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High-temperature studies of surface acoustic wave velocities in silicon by Brillouin scattering
ELSEVIER
Physica B 219&220(1996) 717 719
High-temperature studies of surface acoustic wave velocities in silicon by Brillouin scattering P.R. Stoddart*, J.D. Comins, A.G.
Every
Department of Physics, Universi~ of the Witwatersrand. Johannesburg, WITS 2050, South Afi'ica
Abstract
The angular dependence of surface acoustic wave (SAW) velocities at high temperatures has been investigated by means of Brillouin scattering. The measurements were performed at temperatures up to 800C on the (0 0 1) surface of a silicon single crystal. The bulk elastic constants of silicon have been extracted from the data by performing a least-squares fit of calculated surface wave velocities to the measured angular dependence. There is good agreement between these results and existing ultrasonic measurements of the silicon elastic constants in this temperature range. This demonstrates the value of the technique for measurement of the high-temperature elastic properties of opaque materials.
1. Introduction
Brillouin scattering has proved to be a powerful means of studying the high-temperature elastic properties of transparent materials [-1]. However, in the case of opaque materials, the low signal to noise ratio has hampered the wider application of the technique. In addition, the increase in blackbody radiation with temperature tends to swamp the signal. These problems have become more tractable with the introduction of the tandem Fabry-Perot interferometer [2]. This instrument offers exceptional contrast of better than 1012 and the elimination of overlapping interference orders. Thus, one can probe the elastic properties of opaque samples through the scattering that occurs at or near the surface, while simultaneously reducing the blackbody background. Brillouin scattering is now widely used to study surface excitations in a large variety of opaque materials under ambient conditions. We have exploited these technical developments by extending surface Brillouin scattering to studies of an
* Corresponding author.
opaque material at high temperatures for the first time [3]. The measurements reveal the angular dependence of SAW velocities on the (00 1) surface of an n-type silicon single crystal at temperatures up to 800 C. Both the SAW and pseudo-SAW velocities can be calculated numerically for any direction in a crystal plane, using the computational scheme of Farnell [4]. These calculations require the density and the elastic constants of the material. The only available sources covering all of the temperature range studied here are the ultrasonically measured elastic constants for p-type silicon of Ezz-el-Arab et al. [53 and the thermal expansion derived from the powder X-ray measurements of Dutta [63. Although our sample was n-type material, studies have shown [-7] that the free current carriers have little effect on the silicon elastic constants at the impurity concentrations applicable here ( ~< 1017 cm-3). Thus, it appears reasonable to use these sources for purposes of comparison and evaluation. To this end, we have varied the elastic constants used in calculating the velocities, so as to obtain a best fit to the data. Having extracted the silicon elastic constants from the Brillouin scattering data, these can be compared to the ultrasonic results.
0921-4526/96/$15.00 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 { 9 5 ) 0 0 8 6 4 - 0
718
P.R. Stoddart et al. / Physica B 219&220 (1996) 717-719 5100
-"" 5000
- . . . . Pseudo-SW [ -Rayleigh SW [
"\_1 ...... .
4900
[ooq
4800
4700
J
4600 -45
The scattered light passing through the interferometer was detected by a cooled photomultiplier tube with less than one dark count per second. In order to avoid overloading the photomultiplier, the intense elastic peak was attenuated by means of an acousto-optic modulator. The laser beam was focussed onto the sample by a lens of focal length 120ram and aperture f/5.3. The angle of incidence of the beam on the sample was measured to be 0 = 70.9 ° using a vernier protractor.
3. Results and discussion -30
-15
0
15
30
45
Angle ~ from [100] (degrees)
Fig. 1. Brillouin scattering measurements of surface wave velocity on (001) n-type silicon at room temperature. The curves show the calculated SAW and pseudo-SAW velocities based on ultrasonic measurements of the elastic constants of silicon. The geometry of the backscattering arrangement is illustrated in the inset.
2. E x p e r i m e n t a l m e t h o d
A backscattering geometry was used for measuring the laser light inelastically scattered by SAWs through the ripple mechanism [8]. Spectra were excited using the 514.5 nm line of an argon-ion laser operated in a single mode. The scattered light was analyzed by means of a Sandercock (3 + 3)-pass tandem Fabry-Perot interferometer, similar to that described in Ref. [2]. Referring to the inset in Fig. 1, wavevector conservation gives KII = 2k sin 0 for light of wave vector k backscattered from a surface mode with wavevector Kjl. The angle of incidence 0 is approximately equal to the scattering angle for small collection apertures. The phase velocity V of the surface wave is then related to the spectral frequency shift fB by
The room-temperature measurements of the angular dependence of surface wave velocities on (0 0 1) n-type silicon are shown in Fig. 1. The symmetry of the data allows the [1 00] direction in the sample to be accurately assigned with respect to the laser beam. The SAW and pseudo-SAW velocities were calculated from elastic constant and density data, as outlined above. The discrepancy between the calculations and the measurements can be accounted for by the combined uncertainty in the Brillouin scattering data and the ultrasonically measured elastic constants. Data for the angular dependence of the surface wave velocities were gathered at various temperatures up to 800°C. The results are presented in Fig. 2. It was possible to obtain an excellent fit of the calculated velocities to the measurements by performing a least-squares minimization over a grid of elastic constants. The ultrasonically measured elastic constants provided a starting point for the fitting procedure.
5 . 1 [8~0. 0~ °.C. -.r.-.' .:. . . . ~1 .....
o--
4.8
600 oc
30 oc
--* ~ --~""...... ~ . . . . . 2 j - .-'~''--
""\ ~(~ ,, \
V = 2~f~/Kll.
The angular dependence of V in the crystal plane is measured by rotating the sample around its surface normal, while keeping 0 fixed. A specially designed optical furnace with a rotatable sample holder was assembled for this purpose. The sample temperature was monitored by means of a Pt-Pt(13%Rh) thermocouple positioned close to the sample. Thermal stability of better than I°C could be maintained for long periods by means of a temperature controller.
\
4.7 0
5
10
15
20
25
30
35
40
45
A n g l e ~ from [100] (degrees)
Fig. 2. The temperature dependence of the surface wave velocities in (0 01) silicon.The calculated curves have been fitted to the data by performing a least-squares minimization in terms of the three independent elastic constants.
P.R. Stoddart et al. / Physica B 219&220 (1996) 717 719
and 800~C results, which can be seen in both Figs. 2 and 3. This is possibly the result of a surface coating that developed on the sample before and during the 8 0 0 C measurement series.
170
165 160
4. Conclusion
155
80
150
78 76
~,~
74
6¢ 64
72
62 60 58 56
719
0
200
400
600
800
There is good agreement between the present surface wave velocities measured by Brillouin scattering and theoretical predictions based on thermal expansion and ultrasonic measurements of bulk elastic constants. The elastic constants that give the best fit to the measured velocities generally agree well with the elastic constants found by ultrasonic methods. This agreement is particularly significant with respect to the changes in the elastic constants with temperature. The technique is apparently very sensitive to the surface condition, as illustrated by the unexpectedly large change between 600 and 800°C. These first Brillouin scattering measurements of SAW velocity angular dependence at high temperatures demonstrate the potential value of the technique for future studies of the elastic properties of opaque materials under extreme conditions. The method described here would be of particular relevance for small samples, reactive materials, and thin films, which may be difficult to deal with using conventional high-temperature techniques.
Temperature (°C) Fig. 3. The three independent elastic constants of single crystal silicon have been measured previously at high temperatures by ultrasonic methods. These results are compared here with the elastic constants found by least-squares minimization of the surface wave velocities. Note that the uncertainty in the ultrasonic measurements is not shown.
The "best fit" elastic constants are compared to the ultrasonic results [5] in Fig. 3. The error bars shown reflect the limit of precision of the velocity data, beyond which variations in chi-squared are insignificant. One finds an even larger uncertainty in the calculated velocities due to the standard deviation between a n u m b e r of ultrasonic measurements at room temperature [9]. Ultrasonic measurements are of course very sensitive to small changes in the elastic constants, and it is interesting to note that the Brillouin scattering results generally reflect quite similar changes with temperature. The exception is the unexpectedly large drop between the 600
References [1] E.S. Zouboulis and M. Grimsditch, J. Appl. Phys. 70 (1991) 772. [2] R. Mock, B. Hillebrands and J.R. Sandercock, J. Phys. Instr. 20 (1987) 656. [3] P.R. Stoddart, J.D. Comins and A.G. Every, Phys. Rev. B 51 (1995) 17574. [4] G. Farnell, in: Physical Acoustics, eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1970) p. 109. [5] M. Ezz-el-Arab, B. Galperin, J. Brielles and B. Vodar, Solid State Commun. 26 (1968) 387. [6] B,N. Dutta, Phys. Status Solidi 2 (1962) 984. 1-7] A.E. Kadyshevich, V.M. Beilin, Yu.Kh. Vekilov, O.M. Krasil'nikov and V.N. Podd'yakov, Fiz. Tverd. Tela 9 (1967) 1861. [8] F. Nizzoli and J.R. Sandercock, in: Dynamical Properties of Solids, eds. G.K. Horton and A.A. Maradudin (NorthHolland, Amsterdam, 1990) p. 281. [9] A.G. Every and A.K. McCurdy, in: Numerical Data and Functional Relationships in Science and Technology, ed. D.F. Nelson, Landolt-B6rnstein, New Series Group lII, Vol. 29 (Springer, Berlin, 1992).
Physica B 219&220(1996) 717 719
High-temperature studies of surface acoustic wave velocities in silicon by Brillouin scattering P.R. Stoddart*, J.D. Comins, A.G.
Every
Department of Physics, Universi~ of the Witwatersrand. Johannesburg, WITS 2050, South Afi'ica
Abstract
The angular dependence of surface acoustic wave (SAW) velocities at high temperatures has been investigated by means of Brillouin scattering. The measurements were performed at temperatures up to 800C on the (0 0 1) surface of a silicon single crystal. The bulk elastic constants of silicon have been extracted from the data by performing a least-squares fit of calculated surface wave velocities to the measured angular dependence. There is good agreement between these results and existing ultrasonic measurements of the silicon elastic constants in this temperature range. This demonstrates the value of the technique for measurement of the high-temperature elastic properties of opaque materials.
1. Introduction
Brillouin scattering has proved to be a powerful means of studying the high-temperature elastic properties of transparent materials [-1]. However, in the case of opaque materials, the low signal to noise ratio has hampered the wider application of the technique. In addition, the increase in blackbody radiation with temperature tends to swamp the signal. These problems have become more tractable with the introduction of the tandem Fabry-Perot interferometer [2]. This instrument offers exceptional contrast of better than 1012 and the elimination of overlapping interference orders. Thus, one can probe the elastic properties of opaque samples through the scattering that occurs at or near the surface, while simultaneously reducing the blackbody background. Brillouin scattering is now widely used to study surface excitations in a large variety of opaque materials under ambient conditions. We have exploited these technical developments by extending surface Brillouin scattering to studies of an
* Corresponding author.
opaque material at high temperatures for the first time [3]. The measurements reveal the angular dependence of SAW velocities on the (00 1) surface of an n-type silicon single crystal at temperatures up to 800 C. Both the SAW and pseudo-SAW velocities can be calculated numerically for any direction in a crystal plane, using the computational scheme of Farnell [4]. These calculations require the density and the elastic constants of the material. The only available sources covering all of the temperature range studied here are the ultrasonically measured elastic constants for p-type silicon of Ezz-el-Arab et al. [53 and the thermal expansion derived from the powder X-ray measurements of Dutta [63. Although our sample was n-type material, studies have shown [-7] that the free current carriers have little effect on the silicon elastic constants at the impurity concentrations applicable here ( ~< 1017 cm-3). Thus, it appears reasonable to use these sources for purposes of comparison and evaluation. To this end, we have varied the elastic constants used in calculating the velocities, so as to obtain a best fit to the data. Having extracted the silicon elastic constants from the Brillouin scattering data, these can be compared to the ultrasonic results.
0921-4526/96/$15.00 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 { 9 5 ) 0 0 8 6 4 - 0
718
P.R. Stoddart et al. / Physica B 219&220 (1996) 717-719 5100
-"" 5000
- . . . . Pseudo-SW [ -Rayleigh SW [
"\_1 ...... .
4900
[ooq
4800
4700
J
4600 -45
The scattered light passing through the interferometer was detected by a cooled photomultiplier tube with less than one dark count per second. In order to avoid overloading the photomultiplier, the intense elastic peak was attenuated by means of an acousto-optic modulator. The laser beam was focussed onto the sample by a lens of focal length 120ram and aperture f/5.3. The angle of incidence of the beam on the sample was measured to be 0 = 70.9 ° using a vernier protractor.
3. Results and discussion -30
-15
0
15
30
45
Angle ~ from [100] (degrees)
Fig. 1. Brillouin scattering measurements of surface wave velocity on (001) n-type silicon at room temperature. The curves show the calculated SAW and pseudo-SAW velocities based on ultrasonic measurements of the elastic constants of silicon. The geometry of the backscattering arrangement is illustrated in the inset.
2. E x p e r i m e n t a l m e t h o d
A backscattering geometry was used for measuring the laser light inelastically scattered by SAWs through the ripple mechanism [8]. Spectra were excited using the 514.5 nm line of an argon-ion laser operated in a single mode. The scattered light was analyzed by means of a Sandercock (3 + 3)-pass tandem Fabry-Perot interferometer, similar to that described in Ref. [2]. Referring to the inset in Fig. 1, wavevector conservation gives KII = 2k sin 0 for light of wave vector k backscattered from a surface mode with wavevector Kjl. The angle of incidence 0 is approximately equal to the scattering angle for small collection apertures. The phase velocity V of the surface wave is then related to the spectral frequency shift fB by
The room-temperature measurements of the angular dependence of surface wave velocities on (0 0 1) n-type silicon are shown in Fig. 1. The symmetry of the data allows the [1 00] direction in the sample to be accurately assigned with respect to the laser beam. The SAW and pseudo-SAW velocities were calculated from elastic constant and density data, as outlined above. The discrepancy between the calculations and the measurements can be accounted for by the combined uncertainty in the Brillouin scattering data and the ultrasonically measured elastic constants. Data for the angular dependence of the surface wave velocities were gathered at various temperatures up to 800°C. The results are presented in Fig. 2. It was possible to obtain an excellent fit of the calculated velocities to the measurements by performing a least-squares minimization over a grid of elastic constants. The ultrasonically measured elastic constants provided a starting point for the fitting procedure.
5 . 1 [8~0. 0~ °.C. -.r.-.' .:. . . . ~1 .....
o--
4.8
600 oc
30 oc
--* ~ --~""...... ~ . . . . . 2 j - .-'~''--
""\ ~(~ ,, \
V = 2~f~/Kll.
The angular dependence of V in the crystal plane is measured by rotating the sample around its surface normal, while keeping 0 fixed. A specially designed optical furnace with a rotatable sample holder was assembled for this purpose. The sample temperature was monitored by means of a Pt-Pt(13%Rh) thermocouple positioned close to the sample. Thermal stability of better than I°C could be maintained for long periods by means of a temperature controller.
\
4.7 0
5
10
15
20
25
30
35
40
45
A n g l e ~ from [100] (degrees)
Fig. 2. The temperature dependence of the surface wave velocities in (0 01) silicon.The calculated curves have been fitted to the data by performing a least-squares minimization in terms of the three independent elastic constants.
P.R. Stoddart et al. / Physica B 219&220 (1996) 717 719
and 800~C results, which can be seen in both Figs. 2 and 3. This is possibly the result of a surface coating that developed on the sample before and during the 8 0 0 C measurement series.
170
165 160
4. Conclusion
155
80
150
78 76
~,~
74
6¢ 64
72
62 60 58 56
719
0
200
400
600
800
There is good agreement between the present surface wave velocities measured by Brillouin scattering and theoretical predictions based on thermal expansion and ultrasonic measurements of bulk elastic constants. The elastic constants that give the best fit to the measured velocities generally agree well with the elastic constants found by ultrasonic methods. This agreement is particularly significant with respect to the changes in the elastic constants with temperature. The technique is apparently very sensitive to the surface condition, as illustrated by the unexpectedly large change between 600 and 800°C. These first Brillouin scattering measurements of SAW velocity angular dependence at high temperatures demonstrate the potential value of the technique for future studies of the elastic properties of opaque materials under extreme conditions. The method described here would be of particular relevance for small samples, reactive materials, and thin films, which may be difficult to deal with using conventional high-temperature techniques.
Temperature (°C) Fig. 3. The three independent elastic constants of single crystal silicon have been measured previously at high temperatures by ultrasonic methods. These results are compared here with the elastic constants found by least-squares minimization of the surface wave velocities. Note that the uncertainty in the ultrasonic measurements is not shown.
The "best fit" elastic constants are compared to the ultrasonic results [5] in Fig. 3. The error bars shown reflect the limit of precision of the velocity data, beyond which variations in chi-squared are insignificant. One finds an even larger uncertainty in the calculated velocities due to the standard deviation between a n u m b e r of ultrasonic measurements at room temperature [9]. Ultrasonic measurements are of course very sensitive to small changes in the elastic constants, and it is interesting to note that the Brillouin scattering results generally reflect quite similar changes with temperature. The exception is the unexpectedly large drop between the 600
References [1] E.S. Zouboulis and M. Grimsditch, J. Appl. Phys. 70 (1991) 772. [2] R. Mock, B. Hillebrands and J.R. Sandercock, J. Phys. Instr. 20 (1987) 656. [3] P.R. Stoddart, J.D. Comins and A.G. Every, Phys. Rev. B 51 (1995) 17574. [4] G. Farnell, in: Physical Acoustics, eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1970) p. 109. [5] M. Ezz-el-Arab, B. Galperin, J. Brielles and B. Vodar, Solid State Commun. 26 (1968) 387. [6] B,N. Dutta, Phys. Status Solidi 2 (1962) 984. 1-7] A.E. Kadyshevich, V.M. Beilin, Yu.Kh. Vekilov, O.M. Krasil'nikov and V.N. Podd'yakov, Fiz. Tverd. Tela 9 (1967) 1861. [8] F. Nizzoli and J.R. Sandercock, in: Dynamical Properties of Solids, eds. G.K. Horton and A.A. Maradudin (NorthHolland, Amsterdam, 1990) p. 281. [9] A.G. Every and A.K. McCurdy, in: Numerical Data and Functional Relationships in Science and Technology, ed. D.F. Nelson, Landolt-B6rnstein, New Series Group lII, Vol. 29 (Springer, Berlin, 1992).