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\title{\bf{Hexagonal $ABC$ semiconductors as ferroelectrics} \\[11pt] }
\author{Joseph W. Bennett, Kevin F. Garrity, Karin M. Rabe and David Vanderbilt}
\affiliation{Department of Physics and Astronomy\\
Rutgers University, Piscataway, NJ 08854 }

\date{June 15, 2012}

\begin{abstract}

We use a first-principles rational-design approach to identify a
previously-unrecognized class of ferroelectric materials in the
$P6_{3}mc$ LiGaGe structure type. We calculate structural parameters,
polarization and ferroelectric well depths both for reported and
as-yet hypothetical representatives of this class. Our results provide
guidance for the experimental realization and further investigation of
high-performance materials suitable for practical applications.
\end{abstract}

\pacs{
77.84.-s% Dielectric, piezoelectric, ferroelectric and antiferroelectric
%materials
81.05.Zx, %New materials, theory, and design
77.65.Bn, %Piezoelectric constants
}
\maketitle

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A rapidly developing paradigm for the rational design of functional
materials is based on the first-principles study of large families of
known and as yet unreported compounds.
%~\cite{Jain11p2295,Armiento11p014103, Zhang12p1425,Roy11preprint}.
%~\cite{Fischer06p641, Hautier10p3762, Jain11p2295}.
First-principles calculations of structure and properties are used
first to explore the microscopic origins and establish design
principles for the functional properties of interest, and then to
screen a large number of both equilibrium and metastable phases to
identify promising candidate systems. One recent study showed the
semiconducting members of the $ABC$ half-Heusler family to be
piezoelectric, with a range of piezoelectric properties comparable to
the much-studied $AB$O$_{3}$ perovskite oxides~\cite{Roy11preprint}.

A ferroelectric is a material with a polar phase produced by a
structural transition from a nonpolar high-symmetry paraelectric state,
with an electric polarization that can be switched between two or more
symmetry-related variants by application of an electric
field~\cite{Lines77}. The rational design of new ferroelectric
materials is motivated both by fundamental scientific interest and by
potential technological applications~\cite{Izyumskaya07p111}. New
materials can offer better performance, including reduction in
switching time, in coercive field and in fatigue, operation at higher
or lower temperatures, and the possibility of better integration with
other materials based on structural or chemical compatibility.  New
ferroelectrics with lower band gaps for photoactive
applications~\cite{Bennett10p184106, Gou11p205115} are also of
interest. Additional practical advantages could include decreased
toxicity (e.g., if Pb-free)~\cite{Bennett11p144122}
%
\dvm{tiny change for better grammatical construction.\\
What is Ref. [6]?  I haven't looked it up, but at first glance
it seems no reference would be necessary, so it looks  a little
bit like an excuse to self-cite...}
%
and possible multifunctionality.

Any polar structure (if insulating) could potentially support
ferroelectricity if the barrier to switching is low
enough~\cite{Abrahams68p551,Abrahams96p790, Abrahams88p585,
Abrahams06p26}. We therefore can search for new ferroelectric
semiconductors by targeting intermetallic compounds in polar space
groups and screening both reported and hypothetical compounds to find
insulating representatives with a low barrier to uniform switching
through a nonpolar reference phase, which provides an indication of
the barrier to realistic switching.  $ABC$ compounds with polar space
group $P6_{3}mc$ in the LiGaGe structure type~\cite{Bockelmann74p233,
Burdett90p12, Bojin03p1653, Casper08p035002} are a promising target
class.  This structure, shown in Figure~\ref{fig:displacements}, is a
hexagonal variant of the half-Heusler structure and can be described
as a wurtzite structure ``stuffed'' with a third
cation~\cite{Casper09p1090}.  The {\it Inorganic Crystal Structural
Database} (ICSD)~\cite{Belsky02p364} includes 18 $ABC$ compounds in
this structure type that do not contain an $f$-block element. We can
classify these combinations into the following groups: I-III-IV
(LiGaGe), I-II-V (LiBeSb), XI-III-IV (CuYSn), XI-II-V (AgCaBi) and
II-XII-IV (CaZnSn).
%
\dvm{Does it make sense to mention here how many of these are
metallic and how many insulating?}
%
In addition, we find that at least six entries
(CuScSn, CuYSn, AuYSn, AgCaBi, CaZnSn and CaHgSn) are also reported
with non-polar $P6_{3}/mmc$ symmetry in the ZrBeSi structure type,
which we identify as the nonpolar reference phase.  It has been
previously noted that the $P6_{3}/mmc$ ZrBeSi structure can be
obtained by a symmetry-restoring distortion of the LiGaGe structure in
which the buckling of the atomic planes in the wurtzite structure is
eliminated; this relationship is analogous to that of the wurtzite
structure to the metastable hexagonal structure of ScN
\cite{Farrer02p201203, Ranjan03p257602}.

In this paper, we use first-principles methods to establish a new
class of ferroelectrics in the LiGaGe structure type and to identify
promising candidate materials for further investigation.
Specifically, we compute the structural parameters, band gap,
polarization, and barrier to uniform switching of the eighteen
reported and 72 as-yet-hypothetical $ABC$ compounds in the LiGaGe
structure type. We identify several insulating combinations with
polarization comparable to that of BaTiO$_{3}$ ($\sim$25 C/m$^{2}$)
and uniform switching barriers less than that of PbTiO$_3$.  For all
insulating combinations studied, we find that the band gaps are in the
semiconducting range; the lower band gaps could be useful for
photoactive applications~\cite{Bennett08p17409}. These candidate
ferroelectrics offer promise for experimental investigation and for
the future development of new high-performance materials for practical
applications.

\begin{figure}
     \centering
     \subfigure[$P6_{3}/mmc$]{
          \label{fig:P63mmc}
          \includegraphics[width=1.50in]{fig1a.pdf}}
     \subfigure[$P6_{3}mc$]{
          \label{fig:P63mc}
          \includegraphics[width=1.50in]{fig1b.pdf}}
     \caption{Polar distortion present in $ABC$. Atoms at Wyckoff
     positions $2b$ (dark blue) and $2b^{\prime}$ (gold) (previously in
     one layer in $P6_{3}/mmc$ symmetry) buckle, creating a
     polarization.}
       \label{fig:displacements}
\end{figure}

First principles computations were performed with the ABINIT
package~\cite{Gonze09p2582}. The local density approximation (LDA) and
a $4\times4\times4$ Monkhorst-Pack sampling of the Brillouin
zone~\cite{Monkhorst76p5188} were used for all calculations, except
for the Berry phase polarization~\cite{Resta94p899} calculations, for
which a $8\times8\times8$ grid was used. All atoms were represented by
norm-conserving optimized~\cite{Rappe90p1227} designed
nonlocal~\cite{Ramer99p12471} pseudopotentials, generated with the
OPIUM code~\cite{Opium}. All calculations were performed with a plane
wave cutoff of 50~Ry.

We first consider the eighteen non-rare-earth compounds that have
been experimentally reported in the LiGaGe structure type in the ICSD
\footnote{KSnAs is also reported in the LiGaGe structure type, but on
closer examination we find it to be misassigned. This will be
discussed in detail in a future publication.}.  For each combination,
we optimized the structural parameters for each of the three
structural variants $\ub{A}BC$, $A\ub{B}C$, and $AB\ub{C}$, where the
underscore indicates the element at Wyckoff position $2a$, which
stuffs the wurtzite structure comprised by the other two elements.
The results for the lowest energy structural variant are reported in
Table~\ref{table:exptdata}.  The computed structural parameters
generally show good agreement with experimental values, with
the slight underestimate of lattice constants characteristic of LDA
calculations.

\begin{table}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lllllll}
& $a$ & $c$ & $z_{2b}$ & $z_{2b}^{\prime}$ & $\Delta E$\\
&(\AA)&(\AA)&          &                   & (eV)\\
\ub{Li}GaGe & 4.14 (4.18) & 6.71 (6.78) & 0.314 (0.312) & 0.698 (0.700) & 0.80\\
\ub{Sc}CuSn & 4.35 (4.39) & 6.80 (6.83) & 0.330 (0.326) & 0.729 (0.729) & 0.45 \\
\ub{Y}CuSn  & 4.48 (4.54) & 7.15 (7.27) & 0.320 (0.316) & 0.728 (0.733) & 0.32\\
\ub{Y}CuPb  & 4.51 (4.56) & 7.19 (7.33) & 0.321 (0.323) & 0.729 (0.729) & 0.34\\
\ub{Y}AgSn  & 4.63 (4.68) & 7.24 (7.37) & 0.314 (0.308) & 0.702 (0.719) & 0.39\\
\ub{Sc}AuGe & 4.26 (4.31) & 6.68 (6.85) & 0.304 (0.300) & 0.707 (0.702) & 0.61\\
\ub{Sc}AuSn & 4.51 (4.59) & 7.07 (7.20) & 0.348 (0.340) & 0.730 (0.731) & 0.75\\
\ub{Y}AuSi  & 4.25 (4.29) & 7.36 (7.55) & 0.272 (0.266) & 0.720 (0.725) & 0.05\\
\ub{Y}AuGe  & 4.36 (4.41) & 7.08 (7.31) & 0.281 (0.284) & 0.711 (0.723) & 0.26\\
\ub{Y}AuSn  & 4.61 (4.64) & 7.29 (7.37) & 0.327 (0.321) & 0.726 (0.727) & 0.70\\
\ub{Li}BeSb & 4.09 (4.15) & 6.64 (6.74) & 0.350 (0.341) & 0.733 (0.732) & 0.58\\
\ub{Li}ZnSb & 4.38 (4.43) & 7.08 (7.16) & 0.288 (0.304) & 0.669 (0.689) & 0.80\\
\ub{Li}ZnBi & 4.46 (4.58) & 7.23 (7.38) & 0.277 (0.277) & 0.658 (0.659) & 0.62\\
\ub{Ca}AgBi & 4.73 (4.81) & 7.56 (7.83) & 0.318 (0.312) & 0.715 (0.721) & 0.36\\
\ub{Ca}ZnSn & 4.60 (4.66) & 7.33 (7.63) & 0.313 (0.305) & 0.717 (0.720) & 0.32\\
\ub{Ca}HgSn & 4.75 (4.80) & 7.55 (7.76) & 0.311 (0.305) & 0.712 (0.720) & 0.53\\
\ub{Sr}HgSn & 4.88 (4.89) & 7.80 (8.22) & 0.288 (0.300) & 0.695 (0.720) & 0.36\\
\ub{Sr}HgPb & 4.89 (5.00) & 8.05 (8.17) & 0.289 (0.300) & 0.694 (0.720) & 0.19\\
\end{tabular}
\end{ruledtabular}
\caption{First-principles results for the eighteen LiGaGe-structure
compounds reported in ICSD. The computed ground state structural
parameters are compared to experimental values, given in parentheses
(in some cases, a transformation has been applied to facilitate
comparison). The origin is chosen so that the atom at position $2a$
(underlined) is at $z$=0. $\Delta E$ is the energy difference between
the polar state and the nonpolar high-symmetry reference state.}
\label{table:exptdata}
\end{center}
\end{table}

There are only two insulating compounds in the set of reported LiGaGe
compounds: LiBeSb and LiZnSb; in each the stuffing atom is the
monovalent element Li.  These are of the I-II-V type; the total of 8
$s$ and $p$ valence electrons is expected to improve the likelihood of
band gap formation~\cite{Kandpal06p776}. The band gap for LiBeSb is
1.71 eV (indirect) and for LiZnSb it is 0.67 (direct). The computed
polarizations of 0.59 C/m$^2$ (LiBeSb) and 0.56 C/m$^2$ (LiZnSb) are
larger than that of BaTiO$_{3}$. To assess switchability, we compute
the energy difference between the polar state and the nonpolar
high-symmetry reference state.  While we recognize that ferroelectrics
do not switch by uniform change of the polarization through the
high-symmetry state, the energy barrier for uniform switching can be
used to assess the possibility of realistic switching by comparing
with the values for known ferroelectrics: 0.2~eV for PbTiO$_{3}$ and
0.02~eV for BaTiO$_{3}$ ~\cite{Cohen92p136}. In the present case, this
comparison suggests that the nominal barriers in LiBeSb and LiZnSb
(0.58 eV and 0.80 eV) are too high for switchability to be likely.

To search for candidate LiGaGe-type ferroelectrics with lower
barriers, we consider equiatomic combinations of three distinct
constituent elements $ABC$ with valences given by I-II-V, with I=(Li,
Na, K), II=(Be, Mg, Ca, Sr, Ba, Zn) and V=(P, As, Sb, Bi).  This
generates a total of 72 candidate combinations to be searched, only
two of which are included in Table I.  We optimize the structural
parameters in $P6_{3}mc$ space group for each of the three variants
corresponding to the three choices of element for the $2a$ position.

For 66 of the 72 combinations, the predicted lowest-energy
variant is insulating, with computed band gap ranging from 0.04 eV to
1.81 eV; since DFT tends to underestimate band gaps, we expect that
the actual fraction of insulating compounds will be slightly higher
than our calculations would indicate.  Of these 66 compounds, 49 have
relaxed to the higher non-polar $P6_{3}/mmc$ symmetry; results for
these compounds are given in the supplementary material.
%
\marginpar{Move to discussion?  Of the compounds in our search set
that relax to $P6_{3}/mmc$ symmetry, we find six combinations in ICSD;
all six are a close match to the experimentally determined crystal
structure.\\
DV: To me, it seems OK to put it here.}
%
For the seventeen polar insulating compounds, we also
compute the spontaneous polarization; results for these compounds are
given in Table~\ref{table:hexdata1}. The remaining 6 of the 72
combinations are found to be metallic in the lowest-energy structural
variant: the computed structural parameters and $\Delta E$, the energy
relative to the high-symmetry $P6_{3}/mmc$ phase, for these are given
in Table~\ref{table:hexdata3}.

Thus, we have narrowed the search for new LiGaGe-type ferroelectrics
to seventeen polar insulating combinations.  For this set, we see in
Figure~\ref{fig:fig2} that the polarization has a positive correlation
with $\Delta E$, as would be expected in a simple double-well
model. The eight compounds LiBeP, LiCaBi, NaMgP, NaMgAs, NaZnSb,
NaMgBi, KMgSb, and KMgBi have polarizations comparable to that of
BaTiO$_{3}$ and $\Delta E < $0.25 eV, in the range favorable for
ferroelectric switching, and therefore are promising candidates for
ferroelectricity.\marginpar{Note that I added LiCaBi and LiBeP, but
deleted NaMgSb.}

The ferroelectric double well for NaMgP, shown in the inset of
Figure~\ref{fig:fig2}, is representative of this group. The key to the
switchability of the polarization is that the wurtzite substructure is
not characterized by ideal $sp^3$ bonding like in ZnO, which would
require breaking and reforming of rigid bonds to switch, and is far
from ideal. Rather, the structure should be understood as the buckling
of the flat planes of the $P6_{3}/mmc$ structure, with $sp^2$
bonding. We define a buckling parameter, $d$, as the distance along
the $c$-axis between the inequivalent atoms in the buckled plane
relative to the $P6_{3}/mmc$ structure. This planar buckling gives
rise to the polarization, as the stuffing ion does not displace, but
simply separates the buckled planes. We find that $d$ decreases as the
size of the stuffing ion increases from Li to Na to K, weakening the
interplanar bond so that the barrier to switching is reduced most for
compounds containing K.
%
\dvm{Changed word}
%
Since changing the sense of the buckling does not
involve breaking and reforming of bonds, the barrier to switching can
be low enough for ferroelectricity.\marginpar{Is this reworked
enough?}

\begin{table}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
$ABC$&$a$&$c$&z$^{\rm 2b}$&z$^{\rm 2b^{\prime}}$&$d$&$E_{\rm gap}$&$\Delta E$&$P$\\
     &(\AA)&(\AA)&        &                     &   &(eV)         &(eV)      &(C/m$^2$)\\
\ub{Li}BeP & 3.634 & 5.833 & 0.295 & 0.686 & 0.11 & 1.51(i) & 0.24 & 0.85\\
\ub{Li}MgP & 4.134 & 7.211 & 0.413 & 0.773 & 0.14 & 1.81(i) & 0.42 & 0.91\\
\ub{Li}ZnP & 3.924 & 6.365 & 0.341 & 0.722 & 0.12 & 1.33(i) & 0.65 & 0.84\\
\ub{Li}BeAs & 4.091 & 6.636 & 0.265 & 0.647 & 0.12 & 1.71(i) & 0.30 & 0.67\\
\ub{Li}MgAs & 4.292 & 7.483 & 0.414 & 0.774 & 0.14 & 1.48(d) & 0.44 & 0.59\\
\ub{Li}ZnAs & 4.108 & 6.673 & 0.335 & 0.715 & 0.12 & 0.97(d) & 0.72 & 0.75\\
\ub{Li}BeSb & 4.094 & 6.645 & 0.267 & 0.650 & 0.12 & 1.13(i) & 0.58 & 0.59\\
\ub{Li}ZnSb & 4.376 & 7.081 & 0.288 & 0.669 & 0.12 & 0.67(d) & 0.80 & 0.56\\
\ub{Li}BeBi & 4.179 & 6.806 & 0.262 & 0.643 & 0.12 & 0.45(i) & 0.62 & 0.55\\
Li\ub{Ca}Bi & 4.679 & 7.503 & 0.287 & 0.731 & 0.06 & 0.16(i) & 0.01 & 0.19\\
\ub{Na}MgP & 4.424 & 6.877 & 0.310 & 0.716 & 0.09 & 1.17(d) & 0.20 & 0.49\\
\ub{Na}MgAs & 4.549 & 7.262 & 0.314 & 0.715 & 0.10 & 0.67(d) & 0.23 & 0.48\\
\ub{Na}MgSb & 4.868 & 7.584 & 0.307 & 0.705 & 0.10 & 0.69(d) & 0.29 & 0.43\\
\ub{Na}ZnSb & 4.558 & 7.332 & 0.285 & 0.677 & 0.11 & 0.20(d) & 0.16 & 0.49\\
\ub{Na}MgBi & 4.957 & 7.584 & 0.302 & 0.704 & 0.10 & 0.14(i) & 0.25 & 0.42\\
\ub{K}MgSb & 5.017 & 7.789 & 0.290 & 0.697 & 0.09 & 0.59(d) & 0.08 & 0.40\\
\ub{K}MgBi & 5.092 & 8.005 & 0.291 & 0.698 & 0.09 & 0.15(i) & 0.15 & 0.31\\
\end{tabular}
\end{ruledtabular}
\caption{First-principles results for the lowest-energy variant of
each of the 17 polar insulating $ABC$ combinations in the LiGaGe
structure type. The structure is specified by the lattice constants
and the internal structural parameters for two of the atoms; the
origin is chosen so that the atom at position $2a$ (underlined) is at
$z$=0.  The buckling parameter $d$ is described in the text.  The
computed values for the band gap, polarization and potential well
depth for the polar distortion are included.}
\label{table:hexdata1}
\end{table}

\dvm{Shouldn't the caption of Table II explain the `d' and `i'
notation?}

\begin{table}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
$ABC$&$a$&$c$&z$^{\rm 2b}$&z$^{\rm 2b^{\prime}}$&$d$&$E_{\rm gap}$&$\Delta E$&$P$\\
     &(\AA)&(\AA)&        &                     &   &(eV)         &(eV)      &(C/m$^2$)\\
Li\ub{Mg}Sb & 4.501 & 6.979 & 0.301 & 0.715 & 0.09 & m & 0.10 & 0\\
Li\ub{Mg}Bi & 4.602 & 7.051 & 0.300 & 0.718 & 0.08 & m & 0.42 & 0\\
\ub{Li}ZnBi & 4.461 & 7.233 & 0.277 & 0.658 & 0.12 & m & 0.62 & 0\\
\ub{Na}ZnBi & 4.653 & 7.479 & 0.283 & 0.675 & 0.11 & m & 0.13 & 0\\
\ub{K}SrBi & 5.757 & 7.044 & 0.250 & 0.750 & 0 & m & 0 & 0\\
\ub{K}ZnBi & 4.582 & 10.135 & 0.250 & 0.750 & 0 & m & 0 & 0\\
\end{tabular}
\end{ruledtabular}
\caption{First-principles results for the lowest-energy variant of
each of the 6 metallic $ABC$ combinations in the LiGaGe
structure type. Same as Table~\ref{table:hexdata1}.}
\label{table:hexdata3}
\end{table}

\begin{figure}
 \centering
\includegraphics[width=3.5in]{fig2.pdf}
\caption{Difference in energy between the low ($P6_{3}mc$) and high
($P6_{3}/mmc$) symmetry $ABC$ structures vs polarization for all polar
insulating combinations. Combinations with $A$=Li (black circles) are
less likely to be switchable than those with $A$=Na (red circles) or
$A$=K (green circles). The two reported compounds LiBeSb and LiZnSb
are labeled.The inset shows the characteristic ferroelectric double
well energy of NaMgP as a function of polar distortion obtained by
linear interpolation between the polar $P6_{3}mc$ and nonpolar
$P6_{3}/mmc$ structures.}
\label{fig:fig2}
\end{figure}

A key question is that of the prospects for synthesis of the candidate
compounds in the desired structure.  Of the seventeen compounds we
have identified as polar, fourteen have reported structures in
ICSD. Two are reported as $P6_{3}mc$ (LiBeSb and LiZnSb, as discussed
above), seven as $P4/nmm$ (LiBeP, LiBeAs, NaMgAs, NaMgSb, NaZnSb,
KMgSb, KMgBi), four as $F{\bar4}3m$ (LiMgP, LiZnP, LiMgAs, LiZnAs) and
one as $Pnma$ (LiCaBi). Of the three for which there is no reported
structure, results from a recent theoretical study predict them to be
$P4/nmm$ (NaMgP), $P2_{1}/c$ (NaMgBi), or unstable to decomposition
into binary compounds (LiBeBi)~\cite{Zhang12p1425}. However, it could
still be possible to synthesize at least some of our candidate
LiGaGe-type ferroelectrics as metastable phases, in cases in which the
LiGaGe structure type is sufficiently close in energy to the ground
state. In particular, for NaZnSb the LiGaGe phase is only 0.04~eV per
f.u. higher in energy than the $P4/nmm$ ground state, which makes the
metastable phase quite accessible.
 
\marginpar{Do you have energy difference to the gs values for the seven (eight) candidates?} 
\marginpar{Stability of the lowest energy variant??}  Furthermore, for
NaZnSb the energy difference between the lowest energy variant and the
next (with Zn? as the stuffing atom) is X eV, suggesting that it will
be possible to obtain full chemical ordering.
\marginpar{Fill in the X?}
\jwb{Still calculating a few of these...}
If these compounds are grown as epitaxial films, this would provide an
additional route to engineering the polarization, the switching
barrier, and the relative stability of the LiGaGe phase.  For example,
first-principles calculations show that 3$\%$ tensile strain in the
(0001) plane reduces the energy barrier of the reported compounds
LiZnP by 0.11~eV, LiZnAs by 0.12~eV and LiMgAs by 0.08~eV, to 0.57 eV,
0.60 eV and 0.36 eV, respectively. Strain could also promote a polar
instability in the insulating nonpolar $P6_{3}/mmc$ compounds.  Of the
forty-nine compounds we have identified as nonpolar insulators, six
are reported in ICSD with $P6_{3}/mmc$ symmetry (KZnP, NaBeAs, KZnAs,
LiBaSb, NaBeSb, KZnSb). The previously mentioned theoretical study
found five additional compounds with this structure (NaBeP, KBaSb,
NaCaBi, KBaBi, KZnBi). First principles calculations of the
zone-center phonon frequencies for this set show that in each case the
frequency of the lowest frequency polar mode is below 100~cm$^{-1}$.
However, the coupling of this mode to (0001) epitaxial strain is not
strong enough to produce an instability in the range $\pm$~4~$\%$ in
any of the six compounds we tested (LiBaSb, NaBeSb, NaCaBi, KZnAs,
KZnSb,and KBaSb).

In conclusion, we have used first-principles methods to establish a
new class of ferroelectrics in the LiGaGe structure type and to
identify promising candidate materials for further investigation.
Through targeted synthesis, LiGaGe compounds could potentially be
developed as a valuable class of ferroelectric and piezoelectric
materials; other structure types with substructures related to
wurtzite could similarly yield systems with switchable polarization.
This is a specific application of a larger-scale strategy to identify
new ferroelectrics by targeting polar insulating compounds not
previously recognized as ferroelectric and tuning the composition and
other control parameters, such as epitaxial strain, and/or modifying
the structure by intercalation of atoms to reduce the barrier to
polarization switching.  The identification of ferroelectricity in
classes of materials in which it was previously unrecognized offers
the possibility of optimizing properties and combining polarization
with other functional properties, including magnetism, to produce
multifunctional behavior of fundamental scientific interest and for
groundbreaking technological applications.

\vspace{0.3cm}
\noindent{\bf Acknowledgments}
\vspace{0.3cm}

This work was supported in part by ONR Grant N00014-05-1-0054 and
N000-14-09-1-0300. Calculations were carried out at the Center for
Piezoelectrics by Design.  We thank D. R. Hamann and R. Seshadri for
useful discussions. K. M. R. thanks R. Seshadri for hospitality at
UCSB and the Aspen Center for Physics (grant number) where part of
this work was carried out.

\bibliography{thebibliography}

\end{document}

Any polar structure (if insulating) could potentially be
ferroelectric, if the barrier to switching is low enough. This barrier
can be approximated as the energy difference between a polar and
nonpolar structure if they can be described using a group-subgroup
relation. \jwb{Mention Abrahams strategy of displacements and
predicting $T_{c}$?  Some structures may be refined above $T_{C}$ and
thus be reported as nonpolar.} This relationship to a nonpolar
structure suggests that compounds with the LiGaGe structure type can
be considered as candidate ferroelectrics, subject to two
conditions. First, the compound must have a non-zero band gap, as only
insulators can possess an electric polarization. Because of the
relation to the $P6_{3}/mmc$ high-symmetry reference structure, there
will be two symmetry-related variants with polarization along the
hexagonal $c$ direction. 

\begin{table}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
$ABC$&$a$&$c$&z$^{\rm 2b}$&z$^{\rm 2b^{\prime}}$&$d$&$E_{\rm gap}$&$\Delta E$&$P$\\
     &(\AA)&(\AA)&        &                     &   &(eV)         &(eV)      &(C/m$^2$)\\
Li\ub{Ca}P & 4.141 & 7.169 & 0.250 & 0.750 & 0 & 0.91(i) & 0 & 0\\
Li\ub{Sr}P & 4.266 & 7.813 & 0.250 & 0.750 & 0 & 1.09(i) & 0 & 0\\
Li\ub{Ba}P & 4.405 & 8.502 & 0.250 & 0.750 & 0 & 0.76(i)& 0 & 0\\
Li\ub{Ca}As & 4.294 & 7.271 & 0.250 & 0.750 & 0 & 0.79(i) & 0 & 0\\
Li\ub{Sr}As & 4.408 & 7.963 & 0.250 & 0.750 & 0 & 1.08(i) & 0 & 0\\
Li\ub{Ba}As & 4.567 & 8.619 & 0.250 & 0.750 & 0 & 0.79(i) & 0 & 0\\
Li\ub{Ca}Sb & 4.589 & 7.524 & 0.250 & 0.750 & 0 & 0.46(i) & 0 & 0\\
Li\ub{Sr}Sb & 4.685 & 8.213 & 0.250 & 0.750 & 0 & 0.66(i) & 0 & 0\\
Li\ub{Ba}Sb & 4.811 & 8.902 & 0.250 & 0.750 & 0 & 0.58(i) & 0 & 0\\
Li\ub{Sr}Bi & 4.761 & 8.259 & 0.250 & 0.750 & 0 & 0.49(i) & 0 & 0\\
Li\ub{Ba}Bi & 4.882 & 8.942 & 0.250 & 0.750 & 0 & 0.46(i) & 0 & 0\\

\ub{Na}BeP & 3.628 & 8.652 & 0.250 & 0.750 & 0 & 1.36(i) & 0 & 0\\
\ub{Na}CaP & 4.897 & 5.979 & 0.250 & 0.750 & 0 & 0.42(i) & 0 & 0\\
\ub{Na}SrP & 5.108 & 6.320 & 0.250 & 0.750 & 0 & 0.33(d) & 0 & 0\\
\ub{Na}BaP & 5.327 & 6.669 & 0.250 & 0.750 & 0 & 0.17(d) & 0 & 0\\
\ub{Na}ZnP & 3.985 & 8.282 & 0.250 & 0.750 & 0 & 1.32(d) & 0 & 0\\
\ub{Na}BeAs & 3.799 & 8.659 & 0.250 & 0.750 & 0 & 1.44(d) & 0 & 0\\
\ub{Na}CaAs & 5.054 & 6.197 & 0.250 & 0.750 & 0 & 0.22(d) & 0 & 0\\
\ub{Na}SrAs & 5.271 & 6.473 & 0.250 & 0.750 & 0 & 0.05(d) & 0 & 0\\
\ub{Na}BaAs & 5.512 & 6.807 & 0.250 & 0.750 & 0 & 0.16(i) & 0 & 0\\
\ub{Na}ZnAs & 4.177 & 8.418 & 0.250 & 0.750 & 0 & 0.76(d) & 0 & 0\\
\ub{Na}BeSb & 4.102 & 9.161 & 0.250 & 0.750 & 0 & 0.99(i) & 0 &0\\
\ub{Na}CaSb & 5.314 & 6.527 & 0.250 & 0.750 & 0 & 0.08(d) & 0 & 0\\
\ub{Na}SrSb & 5.541 & 6.766 & 0.250 & 0.750 & 0 & 0.12(d) & 0 & 0\\
\ub{Na}BaSb & 5.782 & 7.087 & 0.250 & 0.750 & 0 & 0.18(i) & 0 & 0\\
\ub{Na}BeBi & 4.191 & 9.122 & 0.250 & 0.750 & 0 & 0.88(i) & 0 & 0\\
\ub{Na}CaBi & 5.417 & 6.564 & 0.250 & 0.750 & 0 & 0.34(i) & 0 & 0\\
\ub{Na}SrBi & 5.604 & 6.845 & 0.250 & 0.750 & 0 & 0.21(i) & 0 & 0\\
\ub{Na}BaBi & 5.852 & 7.375 & 0.250 & 0.750 & 0 & 0.72(i) & 0 & 0\\

\ub{K}BeP & 3.664 & 9.943 & 0.250 & 0.750 & 0 & 0.91(i) & 0 & 0\\
\ub{K}MgP & 4.359 & 9.435 & 0.250 & 0.750 & 0 & 1.16(d) & 0 & 0\\
\ub{K}CaP & 5.066 & 6.253 & 0.250 & 0.750 & 0 & 0.85(i) & 0 & 0\\
\ub{K}SrP & 5.246 & 6.476 & 0.250 & 0.750 & 0 & 0.76(i) & 0 & 0\\
\ub{K}BaP & 5.440 & 6.777 & 0.250 & 0.750 & 0 & 0.51(i) & 0 & 0\\
\ub{K}ZnP & 4.004 & 9.778 & 0.250 & 0.750 & 0 & 0.99(i) & 0 & 0\\
\ub{K}BeAs & 3.861 & 10.122 & 0.250 & 0.750 & 0 & 1.04(i) & 0 & 0\\
\ub{K}MgAs & 4.538 & 9.313 & 0.250 & 0.750 & 0 & 0.94(d) & 0 & 0\\
\ub{K}CaAs & 5.208 & 6.462 & 0.250 & 0.750 & 0 & 0.78(i) & 0 & 0\\
\ub{K}SrAs & 5.387 & 6.642 & 0.250 & 0.750 & 0 & 0.77(i) & 0 & 0\\
\ub{K}BaAs & 5.591 & 6.933 & 0.250 & 0.750 & 0 & 0.51(i) & 0 & 0\\
\ub{K}ZnAs & 4.202 & 9.791 & 0.250 & 0.750 & 0 & 1.02(d) & 0 & 0\\
\ub{K}BeSb & 4.149 & 10.384 & 0.250 & 0.750 & 0 & 0.79(i) & 0 & 0\\
\ub{K}CaSb & 5.487 & 6.753 & 0.250 & 0.750 & 0 & 0.36(i) & 0 & 0\\
\ub{K}SrSb & 5.682 & 6.949 & 0.250 & 0.750 & 0 & 0.48(i) & 0 & 0\\
\ub{K}BaSb & 5.880 & 7.209 & 0.250 & 0.750 & 0 & 0.27(i) & 0 & 0\\
\ub{K}ZnSb & 4.489 & 10.017 & 0.250 &0.750 & 0 & 0.77(d) & 0 & 0\\
\ub{K}BeBi & 4.237 & 10.465 & 0.250 & 0.750 & 0 & 0.69(i) & 0 & 0\\
\ub{K}CaBi & 5.566 & 6.872 & 0.250 & 0.750 & 0 & 0.14(d) & 0 & 0\\
\ub{K}BaBi & 5.956 & 7.287 & 0.250 & 0.750 & 0 & 0.04(d) & 0 & 0\\
\end{tabular}
\end{ruledtabular}
\caption{First-principles results for the lowest-energy variant of
each of the 49 nonpolar insulating $ABC$ combinations in the LiGaGe
structure type. Same description of columns as in Table~\ref{table:hexdata1}.}
\label{table:hexdata2}
\end{table}
