\documentclass[prb,twocolumn,showpacs,amsmath,amssymb]{revtex4}
%\documentclass[prb,galley,showpacs,amsmath,amssymb]{revtex4}
\makeatletter
% Some other (several out of many) possibilities
%\documentclass[preprint,aps]{revtex4}
%\documentclass[preprint,aps,draft]{revtex4}
%\documentclass[prb]{revtex4}% Physical Review B

\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{amsmath}
\usepackage{sidecap}

\def\mi{$-$}
\def\pl{$\phantom{-}$}
\def\mysp{\;\;\;}
\def\vv{$\phantom{\vec{A}}$}

\begin{document}

\title{Theory of prospective tetrahedral perovskite ferroelectrics}
\author{Anindya Roy}
 \email{anindya@physics.rutgers.edu}
\affiliation{
Department of Physics \& Astronomy, Rutgers University,
Piscataway, NJ 08854-8019, USA
}
\author{David Vanderbilt}
% \email{dhv@physics.rutgers.edu}
\affiliation{
Department of Physics \& Astronomy, Rutgers University,
Piscataway, NJ 08854-8019, USA
}

%\date{\today}
\date{December 22, 2010}

\begin{abstract}
%
Using first-principles methods, we predict the energy landscape and
ferroelectric states of double perovskites of the form AA$'$BB$'$O$_6$
in which the atoms on both the A and B sites are arranged in rock-salt order.
%
While we are not aware of compounds that occur naturally in this
structure, we argue that they might be realizable by directed
synthesis.
%
The high-symmetry structure formed by this arrangement
belongs to the tetrahedral $F\bar{4}3m$ space group. If a ferroelectric
instability occurs, the energy landscape will tend to have minima
with the polarization along tetrahedral directions,
leading to a rhombohedral phase, or along Cartesian directions,
leading to an orthorhombic phase.
%
We find that the latter scenario applies to CaBaTiZrO$_6$ and
KCaZrNbO$_6$, which are weakly ferroelectric, and the former one
applies to  PbSnTiZrO$_6$, which is strongly ferroelectric.
%
The results are modeled with a fourth- or fifth-order Landau-Devonshire
expansion, providing good agreement with the first-principles calculations.
%
Computations of zone-center soft modes are also carried out in order
to characterize the polar and octahedral-rotation instabilities
in more detail.
%
Prospects for synthesis of ferroelectric materials belonging to this
class are discussed.
%
\end{abstract}
\pacs{
77.80.bg, % FE and AFE: compositional effects
77.84.Bw, % Dielectric, piezoelectric, ferroelectric, and antiferroelectric
          % materials: elements, oxides, ...
81.05.Zx, % New materials: theory, design, and fabrication 
71.15.Nc} % DFT: Total energy and cohesive energy calculations

\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\marginparwidth 2.7in
\marginparsep 0.5in
\def\dvm#1{\marginpar{\small DV: #1}}
\def\ar#1{\marginpar{\small AR: #1}}
\def\scr{\scriptsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\def\lll{{\langle111\rangle}}
\def\bbb{{\langle\bar1\bar1\bar1\rangle}}
\def\llo{{\langle110\rangle}}
\def\loo{{\langle100\rangle}}

%%========================================================================
\section{Introduction}
\label{sec:intro}
%%========================================================================

Recent interest in novel materials has been stimulated by
unprecedented advances both in experimental materials synthesis
and in first-principles computational methods for predicting
materials properties.  Attention has focused in particular on
functional materials that can be driven between various structural or
electronic phases having distinct properties, as for example
by the application of electric fields, magnetic fields, or strain.

Ferroelectric perovskites constitute a subset of these interesting
compounds, with their switching behavior providing potential
applications in non-volatile memories and their piezoelectric
properties making them attractive as actuators and sensors.
The perovskites
are also of considerable interest in the search for multiferroics
having strongly coupled polar and magnetic properties.
Recent work has shown that many perovskite properties can be tuned
through the application of epitaxial strain.\cite{rabe-ssms05,
ederer-prl05,dieguez-prb05,fennie-prl06,lee-prl10}

Ferroelectric perovskite oxides can generally be classified into
those derived from cubic symmetry, meaning that the actual or putative
high-temperature symmetric phase is cubic, or those derived from
tetragonal symmetry, meaning that the high-symmetry phase is tetragonal.
Most well-known ferroelectrics, including BaTiO$_3$, KNbO$_3$, PbTiO$_3$,
and BiFeO$_3$ belong to the first class, while some layered
ferroelectrics, such as SrBi$_2$Ta$_2$O$_9$,
\cite{stachiotti-prb00,tsai-jpcm03,perez-mato-prb04}
belong to the latter.  It is possible for the compositional ordering
in tetragonal layered systems to break inversion symmetry, as for
example in ``tricolor'' superlattices.\cite{sai-prl00, waru-prl03,
lee-nat05, wu-prl08}
To our knowledge, however, there are no perovskite oxides in which
the chemical composition corresponds to a {\it tetrahedral}
high-symmetry structure.  While some boracites
(M$_3$B$_7$O$_{13}$X, where M is usually a divalent metal and X is
usually a halogen) such as Ni$_3$B$_7$O$_{13}$I are realizations
of this kind of tetrahedral ferroelectric system,\cite{ascher-jap66,
schmid-pss70, nelmes-jphysc74}
it would be very interesting to see the same symmetry class
represented in the better-known cases of perovskite oxides.

The present work is motivated by the idea that perovskite oxides
having a tetrahedral high-symmetry structure might be realized
experimentally and might have interesting ferroelectric or other
physical properties.  With this in mind, we have carried out a
computational study, based on first-principles density-functional
calculations, of perovskites having a tetrahedral compositional
symmetry.  In particular, we focus on AA$'$BB$'$O$_6$ double
perovskites in which both A and B sites exhibit rock-salt order.
While rock-salt order on the B site of a double perovskite is
common, it is quite rare on the A site, where a 50\% mixing of
two atoms typically leads to (001) layered ordering if it
orders at all.  Nevertheless, even if the tetrahedral symmetry
is not realized in the equilibrium phase diagram, we shall argue
that experimental routes to the directed synthesis of such
double-rock-salt tetrahedral AA$'$BB$'$O$_6$ perovskites may
be available.

The paper is organized as follows. Sec.~\ref{sec:tetra-perov}
introduces the structure of the  AA$'$BB$'$O$_6$ perovskite
materials, describes the possible symmetry-determined directions
of the polarization, and discusses possible domain types
and their symmetries.  We then briefly detail our theoretical
methods in Sec.~\ref{sec:methods}.
Sec.~\ref{sec:results} the describes the results of our
first-principles calculations and presents a comparison
with a simple empirical model.  We also discuss
the effects of rotations of oxygen octahedra, and present some
results concerning a Mn-containing double perovskite in which
magnetic ordering is also an issue.
Finally, in Sec.~\ref{sec:dis-sum} we summarize our work
and present our conclusions.

\vspace{1cm}

%%========================================================================
\section{Tetrahedral AA$'$BB$'$O$_6$ double perovskites with
rock-salt order}
\label{sec:tetra-perov}
%%========================================================================

We consider here a class of perovskite ferroelectrics whose
compositional symmetry, and thus the high-temperature symmetric
phase, is tetrahedral, instead of cubic or tetragonal.  The
simplest way to arrive at tetrahedral symmetry in the perovskite
system is to populate both A and B sites with two different kinds
of atoms (A and A$'$, and B and B$'$, respectively) arranged in
rock-salt (three-dimensional checkerboard)
order. The crystal chemistry of the perovskites, while preferring rock-salt
ordering for B sites, resists the same on the A sites, where layered
(001) ordering is preferred instead.\cite{knapp-jssc06,davies-ar08} To our
knowledge, no perovskite oxides exhibiting {\it simultaneous} A- and
B-site rock-salt ordering has been reported.  Nevertheless, we
shall investigate their properties theoretically here.

The structure in question is illustrated in Fig.~\ref{fig:struct},
in which oxygen atoms are suppressed for clarity.  It can be seen
that the point symmetry of each atom is tetrahedral, and since there
is only one formula unit per primitive cell, this also establishes
the crystal point group as tetrahedral, with the $F\bar{4}3m$ space
group (216).
This then identifies the parent high-symmetry structure, and
can be compared with the $Pm3m$ high-symmetry structure that characterizes
most ordinary perovskite ferroelectrics.

\begin{figure}
\includegraphics[width=2.4in]{dd-roy2.eps}
\caption{\label{fig:struct} (Color online) Structure of
  AA$'$BB$'$O$_6$ double perovskites, with A (red circles) and
  A$'$ (magenta squares) atoms forming one rock-salt framework,
  and B (blue diamonds) and B$'$ (green triangles) atoms forming a
  second interpenetrating one.  Oxygen atoms are not shown,
  although one oxygen octahedron is outlined to clarify their
  role in the structure.  Dashed lines illustrate the tetrahedral
  point symmetry.
}
\end{figure}

In such a tetrahedral ferroelectric, the symmetry of the
energy landscape $E({\bf P})$ is such that $E$ will be stationary
with respect to the direction of $\bf P$ for $\bf P$ along
$\lll$, $\bbb$, and $\loo$
directions.  If there are local minima
along these directions, they correspond to the rhombohedral $R3m$
space group (160) in the first two cases, or the orthorhombic $Imm2$
space group (44) in the last case.  This is illustrated in
Fig.~\ref{fig:pol-fig}, where panels (a-c) illustrate the usual
case of cubic perovskites, while panels (d-f) show the corresponding
possibilities in the tetrahedral perovskite system.
If the rhombohedral directions are energetically favored, as
in panels (d-e), then there will be two distinct types of
rhombohedral domains with different energies, and the dependence
of the total energy on $P$ for ${\bf P}=P\hat{n}$ along the body diagonal
$\hat{n}$ will display an asymmetric double-well potential.
This is in sharp contrast with systems like A$_2$BB$'$O$_6$ or
AA$'$B$_2$O$_6$, in which only one sublattice has rock-salt order;
the symmetry is then that of panels (a-c), and the magnitude of
the polarization does not change when the polarization is reversed
during a domain switching event.

\begin{figure}
\includegraphics[width=3.4in]{pol_fig.eps}
\caption{\label{fig:pol-fig} (Color online)
Symmetry-determined possibilities for
  directions of energy minima in the space of polarization,
  represented by red dots. Fig.~(a)-(c) cubic composition leading to
  high-symmetry $Pm3m$ structure, and (d)-(f) tetrahedral composition
  leading to high-symmetry $F\bar{4}3m$ structure. 
}
\end{figure}

Even if these perovskites with double rock-salt do not exist as
equilibrium phases, it may be possible to synthesize them using
atomic-layer epitaxy techniques.  In order to access the desired
compositional order, this would need to be done by epitaxial
growth on the $(111)$ perovskite surface orientation.  As a proof
of principle, Rijnders, Blok and Blank have recently reported such
a growth of SrCaTiMnO$_6$ and CaBiTiFeO$_6$ films on LaAlO$_3$ and
SrTiO$_3$ substrates.\cite{blok-sctm}   Interestingly, a symmetry
analysis shows that a $(111)$ uniaxial strain, of the kind that
results from a lattice mismatch in this kind of epitaxial geometry,
would convert the system to a polar space group by selecting out one of
the body-diagonals as special, resulting in a polarization even
when none is present in the high-symmetry tetrahedral structure.
A related effect, in which improper ferroelectricity can be
induced by octahedral rotations around a body-diagonal direction,
will be discussed in Sec.~\ref{sec:rot}.

%%========================================================================
\section{Computational details}
\label{sec:methods}
%%========================================================================

The calculations are performed using density-functional theory (DFT)
as implemented in the ABINIT code package.\cite{ABINIT}
We use Ceperley-Alder\cite{ceperley-prb78,ceperley-prl80}
exchange-correlation in the
Perdew-Wang\cite{perdew-prb92} parameterization, and Troullier-Martin
norm-conserving pseudopotentials constructed using the FHI98PP
\cite{fuchs-cpc99} code.
A plane-wave cutoff of 70 Hartree is applied. The Brillouin zone of
the 10-atom fcc cell is
sampled by a $6\times6\times6$ Monkhorst-Pack \cite{monkhorst-prb76}
$k$-point mesh, while that for the 20-atom unit cell is
$6\times6\times4$.
(Energy differences between a low-symmetry and a high-symmetry structures
are always computed using an identical cell and k-point mesh.)
A stress threshold of
2$\times$10$^{-2}$~GPa is used for cell relaxation, and forces on ions
are converged below 2.5$\times$10$^{-3}$~eV/\AA.  The electric
polarization is calculated using the Berry-phase
approach. \cite{king-smith-prb93} To avoid any potential confusion in
the branch choice for the polarization of non-orthogonal unit cells,
we also estimated the polarization using the computed Born effective
charge tensors and atomic displacements, finding good agreement.

In addition to ground-state relaxation calculations, we also use
density-functional perturbation theory to compute the frequencies
of the zone-center phonon modes, as an aid in identifying polar or
nonpolar (e.g., octahedral rotation) instabilities.
To this end, the phonon frequencies at the $\Gamma$ point of the
10-atom cell (corresponding to both $\Gamma$ and $R$ points of
a 5-atom cubic cell) were
calculated, and the corresponding soft-mode eigenvectors were
analyzed for any unstable modes having imaginary frequency.
The plane-wave cutoff and other details were the same as for the
ground-state calculations.

%%========================================================================
\section{Results}
\label{sec:results}
%%========================================================================

%%-------------------------------------------
\subsection{High-symmetry states}
%%-------------------------------------------

\begin{table}
\caption{\label{table:compcell} Compounds for which calculations
have been carried out.  Second column gives the acronym that we use
to identify the compound.  Lattice constants are calculated in
the high-symmetry $F\bar{4}3m$ structure and are reported in terms
of an effective 5-atom cubic cell dimension.  ``Average'' refers to
the average computed lattice constant of the ABO$_3$, A$'$BO$_3$,
AB$'$O$_3$, and A$'$B$'$O$_3$ parent materials.}
\begin{ruledtabular}
\begin{tabular}{lccc}
Compound & Alias & \multicolumn{2}{c}{Lattice constant (a.u.)} \\
         &  & Calculated & Average \\
\hline
PbSnTiZrO$_6$ & PSTZ & 7.47 & 7.47 \\
KCaZrNbO$_6$ & KCZN  & 7.55  & 7.55 \\
CaBaTiZrO$_6$ & CBTZ & 7.54 & 7.53 \\
KSrTiNbO$_6$  &  KSTN & 7.35 & 7.33  \\
KBaTiNbO$_6$  &  KBTN & 7.47 &  7.48 \\
SrCaTiMnO$_6$  & SCTM & 7.15 & 7.20 \\

\end{tabular}
\end{ruledtabular}
\end{table}

As mentioned in Sec.~\ref{sec:intro}, rock-salt ordering is not a
common form of ordering on the A site of mixed perovskites, and we are
not aware of any naturally-occurring
AA$'$BB$'$O$_6$ double perovskites that exhibit
rock-salt ordering on both A and B sites.  Therefore, our first step
has been to carry out a theoretical search for potential candidate
materials of this kind.  Even if the double rock-salt ordering is not
the ground-state equilibrium structure for a given material, it could
be a candidate for attempts at directed experimental synthesis.  Thus,
we have carried out calculations for a variety of compounds, placing
each in the high-symmetry $F\bar{4}3m$ structure and minimizing the
energy with respect to the lattice constant.  We look for materials
that are insulating, and select possible candidates showing a range of
ionic sizes, or combinations of cationic charges. For example, we
consider combinations where both parent perovskites have +2 and +4
cations, or those with one parent perovskite having +1 and +5 cations
while the other has +2 and +4 cations.
This search led us to focus on the six candidate materials that are listed
in Table~\ref{table:compcell}.  The second column of this
Table gives the alias by which each compound will be denoted in the
remainder of the paper (``PSTZ'' for PbSnTiZrO$_6$, etc.).

The third column of Table~\ref{table:compcell} presents our
calculated lattice constants for the AA$'$BB$'$O$_6$ double perovskites.
In order to put these in context, we also
calculate the cell constants of all relevant parent
perovskites in their high-symmetry $Pm3m$ structure.  Then, for
each of these six compounds, we average the calculated lattice
constants of the four parent materials
(even if some were occasionally metallic)
and present the result as the last column of Table~\ref{table:compcell}.
In each case we find that the cell constant of the double perovskite
is very close to average of the parents. For PSTZ, for example,
we find that the parents PbTiO$_3$, PbZrO$_3$, SnTiO$_3$ and SnZrO$_3$ have
cell constants of 7.30, 7.70, 7.23 and 7.66 a.u. respectively. The
corresponding arithmetic average is 7.47 a.u., essentially the same
as the calculated value for PSTZ. The fact that the agreement is so
good for
all six cases indicates that the volumes of the parent simple
perovskites essentially determine the cell constants of the double
perovskites in the rock-salt structure.  (Note that the results
reported for SCTM are calculated in the high-symmetry ferromagnetic
spin state, even though the antiferromagnetic structure is lower
in energy; we do this to stay in the spirit of reporting the
high-symmetry behavior here. More realistic spin structures will
be considered in Sec.~\ref{sec:sctm}.)

%%-------------------------------------------
\subsection{Exploration of polar instabilities}
\label{sec:polcalc}
%%-------------------------------------------

\begin{table}
\caption{\label{table:cell_phon} Frequency and
principal character of lowest zone-center mode, and same for next-higher mode,
in the $F\bar{4}3m$ structure for AA$'$BB$'$O$_6$ materials.}
\begin{ruledtabular}
\begin{tabular}{lcccc}
 & \multicolumn{2}{c}{Phonon I} &
                   \multicolumn{2}{c}{Phonon II} \\
AA$'$BB$'$ & $\omega$ (cm$^{-1}$)
& char. & $\omega$ (cm$^{-1}$)& char. \\
\hline
PSTZ & 182$i$ & O only & 160$i$ & A$'$, O \\
KCZN  & 195$i$ & O only & 96$i$ & A$'$, O \\
CBTZ & 147$i$ & O only & 75$i$ & A, O \\
KSTN  &  103$i$  &  O only& & \\
KBTN  &  149$\phantom{i}$ &  A, A$'$, O & &   \\
SCTM  & 172$i$ & O only &  &  \\
\end{tabular}
\end{ruledtabular}
\end{table}

Next we look for ferroelectric instabilities in these systems by
checking the high-symmetry $F\bar{4}3m$ structures to see if there
are any phonon modes with imaginary frequency.  Because of the
high symmetry, all phonon modes at $\Gamma$ have three-fold
degeneracy.  After identifying and discarding the triplet
corresponding to the zero mode (uniform translation), we report the
lowest relevant mode frequencies in the second and fourth columns of
Table~\ref{table:cell_phon}.  We also inspect the mode eigenvectors
and report the character of these modes in the third and fifth
columns of the Table.

We find that KBTN does not exhibit any unstable modes, suggesting that
it is probably stable (or, at least, metastable) in the
$F\bar{4}3m$ structure.
Turning now to the other five materials, we see that there are {\it
two} sets of unstable soft modes for the first three materials,
while there is only one for KSTN and SCTM. Moreover, the most unstable
mode always has character only on oxygen atoms, indicating that it
corresponds to a pattern of octahedral tilts or rotations.  Since this
occurs at the $\Gamma$ point of the 10-atom cell, it corresponds to an
$R$-point instability of the 5-atom parent perovskite.  (We shall
consider rotational instabilities further in Sec.~\ref{sec:rot}.)
However, if we look at the other set of unstable modes for each of the
first three materials, we observe a large contribution coming from the
smaller of the A ions, signaling that they are polar (i.e.,
infrared-active) distortions. We henceforth focus our attention on an
in-depth study of the first three materials, namely CBTZ, KCZN and
PSTZ.

As discussed in Sec.~\ref{sec:tetra-perov}, a ferroelectric
distortion along one of the Cartesian directions leads to a
polarized structure in the $Imm2$ space group, while the evolution
of a polarization along the $\lll$ or $\bbb$ directions
leads to the $R3m$ space group.  For each material we follow the
system into its local symmetry-constrained ground state for each
type of distortion, and we also compute the electric polarization
in this state using the Berry-phase method.  (Polarizations found
by multiplying computed $Z^*$ values times computed displacements
differ only slightly from those calculated using the Berry-phase
approach.)  The results are presented in the first four
columns of Table~\ref{table:ckp}, where the energies are reported
relative to that of the high-symmetry $F\bar{4}3m$ structure.
(Note that $\lll$ denotes the direction from an A atom to a B
neighbor, while $\bbb$ points to a B$'$ neighbor.)

\begin{table*}
\caption{\label{table:ckp}Calculated polarization and
  total-energy reduction (relative to the high-symmetry $F\bar{4}3m$
  structure, per 10-atom cell) for distorted structures of CBTZ, KCZN,
  and PSTZ.  Last four columns present results obtained from the
  models discussed in Sec.~\ref{sec:model} (values in parentheses are
  exact by construction, as they were used as input to the fit).}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{l c c c c c c c}
& & \multicolumn{2}{c}{{\it Ab-initio} results} & \multicolumn{4}{c}
{Landau-Devonshire model} \\
& Space& Polarization & Energy &
  \multicolumn{2}{c}{Fourth order} &
  \multicolumn{2}{c}{Fifth order} \\
Material & group & C/m$^2$ & meV & meV & \% error & meV & \% error \\
\hline
CBTZ & $R3m \lll $ & 0.137 & 10.8 & 11.2 & 3 & (10.8)& (0) \\
& $R3m \bbb$ & 0.136 & 11.5 & 11.1 & 3 & 11.5 & 0 \\
& $Imm2 \loo$ & 0.163 & 16.0 & (16.0) & (0) & (16.0) & (0) \\
\hline
KCZN & $R3m \lll $ & 0.186 & 24.9 & 26.4 & 6 & (24.9) & (0) \\
& $R3m \bbb $ & 0.184 & 27.2 & 26.0 & 4 & 27.5 & 1 \\
& $Imm2 \loo $ & 0.219 & 36.6 & (36.6)& (0) & (36.6) & (0) \\
\hline
PSTZ & $R3m \lll $ & 0.777 & 681.4 & 713.1 & 5 & (681.4) & (0) \\
& $R3m \bbb $ & 0.837 & 794.0 & 799.6 & 1 & 839.5 & 6 \\
& $Imm2 \loo $ & 0.709 & 581.8 & (581.8) & (0) & (581.8) & (0)
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table*}

We see that for CBTZ and KCZN the $Imm2$ structure is energetically
preferred over the $R3m$ structures, and the polarizations are also
larger for the $Imm2$ structure.  In these materials, the energies and
polarizations are also very similar for the structures distorted along
$\lll$ and $\bbb$ directions.  On the other hand, the PSTZ system
behaves very differently. It is strongly polar, with the depth of the
double-well potential, at 0.6-0.8\,eV, being more than an order of
magnitude larger than for the other two materials.
Furthermore, the $R3m$ structure denoted as $\bbb$ is now the favored
structure, being significantly lower in energy than either the $\lll$
$R3m$ structure or the $Imm2$ structure.  This structure also has the
largest polarization, at 0.837\,C/m$^2$.

We thus see that AA$'$BB$'$O$_6$ double perovskites can have 
a rich variety of polar behaviors, ranging from ones that remain
nonpolar like KBTN, to those that are weakly polar like
CBTZ and KCZN, and finally to the case of the strongly polar
PSTZ.  In the next subsection we shall see how this diversity of
behaviors can be captured in a simple analytical model.

Before doing so, we comment briefly on the nature of the ferroelectricity
seen in these tetrahedral ferroelectrics.  For this purpose, we have
inspected the eigenvectors of the ferroelectric soft modes identified
in Table~\ref{table:cell_phon}.  Letting
$\xi_\mu$ be the sum of squares of soft-mode eigenvector components
corresponding to atoms of type $\mu$, expressed as percentages, we find
$\bm{\xi}=(\xi_A,\xi_{A'}, \xi_B,\xi_{B'}, \xi_O)$ = (0.5, 45.7, 1.4,
2.1, 50.3) for PbSnTiZrO$_6$, $\bm{\xi}$ = (0.1, 72.8, 0.2, 0.6, 26.3)
for KCaZrNbO$_6$, and $\bm{\xi}$ = (78.5, 0.5, 0.3, 0.6, 20.1) for
CaBaTiZrO$_6$. In all three cases, the ferroelectricity is found to
be A-site driven, with very little involvement of B cations.
More specifically, it is associated with a displacement of the smaller
of the the A-site cations, coupled with some oxygen motion.
This seems reasonable in retrospect, since the lattice constant of the
overall AA$'$BB$'$O$_6$ material will be expanded by the larger
A atom, leaving a ``rattling cage'' environment for the smaller one.

%%-------------------------------------------
\subsection{Theoretical modeling}
\label{sec:model}
%%-------------------------------------------

The results of our first-principles calculations of the energies
and polarizations of tetrahedral double perovskites can be modeled
by expressing the energy as a polynomial in the components of the
electric polarization, as in Landau-Devonshire theory. Symmetry
considerations exclude certain terms in the expansion, which is
then written as
%
\begin{align}
\label{eqn:expn}
E &= \alpha(P_{x}^{2} + P_{y}^{2} + P_{z}^{2}) + \gamma P_xP_yP_z
      +   \nonumber \\
&\beta(P_{x}^{4} + P_{y}^{4} + P_{z}^{4})
+ \eta(P_{x}^{2}P_{y}^{2} + P_{y}^{2}P_{z}^{2} + P_{z}^{2}P_{x}^{2})
      +  \nonumber \\
&\xi  P_{x}P_{y}P_{z} (P_{x}^{2} + P_{y}^{2} + P_{z}^{2}) + {\cal O}(P^6)
+ \cdots
\end{align}
%
where the energy is measured relative to that of the high-symmetry
$F\bar{4}3m$ space group.
(Note that the $\gamma$ and $\xi$ terms would vanish according
to cubic symmetry.)
We denote the energy and the polarization of the
orthorhombic $Imm2$ space group by $E_{\loo}$ and $P_{\loo}$
respectively.
For the rhombohedral space group $R3m$, we have correspondingly
$E_\lll$, $P_\lll$, $E_\bbb$ and $P_\bbb$.  $P_\lll$ and $P_\bbb$
represent the positive
and negative polarizations of the two minima along the body diagonals.
For ordinary ferroelectric materials these polarization values are
equal in magnitude, and so are their related energies. In the
orthorhombic case we take the $z$ axis to be the symmetry axis,
so that $P_x=P_y=0$ and $P_z=P$.
In the rhombohedral case we keep the Cartesian alignment of the
axes, such that $P_x=P_y=P_z=P/\sqrt{3}$.

Thus, from our {\it ab-initio} calculations we have the calculated
values of the six quantities $E_\lll$, $E_\bbb$, $E_{\loo}$,
$P_\lll$, $P_\bbb$, and $P_{\loo}$ that we can use to determine
the free parameters in Eq.~(\ref{eqn:expn}).
If we truncate Eq.~(\ref{eqn:expn}) at fifth order as shown, we have
six quantities to determine five parameters, thus overconstraining
the solution.  Similarly, if we truncate Eq.~(\ref{eqn:expn}) at fourth
order, we overconstrain more strongly (six constraints and four
parameters).  Going in the other direction to include sixth order
in Eq.~(\ref{eqn:expn}) would be problematic because several
invariants appear at sixth order, so that more than six constraints
would be needed to determine the system of equations.
We therefore attempted the fits with the polynomial truncated
at fourth and fifth order.  When working at fourth order, we
use all values except $E_\lll$ and $E_\bbb$ in the fit, and
then test whether we can successfully predict the values of these
two quantities.  At fifth order, we omit only $E_\bbb$,
and test this value from the fit.

The comparison of the fitted energies with those computed from the
first-principles calculations are presented in Table~\ref{table:ckp}.
Values that were included in the fit, and are therefore exact by
construction, are shown in parentheses.  We see that even the
fourth-order fit gives encouraging agreement, with a worst-case
deviation of about 6\%. However, a closer inspection reveals a
specific feature of the fourth-order calculation that is qualitatively
incorrect, namely that the fitted values of $E_\bbb$ are smaller than
those of $E_\lll$ for CBTZ and KCZN, while the DFT calculations give
the opposite trend. This relates to the fact that these two materials
have an anomalous behavior in that the {\it lower} magnitude of
polarization is associated with the {\it deeper} energy minimum when
comparing the $E_\lll$ and $E_\bbb$ distortions (PSTZ does not show
this anomalous behavior).  It turns out that the fourth-order theory
does not have enough flexibility to reproduce this behavior; at that
level of theory, it can be shown that $|\Delta E_\lll|>|\Delta
E_\bbb|$ if $|P_\lll|>|P_\bbb|$, and vice versa.  This discrepancy is
removed once we go to the fifth-order theory; as can be seen from
Table~\ref{table:ckp}, the relative magnitudes of the energies and
polarizations are now correct for CBTZ and KCZN, and the discrepancy
between the model predictions and {\it ab-initio} calculations
improves substantially.  Thus, we conclude that a fifth-order
expansion is the minimum complexity needed to give a qualitatively
correct description of the energy-polarization relations in the
class of AA$'$BB$'$O$_6$ materials under study here.
We note, however, that there are still quantitative errors for
PSTZ; these can only be removed by going to still higher order,
presumably because the larger magnitude of polarization in
PSTZ accesses higher terms in the Landau-Devonshire expansion.


%%-------------------------------------------
\subsection{Oxygen octahedral rotations}
\label{sec:rot}
%%-------------------------------------------


As we have seen in Table~\ref{table:cell_phon}, most of our investigated
materials show unstable soft modes corresponding to tilts and rotations
of the oxygen octahedra.  In our investigation of the ferroelectric
states in Sec.~\ref{sec:polcalc}, we neglected these modes by relaxing
the systems according to symmetry constraints consistent with polar,
but not rotational, instabilities.  We now consider the effect of these
rotations, which may compete with the ferroelectric distortions in
determining the ground state of the system.

We use PSTZ as a test case for this purpose.  As we saw in
Table~\ref{table:cell_phon}, PSTZ shows two unstable sets of modes in
the high-symmetry $F\bar{4}3m$ structure, a rotational instability
at 182$\,i\,$cm$^{-1}$ and a polar one at 160$\,i\,$cm$^{-1}$.
If we follow the polar mode distortion into the polar
$R3m\bbb$ structure reported previously in
Table~\ref{table:ckp}, we find that it is not a local energy minimum.
Instead, we find that this structure still has an unstable
phonon of frequency $22.7\,i\,$cm$^{-1}$ corresponding to
rotations of the octahedra about the polar $\bbb$ axis
(i.e., like an $R$-point mode of the ideal 5-atom perovskite structure).
Turning on these rotations, the system reaches
a stable $R3$ structure at small rotation angles of $\sim$0.46$^\circ$
and $\sim$0.79$^\circ$ around $\bbb$
for the octahedral rotations
centered on B and B$'$ atoms respectively. The phonon mode that was
previously unstable is now found to have a positive frequency, while
other low-lying mode frequencies and the electric polarization remain
almost unchanged. This new phase is energetically approximately equal
to the $R3m \bbb$
state.  Thus, we find that while the rotations are
present in the ground state, they are small and have little influence
on the properties of the system.

To check whether this structure is truly the global ground state, we
tested what happens if we follow a different route, i.e., starting from the
high-symmetry $F\bar{4}3m$ space group and following the path of the
unstable octahedral rotation (mode frequency 182$i$ in
Table~\ref{table:cell_phon}) without intentionally making any polar
distortion.  However, because it singles out one of the four \{111\}
axes that were previously equivalent, such a rotation immediately
converts the system to the {\it polar} rhombohedral space group $R3$.
Relaxation within this space group is then found to lead back to the
same structure we found before, in which strong
polar distortions predominate over small rotations.
Thus, we again conclude that the rotations are unimportant for PSTZ.

While the calculations above are specific to PSTZ, the symmetry
analysis is more general and hints at the possibility of obtaining
polar samples even if the dominant unstable modes have rotational
character.  In fact, there is an interesting possibility that even if
there are {\it no} unstable polar modes in the high-symmetry
$F\bar{4}3m$ structure, an unstable rotational mode could take the
system into a polar space group.  This would correspond to the
discovery of a new class of improper ferroelectrics, in which the
primary order parameter is the antiferrodistortive rotation, but in
which a polarization necessarily appears because the selection of a
rotation axis in a parent structure without inversion symmetry results in
secondary polar distortions along that rotation axis.
Moreover, it is interesting that {\it electric} fields could, at least
in principle, be used to control the selection of the {\it rotational}
domains in such a material, at least as far as selecting one of the
rotation axes from among the four possible ones.

We already have such cases at our disposal: both KSTN and SCTM
have rotational soft modes at the
high-symmetry phase, but no polar soft modes. In the case of KSTN,
allowing rotation leads to an $R3$ phase which is about 30 meV lower in
energy than the high-symmetry structure, and has a polarization
of 0.0023~C/m$^2$.  This value of polarization is indeed tiny, about
1\% of that seen in the KCZN case, which has a comparable energy
difference between $R3m$ and $F\bar{4}3m$ phases. In the next section
we discuss a similar behavior that emerges from our studies of SCTM.

%%-------------------------------------------
\subsection{Magnetic SCTM structures}
\label{sec:sctm}
%%-------------------------------------------

Recently, Rijnders, Blok and Blank\cite{blok-sctm}
have succeeded in preparing films of the double perovskite SCTM
in rock-salt order using layer-by-layer molecular-beam epitaxy
on the (111) surface of LaAlO$_3$. To our knowledge, this is the
first experimental realization of an AA$'$BB$'$ perovskite system
in the double rock-salt arrangement. 
While their initial characterization of this material does not
appear to show a polar character, we were motivated to extend our
theoretical calculations to this material in order to make contact
with the experiments.  We present our results on SCTM in more detail
in the present section.

The presence of the magnetic Mn ions makes this material distinct
from the others studied so far.  Since Sr and Ca are $2+$ and
Ti is $4+$, we find Mn in its $4+$ oxidation state.
With its $d^3$ filling in an octahedral environment, this
configuration is naturally compatible with a local spin state
in which the majority $t_{2g}$ states are filled and other $d$
states are empty.  To handle the magnetic nature of the Mn atom,
we perform collinear spin-polarized calculations, neglecting
the spin-orbit coupling.  We consider both ferromagnetic (FM)
and antiferromagnetic (AFM) spin arrangements.  The Mn atoms
reside on an FCC lattice, which is capable of exhibiting several
AFM structures, all of which are frustrated to some degree.
\cite{phani-prb80} We perform calculations on two of the most
common AFM variants, the ones of type I and II illustrated in
Fig.~\ref{fig:afm}(a) and (b) respectively.
%
\begin{figure}
\centering
\includegraphics[width=3.4in]{afm_struc.eps}
\caption{\label{fig:afm}(a) Type-I and (b) type-II configuration of SCTM.}
\end{figure}
%
The spins are aligned
ferromagnetically in-plane and antiferromagnetically out-of-plane
with respect to (001) planes in the type-I structure, and with
respect to (111) planes in the type-II structure.

We first consider the FM spin structure, for which some results were
also reported earlier in Tables
\ref{table:compcell}-\ref{table:cell_phon}. The resulting material is
found to be insulating with a relaxed lattice constant of
7.15\,a.u.\ in its high-symmetry structure.  (It also shows a
rotational instability which we have not pursued here.)  However, both
of the AFM structures considered are lower in energy, as shown in
Table~\ref{tab:spin-al}.  The AFM arrangement of spins lowers the
symmetry from $F\bar{4}3m$ to tetragonal $P\bar{4}m2$ and rhombohedral
$R3m$ space groups for type I and II ordering, respectively.  After
relaxation, the new lattice constants still closely resemble that of
the high-symmetry structure (e.g., the $c/a$ for the $p\bar{4}m2$ case
is 0.9996). It is primarily the motion of the ions off the high-symmetry
positions, brought about by the specific spin configurations, that
defines the new space groups. Both these AFM cases are found to be
insulators, with the type-II arrangement being slightly lower in
energy. Table~\ref{tab:spin-al} also provides a concise summary of the
number of parallel nearest-neighbor and next-nearest-neighbor spins
out of all such neighbors.

\begin{table}
\caption{Lattice constants, energy (relative to FM state per 20-atom
  unit cell) and fraction of parallel nearest-neighbor (NN) and
  next-nearest-neighbor (NNN) spins for three magnetic structures of
  SrCaTiMnO$_6$.}
\label{tab:spin-al}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{l c c c c}
& Latt. const. & Energy &  & \\
Structure & (a.u.) & (meV) &  NN & NNN\\
\hline
FM         &  7.15 & \;\;\;\;\,0  & 12/12\;\, & 6/6 \\
AFM (type-I) & 7.15 &  $-$50  & 4/12 & 6/6 \\
AFM (type-II) & 7.15 & $-$70  & 6/12 & 0/6 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}

Next, we find the phonon frequencies at the $\Gamma$ point
of this relaxed type-II AFM cell. There is one set of three soft
modes, but the eigenmode analysis reveals them to be oxygen zone-boundary
phonons, ruling out the possibility of a polar instability.  This
seems to be consistent with the experimental characterization of
Rijnders {\it et al.}\cite{blok-sctm}
However, instead of stopping here, we take the analysis a bit further,
following the spirit of the last section on zone-boundary rotation of oxygen
octahedra. Following the unstable mode in the$R3m$ phase, we arrive at
the polar $R3$ space group, and find it to be lower than the
$R3m$ phase by 319\,meV (per 20-atom unit cell).
As pointed out before, the $R3$ phase is
a polar phase and exhibits an improper polarization induced by rotation
of the oxygen octahedra. The calculated polarization is found to be very
small, about 0.01 C/m$^2$, and points along the $\lll$ direction.
A phonon analysis in this phase reveals that two modes still remain
soft, indicating that the ground state of the system has not yet been
reached. We expect that further relaxation along these soft modes
would lead to a lower-symmetry structure with additional octahedral
rotations, but still with a very small polarization.

While it lies beyond the scope of the present investigation, we note
that it may also be of interest to consider the effects
of epitaxial strain on the SCTM system.  This could provide
more direct contact with the experimental work on epitaxial
growth mentioned earlier, as well as possibly making contact
with recent work \cite{lee-prl10} showing that SrMnO$_3$, which
is one of the parent perovskites of SCTM, can be driven between
FM/FE and AFM/PE states via application of epitaxial strain.
These considerations are left for future investigations.


%%========================================================================
\section{Summary}
\label{sec:dis-sum}
%%========================================================================

In summary, we have carried out a first-principles study of the
properties of prospective AA$'$BB$'$O$_6$
perovskites having double
rock-salt order.  We find several candidate compounds that are
predicted to have ferroelectric instabilities associated with A-site
displacements, with PSTZ (PbSnTiZrO$_6$) being of special interest
because of its large spontaneous polarization and peculiar energy
landscape having four global minima along $\bbb$ directions and four
secondary local minima along $\lll$ directions.  Compounds in this
class may also be capable of exhibiting improper ferroelectricity
based on rotation of oxygen octahedra, again stemming from the lack of
inversion symmetry in the high-symmetry space group. We also predict
that epitaxial strain can, under appropriate conditions, induce a
polarization in an otherwise paraelectric material of this class. The
inclusion of magnetic cations may provide interesting opportunities
for the realization of novel magnetoelectric or multiferroic
materials.

While initial attempts at the synthesis of such materials have not
yet resulted in the demonstration of ferroelectricity, they do
provide an existence proof that such synthesis is possible.
We hope that the crossing of this hurdle will stimulate
attempts at synthesis by other groups, and that eventual success
will lead to novel ferroelectric materials having
interesting and potentially useful physical properties.

\acknowledgments

We wish to thank P. Woodward and K.M. Rabe for useful discussions.
The work was supported by ONR grant N00014-05-1-0054.



\bibliography{aabb}

\end{document}
