NONLINEAR SURFACE ACOUSTIC WAVES IN CUBIC CRYSTALS … · NONLINEAR SURFACE ACOUSTIC WAVES IN CUBIC...

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


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


-0.05

0

0.05

0.1

0 10 20 30 40

Ni η=2.60


-0.05

0

0.05

0.1

0 10 20 30 40

Cu η=3.20


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


-0.05

0

0.05

0.1

0 10 20 30 40

Ni η=2.60


-0.05

0

0.05

0.1

0 10 20 30 40

Cu η=3.20


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


-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


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

)


-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

|

° ° ° °

° ° ° °

°

°

°

°

°


-180

-90

0

90

180

0 10 20 30ar

g(S

lm)


-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

)


−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

|

° ° ° °

° ° ° °


−π

−π/2

0

π/2

π

0 10 20 30

arg(

Slm

)


−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

)


−1

0

1

2

3

0 π 2π

Vxψ


−1

0

1

2

3

0 π 2π

Vxψ


−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)


−1

0

1

2

3

0 π 2π

Vxψ


−2

−1

0

1

2

0 π 2π

Vxψ


−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

)


−1

0

1

2

3

0 π 2π

Vxψ


−1

0

1

2

3

0 π 2πV

xψ


−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)


−1

0

1

2

3

0 π 2π

Vxψ


−2

−1

0

1

2

0 π 2π

Vxψ


−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

)


−1

0

1

2

3

0 π 2π

Vxψ


−1

0

1

2

3

0 π 2πV

xψ


−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

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NONLINEAR SURFACE ACOUSTIC WAVES IN CUBIC CRYSTALS … · NONLINEAR SURFACE ACOUSTIC WAVES IN CUBIC...

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Transcript of NONLINEAR SURFACE ACOUSTIC WAVES IN CUBIC CRYSTALS … · NONLINEAR SURFACE ACOUSTIC WAVES IN CUBIC...