Transcript

OPTICALMATERIALS 1(1992)133-140North-Holland

Diffusion andrecombinationof bipolarplasmasin highly excitedsemiconductors

BerndHönerlageInstitut dePhysiqueet ChimiedesMatértauxdeStrasbourg,Grouped’OptiqueNonlinéaireel d’Optoélectronique,Unite mixte380046 - CNRS-ULP-EJ-JJCS,5, rue de l’Université, F67084Strasbourg,Cedex,France

Received23 December1991; revisedmanuscriptreceived 19 February1992

We studytheoreticallythediffusion of a bipolarplasmageneratedby an inhomogeneouslight excitation.It is shownthatthisdiffusion dependson thecarrierconcentration.In light-inducedgratingexperiments,this leadsto a complex temporalintensityvariationofthegeneratedordersof diffraction.Thereforethiseffecthaslobecarefully checkedbeforeinterpretingquantitativelyexperimentson highly excitedsemiconductors.

1. Introduction pulses,i.e. leadsto pulsediffraction in different or-ders.The intensityof this signaldecreasesas afunc-

The knowledgeof optical nonlinearities,their dy- tion of timedelaysbetweenthepumpand testpulsesnamical behaviourandthe transportpropertiesof duetotwo processes:theradiativeandnon-radiativesemiconductorsareimportantfor the conceptionof decayof the excitedquasiparticlesgeneratedby theoptical devicesbasedon semiconductormaterials, two interferingpumppulsesandtheirdiffusionwhichIn this context,the diffusion propertiesof electron— washesoutthe initial modulation [11.hole plasmasare of outstandinginterest,sincethey Usually, in LIG experiments,oneconsidersplanemainly governthe opticalnonlinearitiesof semicon- waveswhich interfereandexcitehomogeneouslyinductorsat roomtemperature.In laserdiodes,for cx- the z- and y-direction the samples [2]. Then theample, plasmadiffusion affects the stability of the quasiparticledensitiesN1(x, t) are functionsof themodes.To know about the temporal and spatial x-coordinateand of the time only. Theycanbe cal-plasmaprofile underpulsedandcontinuousexcita- culatedfrom rateequationswhich havetypically thetion is thenof technologicalinterest.Its time evo- form [2,3]lution aftercreationby a short,optical light pulsecanbe studiedby different methods. 0N1(x,l)/8t=G1(x, t)—y(N,,IV~)N,(x, I)

In light-inducedgratingexperiments(LIG), which + VD1 VN1 (x, t) , (1)we will discusshere, two coherentpumppulsesofequalintensity,obtainedfrom thesamelight source, where the index i indicatesthe different types ofcoincidentin timeandspace,interfereonthesurface quasiparticlesconsidered.They can be electrons,of a semiconductorcrystal andproducea spatially holes,excitons,etc. G.(x, t) is their generationrate.modulatedintensityof excitation.If the photonen- y(IV~,N~)is a (density dependent)recombinationergy of the light sourceis convenientlychosen,free term,which canleadto a couplingbetweenthe dif-electronsand holes are generated,the density of ferentrateequations(i andj), andthe lastterm de-which is also spatially modulated.As in photore- scribesthe diffusion of the quasiparticlesconsid-fractivematerials, thepresenceofelectronsandholes ered.A quite complexsystemof coupledequationsmodifies the complexindex of refractionandthus describingquasiparticlepopulationsis e.g. given inleadsto thediffraction of a (time-delayed)testpulse. ref. [4].This light-induced grating then generatessignal In the mostsimple case [1] which involves one

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typeofquasiparticlesonly andconstantdiffusionand dencesof y in eq. (1), we haveshown in ref. [61 byrecombinationtermsy andD, thegenerationdynam- varyingA, that underourhigh excitationconditions,ics is replacedby an intial conditionfor the quasi- D cannotbe consideredasa constant.Therefore,aparticledensity: “screeneddiffusion model” wasproposed[6]. In this

model, a bipolar plasma-likediffusion (whereelec-N(x,O)=N0(l+cos2x/A) , (2)

trons andholes diffuse independentlyat high par-whereA is the grating spacing.In this case,eq. (1) tide densities)and an ambipolarpair diffusion atsimplifies to low densitieswereassumedto beresponsiblefor the

non-exponentialdecreaseof the signal.öN(x, t) 0

2N(x, t)= —yN(x,1) +D (3) If the diffusion parameterD is not constant,but

densitydependentassupposedin ref. [61, suchef-and hasan analyticalsolution which is given by fectswill also showup in othertypesof experiments,

namelywherethe plasmaexpansionis studied[9—N(x, t)=N

0[l +exp(—I/TD) cos(2mx/A)] 11]. Thisnonlineartransportpropertiesthen influ-

Xexp( — t/ T,) (4) ence,as we will see,the optical nonlinearresponse[12—16]of the material.Weinvestigatein this pub-

TD is calledthe diffusion time constantand is de- lication thediffusion propertiesof a bipolar plasmafined as in detail andapply the results to recentLIG mea-

TD =A2/4ir2D, (5) surementson CdS [6] at room temperature.

andT1 = 1 /y is the lifetime of the quasiparticles.

According to eq. (4) the amplitude of the mod-ulationof thequasiparticlepopulationvanisheswith 2. Rate equationsof bipolar plasma diffusion and

recombinationa time constant TM = 1/f’ which is given by

[ (4~2IA2 )D + 1 / T

1 . (6) Let usconsidera bipolarplasmacreatedby a light

Let us considernow the casewhenthe complexin- source.If the excitationis homogeneousin thez- anddex of refractionchangeis proportionalto the car- y-directions,in the mostsimpleapproximation,therier density and when the “thin grating approxi- densityof electrons(index e) and of holes (h) aremation” holds, i.e. that governedby therate equations[17]

jr2d/A2n<< 1 , (7) 0fle(X, I) = 81e(X,t)

Yitle(X, t)Y2fleflh8t eox

d is the samplethickness,n the realpart of the re-fractiveindex,and,~ the wavelengthof the fields. If, aflh(X, I) — 8Jh(X, 1)~

YIflh(X, t)Y2fleflh , (10)in addition,the complexindex of refractionchange öt — — eöxis suchthat

wherethe j, are the particle currents,e is the ele-L\n I ~ A/2ird, (8) mentaryelectroncharge, 1 /y~is the electronor the

hole lifetime. Theselifetimes are takenequal in or-thetemporaldecayof the generatedsignalintensity derto maintainchargeneutrality.I~(t)hasa simpleexponentialform [1,3,5]:

Y2 governsthe recombinationof electronsin theI~(t)=I~exp(—2f’t). (9) conductionband with holes in the valencebands.

Then, the currentdensitiesj1aredefinedin termsof

Experimentally,whenwe useCdS platelets[3,6— the mobilities ~, andof the quasichemicalpotentials8] for which the aboveapproximationshold, more ~2,(i= (e,h)), which may be different for electronscomplicatedtime dependencesandnamely a non- andholesthroughoutthe samplesincewe considerexponentialdecayhavebeenobservedunderhigh a non-equilibriumsituation:excitationconditions.While usuallythis non-expo-nential decayis attributed [3] to a densitydepen- j~(x,t)=Ji~n1(x,t) 8fl1(x, t)/öx, (11)

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where8fl,(x, t)/8x gives rise to a driving force and U112(y)= l.2O9y213—0.68O3y213—O.85y2

thustodiffusion.Thechemicalpotentials~2,(x, t), at (17b)their turns,are definedby the quasiparticledensites

for y>~S.5.n,(x, t) and temperatures through the Fermi distri-Usingnoweqs. (11), ( 13), ( 16) and ( 17 ), weob-

bution function. For parabolic bands, characterizedtam for y< 5.5

by their effective masses m1 andtheir degeneracies

g,, we obtain jelekBT~~~X~ t) OØ(x, I)

g, I k2dk Ox lefle(X,t)é

n1=~,—1 (12)

~ exp[fl(h2k2/2m

1—~~)]+l’ 2ith2 \3~2kBT Ofle(X,t)

+~e0.3536(kT) ~fle(X,t)

where8= l/kB Twith T beingthe temperatureof thesystemand ~e0.00495( 2~h2~32kBT 2( t) Otie(x,t)

\,mek8T) ~ Ox

~=ü,(x, t)—eØ(x, t) , (13)(27th2 ~~

9”2 3kBT3( I)

Ø(x, t) is the time-dependentelectrical potential +/Je 0.000125OxmekBTJ g~,functionwhich buildsup from thechargeseparation

due to the different diffusions of the quasiparticles. (18)It gives riseto an electric field Similarly, since

3/2

OØ(x,t)/Ox=—E(x,t). (14) ( 2ith2 ) -~--lkBT

From eqs. (12) to (l4),fl(x, t), n1(x, t) andØ(x, t) [nh~ mhkBT gh]

LLh(X, t) = — U1/2have to be determined selfconsistently, using in ad-—eØ(x,t), (19)

dition Poisson’s equation [19,20]:

820 OE(x, 1) 4mIeI we obtain forjh[tth(x,t)~1e(x,t)]

—— Ox — Oflh 00

(15) ih1hkBT~ —IAhnhe-~---which relatesthe quasiparticledensitymodulation 2~h

2_~~312kT8~h(mhkBTJ g~ Ox

to the electrical potentialwhich builds up. is the — /2h 0.3536static dielectric constantand ~cj the electrical per-mitivity of vacuum.

+~h0.00495( 2~h2~32kBT2OflhIn the “rigid shift approximation”,eq. (12) can \mhkBT) ~T Ox

be expressedby a Fermi integral which can be in-vertedto obtain ~ ~h 0.000l25(2m~

2)9/2 3kBT I 8n~mhkBT —

3—n~---,~—.(20)g~ dxc=kuT U1/2(y,) , (16)with In eqs.(18) and (20), the first two termsgive the

usualdiffusion anddrift currentsdue to the chem-/ 2ith

2 \3~Z2 1 ical potential.All furthertermscanbeconsideredas= ‘~‘~m~k~T) g~ correctionswhich arisein the diffusions due to the

high carrier densities and their influence on theAs discussedin ref. [18] in detail,Ul/

2(y) canbe chemicalpotential.One could now relatethe diffu-approximatedwithin a very good precisionby sion coefficientsD• throughthe Einsteinrelation

UL/2(y)=lny+0.3536y—0.00495y2+0.000125y3, D•=u,kBT/e (21)

(17a)

for y<5.5 andby to ~ but herewe considerratherthe mobilities ~We now introduceeqs. (18) and (20) into the

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continuityequation(10) anduseeq.(15). We then E (x) cannow be determinedby integrationof eq.obtain (l5)as

Ofl~(O~~t)=_y!ne(x,t)_y2nenh E(x,t) ~(~ [h~(t)

~

n�OOfle(x, t) OE(x, t)

+/1eE(x, 1) Ox +Ilefle Ox —e~(t)]exp(inmx/A)

+ ~ekBT{O

2fle(X~ t) 0.3536 ( 2Thh~)3/2 + [ho(t)_eo(I)]x+c(t)). (24)e Ox g~ m~k

8T

[(Ofle’\2 O2flel where the last two terms vanish due to the charge

><[~~) + ‘~e neutrality h0 ( t) = e0(I) Vt and because of energy con-

servation (c(t)=0).— o.004592( 2~h

2~3[2n (~~2+fl2~& Theinitial conditions (eq. (2)) then readg2 \~mkBT)[ ~\~) C Ox2]

h1(0)=h1 (0)=e1(0)=e1(0)=N0/2,

+000~253(2~2 )9~ e0(0)=h0(0)=N0,g~ mekuT

2 hH (0)=e~,1(0)=0. Vi~2. (25)x[3n~(~) + n~(~~)]}, (22) In addition,dueto the specialsymmetry of the prob-

lem, we haveandan equivalentequationfor Oflh(X, t)/Ot, but in h ~ —h ~ e ~ —e ~t~’ Vt (26)which the indices (e, h) haveto be exchangedcorn- “ ‘ “ / ‘ “ / — ‘\ /

paredto eq. (22) andwhere the signsof the terms suchthat only even functions in x are kept in theproportional to E changebecauseof the positive Fourierdevelopment.chargeof the holes. The electronand hole densitiesin eq. (22) and

Poisson’sequation(15) thusleadsto thecoupling their spatial derivativesare now replacedby eqs.betweenbothsetsof integro-differentialequationsin (23). Then productsof the form n~(t,x) nh(t, x)n~andn~in a nontrivialway, sinceeq. (22) involves arise:the explicit knowledgeof the electric field E whichcanbeformally obtainedwhenintegratingeq. (15). ~e n~=(~e~(1) exp( innx/A)

3. Approximativesolutionsof LIG problems ><(~ h,,,(t) exp(imlrx/A)

The system given in eq. (22) is still quitegeneral.It is now approximativelysolved for LIG problems = ~( ~ e,, h~+/(t))exp(i/7rx/A)~by developping first the densities of electrons and /= — ~ ‘~ — 27)holes in Fourier series in the x-direction. The prob-

lem is assumedto be periodicin thex-directionand wherethe summationcanbe changedto give againhomogeneousin the y- and z-directions.The den- a Fourierexpansionof the productsof the periodicsities then read functions. By this method,the Fourier components

can be identified and a set of coupleddifferentialn~(x,t)= ~ e~(t) exp(inmx/A) (23) equationsin time is obtained.Since thelast term in

— the invertedFermiintegral (eq. (17)) is verysmall,and it is neglected.This correspondsto neglecttermsof

00 the order (a1(t) )4 in our development,wherea(t)nh(x,t)= ~ h,,(t)exp(inirx/A),

stands for the Fourier coefficients e,,, or h,,~ In ad-

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Volume 1, number3 OPTICALMATERIALS September1992

dition, in order to extend the polynomial form of our experimental configuration. ~ih and Y~, Y~have toU

1/2(y) to higher y values, we rather use be adjustedto the experimentalresultssince these

values are not precisely known.UI/2(y)~lny+0.3536y—0.003y

2 (28)instead of eq. (1 7a). This is a decent description forU

1/2(y) up to y= 30 with a relativeerror <3%which 4. Theoretical results and discussionisfully sufficientfor ourapproximativetreatmentofLIG signals.If highervaluesof yor abetterprecision We havefirst usedthesetof equations(29) tocal-arerequested,a polynomialdevelopmentof UI/2(y) culate e~(t) and h~(t). Then the first order of dif-shouldbesearchedfor ratherthanto usethesolution fraction of a generatedsignalis proportional [4] togiven in eq. (17b) in order to maintain the possi- [(e1(t))

2+(h1(t))

2] which has been analyzed in ref.bilities of rearrangingthesumsin eq. (27). Wethen [6]. If the system of equations (29) is truncated atobtain for e

1 (1): lI ~4, a convergence has been obtained for I/l = 1,2 provided that N0~7x10’~ cm

3 and A~3~.imarede,(t) [ ILCkBT(l7r’) le,—y

2 ~ e~h~±1 used,which fulfill largely the experimentalcondi-dt [Yi — e~\J1/ J fl~-00 tions.

Now, eqs. (29) canbe rewrittenformally+ ~ (i+ ~ )(h_n+i_e_~+i)en de1/dt=[y, —Dff(l, 1)] e,(t)—y2~

~(o -00’\ —n+1~1

/AekBT(7t)0.3536( 2ith2 \3/2 00

— e A g~ \mCkBT) ~ (30)

whereD~ff(I,t) is aneffectivedensity,andthustime-e~e,,~

1[n(—n+1)+ (_n+1)2]) dependent,diffusion function,characteristicfor theelectrons.

+ /2e1(BT(7t~20.003 / 2ith2 ~ We find

e \Aj g~ \mekBT)D~ff(/, t)=

e A

x{~ei[2(~n(_n+m)e~en÷~) + c~ekuT/~~2O.35362~h2 3/2

e 2 (mekBT)

+ (—m+1)2 fl:-00 enen+,n]}, (29)x (~e~en÷,[n(_n+1)+(_n+1)2])

and the analogousexpressionfor h, when we ex-change in eq. (29) e~by h, and ~ me, g~)by (j~h, j~~k~T/ir\0.003 / 2ith2 \3

m~,ga). e ~ ~ kB T)

Thisset of coupledequationsin e1 and h, cannow

be solved numerical using thedifferentmaterialcon- ><[ ~ e, ,~(2 ~ n ( — n + m)e~e_,, ±

stants. Weare interested in CdS at room tempera-tureandtakethetypical valuesfor thecaseof aniso-tropic two-bandmodel [21,22] in which only the

+(—m+1)2 ~ e~e

conductionband and the A-valence bandare con- n= -00 - + )]}~ (31)sidered: me=0.2, mh=O.9, /1e300 cm2/(Vs),e=8.7. Here,effectivemassvaluesvalid for diffu- In eq. (30), A~(~~00 ~T(I, n)e~) is the couplingsion perpendicularto the cristallographicc-axisare term betweenthe two systemse~and h~.

consideredonly, sincethe excitationis assumedto Similarly, we find for the holesbe homogeneousin y andz directionsandxLc in

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Volume 1, number3 OPTICALMATERIALS September1992

a

r -~ bit

1010 ~

10 2 i0~

,, ,,0.0 0.3 0.5 0.8 0 13 1.5 .8 2.0 2 3 25 00 0.3 0.5 0.8 1.0 13 15 1.8 2.0 2.3 .5

(ns) t (ns)

11

i0-~ C d

* * I * ,

-~

10 ~—

00 0.3 05 08 1.0 1.3 1.5 18 2I (is)

Fig. 1. Signal (first orderof diffraction) decayobserved(crossesfrom ref. [6]) andcalculated(full line) for N0=2x l0’~cm

3 andA= (a) 3 ~.tm,(b) 4 J.tm, (c) 8 ~.tm,(d) 12 tm. The parametersarediscussedin thetext.

dh1/dt= [—y,—D~ff(I,t) ]h1t (A Ae) ~ AN(I, n) e~

eflh~+,—A~~ ~T(1,n)h~, (32)

fl 00 fl 00 = [—Dff(1) +D~~~(I)] e1 . (33)

with Ae=4me/ie/eoandAh=47re~uh/fO. The systemof differential equationsnow reduces

Numerically,we find in all casesstudiede,(t) to

h,(t) with a relative error smaller than l0~. Thisindicates that no effective field canbuild up in CdS de,/dt= [—y~—Dff(l, t)] e,(t)—y

2 ~ e~e~÷1due to diffusion. Then, the systemof differentialequations(30) canbelargelyreducedandcomputer — /~e— Dff( 1, t) + D~ff(1, 1) e,(t) . (34)time savedby simply putting e1=h, and de,/dt= /1e+31,,

dh,/dt. It hasbeencarefullycheckedthat theresultsof thisWith this approximation,thecouplingtermhasto coupledsystemare the sameasfor the full system.

fulfill theconditionEq. (34) showsthatthe electrondiffusion is mod-

ified through the effective hole diffusion and thismodification dependson the electric field which

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Volume 1, number3 OPTICAL MATERIALS September1992

10~ —--~..~- a ~ .

I (is)

Fig. 3. Calculatedtemporal behaviourof the signal intensities~ ~.. for different initial densitiesN

0.

+ 0-

- Usingthe modeldescribedabove,we startwith an-~ ~0 I~ initial density N0=2x i0’~ cm

3, which corre-

~ spondsto our highestexcitation . Then, we adjustthe remainingparametersy,=l/T,, Y2 and ~,, in or-

0 der to fit the room temperaturevaluesof the dif-

fusion coefficientpublishedin ref. [6]. Figure 1 givesthe comparison for A= 3, 4, 8 and 12 ~tm.The over-

~ all agreement is quite good ifwe take y2=0, T, =3.6ns and jo,,= 100 cm2/(Vs). The range of the param-

etervalueswhich canbe usedis in factquite small.

io2 - ______ ________________ Similar fits canbe obtainedif Y2 is increased up toc Y2= 10~12cm3/s which is the maximumvaluefor the

nonlinearrecombinationparameterwhich is corn-patiblewith the dataof ref. [61. This valueis much

10 1~3+++ * * * ~ lower thenthe onereportedin ref. [31,where non-3 * lineardiffusion hasbeenneglected.As seenin fig. 1,

~ * * it is, however,not necessaryto introducea finite

0 value of Y2 if onewantsto explain thenon-exponen-10 tial decreaseobtainedfor smallpath lengthsA. The

value of/I,, is astonishingly high when compared to- mobility values usually assumed [221. One expla-

~. _____ nationmaybethat themobility given aboveis valid0.0 0.5 1.0 15 20 2.5 for directions Ic only, wherethe effectivehole m,,

)ns( massisminimal. In all otherdirections,m,, is bigger

Fig.2. Asfig. I but forA= 3 pmand (a)N0=2x 10’~,(b) 5 x 108 andthe mobility in those directions should then be

and (c) 2x lOIS cm3. smaller. As shown in fig. 2, the parametersgiven

abovealso explain the variation of the signal de-

coupleselectronsand holes, i.e. the correction de- creasewhenwe changethe excitationintensity (N0

pendsmainlyon the differentelectronandholemo- in our case)for a given grating constantA=3 tim.bilities u~and~,, The slight discrepancybetweentheory andexperi-

139

Volume1, number3 OPTICALMATERIALS September1992

ment obviousin fig. 2c mayeitherbe due to a lower Referencesnumberof carriersthanassumedin the calculationor simply be due to the fact that the experimental [1] H.J. Eichler, P. GUntherand D.W. PohI, Laser-inducederrorbarsin fig. 2c aremuch biggerthan in figs. 2a dynamicgratings, SpringerSeriesin OpticalSciences50,

and 2b. (Springer,Berlin, 1986).

The samebehaviouris showntheoreticallyin fig. [2] AL. Smirl, S.C. Moss andJ.R. Lindle, Phys. Rev. B 25(1982)2645.

3 for different N0. Note that the ordinateis nor- [3]H. Saltoand EQ. Gdbel, Phys.Rev. B 31 (1985) 2360.

rnalizeddifferently from onecurve to theotherby a [4] M.J.M. Gomes, R. Levy and B. Honerlage, J. Lumin. 48/

multiplicative factor in order to be ableto compare 49 (1991) 83.

the time decaysfor different excitationdensities. [5] H. Kogelnik, Bell Syst. Tech.J. 48 (1969) 2909.

The systemof differential equationsdiscussedhere [6] B. Kippelen,J.B. Grun,B. Honerlageand R. Levy, J. Opt.Soc. Am. B, 8 (1992) 2363.convergesrapidly for A>3 1.trn and N0<7x l0’~ [7]Y. Aoyagi, Y. Segawaand S. Namba,IEEE J. Quantum

cm—3 as statedbefore.If one of theseconditionsis Electron. QE-22 (1986) 1320.

relaxed, convergencebecomesvery difficult and [8] Ch. Spiegelberg, M. Kretzschmar, J. Puls and F.

many ordersof the Fourier coefficientshaveto be Henneberger,Phys.Stat.Sol. (b) 150 (1988) 769.

considered.In addition, thetimedecreaseisno longer [9] F.A. Majumder, H.-E. Swoboda, K. Kempf and C.Klingshirn, Phys. Rev. B 32 (1985) 2407.

a monotoneousfunction for a given order, but sev- [10] M. Rinker, H.-E. Swoboda, F.A. Majumder and C.

era! relativemaximaandminima canshowup. This Klingshirn. Sol. Stat. Commun. 69 (1989) 887.

behaviouris not only typical for thefirst orderof the [Ii] C. Weber, U. Becker,R. RennerandC. Klingshirn, AppI.generatedsignal, but is accompaniedby a relative Phys.B 45(1988) 113.

increaseof thehigherordersignalswhich mayreach [12] For recentreviews,see for exampleH. Haug, ed.,Opticalnonlinearities and instabilities in semiconductors

the sameintensityasthe first order. This behaviour (AcademicPress.San Diego, 1988). and referencescited

is due to an instability in the systemof differential therein.

equations, which can be at the origin of the chaotic [13]D.A.B. Miller, C.T. Seaion,M.E. Prise and S.D. Smith,

diffraction pattern observed in ref. [23] in two-beam Phys.Rev. Lett. 47 (1981) 197.[14]H.-E. Swoboda, F.A. Majumder, V.G. Lyssenko. C.

experiments. Klingshirn and L. Banyai, Z. Phys.B 70 (1988) 341.

[15] V.D. Egorov, P. Flogel, Hoang Xuan Nguyen and M.Kaschke,Phys.Stat. Sol. (b) 146 (1988) 351.

5. Coaclusion [161F. Henneberger,J. Puls, H. Rossmann,U. Woggon, S.Freundt, Ch. Spiegelberg and A. Schulzgen, Fourth Intern.

Weshow in thispublicationthe importanceof the Conf. on Il-VI Compounds,Berlin, GDR (1990),J. CrystalGrowth 101 (1990) 632.

(density dependent)chemicalpotentialswhich gov-[17] N.W. Ascroft and N.D. Mermin, Solid State Physics,

emthediffusion propertiesof abipolarelectron—hole Saunders CollegePubI. Int., Philadelphia(1988).plasma.Contrarily to otherauthors,we stressthefact [18] R. Zimmerman, Many Particles Theory of Excited

thatdiffusion canalsogive riseto a non-exponential Semiconductors,TeubnerTextezur Physik 18 (Teubner

decay as the well known induced recombination Verlagsgesellschaft,Leipzig, 1988).[19]G.C.Valley,J.Appl.Phys. 59(1986)3363.

terms proportional to Y2 do.[20] F.P. Strohkendl, J.M.C. Jonathan and R.W. Hellwarth,

Optics Lett. 11(1986) 312.[21] B. Honerlage, R. Levy, J.B. Grun, C. Klingshirn and K.

Acknowledgements Bohnert,Phys.Rep. 124 (1985)161.[22] LandoltandBUrnstein,Vol. l7b (Springer,Berlin, 1983).

The authoris grateful to Drs. J.B. Grun, B. Kip- [231 J.M. Hvam,I. Balslev andB. Honerlage,EurophysicsLeit.

pelen and R. Levy for critical readingof the man- 4 (1987) 839.

uscript and many helpful discussions.

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