INFLUENCE OF CARRIER DIFFUSION AND RECOMBINATION ON THE PROTOACOUSTIC SIGNAL OF SEMICONDUCTORS

C.C. Ghizoni and L.C.M. Miranda

Instituto de Estudos AvanCados Centro Tecnico Aeroespacial 12200 S.J.Campos, SP, Brazil

Abstract

A model for the photoacoustic signal gene5

presented and discussed. The theoretical predic

LE = - -+D n - a2n + ,- I) ’I e B ~ ejwt at T hv ax

ation in semiconductors taking into account the effects of carrier diffusion and recombination is

tions were experimentally verified for the cases of InSb and Si samples.

(-l! < x < 0)

where D and T are the carrier diffusion coef- ficient and recombination time, respectively, and n is the quantum efficiency such that if each

1. Theory absorbed quantum produces one electron-hole pair then q = 1. Equation (2) together with the bond- ary conditions, ? Dan/& = v n(x,t) with x = O and - e , v being the surface recimbination velocity5, complztely specify the carrier concentration. We note that here n(x,t), D and T denote the corre- sponding quantities for the excess carrier. Hence,

In this paper we extend the RG model’ for the description of the PA effect in semiconduc- tors by taking into account two so far neglected important features of the behavior of carriers under time-varying excitation, namely,the carrier diffusion and recombination. for an n-type semiconductor, n is the excess hole

The introduction of the carrier diffusion and recombination in the theoretical description of the PA effect in semiconductors can be done as follows. The incident light pulse creates an ex- cess carrier concentration n(x,t). These excess carriers diffuse through the sample and re-estab lish equilibrium by disposing of the energy in excess both by emitting radiation and by gener- ation of heat. The PA technique responds only to the fraction of this energy that is converted into heat. The heat generation during this process of reestablishment of equilibrium i s essentially due to two sources; first, heat is generated within the conduction band due to the electron-lattice collision processes which is of the order of lo-’’ SEC‘. The second type of heat generation processes are the band-to-band non-radiative transition4 such as the Auger effect, surface recombination and the recombination through traps. The time scale for these processes varies from a few millisecond up to 0.1 microsecond. Since, in the case of semiconductors, the intraband electron- -lattice relaxation processes are much faster than the band-to-band non-radiative recombination, we may safely assume that the heat is instantaneously produced within the conduction band, so that the thermal power density produced at a given point in the semiconductor is given by the rate of change of the excess carrier, as,

an at Qs(x,t) = hv - (x,t)

Here and in what follows we shall use the same one dimensional configuration and notation as in Ref. 1. ??le excess carrier concentration n(x,t) is obtained from the solution of the carrier dif- fusion equation ’-“, namely,

concentration, D = D fusion coefficient, !nt .rB=,h bination time; in the case opnear-intrinsic materials, D and are the effective diffusion coefficient and effective mobility, respectively. We refer to Ref. 4 for a more detailed discussion on the diffusion of carriers in semiconductors. Furthermore, in the limit of infinite recombi- nation time (T + m) and of no carrier diffusion (D = 0), Eqs. (1) and ( 2 ) reduces to the RG source term.

= k Tu / e is the hole dif- is the role recom-

Solving Eq. (2) and using expression (1) as the source term in the diffusion equations governing the temperature distribution in the PA cell, the amplitude 8 of the gas periodic tempera ture at the solid-gas interface at x = 0 can be found. Once we know the temperature in the gas, we can then assume the RG acoustic piston model to evaluate the pressure fluctuation in the cell. A s the expression found for e is, in the general case, quite cumbersome we shall consider in what follows only the special case of an optically opaque and thermally thick sample. In this case, the thermal diffusion length is smaller than the sample length but larger than the optical penetration depth. Hence, setting in the expression for 8 , exp(-BL)-O, exp(-uLs)=O, and Irl>>l with absorption coefficient, os= (1 + j)as where a

being the optical

= (w/2aS)’/2 and r = f3/as, one gets:

Here a’ =(l+ jwT)/Dr, and a fusivity of the sample. In zrriving at Eq. (3) we have also assumed that vs << vd(vd = Da), wich is

is the thermal dif-

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well satisfied for most materials. For instance, for a typical n-type Ge with a recombi nation time of msec and carrier diffusion coef- ficient of 100 cm*/sec, the surface recombination velocity estimated from photo-conductivity exper- i m e n t ~ ~ is of about 50 cmisec. Hence, for all chopping frequencies of interest (f> 5 Hz) one can safely assume that the condition vd > > v is satisfied. Furthermore, Eq. (3) tells us that’ the amplitude of the gas temperature fluctuation depends also on the relative size of the carrier (a) and the thermal (a) diffusior? coefficients, given bv

- 1 Here f, = ( 2 n ~ ) a <<D). In the case of a ; > o (which also egtails that La > > 1) fluctuation:

and f, = (2as/D)f2 (<<f2 since

one get; for the pressure

I.e., for f < < f l the PA signal of an optically opaque and thermally thick semiconductor should exhibit a saturation, proportional to the carrier recombination time. This saturation of the PA signal is not predicted by the RG theory. Physically, it is understood as follows.The amount of kent generated in the sample depends not only on the photo-injected excess carrier concentration but also on the time they stay at the conduction band. If the carrier recombination time is ex- tremely small (T + 0), of course, there will be no finite concentration and the acoustic signal vanishes; i.e., as soon as the electron reaches the conduction band it recombines with the hole left behind in the valence band thereby producing no heat. With increasing carrier recombination time, one favors the light into heat conversion and, consequently, the PA signal increases. On the other hand, in the case where a i< o one gets:

J2 K

I.e., for fl<< f<< f,, . (6 ) tells us that the PA signal exhibits and varies a s (r/D) 2 for La>>l and proportional to T if i a < < l . On further increasing the chopping frequency sufh that f >> f,, the acousti signal varies as (D’2fI-l for La>>l, and as &a<<l. Summing up these results one may say that whenever the PA signal exhibits a frequency dependence, its dependence on the sample’s thermal properties i s inversely proportional to the peat capacity of the sample (a /ks) and not as as ’2/k as predicted by the RG thzory. Furthermore, the PA signal now exhibits a further dependence on the carrier diffusion coefficient D. For instance, the f-’-frequency dependence of the acoustic signal of an optically opaque and thermally thick semiconduc

f-’: frequency dependence

for

tor can be used to determine the carrier diffusion coefficient. Physically, this dependence on D-y2 can be understood as follows: for very large carrier diffusion coefficient, the carrier dif- fusion time is small. This entails that the amount of heat generated within a thermal diffusion length is small. A s the acoustic signal of a thermally thick sample is essentially determined by the amount of heat within a thermal diffusion length from the surface, one should then expect weak acoustic signals for samples with large carrier diffusion coefficients.

2. Experiment

TO ultimately test the above theory we have performed measurements of the PA signal of an n-type InSb sample ( a = 1 mm; as = 0.25 cm2/s; k =0.15 W/cm K and of a n-type Si sample(es=1.5W a s = 0.77 cm2/s; ks 7 1.30 W/cm°K) in a wide range 02 chopping frequencies. The chopping frequency range as well as the light source used (an Spectra Physics Argon Laser) ensured us that the condition of optical and thelmal opaqueness for both samples. The PA cell consisted of a brass cylindrical cell (P. = 2.5 mm; a = 0.22 cm2/s) with a condenser migrophone mouneed on its wall. The microphone used was a BK4134 capable of providing a flat frequency response of 1.16 mV/pbar from 3.9 Hz to 20 k Hz. The laser beam had a diameter of 1.25 mm and its power was constantly monitored in each measurement. The experimental data on the photo-

PA acoustic signal normalized to the laser power S (in mV/W), as a function of the chopping frequency for the InSb and the Si samples was fitted to the full expression obtained from Eqs. (1) and (2),for an optically opaque sample. In fitting the exper- imental data we have used the known values of the carrier diffusion coefficient as given by the Einstein relation D = kgTp!e. In the case of the n-type InSb sample, D is given by the excess hole Dh= 25 cm2/s, whereas for the n-type Si, D is given by the excess hole Dh = 12.9 cm’is. Thus,our fitting parameters are ?,U and T. In Fig. 1, we plot the normalized PA sig:al SPA for both samples as a function of the chopping frequency. It is also depicted by the solid curves the results of our fitting of the experimental data to the theoretically predicted expression for the acoustic signal,

602 - 1982 ULTRASONICS SYMPOSIUM

Fig.1: Photoacoustic signal normalized to the laser power, SpA(mV/W), as a function of the chopping frequency for n-type InSb and Si samples.

The results we got for the fitting parameters are also listed in Table I.

T V Sample b [msec] [cm/Zec]

0.10 3.4 576 .1 0 . 0 6 InSb (n-type)

TABLE I: Values of the quantum efficiency r l , recombination time T, the surface recombination velocity v s , and the backing parameter b, obtained from the fitting of the experimental data.

The extremely good agreement between the theory and the experimental results strongly supports our claim for taking into account the carrier diffusion and recombination effects when describing the PA signal of such materials.

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3 . References

A. Rosencwaig and A . Gersho, J. Appl. Phys. 4 7 , 64 ( 1 9 7 6 ) . - W. Shockley and W.T. Read, Phys. Rev. 87, 835 ( 1 9 5 2 ) .

R.N. Hall, Phys. Rev. - 8 7 , 387 ( 1 9 5 2 ) .

R.A. Smith, in "Semiconductors" (Cambridge University Press; Cambridge, 1 9 6 4 ) p. 2 3 4 .

W. Van Roosbroeck and W. Shockley, Phys. Rev. 9 4 , 1558 ( 1 9 5 4 ) . -

1982 ULTRASONICS SYMPOSIUM - 603