An Analytical Solution for Curved Piezoelectric ...

IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy conTrol, vol. TBc, no. TBc, TBc TBc 1

0885–3010 © TBc IEEE

DR

AFT

An Analytical Solution for Curved Piezoelectric Micromachined Ultrasonic

Transducers With Spherically Shaped DiaphragmsFiras sammoura, sina akhbari, and liwei lin

Abstract—An analytical solution for piezoelectrically ac-tuated spherically shaped diaphragms has been developed to study their dynamic behavior with targeted applications in piezoelectric micromachined ultrasonic transducers (pMUT). The analytical model starts with a curved pMUT composed of a piezoelectric diaphragm with a nominal radius in size, a radius of curvature in shape, and under both possible ac-tuation sources of radial pressure and electric potential. The diaphragm has the piezoelectric material polarized in the di-rection perpendicular to its surface and sandwiched between two metal electrodes. When an electric field is applied between the two electrodes, the in-plane piezoelectric strain can cause larger out-of-plane deflections than a flat unimorph piezoelec-tric diaphragm because of the diaphragm’s spherical curvature with a clamped periphery for high electromechanical coupling factor. Key performance parameters, including mechanical mode shapes, resonant frequencies, dynamic responses, and displacements, with respect to the curvature and size of the diaphragm have been investigated. Both analytical derivations and numerical simulations using finite element analysis have been performed for the optimal design of the electromechani-cal coupling factor, with varying factors such as mechanical resonant frequency, radius of curvature, nominal radius, and thickness. As such, this work provides theoretical foundations for the design of curved pMUTs with high electromechanical coupling factor compared with planar-shape pMUTs.

I. Introduction

Ultrasonography has been widely used in medical diagnostic apparatuses by detecting echoes of high-

frequency sound waves to map images of soft tissues, ten-dons, muscles, blood vessels, kidneys, liver, etc. [1] for ap-plications in therapeutic medicine [2], medical procedures [3], and early detection of diseases such as cancer tumors [4]. The state-of-the-art ultrasonic devices are built us-ing piezoelectric materials such as lead zirconate titanate (PZT) and operate in the thickness mode, where the res-onant frequency of the transducer is directly correlated

to the thickness of the diaphragm via well-controlled me-chanical polishing processes [5]. Tedious dicing and assem-bly procedures are followed to construct ultrasonic trans-ducers manually with an application-specific integrated circuit (asIc) chip. Typical ultrasonic transducers have low electromechanical coupling efficiency and bandwidth because of their inherently high acoustic impedance, which is difficult to match with the external environment [6]. as such, it is difficult to build high-resolution, high-frequency, and low-power ultrasonic sensors cost-effectively for con-sumer markets such as hand-held devices in photoacoustic imaging [7], portable healthcare [8], minimally invasive monitoring [9], underwater exploration [10], sign language interpretation [11], etc.

With the advancement of microelectromechanical sys-tems (MEMs), ultrasonic devices operating based on the plate flexural mode have shown remarkable improvements in bandwidth, cost, and yield over conventional thickness-mode PZT sensors [12]. MEMs fabrication technologies have been utilized to realize both capacitive (cMUTs [13]) and piezoelectric (pMUTs [14]) micromachined ul-trasonic transducers with mechanical impedance closely matched to that of the imaging medium, resulting in im-proved bandwidth and system efficiency. conventional pMUTs have unimorph plate structures composed of a piezoelectric layer deposited on a structural layer, where the strain mismatch between the two layers resulting from the voltage-induced piezoelectric stress causes the flexural vibration. The MEMs-compatible interconnect technol-ogy based on through-silicon vias (TsVs) allows for the direct integration of the cMUT/pMUT arrays with asIcs for robust 3-d imaging systems [15]. For example, air-borne volumetric imaging was demonstrated using a 26 × 26 cMUT array at 89 kHz [16], and a one-dimensional, 7-element pMUT array at 190 kHz [17]. real-time inter-cardiac (IcE) and intravascular (IVUs) ultrasound for-ward-looking imaging systems were implemented using a 64-element cMUT ring array at 10 MHz [18] and 5 × 5 pMUT arrays between 4 and 11 MHz [19]. Ultrasound ges-ture recognition has recently gained attention as a means of capturing touchless communications between human beings and electronic gadgets [20]. among these demon-strations, cMUTs require high dc polarization voltages and small separation gaps, but they typically have higher electromechanical coupling factors than those of pMUTs [21]. nevertheless, recent works on pMUTs with curved

t06121

Manuscript received october 21, 2013; accepted May 12, 2014. This project is supported in part by the Masdar Institute of science and Tech-nology, abu dhabi, grant number 11raMa3, and core funds from the Berkeley sensors and actuator center.

The authors are with the department of Mechanical Engineering, University of california at Berkeley, Berkeley, ca (e-mail: [email protected]).

F. sammoura is also with the department of Electrical Engineering and computer science, Masdar Institute of science and Technology, abu dhabi, UaE.

doI http://dx.doi.org/TBc

IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy conTrol, vol. TBc, no. TBc, TBc TBc2

diaphragm designs have shown improved electromechani-cal coupling with higher mechanical deformation per unit input energy compared with the conventional unimorph plate structures. dome-shaped single-layer pMUTs use in-plane piezoelectric strain, which leads to large out-of- plane deflection because of their spherical curvature and clamped periphery. For instance, Morris et al. analyzed and measured the actuation characteristics of composite piezoelectric diaphragms with an initial curvature actuat-ed by a static pressure [22]. Hajati et al. reported a dome-shaped pMUT device with improved electromechanical coupling factor at 45% [23]. We have demonstrated highly responsive curved pMUTs based on cMos-compatible fabrication process using aluminum nitride (aln) as the transduction material, where silicon wet etching was used to form the curved device structural base [24]. However, there have been no detailed theoretical analyses on the fundamentals of curved pMUTs.

Previously, extensive theoretical modeling work has been developed to extract the lumped-element parameters for the thickness-mode piezoceramics. Krimholtz el al. has pioneered this effort using the constitutive relations of piezoelectric materials to allow for robust construction of beamformer arrays with matching layers for improved efficiency, and effective interface with electronic circuitry [25]. cMUT elements were described by a 1-d model, and the predicted results fitted well with FEM simulation [26]. recently, we have modeled planar pMUTs of multiple ring electrodes and residual stress with equivalent parameters in a circuit model using Green’s function approach with good consistency with experiments in the prediction of deflection and bandwidth [27]–[31]. These analyses have been used to optimize the electrode designs for maximum acoustic pressure per unit input voltage [32]. Furthermore, we have also developed techniques to cancel the zero-bias deflection, suppress the higher harmonics, and double the output acoustic pressure under low input bias voltage us-ing pMUTs with annular and circular electrodes [33], [34].

several relevant works have been reported on the me-chanical and electrical actuation of spherically shaped diaphragms. Peng et al. analyzed single-layer pMUTs with curvature formed by a pressure differential across the membrane using finite element modeling (FEM) [35]. Toda and Tosima introduced the differential equations in cylindrical coordinates to calculate the vibration modes of clamped, cylindrical transducers with single and double curvatures under piezoelectric actuation [36]. Berry intro-duced the stress function and developed two fourth-order differential equations for thin, elastic, spherical shells un-der static radial stress [37]. naghdi and Kalnins intro-duced the free vibration equations of axisymmetric elastic spherical shells and solved for the lowest natural frequency as a function of the thickness for the special case of a hemispherical bell anchored at its apex with free edges [38]. In contrast to these previous works, this work pres-ents several new results: 1) theoretically derived differen-tial equations governing forced vibration of a spherical piezoelectric shell polarized in a direction perpendicular to

its curvature; 2) closed-form solutions for the forced vibra-tion equations under both radial pressure and electric po-tential with clamped boundary conditions; and 3) explicit predictions of the axisymmetric radial displacement shape function of a curved pMUT with respect to the tangen-tial angular position, structural layer thickness, radius of curvature, and nominal radius of the spherical diaphragm. The developed theoretical model is utilized for design op-timizations to achieve maximum radial deflection per unit input voltage. FEM simulations are used to validate the theoretical model in both displacements per unit voltage and mechanical resonant frequencies. as such, this work provides fundamental understanding of the operation of curved pMUTs as well as the design of the next generation ultrasonic transducers based on curved pMUTs.

II. Theoretical analysis

A. Overview of the Curved PMUT

Fig. 1(a) is a 3-d schematic diagram of a curved pMUT which is composed of a piezoelectric layer of thickness h sandwiched between two thin metal electrodes. The curved pMUT has a nominal radius r and a radius of cur-vature Rc, where the piezoelectric layer is polarized in the direction perpendicular to the surface of the diaphragm. The conceptual diagram shown in Fig. 1(b) with a cross-sectional view of the diaphragm illustrates the operation of the pMUT in the transmission mode. When an ac volt-age of magnitude Vr is applied between the top and bot-tom electrodes, an in-plane tension is developed in the piezoelectric layer. The diaphragm is made up of piezo-electric material that serves both the piezoelectric and the structural functions. The in-plane strain can be converted to vertical mechanical motion by the dimensional varia-tions in the planar direction, which is constrained by the periphery of the curved diaphragm for high electrome-chanical coupling. This is in contrast to flat unimorph pMUTs composed of piezoelectric and structural layers, where the planar strain mismatch between the two layers causes out-of-plane motion. The out-of-plane vibration of the curved pMUT causes the transmission of an acoustic wave of pressure pr. In the receive mode, as demonstrated in Fig. 1(c), an impinging acoustic wave of pressure pr will result in the flexural motion of the piezoelectric dia-phragm and the consequent separation of charge resulting from the piezoelectric effect to be collected in the charge amplifier as Ir.

B. Strain-Displacement Relations in Spherical Coordinates

Fig. 2(a) shows a curved pMUT as a part of a spherical shell with a radius of curvature Rc along with an infini-tesimal element in spherical coordinate system (r, θ, ϕ). a magnified view of the volume element is shown in Fig. 2(b) along with the stress couples and stress resultants, which are the forces and moments acting at the edge, re-

sammoura et al.: analytical solution for curved piezoelectric micromachined ultrasonic transducers 3

spectively [39]. Because the thickness of the curved pMUT is small compared with the other surface dimensions (h/Rc ≪ 1), love’s first approximation theory is appropriate for the geometrical and dynamic analysis of this device. The transverse normal, or the straight line normal to the undeformed middle surface, remains straight and normal to the deformed middle surface such that the transverse shear strains are infinitesimal (εrθ ≈ 0 and εrϕ ≈ 0) and all nonlinear terms can be neglected. It is also assumed that the transverse normal is inextensible and the transverse normal strain in negligible (εrr ≈ 0). The total strain of a spherical shell εij in the i- and j-directions can be decom-posed into membrane strains εij

0 and flexural strains εij1

as [39]:

ε ε ζε θ φij ij ij i j= + ∈ 0 1, where , , , (1)

where ζ is the radial position measured from the center of the pMUT diaphragm as shown in Fig. 2(b). By apply-ing the love–Kirchhoff’s assumption, strain in spherical coordinates can be expressed as a function of the displace-ment vector components and their derivative in spherical coordinates [39]:

ε φφφφ0 1

=∂∂+

R

uw

c (2)

ε φ θ φθθθ

φ0 1 1=

∂∂+ +

R

uu w

c sin cot (3)

21 10ε φ φ φ θθφ

θθ

φ=∂∂− +

∂∂

R

uu

uc

cot sin (4)

ε φ φφφφ1

2

2

21

=∂∂−∂∂

R

u w

c (5)

ε φ θ φ θφ φ φθθ

θφ

12 2

2

21 1 1

=∂∂−

∂∂+ −

∂∂

R

u wu

w

c sin sincot cot

(6)

2

1 2 2

1

12

2

2ε φ φ φ θ φφφ θθφ

θθ=

∂∂

− −∂∂ ∂

+∂∂

+

Ru

uw w

c

cot sincos

sin

siin ,φ θφ∂∂

u (7)

where uθ, uϕ, and w are the displacement vector compo-nents in the θ-, ϕ-, and r-directions, respectively.

C. Stress Resultants and Stress Couples in Spherical Coordinates

The piezoelectric material is modeled as an isotropic material, where the young’s modulus Y0 and the Poisson’s ratio v are the only two independent variables required to represent the mechanical properties. Using love’s ap-proximation, the transverse shear stresses are negligible (σrθ ≈ 0 and σrϕ ≈ 0), and the transverse normal stress is small compared with the other normal stresses (σrr ≈ 0). Because the external electric field Er is applied along the polarization direction, the transverse piezoelectric charge constants drθ and drϕ are assumed to be equal with the magnitude of d31. The constitutive equations for a lin-ear isotropic piezoelectric medium relating the tangential stresses of a curved pMUT with constant radius of curva-ture can be written as [40]:

σ ε εφφ φφ θθ=−

+ − +Yv

v v d E02 311

1[ ( ) ]r (8)

σ ε εθθ θθ φφ=−

+ − +Yv

v v d E02 311

1[ ( ) ]r (9)

σ εφθ φθ=+Yv

01 . (10)

The stress resultants (Nθθ, Nϕϕ, and Nθϕ) in the θ-, ϕ-, and ϕθ-directions, respectively, can be related to the strains and the curved pMUT material and geometric properties

Fig. 1. (a) 3-d schematic of a curved PMUT based on a cMos-compat-ible fabrication process, (b) conceptual diagram showing the cross-sec-tional view of a curved PMUT illustrating its operation in the transmis-sion mode. The curved diaphragm promotes the conversion of in-plane strain to vertical mechanical motion for larger displacement for a high acoustic pressure, pr, and (c) the curved pMUT in reception mode, where an echo pulse with the acoustic pressure, pr, results in mechanical strain on the curved diaphragm and causes charge separation due to the piezo-

electric effect, which is collected at the charge amplifier as Ir.


by integrating the tangential stresses along the thickness of the curved pMUT as [39]

NNN h

φφ

θθ

θφ

φφ

θθ

θφ

σσσ

=

− /22

2

02

1

1

1 01 0

0 01

2

h

R d

Y hv

vv

v

/

c

=

∫ +

− −

ζζ

( )

−−

εεε

φφ

θθ

θφ

0

0

0

0 31

21

110

Y d Vvr

.

(11)

The stress couples (Mθθ, Mϕϕ, and Mθϕ) can be formulated as well by integrating the infinitesimal stress couple of the respective tangential stress at a distance ζ from the middle surface along the pMUT thickness as

MMM h

φφ

θθ

θφ

φφ

θθ

θφ

σσσ

=

− /22

2

1

1

1 01 0

0 01

2

h

R d

Dv

vv

/

c∫ +

=−

ζζ ζ

εφφ

( )εεεθθ

θφ

1

12

,

(12)

where D is the flexural rigidity defined as [27]

DY hv

=−0

3

212 1( ).

(13)

D. Dynamic Stress Equations for Piezoelectric Spherical Shells

The active piezoelectric layer introduces surface forces in the θ- and ϕ-directions, which are proportional to the applied voltage and piezoelectric charge constant as shown in (11). The motion of the curved pMUT in both transmit and receive modes is rotationally symmetric around the z-axis, as shown in Fig. 2(b). as such, axisymmetric con-ditions prevail and ∂/∂θ = 0. Using love’s approximation, the dynamic stress equations of a shell with constant ra-dius of curvature [38], [39] and in-plane piezoelectric forces can be expressed as

∂∂

+∂∂

− +

=∂∂

′ ′φ φ θ φ φ

ρ φ

φφ θθ

φ

θφ φφ(sin ) ( ) cos sin

sin

N N N Q

hRutc

2

2

(14)

∂∂

+∂∂

+ +

=∂∂

′θ φ φ φ φ

ρ φ

θθ

θ

θφ θφ θθ( ) (sin ) cos sin

sin

N N N Q

hRutc

2

2

(15)

∂∂

+∂∂

− +

=∂∂−

′ ′φ φ θ φ

ρ φ φ

φφ θθ θθ φφ(sin ) ( ) sin ( )

sin sin

Q Q N N

hRwtc

2

22YY d V

v R prr

0 311− − c sinφ

(16)

∂∂

+∂∂

− − =φ φ θ φ φφφ θφ θθ φφ(sin ) ( ) cos sinM M M R Qc 0

(17)

∂∂

+∂∂

+ − =φ φ θ φ φθφ θθ θφ θθ(sin ) ( ) cos sin ,M M M R Qc 0

(18)

where ρ is the density of the piezoelectric material; pr is the received echo or the transmitted acoustic pressure per-pendicular to the curvature of the diagram; Qθθ and Qϕϕ , are transverse shear stress resultants; and N θθ′ and N φφ′ are the modified stress resultant in the θ- and ϕ-directions, respectively:

NY hv

vφφ φφ θθε ε′ =−

+02

0 0

1[ ] (19)

Fig. 2. (a) a curved pMUT of nominal radius r and radius of curva-ture Rc in spherical coordinates, and (b) the stress couples and stress resultants on an infinitesimal element extracted from the curved pMUT diaphragm in spherical coordinates.


NY hv

vθθ θθ φφε ε′ =−

+02

0 0

1[ ]. (20)

With the introduction of the modified shear resultant stresses N θθ′ and N φφ′ , the dynamic stress equations for the spherical shells with piezoelectric active layer as in (14), (15), (17), and (18) are reduced to the same equa-tions with the absence of surface loads as described in [38]. It is interesting to observe that the piezoelectric surface forces are transformed into radial forces as shown in the extra term on the right-hand side of (16), which can be treated as a radial pressure imposed because of the piezo-electric phenomenon.

E. The Curved pMUT Vibration Differential Equations

The compatibility equation can be derived by the elimi-nation of uϕ from (2) and (3) and replacing the strains with their resultant stress equivalent using (19) and (20):

tan ( ) (sec ) ( sec )

tan [

,φ φ φ

φ

θθ φφ θθ φφφN vN v N v N

Y hRc

′ ′ ′ ′− + + − +

=

2 2

0

1

ww w, tan ]φ φ+

(21)

as proposed by [38], we define a new stress function F(ϕ, t), which satisfies the following relations:

N F Fv RY

Ft

DR R

wφ φφρ′ = + −+ ∂

∂+ ∇ +

cot

( ),

1 22

0

2

22

2c

c c

(22)

N F Fv RY

Ft

DR R

wθ φφρ′ = + −+ ∂

∂+ ∇ +

,

( ).

1 22

0

2

22

2c

c c

(23)

Using (2), (3), (14), (22), and (23), and eliminating uϕ, N θθ′ , and N φφ′ , F and w are governed by the following dif-ferential equation:

dd

2

2 0Γt= , (24)

where Γ is defined as

Γ = ∇ + − −

− ∂∂

+ − ∇ +

R F v Fv RY

Ft

vDR w

Rw

cc

c c

2 22 2

0

2

2

22

11

12

( )( )

( )

ρ

−Y hR w0

c.

(25)

another differential equation relating F and w can be de-rived by substituting N θθ′ and N φφ′ as defined in (22) and (23) into the compatibility equation as in (21), which leads to

tan .,φ φΓ Γ+ = 0 (26)

In general, Γ can be set to zero and the first differential equation in F and the radial displacement w is

R F v F

v RY

Ft

vDR w

Rw

cc

c c

2 22 2

0

2

2

22

11

12

∇ + − −− ∂

∂

+ − ∇ +

( )( )

( )

ρ

− =Y hR w0 0c

. (27)

To obtain the second differential equation in F and w, (2), (3), (14), and (17) are first combined to express the shear stress resultant Qϕϕ as a function of uϕ and w:

Qv

RDY

ut

DR w

Rwφφ

φ

φρ=

− ∂∂

− ∇ +

( ).

,

1 22

0

2

22

2c c c

(28)

By substitution of (28) into (16) and using (22) and (23), the second differential equation in F and w can be ob-tained:

R F Fv RY

Ft

RYh

vFt

DR

cc c

c

2 22

0

2

2

2

0

2 2 2

2

2

2 21

12 1∇ + −+ ∂

∂+ +

∂ ∇∂

+ ∇

ρρ

( )( )

∇∇ +

+ ∇ +

+∂∂=

22

22

2

2

22

2w

Rw

DR w

Rw

hRwt

R p

c c c

cρ ζζ + −210 31Y d Vvr .

(29)

F. Harmonic Forced Vibration of a Clamped Curved pMUT

The harmonic solution of the two differential equations in F and w in (27) and (29) at the operation frequency of ω is

wF

w eF e

j t

j t =

Re*

*

ω

ω (30)

where w* and F* are the magnitudes of the radial dis-placement w and the stress function F. By eliminating F* from (27) and (29), the most general form of the governing vibration equation for a curved pMUT becomes:

∇ + ∇ + ∇ +

= − −−

62

43

24

5 50 312

1

w d w d w d w

d R p dY d V

vr

* * * *

( ) ,c r

(31)

where d2, d3, d4, and d5 are functions of the frequency of operation ω and the curved pMUT material properties and dimensions:

dR

v2 22 21

4 1= + −c[ ( ) ]Ω (32)

dv

h RhYR D3

2

2 22 0

2212 1

1 1=−

− = −( )

( ) ( )c c

Ω Ω (33)


dhYDR

v v404

2 2 42 1 3 1= + + + −c[ ( ) ( ) ]Ω Ω (34)

dDR

v v5 32 21

1 1= − + −c[( ) ( ) ]Ω (35)

Ω22 2

0=ρω RY

c . (36)

The frequency term Ω2 is equivalent to the frequency term β4 used for flat pMUTs [27]. The free vibration equation can be simplified into the product of three legendre dif-ferential equations in spherical coordinates, and the gen-eral solution of the radial displacement wα* of each equa-tion is

w A P B Ql lα α αα αφ φ α* (cos ) (cos ), , , ,= + = 1 2 3 (37)

where Plα and Qlα are the legendre functions of the first and second kind of order lα, respectively, and lα can be calculated as [38]

lα αλ α= +

− =

14

12 1 2 3

1 2/

, , , , (38)

λ λ λ3 2 2 2 2

22

2

4 1 12 1 1

12 1

− + − + − −

− −

[ ( ) ] ( ) ( )

( )

v vRh

vR

Ω Ωc

chh v v + + + − =2

2 2 42 1 3 1 0[ ( ) ( ) ] .Ω Ω

(39)

The specific solution has a portion resulting from the ra-dial pressure and another resulting from the applied volt-age that can be simplified as

wv vv v

Rh Y Rs

cc

* ( ) ( )[ ( ) ( ) ]

= −− + −

+ + + −

1 12 1 3 1

12 2

2 2 40

ΩΩ Ω

ppY d V

vrr−

−

21

0 31( ) .

(40)

The specific solution is a function of the material proper-ties (Y0, d31, ν, ρ), diaphragm geometry (Rc, h) and exter-nal stimuli (ω, πρ, ςρ). The final solution is the superposi-tion of the general and specific ones:

w w w w ws* * * * *.= + + +1 2 3 (41)

G. Boundary Conditions

Because the curved pMUT is clamped on its edges, it cannot translate in the r- and ϕ-directions. In addition, the diaphragm cannot undergo any rotation around the ϕ-axis. The boundary conditions can be summarized as

w *φ φ= =

00 (42)

uφ φ φ*==

00 (43)

dwd

*,φ φ φ=

=0

0 (44)

where ϕ0 is the angle between the z-axis and a point on the pMUT, as shown in Fig. 3. The legendre’s function of the second kind Qlα tends to be infinity at the curved pMUT apex point (ϕ = 0). To eliminate this singularity, the coefficients B1, B2, and B3 as in (37) are set to zero. Using the boundary conditions in (42)–(44) along with the expressions for uφ* as derived in the appendix , the coeffi-cients A1, A2, and A3 as in (37) can be solved using the following set of three equations:

A P wlαα

αφ(cos ) *

01

3

=∑ = − s (45)

A l P Pl lα αα

φ φ φ φα α

( )[csc (cos ) cot (cos )]+ − =+=∑ 1 00 1 0 0 0

1

3

(46)

Av v

lvhR

α

λ

αα

α

α

λ

−+

=

( ) +

− + − + −+

×

∑2

12 1

2

2 21

3 1

1 11( )

[ ( ) ( ) ]( )

Ω

[[csc (cos ) cot (cos )] .φ φ φ φα α0 1 0 0 0 0P Pl l+ − =

(47)

The general solution described in (37) can now be explic-itly expressed. Because the diaphragm displacement equa-tion is the summation of the general solution and the spe-

Fig. 3. a 2-d schematic of the axisymmetric curved pMUT with clamped boundary condition, in a spherical shell of center O and radius Rc. The radial and tangential displacements at a point B with an angular posi-tion ϕ from the shell axis are denoted as w(ϕ) and uϕ(ϕ), respectively. The apex point is denoted as A.


cific solution as in (40), the diaphragm dynamic response can be plotted and the resonant frequency extracted.

III. Model analysis and Verification

The model developed in the previous section is used to further analyze a clamped curved pMUT [24]. The thick-nesses of the metal layers are ignored, and the piezoelectric material, aln, with the isotropic material properties as shown in Table I (theoretical analysis) and the anisotropic material properties as shown in Table II (FEM simula-tion) is used as the piezoelectric and the structural layer. The mode-shape function, the dynamic response, and the effect of the diaphragm curvature on the displacement amplitude and resonant frequency are extracted from the theoretical model and verified with FEM simulations us-ing comsol (comsol Inc., Burlington, Ma). In these stud-ies, we used the exemplary model of a curved pMUT com-posed of a 2-µm-thick aln layer, with a nominal radius of r = 70 µm and a radius of curvature Rc = 1168 µm. Fig. 4 shows a 3-d model of a curved pMUT with clamped boundary conditions. The pMUT diaphragm is assigned to a user-defined orthogonal coordinate with unit vectors x1, x2, and x3, where x3 is aligned perpendicular to the surface to pole the aln material along its thickness.

A. Mode Shape Function

as shown in Fig. 5, the mode shape function, or the radial displacement amplitude w(ϕ) versus the tangential angular position ϕ, is plotted using the generalized vi-

bration equation as derived in (37) and (40)–(41) along with the clamped boundary condition as in (45)–(47). The radial displacement is normalized with respect to the maximum displacement at the center of the diaphragm. The theoretical prediction matches well with the simu-lated value and the difference is less than 3%, which is due to the fact that aln is modeled as an isotropic mate-rial but is anisotropic in reality. The mode shape shows an inflection point at an angular position 85% away from the center of the curved pMUT diaphragm versus 60% for its planar counterpart [32], which is expected to lead to further enhanced volumetric velocity and acoustic pres-sure emission.

B. Dynamic Response

The frequency response is plotted in Fig. 6 for the exemplary model. The analytical model and simulation results on the center diaphragm displacements at low fre-quencies are reported as 1.41 nm/V and 1.29 nm/V, re-spectively, indicating a difference of less than 8.4%. The theoretically calculated resonant frequency is 2.80 MHz, which matches closely to the simulated value of 2.85 MHz. Therefore, the theoretical model and analyses could be useful tools to design curved pMUTs to accurately predict their dynamic behaviors.

C. Effect of Diaphragm Curvature

The effect of the diaphragm curvature on the displace-ment and resonant frequency is studied with respect to its nominal radius and thickness as shown in Fig. 7 based

TaBlE I. The Material Properties of aluminum nitride Used in the curved pMUT Theoretical analysis.

Parameter description Value Unit

ρ Plate density 3300 kg/m3

Y0 young’s modulus 348 GPaηY0 young’s modulus loss factor 0.01 —

ν Poisson’s ratio 0.30 —d31 Piezoelectric charge constant 2.20 pm/Vε dielectric constant 9 —

TaBlE II. The Material Properties of aluminum nitride Used in the curved pMUT comsol simulation.

Parameter description Value Unit

ρ Plate density 3300 kg/m3

cE Elastic matrix

410 149 99 0 0 0149 410 149 0 0 099 149 389 0 0 00 0 0 125 0 00 0 0 0 125 00 0 0 0 0 125

GPa

ηcE loss factor for elasticity matrix 0.01 —

dET coupling matrix0 0 0 0 3 84 00 0 0 3 84 0 01 73 2 78 5 49 0 0 0

−−

− −

..

. . .pm/V

ε dielectric constant 9 —


on the exemplary model. The static center displacement per unit input voltage increases rapidly with the radius of curvature, reaches an optimum value, and then gradu-ally drops to zero as the curvature becomes large (planar diaphragm). For a 2-µm-thick aln diaphragm, the static center displacement is 0.94 nm/V for a diaphragm with 500 µm in radius of curvature, whereas the maximal static displacement per unit input voltage reaches 1.41 nm/V

for the diaphragm with 1168 µm in radius of curvature. For larger radii of curvature, both the analytical model and simulation results predict further reductions of static center displacements. Furthermore, it is observed that the static center displacement decreases with increased dia-phragm thickness, and the maximum static displacement is 1.41 µm/V and 0.35 µm/V for the 2-µm- and 4-µm-thick aln diaphragms with radii of curvature at 1168 and 548 µm, respectively.

Fig. 8 shows the resonant frequency versus the dia-phragm radius of curvature from both analytical model and numerical simulation results with very good consis-tency between the two predictions. In general, the me-chanical resonant frequency decreases when the radius of curvature of the diaphragm increases or the thickness of

Fig. 4. a comsol 3-d model of a curved pMUT with clamped boundary conditions. The pMUT diaphragm is composed of an aln film of thick-ness h, radius of curvature Rc, nominal radius r, and apex angle ϕ0. The lower surface of the piezoelectric diaphragm is at electrical ground while a potential V is applied to its top surface. The piezoelectric material is assigned to a user-defined orthogonal coordinate system with unit vec-tors x1, x2, and x3, where x3 is aligned along the radial direction such that aln is poled along the thickness direction.

Fig. 5. comparison of the mode-shape function as predicted by the ana-lytical model and FEM simulation results. The normalized radial dis-placement w(ϕ) for the curved pMUT is plotted as a function of the tangential angular position ϕ. In this study, the curved pMUT is com-posed of a 2-µm-thick aln with the material properties shown in Table I, and the nominal radius r and the radius of curvature Rc are 70 µm and 1168 µm, respectively.

Fig. 6. comparison of the center displacement of the clamped curved pMUT per unit input voltage versus the excitation frequency from ana-lytical model and simulation results using the example diaphragm de-scribed in Fig. 5.

Fig. 7. static center displacement per unit input voltage versus the ra-dius of curvature, Rc, of the curved pMUT diaphragm with different thicknesses from 2 to 4 µm. The nominal radius, r, is 70 µm in all cases. Both analytical and numerical simulations results are consistent and optimal displacement magnitudes are found at 1.41, 0.62, and 0.35 nm at the optimal radius of curvature for the 2-, 3-, and 4-mm-thick dia-phragms, respectively.


the diaphragm decreases. Furthermore, when the radius of curvature is small (highly curved structures), the resonant frequency varies a lot because of small variations in the radius of curvature. For instance, the slope of the resonant frequency versus radius of curvature drops from 10 800 to 61.2 Hz/µm for the 2-µm-thick diaphragm around radii of curvature at 450 µm and 4900 µm, respectively. In addi-tion, the slope of the resonant frequency versus radius of curvature curve slightly decreases as the thickness of the diaphragm increases. For example, the slope of the reso-nant frequency versus radius of curvature slope decreases from 10.8 kHz/µm to 9.49 kHz/µm as the diaphragm thickness increases from 2 µm to 4 µm when the radius of curvature of the diaphragm is 450 µm.

Fig. 9 is the study of static center displacement versus radius of curvature under different nominal diaphragm ra-dii of 50, 70, and 90 µm, respectively. The thickness, h, is 2 µm in all cases. results show the optimal static cen-

ter displacement increases from 0.72 to 2.32 nm/V when the nominal radius of the diaphragm increases from 50 to 90 µm, respectively. Fig. 10 shows both analytical and simulation results of the mechanical resonant frequency versus the radius curvature of the curved diaphragm for the exemplary model under different nominal diaphragm radii of 50, 70, and 90 µm, respectively. It is found that the resonant frequency decreases as the radius of curvature of the diaphragm increases and reaches 1.30 MHz at a radius of curvature of 8100 µm, which is close to the 1.26 MHz value predicted for a planar pMUT. as the nominal ra-dius increases, the diaphragm resonant frequency becomes more sensitive to changes in the radius of curvature. For example, at radius of curvature of 811 µm, the slopes of the resonant frequency versus radius of curvature are pre-dicted as 2.99 kHz/µm and 2.00 kHz/µm for nominal ra-dii of 90 µm and 50 µm, respectively.

at very large radii of curvature, the resonant frequency is a strong function of the diaphragm thickness and nomi-nal radius, whereas it is a weak function of the radius of curvature. at very small radii of curvature, the resonant frequency is mainly decided by the diaphragm curvature, and it is weakly dependent on the diaphragm nominal radius and thickness.

IV. conclusion

a theoretical model for spherically shaped piezoelec-tric diaphragm using love’s first approximation theory has been developed with good consistency from the nu-merical simulation results. The stress resultant and stress couples have been formulated by integrating the tangen-tial stresses over the cross-sectional area of the diaphragm using the stress–strain relations in spherical coordinates. The harmonic forced vibration has been described by a sixth-order differential equation in terms of the frequency of operation, radial pressure, external electric potential,

Fig. 8. resonant frequency, fr, versus radius of curvature of the curved pMUT diaphragm with different thicknesses from 2 to 4 µm. The nomi-nal radius, r, is 70 µm in all cases.

Fig. 9. study of the static center displacement versus radius of curvature under different nominal diaphragm radii of 50, 70, and 90 µm. The thick-ness, h, is 2 µm in all cases.

Fig. 10. study of the resonant frequency versus radius of curvature under different nominal diaphragm radii of 50, 70, and 90 µm. The thickness, h, is 2 µm in all cases.


radius of curvature, piezoelectric charge constant, young’s modulus, thickness, and density of the diaphragm. The radial displacement with respect to the tangential angu-lar position has been explicitly solved for the exemplary model using legendre’s functions for detailed analyses in this work.

The mode shape function of the curved pMUT has been extracted and the results show that the inflection point was located at 85% away from the center of the dia-phragm. The static displacements and resonant frequen-cies calculated using the developed analytical solutions have matched well with the FEM using comsol.

results show that the static center displacement per unit input voltage increases with the radius of curvature, reaches an optimum value, and then gradually decreases to zero for the case of planar diaphragm. results also demon-strate that the resonant frequency decreases with increas-ing radius of curvature of the diaphragm, increases with increasing thickness of the diaphragm, and decreases with increasing nominal radius of the diaphragm. The sensitiv-ity of the resonant frequency with respect to the radius of curvature of diaphragm decreases as the diaphragm thick-ness increases or the nominal radius decreases. as such, the demonstrated theoretical model could be a crucial tool in designing robust curved pMUTs for ultrasonic sensors.

appendix solution of the Tangential displacement u*φ

Using the strain displacement relation in (3) and the modified stress resultants as defined in (19) and (20), the tangential displacement uφ* can be related to the radial displacement as

uRY h N vN wφ θ φφ= − −

′ ′tan ( ) .c

0 (a1)

With the help of the stress resultant equivalent expres-sions in terms of F and w as in (22) and (23), the first term of the right hand side of (a1) can be rewritten as

RY h N vN

RY h R F v F

v Fv

c cc

0 0

2 2 1

11

( ) ( )cot

( )(

,θ φ φφ

ρ

′ ′− =∇ − +

+ − −− 22 2

0

2

2

221

2

)

( ) .

RY

Ft

vDR R

w

c

c c

∂∂

+ − ∇ +

(a2)

The tangential displacement uϕ can be related to the stress resultant F with a simple expression using (27), (a1), and (a2):

u vRY h Fφ φ= − +( ) .,1

0

c (a3)

By further manipulating (27), (30), and (36), the stress function F* can be related to the radial displacement w* as

FY hR v v

wvhR* ( ) *

[ ( ) ( ) ].=

( ) +

− + − + −

−+02

12 1

2

2 2

1

1 1c

c

λ

λ Ω (a4)

Because the radial displacement is the superposition of the general and specific solutions as in (40), the magni-tude of the stress function can be expressed as

FY hR v v

wvhR* ( ) *

[ ( ) ( ) ]=

( ) +

− + − + −

−+

=

02

12 1

2

2 2

1

1 1c

c

λ

αα

α

α

λ Ω11

3

212 1

2

2 2

1

1 1

∑

+( ) +

− + −

−+

cs

( ) *

[( ) ( ) ].

vhR

v vw

Ω

(a5)

The specific solution of the radial displacement is indepen-dent of the tangential angular position. Upon performing the absolute derivative with respect to ϕ using the respec-tive identities for the legendre’s functions, the following relation is derived:

dd c

cF Y hR v v

vhR

*( )

[ ( ) ( ) ]φ λ

λ

αα

α

=( ) +

− + − + −

−+

=

02

12 1

2

2 2

1

1 1 Ω11

3

11

∑× +( ) −+ A l P Pl lα α φ φ φ φ

α α[csc (cos ) cot (cos )].

(a6)

The tangential displacement as a function of the material properties and dimensions, tangential angular position, and frequency of operation can be stated as

u vv v

vhR

φ

λ

αα

α

λ* ( )( )

[ ( ) ( ) ]= − +

( ) +

− + − + −

−+

=

11

1 1

212 1

2

2 21

c

Ω

33

11

∑× +( ) −+ A l P Pl lα α φ φ φ φ

α α[csc (cos ) cot (cos )].

(a7)

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Firas Sammoura is currently an assistant Pro-fessor in the department of Electrical Engineering and computer science at the Masdar Institute of science and Technology, abu dhabi. He is also a visiting scholar in Prof. liwei lin’s lab in the me-chanical engineering department of the University of california, Berkeley. Firas sammoura received his B.E. degree in mechanical engineering from the american University of Beirut in June 2001 with high distinction. From 2001 until 2006, he was a graduate student researcher at the Univer-

sity of california, Berkeley. In May 2006, he earned his Ph.d. degree in the field of micro-electromechanical systems (MEMs). His dissertation focused on building plastic millimeter-wave systems for radar applica-tions at 95 GHz. Prior to joining the Masdar Institute, dr. sammoura was a visiting scientist in the MI/MIT cooperative program at the Mas-sachusetts Institute of Technology, where he started a new research pro-gram in the field of piezoelectric micromachined ultrasonic transducers (pMUTs) in collaboration with Prof. sang-Gook Kim in the mechanical engineering department. From January 2007 until January 2011, he was senior device characterization engineer in the advanced development


Group at the MEMs/sensors division of analog devices in Wilmington, Ma. From March 2004 until May 2006, he also worked as a student re-searcher with Hitachi Global storage Technologies at the IBM almaden research center, where he did proprietary research in MEMs applica-tions for the hard drive disk industry. dr. sammoura won the spot award for solving the stiction problem that plagued consumer inertial MEMs accelerometers. He has several pending and issued patents in the field of microwave engineering and MEMs fabrication, design, and char-acterization.

Sina Akhbari is currently a Ph.d. student at the department of Mechanical Engineering at Uc Berkeley. He received his B.sc. degree in mechani-cal engineering from the sharif University of Tech-nology, Tehran, Iran, in 2012. He has been work-ing on design and fabrication of MEMs gyroscopes and piezoelectric micromachined ultrasonic trans-ducers (pMUTs) as part of his Ph.d. research.

Liwei Lin currently serves as the chancellor’s Professor of the department of Mechanical Engi-neering at the University of california, Berkeley, and co-director of the Berkeley sensor and ac-tuator center. Professor lin received his B.s. de-gree in 1986 in power mechanical engineering from the national Tsing Hua University in Taiwan, and then received his M.s. and Ph.d. degrees from Uc Berkeley in mechanical engineering in 1991 and 1993, respectively. after graduation, Professor lin held the position of senior research scientist at

BEI Electronics Inc. He also served as associate professor at the na-tional Taiwan University and assistant professor at the University of Michigan before joining the faculty at Uc Berkeley in 1999. Professor lin also held the position of vice chair of graduate study within the me-chanical engineering department from 2006–2009. Professor lin’s re-search interests and activities at Uc Berkeley include MEMs, nEMs, nanotechnology, design and manufacturing of microsensors and microac-tuators, development of micromachining processes by silicon surface/bulk micromachining, micro molding processes, and mechanical issues in MEMs such as heat transfer, solid/fluid mechanics and dynamics. Pro-fessor lin is the co-inventor listed on 14 patents in MEMs and has au-thored or coauthored 90 journal publications and more than 130 refereed conference proceedings. since 1996, he has graduated 17 Ph.d. and 25 M.s. students.

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