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:

sij
 
 
cijklekl+dijklmneklemn
 
cijkl 
 
®
 
 3 Second Order Elastic (SOE) constants 
 
dijklmn 
 
®
 
 6 Third Order Elastic (TOE) constants 
 
 
 
Data for Si elastic constants:  
Diamond Cubic Structure:       Crystal Cut in Experiment:
 
 
 


NONLINEAR THEORY
 
Approach:  Hamiltonian mechanics formalism 
(Hamilton, Il'inskii, Zabolotskaya, 1996)
 

Velocity waveforms in solid:

vj(x,z,t)
 
 
¥  
å  
n = -¥ 
vn(x) unj(z) ein(kx-wt) 
 
unj(z)
 
 
 
å  
s = 1 
q(s)j einkl3(s)z
 
 
 
Coupled spectral evolution equations:
dv
dx
+an vn n2
2rc4
å  
l+m = n 
lm 
|lm|
Slmvlvm
 
 
 
 
 
vn 
 
®
 
nth harmonic amplitude
 
Slm 
 
®
 
nonlinearity matrix elements
 
an 
 
®
 
weak attenuation
 
 

 


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:

Sn1 n2
å 
s1,s2,s3 = 1 
[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)]*
 
 
 
 
l3(s) 
 
®
 
eigenvalues of linear problem
 
qj(s) 
 
®
 
eigenvectors of linear problem
 
d'ijklmn 
 
®
 
SOE and TOE constants
 
 
 
Selected matrix elements for Si on (001) plane:

SHOCK FORMATION DISTANCE
 
Estimate of shock formation distance:

x  
 

|bx| ex k
,       bx 4S11 
rc2
 
 
 
where ex = vx0/c, and vx = vx0sinwt at x =  0.
 
Example:  w/2p   =   50 MHz   
  vx0     =   36 m/s    (ex »  0.007)
 Shock formation distance for Si on (001) plane:

SIMULATIONS WITH SINUSOIDS

CONCLUSION

Summary:

Results:

Supplements:


NONLINEARITY MATRIX & LINEAR THEORY
 

The nonlinearity matrix is given by

Sn1 n2
å 
s1,s2,s3 = 1 
[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)]*
 
 
 
where qi(s) = Csai(s) and
d'ijklmn = dijklmn+cijlndkm +cjnkldim+cjlmndik .
 
 
 
To compute this expression, the linear problem must first be solved.

Start with linearized wave equation 

r 2u
t2
¶sij 
xj
= cijkl 2 u
xj xl
 .
 
 
 
(1)
Next assume SAW solution of form 
ui  
å  
s = 1 
Cs ai(s) eik(ls ·r-wt)
 
 
 
(2)
where ls = {1,0,z}. Substitute Eq. (2) into Eq. (1) to yield 
rc2 ai  
c  
 
ijkl lj ll ak  .
 
 
 
(3)
Solve Eq. (3) subject to the stress-free surface bound. cond.
si3 |x3 = 0 = 0  .
 
 
 
(4)
Substituting Eq. (2) into Eq. (4) yields 
ik  
c  
 
i3kl  
å  
s = 1 
Cs ak(s)(c) ll(s) = 0  .
 
 
 
(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:

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.