Carrier relaxation in quantum wires : consequences for quantum wire laser I performance

Quantum wire lasers are expected to require very low threshold currents owing to the nature of the 1 D density of states which develops a sharp peak at the band edge and ensures superior laser characteristics. However, carrier relaxation processes in quasi-lD structures may be much slower than in bulk material owing to reduction in the momentum space. For very long relaxation times, these equilibrium processes are expected to limit the maximum modulation frequency of the quantum wire lasers. We perform a Monte Carlo simulation of electron relaxation in quantum wires with the inclusion of the electron-buikiike polar optical and acoustic phonon, electron-electron and electron-hole interactions as well as Thomas-Fermi screening. We find that for a carrier density of lq” ~ m ~ , t h e electron relaxation time ranges from 120 ps for the 100 A x 100 A wire to 30 p s for the 200 A x 200 A wire. Since the threshold current in a quantum wire laser increases with the wire cross section, within the limits of our relaxation model, this indicates possible existence of a trade-off between speed and efficiency in a quantum wire laser. W e also analyse the effects of carrier relaxation on’gain compression in quantum wire lasers by solving the Boltzmann equation using a novel Monte Carlo technique. A spectral hole forms in the carrier distribution at high injected currents with the resulting decrease in the slope of the light-current characteristic. The effect of a non-Fermi-Dirac distribution of electrons is found to result in a suppression of the peak gain as compared with the peak gain calculated using the equilibrium distribution. In recent years, considerable attention has been devoted to the applications of quasi-ID electronic structures in constructing very low-threshold and very high-speed semiconductor lasers. The improvement in threshold current and speed is expected to stem from the sharp peak in the density of states that occurs near the band edge in quantum wire structures [1,2]. However, it should be kept in mind that the reduction in the momentum space to one dimension leads to qualitatively new phenomena in the carrier relaxation process which may increase the equilibration time to such an extent that high-speed operation of quantum wire lasers can be made problematic [3,4]. Intersubband electron-electron scattering is prohibited, and thermalization of the electron gas, which loses its energy primarily by emitting monoenergetic polar optical phonons (POP), is exceedingly slow, leading to bottlenecks in the relaxation process [4]. Thermalization in the photoexcited electron gas with the inclusion of electron-electron and polar optical phonon scattering has been studied [SI, but relaxation of carriers injected into the active region and also undergoing acoustic phonon and electron-hole scattering events has not been investigated to the best of our knowledge. Here 0268-1242/94/050878+04 $19.50 @ 1994 IOP Publishing Ltd we study quantitatively the physics of the equilibration process using the Monte Carlo simulation technique and then proceed to examine the consequences of the slow carrier relaxation processes for the performance of quantum wire lasers. We consider in particular the effect of electron relaxation on the nonlinear gain characteristics and present qualitative results for gain compression due to spectral hole burning. The relaxation processes in the valence band are expected to be much faster than those in the conduction band owing to the higher valence band density of states; therefore, in this paper we are concerned solely with electron relaxation. We simulate the relaxaiion process by injecting electrons in a thermal distribution at the potential well edge and following their progress as they come to thermal equilibrium with the lattice maintained at a constant temperature of 300 K. The conduction band 1D states are calculated in the effective mass approximation by numerically discretizing Schrodinger’s equation on a uniform mesh in the wire region and converting derivatives into finite differences, and solving the resulting matrix equation iteratively. For the 100 8, x 100 8, wire, two subbands lie below the potential well, the first one Carrier relaxation in quantum wires at 0.05 eV and the second one at 0.11 eV, while there are six subbands below the conduction band discontinuity in the 200 8, x 200 8, wire. The valence band states are found by diagonalizing the four-band Kohn-Lnttinger Hamiltonian by the same finite difference approach. The major energy loss process is found to be emission of polar optical phonons. The only intersubband electronelectron scattering process allowed by the requirement of simultaneous conservation of energy and momentum in a quasi-1D system is an exchange of states between the interacting electrons. This scattering process cannot alter the shape ofthe distribution function. In the virtual absence of electron-electron scattering, it becomes necessary to include inelastic acoustic phonon scattering as well as electron-hole scattering into the Monte Carlo simulation in order to describe the thermalization processes in a quantum wire. Since we consider quantum wires with cross sections from 1008, x 1008, up, we assume that the coupling between the confined electronic states and localized interface phonon modes is much weaker than that between electrons and bulk-like confined phonon modes and include only the latter interaction in our calculations. The coupling coefficient for the POP scattering is obtained from the Frohlich Hamiltonian [SI, and that for AP (acoustic phonon) scattering from the deformation potential theory. The details of the simulation have been presented elsewhere [4], and the full formalism is forthcoming [7]. In order to estimate the effect of electron-hole scattering on the evolution of the electron distribution, we adopt the assumption oi the equiiibrium hoie distribution because the full relaxation problem involving two types oi carriers is too complex to be solved in practical simulation times. Moreover, the hole-phonon coupling is expected to be significantly stronger than the electron-phonon coupling with the rate of energy loss for holes far greater than that for electrons. We also approximate the valence band structure as parabolic, which is equivalent to setting the off-diagonal terms of the Koh-Luttinger Hamiltonian to zero. This assumptechnique while the consequences for the electron-hole scattering processes are believed to be secondary. Electrons and holes interact through the Coulombic attraction, and the scattering rate can be computed most are incorporated in the static longwave limit. The average energy of an ensemble of 5000 electrons as a function of time after injection is shown in figure 1 for the lOOA x 1008, GaAs/Al,,,Ga,,, wire for two carrier concentrations. Whi!e the icitial stage in the electron relaxation process can be modelled as exponential decay with a characteristic time of several picoseconds, the rate of energy loss is reduced drastically after 10 ps. This phenomenon can be accounted for by the the effective band edge can give up energy only by emitting acoustic phonons, which is a statistically very slow process. Note that the multiple subband occupation increases the mean energy for the equilibrium distribution . . .: .. ---..I1:. ......... ,IC .L .. _ P &L . ̂^, ..~. .&: ... 1 LlUU ICSUILS 111 a IIlaJUr SllIlpllllCaLlUU U1 LllC tidlCUl~tlUILdl a " 4 1 . , :.. +hn n-.. -..-,.4--+A. rQ1 CnmP..irn nffn..+r CLL"UJ 111 L U I Y V l l l appLw"""aLLw" LY,. U"'II"Y& C L L b U L U fact that the carriers within the POP energy (36 meV) of > . 3