S. ZEKENG et al. : Raman Determination of the Jj’aust-Henry Coefficient of GaAs 65

phys. stat. sol. (b) 150, 65 (1988)

Subject classification: 63.20 and 78.30; S7.12

arcape Recherches Physiques et Mathiaux, Centre de Recherches Nucldaires associd au C.N. R.S. el Universite’ Louis Pasteur, 8trmbwurgl)

Raman Determination of the Faust-Henry Coefficient of GaAs in the 1.9 to 2.7 eV Range at Ordinary and Low Temperatures

BY S. ZEKENG, €3. PREVOT, and C. SCHWAB

The Faust-Henry coefficient of GaAs is determined a t ordinary and low (20 K) temperatures by using Raman scattering by the TO and LO first order mode vibrations excited by photons with energies located between 1.9 and 2.7 eV. After careful normalization of the experimental results with respect to optical as well as instrumental conditions, the Faust-Henry coefficient is found to be nearly constant over the energy range investigated with a mean value of C = -0.48 f 0.04 a t 300 K and a slightly lower value (C = -0.63 f 0.04) a t low temperature. The pure electro- optic contribution to the Raman efficiency of GaAs is also deduced.

On a determid le coefficient de Fanst-Henry de GaAs 8, temperature ordinaire et 8, basse tem- perature (20 K) en utilisant la technique de diffusion Raman par 1es modes optiques du ler ordre et pour des photons incidents d’hnergie comprise entre 1,9 et 2,7 eV. Lors de l’exploitation den rbsultats exphrimentaux, il a 8th tenu compte des conditions optiques ainsi que des conditions instrumentales. Dam le domaine htudih, la valeur du coefficient de Faust-Henry a Bt6 t row& quasiment constante et Bgale t i -0,48 0,04 8, basse tempkrature. La contribution purement Blectro-optique h la diffusion Raman dens GaAs a Bt6 Bgalement d6duite.

0,04 h 300 K et 8, -0,53

1. Introduction Owing to its technological importance in the field of optoelectronic and high-speed microelectronic devices, GaAs is probably among the members of the 111-V family materials, the compound for which the highest number of Raman investigations has been devoted (for a review see [l a to 1 d]). As a direct consequence, the mechanisms which participate to the Raman scattering (RS) efficiency in GaAs are well established since a long time [2]. For example, in the case of an “allowed” scattering by a polar vibration, we know that the deformation potential (or lattice) mechanism contributes to both the transverse optical (TO) and the longitudinal optical (LO) first-order modes, while an electro-optic mechanism (or iiiterband Prohlich interaction) contri- butes additively only to the LO mode. The ratio of the lattice to the electronic contri- butions is known as the Paust-Henry coefficient C [3]; it determines the relative magnitude of the two contributions. It turns out that the value of this coefficient is important when one is dealing with Raman intensities in spontaneous Raman scat- tering by phonons, by coupled phonon-plasmon modes [4] or by polaritons [5 ] . Furthermore, the Faust-Henry coefficient stands also for the ratio of the pure mechanical to the pure electronic mntributions in the non-linear optical coefficient dEO,, of the linear electro-optic effect [el.

l) 23, rue de Loess, F-67037 Strasbourg CBdex, France.

6 physica (h) 150/1

66 S. ZEKENQ, B. PREVOT, and C. SCHWAB

Surprisingly, there are only few experimental data concerning the Faust-Henry coefficient in GaAs although RS is used in many instances for diagnostic purposes [7]. Since the first values quoted by Mooradian and McWhorter [8]: C = -0.46 and by Johnston and Kaminow [9] C = -0.59 and a more recent one.[lO] giving C = = -0.51, all these values being obtained a t ;li = 1.06 ym (300 K) where 4 is the excitiug laser wavelength, there are only two determinationsin the visible: C = -0.51 for 1, = 514.5 nm [ll] and our previous determination [12] leading to a mean value of C = -0.48 for 488 nm 5 Li 5 647 nm.

In this paper, we present a determination of the Faust-Henry coefficient performed in the 1.9 to 2.7 eV region, i.e. for exciting photon energies located between the E, + A , (1.8 eV) and El (2.9 eV) energy gaps where there are a priori no reasons for C to exhibit strong dispersive effects. I n fact, this study is an outgrowth of a previous one [12] by extension of the energy range and inclusion of low t,emperature data.

2. Experimental Details

The Raman experiments were performed on a standard polished wafer of liquid encapsulated Czochralski (LEC) semi-insulating (Cr-doped) GaAs. As often encounter- ed with device-devoted wafers, the surface was oriented 6" off a [OOl] plane toward a [110] cleavage edge. All spectra were taken with the laser-beam polarized within the incident plane and directed a t 77" from the normal; this strictly corresponds to the Brewster geometry for GaAs a t iii = 496 nm [13]. The scattered signal was ob- served normal to the sample. The sample setting was chosen to maximize the angle between the incoming wave-vector kl (inside the crystal) and the (001) crystallo- graphic axis in order to make the TO and LO peaks observable with comparable intensities [12]. The scattered light was analyzed with a double monochromator equipped with a cooled RCA Quantacon photomultiplier and photon counting elec- tronics; during all this work, the instrumental resolution was fixed a t a value of 3 cm-1. The scattered light was not analyzed in polarization, but in the further analysis of the signal, the anisotropy factor of the monochromator expressed by z = 1,/111 where 1, and 111 stand, respectively, for the scattering efficiency of the instrument for light polarized perpendicular and parallel to the grooves of the gratings, was taken into account. Measurements were obtained using the eight available lines of an Ar-ion laser and five selected lines delivered by a Rhodamine 6G Dye laser. I n order to avoid any heating of the sample, the laser power was limited to a value of 100 mW on the sample and the beam was focussed with a cylindrical lens. For the low temperature measurements, the sample was glued with silver paste on the cold finger of a continuous-flow cryostat. During the low temperature runs, the sample holder was maintained at 7 + 1 K. However, due to radiation losses, the temperature of the GaAs specimen was a t a slightly higher value, but not exceeding 2 0 K as indicated by the peak positions and linewidths.

3. Determination of the Faust-Henry Coefficient by RS

The determination of the Faust-Henry coefficient has been developed in details in our previous paper [12], thus we give here only the main points together with the relevant expressions.

The starting point is the ratio R(wi) of the TO over LO mode intensities observed outside the crystal for an incident frequency wi

Raman Determination of the Faust-Henry Coefficient of GaAs 67

Here R,,(wi) represents the bulk value of R(w,) , i.e. inside the crystal, T(wi) stands for a transmission factor for the incident and scattered light beams, i t incorporates also some other corrections [14]. The geometrical factor Q(wJ is the ratio of the an- isotropic factors of the pure transverse component over the pure longitudinal one of the given phonon species. We showed that these quantities require only the knowledge of the refractive index a t wi and three angular data characterizing the crystal orienta- tion within the laboratory system [12].

The expressions of the scattering efficiency for a Stokes event by a TO or an LO phonon allow us to write [la]

R,(w,) = B(l - A/C)l12 , where C is the Faust-Henry coefficient defined by [4]

c = (&/%+) (ax/au), ( a x / a z q i l . (3) e! being the dynamical charge of the vibration of frequency wT and its reduced mass, (ax/aU), and (ax/aE), are the susceptibility derivatives due respectively to the atomic displacements and to the local electric field associated with the LO vibration. In (2), A and B have the following meanings:

[ G b T ) + 1 ] ["'"LIL+ 11-1 , (4)

2 2 A =wL -wT B =

w; ' O T

where wT and % are the frequencies of the TO and LO phonons and G(w) the Bose- Einstein statistical factor relative to w. From the above expressions, the determination of 6' is straightforward

It has been shown that the pertinent sign of C ( + or -) could be decided from the independent determination of r&, the strain free electro-opt,ic coefficient [9]. In the case of GaAs, C- is the correct solution as in most sphalerite structure materials [14].

4. Methodology and Results The evaluation of C is performed in two steps: the R(w,) values are first determined experimentally, then they are normalized by the corresponding T(wi) and G(wi) factors. For a fixed incident photon energy, the Raman spectra were repeatedly recorded as many times as necessary to obtain an accuracy better than 6% on R. Between two successive recordings, the crystal setting was checked together with the optical alignments in order to avoid possible systematic errors. The counting time per channel was chosen to minimize as far as possible the statistical uncertainties; with the Ar-laser lines, accumulation times of 20 s per channel were sufficient, while longer periods were needed when using lines generated from the dye laser. This is a direct consequense of an overall lowering of the Raman efficiency of GaAs for incident photon wavelengths increasing from the blue region (470 nm) towards the yellow one (= 600 nm). Fig. 1 displays some typical spectra recorded a t two temperatures. For obtaining the peak intensities, no correction for the instrumental response was applied, since, to first order, such corrections are believed to cancel if applied to each member of a ratio. In order to evaluate the A and B factors defined in (4) the mean values of wT and wL were determined from all the spectra obtained in this study. These frequencies and the resulting A and B quantities are given in Table 1 for the two relevant temperatures. 6.

68 S. ZEKENG, B. PEEVOT, and C. SCHWAB

I t

" u c b c p o 3 0 . 7 ; I

320 -

240

160

80

0

I

250

60

40

20

0 270 230 310 260 280 300

Raman shif? (cm-') - Fig. 1. Typical Raman spectra of GaAs obtained with two different photon excitations a t room temperature (upper part) and low temperature (lower part). All spectra, were recorded using the same sample setting. The differences observed in the counting rates in the TO peak are well accounted for either by the Bose-Einstein statistical factor for the temperature variation, or by the dispersion properties of the scattering efficiency (see also Fig. 5)

The T ( w J values necessitate only the knowledge of n, and k,, the index of refrac- tion, and the absorption coefficient of GaAs at w,, where OL = i, ST and SL are the incident, transverse, and longitudinal frequencies. The n and k values at 300 K were taken from the work of Aspnes and Studna [13]; at 20 I(, we used those recently published by Lautenschlager et al. [15]. As shown in Fig. 2, T(wi) is almost constant and equal to one between 458 and 641 nm (i.e. 2.7 and 1.9 eV) a t room temperature, while at low temperature T(wi) is slowly decreasing with increasing wavelength.

Tab le 1 GaAs frequencies and resulting quantities used in this study. The absolute accuracy on the 0 ' s is & 0.2 cm-l; and better than 2 0.001 for A and B

T WT WL A B (K) (cm-1) (cm-l)

300 268.5 291.4 0.178 1.128 N 20 272.0 296.4 0.179 1.129

Raman Determination of the Faust-Henry Coefficient of GaAs 69

Fig. 2. The transmission factor T(q) (open symbols - left-hand scale) and the geometrical factor G(ui) (full symbols - right-hand scale) used in this study. Notice that the T(oi ) ’s are functions of the optical properties of GaAs, while the a(wi)’s are more complicatd functions of the scattering configuration and of the spectrometer properties. 0, T = 300 K, A, A T = 20 K

The most important factor entering (1) is G(q) = (IT/IL)ol which, in our case, must include the anisotropy factor -c of the instrumental response relative to the polarization of the scattered light. It follows that G(w,) can be written under the form

where 1,: and Iv stands for the horizontally and. vertically, respectively, polarized contribution of the scattered light by a transverse (T) or a longitudinal (L) mode. These quantities are obtained in a straightforward manner by making the appropriate transformation of the Raman tensor from its expression in the crystallographic system to the present laboratory system, and then by dealing separately with the T and L components of the given symmetry species (here the F,, one). Obviously, they are func- tions of the external angles - and thus of n,, - therefore the G(q)’s are to be evaluated for each sample temperature. They are shown in Fig. 2. Finally Fig. 3 reports the values of R,,(w,) obtained a t two temperatures, including at 300 K the value quoted previously for A, = 647 nm [12] but slightly revised to take into account small differ- ences in the n and k values used in the two papers. Fig. 4 shows the final results for the dispersion of the Faust-Henry coefficient of GaAs.

5. Discussion

The frequency dependence of C is directly correlated with those of the atomic displace- ment and electric field contributions. The former contribution due to the deformation potential mechanism, has been determined experimentally in GaAs a t 300 K and the results have been successfully explained with a theory based on a simple band- structure model [16]. From this study, i t appears that the single element of the

70 S. ZEEBNG, B. PREVOT, .and C. SCHWAB

- photon energy (eV) 26 24 22 zo

1 300K 08

I I 1 I

450 500 550 600 650 AInrn) -

Fig. 3

I I

450 500 550 600 650 dInm) --

Fig. 4 Fig. 3. The intrinsic value of R,(T/L) (= RITB) in GaAs as a function of the wavelength of the incident light beam for two significant sample temperatures. The full lines are representative of the mean values; the standard deviation is shown by the small vertical bar

Fig.4. Values of the Faust-Henry coefficient of GaAs a t two temperatures between 450 and 650 nm, i.e. for incident photons in the 1.9 to 2.7 eV energy range. The horizontal lines stand for the mean values of C

atomic displacement Raman tensor ( ax/i3U)E shows a monotonic behaviour for the incident light energy E, varying between 1.9 and 2.7 eV, provided El is kept away from the resonance regions. This is the case of the present study. Further, since our data have been obtained in "allowed" scattering conditions, a similar behaviour is expected for the corresponding electro-optic Raman contribution (ax/aE),. We are thus justified to assume that C has a quasi-constant value in the 1.9 to 2.7 eV energy range. From Fig. 4, we determine a mean value C = -0.48 + 0.04 a t 300 K and C = -0.63

The present room temperature determination agrees well with results reported previously either in the visible [ll, 121 or, within errors, in the near infra-red [S, lo].

0.04 a t 20 K.

i Fig. 5. Comparison of the two channels re- sponsible for the Raman scattering efficiency in GaAs a t 300 K. The mechanical contribution

electro-optic one (full symbols) was obtained in this paper. Both mechanisms are expressed in the same units, but they have opposite signs

0 was taken from [16] (open symbols), while the 0 00

- - - m m J , 550 600 650

h Inm) -

Raman Determination of the Faust-Henry Coefficient of GaAs 71

Only the data of Johnston and Kaminow [9]: C = -0.59 seems to deviate signifi- cantly. However, using their value of R,(T/L) , i.e. 0.65, and our A and B factors defined by [4], one obtains C = -0.56. Furthermore, if their experimental uncer- tainties are taken into account, the final result can be expressed as C = -0.56 0.25. This points out the main limitations in such a determination. They include the ex- perimental methodology as well as the surface quality of the sample. Under these conditions, we believe that the actual accuracy (8 to 9% for C) could bo hardly im- proved.

From Fig. 4, it is concluded that C has a lower value a t 20 K than a t room tempera- ture. A similar behaviour was reported in the early paper of Mooradian and McWhorter [8] who have obtained C = -0.51 for T + 0 K and Ai = 1.06 pm; further from the theoretical fit of the coupled plasmon-LO phonon in n-type GaAs, Abstreiter et al. [17] havefound C = -0.54for T = 80 K and li = 568 nm. It is, however, recognized that the difference between the low and high temperature values is of the same order as the experimental accuracy.

Starting from the definition of C given in (3), the knowledge of C and of its purely mechanical component [16] allows for the determination of the electro-optic contribu- tion to the longitudinal signal. The dispersions of the two contributions are shown in Fig. 5. For the sake of comparison, they are displayed in the same units, but they have opposite signs.

As noted in the introduction, the difference between the intensities of the two first- order modes in non-centrosymmetric crystals can be ascribed to the electro-optic effect arising from the electric field associated with the LO vibration. Thus there exists a close correlation between the electro-optic coefficients and the first order R.S. efficiencies. Following Johnston [ 181 the constant-strain (or clamped) electro-optic coefficient for a crystal with high symmetry can be written as:

-n4rpS1 = 4&0,, = 4(&, + d,E,) = 4(1 + C) d F 4 , (7)

where dEO,, is the non-linear optical coefficient of the linear electro-optic (or Pockels) coefficient ; d:4 and dF4 are respectively the purely ionic and electronic contributions. C is the ratio of these two quantities and corresponds to the Faust-Henry coefficient ; n is the refractive index of the source wavelength. For obvious reaBons, the linear electro-optic properties of GaAs have been determined in the transparent region of the spectrum, i.e. for photon energies below the fundamental edge E,,. Published values are around -1.5 x 10-12 mV--l [lo]. On the other hand, in the non-dispersive region, dF4 is equivalent to @pa, the second harmonic generation coefficient, which is of the order of 1 x rnV-1 for GaAs [lo]. With these values and taking n = 3.38 (2 = 1.5 pm), C = -0.51 is found from relation (7), a value which compares well with our present determination. However, this might be considered as fortuitous since these C values are obtained a t different photon energies located on both sides of the fundamental gap. Furthermore, when the photons have energies comparable or higher than E,, r h and d f F G become complex quantities with strongly varying values, so relation (7) certainly no more holds.

AS a final remark, i t might be observed from Fig. 3, that R, for ili = 472 nm lies significantly outside (z 10%) the mean value of all the other determinations made a t 300 K. Indeed, in spite of the special attention paid for that point, a higher absolute value was systematically recorded. This results in a singular data either for C (Fig. 4) or for the electro-optic contribution (Fig. 5). Any convincing explanation of such a behaviour was found neither in the corresponding T and G factors - see Fig. 2 - nor in the electronic band structure of GaAs.

72 S. ZEKENQ et al.: Raman Determination of the Faust-Henry Coefficient of GaAs

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(1987).

(Received June 24,1988)