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            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
 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 [2]). 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
 quantum well, emitting at approximately 360 nm and compare                          of gain in the GaN material system [3], 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-
 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 [5].
 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 [6]; 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 [7]. 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
 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

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     198                                                                                    IEEE PHOTONICS TECHNOLOGY LE'ITERS, VOL. 8, NO. 2, FEBRUARY 1996


                            GalnP-AIGalnP                                                                                                          GaN-AlGaN          ,'           -

           -   3000

           B   2000
         s     1000

                                                                                                       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.

      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 [9], 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 [2]
      and 11.75 [9] 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
       [7].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

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    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 [14]). 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 [15] 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
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                                                                                              P. W. A. MscIlroy, A. Kurobe, and Y. Uematsu, “Analysis and appli-
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