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3~ Solid State Communications, Vol.59,No.]O, pp.661-663, 1986.
~ Printed in Great Britain.
~"
0038-1098/86 $3.00 + .00
Pergamon Journals Ltd.
STEADY STATE OF PHOTOEXCITED PLASMA IN SEMICONDUCTORS
Tânia TomeDepartamento de Física do Estado Sólido
Instituto de Física "Gleb Wataghin"Universidade Estadual de Campinas (UNICAMP)
13100 - Campinas, S.P., Brasil
(RECEIVED JULY 3th 1986 BY R.C.C.LEITE)
The nonequilibrium macroscopic state of a polar semiconductor underconstant laser illumination is studied. The values of the relevant steady-state thermodynamic parameters are determined. They are the solutions ofgeneralized transport equations derived using the nonequilibrium statisticaloperator method. Numerical results are obtained for the case of GaAs.
We consider an intrinsic direct-gap polar semiconductor sample incontact with a thermal bath and undercontinuous laser light illuminationwith power flux IL and photon energyhWL (larger than the energy gap).Electron-hole pairs are created bydirect absorption of one photon."Theyform, together with the latticebackground, a highly excited plasma insemiconductors (HEPS); we assume thatthe experimental conditions are suchthat the carriers are on the metallicside of Mott transition. These kind ofsituations appear in semiconductordevices and quantum generators (highlevels of excitation can also be createdby strong fields), and laser annealingtreatments. After a transient regimethis open system is expected to go intoa steady state, which we attempt to'characterize here. For this purpose weresort to the use of the nonequilibriumstatistical operator (NSO) method(l) inZubarev's approach(2).
According to the method the NSOp(t) depends on a set of dynamical
quantities {Pj}, j = 1,2,...,n,whoseaverage values, Qj(t) = Tr{Pj p(t)},are the basis set of thermodynam~calvariables to be used for the descriptionof the macroscopic states of thenonequilibrium system. The choice ofthe basis set of macrovariables is notunique(l) but depends on eachparticular problem. To deal with thepresent problem we take, as done forstudies of pump-probe experiments(3,4),the set composed of, Hc, the energyoperator for carriers (in Bloch bandsplus Coulomb interaction; they aretreated in Landau's quasi-particlepicture, and the effective massapproximation and the electron-holerepresentation areused), HLO and HA'the hamiltonians of the longitudinaloptical and acoustical phonons,respectively, and the number operatorsfor electrons Ne, and for holes Nh' We
call Ec(t), ELO(t), EA(t), n(t)n (t) the five correspondingmacrovariables, where n(t) is thedensity of carriers, and the othersare the energy densities of thedifferent subsystems. Further, Fj(t),the Lagrange multipliers in NSO methodand intensive variables in generalizedthermodynamics,(l) are written Bc(t),BLO(t), BA(t}, -Bç(t}Pe(t),-Bc Ph(t),defining the quas~-temperatureskT~(t) = B~l(t) for the subsystems,and the quasi-chemical potentials forelectrons, Pe(t), and for holesPh(t). The total hamiltonian of thesystem is H = Ho + H', where Ho iscomposed of the three hamiltonian ofthe free subsystems, and H' comprisesthe interactions between carriers andphonons, anharmonic interactions, thecoupling of carriers with laser andradiation fields and the interactionof the subsystems with a thermal bath.The semiconductor sample is an opensystem coupled to the laser and to thethermal bath (the latter taken attemperature To), with the last twoconsidered as ideal reservoirs, i.e.they are assumed to'remain unchangedduring the experiment: This allows usto apply Zubarev's method to the opensemiconductor system.(5)
It should be noted that the
quantities Pj' j = 1 to 5, commutebetween themselves and with Ho. andthen the NSO method produces "
generalized transport equations of theform(7}
(2) (3)J. (t)+J. (t)+...J J
(1)
where the right hand side is a seriesexpansion of collision operators ofincreasing order in the interactionoperatorH' . For the presentcase
Jl°) and Jjl) are null,and we truncate
66]
662 STEADY STATE OF PHOTOEXCITED PLASMA IN SEMICONDUCTORS VoI. 59, No. ]0
tne series retaining only the termJ~2) (the quasi-linear approximation(6»J (7)
given by
J
o E:t'
-oodt'e .Tr{!H'(t:'),!H',P.]]p (t)},
J cg
(2)
where E:(>O) goes to zero and p is ac.oarse-grained operator definedc§s theprojection of the NSO over the subspace
of quantities Pj excluding dissipativeeffects.(1,8}
In the steady state dQj/dt = O,e2}
!o O
].
and therefore Jf Fl,...,Fn = O, J = 1to 5, is a set of five algebraicnonlinearcoupled equations, whose solution providesthe stationaryvalues F9 of the .thermodynamic variablesJon which each
J12) explicitlyd~~ends;detailedexpressions for J~ ) are given inreferences 3 and 4. Using computacionaliterative methods we obtain numericalsolutions for the specific case ofGaAs, choosing hWL = 2.4 eV, andTo = 300K, and for laser powers up to500 KU cm-2. Taking into account thatfor laseT power below 500 KW cm~2 ~heacoustic phonons are only slightlyheated,(9) T~ is taken constantly equalto To = 300K and the correspondingequation for the energy of the A phononsis dropped out. AlI other stationaryquasi-temperatures are found to be verynear To. Using the values of T~(= 300K) and of the quasi-chemicalpotentials we find the dependence ofthe carrier concentration on IL shown infigure 1. We have also solved theevolution equations (1) starting at timeto (~ 0.5 ps) after initial applicationof the c.w. Iaser illumination, usingas initial conditions carrier quasi-temperatures of near 3000 K (obtainedfrom the energy conservation kT~(to)hWL-Eg) and the phonons at bathtemperature, and neto) calculatedusing the values of the laser intensityand of the absorption coefficient at thegiven laser frequency. Our resultsreconfirm the fact that the quasi-temperatures of carriers and opticalphonons become equalized in a lapse ofa few picoseconds.(3) Further, undercontinuous illumination withIL < 500 KW cm-2 alI quasi-temperaturesattain a stationary value very near300 K in less than 30 ps. The carrierconcentration becomes stationary inmuch longer times as shown in figure 1(right hand ordinate); these transienttimes, Tst, are defined by the lapse thatit takes the concentration to get a value10% smaller than that in the steady state.
In conclusion, we have found forHEPS in GaAs, and we expect the resultsto be similar for most intermediate tostrong polar semiconductors, that forc.w. laser powers below 500 KW cm-2 auniform stationary space distribution
15
.'-....."-.
---
---o -.
III
55010 20 30 40
LASER POWER (kW crii2)
Fig. 1. Dependence of the stationaryvalues of the concentration of carriers
(full line), and the logarithm of thetransient time (dot-dash line) on laser
power flux IL.
of photoinjected electron-hole pairssets in on a larger than nanosecondtime scale. Their concentration go from5xl017 cm-3 for lL = 3 KW cm-2 to1.7xl019 cm-3 for IL = 500 KW cm-2, whilethere occurs no appreciable heating ofthe system. The curve nO vs.IL in figure1 is the one that comes out continuouslyfrom the equilibrium state withincreasing photoexcitation. The highlynonlinear character of Eqs. (1) mayresult in the existence of branchingpoints with the emergence of additionalsolutions. preliminary results(lO)indicate that the uniform steady-stateHEPS may be bnstable~ at a thresholdpower of roughly5 KW cm-2 againsttheformation of a stationary charge densitywave of wavenumberQ = 0.3 ;m-l.Thepositioning of the branching point,occuringat T~o = 300 K, .nO = 1.Oxl018 cm-3, is obtained bylooking for the zeros of the static andwavevector dependent dielectric .functionE:(Q;w =O) in the uniformstationarycarrier system. The imaginary part of E:is identically nu 11, and the real partvs.Q, for a couple of values of IL, isshown in figure 2. The instability isdue to the appearence of a soft-plasmon:beyond the critical point an orderedstructure builds up spontaneously fromfluctuations. This is a striking case ofa dissipative structure arising in opensemiconductors systems which can bemantained in such steady low-entropystate by a flow of energy from theoutside.(ll) The formation of a
_ 5/f)
'Eu
CD 4Õ
><
Z 3OI-<ta:I- 2ZwUZOu
<liQ.--'"p
10 t:>O-I
VoI. 59, No. ]0
1.6
STEADY STATE OF PHOTOEXCITED PLASMA IN SEMICONDUCTORS 663
>< 1.2
zO~uz::>IJ...Ua:~uw-'wa
0.8(I)
0.4(2)
o
-0.40.2 0.3 0.4 0.5 0.6 0.7
SCOW WAVENUMBER(em-I)
Fig. 2. Static and wave-vector dependentdie1ectric function versus SSCDW wave-
vector for 1aser_~ntensities:(1) IL = 3 KW cm , and
-2(2) IL = 5 KW cm .
1) A.C. A1garte,A.R. Vasconce11os,R. Luzzi and A.J. Sampaio, Rev.Brasil. Fis. 15, 106 (1985); in thisartic1e is given a brief discussionof the main difficu1ties in thetheory of irreversib1e processes inmany-body systems, and a unifyingscheme for the construction ofnonequi1ibrium statistica1 operatorsis presented.
2) D.N. Zubarev, NeravnovesnaiaStatiseheskaia Termodinamika. (Izd.Nauka, Moskwa, 1971) [Eng1ish Trans1.;Nonequilibrium StatistiealThermodynamies (Consu1tants Bureau,New York, 1974).Ch. IV, p. 237ff].
3) A.C. A1garte and R. Luzzi, Phys.Rev. B27, 7563 (1983);R. Luzzi andA.R. Vasconce11os, in SemieonduetorsProbed by Ultrafast LaserSpeetroseopy, edited by R.R. A1fano(Academic,New York, 1984).Vol. 1,pp. 135-169.
4) A.C. A1garte, A.R. Vasconce11os,R. Luzzi and A.J. Sampaio, Rev.Brasil. Fis. xx, xxx (1986).
5) J. Seke, Phys:-Rev. A21, 2156(1980). -
steady-state charge density wave (SSCDW)
in the homogeneous inverted popu1atio~~of e1ectrons and ho1es a10ng diode junctions )cou1d be a possib1e mechanism to look for
connected with fi1ame?tation effects in1aser semiconductors. 13) A1so the SSCDWmay be 1inked with the mechanismproposed by Van Vechten(141 for anatherma1 annea1ing of ion-imp1antedsemiconductors interacting with a longand intense 1aser pulse: the pumpedenergy wou1d be used to produce orderin the far-from-equi1ibrium systeminstead of therma1 heating 1eading tome1ting fo11owed DY recrysta11ization.
Fruitfu1 discussions withR. Luzzi and A.R. Vasconce11os, as we11as their critica1 reading of themanuscript, are gratefu11y acknow1edged.The author aIso thanks P. Sakanaka forhis va1uab1e advice on computationa1techniques. The author is a Fundação deAmparo a Pesquisa do Estado de são Paulo(FAPESP) pre-doctora1 fe11ow.
REFERENCES
6) V.P. Ka1ashnikov, Teor. Mat. Fiz.35, 127 (1978) [Theor. Math. Phys.(USSR) 35, 362 (1978)].
7) D.N. Zubarev, in reference 2, Ch.IV, Section 25.
8) R. Luzzi and A.R. Vasconce11os, J.Stat. Phys. 23, 539 (1980).
9) A.R. Vasconcel1os and R. Luzzi,Phys. Status Solidi (b) 126, 63(1984). -
10) T. Tome, Ph.D. thesis (UNICAMP, indeve1opment).
11) I. prigogine, Nature 246, 67 (1973);R. Landauer, IBM J. R~ Dev. 5, 183(1961); Nico1is and Prigogine,-Selforganization in NonequilibriumSystems (Wi1ey - Interscience, NewYork, 1977).
12) A. Yariv and R.C.C. Leite,App1.Phys. Lett. 2, 55 (1963~.
13) M.S. Abrahams, A.K. Jonscher andM.R. Boy1e, Proe. IPPS Symposium onGaAs (Institute of Physics'and thePhysica1 Society, London , 1966).
14) J.A. Van Vechten, Solid StateCommun. ~, 1835 (1981).