Inclusion of Bandstructure and Many-Body Effects in a Quantum Well Laser Simulator

A self-consistent eight band k.p calculation, which takes into account strain and includes Hartree, exchange, and correlation terms (determined from a local density approximation) is incorporated into a QW laser simulator (MINILASE-II). The computation is performed within the envelope function approximation for a super- lattice, in which all spatially varying terms of the k.p Hamiltonian, including the exchange and correlation energies are expanded in plane waves. The k.p eigenvalue equation, and Poisson’s equation are solved iteratively until self-consistency is attained. Results from the k.p calculation are exported to MINILASE-II via a density of states and an energy dependent optical matrix element factor, renormalized by a Coulomb enhancement factor to account for electron-hole attraction. Results are presented for the gain spectrum and modulation response for a Ga0.8 In0.2 As/A10.1 Ga0.9 As quantum well laser with and without the inclusion of the Coulomb enhancement factor.

structures), quantum well (QW)capture, and the photon rate equations for arbitrary two dimensional geometries. Many important effects arise from the bandstructure near the quantum well.
However, it is computationally impractical to perform an accurate self-consistent bandstructure calculation within the laser simulator at each iteration. The purpose of this paper is to explain how we connect a separate eight band k.p superlattice calculation, including many-body * Corresponding author. effects, with MINILASE-II and to present results for the gain and modulation response for a strained-layer Ino.2Gao.8As/Alo.lGao.9As quantum well laser.
The k.p calculation involves diagonalizing a Hamiltonian which at the center of the Brillouin zone can be expressed as 464 F. OYAFUSO et al.
where the indices run over the bands in our basis set B, Pne are the momentum matrix elements, and 'ne are renormalization constants describing the contribution from bands not contained in B. To characterize the bandstructure of crystals with zincblende symmetry near 1 , it is sufficient to include the heavy hole (HH) and light hole bands (LH) (I), the split-off bands (SO) (I,) and the lowest conduction bands (1,). n(strain), a kindependent term, describes the strain, which is assumed to be confined to the well region. The resulting 88 Hamiltonian has been described in the literature [2,3].
For the superlattice calculation, z is taken to be the growth direction and the usual substitution kzOz/i is made to obtain the effective mass Eq. [4], (2) where kq is the wave number for the 1D Brillouin zone of the superlattice, and b(m null) (z), the superlattice envelope functions, are related to the wave functions to second order by The band parameters which enter the Hamiltonian are allowed to have different values in the well and the barrier regions [5], and the resulting operator is Fourier transformed. For a superlattice period of 500 ,, typically about thirty Fourier components are required to ensure convergence of the ground state eigenenergies to within teV. The figures throughout this paper were generated for a superlattice consisting of A10.1Ga0.9As barriers of 400 ,, and In0.2Ga0.8As wells of 80,. The barriers are wide enough to decouple adjacent wells so that the superlattice is in practice a collection of independent quantum wells. Most band parameters were determined from [6] except for the conduction band and valence band deformation potentials used to determine band edge shifts due to strain and the fraction conduction band discontinuity, which are not well known. We assume a fraction conduction band discontinuity of 0.7 and 11 eV for the difference in conduction band and valence band deformation potentials.
To account for carrier-carrier interactions additional terms which depend on electron and hole densities are added as diagonal terms in the k-p Hamiltonian. The direct Coulomb interaction gives the Hartree potential Vn which is determined by solving the 1D Poisson equation: where the envelope functions 4j have been normalized to the superlattice period L. The Fermi distribution f(E, #) depends on the energy dispersions found from the k.p calculation and on the quasi-chemical potential #, determined from the input parameter n2z, the carrier density per unit surface in one period of the superlattice.
The interaction of a carrier with its exchangecorrelation hole lowers its energy and results in a narrowing of the bandgap. This effect is taken into account in the local density approximation. We use an interpolated expression derived by Hedin and Lundquist [7] for three dimensional systems, and treat electrons and holes as separate plasmas.
The bandgap renormalization (BGR) obtained is in qualitative agreement with experiment and can be improved by using an expression more appropriate for 2D systems.
Because these additional terms depend on the eigenfunctions determined from the k.p calcula- To assure convergence, an underrelaxation scheme is used for the exchange and correlation potentials. The approach is illustrated in the flow chart in Figure 1.
Once convergence has been attained, optical matrix elements (OME) given by Mnn'--(@nlll plbn,ll) are computed for later use in MINILASE II. The square of the OME gives the strength of the electron-hole coupling through the photon interaction and therefore enters the expressions for spontaneous and stimulated emission. Figure 2a shows the OME dispersion for TE polarization and two directions of kll, parallel to and perpendicular to .A t 1 , CB1 couples only to HH1 and   derived in the literature [9] and is given by Here f, and fp are Fermi occupation factors,is the carrier-carrier scattering time, assumed to be 100 fs, and V () is the 2D screened Coulomb potential which in the static long wavelength limit of RPA is given by V(S)(q) 27re 2 eoq + n/q where O#p is the inverse screening length. Figure 2b shows the renormalization factor for the CB1-HH1 transition between the lowest conduction subband and lowest valence subband for several carrier densities above threshold. In this calculation, an axial average of the matrix elements is performed, so that the OME may be treated strictly as a function of energy rather than in-plane wave vector kll. We see that the modest enhancement of the OME decreases as the carrier density increases. This result stems from the fact that screening is enhanced at higher densities. The results of these calculations are exported to MINILASE II in three forms: the spatial profile of the envelope functions, a density of states and an optical matrix element factor Pn(ft) (associated with each conduction band energy grid point n) given by d2kll Pn() . . i27r) 2 I" Mij(kll)12rln(E(kll)) 6(Eic(kll) EJ(kll a) (8) The direction of the light polarization is , and r/n is a hat function with support on the energy range defined by grid points n-1 and n + and whose area is normalized to unity. The sum extends over all conduction subbands and valence subbands j. The gain coefficient is then given by g()(f't) -Z(L(En)-qt--j(E" n '-')-1) n (9) where/2 is a Lorentzian broadening function and jT and jT are the non-equilibrium electron and hole distribution functions, respectively. Minilase II incorporates the k.p data in an iterative manner. That is, the simulator first solves the carrier transport equations, Poisson's equation and the photon rate equations through Newton's method.
The resulting carrier densities are used to recompute the band edges from an interpolated expression for the bandgap renormalization. Then, files appropriate for the computed density are read from disk. Since the exchange-correlation terms are added as k-independent diagonal elements, their inclusion does not significantly alter the effective masses and therefore the densities of state, except for shifting the band-edges.
We now briefly describe a few results obtained from MINILASE-II for an operating regime beyond the lasing threshold. Figure 3(a) shows the optical gain spectra with and without the Coulomb enhancement factor for the same applied current. In steady state, the maximum height of the gain is pinned by the losses in the laser which is the same in each case. There are two salient features that distinguish the gain curve without the Coulomb enhancement from the other curve. These are the reduced threshold for the onset of gain and the larger transparency point. Both features stem from the smaller effective matrix element in the absence of Coulomb enhancement that requires greater pumping of the laser to achieve a fixed gain. Thus, more carriers are required which results in a reduction in the band gap and implies a greater separation of the electron and hole quasi-fermilevels and thus a larger transparency point. Figure 3(b) displays the laser's response to a small square pulse in the applied voltage. We note that the frequency peak is blue shifted when a larger effective matrix element is used. This may be explained as follows. The photon rate equation has the form (G-I/7-.)S + R where u is a mode index, S is the photon population, G is the modal gain, -. is the photon lifetime and R is the spontaneous emission rate, which is much smaller than the other terms of the right hand side under lasing conditions. Since the losses 1/are fixed, the gain is roughly proportional to the time derivative of the photon density in the cavity which implies a more swiftly responding laser. In conclusion, we described how an accurate k.p calculation was connected to a sophisticated laser simulator and demonstrated how the larger effective matrix element that stems from the Coulomb enhancement affects the gain and resonance frequency in the modulation response of a laser.
at Urbana-Champaign. His thesis involves the study of the temperature dependence of threshold current densities in quantum well lasers. Karl Hess has dedicated the major portion of his research career to the understanding of electronic current flow in semiconductors and semiconductor devices with particular emphasis on effects pertinent to device miniaturization. His theories and use of large computer resources are aimed at complex problems with clear applications and relevance to miniaturization of electronics. He is currently the Swanlund Professor of Electrical and Computer Engineering, Professor of Physics, Adjunct Professor for Supercomputing Applications and a Research Professor in the Beckman Institute working on topics related to Molecular and Electronic Nanostructures. He has received numerous awards, for example the IEEE