Quantum Well Laser Simulator Eight Band k.p Modelband k.p model is used to determine the electronic band structure for a strained-layer In.2Ga.sAs/Al.lGa.9As system. The k.p calculation

VLSI DESIGN1998, Vol. 6, Nos. (1-4), pp. 367-371Reprints available directly from the publisherPhotocopying permitted by license only

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Gain Calculation in a Quantum Well Laser SimulatorUsing an Eight Band k.p Model

F. OYAFUSO, P. VON ALLMEN, M. GRUPEN and K. HESS

Beckman Institute, University ofIllinois, Urbana, IL 61801

Effects of non-parabolicity and band-warping of the energy dispersion are entered in a quan-tum well laser simulator (MINILASE-II), which self-consistently solves SchSdinger’s equa-tion, Poisson’s equation, the drift diffusion equations, and the photon rate equations. An eightband k.p model is used to determine the electronic band structure for a strained-layerIn.2Ga.sAs/Al.lGa.9As system. The k.p calculation is performed independently of the lasersimulator, and exported to MINILASE-II in the form of a density of states and an energy-dependent averaged momentum matrix element. The results obtained for the gain and modu-lation response are compared to those obtained from a parabolic band model with a constantmatrix element.

Keywords: gain, k. p, laser, quantum well, modulation response, bandstructure

In recent years, growing computational resourceshave made it possible to develop sophisticated semi-conductor laser simulators, such as MINILASE-II ],that provide a better understanding of the physicalprocesses involved. MINILASE-II solves simultane-ously the drift diffusion equations, including thermi-onic emission and quantum well (QW) capture,Poisson’s equation and the photon rate equations forarbitrary two dimensional geometries. An accuratedescription of the gain still requires knowledge of theQW energy bandstructure. In this paper we explainhow we connect an accurate description of the elec-tronic bandstructure obtained from a multiband k.ptheory [2] with the simulator MINILASE-II and dis-cuss the resulting gain and modulation response of astrained-layer In.2Ga.sAs/Al.lGa.9As quantum welllaser.The k.p Hamiltonian at the center of the Brillouinzone is written as follows,

367

+Hnl(kll,kz) En(O) + 2m 5,t (1)

a (strain)/(kll / kz6z) Pnl / .ctf Fnl kak +ntwhere the indices n over e bds in our basis setB, Pnl e the momentum matrix elements, F erenoalization constants describing e conibutionfrom bands not contained in B, and H(strain) describesthe strain, assumed to be confined to the well region.e number of bds included in B deteines thefraction of e Bfillouin zone at is accuratelydescribed. Since lasing involves trsitions betweenstates ne F, it is sufficient for III-V compounds wizincblende symmetry to include the heavy hole (HH)and light hole bands (LH) (F), e split-off bands(SO) (F) and e lowest conduction bds (F).e resulting 8 x 8 Hamiltoni has been described inthe literature [3, 4].

368 E OYAFUSO et al.

The quantum well layers are assumed to be grownin the z direction and the usual substitution kz - Oz/iis made to obtain the effective mass equations,

:] a’(nkllkq) (kll kq). Hml(k zl’,vl (z) E(n) (z), (2)where kq is the wave number for the 1D Brillouinzone of the superlattice, and the band parameters takedifferent values in the well and the barrier regions [2].

(k kq) (Z) are the superlattice envelope functionswhich are expanded in plane waves in the z-direction. The wave functions, needed to computethe k-dependent optical matrix elements, Mnn,=< nklllPln’kll >’ have the following form:nkll exp(ikll, r)

mB lB

In principle, Poisson’s equation, the .p equations,and the rate equations are coupled together in a com-pletely self-consistent manner. However, it is imprac-tical to perform a complete .p calculation withinMINILASE-II at each iteration for the electric fieldand the distribution functions. We therefore assumeflat bands for the .p calculation, and the resultingdensity of states (DOS) and optical matrix elements(OME) are exported to the laser simulator. MINI-LASE-II then self-consistently solves the Schr6dingerequation for the QW, Poisson’s equation including theunconfined carriers, and the rate equations, in thepresence of the external electric field. More in detail,the DOS are pinned to the lowest CB and VB sub-bands edges to evaluate the QW contribution, Pw, tothe charge density. For example, for the CB, the elec-tron density is given by

oow,c (0) + I*cl

where gc(n) is the total DOS for all the conductionsubbands, obtained from the k-p calculation, c is thefirst eigenstate of the Schr6dinger equation solvedwithin MINILASE-II, using an effective mass deter-

mined from the .p calculation, and E,(O) andare the first subband edges computed with the

model and in MINILASE-II, respectively. The sum-mation is made over an energy grid determined in

MINILASE-II, and )z is the non-equilibrium distribu-tion function also obtained within MINILASE-II. Wehave assumed in eq (4) that the subband dispersionsand the separation between subband edges do notdepend on the electric field in the QW, since in thelasing regime the unconfined carriers efficientlyscreen the applied electric field. Also, the higher sub-bands are assumed to be almost empty so that it is suf-ficient to use only the wave function of the firstsubband in 9QW. We can easily improve the expres-sion for 9QW by adding a summation over the numberof subbands and by taking a different DOS for eachsubband.

An unbroadened gain coefficient is obtained fromthe following expression,

g(O) () (j(en) + j(en ) 1)Pn(Y), (5)

where

d k[I 12Pn(-’2)-- (2t): I’Mij(KIItJTl (E{.(kll))6(E/.(kll) EvJ (kll) a)

(6)

is a matrix element factor calculated separately fromand is an input for MINILASE-II. The direction of the

light polarization is 6, 3Ze and ,/h are the non-equilib-rium electron and hole distribution functions, respec-tively, and tin is a hat function with support on the

energy range [En_l, En+l]. The sum extends over allconduction subbands and valence subbands j. Broad-ening is included in a phenomenological way assum-ing it is caused by a scattering rate of "c =0.1 ps.The k.p calculation is performed for a superlatticeconsisting of AI.IGa.9As barriers of 100 , andIn.2Ga.sAs wells of 80 . The barriers are wideenough to decouple adjacent wells so that the super-lattice is in practice a collection of independent QW.The band parameters, Pnl and F are obtained fromphysical parameters (Table I) with the proceduredescribed in ref. [2, 5]. We assume a fraction conduc-tion band discontinuity of .70. Figures (a)-I (b) showthe unbroadened DOS for the conduction and the

GAIN CALCULATION IN A QUANTUM WELL LASER SIMULATOR USING A EIGHT BAND k.p MODEL 369

material

TABLE Material Parameters used in the k.p calculation

1 2 3 mc/m E(eV) A0 (eV) AEhydro(eV AEshear(eV

A1. Ga.9As 6.49 1.95 2.73 0.0713 1.5396 0.3342

In.2Ga.sAs 9.56 3.34 4.14 0.0551 1.1351 0.3488 0.1826 -0.0539

valence subbands. We recognize the steps typical of a2D confined system. The structures on the plateau aredue to the non-parabolicity of the subband dispersion.Figures l(c)-l(d) show the OME dispersion for TEpolarization in the [110] direction and two directionsof k[, parallel to and at 45. The strongest couplingnear k 0 is between CB 1 and HH as is expected forthis polarization from the form of the HH Bloch func-tion. The OME decreases as the k component parallelto increases (for k 0, k is characterized by a finitekz), and the decrease is fastest for k[ parallel to [6].

We now describe the results obtained from MINI-LASE-II for an operating regime beyond the lasingthreshold. Figure 2(a) shows the optical gain spectrafor various implementations of the DOS and OME.The maximum height of the gain is pinned by thelosses in the laser which are the same for all threeimplementations. For the curve labeled "KP", theDOS and the OME are imported from the k.p calcula-tion in the manner described above. For the othercurves the DOS are determined by the effectivemasses of CB and HH1 at F, and the OME are con-

1810

1.3 1.35 1.4 1.45 1.5Energy (eV)

18x 1o

10(b)

-06 -0.04 -0.02 0 0.02 0.)4Energy (eV)

I00’

80

(c)

0,02 0.04 0.06 0.08 0.1lkl/(2rda)

80 \- CB1-LH1o o.6 o.b4 0.o6 0.08 o.

Ikl/(2rda)

FIGURE (a) and (b) DOS for the CB and VB subbands, respectively. (c) and (d) optical matrix elements for k[] in the [110] and [100]directions respectively. Transitions between CB and the valence bands corresponding to bound states at F are shown. The polarization isin the [110] direction

370 F. OYAFUSO et al.

stants derived from averaging the OME for a CB-HHtransitions over the directions of k in a small neigh-borhood of F. For the curve labeled "2D", the OME(1-2212) p2 is averaged over kll, where P =<SlpxlX > is the band parameter used for the k.p calcu-lation. The OME for the curve labeled "3D"

/i 3.12 2p2 is obtained from an average over alldirections of k. For the last curve, labeled "KANE", a3D average over k is used, and P is obtained from a 3-band Kane model [7], coupling the CB, LH and SObands. The resulting value of P, 9.26 eV-, is smallerthan the value used for the k.p calculation, 10.23 eV-. Fig. 2(a) shows that for increasing constant OME,both the lasing frequency and transparency energy arereduced, since a larger OME requires a smaller sepa-ration between the electron and hole quasi-Fermi lev-els to achieve a fixed gain. The 2D averageoverestimates the "KP" result because the k-depend-ent expression used for the average is valid only in asmall neighborhood of F, and a 3D average underesti-mates the "KP" result.

Figure 2(b) displays the laser’s response to a smallsquare pulse in the applied voltage. The three imple-mentations with the same DOS (KANE, 3D and 2D)give a peak shifted to larger frequencies when theOME increases. We found the population inversion tobe identical for the three cases. Thus, the resonancefrequency is primarily proportional to the time deriva-tive of the photon density in the cavity which is pro-portional to the gain. This proportionality explains thefrequency shift.As in the case of the gain curves, one might expect

the peak of the "KP" curve to lie in between the "2D"and "3D" results. However, the frequency peak isshifted to a smaller frequency than expected. Thisshift is due to the larger DOS for the "KP" implemen-tation relative to the other implementations, resultingin a smaller population inversion factor. The height ofthe resonance in the three constant OME implementa-tions decreases with increased OME because the laseris increasingly overdamped. The height of the reso-nance is larger for the "KP" implementation becausethe capture process is more efficient in the VB due tothe smaller quasi-Fermi level and the larger DOS.

5O00

4OOO

_,300

2000

1000

o1.24

(a)

1.25 1.26 1,27 1.28 1.29 1.3Energy (eV)

5 10 15 20 25Frequency (GHz)

FIGURE 2 (a) Gain and (b) modulation response for four opticalmatrix elements. KANE, 3D, and 2D are constant matrix ele-ments. K.P is the k-dependent matrix element determined from theeight band k.p calculation

In conclusion, we described how an accurate k.pcalculation was connected to a sophisticated lasersimulator and found that the transparency energy inthe gain spectra and the resonance frequency in themodulation response are affected if compared to lessaccurate models for the DOS and the OME.We acknowledge the support of NSF through

NCCE and of the Office of Naval Research.

References1] M. Grupen, K. Hess, and L. Rota, "Simulating Spectral Hole

Burning and the Modulation Response of Quantum WellLaser Diodes", Proc. of the SPIE Conf, 6-9 February 1995,San Jose, California.

[2] E von Allmen, "Conduction Subbands in a GaAs/AlxGal_xAsQuantum Well: Comparing Different k.p models", Phys. Rev.B 46, 15382, December 15, 1992.

GAIN CALCULATION IN A QUANTUM WELL LASER SIMULATOR USING A EIGHT BAND k.p MODEL 371

[3] M. Cardona, N. E. Christensen, and G. Fasol, "RelativisticBand Structure and Spin Orbit Splitting of Zinc-blende-typeSemiconductors", Phys. Rev. B 38, 1806, July 15, 1988.

[4] E. E O’Reilly, "Valence Band Engineering in Strained-LayerStructures", Semicond. Sci. Technol. 4, 121, 1989.

[5] C. Hermann and C. Weisbuch, "k.p Perturbation Theory inIII-V Compounds and Alloys: a Reexamination", Phys. Rev.B 15, 823, January 15, 1977.

[6] Corzine, S. W., Yan R., Coldren L., "Optical Gain in III-VBulk and Quantum Well Semiconductors" in Quantum WellLasers (edited by Zory, ES.), Academic Press, 1993.

[7] E. O. Kane, "Band Structure of Indium Antimonide",J. Phys. Chem. Solids 1,249, 1957.

BiographiesFabiano Oyafuso received B.A. degrees in mathe-matics (with honors) and physics from UC Berkeleyin 1992. He is currently pursuing the Ph.D. degree inphysics at the University of Illinois at Urbana-Cham-paign. His thesis involves the study of the temperaturedependence of theshold current densities in quantumwell lasers, email: [email protected]

Paul yon Alilnen obtained his B.S. and Ph.D. inphysics from the Swiss Federal Institute of Technol-ogy in Lausanne. He joined the Zurich IBM researchlaboratory as a postdoctoral research associate in1990. Since 1992 he has been an invited scholar at theBeckman Institute for Advanced Science and Tech-nology at the University of Illinois in Urbana-Cham-paign. His interests range from subband structuresand many-body effects in confined electron systemsto the dynamical properties of nanostructures and sur-

faces. His main field of interest is presently the studyof the desorption mechanism of hydrogen and deute-rium from a silicon surface and the related isotopeeffect which has important consequences for theresistance of MOS-transistors against hot carrier deg-radation, email: [email protected] Grupen received his B.S. from Penn State

University in Engineering Science in 1985. He thenattended UCLA where he received an M.S. in Elec-tron Device Physics in 1989. In 1994, he received aPh.D. in Computational Electronics at the Universityof Illinois, where he currently holds a post-doctoralposition, email: [email protected]

Karl Hess has dedicated the major portion of hisresearch career to the understanding of electronic cur-rent flow in semiconductors and semiconductordevices with particular emphasis on effects pertinentto microchip technology. His theories and use of largecomputer resources are aimed at complex problemswith clear applications and relevance to miniaturiza-tion of electronics. He is currently the Swanlund Pro-fessor of Electrical and Computer Engineering,Adjunct Professor for Supercomputing Applicationsand a Research Professor in the Beckman Instituteworking on topics related to Molecular and ElectronicNanostructures. He has received numerous awards,for example the IEEE David Sarnoff Field Award forelectronics in 1995. email: [email protected]

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Quantum Well Laser Simulator Eight Band k.p Modelband k.p model is used to determine the electronic band structure for a strained-layer In.2Ga.sAs/Al.lGa.9As system. The k.p calculation