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Hybrid Surface-Bulk Mode in Periodic Gratings

Natalya Naumenko1, Benjamin Abbott2

1Moscow Steel and Alloys Institute, 117936, Leninski prosp.,4. Moscow, Russia

2 SAWTEK Inc., 1818 South Hwy 441, Apopka, FL 32703

Abstract – We report on a SAW/BAW hybrid mode (SBH),

which arises from surface skimming bulk wave due to the

trapping of its energy by a counter-propagating surface wave

(SAW) or leaky surface wave (LSAW), in periodic grating

structures. The SBH is the result of an interaction between

surface and bulk waves and exists within limited frequency

region, outside which it is decomposed into counter-

propagating SAW and BAW. Different examples of SBH

mode illustrate interactions of BAW with pure SAW, leaky

and high-velocity leaky waves. The behavior of the SBH mode

was numerically investigated using harmonic admittance

function. Calculation of velocity dispersion revealed the

existence of additional branch of dispersion curve, which

refers to SBH mode. A stopband occurs due to SAW/BAW

interaction and manifests itself by a resonance of harmonic

admittance at its lower edge, which can be considered as the

cut-off frequency of bulk wave radiation.

I. INTRODUCTION

One of the most important requirements of modern

SAW filters, especially for RF applications, is low

insertion loss, which can be provided by using resonator

structures on piezoelectric substrates with strong

piezoelectric coupling. High propagation velocity of

acoustic wave is also desirable for high frequency devices.

Therefore leaky (or pseudo-surface) acoustic waves

(LSAW), existing in rotated Y-cuts of lithium niobate (LN)

and lithium tantalate (LT)

electromechanical coupling coefficient, are the good

choices of substrate material in low loss filters.

Rotated Y-cuts of LT, X propagation, with rotation

angles from 36° to 42° and rotated Y-cuts of LN with

rotation angles from 41° to 64° are most widely used.

Though any LSAW exhibits nonzero propagation loss

caused by bulk wave radiation, with proper choice of

rotation angle and metal electrode thickness in the grating,

this loss can be minimized. Propagation velocity of LSAW

in the grating also depends on orientation and electrode

thickness and its maximum value tends to that of the fast

shear surface skimming bulk wave with reducing metal

thickness and rotation angle approaching 37° for LT and

41° for LN. In mentioned orientations, fast shear bulk

wave becomes SH-polarized and satisfies stress-free

and having high

mechanical boundary conditions without a metal grating

(In SAW theory such a bulk wave is called exceptional)

[1]. As a result, the bulk wave is strongly generated by the

grating and interacts with the leaky wave, thus giving

parasitic response and degradation of SAW filter

performance.

The problem of interaction between LSAW and fast

shear bulk wave in 36YX cut of LT was recently

investigated by various researchers [2-4] using different

numerical techniques. In particular, Fusero et al [2]

examined harmonic admittance as function of frequency

and wave number, and arrived to the conclusion that

additional “PSAW-like” mode exists in this orientation

when electrode thickness is sufficiently large.

A detailed analysis of the new mode was performed to

verify its existence. As a result, the mechanism of its

forming in the grating was found. The results of a thorough

numerical investigation of the new mode are presented,

and a physical interpretation is suggested. We found that

the SAW/BAW Hybrid mode (SBH) arises from the surface

skimming bulk acoustic wave (BAW) due to the trapping

of its energy by the counter-propagating surface wave. The

mechanism of its forming from the bulk wave is

fundamentally different from that of other known surface

and leaky waves with quasi-bulk structure. SBH mode

exists only in a grating and within limited frequency range.

Outside this range, SBH mode is decomposed into pure

bulk wave and counter-propagating SAW.

Different examples of SBH, which illustrate interaction

of BAW with pure SAW, STW, and leaky waves, (low-

and high-velocity), are described. The lower edge of the

stopband, which occurs due to SAW/BAW interaction, was

found to manifest itself as a resonance of the harmonic

admittance which is indicative of spurious resonances in

SAW filter frequency response.

II. HARMONIC ADMITTANCE AS METHOD OF ANALYSIS

In recent years, rapid progress has been made in the

development of efficient techniques for rigorous analysis

of SAW and LSAW propagation characteristics in grating

structures. In particular, harmonic admittance is very

useful for both the simulation of resonator SAW filters and

Proc. 2001 IEEE Ultrasonics Symposium, Atlanta, USA, Oct.7-10, 2001, pp. 243-248

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for extraction of wave characteristics in infinitely periodic

electrode structures.

The concept of harmonic admittance was first

introduced by Blotekjaer et al [5,6] in 1973. However,

Blotekjaer used an approximation for the piezoelectric

substrate's effective permittivity. Milsom et al [7]

introduced the rigorous evaluation of the effective

permittivity for piezoelectric substrates, which Zhang et al

[8] applied to the evaluation of the harmonic admittance

for periodic massless metal

contributions of Ventura et al [9] the concept of harmonic

admittance is now applicable to massive metal gratings.

The harmonic admittance represents the admittance of

an individual electrode in a periodic array of electrodes to

which a harmonic excitation is applied. Consider the

periodic array in Fig 1a. The applied voltages, Vn, are

)2exp(

nsVV

Tn

These harmonic excitations vary spatially with the

normalized wavenumber, s=p/V. V is the period of the

harmonic excitation. In Fig.1a, each electrode's current is

proportional to the applied voltage and the harmonic

admittance function.

),(

sfYVI

nn

(2)

The harmonic admittance function, Y(f,s), is a powerful

tool for evaluating the spectral and frequency dependences

of SAW interdigital transducers. It may be used to

characterize sequences of electrodes with harmonic

excitations as well as finite length excitations. In

transducers with finite length excitations, the complete

spectrum of spatial harmonics must be considered, 0 s<1.

To evaluate the admittance of a finite length excitation

Fourier analysis is used.

gratings. With the

)2 exp(

fj

(1)

Consider the electrode sequence shown in Fig 1b, which

is assumed to have a finite excitation length. The spectral

representation of the transducer's voltage is given by the

Fourier transform of the electrode voltages.

The individual electrode voltages are dependent upon the

applied transducer voltage, VT, and each electrodes

polarity, pn.

pVv

(4)

)2exp(),(

nsjvsfV

n

n

(3)

nTn

The spectral representation of the electrode currents is

given by the product of the spectral voltage and the

harmonic admittance.

,(),(

sfVsfI

The total transducer current, IT, is given by the

summation of the individual electrode currents.

By applying Fourier analysis, the total transducer

current may be represented by the integral of the spectral

representation of the transducer's current.

00

Given the voltage applied to the transducer is VT, the

transducer's admittance, YT(f), is

0

V

T

Therefore, the admittance of a transducer with a finite

length excitation is dependent upon the complete spectrum

of the harmonic admittance, 0s<1. Thus all resonances

present in the harmonic admittance function manifest

themselves in the finite transducer's admittance. The

resonances may have a negative impact on the performance

of SAW filters. Therefore, it is important to give

consideration to these undesired resonances. Particularly

those in close proximity to the resonances of the desired

surface modes. In the following section the SBH

resonances occurring between surface and bulk modes are

investigated. These hybrid resonances can be particularly

harmful to the performance of SAW filters due to their

close proximity to the resonances of the pure surface

modes.

Harmonic admittance can be regarded as an example of

a wavenumber-dependent function, which relates the

electric potential to the normal component of electric

)

,()

sfY

(5)

n

n

nT

piI

(6)

11

),(),(),(

dssfYsfVdssfIIT

(7)

1

),(),(

1

)(

dssfYsfVfY

T

(8)

Fig.1. Periodic array of electrodes with a) harmonic

excitation, b) finite excitation

a

b

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displacement on the surface and contains information

about all acoustic modes in a given piezoelectric substrate.

Other examples of such a function are electric surface

impedance [10] and effective dielectric permittivity [7],

which were first introduced for semi-infinite piezoelectric

medium and appeared to be very efficient approach to the

analysis of acoustic waves in other structures, including

periodic grating. The presence of velocity dispersion

makes analysis of wave propagation more complicated.

However, the basic relations obtained for semi-infinite

medium were successfully applied to the analysis of

grating structures [5,6,10].

In particular, the concept of effective piezoelectric

coupling coefficient, first introduced for non-dispersive

piezoelectric medium [10], can be generalized for

dispersive wave propagation problems. This coefficient

can be defined in terms of the velocity difference obtained

by changing electrical condition from short-circuited to

open-circuited, and characterizes power transfer from

electromagnetic source to acoustic waves.

For any frequency f, the following approximation is

valid, unless harmonic admittance is perturbed by another

acoustic mode,

s

jC

sfY

),(

ss

s

0

, (9)

where s0, s and C are functions of frequency. With s=s

being normalized wavelength of acoustic mode in short-

circuited grating, the effect of changing electric boundary

condition upon the solution of dispersive relation can be

described by

fs

1

1

2

2

s

)(

)]()([

2)(

ss

s

Yjfsfs

fK

o

(10)

assuming C=1 in a narrow frequency interval. With (9)

generalized for complex-valued s, K2(f) also becomes

complex, and s must be determined as an eigenmode of

dispersion relation for short-circuited grating.

Though derived in different way, K2(f) defined by (10)

is analogous to the effective electromechanical coupling

factor of the grating mode introduced by Hashimoto et al

[11] and shows frequency-dependent efficiency of

transformation of applied electric voltage into acoustic

power radiated in forward and backward directions. As

will be apparent from the following discussion, analysis of

this function helps to understand the effect of SBH mode

on SAW device performance and to see how the energy is

transferred between different modes in the grating.

Numerical technique, which combines matrix formalism

for finding discrete Green functions with BEM analysis of

electrode region, was used to calculate harmonic

admittance and velocity dispersion in the grating.

III. RESULTS AND DISCUSSION

To understand the nature of “PSAW-like” mode found

by Fusero et al [2], we analyzed the same example, 36YX

cut of LT, with a short-circuited infinite periodic two-

electrode-per wavelength grating structure, with Al

electrodes of thickness h/=0.04, and metallization ratio

w/p=0.5 (where is LSAW wavelength). First, the

harmonic admittance Y(f,s) was calculated as a function of

frequency, at fixed value of normalized wavelength s.

Under the Bragg reflection condition, s=0.5, only one

resonance of Y(f) occurs at the lower edge of the stopband.

Due to the symmetry of analyzed orientation and electrode

structure, there is no resonance at the opposite edge.

However, as was previously reported [2-4], with small

detuning from Bragg condition, Y(f) exhibits second

resonance at frequencies approximately corresponding to

the upper edge of the stopband .

Then, following the method suggested in [2], we

analyzed harmonic admittance as function of normalized

wavelength, at different frequencies around the upper edge

of LSAW stop band. The results are shown in Fig.2. We

used normalized frequency f’=fp/Vbulk, where p is the

period of the grating and Vbulk=4185,94 m/s is the fast shear

bulk wave velocity (calculations were made with material

constants of LT reported in [12]). Hence, s=f’ corresponds

to V=Vbulk. Two resonances of Y(s) appear in a narrow

frequency interval between f’=0.4970 and f’=0.4971 and

move with frequency. Both resonances are characterized

by nonzero propagation loss, Re(Y)0. The resonances

merge at f’ about 0.49711. The resulting maximum of

Re(Y) reduces rapidly with further increasing frequency

and vanishes at f’ about 0.49715.

Assuming that Re(Y) is negligible, each resonance in

Fig.2 represents a mode in the grating with velocity

V=f’Vbulk/s. Therefore, the velocity dispersion function can

be derived for each mode from the harmonic admittance

data. Both modes are confined in the velocity interval

(Vmax-LSAW, Vbulk), where Vmax-LSAW is the maximum LSAW

velocity in the short-circuited grating. Furthermore,

according to the behavior of Im(Y), these modes are

counter-propagating.

Direct calculation of the velocity dispersion function

was performed. The frequency interval around f’=0.497

was carefully examined and revealed that there exists an

additional branch of dispersion curve. In Fig.3, the

calculated velocity dispersion is shown for forward and

backward LSAW modes, with normalized wave numbers

s+ and s-=1-s+, respectively. An additional branch arises

from Vbulk at f’=0.4970 and merges with backward LSAW

at f’=0.49711. These results are consistent with the

behavior of harmonic admittance in Fig.2 and reveal the

nature of the new branch of the dispersion curve as a result

of interaction between the counter-propagating LSAW and

BAW. Hence the new mode can be referred to as

SAW/BAW Hybrid mode (SBH). It exists within limited

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frequency interval, outside which it decomposes into pure

BAW and counter propagating LSAW. The stopband

occurs at the frequencies higher than f’=0.4971 due

LSAW/BAW interaction, and Im(V) increases in the

stopband indicating Bragg reflection. Thus the lower edge

of SBH stopband can be considered as the cut-off

frequency of bulk wave radiation. In the absence of SBH

mode, the cut-off frequency is higher and can be

determined as an intersection between the velocity of

backward LSAW and Vbulk .

Fig.4 illustrates how SBH mode changes with electrode

thickness. With thicker electrodes, SBH mode exists in

wider frequency interval. For h/=0.06 the stopband of the

SBH mode merges with the stopband of LSAW. As a

result, Im(V) does not vanish at the upper edge of LSAW

stopband, which makes the rigorous evaluation of this

important characteristic more difficult.

In addition to Re(V) and Im(V), the effective coupling

coefficient K2 of the SBH mode was calculated using (10).

For each electrode thickness analyzed, at the lower edge of

the stopband, which corresponds to SAW/BAW Bragg

reflection, Re(K2) reaches maximum value. Consequently,

SBH mode is expected to manifest itself by spurious

resonance at this frequency. When h/=0.04, Re(K2)

crosses zero value at f’ about 0.49712 and becomes

negative with further increasing frequency. This can be

interpreted as vanishing radiation in forward propagation

direction and further growth of radiation in backward

direction. However, these numerical results should be

treated with caution because approximation (9) used for

definition of coupling coefficient (10) can be invalid here

due to the interaction between LSAW and SBH mode.

Apparently, analytical consideration is required to

understand the mechanisms of energy transfer between

different modes in the grating.

With increasing electrode thickness, the spurious

resonance caused by SBH mode is expected to become

weaker. It is also expected to decrease with increasing

rotation angle of LT orientation. Fig.5 shows calculated

velocity dispersion of LSAW and SBH mode in two

orientations, 36YX and 42YX. With thick electrodes,

h/=0.1, the stopbands of LSAW and SBH mode overlap.

With increasing rotation angle, LSAW/BAW interaction

becomes weaker. If rotation angle exceeds 44, SBH mode

does not occur.

Other examples of the SBH were also found [13]. In

128YX cut of LN, the SBH mode is a result of interaction

between a pure SAW and slow shear bulk wave. Since the

latter becomes exceptional in 131YX cut [1], it is strongly

radiated in 128YX cut. In this case, SBH is pure surface

mode, having zero propagation loss. In 36Y, X+90 cut of

Magnitude

0.4965 0.49750.4985 0.4995

Magnitude

Fig.2 Harmonic admittance calculated for LSAW in 36YX

LT, with short-circuited Al grating, h/=0.04, w/p=0.5, as

function of normalized wavelength s=p/ for. Normalized

frequency f’=fp/Vbulk changes from 0.49700 to 0.49711.

3960

4000

4040

4080

4120

4160

4200

0.4750.485 0.495

Velocity (m/s)

0

50

100

150

200

4160

4170

4180

4190

0.49700.4971 0.4972

Velocity (m/s)

0

10

20

30

40

50

Fig.3. Velocity dispersion of LSAW (a) in 36YX cut of LT,

h/=0.04, with enlarged fragment (b) showing SBH mode which

appears in close proximity to the upper edge of the stop band.

Dashed lines refer to backward propagating modes.

f’=0.49710

f’=0.49710

f’=0.49705

f’=0.49705

f’=0.49700

f’=0.49700

f’=0.49711

f’=0.49711

p/

Re(Y)

Im(Y)

Vbulk

Backward propagating

LSAW

SBH mode

Re(V)

Im(V)

Im(V)

Vbulk

Fig.3b

Normalized frequency fp/Vbulk

(a)

(b)

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quart, SBH mode is a result of interaction between STW

and fast shear bulk wave (Fig.6). In this example, the lower

edge of SBH stopband nearly coincides with the upper

edge of STW stopband and the effective coupling of SBH

mode changes sign at the same frequency, dividing it into

the forward and backward propagating modes and

indicating that there is a transfer of energy between two

modes, STW and SBH, with frequency. More detailed

analysis of this and other examples, including analytical

approach, could give a further insight of the nature of SBH

mode and its role in the transformation of the wave

structure from SAW to LSAW (or from LSAW to high-

velocity LSAW) when the propagation velocity crosses

that of the bulk wave.

An example of SBH having HVPSAW nature, which

arises from quasi-longitudinal bulk wave, has been found

in LBO cut with Euler angles (45,46,90) (Fig.7).

All types of SBH modes considered above have some

typical common features. The fundamental feature, which

distinguishes SBH mode from other known quasi-bulk

waves, is that it exists only in a grating and in limited

frequency interval of SAW/BAW interaction. At low

frequencies, the velocity of SBH mode approaches to that

of the bulk wave. The bulk wave, which gives rise to SBH

mode, must be close to exceptional one. Therefore,

analysis of exceptional wave lines [1] can help predict the

5030

Velocity (m/s)

5050

5070

5090

Velocity (m/s)

-20

0

20

40

0.49730.4975 0.49770.4979

Effective

coupling

STW

SBH mode

Fig.6. Velocity dispersion and Re(K2) in 36˚Y,X+90˚ cut of

quartz, including STW and SBH mode, h/Λ=0.04, w/p=0.9.

4160

Velocity (m/s)

4170

4180

4190

Velocity (m/s)

0

20

40

60

0.49700.4972 0.49740.4976

Effective

coupling

Fig.4. Velocity dispersion and effective coupling coefficient

near the upper edge of LSAW stopband in 36YX cut of LT, with

Al electrodes of different thickness. SBH velocity and backward

LSAW velocity are shown with solid and dashed lines,

respectively.

4150

4160

4170

4180

4190

0.4970.4980.499

Velocity (m/s)

36-YX

42-YX

Fig. 5. Velocity dispersion of LSAW propagating in two cuts of

LT with Al grating, h/=0.1, w/p=0.5 (a).

Forward SBH

Backward SBH

Backward STW

Forward STW

SBH mode

STW

Im(V)

Re(V)

Re(K2)

Normalized frequency fp/Vbulk

Normalized frequency fp/Vbulk

Vbulk

LSAW

SBH mode

h/=0.06

h/=0.04

h/=0.04

h/=0.05

h/=0.06

h/=0.05

h/=0.04

h/=0.05 h/=0.06

Im(V)

Re(V)

Normalized frequency fp/Vbulk

Re(K2)

Im(K2)

LSAW

SBH mode

K2

Vbulk

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existence of SBH mode and, if necessary, change crystal

orientation to reduce spurious resonance caused by this

parasitic mode.

IV.CONCLUSIONS

Rigorous numerical analysis of “PSAW-like” mode,

recently found to exist in 36YX cut of LT with

sufficiently thick periodic structure, was performed and

revealed that this SAW/BAW hybrid mode is a result of

trapping of surface skimming BAW by counter-

propagating LSAW. Different examples of the SBH were

found and investigated. The typical features of the SBH

behavior can be generalized as follows.

1) SBH occurs in certain crystal orientations providing

that, in the absence of a grating, one of surface skimming

bulk waves is nearly exceptional. With deviation from this

selected orientation, the SBH mode gradually disappears.

2) The SBH mode can be pure SAW, LSAW or high

velocity LSAW, dependent on whether the generating

BAW is slow quasi-shear, fast quasi-shear or quasi-

longitudinal.

3) In contrast to other quasi-bulk waves, the SBH mode

exists only in grating structures, within limited frequency

range, outside which it decomposes into the pure bulk

wave and counter-propagating SAW or LSAW.

4) Due to Bragg reflection condition fulfilled for

SAW/BAW interaction, a stopband occurs for the SBH

mode. The lower edge of this stopband can be regarded as

the cut-off frequency for bulk wave radiation and manifests

itself by a resonance of harmonic admittance.

5) Finally, the existence of the SBH mode can play an

important role in transformation of the wave structure from

pure SAW to LSAW and from LSAW to high-velocity

LSAW, with increasing frequency. Analysis of energy

transfer between SAW (or LSAW) and SBH mode could

reveal the mechanism of such transformation.

REFERENCES

[1] N. F. Naumenko, "Application of exceptional wave

theory to materials used in surface acoustic wave

devices," J. Appl. Phys, vol.79, pp. 8936-8943, 1996.

[2] Y. Fusero, S. Ballandras, J. Desbois, J. M. Hode, and

P. Ventura, “SSBW to PSAW conversion in SAW

devices using heavy mechanical loading,” 2000 IEEE

Ultrason. Symp. Proc., pp.163-166.

[3] K. Hashimoto, M. Yamaguchi, G. Kovacs, K. Wagner,

W. Ruile, and R. Weigel, “Effects of bulk wave

radiation on IDT admittance on 36YX-LiTaO3,” 2000

IEEE Ultrason. Symp. Proc., pp.253-258.

[4] J. Koskela, J. V. Knuuttila, V. P. Plessky, and M. M.

Salomaa, “Acoustic loss mechanisms in leaky SAW

resonators on lithium tantalate,” 2000 IEEE Ultrason.

Symp. Proc., pp.209-213.

[5] K. Blotekjaer, K. A. Ingebrigtsen, and H. Skeie, "A

method for analyzing waves in structures consisting of

metal strips on dispersive media," IEEE Trans.

Electron Devices, vol. ED-20, pp. 1133-1138, 1973

[6] K. Blotekjaer, K. A. Ingebrigtsen, and H. Skeie,

"Acoustic surface waves in piezoelectric materials

with periodic metal strips on the surface," IEEE Trans.

Electron Devices, vol. ED-20, pp. 1139-1146, 1973.

[7] R. F. Milsom, N. H. C. Reilly, and M. Redwood,

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1977.

[8] Y. Zhang, J. Desbois, and L. Boyer, "Characteristic

parameters of surface acoustic waves in a periodic

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Trans. Sonics Ultrason., vol. SU-40, pp. 183-192,

1993.

[9] P. Ventura, J. M. Hode, M. Solal, J. Desbois, and J.

Ribbe, "Numerical methods for SAW propagation

characterization," 1998 IEEE Ultrason. Symp. Proc.,

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[10] K. A. Ingebrigtsen, “Surface waves in piezoelectrics,”

J.Appl.Phys., vol.40, pp.2681-2686, 1969.

[11] K. Hashimoto, and M. Yamaguchi, “Analysis of

excitation and propagation of acoustic waves under

periodic metallic-grating structure for SAW device

modeling, 1993 IEEE Ultrason. Symp. Proc., 143-148.

[12] R. M. Taziev, and I. B. Yakovkin, “Fast algorithm for

correction of material constants of piezoelectric

crystals on SAW velocity material data”, 1994 IEEE

Ultrason. Symp. Proc., pp.415-419.

[13] N.F.Naumenko, and B.P.Abbott “SAW/BAW Hybrid

Mode Propagating on Substrate with Periodic

Grating”, Proc. Int. Symp. Theoretical Electrical

Engineering, Linz, 19-22 August, 2001.

6600

6700

0,4950,4960,4970,498

Fig.7. Real part of velocity versus normalized frequency,

calculated for LBO cut with Euler angles (45, 46, 90), including

HVPSAW and SBH modes; h/Λ=0.04,w/p=0.9.

Vbulk

HVPSAW

SBH mode

Normalized frequency fp/Vbulk

s–

s+