High Speed Quantum-Well Lasers and Carrier Transport Effects

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1990 IEEE J OURNAL OF QUANTUM EL ECTRONICS, VOL . 28, NO I O OCTOBER 1992

HighSpeed Quantum-Well Lasers and CarrierTransport Effects

Radhakrishnan Nagarajan, M asayuki I shikawa, Toru Fukushima, Randall S . Geels, Member, IEEE,and J ohn E. Bowers, Senior Member, IEEE

Abstract-Carrier transport can significantly affect the highspeed properties of quantum-well asers. We have developed amodel and derived analytical expressions for the modulationresponse, resonance frequency, damping rate, and K factor toinclude these effects. We show theoretically and experimentallythat carrier transport can lead to significant ow frequency par-asitic-like rolloff that reduces the modulation response by asmuch as a factor of six in quantum-well asers. We also showthat, in addition, it leads to a reduction in the effective differ-ential gain and thus the resonance frequency, while the nonlin-ear gain compression factor remains largely unaffected by it.We present the temperature dependence data for the K factoras further evidence for the effects of carr ier transport. I n thepresence of significant transport effects, we show that the reallimit to the maximum possible modulation bandwidth is muchlower than the one predicted by the K factor alone. The dif-ferences between the modulation response and relative inten-sitv noise measurements are discussed.

I . INTRODUCTIONHE high speed dynamics of semiconductor lasers haveT een conventionally modeled using a set of two cou-pled first-order linear differential equations; one for thecarrier density and the other for the photon density in thecavity. Using this approach, the small -signal response ofthe laser can be written as a second-order transfer func-tion, and the modulation bandwidth of such a system isdetermined by its resonance frequency f , and the damping

rate y. From the small-signal analysi s of these rate equa-tions, the resonance frequency can be written in terms ofmore fundamental laser cavity parameters; f , = (1/2a)Jv,g'sI7,,where ug is the group velocity in the cavity,g' is the differential gain, S is the photon density, and rpis the photon lifetime [13, [2].In this analysis, the intrinsic damping mechanism in theM anuscript received J anuary 2, 1992; revised March 19, 1992. This workwas supported by DA RPA and Rome Laboratories (Hanscom AFB ).R. Nagarajan and J. E . Bowers are with the Department of E lectrical andComputer Engineering, University of Cal if omia, Santa Barbara, C A 93 106.M . I shikawa was with the Department of E lectrical and Computer En-gineering, University of Cal if omia, Santa Barbara, CA 93106, on leavefrom Toshiba Corp., Tokyo, J apan.T. Fukushima was with the Department of Electrical and Computer En-

gineering, University of Cal if ornia, Santa Barbara, CA on leave from Fu-rukawa Electric Co., T okyo, J apan.R. S. Geels was with the Department of Electrical and Computer Engi-neering, University of Cal if omia, Santa Barbara, CA 93106. He is nowwith Spectra Diode Laboratories, San J ose, CA 95134.IEEE Log Number 9202381.

laser is the cavity loss rate determined by the photon life-time. Further, the optical gain is also photon density de-pendent, i .e. , it saturates at high photon density l evels,and this provides an additional source of damping [l],121. This nonlinear photon density dependence of opticalgain is introduced into the rate equation formali sm via aphenomenological gain compression factor E . The gain orthe differential gain in the laser cavity is then written asg' = g,/(l +E S)and in the case of small E S,keepingonly the linear term of the Taylor expansion, it is alsosometimes written as g' = g,(l - E S)where g, is thedifferential gain component determined solely by materialparameters and may also be carrier density dependent.The physical origins of the gain compression factorhave been attributed to various phenomena l ike cavitystanding wave dielectric grating [3], spectral hole burning141-[6], transient carrier heating [7], [8], or some com-bination of the last two [9]. The damping rate y varieslinearly withf : , and the proportionality constant is calledthe K factor [lo]; K =4n2(7, + E/uggo).The maximumpossible intrinsic modulation bandwidth is determinedsolely by this factor, f,,, =& (27r/K ), and thus K isoften taken as a figure of merit for high speed semicon-ductor lasers. L asers with smallerK factors have also beenshown to have a lower level of relative intensity noise(K IN) at low frequencies [111. In practice, the maximumpossible modulation bandwidth in lasers with bulk activeareas is usually limited by RC parasitics, device heating,and maximum power handling capability of the laser, es-pecially the facets. We will show in this paper that, par-ticularly in quantum-well lasers, carrier transport (diffu-sion, tunneling, and thermionic emission) is also animportant and often the dominant limit.Conventionally one used the expression forf , as a guidefor high-speed structures, and designed lasers with tightoptical mode confinement for high photon densities, shortcavity length for short photon lifetimes, and p-doped ac-tive areas [12], [13] for high differential gain. L asers withquantum-well active areas have been theoretically and ex-perimentally shown to have enhanced differential gainover the bulk lasers. A dditional enhancements can be ob-tained with the inclusion of strain 1141-[16], p doping ormodulation doping [171. This led to theoretical predic-tions of increased modulation bandwidth in quantum-welllasers [181. I t is only recently that the modulation band-width in quantum-well lasers have been comparable to the

0018-9197/92$03.000 1992 IEE E


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NAGARAJAN et al.: CARRIER TRANSPORT EFFECTS 1991

best bulk lasers, with the quantum-well lasers in theInGaAs-GaAs system [191, [20] presently performingbetter than those in the GaAs-A lGaAs [21] and the InPsystems [22], [23]. Further there has been a large vari-ation in the K factors reported in print for quantum-welllasers; 0.13 to 2.4 ns [ l l ] , [22], [24]-[27] implying amaximum possible intrinsic modulation bandwidth rang-ing from 68 to 4 GH z, while K values f or the bulk lasershave been in the narrow range of 0.2 to 0.4 ns [6]. Al-though the E values quoted for lasers in the InGaAs-InP[6], [1 1, [24], [27] system on the average are higher thanthe ones that we have reported for the InGaAs-GaAs sys-tem [26], they are by no means anomalously high or en-hanced in quantum-well lasers as they were initially spec-ulated to be [25]. Recent experimental reports have shownthat E is independent of the laser structure, and is evenunaffected by the inclusion of strain, compressive or ten-sile, in the quantum wells [22], [241, 1261, [271.The higher K values in some quantum-well lasers overthe bulk ones have been explained by either an increasein the nonlinear gain compression factor E or a reductionin the differential gain. I t has been theoretically proposedthat E is enhanced by quantum confinement [28], and theinclusion of strain increases it even more [29]. In [30],the authors have used a spectral hole burning model forthe computation of the intrinsic E and its variation withquantum confinement, and optimized quantum-well struc-tures for high speed operation. A well-barrier hole burn-ing model [31] was recently proposed, and it concludesthat there is an additional contribution to the intrinsic Ewhich is structure dependent. T he authors have proposedthat this enhancement in E is responsible for the variationin the K factors that have been published. Recently, wehave proposed a model for high speed quantum-well la-sers based on carrier transport [32], and numerically cal-culated the effects of carrier transport on the modulationbandwidth and obtained good agreement with the mea-surements done on single and multiple quantum-well la-sers. In [33], the authors have measured the spontaneousemission spectra from different types of lasers, and re-ported enhanced light emission from the confinement lay-ers, relative to the quantum-well active area, of morestrongly damped lasers. For the case of single quantum-well lasers, we have derived analytic expressions for theeffects of carrier diffusion and thermionic emission on themodulation response. We have also confirmed this modelby an experiment where the modulation response of threelaser structures with different carrier diffusion times acrossthe separate confinement heterostructures were measuredWI.In this paper, we wil l present the details of our carriertransport model, and deriveanalytical expressions for theresonance frequency, damping rate, and K factor in thecase of single quantum-well (SQW ) lasers to include theeffects of carrier transport. W e will use anumerical modelin the case of multiple quantum-well (M QW) lasers. Wewill show theoretically and experimentally that E is fairlyindependent of the laser structure, and that carrier trans-

port can have a significant effect on the modulation prop-erties of high speed lasers via a reduction of the effectivedifferential gain and not an enhancement of E. In addition,we will also show that the carrier transport across the SCHcan lead to severe low frequency rolloff. In the presenceof such rolloff and even when all other extrinsic devicelimits are absent, the K factor is no longer an accuratemeasure of the maximum possible intrinsic modulationbandwidth in quantum-well lasers. W e will also presentthe temperature dependence of the K factor as further evi-dence for the significant effects of carrier transport. Fi-nally, we wil l use our model to present some design cri-teria for optimizing quantum-well lasers for high speedoperation. Among the parameters that can be vari ed, arethe width and number of the quantum wells, width andheight of the barriers, types of compositional grading inthe separate confinement heterostructure (SCH) to tailorthe built-in electric field profile, the amount of strain, andthe type and amount of doping in the active area. Giventhe range of possible variations in structure and their ef-fects on the various l aser parameters, optimizing thequantum-well laser for high speed operation i s very in-volved.

11. CARRI ER RA NSPORTN THE SEPARATECONFINEMENTETEROSTRUCTUREFig. 1 shows a typical separate confinement hetero-structure (SCH ) SQW. W e will model the carrier trans-port and capture in such a structure and the effect this hason the modulation response of the laser. This is also thestructure that is used in the SQW and MQW laser exper-iments.Electron and hole transport f rom the doped claddinglayers to the quantum well consists of two parts [35], [36].First is the transport across the SCH. T his is governed by

the classical current continuity equations which describethe diffusion, recombination, and, in the presence of anyelectric field, drift of carriers across the SCH. T he secondpart is the carrier capture by the quantum well. This is aquantum mechanical problem which has to take into ac-count the relevant dynamics of the phonon scatteringmechanism which mediates this capture. This scatteringprocess is a function of the initial and final state wave-functions, the coupling strength of the transition, and thephonon dispersion in the material.The next transport mechanism which is significant indevices operating at room temperature is the thermally ac-tivated carrier escape from the quantum well or ther-mionic emission. A lthough this process is in oppositionto carrier capture and degrades the overall carrier captureefficiency of a SQW structure, it is essential for carriertransport between quantum wells in a MQW structure.Another transport process of interest in a MQW system istunneling between the quantum wells. Thermionic emis-sion is a strong function of barrier height while tunnelingis sensitive to both barrier height and width. T he barriersin a MQW structure would have to be designed such that


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1992

0I

Injection

ElecnonIn ection-Right SC H

x =- L , x =o X = L sFig. 1. Schematic diagram of a single quantum-well laser with a separateconfinement hetemstructure used in the camer transport model .

the quantum wells efficiently capture and contain the car-riers for laser action in a two-dimensionally confined sys-tem, without sacrificing the transport (leading to carriertrapping) across the structure adversely.

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28. NO . I O OCTOBER 1992

and diffusion components are given by

A. Transport Across the SCHFor small-signal modulation experiments, the laserdiode is forward biased and a small microwave signal isimposed upon this dc bias. The response of the laser isthen measured using a high speed photodiode. Since themodulating signal is very small, the device is essentiallyat constant forward bias. Further, the SCH is not com-positionally graded, and this excludes any built-in fieldsin the SCH to aid carrier transport.Under normal operation, the semiconductor laser ises-sentially a forward biased p-i-n diode, with the left andright claddings doped p and n, respectively, and the SCHregion nominally undoped (Fig. 1).In the figure, the elec-trons are injected from the right and the holes from theleft and across the SCH. The important difference is thequantum well in the middle of the SCH where the injectedcarriers recombine. A lthough the carrier injection is fromthe opposite ends of the SCH, any physical separation ofthe two types of charges across the quantum-well layerwould lead to very large electric fields between them. Thelaser normally operates under high forward injectionwhere the carrier density levels are about 10" ~ m - ~ ,ndsolving the Poisson equation in one dimension, assumingthat the SCH layers on right- and left-hand side of thequantum well are uniformly pumped with electrons andholes, respectively, leads to electric fields in excess of lo6V /cm between the carrier distributions. Since the carriers

are highly mobile, they would redistribute themselves un-til the charge neutrality condition is satisfied throughoutthe SCH. Further, at this level of carrier injection, unin-tentional background doping levels are negligible.This laser structure can then be analyzed like a heavilyforward biased p-i-n diode [37]. The equations for theelectron and hole current densities, including both the drift

J p = qDp($ $)In this expression we have used the Einstein relation;(Did = (kT/q) .The continuity conditions are

(4)where U(n,p) is the net recombination rate. A ssuminghigh injection conditions, n =p , and charge neutrality,(aE/ax) = 0, (3) and (4) can be conbined. T he -fieldterm can be eliminated to give the following equation un-der steady-state conditions, i.e., (&/at) = cap/&) =0.

U(p)=0.d2p----Dn +Dpah2 2DnD,In the above equation, if an ambipolar diffusion coeffi-cient, D, =(2DnD,/Dn+D,), is introduced, and if therecombination rate U is detemined by an ambipolar life-time r , then the equation modifies to

ah2 L2,,where La,= J D , ~,s the ambipolar diffusion length. Inthe GaAs system D, >>Dp, and this leads to an ambi-polar diffusion coefficient D, = 20,. In other words thecarrier transport proceeds as if it is purely diffusion, butwith an effective diffusion coefficient which is twice thenormal hole diffusion coefficient. Subpicosecond lumi-nescence spectroscopy experiments done in graded andungraded SCH GaAs-A1,Gal - As quantum-well struc-tures have revealed evidence for ambipolar transport ofcarriers in structures similar to the one that we are mod-elling here [38], [39].B. Carrier Distribution in the SCH

The steady-state carrier distribution in the SCH regionis obtained by solving the diffusion equation ( 5 ) with ap-propriate boundary conditions. The general solution to thissecond-order differential equation isp(x) =Aetx/Lo +BePXILaO

where A and B are constants to be determined from therelevant boundary conditions. At x = -L , , a constantsupply of carriers is established by the dc current biasflowing into the SCH. For a dc current of I,,the requiredboundary condition is


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wherea is the cross-sectional area of the laser diode. Thesecond boundary condition is that at x =0, i.e., at theposition of the quantum well, the carrier density is givenby the steady-state carri er density in the well Pw. The car-rier density in the quantum well is determined by thesteady-state Carrier-photon dynamics within the quantumwell . This is modeled later using a set of rate equationsfor the carrier and photon densities.The coefficients A and B are then

much longer than the SCH layer ambipolar transport timegiven by, rs= ( L f /2DO) .his is similar to the base layerof a BJ T, where the bandwidth l imit due to the minoritycarrier transport time is called the alpha cutoff f requency,and is given by f T = 1 (27rrs).The transport time acrossthe SCH has been directly determined from gain recoverymeasurements in quantum-well optical ampli fiers, and hasbeen shown to experimentally vary as the square of SCHwidth [41], 42]as from (6).The expression for 7s ,derived above, can also be writ-ten as,

The current flowing into the quantum well under steady- This means that the total ambipolar diffusion time can beviewed as an average of the individual hole and electrondiffusion times. This is a rather useful way of concep-tualizing the ambipolar carrier transport in a quantum-wellstructure and will be used extensively later.state conditions I w is given by

= I s sech (k) 4Vw@' tanh (5)where Vw is the volume of the quantum well andL w is thewidth of the quantum well; [email protected] . SCH Transport F actor

We define a differential SCH transport factor asCHnal-ogous to the common base current gain of a bipolar junc-tion transistor (BJT) [40] (3l Wa4QSCH =- sechWe are interested in the small-signal value of cySCH. Theexpression can be derived by substituting La, by Lawhichis given by Ia =41 + j a r ,The small-signal expression simplifies to

1cosh ((L ,'/Du7,)'/* l+w7,)-%CH, smallsignal -

In the final expression on the right, we have assumed thatthe width of one side of the SCH is much smaller than theambipolar diffusion length, i.e., Ls


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1994 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28, NO. 10, OCTOBER 1992

surement that places the electron scattering time in GaAsfrom the bottom of the L to J? valley at around 2 ps [44].The laser is operating at room temperature, and the small -est SCH width we have expFrimentally investigated is 700A and the largest is 3000 A . Under these conditions, bal-listic electron transport should be negligible for the longSCH devices, and may be present in the short SCH ones.Furthermore, at the high carri er densities encountered inlasers, the electron-electron scattering times are very short(=10 fs) [45], and this would adversely affect ballistictransport. Even for short SCH devices, ball istic transportof electrons would not alter the final result, because thereal limit to the high speed performance of these devicesis hole and not electron transport.Any ballistic transport of holes on the other hand couldin principle influence the results of our model. Due totheir much larger mass and a lower valence band discon-tinuity in this material system and therefore lesser excessenergy, and in the presence of high injected carrier den-sities, the holes would thermalize quickly on crossing intothe SCH. Thus for the wide SCH devices, near ballisticor hot carrier effects of holes are not significant at roomtemperature, especially over a distance of 3000 A . Forthe short SC H devices this again may not pose a problem,because we believe that in our present structures the RCparasitic and device heating limits are lower than the car-rier transport limit.E.

Calculating the carrier capture time by the quantum wellinvolves the computation of the probability per unit timethat a carrier in some initial state in the SCH emits a lon-gitudinal optical (LO) phonon, and ends up in some finalstate within the quantum well as dictated by energy andmomentum conservation conditions. Early theoreticalworks based on semiclassical treatment argued that thequantum-well width had to be of the order of or largerthan the L O ehonon scattering limited electron mean freepath (=100A) for the quantum well to efficiently capturecarriers, and act as a center for radiative recombination[46l, [471.More recent quantum mechanical calculations have re-laxed this condition, but have predicted strong oscilla-tions in the capture time as a function of the quantum-well width. As the well width increases, first the stateswithin the quantum well become more tightly bound, andsecond, the virtual states above the quantum well becomebound to the well. These result in the final energy stateswithin the quantum well moving in and out of the reachof any state in the SC H separated by a phonon energy,from which a carrier can scatter [48]-[50]. Theoretically,calculated capture times have been found to oscillate be-tween 10ps and l ns for quantum wells with an ungradedSCH structure [48], and between 20and 200 ps for quan-tum wells with a linearly graded SCH [49]. Neither theoscillations nor the capture times of such a large magni-tude have been experimentally observed [36]. The dis-

Carrier Capture Time of the Quantum Well

persion of the LO phonons were neglected in these ini tialcalculations, and it has been speculated that the reso-nances resulting from the variations in the confinedphonon modes, as a function of well width, could even-tually modify these theoretical results [36].Recent subpicosecond time resolved measurements ofthe barrier luminescence decay in the GaAs-A lGaAs sys-tem, have determined the quantum carrier capture time tobe 650 fs for the holes and 1.2 ps for the electrons, andbe independent of quantum-well width [36], [ 5 11.F . Egects of a Graded SCH L ayer on Carrier Transport

The use of a linearly or parabolically graded index(L -GRIN or P-GRI N) SCH layer has been experimentallyshown to enhance the carrier collection efficiency of thequantum well [35], [36]. In the GaAs-A1,Gal -,A s sys-tem, this grading is achieved during crystal growth bycontinually varying the A1 mole fraction x to introduce aquasi-electric field within the SCH. At low temperatures(6 K ), quantum wells with the L -GRI NSCH have beenshown to exhibit almost 100% collection efficiency whilethose with an ungraded SCH only about 50% [35]. Timeresolved photoluminescence experiments have shown thatthe decay time of the luminescence from a L-GRI NSCHcan be as much as 11 times faster than the ungraded SCHluminescence at low temperatures [36]. The performanceof the P-GRINSCH quantum well is in between these twolimits, probably due to the electric field going to zero atthe quantum-well interface.At low temperatures, the main transport limit in theL -GRI NSCH is the carrier capture by the quantum well .At room temperature the drift time in the L -GRINSCHbecomes important compared to the quantum-well capturetime. This is in contrast to the ungraded SCH where thevariation in transport time is essentially due to the vari-ation of mobil ity with temperature. T hus at room temper-ature the L -GRI NSCH has been predicted to be only twiceas fast as the ungraded SCH [36]. This would not lead toany improvement in the high speed performance of L -GRINSCH lasers because the width of the L -GR INSCHneeded to maximize the optical confinement factor is abouttwice that for an ungraded SCH, and for any shorterwidths the confinement factor decreases rather rapidly.Further, there are some recent experiments which havedetermined that the gain recovery times, which are a mea-sure of the carrier transport/dwell times in the SCH, in L -GRINSCH quantum-well optical amplifiers may actuallybe about 3 to 4 times larger than in ones with ungradedSCH of comparable width [42]. Thus, at room tempera-ture there may not be any real advantage in using a L-GRINSCH for the design of a high speed quantum-welllaser.G. Thermionic Emission

Assuming that the carriers in the barriers have bulk-likeproperties and obey Boltzmann statistics, Schneider et al.


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[52], have derived the thermionic emission lifetime 7,from a quantum well:

where EB is the effective barrier height, m* is the carriermass, kB is the Boltzmann constant, and T is temperaturein K . Thermionic emission is a sensitive function of bar-rier height and temperature.H . Tunneling

In a MQW structure there are a number of wells eachseparated by barriers of energy height EB and thicknessLB. In this case the electron and hole wave functions areno longer completely localized within the individualquantum wells, and for any finite LB there will be cou-pling between the wells. For a system of two symmetricwells, the interwell coupling would remove the energydegeneracy which exists in the limit of infinite LB. Thelowest odd (E,) and even (E,) bound states of such a twowell system is separated by some small energy AE = E ,- E, (E , >E ,) [53], [54]. The overall linear superposi-tion wave function of this system corresponds to an elec-tron or a hole oscillating between the wells at a frequencygiven by (AE /h )where h is the Planck constant. T he tun-neling time, defined as the one half period of the oscilla-tion, is given by

h7l =-

Although this result has been derived for a two well sys-tem, it wil l generally be true for a MQW system in thelimit of weak interwell coupling [53].

2 AE '

I . Multiple Quantum- Well StructureThere are additional complications due to carrier trans-port between the various wells in the MQW system. Againcharge neutrality is assumed tohold in the entire intrinsicSCH region, and holes dominate the carrier dynamics. Theanalyses for carrier transport across the SCH and captureby the first quantum well is the same as for the SQW case.Due to their very small capture time we can assume thatall the holes are captured by the first quantum well, andsubsequent transport across the MQW structure is eithervia thermionic emission, then diffusion across the barrierand capture by the next quantum well or tunneling throughthe barrier. For a well designed barrier, the subsequentdiffusion and capture times are negligible compared to theinitial thermionic emission time. Thermionic emission and

tunneling are competing processes, and the faster one willdominate. If a barrier transport time 7b is defined, then1 1 1_ - -- +- .76 7e 7~

The barrier transport time 76 corresponding to the elec-trons and holes would be different. Since the ambipolar

- 100z; 0c)bE$ 1

F

aEffective BarrierTransport Time.\U

0 5 0 100 1 5 0 2 0 0 2 5 0 300Barrier Width (A)

Y ig. 2. Variation of the electron and hole transport times between two 80A In,,Gq.,As quantum wells with bamer thickness, for a GaAs bamer.The effective transport time is the average of the individual transport times.

transport times can be viewed as an average of the re-spective hole and electron transport times, a quantitycalled the effective barrier transport time can be definedas follows:7 6 ,holes +76,electrons

7, =We have originally proposed a rate equation model forthe MQW laser in [32], based on the derivations pre-sented in detail here. I n this paper we are not presentingthe model again, rather we will be quoting some of theprevious results or presenting some new ones based on themodel.Fig. 2 shows the variation of the carrier transport times,at room temperature, for different barrier thickness for aI I I ,,~G~~.~A Suantum well an! a GaAs barrier. F or suffi-ciently small widths (


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1996 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28, NO . IO, OCTOBER 1992

for the SQW case, whi le the MQW case is analyzed viaa numerical model.A. Rate Equations

The rate equations for the carrier density in the quan-tum well (Pw)and the SCH layer (PE) nd the photon den-sity in the cavity (S)are written as(7)p B - I + pw(VW/vSCH)dt qVSCH 7s re

(9)where r, is the bimolecular recombination l ifetime, rp sthe photon lifetime, r is the optical confinement factor,G(P) is the carrier density dependent gain, vg is the modevelocity, E is the gain compression factor, Vw is the vol-ume of the quantum well, and VsCH is the volume of theSCH. We have neglected the spontaneous emission feed-back term because it does not have a significant effect onthe modulation response. In previous work [34], we hadused two independent equations for the carrier density inthe SCH; one for the layer to the right of the quantumwell and the other for the layer to the left. The final resultsof our previous work are not altered in any way if onlyone equation is used for the carrier density in the barrier.The optical gain G(P) is a function of both the electronand hole carrier density within the quantum well. Sincecharge neutrality has been assumed, the electron and holedensities are equal to one another in the quantum well . Asin Section 11, we have ignored carrier recombination inthe SCH layer.In the rate equations, the terms with 7, represent thelossof carriers from the quantum well and the gain by theSCH layer, and the terms with rsvice versa. Although inprinciple r, would have to include the finite capture time

tions in terms of volume carrier densities. The volumeratios VscH/ Vw and Vw/ VsCH have then been used in therate equations to account for the fact that PB and Pw arenormalized with respect to two different volumes.The small -signal solution of the equations is done byfirst making the following substitution: I =Io +i ,jut, PB= PE , +pBeJ wr,w=,Pwo+pweJW', = So + s eJ utand G = Go +gopwelu'. Although G(P) is a sublinearfunction of the carrier density, it can always be linearizedabout some steady-state operating point, where go is thedifferential gain at that steady-state carrier density. Forthe small-signal carrier density variation pw,go is a con-stant because the steady-state carri er density in the quan-tum well is clamped at the steady state value of P ,. Thego is then different for different values of Pwo, and thisaccounts for the saturation of gain at high carrier densitiesin quantum-well lasers.B. Modulation Response

After the small-signal quantities are substituted into (7),(8), and (9), the steady-state quantities are set to zero.The resulting small -signal equations are

r v ,goso S Sj u s - 1+SoPw +The steady-state solution to the photon density equationgives the basic gain-loss relationship in the laser cavity;I'ugGo/(l +So)= l / rp , his set of small -signal equa-tions can be reduced by eliminating pB and pw to give arelationship between the modulating current i and the op-tical output s.

~ ~ ~of the quantum well, we have neglected it here because itis


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The expressions for A , us, and y arer(ggo/x) o4VW(l +So)rp(l +So)(~gg0lX)So fS0/7p 1(1 +So) (1 + So) xrn

A =U; = ( V & J I X ) S O( + - vg;or)

+ +-=where a transport factor, x = 1+ r,/r,, has been intro-duced into the equations. In the modulation response, U,and y are commonly referred to as the resonance fre-quency and the dampin rate, respectively. T he originalexpressions for y and U, are recoverable in the limit of x= 1, i.e., in the limit of short transport time across theSCH.From these analytic expressions a relationship for theK factor can be derived. The K factor was originally de-fined as y = Kfs [lo]. However, experimentally, a dcoffset is always observed and therefore a better definitionis y =K fs +yo[34]. From this definition,

9

Therefore, depending on the laser structure, the K factorcan be significantly affected by transport.From these analytic expressions a number of effects oftransport are evident.1) The low frequency rolloff in the modulation re-sponse which is only due to the transport time acrossthe SCH [31], [32], [34]. This is a parasitic-like

rolloff which would significantly limit the -3 dBmodulation bandwidth for large r,, and is indistin-guishable from the parasitics.2) The effective differential gain has been reduced tog , / x . This reduction is not responsible for the roll-off mentioned above in l ) , and is present even inthe absence of a significant rolloff. This results inthe reduction of the resonance frequency for thesame power levels.3) The effective bimolecular recombination lifetimehas been increased to xrn.This leads to a structuredependence of yo.4) The gain compression factor remains unmodified.This is in contrast to [311where the well barrier holebuming model has been used to explain the degra-dation of K factor via an enhancement in e.I ff ? and y remain linear functions of power, the max-imum possible -3 dB modulation bandwidth in a laser isgiven by [lo], maX = h (27rlK ). However, as men-tioned previously, in practice the maximum modulationbandwidth is often limited by RC parasitics, device heat-

h5Exact Solution5 Analytical Solution

0$ - 5d -1 0f.

- 1 5-2 0

Width of SCH=3000 A

0 4 8 1 2 16Frequency (GHz)

Fig. 3. Comparison of the exact and analytical solutions to the modulationresponse for an 80 A In,,2G%,,As SQW laser with 300 pm cavity lengthat40mWof output power. The modulation r esponse has been normalizedwith respect to 1 M(0) .

ing, and maximum power handling capability of the laser.Further, in light of point 1) above, the K factor is not areliable indicator of the modulation performance in thepresence of severe low frequency rolloff due to carriertransport across the SCH; this latter would then determinetheintrinsic limit to the maximum modulation bandwidth.If the assumption to derive (10) is not made, then sim-ple analytic expressions for the resonance frequency anddamping rate are no longer possible. T he complete M(u)function is

A, = ( 1 +&)7P

A I = v,goS, (1 +; +$ 1 -I- +:17,

+- 1 +So)ESOTS+vggoSors+-

TPA3= r,(1 +do). (12)

The denominator is a third-order polynomial and can besolved numerically. Fig. 3 compares the modulation re-sponse computed using the analytic solution given by (10)and the exact solution given by (12) for the case of thewidest SCH devices used in ourexperiments. The agree-ment is good and we will present more evidence later toshow that the analytic solutions are in fact valid.C. Relative Intensity Noise (RI N) Spectrum

The relative intensity noise spectrum can be derivedusing the same formalism. T he dif ference in this case isthat the driving force is no longer the input current, butrather the real part of the Langevin force F , of the field


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1998 I E E E J OURNAL OF QUANTUM ELECTRONICS. VOL. 28, NO . IO , OCTOBER 1992

due to the spontaneous emission which is assumed to beuncorrelated white Gaussian noise [56]. ncluding thisterm, the small-signal equations are.8 - 10p.%Eq 0.B & - 5-mC O

-1 0ocrlE - 1 5

-2 00 4 8 1 2 16

Frequency (GHz)Fig. 4. Noise spectrum and the modulation response for the same laser asin Fig. 3. The noise spectrum indicates a much larger modulation band-width than is physically possible. The modulation response has been nor-malized with respect to IM(0) , and the noise spectrum has been normal-ized such that it may be conveniently plotted on the same scale as themodulation response.We will eliminatepB andpWand obtain the expressionfor the Fourier component of the amplitude fluctuationsR(w) due to F ,. Again neglecting the frequency depen-dence of rsas previously, the resulting expression is

The other terms of the expression are the same as themodulation response function. The y and y* terms arealmost identical at sufficiently low powers and small E.The relative intensity noise (R IN) spectrum is the powerspectral density of R(w).The RIN spectrum can be writtenasy** + w2

( U;- w2)2 +w2y2RIN ( U)= Cwhere C is the term resulting from F ,, and gives the in-trinsic Schawlow-Townes linewidth which is due to spon-taneous emission events [56].The additional low frequency rolloff present in theexpression for the modulation response is absent here. Forlarge values of rsthe modulation bandwidth could be se-verely reduced, although the K factor values, which areaffected by x alone and not the low frequency rolloff, asdetermined from the noise measurements could still beoptimistic. It is generally believed that the noise spectrummeasurement is aparasitic f ree means of determining theactual or potential modulation performance of a laser. Inthe presence of significant transport effects this is nolonger true. Fig. 4shows the modulation response cal-culated using (12), and the noise spectrum calculatedusing (13) at a powerJeve1 of 40mW for a device with aSCH width of 3000 A. The peak of the noise spectrum,which is a good indication of the resonance frequency forthe case of small damping, occurs at 10.9GHz implyinga maximum possible modulation bandwidth of 16.9 GHz[2] at that power level. This is clearly a wrong conclusion

as theactual bandwidth limited by carrier transport in thiscase is only 4.5GHz.Without any simplifying assumptions, R(w) s given byBo + B l w - B2w2R(w) =F , (13)+ A l w - ,W* - ~ , w ,

where the numerator terms are given by

B2 = 7,(1 + So).The denominator terms remain unchanged. To obtain yand U, n the conventional sense of their definitions, i.e.,with respect to a second order polynomial in the denom-inator, one would have to fi rst compute the RIN spectrumusing the expression given above, and then numericallyfit it to a conventional form of the expression and extractthe relevant parameters.

IV . MATERIAL ARAMETERSThe main material parameters that are required for thecomputation of the optical gain and the transport times arethe hole and electron mobilities, bandgaps, and effectivemasses in the GaA s-A lGaAs and the strained InGaAs-GaAs system. We have also measured the noise spectraat various temperatures above and below room tempera-ture to confirm the mechanisms that limit the high speeddynamics in the quantum-well lasers. T o model these ef-

fects we also need accurate temperature dependence datafor the various parameters.A . Carrier Mobili ty and its Temperature Dependence

At thermal equil ibrium the diffusion coefficient D is re-lated to the carrier mobility via the Einstein relation. Innondegenerate semiconductors the following relation

I


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NA GA RA JA N et al.: CARRIER TRANSPORT EFFECTS 1999

holds;

where the subscript d refers to the fact that the mobilityused is the drift (conductivity) mobility [57]. Since mostof the transport parameters reported in literature are typ-ically from Hall measurements, the resulting mobilityis the Hall mobility pH. The two mobil ities are related toone another via the Hall factor rH.

PHrHpd = -.

We have taken the temperature dependence of hole mo-bility to have the following form300 (K ) P = P300K (m)

For holes in GaAs, at sufficiently large temperatures,i.e., in the region of interest to us (100 K to 400 K ), Hill[58] has suggested that /3 = 2.41 would be a good fit tothe temperature dependence of mobility, while Wiley [59]from his earlier work and an extensive review later, sug-gests a 0 of 2.3. This is somewhat higher than the tem-perature dependence of T -2.1 suggested by Sze [57]. Anumerical fi t to the later work by M asu et al., 1601 givesa temperature dependence of T-2.3. he Hall factor is alsotemperature dependent, and after accounting for i t, A d-ams [61] suggests a /3 of 2.2. We have assumed thisvalue of /3 also for A lo,lG ao,9A s,which forms the SCHlayer. We have taken the Hall mobil ity for holes at roomtemperature in A lo,1Gao,9Aso be 250 cm2/Vs [60], andthe Hall factor to be 1.25 [43] resulting in a value of P3mK=200 cm2/V s.The electron mobility is more complicated. No simplepower fit would be suitable to model the temperature de-pendence of electron mobility in the range of our interest[43], [62]. We have done a polynomial fit to the data re-ported by Bhattacharya et al., [63] and obtained the fol-lowing expression for the temperature dependence ofelectron mobili ty in A lo. Gao,9A s;

We have taken the Hall factor to be 1.175 [43], and in-dependent of temperature.B. Optical Gain and its Temperature DependenceWe have theoretically calculated the carrier density de-pendent optical gain including valence band mixing ef-fects using the model presented in [14]. A conductionband offset of 60% was used, and rest of the material pa-rameters were taken from [14], and used throughout thispaper for consistency.

We have also taken the band offset ratio to be temper-ature independent and the bandgaps of the various con-stituent layers to have the same temperature dependence.This assumption would ensure that the energy offsets be-tween the various layers, or in other words the potentialbarrier heights, in the laser structure are independent oftemperature although the absolute bandgaps are changing.Fig. 5 shows the theoretically calculated gain-thresholdcurrent density relation presented together with the ex-perimental data for threshold gain obtained from broadarea measurements on three wafers of SQW lasers. Ex-cept at very low current densities there is a good agree-ment between the model and the measured gain values.

V . EXPERIMENTND DISCUSSIONWe have carried out a series of experiments to verifythe predictions of our model, utilizing systematically var-ied SQW and MQW laser structures. The experimentsused Ino.2Gao,8As-GaAs strained quantum-well lasers.The wafers were grown by molecular beam epitaxy(M BE), and processed into polyimide buriedo idge wave-guide lasers. T he quantum wells were 80 A thick, andthe barriers for t MQW sample were composed of GaAsand were 80 A in thickness. The SCH region wasA lo.lGao,9As, and the p and n cladding layers wereAlo,6Gao.4As. he wavelength of operation of the lasers is980 nm. T he details of the crystal growth and device pro-cessing have been published [19], [26].Experimental results pertaining to two sets of devicesare considered here. T he static characteristics of these de-vices are given in Table I. First is a set of three samples(A, B, and C) which were consecutively grown and iden-tically processed together. These samples differ only inthe width of their SCH layers. The results from this setwill be used to directly demonstrate the effects of carrier

transport on the high speed dynamics of quantum-well la-sers.The second set consists of sample D which is a SQWlaser, and Ewhich is MQW laser. These two samples wereprocessed together, and their data will be used later toshow the differences in high speed characteristics betweenSQW and MQW devices.The internal loss is somewhat higher for samples A , B,and C because the cladding layers in these samples hadbeen doped to a higher level than samples D and E. Thishelped to reduce the series resistance, and hence the de-vice parasitics for this set of lasers.A. Experimental Results

Among the first set of l asers, sample C had the widestSCH region, and consequently the lowest optical confine-ment factor. Despite this, i t also had the lowest broad areathreshold current density, because it had the lowest inter-nal loss. This is signif icant because, due to gain saturationwith increasing carrier density in SQW lasers, sample Calso had the largest static differential gain. This is illus-

T


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2000 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28, NO. IO , OCTOBER 1992TA B LE I

Conf inement Broad Area InternalWidth of SCH Factor Threshold Current LossL3 r Density %Sample (One Side) (A ) ( W) (A/") (cm-')

A (1 QW) 760 2.9 140 21.8B (1 QW) 1500 2.7 129 11.8C (1 QW) 3000 1.9 93 4.3D (1 QW) 900 3.0 125 6.9E (3 QWs) 700 9.2 247 14.4

3000M.5;000d"

1000

00 Experiment

I . . . . I . . . . I . . . . I . . . . I . . . . l . . . .

0 100 200 300 400Current Density (A/cmZ)Fig. 5. Theoretical and experimental variation of material gain with cur-rent density for an 80 A In,,,Ga,,,,As-GaAs quantum well . The experi-mental data are from the broad area measurements on samplesA , B, and C(see Table I).

trated in Fig. 5, where the operating points of laser sam-ples A, B, and C for a 300pm cavity length are shown.The 300 pm cavity length samples, all of which had2.5 pm wide ridges, were used for the modulation band-width measurement. For this measurement, each individ-ual laser diode was soldered to a copper heatsink andbonded to a 50Q microstrip transmission line on an alu-mina substrate with a gold mesh. The other end of themicrostrip was attached to the center of a K -connectorlaunch. T he modulation response of this microwave pack-age was then measured using a HP 8510 vector networkanalyzer and a high speed photodetector 1641.Fig. 6(a) and (b) shows the CW modulation responseof samples A and C. Although the devices were identicalexcept for the SCH width, the modulation response ofsample C with the widest SCH region was completelydamped at about half the output power level of sample A,and the bandwidth of sample A was six times that of sam-ple C at higher power levels. The maximum CW modu-lation bandwidth for sample A was 18.1GHz, which isthe largest reported to date in SQW lasers. Sample B hadalmost as good a performance with a maximum modula-tion bandwidth of 17.7 GHz at an output power level of41.9 mW.

The damped response of sample C showed all the ef-fects predicted by the analytic model. The response issimlar to one of a device which is limited by a low fre-quency RC li ke parasitic rolloff. In this case it cannot beattributed to device parasitics because both devices hadbeen identically processed. To ascertain this, the para-

10

6hmUvi3 - 2- 6

-100 4 8 12 16 20Frequency (GHz)

(a)10

2.4 mW Samde C

0 4 8 12 16 20Frequency (GHz)

(b)Fig. 6. Modulation response for (a) a narrow SCH (sample A ) and (b) awide SCH (sample C) SQW laser. The modulation response for the wideSCH laser, at comparable power levels, shows the detrimental effects ofthe low frequency rolloff due to carrier transport.

sitics, including the bond wire inductance, were extractedfrom the S 11parameter measured at the input port to thelaser mount, and in addition the series resistance wasmeasured using the H P 4145B semiconductor parameteranalyzer. T he series resistance of the devices is typically4 Q, and the capacitance is 1.6pF. Thus, theRCproductis only 6.4 ps, i .e., the rolloff frequency due to the deviceparasitics is about 25 GHz.The low frequency rolloff in sample C is, as predictedin Section 111, due to the transport across the SCH. Fromthe fit to the low frequency rolloff in the modulation re-sponse of sample C, the va!ue of 7, was determined to beabout 54 ps for the 3000 A SCH width. Using the mo-bility values that we have taken from published sourcesand quoted in Section IV , the calculated value for 7$ s 46ps which is close to the experimental result.By fitting the modulation response and the relative in-tensity noise spectra of these lasers to the expressionsgiven in Section 111, we computed the differential gain andthe gain compression factor E . As shown in Fig. 7, theresults from the modulation response agree with the dataextracted from the noise spectra measurements. Fig. 7shows that the gain compression factor slightly increasesfor larger SCH widths, but given the scatter in the data,this variation of e with SCH is not significant, and thisvariation alone cannot explain the severe low frequencyrolloff in sample C with the widest SCH region.


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NA GA RA JA N er al.: CARRIER TRANSPORT EFFECTS 2001l w '

hN5e.-d t iA Noi se Spectra

0 M odulation Response

0 1000 2000 3000 4000

1 0h-r?U 8CL0- 63v

0 1000 zoo0 3000Width of SC H (A) Width of SC H (A)

hThe variation of the differential gain extracted from themodulation response and noise spectra measurements,which can be thought of as an effective differential gain,is in contrary to the results from the threshold gain mea-surements which predict an increasing differential gainvalue with increasing width of the SCH layer (Fig. 5).Sample C with the lowest threshold current density hadthe lowest effective differential gain. This is again due tocarrier transport which reduces the differential gain fromgoto go/x (Section 111). This reduction in the differentialgain is also responsible for the reduction in the resonancefrequency [Fig. 6(a) and (b)] at high power levels in sam-ple C compared to sampleA .The last effect of Carrier transport predicted in Section

0 1 2 3 4 5 6 7 8Power ( mw / *)

Fig. 9. Comparison of the camer transport model and experiment for thevariation of modulation bandwidth with the square root of power. Thedashed line is the result from the analytic solution and the bold line that ofI11 was the reduction of yowith increasing SCH width,and this is shown in Fig. 8. Increasing the width of SC H,increases x which decreases yowhich is given by 1/xr,[(I 111.B. Comparison to the Model

Fig. 9 shows the variation of the - 3 dB modulationbandwidth with the square root of optical power for sam-ples B and C. There is good agreement between the ex-periment and model over a large range of optical power,and the model accurately predicts the discontinuity,caused by carrier transport, in the -3 dB bandwidth forsample C . We have taken E to be 1.5 x lo-'' cmP3throughout. The bold l ine gives the solution to (12) andthe dashed line shows the palyti cal results from (10). Forthe SCH width of01500 A both equations give identicalresults. At 3000 A , except in the vicinity of the powerlevel at which the low frequency rolloff causes a "jump"in the curve, the bandwidth values predicted by bothexpressions are almost the same. This shows that the as-sumption made to obtain (10) in Section I11 is valid evenfor the devices with the widest SCH layers.Fig. 10shows that the model predicts the existence ofan optimum SCH width for the maximum modulationbandwidth in SQW lasers. Here again the predictions of(10) are very close to the results from the exact expres-sion. This optimum width also corresponds roughly to the

the exact expression. Severe low frequency rolloff due to carrier transportcauses the discontinuity in the curve for sample C at high power levels.

B

0 1000 2000 3000 4000Width of SC H (A)

Fig. 10. Comparison of the calculated and measured variation in modu-lation bandwidth with SCH width. The dashed li ne is the result from theanalytic solution and the bold line that of the exact expression. A s in Fig.9,carrier transport effects dominate at sufficiently high power levels.

point at which the optical confinement factor is a maxi-mum. For a narrow SCH, the bandwidth drops off due todecreasing confinement factor resulting in a larger thresh-old gain and thus a lower differential gain. A t larger SCHwidths, the combination of a decreasing confinement fac-tor and an increasing carrier transport time across the un-doped regions of the SC H, limits the modulation band-


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2002 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28, NO. I O OCTOBER 1992

0.8 . . . . I . . . . I . . . . I . . . . I . . . .Sample DSCHWidth =900A

0.6 -2vL5 0. 4 -k /-+

0.2 1

0.0 " ~ ' l ' ' ~ ' l ' . . " ~ ~ ' " . ' " -

2 5 :

15 :

3 QW s - Sample E(&SQW - Sample D A

0 200 400 600 800 1000Cavity Length (pm)

Fig. 11. Comparison of the calculated and measured variation in modu-lation bandwidth with cavity length for SQW and M QW lasers. The SQWlasers exhibit a broad maximum for the cavity length dependenceof mod-ulation bandwidth whil e MQW lasers are generally more sensitive to cavitylength variations.

width. A t sufficiently high powers for wide SCH devices,the characteristic drop due to carrier transport, appearsagain in the modulation bandwidth curve.Fig. 11 shows the cavity length dependence of modu-lation bandwidth for samplesD and E. The parasitics weremore significant in this set of devices, and the RC productwas about 14.5ps [19]. The effects of the parasitics havebeen deconvolved from the experimental data presentedin Fig. 11. MQW lasers generally have larger modulationbandwidths than SQW lasers, and this has been broughtout by the trends in the experiment and the results fromthe model. A s the cavity length is reduced, the thresholdgain increases leading to a decrease in the differential gain.This is balanced by the reduction in the photon l ifetime,and this results in an optimum cavity length for the max-imum modulation bandwidth. The optimum length for aSQW laser is around 300pm, and this has also been borneout accurately by the model. The optimum point is some-what smaller for the MQW case, and beyond the cavitylength limit that can be cleaved with a reasonable yield inpractice. Herein lies the practical advantage of a quan-tum-well laser with a smaller number of wells; they canbe easily fabricated for high speed applications. Further,longer cavity length lasers can be more effectively heat-sunk, and therefore suffer less from the adverse effects ofdevice heating.

VI. TEMPERATUREEPENDENCEATAThe various carrier transport times in a quantum welllaser are a sensitive function of temperature. A s a furtherproof of validity of the present model, we have measuredthe noise spectra of the lasers at various temperatures be-tween -65C and +5O" C, and extracted the temperature

dependence of K factor.In semiconductor lasers, even in the absence of trans-port effects, the K factor would increase with temperatureif the gain compression factor is temperature independent.This is because the gain, and thus the differential gain,decreases as the temperature increases. In the presence of

Sample C Anal ytic. SCHWidth =3000 A 00. 6 -

L2.%v3 0. 4I% Transport.

0.2 1 ,0.0

150 200 2 50 300 350 40 0Temperature(K )

(b)Fig. 12. Temperature dependenceof K factor for (a) a narrow SCH (sam-ple D) and (b) a wide SCH (sample C) SQW laser. A large contribution tothe significant variation of K factor with temperature for wide SCH devicescomes from the temperature dependenceof the transport parameters.

transport effects, this increase would be more pronouncedbecause the transport factor x also increases with temper-ature leading to a further reduction in the differential gain[see (1 )] .Fig. 12(a) shows the variation of K factor with temper-ature for sampde D, which is a SQW laser with a SCHwidth of 900 A , and Fig. 12(b) that of sample C,owhichis also a SQW laser but with a SCH width of 3000A [65].TheK factor for sampleC is much more sensitive to tem-perature variations than sampleD.In Fig. 12(a) and (b), together with the experimentaldata we have presented results f rom the model for threecases; the first curve was computed using the analyticexpression in (1 ), the second using the solution to (12),and the third using (11) but neglecting transport effectsand considering only the variation in differential gain withtemperature. For the case of sample D, where the trans-port effects are not significant, all three curves lie closeto one another and do not show much variation with tem-perature. For sample C , variations in differential gain withtemperature alone is clearly insufficient to explain the Kfactor trend with temperature. The enhanced sensitivityof the K factor in this case arises from the additional tem-perature dependence of the transport parameters.Fig. 13 shows the variation of r,, r,, and x with tem-perature for sample C with the widest undoped SCH re-gion. T he thermionic emission time decreases exponen-


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E

L# /. . . . . . . , . . . .F2.01.8

hPV

1.6 2+1. 4 r(

?-?v

1.2

1.o1 50 200 2 5 0 3 0 0 3 5 0 40 0

Temperature (K )Fig. 13. Calculated temperature dependenceof the transport times and thetransport factor x for the wide SCH sample. A s the temperature ncreases,the thermionic emission lifetime fr om the quantum well is considerablyreduced, and the transport time across the SCH is somewhat enhanced,leading to a significant increase in the transport factor x for wide SCHdevices.tially with temperature, and is the major contributor to theincrease in x. The SCH diffusion time is temperature de-pendent via the temperature dependence of the ambipolardiffusion coefficient. Generally the camer mobilities de-crease as the square of the temperature, and the diffusioncoefficient increases linearly with temperature leading toa linear increase in T~with temperature. Both the decreasein 7 , and the increase in r , lead to an increase in x, andthus an enhanced reduction in K . This clearly demon-strates the significant role played by camer transport inthe modulation dynamics of semiconductor lasers.

VI I . HIGHSPEED ASERDESIGNIn general, one can vary a number of parameters tooptimize the high speed performance of quantum-well la-sers, and there is usually a useful range over which eachparameter can be varied effectively. In this section we wil luse the carrier transport model to discuss such an opti-

mization process. A lthough the process described herewill use the InGaAs-GaAs system as the example, it is ingeneral applicable to quantum-well lasers from any ma-terial system. A nother material system of current interestis the InGaAs-InP system which is commonly used forthe fabrication of semiconductor lasers operating in the1.3 to 1.55 pm wavelength region. A lthough we will notdeal wi th the InGaAs-InP system in detail , at the end ofthis section we will highlight the major differences in car-rier transport between the InGaAs-InP and the InGaAs-GaA s systems.In quantum-well lasers there exists an optimum cavitylength for the maximum modulation bandwidth. This isquite unlike lasers with bulk active areas where shortercavity lengths generally lead to larger modulation band-widths due to the decrease in the photon lifetime. I n quan-tum-well lasers the differential gain is strongly carrier-density dependent, and this works in opposition to the ef-fects of a reduced photon lifetime. These tradeoffs are il-lustrated in Fig. 14, which shows the optimal cavity lengthfor devices with a different number of quantum wells.

0 20 0 400 600 80 0Cavity Length ( pm

Fig. 14. Variationof modulation bandwidth as a function of cavity length.This has been calculated for devices with 1000A SCH, 80A In, ,Asquantum well , 80 A GaAs barrier and 2.5 pm ridge operating at 20 mWof output power. MQW lasers generally have a larger bandwidth than SQWlasers but are a lot more sensitive to cavity length variations.

Generally, MQW lasers have a larger modulation band-width than SQW lasers, but the cavity length in the MQWcase has to be sufficiently short to take full advantage ofthis. The optimal cavity length for the SQW case is 300pm, and decreases to 60pm in the case of five quantumwells. T he locus of the optimal cavity lengths is also in-dicated in Fig. 14. For three or more quantum wells theoptimal cavity length is below 100 pm. It is difficult tocleave lasers below this limit, and one may have to resortto dry etching techniques to fabricate lasers with very shortcavity lengths [20]. Further, lasers with short cavitylengths are more prone to thermal heating, and may notperform as well as theoretically expected. To overcomethis practical problem one may have to use MQW laserswith nonoptimal cavity lengths.Fig. 15(a) and (b) presents two cases for the optimi-zation of SCH widths. First is the case where the claddingloss is dominant. This is true in the caseof samplesA , B ,and C where the total internal loss increased inversely withthe square root of the SCH width. In this case the SCHwidth is the dopi ng offset between the quantum-well ac-tive area, and the point from which the claddings are beingdoped. For carrier transport, it is the width of the undopedregion between the quantum well and the cladding layerthat is of significance, and in most cases, like ours, it isthe width of the SCH. Fig. 15(b) is a more typical casewhere the doping levels in the claddings are not very high,and consequently the loss in the active area dominates. Inthis case, the total internal loss can be written as a func-tion of the optical confinement factor. This is true forsamples D and E . The functional relations for the totalintemal loss have been given as insets in the figures them-selves. The various parameters in the relationships, as ,ab, aactive, and cycladding were experimentally determined.In both cases, there exists an optimum value of SCH widthfor maximum modulation bandwidth. The bandwidths aresomewhat higher in Fig. 15(a), because the higher inter-nal loss in this case also leads to a reduced photon lifetimewhich enhances the modulation performance. This en-hancement comes at the expense of a larger operating cur-


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2004

3 53025201 5

105

0 5 0 0 1000 1500 2000 2500Width of SCH (A)

(a)h

30 atot= aactivc +I 1 adadding9

U

0 50 0 1000 1500 2000 25002 5 f a . . . . . I . . I - . - :Width of SCH (A)(b)Fig. 15. (a) Variation of modulation bandwidth with SCH width for thecase when the cladding lossis dominant. This is calculated fordevices with300pmcavity l ength and other device parameters as specif ied for Fig. 14 .The values fora, and ab n the expression given as the inset in the figurewere obtained by f itting the internal lossvariation with the SCH width forsamplesA , B , and C. (b) Same plot as in (a) but forthe case when the lossin the quantum well active area is dominant. The values fora,,,,,,and cyclad.

d,ng are 124 cm-' and 3.3 cm- I , respectively. These values were obtainedby fitting the internal loss data for samplesD and E.

rent density, and thus the devices would be more prone tothe effects of thermal heating.The SCH could be partly doped or doped all the wayup to the quantum wells to help decrease the transporttime. Depending on the doping level, this may increasethe internal loss considerably, leading to a reduction inthe differential gain and an increase in the operating cur-rent density.The optimum design for the SCH is then one in whichthe quantum well is placed asymmetrically in the struc-ture, i.e., closer to the p cladding. This would minimizethe hole transport time, and the width of the SCH on then cladding side can then be optimized for maximum op-tical confinement factor. One drawback of this structurewould be the asymmetric optical mode pattern along thetransverse direction.Fig. 16shows the modulation bandwidth dependenceon the barrier width for the MQW case. In a MQW struc-ture, the barri ers must be of some minimum thickness toensure that the camers are two-dimensionally confined tothe quantum wells, and the interwell coupling which leadsto a broadening of the quantized energy levels is mini -mized. While taking advantage of the two-dimensional

IEEE JOURNAL OF QUANTUM ELECTRONICS. VOL. 28, NO . 10, OCTOBER 1992

2 o

1 5i5 0 100 150 200 250 300Barrier Width (A )Fig. 16. Variation of modulation bandwidth, from transport considera-tions alone, as a function of he barrier thickness for the device parametersspecified in Figs. 14 and 15.confinement, one has to also ensure that carrier transportacross the structure is not hampered leading to poor highspeed performance. Further, the use of thick barriers, es-pecially together with a large number of quantum wells,will result in carrier trapping leading to unequal carrierconcentrations in the dif ferent wells. Fig. 16considersonly the carrier transport effects, and thus clearly no min-imum limit for the barrier width exists. A lthough there issome bandwidth penalty due to increasing barrier thick-ness, it is not very significant in the range we have con-sidered here. This partly explains the success of highspeed MQW structures with thick barriers [20].In SQW lasers, where the problem of carrier transportbetween quantum wells does not arise, thermionic emis-sion time will have to be maximized in order to minimizethe value of x. There is a limit to which one can increasethe bandgap of the SCH layer to increase T, , before theresulting decrease in the optical confinement factor be-comes unacceptable. T he adverse effect on x is also ap-plicable to the case of MQW lasers, when one considerslowering the barrier height to facilitate carrier transportbetween the various quantum wells. Recently in theInGaAs-GaAs system, where the carrier transport be-tween the quantum wells is not as serious a problem as inthe InGaAs-InP system, large bandwidth enhancementshave been reported in MQW lasers with increased barrierheights [66]. A compromise in design can be achieved inMQW lasers by having a higher energy barrier betweenthe first and the last quantum wells and the SCH layer, tominimize the overall x, and a lower energy barrier be-tween the quantum wells, to facilitate carrier transport.Conventionally, the maximum possible modulationbandwidth in a semiconductor laser was determined bythe K factor. In principle, if one could drive the laser toarbitrarily high output power levels without the deterio-rating effects of device heating, then the maximum pos-sible -3 dB bandwidth, in the absence of device para-sitics, is given by this K factor limit. In the presence ofcarrier transport effects, the K factor is not only reduceddue to the reduction in the effective differential gain, butalsono longer determines the maximum modulation band-width, even in an otherwise perfect device.


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NA GA RA JA N et al.: CARRIER TRANSPORT EFFECTS 2005

605 040302 0100

- ransport L imit (Modulation)

0 1000 2000 3000 4000Width of SCH (A)

Fig. 17. Comparison of maximum possible modulation bandwidth inferredfrom the modulation response and intensity noise measurements. As thetransport effects become significant, the K factor inferred from noise spec-tra measurements are not indimrive of the maximum possible modulationbandwidth. T he real limit i s much reduced, and is determined by the lowfrequency rolloff due to the transport delay.

the same problem, the implications are dif ferent. The firstimplies a small thermionic emission time, and thus a largex, eading to a reduction of the effective differential gain.The second implies a large barrier transport time leadingto hole trapping, a large x and poor operation of a MQWstructure. Since the carrier transport in this case is alsoambipolar, the average of these two effects will governthe dynamics of lasers in this system.Although the interaction between these two effects hasnot been considered in literature, it is generally believedthat the M QW InGaAs-InP lasers suffer from the detri-mental effects of hole trapping [67].One solution to the carrier transport problems encoun-tered in high speed lasers in the InGaAs-InP system, isto use InA lGaAs instead of I nGaAsP for the barrier, SCHand the cladding regions. The conduction band offset ratioin the InGaAs-InAlGaAs system is the same as the In-GaAs-GaAs system. Recently InGaAs-InAlGaAs lasersoperating at 1.55 pm have been shown to have a smallerK factor and a smaller gain compression factor comparedto lasers in the lnGaAs-lnGaAsP system 16*]*

Fig. 17shows the maximum possible modulation band-width as a function of SCH width for a 300 pm cavitylength SQW laser. T he dashed line shows the K factorlimit which is commonly inferred from noise spectra mea-surements. The bold line is the real limit determined bycarrier transport across the SC H; this i s the maximum thatis possible before the low frequency rolloff due to trans-port becomes severe enough to cause a sudden drop (seeFig. 9) in the variation of modulation bandwidth with in-creasing optical power.Herein lies the real danger of relying on K factors andRIN measurements to predict modulation bandwidths. Asmall K factor is indeed an indication of the good deviceand material quality, but one has to be careful as to themaximum modulation bandwidth limits implied by thisquantity. Only in devices where the carrier transport ef-fects are minimal, can one truly use this as the limit.I nGaAs-lnP SystemThe first difference in this system is that the total changein the confinement energy that can be obtained in goingfrom InP to InGaAs, under lattice matched conditions, isonly 0.75 eV compared to 1.59 eV difference in the band-gaps of GaAs and AlA s. InGaAs has a smaller bandgapthan GaAs, thus offering an even larger energy range. Therefractive index difference obtainable for the same com-position range is also larger in the InGaAs-GaAs system.These factors lead to more efficient design of the quantumconfinement and waveguiding structures in the InGaAs-GaAs system as compared to the InGaAs-InP system.The second difference in the InGaAs-InP system is thatthe conduction band offset is 40% and thus is smallerthan the valence band offset. This, together with the factthat the energy range for the design of quantum structuresin the InGaAs-InP system is small, leads to two prob-lems. T he first is that the electrons are not sufficientlyconfined to the quantum wells, and the second is that theholes are rather easily trapped in the quantum wells. A l-though this may seem like two different ways of stating

VIII. CONCLUSIONIn summary, we have shown that the conventional pa-rameters of high speed design in semiconductor lasers arenot sufficient for the optimization of high speed quantum-well lasers. The cladding, separate confinement hetero-structure, and barrier layers in a quantum-well laser haveto be carefully designed to not only maximize the opticaland carrier confinement, but also to optimize the variouscarrier transport times.The transport time 7$ across the SCH has to be mini-mized. Long carrier transport delay in the undoped re-gions of the laser, as in the SCH, introduces a low fre-quency parasitic-like rolloff which severely limits themaximum possible modulation bandwidth. A narrow SCH

is desirable towards this end, but this will increase theinternal loss in the case where the cladding loss is domi-nant, and decreasing the SCH width below an optimumpoint will also lead to a sharp drop in the optical confine-ment factor. These effects will result in a higher carrierdensity at threshold, and consequently a lower differentialgain as a result of rapid gain saturation in quantum-welllasers. These constraints lead to an optimum width for theSCH.The transport factor x given by (1 + ~ , / 7 , )has to beminimized. A large x leads to a decrease in the effectivedifferential gain, and thus a decrease in the resonance fre-quency and an increase in the K factor. So, in addition tominimizing 7,, the thermionic emission lifetime 7, has tobe maximized. This can be done by having deeper andwider quantum wells. A lthough this is true for a SQWlaser, in a M QW structure 7, cannot be increased arbi-trarily by increasing the depth of the quantum wells or thewidth of the barriers, because it will lead to carrier trap-ping in the quantum wells, and poor carrier transport be-tween the quantum well s. T hese compromises lead to an


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2006 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28. NO. 10. OCTOBER 1992

optimum barrier height and width to maximize the mod-ulation bandwidth in MQW lasers.Modulation bandwidth in quantum-well lasers is also asensitive function of the cavity loss because of the rapidgain saturation with increasing carrier density. Thus, inquantum-well lasers there is an optimum cavity length forthe maximum modulation bandwidth below which the re-duction in photon lifetime can no longer offset the loss indifferential gain. This optimum value is dependent on thenumber of quantum wells, and for a SQW laser or laserswith a small number of quantum wells, good high speedperformance can be obtained over a large range of cavitylengths.In the presence of transport effects, the K factor ex-tracted from RI N measurements cannotbe used to predictthe modulation bandwidth of the quantum-well lasers. Themaximum modulation bandwidth in this case is limited bythe low frequency rolloff due to transport, which does notaffect the RIN spectrum. The maximum modulation band-width occurs at a power level where the initial rolloff dueto camer transport is so severe that the actual modulationresponse barely touches the -3 dB point before the res-onance peak in the intrinsic laser response causes it to turnback up again, before finally rolling off at a higher fre-quency.Thus, careful device design is required to take advan-tage of the enhancement in differential gain obtained bythe use of quantum wells in the active area. A mere re-placement of the active area in a bulk laser by a quantumwell is not sufficient.

ACKNOWLEDGMENTWe would like to thank S. Corzine for the computerprogram used to calculate the carrier dependent gain, andProfessor H. K roemer and Professor U. Mishra for theirinsightful comments on the various aspects of carrier

transport in semiconductor lasers. W e would also like tothank Professor E. Hu and T. Ishikawa for meticulouslyreading through the manuscript, and for pointing out partsof the paper that lacked sufficient clarity.REFERENCES

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Radhakrishnan Nagaraj anwas born in 1962. Hereceived the B.Eng. (1st Class Honours) degreein electrical engineering from the National Uni-versity of Singapore in 1986, where he was alsothe recipient of the Government of SingaporeM erit Scholarship. A fter the completion of thismilitary service in 1987, he was awarded the Hi-tachi Foundation Scholarship to pursue his grad-uate study at the University of Tokyo, T okyo, J a-pan, where he received the M.Eng. degree inelectronic engineering in 1989. His thes

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High Speed Quantum-Well Lasers and Carrier Transport Effects