Dependence of surface acoustic wave nonlinearity on propagation
direction in crystalline silicon
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 787121063, USA
^{f }Institute of Physical Chemistry,
University of Heidelberg,
69120 Heidelberg, Germany
^{f }General Physics Institute,
Russian Academy of Sciences,
117942 Moscow, Russia
Abstract
Nonlinear distortion of a surface acoustic wave in a crystal depends
not only on the second and thirdorder elastic constants, but also on
the plane and direction of propagation. A theory developed recently for
nonlinear surface waves in arbitrary anisotropic solids reveals a
strong dependence on propagation direction in the (001) plane of
crystalline silicon. Specifically, the longitudinal velocity waveform
develops compression shocks in some directions, rarefaction shocks
in others. Measurements of lasergenerated pulses support these
predictions.
Note:
A Postscript version of each figure in the text may
be downloaded by selecting the figure.
Theory and Experiment
The anisotropy of the elastic properties of crystalline materials
causes the propagation of surface acoustic waves
(SAWs) to vary as a function of direction.
The theory employed here was developed by Hamilton et
al. [1] for arbitrary crystalline symmetries and follows
the approach used by Zabolotskaya [2] for
nonlinear Rayleigh waves in isotropic media.
The resulting model equations for the jth vector component (j = x,y,z) of
particle velocity are given by
v_{j}(x,z,t) = 
¥ å
n = ¥

v_{n}(x)u_{nj}(z)e^{in(kxwt)} , 
dv_{n}
dx

+a_{n} v_{n} = 
n^{2}w
2rc^{4}


å
l+m = n


lm
lm

S_{lm}v_{l}v_{m} , 

where x is the direction of propagation, z is the coordinate normal
to the tractionfree surface of the solid, w is the fundamental
angular frequency, k is the corresponding wavenumber in the
expansion, r is the density, c is the SAW speed,
u_{nj}(z) are eigenfunctions of the linear problem, and
S_{lm} is a nonlinearity matrix that is known explicitly in terms of
the second and thirdorder elastic constants of the material.
The coupled equations for v_{n}(x) are solved numerically, with the
ad hoc absorption coefficients a_{n} = n^{2}a_{1} introduced for
numerical stability when shocks are formed.
One of the advantages of the theory is that it has no adjustable parameters,
and no curve fitting need be employed. For all calculations
below, the elastic constants for silicon were taken
from the measurements reported by McSkimin and Andreatch [3].
However, neither the theoretical nor experimental techniques described below
are limited to this material.
Figure 1: Nonlinearity matrix elements S_{11}, S_{12}, S_{13} 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 secondorder
elastic constant c_{44}.
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 S_{11}, S_{12}, S_{13} 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^{°} < q < 32^{°}), 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 S_{11}, S_{12}, S_{13} 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 3040 ns duration
with peak strains up to 0.005. The effects of diffraction were
determined to be insignificant.
Results
Figure 2 shows SAW pulses in the
directions q = 0^{°} [(a)(d)] and q = 26^{°} [(e)(h)]
with respect to the
á100ñ direction. The top row of plots shows measured
waveforms at the probe location close to the source, while the bottom row
shows measured (solid) and predicted (dashed) waveforms at the remote location
14.6 mm away. The left column of each set gives the directly measured
vertical velocity v_{z}, while the right column gives
the longitudinal velocity v_{x} computed from linear theory.
In both cases, the absorption coefficient at the peak frequency
was selected to be (100[`(x)])^{1}, where [`(x)] is
the estimated shock formation distance for that case.
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 wellknown 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
Yu. A. Il'inskii and E. A. Zabolotskaya are thanked for helpful
discussions of this work. The experiments were performed while one of
the authors (AL) was on leave at the University of Heidelberg with
financial support from VolkswagenStiftung, Deutche
Forschungsgemeinschaft, and the Russian Foundation for Basic Research.
The theoretical work was supported by the U. S. Office of Naval
Research.
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. 106111.
 [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. 15571558.
Ronald Kumon /
UTAustin /
Created 09 Jul 1999 / Updated 09 Jul 1999
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