# Pyroelectric response of ferroelectric nanoparticles: size effect and electric energy harvesting

**ABSTRACT** The size effect on pyroelectric response of ferroelectric nanowires and nanotubes is analyzed. The pyroelectric coefficient strongly increases with the wire radius decrease and diverges at critical radius Rcr corresponding to the size-driven transition into paraelectric phase. Size-driven enhancement of pyroelectric coupling leads to the giant pyroelectric current and voltage generation by the polarized ferroelectric nanoparticles in response to the temperature fluctuation. The maximum efficiency of the pyroelectric energy harvesting and bolometric detection is derived, and is shown to approach the Carnot limit for low temperatures. Comment: 17 pages, 4 figures, 1 Appendix

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**ABSTRACT:**We predict that ferroelectric phase can be induced by the strong intrinsic surface stress inevitably present under the curved surface in the high aspect ratio cylindrical nanoparticles of nonferroelectric binary oxides (BaO, EuO, MgO, etc). We calculated the sizes and temperature range of the ferroelectric phase in BaO nanowires. The analytical calculations were performed within Landau-Ginzburg-Devonshire theory with phenomenological parameters extracted from the first principle calculations [E. Bousquet et al, Strain-induced ferroelectricity in simple rocksalt binary oxides. arXiv:0906.4235v1] and tabulated experimental data. In accordance with our calculations BaO nanowires of radius ~(1-10) nm can be ferroelectric at room temperature (with spontaneous polarization values up to 0.5 C/m2) for the typical surface stress coefficients ~ (10-50) N/m. We hope that our prediction can stimulate both experimental studies of rocksalt binary oxides nanoparticles polar properties as well as the first principle calculations of their spontaneous dipole moment induced by the intrinsic stress under the curved surface. Comment: 13 pages, 3 figure, 2 tables, 1 appendix10/2009; -
##### Conference Paper: DLTS study of proton and electron irradiated n+p InP MOCVD mesa diodes [solar cells]

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**ABSTRACT:**A study of proton irradiated InP junctions is described. Results are presented that show that the deep level transient spectroscopy (DLTS) spectra produced by 1 MeV electrons and 3 MeV protons in InP mesa diodes made using metalorganic chemical vapor deposition (MOCVD) are essentially the same. The results also show that there are some differences in the annealing behavior of the defects, especially following minority carrier injection at low temperaturesIndium Phosphide and Related Materials, 1991., Third International Conference.; 05/1991

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Pyroelectric response of ferroelectric nanoparticles:

size effect and electric energy harvesting

A.N. Morozovska a*, E.A. Eliseev b, G.S. Svechnikov a, and S.V.Kalininc†

aV. Lashkarev Institute of Semiconductor Physics, National Academy of Sciences of Ukraine,

41, pr. Nauki, 03028 Kiev, Ukraine

bInstitute of Problems of Materials Science, National Academy of Sciences of Ukraine,

3, Krjijanovskogo, 03142 Kiev, Ukraine

cOak Ridge National Laboratory, Oak Ridge, TN 37831

Abstract

The size effect on pyroelectric response of ferroelectric nanowires and nanotubes is analyzed. The

pyroelectric coefficient strongly increases with the wire radius decrease and diverges at critical radius Rcr

corresponding to the size-driven transition into paraelectric phase. Size-driven enhancement of

pyroelectric coupling leads to the giant pyroelectric current and voltage generation by the polarized

ferroelectric nanoparticles in response to the temperature fluctuation. The maximum efficiency of the

pyroelectric energy harvesting and bolometric detection is derived, and is shown to approach the Carnot

limit for low temperatures.

Keywords: pyroelectric response, ferroelectric nanowires, size effects, surface energy, size-driven phase

transitions

I. Introduction

The unique features of nanosized piezo-, pyro- and ferroelectrics enable a broad spectrum of

thermo-electrical, electro-mechanical, electronic and dielectric properties for sensors and actuators,

compact electronics, pyrosensors and thermal imaging [1, 2]. Piezoelectric nanowires have been studied

as potential strain – based energy harvesting devices, in particular as a direct current generators [3, 4, 5].

* morozo@i.com.ua

† sergei2@ornl.gov

Page 2

2

In these, piezoelectric nanowires array is aligned normally to substrate and sandwiched between the

bottom substrate electrode and the top ratchet-like electrode. The acoustic excitation of piezoelectric

nanowires leads to their bending and results in charge generation. The output d.c. current value is in the

range of several pA to nA, which is enough for different nanoscale devices power supply. However, the

presence of moving parts may result in rapid degradation of the structure.

An alternative approach for ferroelectric-based energy harvesting is based on the pyroelectric

properties. Early attempts [6, 7] showed the efficiency of pyroelectric energy conversion is about ten

percents for polymeric and ceramic bulk ferroelectrics. Recently Mischenko et al. [8, 9] demonstrated the

giant electrocaloric effect, i.e. the temperature change in response to electric field applied under adiabatic

conditions, in PbZr0.95Ti0.05O3 films [8] and relaxor ferroelectric 0.9 PbMg1/3Nb2/3O3 − 0.1 PbTiO3 [9]

near the ferroelectric Curie temperatures (correspondingly 222oC and 60oC). Mischenko et al. pointed out

that the direct electrocaloric effect and pyroelectric effects are strongly enhanced in the vicinity of phase

transitions, potentially enabling efficient Peltier-type devices and efficient energy harvesting approaches.

The rapid progress in synthesis of ferroelectrics nanoparticles, in particular vertical arrays of free-

standing tubes [10], wires [11] and rods in porous template [12, 13], demonstrated their enhanced polar

properties and unusual domain structure. Then possibility to control the temperature of the phase

transitions in ferroelectric nanoparticles due to the size-driven phase transition has been studied

theoretically [14, 15, 16, 17, 18].

The size effect can be used to tune the phase transition temperatures in ferroelectric

nanostructures, thus enabling the systems with tunable giant pyroelectric response. However, unlike the

well-studied theoretically ferroelectric and dielectric properties, dynamic behavior of pyroelectric

response of ferroelectric nanostructures has not been considered, and a small number of existing

treatments have been limited to pyroelectric coefficient calculations in thin epitaxial films [19, 20] and

rods [21]. Here, we analyze in details the dynamic pyroelectric response of polarized ferroelectric

nanotubes and wires within Landau-Ginzburg-Devonshire phenomenological theory, and derive the

power spectrum and efficiency of idealized devices (arrays of ferroelectric nanowires or nanotubes fixed

Page 3

3

in a flat capacitor). We demonstrate that the devices are prominent candidates for the harvesting of

electric energy from different heat sources.

II. Basic equations

Bound charge excess Qpr (typically called “pyroelectric” charge) appears at the polar faces of ferroelectric

in response to the time-dependent thermal flux via the temperature variation

dtTdδ

(see Fig. 1a).

Hereinafter the variation δT(t) is regarded spatially quasi-homogeneous across the nanoparticle and small

in comparison with ambient temperature T0, namely:

()() t,TTtT,

0

rr

δ+=

and

() t,TT

0

r

δ >>

. The

temperature fluctuations

dtdT

have a known frequency spectrum

( )

ω

()

∫

0

∞

ω

•

=δ

tie dtdT dtT

.

The distinctive feature of the nanosized systems pyroelectric response is the spatial inhomogeneity

of the pyroelectric charge originated from the spatial inhomogeneity of their spontaneous polarization

distribution P3 related with the surface influence on elementary dipoles correlation (see Fig. 1a).

Pyroelectric coefficient is given by expression [1]:

( )

0

T

0

33

3

TT

ij

ij

T

u

u

P

T

P

=

∂

∂

⋅

∂

∂

+

∂

∂

=Π

. (1)

The first term

TP ∂∂3

originated from the primary pyroelectric effect related to spontaneous polarization

changes, the second term ()()

TuuP

ij ij

∂∂∂∂3

originated from the secondary pyroelectric effect related

with the possible temperature dependence of mechanical strains uij via thermal expansion. In Eq.(1) we

neglected the ternary pyroelectric effect originated from inhomogeneous temperature distribution that

leads to polarization variation as

()

0 T

=

T

j

T

kl ijkli

xTafP

∂∂=δ

(fijkl is the flexoelectric effect tensor, thermal

expansion coefficients are denoted as

T

ij a ). Under low enough

jxT ∂∂

, the ternary effect is negligibly

small in pyroelectrics and ferroelectrics in comparison with the primary and even secondary effects.

Below we calculate pyroelectric response of ferroelectric nanoparticles within Landau-Ginzburg-

Devonshire phenomenological theory.

Page 4

4

y

x

z

R

Electrode 1 with area Se

l

Ze

C

EMF

Pyroelectric-EMF

Upr

Jpr

N

Electrode 2 with area Se

/Substrate

Temperature

variation dT/dt

Diode

(if any)

Pyro-current

Jpr

N

r

Rectification

(if any)

r

P3

(a)

(b)

(c)

P3

Upr≠ ≠ 0

U = 0

t1

P3

t2

T = T0

T = T0+δT

+Qpr

Qpr

t1

t2

x

R

−R

0

-Qpr

FIG. 1. (Color online) (a) Generation of the pyroelectric charge Qpr in the electrodes around a

ferroelectric nanorod with inhomogeneous spontaneous polarization P3 under its temperature variation on

δT(t). The lower plot schematically shows the charge Qpr profile across the rod at time moments t1 and t2.

(b) Pyroelectric electromotive force (EMF) and current generation by ferroelectric nanostructures: a

single polarized rod (with or without rigid core) and nanowires vertically-aligned array in contact with the

electrode plates. (c) Equivalent circuit of operating pyroelectric electromotive force: C is the effective

capacitance of the capacitor, r is the diode (if any appeared due to the possible rectification effect at the

ferroelectric-semiconductor-electrode interface), Ze is the external load impedance, at that

( )

ω

pre pr

n

pr

CUiZUJ

ω−≈

for small internal resistance of current source.

Correct phenomenological description of nanosized system requires the consideration of

appropriate surface energy. Including the surface energy term FS, Landau-Ginzburg-Devonshire free

energy F depends on the chosen order parameter – spontaneous polarization component P3 and

mechanical strains uij as [15]:

Page 5

5

()

+−

+−∇+

γ

6

+

β

4

+

α

2

+

α

=

∫

V

∫

S

klij

ijkl

2

ij ij

d

3

e

S

uu

c

Puq

E

EPP

g

2

PPPrdPrdF

22

2

3 333

2

3

6

3

4

3

2

3

32

3

2

(2)

Typically the surface energy coefficient

S

α is regarded positive, isotropic and weakly temperature

dependent, thus the terms ~P34 can be neglected in the surface energy expansion. Integration in the first

and the second terms of Eq. (2) is performed over the system surface S and volume V correspondingly.

Expansion coefficient β > 0 for the second order phase transitions, γ > 0 and the gradient coefficient

g > 0. Coefficient

()

CT

TTT

−α=α

)( , T is temperature; TC is Curie temperature of bulk material. The

stiffness tensor cijkl is positively defined, qijkl stands for the electrostriction stress tensor. Ee is the external

electric field. Considering high aspect ratio cylindrical nanoparticles (high aspect ratio nanoellipsoids,

nanotubes or nanowires with length l much higher than radius R) with spontaneous polarization directed

along the cylinder axes z we will neglect the effects of depolarization field Ed in Eq.(1).

Minimization of the free energy on polarization and strain components gives the equations of

state. These equations are supplemented with Maxwell equations for electrostatic electric field and

compatibility conditions for strain and equilibrium conditions

0

=∂σ∂

i ij

x

for stress components.

The intrinsic surface stress

S

αβ

µ exists under the curved surface of solid body and determines the

excess pressure on the surface [14-17]. The surface stress tensor

S

αβ

µ is defined as the derivative of the

surface energy on the deformation tensor. Intrinsic mechanical stress under curved surface is determined

by the tensor of intrinsic surface stress as

α αα

µ−=σ

Rnn

S

j

S

jkk

, where

α

R are the main curvatures of

surface free of facets and edges in continuum media approximation,

k n are the components of the

external normal.

The strain field inside cylindrical ferroelectric nanoparticles is rather complicated because of the

spatially distributed polarization. Using the Saint-Venant principle one could get the quantitatively correct

physical picture (except the immediate vicinity of the faces z = 0 and z = l) and derived appropriate

analytical expressions for the strain field uij in nanoparticles (see Appendix).

Variation of the free energy functional (1) leads to the Euler-Lagrange equations:

Page 6

6

=

∈

∂

∂

λ+

=∆−γ+β+α

0

,

3

3

03

5

3

3

33

S

P

n

P

EPgPPP

RR

r

(3)

Where ∆ is Laplace operator. Note that the polarization relaxation time is extremely small (about 10-10 s)

in comparison with the temperature rates

dt dT

~ 0.01-1 K/s. Hence, we omit the time derivatives in the

Euler-Lagrange equation (3). Extrapolation length

S

g α=λ

is positive, n is the outer normal to the

surface S.

For tetragonal ferroelectric and cubic elastic symmetry groups coefficients α and β are

renormalized by thermal expansion, surface tension and strains as:

()()

−+

α

−

µ

+−+−α=α

12

11

12

11

2

2

12

R

0 33

21

4

q

c

c

q

R

ru

Q

TTaqTT

T

c

S

T

ij ijCTR

, (4a)

()

( )()

−

+−

+−

c

+

−−β≈β

2

2

12 1112 11

2

1211 12

2

1112

c

2

1112

c

11

2

2

11

2

12

1

24

22

R

r

c

qcqqcqcc

R

r

c

q

R

. (4b)

The tube outer radius is R, the inner radius is r (see Fig. 1b); uc is the strain at the interface

r

=ρ

(if any).

For the practically important case of the ferroelectric tube deposited on a rigid dielectric core, the tube

and core lattices mismatch or the difference of their thermal expansion coefficients determines uc value

(allowing for the possible strain relaxation for thick tubes). Parameter

( )()

12 1112 11

11

2c

1212 11

−

12

ccc

q

+

cqc

Q

−

=

stands

for the stress electrostriction coefficient. The second terms in Eqs.(4) originated from thermal expansion

~

T

ij a , the third terms originated from intrinsic surface stress ~

S

µ , the third terms are the strains induced

by core ~ uc, the last terms are the spontaneous strains created by inhomogeneous polarization.

Using direct variational method [14-15], the approximation for the averaged spontaneous

polarization

3 P and dielectric susceptibility

33

χ were derived as:

()

()

r

()

0

2

R

0

,

03

)(4

),(2

β

,,

TRT

TrRT

TrRP

crTR

cr

γα

T

+−+β

−α

=

, (5a)

()

()

4

3

2

30

0 33

53),(2

1

,,

PPTrRT

TrR

crT

γ+β+−α

=χ

. (5b)

Page 7

7

Hereinafter a dash over the letter stands for the averaging over the nanoparticle volume. Corresponding

temperature of the size-driven transition to the paraelectric phase acquires the form:

()()

()

−

()

1

2

0

2

2

12

11

12

11

2

2

12

2

r

2

α

21

4

,

−

+π

−

+−λ−

−+

α

+

α

µ

R

−=

rRk

rRR

rR

g

q

c

c

q

R

ru

Q

TrRT

T

T

c

T

S

C cr

. (6)

Constant k0 = 2.408 is the minimal root of Bessel function J0(k). By changing the wire radius one can tune

the transition temperature in the wide range. The first term in Eqs.(6) is the bulk transition temperature,

the second one is the contribution of intrinsic surface stress ~

S

µ , the third term is the effect of mismatch

strain uc, the last term ~g originated from correlation effects.

The size-dependent pyroelectric coefficient is calculated from Eq.(1) as:

( )

T

0

(

(

)

)

(

)

)

()

(

α

+

− γα+β−

− γα+β+βα

−≈Π

T

T

ij

ij

crTR cr

crTRRT

T

a

q

TTT

TT

33

0

2

0

0

2

3

1

48

4

. (7)

For the sake of simplicity we considered the case of the nanowire without core (i.e. r = 0). The

pyroelectric coefficient increases with the wire radius R decrease and diverges at critical radius Rcr

corresponding to the size-driven transition into paraelectric phase and then drops to zero in paraelectric

phase (see Figs. 2a,b). The value of Rcr is determined from the condition of susceptibility divergence

()

∞→χ

TRcr,

33

. Thus it is possible to tune the pyroelectric coefficient value by varying the nanowire

radius R and ambient media characteristics responsible for surface tension coefficients µS and surface

energy coefficient αS (since λ-1~αS).

The physical characteristics of the nanowire array for should be averaged over their radii R with

corresponding distribution function f(R) as

∫

min

R

=

max

R

)()( dRRfRFF. The averaging leads to the noticeable

smearing of the size dependences of pyroelectric coefficient and susceptibility, at that the divergences at

critical radius transform into maxima and to the appearance of the dispersion of pyroelectric coefficient

and susceptibility maxima position corresponding to different halfwidth δR of the sizes distribution

functions (see Fig. 2c). The smearing and dispersion increase with relative halfwidth

RR

δ

increase

(compare the curves 0, 1, 3 and 4).

Page 8

8

0

5

10

15

1

2

3

3

2

1

bulk

4

Rcr

0

5

10

0

1

2

3

4

3

2

1

Rcr

bulk

0

(c)

Radius 〈R〉 (nm)

Π3 (mC/m2K)

(b)

(b)

Radius R (nm)

(a)

〈Π3〉 (mC/m2K)

0 12

0

2

4

6

4 3

2

1

0

R/〈R〉

f(R)

(d)

05

10

15

1

2

3

3

2

1

Rcr

bulk

R (nm)

FIG. 2. (Color online) Pyroelectric coefficient Π3 vs. nanowire radius R for different surface tension

coefficients µS = 0, 1, 10 N/m and fixed length λ = 0 (curves 1-3 in plot (b)); different length λ = 0, 1, 3,

10 nm and fixed surface tension coefficients µS = 1 N/m (curves 1-4 in plot (a)). (c) Averaged

pyroelectric coefficient

3

Π

vs. the average radius 〈R〉 for different relative halfwidth

RR

δ

= 0.01,

0.25, 0.5, 1, 2 (curves 0-4) of the size distribution function f(R) shown in plot (d). Surface tension

coefficient µS = 1 N/m, length λ = 1 nm. Horizontal lines indicate Π3 of bulk material PbZr0.4Ti0.6O3.

Material parameters of PbZr0.4Ti0.6O3: αT = 4.25⋅105 m/(F K), TC = 691 K, βR = 1.44⋅108 m5/(C2F),

γ = 1.12⋅109 m9/(C4F), Q12 = −0.0295 m4/C2, room temperature T = 300oK, gradient term coefficient

g = 10−9 m3/F.

Here, we consider the maximal efficiency of pyroelectric energy harvesting device formed by

ferroelectric nanowires arranged in vertical array. The pyroelectric current

( )

t

Ππ≡π=

dt

dT

R

dt

Pd

RJpr3

232

generated by a single ferroelectric nanowire in response to the

temperature variation

dtdT

has the following power spectrum:

() ( )

ω

()

2

03

,,

~

Jpr

RTRTR

π⋅Π⋅δ=ω

•

. (8)

Page 9

9

The total current

( ) tJn

pr

and voltage )(tUpr

the produced on external loading

e

Z by pyroelectric capacitor

C filled with the array of N almost identical nanowires vertically-aligned with respect to the electrodes

have the following power spectrum:

( )

ω

( )

i

ω

()

( )() rZnlRC

T

+

R nST

−

J

e

e

n

pr

Π⋅

,

⋅ωδ

1

=

•

,

,

~

03

,

( )

ω

( )

ω

e

n

pr pr

ZJU

⋅=

~~

. (9)

The fraction of nanowires in capacitor is defined as

e SRNn

2

π=

, Se is the electrodes area (see

Fig. 1b,c). The effective capacity of the system was estimated as:

()()

lSnRnlRC

e

eff

33

,,,

0εε≈

(ε0 is

universal dielectric constant). For

)

R

considered geometry effective dielectric permittivity

()() ( )

(

1

nnnR

e

eff

33 33

1

0

1,

χ⋅ε++−ε=ε

−

coincides in both self-consistent Bruggeman and Maxwell-

Garnett approximations,

eε is the ambient dielectric permittivity, dielectric susceptibility of a single

nanorod

33

χ is given by Eq.(5b). The external load has complex impedance

()

eeee

CLirZ

ω−ω−=

1 .

Let us consider typical exponentially vanishing fluctuation of the rod temperature [22]:

()()

( )

ω

ωτ−

δ

−=δ→

≤

>τ−τ

. 0

δ−

, 0

=δ

•

i

T

T

t

ttT

T

dt

d

1

, 0, exp

0

0

. (10)

For the case of active load resistance

ee

rZ ≡

one obtains the time dependences from Eqs.(9) and (10) as

( )

t

()

()()

)

()

−

()

(

τ+

τ−−+−

Π⋅⋅δ=

rrC

trrCt

TR nSTJ

e

e

e

n

pr

expexp

,

030

and

( ) tJrtU

n

pre pr

=

)(

. The current

)(tJn

pr

and voltage )(tUpr

produced by pyroelectric nanowires in response to the temperature fluctuation (10) are

shown in Figs. 3 for different radius R and fraction n respectively. It is clear from the Fig. 3a that both

pyroelectric voltage and corresponding current Jpr ≈ Upr/re increase with nanowire radius decrease up to

the critical value Rcr ≈ 5.8 nm for chosen material parameters and high external resistance re. Fig. 3b

demonstrates the increase of pyroelectric response with increase of nanowire fraction n. Additional

calculations show that the increase of voltage Upr with radius R decrease is monotonic at constant n.

Typical form of pyroelectric current impulse in the case of small external resistance re is shown in Fig. 3c

for different nanowire radius R and fixed fraction n. As anticipated the values of Upr and Jpr increase

linearly with the temperature rate

dtdT

increase.

Page 10

10

10 -6

10 -4

Time t (sec)

10 -2

1

10 2

10 -2

0.1

1

10

10 2

R = 6 nm

R = 8 nm

R = 10 nm

R = 20 nm

dT/dt

10 -2

0.1

Time t (sec)

1

10

10 2

10 -4

10 -3

10 -2

0.1

R = 6 nm

R = 8 nm

R = 10 nm

R = 20 nm

dT/dt

10 -2

0.1

1

10

10 2

10 3

10 -4

10 -3

10 -2

0.1

n=45⋅10−4

n=45⋅10−3

n=0.045

n=0.45

dT/dt

Time t (sec)

Voltage Upr (V)

(c) re = 10 GΩ

n = 0.45

Voltage Upr (V)

(d) re = 10 GΩ

R = 6 nm

(a) re = 1kΩ, n = 0.45

Current Jpr (pA)

10 -6

10 -4

10 -2

1

10 2

10 -2

0.1

1

10

10 2

n=45⋅10−4

n=45⋅10−3

n=0.045

n=0.45

dT/dt

(b) re = 1 kΩ, R = 6 nm

Current Jpr (pA)

Time t (sec)

10 -2

0.1

1

10

10 2

10 -2

0.1

1

Time t (sec)

dT/dt (K)

t

dT/dt

Zoom

t

Pr-response

FIG. 3. (Color online) Pyroelectric current

)(tJn

pr

(a,b) and voltage )(tUpr

(c,d) vs. time t for different

nanowire radius R = 6, 8, 10, 20 nm and fixed fraction of nanowires n = 0.45 (plots a,c); different fraction

of nanowires n = 45⋅10−4, 4.5⋅10−3, 4.5⋅10−2, 0.45 and fixed radius R = 6 nm (plots b,d), small load

resistance re ≤ 1 kΩ (a,b) and high resistance re = 10 GΩ (c,d). Central inset shows the temperature

variation

dt dT

and its conversion into pyroelectric response. Wires length l = 1 µm, electrode area

Se = 0.25 mm2, temperature variation amplitude δT0 = 1 K, relaxation time τ = 10 s, λ = 0, µS = 0, εe = 1.

Other parameters are the same as in Fig. 2.

Page 11

11

The efficiency η of the power converter is defined as the ratio of electrical work given by the

system to the absorbed heat energy. Adopting the calculations made in Ref. [23] for a bulk material to the

case of nanoparticle, we obtained the estimation for the actual temperature range

cr

TTT

<<

0

:

()

()

T

()

()

(

)

TTTCT

TT

TCPT

PTT

TrR

crTRRTP cr

cr

crP crT

cr

+

T

− γα+β+βα⋅+

−

≈

−α

−α

=η

−

4)( 5 . 0

,,

22

2

3

2

3

. (11)

One should take into account the temperature dependence of lattice specific heat, e.g. the Debye law

()

∫

0

θ

−

−⋅

θ

=

T

xx

BP

dxexe

T

CTC

2

4

3

13)( (θ is characteristic Debye temperature). Expression for

polarization

()

TrRP

,,

3

was taken from Eq.(5a) neglecting the small contribution of thermal expansion

for the sake of simplicity. ),(

rRTcr

is given by Eq.(6). Estimations show that in the vicinity of the size-

driven phase transition point ),(

rRTT

cr

≈

and for materials with β < 0, the efficiency tends to the

maximal efficiency of Carnot circle: ()

),(1,,

rRTTTrR

cr

−→η

.

Contour map of efficiency in coordinates temperature-radius and its radius dependence are shown

in Figs. 4a,b. The efficiency is about several percents at room temperatures. Such low efficiency prevents

direct application of ferroelectric nanowires as the heat power converter at normal conditions. Only at low

temperatures the heat conversion into electric power by nanowires may be reasonable.

The current power frequency spectrum

( )

ω

n

pr

J~

generated it response to the temperature variation

( )

ωδ

•

T

is shown in Figs. 4c for different wire radius R. It is clear that total pyroelectric charge

() 0

~

J

=ω=

n

pr pr

Q

as well as the broadest spectrum correspond to the wire radius close to the critical one.

The total charge

pr

Q decreases and tends to constant value with the wire radius increase (compare

different curves in plot c).

Page 12

12

10

20

0

100

200

300

100

05 10 15

1

10

(b)

Radius R (nm)

Temperature T (K)

Radius R (nm)

(a)

Efficiency η (%)

1%

5%

10%

20%

50%

10 K

30 K

300 K

100 K

10 - 4

10 -3

Frequency w (Hz)

10 -2

0.1

1

0.01

0.1

1

R = 6 nm

R = 8 nm

R = 10 nm

R = 20 nm

(c) re = 1kΩ

n = 0.45

T = 300 K

Current pulse spectrum (nC)

Paraelectric phase

Paraelectric phase

FIG. 4. (Color online) (a,b) Efficiency of the power converter vs. temperature and wire radius for

parameters λ = 1 nm, µS = 1 N/m, θ = 400 K, CB = 3⋅106 J/(K⋅m3). (a) Contour map of η values in

coordinates temperature-thickness. Different curves correspond to the fixed efficiency values 1, 5, 10, 20

and 50 %. (b) Efficiency vs. radius for different temperatures T = 10, 30, 100, 300 K (figures near the

curves). (c) The current

( )

ω

n

pr

J~

power spectrum at T = 300 K. Other parameters are the same as in Figs. 2

and 3a.

Discussion

Using concrete example of nanotubes and nanowires, we consider the influence of size effect on

pyroelectric response of ferroelectric nanoparticles within phenomenological theory.

Page 13

13

We obtained that pyroelectric coefficient increases with the wire radius decrease and diverges at

critical radius Rcr corresponding to the size-driven transition into paraelectric phase. Our analytical results

predict that it is possible to tune pyroelectric coefficient value by varying the nanowire radius R and

ambient media (e.g. template material, gas or gel), since the ferroelectric-ambient interface determines the

surface energy coefficient αS. The strong size effect on pyroelectric response should appear for arbitrary

nanoparticle shape in the case when any of its sizes approach the critical ones.

We calculated that pyroelectric voltages Upr ~ 0.1 V (in open-circuit mode) and direct current

density jpr ~ 0.5 nA/mm2 (in short-circuit mode) can be generated by polarized Pb(Zr,Ti)O3 ferroelectric

nanowires array in response to a temperature variation with rate dT/dt ~ 0.1 K/s. The advantage of the

proposed ferroelectric nanowire-based device is the absence of moving parts possible due to their

pyroelectric response. The appropriate choice of the electrodes (e.g. silicon covered Au or Pt, or LSMO)

and ferroelectric-semiconductor with definite electronic properties (e.g. donor-doped BaTiO3, BiFeO3,

Pb(Zr,Ti)O3, S2P2(S,Se)6) make it possible to design Schottky barrier at the ferroelectric-semiconductor-

metal interface. Rectification effect of the Schottky barrier allows the application of ferroelectric

nanowires array fixed between the flat electrodes as the direct current generator.

Due to the size effect ferroelectric nanowires can successfully operate in the high-sensitive

pyroelectric sensors, if the halfwidth ∆R of the nanowire radius distribution function is small enough:

∆R << 〈R〉. The scattering in the radius R unavoidably leads to noticeable diffuseness of the transition

temperature Tcr and strongly smears the pyroelectric coefficient size dependence.

The absence of moving parts makes capacitors filled with ferroelectric nanowires or nanotubes

suitable for the harvesting of electric current and voltage from different heat sources.

The efficiency of pyroelectric nanoparticles used as the heat power converters into electric power

is relatively low at room temperatures (about several %). However at temperatures close to the size-

driven transition temperature Tcr the efficiency tends to the maximal Carnot cycle efficiency, the latter

can be increased at low temperatures (in particularly in the outer space).

Page 14

14

Acknowledgements

Research sponsored by Ministry of Science and Education of Ukrainian and National Science

Foundation (Materials World Network, DMR-0908718). EEA, ANM and GSS gratefully acknowledge

financial support from National Academy of Science of Ukraine and Russian Academy of Science, joint

Russian-Ukrainian grant NASU N 17-Ukr_a (RFBR N 08-02-90434). The research is supported in part

(SVK) by the Division of Scientific User Facilities, DOE BES. Authors acknowledge multiple

discussions with Profs. S.L. Bravina and N.V. Morozovskii.

Appendix

Using the Saint-Venant principle one could get the quantitatively correct physical picture (except

the immediate vicinity of the faces z = 0 and z = l) and derived appropriate analytical expressions for the

strain field in nanoparticles of tetragonal ferroelectric and cubic elastic symmetry

()

−−+

−+

−+

µ

R

ν−

Y

−−≈+

2

3

2

2

2

3

11

12

2

3

2

2

12

2

2

11

12

0 11 22 11

11221

1

22

P

R

r

P

c

q

P

R

r

Q

R

r

c

c

uTTauu

c

S

T

, (A.1a)

()

2

3 11

2

2

2

2

0 3333

12

PQ

R

r

R

r

u

RY

TTau

c

S

T

−++

µν

+−≈

,

0

13, 12, 23

=

u

. (A.1b)

Thermal expansion coefficients are denoted as

T

ij a hereinafter. The tube outer radius is R, the inner radius

is r (see Fig. 1b); uc is the strain at the interface

r

=ρ

(if any). For the practically important case of the

ferroelectric tube deposited on a rigid dielectric core, the tube and core lattices mismatch or the difference

of their thermal expansion coefficients determines uc value (allowing for the possible strain relaxation for

thick tubes). Y is the Young module, ν is the Poisson coefficient. Parameters

( )()

1211 12 11

11

2c

121211

−

12

ccc

q

+

cqc

Q

−

=

and

()

( )()

121112 11

1212

c

11

c

12

c

11

c

11

2

2

+

cqqcc

Q

−

−+

=

stand for the stress electrostriction coefficients. The first terms in

Eqs.(A.1) originated from thermal expansion ~

T

ij a , the second terms originated from intrinsic surface

stress ~

S

µ , the third terms are the strains induced by core ~ uc, the last terms are the spontaneous strains

created by inhomogeneous polarization. Hereinafter a dash over the letter stands for the averaging over

the nanoparticle volume.

Page 15

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- Available from Sergei V Kalinin · May 29, 2014
- Available from arxiv.org