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Ultrafast relaxation of hot photoexcited carriers in GaAs
Solid-State Electronics Vol 31, No. 3/4, pp. 401 406, 1988 Printed in Great Britain
0038-1101/88 $3.00 + 0.00 Pergamon Journals Ltd
ULTRAFAST RELAXATION OF HOT PHOTOEXCITED CARRIERS IN GaAs
D. K. Ferry*, M. A. Osman**,
R. Joshi*,
and M.-J. Kann*
*Center for Solid State Electronics Research Arizona State University, Tempe, AZ 85287 USA **Scientific Research Assoc., Inc., Glastonbury, CN 06033 USA
ABSTRACT The roles of carrier-carrier interactions and non-equilibrium phonons on the u l t r a f a s t r e l a x a tion of p h o t o e x c i t e d c a r r i e r s in GaAs are examined. At low carrier concentrations, the e-ph interaction is the main energy loss channel for hot electrons, while at high carrier c o n c e n t r a tions, the e-h interaction is the primary energy loss channel. This latter result follows from the high e-h scattering rate, the screening of the e-ph interaction, and the high e f f i c i e n c y of h o l e - p h o n o n s c a t t e r i n g through the unscreened deformation potential interaction. The electron energy-loss rates through the e-h interaction increases as the e x c i t a t i o n e n e r g i e s and intens i t i e s are increased. In two-dimensional systems, the e~h interaction further complicates the problem since the transverse o p t i c a l m o d e s out are also d r i v e n out of e q u i l i b r i u m by t h e i r interaction with the holes.
INTRODUCTION P r o g r e s s in the generation of ultrashort pulses and their application to study the phenomena on a picosecond time scale has made it possible to investigate processes such as c o o l i n g rates of p h o t o e x c l t e d carriers (Shah et al., 1985; H~pfel et al., 1986a), the lifetimes of phonons (Kash et al., 1985), the screening of optlcal-phonon-carrler interactions (Collins and Yu, 1984), and velocity overshoot in semiconductors (Shank et al., 1981), which all occur on the sub-plcosecond time scale. Several models have been developed to explain the role of processes such as screening of LO p h o n o n - c a r r l e r i n t e r a c t i o n s (Algarte, 1985), hot phonons (P~tz and Kocevar, 1980; Luzzl and Vasconcellos, 1984), and e-h interactions (Asche and Sarbel, 1984; Osman et al., 1986) in the o b s e r v e d c o o l i n g rates of the excited carriers. Although carriers are generated monoenergetically, and the time scales involved are very short for a s t a n d a r d d i s t r i b u t i o n to be defined, m o s t e a r l y models assumed either Fermi-Dirac or Maxwell-Boltzmann statistics, and the electrons and holes were assumed to have equal temperatures even though i n i t i a l l y the e n e r g i e s of the electrons and holes are quite different. These approaches are valid only on time scales that are large compared to the relaxation time for micro-informatlon, which is the time for the s y s t e m to lose the m e m o r y of the initial distribution and to reach a quasi-equillbrlum state. This inherently assumes that the system has thermalized through the carrler-carrier i n t e r a c t i o n processes, at which point the form for the non-equilibrlum distribution function can be defined. Because the initial rapid t h e r m a l l z a t l o n of the p h o t o - e x c l t e d e l e c t r o n - h o l e p l a s m a o c c u r s through carrier-carrler i n t e r a c t i o n s , the u n d e r s t a n d i n g of the cooling process requires the knowledge of how this thermalizatlon proceeds on the sub-plcosecond time scale. On the other hand, the recent experiments using picosecond and sub-plcosecond laser pulses have revealed new information about the nature of the electron~hole interaction and how it influences the transport properties of semiconductors. H~pfel et al. (1986a, 1986b) have observed that the energy-loss rates for the photoexcited e l e c t r o n s was d o u b l e d in the p r e s e n c e of a cold hole plasma and that minority electrons in quantum well structures exhibit absolute negative mobility at low temperatures and low electric fields. Similarly Degani et al. (1981) h a v e o b s e r v e d an i n c r e a s e in the v e l o c i t y and no n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e effect for the minority electrons. Instead, the velocity continued to increase throughout the range of electric f i e l d s used in t h e i r e x p e r i m e n t . All of the above observations have been attributed to the energy exchange between the minority electrons and holes which tends to t r a n s f e r the e n e r g y f r o m the hot electrons to the holes. The role of the e-h interaction in the ultrafast relaxation of hot photoexclted carriers has now b e e n e x a m i n e d at d i f f e r e n t e x c i t a t i o n levels and energies using the ensemble Monte Carlo approach (Osman e t a__l., 1986, 1987a, 1987b). From this study, we find that the p r e s e n c e of b o t h the e l e c t r o n s and the h o l e s acts to s c r e e n the polar optical phonons and reduce this latter interaction at high densities. The p a t h of e n e r g y decay is then t h r o u g h the e l e c t r o n - h o l e interaction. T h i s transfers the electron energy to the holes, where it in turn is lost to the lattice through interaction with the three o p t i c a l modes. In b o t h cases, h o w e v e r , the polar p h o n o n s can be driven out of equilibrium through the large emission rates, and this acts to further slow the d e c a y of e n e r g y loss to the l a t t i c e . In a d d i t i o n , at l e a s t in a twodimensional system, the t r a n s f e r of the e l e c t r o n e n e r g y to the l a t t i c e via the h o l e and
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403
transverse optical modes can even drive these latter phonons out of equilibrium. Consequently, the total process of energy decay must be looked at hollstically, which can only effectively be done on the sub-picosecond scale through the capabilities of the ensemble Monte Carlo process.
THE APPROACH
The dynamics of electrons and h o l e s was s t u d i e d on the s u b - p l c o s e c o n d time scale u s i n g the Ensemble Monte Carlo (EMC) approach, which has been shown to be suitable to study the fast transients (Zimmermann et al., 1983). On the longer time s c a l e s (t>2 ps), the EMC techn i q u e is not e f f i c i e n t , and an a n a l y t i c a l f o r m u l a t i o n in terms of quasi-Ferml functions is usable. It is found that these quasi-equillbrlum (though far from equilibrium in truth) forms are quite well established by the carrier-carrier interactions on this time scale. By fitting the parameters in these functions to values deduced from the EMC c a l c u l a t i o n s , the t r a n s i t i o n can be c a r r i e d out s m o o t h l y . The m o d e l s take into a c c o u n t the e-e, e-h, h - h scattering, carrler-phonon, and carrler-impurlty scattering. Furthermore, the screening of the e-ph and hph i n t e r a c t i o n s in p o l a r s e m i c o n d u c t o r s is i n c l u d e d s e l f - c o n s l s t e n t l y in the static, long wavelength limit of the random phase approximation. The conduction band model consists of three nonparabolic valleys, while the valence band consists of a parabolic heavy-hole band (the lighthole is included in the model but ignored in this investigation). The Inter-valence b a n d holeh o l e and s c r e e n e d l l g h t - h o l e - p h o n o n scattering are built into the model, but as mentioned the role of the light holes has been i g n o r e d in the p r e s e n t c a l c u l a t i o n b e c a u s e of their small population (less than 10% of the holes).
CARRIER-CARRIER
INTERACTIONS
Effect of Carrier-Phonon Screenln~ The cooling rates of the photoexcited electrons were examined in the absence of the c-c interactions u s i n g b o t h the D e b y e - H ~ c k e l and the self-consistent screening models. This was done to compare the results of the simulation to previous models which either assumed equal electron and h o l e t e m p e r a t u r e s or totally ignored the presence of holes on account of the small fraction of energy they receive in the p h o t o e x c i t a t i o n process. In Fig. I, we i l l u s t r a t e the r e s u l t s . C u r v e I c o r r e s p o n d s to the situation where the screening is totally ignored, which predicts a very fast cooling rate in contrast to the observed slow cooling rates at high excitation levels. W h e n the s c r e e n i n g of the e - p h i n t e r a c t i o n is taken into account, while the presence of the photo-excited holes is ignored, the c o o l i n g r a t e is s l i g h t l y r e d u c e d as s h o w n by curve 2. H o w e v e r , w h e n the p r e s e n c e of the p h o t o - e x c i t e d holes in included, using the same screening model, the cooling rate is significantly d e c r e a s e d , w h i c h is s h o w n by c u r v e 3. Immediately after excitation, the distribution function is significantly different from the Maxwelllan upon which the Debye-H~ckel model is based. We have examined the importance of the d l s t r l b u t l o n in screening by using the self-conslstent screening model discussed above. The cooling rates shown by curves 4 and 5 in Fig. I correspond to situations where the presence of the holes is e i t h e r i g n o r e d (curve 4) or i n c l u d e d (curve 5). F r o m these c o o l i n g rates, it is obvious that the screening length for the e-h plasma is controlled by the holes which are initially g e n e r a t e d at lower energies, and also thermallze at a faster rate due to the strength of the h-h interaction. Furthermore, the fact that the decrease in the cooling rate is smaller for the s e l f - c o n s l s t e n t s c r e e n i n g m o d e l r e f l e c t s the e v o l u t i o n of the distribution function of the carriers. On the other hand, the Debye-H~ckel model assumes an equilibrium distribution, so that the screening is d o m i n a t e d by the cold (low energy) c a r r i e r s , w h i c h do not exist in the first stages of the cooling of photo-exclted plasmas. The Effect of Electron-Hole
Interaction
The investigation of the cooling rates in the previous section a s s u m e d that the e l e c t r o n s and h o l e s cool i n d e p e n d e n t l y . Thus, there was no c o u p l i n g b e t w e e n the c o o l i n g rates of the electrons and the holes, and no energy transfer from one system component to the other. Because the initial energy of the electron is about seven times that of the hole, the e-h plasma forms a far-from-equillbrlum plasma in w h i c h one c o m p o n e n t is s i g n i f i c a n t l y h o t t e r than the other. C o n s e q u e n t l y , one e x p e c t s energy transfer from the hot electron system to the relatively cold hole system, and the role of e-ph and h - p h i n t e r a c t i o n s to vary a c c o r d i n g l y . The r e l a t i v e importance of the e-h and e-ph interactions as energy transfer channels depends upon the excitation level and energy. The detalls of these cooling processes were investigated over a range of densities. In Fig. 2, we show the results of simulations for a relatively high excited density of carriers. C u r v e I in Fig. 2 c o r r e s p o n d s to the situation in which only phonon relaxation processes are considered and it is obvious that the electrons cool faster. Curve 2 shows the cooling p r o c e s s w h e n the e l e c t r o n - e l e c t r o n and hole-hole interactions were included in addition to the optical phonon interaction. The interaction among the electrons q u i c k l y r e d i s t r i b u t e s the e l e c t r o n s into h i g h and low e n e r g y r e g i o n s . Thus, many electrons end up in regions where they can not emit optical phonons, while those which end up in the higher energy regions essentially have the same probability for emission of optical phonons. Those electrons which have energies below the phonon emission threshold can cool f u r t h e r only by g i v i n g up some of their e n e r g y to o t h e r e l e c t r o n s , to the holes, or to gain energy through the e-e process sufficient to emit a phonon. Thus, we end up at a s i t u a t i o n w h e r e m o r e e l e c t r o n s are j u s t b e l o w the p h o n o n emission
404
threshold. The s l o w i n g b e c o m e s m o r e s i g n i f i c a n t at h i g h concentrations, phonons are strongly screened and the e-e scattering is more frequent.
where the optical
The case w h e r e only the e-ph and e-h interactions were active was then ~nvestigated to understand how these two energy loss mechanisms contribute to the c o o l i n g of the e l e c t r o n s . From examining curve 3, it is obvious that the electron-hole interaction enhances the cooling rate at a h~gh carrier concentration, since the electrons channel their e x c e s s e n e r g y to the l a t t i c e t h r o u g h the holes. On the other hand, at low concentrations, the e-h scattering slows down the cooling rate by gradually shifting the electron population to lower energy states. Finally, the m o r e r e a l i s t i c situation where all the scattering mechanisms e-e, e-h, h-h, e-ph and h-ph are present was investigated. At low concentrations, the cooling of the e l e c t r o n s is not a f f e c t e d by the added complexity, and the resulting cooling rate is identical to the situation where o n l y the e-h and h - p h i n t e r a c t i o n s were present, r e f l e c t i n g the fact that e-h s c a t t e r i n g is efficient in moving electrons to lower states even at this low concentration. At high concentrations, the e-h interaction is the most important factor that determines the cooling rate. The p r e s e n c e of the e-e scattering slows down the cooling rate slightly during the first two plcoseconds, during which the electron therma]Ization takes place through the e-e and e-h collisions (curve 4 in Fig. 2). However, for the holes, the presence of the h-h scattering, together with the e-h scattering leads to a faster cooling rate because of the dominance of the h-h collisions.
NON-EQUILIBRIUM
PHONONS
We can also i n c l u d e the non-equilibrium phonons into the calculation. The phonon distribution ~s given d i r e c t l y by a d e t a i l e d b a l a n c e of the e m i s s i o n and a b s o r p t i o n e v e n t s d u r i n g the simulation. The d y n a m i c s are r e c o r d e d on an a d d i t i o n a l grid in q-space with the excess LO phonon population of each mode decaying with a characteristic lifetime. This l a t t e r v a l u e has been taken to be 7 ps. The basic model is described in Lugli et al. (1987). In Fig. 3, we show the relaxation of the carrier energy with both the c a r r i e r - c a r r i e r i n t e r a c t i o n s and the none q u i l i b r i u m p h o n o n s i n c o r p o r a t e d into the model. The deviation of the phonon population from its equilibrium value further slows the energy relaxation process. On the o t h e r hand, it also i n c r e a s e s the t r a n s f e r of e n e r g y from the electrons to the holes, heating this latter system further.
ANALYTICAL SOLUTIONS Several authors (Asche and Sarbel, 1984; P~tz and Kocevar, 1980) have used a n a l y t i c a l f o r m u l a tions to investigate the relaxation of hot photo-excited carriers. Generally, these approaches assume that a quasi-equillbrium F e r m l - D i r a c d i s t r i b u t i o n e x i s t s for both the e l e c t r o n s and holes, and may even assume that these two types of carriers have the same temperature. The use of quasi-equilibrium forms for n o n - e q u i l i b r i u m c a r r i e r s has a quite old b a s i s (Fano, 1957; Zubarev, 1974) and generally depends upon mlcro-interactions among the carriers establishing a pseudo-equilibrium (Bogoliubov, 1946). These m l c r o - l n t e r a c t i o n s , in the p r e s e n t case, a r i s e from the e-e, h-h, and e-h scattering. The first two of these are quite important for establishing local temperatures for the electrons and holes. However, we find that it takes a very long time (>0.2 ns) for the latter interaction to bring these two temperatures into equality. The use of the quasi-equillbrium distributions is based upon the method of q u a s i - i n v a r i a n t s of the motion as developed in classical mechanics. These can be e.g. the drift velocity, the local temperature, the quasi-Fermi level, and others, and the distribution function can be w r i t t e n in a p a r a m e t e r i z e d form in terms of these quantities. While early treatments assumed that all of these quantities started with their equilibrium v a l u e s at t=0, c o r r e s p o n d i n g to the initial a r r i v a l of the first photons. However, the nature of the far-from-equillbrium plasma produced by the laser puise is such that this initialization leads to e r r o n e o u s r e s u l t s by i m p r o p e r l y t r e a t i n g the d e t a i l s of the r a n d o m i z a t i o n processes. With the advent of highly accurate EMC techniques, this initialization no longer needs to be used. The EMC t e c h n i q u e s are quite accurate, but consume considerable computer resources for times longer than 2-3 ps. On the other hand, these EMC c a l c u l a t i o n s show that the d i s t r i b u t i o n f u n c t i o n s e t t l e s into good quasiequilibrium shapes by about 2 ps. Thus, it is possible to initialize the analytical formulation not at t=O, but at t=2 ps, and take values for the quasi-invariants from the EMC results. The a n a l y t i c a l formulation is quite efficient in terms of computer resources and will allow studies of the detailed approach to equilibrium at long t~mes, e.g. t>1 ns. We have d e v e l o p e d an a n a l y t i c a l approach for studying the time evolution of the laser-excited plasma in a quasi-two-dimenslonal quantum well in GaAs. The phonons, h o w e v e r , are still cons i d e r e d to be three dimensional in nature. All of the ~nter-carrier interactions and screening are included in the model in a self-consistent manner in order to treat the variation in screening that rises as the distribution functions evolve. So far, we treat only the lowest subband for each carrier type, but the a p p r o a c h is being e x t e n d e d to include m u l t i p l e s u b b a n d s and i n t e r - s u b b a n d scattering. The role of the electron-hole interaction on carrier thermalization and non-equilibrlum phonons is being studied with this t w o - t e m p e r a t u r e model. In Fig. 4, we show the r e s u l t s for both the c a r r i e r r e l a x a t i o n and the non-equilibrlum phonon population. These results are for the usual initial conditions. While the expected significant b u i l d - u p of the LO phonons appears, there is also a weaker build-up of the TO modes. The former phonons are driven out of equilibrium due to their i n t e r a c t i o n w i t h the e l e c t r o n s . The l a t t e r modes,
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however, are driven out of e q u i l i b r i u m by the holes, primarily because of the electron-holelattice route of energy relaxation.
CONCLUSIONS A detailed ensemble Monte Carlo model for the analysis of the u l t r a f a s t r e l a x a t i o n of photoe x c i t e d c a r r i e r s in GaAs has been presented. The screening of the e-ph interaction leads to a significant reduction in the scattering rates, especially at high carrier concentrations and low temperatures. The screening is included in a self-conslstent, but long wavelength, approxlmation to take advantage of the built-ln, time-evolving distribution function inherent in the EMC approach. The model does not make any assumptions on the form of the distribution functions or on the individual or average energies of the electrons and holes. The e-h i n t e r a c t i o n slows down the c o o l i n g rate of the Dhoto-exclted electrons at low excitation levels by shifting the electron population to e n e r g i e s b e l o w the p h o n o n e m i s s i o n threshold. The e-e i n t e r a c t i o n r e s u l t s in the same behavior. On the other hand, at high concentrations, the e-h interaction enhances the cooling rates by transferring the e l e c t r o n energy to the holes. These results s u g g e s t that at low densitles, the energy flow from the electrons to the lattice is primarily through the e-ph interation, while at high concentrations it is primarily through the e-h intera c t i o n and then t h r o u g h the h-pn interaction. This latter p r o c e e d s primarily through the unscreened TO [Bformation potential interaction. The p r e s e n c e of the e-e and h-h interactions is necessary to achieve the rapid randomization of energy and momentum in the resPective distributions. At higher laser energies, the loss rate of the e l e c t r o n s t h r o u g h the e-h interaction is further increased, a result that is also achieved by increasing values of the electric field. The p o s s i b i l i t y of n o n - e q u i l l b r l u m p h o n o n s also slows the decay and affect the relaxation process. The authors express their appreciation for many s t i m u l a t i n g d i s c u s s i o n s w i t h Drs. Lugli and R e g g i a n ~ of the University of Modena, Drs. Ravaioli and P~tz of the University of Illinois, and Dr. Sankey of ASU. This work was supported by the Office of Naval Research.
REFERENCES Algarte, A. C. S. (1985). Phys. Rev. B, 32, 2388. Asche, M., and O. G. Sarbei (1984). Phys. Stat. Sol. (b), 126, 607. Bogoliubov, N. N. (1946). Problemi dinam, teorll u star. Fiz., Moscow. Collins, C. L., and Y. Yu (1984). Sol. State Commun., 51, 123. DeganJ, R. J., Leheny, R. F., Nahory, R. E., and J. P. Heritage (1981). Appl. Phys. Letters, 39, 569. Fano, U. (1957). Rev. Mod. Phys., 29, 74. H~pfel, R. A., Shah, J.. and A. C. Gossard (1986a). Phys. Rev. Letters, 56, 2045. H~pfel, R. A., Shah, J., Wolf, P. A., and A. C. Gossard (1986b). Phys. Rev. Letters, 56, 2736. Kash, J. A., Tsang, J. C., and J. Hvam (1985). Phys. Rev. Letters, 54, 2151. Lugl~, P., Jacoboni, C., Reggianl, L., and P. Kocevar (1987). ARpI. Phys~ Letter s , 50, 1251. Luzzl, R. and A. R. Vasconcellos (1984). Semiconductors Probed by Ultrafast Laser Spectroscopy, Vol. I, Ed. by R. R. Alfano. Academic Press, Orlando, 135. Osman, M. A. and D. K. Ferry (1987a). J. Appl. Phys., xx, in press. Osman, M. A. and D. K. Ferry (1987b). Submitted for publication. Osman, M. A., Ravaioli, U., and D. K. F e r r y (1986). Proc. 18th Intern. Conf. Physics of Semiconductors, Ed. by O. Engstrom. World Scientific Press, Singapore, 1311. PDtz, W. and P. Kocevar (1980). Phys. Rev. B, 28, ?040. Shah, J., Pinczuk, A., Oossard, A. C., and W. Wiegmann (1985). Phys. Rev. Letters, 54, 2045. Shank, C. V., Fork, R. L., Oreene, B. I., Reinhart, F. K., and R. A. Logan (1981). Appl. Phys. Letters, 38, 104. Z~mmermann, J., Lugli, P., and D. K. Ferry (1983). Sol. State Electron., 26, 233. Zubarev, D. N. (1974). Nonequlllbrium S t a t i s t i c a l T h e r m o d y n a m i c s . C o n s u l t a n t s Bureau, New York.
0038-1101/88 $3.00 + 0.00 Pergamon Journals Ltd
ULTRAFAST RELAXATION OF HOT PHOTOEXCITED CARRIERS IN GaAs
D. K. Ferry*, M. A. Osman**,
R. Joshi*,
and M.-J. Kann*
*Center for Solid State Electronics Research Arizona State University, Tempe, AZ 85287 USA **Scientific Research Assoc., Inc., Glastonbury, CN 06033 USA
ABSTRACT The roles of carrier-carrier interactions and non-equilibrium phonons on the u l t r a f a s t r e l a x a tion of p h o t o e x c i t e d c a r r i e r s in GaAs are examined. At low carrier concentrations, the e-ph interaction is the main energy loss channel for hot electrons, while at high carrier c o n c e n t r a tions, the e-h interaction is the primary energy loss channel. This latter result follows from the high e-h scattering rate, the screening of the e-ph interaction, and the high e f f i c i e n c y of h o l e - p h o n o n s c a t t e r i n g through the unscreened deformation potential interaction. The electron energy-loss rates through the e-h interaction increases as the e x c i t a t i o n e n e r g i e s and intens i t i e s are increased. In two-dimensional systems, the e~h interaction further complicates the problem since the transverse o p t i c a l m o d e s out are also d r i v e n out of e q u i l i b r i u m by t h e i r interaction with the holes.
INTRODUCTION P r o g r e s s in the generation of ultrashort pulses and their application to study the phenomena on a picosecond time scale has made it possible to investigate processes such as c o o l i n g rates of p h o t o e x c l t e d carriers (Shah et al., 1985; H~pfel et al., 1986a), the lifetimes of phonons (Kash et al., 1985), the screening of optlcal-phonon-carrler interactions (Collins and Yu, 1984), and velocity overshoot in semiconductors (Shank et al., 1981), which all occur on the sub-plcosecond time scale. Several models have been developed to explain the role of processes such as screening of LO p h o n o n - c a r r l e r i n t e r a c t i o n s (Algarte, 1985), hot phonons (P~tz and Kocevar, 1980; Luzzl and Vasconcellos, 1984), and e-h interactions (Asche and Sarbel, 1984; Osman et al., 1986) in the o b s e r v e d c o o l i n g rates of the excited carriers. Although carriers are generated monoenergetically, and the time scales involved are very short for a s t a n d a r d d i s t r i b u t i o n to be defined, m o s t e a r l y models assumed either Fermi-Dirac or Maxwell-Boltzmann statistics, and the electrons and holes were assumed to have equal temperatures even though i n i t i a l l y the e n e r g i e s of the electrons and holes are quite different. These approaches are valid only on time scales that are large compared to the relaxation time for micro-informatlon, which is the time for the s y s t e m to lose the m e m o r y of the initial distribution and to reach a quasi-equillbrlum state. This inherently assumes that the system has thermalized through the carrler-carrier i n t e r a c t i o n processes, at which point the form for the non-equilibrlum distribution function can be defined. Because the initial rapid t h e r m a l l z a t l o n of the p h o t o - e x c l t e d e l e c t r o n - h o l e p l a s m a o c c u r s through carrier-carrler i n t e r a c t i o n s , the u n d e r s t a n d i n g of the cooling process requires the knowledge of how this thermalizatlon proceeds on the sub-plcosecond time scale. On the other hand, the recent experiments using picosecond and sub-plcosecond laser pulses have revealed new information about the nature of the electron~hole interaction and how it influences the transport properties of semiconductors. H~pfel et al. (1986a, 1986b) have observed that the energy-loss rates for the photoexcited e l e c t r o n s was d o u b l e d in the p r e s e n c e of a cold hole plasma and that minority electrons in quantum well structures exhibit absolute negative mobility at low temperatures and low electric fields. Similarly Degani et al. (1981) h a v e o b s e r v e d an i n c r e a s e in the v e l o c i t y and no n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e effect for the minority electrons. Instead, the velocity continued to increase throughout the range of electric f i e l d s used in t h e i r e x p e r i m e n t . All of the above observations have been attributed to the energy exchange between the minority electrons and holes which tends to t r a n s f e r the e n e r g y f r o m the hot electrons to the holes. The role of the e-h interaction in the ultrafast relaxation of hot photoexclted carriers has now b e e n e x a m i n e d at d i f f e r e n t e x c i t a t i o n levels and energies using the ensemble Monte Carlo approach (Osman e t a__l., 1986, 1987a, 1987b). From this study, we find that the p r e s e n c e of b o t h the e l e c t r o n s and the h o l e s acts to s c r e e n the polar optical phonons and reduce this latter interaction at high densities. The p a t h of e n e r g y decay is then t h r o u g h the e l e c t r o n - h o l e interaction. T h i s transfers the electron energy to the holes, where it in turn is lost to the lattice through interaction with the three o p t i c a l modes. In b o t h cases, h o w e v e r , the polar p h o n o n s can be driven out of equilibrium through the large emission rates, and this acts to further slow the d e c a y of e n e r g y loss to the l a t t i c e . In a d d i t i o n , at l e a s t in a twodimensional system, the t r a n s f e r of the e l e c t r o n e n e r g y to the l a t t i c e via the h o l e and
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transverse optical modes can even drive these latter phonons out of equilibrium. Consequently, the total process of energy decay must be looked at hollstically, which can only effectively be done on the sub-picosecond scale through the capabilities of the ensemble Monte Carlo process.
THE APPROACH
The dynamics of electrons and h o l e s was s t u d i e d on the s u b - p l c o s e c o n d time scale u s i n g the Ensemble Monte Carlo (EMC) approach, which has been shown to be suitable to study the fast transients (Zimmermann et al., 1983). On the longer time s c a l e s (t>2 ps), the EMC techn i q u e is not e f f i c i e n t , and an a n a l y t i c a l f o r m u l a t i o n in terms of quasi-Ferml functions is usable. It is found that these quasi-equillbrlum (though far from equilibrium in truth) forms are quite well established by the carrier-carrier interactions on this time scale. By fitting the parameters in these functions to values deduced from the EMC c a l c u l a t i o n s , the t r a n s i t i o n can be c a r r i e d out s m o o t h l y . The m o d e l s take into a c c o u n t the e-e, e-h, h - h scattering, carrler-phonon, and carrler-impurlty scattering. Furthermore, the screening of the e-ph and hph i n t e r a c t i o n s in p o l a r s e m i c o n d u c t o r s is i n c l u d e d s e l f - c o n s l s t e n t l y in the static, long wavelength limit of the random phase approximation. The conduction band model consists of three nonparabolic valleys, while the valence band consists of a parabolic heavy-hole band (the lighthole is included in the model but ignored in this investigation). The Inter-valence b a n d holeh o l e and s c r e e n e d l l g h t - h o l e - p h o n o n scattering are built into the model, but as mentioned the role of the light holes has been i g n o r e d in the p r e s e n t c a l c u l a t i o n b e c a u s e of their small population (less than 10% of the holes).
CARRIER-CARRIER
INTERACTIONS
Effect of Carrier-Phonon Screenln~ The cooling rates of the photoexcited electrons were examined in the absence of the c-c interactions u s i n g b o t h the D e b y e - H ~ c k e l and the self-consistent screening models. This was done to compare the results of the simulation to previous models which either assumed equal electron and h o l e t e m p e r a t u r e s or totally ignored the presence of holes on account of the small fraction of energy they receive in the p h o t o e x c i t a t i o n process. In Fig. I, we i l l u s t r a t e the r e s u l t s . C u r v e I c o r r e s p o n d s to the situation where the screening is totally ignored, which predicts a very fast cooling rate in contrast to the observed slow cooling rates at high excitation levels. W h e n the s c r e e n i n g of the e - p h i n t e r a c t i o n is taken into account, while the presence of the photo-excited holes is ignored, the c o o l i n g r a t e is s l i g h t l y r e d u c e d as s h o w n by curve 2. H o w e v e r , w h e n the p r e s e n c e of the p h o t o - e x c i t e d holes in included, using the same screening model, the cooling rate is significantly d e c r e a s e d , w h i c h is s h o w n by c u r v e 3. Immediately after excitation, the distribution function is significantly different from the Maxwelllan upon which the Debye-H~ckel model is based. We have examined the importance of the d l s t r l b u t l o n in screening by using the self-conslstent screening model discussed above. The cooling rates shown by curves 4 and 5 in Fig. I correspond to situations where the presence of the holes is e i t h e r i g n o r e d (curve 4) or i n c l u d e d (curve 5). F r o m these c o o l i n g rates, it is obvious that the screening length for the e-h plasma is controlled by the holes which are initially g e n e r a t e d at lower energies, and also thermallze at a faster rate due to the strength of the h-h interaction. Furthermore, the fact that the decrease in the cooling rate is smaller for the s e l f - c o n s l s t e n t s c r e e n i n g m o d e l r e f l e c t s the e v o l u t i o n of the distribution function of the carriers. On the other hand, the Debye-H~ckel model assumes an equilibrium distribution, so that the screening is d o m i n a t e d by the cold (low energy) c a r r i e r s , w h i c h do not exist in the first stages of the cooling of photo-exclted plasmas. The Effect of Electron-Hole
Interaction
The investigation of the cooling rates in the previous section a s s u m e d that the e l e c t r o n s and h o l e s cool i n d e p e n d e n t l y . Thus, there was no c o u p l i n g b e t w e e n the c o o l i n g rates of the electrons and the holes, and no energy transfer from one system component to the other. Because the initial energy of the electron is about seven times that of the hole, the e-h plasma forms a far-from-equillbrlum plasma in w h i c h one c o m p o n e n t is s i g n i f i c a n t l y h o t t e r than the other. C o n s e q u e n t l y , one e x p e c t s energy transfer from the hot electron system to the relatively cold hole system, and the role of e-ph and h - p h i n t e r a c t i o n s to vary a c c o r d i n g l y . The r e l a t i v e importance of the e-h and e-ph interactions as energy transfer channels depends upon the excitation level and energy. The detalls of these cooling processes were investigated over a range of densities. In Fig. 2, we show the results of simulations for a relatively high excited density of carriers. C u r v e I in Fig. 2 c o r r e s p o n d s to the situation in which only phonon relaxation processes are considered and it is obvious that the electrons cool faster. Curve 2 shows the cooling p r o c e s s w h e n the e l e c t r o n - e l e c t r o n and hole-hole interactions were included in addition to the optical phonon interaction. The interaction among the electrons q u i c k l y r e d i s t r i b u t e s the e l e c t r o n s into h i g h and low e n e r g y r e g i o n s . Thus, many electrons end up in regions where they can not emit optical phonons, while those which end up in the higher energy regions essentially have the same probability for emission of optical phonons. Those electrons which have energies below the phonon emission threshold can cool f u r t h e r only by g i v i n g up some of their e n e r g y to o t h e r e l e c t r o n s , to the holes, or to gain energy through the e-e process sufficient to emit a phonon. Thus, we end up at a s i t u a t i o n w h e r e m o r e e l e c t r o n s are j u s t b e l o w the p h o n o n emission
404
threshold. The s l o w i n g b e c o m e s m o r e s i g n i f i c a n t at h i g h concentrations, phonons are strongly screened and the e-e scattering is more frequent.
where the optical
The case w h e r e only the e-ph and e-h interactions were active was then ~nvestigated to understand how these two energy loss mechanisms contribute to the c o o l i n g of the e l e c t r o n s . From examining curve 3, it is obvious that the electron-hole interaction enhances the cooling rate at a h~gh carrier concentration, since the electrons channel their e x c e s s e n e r g y to the l a t t i c e t h r o u g h the holes. On the other hand, at low concentrations, the e-h scattering slows down the cooling rate by gradually shifting the electron population to lower energy states. Finally, the m o r e r e a l i s t i c situation where all the scattering mechanisms e-e, e-h, h-h, e-ph and h-ph are present was investigated. At low concentrations, the cooling of the e l e c t r o n s is not a f f e c t e d by the added complexity, and the resulting cooling rate is identical to the situation where o n l y the e-h and h - p h i n t e r a c t i o n s were present, r e f l e c t i n g the fact that e-h s c a t t e r i n g is efficient in moving electrons to lower states even at this low concentration. At high concentrations, the e-h interaction is the most important factor that determines the cooling rate. The p r e s e n c e of the e-e scattering slows down the cooling rate slightly during the first two plcoseconds, during which the electron therma]Ization takes place through the e-e and e-h collisions (curve 4 in Fig. 2). However, for the holes, the presence of the h-h scattering, together with the e-h scattering leads to a faster cooling rate because of the dominance of the h-h collisions.
NON-EQUILIBRIUM
PHONONS
We can also i n c l u d e the non-equilibrium phonons into the calculation. The phonon distribution ~s given d i r e c t l y by a d e t a i l e d b a l a n c e of the e m i s s i o n and a b s o r p t i o n e v e n t s d u r i n g the simulation. The d y n a m i c s are r e c o r d e d on an a d d i t i o n a l grid in q-space with the excess LO phonon population of each mode decaying with a characteristic lifetime. This l a t t e r v a l u e has been taken to be 7 ps. The basic model is described in Lugli et al. (1987). In Fig. 3, we show the relaxation of the carrier energy with both the c a r r i e r - c a r r i e r i n t e r a c t i o n s and the none q u i l i b r i u m p h o n o n s i n c o r p o r a t e d into the model. The deviation of the phonon population from its equilibrium value further slows the energy relaxation process. On the o t h e r hand, it also i n c r e a s e s the t r a n s f e r of e n e r g y from the electrons to the holes, heating this latter system further.
ANALYTICAL SOLUTIONS Several authors (Asche and Sarbel, 1984; P~tz and Kocevar, 1980) have used a n a l y t i c a l f o r m u l a tions to investigate the relaxation of hot photo-excited carriers. Generally, these approaches assume that a quasi-equillbrium F e r m l - D i r a c d i s t r i b u t i o n e x i s t s for both the e l e c t r o n s and holes, and may even assume that these two types of carriers have the same temperature. The use of quasi-equilibrium forms for n o n - e q u i l i b r i u m c a r r i e r s has a quite old b a s i s (Fano, 1957; Zubarev, 1974) and generally depends upon mlcro-interactions among the carriers establishing a pseudo-equilibrium (Bogoliubov, 1946). These m l c r o - l n t e r a c t i o n s , in the p r e s e n t case, a r i s e from the e-e, h-h, and e-h scattering. The first two of these are quite important for establishing local temperatures for the electrons and holes. However, we find that it takes a very long time (>0.2 ns) for the latter interaction to bring these two temperatures into equality. The use of the quasi-equillbrium distributions is based upon the method of q u a s i - i n v a r i a n t s of the motion as developed in classical mechanics. These can be e.g. the drift velocity, the local temperature, the quasi-Fermi level, and others, and the distribution function can be w r i t t e n in a p a r a m e t e r i z e d form in terms of these quantities. While early treatments assumed that all of these quantities started with their equilibrium v a l u e s at t=0, c o r r e s p o n d i n g to the initial a r r i v a l of the first photons. However, the nature of the far-from-equillbrium plasma produced by the laser puise is such that this initialization leads to e r r o n e o u s r e s u l t s by i m p r o p e r l y t r e a t i n g the d e t a i l s of the r a n d o m i z a t i o n processes. With the advent of highly accurate EMC techniques, this initialization no longer needs to be used. The EMC t e c h n i q u e s are quite accurate, but consume considerable computer resources for times longer than 2-3 ps. On the other hand, these EMC c a l c u l a t i o n s show that the d i s t r i b u t i o n f u n c t i o n s e t t l e s into good quasiequilibrium shapes by about 2 ps. Thus, it is possible to initialize the analytical formulation not at t=O, but at t=2 ps, and take values for the quasi-invariants from the EMC results. The a n a l y t i c a l formulation is quite efficient in terms of computer resources and will allow studies of the detailed approach to equilibrium at long t~mes, e.g. t>1 ns. We have d e v e l o p e d an a n a l y t i c a l approach for studying the time evolution of the laser-excited plasma in a quasi-two-dimenslonal quantum well in GaAs. The phonons, h o w e v e r , are still cons i d e r e d to be three dimensional in nature. All of the ~nter-carrier interactions and screening are included in the model in a self-consistent manner in order to treat the variation in screening that rises as the distribution functions evolve. So far, we treat only the lowest subband for each carrier type, but the a p p r o a c h is being e x t e n d e d to include m u l t i p l e s u b b a n d s and i n t e r - s u b b a n d scattering. The role of the electron-hole interaction on carrier thermalization and non-equilibrlum phonons is being studied with this t w o - t e m p e r a t u r e model. In Fig. 4, we show the r e s u l t s for both the c a r r i e r r e l a x a t i o n and the non-equilibrlum phonon population. These results are for the usual initial conditions. While the expected significant b u i l d - u p of the LO phonons appears, there is also a weaker build-up of the TO modes. The former phonons are driven out of equilibrium due to their i n t e r a c t i o n w i t h the e l e c t r o n s . The l a t t e r modes,
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CONCLUSIONS A detailed ensemble Monte Carlo model for the analysis of the u l t r a f a s t r e l a x a t i o n of photoe x c i t e d c a r r i e r s in GaAs has been presented. The screening of the e-ph interaction leads to a significant reduction in the scattering rates, especially at high carrier concentrations and low temperatures. The screening is included in a self-conslstent, but long wavelength, approxlmation to take advantage of the built-ln, time-evolving distribution function inherent in the EMC approach. The model does not make any assumptions on the form of the distribution functions or on the individual or average energies of the electrons and holes. The e-h i n t e r a c t i o n slows down the c o o l i n g rate of the Dhoto-exclted electrons at low excitation levels by shifting the electron population to e n e r g i e s b e l o w the p h o n o n e m i s s i o n threshold. The e-e i n t e r a c t i o n r e s u l t s in the same behavior. On the other hand, at high concentrations, the e-h interaction enhances the cooling rates by transferring the e l e c t r o n energy to the holes. These results s u g g e s t that at low densitles, the energy flow from the electrons to the lattice is primarily through the e-ph interation, while at high concentrations it is primarily through the e-h intera c t i o n and then t h r o u g h the h-pn interaction. This latter p r o c e e d s primarily through the unscreened TO [Bformation potential interaction. The p r e s e n c e of the e-e and h-h interactions is necessary to achieve the rapid randomization of energy and momentum in the resPective distributions. At higher laser energies, the loss rate of the e l e c t r o n s t h r o u g h the e-h interaction is further increased, a result that is also achieved by increasing values of the electric field. The p o s s i b i l i t y of n o n - e q u i l l b r l u m p h o n o n s also slows the decay and affect the relaxation process. The authors express their appreciation for many s t i m u l a t i n g d i s c u s s i o n s w i t h Drs. Lugli and R e g g i a n ~ of the University of Modena, Drs. Ravaioli and P~tz of the University of Illinois, and Dr. Sankey of ASU. This work was supported by the Office of Naval Research.
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