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Appl Phys A (2009) 97: 617–626
DOI 10.1007/s00339-009-5261-8
Thermodynamic modeling of critical properties of ferroelectric
superlattices in nano-scale
Yue Zheng · C.H. Woo
Received: 17 February 2009 / Accepted: 23 April 2009 / Published online: 13 May 2009
© Springer-Verlag 2009
Abstract Modeling nano-scale ferroelectric superlattices
using the Landau free-energy functional approach requires
incorporating contributions from the interfacial and depolarization field effects. The choice of the order parameter then
becomes a vital issue. In this paper, we compare the predictions of models using the spontaneous polarization as order parameter (SPOP approach) with models using the total
polarization as order parameter (TPOP approach). We have
comprehensively calculated the critical properties of nanoscale ferroelectric superlattices, such as the phase-transition
temperature, critical thickness and Curie–Weiss-type relation using both approaches. We found that all the SPOP
results are in excellent agreement with experimental measurements and first-principle calculations in all cases studied
here. The TPOP approach, on the other hand, much overestimates the depolarization by underestimating the effect of the
dielectric screening and produces results that deviate significantly from the experimental ones. Our results also traced
the dependence of the critical properties on the thicknesses
of the constituent layers of the ferroelectric superlattices to
the interfacial and depolarization field effects.
PACS 77.80.-e · 77.80.Bh · 77.84.Dy · 77.80.-e · 68.03.Cd
1 Introduction
Recent research of artificially structured ferroelectrics shows
a richness of novel physical properties [1–21]. Ferroelectric bilayers, multilayers and superlattices, such as
Y. Zheng · C.H. Woo ()
Department of Electronic and Information Engineering, The
Hong Kong Polytechnic University, Hong Kong, Hong Kong
e-mail: [email protected]
Fax: +852-2-3654703
KNbO3 /KTaO3 , PbTiO3 /SrTiO3 , BaTiO3 /SrTiO3 , PbTiO3 /
PbZrO3 , PbTiO3 /BaTiO3 show phase-transition temperatures, susceptibility, and polarization determined by the
structural periodicity, misfit strain, boundary conditions,
electrostatic interactions etc. [1, 2]. Using ultraviolet Raman
spectroscopy, Tenne et al. [3] found that the transition temperature of ferroelectric ultrathin BaTiO3 /SrTiO3 superlattices grown on SrTiO3 can change by up to 500 K, depending on the thickness of the BaTiO3 and SrTiO3 layers. This
illustrates the important roles of electrical and mechanical
boundary conditions on the critical properties of a ferroelectric superlattice. Dawber et al. [4] studied PbTiO3 /SrTiO3
superlattices grown on SrTiO3 by X-ray diffraction and
piezoelectric atomic force microscopy and found that the polarization and phase-transition temperature were functions
of the thicknesses of PbTiO3 and SrTiO3 . At the same time,
using both experiments and first-principle calculations [5],
they also found that the ferroelectricity of the superlattice
will disappear when the thickness or ratio of the PbTiO3
layer is below some critical value.
First-principle calculations, atomistic simulations and
thermodynamic models have contributed significantly to understanding the effects of artificial structures on the properties of ferroelectrics. Thermodynamic models formulated
based on the Landau free-energy expansion are popular and
well suited to studies involving properties on the ferro/paraelectric transition or near it [22, 23]. The simplicity of the
model and easy experimental comparison are among the attractive features. Indeed, it is the general validity of thermodynamics from which the strength of the model is derived.
Despite inaccuracies due to the neglect of stochastic fluctuations of a thermodynamic system [22], this approach has
been proven powerful and effective, and widely used for
investigating the properties of ferroelectrics in different dimension scales [1, 23–26].
618
Yet, when surface effects from polarization gradients,
near-surface lattice relaxation [27], and depolarization field
Ed are involved, the accuracy of the Landau approach suffers, particularly when Ed is significant such as under opencircuit boundary conditions. Very often [7], the depolarization field calculated based on the usual expression Ed =
−P T /ε0 completely overwhelms the total free energy of a
typical ferroelectric thin film of PbTiO3 or BaTiO3 . Here P T
and ε0 are the total polarization and permittivity of the vacuum, respectively. Indeed, the depolarization field in such
cases reaches strengths of ∼105 kV/cm, which are two orders of magnitude larger than the sum of all other contributions to the total free energy [12, 13, 24]. The consequences
include a downward shift in the phase-transition temperatures of an unphysical amount of ∼104 K [22, 23, 26], or
an 180◦ domain shrinkage of over 100% at very low temperature [24]. Even under short-circuit boundary conditions
(i.e. Ed = −(P T − q)/ε0 [6–13], where q is the induced
surface charge density in the electrodes), this depolarization
contribution to the total free energy is also about two orders
of magnitude larger than the sum of all the other contributions. This is true even when the deviation from uniformity
P T (z)−P T is only 1% of the polarization for bulk BaTiO3
at room temperature (∼0.25 C/m2 ) [12]. Due to the absolute
dominance of this term, a minimized total free energy cannot allow for a relative deviation of P T stronger than 10−4
from a uniform distribution, even in the presence of lattice defects, such as dislocation, cracks, interface and surfaces [6–13]. In order to minimize the non-uniformity of
the polarization distribution under boundary conditions imposed by the presence of defects with significant surface effects, interface effects, the action of the Ginzburg term [1,
6, 7, 22, 27–31] (i.e. the gradient term), would force the
polarization field to deviate significantly from zero. In this
way, the effect of the depolarization would be so overpowering that the permanent polarization that determines its ferroelectric character would be practically eliminated, buried
under its dielectric properties. The absolute dominance of
the depolarization field could lead to the quenching of domain patterns and other properties in the presence of strain
fields created by interfacial dislocations, point defects, composition gradient, bilayers, multilayers or superlattices, etc.
As a result, there has not been any successful study of multilayered structures using the Landau approach thus far. Indeed, any study in this direction cannot be very meaningful
without a good understanding of the problem and a satisfactory method of resolution.
Woo and Zheng (WZ) [26] investigated the source of the
foregoing difficulties and suggested that the problem is the
result of a serious underestimation of the dielectric screening of the electrostatic interaction between the spontaneous
polarization and the depolarization field Ed , and that the
simple solution is to use the spontaneous polarization as the
Y. Zheng, C.H. Woo
order parameter. This suggestion has been adopted by several recent works [29–33].
Indeed, although the order parameter P T used in the
Landau free-energy expansion is supposed to represent the
total polarization [6–13], an approach that we may call
the Total-Polarization-Order-Parameter (TPOP) approach,
in reality the order parameter at this stage is only an unknown of the Euler–Lagrange (E–L) equation derived from
the free-energy functional through the variational principle
(see e.g., the derivation of the time-dependent Ginzburg–
Landau equation in [27]). The dynamic characteristic of
P T , such as its instability that marks the phase transition,
is totally determined by the form and the parameters of the
E–L equation [32–34]. Thus, in order for the polarization
effects of the depolarization field to enter into the dynamic
character of the E–L equation, it must be included explicitly in the free-energy functional, and it cannot be done implicitly through the order parameter. In the TPOP approach,
the polarization induced by an applied field is only implicitly included in the free-energy functional. Its contribution
is thus not recognized and is independent of what the assigned meaning of P T is supposed to be; its effect will not
be present in the dynamical behavior of its solution P T . As a
result, there is a lack of the dielectric screening in the effect
of the depolarization field.
Using the spontaneous polarization alone as the order parameter in the Landau free-energy expansion and including
the contribution from the induced polarization explicitly in
the free-energy functional, WZ [26] have shown that its effects enter the associated E–L equation properly, and the difficulties encountered in the TPOP approach can be avoided.
For easy description, and to highlight the substantial difference between the results of the two approaches, particularly
in regard to the critical properties, we name this approach
the SPOP (Spontaneous-Polarization-Order-Parameter) approach.
In the present paper, we further compare the SPOP and
the TPOP calculations of the critical properties of ferroelectric superlattices. The results will be compared and discussed in relation to experimental observations and ab initio
calculations.
2 Formulation
2.1 The free-energy function of ferroelectrics
The formulation of the free-energy functional in the SPOP
approach has been briefly described by WZ [26]. To elaborate on the finer points for the readers’ benefit, we describe
the formulation in fuller details in the following.
In the SPOP approach, the spontaneous polarization field
P (simply referred to as the polarization in the following) is
Thermodynamic modeling of critical properties of ferroelectric superlattices in nano-scale
619
used as the order parameter. As mentioned, P is the polarization from the permanent electric moment formed from spontaneous atomic displacements generated in a dielectric when
going through a ferroelectric phase transformation. We also
consider another component of the polarization PE , which
is the result of mechanisms such as the electronic polarization, and other non-permanent displacements of the ionic
charge distribution, induced by an applied electric field E,
and which account for the dielectric screening of the effect
of E.
This description is consistent with Lines and Glass [24]
who observed that the total dielectric susceptibility χ T of
the ferroelectrics can be expressed in general as the sum of
linear and non-linear parts:
D = ε0 E + PT = ε0 E + PE + P
T
∂χ T
∂χ
χ =χ +
E+
σ
∂E σ ,T
∂σ E,T
T
∂χ
higher-order
+
T+
non-linerities
∂T E,σ
In (2) and (3), the induced component of the total polarization is explicitly taken into account. We consider as our
thermodynamic reference a crystal of infinite extent (surfaceless) without an applied field, i.e., electric, magnetic, or
mechanical (internal or external) [23–25]. The general freeenergy functional of ferroelectrics can be written as
T
0
≡ χ 0 + χ (T , E, σ )
= ε0 E + χ b E + P = ε b E + P,
(2)
where ε b is the background dielectric constant. Note that in
WZ [26], we have used the symbol ε d to denote this quantity. Indeed, in the absence of an external electric field, the
total electric field has only one contribution: that is the depolarization field Ed , i.e., E = Ed , the electric displacement
is given by
D = ε0 Ed + PT = ε0 Ed + PE + P
= ε0 Ed + χ b Ed + P = ε b Ed + P.
(1)
according to which the dependence of the susceptibility,
Curie–Weiss relation, etc. on the temperature T , stress σ
and electric field E are described by the non-linear component χ . The total polarization PT of the ferroelectrics in this
description is a sum of two components, one due to a permanent electric moment, which can be identified with P and
whose relation to T , σ and E can be expressed via χ . This is
embedded in a background material with dielectric response
to E, which is explicitly describable by a susceptibility χ 0
independent of P, and which creates an induced polarization
PE proportion to E. The sum of PE and P constitutes the total polarization PT . To a good approximation, one may consider χ 0 as the dielectric susceptibility of the ferroelectrics
in the paraelectric state far away from the phase transition
(e.g. ∂χ T /∂T = ∂χ /∂T = 0 in (1) at T Tc for E = σ = 0,
and χ T = χ 0 for T Tc ). It should thus be accurately measured at T Tc for E = σ = 0 [24, 35], where Tc is the
phase-transition temperature. To be consistent with the notation, we identify χ 0 with the background susceptibility χ b
or χ d used in our previous works. Similar discussions have
also been reported by other authors [11, 24, 26, 34, 36]. We
note that the response of P to an applied electric field E, and
for that matter, the non-linear component of the total susceptibility, is implicit in this formulation and is only realizable through the thermodynamic relation between PT and E
via the total free-energy functional, after the associated E–L
equation is solved [26].
At constant applied stress and temperature, the electric
displacement field D can thus be expressed in terms of the
spontaneous polarization as [11, 24, 26, 35, 37],
(3)
g(P, E, σ ) = g(P, E, σ )|E=0,σ =0 −
= g0 −
σ
E
u dσ −
0
D dE
0
E
u dσ −
0
σ
D dE,
(4)
0
where u and σ are the strain and stress tensor fields, respectively. Here we note that g0 is the free-energy functional of
the field-free, a uniform and infinite crystal (surfaceless and
thus with no depolarization charges) we use as reference.
In the following discussion, we limit ourselves to consider the Gibb’s free-energy density (including those of the
applied fields) in an unstressed sample at constant temperature. If the electric field in the ferroelectrics is not zero, the
thermodynamic potential can be obtained by starting from
the relation dg/dE = −D, where D is given by (1). In this
case, it follows from (2) and the fact that P has no explicit
functional dependence on E that
g(P, E) = g(P, E)|E=0 − P · E −
E
εb · E dE
0
1
= g0 − P · E − ε b E2 .
2
(5)
We note that (5) is the same as the equation derived by Landau and Lifshitz [23]. Being the free-energy functional of
the field-free (i.e., E = 0), uniform and infinite (surfaceless)
crystal, g0 can be expanded near the phase-transition instability in terms of P in the form of the Landau free-energy
functional. Then for the case in which E and P are parallel
and are both spatially uniform, substituting E = Eext + Ed
in (5) leads to
620
g(T , E) =
Y. Zheng, C.H. Woo
α0
β
γ
(T − Tc0 )P 2 + P 4 + P 6
2
4
6
1
− P (Eext + Ed ) − εb (Eext + Ed )2 ,
2
(6)
where α0 = 1/(ε0 C0 ), C0 being the Curie–Weiss constant
and ε0 the permittivity of vacuum. Tc0 is the Curie temperature and β and γ are the expansion coefficients of the free
energy of the reference crystal. It can be seen that the corresponding E–L equation will give the correct phase-transition
characteristics for E = 0.
Instead of (6), a number of authors prefer to express the
free-energy functional and depolarization field in terms of
the total polarization and adopt the TPOP approach [6–13],
in which the free-energy functional is written as
g(T , E) =
B
C
A
(T − Tc0 )P T2 + P T4 + P T6
2
4
6
1
− P T Eext − P T Ed ,
2
(7)
where the effect of the extrapolation length has been neglected. It can be seen immediately that both the free energies of the reference state and the electrostatic contributions in (6) and (7) are different. Although the free energy
expressed by (7) has the correct form in the limit of E → 0,
yet (7) no longer satisfies ∂g/∂E = −D, as it should. As
mentioned, in order for the effect of the applied field E to
be fully reflected, it should be explicitly excluded from the
order parameter when deriving the E–L equation from the
free-energy functional via the variational principle. Indeed,
it can clearly be seen that the dielectric screening effect due
to P E is totally missing from the E–L equation derived from
the free-energy expression in (7). Furthermore, ignoring the
interrelation between Ed and P T could also produce errors
in evaluating
the contributions from the internal energy of
the field D(E) dE in (5), and from the missing interaction
term with Eext in the last term of (6).
More importantly, according to (1), the application of E
produces an indirect effect on P when the variational minimum of the free-energy functional is sought. The total susceptibility χ T is then a sum of the linear part χb and nonlinear part χ , which measures this indirect effect of E on
P , and which can be obtained by solving the corresponding E–L equation. Accordingly, the dielectric properties of
ferroelectrics are governed by a Curie–Weiss-type relation
expressed as [24, 26, 28, 34],
α
0
+ χb , for T < Tc ,
T
c|
(8)
χ = χ + χb = 2|Tα−T
0
for T > Tc .
|T −Tc | + χb ,
It also gives the correct limit of the total susceptibility far away from its phase-transition temperature Tc , i.e.
χ T |T →∞ = [ α0 (T1−Tc ) + χb ]|T →∞ = χb , which should
be experimentally determined [11, 24]. For example, the
background susceptibility χb or dielectric constants εb of
PbTiO3 , BaTiO3 , SrTiO3 , CaTiO3 are of the order of 50ε0
in the limit of infinite temperature [24, 35, 36, 43], which
are also used in our following calculations. The corresponding relation obtained using the TPOP approach of (7), on the
other hand, the limit of the total susceptibility tends to zero,
i.e. χ T |T →∞ = α0 (T1−Tc ) |T →∞ = 0.
2.2 The free-energy functional of ferroelectric superlattices
In what follows, we consider the phase-transition characteristics of the ferroelectric superlattice (FS) structure made
from multilayer thin films (MTF) sandwiched between electrodes under short-circuit boundary conditions and grown
on a substrate (Fig. 1). The MTF is made of alternating layers of different materials A and B of thicknesses hA and
hB , respectively, with total thickness large compared with
the thickness L of a single period of the superlattice (i.e.,
hA + hB ) (Fig. 1b). We will use periodic boundary conditions and neglect the effects of the top and bottom interfaces
between the MTF and electrodes. In most existing superlattices grown on compressive substrate (e.g., BaTiO3 /SrTiO3 ,
PbTiO3 /SrTiO3 and BaTiO3 /PbTiO3 on SrTiO3 ), all vector
fields in the MTF are directed in the z-direction. Accordingly, axial symmetry is assumed in this work to simplify
the formulation and computation. In the STOP approach, the
depolarization fields Edi (i = A, B) in i layers are related to
Pi in the absence of Eext , and they can be obtained using the
Maxwell equations [7, 18, 19, 34, 37, 38]:
EdA = −
1 PA − P εbA
1 hA
PA −
P dz ,
=−
εbA
L −hB
1
1 PB − P εbB
1 hA
1
PB −
P dz ,
=−
εbB
L −hB
(9)
EdB = −
where εbi are the background dielectric constants of i layers.
The total free energy per unit area of a single period of
the superlattice is defined as F = FA + FB , where FA and
FB are the free energy per unit area of the layers A and B, respectively. For each layer, the free energy can be expressed
as the sum of FP , the energy contributions from the polarization, FS , the elastic energy, FC , the energy of the electrostrictive coupling between polarization and strain, FD ,
the energy of the depolarization field, and FI , the energy
due to the superlattice interfaces [14–21]. The free-energy
formulation based on (4) may be generalized accordingly.
In addition, the Landau–Ginzburg expansion can be used
Thermodynamic modeling of critical properties of ferroelectric superlattices in nano-scale
621
Fig. 1 Schematics of a an
epitaxial superlattice on a
substrate, and b the
single-period superlattice A and
B
to take into account the spatial dependence of P . Thus, for
a thick compressive substrate, FA and FB can be approximately expressed as [14–21]
hA γA 6 DA ∂PA 2
∗ 2
∗ 4
αA PA + βA PA +
P +
FA =
6 A
2
∂z
0
2
umA
1 2
dz
PA − P 2 +
+
2εbA
s11A + s12A
1
−1 2
+ D A δA
PA (0) + PA2 (hA )
2
1 − ξ PA (0) · PB (0) + PA (hA ) · PB (−hB ) ,
2
0 γB
DB ∂PB 2
FB =
αB∗ PB2 + βB∗ PB4 + PB6 +
6
2
∂z
−hB
2
umB
1 2
dz
PB − P 2 +
+
2εbB
s11B + s12B
1
+ DB δB−1 PB2 (0) + PB2 (−hB )
2
1 − ξ PB (0) · PA (0) + PB (−hB ) · PA (hA ) ,
2
(10)
∗ , β ∗ , α ∗ and
where the renormalized Landau coefficients αA
A B
∗
βB can be expressed as
∗
αA
=
α0A
2Q12A umA
,
(T − Tc0A ) −
2
s11A + s12A
βA∗ =
Q212A
βA
+
,
4
s11A + s12A
αB∗
α0B
2Q12B umB
(T − Tc0B ) −
=
,
2
s11B + s12B
βB∗ =
1/(ε0 CB ), Ci are the Curie constant of layer i. Tc0i are the
Curie–Weiss temperatures of their bulk materials. Q12i are
the electrostriction tensors. s11i and s12i are components of
the elastic compliance tensor. umi are the misfit strains between i layers and substrate. The δi are the so-called extrapolation lengths, which describe surface relaxation effects
[7, 8, 15, 16, 38, 39]. Together with the short-range coupling strength ξ between layer A and layer B [14–19], they
describe the change to the free energy due to the presence
of the interface. Note that the depolarization field and free
energy of the TPOP approach have the same form as (10)
and (11), but with εbi and Pi replaced by the permittivity
of free space, ε0 , and the total polarization, PiT [6–13], respectively. In TPOP calculations, the absolute dominance of
the depolarization field will lead to the quenching of critical and any other properties in the presence of PTO/STO
superlattice and interfaces.
The dynamic behavior of Pi is governed by the E–L
equation associated with F in (10), which in the present
case takes the form of the time-dependent Landau–Ginzburg
(TDLG) equation or Landau–Khalatnikov (LK) equation [3,
18, 19, 27, 28, 37, 38],
δF
∂PA (z, t)
= −MA
∂t
δPA (z, t)
∗
= −MA 2αA
PA + 4βA∗ PA3 + γA PA5
− DA
(11)
Q212B
βB
,
+
4
s11B + s12B
where α0i , βi and γi are the expansion coefficients of
the free energy of layer i. α0A = 1/(ε0 CA ) and α0B =
∂ 2 PA
PA
1
+
−
P
,
εbA εbA
∂z2
δF
∂PB (z, t)
= −MB
∂t
δPB (z, t)
= −MB 2αB∗ PB + 4βB∗ PB3 + γB PB5
∂ 2 PB
PB
1
− DB
+
−
P .
εbB εbB
∂z2
(12)
622
Here MA and MB are the kinetic coefficients that describe
the time delay in the evolution of the polarization
h fields.
P is the average P which can be expressed as L1 −hAB P dz.
Following recent works [3, 18, 19], we assume that the
interface between A and B is coherent. The interfacial effects approximately represented in (10) can be replaced in
(12) with the following boundary conditions at z = −hB , 0
and hA [14–17]:
∂PB (z, t) = δB−1 PB (−hB , t) − DB−1 ξ PA (hA , t),
∂z
z=−hB
∂PB (z, t) −1
−1
− = −δB PB (0, t) + DB ξ PA (0, t),
∂z
z=0
(13)
∂PA (z, t) −1
−1
=
δ
P
(0,
t)
−
D
ξ
P
(0,
t),
A
B
A
A
+
∂z
z=0
∂PA (z, t) −1
−1
= −δA
PA (hA , t) + DA
ξ PB (−hB , t).
∂z
z=hA
3 Results and discussions
In the following, we report the results of our study on the
specific example of PbTiO3 /SrTiO3 superlattice (PTO/STO)
grown on a STO substrate, which have been successfully
fabricated and characterized recently. Material constants,
such as the electrostrictive and elastic coefficients of PTO
[20, 40] and STO [41], are well known. The susceptibility and dielectric constants of background materials in
(8) and (9) [35, 37, 38, 43, 44], the extrapolation lengths
[40, 42], and the interfacial short-range coupling coefficients
in (10) [14–19] can be approximately obtained from previous work. Note that entirely accurate parameters for different ferroelectric materials, such as the dielectric constant of
background materials and extrapolation lengths, need future
work based on experiments [24, 35, 43], first-principle calculations [1, 34] or molecular dynamics [45, 46] to finish.
All of the parameters are listed in the footnote.1
Analytic expressions have been derived for the Curie
temperature and critical thickness of single ferroelectric thin
1 The
list of the parameters (IS units, the temperature T in
K). For PTO/STO: α0A = 7.6 × 105 , βA = −2.92 × 108 ,
γA = 1.56 × 109 , DA = 2.7 × 10−9 , Tc0A = 763 K, Q12A = −0.026,
s11A = 8.0 × 10−12 , s12A = −2.5 × 10−12 , α0B = 7.3 × 105 ,
βB = 1.7 × 108 , DB = 2.7 × 10−9 , Tc0B = 35.5 K, Q12B = −0.015,
s11B = 3.52 × 10−12 , s12B = −0.85 × 10−12 , ε0 = 8.85 × 10−12 F/m,
δA = δB ≈ 7 nm, ξ = 0.172, umA = −0.016, umB = 0. The
“linear-part” background susceptibilities and dielectric constants
of the ferroelectrics have been discussed in previous works by using
thermodynamic theory [26–34], phase field simulations [24, 37,
38, 44], experiments [11, 24, 35, 36, 43], and should be accurately
measured in the paraelectric state far away from the phase transition
[24, 35, 43]. The lattice constants in the paraelectric state are
as = 3.905 and ap = 3.969 for STO and PTO, respectively [4, 5].
Y. Zheng, C.H. Woo
Fig. 2 The phase-transition temperature as a function of the relative
thickness of the PTO layer
films using the TDLG [27, 28], with results showing that
the phase-transition temperatures are sensitive to the film
thickness, surface boundary conditions and the depolarization field. However, for the MTF considered here, the mathematical analysis involved would be much more complex.
We thus adopt in the present work a numerical approach
[17–19] in which we follow the near-phase-transition isothermal evolution of the polarization in a small region at a random location [34, 35]. In the ferroelectric regime, an initially
local polarized region anywhere in the MTF will spread
and grow into a stable polarization configuration over the
whole region across the interfaces. On the other hand, in the
paraelectric region, the same initially local polarized region
anywhere in the MTF is unstable and will shrink away. In
this way [17–19], [34, 35], the phase-transition temperature,
stable polarization, susceptibility, switching behavior, and
other properties of the superlattice can be determined.
3.1 The phase-transition temperature of PTO/STO
superlattice
The transition temperature of ferroelectric ultrathin superlattices is known to depend significantly on the thicknesses
of the component layers [3, 4]. In the present study, the
ferro/paraelectric transition temperature Tc of the FS is calculated as a function of the thicknesses of the STO and PTO
layers. We note here that the Curie temperatures of STO and
PTO are 35.5 K and 763 K, respectively. In Fig. 2, the calculated Tc is plotted against the relative thickness of the PTO
layer in a FS in which the total thickness of the PTO and
STO layers are kept constant at 10 nm. It can be seen that
Tc increases with the PTO thickness and decreases with increasing STO thickness. It can also be seen that a very significant shift of up to a thousand degrees of Tc is achievable
by varying the relative thicknesses of STO and PTO.
Thermodynamic modeling of critical properties of ferroelectric superlattices in nano-scale
Figure 2 shows that a minimum relative thickness (MRT)
of the PTO layer exists, which in the present case is ∼17%,
below which ferroelectricity disappears permanently. For
comparison, the corresponding experimental measurements
using X-ray diffraction [4] is also shown. It can be seen that
there is excellent agreement between the experiment and
calculated results. This agreement is even more remarkable
when we consider that there is no fitting parameter in the
calculation. We note that this dependence on the PTO relative fraction can provide a convenient means of control of
Tc via the relative thicknesses.
One may in addition be interested in comparing the foregoing results with those obtained from a similar treatment
based on the TPOP approach. The results are shown in Fig. 2
as red triangles, which can be seen to be shifted downwards
by ∼500 degrees from experimental results and from results
calculated following the SPOP approach. At the same time,
the MRT of the PTO layer obtained is over 400% larger than
that obtained from the SPOP approach. As explained earlier, these results represent a significant under-estimation of
the total polarization due to an insufficient screening of the
depolarization effects in the TPOP approach.
Indeed, even if the deviation from uniformity is as small
as ∼0.4% of the polarization in bulk PbTiO3 at room temperature (∼0.7 C/m2 ) under short-circuit boundary conditions, the total free energy is still dominated by the contribution from the depolarization field, which is about two orders of magnitude larger than the sum of all other contributions in the total free energy. This dominance of the depolarization field may cause a total quenching of the properties
of individual thin film layers. Similar results are obtained
in other theoretical calculations on BaTiO3 /SrTiO3 bilayers
using the TPOP approach [6], where the MRT obtained for
BTO is over 85%. Indeed, we have also repeated such calculations and confirm the results. The TPOP results obviously
deviate significantly from the experimental ones of Tenne et
al. [3], where the ferroelectricity of BaTiO3 /SrTiO3 superlattice is measurable even when the relative BTO thickness
is below 30%.
3.2 Polarization and critical thickness of PTO/STO
superlattice
The variation of the critical properties of the FS as a function of the relative thicknesses of the STO and PTO layers
suggests that it is an effect coming from the effects of the
interface and depolarization field. Similar effects are also
expected to affect the total polarization P T , calculated as
the sum of the local spontaneous polarization Pi and that
induced by the depolarization field, i.e. PiE = χbi Edi .
The behavior of P T at 0 K is plotted in Fig. 3 as a function of np /ns where np and ns are the thicknesses of the
PTO and STO layers, respectively, measured in terms of
623
Fig. 3 The average total polarization as a function of the ratio np /ns
of the PTO and STO layers
number of atomic planes. To facilitate comparison with the
results of ab initio calculations [5], ns is kept constant at a
value of 3, and np varies between 1 and 48. The excellent
agreement between the two results shown in Fig. 3 further
reinforces the credibility of the SPOP approach.
Our results in Fig. 3 also show the increasing strength
of the interfacial depolarization charges as the PTO thickness decreases. Below a critical ratio of np /ns the FS cannot sustain any ferroelectric characteristics any more. Consistent with the results of the ab initio calculations [5], the
critical thickness of the PTO layer is hcA ≈ 1.2 nm.
For completeness, we present the results obtained using
the TPOP approach in Fig. 3 as red triangles. Comparison
with results from the ab initio calculations of [5] shows substantial disagreement. They show that the ferroelectricity of
FS disappears at a relative thickness of the STO layer of
about 15% (instead of 50%), and the corresponding critical thickness calculated using TPOP is much larger, around
hcA ≈ 6.8 nm (instead of hcA ≈ 1.2 nm). This is consistent
with a significantly overestimated depolarization field without the inadequate screening. The modification adopted by
using the electrostatic model [5] helps to push the results
into better agreement in the regime of small np /ns . However, the polarization does not seem to help when the FS is
dominated by the PTO layer.
We have also calculated the average spontaneous polarization at 300 K as a function of the relative thickness of
the PTO layer for a fixed L = 10 nm (Fig. 4a). The data
points are derived from those in the inset in Fig. 4a, which
shows the through-thickness (i.e., as a function of z) values
of PA and PB for various relative thicknesses of the PTO
versus STO layers. It is clear that as the PTO layer gets thinner, the spontaneous polarizations decrease. For PTO in the
PTO/STO superlattice the MRT at 300 K is ∼28%. Similar
results have been discussed in [18, 19].
624
Y. Zheng, C.H. Woo
Fig. 5 The Curie–Weiss-type relation of the PTO/STO superlattice with different relative thicknesses of the PTO layer for a fixed
L = 10 nm
larization charges induced by the spontaneous polarization
of PTO layer under the short-circuit boundary conditions.
Based on results in the inset of Fig. 4b, the calculated relation between P and L at 300 K is shown as a squares dash
line in Fig. 4b. It can be seen that there is a critical value of
Lc ≈ 4 nm, below which the spontaneous polarization disappears permanently.
Fig. 4 a The relation between P and the relative thickness of the
PTO layer for a fixed L of 10 nm. The inset shows the distribution
of PA and PB in a single-period superlattice with the relative thickness of the PTO layer from 72% (shown as “
”) to 8% (shown as
”). b The relation between P and L (e.g. squares dash line
“
“
”). The inset shows the distribution of PA and PB in a single-period superlattice
To consider the effects of the interfacial and depolarization field effects, we also plot the sum P of the average
spontaneous polarizations PA and PB in individual STO
and PTO layers as a function of the single-period thickness L(= hA + hB ) at 300 K for equal thicknesses of the
two layers in Fig. 4b. Here PA and PB are expressed
h
0
as 0 A PA dz/ hA and −hB PB dz/ hB , where the throughthickness spontaneous polarizations PA and PB are shown
in the inset. It can be seen that P , as well as PA and PB ,
all decrease with decreasing L as the interface effects grow
stronger with decreasing L.
According to (9) to (13), if the depolarization field is absent (i.e., if EdA = EdB = 0), the average spontaneous polarization PB of the STO layer is near zero at room temperature. Noting that bulk PTO and STO are respectively in
the ferroelectric (PA = 0) and paraelectric (PB = 0) states
at room temperature, the increase of PB and the decrease
of PA shown in Fig. 4b is mainly the result of the depo-
3.3 The Curie–Weiss-type relation of PTO/STO
superlattice
As mentioned, the application of the applied electric field E
affects the free-energy functional and produces an indirect
effect on the function P when the variational minimum is
sought. Accordingly, the dielectric properties of FS are governed by a Curie–Weiss-type relation that can be expressed
as [24, 26–28, 31]
ε C
0
−Tc | + χb , for T < Tc ,
T
χ = χ + χb = 2|T
(14)
ε0 C
for T > Tc .
|T −Tc | + χb ,
Here the average Curie constant C and background susceptibility χb of the FS are
hB CB
hA CA
+
and
hA + hB hA + hB
hA χbA
hB χbB
+
.
χb =
hA + hB hA + hB
C =
(15)
To complete the present study, we calculate χT in the
PTO layer for different temperatures, and the results are
plotted in Fig. 5 against T − Tc where Tc is taken from
Fig. 2. It can clearly be seen that the susceptibility is controllable very well via the relative thicknesses of the layers.
Equation (11) also gives the correct limit of the susceptibility far away from Tc , i.e. χ T |T →∞ = χb , which should
Thermodynamic modeling of critical properties of ferroelectric superlattices in nano-scale
be investigated and confirmed by experiments. Moreover, it
is known that interfaces may carry charges trapped on interface states [36]. These may generate fields interacting with
the polarization, strain field and also affect the capacitance
of the superlattice.
4 Conclusions
In summary, modeling nano-scale ferroelectric superlattices
using the Landau free-energy functional approach requires
incorporating contributions from the interfacial and depolarization field effects. We have shown that the derivation of the
free-energy functional based on the Landau expansion has to
be performed with a formulation in which the spontaneous
polarization is the order parameter (SPOP approach). Formulations based on the total polarization as the order parameter (TPOP approach) likely overestimate the depolarization significantly by underestimating the dielectric screening. Based on SPOP approach, we have comprehensively
calculated and discussed the critical properties of nano-scale
ferroelectric superlattices, such as the phase-transition temperature, critical thickness and Curie–Weiss-type relation,
as functions of relative thicknesses of the constituent layers.
Our results are in good agreement with experimental measurements and first-principle calculations. At the same time,
we also show that the interfacial and depolarization field effects are the main reasons for the dependence of the critical
properties on the thicknesses of the constituent layers of the
ferroelectric superlattices.
Acknowledgements This project was supported by grants from the
Research Grants Council of the Hong Kong Special Administrative
Region (G-YX0T, N_534/08, 5305/07E). Y. Zheng is also grateful
for support from the National Science Foundation of China (Nos.
10732100, 10831160504).
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