Viscosity sensor using ZnO and AlN thin film bulk acoustic resonatorswith tilted polar c-axis orientations

Lifeng Qin,1 Qingming Chen,2 Hongbin Cheng,1 Qian Chen,1 Jing-Feng Li,3

and Qing-Ming Wang1

1Department of Mechanical Engineering and Materials Science, University of Pittsburgh,Pittsburgh, Pennsylvania 15261, USA2School of Materials Science and Engineering, Kunming University of Science and Technology,Kunming, Yunnan, People’s Republic of China3Department of Materials Science and Engineering, Tsinghua University, Beijing 10084, China

(Received 2 September 2011; accepted 21 September 2011; published online 9 November 2011)

We present a theoretical analysis of liquid viscosity sensors using ZnO and AlN thin film bulk

acoustic wave resonators (FBARs) with tilted polar c-axis orientations. Besides the thickness

longitudinal mode, the tilted c-axis orientation induces thickness shear mode in the resonator,

which allows resonators operated in liquid medium without significant damping for sensory

application. The equation for predicting electric impedance of shear mode film bulk acoustic wave

resonators (FBARs) with a viscous liquid loading was derived from the basic piezoelectric

constitutive equations. The viscosity sensitivity of shear mode ZnO and AlN resonators was

achieved by calculation of resonant frequency shift due to viscous liquid loading. In the simulation,

it is assumed that all the resonators have 2 lm thickness and 300 lm�300 lm electrode area; three

different liquids (water, acetone, and olive oil) were chosen as the liquid loadings; and different tilt

c-axis angles for both ZnO FBARs and AlN FBARs have been examined. It was found that the

sensitivities of shear mode resonators to the three liquid loading are very close, and do not change

much with the c-axis tilt angle with a value rang from 0.91e-3 to 0.97e-3 (kg m�3 Pa�S)�0.5 for

ZnO FBARs and from 1.12e�3 to 1.13 e�3 (kg m�3 Pa�S)�0.5 for AlN FBARs. When the

resonator’s mechanical quality factor (Q) is changed from 50 to 10 000, viscosity sensitivities are

almost same. However, Q has a great effect on resonator impedance; and if Q is too low or the

viscosity of the liquid is high, the maximum phase angle of the resonator will be less than 0, which

makes excitation of the oscillation difficult if an oscillator circuit is used for sensor measurement.

The results can be used for design and application of ZnO or AlN FBARs to monitor liquid

viscosity. VC 2011 American Institute of Physics. [doi:10.1063/1.3657781]

I. INTRODUCTION

Viscosity measurement is very important in many labo-

ratory research analysis and industry applications. For

example,1–3 to prevent engine failure, changing the engine

oil of automobiles is required after a certain period of time

due to its deterioration during use. On the other hand, for ec-

ological and economical consideration, it needs to avoid an

unnecessary oil change. Hence, to optimize the oil change

interval, it is necessary to monitor the oil condition.

Moreover, the engine oil condition provides insight into

the actual state of the engine, providing early detection of pos-

sible engine failures. For the characterization of the oil condi-

tion, viscosity is one of the most important parameters that

need to be considered. There are various kinds of methods for

viscosity measurement. Tube viscometers, such as the Ubbe-

lohde viscometer, consist of a U-shaped glass tube, where

there are two glass bulbs joined by a capillary tubing. Liquid

is drawn into the upper bulb by suction, then flows down

through the capillary into the lower bulb. The time for the liq-

uid to pass between two marks (above and below the upper

bulb) is used for viscosity measurement since it is

proportional to the viscosity. Rotational systems, such as the

Brookfield viscometer, measure viscosity by sensing the

torque required to rotate a spindle at constant speed while

immersed in fluid. The torque is linear to the viscous drag on

the spindle and thus to the liquid viscosity. Oscillating piston

viscometers are comprised of a measurement chamber and

magnetically influenced piston. The piston is driven by elec-

tronics into oscillatory motion in the measurement chamber

with a controlled magnetic field. Due to the piston travel a

shear stress is imposed on the liquid, and its viscosity is deter-

mined by measuring the travel time of the piston. These sen-

sors and systems have various advantages and have been

widely used in many different applications. However, they

usually have large physical size and require high volume of

liquid sample or with high cost, and complex auxiliary elec-

tronic parts. Therefore, there is still a need for compact sen-

sors that are low-cost, simple in operation and applicable for

in situ viscosity monitoring.

In contrast, due to their small size and the absence of

macroscopically moving parts, recently bulk acoustic wave

(BAW) resonators, such as quartz crystal microbalance

(QCM) appear ideal for this purpose when compared to con-

ventional viscometers.1–4 QCM has been widely used in many

physical and chemical sensing applications due to its high

0021-8979/2011/110(9)/094511/11/$30.00 VC 2011 American Institute of Physics110, 094511-1

JOURNAL OF APPLIED PHYSICS 110, 094511 (2011)

sensitivity, simple structure, and easy interconnection with

electronic measurement systems.5 For sensor application, usu-

ally special material which is sensitive to the target is coated on

the resonator surface. Surface changes due to physical/chemical

adsorption and absorption will induce a linear resonant fre-

quency shift for the small load, which can be described by the

Sauerbrey equation6 Df ¼ ð�2f0Dm=Affiffiffiffiffiffiffiffiffiffiqqlqp Þ, where Df is

the change in frequency, f0 is the fundamental resonant fre-

quency of the resonator, Dm is the mass change, A is the elec-

trode area, qq is the density of the quartz, and lq is the shear

modulus of quartz. For liquid viscosity sensing, Kanazawa and

Gordon7 built up a physical model coupling the shear wave in

the quartz to the damped shear wave in the fluid, which

described the relation between Df and liquid viscosity, gl,

Df ¼ �f3=20

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiglql=plqqq

q, where ql is the density of the liquid.

Commonly, a BAW resonator consists of one piezoelec-

tric layer with two electrodes deposited on both sides. The

resonant frequency f0 is inversely proportional to the resona-

tor thickness h, following the equation f0 ¼ �s=2h, where �s

is the acoustic wave velocity and determined by the material

property of the piezoelectric layer; for QCM, �s is equal toffiffiffiffiffiffiffiffiffiffiffiffilq=qq

q. On the other hand, the sensitivity of the resonator

sensor usually can be increased with resonant frequency.

Hence, choosing piezoelectric material with high acoustic

wave velocity or reducing resonator thickness can be used to

improve sensor sensitivity. However, high-frequency QCM

is hardly achieved due to the difficulty in the fabrication of

ultrathin quartz plate.8 With the improvement of the fabrica-

tion process and material growth techniques, now it is possi-

ble to make the resonator in a very small size. Recently, ZnO

and AlN FBARs can be successfully achieved with thickness

down to tenth of micrometers, which brings the resonant fre-

quency to tens of gigahertz and thus to higher sensor

sensitivity.9–12

ZnO and AlN thin films used for FBAR fabrication are

usually with normal-plane c-axis orientation, which makes

resonator operated in thickness longitudinal mode. In liq-

uid, acoustic wave energy of the longitudinal mode is radi-

ated into liquid through compressional motion, resulting in

resonator damping, while damping of the shear mode

acoustic wave is not severe. Hence, shear wave FBARs are

preferred for liquid application. Fortunately, the shear

mode can be excited beside the longitudinal mode when the

c-axis of AlN or ZnO thin films is tilted to a certain

angle.13–19 In our previous research,20,21 we have theoreti-

cally analyzed the performance of dual mode FBARs based

on the c-axis tilted ZnO and AlN thin films and their sensi-

tivities for mass sensors application in air. However, the

theoretical analysis of c-axis tilted ZnO and AlN FBARs is

less, and there has not been a thorough study on shear mode

ZnO and AlN FBARs for viscosity sensor application in liq-

uid. For this purpose, we derived the electric impedance

expression of shear mode ZnO and AlN FBARs with a liq-

uid loading layer by the basic piezoelectric equations. Then

the resonant frequency shift for different liquid loading was

examined. Finally the viscosity sensitivity of shear mode

ZnO and AlN FBAR was calculated and discussed.

II. THEORY

A. Dual mode ZnO and AlN FBARs

Due to the crystal orientation dependence of material

properties including elastic constants, piezoelectric constants

and dielectric constants, thickness shear mode will be pro-

duced in the resonator except for thickness longitudinal

mode when the c-axis of ZnO or AlN thin film is tilted, as

mentioned before. The existence of thickness shear mode

makes FBAR possible for sensor application in liquid, and

the theoretical analysis of dual mode FBARs using c-axis

tilted ZnO or AlN is the basis for analytical study of shear

mode ZnO or AlN FBAR with liquid loading. Here, we start

from the impedance expression of c-axis tilted ZnO or AlN

FBARs without liquid loading and then to viscous liquid

loading. Following a similar procedure,22 the impedance

equation for the c-axis tilted ZnO or AlN FBARs without

loading could be derived.20 Figure 1(a) shows FBAR based

on ZnO or AlN thin film with normal c-axis, and (x1, x2, x3)

is the original material coordinate system; Fig. 1(b) shows

the schematic of FBAR based on c-axis tilted ZnO or AlN

thin film, where the top and bottom electrode are ignored for

model simplification. A rectangular Cartesian coordinate

system (x01, x02, x03) is chosen with the top electrode on x03 ¼ hand the bottom electrode on x03 ¼ 0. The c-axis of ZnO or

AlN film is tilted at an angle h to x03. The coordinate system

FIG. 1. The schematic of FBARs based on c-axis tilted ZnO or AlN thin

film and coordinate systems: (a) FBAR based on ZnO or AlN thin film with

normal c-axis and coordinate system (X1, X2, X3), (b) FBAR based on ZnO

or AlN thin film with tilted c-axis and coordinate system (X10, X2

0, X30), (c)

the relation of (X1, X2, X3) and (X10, X2

0, X30), (d) ZnO material properties

in (X1, X2, X3), and (f) AlN material properties in (X1, X2, X3).

094511-2 Qin et al. J. Appl. Phys. 110, 094511 (2011)

(x01, x02, x03) can be treated as the result of rotation of (x1, x2,

x3) about x2 with an angle h, as shown in Fig. 1(c).

In the coordinate system (x01, x02, x03), we have the follow-

ing acoustic field equations and piezoelectric constitution

equations:23

@T1

@x01

þ @T6

@x02

þ @T5

@x03

¼ q@2u1

@t2;

@T6

@x01

þ @T2

@x02

þ @T4

@x03

¼ q@2u2

@t2;

@T5

@x01

þ @T4

@x02

þ @T3

@x03

¼ q@2u3

@t2;

(1)

S1 ¼@u1

@x01

; S2 ¼@u2

@x02

; S3 ¼@u3

@x03

; S4 ¼@u3

@x02

þ @u2

@x03

;

S5 ¼@u3

@x01

þ @u1

@x03

; S6 ¼@u2

@x01

þ @u1

@x02

; (2)

Tp ¼ cEpq0Sq � e

0

kpEk; (3)

Di ¼ e0

iqSq þ eSikEk; (4)

where Tp, Sq, Di, and Ek are the components of stress, strain,

electric displacement, and electric field intensity, cEpq are the

elastic stiffness constants under constant electric field

FIG. 3. (Color online) Simulation of admittance (Y¼GþjB) spectrum of FBAR based on c-axis tilted ZnO thin film. G: (a) and (b); B: (c) and (d).

FIG. 2. (Color online) The schematic of the viscosity sensor based on ZnO

or AlN FBAR.

TABLE I. Parameters for simulation.

Piezoelectric

layer

Mechanical

quality factor

Thickness

(lm)

Electrode

area (lm�lm)

ZnO 350 2 300�300

AlN 400 2 300�300

Liquid loading

Temperature

(� C)

Density

(kg/m3)

Viscosity

(mPa�S)

Acetone 20 792.5 0.31

Water 20 1000 1

Olive oil 20 920 84

094511-3 Qin et al. J. Appl. Phys. 110, 094511 (2011)

intensity, e0kp are the piezoelectric stress constants, eSik are the

permittivity constants under constant strain, q is the density

of AlN, and ui is the displacement in the direction of x0i(i; k ¼ 1; 2; 3 and p; q ¼ 1; 2; 3; 4; 5; 6). The resonator vibra-

tion can be treated as one dimension problem considering its

high ratio of lateral dimensions to the thickness of the reso-

nator. Hence, we can assume

@Ti

@x01

¼ @Ti

@x02

¼ 0;

E1 ¼ E2 ¼ 0; E3 6¼ 0;

@D3

@x03

¼ 0:

(5)

Under a sinusoidal excitation, the voltage V and current Iacross the FBAR can be determined by

V ¼ðh

0

E3dx03; I ¼ jxAD3; (6)

where A is electrode area and x is angle frequency. For trac-

tion forces at the boundary (x03 ¼ 0; h) we have

T5ð0Þ ¼ T3ð0Þ ¼ T5ðhÞ ¼ T3ðhÞ ¼ 0: (7)

Also we found

cE0

34 ¼ cE0

45 ¼ e034 ¼ 0: (8)

From equations (1) to (8), the impedance of dual mode

FBAR can be solved,

Z ¼ 1

jxC0

ð1� ðkLÞ2 tanðcL=2ÞcL=2

� ðkSÞ2 tanðcS=2ÞcS=2

Þ; (9a)

C0 ¼eS0

33A

h; (9b)

cL ¼xh

vðLÞ; cS ¼

xh

vðSÞ; (9c)

vðLÞ ¼ cE033 þ cE0

55

2qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficE0

33 � cE055

2q

!2

þ cE035

q

!2vuut

264

375

1=2

; (9d)

vðSÞ ¼ cE033 þ cE0

55

2q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficE0

33 � cE055

2q

!2

þ cE035

q

!2vuut

264

375

1=2

; (9e)

FIG. 3. (Color online) (continued)

094511-4 Qin et al. J. Appl. Phys. 110, 094511 (2011)

cE033 ¼ cE0

33 þe033

� �2

eS033

; cE035 ¼ cE0

35 þe033e

035

eS033

;

cE055 ¼ cE0

55 þe035

� �2

eS33

; (9f)

ðkLÞ2 ¼ ðeLÞ2

eS033qðvðLÞÞ

2; ðkSÞ2 ¼ ðeSÞ2

eS033qðvðSÞÞ

2; (9g)

eL ¼ e0

35 sinðaÞ þ e0

33 cosðaÞ;eS ¼ e

0

35 cosðaÞ � e0

33 sinðaÞ;(9h)

a ¼ 1

2arctan

2cE035

cE033 � cE0

55

!: (9i)

Here, f is frequency, vðLÞ and vðSÞ are the acoustic velocities

for longitudinal mode and shear mode, k2L and k2

S are defined

as the electromechanical coupling coefficient of longitudinal

and shear mode, respectively.

B. Shear mode ZnO or AlN FBAR with liquid loading

In impedance equation (9), there are two main items:

one is ðkLÞ2 tanðcL=2Þ=cL=2½ � for the longitudinal mode, and

the other one is ðkSÞ2 tanðcS=2Þ=cS=2½ � for the shear mode. It

is found that the coupling of longitudinal and shear mode is

very weak, so the resonator can be approximated to a simple

combination of two single modes.20,21 Considering the weak

coupling of the shear and longitudinal mode and the heavy

damping of the longitudinal in liquid, we only consider the

shear mode for viscous liquid loading, and the longitudinal

mode is ignored for simplification of the model.

As mentioned before, the existence of the thickness

shear mode makes ZnO and AlN FBARs possible for sensor

application in liquid. In addition, we know that for the acous-

tic sensor, the higher resonant frequency usually means

higher sensor sensitivity. Considering FBAR whose opera-

tion frequency can be as high as tens of GHz, it is expected

that sensor sensitivity will be greatly improved. Figure 2

shows the schematic of the viscosity sensor based on ZnO or

AlN FBAR. For analyzing sensor performance of shear

FBAR for viscosity measurement, the impedance expression

of shear mode FBAR with liquid loading is needed to know.

Assuming the interface of the liquid and resonator is at

x03 ¼ 0, now the boundary condition (7) changes to

T5ðhÞ ¼ T3ðhÞ ¼ T3ð0Þ ¼ 0;

T5ð0Þ ¼ �jxZLiquidu1ð0Þ;

ZLiquid ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijqLiquidxgLiquid

q;

(10)

FIG. 3. (Color online) (continued)

094511-5 Qin et al. J. Appl. Phys. 110, 094511 (2011)

where qLiquid and gLiquid are the density and viscosity of liq-

uid. The impedance of shear mode ZnO or AlN FBARs with

liquid loading finally can be solved,

Z ¼ 1

jxC0

ð1� ðkSÞ2

cS

2 tanðcS=2Þ � jZLiquid

ZS

1� ZLiquid

ZScotðcSÞ

0BB@

1CCA: (11)

III. RESULTS AND DISCUSSION

A. Simulation of impedance spectra of shear modeZnO or AlN FBARs with liquid loading

Based on equation (11) and material properties of ZnO

(Refs. 24 and 25) and AlN (Ref. 26) in (x1, x2, x3) shown in

Figs. 1(d) and 1(e), the resonator spectrum with liquid load-

ing can be simulated. Table I lists the parameters used in the

simulation. Figures 3 and 4 show the calculated impedance

spectra of ZnO and AlN FBARs with different liquid load-

ing. In the case of no liquid loading, it can be seen that the

resonant frequencies are a little different for different c-axis

tilt angle. The resonant frequencies for ZnO FBARs are in

the range of 0.68-0.78 GHz, and the resonant frequencies of

AlN FBARs are in the range of 1.5–1.6 GHz. This can be

explained by the following fact: as shown before, the reso-

nant frequency for an ideal resonator is determined by

f0 ¼ ts=2h, and the acoustic wave velocity ts is determined

by material properties. On the other hand, due to the crystal

orientation dependence of material properties including elas-

tic constants, piezoelectric constants, and dielectric con-

stants, the acoustic velocity changes with c-axis orientation.

Hence the resonant frequencies of ZnO or AlN FBARs are

different, although they have the same thickness.

When the resonator is loaded with viscous liquid, we

can clearly see that the resonant frequencies for ZnO or AlN

FBARs decrease, and the resonant peak becomes small and

broadened, indicating that the mechanical quality factor of

the resonator is reduced due to the energy loss in liquid. In

addition, it can be found that liquid effect of olive oil on the

resonator is much bigger than that of water and acetone,

which is attributed to the higher acoustic loading of olive oil

than water and acetone.

B. Viscosity sensitivities of shear mode ZnO or AlNFBARs

To have an overall understanding of ZnO and AlN

FBARs for the viscosity sensor, we gradually increase the

loadingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqliquidgliquidp

and investigate the change of the

FIG. 3. (Color online) (continued).

094511-6 Qin et al. J. Appl. Phys. 110, 094511 (2011)

FIG. 4. (Color online) Simulation of admittance (Y¼GþjB) spectrum of FBAR based on c-axis tilted AlN thin film. G: (a) and (b); B: (c) and (d).

094511-7 Qin et al. J. Appl. Phys. 110, 094511 (2011)

FIG. 4. (Color online) (continued).

094511-8 Qin et al. J. Appl. Phys. 110, 094511 (2011)

resonant frequency, which corresponds to maximum conduct-

ance and usually adopted for sensor output. Figures 5 and 6

show the relative resonant frequency change of ZnO or AlN

FBARs with liquid loading for some specific tilt angles and

different mechanical quality factor. As shown in Figs. 5 and 6,

the resonant frequency of ZnO or AlN FBAR linearly

decreases with the liquid loadingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqliquidgliquidp

when the load-

ing is small, and changes slowly with further increased load-

ing. Also, we can see that the resonant frequency shift curves

of ZnO FBARs or AlN FBARs almost coincide for different

c-axis tilt angle and Q in the linear area, which becomes dis-

persed in a nonlinear area, especially for ZnO FBARs.

The viscosity sensitivity (S) of the shear mode ZnO or

AlN FBARs is defined below,

S ¼ 1

f0

df

dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqLiquidgLiquidp

����������

¼ 1

f0

limDg!0

DfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqLiquidðgLiquid þ DgLiquidÞ

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqLiquidgLiquidÞ

q�������

�������:(12)

Here we assume that the liquid density does not change. f0 is

the initial resonant frequency and Df is the resonant fre-

quency shift due to viscosity change DgLiquid . Tables II and

III list the viscosity sensitivities of the shear mode FBARs

based on 2 lm ZnO and AlN thin films with c-axis tilt angle

in the range of 0�–90� and under three different liquid load-

ings, where the viscosity sensitivity was calculated by calcu-

lating the resonant frequency shift for 1% viscosity change,

and resonator Q is set to 350 for ZnO and 400 for AlN. Here

we can see that the viscosity sensitivities of ZnO or AlN

FBARs are very similar with different tilt angle and liquid

loading. This is due to the fact that according to equation

(12), the viscosity sensitivity is determined by the slope of

the resonant frequency versusffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqliquidgliquidp

, and as shown in

Figs. 5 and 6 the slopes of both ZnO and AlN FBARs for dif-

ferent angles and Q(>50) are almost the same in the linear

loading area, which covers these three different loadings.

Hence, we can conclude similar viscosity sensitivities other

than the case of Q¼ 350 for ZnO and 400 for AlN, only if

the viscosity loading is in the linear area and the Q is not less

than 50.

As shown in above discussion, if we measure the imped-

ance spectrum of the resonator using the impedance ana-

lyzer, then we can determine the resonant frequency as the

sensor output. The effect of Q(>50) in the sensor sensitivity

for viscosity measurement is small for both ZnO and AlN

FBARs. However, if an oscillator circuit is used for measure-

ment, high Q is required. With the decrease of Q, the maxi-

mum impedance phase rotation (umax) of the resonator is

reduced, and if phase rotation does not reach 0, excitation

FIG. 5. (Color online) Relative resonant frequency change of ZnO FBARs

as a function of liquid loading and resonator mechanical quality factor.

FIG. 6. (Color online) Relative resonant frequency change of AlN FBARs

as a function of liquid loading and resonator mechanical quality factor.

TABLE II. Viscosity sensitivity of 2 lm shear mode ZnO FBARs.

Viscosity sensitivity S (10�3 kg m�3 Pa�S)�0.5

c-axis (�)Liquid 15 30 43 60 75 90

water 0.95 0.95 0.93 0.91 0.94 0.97

Acetone 0.95 0.95 0.93 0.91 0.94 0.97

Olive oil 0.95 0.95 0.93 0.92 0.94 0.97

TABLE III. Viscosity sensitivity of 2 lm shear mode AlN FBARs.

Viscosity sensitivity S (10�3 kg m�3 Pa�S)�0.5

c-axis (�)Liquid 15 30 46.1 60 75 90

water 1.12 1.13 1.12 1.11 1.11 1.13

Acetone 1.12 1.13 1.12 1.11 1.12 1.13

Olive oil 1.13 1.13 1.12 1.11 1.12 1.13

094511-9 Qin et al. J. Appl. Phys. 110, 094511 (2011)

with an oscillator becomes much more difficult.8,27 For

example, for ZnO FBARs with c-axis tilt angle of 90�, the

minimum Q (Qmin) to satisfy umax ¼ 0 for no loading and

water loading is 36 and 86, separately (shown in Fig. 7); for

AlN FBARs with c-axis tilt angle 90�, Qmin is 100 and 131

for nonloading and olive oil loading separately (shown in

Fig. 8). Hence, for sensor application, the resonator with

high Q is always preferred.

For comparison, we also calculated the viscosity sensitiv-

ity of a 6 MHz quartz thickness shear mode resonator, which

has a value around 1.43 e-4 (kg m�3 Pa�S)�0.5. The viscosity

sensitivities of the shear mode 2 lm ZnO FBARs and AlN

FBARs are around 8 and 7 times as the 6 MHz quartz resona-

tor separately, which seems a little low considering the huge

difference of their resonant frequencies to QCM. One possible

reason is the definition of viscosity sensitivity, where the

FIG. 7. (Color online) Impedance/phase

spectrum of ZnO FBARs as function of

resonator mechanical quality factor.

FIG. 8. (Color online) Impedance/phase

spectrum of AlN FBARs as a function of

resonator mechanical quality factor.

094511-10 Qin et al. J. Appl. Phys. 110, 094511 (2011)

relative frequency shift is used instead of absolute frequency

shift, hence lowering the effect of operation frequency on the

value of viscosity sensitivity during calculation.

IV. CONCLUSIONS

In summary, viscosity sensitivity of ZnO and AlN FBARs

with c-axis tilt angle has been theoretically studied. The im-

pedance expression of shear mode ZnO and AlN FBARs with

a liquid layer has been derived from the basic piezoelectric

constitutive equations. Viscosity sensitivities were achieved by

the calculation of the relative resonant frequency shift for 1%

viscosity change. The viscosity sensitivities of shear mode

2 lm ZnO and AlN FBARs have a value in the range of

0.91 e-3 to 0.97 e-3 (kg m�3 Pa�S)�0.5 and in the range of

1.12 e�3 to 1.13 e-3(kg m�3 Pa�S)�0.5 separately, and they do

not change much with c-axis tilt angle, the mechanical quality

factor of the resonator and the different liquid loading only if

the viscosity loading is in linear area and the Q is not less than

50. However, the mechanical quality factor Q has a great effect

on resonator impedance, and high Q is required for sensor

application if an oscillator circuit is used for measurement. It is

found that the viscosity sensitivity of 2 lm ZnO and AlN

FBAR has not one order of magnitude higher than 6 MHz

QCM, which may be due to the adopted relative frequency

change in the viscosity sensitivity definition. The simulation

results can be used for FBARs design and sensor application.

ACKNOWLEDGMENTS

The authors would like to acknowledge the financial

support by the U.S. National Science Foundation under

Award No. ECCS-0925716. This work is also partially sup-

ported by the Natural Science Foundation of China (NSFC)

under Project No. 51028202.

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