Surface electrical charge and field analysis of single-electrode type interdigital transducers

with bus-bar

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2007 J. Phys. D: Appl. Phys. 40 4476

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 40 (2007) 4476–4481 doi:10.1088/0022-3727/40/15/015

Surface electrical charge and fieldanalysis of single-electrode typeinterdigital transducers with bus-barZhaohong Wang, Tiantong Tang, Shi Chen and Bing Chen

Department of Electronic Science and Technology, Xi’an Jiaotong University, Xi’an 710049,People’s Republic of China

E-mail: zhwang@mail.xjtu.edu.cn

Received 6 February 2007, in final form 4 June 2007Published 13 July 2007Online at stacks.iop.org/JPhysD/40/4476

AbstractA single-electrode type interdigital transducer (SIDT) is extensively used toexcite surface acoustic waves (SAWs) on piezoelectric substrates because ofits simple structure and fabrication process. In a SIDT a bus-bar is used forfeeding in electrical voltage. However, systematic analyses about bus-bareffects of SIDT are still sparse among the existing studies concerning thepropagation of acoustic wave of SIDT. In this paper, a two-dimensionalinterface Green’s function method is adopted for calculating the electricalcharge density on the electrode surfaces and acoustic wave fields in a Y-cutZ-propagation LiNbO3 substrate excited by SIDTs with a bus-bar. Thecomputation results show that the charge density associated with SAWcannot be neglected; the bus-bar has essential effects on the excited SAWfields and the SAW field generally diverges strongly due to the influence ofthe bus-bar. Approaches for reducing this effect are discussed.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Surface acoustic waves (SAWs) on piezoelectric crystals areusually excited by interdigital transducers (IDTs). The IDTelectrodes, consisting of two alternating sets of fingers, areconnected to an electric power supply through bus-bars. Asingle-electrode type interdigital transducer (SIDT) is widelyused to excite SAWs in electronic and optical communicationdevices such as convolvers, acousto-optical switches andacousto-optical filters because of its simple structure and lowdemand for fabrication equipments [1–3].

Although there has been a lot of research on the analysisof IDTs, an effective rigorous calculation for the excitationspectral characteristic and the wave field distribution isstill hardly accomplished. The analytical solution of theelectrostatic charge in the frequency domain can be obtainedonly for the simple periodic system of infinitely long stripfingers without a bus-bar [4, 5]. However, the dimensions ofpractical IDTs are finite and the bus-bar for feeding in electricalpower is needed; their SAW field and electrical charge density

distribution on the surface of the electrodes cannot be expressedby analytical solutions.

The charge density distribution for IDTs without a bus-bar was calculated utilizing a one-dimensional (1D) Green’sfunction [6], where the electric charge densities consist of twoparts related to the electrostatic and the piezoelectric effectsof the acoustic wave specifically. In most of the publishedresearch about SIDT, the bus-bar is neglected [7, 8]. Baghai-Wadij and co-workers calculated the input impedance and theacoustic field of the SIDT with a bus-bar [9–14]. However, intheir works the electrical field analysis is totally electrostaticand the influence of the charge associated with the acousticwave is not considered.

In order to consider both the electric and the elasticacoustic field effects in IDT calculation, a two-dimensional(2D) interface Green’s function was presented via aneffective dielectric coefficient [17]. In this approachthe three-dimensional (3D) complex coupled electric-elasticfield problem is simplified to a 2D electric one on theinterface. However, this Green’s function is very complicated

0022-3727/07/154476+06$30.00 © 2007 IOP Publishing Ltd Printed in the UK 4476

Surface electrical charge and field analysis of single-electrode type interdigital transducers with bus-bar

and still no practical calculation based on it has beenpublished so far.

In this paper, the charge density distribution on theelectrode surface and the SAW field distribution for apractical SIDT with a bus-bar are calculated considering bothelectrostatic and piezoelectric actions using the 2D interfaceGreen’s function. The computer program was written inFORTRAN. Then the acoustic wave fields of practical SIDTsdeposited on the Y-Z lithium niobate (LiNbO3) substrate arecalculated. The acoustic wave fields of a SIDT with the bus-barare calculated and these fields are compared with that withoutthe bus-bar to explore the influence of the bus-bar.

2. The charge density distribution for an IDT withbus-bar

We first consider a SIDT consisting of a semi-infinitedielectric material and a set of infinitely thin masslessdeposited electrodes. The SIDT has bus-bar and certainelectrode dimensions as shown in figure 1(a). We areinterested in the physical state at the surface, which islocated at z = 0. The frequency (ω)-factor is omittedhereafter.

The electric potential φ(x, y) on the surface can beexpressed by the convolution of the surface charge densityσ(x ′, y ′) with a 2D interface Green’s function G(x−x ′, y−y ′):

φ(x, y) =∫ +∞

−∞

∫ +∞

−∞G(x − x ′, y − y ′)σ (x ′, y ′) dx ′ dy ′.

(1)

Green’s function G(x, y) can be written as a sum of three termsfor the physical understanding and the theoretical analysis [16]:

G(x, y) = Ge(x, y) + GSAW(x, y) + Grest(x, y). (2)

Here, Ge(x, y) is the electrostatic part of Green’s function,GSAW(x, y) denotes the SAW parts of Green’s function,Grest(x, y) is related to the bulk acoustic waves (BAWs)and the asymptotic contribution at k → ∞. The electricalpotential can be calculated by the wavenumber domainGreen’s function G(kx, ky) and the wavenumber domaincharge density σ (kx, ky):

φ(x, y) =(

1

2π

)2 ∫ +∞

−∞

∫ +∞

−∞G(kx, ky)σ (kx, ky)

×e−jkxxe−jkyy dkx dky. (3)

In order to calculate σ (kx, ky) by equation (3), it is moreconvenient to set the wavenumber domain coordinates to thepolar system (k, θ ). Setting kx = k cos θ and ky = k sin θ weobtain

φ(x, y) =(

1

2π

)2 ∫ π

0

∫ +∞

−∞G(k, θ)σ (k, θ)

×e−jkx cos θe−jky sin θ k dk dθ. (4)

Here, G(k, θ) consists of three parts also:

G(k, θ) = Ge(k, θ) + GSAW(k, θ) + Grest(k, θ). (5)

For the electrostatic Green’s function in the wavenumberdomain, an analytical formulation has been given [14]:

Ge(k, θ) = 1

|k|εeff,∞(θ), (6)

where εeff,∞(θ) denotes the effective permittivity of thepiezoelectric dielectric at k → ∞. Considering the acousto-electric coupling effects,

εeff,∞(θ) = ε0+ε33

√ε2r1 ·cos2 θ + 2εr12 cos θ sin θ + ε2

r2 ·sin2 θ

with

ε2r1 = ε11ε33 − ε2

13

ε233

,

εr12 = ε12ε33 − ε13ε33

ε233

,

ε2r2 = ε22ε33 − ε2

23

ε233

.

Here, εij (i, j = 1, 2, 3) denotes the relative dielectriccoefficient of the piezoelectric material. For the inversetransform of the electrostatic part Ge(k, θ) in the polarcoordinates (r, ϑ), a very handy formulation has been obtainedin [17]:

Ge(r, ϑ) = 1

2πrεeff,∞(ϑ +

π

2

) . (7)

The electrostatic potential decays with r−1 and also theanisotropy of the substrate appears in the factor εeff,∞(ϑ + π

2 )

according to the above equation.The SAW part of the 2D interface Green’s function

GSAW(k, θ) can be written as [15]

GSAW(k, θ) = Gs(θ) · 2k0(θ)

k2 − k20(θ)

, (8)

where Gs(θ) is the piezoelectric coupling factor and k0(θ) isthe wavenumber at the free surface of the substrate for SAW.Here,

Gs(θ) =[vm − v0

v0εeff,∞

]θ

, (9)

k0(θ) = ω

v0(θ), (10)

where v0 and vm denote the surface wave velocity on thefree surface and the metallized surface, respectively. In theequation, the polar angle θ is referred to the x-axis and ω

is the angular frequency. Using equation (8) and applyingCauchy’s residue theorem, we obtained the inverse transformof GSAW(k, θ) as

GSAW(r, ϑ) = j1

2π

∫ π

0Gs(θ)ek0(θ)r|cos(θ−ϑ)| dθ. (11)

The expressions for Grest(k, θ) are available in the formof numerical data only. The influence of asymptotic behaviourcan be omitted for acoustic field calculation in the regionfar departed from the finger electrodes [18]. What is more,the bulk wave effect is small enough if the directions ofthe substrate’s cut and the wave’s propagation are suitablyselected for an IDT operated near its centre frequency [6].

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Z Wang et al

(a) (b) (c)

Figure 1. SIDTs (� = 2a = 20.6 µm). (a) Practical SIDT with bus-bar, (b) ideal IDT without bus-bar, (c) the electric field in a period.

Therefore, the parts of Green function related to BAWs andasymptotic contributions are ignored in this work hereafter.

Adopting the electrostatic and the SAW parts of the 2Dinterface Green’s function given by equations (6) and (8),respectively, and using the boundary condition of the surfaceelectrical potential φ(x, y) (in figure 1 the applied voltage is±1 V on electrodes), the electric charge density σ is calculatednumerically by the method of moment (MoM) [16].

In the MoM, appropriate basis functions, such as impulse,rectangular or triangle ones, are generally used to approximatethe charge density distribution. In this work we chosethe rectangular basis function. In order to overcome thesingularities of the charge density appearing at the finger-edgeand to reduce the demand of computer memory, we use a non-equidistance discretization of the electrodes. A large enoughdiscretization number is used to obtain numerical results withhigh precision. A further discussion on the calculation ofsurface electrical charge can be found in [19].

The surface electrical charge density of a SIDT issimulated numerically. The SIDT is shown in figure 1(a),where L is the numerical aperture and d is the distancebetween the bus-bar and the end of the parallel electrodes.The period � of the SIDT is equal to the wavelength λ ofSAW. The computed distributions of electric charge densityσ(x, y) are shown in figures 2. The charge density is ofcomplex nature; its real part is shown in figure 2(a) and theimaginary one is shown in figure 2(b). Obviously, the real parthas singularities appearing at the finger-edge. The distributionof pure electrostatic calculation is shown in figure 2(c) forcomparison. We observed that the pure electrostatic electriccharge density shown in figure 2(c) is almost identical tothat shown in figure 2(a). It is obvious that the real partof the complex charge density is mainly associated with theelectrostatic action and the imaginary part has to be associatedwith the piezoelectric action. The magnitude of the imaginarypart of σ(x, y) is close to 10% of the real part. Therefore,it is necessary to consider the piezoelectric action in the IDTcalculation.

3. Distributions of acoustic fields

We consider the surface charge as an excitation source forgenerating SAW [20]. In those regions departed far from the

electrodesGe(x, y) � GSAW(x, y)

because the electrostatic decays faster than the surface wave[17]. Grest(k, θ) is omitted as well; hence the potentialφ(x ′, y ′) associated with SAW is approximately

φ(x ′, y ′) =(

1

2π

)2 ∫ π

0

∫ +∞

−∞GSAW(k, θ)σ (k, θ)

×e−jkx ′ cos θe−jky ′ sin θ k dk dθ. (12)

The amplitude of the electric field intensity is proportionalto that of the potential for a z-direction propagated wave.Hence the amplitude of SAW intensity is determined by thatof potential φ(x ′, y ′).

The computed equi-amplitude contours of potentialφ(x ′, y ′) of the SIDT are shown along the +x direction infigure 3. We observed that the acoustic wave field is divergent.Different from that without bus-bar, the divergence of the SAWfield of the SIDT with bus-bar is larger than that without bus-bar. This can be explained using figure 1(c), because the wavefield along the SAW propagation direction is generated mainlyby the parallel parts of the electrodes and the field generatedby the interaction of bus-bar and the end of parallel electrodesleads to a wave propagated in the lateral direction. Structureswith different bus-bar widths b are computed. We observedthat the acoustic fields are similar for different bus-bar widths.

Therefore, the distance d (shown in figure 1(a)) can beincreased in order to reduce the influence of bus-bar. Thecontour line of potential distributions for d = λ/2 is shown infigure 3(c), and the axial distributions of potential for differentd are shown in figure 4, where the distribution for SIDT withoutbus-bar is also shown for comparison. Comparing curves infigure 4 and comparing figure 3(c) with figures 3(a) and (b), itis observed that the influence of bus-bar is reduced when thedistance d is increased. The bus-bar effect can be reduced aswell by modifying the shape of the bus-bar. For example, therectangular form can be substituted by arc or polygonal ones.In addition, the diffraction effect becomes small by increasingthe numerical aperture or increasing the pairs of electrodes(shown in figure 5).

When the acoustic wave interacts with the optical wavefield, the optical wave is diffracted and the diffraction lightbeam is decided by the form of the acoustic beam. In a lot

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Surface electrical charge and field analysis of single-electrode type interdigital transducers with bus-bar

(a)

(b)

(c)

Figure 2. Charge density on negative voltage electrode, shown infigure 1(a), bus-bar included. (a) The real and (b) the imaginaryparts of charge density distribution taken into electrostatic and SAWinteraction. (c) Pure electrostatic charge density.

of research, the width of the acoustic beam is assumed fixedand equal to the numerical aperture of SIDT. Acousto-opticinteraction is dependent strongly on the actual form of theacoustic beam. Therefore, it is indispensable to calculate theactual acoustic field distribution for evaluating the expansionof the acoustic beams.

In a paper published earlier by the authors, we observedthat focused acoustic wave fields are generated by curvedIDTs [19]. SIDT can be made in a curved form with a small

0

x/Ly/

L

-1

1

0

x/L

y/L

-1

1

02 4 6 8 10 12 14

2 4 6 8 10 12 14

2 4 6 8 10 12 14

y/L

x/L

-1

1

Figure 3. Contour line plots of the calculated potential distributionof the SIDT. (a) Potentials of ideal SIDT without bus-bar,(b) practical SIDT with bus-bar (d = λ/4), (c) practical SIDT withbus-bar (d = λ/2).

curve angle to compensate for the diffraction, hence to generatea SAW propagated roughly in a parallel way. A curved IDTwith concentric circular electrodes is shown in figure 6(a).The IDT’s parameter is shown for five pairs of electrodes,curved angle θ = 1◦ and curved radius R = 50λ, and thecorresponding numerical result of the potential distribution isshown in figure 6(b). The divergence of its acoustic field isvery slow, and the divergence can be neglected for a certainpropagation distance.

4. Conclusions

In this paper, the electric charge densities on the IDT electrodeswith bus-bar originating from the electrostatic field and thegenerated SAWs were calculated numerically first using a 2Dinterface GFM. Then the charge density distribution as the

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Z Wang et al

0.32 4 6 8 10 12 14

0.4

0.5with bus-bar (d=λ/4)with bus-bar (d=λ/2)

without bus-bar

|φ|

x/L

Figure 4. Distribution of potential along y = 0. (dash lined is thecorresponding potential of ideal SIDT without bus-bar, the dottedline is the corresponding potential for d = λ/4, the solid line is thecorresponding potential for d = λ/2).

x/La

y/L

a

-1

2 4 6 8 10 12 14 16

1

020 40 60 80

x/L

y/L

-1

1

(a)

(b)

Figure 5. Contour line plots of the calculated potential distributionof the SIDT (d = λ/4). (a) SIDT with larger numerical apertureLa = 2L. (b) SIDT with 90 pairs of electrodes.

distributed source of SAW is used to calculate the SAW field.An SIDT with bus-bar is calculated. A computer program waswritten, and the simulation results for the SIDT deposited onthe Y-Z LiNbO3 substrate are given. The charge densitiesrelated to the electrostatic action are larger in magnitude,but with the view of generating SAWs, the charge densitiesassociated with the SAW are essential and cannot be neglected.The acoustic wave field of SIDT has a divergent feature dueto bus-bar and diffraction effects. Several methods, such asincreasing the distance between the bus-bar and the fingerelectrodes, changing the shapes of bus-bar, increasing the

02 4 6 8 10 12 14

x/Wc

y/W

c

(a)

(b)

Figure 6. Curved IDT and corresponding acoustic field distribution.(a) Curved IDT with concentric circular, (b) acoustic fielddistribution (the solid line is the contour line and the dotted line isthe acoustic field).

numerical aperture and increasing the electrode numbers, canbe adopted to reduce these effects. Roughly parallel SAWbeams can be generated by small curved angle IDTs withconcentric circular electrodes.

References

[1] Auld B A 1973 Acoustic Fields and Waves in Solids vol I (NewYork: Wiley)

[2] Hashimoto K Ya 2000 Surface Acoustic Wave Devices inTelecommunications (Berlin: Springer)

[3] Herrmann H, Schafer K and Sohler W 1994 Polarizationindependent, integrated optical, acoustically tunablewavelength filtered/switches with tapered acousticaldirectional coupler IEEE Photon. Technol. Lett.6 1335–7

[4] Engan H 1975 Surface acoustic wave multielectrodetransducers IEEE Trans. Son. Ultrason. 22 395–401

[5] Danicki E J 2004 Electrostatics of interdigital transducersIEEE Trans. Ultrason. Ferroelectr. Freq. Control 51 444–52

[6] Wagner K C et al 1987 Interdigital transducer interaction withSAW and BAWs IEEE Ultrasonics Symp. (New York)pp 149–53

[7] Bu G et al 2004 Guided-wave acousto-optic diffraction inAlxGa1−xN epitaxial layers Appl. Phys. Lett. 85 2157–9

[8] Anna Maria Matteo, Chen S Tsai and Do N 2000 Collinearguided wave to leaky wave acoustooptic interactions inproton-exchanged LiNbO3 waveguides IEEE Trans.Ultrason. Ferroelectr. Freq. Control 47 16–28

[9] Visintini G, Baghai-Wadji A R and Manner O 1992 Modulartwo-dimensional analysis of SAW filters: I. Theory IEEETrans. Ultrason. Ferroelectr. Freq. Control 39 61–71

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[10] Zidek H P, Baghai-Wadji A R and Seifert F J 1992 Full-wave3D analysis of singly-and doubly periodic SAW-structuresIEEE Ultrasonics Symp. (Tucson, AZ) pp 11–14

[11] Baghai-Wadji A R and Maradudin A A 1990 Rigorousanalysis of surface transverse waves in periodic arrays witharbitrary electrode profiles IEEE Ultrasonics Symp.(Honolulu, HI) pp 425–8

[12] Zidek H P, Baghai-Wadji A R and Manner O 1993 Full-wave3D analysis of wave scattering on SAW-structures withfinite aperture IEEE Ultrasonics Symp. (Baltimore, MD)pp 149–52

[13] Manzuri-Shalmani M T and Baghai-Wadji A R 2003Elemental field distributions in corrugated structures withlarge-amplitude gratings Electron. Lett. 39 1690–1

[14] Baghai-Wadji A R, Selberherr S and Seifert F 1986 Rigorous3D Electrostatic Field of SAW Transducers with closedform IEEE Ultrasonics Symp. (Williamsburg, VA)pp 23–8

[15] Wagner K C and Visintini G 1989 Simulation of bulk waveeffects in SAW devices using green’s function and angular

spectrum of waves IEEE Ultrasonics Symp. (Montreal,Que) 103–6

[16] Milsom R F, Reilly N H C and Redwood M 1977 Analysis ofgeneration and detection of surface and bulk acoustic wavesby interdigital transducers IEEE Trans. Son. Ultrason.24 147–66

[17] Danicki E 1988 Green’s function for anisotropic dielectrichalfspace IEEE Trans. Ultrason. Ferroelectr. Freq. Control35 643

[18] Laude V, Jerez-Hanckes C F and Ballandras S 2006 Surfacegreen’s function of a piezoelectric half-space IEEE Trans.Ultrason. Ferroelectr. Freq. Control53 420–8

[19] Wang Z et al 2006 Field analysis and calculation of interdigitaltransducers with arbitrary finger shapes J. Phys. D: Appl.Phys. 39 4902–8

[20] Baghai-Wadji A R, Selberherr S and Seifert F J 1986Two-dimensional Green’s function of a semi-infiniteanisotropic dielectric in the wavenumber domain IEEETrans. Ultrason. Ferroelectr. Freq. Control 33 315–17

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