Physica 1448 (1987) 376-390North-Holland, Amsterdam

POLAR SEMICONDUCTORS UNDER CONTINUOUS PHOTOEXCITATION

'Tânia TOMÉ, Áurea R. VASCONCELLOS and R. LUZZIDepartamento de Fisica do Estado Sólido, Instituto de Fisica "Gleb Wataghin", UniversidadeEstadual de Campinas.(UNICAMP), 13100-Campinas,S,P., Brazil

Received 25 July 1986

The nonequilibrium thermodynamic state of a highly excited plasma in direct-gap polar semiconductors underconditions of continuous laser iIIumination is studied using the nonequilibrium statistical operator method. We obtain theset of coupled nonlinear generalized transport equations for the basic set of thermodynamic variables chosen for thedescription of the macroscopic state of the system. These equations are solved for the specific case of GaAs, to describe thetransient regime and to obtain the steady-state values of the macrovariables in terms of the laser power. The rates ofenergy pumping and of energy transfer through the different relaxation channels are also given. Finally, we comment onthe stability of the uniform steady-state solution.

1. Introduction

Semiconductor systems can be strongly departed from equilibrium by the action of high fields orintense laser illumination. The latter produces high levels of nonthermal carriers (electron-hole pairs)in intrinsic semiconductors, which move in the (also nonthermal) lattice background giving rise to theso-called nonequilibrium plasma in semiconductors, a particular case of solid state plasma [1]. This is anattractive subject for study because of the related technological interest for application in devices andthe fact that highly excited plasma in semiconductors (HEPS) offer an excellent testing ground fortheoretical ideas in the are a of far-from-equilibrium systems. A large amount of experimental andtheoretical research has been done on HEPS. On the experimental side a particularly useful techniqueis the ultrafast laser spectroscopy which allows for the analysis of relaxation processes and nonlineartransport in the picosecond and even femtosecond scales [2]. On the theoretical side, efforts have beendevoted to improve ways to study the macroscopic state of nonequilibrium systems with very manydegrees of freedom. Several methods and approaches in nonequilibrium statistical mechanics have beendeveloped in an effort to provide satisfactory answers to the most relevant questions in the field ofirreversible processes in complex systems [3]. They are required to improve upon the limitations in thedomain of validity of the usual linear and nonlinear response function theory, no longer applicable tosituations involving large thermal perturbations and feedback mechanisms, as is the case for HEPS.

From the existing theoretical approaches to the question of the dynamics of relaxation processes infar-from-equilibrium many-body systems, the nonequilibrium statistical operator (NSO) method seemsto offer a very effective technique to deal with a large c1ass of physical problems. It leads to theconstruction of a nonlinear transport theory considered to be a far-reaching generalization of theChapman-Enskog method in the kinetic theory of gases. The advantages of the NSO method reside inthe possibility to describe a large c1ass of experimental situations where the system can be arbitrarilyaway from equilibrium, and with nonlinear, nonlocal, and retardation (memory) effects incorporated

" from the outset. We have used the NSO method in Zubarev's form [4] and adapted for the treatment of

t' open systems [5] to discuss ultrafast optical responses [6] and ultrafast transient transport [7] in HEPS.,- The same theoretical approach is used to study the transient evolution and the ensuing stationary

condition of the macroscopic state of direct-gap polar semiconductors under continuous laser illumina-

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T. Tomé et aI. / Polar semiconductorsunder photoexcitation 377

tion, and the results are reported in section 3. This kind of situation may arise in quantum generatorsand semiconductor devices, and, particularly, in experiments involving laser-induced damage andannealing [8]. In the next section the NSO method is briefty reviewed and from it we derive the set of

.". coupled nonlinear transport equations for the set of relevant thermodynamic variables that describe themacroscopic state of the HEPS. These are the quasi-temperatures of carriers and of the various phononbranches, and the concentration of carriers. Their evolution in time is derived according to the NSOmethod from the transport equations for the energy densities of the phonons and of the excited carriersystems. They are solved numericalIy for the case of GaAs at room temperature under constant laserillumination with powers up to 500 kW cm-2. At the steady state the quasi-temperatures of theelementary excitations become very nearly equal to the bath temperature. On the other hand, carrierconcentration at the steady-state increases with the laser power up to, roughly, 1019cm-3 for500 kW cm-2. The stationary values of the effective temperatures folIow in a few tens of picosecondsafter application of the laser light, while the carrier concentration shows a transient time ofnanoseconds.

The evolution equations for the variables that describe the macroscopic state of the HEPS underconsideration are highly nonlinear, and thus the steady-state may display multiple solutions [9]. Theuniform steady-state (space and time constancy of quasi-temperatures and carrier density) is theso-calIed thermodynamic branch of solutions [10], i.e. the one that emerges continuously from theequilibrium state with increasing photoexcitation. The detailed analysis of the stability of this solution isunder way; preliminary results [11] show that beyond a certain threshold of the laser power (roughly:o;:;10 kW cm-2) a branching point of the solutions occurs and the uniform state becomes unstableagainst the formation of a carrier superlattice. This is a striking case of a dissipa tive structure [10],which may stabilize in the open carrier system while it remains in a far-from-equilibrium macroscopicstate subject to nonlinear kinetic laws like eq. (11) in the folIowing section. Semiconductor laserfilamentation could be a manifestation of this instability in the inverted populations of electrons andholes in a uniform spatial distribution [12]. Formation of spatial ordering in the HEPS may also be amechanism for athermal recrystalIization in ion-implanted semiconductors, as discussed in the lastsection.

2. Theoretical background

2.1. The nonequilibrium statistical operator method

The NSO method is based on the construction of an ensemble of replicas of the system distributedover alI microscopic states in accordance with the initial specifications and constraints imposed on thesystem by the experimental conditions. To the representative ensemble one associates a distributionfunction (statistical operator for the quantal case) p(t), and the average over the ensemble of adynamical quantity A:

(Alt) = Tr{Ap(t)} ,

is placed in correspondence with the result of a measurement, a(t) of A, performed on the actualphysical system. The NSO, p( t), is a functional of a reduced set of dynamical quantities P l' . . . , P n'where n is much smalIer than the number of degrees of freedom of the physical system. The averagevalues of the Pj (j = 1, 2,' . . , n) over the nonequilibrium ensemble, Q/t) = Tr{PjP(t)}, are the set ofnonequilibrium thermodynamic variables, or macrovariables for short, that describe the macroscopicstate of the system. It ought to be stressed that this contracted macroscopic description is based on

378 T. Tomé et ai. / Polar semiconductors under photoexcitation

Bogoliubov's assertion that such contraction is possible on time scales larger than a certain time formicroinformation, TIL'which characterizes the lifetime for correlations that can then be ignored [13].Thus, the contraction process is dependent on the separation from the total Hamiltonian of stronginteractions whth certain symmetries, i.e. those related to the fast relaxing processes [14]. To presentthere seems to be no wholly satisfactory theory to generate a unique selection of the basic set ofvariables; the modelling of a concrete problem amounts to a judicious choice of the variables and theform of their evolution equations [15].

Different forms of NSOs have been constructed by several authors using either heuristic approaches[16-19], or projection operator techniques [20-22]. All of these results can be put under a unifieddescription [23] using Jaynes' maximum entropy formalism [24]. This is done maximizing Gibbs(statistical) entropy:

SG(t) = -Tr{p(t) ln p(t)} , (1)

with p(t) normalized at alI times, i.e. Tr{p(t')} = 1, and subjected to the constraints Q/t') =Tr{PiP(t')}, to ~ t' ~ t, for a given basis of dynamical quantities Pi, j = 1,2,.. . , n, to be used for thecontracted description of the macroscopic state of the system after the initial time to. We obtain theNSO

I

Pw(t) = exp{! dt' w(t, t'; to) ln p(t', t' - t)} ,10

(2)

where

(3)

is called the coarse-grained statistical operator. For simplicity we neglect the possible spatial depend-ence of P, Q, and F, and operators Pare given in the Heisenberg representation. In eq. (3), 4> ensuresthe normalization of p, and the Fj(t), j = 1, 2,. . . , n, are the Lagrange multipliers defined by therelations

(4)

Finally, w(t, t'; to) is an auxiliary function which allows, firstly, to make possible to identify the F/t)with the intensive variables conjugated to Q /t) in the sense of Generalized Thermodynamics [10], and,secondly, to include irreversible evolution of the macroscopic state of the system from an initialcondition fixed by Pw(to)= p(to). The normalization of the coarse-grained statistical operator, togetherwith eq. (4) implies in the normalization of the NSO. Zubarev's NSO [4] is retrieved as a particular caseof the general method of construction we have just summarized by the choice w(t, t'; to) = exp{ s(t' -t)}, to~ -00, and s( > O)is an infinitesimalreal number which goes to zero after the trace operation inthe calculation of averages have been performed.

The NSO of eq. (2) can be separated in two parts

Pw(t) = p(t) + p'(t) , (5)

where p(t) is a non-dissipative term (which defines the instantaneous mean values of the operators Pi,

T. Tomé et ai. / Polar semiconductorsunderphotoexcitation 379

as given by eq. (4)), and p' carries the information on the microscopic dynamics relevant to thedescription of the irreversible evolution of the system.

Connection with irreversible thermodynamics is made through the definition of the coarse-grainedentropy

[f(t) = -Tr{pw(t) ln p(t)} = -Tr{p(t) ln p(t)} , (6)

where the last equality is a result of eq. (4). This function is next identified with the phenomenologicalentropy of generalized thermodynamics [10], having the property that the rate of entropy production isgiven by

d n d(T(t) = -d [f(t) = L Fi(t) -d Q /t) ,

t i= 1 t(7)

where

a[f(t)

Fj(t) = aQ/t)(8)

defines each Fi as thermodynamically conjugated to Q rA nonlinear transport theory follows from the NSO formalism through the construction of the

generalized transport equations (equations of evolution for macrovariables Q i)

(9)

where H is the total Hamiltonian. Alternatively, eq. (9) can be transformed into a set of coupledequations for the intensive thermodynamic variables Fj(t) using the relation

(10)

where Cfk1(t) = -a2[f(t)laQ/t) aQk(t) is the inverse of the correlation matrix of quantities Pi in thecoarse-grained ensemble,

(11)

The use of this thermodynamically equivalent representation is convenient from a practical point ofview since the right-hand side of eqs. (9) is explicitly dependent on the F/t), and only implicitlydependent on the Q ;(t) through eq. (8).

Using the separation of the NSO given by eq. (5), and assuming that the total Hamiltonian H can besplit into two parts Ho + H', such that for the set of operators Pi it is satisfied that [Pi' Ho] = I:k aikPk,where aik are real coefficients, the right-hand side of eq. (9) can be written in the form of a series ofcollision operators of an ever increasing order in the interaction strengths [25]. Each of these collisionoperators involve correlation functions in the coarse-grained statistical ensemble, and, for the particularcase of Zubarev's method, we have [4,25,26]

(12)

380 T. Tomé et ai. / Polar semiconductorsunderphotoexcitation

(O) _ 1 -lj (t) - iA Tr{[Pj' Ho]p(t)} ,

l~I)(t) = i~ Tr{[Pj, H']p(t)} ,

(13a)

(13b)

O

l~2)(t) = c~r f dt' eE1'Tr{[H'(t'), [H', Pj]]p(t)}

(13c)

The set of nonlinear transport equations (12) are of first order in the time derivative, and thereforeits solution requires to provide initial conditions, i.e. the values Q j(to), or F/to), (and aiso boundaryconditions if space dependence is considered). Thus we see a posteriori that the NSO constructed usingthe MEF with information gathered on the time interval (to, t), can be obtained with only theknowledge of the values of the basic set of variables at a unique time to' The subsequent values for t> toare consistently given by the formalism through the nonlinear transport theory it provides. Next weapply the results summarized in this subsection to the case of HEPS under continuous illumination.

2.2. Transient and steady states in HEPS

Consider an intrinsic direct-gap polar semiconductor under illumination from a continuous lasersource, with electron-hole pairs created in the processes of absorption of one photon. The excessenergy optically pumped into the electron system is mainly transferred to the lattice via relaxationprocesses mediated by the deformation potential and the Frõhlich interactions. The number of pairsvaries in recombination processes, and anharmonic effects redistribute the excess energy among thedifferent phonon modes. The highly excited photogenerated carrier system is brought into internalequilibrium in a fraction of a picosecond due to the Coulomb interaction, acquiring a distribution inenergy space characterized by a quasi-temperature, {3;I(t) = kT~(t), much larger than the latticetemperature, and by quasi-chemical potentials JLe(t) and JLh(t), which are dependent on T~ and thecarrier concentration n(t) [6,28,29]. Next, in moderate to strong polar semiconductors mutualthermalization of carriers and longitudinal-optical (LO) and transverse-optical (TO) photons sets invery rapidly (picosecond time scale), being followed by energy relaxation to the acoustic (A) phononsvia the anharmonic interaction in, typically, a tens of ps time scale. The carrier concentration varies as aconsequence of the interaction with the laser photon field, which builds it up in absorption processes,while recombination processes reduce it. We consider only radiative spontaneous recombination,neglecting induced recombination, non-radiative recombination, self-absorption, and the Auger effect,as well as diffusion from the volume of laser light focalization. Furthermore, we assume a constant laserlight intensity throughout the focused spot. The sample is in contact with a thermal bath at temperatureTo, and IL indicates the laser power ftux, and wL the laser photon frequency.

We take the system Hamiltonian to be composed of (a) the electron energy operator (Bloch bandsHamiltonian plus Coulomb interaction; the electrons are treated as Landau's quasi-particles in therandom phase approximation (RPA), and the effective mass approximation in the electron-holerepresentation is used), (b) the Hamiltonians of the phonon and photon free fields and (c) the

T. Tomé et ai. I Polar semiconductors under photoexcitation 381

interaction energy operators between these different subsystems. They are well-known and thereforewe do not write them down here [30,31].

Next, to apply the statistical procedure of section 2.1 to the nonequilibrium system being'.. . considered, we need to define the basic set of macrovariables. First we consider the semiconductor

sample as an open system interacting with the laser source and the thermal bath, both taken as idealreservoirs, i.e. it is assumed that they remain in near-stationary conditions (characterized by IL and To)sustaining negligible modifications as a consequence of the coupling with the semiconductor. The totalNSO is then the product of the constant statistical opera tor for the external sources and reservoirs withthe NSO of the time-evolving open system [5]. We take the latter to be dependent on the set ofdynamical quantities composed of the Hamiltonian of the carriers, Hc, the Hamiltonian of the opticalphonons, HLO and HTO' the Hamiltonian of the acoustic phonons, HA, and the number operators forelectrons, Ne, and for holes, Nh, i.e. {P) =={Hc, HLO' HTO' HA, Ne, Nh}. We introduce for the set ofintensive variables {F/t)} =={/3c(t), /3LO(t), /3TO(t), /3A(t), -/3c(t)/Le(t), -/3c(t)/Lh(t)}, defining theinverse of quasi-temperatures of carriers, T~(t), and of phonons, T~o(t), T~o(t), T1(t), and thequasi-chemical potentials for electrons and holes, /Le(t) and /Lh(t). The associated extensive nonequilib-rium thermodynamical variables are then {Q/t)} =={Ec(t), ELO(t), ETO(t), EA(t), n(t), n(t)}, i.e. theenergies of the carriers and phonons subsystems and the carrier concentration (the same for electronsand holes since. they are produced and annihilated in pairs). The coarse-grained statistical opera to r forthe semiconductor system with the above selection of macrovariables, according to eq. (3), is

p(t) = exp{ -<p(t) - /3c(t)Hc - /3LO(t)HLO- /3To(t)HTO - /3A(t)HA

+ /3c(t)/Le(t)Ne + /3c(t)/Lh(t)Nh} . (14)

The total Hamiltonian is separated into Ho, containing the Hamiltonians of the free subsystems ofcarriers and phonons, plus H', composed of all the interactions between the subsystems .and theinteraction of the carriers with the laser and recombination radiation fields. Heat diffusion effects fromthe volume of laser focalization will be included in a phenomenological way in the transport equations.The basic set of dynamical quantities commute between themselves, and also with Ho. Hence, thecorresponding collision operators J~O)and J~l) of eqs. (13a) and (13b) are, for the present case, null.Next we solve the six generalized transport equations in the quasi-linear approximation, Í.e. retainingonly J~2), which then reduces to the calculation of the first term on the right-hand side of eq. (13c). Inthe quasi-linear approximation the scattering operators are cast in the form of instantaneousBoltzmann-like collision terms. They have the form that would be obtained using the Bom approxima-tion in the perturbation theory but with the equilibrium distributions replaced by those characterized bythe thermodynamic parameters F already listed.

Omitting the details of the calculations we only write down the final form of the nonlinear transportequations, beginning with that for the carrier energy:

(15)

where the different terms on the right-hand side are the contributions to the rate of variation of thecarrier energy due to interactions with the laser source, radiative recombination, and interactions withthe longitudinal-optical (LO), transverse-optical (TO), and acoustic (A) phonon fields, respectively.The five terms are

(16a)

382 T. Tomé et ai. / Polar semiconductors under photoexcitation

:t Ec,R(t) = - 2; L IUR(k,q)12(€~+ 6~)f~(t)f~(t)B (6~ + 6~_ 1í~/;),~q 6.

dd EC,y(t) = 2",'TT'L IU~(k, qW(6~+q - 6:){v~(t)f~(t)[1- f~+q(t)]t n k,q

-[1 + v~(t)][1- f~(t)]f~+q(t)}B(6~+q - 6~-1íw~).

(16b)

(16c)

ln the above equations, a = e or h, y = LO, TO, or A,

(17a)

(17b)

and the matrix elements are

(18a)

(18b)

where we have used the RPA dielectric function

6(q, t) = 1 + [q~(t)/q2],

q~(t) = 4'TT'e2L1

âf~~)I

.60V a,k â6k

The time dependence of the dielectric function is a result of its dependence on the instantaneousmacroscopic state of the system. Eo is the forbidden energy gap, 60and 6. the static and high-frequencydielectric constants, WLQthe dispersionless frequency of the LO phonons, and V the active volume ofthe samp1e, 6~= Eo + (1í2e/2me), 6~= 1í2e/2mh are the energy dispersion relations for the electronsand the holes with effective masses me and mh, respectively. Moreover,

E2 1íluTO (k )1

2= TO,aa , q 2gVwTO '

E2 1íIUA (k )12 = A,a q

a , q 2gVs'

(19a)

(19b)

where g is the density of the material, s the speed of sound, a Debye model was used to describe the Aphonons (then w~ = sq), wTOis the dispersionless frequency of the TO phonons, and Ey,a the strengthsof the deformation potential interactions.

The rate of carrier energy variation due to interaction with a laser source of frequency WLand fluxintensity IL(t), eq. (16a), is written in the form

(20a)

(20b)

T. Tomé et ai. I Polar semiconductorsunderphotoexcitation 383

where aI and a2 are the absorption coefficients of one and two photons, respectively; ilL in eq. (16a) iswLor 2wL, respectively, for the two cases (20a) and (20b), and UL is the corresponding matrix element.To perform numerical calculations we use the experimental values of the absorption coefficients.

.. Finally, r and t are the Fermi distributions with energies Ee,h= (mx/me.h)(hilL - Ea), wherem;1 = m;1 + m~l, and for temperature T~(t) and chemical potentials /-Le(t)and J.Lb(t).

We also have

d d ddt ELO(t) = - dt Ec.LO(t) + dt ELO,A (t) ,

(21)

where

(22)

is the rate of energy transfer of LO phonons to A phonons due to anharmonic processes: 'T is aphenomenological relaxation time, and vLO(t, T~) = {exp[J3A(t)hw~o] -I} -1.

For the rate of variation of energy of the TO and A phonons we find

(23)

(24)

(25)

(26)

where B refers to the thermal bath at temperature To; eq. (26) accounts for heat diffusion from theA-phonon system to the thermal bath, and we have introduced in it a phenomenological relaxation timefor heat diffusion which depends on the diffusion coefficient and the dimensions of the surface of theactive volume of the sample [30], and it should be estimated for each case when performing specificnumerical calculations. The other phenomenological relaxation times, 'TLOAand 'TTOA' can be evaluatedfrom linewidths of the Raman lines. ' .

The rate of change of the carrier density is

d dI

-1 d-d n(t) = -d n(t) + (h~ - Ea) -d Ec L(t) ,t t R t '

(27)

where

dI

21T '" I R1

2 e h ( e h hcq)d n(t) =- _h L..- U (k, q) !k(t)f k(t)8 Ek + Ek - 1i2 .t R k,q Eco

(28)

To obtain the intensive nonequilibrium thermodynamic parameters Fj' eqs. (15), (21), (23), (25) and(27) must be complemented with eq. (10), or better with their reversed form

384 T. Tomé et ai. / Polar semiconductors under photoexcitation

d . ~ .-d Ee(t) = -(He; Helt)/3e+ L.J(He; Nalt)(JLa/3e+ P-a/3e),t a

d .dt ELO(t) = -(HLO; HLOlt)/3LO,

d .dt ETO(t) = -(HTO; HTOlt)/3TO,

d .dt EA(t) = -(H A; HAlt)/3A ,

:t n(t) = _(Na; Helt)J3e+ (Na; Na It)(JLaJ3e+ P-a/3J ,

(29a)

(29b)

(29c)

(29d)

(2ge)

where we have introduced the correlation matrix C(t) with elements

(H e; Helt) = 2: (E;)2t;(t)[l- t:(t)] ,a,k

(30a)

(HLO; HLOIt) = 2: h2w~ov~o(t)[l + v~°(t)],q(30b)

(30c)

(H A; HAIt) = 2: (hsq)2v~(t)[l + v~(t)] ,q

(30d)

(Na; Nalt) = 2: t;(t)[l- [;(t)],q

(30e)

(He; Nalt) = (Na; Helt) = 2: E;[;(t)[l- [;(t)],k

(30f)

and all cross-correlation functions other than (30f) are null.Finally, the concentration of carriers is related to the quasi-chemical potentials by [34]

(31)

where

and F1/2 are the Fermi functions of index half.We proceed next to obtain numerical solutions for the case of gallium arsenide samples.

3. Uniform electron-hole fluid in GaAs

A GaAs sample is assumed to be in contact with a thermal bath at room temperature (To = 300K),and ilIuminated by a constant laser light with power flux IL (to be varied) and photon energyhWL = 2.4 eV.The one-photon absorption coefficient is -1 x 104em-1; this calculated value is in goodagreement with the experimental data [33]. The parametric relaxation time used in the calculations is

T. Tomé et ai. / Polar semiconductorsunderphotoexcitation 385

7AN -10 ps [32]. We assume that the aeoustie phonons remain eonstantly at room temperature; [35] allother parameters, sueh as effeetive masses, dieleetrie eonstant, interaetion strengths, ete. are obtainedfrom the eurrent literature [36].

.. We obtain the evolution in time of the six thermodynamieal variables, i.e. the solutions of thesystem of eoupled integrodifferential equations obtained in the last seetion, whieh show that, asexpeeted, a steady state appears after a transient time. The values of the variables in the steady stateare reeonfirmed solving the stationary equations dQ / dt = J~2) = O, j = 1 to 6. The solution of thekinetic equations requires to give initial eonditions; for this purpose the initial time to is taken as thedelay time after applieation of the laser light sueh that the earrier eoneentration is roughly 1014 to1015 em -3 (n(to) = a1ILto/hwL). Sinee the excitation energy is hWL - Ea = 0.88 eV, giving an effeetiveearrier temperature of roughly 3000 K, at those eoneentrations the exeited pairs should form a doubleFermi liquid on the metallie side of the Mott transition, being nearly in internal equilibrium, and thenbe suseeptible to the theoretieal treatment of seetion 2.

For laser powers in the range 5 to 500 kW em -2 we take to =0.5 ps; up to this delay time and withthe thermal bath temperature at 300 K only a small fraetion of optieal phonons in exeess of equilibriumis present at to. Thus, we take 300 K for the initial quasi-temperature of phonons, and -3000 K for thatof earriers (3kT~(to) = hWL - Ea). It should be notieed that the solution of the kinetie equations isweakly dependent on the initial eonditions when the results are analyzed in a larger than pieoseeondtime seale, as we do here. Finally, given T~(to) and n(to), the initial values of the quasi-ehemicalpotentials follow from eq. (31). Next, we solved eomputationally the system of eoupled differentialequations for laser powers of up to 500 kW/em -2.

The effeetive temperatures of earriers and optieal phonons after rapid equalization, that takes plaeein a few ps as already demonstrated in earlier studies of ultrafast laser speetroseopy [6], deerease tonearly room temperature in a lapse of time of the order of 20 to 40 ps after the initial applieation of thelaser illumination for IL in the range 10 to 500kW em-2 (see figo1). On the other hand, the earriereoneentration grows in time attaining a stationary value after transient times of orders of magnitudelarger than those for T~. Fig. 2 shows the evolution of n(t) for three values of the laser power, and figo

Fig. 1. Time evolution of the quasi-temperature of carriers for three different values of laser power.

2000 1+\ ( I) 12.5 kW cm-2

II\(2) 50 kWcm-2

-(3) 500 kWcm-2lJJ

a::=>I-

1000lJJc..:ElJJI-I

<f)<t

500=>o

300

I I I I I I I5 10 15 20 25 30

TI ME(picoseconds)

386 T. Tomé et ai. / Polar semiconductorsunder photoexcitation

104 106 108 1010

TI ME (picoseconds)

Fig. 2. Time evolution of the concentration of carriers for three different values of laser power.

3 the transient times for laser power up to 0.5 MW em-2; these transient times are defined as the lapseof time between the switehing on of the laser light and the moment when n differs by 10% of itsstationary value.

Fig. 4 provides the values of the earriers eoneentration in the steady state, nO,for laser powers up to50 kW em-2. At low laser powers, IL ~ 10 kW em -2 produeing values of nO smaller than 1018em-3, avery near dependenee of n on the square root of IL is observed, roughly given by nO= 5 x 1017I~/2em-3;

this is a resuIt that in eq. (28) the earrier population funetions of e1' (17a) approximate the classieMaxwell distribution. Carriers eoneentrations larger than 1018em- follow for laser power IL p10 kW em-2. Stretehing out the ealculations well beyond this point, a value no = 1019em-3 is obtained

100 200 300 400

LASER POWER (kwcni'2)

Fig. 3. The transient time for carriers concentration to attain the steady-state value.

5 f- LASER POWER / (I)

(I) 50 k W cm-2/

(2) 25 k W cm-2I "

'" 41 (3) 12.5 k W cm-2'eu

CI)

-Q.!5z 3oI-<ta::I-

2uzou

1210

-;;; 010IW::Ei=I-ZW(/)Z

105<ta::I-

T. Tomé et ai. I Polar semiconductors under photoexcitation 387

6

.i.

10 20 30 40 50

LASER POWER ( kWcm-2)

Fig. 4. Concentration of the photo-injected electron-hole pairs in the steady state.

Fig. 5. The rates of energy pumping (fullline), of energy loss in recombination processes (dashed line), and of energy transferfrom carriers to optical phonons (dot-dashed line) and to acoustical phonons (double dot-dashed line). (Note the different scalefactors indicated in the upper left comer.)

5'"'e'!lu 4Q><

3I-«g: 2zwuzou

(x 1018)

'i r --- ,.,0",

/e /

_U4 _._ (x 1017) /I /(J) 14 ./

i -..- (x IO )

/ .

/./''///

a:.

3<f)z .'/«a:I- -/>- .(f(!)5 2 /z ;/w ;/I.J... ;/ .....--..o :/ -.......--..<f)w .:/ --"I- /7/ ..--"

1. -

.4 ..-"..-/ .'/'

Ç/ ././

/'I

o10 20 30 40 50 60 70

LASER POWER (kWcm-2)

388 T. Tomé et ai. / Polarsemiconductorsunderphotoexcitation

zof= - 8<1 (/)~.=.mW::2::2

8 f= 6WQ:

10

... '..

410 20 30 4 O

LASERPOWER(k W cm-2)

Fig. 6. The recombination times in the steady state for increasing values of laser power.

for IL =200 kW em -2, and for IL = 500 kW em-2 we get nO= 2 x 1019em -3. The parabolie dependeneebetween nO and IL obtained for IL ~ 10 em-2 is no longe r satisfied and in the range 50 ~ IL ~500 kW em -2 we find the approximate relationship nO=0.8(1 + 0.9I~2) x 1018em-3.

Fig. 5 shows, for different values of the laser power and in the steady-state, the rates of energytransfer along various ehannels of relaxation and pumping. For laser powers in the range 10 to50 kW em -2, a fairly linear dependenee of the rates of energy transfer from earriers to TO, LO, and Aphonons, as welI as for the rate of energy pumping, eould be noted. The latter varies between0.1 mJps-1 em-3 (or 0.14meVps-1 per pair) and 0.5 mJps-1 em-3 (or 0.52 meV pS-1 per pair) in thatrange. Most of that energy is transferred in reeombination processes at a rate roughly 5% smalIer thanthat of pumping. To attain the steady state, that differenee is taken eare of by the energy transfer to LOphonons, and, to an almost negligible degree, to the A phonons. [Note the different sealing faetors forthe different ehannels used in figo 5.]

The rate of pair produetion (not shown here) is praetiealIy linear with laser power, and from theangular eoeffieient we obtain the one-photon absorption eoeffieient Cl'L(wL) = 1 x 104 em-I; the photo-exeited earriers population inversion does not modify in the region of interest the value of theabsorption eoeffieient experimentalIy determined in near equilibrium eondition (the measured value isin very good agreement with that obtained by direet ealculation). Evidently, in the steady state thereeombination rate equals th~ produetion rate, and the reeombination times, defined by TR=-n/(dn/dt)R' are found to vary from -7ns at IL=10kWem-2 (no=1.7x1018em-3) to -5ns atIL = 50 kW em -2 (nO=5.7 x 1018em -3); see figo6. We reealI that in our ealculations we have negleetedthe indueed reeombination, selfabsorption, and the Auger effeet, as welI as assisted and nonradiativereeombination effeets.

4. Concluding remarks

We have studied a GaAs sample under illumination from a eontinuous laser souree and in eontaetwith a thermal bath at temperature To = 300 K. AlI the ealculations were performed using thenonequilibrium statistieal opera tor method in Zubarev's form and adapted to treat open systems. The . .solutions of the system of transport equations for the thermodynamie variables of the system, fordifferent laser powers IL, are obtained by an iterative numerieal ealculation. We have varied IL from5 KW em-2 to 500 KW em-2 and for eaeh IL we have obtained the time evolution of the maerovariablestowards the uniform stationary states, eharaeterized by spaee and time eonstaney of effeetive tempera-tures and earrier density.

T. Tomé et aI. I Polarsemiconductorsunderphotoexcitation 389

At the steady state the effective temperatures of the elementary excitations are very nearly equal tothe bath temperature. On the other hand, the concentrations of carriers at the steady state increaseswith the laser power up to, roughly, 1019em -3 for IL - 500 KW em -2. The stationary values of the

., effective temperatures are attained in a few tens of picoseconds after application of the laser light . Thisis a result of the rapid relaxation of the carriers excess energy to the optical phonon system with finalstationary quasi-temperatures very near the bath temperature. Different1y, the concentration of carriersshows a transient time of nanoseconds that decreases with increasing laser powers. The figurespresented in the last section provide a good description of the macroscopic state of the highly excitedplasma in polar inverted-band semiconductors. However the numerical calculations refers to the specificcase of GaAs, we expect that qualitatively and semiquantitatively those results may represent thesituation to be expected in most intermediate to strong polar semiconductors.

As already noted in the Introduction, we have obtained the so-called thermodynamic branch ofsolutions [10] of the highly nonlinear generalized transport equations, and an instability at a possiblebranching point of the solutions [9] cannot (and must not) be ruled out without further analysis. Wehave shown, using linear stability analysis [9], that for a description of the macroscopic state of theHEPS in the Gibbs space of the six chosen macrovariables the thermodynamic branch is stable.However, this analysis must be extended to include the Gibbs spaces of enlarged dimensions. Extendingthe thermodynamic fluctuation theory to steady states far from equilibrium, we have been able to testthe existence of instabilities of the uniform steady state in HEPS against the formation of spatialordering. Preliminary results [11] indicate that a plasmon mode in the carriers system becomes "soft",leading to the formation of a steady-state charge density wave, i.e. a dissipa tive (spatially ordered)structure in Prigogine's sense [10]. Filamentation in semiconductor lasers could be a manifestation ofthis instability [12,37]. This instability is also akin to the one proposed by Van Vechten [38] in indirectnonpolar semiconductors, in the debate on thermal vs. athermal laser annealing of ion-implantedsemiconductors. The controversy for that case seems to be definitely settled in favor of the thermaleffect. Our results may feed the conjecture that the self-organizing effect in far-from-equilibriumconditions may eventually be possible in direct-gap polar semiconductors. To present our treatment hasbeen based on an itinerant plane-wave model for the carrier system, and will be reported in the nearfuture; a more elaborate study using the chemical-bond approach is under way.

Acknowledgements

One of us (T.T) gratefully acknowledges a pre-doctoral fellowship from Fundação de Amparo aPesquisa do Estado de São Paulo. We also thank Prof. Dr. Paulo Sakanaka for his valuable advice withthe computational work. Miss Rosa Yukiko Kawaguchi, secretary of the Interdepartmental Center forTheoretical Physics to which the authors are associated, is kindly appreciated for the ever excellent andaccurate typing of manuscripts.

References

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390 T. Tomé et ai. I Polar semiconductors under photoexcitation

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