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Ge superlattices determined by Brillouin light scattering
Thin Solid Films 317 Ž1998. 255–258
Elastic behaviour of SirGe superlattices determined by Brillouin light scattering A. de Bernabe´ ) , R. Jimenez, M. Garcıa-Hernandez, C. Prieto ´ ´ ´ Instituto de Ciencia de Materiales de Madrid (C.S.I.C.), Cantoblanco, Madrid E-28049, Spain
Abstract The elastic properties of SirGe strained-layer superlattices have been studied by means of Brillouin light scattering spectroscopy. The superstructures, grown along the w001x direction, have composition modulation wavelengths ranging from 1.4 to 5.6 nm and SirGe ratios: 1:1, 3:1, 4:1 and 1:4. The study has been carried out in three different stages: Ži. the checking of the non-influence from the substrate on the determination of the elastic properties; Žii. the angular dependence of the surface sound velocity within the surface plane; and Žiii. the variation of the surface sound velocity with the modulation wavelength. The obtained data of hypersonic surface waves are in good agreement with those calculated from previous results reported in Si and Ge pure single-crystals. No evidence of anomalous supermodulus effect has been found in these superlattices. q 1998 Elsevier Science S.A. Keywords: SirGe superlattices; Brillouin light scattering; Surface wave’s
1. Introduction Brillouin light scattering spectroscopy ŽBLS. has been proven to be a very efficient tool to measure elastic properties of opaque materials w1x. Compared to other elastic probing techniques BLS presents many advantages. The low penetration of light inside non-transparent materials allows to obtain information arising only from the sample contribution. In addition, no mechanical contacts are needed enabling thus to perform experiments with no sample damage. Finally, the use of a laser spot combined with a focusing lens permits to study small size samples and local effects. BLS is an adequate technique to study the so called ‘supermodulus effect’. Such effect appears in superlattices ŽSLs. when the modulation wavelength L Žbilayer thickness. is about 2 nm. It consists of an increase of the biaxial modulus Ž Y b . along with a softening of some of the elastic constants. This anomalous elastic effect would allow to obtain materials with a Young modulus higher than those of metals they are made of. The aim of the present work is to check the existence of any elastic anomaly in Si m rGe m and Si m rGe n SLs, where subscripts m and n refer to the number of deposited monolayers. The presence of any anomalous elastic be-
haviour in these SLs could be very helpful to understand the nature of the supermodulus effect.
2. Theoretical aspects In the absence of body forces and piezoelectric effects the wave equation for the displacement in a perfectly elastic, homogeneous, anisotropic medium can be written as:
r
E 2 uj E t2
s Ci jk l
E 2 uk E xiE xl
Ž i , j,k ,l s 1,2,3 .
Ž 1.
where u i are the displacement components along the Cartesian axes Ž x i ., Ci jk l the stiffness tensor and r the medium density. In opaque materials, due to the optical absorption, the system can be considered as a semi-infinite medium Žprovided that the sample is larger than the light penetration depth.. Solutions to the secular equation must, then, vanish for x ™ y` as well as the stress tensor, si j , must be zero at the surface. Solutions to the wave equation take the form w2x 3
ui s
Ý Cn a iŽ n.exp
iK Ž l 1 x 1 q l 2 x 2 q l 3Ž n. x 3 y Õt .
Ž 2.
ns1
)
Corresponding author.
0040-6090r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved. PII S 0 0 4 0 - 6 0 9 0 Ž 9 7 . 0 0 6 4 1 - X
where 1 i are the direction cosines of the propagating phonons, Cn are weighting factors determined from the
A. de Bernabe´ et al.r Thin Solid Films 317 (1998) 255–258
256
boundary conditions and a iŽ n. are some exponentially damping terms due to the optical absorption. In superlattices and stratified media additional boundary conditions are required at each interface.
si3 Ž A . s si j Ž B . i s 1,2,3
Ž 3.
u i j Ž A . s u i j Ž B . i , j s 1,2
Ž 4.
where A and B stand for materials A and B. Brillouin light scattering spectroscopy enables the study of elastic waves through a coupling between an incident electromagnetic field and thermal fluctuations inside materials. That coupling mechanism, called Brillouin– Mandelstann light scattering, can be seen as a photon–phonon interaction, where energy and momentum should be conserved:
power of about 100 mW. The backscattering geometry was used. Samples were placed on a double goniometer which allowed the variation of both, the incidence angle, u , and the azimuthal one, a Žangle between the phonon propagation direction and the w110x direction of the SL.. The incidence polarization direction was chosen to be in-plane Ž p-polarization. while no polarization analysis of the scattered light was made.
4. Results and discussion
3. Experimental
The first step to ensure the reliability of the present work results, is the check of the possible substrate influence on the obtained phase velocities Ž Õp .. The general procedure to determine the extent of the contribution arising from the substrate, consists of plotting the phase velocities vs. q = h, where q is the phonon wavevector and h is the sample thickness w6x. The variation of q comes just from a change in the incidence angle, u , and thus in the penetration depth of the incident light. As u decreases, the light penetration increases, yielding a contribution from the substrate if the sample thickness is not sufficient. If Õp remains constant Žprovided that the phase velocity of the substrate is different from the sample’s. there will exist no contribution arising from the substrate. In the present work the phase velocity has been determined, for each sample, at three different angles within the range 458–708. The completely flat dependence of Õp vs. q = h indicates that there is no influence of the substrate on the obtained velocities. Fig. 1 shows a typical Brillouin spectrum corresponding to the Si 15 rGe15 SL at an incidence angle of u s 61.48. The surface phonon propagates within the Ž001. plane and along the w110x direction. The elastic line has been filtered and it is represented in a different scale. A ramp calibration system has been used to avoid the appearance of the ‘ghost’ peaks.
Samples were prepared by Prof. Wang’s Group at UCLA using Molecular Beam Epitaxy. Their growth procedure and characterization have been reported elsewhere w3,4x. Measurements have been performed on a set of SirGe superlattices with modulation wavelengths ranging from 1.4 to 5.6 nm and different SirGe ratios: Si m rGe m ŽSi 5 rGe 5 , Si 10 rGe 10 , Si 15 rGe 15 and Si 20 rGe 20 ., Si 3 m rGe m ŽSi 12 rGe 4 ., Si 4 m rGe m ŽSi 32 rGe 8 and Si 16 rGe 4 . and Si m rGe 4 m ŽSi 4rGe16 .. The number of bilayers was calculated to keep fixed the total sample thickness around 300 nm. The interferometer used was a 3 q 3 pass tandem Fabry–Perot ´ w5x with a selected free spectral range of 30 GHz. A 2060 Beamlok Spectra Physics Argon laser was used as monochromatic source of light. It is provided with an intracavity single mode and single frequency z-lok etalon, which is temperature stabilized. The selected wavelength, 514.5 nm, provided an incident
Fig. 1. Brillouin spectrum corresponding to the Si 15 rGe15 SL at an incidence angle u s61.48. The surface phonon propagates within the Ž001. plane and along the w110x direction.
K s y K i s "q Ž v . ; v s y v 1 s "v ph
Ž 5.
K i and K s Ž v i , v s . are the wavevectors Žfrequencies. of the incident and scattered photons respectively. In opaque materials, however, momentum is only conserved within the surface plane and it is necessary to substitute momenta in Eq. Ž5. by their projections on the surface K s sin us y K i sin u i s "q x Ž v .
Ž 6.
yielding, for the surface acoustic wave velocity in backscattering geometry Ž u i s us .
l Dv VSAW s
2sin u
Ž 7.
where l is the wavelength of the incident light, D v the observed frequency shift and u i Ž us . the angle between the normal to the surface and the incidence Žscattered. direction.
A. de Bernabe´ et al.r Thin Solid Films 317 (1998) 255–258
Fig. 2. Angular dependence of the phase velocity for different periodicities of the SirGe SLs. The upper part shows the Si 4 m rGe m SLs while the lower the Si m rGe m ones.
The angular dispersion for some of the SLs studied in this work are plotted in Fig. 2. The upper part, Ža., compares the two Si 4 m rGe m SLs while the lower one, Žb., shows all the Si m rGe m samples. The angular dispersion behaviour obtained is in good agreement with those previously reported in pure Si w1x and Ge w7x, as well as with other results obtained for different SL periodicities w8x. All the observed behaviours show a well defined four-fold axis, corresponding to an unaltered cubic structure in the xy-plane. In Fig. 2, nevertheless, it should be remarked that all the experimental points do not correspond to surface acoustic waves. In cubic crystals there is a surface wave propagating in the Ž001. plane and along the w100x direction. As the propagation direction deviates from w100x, a bulk-type solution of the wave equation becomes more and Table 1 Experimental C44 elastic constants obtained for different periodicities of the SirGe SLs Žw8x and present work. compared with the calculated results from the Elasticity Theory using Grimsditch–Nizzoli approach w10x and taking values for bulk Si and Ge from Refs. w11,12x respectively
Si m rGe m Ž ms 5, 10, 15, 20. Si 2 rGe1 Si 12 rGe 4 Si 16 rGe 4 Si 32 rGe 8 Si 5.5 rGe1 Si 4 rGe16 Si Ge
C44 ŽGPa. Experimental
C44 ŽGPa. Elasticity theory
69.6"0.6 68.1"2.0 w8x 75.4"0.5 76.6"0.5 74.8"0.5 73.0"2.2 w8x 64.2".6 79.6 w11x 66.8 w12x
72.6 74.8 76.0 76.7 76.7 77.3 69.0 y y
257
more important until at 458 Žalong w110x. the solution has completely varied into a bulk wave w2x. An interesting remark should be made on the difference between the phase velocities obtained for the Si 16 rGe 4 sample and the Si 32 rGe 8 one. On the basis of the Elasticity Theory, this change in the phase velocity could not be explained, since it only depends on the fraction of Si and Ge. However, the previous structural characterization by X-ray diffraction w4x yields a higher density of dislocation in the Si 32 rGe 8 SL. That could account for the observed decrease on the phase velocity as happens with Si during its amorphization w9x. Owing to the stratified character of SLs, additional boundary conditions are required for solutions to the wave equation ŽEqs. Ž5. and Ž6... They may be satisfied intro. which ducing an effective elastic constant tensor Ž Cieff. j can be calculated according to the method described by Grimsditch and Nizzoli w10x and using values from the bulk elastic constants of each component. Usually, the obtained Cieff. j tensor is complicated, however in our supereff. structures the C44 elastic constant can be easily calculated from: 1 eff . C44
s
fA A C44
q
fB B C44
Ž 8.
where fA and f B are the fraction of materials A and B. Taking the density r from the linear interpolation between those of pure Si and Ge, we obtain: rm r m s 3.82 g cmy3 , r 3 m r m s 3.08 g cmy3 , r4 m r m s 2.93 g cmy3 , rm r4 m s 4.72 g cmy3 , for the Si m rGe m , Si 3 m rGe m , eff. Si 4 m rGe m and Si m rGe 4 m SLs respectively. The C44 elastic constant is then calculated from the obtained phase velocity and the density. The results derived from our work are compared in Table 1 with those resulting from calculations using the theoretical approach by Grimsditch and Nizzoli and the bulk elastic constants reported from ultrasonic measurements for pure Si and Ge w11,12x. The comparison is extended to previous experimental results obtained in SLs with different SirGe ratios and periodicities w8x. Provided that velocities obtained by BLS are 2–5% lower than those measured by ultrasounds w13x, the experimental values reported here are in good agreement with the theoretical predictions. Though, no evidence of supermodulus effect was observed, as can be concluded from Fig. 2.
5. Conclusions Brillouin light scattering spectroscopy has been used to test the possible existence of any anomalous elastic behaviour of Si m rGe m and Si m rGe n superlattices. A previous check of the substrate contribution was made in order to assure the reliability of the results. These SLs did not show any elastic anomaly, what can be explained on the
258
A. de Bernabe´ et al.r Thin Solid Films 317 (1998) 255–258
basis of the latest theories about the supermodulus effect. In addition, the angular dispersion behaviour of these SLs is in good agreement with what would be expected from those of Si and Ge pure single crystals. Acknowledgements We gratefully acknowledge Prof. K.L. Wang for providing the samples. This work has been supported by the CICyT under Contract No. MAT94r-0722. References w1x J.R. Sandercock, Solid State Commun. 26 Ž1978. 547. w2x G.W. Farnell, in: W.P. Mason ŽEd.., Physical Acoustics, Vol. 6, Academic Press, New York, 1968, pp. 109–165.
w3x V. Arbet, S.J. Chang, K.L. Wang, Thin Solid Films 183 Ž1989. 57. w4x P.M. Adams, R.C. Bowman, C.C. Ahn, S.J. Chang, V. Arbet-Engels, M.A. Kallel, K.L. Wang, J. Appl. Phys. 71 Ž1992. 4305. w5x J.R. Sandercock, in: M. Cardona, G. Guntherdot ŽEds.., Light ¨ Scattering in Solids, Vol. III, Springer-Verlag, Berlin, 1982, pp. 173–206. w6x G.W. Farnell, E.L. Adler, in: W.P. Mason, R.N. Thurston ŽEds.., Physical Acoustics, Vol. IX, Academic Press, New York, 1972, pp. 109–165. w7x M.W. Elmiger, J. Henz, H. v. Kanel, M. Ospelt, P. Wachter, Surf. ¨ Interf. Anal. 14 Ž1989. 18. w8x M. Mendik, M. Ospelt, H. von Kanel, P. Wachter, Appl. Surf. Sci. ¨ 50 Ž1991. 303. w9x R. Bhadra, J. Pearson, P. Okamoto, L. Rehn, M. Grimsditch, Phys. Rev. B 38 Ž1988. 12656. w10x M. Grimsditch, F. Nizzoli, Phys. Rev. B 33 Ž1986. 5891. w11x H.J. McSkimin, P. Andreatch Jr., J. Appl. Phys. 35 Ž1964. 2161. w12x H.J. McSkimin, P. Andreatch Jr., J. Appl. Phys. 34 Ž1963. 651. w13x V.R. Velasco, F. Garcıa-Moliner, Solid State Commun. 33 Ž1980. 1. ´
Elastic behaviour of SirGe superlattices determined by Brillouin light scattering A. de Bernabe´ ) , R. Jimenez, M. Garcıa-Hernandez, C. Prieto ´ ´ ´ Instituto de Ciencia de Materiales de Madrid (C.S.I.C.), Cantoblanco, Madrid E-28049, Spain
Abstract The elastic properties of SirGe strained-layer superlattices have been studied by means of Brillouin light scattering spectroscopy. The superstructures, grown along the w001x direction, have composition modulation wavelengths ranging from 1.4 to 5.6 nm and SirGe ratios: 1:1, 3:1, 4:1 and 1:4. The study has been carried out in three different stages: Ži. the checking of the non-influence from the substrate on the determination of the elastic properties; Žii. the angular dependence of the surface sound velocity within the surface plane; and Žiii. the variation of the surface sound velocity with the modulation wavelength. The obtained data of hypersonic surface waves are in good agreement with those calculated from previous results reported in Si and Ge pure single-crystals. No evidence of anomalous supermodulus effect has been found in these superlattices. q 1998 Elsevier Science S.A. Keywords: SirGe superlattices; Brillouin light scattering; Surface wave’s
1. Introduction Brillouin light scattering spectroscopy ŽBLS. has been proven to be a very efficient tool to measure elastic properties of opaque materials w1x. Compared to other elastic probing techniques BLS presents many advantages. The low penetration of light inside non-transparent materials allows to obtain information arising only from the sample contribution. In addition, no mechanical contacts are needed enabling thus to perform experiments with no sample damage. Finally, the use of a laser spot combined with a focusing lens permits to study small size samples and local effects. BLS is an adequate technique to study the so called ‘supermodulus effect’. Such effect appears in superlattices ŽSLs. when the modulation wavelength L Žbilayer thickness. is about 2 nm. It consists of an increase of the biaxial modulus Ž Y b . along with a softening of some of the elastic constants. This anomalous elastic effect would allow to obtain materials with a Young modulus higher than those of metals they are made of. The aim of the present work is to check the existence of any elastic anomaly in Si m rGe m and Si m rGe n SLs, where subscripts m and n refer to the number of deposited monolayers. The presence of any anomalous elastic be-
haviour in these SLs could be very helpful to understand the nature of the supermodulus effect.
2. Theoretical aspects In the absence of body forces and piezoelectric effects the wave equation for the displacement in a perfectly elastic, homogeneous, anisotropic medium can be written as:
r
E 2 uj E t2
s Ci jk l
E 2 uk E xiE xl
Ž i , j,k ,l s 1,2,3 .
Ž 1.
where u i are the displacement components along the Cartesian axes Ž x i ., Ci jk l the stiffness tensor and r the medium density. In opaque materials, due to the optical absorption, the system can be considered as a semi-infinite medium Žprovided that the sample is larger than the light penetration depth.. Solutions to the secular equation must, then, vanish for x ™ y` as well as the stress tensor, si j , must be zero at the surface. Solutions to the wave equation take the form w2x 3
ui s
Ý Cn a iŽ n.exp
iK Ž l 1 x 1 q l 2 x 2 q l 3Ž n. x 3 y Õt .
Ž 2.
ns1
)
Corresponding author.
0040-6090r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved. PII S 0 0 4 0 - 6 0 9 0 Ž 9 7 . 0 0 6 4 1 - X
where 1 i are the direction cosines of the propagating phonons, Cn are weighting factors determined from the
A. de Bernabe´ et al.r Thin Solid Films 317 (1998) 255–258
256
boundary conditions and a iŽ n. are some exponentially damping terms due to the optical absorption. In superlattices and stratified media additional boundary conditions are required at each interface.
si3 Ž A . s si j Ž B . i s 1,2,3
Ž 3.
u i j Ž A . s u i j Ž B . i , j s 1,2
Ž 4.
where A and B stand for materials A and B. Brillouin light scattering spectroscopy enables the study of elastic waves through a coupling between an incident electromagnetic field and thermal fluctuations inside materials. That coupling mechanism, called Brillouin– Mandelstann light scattering, can be seen as a photon–phonon interaction, where energy and momentum should be conserved:
power of about 100 mW. The backscattering geometry was used. Samples were placed on a double goniometer which allowed the variation of both, the incidence angle, u , and the azimuthal one, a Žangle between the phonon propagation direction and the w110x direction of the SL.. The incidence polarization direction was chosen to be in-plane Ž p-polarization. while no polarization analysis of the scattered light was made.
4. Results and discussion
3. Experimental
The first step to ensure the reliability of the present work results, is the check of the possible substrate influence on the obtained phase velocities Ž Õp .. The general procedure to determine the extent of the contribution arising from the substrate, consists of plotting the phase velocities vs. q = h, where q is the phonon wavevector and h is the sample thickness w6x. The variation of q comes just from a change in the incidence angle, u , and thus in the penetration depth of the incident light. As u decreases, the light penetration increases, yielding a contribution from the substrate if the sample thickness is not sufficient. If Õp remains constant Žprovided that the phase velocity of the substrate is different from the sample’s. there will exist no contribution arising from the substrate. In the present work the phase velocity has been determined, for each sample, at three different angles within the range 458–708. The completely flat dependence of Õp vs. q = h indicates that there is no influence of the substrate on the obtained velocities. Fig. 1 shows a typical Brillouin spectrum corresponding to the Si 15 rGe15 SL at an incidence angle of u s 61.48. The surface phonon propagates within the Ž001. plane and along the w110x direction. The elastic line has been filtered and it is represented in a different scale. A ramp calibration system has been used to avoid the appearance of the ‘ghost’ peaks.
Samples were prepared by Prof. Wang’s Group at UCLA using Molecular Beam Epitaxy. Their growth procedure and characterization have been reported elsewhere w3,4x. Measurements have been performed on a set of SirGe superlattices with modulation wavelengths ranging from 1.4 to 5.6 nm and different SirGe ratios: Si m rGe m ŽSi 5 rGe 5 , Si 10 rGe 10 , Si 15 rGe 15 and Si 20 rGe 20 ., Si 3 m rGe m ŽSi 12 rGe 4 ., Si 4 m rGe m ŽSi 32 rGe 8 and Si 16 rGe 4 . and Si m rGe 4 m ŽSi 4rGe16 .. The number of bilayers was calculated to keep fixed the total sample thickness around 300 nm. The interferometer used was a 3 q 3 pass tandem Fabry–Perot ´ w5x with a selected free spectral range of 30 GHz. A 2060 Beamlok Spectra Physics Argon laser was used as monochromatic source of light. It is provided with an intracavity single mode and single frequency z-lok etalon, which is temperature stabilized. The selected wavelength, 514.5 nm, provided an incident
Fig. 1. Brillouin spectrum corresponding to the Si 15 rGe15 SL at an incidence angle u s61.48. The surface phonon propagates within the Ž001. plane and along the w110x direction.
K s y K i s "q Ž v . ; v s y v 1 s "v ph
Ž 5.
K i and K s Ž v i , v s . are the wavevectors Žfrequencies. of the incident and scattered photons respectively. In opaque materials, however, momentum is only conserved within the surface plane and it is necessary to substitute momenta in Eq. Ž5. by their projections on the surface K s sin us y K i sin u i s "q x Ž v .
Ž 6.
yielding, for the surface acoustic wave velocity in backscattering geometry Ž u i s us .
l Dv VSAW s
2sin u
Ž 7.
where l is the wavelength of the incident light, D v the observed frequency shift and u i Ž us . the angle between the normal to the surface and the incidence Žscattered. direction.
A. de Bernabe´ et al.r Thin Solid Films 317 (1998) 255–258
Fig. 2. Angular dependence of the phase velocity for different periodicities of the SirGe SLs. The upper part shows the Si 4 m rGe m SLs while the lower the Si m rGe m ones.
The angular dispersion for some of the SLs studied in this work are plotted in Fig. 2. The upper part, Ža., compares the two Si 4 m rGe m SLs while the lower one, Žb., shows all the Si m rGe m samples. The angular dispersion behaviour obtained is in good agreement with those previously reported in pure Si w1x and Ge w7x, as well as with other results obtained for different SL periodicities w8x. All the observed behaviours show a well defined four-fold axis, corresponding to an unaltered cubic structure in the xy-plane. In Fig. 2, nevertheless, it should be remarked that all the experimental points do not correspond to surface acoustic waves. In cubic crystals there is a surface wave propagating in the Ž001. plane and along the w100x direction. As the propagation direction deviates from w100x, a bulk-type solution of the wave equation becomes more and Table 1 Experimental C44 elastic constants obtained for different periodicities of the SirGe SLs Žw8x and present work. compared with the calculated results from the Elasticity Theory using Grimsditch–Nizzoli approach w10x and taking values for bulk Si and Ge from Refs. w11,12x respectively
Si m rGe m Ž ms 5, 10, 15, 20. Si 2 rGe1 Si 12 rGe 4 Si 16 rGe 4 Si 32 rGe 8 Si 5.5 rGe1 Si 4 rGe16 Si Ge
C44 ŽGPa. Experimental
C44 ŽGPa. Elasticity theory
69.6"0.6 68.1"2.0 w8x 75.4"0.5 76.6"0.5 74.8"0.5 73.0"2.2 w8x 64.2".6 79.6 w11x 66.8 w12x
72.6 74.8 76.0 76.7 76.7 77.3 69.0 y y
257
more important until at 458 Žalong w110x. the solution has completely varied into a bulk wave w2x. An interesting remark should be made on the difference between the phase velocities obtained for the Si 16 rGe 4 sample and the Si 32 rGe 8 one. On the basis of the Elasticity Theory, this change in the phase velocity could not be explained, since it only depends on the fraction of Si and Ge. However, the previous structural characterization by X-ray diffraction w4x yields a higher density of dislocation in the Si 32 rGe 8 SL. That could account for the observed decrease on the phase velocity as happens with Si during its amorphization w9x. Owing to the stratified character of SLs, additional boundary conditions are required for solutions to the wave equation ŽEqs. Ž5. and Ž6... They may be satisfied intro. which ducing an effective elastic constant tensor Ž Cieff. j can be calculated according to the method described by Grimsditch and Nizzoli w10x and using values from the bulk elastic constants of each component. Usually, the obtained Cieff. j tensor is complicated, however in our supereff. structures the C44 elastic constant can be easily calculated from: 1 eff . C44
s
fA A C44
q
fB B C44
Ž 8.
where fA and f B are the fraction of materials A and B. Taking the density r from the linear interpolation between those of pure Si and Ge, we obtain: rm r m s 3.82 g cmy3 , r 3 m r m s 3.08 g cmy3 , r4 m r m s 2.93 g cmy3 , rm r4 m s 4.72 g cmy3 , for the Si m rGe m , Si 3 m rGe m , eff. Si 4 m rGe m and Si m rGe 4 m SLs respectively. The C44 elastic constant is then calculated from the obtained phase velocity and the density. The results derived from our work are compared in Table 1 with those resulting from calculations using the theoretical approach by Grimsditch and Nizzoli and the bulk elastic constants reported from ultrasonic measurements for pure Si and Ge w11,12x. The comparison is extended to previous experimental results obtained in SLs with different SirGe ratios and periodicities w8x. Provided that velocities obtained by BLS are 2–5% lower than those measured by ultrasounds w13x, the experimental values reported here are in good agreement with the theoretical predictions. Though, no evidence of supermodulus effect was observed, as can be concluded from Fig. 2.
5. Conclusions Brillouin light scattering spectroscopy has been used to test the possible existence of any anomalous elastic behaviour of Si m rGe m and Si m rGe n superlattices. A previous check of the substrate contribution was made in order to assure the reliability of the results. These SLs did not show any elastic anomaly, what can be explained on the
258
A. de Bernabe´ et al.r Thin Solid Films 317 (1998) 255–258
basis of the latest theories about the supermodulus effect. In addition, the angular dispersion behaviour of these SLs is in good agreement with what would be expected from those of Si and Ge pure single crystals. Acknowledgements We gratefully acknowledge Prof. K.L. Wang for providing the samples. This work has been supported by the CICyT under Contract No. MAT94r-0722. References w1x J.R. Sandercock, Solid State Commun. 26 Ž1978. 547. w2x G.W. Farnell, in: W.P. Mason ŽEd.., Physical Acoustics, Vol. 6, Academic Press, New York, 1968, pp. 109–165.
w3x V. Arbet, S.J. Chang, K.L. Wang, Thin Solid Films 183 Ž1989. 57. w4x P.M. Adams, R.C. Bowman, C.C. Ahn, S.J. Chang, V. Arbet-Engels, M.A. Kallel, K.L. Wang, J. Appl. Phys. 71 Ž1992. 4305. w5x J.R. Sandercock, in: M. Cardona, G. Guntherdot ŽEds.., Light ¨ Scattering in Solids, Vol. III, Springer-Verlag, Berlin, 1982, pp. 173–206. w6x G.W. Farnell, E.L. Adler, in: W.P. Mason, R.N. Thurston ŽEds.., Physical Acoustics, Vol. IX, Academic Press, New York, 1972, pp. 109–165. w7x M.W. Elmiger, J. Henz, H. v. Kanel, M. Ospelt, P. Wachter, Surf. ¨ Interf. Anal. 14 Ž1989. 18. w8x M. Mendik, M. Ospelt, H. von Kanel, P. Wachter, Appl. Surf. Sci. ¨ 50 Ž1991. 303. w9x R. Bhadra, J. Pearson, P. Okamoto, L. Rehn, M. Grimsditch, Phys. Rev. B 38 Ž1988. 12656. w10x M. Grimsditch, F. Nizzoli, Phys. Rev. B 33 Ž1986. 5891. w11x H.J. McSkimin, P. Andreatch Jr., J. Appl. Phys. 35 Ž1964. 2161. w12x H.J. McSkimin, P. Andreatch Jr., J. Appl. Phys. 34 Ž1963. 651. w13x V.R. Velasco, F. Garcıa-Moliner, Solid State Commun. 33 Ž1980. 1. ´