Vol. 5, No. 4, 1989
MONTE CARLO SIMULATION OF INTERSUBBAND
IN SEMICONDUCTOR QUANTUM WELLS
S.M. Goodnick Center for Advanced Materials Research, Oregon State University Corvallis, Oregon 97331
P. Lugli di Ingegneria Mecanica, II Universita di Roma Via 0. Raimondo, 00173 Roma, Italy (Received 12 August 1988)
We have used an ensemble Monte Carlo simulation of semiconductor quantum wells to simulate recent time resolved optical experiments of intersubband relaxation. In the present work we model a single quantum well electron system with the inclusion of intra- and intersubband carriercarrier scattering and nonequilibrium polar optical phonons. Simulated results with and without these basic mechanisms show that the experimentally measured long time constant pile-up of carriers in upper subbands in wide quantum wells appears to be associated with the thermal tail of the total carrier distribution function which remains hot for long times after the pump pulse due to hot phonon effects.
Time resolved optical studies of bulk GaAs and GaAs/AlGaAs quantum wells have provided a wealth of information on the dynamics of hot carrier relaxation in semiconductors. Such experiments have been used to illuminate effects such as nonequilibrium phonons'** and electron-electron relaxation in quantum well systems.3 Recently, time resolved techniques have been employed to study the relaxation of carriers through intersubband scattering.4*5 In such experiments, a short time duration pump pulse is used to generate carriers in upper subbands of a quantum well system, and a subsequent optical probe is used to study the relative population of carriers in the upper subbands as a function of time. Oberli et ale4 used intersubband Raman scattering to look at the relaxation time for 116A and 2ISA wells using a pump pulse which generates carriers in the first and second subbands simultaneously at energies just above the second subband edge. In the narrow well, the observed depopulation of the second subband was extremely rapid, less than the time resolution of the exoeriment. Such a rapid deoooulation is expected iince the intersubbahd scat-' tering rate due to polar optical phonons (POP) Js relatively quick, on the order of-a picosecond.b For the wider well, the separation between the first and second subbands was reported to be 27meV which is less than the optical phonon energy (35meV). Thus, carriers at the bottom of the second subband are forbidden from making transitions via the POP mechanism, leading to the possibility of a 'bottleneck' in the depopulation of the upper level." Long time constants for the carrier population in the second subband were indeed observed, on the order of several hundred picoseconds. Seilmeier et a1.5 used time resolved
infrared absorption to study intersubband relaxation in 50A wells where the intersubband seoaration is large. In contrast to tCn results of Oberli et al., they observed a relatively slow relaxation rate in their stud'es with time constants of 11-14ps, much longer than the expected rate due to intersubband POP scattering rate. In order to more fully interpret-these experiments, we have performed Monte Carlo simulations of the above experiments. Here we model only the electron system and defer stu?j of electron-hole interactions to future work. The model we use is discussed in detail elsewhere'-' where the main mechanisms of concern in the present paper are the inclusion of inter- and intrasubband e-e scattering, and bulk-like POP scattering including hot phonon effects. A simple square well potential is assunec! for subband wavefunctions and energies. For e-e scatterina. a static screened interaction is employed using'the long wavelength limit of the RPA dielectric matrix which changes continually during the simulation through changes in the carrier distribution function.* For intersubband e-e scattering, we use the same dielectric matrix as for intra-band scattering which overestimates screening. Thus the rate of intersubband e-e scattering is expected to be even greater than modeled here. For noneauilibrium ohonons in the Monte Carlo simulatidn,' the population of phonons with momentum parallel to the well is calculated from a detailed balance of emission and absorption events during the simulation as well as decay due to a phenomenological time constant taken as 7ps from Raman experiments.' The phonon wavevector normal to the well is constrained to wavevectors less
than the inverse width of the quantum well system which corresponds to the width of the phonon 'wave packet' in momentum space." To simulate optical excitation, we add particles to the simulation centered around the excess electron energy in the conduction band accordinq to the ternDora distr:bution of the experimental laser pulse.g Deqeneracy of the final states is also included through a rejection techniaue.g in the present paper, we report on simulations of carrier relaxation corresponding to the exDeriments of Oberli et a1.4 Here we assume a total carrier density of 4x10"/cm2 after injection at a lattice temperature of SK. A pulse duration of lps is assumed in all the simulated results with the pulse peaking lps into the simulation. The injection energies are chosen so that carriers are excited below the ootical ohonon energy in the upper subband in accordance with the experimental studies." Well widths of 120A and 230A were used in our calculations in order to match the experimental intersubband separations.
Vol. 5, No. 4, 1989
‘i 40 PS
Figure 2. Total energy distribution function for carrier relaxation in a 230A well at various times after the peak of the pump pulse at t-lps. Subband
cl.7J 0.6 0.5 0.4 0.3 0.2 0.1 O0
Figure 1. Fraction of the total carrier population residing in subbands 1 and 2 for two different well widths.
Figure 1 shows the comparison of the relative subband DoDulations for both the 120A and the 230A well. The'result here is similar to that found by Oberli et al., at least for the time scale that can reasonably be modeled using the Monte Carlo method. The population in the upper subband of the 12OA well decays quite rapidly and is completely empty after about 1Ofs. The 230A well on the other hand maintains a sizable population in the upper subband even after 40~s. The auestion is whether a bottleneck occurs in the intersubband transfer due to the suppression of POP scattering. To study this, we look at the total di stribution of the electrons in the 230A well at various times after ohotoexcitation into the second subband as shown in Fig. 2. The kink in the distribution function at 27meV corresponds to the position of the second subband for this particular well. If there is a true bottleneck occurring, then the occupancy of the second
Figure 3. Average energy and fraction of carriers residing in the second subband for the 230A well with and without the effect of hot Dhonons.
subband should be greater than that of the first subband. However, as seen from Fig. 2, the relative occupancy of the first and second subbands are virtually the same at all times. The occupancy of the second subband appears to be due to the thermal tail of the distribution function from the first subband. To understand the origin of this long tail, we selectively 'turn off' the effect of nonequilibrium phonons by forcing them to remain in equilibrium, and compare to the previous result. This comparison is shown in Fig. 3. We first note that the average energy of the carriers decreases rapidly and quickly reaches its equilibrium value of half the Fermi energy at SK. As is well known, the effect of nonequilibrium phonons is to slow down the dissipation of energy by keeping the excess electron energy within the coupled
Vol. 5, No. 4, 1989
electron-phonon system through incraased absorption.13 This effect is reflected in the relative occupancy of the second subband. As shown in Fig..3, the occupancy of the second subband decays rapidly without the presence of hot phonons to maintajn a thermal tail above the second subband energy. There are several competing effects which tend to prevent a bottleneck from forming due to polar optical phonon suppression. One is inelastic e-e scattering which allows some of the electrons to be promoted above the effective threshold for scattering from subband 2 to subband 1. Another factor is e-e intersubband scatterina which is always present. In the case of the"230A well, the total number of scatterin* events due to intersubband e-e scattering was about a tenth that due to POP intersubband scattering. Thus the contribution is relatively weak, although as mentioned previously, a more complete screening model for this mechanism could result in an increased intersubband scattering rate. Finally, in order for electrons to scatter from t;.e lowest subband to the second subband via ohonon emission, they have to have an energy at ieast one phonon energy (35meV) above the bottom of the second subband. Thus there is a barrier to repopulation of the second subband which opposes the formation of a bottleneck. We have additionally run simulations of the experiments of Seilmeier et al.' and find no decreased intersubband relaxation rate as was measured experimentally. However, in their experiments, the second subband in 5OAauantum wells is close'to the top of the barrier, so real space transfer effects could play an important role which was not considered here.
Acknowledgement - The authors would like to thank D. Oberli and J. Shah for useful discussions related to this work. This work was partially supported by the Office of Naval Research.
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