FIGURES OF MERITS OF PIEZOELECTRIC MATERIALS IN ENERGY

HARVESTERS

R. Xu1*

, S.G. Kim1

1Mechanical Engineering Department, Massachusetts Institute of Technology (MIT), Cambridge, MA,

USA

Abstract: Piezoelectric material selection is crucial in the design of piezoelectric micro energy harvesters.

Different figures of merit (FOMs) have been sought to compare and to form a preference of piezoelectric

materials depending on the applications. FOMs may include the operational bandwidth and the cost in addition to

the energy conversion efficiency at MEMS-scale. This paper reviews only FOMs that are related to energy

conversion efficiency. The FOMs for MEMS harvesters should be much different from the FOMs for bulk

piezoelectric material harvesters. After analyzing composite beam based MEMS energy harvesters, accurate

evaluation on the holistic efficiency is obtained. Also a new FOM is proposed to represent the energy conversion

capacity of the piezoelectric materials, which existing FOMs do not take into account.

Keywords: piezoelectric, energy harvesting, figure of merit (FOM)

INTRODUCTION A variety of performance metrics have been used to

compare piezoelectric materials on diverse

applications. For actuating and sensing, the

piezoelectric strain constant and piezoelectric voltage

constant are appropriate criterion while the output

voltage, power density, cost and operational

bandwidth are for energy harvesting. Among them,

energy conversion efficiency is the most important

one when designing piezoelectric MEMS energy

harvesters for real applications.

Various FOMs have been generated for evaluating

the energy conversion efficiency of piezoelectric

materials: the widely used coupling factor, which is

the ratio of stored electrical energy to input

mechanical energy (we will name it as FOM1 in this

paper) [1], the product of two piezoelectric constants,

d and g, which leaves out the input and measures the

stored electrical energy when the piezoelectric

materials are at the same stress (FOM2 in this paper)

[2], and the maximum power condition when the

piezoelectric layer is much thinner than the elastic

beam layer of a linear cantilever structure (FOM3 in

this paper) [3,4], a FOM (FOM4) based on on-

resonance and off-resonance [5].

MEMS scale energy harvesters typically have a

composite beam structure, which consists of a thin

film piezoelectric layer, an elastic substrate and other

layers of different functions. The evaluation of thin

film piezoelectric element’s performance from the

composite beam can be an issue, since the elastic

layer dominates the energy distribution in the beam.

Coupling factor cannot be applied directly, and some

have proposed a new FOM (FOM3) to characterize

the thin film piezoelectric materials [3,4].

To characterize the thin film of piezoelectric

material in a composite beam, it is necessary to study

the distribution of elastic energy in each layer of the

composite beam. This paper provides an accurate

measure of the devices’ performance (FOM5) in this

way. The output power of an energy harvester not

only depends on the efficiency of energy transduction

but the energy capacity of the piezoelectric material.

Therefore, a new FOM (FOM6), which combines the

transduction efficiency and the energy conversion

capacity, is proposed for comparing piezoelectric

materials for energy harvesting application in a more

comprehensive way.

FOMS OF PIEZOELECTRIC MATERIALS FOM1: Energy Transduction Rates

Electromechanical coupling factor is usually

adopted to assess piezoelectric materials’ transduction

efficiency. The coupling factor is defined as:

(1)

k2 = stored electrical energy / input mechanical

energy (2)

where d and e are the piezoelectric coefficients, Ey is

Young’s modulus, and εe is the dielectric permittivity. The physical meaning of the coupling factor is

illustrated graphically in Fig. 1 [1]. When a

compressive stress T is applied to a short-circuited

piezoelectric element, the induced strain increases

along the slope s33

E . Then remove the stress and keep

the element open-circuited, the strain decreases along

s33

D . Here, s33 is the elastic compliance for stress in

FOM1= k2=d2Ey

εe

=e2

εeEy

978-0-9743611-9-2/PMEMS2012/$20©2012TRF 464 PowerMEMS 2012, Atlanta, GA, USA, December 2-5, 2012

direction 3 and accompanying strain in direction 3.

The superscript E and D denote under constant

electric field (short circuit) and under constant electric

displacement (open circuit) respectively. If connecting

an electric load, the induced electrical energy will do

work to the load and the cycle completes. From the

energy point of view, W1 + W2 is the total input

mechanical energy density, W1 is the work done on

the electrical load. Coupling factor is the ratio of the

effective stored electrical energy W1 to the total input

mechanical energy W1 + W2.

Fig1. Graphical illustration of coupling factor [1].

The coupling factor FOM can be derived by

evaluating Eq. 2. When a mechanical stress is applied

to a piezoelectric element, the energy density of the

input mechanical strain energy is,

Em=1

2ε2Ey (3)

The energy density of the stored electrical energy is,

Ee=1

2εeE2 (4)

Therefore, the coupling factor can be obtained by

using its definition:

k2=Ee

Em

=εeE

2

ε2Ey

(5)

Substituting and into Eq. 5, the

coupling factor becomes,

k2= εeEyg

2

(6)

Since , Eq. 6 can be rewritten as

k2=d2Ey

εe

(7)

FOM2: Piezoelectric Coefficients Product

Priya et al. proposed a FOM by eliminating the

input mechanical energy in coupling factor [2]:

(8)

FOM2 is the product in the electrical energy

density formula [2]:

(9)

where F is the force, A is the area and ue is the area of

W1 in Fig.1:

(10)

From Eq. 9 and Eq.10, it is obvious that the

electrical energy density in the piezoelectric materials

is proportional to FOM2. FOM2 is a straightforward

measure to compare the electric energy generated by

various piezoelectric materials at the same mechanical

stress applied. It should be noted that when the same

amount of mechanical energy is injected instead of the

same stress, even if the FOM2 of a material is higher,

the generated electrical energy might be smaller if the

transduction rate of the material is low.

FOM3: Piezoelectric Thin Film Based Beam

Some have argued that the FOM1 described above

works well when the passive elastic layer can be

neglected, and if the piezoelectric layer is thin

compared with the passive elastic layer, FOM1 is not

accurate. Therefore, another FOM has been used to

measure the efficiency of piezoelectric element in an

energy harvester when the passive elastic layer is

much thicker than piezoelectric layer [3,4]:

FOM 3 =e2

εe

(11)

The origin of this FOM comes from Roundy’s

formula for maximum power that can be extracted

from the piezoelectric vibrational harvester [6],

Pmax

=1

4

e31

2

ε0ε33

T

1−νEy

m

ω0

Qtot

2a0

2 (12)

In [3,4], it is claimed that for thin film

piezoelectrics, Young’s modulus and Poisson’s ratio

of the substrate material are to be considered instead

of those of the piezoelectric film itself, the term

e31

2/ ε

0ε33

T therefore can be used as a FOM.

FOM4: On-Resonance and Off-Resonance

Oliver et al. [5] have proposed a dimensionless

FOM for piezoelectric material in energy harvesting

devices as,

FOM 4 =k31

2iQm

s11

E

on−resonance

d31

i g31

tanδ

off −resonance

(13)

where k31 is the transversal electromechanical

coupling factor, Qm is the mechanical quality factor,

s11E is the elastic compliance at the constant electric

field, d31 is the transversal piezoelectric strain constant,

g31 is the transversal piezoelectric voltage constant,

E = gσ σ = εEy

g = d / εe

FOM2 =d2

εe=

e2

Ey

2εe

d ⋅g

ue =1

2d i g( ) i

F

A

2

ue=1

2sE− s

D( ) iT 2

465

and tanδ is the loss factor. The FOM is a product of

two FOM’s representing off-resonance and on-

resonance conditions and is dimensionless.

FOM5: Piezoelectric Materials in Composite

Beams

To characterize thin piezoelectric element based

energy harvesting device, we have considered the

strain energy distribution in the composite beam, and

generated a new FOM:

FOM5 =total stored electrical energy / total input

mechanical energy

=total stored electrical energy / (mechanical

energy input to elastic layer + mechanical energy

input to piezoelectric layer) (14)

To calculate FOM5, we need to know the strain

energy in the whole beam and that in the piezoelectric

layer. The strain energy density in i-th layer of a

general multi-layer cantilever beam is,

(15)

where σ is the stress and Ei is the Young’s modulus of

i-th layer. The strain energy in i-th layer can be then

obtained,

(16)

where W is the width of the beam. The stress is

linearly distributed with respect to neutral axis:

(17)

where y is the vertical coordinate with respect to

neutral axis, ρ is the radius of curvature of the beam.

The moment generated by the force at the tip and the

radius of curvature has the relationship,

(18)

where EI is the product of effective Young’s modulus

and moment of inertia of the beam, x is the

longitudinal location along the beam, and F is the

concentrated force at the free end of the cantilever

beam. Substituting Eq.17 and Eq.18 to Eq.16, and

perform the integration, the strain energy in the i-th

layer can then be evaluated as,

(19)

The total strain energy in the beam is the sum of the

strain energy in all layers:

(20)

The FOM5 is,

FOM 5 =k2iupiezo

ui∑ (21)

To see how the elastic layer changes FOM5, we

may still use the double layer cantilever beam with

varying the elastic layer. Fig. 3 illustrates the variance

of FOM5 with increasing thickness of elastic layer for

six commonly used piezoelectric materials. The

material properties used for simulation are listed in

Table 1.

Fig2. Comparison of FOM5 with increasing thickness

of elastic layer for six different piezoelectric

materials.

Table 1. Piezoelectric material properties [7 – 11].

d33 (m/V) E (Pa) εr εy

PVDF -3.30e-11 2.00e+09 12 3.00%

AlN 3.40e-12 4.20e+11 10.4 0.15%

BaTiO3 1.49e-10 6.70e+10 1700 /

PZT 3.60e-10 6.30e+10 1700 0.20%

PZN-PT 2.00e-09 9.26e+09 5200 1%

PMN-PT 2.82e-09 8.36e+09 8200 0.12%

FOM6: Incorporating Energy Capacity

Besides transduction rate, energy capacity of

piezoelectric materials can significantly determine the

performance of the piezoelectric energy harvesters.

Since the maximum stored electrical energy of a

piezoelectric element not only depends on the

conversion efficiency but also the maximum amount

of mechanical energy the element can absorb, more

specifically, it is the product of the maximum input

mechanical energy and the energy conversion rate.

When an external energy source input energy to a

piezoelectric element, if assuming the element goes

through the ideal cycle, the input energy will be

transferred into the strain energy of the piezoelectric

element. The maximum strain energy that a

piezoelectric element can capture is limited by its

ui=1

2

σ2

Ei

ui =1

2h+ hi−1∑

h+ hi∑∫

0

L

∫σ 2

Ei

Wdydx

σ =yEi

ρ

M =EI

ρ= Fx

ui =WL

3EiF

2

6 EI( )2

y3

3

h+ hi−1∑

h+ hi∑

Ustrain

= ui

i

∑

466

maximum strain and Young’s modulus. The yield

strain should be the upper limit of the deformation to

keep the element from significant fatigue and

degradation. Assuming any strain level can be reached

in the piezoelectric element, and then the maximum

mechanical energy in a piezoelectric element is,

(22)

where uy is the maximum strain energy density, is

the stress, is the yield strain, Ey is the Young’s

modulus. With the maximum strain energy density,

we can define a new FOM – the maximum stored

electrical energy density of the piezoelectric material

after transduction, which is the product of maximum

strain energy and the conversion rate:

FOM 6 = k2iuy =

d2Ey

2ε y2

2εe

=e2ε y2

2εe

(23)

where d and e are the piezoelectric coefficients, and εe

is the dielectric permittivity.

Table 2. Summary of five main FOMs discussed in

this paper. (FOM2, FOM3 FOM5, and FOM6 are normalized

to be easily compared.) FOM1 FOM2 FOM3 FOM5* FOM6

k2iupiezo

ui∑

PVDF

0.0205

0.0936

0.0012

0.0013 0.0495

AlN

0.0527

0.0011

0.6480

0.4226 0.0669

BaTiO3

0.0989

0.0135

0.1937

0.1927 /

PZT

0.5424

0.0786

1.0000

1.0000 0.1835

PZN-PT

0.8044

0.7932

0.2180

0.2353 1.0000

PMN-

PT

0.9157

1.0000

0.2240

0.2421 0.0148

* FOM5 is coupled with device dimensions. Here, the FOM5s are

based on the cantilever beam used in this paper and the thickness

of the elastic layer is 20 times of that of the piezoelectric layer.

DISCUSSION AND CONCLUSION Table 2 summarizes five FOMs with typical values

for six widely used piezoelectric materials. This table

may serve as a guide for selecting piezoelectric

materials when designing MEMS energy harvesters.

FOM1 (coupling factor) and FOM2 indicate energy

conversion efficiency based on different inputs. When

piezoelectric materials are injected to the same

amount of mechanical energy, which is often the case

for energy harvesting, FOM1 is the right figure of

merit. While, when the materials are subjected to the

same stress, FOM2 can be used to compare the

generated electrical energy.

Both FOM3 and FOM5 are for MEMS energy

harvesters with a thin piezoelectric film and a much

thicker elastic layer, and FOM5 is the exact solution.

Table 2 shows that FOM3 agrees quite well with

FOM5. FOM3 involves simpler computation, which

makes it a convenient measure of composite beam

based energy harvesters. However, if a more accurate

evaluation is needed, FOM5 should be considered.

FOM6 provides a new perspective to designers. Not

only the energy conversion rate, but also the energy

conversion capacity is revealed by this new FOM.

Besides, the simplicity of calculation is preserved,

which makes FOM6 a desirable tool for future

material comparison.

This paper reviewed several FOMs for MEMS

energy harvesters that are related to energy conversion

efficiency. Characterization of piezoelectric thin film

based energy harvesters can be successful by selecting

proper FOM during the design stage and choosing the

right active material.

Acknowledgement Authors are grateful to MIT-

INL (International Iberian Nanotechnology

Laboratory) Program for the support of this study.

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uy = σ ε( )dε = Eyε dε =1

2Eyε y

2

0

ε y

∫0

ε y

∫σ

εy

d2Ey

εe

d2

εe

e2

εe

d2Ey

2ε y2

2εe

467