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OPTICAL MATERIALS 1(1992)133-140North-Holland

Diffusion and recombination of bipolar plasmas in highly excitedsemiconductorsBernd HnerlageInstitut de Physique et Chimie des Matrtaux de Strasbourg, Groupe dOptique Nonlinaire el dOptolectronique, Unite mixte380046 - CNRS-ULP-EJ-JJCS, 5, rue de lUniversit, F 67084 Strasbourg, Cedex, France

Received 23 December 1991; revised manuscript received 19 February 1992

We study theoretically the diffusion of a bipolar plasma generated by an inhomogeneous light excitation. It is shown that thisdiffusion depends on the carrier concentration. In light-induced grating experiments, this leads to a complex temporal intensityvariation ofthe generated ordersof diffraction. Therefore this effect has lobe carefully checked before interpreting quantitativelyexperiments on highly excited semiconductors.

1. Introduction pulses, i.e. leads to pulse diffraction in different or-ders. The intensity of this signal decreases as a func-

The knowledge of optical nonlinearities, their dy- tion of time delays between the pump and test pulsesnamical behaviour and the transport properties of due to two processes: the radiative and non-radiativesemiconductors are important for the conception of decay of the excited quasiparticles generated by theoptical devices based on semiconductor materials, two interfering pump pulses and their diffusion whichIn this context, the diffusion properties of electron washes out the initial modulation [11.hole plasmas are of outstanding interest, since they Usually, in LIG experiments, one considers planemainly govern the optical nonlinearities of semicon- waves which interfere and excite homogeneously inductors at room temperature. In laser diodes, for cx- the z- and y-direction the samples [2]. Then theample, plasma diffusion affects the stability of the quasiparticle densities N1(x, t) are functions of themodes. To know about the temporal and spatial x-coordinate and of the time only. They can be cal-plasma profile under pulsed and continuous excita- culated from rate equations which have typically thetion is then of technological interest. Its time evo- form [2,3]lution after creation by a short, optical light pulse canbe studied by different methods. 0N1(x, l)/8t=G1(x, t)y(N,,IV~)N,(x, I)

In light-induced grating experiments (LIG), which + VD1 VN1 (x, t) , (1)we will discuss here, two coherent pump pulses ofequal intensity, obtained from the same light source, where the index i indicates the different types ofcoincident in time and space, interfere on the surface quasiparticles considered. They can be electrons,of a semiconductor crystal and produce a spatially holes, excitons, etc. G.(x, t) is their generation rate.modulated intensity of excitation. If the photon en- y(IV~,N~)is a (density dependent) recombinationergy of the light source is conveniently chosen, free term, which can lead to a coupling between the dif-electrons and holes are generated, the density of ferent rate equations (i andj), and the last term de-which is also spatially modulated. As in photore- scribes the diffusion of the quasiparticles consid-fractive materials, the presence ofelectrons and holes ered. A quite complex system of coupled equationsmodifies the complex index of refraction and thus describing quasiparticle populations is e.g. given inleads to the diffraction of a (time-delayed) testpulse. ref. [4].This light-induced grating then generates signal In the most simple case [1] which involves one

0925-3467/92/$05.00 1992 Elsevier Science Publishers B.V. All rights reserved. 133

Volume 1, number 3 OPTICAL MATERIALS September 1992

type ofquasiparticles only and constant diffusion and dences of y in eq. (1), we have shown in ref. [61 byrecombination terms y and D, the generation dynam- varying A, that under our high excitation conditions,ics is replaced by an intial condition for the quasi- D cannot be considered as a constant. Therefore, aparticle density: screened diffusion model was proposed [6]. In this

model, a bipolar plasma-like diffusion (where elec-N(x,O)=N0(l+cos2x/A) , (2)

trons and holes diffuse independently at high par-where A is the grating spacing. In this case, eq. (1) tide densities) and an ambipolar pair diffusion atsimplifies to low densities were assumed to be responsible for the

non-exponential decrease of the signal.N(x, t) 0

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

density dependent as supposed in ref. [61, such ef-and has an analytical solution which is given by fects will also show up in other types of experiments,

namely where the plasma expansion is studied [9N(x, t)=N

0[l +exp( I/TD) cos(2mx/A)] 11]. This nonlinear transport properties then influ-X exp ( t/ T,) (4) ence, as we will see, the optical nonlinear response

[1216]of the material. We investigate in this pub-TD is called the diffusion time constant and is de- lication the diffusion properties of a bipolar plasmafined as in detail and apply the results to recent LIG mea-

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

and T1 = 1 /y is the lifetime of the quasiparticles.

According to eq. (4) the amplitude of the mod-ulationof the quasiparticle population vanishes with 2. Rate equations of bipolar plasma diffusion and

recombinationa time constant TM = 1/f which is given by[ (4~2IA

2 )D + 1 / T1 . (6) Let us consider a bipolar plasma created by a light

Let us consider now the case when the complex in- source. If the excitation is homogeneous in the z- anddex of refraction change is proportional to the car- y-directions, in the most simple approximation, therier density and when the thin grating approxi- density of electrons (index e) and of holes (h) aremation holds, i.e. that governed by the rate equations [17]jr2d/A

2n

Volume I, number 3 OPTICAL MATERIALS September 1992

where 8fl,(x, t)/8x gives rise to a driving force and U112(y) = l.2O9y2130.68O3y213O.85y2

thus todiffusion. The chemical potentials~2,(x, t), at (1 7b)their turns, are defined by the quasiparticle densites

for y>~S.5.n,(x, t) and temperatures through the Fermi distri- Usingnow eqs. (11), ( 13), ( 16) and ( 17 ), weob-bution function. For parabolic bands, characterized

tam for y< 5.5by their effective masses m

1 and their degeneraciesg,, we obtain jelekBT~~~X~ t) O(x, I)

g, I k2dk Ox lefle(X,t)

n1=~,1 (12)

~ exp[fl(h2k2/2m

1~~)]+l 2ith2 \3~2kBT Ofle(X,t)

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

where 8= l/kB T with T being the temperature of thesystem and ~e0.00495( 2~h2~32kBT 2( t) Otie(x,t)

\,mek8T) ~ Ox

~=,(x,t)e(x, t) , (13)(27th2 ~~

92 3kBT3( I)(x, t) is the time-dependent electrical potential +/Je 0.000125 OxmekBTJ g~,function which builds up from the charge separation

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

3/2

O(x,t)/Ox=E(x,t). (14) ( 2ith2 ) -~--lkBTFrom eqs. (12) to (l4),fl(x, t), n1(x, t) and(x, t) [nh~ mhkBT gh]

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

dition Poissons equation [19,20]:820 OE(x, 1) 4mIeI we obtain forjh

[tth(x,t)~1e(x,t)] Ox Oflh 00

(15) ih1hkBT~ IAhnhe-~---which relates the quasiparticle density modulation 2~h

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

to the electrical potential which builds up. is the /2h 0.3536static dielectric constant and ~cjthe electrical per-mitivity ofvacuum.

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

be expressed by a Fermi integral which can be in-verted to obtain ~ ~h 0.000l25( 2m~

2)9/2 3kBT I 8n~mhkBT

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

usual diffusion and drift currents due to the chem-/ 2ith

2 \3~Z2 1 ical potential. All further terms can be considered as= ~~m~k~T) g~ corrections which arise in the diffusions due to the

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

2(y) canbe chemical potential. One could now relate the diffu-approximated within a very good precision by sion coefficients D through the Einstein relationUL/2(y)=lny+0.3536y0.00495y

2+0.000125y3, D=u,kBT/e (21)(17a)

for y< 5.5 and by to ~ but here we consider rather the mobilities ~We now introduce eqs. (18) and (20) into the

135

Volume 1, number 3 OPTICAL MATERIALS September 1992

continuity equation (10) and use eq. (15). We then E (x) can now be determined by integration of eq.obtain (l5)asOfl~(O~~t)=_y!ne(x,t)_y2nenh E(x,t) ~(~ [h~(t)

~

nOOfle(x, t) OE(x, t)

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

+ ~ekBT{O2fle(X~ t) 0.3536 ( 2Thh~)3/2 + [ho(t)_eo(I)]x+c(t)). (24)

e Ox g~ m~k8T

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

>

Volume 1, number 3 OPTICAL MATERIALS September 1992

dition, in order to extend the polynomial form of our experimental configuration. ~ih and Y~,Y~have toU1/2(y) to higher y values, we rather use be adjusted to the experimental results since these

values are not precisely known.UI/2(y)~lny+0.3536y0.003y

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

1/2(y) up to y= 30 with a relative error

Volume 1, number 3 OPTICAL MATERIALS September 1992

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)

11i0-~ C d

* * I * ,-~

10 ~00 0.3 05 08 1.0 1.3 1.5 18 2

I (is)

Fig. 1. Signal (first order of diffraction) decay observed (crosses from ref. [6]) and calculated (full line) for N0=2x l0~cm3 and

A= (a) 3 ~.tm,(b) 4 J.tm, (c) 8 ~.tm,(d) 12 tm. The parameters are discussed in the text.

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/eo and Ah=47re~uh/fO. The system of differential equations now reduces

Numerically, we find in all cases studied e,(t) toh,(t) with a relative error smaller than l0~.Thisindicates that no effective field can build up in CdS de,/dt= [y~Dff(l, t)] e,(t)y

2 ~ e~e~1due to diffusion. Then, the system of differentialequations (30) canbe largely reduced and computer /~e Dff( 1, t) +D~ff(1, 1) e,(t) . (34)time saved by simply putting e1=h, and de,/dt= /1e +31,,dh,/dt. It has been carefully checked that the results of this

With this approximation, the coupling term has to coupled system are the same as for the full system.fulfill the condition

Eq. (34) shows that the electron diffusion is mod-ified through the effective hole diffusion and thismodification depends on the electric field which

138

Volume 1, number 3 OPTICAL MATERIALS September 1992

10~ --~..~- a ~ .

I (is)Fig. 3. Calculated temporal behaviour of the signal intensities

~ ~.. for different initial densities N0.

+ 0-- Usingthe model described above, we start with an

-~ ~0 I~ initial density N0=2x i0~ cm3, which corre-

~ sponds to our highest excitation . Then, we adjustthe remaining parameters y,=l/T,, Y2 and ~,, in or-

0 der to fit the room temperature values of the dif-

fusion coefficientpublished in 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-eter values which can be used is in fact quite small.

io2 - ______ ________________ Similar fits can be obtained if Y2 is increased up toc Y2= 10~12cm3/s which is the maximum value for the

nonlinear recombination parameter which is corn-patible with the data of ref. [61. This value is much

10 1~3+++ * * * ~ lower then the one reported in ref. [31,where non-3 * linear diffusion has been neglected. As seen in fig. 1,

~ * * it is, however, not necessary to introduce a finite

0 value of Y2 if one wants to explain the non-exponen-10 tial decrease obtained for small path lengths A. The

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

~. _____ nation may be that the mobility given above is valid0.0 0.5 1.0 15 20 2.5 for directions Ic only, where the effective hole m,,

)ns( mass is minimal. In all other directions, m,, is biggerFig. 2. As fig. I but forA= 3 pm and (a) N

0=2 x 10~,(b) 5 x 108 and the mobility in those directions should then beand (c) 2x lOIS cm

3. smaller. As shown in fig. 2, the parameters givenabove also explain the variation of the signal de-

couples electrons and holes, i.e. the correction de- crease when we change the excitation intensity (N0

pends mainly on the different electron and hole mo- in our case) for a given grating constant A=3 tim.bilities u~and ~,, The slight discrepancy between theory and experi-

139

Volume 1, number 3 OPTICAL MATERIALS September 1992

ment obvious in fig. 2c may either be due to a lower Referencesnumber of carriers than assumed in the calculationor simply be due to the fact that the experimental [1] H.J. Eichler, P. GUnther and D.W. PohI, Laser-inducederror bars in fig. 2c are much bigger than in figs. 2a dynamic gratings, Springer Series in Optical Sciences 50,and 2b. (Springer, Berlin, 1986).

The same behaviour is shown theoretically in fig. [2] AL. Smirl, S.C. Moss and J.R. Lindle, Phys. Rev. B 25(1982) 2645.3 for different N0. Note that the ordinate is nor- [3]H. Salto and EQ. Gdbel, Phys. Rev. B 31 (1985) 2360.rnalized differently from one curve to the other by a [4] M.J.M. Gomes, R. Levy and B. Honerlage, J. Lumin. 48/multiplicative factor in order to be able to compare 49 (1991) 83.the time decays for different excitation densities. [5] H. Kogelnik, Bell Syst. Tech. J. 48 (1969) 2909.

The system of differential equations discussed here [6] B. Kippelen, J.B. Grun, B. Honerlage and R. Levy, J. Opt.Soc. Am. B, 8 (1992) 2363.converges rapidly for A>3 1.trn and N0