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 sizedriven transition into paraelectric phase. Sizedriven 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
 [Show abstract] [Hide abstract]
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 LandauGinzburgDevonshire theory with phenomenological parameters extracted from the first principle calculations [E. Bousquet et al, Straininduced ferroelectricity in simple rocksalt binary oxides. arXiv:0906.4235v1] and tabulated experimental data. In accordance with our calculations BaO nanowires of radius ~(110) nm can be ferroelectric at room temperature (with spontaneous polarization values up to 0.5 C/m2) for the typical surface stress coefficients ~ (1050) 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;  SourceAvailable from: Ivan StarkovSergey Karmanenko, Alexander Semenov, Antonina Dedyk, Andrey Es’kov, Alexey Ivanov, Pavel Beliavskiy, Yulia Pavlova, Andrey Nikitin, Ivan Starkov, Alexander Starkov, Oleg PakhomovElectrocaloric Materials, Edited by Correia, Tatiana and Zhang, Qi, 01/2014: pages 183223; Springer Berlin Heidelberg., ISBN: 9783642402630

Conference Paper: DLTS study of proton and electron irradiated n+p InP MOCVD mesa diodes [solar cells]
[Show abstract] [Hide abstract]
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
Page 1
1
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 sizedriven transition into paraelectric phase. Sizedriven 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, sizedriven phase
transitions
I. Introduction
The unique features of nanosized piezo, pyro and ferroelectrics enable a broad spectrum of
thermoelectrical, electromechanical, 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 ratchetlike 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 ferroelectricbased 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 Peltiertype 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 sizedriven 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
wellstudied 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 LandauGinzburgDevonshire 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 timedependent thermal flux via the temperature variation
dtTdδ
(see Fig. 1a).
Hereinafter the variation δT(t) is regarded spatially quasihomogeneous 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 LandauGinzburg
Devonshire phenomenological theory.
Page 4
4
y
x
z
R
Electrode 1 with area Se
l
Ze
C
EMF
PyroelectricEMF
Upr
Jpr
N
Electrode 2 with area Se
/Substrate
Temperature
variation dT/dt
Diode
(if any)
Pyrocurrent
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 verticallyaligned 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
ferroelectricsemiconductorelectrode 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, LandauGinzburgDevonshire 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 [1417]. 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 SaintVenant 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 EulerLagrange 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 1010 s)
in comparison with the temperature rates
dt dT
~ 0.011 K/s. Hence, we omit the time derivatives in the
EulerLagrange 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 [1415], 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 sizedriven 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 sizedependent 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 sizedriven 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 λ1~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 13 in plot (b)); different length λ = 0, 1, 3,
10 nm and fixed surface tension coefficients µS = 1 N/m (curves 14 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 04) 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 verticallyaligned 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 selfconsistent 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
Prresponse
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 temperatureradius 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 temperaturethickness. 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 sizedriven 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 ferroelectricambient 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 opencircuit mode) and direct current
density jpr ~ 0.5 nA/mm2 (in shortcircuit 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 nanowirebased 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 ferroelectricsemiconductor with definite electronic properties (e.g. donordoped BaTiO3, BiFeO3,
Pb(Zr,Ti)O3, S2P2(S,Se)6) make it possible to design Schottky barrier at the ferroelectricsemiconductor
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 highsensitive
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, DMR0908718). EEA, ANM and GSS gratefully acknowledge
financial support from National Academy of Science of Ukraine and Russian Academy of Science, joint
RussianUkrainian grant NASU N 17Ukr_a (RFBR N 080290434). 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 SaintVenant 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.
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