Hot phonon dynamics

Solid-State Electronics Vol. 31, No. 3/4, pp. 667-672, 1988 Printed in Great Britain

0038-1101/88 $3.00+ 0.00 Pergamon Journals Ltd

HOT PHONON DYNAMICS

Paolo Lugli Dipartimento di Fisica e Centro Interuniversitario di Struttura della Materia dell'Universit£ di Modena, Via Campi 213/A, 41100 Modena, Italy ABSTRACT The present paper illustrates a series of theoretical results on non-equilibrium phonon effects based on a novel Monte Carlo algorithm. No assumptions on the form of the phonon or the electron distributions are required. The main emphasis is given to the study of LO phonon perturbation as a result of the relaxation of photoexcited carriers in polar semiconductors. Good agreement if found with available experimental results from time resolved luminescence and Raman measurements. KEYWORDS Semiconductors, non-equilibrium phonons, Monte Carlo method. INTRODUCTION In the presence of strong external perturbations (laser excitation, applied electric fields), the energy transfered to a semiconductors drives the carrier system out of equilibrium. Such a situation is usually referred to as hot carrier condition. If the main dissipation channel for the carriers is via phonon emission, then a non equilibrium (hot) phonon population can be found as a result of the energy transfer to the lattice. The presence of phonon amplifications will ultimately depend on the rate at which carriers supply energy to the phonons compared to the rate at which the phonons dissipate their excess energy to the thermal bath. While Raman measurements have been able to detect hot phonon distributions for the case of picosecond (Von der Linde and others, 1980) and subpicosecond excitations (Kash and others, 1985), only indirect and non conclusive evidences exist today on the possibility for the phonon disturbaces to feed back into the carrier systems and modify the overall energy transfer process ( Ryan and others, 1984; J. Shah. and others, 1985). The purpose of the present contribution is to use a Monte Carlo simulation, to solve directly the coupled Boltzmann equations for phonons and carriers. The advantages of the Monte Carlo technique are that it provides a very accurate microscopic description of the physical processes, and it does not require any assumption on the electron nor on the phonon distribution functions. Furthermore, a direct evaluation of the characteristic times for the various scattering mechanisms involved in the interaction process is possible. The dynamics of the carrier-phonon system can be extremely complex. For instance, in the case of laser excitation in GaAs, six types of interacting carriers (I', L and X electrons, light, heavy, and splitoff holes) might be present. Depending on the range of excited densities and on the lattice temperature, different scattering mechanisms can be of importance. Phenomena such as screening of the carrier-phonon interaction, and carrier-carrier scattering are thought to dominate the dissipation process at high carrier concentrations. It is clear that an exact theoretical treatment of such a system is today out of range, althought a considerable effort is spent in this direction. On the other side, experimental measurements usually give an integrated view that might not allow to isolate the effects of specific phenomena. With this in mind, the present communication will briefly outline in Sec. 2 the scenario we are working in, and some of the theoretical approaches that have been pursued. Sec. 3 will present the details of the Monte Carlo algorithm, applied then to a simple model for laser excitation in bulk GaAs and GaAs/AIGaAs quantum wells. A more realistic case will be examined in Sec. 4, and a direct comparison with available experimental results will be given. Throughout the paper, the Monte Carlo analysis will only be limited to situations in which the adopted physical model is justified.

667

668

THE TRANSPORT MODEL The dynamical evolution of the carrier phonon system can be adeguately described by the coupled Boltzmann equations:

d/ _ df l dt

+ df

~-vh +

dt Ifi¢ld

df ~

df ~ c + ~ ~-i,'~v -

dNq _ dNq dNq dt dr- vu-~ + dt ph-vh

(1)

where f and Nq are respectively the carrier and the phonon distribution functions. A decisive simplification of the phonon equation comes from the possibility to use a relaxation time for the phonon-phonon interactions, in the form:

dN(q) vu-vn -

N(q) - No

,~t

T,,h

where is No is the thermal Plank distribution No

=

(6 t~w/kDTL

,

(2)

1) -1 . The relaxation time approximation is

justified by the fact that the phonon-phonon interactions are dominated by the decay of the LO phonons into pairs of electronically non active phonons from zone-boundary modes. Values of the phonon lifetime rvh are generally of the order of 10 ps, with a weak temperature dependence (Klemens, 1966). Several theoretical approaches have been presented in the literature for the solution of Eqs. 1 and 2. Some are based on a direct numerical solution of the coupled equations (Collet and others), others assume a given form for the distribution functions (Poetz and Kocevar, 1983; Cai and others, 1987). In the first attempt to use a Monte Carlo (MC) technique in the study of phonon perturbations, an iterative procedure was developed that coupled a heated and drift Maxwellian (HDM) scheme (Koeevar, 1985) to a one particle MC simulation (Bordone and others, 1986). Starting the MC simulation with the equilibrium phonon distribution, the values of the carrier mean energy and drift velocity were determined and substitute into the asymptotic steady-state solution of the phonon Boltzrnann equation in the HDM scheme. The new scattering rates, accounting for the phonon perturbation, were then calculated numerically and the MC algorithm repeated until convergence was achieved. A semiconductor model parameterized for the case of holes in Ge was used. Figure 1 shows the non equilibrium phonon distribution for a carrier concentration of 1013cm 3 and an electric field of 2.6 V / c m at 4.2 K. In the presence of an electric field, the phonon population increases significantly at phonon wavevectors parallel to the field, and is found to broaden significantly at increasing field. The results for the dependence on electric field strength of the drift velocity and mean energy are reported in Figs. 2 and 3. The calculations performed in presence of the phonon perturbation (continuous curves) are compared with those for phonon equilibrium (dashed curves). The effect of the phonon disturbance leads to an increase of both drift velocity and mean energy at low fields and to a decrease of these quantities at higher fields. The reason of this behavior is associated with the competing roles of the anisotropic and of the isotropic contribution to the phonon amplification. When the carriers reabsorb the emitted phonons before these have the possibility to thermalize and randomize themselves, the phonon-gas drags the carriers whose drift velocity and mean energy increase above their unperturbed value (directional effect of the phonons). When, on the contrary, a net isotropic amplification dominates, the phonon gas most effectively randomizes the motion of the carriers and therefore the average quantities are reduced with respect to their unperturbed values. The method just described was still lacking consistency, since it relied on the assumption of a HDM electron distribution. The full consistent Monte Carlo simulation, developed for the case of photoexcitation in GaAs, will be presented in the next section.

Fig. 2 Drift velocity as functions of the electric field strength for holes in Ge at 4.2 K. Dashed (continuous) curves refer to an unperturbed (perturbed) phonon distribution function.

Fig. 1 3-dimensionM plot of the perturbed phonon distribution, as a function of the amplitude of the phonon momentum (x = h q / m s ) and of the cosine of the angle between q and the electric field.

""

~--~-'~ i,"

lO

/j"

i

E

L

,

Fig. 3 Mean energy as functions of the electric field strength in the same conditions as Fig. 2.

o!........f ,o" 2" ol

1

E(V cm'l;,

/

i lo

669 THE MONTE CARLO ALGORITHM We present a novel Ensemble Monte Carlo algorithm specialized here to InP

the study of non-equilibrium-phonon effects on the relaxation rate of photoexcited electrons in polar semiconductors. The straightforward exten-

eg(eV)

1.34

1987). The novel procedure allows to follow the time evolution of the

m~/mo

0.078

phonon distribution. Preliminary results have been already presented by

m~l~o

0.4

Lugli and others (1987). A three-valley non parabolic model includes the

mX/~o

sion to the field dependent case is presented in a separate paper (Rieger,

following scattering mechanisms: a) dissipative acoustic phonons with deformation potential coupling; b) polar optical phonons; c) ionized impuri-

AI'-L(eV)

ties, treated in the Conwell-Weisskopf formalism; d) non-equivalent and

Ar_x(eV )

equivalent intervalley phonons; e) electron-electron interaction between F-

0.61

coo

9.52

valley electrons, treated as in Lugli and Ferry (1985). A list of the most important parameters is given in Tab. 1. Throughout the paper, nominally undoped materials (with a residual impurity concentration around 1014 c m - 3 ) will be considered. Screening of the polar optical phonon

~o

12.35

DrL(~-)

2.5 x 108

and electron-hole scattering are not considered in the simulation. Such an approximation is justified for the carrier concentrations considered in this

eV DLX(~)

work (typically around 5 x lOZ%m - 3). The laser excitation is reproduced

D aC~crnJ (ev ~

7 x

by adding particles to the simulation, distributed in time as the lineshape of the laser pulse, as shown in the insert of Fig. 4. The initial energy of

TirL(K)

492

the photoexcited electron depends on the details of the absorption pro-

T~rx(K)

cesses. In a finite difference scheme, Eq. (1) for the phonon evolution can

TiLx(K)

eV

Drx(~)

be written in the form:

Tpop(K) At

N q(jAt) = g q ( ( i - 1)At) + 6Nq(iAt) ~-ph - (g~(j,~t) - N o ) ~

(3)

108

498

Tab. 1 Monte Carlo Parameters

The simulation is subdiveded in time intervals At (with a t typically much shorter than the average scattering time for the LO phonon scattering), and the following procedure is set up to account for the LO-phonon disturbances: [[tl

i) the time evolution of the LO-phonon distribution Nq is calculated as a 2.4

function of wavevector q from the Monte Carlo simulation, by setting up a histogram hq defined over a grid in q-space of mesh size Aq. After each scattering event involving an LO phonon, the histogram is updated. In the absence of external d.c. fields, because of the full spherical symmetry

t:r 1.6 z O.8

only the amplitude of q is relevant, thus reducing the complexity and the storage requirements of the simulation. ii) At fixed times T = j A t during the simulation, Nq is calculated from Eq.

0

I

2

3

q(lo

4

6crn-1 ]

(3), where the term 6 N q ( j A t ) = A Ahq gives the dynamical contribution of the electronic processes to the phonon distribution during the time step At. A is a normalization factor accounting for the density of states in q -space and for the concentration of excited electrons n, and it is given by A -- q~2~. AqN " iii) to account for the modifications induced by the phonon disturbance on the rate of electron-phonon scatterings, the integrated scattering probabilities for LO-phonons are calculated and tabulated at the beginning of the simulation using an artificially high value Nm~= for the phonon distribution. The choice of the final state of each scattering process involving an LO phonon is made using a rejection technique which compares the actual value of the differential scattering rate with the maximized one. In this way, we are able to discriminate between the scatterings t h a t can

Fig. 4. Non-equilibrlum LO phonon distribution functions at three different delay times as a function of the phonon wavevector. The insert shows the shape of the laser pulse.

670

be attributed to the enhanced phonon distribution versus those induced by the initial maximization [ which are treated as self-scatterings in the simulation). The algorithm just outlined has been applied to a simple case, where electrons are excited at 0.25 eV above the bottom of the conduction band (with a monoenergetic distribution). The lattice temperature is 77 K. The time evolution of the perturbed phonon distribution is shown in Fig. 4 for an excited carrier density of 5xl016 e m - 3. The lineshape of the laser

Z 01

pulse is shown in the insert (halfwidth = 0.8 ps). The LO distribution is driven out of equilibrium even during the excitation, due to the fast power

"~

dissipation of the high-energy photoexcited electrons. The maximum is

. . . . . . . . .

o •

20~

2

reached at a delay time of 1 picosecond for wavevectors of about 6x105

4

6

o

11o o

"x.

"x--x

8

10

tips}

c m - z. The small q values that are amplified during and immediately after the excitation are due to the polar nature of the e-phonon coupling. At longer times, the phonon distribution relaxes towards its equilibrium value as a result of two distinct processes, phonon reabsorption and phonon-

Fig. 5. Time evolution of four different modes as a function of delay time.

phonon interaction. The first one is due to the fact that the group velocity of optical phonons is very small (less than 1 0 3 c m / s ) , implying that the phonons cannot drift away of the excitation volume during their lifetime. Therefore, if the phonon lifetime is long enough and the coupling with the

PHONON OCCUPATION FOR VARIOUS TIMES DURING PHOTOE×CITATION

carriers sufficiently strong, emitted phonons can be reabsorbed. It is important to notice that modes of different wavevector evolve in time

2

in different ways, as indicated in Fig. 5 . Those with the smaller q (6x10 ~

e r a - z) exhibit an exponential decay, immediately after the end of the exci-

It

.

,. I p ~

!

, .o~

"

tation, with a characteristic decay time of 7 ps. At intermediate q's (8 and 10xl05 c m - z ) the phonon distribution decay much faster at short times

:1:109 m

(up to 5 and 8 ps delay), approaching then the exponential behavior. The amplification of these large-q phonons is not as pronounced as that of the small-q ones. The time evolution of the phonon distribution reflects the microscopic details of the cooling processes in the coupled electron-phonon system. While phonon-phonon processes are always active, and their effect is independent of wavevector, phonon reabsorption varies drastically as a

Fig. 6. LO Phonon distribution as a function of total parallel momentum (for q= = O) for times during and after the laser excitation.

function of time and wavevector. In fact, the very rapid changes in the electron distribution function modify the range of phonon transitions that are allowed by energy and momentum conservation. At high energy, electrons can emit phonons with very small q, but as they cool the minimum allowed q shifts at higher values. Such a shift appears in Fig. 4, hidden though by the strong initial amplification. Furthermore, an electron will not be able to reabsorb the earlier emitted phonons once it goes below a

~000

8 ~ff2,s,~olllcm~

~00

x ninl~Zh,1011/cm2 NO HOT PHONONS

certain energy. This simple analysis explains why the phonons with small q-vector excited during the first stages of the electron relaxation (up to

4O0

2 ps delay time) cannot be reabsorbed, and decay exponentially via non-

2

electronic processes. On the other side, both the reabsorption and the phonon-phonon terms will contribute to the damping of phonons of larger

Timelps)

wavevector in the first few picoseconds, leading to their faster decay over this time interval. The electron relaxation rate is drastically reduced because of the presence of non-equilibrium phonons. The phenomenon is mainly due to the reabsorption of the LO phonons that have been emitted in the first stage of the relaxation without having had enough time to decay. The effect of phonon reabsorption grows with time as the electrons populate the low

Fig. 7. Average electron energy as a function of time for two different injection densities.

671

energy regions below the threshold for optical emission. A detailed dis-

/

cussion, and a comparison with the results of other theoretical models can be found in Lugli and others (1987) and Lugli (1987). The full Monte Carlo simulation for electrons and phonons has been performed also for a GaAs-A1GaAs q u a n t u m well (Goodnick and Lugli, 1986; Lugli and Goodnick, 1987). In such a system, recent results obtained with time resolved photoluminescence have shown dramatically reduced energy relaxation rates for photo-excited electrons (Ryan and others , 1986; Xu and Tang, 1984). A single quantum well of GaAs-A1GaAs (tw = 150 ~, CB = 0.28 eV) has been considered, with subband energies given by the solution of the one dimensional wave equation for a square well potential. The bands are assumed parabolic. The scattering rates of the quantized

Fig. 8 GaAs Band Structure

2D electrons with bulk unscreened LO-phonon (both intra- and intersubband) are calculated numerically without the use of m o m e n t u m conserving

fF(av)

approximations. Intervalley transfer to the L-valleys (also quantized), 2D electron-electron scattering (Goodnick and Lugli, 1987), and degeneracy effects due to the Pauli exclusion principle are are included as well. The cooling of photoexcited electrons in an n-type GaAs-A1GaAs quantum well at low temperature (5 K) has been considered. A background density of 2.5x1011 c m - 2 is used. The injected density is 5x1011 cm -2 The GaAs

10 '~~p

parameters are the same as for the bulk case. The width of the simulated laser pulse is about 1 ps, during which time carriers are added to the simulation with an initial energy of 0.25 eV above the b o t t o m of the

GaAs', k .

lowest subband. The time evolution of the phonon distribution at qz=0 (intra subband scattering) shown in fig. 6. The same, although reduced, features are found also at qz=0. LO-phonon emission during the pulse

1

"~\

0 0.1 0.2 0.3 0.4 I

and immediately after creates a large population of phonons at small q's. At longer times, phonon reabsorptions and phonon-phonon losses drive

E (eV)

the distribution back to equilibrium. The secondary peak that develops at later times in the phonon distribution is due to phonon emission by electrons t h a t have already relaxed to lower energy. Fig. 7 shows the evolution of the electron total energy (kinetic plus potential) as a function of time during and after the pulse. The excited electrons lose energy mainly through the interaction with the background electrons and through

Fig. 9 Distribution function of I" valley electron 2 ps after excitation for an electron density of 5xl01Scm-S, at 300 K

Nq-No

the emission of LO phonons. Without hot phonons (x), the hot electrons are found to reach equilibrium in about 3 ps. In contrast with the case

[•lnP

of an unperturbed phonon distribution, a much slower relaxation is found when non-equilibrium phonons are accounted for (o , 4). The two cases in fig. 7 correspond to two different injection densities. It can be added that no real dependence on well width has been found in the Monte Carlo result. The reduction of the electron cooling rate is due to the reabsorption

GaAs x

of non-equilibrium phonons which build up during the initial pumping and

-~

the first stage of the electron relaxation. The reduction in the electron relaxation rate is stronger when a considerable number of electrons have relaxed to the low energy region below the emission threshold. Shortly after the end of the laser pulse (t = 1.6 ps), the strong intercarrier scattering creates a broad distribution where the subband minima (indicated by arrows) clearly appear. Within each subband, the distribution function starts to exhibit a Fermi-like appearance which is fully established at longer times.

S.S.E. 31/~-4--X

0.10

|-

1

I% I 2

~II :3

qOO%m-1)

Fig. 10 Non equilibrium LO phonon distribution in the conditions of Fig. 9

672

REALISTIC CASE A more realistic case has also been studied. Fig. 8 shows the details of the band structure of GaAs. The laser excitation creates three electron distributions, corresponding to the three hole bands. For a photon energy of 2.04 eV, about 70 percent of the electrons are excited above the F - L separation. The Monte Carlo results (using the parameters of Tab. 1) show that the electrons transfer very rapidly into the L valley, returning then slowly to the central valley. This effect is responsabile for a very slow rise of the integrated luminescence spectrum. Fig. 9 shows the distribution function of the F electrons in GaAs (dashed curve) and InP (eontinous curve) after 2 ps form the exitation of a 400 fs laser pulse. It can be noticed that electron-electron scattering is very efficient in driving the distribution towards a maxwellian. The InP population is much higher, since there is there no significant transfer to the L valley. As a consequence, the phonon perturbation is much stronger, as seen in Fig. 10. In fact, due to the much larger effective mass, electron in the L valley emit large-q LO phonons, which do not contribute significantly to the phonon amplification. A reduction of the electron cooling rate is found even at room temperature in InP for a density of 5 x 1017cm - 3 , while lower temperatures are needed to detect a similar effect in GaAs. The Monte Carlo data, t h a t will be presented in a forthcoming publication, agree very well with the luminescence and raman data. In summary, a full Monte Carlo technique for the study of electron and phonon dynamics in bulk GaAs and InP, and in GaAs-A1GaAs quantum wells has been presented. For the case of laser excitation, a strongly perturbed phonon distribution is found in the first picoseconds after the laser pulse, which is responsible for a reduction of the cooling rate of the photoexited carriers. Good agreement with time resolved luminescence and Raman data has been found. ACKNOWLEDCMENTS I would like to t h a n k Dr. S. M. Goodnick, Dr. P. Koeevar, Dr. C. Jacoboni, Dr. L. Reggiani, and Dr. J. Shah for many useful discussions and for their contribution to part of the work presented here. The financial support of the European Research Office, the italian National Research Council, and the Computer center of the Modena University is gratefully acknowledged. REFERENCES Bordone, P., C. Jacoboni, P. Lugli, L. Reggiani, P. Kocevar (1987). J. Appl. Phys. 61, 1460-1468. Cai, W., C. M. Marchetti, M. Lax (1987). Phys. Rev. B 35, 1369-1373. Collet, J., A. Cornet, M. Pugnet, A. Amand (1982). Solid St. Comm. 42,883-887. Collins C.L., and P.Y. Yu (1984). Phys. Rev. B30, 4501-4515. Goodnick,S.M., and P. Lugli (1986). Proe. of XVIII Int. Conf. Phys. Sem., Stockholm, Sweden, 1527-1530. Goodnick, S.M., and P. Lugli (1987). Phys. Rev. Lett. Accepted for publication. Kash, J.A., and J.C. Tsang and J.M. Hvam (1985). Phys. Rev. Lett. 54, 2151-2154. Klemens, P.G. (1966). Phys. Rev. 148, 845-848. Kocevar, P. (1985). Physica 1 3 4 B , 155-163. Lugli P., and D.K. Ferry (1985). Physica 134B, 364-368. Lugli P., and S. M. Goodnick (1986). "High Speed Electronics", Eds. B. Kallback and H. Benekin, Springer Verlag, Berlin , 116-119. Lugli, P., C. Jacoboni, L. Reggiani, and P. Kocevar (1987). Appl. Phys. Lett., 50, 1521-1524. Lugli, P. (1987). Physica Scripta. Accepted for publication Poetz, W. and P. Kocevar. (1983). Phys. Rev. B28, 7040-7049. Rieger M., P. Kocevar, P. Bordone, P. Lugli, and L. Reggiani (1987) these proceedings. Ryan, J.F., R. A. Taylor, A. J. Turbefield, A. Manciel, J. M. Worlock, and A.C. Gossard (1984). Phys. Rev. Lett. 53, 1841-1844. Ryan, J.F., R. A. Taylor, A. J. Turbefield, and J. M. Worlock (1986). Surf. Sci. 170, 511-515. Shah, J., A. Pinczuk, A. C. Gossard, and W. Wiegmann (1985). Phys. Rev. Lett. 54B, 2045-2048 Von der Linde, D., J. Kuhl and H. Klingenberg (1980). Phys. Rev. Lett. 44, 1505-1508. Xu, Z.Y., and C. L. Tang (1984). Appl. Phys. Lett. 44, 692-695.