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\title{Theoretical phase diagram of ultrathin films of
incipient ferroelectrics}


\author{A. R. Akbarzadeh$^{1,2}$, L. Bellaiche$^{2}$, Jorge
\'I\~niguez$^{3}$, and David Vanderbilt$^{4}$}

\address{$^{1}$Department of Materials Science and Engineering,
University of California, Los Angeles, P.O. Box 951595, Los
Angeles, California 90095-1595, USA}
\address{$^{2}$Physics Department, University of Arkansas,
Fayetteville, AR 72701, Arkansas, USA}
\address{$^{3}$Institut de Ciencia de Materials de Barcelona
(ICMAB-CSIC), Campus UAB, 08193 Bellaterra, Spain}
\address{$^{4}$Department of Physics and Astronomy, Rutgers
University, Piscataway, New Jersey 08854-8019, USA}


\date{\today}

%\maketitle

\begin{abstract}
We have used a first-principles-based scheme to compute the
temperature-versus-misfit strain ``Pertsev'' phase diagram of
ultrathin films of {\sl incipient} ferroelectric KTaO$_3$. Our results
suggest that, at variance with the bulk material, KTaO$_3$ ultrathin
films cannot be described as quantum paraelectrics. Rather, the
behavior of the films is largely determined by surface/interface
effects that favor ferroelectricity and the imperfect screening of the
depolarizing fields. This leads to Pertsev phase diagrams that are
qualitatively similar to those of {\sl normal} ferroelectrics such as
BaTiO$_3$.
\end{abstract}

%\pacs{68.55.-a,77.22.Ej,77.80.Bh,77.84.Dy,81.30.Dz}

\narrowtext

\maketitle

\marginparwidth 2.7in \marginparsep 0.5in

The need for miniaturized devices and the quest for knowledge in
nanoscience have led to a flurry of activities in ferroelectric thin
films (see Refs.~\onlinecite{Dawber,ScottScience,Igorreview} and references
therein).  As a result, the influence of surface/interface
effects and boundary conditions on their
properties is now well documented.  For
instance, studies revealed that the strain arising from the
substrate can lead to ferroelectric phases absent in the
bulk~\cite{Bo,pertsev,dieguez212101}, and that residual depolarizing fields can
 generate anomalous effects -- such as stripe domains with remarkably small
periods~\cite{Streiffer,IgorPRL,Inna}.

An important issue that remains to be fully understood concerns the
properties of ferroelectric films made of incipient
ferroelectrics, i.e., materials such as SrTiO$_3$ and KTaO$_3$ for
which the bulk phase ferroelectricity is suppressed by quantum
fluctuations~\cite{viana94,KTa04,slace94}. The pioneering work in
Ref.~\onlinecite{Haeni:Nature758} revealed the existence of
room-temperature ferroelectricity in {\it strained} 500\,\AA-thick SrTiO$_3$
films, and can thus be taken to suggest that the coupling between
strain and dipoles tends to prevail over quantum effects in
nanostructures. One may thus wonder what the effect (if any) of
quantum zero-point vibrations is in ultrathin films made of incipient
ferroelectrics. In particular, the following questions are, to the
best of our knowledge, currently unanswered: 
(1)  Can quantum fluctuations suppress some
phases in these films, and, if so, which ones?
(2) For zero misfit strain, are the films
paraelectric or ferroelectric? (3) What are the
consequences and signatures of the zero-point motion of the ions in a
film experiencing a depolarizing field?

The aim of this Letter is to answer these questions by predicting 
the temperature-versus-misfit strain phase diagrams of
KTaO$_3$ ultrathin films under different electrical boundary
conditions. 

Here, we use a first-principles-based effective Hamiltonian 
approach to study KTaO$_3$ ultrathin films grown along the [001] 
pseudo-cubic direction (chosen to be along the z-axis), having K-O 
terminated surfaces/interfaces, and being 28\,\AA\ thick. The films are modeled by
10$\times$10$\times$7 supercells that are periodic along the x- and
y-axes (which lie along the [100] and [010] pseudo-cubic directions,
respectively) and contain seven TaO$_2$ (001) layers stacked along the
non-periodic z-axis. We make use of an effective-Hamiltonian expansion
$\mathcal{E}_{\rm Heff}({\mathbf u_{\rm i}},
{\mathbf v_{\rm i}},\eta)$,
where the ${\mathbf u_{\rm i}}$ are the local soft-mode amplitudes
in the unit cell {\rm i} of
the film (such that $Z^* \mathbf{u_{\rm i}}$ yields the local
electrical dipole), ${\bf \{ v_{\rm i}\}}$ are the
inhomogeneous strains within the supercell~\cite{zhon95}, and $\eta$ is
the homogeneous strain tensor.  The latter is relevant to
mechanical boundary conditions since epitaxial (001) films are
associated with the freezing of some components of $\eta$, i.e., 
$\eta_6$\,=\,0 and $\eta_1$\,=\,$\eta_2$\,=\,$\delta$, with
$\delta$ being the value forcing the film to adopt the in-plane
lattice constant of the
substrate~\cite{pertsev,dieguez212101,IgorPRL}. In practice,
$\delta$=$(a_{\rm sub}-a_{\rm KTa})/a_{\rm KTa}$, where $a_{\rm sub}$
is the in-plane lattice parameter of the substrate and $a_{\rm
KTa}$\,=\,3.983\,\AA\ is the 0\,K cubic lattice constant of bulk
KTaO$_3$ used in Ref.~\onlinecite{KTa04}. 

The analytical expression \cite{zhon95} and first-principles-derived
parameters \cite{KTa04} for this $\mathcal{E}_{\rm Heff}$, the intrinsic 
effective-Hamiltonian energy of the film, are those of {\it bulk} KTaO$_3$,
except that here we use the formula for the dipole-dipole interactions for
thin films under ideal open-circuit (OC) boundary conditions derived in
Refs.~\onlinecite{InnaPRB} and \onlinecite{Ivancondmat}. Such
electrical boundary conditions naturally lead to the existence of a
maximum depolarizing field inside the film (denoted by $\mathbf E_{\rm
d}$) when the dipoles point along the [001] direction.  To treat more
general electrical boundary conditions in which there is partial
screening of this depolarizing field, we actually minimize the total-energy
function
%
\begin{equation}\label{eq:eq1}
\mathcal{E}_{\rm tot}=\mathcal{E}_{\rm Heff}({\mathbf u_{\rm i}},{\mathbf v_{\rm i}},\eta)+
   \beta \mathbf{E_{\rm d}} \cdot \sum_{\rm i} Z^* \mathbf{u_{\rm i}} \;,
\label{eq:1}
\end{equation}
%
where $\beta$ is a screening parameter defined such that
$\beta$\,=\,0 corresponds to ideal OC 
conditions, an increase in $\beta$ lowers the magnitude of the resulting 
depolarizing field, and $\beta$\,=\,1 corresponds to ideal short-circuit (SC) 
conditions for which the depolarizing field vanishes.  Note that 
$\mathbf E_{\rm d}$ is derived at an atomistic level~\cite{InnaPRB}, 
and is self-consistently updated during the simulations.
Also note that, as in Ref. \onlinecite{Bo}, our $\mathcal{E}_{\rm tot}$ assumes
a simple truncation, at the surface/interface layers, of the
interactions existing in bulk KTaO$_3$.  Such approximation results in 
an enhancement of the surface/interface polar modes for electrical boundary
conditions close to SC, because the truncation removes short-range
interactions that oppose the onset of local dipoles. Interestingly, such enhancement 
of the surface/interface polar modes has been reported in some low-dimensional 
systems~\cite{footnotetrunc} (see, e.g, the {\sl ab initio} results for 
PbTiO$_3$ films sandwiched by Pt electrodes~\cite{sai} and the 
experimental and first-principles works in ferroelectric  BaTiO$_3$ 
nanowires~\cite{spanier06}). 
%
%\vspace{-1.5 cm}
\begin{figure}[htb]
 \includegraphics[height=0.25\textheight, width=0.45\textwidth]{fig1.eps}
%\vspace{-1.5 cm}
\caption{Temperature \emph{versus} misfit strain diagram of a
28\,\AA-thick KTaO$_3$ ultrathin film under ideal SC conditions
($\beta=1$). Solid and open symbols refer to CMC and PI-QMC
simulations, respectively. Serving as guides for the eyes, the
solid and dashed lines are linear fits of the CMC and PI-QMC data,
respectively.  The inset shows a magnified view of the region in which
the {\sl r} phase occurs. PI-QMC simulations performed at temperatures higher than
100\,K, at 0.5\,\% and 1.3\,\% misfit tensile strains, are indicated
with arrows in Fig.~1, and reveal that the high-temperature {\sl
p}--to-{\sl c} and {\sl p}--to-{\sl aa} phase transitions are
unaffected by quantum fluctuations.}\label{f:Fig1}
\end{figure}

As in Ref.~\onlinecite{KTa04}, $\mathcal{E}_{\rm tot}$ is used in two
different kinds of Monte-Carlo (MC) simulations: classical Monte Carlo
(CMC)~\cite{Metropolis}, which does not take into account zero-point
vibrations, and path-integral quantum Monte Carlo
(PI-QMC)~\cite{zhong,jorge,Ceperely95}, which includes the
quantum-mechanical zero-point motions.  We typically used 30,000 MC sweeps to
thermalize the system and 70,000 more to compute averages, except at
low temperature in PI-QMC where more statistics are needed.

Figure~1 displays the ``Pertsev'' (that is, the temperature
\emph{versus} misfit strain $\delta$) phase diagram for the
28\,\AA-thick KTaO$_3$ thin film under ideal SC conditions, as
resulting from CMC  and PI-QMC 
simulations.  One can clearly see that both kinds of simulations
generate the four phases that also appear in the Pertsev diagram of
(001) BaTiO$_3$ ultrathin films~\cite{Bo,dieguez212101}. These phases
are: a paraelectric {\sl p} state at high temperatures; a
ferroelectric tetragonal {\sl c} phase for intermediate temperatures
and compressive and weakly tensile strains, having a polarization
lying along the [001] growth direction; a ferroelectric orthorhombic
{\sl aa} phase for low temperatures and large tensile strains, in
which the polarization is parallel to the in-plane [110] direction;
and a ferroelectric monoclinic {\sl r} phase for the lowest
temperatures and intermediate tensile strains, for which the
polarization direction continuously rotates from [001] to [110] as
$\delta$ increases.

Figure~1 further displays four remarkable features. (1) Quantum
effects begin to appear at temperatures below $\approx$\,100\,K. As a
result, the {\sl
c}--to--{\sl r} and, to a lesser extent, the {\sl aa}--to--{\sl r}
phase boundaries are the {\it only} boundaries affected by quantum
fluctuations. Essentially, there is a small reduction of the strain range in
which the {\sl r} phase occurs: in the low-temperature limit, it
passes from about +0.4\,\%$\leq\,\delta\,\leq$\,+1.3\,\% at a
classical level to about +0.5\,\%\,$\leq\,\delta\,\leq$\,+1.3\,\% when
quantum fluctuations are included. (2) As in BaTiO$_3$ ultrathin
films~\cite{Bo,dieguez212101}, the {\sl p}, {\sl c}, {\sl aa}, and
{\sl r} phases {\sl meet} at a single four-phase point, which occurs
here at a temperature of about 160\,K and a {\it tensile} strain of
about 1.12\,\% for both CMC and PI-QMC simulations~\cite{footnote1}.
(3) The phase diagram is asymmetric with respect to zero misfit
strain.  Such an asymmetry is hinted in experiments on
Ba$_{0.5}$Sr$_{0.5}$TiO$_3$ films~\cite{rios} and is 
related to the enhancement of the z-component of the local dipoles
at the surfaces/interfaces \cite{footnote2}. 
(4) For zero misfit strain, the paraelectric--to--\emph{c}
transition occurs at about 460\,K for both CMC and PI-QMC simulations.
This high transition temperature ($T_{\rm C}$) 
 starkly contrasts with the behavior of the bulk material,
for which the paraelectric--to--ferroelectric transition occurs at
about 30\,K at a classical level and vanishes when quantum effects are
considered~\cite{KTa04}. The physical origin of such a high $T_{\rm C}$ lies
on the above mentioned dipole enhancement at the surface/interface of
the films. The striking consequence of such an enhancement is that
these ultrathin films are no longer incipient ferroelectrics, but
display a Pertsev phase diagram that is in essence identical to that
of a normal ferroelectric like BaTiO$_3$.
%
%\vspace{-1.5 cm}
\begin{figure}[htb]
 \includegraphics[height=0.25\textheight, width=0.45\textwidth]{fig2.eps}
% \vspace{-1.5 cm}
\caption{Temperature \emph{versus} misfit strain diagram of a
28\,\AA-thick ultrathin film KTaO$_3$ under a residual depolarizing
field ($\beta$=0.96). Solid and open symbols refer to CMC and
PI-QMC simulations, respectively.  The solid and dashed lines
are linear fits of the CMC and PI-QMC data, respectively, and are
guides for the eyes. The inset is a magnification of the region
in which the {\sl r} phase occurs, with the PI-QMC data point for
the {\sl c}--to--{\sl r} transition temperature at zero misfit strain
being a guess resulting from the interpolation of the PI-QMC
simulations for the lowest computed temperatures.}\label{f:Fig2}
\end{figure}

We now discuss how depolarizing fields, always present in reality as, e.g., 
the metallic electrodes are never ideal \cite{JavierNature}, affect the Pertsev phase
diagram of KTaO$_3$ ultrathin films. Here, we have
chosen $\beta$\,=\,0.96 in Eq.~(1), consistent with the {\sl ab initio} studies 
of Sai {\sl et al.}~\cite{sai},  which found this kind of cancellation of the 
depolarizing field in some films. The resulting Pertsev diagram is shown in
Fig.~2, for both CMC and PI-QMC simulations. By comparing the CMC
results from Figs.~1 and 2, it is clear that the depolarizing field
tends to suppress the z-component of the polarization, as expected.
Indeed, this effect dramatically decreases the {\sl p}--to--{\sl c}
transition temperatures; for instance, the transition occurs around
60\,K at zero misfit strain, i.e., the $T_{\rm C}$ is 400\,K lower
than the one obtained for ideal SC. It also considerably extends the
region of small tensile strain associated with the $aa$ phase 
and significantly narrows the {\sl r-phase}
region. As a result, the phase diagram becomes more symmetric around the zero
misfit strain and the four-phase point of Fig.~1 shifts towards lower
temperature (about 50\,K for CMC simulations~\cite{footnote1}).  As
shown in Fig.~2, such a small transition temperature implies that the
quantum fluctuations (which are appreciable only at temperatures below
120\,K, and become more pronounced as the temperature decreases) have
two effects that do {\it not} occur in the film under SC
conditions.  First of all, the zero-point vibrations now affect the
{\sl p}--to--{\sl c} and {\sl p}--to--{\sl aa} phase boundaries (for
small strain) -- in addition to the boundaries involving the {\sl r}
phase. Secondly, quantum fluctuations significantly reduce the
stability range of the {\sl r-phase}. This suggests that, for slightly
worse electrodes (i.e., for $\beta\,<$\,0.96), quantum effects could
well lead to the disappearance of the four-phase point in favor of a
{\it three}-phase point in which ``only'' the {\sl p}, {\sl c}, and
{\sl aa} meet.
%

In summary, we have used a first-principles-based scheme and found that
quantum fluctuations have a negligible effect for {\it ideal} electrodes: the 
phase diagram closely resembles what is found in
films made of {\sl strong} ferroelectrics, e.g. BaTiO$_3$. We have also simulated realistic (imperfect)
electrodes and found that the corresponding depolarizing fields have a
great impact on the calculated phase diagram; in particular, the
transition temperatures of the {\sl c} and {\sl r} phases are
significantly reduced. As a result, quantum mechanical effects become
more important in this case, and may alter the phase diagram
qualitatively, e.g., by suppressing the {\sl r} phase.

We acknowledge useful discussions with George A. Samara and Kevin Leung.  This work
is supported by ONR grants N00014-01-1-0365, N00014-04-1-0413,
 and N00014-05-1-0054, by NSF grant DMR-0404335,
and by DOE grant DE-FG02-05ER46188. J.I. thanks support from the
Spanish Ministry of Science and Education (FIS2006-12117-C04-01)
and FAME-NoE.
%
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\end{thebibliography}

\end{document}