# HighSpeedQuantum-WellLasersandCarrier※

1990 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28, NO IO, OCTOBER 1992 High Speed Quantum-Well Lasers and Carrier Transport Effects Radhakrishnan Nagarajan, Masayuki Ishikawa, Toru Fukushima, Randall S . Geels, Member, IEEE, and John E. Bowers, Senior Member, IEEE Abstract-Carrier transport can significantly affect the high speed properties of quantum-well lasers. We have developed a model and derived analytical expressions for the modulation response, resonance frequency, damping rate, and K factor to include these effects. We show theoretically and experimentally that carrier transport can lead to significant low frequency par- asitic-like rolloff that reduces the modulation response by as much as a factor of six in quantum-well lasers. We also show that, 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 factor as further evidence for the effects of carrier transport. In the presence of significant transport effects, we show that the real limit to the maximum possible modulation bandwidth is much lower 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. INTRODUCTION HE high speed dynamics of semiconductor lasers have T been conventionally modeled using a set of two cou- pled first-order linear differential equations; one for the carrier density and the other for the photon density in the cavity. Using this approach, the small-signal response of the laser can be written as a second-order transfer func- tion, and the modulation bandwidth of such a system is determined by its resonance frequency f , and the damping rate y. From the small-signal analysis of these rate equa- tions, the resonance frequency can be written in terms of more 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 rp is the photon lifetime [ 13, [2]. In this analysis, the intrinsic damping mechanism in the Manuscript received January 2, 1992; revised March 19, 1992. This work was supported by DARPA and Rome Laboratories (Hanscom AFB). R. Nagarajan and J. E. Bowers are with the Department of Electrical and Computer Engineering, University of Califomia, Santa Barbara, CA 93 106. M. Ishikawa was with the Department of Electrical and Computer En- gineering, University of Califomia, Santa Barbara, CA 93106, on leave from Toshiba Corp., Tokyo, Japan. T. Fukushima was with the Department of Electrical and Computer En- gineering, University of California, Santa Barbara, CA on leave from Fu- rukawa Electric Co., Tokyo, Japan. R. S. Geels was with the Department of Electrical and Computer Engi- neering, University of Califomia, Santa Barbara, CA 93106. He is now with Spectra Diode Laboratories, San Jose, 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 levels, and this provides an additional source of damping [l], 121. This nonlinear photon density dependence of optical gain is introduced into the rate equation formalism via a phenomenological gain compression factor E . The gain or the differential gain in the laser cavity is then written as g = g,/(l + ES) and in the case of small ES, keeping only the linear term of the Taylor expansion, it is also sometimes written as g = g,(l - ES) where g, is the differential gain component determined solely by material parameters and may also be carrier density dependent. The physical origins of the gain compression factor have been attributed to various phenomena like cavity standing wave dielectric grating [3], spectral hole burning 141-[6], transient carrier heating [7], [8], or some com- bination of the last two [9]. The damping rate y varies linearly with f:, and the proportionality constant is called the K factor [lo]; K = 4n2(7, + E/uggo). The maximum possible intrinsic modulation bandwidth is determined solely by this factor, f,,, = 0.13 to 2.4 ns [ll], [22], [24]-[27] implying a maximum possible intrinsic modulation bandwidth rang- ing from 68 to 4 GHz, while K values for the bulk lasers have 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 11, [24], [27] system on the average are higher than the 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 shown that E is independent of the laser structure, and is even unaffected 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 over the bulk ones have been explained by either an increase in the nonlinear gain compression factor E or a reduction in the differential gain. It has been theoretically proposed that E is enhanced by quantum confinement [28], and the inclusion of strain increases it even more [29]. In [30], the authors have used a spectral hole burning model for the computation of the intrinsic E and its variation with quantum confinement, and optimized quantum-well struc- tures for high speed operation. A well-barrier hole burn- ing model [31] was recently proposed, and it concludes that there is an additional contribution to the intrinsic E which is structure dependent. The authors have proposed that this enhancement in E is responsible for the variation in the K factors that have been published. Recently, we have 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 modulation bandwidth and obtained good agreement with the mea- surements done on single and multiple quantum-well la- sers. In [33], the authors have measured the spontaneous emission 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 more strongly damped lasers. For the case of single quantum- well lasers, we have derived analytic expressions for the effects of carrier diffusion and thermionic emission on the modulation response. We have also confirmed this model by an experiment where the modulation response of three laser structures with different carrier diffusion times across the separate confinement heterostructures were measured WI. In this paper, we will present the details of our carrier transport model, and derive analytical expressions for the resonance frequency, damping rate, and K factor in the case of single quantum-well (SQW) lasers to include the effects of carrier transport. We will use a numerical model in the case of multiple quantum-well (MQW) lasers. We will show theoretically and experimentally that E is fairly independent 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 effective differential gain and not an enhancement of E. In addition, we will also show that the carrier transport across the SCH can lead to severe low frequency rolloff. In the presence of such rolloff and even when all other extrinsic device limits are absent, the K factor is no longer an accurate measure of the maximum possible intrinsic modulation bandwidth in quantum-well lasers. We will also present the temperature dependence of the K factor as further evi- dence for the significant effects of carrier transport. Fi- nally, we will use our model to present some design cri- teria for optimizing quantum-well lasers for high speed operation. Among the parameters that can be varied, are the width and number of the quantum wells, width and height of the barriers, types of compositional grading in the separate confinement heterostructure (SCH) to tailor the built-in electric field profile, the amount of strain, and the type and amount of doping in the active area. Given the range of possible variations in structure and their ef- fects on the various laser parameters, optimizing the quantum-well laser for high speed operation is very in- volved. 11. CARRIER TRANSPORT IN THE SEPARATE CONFINEMENT HETEROSTRUCTURE Fig. 1 shows a typical separate confinement hetero- structure (SCH) SQW. We will model the carrier trans- port and capture in such a structure and the effect this has on the modulation response of the laser. This is also the structure that is used in the SQW and MQW laser exper- iments. Electron and hole transport from the doped cladding layers to the quantum well consists of two parts [35], [36]. First is the transport across the SCH. This is governed by the classical current continuity equations which describe the diffusion, recombination, and, in the presence of any electric field, drift of carriers across the SCH. The second part is the carrier capture by the quantum well. This is a quantum mechanical problem which has to take into ac- count the relevant dynamics of the phonon scattering mechanism which mediates this capture. This scattering process is a function of the initial and final state wave- functions, the coupling strength of the transition, and the phonon dispersion in the material. The next transport mechanism which is significant in devices operating at room temperature is the thermally ac- tivated carrier escape from the quantum well or ther- mionic emission. Although this process is in opposition to carrier capture and degrades the overall carrier capture efficiency of a SQW structure, it is essential for carrier transport between quantum wells in a MQW structure. Another transport process of interest in a MQW system is tunneling between the quantum wells. Thermionic emis- sion is a strong function of barrier height while tunneling is sensitive to both barrier height and width. The barriers in a MQW structure would have to be designed such that 1992 0 I Injection Elecnon Injection - Right SCH x=-L, x = o X=Ls Fig. 1. Schematic diagram of a single quantum-well laser with a separate confinement 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 carrier trapping) across the structure adversely. IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 28. NO. IO, OCTOBER 1992 and diffusion components are given by A. Transport Across the SCH For small-signal modulation experiments, the laser diode is forward biased and a small microwave signal is imposed upon this dc bias. The response of the laser is then measured using a high speed photodiode. Since the modulating signal is very small, the device is essentially at constant forward bias. Further, the SCH is not com- positionally graded, and this excludes any built-in fields in the SCH to aid carrier transport. Under normal operation, the semiconductor laser is es- sentially a forward biased p-i-n diode, with the left and right claddings doped p and n, respectively, and the SCH region nominally undoped (Fig. 1). In the figure, the elec- trons are injected from the right and the holes from the left and across the SCH. The important difference is the quantum well in the middle of the SCH where the injected carriers recombine. Although the carrier injection is from the opposite ends of the SCH, any physical separation of the two types of charges across the quantum-well layer would lead to very large electric fields between them. The laser normally operates under high forward injection where the carrier density levels are about 10“ ~ m - ~ , and solving the Poisson equation in one dimension, assuming that the SCH layers on right- and left-hand side of the quantum well are uniformly pumped with electrons and holes, respectively, leads to electric fields in excess of lo6 V/cm between the carrier distributions. Since the carriers are highly mobile, they would redistribute themselves un- til the charge neutrality condition is satisfied throughout the SCH. Further, at this level of carrier injection, unin- tentional background doping levels are negligible. This laser structure can then be analyzed like a heavily forward biased p-i-n diode [37]. The equations for the electron and hole current densities, including both the drift Jp = 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. Assuming high injection conditions, n = p , and charge neutrality, (aE/ax) = 0, (3) and (4) can be conbined. The €-field term can be eliminated to give the following equation un- der steady-state conditions, i.e., ( Vw = @Lw. C. SCH Transport Factor We define a differential SCH transport factor asCH anal- ogous to the common base current gain of a bipolar junc- tion transistor (BJT) [40] (3 alW a4 QSCH = - = sech We are interested in the small-signal value of cySCH. The expression can be derived by substituting La, by La which is given by I La = 4 “ . 1 +jar, The small-signal expression simplifies to 1 cosh ((L, /Du7,) /* Jl+jw7,) - %CH, small signal - In the final expression on the right, we have assumed that the width of one side of the SCH is much smaller than the ambipolar diffusion length, i.e., Ls 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 frequency given by (AE/h) where h is the Planck constant. The tun- neling time, defined as the one half period of the oscilla- tion, is given by h 7l = - Although this result has been derived for a two well sys- tem, it will generally be true for a MQW system in the limit of weak interwell coupling [53]. 2 AE I. Multiple Quantum- Well Structure There are additional complications due to carrier trans- port between the various wells in the MQW system. Again charge neutrality is assumed to hold in the entire intrinsic SCH region, and holes dominate the carrier dynamics. The analyses for carrier transport across the SCH and capture by the first quantum well is the same as for the SQW case. Due to their very small capture time we can assume that all the holes are captured by the first quantum well, and subsequent transport across the MQW structure is either via thermionic emission, then diffusion across the barrier and capture by the next quantum well or tunneling through the barrier. For a well designed barrier, the subsequent diffusion and capture times are negligible compared to the initial thermionic emission time. Thermionic emission and tunneling are competi