DEPENDENCE OF SURFACE
WAVE NONLINEARITY
ON PROPAGATION DIRECTION
IN CRYSTALLINE SILICON
R. E. Kumon, M. F.
Hamilton,
Yu. A. Il'inskii,
and E. A. Zabolotskaya
Department of Mechanical
Engineering
The University of Texas
at Austin
Paper 3pPA1
136th Meeting of the
Acoustical Society of America
Norfolk, Virginia
12-16 October 1998
J. Acoust. Soc. Am.
104, 1815(A) (1998)
ANISOTROPY IN CRYSTALLINE
SILICON
Stress-strain relation for
cubic crystal:
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| 3
Second Order Elastic (SOE) constants |
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| 6
Third Order Elastic (TOE) constants |
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Data for Si elastic constants:
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McSkimin, H. J. and Andreatch, Jr.,
P., J. Appl. Phys. 35,
3312-3319 (1964).
| Diamond Cubic Structure: |
Crystal Cut in Experiment: |
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NONLINEAR THEORY
| Approach: |
Hamiltonian mechanics formalism |
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(Hamilton, Il'inskii, Zabolotskaya,
1996) |
Velocity waveforms in solid:
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¥
å
n = -¥ |
vn(x)
unj(z) ein(kx-wt) |
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3
å
s = 1 |
q(s)j
einkl3(s)z |
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Coupled spectral evolution equations:
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dvn
dx |
+an
vn = |
n2w
2rc4 |
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å
l+m = n |
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lm
|lm| |
Slmvlvm |
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| nonlinearity
matrix elements |
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COMPARISON WITH EXPERIMENT
[from Kumon et al.,
Seattle ICA/ASA, June 1998]
Experiment: Laser-excited
pulses in Si on (111) plane in <112>
Velocity waveform at x
= 5 mm from source:
Velocity waveform at x
= 21 mm from source:
NONLINEARITY MATRIX
Nonlinearity matrix elements:
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| Sn1 n2 = |
3
å
s1,s2,s3 = 1 |
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[1/2] d'ijklmn qi(s1)
qk(s2) [qm(s3)]*
lj(s1) ll(s2)
[ln(s3)]*
n1 l3(s1)
+ n2 l3(s2)
- (n1 + n2) [l3(s3)]* |
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| eigenvalues
of linear problem |
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| eigenvectors
of linear problem |
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Selected matrix elements for
Si on (001) plane:
SHOCK FORMATION DISTANCE
Estimate of shock formation
distance:
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x
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= |
1
|bx|
ex k |
,
bx = |
4S11
rc2 |
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where ex
= vx0/c, and
vx = vx0sinwt
at x = 0.
| Example: |
w/2p |
= |
50 MHz |
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vx0 |
= |
36 m/s |
(ex
» 0.007) |
Shock formation distance
for Si on (001) plane:
SIMULATIONS WITH
SINUSOIDS
CONCLUSION
Summary:
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Thorough theoretical study of
nonlinear properties of SAWs in (001) plane of crystalline silicon.
Results:
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Nonlinearity matrix properties
divide the waveform
distortion into three regions:
| Region |
Angular range |
Waveform Behavior |
| I |
0° <
q < 21° |
Steepens ``backward'' |
| II |
21° <
q < 32° |
Steepens ``forward'' |
| III |
32° <
q < 45° |
Steepens ``backward'' |
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Wave propagation for
q @
21° and
q @ 32°
is linear even for finite amplitude SAW.
Supplements:
NONLINEARITY MATRIX
& LINEAR THEORY
The nonlinearity matrix is
given by
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| Sn1 n2 = |
3
å
s1,s2,s3 = 1 |
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[1/2] d'ijklmn qi(s1)
qk(s2) [qm(s3)]*
lj(s1) ll(s2)
[ln(s3)]*
n1 l3(s1)
+ n2 l3(s2)
- (n1 + n2) [l3(s3)]* |
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where qi(s)
= Csai(s)
and
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| d'ijklmn =
dijklmn+cijlndkm
+cjnkldim+cjlmndik
. |
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To compute this expression,
the linear problem must first be solved.
Start with linearized wave
equation
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| r |
¶2ui
¶t2 |
= |
¶sij
¶xj |
= cijkl |
¶2
uk
¶xj
¶xl |
. |
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(1)
Next assume SAW solution of
form
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| ui
= |
3
å
s = 1 |
Cs ai(s)
eik(ls ·r-wt) |
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(2)
where ls
= {1,0,z}. Substitute
Eq. (2) into Eq. (1)
to yield
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| rc2
ai = |
~
c
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ijkl lj
ll ak
. |
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(3)
Solve Eq. (3)
subject to the stress-free surface bound. cond.
(4)
Substituting Eq. (2)
into Eq. (4) yields
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| ik |
~
c
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i3kl |
3
å
s = 1 |
Cs ak(s)(c)
ll(s) = 0 . |
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(5)
These equations can be solved
numerically for li(s),
ai(s),
and Cs.
RESULTS FROM LINEAR
THEORY
Description of acoustic modes
for Si with SAW on (001) plane:
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Three bulk modes, one surface
mode
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Surface mode ®
exceptional bulk shear wave
as q®
45°
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Change in wave speeds is relatively
small (5% to 20%)
for 0°
< q < 45°
Relative velocity vs.
angle for all modes: 
WAVEFORMS: REGION
I
WAVEFORMS: REGION
II
WAVEFORMS: REGION
III
File translated from TEX
by TTH, version 0.99.