is shorted by the zinc diffused caseswhere carriers are treated as a perturbation and where stripe, so the electrical model omits the top n-layer and con- carriers define the lateral opticalmode structure. Lasers of siders the zinc diffusion as a 2 pm wide stripe contact. Refrac- the first type analyzed include the embedded laser [I41 and tive indexes are given instead of relative permittivities, where the channeledsubstrate laser [9] , [ 101 . Only one laser E = n2. The substrate and contact layer may be omitted from structure of the second typehas beenanalyzed, the beryl- the waveguide problemwith the result thatthe effective lium implanted laser [ 111 . Specific results are presented here permittivity is real. for the laser structure of Burnham [lo] , which as been ana- The solution of the equations forelectron and hole densities lyzed in simplified form by Streifer [6] , [ 151 . Unfortunately, as a function of voltage difference across the heterojunction that analysis neglects the effect mentioned in connection with is shown in Fig. 4. Note that since the Fermi functions ap- (24) and (25) and as a result the solution to the diffusion propriate to degeneratesemiconductors are used,the curves equation presented there is incorrect, as it does not conserve begin to bend over at high injection levels. carriers. This device has an obvious mirror symmetry, and this will The structure of the device is shown in Fig. 3. The material be exploited to ease the calculation. However, itmust be and structural parameters assumed for the device are listed in remembered that with this simplification all currents and out- Tables I and 11, respectively. The material parameters used are put powers should be doubled. WILT AND YARIV: DOUBLE-HETEROSTRUCTURE LASER 1947

EffectivePermittivity Profile 12.6 c (In? Eeff = 01 IO‘OZ2[20

Ni = 3 x I0”crri’

N, = 4.7xlO”cm~‘

N, = 7xlO”cm-’

10 I/ Y (pm) ‘00.80 1.00 i.;o I.hO IBO 1.80 2.bo Junctlon Voltage (Volts) Fig. 6. The effective permittivity profile for the device. Use has been Fig. 4. The electron and hole densities in the active layer as a function made of the device symmetry. of the voltage acrossthe heterojunction.

Symmetry axis

Far IntensityField Far Fleld Intensity 1 Near Field IntensityIntensityFieldFieldNearNear

036912 Y (pm) Y (pm)

For FieldIntensity Far FleldIntensity

Nearnrde: Field intensityaxi:b+ Nearrd Fieldo;r Intensity, ,

Symmetry Symmetry Symmetry OllS A3

36912 Y (pm) Y (pm) Fig. 7. Thefour lowest lateralmodes (Y)of the device ,and their Fig. 5.’ The finite-element model constructed for the ohmic conddc- correspondingfar-field patterns. Use has been made of the device tion problems. Use has been made of the device symmetry. symmetry.

Thefinite-element model used forthe calculation of the where t is the active layer thickness and y is the lateral dis- Green’s functions (47) and (48) is shown in Fig. 5. The use tance measured from the center of the stripe, both measured of a large number of elements for the modeling of the sub- in pm.The effective permittivity profile forthe deviceis strate is not necessary but does give the device a reasonable shown in Fig. 6 and the lowest f0u.r lateral modes and their series resistance. In most situations, assuming the substrate- corresponding far-field patterns are shown in Fig. 7. Since epilayer interfaceto be equipotential is a good approximation. the waveguiding propertiesof this device aregeometrically The geometric model of the device (see Fig. 3) is used for determined, (40) isused to determinemodal gains forthe the calculationof the effective permittivity (35), and the device. lbwest four lateral optical modes (Y)of the device are cal- Thesolutions for the static device behaviorwith pump culated as described. The active layer thickness is assumed to current as a parameter are shown in Figs. 8-1 1. Fig. 8 shows vary as thecurrent versus voltage characteristics ofthe device and clearly shows the saturation of the diode junction voltage at t = 0.08 + 0.2 exp - 0.0732 y2 (49) lasing threshold, which can be seen to occur at approximately 1948 IEEE JOURNAL OF QUANTUMELECTRONICS,NO. QE-17, VOL. 9, SEPTEMBER 1981

2.4 m- - 0 C X : ro a V lasing threshold ‘E 1.6 0 m I Q -I 0 Io 0.8

1.38 1.48 1.581.78 1.68 DeviceVoltage (volts)

3.2 1

60*Or

0 00 - O0 00

losing 00 threshold 00

)oOOoo

i4! 20 - oo 0” 0 - , I 0 I I I I L,-- 1.38 1.48 1.581.78 1.68 0 6 12 18 24 Device Voltage (volts) Y(pm) Fig. 8. The current-voltage characteristics of the device. Toobtain Fig. 9. The lateral carrier density profiles for the device in operation. total device current, the scale should be doubled. Total device current is varied as a parameter from 4 to 100 mA with a step of4 mA. 31 mA oftotal device current. Above threshold,further In addition, the above threshold analysis in [6], although for increases in devicevoltage are due to the finite deviceresis- adifferent structure, shows a different type of spatial hole tance,here approximately 2.4 a. The carrier profiles for burning than this model.In that calculation, spatial hole the device with the pump current as a parameter are shown burning was found to significantly lower the carrier popula- in Fig. 9. The saturation of the carrier populations at lasing tionat the center of the stripe under lasing conditions.In threshold and the effects of spatial hole burning can be seen. this model, the carrier population at the center of the stripe This is a different effect fromthe “diffusion focusing” de- is nearly constant above threshold and lateral mode switching scribed in [ 151 . The light versus current characteristics of the results from the increase in the carrier population outside the device are shown in Fig. 10, where stimulated power output lasing mode. Ths difference can be attributed directly to the to each mode as well as modal gains are plotted as functions p-n junction boundary conditions appliedin the two models. ofpump current. The total power output as afunction of pump current is shown in Fig. 11. The effect of spatial hole VI. SUMMARYAND CONCLUSIONS burning can be seen to eventually let higher order modes of In summary,a model of the double-heterostructure laser the structure emit stimulated power. The kink associated with has been presented that correctly treats the diode junction of thefirst-order mode beginning to lase atapproximately 52 the device. It is valid above threshold and is capable of treat- mA totalcurrent and 20 mW total power output is clearly ing a large number of the device geometries in use. With this visible. These results are in agreement with the experimental model, the quantitativebehavior of devices can,be investi- results for this device. gated above lasing threshold and compared and optimized. To compare with the results presented in [6] and [15] , the sheet resistance for the p-layers assumed here is approximately REFERENCES 500 a. Thecalculated threshold current in [15]for this [ 11 For some examples and more references, see Chapter 7 of H. C. sheet resistance and a 2 ym wide stripe contact is 53.7 mA. In Casey, Jr. and M. B. Panish, Heterostructure Lasers: Part B, this model,the injected carrier profile at threshold falls to Materialsand Operating Characteristics, New York: Academic, 1978. half of its value at the center of the stripe at a lateral distance [2] J. Buus, “A model for the static properties of DH lasers,” ZEEE of 10 pm. In comparison, [15] yields 6 pm for this distance. J. Quantum Electron., vol. QE-15, pp. 134-149, Aug. 1979. WILT AND YARIV: DOUBLE-HETEROSTRUCTURE LASER 1949

[3] K.A. Shore, T. E. Rozzi, and G. El. in’t Veld, “Semiconductor l6 r laseranalysis: Generalmethod for characterisiig devices of various cross sectional geometries,” ZEE Proc., pt.- I, vol. 127, PP. 221-229,1980. [4] N. Chinone,-“Nonlinearityin power-output-current characteristics of stripegeometry injection lasers,” 3. Appl. Phys., vol. 48, pp. 3237-3243,1977. [5] R. Lang,“Lateral transverse modeinstability and its stabiliza- A tionin stripe geometry injection lasers,” IEEE J. Quantum Electron., vol. QE-15, pp. 718-726, Aug. 1979. no A [6] W. Streifer, D. R. Scifres, and R. D. Burnham, “Above threshold analysis of double heterostructure lasers with laterally tapered A no active regions,” Appl. Phys. Lett., voL 37, pp. 877-879, 1980. [7] P. M. Asbeck, D.A. Cammack, J. J. Daniele, and V. Klebanoff, “Lateral mode behavior in narrow !stripe lasers,” IEEE J. Quan- tum Electron.,vol. QE-15, pp. 727-733, Aug. 1979. A 01 0 I I I [ 81T. Tsukada, “GaAs-GaxAll -,As buriedheterostructure injec- 20 40 60 80 tion lasers,” J. Appl. Phys., vol. 45, pp. 4899-4906, 1974. Current (mA) [9] K. Aiki, M. Nakamura, T. Kuroda, and J. Umeda, “Channelled- substrateplanar structure (A1Ga)As injection lasers,” Appl. Phys. Lett., vol. 30, pp. 649-651, 1977. 70 r [lo] R.D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, "Nan- planar large optical cavity GaAs/GaAIAs semiconductor laser,” AppL Phys. Lett., vol. 35, pp. 734-736, 1979. 111 N. Bar-Chaim, M. Lanir, S. Margalit, I. Ury, D. Wilt, M. Yust, - 0 and A. Yariv, “Be implanted (GaAI)As stripe geometry lasers,” P Appl. Phys. Lett., vol. 36, pp. 233-235,1980. !? !? 121 Y. Yonezu, I. Sakuma, K. Kobayashi, T. Kamejima, M. Ueno, and Y. Nannichi, “A GaAs-A1,Gal_,As double heterostructure planar -8 0 01h order mode stripelaser,” JQpQn. J. Appl. Phys., vol. 12, pp. 1585-1592, 0 = Is1 ordermode 1973. B L, = 2nd order mode 131 D. Botez, “CW high-powersingle-mode lasers using constricted t = 3rd order mode double heterostructures with a large optical cavity (CDH-LOC),” -6 Top.Meet. Integrated Guided Wave Optics, Incline ‘Village. NV, Jan. 1980. uauer MC2. [ 141 J. Katz, S. Margak, D. Wilt, P. C. Chen, and A. Yariv, “Single growth embedded epitaxyAlGaAs injection lasers with extremely - 90 I I I I lowthreshold currents,” Appl. Phys. Lett., vol. 37, pp. 987- 0 20 40 60 80 989,1980. Current (mA) [15] W. Streifer, R.D. Burnham,and D. R. Scifres, “Analysis of diode lasers with lateral spatial variations in thickness,” Appl. Fig. 10. The light versus current characteristics and modal gains for the Lett., vol. 37, pp. 121-123, 1980. device. Eachlateral mode is plottedseparately. To obtaintotal Phys. device current and power, thescales should be doubled.

0 0 0 0 0 0 0 0 0 0 0 0

0 0 nl I I I 20 40 60 80 Current (mA1 Fig. 11. Thelight versus currentcharacteristics for the device, with output powers in the lateral modes summed. To obtain total device Amnon Yariv (S’56-M’59-F’70), for aphotograph and biography, current and power, thescales should be doubled. see p. 1394 of the August 1981 issue of this JOURNAL.