Modeling Of Gain, Differential Gain, Index Change, And Linewidth ...

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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 10, NO. 10, OCTOBER 1998

Modeling of Gain, Differential Gain, Index Change, and Linewidth Enhancement Factor for Strain-Compensated QW’s G. L. Tan and J. M. Xu

Abstract— The gain, differential gain, index change, and linewidth enhancement factor for strain-compensated InGaAs–GaAsP–InGaP quantum wells (QW’s) are modeled. The model we have developed builds upon the model-solid theory for determining the band offsets, the k 1 p method for calculating the matrix elements of dipole moment, and the density matrix approach for computing the complex susceptibility of strain compensated QW’s. We also incorporate bandgap renormalization. The calculated results based on the model are consistent with available experimental results in the literature. It is shown that InGaAs–GaAsP–InGaP strain-compensated QW’s could offer much higher gain, higher differential gain, and lower linewidth enhancement factor than AlGaAs–GaAs conventionally compressively strained QW’s, but more because of its larger band offset than anything else. Index Terms— Linewidth enhancement factor, many-body effects, optical gain, quantum-well lasers, strain-compensated quantum wells.

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TRAIN-COMPENSATED quantum-well (SC-QW) lasers are being developed in the hopes of improved performance [1]–[2]: lower threshold current, higher modulation bandwidth, and lower frequency chipping. However, few experimental reports [3] exist to allow for reliable extraction of key parameters, such as gain, differential gain, and linewidth enhancement factor, thus, hindering device design and optimization. It is, thus, more important than ever to be able to calculate these important material parameters from first principles. In this letter, we present the first calculation of these parameters based on the density matrix method and the model-solid theory [4]–[8]. Two major obstacles for such calculation are the determination of the strain-dependent band offsets at the interface of the ternary and quaternary materials and the extraction of the strain-dependent anisotropic masses. These obstacles are overcome by incorporating the modelmethod [9] solid theory [7], [8] and the approximate into our model. The density matrix method [6] is then used to calculate the complex susceptibility of the SC-QW directly, hence the gain, index change, and linewidth enhancement factor ( factor) can be computed simutaneously avoiding the numerical calculation difficulties associated with performManuscript received March 20, 1998; revised June 18, 1998. This work was supported in part by the Ontario Laser and Lightwave Research Centre and by Nortel Technology. The authors are with the Department of Electrical Engineering, University of Toronto, Toronto, ON, M5S 1A4, Canada. Publisher Item Identifier S 1041-1135(98)07106-7.

ing Kromers–Kronig transformation over limited range. Also many-body effects are considered by incorporating bandgap renormalization [10]–[11]. The optical susceptibility of a quantum-well (QW) laser can be derived using the density matrix formalism [9]:

(1) is the photon energy, is the transition energy, Here, and the index denotes the th energy level for conduction is the and valence band (assuming -selection rule). corresponding momentum matrix element for the transition from the th subband in the conduction band to the th light hole (LH) or heavy-hole (HH) subband. The index denotes the hole subband type (i.e., light or heavy). and are the corresponding Fermi functions for the th and conduction and valence subband, respectively. are quasi-Fermi energy levels for electron and hole. is (the band the density of state. Noted that when and are substituted by (bulk gap of the barrier), (bulk density of state), momemtum matrix element) and , where , , respectively. are the free electron mass, vacuum wavenumber, and and susceptibility, respectively. From the optical susceptibility , and the carrierthe gain is calculated by induced index change with respect to bulk material index by , where is the bulk refractive index of the QW. Since experimental measurements for the band offsets and the effective masses in strained QW’s remain very difficult, scarce, and uncertain, a theoretical approach must be used to obtain physically reasonable estimations of material parameters for strained QW’s. For our model, following the approach of the model-solid theory [7], the average valence-band energy and hydrostatic deformation potentials are obtained for the constituent binary compounds. The bandgap, spin-orbit splitting, and the parameters for calculating the shear deformation potential are obtained from the corresponding experimental data of the binary compounds. Then, an interpolation scheme [8] of expanding the material parameters of quaternary alloys in a product of polynomials, in fractional constituent ratio

1041–1135/98$10.00  1998 IEEE

TAN AND XU: MODELING OF GAIN, DIFFERENTIAL GAIN, INDEX CHANGE, AND LINEWIDTH ENHANCEMENT FACTOR

BAND OFFSET PARAMETERS

FOR

TABLE I STRAIN-COMPENSATED

AND

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STRAINED QW’S

and , is used to determined the band parameters for lattice matched and strained heterostructures. With this method, the band offsets of strained QW’s in ternary and quaternary system are obtainable from a common basis established from first principles [7], [12] rather than from empirical extrapolations of rarely available, scattered, and often indirect experimental data for quaternary compounds. The strain-dependent, anisotropic effective masses of the HH and LH bands are determined by an approximative method similar to that used in [9]. The bulk hole dispersion relation is fitted with an anisotropic parabolic expression based on the 4 4 strain modified Luttinger–Kohn Hamiltonian. The effective hole mass obtained by this procedure is an average effective mass over a small range of near the band edge. We for the calculation in chose the range of 0 to 0.047 is the lattice constant of the substrate, this work, where values contributes more significantly since this range of to the integral of (1) as confirmed by comparisons of this approximation to evaluations of more rigorous expressions [4]. The effective mass of the conduction band is determined via linear interpolation of the results from the self-consistent ab initio calculation for corresponding binary constituents [7] and is independent of strain on the basis of a common assumption that the conduction band is free of shear deformation potential. We have also included in our model bandgap renormalization [10], [11] to account for many-body effects. Two terms contribute to the bandgap shift from renormalization. Debye comes from shift or Coulomb-hole (CH) self energy the screening of a repulsive interaction energy leading to a lowering of the conduction electron energy. For quantum wells, the shifts are calculated by (2) is the Bohr radius, where is the Rydberg energy, is the inverse static screening length, is the reduced mass, is the background dielectric constant, and is a constant between 1 and 4. The other contributing factor is the Screened, which comes from Hartree–Fock exchange (SX) shift

Fig. 1. Comparison of gain and differential gain spectra strain-compensated and a compressively strained QW laser.

for

a

Fig. 2. The linewidth enhancement factor spectra for the strain-compensated QW at several carrier concentrations. The symbol denotes the experimental results.

2

correction [6], [10], given by

(3) is the carrier density and is the width of quantum where well. For a transition between continuous spectra we use the bulk formula of shifts [6] instead of the above (2) and (3). Therefore, when considering many-body effects, the bandgap Because the shifts depend becomes on the carrier density, the bandgap is also dependent on carrier density. The authors in [3] compared their strain compensated (SC) InGaAs–GaAsP–InGaP laser with a conventionally strained InGaAs–GaAs laser and found that the SC laser has a higher gain, a higher differential gain and a lower linewidth enhancement factor. To evaluate our model, we used their two QW strainstructures as examples: an In Ga As-GaAs P compensated QW laser with 60 A of well width and a conventional In Ga As-GaAs compressively strained QW laser with the similar layer structure (same number of wells

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Fig. 3. Maximum gain and differential gain versus carrier concentration.

and thicknesses). Table I shows the calculated band offsets for denotes parallel to the hetboth structures. The superscript erointerface and denotes perpendicular to the heterointerface. We observed that the bandgap difference between the tensile barrier and the compressively strained QW is greater than the conventionally strained QW structure leading to a factor of 2 increase in the HH band offset. More specifically, the HH band shifts above the LH band in the compressively strained QW layer while the reverse occurs in the tensile barrier thus also contributing to the increase in the HH band offset. Moreover, the effective mass along the heterointerface with the tensile barrier has a much smaller value than in the purely compressively strained QW. Fig. 1 shows the comparisons of gain spectra for the strain-compensated (SC) and conventionally strained QW lasers for carrier concen10 cm . Also shown is the trations of 2, 3, and 4 comparison of the differential gains at carrier concentrations corresponding to a modal gain of 20 cm . We observe that the SC structure’s maximum gain is almost double that of the strained structure and with its differential gain increased by 60%. The calculated peak gain wavelength is consistent with the experimental lasing wavelength of 0.97 m [3]. Fig. 2 shows SC structure’s linewidth enhancement factor ( factor) spectra for several carrier concentrations with the symbol denoting the measured results from [3]. As shown, the calculated spectra obtained below threshold are consistent with the measured results. Proceeding further, Fig. 3 gives a comparison of the maximum gain and differential gain versus carrier concentration. We see from Fig. 3 that due to the high gain of the SC structure, its carrier concentrations are far smaller than for the conventionally strained structure in the typical range of 2000–4000 cm (corresponding to a the modal gain range of 20–40 cm ). This results in a lower current and current density to produce the same optical power. The SC structure has 10 , which is similar to a higher differential gain of the experimentally estimated value of 2.1 10 [3]. Because the relaxation oscillation frequency of a laser is proportional to the square root of the differential gain at a given optical power, the SC structure offers a higher relaxation oscillation

IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 10, NO. 10, OCTOBER 1998

frequency and bandwidth. The linewidth enhancement factor is a key parameter that determines the spectral width of a laser under both CW operation and high-frequency modulation. It characterizes the spectral width broadening due to fluctuations in the carrier density that changes both the real and imaginary part of the index. From calculation, the SC structure has a slightly large index change, but due to its greater differential gain, it still has a lower factor in a typical modal gain range. From these results, it is tempting to assert that the performance improvements of an SC InGaAs–GaAsP–InGaP QW laser over a conventional compressive strained InGaAs–GaAs QW laser can be attributed to the presence of strain compensation. However, further analysis comparing the SC InGaAs–GaAsP QW with one in which the strain in the barrier being artificially removed suggest that the compensating tensile strain itself plays a small role on the gain spectrum with most of the effect coming from the band offset differences. In summary, a theoretical model for gain, differential gain, index change, and linewidth enhancement factor in strain and strain compensated QW’s is developed from the model-solid method, and models of bandgap theory, the approximative renormalization. Calculated results using the examples of a strain-compensated and a comparable compressively strained QW laser are consistent with the reported experimental findings. ACKNOWLEDGMENT The authors wish to thank H. Chik for his assistance in the preparation of this manuscript. REFERENCES [1] H. Han, P. N. Freeman, W. S. Hobson, N. K. Dutta, J. D. Wynn, and S. N. G. Chu, “High-speed modulation of strain-compensated InGaAsGaAsP-InGaP multiple-quantum-well lasers,” IEEE Photon. Technol. Lett., vol. 8, pp. 1133–1135, 1996. [2] H. S. Cho, D. H. Jang, J. K. Lee, K. H. Park, J. S. Kim, S. W. Lee, H. M. Kim, and H. M. Park, “High-performance strain-compensated multiple quantum well planar buried heterostructure laser diodes with low leakage current,” Jpn. J. Appl. Phys., vol. 35, pp. 1751–1757, 1996. [3] N. K. Dutta, W. S. Hobson, D. Vakhshoori, H. Han, P. N. Freeman, J. F. de Jong, and J. Lopata, “Strain compensated InGaAs-GaAsP-InGaP laser,” IEEE Photon. Technol. Lett., vol. 8, pp. 852–854, 1996. [4] T. C. Chong and C. G. Fonstad, “Theoretical gain of strain-layer semiconductor lasers in the large strain regime,” IEEE J. Quantum Electron., vol. 25, pp. 171–178, 1989. [5] G. Tan, K. Lee, and J. M. Xu, “Finite element light emitter simulator (FELES): A new 2D software tool for laser devices,” Jpn. J. Appl. Phys., vol. 32, pp. 583–589, 1993. [6] W. W. Chow, S. W. Koch, and M. Sargent, III, Semiconductor-Laser Physics. Berlin, Germany: Springer-Verlag, 1994. [7] C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B, vol. 39, pp. 1871–1883, 1989. [8] M. P. Krijn, “Heterojunction band offsets and effective masses in III–V quaternary alloys,” Semiconduct. Sci. Technol., vol. 6, pp. 27–31, 1991. [9] Z. M. Li, M. Dion, Y. Zou, J. Wang, M. Davies, and S. P. McAlister, “An approximate k 1 p theory for optical gain of strained InGaAsP quantum-well lasers,” IEEE J. Quantum Electron., vol. 30, pp. 538–546, 1994. [10] H. Haug and S. W. Koch, “Semiconductor laser theroy with many-body effects,” Phys. Rev. A, pp. 1887–1898, 1989. [11] D. Ahn and L. Chuang, “The theory of strained-layer quantum-well laser with bandgap renormalization,” IEEE J. Quantum Electron., vol. 30, pp. 350–365, 1994. [12] M. L. Xu, G. L. Tan, J. M. Xu, M. Irikawa, H. Shimizu, T. Fukushima, Y. Hirayma, and R. S. Mand, “Ultra-high differential gain in GaInAsAl-GaInAs quantum wells: Experiment and modeling,” IEEE Photon. Technol. Lett., vol. 7, pp. 947–949, 1995.