NONLINEAR SURFACE ACOUSTIC WAVES
IN CUBIC CRYSTALS
Ronald E. Kumon∗ and Mark F. Hamilton
Department of Mechanical Engineering
The University of Texas at Austin
Austin, Texas, U.S.A.
∗currently at theNational Institute of Standards and Technology
Boulder, Colorado, U.S.A.
1
OUTLINE
• Background
– Nonlinear surface waves
– Surface waves in crystals
– Nonlinear theory
• Results in (001) plane
– Simulations
– Experiment
• Results in (111) plane
– Simulations
– Experiment
– Approximate Method
• Summary
2
NONLINEAR SURFACE WAVES
Schematic Diagram:
kx
z
Solid
∼λ
z/λe
R
Rayleigh Wave
Retrograde
Prograde
Waveform Distortion:(in isotropic media or certain mirror planes of cubic crystals)
-2
-1
0
1
2
0 π 2π
V x
ωτ
X=0
X=1
X=2
-5
-4
-3
-2
-1
0
1
2
0 π 2π
V z
ωτ
X=0
X=1
X=2
3
SURFACE WAVES IN CRYSTALS
Schematic Diagram:
∼λ
k
Retrograde
x
z
Generalized Rayleigh Wave
Retrograde
Prograde
Solid
ez/λ
Typical Planes for Crystal Cuts:
θ
y
x
z
<100>
(001) plane
y
x
z
θ
(111) plane
<112>
4
NONLINEAR THEORY
Approach: Hamiltonian mechanics formalism(Hamilton, Il’inskii, Zabolotskaya, 1996)
Velocity waveforms in solid:
vj(x, z, t) =∞∑
n=−∞vn(x)unj(z)e
in(kx−ωt)
unj(z) =3∑s=1
β(s)j einkλ
(s)3 z
Coupled spectral evolution equations:
dvndx
+ αnvn = −n2ωc44
2ρc4
∑l+m=n
lm
|lm|Slmvlvm
vn → nth harmonic amplitude
Slm → nonlinearity matrix elements
αn → weak attenuation
Observation:
(001) plane ⇒ Slm real valued
(111) plane ⇒ Slm complex valued
5
NONLINEARITY MATRIX FOR SI (001)
Nonlinearity matrix elements:
Slm =−1
c44
3∑s1,s2,s3=1
12d′ijpqrsβ
(s1)i β
(s2)p [β
(s3)r ]∗λ(s1)
j λ(s2)q [λ
(s3)s ]∗
lλ(s1)3 +mλ
(s2)3 − (l +m)[λ
(s3)3 ]∗
λ(s)i → eigenvalues of linear problem
β(s)j → eigenvectors of linear problem
d′ijpqrs → 2nd and 3rd order elastic constants
Selected matrix elements for Si in (001) plane:
-0.02
0
0.02
0.04
0.06
0 5 10 15 20 25 30 35 40 45
S
S
S
S
13
12
11
lm
Angle θ from [degrees]⟨100⟩
Nonlinearity Matrix Elements for Si in (001) Plane⟨
⟨⟨
⟨
I II III
6
SIMULATIONS WITH SINUSOIDS FOR SI (001)
-1
-0.5
0
0.5
1
-π 0 π
0°
Vx
ωτ
-1
0
1
2
3
4
-π 0 π
0°
Vz
ωτ
-1
-0.5
0
0.5
1
0 π 2π
26°
Vx
ωτ
-4
-3
-2
-1
0
1
0 π 2π
26°
Vz
ωτ
-0.02
0
0.02
0.04
0.06
0 15 30 45
S11
° ° °°
⟨
X=1
X=2
X=0X=0X=1X=2
X=1X=0
X=2
X=0
X=1
X=2
Si
planein (001)
7
EXPERIMENT
Approach: Laser-excited photoelastic SAW generation(Lomonosov and Hess, 1996)
++ --
xy
z
crystal
splitphotodiodes
beamsprobe
absorbinglayer
cw
excitationpulsed laser
• Pulse detection:
Probe beam deflection proportional to vertical vel.
Temporal resolution: 1 ns
• Beam locations:
1st probe beam: 5 mm from source
2nd probe beam: 10–15 mm from 1st probe beam
• Resulting SAW pulses:
Duration: 20 to 40 ns
Peak strain: 0.005 to 0.010 (near fracture)
8
EXPERIMENT FOR SI (001)
Longitudinal velocity waveforms at close location:
-40-40-30-30-20-20-10-10
001010202030304040
00 2020 4040 6060 8080
vv xx [m/s
]
[m/s
]
t [ns]t [ns]
26°0° 1111 S > 0S < 0
⟨ ⟨
Longitudinal velocity waveforms at remote location:
-40-40-30-30-20-20-10-10
experimenttheory
001010202030304040
00 2020 4040 6060 8080
vv xx [m/s
]
[m/s
]
t [ns]t [ns]
0° 26°
9
NONLINEARITY MATRICES IN (001) PLANE
0
0.01RbCl
S
S
11
11
S
S
12
12
S
S
13
13
-0.02
-0.01
0
0.01KCl
-0.02
0
0.02
0.04
0.06
0.08NaCl
0
0.01CaF2
-0.01
0
0.01 SrF2
-0.06
-0.04
-0.02
0BaF2
0
0.05
0.1
0.15
0 10 20 30 40
Al
Angle from ⟨100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Ni
Angle from ⟨
⟨⟨
⟨
100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Cu
Angle from ⟨100⟩ [deg]
0.05
0.1
0.15C
-0.02
0
0.02
0.04
0.06
0.08Si
-0.02
0
0.02
0.04
0.06
0.08Ge
0
0.01RbCl η=0.312
S11S12S13
-0.02
-0.01
0
0.01KCl
-0.02
0
0.02
0.04
0.06
0.08NaCl
0
0.01CaF2 η=0.373
-0.01
0
0.01 SrF2 η=0.803
-0.06
-0.04
-0.02
0BaF2 η=1.02
0
0.05
0.1
0.15
0 10 20 30 40
Al η=1.22
Angle from ⟨100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Ni η=2.60
Angle from ⟨100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Cu η=3.20
Angle from ⟨100⟩ [deg]
0.05
0.1
0.15C η=1.26
-0.02
0
0.02
0.04
0.06
0.08Si
-0.02
0
0.02
0.04
0.06
0.08Ge
0
0.01RbCl η=0.312
S11S12S13
-0.02
-0.01
0
0.01KCl
-0.02
0
0.02
0.04
0.06
0.08NaCl
0
0.01CaF2 η=0.373
-0.01
0
0.01 SrF2 η=0.803
-0.06
-0.04
-0.02
0BaF2 η=1.02
0
0.05
0.1
0.15
0 10 20 30 40
Al η=1.22
Angle from ⟨100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Ni η=2.60
Angle from ⟨100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Cu η=3.20
Angle from ⟨100⟩ [deg]
0.05
0.1
0.15C η=1.26
-0.02
0
0.02
0.04
0.06
0.08Si
-0.02
0
0.02
0.04
0.06
0.08Ge
0
0.01RbCl
S
S
11
11
S
S
12
12
S
S
13
13
-0.02
-0.01
0
0.01KCl
-0.02
0
0.02
0.04
0.06
0.08NaCl
0
0.01CaF2
-0.01
0
0.01 SrF2
-0.06
-0.04
-0.02
0BaF2
0
0.05
0.1
0.15
0 10 20 30 40
Al
Angle from ⟨100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Ni
Angle from ⟨
⟨⟨
⟨
100⟩ [deg]
-0.05
0
0.05
0.1
0 10 20 30 40
Cu
Angle from ⟨100⟩ [deg]
0.05
0.1
0.15C
-0.02
0
0.02
0.04
0.06
0.08Si
-0.02
0
0.02
0.04
0.06
0.08Ge
Other crystals investigated:RbCl, BaF2, CaF2, SrF2, Al, Ni, Cu, C (diamond)
10
OTHER EFFECTS IN (001) PLANE
Si (001), θ ≈ 21◦: Slm ≈ 0
−1
−0.5
0
0.5
1
0 π 2π
°
Vx
ωτ
−1
−0.5
0
0.5
1
0 π 2πV
zωτ
0
0.5
1
0 2 4 6 8 10
20.8°
VV
nn
X
−0.02
0
0.02
0.04
0.06
0 15 30 45
Slm
° ° ° °
0
0.5
1
0 2 4 6 8 10
26°
X
Propagation Curves
n=2n=3
n=4n=5
n=1
⟨
n=2, 3, 4, 5
n=1
X=0, 1, 2X=0, 1, 2
Nonlinearity Matrix
Propagation in the (001) plane of Si
S11
⟨
S
⟨
12
S
⟨
13
⟨Angle from ⟩100
||
||
Waveforms at 20.8
KCl (001), θ ≈ 3◦: S11 > 0, S12 ≈ 0, S13 < 0
−1
−0.5
0
0.5
1
0 π 2π
°
Vx
ωτ
−2
−1
0
1
0 π 2π
Vz
ωτ
−0.02
−0.01
0
0.01
0 15 30 45
Slm
° ° ° ° 0
0.5
1
0 2 4 6 8 10
3.2°
Vn
X
X=0
X=2X=1
X=0
X=2
X=1⟨
S11
⟨
S12
⟨
S
⟨
13
n=1
2
5
3
4
Propagation in the (001) Plane of KCl
⟨Angle from ⟩100
Nonlinearity Matrix Propagation Curves
||
Waveforms at 3.2
KCl (001), θ ≈ 10◦: |S11| < |S12| < |S13|
-1
-0.5
0
0.5
1
-π 0 π
Vx
ωτ
-1
0
1
2
3
4
5
-π 0 π
Vz
ωτ
-0.02
-0.01
0
0.01
0 15 30 45
Slm
Angle from ⟨
⟨
100⟩
0
0.5
1
0 2 4 6 8 10
10
° ° ° °
°
°
Vn
X
5432
Propagation Curves
S11
⟨
S12
⟨
S
⟨
13
Nonlinearity Matrix
Propagation in the (001) Plane of KCl
Waveforms at 10
n=1
X=0 21
X=0
21
||
11
NONLINEARITY MATRICES IN (111) PLANE
0
0.05
0.1
0.15
0.2KCl
0
0.05
0.1 NaCl
−π
−π/2
0
π/2
π
⟩⟩
⟩
S11S12S13
−π
−π/2
0
π/2
π 0
0.01
0.02
0.03 SrF2η=0.803
0
0.02
0.04
0.06BaF2
η=1.02
−π
−π/2
0
π/2
π
−π
−π/2
0
π/2
π
0
0.05
0.1 Si
0
0.05
0.1 Ge
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
0
0.05
0.1
0.15Niη=2.60
0
0.1
0.2
0.3 Cuη=3.20
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
ψlm
[rad
] |S
lm|
⟨
ψlm
[rad
] |S
lm|
⟨
0
0.05
0.1
0.15
0.2KCl
0
0.05
0.1 NaCl
−π
−π/2
0
π/2
π
⟩⟩
⟩
S11S12S13
−π
−π/2
0
π/2
π 0
0.01
0.02
0.03 SrF2η=0.803
0
0.02
0.04
0.06BaF2
η=1.02
−π
−π/2
0
π/2
π
−π
−π/2
0
π/2
π
0
0.05
0.1 Si
0
0.05
0.1 Ge
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
0
0.05
0.1
0.15Niη=2.60
0
0.1
0.2
0.3 Cuη=3.20
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
−π
−π/2
0
π/2
π
0 10 20 30θ [deg]
ψlm
[rad
] |S
lm|
⟨
ψlm
[rad
] |S
lm|
⟨
12
SIMULATIONS WITH SINUSOIDS FOR SI (111)
Velocity waveform in direction 0◦ from 〈112〉: 0
0.02
0.04
0.06
0.08
0.1
0.12
0 10 20 30
|Slm
|
° ° ° °
° ° ° °
°
°
°
°
°
Angle from ⟨112⟩
-180
-90
0
90
180
0 10 20 30
arg(
Slm
)
Angle from ⟨112⟩
-1
0
1
2
3
0 π 2π
Vx
ωτ
-1
0
1
2
3
0 π 2π
Vz
ωτ
⟨
⟨
11S
⟨
S
⟨
12
S
⟨
13
11S
⟨ S
⟨
12S
⟨
13
X=0
X=2
X=1
X=1
X=2
X=0
0
0.02
0.04
0.06
0.08
0.1
0.12
0 10 20 30
|Slm
|
° ° ° °
° ° ° °
°
°
°
°
°
Angle from ⟨112⟩
-180
-90
0
90
180
0 10 20 30ar
g(S
lm)
Angle from ⟨112⟩
-1
0
1
2
3
0 π 2π
Vx
ωτ
-1
0
1
2
3
0 π 2π
Vz
ωτ
⟨
⟨
11S
⟨
S
⟨
12
S
⟨
13
11S
⟨ S
⟨
12S
⟨
13
X=0
X=2
X=1
X=1
X=2
X=0
Nonlinearity matrix elements:
0
0.02
0.04
0.06
0.08
0.1
0.12
0 10 20 30
|Slm
|
° ° ° °
° ° ° °
Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vx
ωτ
−1
0
1
2
3
0 π 2π
Vz
ωτ
⟨
⟨
11S
⟨
S
⟨
12
S
⟨
13
11S
⟨ S
⟨
12S
⟨
13
X=0
X=2
X=1
X=1
X=2
X=0
0
0.02
0.04
0.06
0.08
0.1
0.12
0 10 20 30
|Slm
|
° ° ° °
° ° ° °
Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vx
ωτ
−1
0
1
2
3
0 π 2π
Vz
ωτ
⟨
⟨
11S
⟨
S
⟨
12
S
⟨
13
11S
⟨ S
⟨
12S
⟨
13
X=0
X=2
X=1
X=1
X=2
X=0
13
EXPERIMENT FOR SI (111)
Propagation in direction 0◦ from 〈112〉Velocity waveforms at close location:
-60
-40
-20
0
20
40
0 10 20 30 40 50
v z [m
/s]
t [ns]
-40
-20
0
20
40
60
0 10 20 30 40 50
v x [m
/s]
t [ns]
-60
-40
-20
0
20
40
0 10 20 30 40 50
v z [m
/s]
t [ns]
-40
-20
0
20
40
60
0 10 20 30 40 50
v x [m
/s]
t [ns]
Velocity waveforms at remote location:
-60
-40
-20
0
20
40
0 10 20 30 40 50
v z [m
/s]
t [ns]
-40
-20
0
20
40
60
0 10 20 30 40 50
v x [m
/s]
t [ns]
theoryexperiment
theoryexperiment
-60
-40
-20
0
20
40
0 10 20 30 40 50
v z [m
/s]
t [ns]
-40
-20
0
20
40
60
0 10 20 30 40 50
v x [m
/s]
t [ns]
theoryexperiment
theoryexperiment
14
PHASE OF MATRIX ELEMENTS
Define the matrix element transformation
Sψlm = Slmei(sgn n)ψ , n = l +m .
The evolution equations with the transformed matrix are
dvψndx
= −n2ω0c44
2ρc4
∑l+m=n
lm
|lm|Sψlmv
ψl v
ψm
= −n2ω0c44
2ρc4
∑l+m=n
lm
|lm|Slmei(sgn n)ψvψl v
ψm
If the transformed spectral amplitudes are
vψn = vne−i(sgn n)ψ ,
then it follows that
dvndx
= −n2ω0c44
2ρc4
∑l+m=n
lm
|lm|Slmvlvm .
Result: Equations relate phase ψ of matrix elementsto spectral amplitudes (and thus waveforms).
15
PHASE TRANSFORMED WAVEFORMS
Consider artificial example of Rayleigh waves in steel
Slm → positive, real valued matrix elements
vn → corresponding spectral components
under the transformation vψn = vne−i(sgn n)ψ .
Resulting longitudinal velocity waveforms:
−2
0
2
4
6
0 π 2π
ψ=0
−2
0
2
4
6
0 π 2π
ψ=π/8
−2
0
2
4
6
0 π 2π
ψ=π/4
ωτ
−2
0
2
4
6
0 π 2π
ψ=3π/8
−2
0
2
4
6
0 π 2π
ψ=π/2
−2
0
2
4
6
0 π 2π
ψ=5π/8
ωτ
−2
0
2
4
6
0 π 2π
ψ=3π/4
−2
0
2
4
6
0 π 2π
ψ=7π/8
−2
0
2
4
6
0 π 2π
ψ=π
ωτ
Vxψ
Vxψ
Vxψ
16
APPROXIMATE AND FULL SOLUTIONS
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
° ° ° ° ° ° ° °Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vxψ
Steel with ψ = 0.59π
−2
−1
0
1
2
0 π 2π
Vxψ
Steel with ψ = −0.83π
−1
0
1
2
3
0 π 2π
Vx
Si (111) 0
Si (111) 0
°
°
ωτ
−2
−1
0
1
2
0 π 2π
Vx
KCl (111) 0
KCl (111) 0
°
°
ωτ
⟨ ⟨S
⟨
11S
⟨
12S
⟨
13
S
⟨
11S
⟨
12S
⟨13
1
X=0
2
1
X=0
2
1
2
X=0
1
2
X=0
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
° ° ° ° ° ° ° °Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vxψ
Steel with ψ = 0.39π
−1
0
1
2
3
0 π 2π
Vxψ
Steel with ψ = 0.47π
−1
0
1
2
3
0 π 2π
Vx
KCl (111) 20
KCl (111) 20
°
°
ωτ
−1
0
1
2
3
0 π 2π
Vx
KCl (111) 28
KCl (111) 28
°
°
ωτ
S
⟨
11S
⟨
12S
⟨13
S
⟨
11S
⟨
12S
⟨
13
X=0
1
2 2
1X=0
⟨⟨
X=0
1
2
X=0
1
2
Approximation:
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
° ° ° ° ° ° ° °Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30ar
g(S
lm)
Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vxψ
Steel with ψ = 0.59π
−2
−1
0
1
2
0 π 2π
Vxψ
Steel with ψ = −0.83π
−1
0
1
2
3
0 π 2π
Vx
Si (111) 0
Si (111) 0
°
°
ωτ
−2
−1
0
1
2
0 π 2π
Vx
KCl (111) 0
KCl (111) 0
°
°
ωτ
⟨ ⟨S
⟨
11S
⟨
12S
⟨
13
S
⟨
11S
⟨12
S
⟨
13
1
X=0
2
1
X=0
2
1
2
X=0
1
2
X=0
−π
−π/2
0
π/2
π
0 10 20 30ar
g(S
lm)
° ° ° ° ° ° ° °Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vxψ
Steel with ψ = 0.39π
−1
0
1
2
3
0 π 2πV
xψ
Steel with ψ = 0.47π
−1
0
1
2
3
0 π 2π
Vx
KCl (111) 20
KCl (111) 20
°
°
ωτ
−1
0
1
2
3
0 π 2π
Vx
KCl (111) 28
KCl (111) 28
°
°
ωτ
S
⟨
11S
⟨12
S
⟨
13
S
⟨
11S
⟨
12S
⟨
13
X=0
1
2 2
1X=0
⟨⟨
X=0
1
2
X=0
1
2
Full Solution:
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
° ° ° ° ° ° ° °Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30ar
g(S
lm)
Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vxψ
Steel with ψ = 0.59π
−2
−1
0
1
2
0 π 2π
Vxψ
Steel with ψ = −0.83π
−1
0
1
2
3
0 π 2π
Vx
Si (111) 0
Si (111) 0
°
°
ωτ
−2
−1
0
1
2
0 π 2π
Vx
KCl (111) 0
KCl (111) 0
°
°
ωτ
⟨ ⟨S
⟨
11S
⟨
12S
⟨
13
S⟨
11S
⟨
12S
⟨
13
1
X=0
2
1
X=0
2
1
2
X=0
1
2
X=0
−π
−π/2
0
π/2
π
0 10 20 30ar
g(S
lm)
° ° ° ° ° ° ° °Angle from ⟨112 ⟩
−π
−π/2
0
π/2
π
0 10 20 30
arg(
Slm
)
Angle from ⟨112 ⟩
−1
0
1
2
3
0 π 2π
Vxψ
Steel with ψ = 0.39π
−1
0
1
2
3
0 π 2πV
xψ
Steel with ψ = 0.47π
−1
0
1
2
3
0 π 2π
Vx
KCl (111) 20
KCl (111) 20
°
°
ωτ
−1
0
1
2
3
0 π 2π
Vx
KCl (111) 28
KCl (111) 28
°
°
ωτ
S⟨
11S
⟨
12S
⟨
13
S
⟨
11S
⟨
12S
⟨
13
X=0
1
2 2
1X=0
⟨⟨
X=0
1
2
X=0
1
2
17
SUMMARY
Results:
• Investigations of nonlinear SAWs in cubic crystals for:
– Different materials, cuts, directions
• Nonlinearity matrix is useful for characterizingwaveform distortion.
• In (001) plane:
– Sensitive dependence of nonlinearity on direction
– Compression and rarefaction shocks in same plane
– Shock suppression in certain directions
– Agreement between measured and calculated waveforms
• In (111) plane:
– Asymmetric waveform distortion
– Low frequency oscillations due to phase differences
– Agreement between measured and calculated waveforms
– Development of approximate method which works wellwhere phases between elements are similar
18