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% --------SYMBOL DEFINITIONS ---------
\def\AO{$A_2$O$_3$}
\def\ABO{$AB$O$_3$}
\def\ABB'O{$A_2BB'$O$_6$}
\def\BTO{BaTiO$_3$}
\def\LNO{LiNbO$_3$}
\def\LTO{LiTaO$_3$}
\def\ZSO{ZnSnO$_3$}
\def\MTO{MnTiO$_3$}
\def\FTO{FeTiO$_3$}
\def\NTO{Ni$_3$TeO$_3$}
\def\LZTO{Li$_2$ZrTeO$_6$}
\def\LHTO{Li$_2$HfTeO$_6$}
\def\MFWO{Mn$_2$FeWO$_6$}
\def\MWO{Mn$_3$WO$_6$}
\def\ZFOO{Zn$_2$FeOsO$_6$}
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\begin{document}

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\title{Supplement: Ferroelectricity in corundum derivatives}

\author{Meng Ye}
\email{mengye@physics.rutgers.edu}
\affiliation{
Department of Physics \& Astronomy, Rutgers University,
Piscataway, New Jersey 08854, USA}

\author{David Vanderbilt}
\affiliation{
Department of Physics \& Astronomy, Rutgers University,
Piscataway, New Jersey 08854, USA}

%\date{\today}

\maketitle

The properties of FE materials are sensitive to atomic displacements
and strain, so it is essential to start our calculation with accurate
structural parameters. The lattice constants and Wyckoff positions
obtained from our calculations are summarized in the Tables~\ref{tab:StruABO}
and \ref{tab:StruABB'}, with experimental results provided for
reference. Our structure parameters are very close to the experimental
results.

To investigate the magnetic structures of \MFWO\ and \MWO, four different
types of unit cell are considered in the calculation. We adopt a notation
like ``$udu$" to describe the possible spin structure, where ``$u$" is
spin-up, ``$d$" is spin-down, and the spins are given on atom $\beta$,
$\delta$ and $\gamma$, in that order. The four possible states (not
counting those that are trivially related by a global spin reversal)
are $uuu$, $uud$, $udu$, and $udd$. The energy of each fully-relaxed
magnetic structure is listed in Table~\ref{tab:magGS}. Of those, the
most stable state for both \MFWO\ and \MWO\ is $udu$. However, for \MWO,
the energy difference between the $uud$ and $udu$ states is tiny, so
we considered the polarization reversal for both magnetic states.

The magnetic ground states of \MFWO\ and \MWO\ can be understood by
analyzing the super-exchange interactions between $A_1$, $A_2$ and
$B$ cations (also shown as $\beta$, $\gamma$ and $\delta$ cations).
The magnetic moments are
coupled through the oxygen octahedra, and there are three independent
coupling constants. The moments on $A_2$ and $B$ sites are coupled through
face-sharing and corner-sharing oxygen octahedra with strength $J_{A_2B}$;
the $A_1$ and $B$ moments are coupled through edge-sharing octahedra
with strength $J_{A_1B}$; and the $A_1$ and $A_2$ moments are coupled
through corner-sharing octahedra with strength $J_{A_1A_2}$. Then the
magnetic energy $E_{\rm{mag}}$ can be written as
\beq
E_{\rm{mag}} = J_{A_1B} \hat S_{A_1}\cdot \hat S_B +
J_{A_2B} \hat S_{A_2}\cdot \hat S_B + J_{A_1A_2} \hat S_{A_1}\cdot \hat S_{A_2}\,,
\label{eq:magGS}
\eeq
where $\hat S$ represent the spin on each site. Substituting the
energy of different magnetic orderings in Table \ref{tab:magGS}
into Eq.(\ref{eq:magGS}), we find that the coupling constants are all
positive. This result implies that the three magnetic moments all favor
AFM coupling. However, it is impossible to make three collinear spins
couple antiferromagnetically, and this frustration implies that one
pair must be
ferromagnetically coupled.

In Table \ref{tab:EX} we list the relative spin orientations of the magnetic ions.
Since the face-sharing coupling $J_{A_2B}$ is the strongest, it is not surprising
that the $A_2$ and $B$ moments couple antiferromagnetically; the competition
between $J_{A_1B}$ and $J_{A_1A_2}$ then determines the magnetic ground
state. In \MWO, these two couplings are comparable, so the energy
difference between the $uud$ and $udu$ states is tiny. In the case of \MFWO,
the magnetic moment on the $B$ cation is smaller, so the coupling $J_{A_1B}$
is weaker than $J_{A_1A_2}$. Therefore, the lowest-energy state is $udu$.

For the ordered-LNO materials the two $A$ cations are inequivalent,
and one of them is already closer to the oxygen plane in the ground
state ($\xi_{1 \rm S}\!\neq\xi_{2 \rm S}$).  Therefore it is
energetically favorable for this particular $A$ cation to migrate
first in the reversal path, which causes either the $B$ or $B'$ cation
to be sandwiched between two $A$ cations in the midpoint structure.
If $\xi_{1 \rm S}>\xi_{2 \rm S}$, the sandwiched cation is $B$;
otherwise it is $B'$. By calculating the energy
difference $\Delta E\!=E_B-E_{B'}$ between the $B$ and $B'$ sandwiched
midpoint structures, we find that the cations that are sandwiched in
the midpoint structures are Zr for \LZTO, Hf for \LHTO, W for \MFWO\
and \MWO, and Os for \ZFOO. These results are consistent with the
analysis of $\xi_{1 \rm S}$ and $\xi_{2 \rm S}$, as shown in
Table~\ref{tab:sandwich}. We find that the energy difference between
the two distinct midpoint structures is mainly due to the difference
$\Delta E^{\rm M}\!=E_B^{\rm M}-E_{B'}^{\rm M}$ between the Madelung
energies of the two structures in a simple point-ion model, as
shown in Table.~\ref{tab:sandwich}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tables
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}
\begin{center}
\begin{ruledtabular}
    \caption{Rhombohedral structural parameters of LNO-type \ABO\
    corundum derivatives \LNO, \LTO, \ZSO, \FTO\ and \MTO\ from
    first-principles calculations and experiments.
    The Wyckoff positions are 2a for $A$ and $B$ cations, and 6c for
    oxygen anions (note that $A_x\!=A_y\!=A_z$ and $B_x\!=B_y\!=B_z$).
    The origin is defined by setting the Wyckoff position $B_x$ to zero.}
    \begin{tabular}{ccccccccc}
    &&\multicolumn{2}{c}{Lattice constants} &\multicolumn{5}{c}{Wyckoff positions}\\
    &&\multicolumn{1}{c}{a ({\AA})} &\multicolumn{1}{c}{$\alpha$ ($^\circ$)} &\multicolumn{1}{c}{$A_x$} &\multicolumn{1}{c}{$B_x$}
    &\multicolumn{1}{c}{O$_x$} &\multicolumn{1}{c}{O$_y$} &\multicolumn{1}{c}{O$_z$}\\\hline
    \multirow{2}{*}{\LNO} &Calc.              &5.486 &56.0 &0.282 &0.000 &0.360 &0.719 &0.112 \\
                          &Exp. \cite{LNOlatt}&5.494 &55.9 &0.280 &0.000 &0.359 &0.720 &0.111 \\ \hline
    \multirow{2}{*}{\LTO} &Calc.              &5.476 &56.2 &0.284 &0.000 &0.365 &0.726 &0.119 \\
                          &Exp. \cite{LTOlatt}&5.473 &56.2 &0.291 &0.000 &0.368 &0.732 &0.124 \\ \hline
    \multirow{2}{*}{\ZSO} &Calc.              &5.584 &56.5 &0.283 &0.000 &0.392 &0.709 &0.104 \\
                          &Exp. \cite{ZSOlatt}&5.569 &56.4 &0.286 &0.000 &0.381 &0.721 &0.111 \\ \hline
    \multirow{2}{*}{\FTO} &Calc.              &5.434 &56.5 &0.290 &0.000 &0.364 &0.721 &0.104 \\
                          &Exp. \cite{FTOlatt}&5.458 &56.0 &0.287 &0.000 &0.364 &0.720 &0.109 \\ \hline
    \multirow{2}{*}{\MTO} &Calc.              &5.481 &56.6 &0.279 &0.000 &0.348 &0.721 &0.120 \\
                          &Exp.\cite{MTOlatt} &5.455 &56.8 &0.276 &0.000 &0.345 &0.731 &0.128 \\
    \end{tabular}\label{tab:StruABO}
\end{ruledtabular}
\end{center}
\end{table}
%
\begin{table}
\begin{center}
\begin{ruledtabular}
    \caption{Rhombohedral structure parameters of ordered-LNO \ABB'O
    corundum derivatives \LZTO, \LHTO, \MFWO, \MWO\ and \ZFOO\ from
    first-principles calculations and experiments.
    Wyckoff positions are 1a for $A_1$, $A_2$, $B$ and $B'$ cations,
    and 3b for O$_1$ and O$_2$ anions.
    The origin is defined by setting the Wyckoff position $B'_x$ to zero.
    For ordered-LNO \MWO\ and \ZFOO, no experimental results are available.}
    \begin{tabular}{lcccccccccccccc}
    &&\multicolumn{1}{c}{Magnetic} &\multicolumn{2}{c}{Lattice constants} &\multicolumn{10}{c}{Wyckoff position}\\
    &&\multicolumn{1}{c}{order}&\multicolumn{1}{c}{a ({\AA})} &\multicolumn{1}{c}{$\alpha$ ($^\circ$)}
    &\multicolumn{1}{c}{$A_{1x}$} &\multicolumn{1}{c}{$A_{2x}$} &\multicolumn{1}{c}{$B_x$} &\multicolumn{1}{c}{$B'_x$}
    &\multicolumn{1}{c}{O$_{1x}$} &\multicolumn{1}{c}{O$_{1y}$} &\multicolumn{1}{c}{O$_{1z}$}
    &\multicolumn{1}{c}{O$_{2x}$} &\multicolumn{1}{c}{O$_{2y}$} &\multicolumn{1}{c}{O$_{2z}$}\\ \hline
    \multirow{2}{*}{\LZTO}&Calc.                 &---   &5.526 &56.1 &0.291 &0.781 &0.504 &0.000 &0.366 &0.745 &0.111 &0.628 &0.219 &0.895\\
                          &Exp. \cite{LiZr/HfTeO}&---   &5.497 &56.1 &0.298 &0.768 &0.507 &0.000 &0.390 &0.729 &0.133 &0.621 &0.235 &0.893\\\hline
    \multirow{2}{*}{\LHTO}&Calc.                 &---   &5.480 &56.3 &0.289 &0.781 &0.504 &0.000 &0.366 &0.743 &0.115 &0.629 &0.222 &0.889\\
                          &Exp.                  &---   & NA   & NA  & NA   & NA   & NA   & NA   & NA   &  NA  &  NA  &  NA  &  NA  &  NA \\\hline
    \multirow{2}{*}{\MFWO}&Calc.                 &$udu$ &5.531 &57.3 &0.286 &0.787 &0.506 &0.000 &0.356 &0.744 &0.109 &0.632 &0.207 &0.884\\
                          &Exp. \cite{Mn2FeWO}   & NA   &5.562 &56.9 &0.278 &0.785 &0.493 &0.000 &0.347 &0.745 &0.102 &0.631 &0.197 &0.885\\\hline
    \multirow{3}{*}{\MWO} &Calc.                 &$udu$ &5.607 &56.7 &0.287 &0.790 &0.508 &0.000 &0.356 &0.742 &0.102 &0.631 &0.199 &0.893\\
                          &Calc.                 &$uud$ &5.613 &56.6 &0.283 &0.779 &0.493 &0.000 &0.384 &0.693 &0.124 &0.595 &0.234 &0.848\\
                          &Exp. \cite{Mn3WO}     & NA   &5.605 &56.7 &0.283 &0.788 &0.510 &0.000 &0.351 &0.744 &0.104 &0.631 &0.190 &0.901\\\hline
    \multirow{2}{*}{\ZFOO}&Calc.&FiM   &5.410 &56.7 &0.284 &0.783 &0.504 &0.000 &0.376 &0.732 &0.114 &0.619 &0.223 &0.885\\
                          &Exp.                  & NA   & NA   & NA  & NA   & NA   & NA   & NA   & NA   &  NA  &  NA  &  NA  &  NA  &  NA \\
    \end{tabular}\label{tab:StruABB'}
\end{ruledtabular}
\end{center}
\end{table}
%
\begin{table}
\begin{center}
\begin{ruledtabular}
    \caption{Magnetic energy of different magnetic states relative to
    the lowest-energy state in \MFWO\ and \MWO, in units of meV per
    formula unit (f.u.).}
    \begin{tabular}{ldddd}
            &\multicolumn{4}{c}{Energy (meV/f.u.)}\\
            &\multicolumn{1}{c}{$uuu$} &\multicolumn{1}{c}{$uud$} &\multicolumn{1}{c}{$udd$} &\multicolumn{1}{c}{$udu$}\\ \hline
    \MFWO   &90.2  &32.1 &39.5  &0.0\\
    \MWO    &101.8 &1.0  &19.2  &0.0\\
    \end{tabular}\label{tab:magGS}
\end{ruledtabular}
\end{center}
\end{table}
%
\begin{table}
\begin{center}
\begin{ruledtabular}
    \caption{Relative spin direction between different magnetic ions
    in \MFWO\ and \MWO.\hfill}
    \begin{tabular}{lcccc}
    &\multicolumn{1}{c}{$uuu$} &\multicolumn{1}{c}{$uud$} &\multicolumn{1}{c}{$udd$} &\multicolumn{1}{c}{$udu$}\\ \hline
    $A_1$ and $B$   & FM & AFM & FM  & AFM \\
    $A_2$ and $B$   & FM & AFM & AFM & FM  \\
    $A_1$ and $A_2$ & FM & FM  & AFM & AFM \\
    \end{tabular} \label{tab:EX}
\end{ruledtabular}
\end{center}
\end{table}
%
\begin{table}
\begin{center}
\begin{ruledtabular}
    \caption{Ground state and midpoint structures of ordered-LNO
    candidates and the energy differences between $B$ and $B'$
    sandwiched midpoint structures. The distances between $A$ cations
    and the oxygen planes in the ground state are characterized by
    $\xi_{1 \rm S}$ and $\xi_{2 \rm S}$. The energy difference between
    the $B$ and $B'$ sandwiched midpoint structures is $\Delta E$. The
    Madelung energy difference between the $B$ and $B'$ sandwiched
    midpoint structures is $\Delta E^{\rm M}$. }
    \begin{tabular}{lcccccdd}
    \multicolumn{1}{c}{Ordered-LNO} &\multicolumn{1}{c}{$B$} &\multicolumn{1}{c}{$B'$} &\multicolumn{1}{c}{Sandwiched cation} &\multicolumn{1}{c}{$\xi_{1 \rm S}$ (\AA)} &\multicolumn{1}{c}{$\xi_{2 \rm S}$ (\AA)} &\multicolumn{1}{c}{$\Delta E^{\rm M}$ (meV)} &\multicolumn{1}{c}{$\Delta E$ (meV)}  \\ \hline
    \LZTO       & Zr & Te & Zr &0.607 &0.559 & -186 & -48 \\
    \LHTO       & Hf & Te & Hf &0.582 &0.550 & -94  & -29 \\
    \MFWO       & Fe & W  & W &0.623 &0.707 & 2085 & 225 \\
    $uud$ \MWO  & Mn & W  & W &0.635 &0.799 & 1666 & 266 \\
    $udu$ \MWO  & Mn & W  & W &0.642 &0.790 & 1499 & 290 \\
    \ZFOO       & Fe & Os & Os &0.565 &0.574 & 382  & 102 \\
    \end{tabular}\label{tab:sandwich}
\end{ruledtabular}
\end{center}
\end{table}

\begin{thebibliography}{99}
\bibitem{LNOlatt} S. C. Abrahams, W. C. Hamilton, J. M. Reddy, J. Phys. Chem. Solids {\bf 27}, 1013 (1966).

\bibitem{LTOlatt} M. Ohgaki, K. Tanaka, F. Marumo, Mineralogical Journal (Japan) {\bf 14}, 373 (1989).

\bibitem{ZSOlatt} Y. Inaguma, M. Yoshida, T. Katsumata, J. Am. Chem. Soc. {\bf 130}, 6704 (2008).

\bibitem{FTOlatt} K. Leinenweber, J. Linton, A. Navrotsky, Y. Fei, J.B. Parise, Phys. Chem. Miner. {\bf 22}, 251 (1995).

\bibitem{MTOlatt} A. M. Ar\'{e}valo-L\'{o}pez and J. P. Attfield, Phys. Rev. B {\bf 88}, 104416 (2013).

\bibitem{LiZr/HfTeO} J. Choisnet, A. Rulmont, P.Tarte, J. Solid State Chem. {\bf 75}, 124 (1988).

\bibitem{Mn2FeWO} M.-R. Li, M. Croft, P. W. Stephens, M. Ye, D. Vanderbilt, M. Retuerto, Z. Deng, C. P. Grams, J. Hemberger, J. Hadermann, W.-M. Li, C.-Q. Jin, F. O. Saouma, J. I. Jang, H. Akamatsu, V. Gopalan, D. Walker, and M. Greenblatt, Adv. Mater. {\bf 27}, 2177 (2015).

\bibitem{Mn3WO} Private communication with Prof. Martha Greenblatt and Dr. Manrong Li.

\end{thebibliography}
\end{document}
