IEEE PHOTONICS TECHNOLOGY LE'lTERS, VOL. 8, NO. 2, FEBRUARY 1996 197 Calculated Threshold Currents of Nitride- and Phosphide-Based Quantum-'Well Lasers P. Rees, C. Cooper, P. M. Smowton, P. Blood, imd J. Hegarty Abstract- We have calculated the room temperature reduced the confinement factor of a quantum well of a given gain-current characteristics for a 360 nm wavelength, 80 A width increases thus reducing the gain requirement for a laser GaN- A10.14Ga0.86N and a red-emitting, 80 A Ga0.511n~.49P- of given cavity length and facet reflectivity. All these factors (A10.44Ga056)O . S I InO.49P quantum well laser structures, including many body effects. Although the carrier density and suggest that the performance expected of a ID-V blue laser is spontaneous current are much higher (by a factor of 4 and 3, not a simple extrapolation from red-emitting devices operating respectively) in the nitride structures for a given local gain, at a longer wavelength. the higher confinement factor at short wavelengths means the In this paper we compare threshold currents of nitride intrinsic threshold current of these devices i predicted to be s approximately twice that of red lasers with the same optical loss. and phosphide lasers, both calculated including many-body effects, and discus optimisation of the cavity length of these devices. Khan et al. [l] have observed photoluminescence quantum wells (on sapphire sub- from G ~ N - & . I ~ G % , ~ ~ N T HE 111-V nitride material systems are attracting much attention for their potential as optoelectronic devices at blue and ultraviolet wavelengths. Recent advances in growth strates) of various widths and we will use this well structure for our calculations. All computations are for a hypothetical, procedures have allowed GaN-AIGaN quantum well structures unstrained, cubic structure due to the lack of detailed band to be grown [ll and GaN to be deposited on a variety of structure information and strain parameters in GaN. We expect substrates with an improving crystalline quality (for a review the effects of stIain to be similar to that in other material see reference ). Although some work has been published systems therefore our comparison of unstrained structures on optical gain in bulk material, as yet very little work has should give a good indication of the important distinctions been reported on the gain-current characteristics of quantum between the short and longer wavelength devices. well devices. In this letter we report results of calculations of Previously we have shown that the inclusion of many the gain-current characteristics for a 808, GaN-Alo.14Gao.86N body effects is a essential consideration in the calculation n quantum well, emitting at approximately 360 nm and compare of gain in the GaN material system , nevertheless our the results with those of a similar structure in the AlGaInP calculations suggest that excitons are screened-out at typical material system, emitting in the red, to provide a contrast carrier densities I L a~ laser and do not contribute significantly for assessing the feasibility of such a device. The AlGaInP to the gain in GaN so we use a model in which Coulomb material system is chosen for comparison having the largest enhancement is included in recombination from an electron bandgap of well-understood 111-V semiconductors. hole plasma. Coulomb enhancement has not been included There are a number of fundamental differences between in previous calculations for phosphide-based lasers and to lasers at short wavelengths in the blue and present-day devices enable us to make meaningful comparisons we include it here operating in the red and near infra-red. The carrier density for the first time. We calculate the many-body gain using needed to achieve transparency is significantly higher due to the matrix inversion method to solve the equation for the the higher electron and hole effective masses, and the radiative intraband polarisation [ ] The calculation uses parabolic sub- 4. recombination matrix element, which is inversely proportional bands, and strict k selection is adopted for optical transitions to the material band gap, is smaller than in narrower gap as we are considering an undoped active region with the semiconductors. Although the lower matrix element reduces optical transition matrix element as described by Kane . the recombination rate per carrier, the higher threshold carrier The dephasing time, which we assume to be given by the density leads to a higher intrinsic recombination current at a carrier-carrier scattering lifetime, is calculated as described in given local gain. Finally we note that as the wavelength is reference ; wc have not included carrier-phonon scattering. The masses for the electron and heavy hole in GaN were taken Manuscript received July 18, 1995; revised October 27, 1995. P. R e s as 0.2 and 0.8 respectively . Our calculations ignore the light was supported in part by the EC under the Human Capital Mobility research hole band because we lack any value for its effective mass. We training program. C. Cooper and P. M. Smowton were supported in part by have used a fixed ratio of 0.7 :0.3 conduction :valence band the EPSRC. P. Rees and J. Hegarty are with the Physics Department, Tiiy College, rnt offset ratio [ ] The calculations for both material systems 8. Dublin 2, Ireland. are identical though the resulting Coulomb enhancement is C. Cooper, P. M. Smowton, and P. Blood are with the Department of smaller in GaInP. Physics and Astronomy, University of Wales Cardiff, P.O. Box 913, Cardiff CF2 3YB, UK. Fig. 1 shows the calculated TE peak local gain (9) Publisher Item Identifier S 1041-1135(96)01259-1. versus carrier density ( n ) for an 808, GaN-Alo.14Gao.86N 1041-1135/96$05.00 0 1996 JEEE Authorized licensed use limited to: TRINITY COLLEGE DUBLIN. Downloaded on February 2, 2009 at 12:44 from IEEE Xplore. Restrictions apply. 198 IEEE PHOTONICS TECHNOLOGY LE'ITERS, VOL. 8, NO. 2, FEBRUARY 1996 4000 GalnP-AIGalnP GaN-AlGaN ,' - .- c - 3000 f GaN-AIGaN P E v C B 2000 E! x s 1000 " 0 0 1000 2000 3060 0 10 20 30 40 Spontaneous recombination current (Acm") Carrier denisty ( ~ l O ' ~ c m ' 3 ) Fig. 2. Peak TE gain (solid lines) and peak modal TE gam (dotted lines) versus spontaneous recombinahon current calculated for a 808, Fig. 1. Peak TE gam versus carrier density calculated for an 80 8, Gd'J-Alo lrGao86N and an 80 A ( d o 4 4 G % 5 6 ) 0 5 i h 4 9 P - Ga-AIo l4G% 86N and a 808, (MO 44Gao 56)O 5 i h O rep-Gao 5 l h O 49P Gao 51Ino 4 g P quantum well, including Coulomb enhancement i both cases. quantum well, mcluding Coulomb enhancement. n quantum well and for an unstrained, red-emitting 80A Gao.slIno49P-(AlO 44Gao56)o 51hO.49P quantum well chosen 1.25~ 1 GalnPAlGalnP because, with the band offset ratio of 0.7:0.3 , both structures have similar conduction and valence band well depths. (For convenience, n is expressed per unit volume and is calculated as the carrier density per unit area multiplied by the well thickness.) The calculation for the phosphide structure used electron and hole effective masses of 0.11 and 0.43 mo, m the light holes being omitted from the calculation as for the GaN well to give a more meaningful comparison. The values 0.25 of the dielectric constant for GaN and GaInF' were 9.5  and 11.75  respectively. 0 0 10 20 30 40 It is clear from Fig. 1 that the gain due to the lowest el-hhl transition saturates at a lower value for the GaN well; this is Carrier density ( ~ l O ' ~ c m - ~ ) because the matrix element for optical transitions is inversely proportional to the bandgap and the electron effective mass Fig. 3. The radiative recombination coefficient, B , calculated for an 80 8, Gfl-Alo i4GW s 6 N and a 808, (Alo s4Gao 5610 sirno 4 9 P 4 a o s i h o 49p .This could cause devices having a high cavity loss to quantum well over the carrier densities given in Fig. 1. operate on a higher order transition (n > 1) to achieve the high level of gain required, leading to a shift in wavelength and higher threshold current. The carrier density required to ratio of threshold currents is reduced compared to the carrier achieve the same local gain is a factor of 4 higher for the GaN densities by the lower matrix element in the nitrides. well compared with the GaInP well. The spontaneous recombination current per unit volume is We have obtained the spontaneous recombination current often approximated by the relation Jspon eBnp, where n = by integrating the spontaneous emission spectrum derived and p are the electron and hole volume densities (n = p in an from the absorption spectrum using detailed balance argu- undoped active region) and B is the radiative recombination ments [lo]. Plots of the calculated peak TE? gain verses coefficient. In Fig. 3 we show the values of B obtained spontaneous recombination current per unit area ( J ) for from the results of the calculation for the values of carrier a SOA (A10 44Gao 56)o SlInO 49P-Ga~.slIn~.49P a 80A and density shown in Fig. 1. For the carrier densities quoted GaN-Alo 14Ga0.86Nquantum well are shown in Fig. 2. The above the corresponding values of B are 1.10 * and transparency current densities are 390 and 1210 Acm-2 0.29 * cm3 s-l for GaInP and GaN respectively. As respectively, and expressing these curves in a logarithmic expected the value of B is lower in the nitride well because form [11,12] the scaling constants are Jt = 830 and 2030 of the smaller matrix element, though the dependence upon A cm-', and gt = 1490 and 1200 cm-', respectively. carrier density is not as strong as in the phosphide structure. For a typical threshold gain of 1250 cm-' the value of the From Fig. 2, it is clear that to achieve a specific value of lo- spontaneous recombination current is 750 AcmV2 for GaInP cal gain the spontaneous recombination current is significantly and 2100 Acm-2 for GaN, a factor of 3 greater. The carrier higher for the GaN well. However a more useful comparison is densities required to achieve this gain are 6 x 10l8cm-3 and the threshold current for lasers with the same optical loss, i.e., 24 x 10" cm-3 for GaInP and GaN respectively (Fig. 1); the the current for the same modal gain, G. (G = (Fg), where r is Authorized licensed use limited to: TRINITY COLLEGE DUBLIN. Downloaded on February 2, 2009 at 12:44 from IEEE Xplore. Restrictions apply. REES et al.: CALCULATED THRESHOLD CURRENTS OF NITRIDE- AND PHOSPHIDE-BASED QW LASERS 199 the optical confinement factor). A typical waveguide cladding vices, however due to the higher confinement factor at short in AlGaInP lasers is (Alo.70Gao.30)o.slIno.~~P giving an [131 wavelength the threshold current density for a given cavity optimum waveguide width of 235 nm and r = 0.024 per u loss is only about twice that of a phosphide 650 nm laser. O r well at an operating wavelength of 650 nm (using refractive calculations suggest that, for nitride lasers, operation on the indices of 3.35 and 3.26 ). Although data is not available lowest pair of sub-bands is only possible for devices requiring for the refractiveindices of the cubic phase of AlGaN, an index a modal gain less than about 45 cm-l, and this consideration step of 0.09 should be possible since the refractive indices may dictate the choice of length rather than the more usual obtainable for different compositions are proportional to the optimisation proc!ess. The transparency current is the major band gaps available  and the range of band gaps available contribution to the total intrinsic threshold current in nitride from the cubic phase of AlGaN is significantly greater than devices so there is no direct advantage in using more than from AlGaInP. For AlGaN, the optimised waveguide width one well, however multiple well devices may be necessary is 149 nm for a similar waveguide step giving I’ = 0.036 to retain operation on the n = 1 sub-bands and to reduce per well, which is considerably greater than for the AlGaInP extrinsic leakage currents. structure as the 80 A AlGaN well overlaps with more of the lowest optical mode at shorter wavelengths and less local gain ACKNOWLEDGMENT is required to overcome the cavity losses. Fig. 2 shows a plot of The authors would like to thank Dr. I. Galbraith and Prof. the calculated modal gain versus spontaneous current density. W. W. Chow for help and useful discussions. For a modal gain of 40 cm-I the threshold current densities are 940 and 1870 AcmF2 for the phosphide and nitride devices REFERENCES respectively,a ratio of about a factor 2. The effect of the higher M. A. Khan, R. A. Skogman, and 3. M. 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Uematsu, “Analysis and appli- for real devices these threshold currents must be scaled by the cation of thcoretical gain curves to the design of multi-quantum-well internal quantum efficiency. If there is a significant thermally lasers,” ZEEE J. Quantum Electron, QE-21, pp. 1958-1963, 1985. P. Blood, in Physics and Technology of Hetrostructure Devices, D. V. activated leakage current which is sensitive to the position of Morgan and R. H. Williams, Eds. Stevenage, UK Peter Perigdnus, the quasi-Fermi levels then it may be desirable to increase 1991, ch. 7 the number of wells to lower the local gain requirement per H. D. Summers, P. Blood, and P. Rees, “Gain-current characteristicsof strained AlGaInP quantum well lasers,” Appl. Phys. Lett., vol. 63, no. well (see, e.g., ). Similarly, a multiple well laser may be 20, pp. 2792-2194, 1993. desirable to maintain the lasing process on the lowest pair to Y. Kaneko and K. 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