Dependence of surface acoustic wave nonlinearity on propagation direction in crystalline silicon

R. E. Kumon^*, M. F. Hamilton^*,
P. Hess^f , A. Lomonosov^f , and V. G. Mikhalevich^f

^*Dept. of Mech. Engineering, Univ. of Texas at Austin, Austin, Texas 78712-1063, USA
^f Institute of Physical Chemistry, University of Heidelberg, 69120 Heidelberg, Germany
^f General Physics Institute, Russian Academy of Sciences, 117942 Moscow, Russia

Abstract

Note: A Postscript version of each figure in the text may be downloaded by selecting the figure.

Theory and Experiment

Figure 1: Nonlinearity matrix elements S₁₁, S₁₂, S₁₃ for crystalline silicon in the (001) plane as a function of direction. Because of the symmetries of this cut, the matrix elements are symmetric about q = 45^° and periodic every 90^°. The matrix elements S_lm are normalized by the second-order elastic constant c₄₄.

Figure 2: Comparison of experiment (solid lines) and theory (dashed lines) for surface acoustic wave pulses propagating in the (001) plane of crystalline silicon in the directions q = 0^° (Region I) and q = 26^° (Region II) from á100ñ, from the location close to the excitation region (upper row) to the remote location 14.6 mm away (lower row). Here, v_z and v_x are the vertical (left columns) and longitudinal (right columns) velocity waveforms.

The nonlinearity matrix elements indicate the coupling strength between pairs of harmonics, e.g., S_lm describes how energy is transferred from the lth and mth harmonics to the (l+m)th harmonic. The elements S₁₁, S₁₂, S₁₃ are plotted in Figure 1. The nonlinearity divides into three distinct regions in terms of the angle q between the direction of propagation and á100ñ direction. In Region I (0^° £ q < 21^°), the nonlinearity is ``negative.'' This means that positive segments of the longitudinal particle velocity waveform steepen backward in space, and negative segments steepen forward (i.e., opposite what a sound wave does in air or water). In Region II (21<sup>°</sup> < q < 32<sup>°</sup>), the nonlinearity is ``positive,'' with waveform distortion the reverse of that in Region I. In Region III (32^° < q £ 45^°), the nonlinearity is again ``negative,'' although the surface wave behaves increasingly like a shear wave for q®45^°, and the nonlinearity is very weak. Finally, Figure 1 indicates that S₁₁, S₁₂, S₁₃ go to zero at q @ 21^° and q @ 32^°. Additional calculations show that all elements are close to zero at these angles, and hence propagation is expected to be nearly linear in both these directions even for finite amplitude waves.

To generate SAWs with a sufficiently large amplitude to exhibit nonlinear effects, a pulsed laser technique was employed. Using this technique, several authors have reported data with significant harmonic generation and shock formation in both isotropic [4,5,6] and crystalline [6,7] media. Additional measurements are reported here for two directions in the (001) plane of crystalline silicon. Excitation was accomplished using a Nd:YAG laser of wavelength 1064 nm, pulse duration 7 ns, and energy up to 50 mJ that was focused into a thin strip 6 mm by 50 mm. A strongly absorbing carbon layer in the form of an aqueous suspension was placed on the surface in the excitation region to intensify SAW excitation. The transient SAW waveforms were measured absolutely with a calibrated probe beam deflection setup using stabilized cw Nd:YAG laser probes of wavelength 532 nm and power 40 mW that were focused into spots 4 mm diameter and located 14.6 mm apart. The reflected signals were detected by two photodiodes, the output of which is proportional to the vertical velocity component v_z at the surface. The detection bandwidth was limited to 500 MHz. The resulting SAW pulses were typically 30-40 ns duration with peak strains up to 0.005. The effects of diffraction were determined to be insignificant.

Results

A comparison of experiment and theory verify that at least two distinct regions of nonlinearity exist in this surface cut. First, consider the q = 0^° case. In Region I, the coefficient of nonlinearity is negative. Hence in the v_x waveform the peaks travel slower than the SAW speed, and troughs travel faster. This behavior can be most easily seen in (b) and (d), where the trough becomes shallower as it advances and the pulse evolves into an inverted N wave. It is also seen in the pulse lengthening of the waveforms as the trough and peak move apart in time. In contrast, consider the q = 26^° case. In Region II, the coefficient of nonlinearity is positive. Hence in the v_x waveform the peaks travel faster than the SAW speed, and troughs travel slower. This behavior can be most easily seen in (f) and (h), where the nonlinearity causes the waveforms to develop into a sawtooth wave. This evolution is similar to that in a fluid, except for the cusping near the shock front, which is a well-known characteristic of nonlinear SAWs [2].

In conclusion, these results demonstrate that (1) the nonlinearity of SAWs in the (001) plane of crystalline silicon is strongly dependent upon the direction of propagation, and (2) at least two distinct regions of nonlinearity exist in this surface cut. These are also the first reported measurements of such angular dependence.

Acknowledgments

References

[1]

Hamilton, M. F., Il'inskii, Yu. A., and Zabolotskaya, E. A., J. Acoust. Soc. Am. 105, 639 (1999).

[2]

Zabolotskaya, E. A., J. Acoust. Soc. Am. 91, 2569 (1992).

[3]

McSkimin, H. J. and Andreatch, Jr., P., J. Appl. Phys. 35, 3312 (1964).

[4]

Lomonosov, A., et al., J. Acoust. Soc. Am. 105, 2093 (1999).

[5]

Kolomenskii, Al. A., et al., Phys. Rev. Lett. 79, 1325 (1997).

[6]

Lomonosov, A. and Hess, P., in Proc. 14th ISNA, ed. by Wei, R. J. (Nanjing University Press, Nanjing, 1996), pp. 106-111.

[7]

Kumon, R. E. et al., in Proc. 16th ICA & 135th ASA Meeting, ed. by Kuhl, P. K. and Crum, L. A. (ASA, Woodbury, New York, 1998), Vol. 3, pp. 1557-1558.