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\title{\bf{Half-Heusler semiconductors as piezoelectrics} \\[11pt] }
\author{Anindya Roy, Joseph W. Bennett, Karin M. Rabe* and David
Vanderbilt} \affiliation{ Department of Physics and Astronomy\\
Rutgers University, Piscataway, NJ 08854\\* Corresponding Author:
rabe@physics.rutgers.edu }
\date{July 25, 2011}

\maketitle

{\bf One of the central challenges in materials science is the design
of functional and multifunctional materials, in which large responses
are produced by applied fields and stresses. A rapidly developing
paradigm for the rational design of such materials is based on the
first-principles study of a large materials family, the perovskite
oxides being the prototypical case. Specifically, first-principles
calculations of structure and properties are used to explore the
microscopic origins of the functional properties of interest and to
search a large space of equilibrium and metastable phases to identify
promising candidate systems.  In this paper, we use a first-principles
rational-design approach to demonstrate semiconducting half-Heusler
compounds as a previously-unrecognized class of piezoelectric
materials, and to provide guidance for the experimental realization
and further investigation of high-performance materials suitable for
practical applications. }

\vspace{0.3cm} % to separate the abstract visually...

The $ABC$ half-Heusler compounds (with MgAgAs structure type, also
called semi-Heusler or Juza-Nowotny compounds~\cite{Nowotny50p488})
are a large and varied family, with almost 150 distinct $ABC$
half-Heusler compounds reported in the {\it Inorganic Crystal
Structural Database} (ICSD)~\cite{Belsky02p364}.  The half-Heusler
structure, shown in Fig.~\ref{fig:Figure1}, has $F{\bar4}3m$
symmetry and can be viewed as a rocksalt lattice formed from two of
the three constituent atoms (at Wyckoff positions 4a and 4b), with
the third atom filling half of the tetrahedral interstitial sites
(either Wyckoff position 4c or 4d); it is related to the $A_2BC$
Heusler structure by removal of one of the $A$ sublattices, and can
alternately be viewed as a stuffed $AC$ or $BC$ zincblende
structure~\cite{Kandpal06p776}.

Following an initial wave of interest stimulated by the observation of
half-metallic ferromagnetic behavior in half-Heusler
compounds~\cite{deGroot83p2024}, there has been a resurgence of
interest in these compounds as materials that can display topological
properties~\cite{Chadov10p541, Lin10p546} or be tailored for uses as
diverse as components in spintronic devices~\cite{Felser07p668} and
high-performance thermoelectrics~\cite{Shen01p4165, Nolas06p199,
Balke11p702}.  The half-Heusler family is regarded as especially
versatile as it displays a wide range of functionality, lattice
constants, and band gaps.

\begin{figure}
\includegraphics[width=3.0in]{Figure1.eps}
\caption{{The $ABC$ half-Heusler structure type: $A$ (green) and
$B$ (orange) are arranged in a rocksalt lattice, with the tetrahedral
coordination of $C$ (blue) by $A$ shown.}} \label{fig:Figure1}
\end{figure}

The semiconducting half-Heusler compounds are of particular
interest. True gaps on the order of 1\,eV were predicted in
Li-containing half-Heusler intermetallic compounds in first-principles
calculations by Wood {\em et al.}~\cite{Wood85p2570} in 1985. The most
well-studied of these compounds include LiMg$X$ and LiZn$X$ (where
$X$=N, P and As), LiMgSb, and the cell-doubled compounds
Li$_{3}$AlN$_{2}$, Li$_{3}$GaN$_{2}$, and Li$_{3}$GaP$_{2}$.  Since
the identification of NiSnZr as a semiconducting half-Heusler compound
through a combination of experimental observations and
first-principles investigations~\cite{Aliev87p684, Ogut95p10443}, many
additional semiconducting systems have been
identified~\cite{Kandpal06p776, Kieven10p075208, Gruhn10p125210}. Some
of these compounds, such as LiMgN and LiMgP, have quite substantial
gaps that may make them suitable for solar cell
applications~\cite{Casper09p1090, Kuriyama02p2019, Kuriyama98p429}.

As insulators, the semiconducting half-Heuslers can also exhibit
functional properties associated with electric polarization, but
these properties have received very little attention. In fact, while
the $F\bar{4}3m$ space group of the half-Heusler structure allows a
nonzero $d_{14}$ piezoelectric response, no measurements of
piezoelectricity in these systems have yet been reported in the
literature.

In this paper, we use first-principles methods to predict the
piezoelectric response and related properties of half-Heusler
compounds. We present these predictions first for compounds already
reported in the half-Heusler structure. We then perform a
high-throughput analysis of a much larger set of candidate
combinations, identifying high-performance compounds for practical
application. Through targeted synthesis, which might include
compositional substitution, epitaxial growth or artificial
structuring, half-Heusler compounds could thus be developed as a
valuable class of piezoelectric materials.

First-principles calculations are performed with the ABINIT
package~\cite{Gonze02p478,Gonze05p558,Gonze09p2582} using the local
density approximation (LDA) and an 8$\times$8$\times$8 Monkhorst-Pack
sampling of the fcc Brillouin
zone~\cite{Monkhorst76p5188}. Norm-conserving Troullier-Martins
pseudopotentials are employed, with a plane-wave cutoff of 30\,Ha.
For selected compounds, the results were verified using optimized
norm-conserving pseudopotentials~\cite{Rappe90p1227,Ramer99p12471}
generated using the OPIUM code~\cite{opium}.

For the high-throughput search, we consider combinations of three
distinct elements $ABC$. We limit the search to combinations with a
total of 8 $s$ and $p$ valence electrons, since we expect this to
improve the likelihood of band gap formation~\cite{Kandpal06p776}.
We also require that there be at least one and at most two $p$-block
elements among the three constituents, with the remainder coming
from the $s$ and/or $d$ blocks of the periodic table. We use Roman
numerals to denote the valences of the constituent elements
(including $d$ electrons) to classify the combinations into families
with 8 valence electrons (I-I-VI, I-II-V, I-III-IV, II-II-IV, and
II-III-III), 18 valence electrons (XI-I-VI, XI-II-V, XI-III-IV,
I-XII-V, II-XII-IV, III-XII-III, X-II-VI, X-III-V, and X-IV-IV), or
28 valence electrons (X-XII-VI, XI-XI-VI, XI-XII-V and XII-XII-IV).
Among these families, we consider members constructed from the
selections I=(Li, Na), II=(Be, Mg, Ca, Sr, Ba), III=(B, Sc, Y, Al,
Ga), IV=(C, Si, Ge, Sn, Pb, Ti, Zr, Hf), V=(N, P, As, Sb, Bi),
VI=(O, S, Se, Te), X=(Ni, Pd, Pt), XI=(Cu, Ag, Au), and XII=(Zn,
Cd).  This generates a total of 792 candidate combinations to be
searched.

We first consider the 36 combinations in our search set that have been
experimentally reported in the ICSD. For each combination, we optimize
the lattice constant for each of the three structural variants
$\ub{A}BC$, $A\ub{B}C$, and $AB\ub{C}$, where the underscore indicates
the unique element that is tetrahedrally coordinated by both of the
two other elements. First-principles results show that 23 of them,
listed in Table~\ref{table:Table1}, are
insulating.\footnotetext[1]{The thirteen metallic combinations are
LiAlSi, LiGaSi, LiAlGe, BeAlB, CuMgSb, AgMgAs, CuMgBi, AuCaBi, NiYBi,
CuCdSb, AgCdSb, AuCdSb, and AuYPb.} The predicted lowest-energy
structural variant is indicated in the first column.  We find that for
the five compounds (LiZnAs, AuScSn, NiScSb, PdScSb and PtYSb) for
which a refined structure, including R value and temperature factors,
is available in ICSD, our prediction agrees with the experimentally
observed variant. Most of the computed equilibrium lattice constants
given in Table~\ref{table:Table1} are in excellent agreement with
experiment. The computed gaps range from 0.15\,eV for NiScBi to
1.42\,eV for PtZrSn.

For each compound, we perform a linear-response calculation using
density-functional perturbation theory (DFPT)~\cite{Gonze97p10355}, as
implemented in ABINIT, to compute the dynamical charges and
zone-center phonon frequencies and eigenvectors, from which the
free-stress static dielectric constant $\epsilon_{0}$ can be
obtained~\cite{born54p268}.

We also carry out calculations of the strain response
\cite{Wu05p035105,Hamann05p035117} to obtain the $C_{44}$ elastic
constant and $d_{14}$ piezoelectric coefficient. All this information
can be combined to obtain the electromechanical coupling coefficient
$k_{14}$ as described in Ref.~\onlinecite{Wu05p035105} using the
formula $k_{14} = |d_{14}| \sqrt{C_{44}/(\epsilon_{\rm
fs}\epsilon_0)}$, where $\epsilon_ {\rm fs}$ is the permittivity of
free space, and $\epsilon_0$ is our computed dimensionless free-stress
dielectric constant.

The results are included in
Table~\ref{table:Table1}.  The computed values of the piezoelectric
constant $d_{14}$ mostly fall in the range from 2.0 to 14\,pC/N. All
but five compounds have $d_{14}>2.6$\,pC/N, the experimentally
measured piezoelectric coefficient of GaAs. The highest values found
are for NiYSb, PdYSb and PtYSb, and can be associated with their small
bandgap, large $\epsilon_0$ and low elastic constant. These three
compounds also have the highest values of electromechanical coupling
$k_{14}$, as shown in Table I, arising from their large $d_{14}$ and
low elastic constant, though limited by the size of $\epsilon_0$.

\begin{table}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{l d d d d d d d}
$ABC$ & \mcc{$a_{\rm expt.}$} & \mcc{$a$}
 & \mcc{$E_{\rm gap}$} & \mcc{$d_{14}$}
 & \mcc{$C_{44}$} & \mcc{$k_{14}$} & \mcc{$\epsilon_0$} \\
 & \mcc{(\AA)} & \mcc{(\AA)} & \mcc{(eV)} &
    \mcc{(pC/N)} & \mcc{(10$^{11}$Pa)}& & \\
\hline
LiMg\ub{P} & 6.02 & 5.94 & 1.38 & 3.2 & 0.46 & 0.07 & 11.33 \\
LiMg\ub{As} & 6.19 & 6.10 & 1.15 & 2.2 & 0.41 & 0.04 & 12.61 \\
LiMg\ub{Bi} & 6.74 & 6.52 & 0.69 & 0.3 & 0.31 & 0.00 & 15.76 \\
LiZn\ub{N} & 4.87 & 4.85 & 0.72 & 7.6 & 0.82 & 0.18 & 15.39 \\
LiZn\ub{P} & 5.78 & 5.64 & 1.11 & 12.1 & 0.57 & 0.24 & 15.92 \\
LiZn\ub{As} & 5.94 & 5.81 & 0.61 & 13.5 & 0.58 & 0.24 & 20.08 \\
LiCd\ub{P} & 6.09 & 6.02 & 0.51 & 0.4 & 0.44 & 0.01 & 19.18 \\
\ub{Au}ScSn & 6.42 & 6.41 & 0.22 & 10.2 & 0.56 & 0.17 & 22.57 \\
\ub{Ni}TiSn & 5.94 & 5.82 & 0.50 & 0.2 & 0.90 & 0.00 & 29.04 \\
\ub{Ni}ZrSn & 6.11 & 6.13 & 0.60 & 13.7 & 0.67 & 0.22 & 28.38 \\
\ub{Ni}HfSn & 6.08 & 6.06 & 0.52 & 10.9 & 0.79 & 0.19 & 28.38 \\
\ub{Pd}ZrSn & 6.32 & 6.35 & 0.56 & 13.1 & 0.61 & 0.21 & 25.02 \\
\ub{Pd}HfSn & 6.30 & 6.29 & 0.51 & 10.8 & 0.71 & 0.19 & 24.71 \\
\ub{Pt}TiSn & 6.16 & 6.08 & 1.09 & 0.7 & 0.94 & 0.02 & 21.92 \\
\ub{Pt}ZrSn & 6.32 & 6.35 & 1.42 & 9.3 & 0.74 & 0.19 & 20.00 \\
\ub{Pt}HfSn & 6.31 & 6.29 & 1.40 & 7.3 & 0.85 & 0.16 & 19.28 \\
\ub{Ni}ScSb & 6.06 & 6.04 & 0.30 & 12.6 & 0.70 & 0.23 & 22.96 \\
\ub{Ni}ScBi & 6.19 & 6.30 & 0.15 & 4.1 & 0.57 & 0.06 & 27.99 \\
\ub{Ni}YSb & 6.31 & 6.43 & 0.28 & 242.3 & 0.05 & 0.77 & 23.30 \\
\ub{Pd}ScSb & 6.31 & 6.29 & 0.31 & 10.9 & 0.59 & 0.19 & 21.12 \\
\ub{Pd}YSb & 6.53 & 6.63 & 0.19 & 33.0 & 0.25 & 0.35 & 21.81 \\
\ub{Pt}ScSb & 6.31 & 6.27 & 1.04 & 9.9 & 0.72 & 0.19 & 20.31 \\
\ub{Pt}YSb & 6.54 & 6.59 & 0.34 & 58.2 & 0.21 & 0.50 & 24.50 \\
\end{tabular}
\end{ruledtabular}
\caption{Properties of experimentally synthesized half-Heuslers,
  grouped by family. Experimental lattice constants are from the
  ICSD. Also presented are the theoretical lattice constant $a$, band
  gap $E_{\rm gap}$, piezoelectric coefficient $d_{14}$,
  electromechanical coupling coefficient $k_{14}$, and free-stress static
  dielectric constant $\epsilon_0$.}
\label{table:Table1}
\end{center}
\end{table}

It is remarkable that no piezoelectric response data for any
half-Heusler compound has yet been reported. With a single-crystal
sample of sufficiently low conductivity, the piezoelectric coefficient
should be readily measurable for most if not all of these compounds.
Moreover, measurements of the dielectric response and elastic
coefficients, which also have not been reported to date, would provide
an additional test of these theoretical predictions and a more
complete characterization of the polarization-related properties of
these otherwise much-studied compounds.

Next, we consider the properties of the full set of 792 hypothetical
and real $ABC$ combinations identified earlier for study. As before,
for each combination we optimize the lattice constant for each of
the variants $\ub{A}BC$, $A\ub{B}C$, and $AB\ub{C}$. Choosing the
variant having the lowest total energy, we determine whether our LDA
calculations predict it to be insulating. Of the 792 combinations,
we find 234 insulators having either 8 or 18 valence electrons,
while the compounds containing 28 valence electrons are all found to
be metallic. For the insulators, we perform linear-response
calculations using ABINIT as described above, except that we also
compute the $C_{11}$ and $C_{12}$ elastic constants.  This allows us
to screen for local elastic stability by requiring that the elastic
constants satisfy $C_{11}+ 2C_{12} > 0$, $C_{44} > 0$, $C_{11} -
C_{12} > 0$, as described in Ref.~\onlinecite{Karki97p8579} and
references therein.
%$C_{11}, C_{44} > 0$ and $C_{12}C_{21}< C_{11}^{2}$. 
Furthermore, we calculate phonon frequencies at three additional
high-symmetry points ($X, L$ and $W$) and eliminate combinations which
exhibit any unstable modes.  This reduces the combinations further
from 234 down to 189.

We thus arrive at 189 combinations that are predicted to be insulating
and locally stable in the lowest-energy variant of the three possible
$ABC$ half-Heusler structures. Since DFT tends to underestimate band
gaps, we expect that the actual fraction of insulating structures will
be slightly higher than our calculations would indicate.  The computed
band gaps and lattice parameters for these 189 compounds are shown in
Figure~\ref{fig:Figure2}. We expected no particular correlation
between lattice parameter and band gap, and indeed we find none.  Both
quantities are rather broadly distributed, suggesting that there could
be considerable flexibility in choosing materials over a substantial
range of desired gap or lattice constant.

For these 189 compounds, we also compute $d_{14}$,  $k_{14}$, and
$\epsilon_{0}$ using the same methods as before.  To give a sense of
how the range of properties in the full set of known and hypothetical
compounds compares with that of the subset of known compounds, we
present a scatter plot of $k_{14}$ vs log $d_{14}$ in Figure
~\ref{fig:Figure3}.

It can be seen that there are hypothetical compounds with $k_{14}$ and
$d_{14}$ values well above those of known compounds (some have
$k_{14}$ close to one and $d_{14}$ up to several hundred).  We also
find that these two quantities show the expected positive
correlation. The twelve compounds with the highest values of $d_{14}$
are listed in Table~\ref{table:Table2}. As $d_{14}$ is determined both
by the polarization-strain response $e_{14}$ and the shear elastic
constant, large values can arise either from the former being large,
the latter being small, or both; this is reflected in the examples in
Table~\ref{table:Table2}, where all $C_{44}$ are small.

\begin{figure}
\includegraphics[width=3.0in]{Figure2.eps}
\caption{{Cell parameters (in \AA) of insulating $ABC$ combinations as
well as their range of band gaps (in eV) are depicted as open black
circles. Known combinations, from Table 1, are highlighted as filled
red circles.}}
\label{fig:Figure2}
\end{figure}

\begin{figure}
\includegraphics[width=3.0in]{Figure3.eps}
\caption{{Electromechanical coupling factor $k_{14}$ of insulating
$ABC$ combinations as a function of piezoelectric constant
$d_{14}$. Known combinations, from Table 1, are highlighted as filled
red circles}}
\label{fig:Figure3}
\end{figure}

\begin{table}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{l d d d d d d}
$ABC$ & \mcc{$a$} & \mcc{$E_{\rm gap}$} & \mcc{$d_{14}$} &
\mcc{$C_{44}$} & \mcc{$k_{14}$} & \mcc{$\epsilon_0$} \\
  & \mcc{(\AA)} & \mcc{(eV)} & \mcc{(pC/N)} & \mcc{(10$^{11}$Pa)} &
 & \\
\hline
SrCd\ub{Si} & 7.01 & 0.11 & 2878.8 & 0.01 & 0.97 & 24.98 \\
SrAl\ub{Ga} & 6.87 & 0.13 & 1122.3 & 0.01 & 0.86 & 48.85 \\
NaZn\ub{P} & 6.07 & 0.37 & 344.1 & 0.06 & 0.90 & 17.79 \\
Na\ub{B}Ge & 5.56 & 0.43 & 304.2 & 0.03 & 0.79 & 18.12 \\
MgSr\ub{Si} & 7.09 & 0.87 & 280.4 & 0.04 & 0.80 & 20.63 \\
\ub{Ni}YSb & 6.43 & 0.28 & 242.3 & 0.05 & 0.77 & 23.30 \\
MgSr\ub{Ge} & 7.11 & 0.36 & 230.4 & 0.05 & 0.75 & 21.95 \\
CuY\ub{Si} & 6.36 & 0.50 & 217.0 & 0.05 & 0.76 & 19.87 \\
NaMg\ub{N} & 5.46 & 0.72 & 190.6 & 0.09 & 0.87 & 11.33 \\
BeSc\ub{B} & 5.39 & 0.33 & 133.4 & 0.13 & 0.72 & 23.84 \\
MgSr\ub{Sn} & 7.41 & 0.72 & 127.2 & 0.07 & 0.58 & 25.56 \\
LiCa\ub{P} & 6.63 & 1.73 & 123.2 & 0.07 & 0.72 & 10.82 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\caption{Top twelve half-Heusler compounds, ranked according to
  piezoelectric coefficient $d_{14}$.}
\label{table:Table2}
\end{table}

As a guide for experimental investigation of piezoelectric
half-Heusler compounds, we also present an alternative selection of
compounds chosen according to relevant practical considerations. We
filter the list to eliminate compounds with toxic (Pb, Cd, As) or
expensive (Be, Pd, Pt) elements. In addition, we require a DFT band
gap above 0.4\,eV to favor low sample conductivity. Lastly, to
promote cation ordering into the lowest-energy variant, we require
$\Delta E>0.15$\,eV, where $\Delta E$ is the difference in energy
between the lowest-energy and next-lowest-energy variants. Of the
remaining 65 compounds that satisfy these criteria, 59 have
$d_{14}>2.6$\,pC/N, the experimental value of GaAs. The fifteen with
the largest $d_{14}$ values are presented in Table III.

\begin{table}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{l d d d d d d}
$ABC$ & \mcc{$a$} & \mcc{$E_{\rm gap}$} & \mcc{$d_{14}$} &
\mcc{$C_{44}$} & \mcc{$k_{14}$} & \mcc{$\epsilon_0$} \\
 & \mcc{(\AA)} & \mcc{(eV)} & \mcc{(pC/N)} & \mcc{(10$^{11}$Pa)} &
 & \\
\hline
LiY\ub{C} & 5.83 & 0.72 & 58.5 & 0.17 & 0.52 & 17.54 \\
CuY\ub{Ge} & 6.36 & 0.47 & 55.3 & 0.19 & 0.49 & 20.18 \\
AgY\ub{Si} & 6.55 & 0.56 & 47.3 & 0.14 & 0.40 & 18.79 \\
MgCa\ub{Si} & 6.79 & 0.62 & 44.0 & 0.19 & 0.43 & 18.17 \\
MgCa\ub{Ge} & 6.81 & 0.57 & 42.6 & 0.19 & 0.41 & 19.25 \\
NaSc\ub{C} & 5.77 & 0.61 & 38.2 & 0.32 & 0.44 & 21.12 \\
LiY\ub{Si} & 6.58 & 0.54 & 38.2 & 0.17 & 0.35 & 19.91 \\
LiY\ub{Ge} & 6.58 & 0.52 & 37.5 & 0.17 & 0.34 & 20.41 \\
ZnSc\ub{B} & 5.63 & 0.28 & 36.9 & 0.32 & 0.41 & 23.70 \\
AgY\ub{Ge} & 6.56 & 0.52 & 36.2 & 0.17 & 0.34 & 19.35 \\
MgZn\ub{C} & 5.22 & 0.80 & 35.5 & 0.50 & 0.51 & 20.14 \\
NaAl\ub{Si} & 6.26 & 0.34 & 35.4 & 0.27 & 0.41 & 19.25 \\
\ub{Ni}ScP & 5.79 & 0.49 & 34.1 & 0.30 & 0.41 & 19.75 \\
CaSr\ub{Ge} & 7.57 & 0.30 & 33.9 & 0.12 & 0.25 & 24.27 \\
MgCa\ub{Sn} & 7.10 & 0.34 & 32.6 & 0.19 & 0.31 & 21.78 \\
\end{tabular}
\end{ruledtabular}
\caption{Best fifteen remaining half-Heusler compounds, ranked
according to piezoelectric coefficient $d_{14}$.  None of the initial
top twelve presented in Table II remain after considering band gap,
stability, cost and toxicity.}
\label{table:Table3}
\end{center}
\end{table}

Further investigation of the hypothetical half-Heusler
piezoelectrics hinges on the possibility of experimentally realizing
the desired compounds in the half-Heusler structure. Additional
information about bulk equilibrium $ABC$ phases can be obtained from
the ICSD. In the cases where no phase is reported, it could be that
either no stable bulk phase exists with that stoichiometry, or simply
that the relevant composition has not been studied. $ABC$
combinations are also reported in several structures other than the
half-Heusler structure. Specifically, we find that nine combinations
listed in Table III are reported in the ICSD with other structures:
MgSrSi, MgCaSi and MgCaGe are reported with structures in the $Pnma$
space group, while NaBaP, AgYSi and LiYSi are reported to have
structures with $P{\bar6}2m$ symmetry. The latter space group is
piezoelectric, so these compounds can offer an additional route to
new piezoelectrics. Moreover, 13 compounds of the original 792
combinations are reported in two different space groups, namely
LiMgN, LiSrP, LiBaP, LiAlGe, CuMgAs, AgCaBi, CuScSn, CuYSn, AuScSn,
CaZnSn, PdTiGe, NaAgO and LiAuS. Of these, only AuScSn is reported
in the half-Heusler structure, while none are found in either Tables
II or III.

This structural variety should encourage interest in further
exploration of these systems for piezoelectricity and other
functional behavior.  Polymorphism suggests the possibility of
intentionally stabilizing one of several competing structures,
or of switching between them at will, via the application of
external stresses, fields, or other perturbations.

First-principles calculations of the total energy of alternative
structures could be used both to predict ground state structures and
to identify systems with piezoelectric structures as low-energy
alternatives that would be suitable candidates for stabilization
through compositional substitution or epitaxial strain.  Furthermore,
half-Heusler systems offer the opportunity to combine piezoelectricity
with other functional properties, including magnetism, to produce
multifunctional behavior of fundamental scientific interest and for
groundbreaking technological applications.

In summary, we have used a first-principles rational-design
approach to demonstrate semiconducting half-Heusler
compounds as a previously unrecognized class of piezoelectric
materials. We have presented these predictions first for compounds
already reported in the half-Heusler structure, and then for
a much larger set of candidate combinations that were generated and
screened for high performance via a high-throughput analysis.
We hope that our results may
provide guidance for the experimental realization and further
investigation of high-performance materials suitable for practical
applications.
We also suggest that the combination of piezoelectric
properties with other characteristic properties of Heuslers,
especially magnetic properties, may offer inviting avenues for
further development of multifunctional materials.

\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, C. Felser, I.
Takeuchi, and S.-W. Cheong for useful discussions.

\vspace{0.3cm}
\noindent{\bf Competing Financial Interests}
\vspace{0.3cm}

There are no competing financial interests.  

\bibliography{thebibliography}% Produces the bibliography via BibTeX.

\end{document}
