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\begin{document}

\title{Antiferroelectric topological insulators in $ABC$ compounds}

\author{Bartomeu Monserrat}
\email{monserrat@physics.rutgers.edu}
\affiliation{Department of Physics and Astronomy, Rutgers University,
  Piscataway, New Jersey 08854-8019, USA}
\affiliation{TCM Group, Cavendish Laboratory, University of Cambridge,
  J.\ J.\ Thomson Avenue, Cambridge CB3 0HE, United Kingdom}
\author{Joseph W. Bennett}
\affiliation{Department of Physics and Astronomy, Rutgers University,
  Piscataway, New Jersey 08854-8019, USA}
\author{Karin M. Rabe}
\affiliation{Department of Physics and Astronomy, Rutgers University,
  Piscataway, New Jersey 08854-8019, USA}
\author{David Vanderbilt}
\affiliation{Department of Physics and Astronomy, Rutgers University,
  Piscataway, New Jersey 08854-8019, USA}

\date{\today}

\begin{abstract}
We introduce antiferroelectric topological insulators as a new class of functional material in which an electric field can be used to control topological order and induce topological phase transitions. Using first principles methods, we predict that several members of an $ABC$ family of orthorhombic compounds are antiferroelectric topological insulators. We also show that epitaxial strain and hydrostatic pressure can be used to tune the topological order and the band gap of these $ABC$ compounds. Antiferroelectric topological insulators could enable applications in electronics and spintronics.
\end{abstract}

\pacs{}

\maketitle

% Electric field control of topological insulators
Topological insulators and related materials~\cite{z2_topological_insulators,topological_insulators_review,topological_insulators_review2} exhibit interesting properties such as robust edge currents or spin-momentum locking that make them promising candidates for technological applications. In this context, the control of topological order by means of electric fields is an interesting route to applications of these materials in electronics and spintronics~\cite{murakami_ferro_topo_superlattice,efield_ti_zunger,narayan_abc_feti,rappe_feti,picozzi_topo_hyperferroelectrics}. 

% Overview of previous work on electric field control of topological insulators
A promising direction is the use of ferroelectrics, which exhibit two states of opposite polarity ($\mathbf{P}$) that can be controlled by an electric field ($\mathbf{E}$) as shown schematically in Fig.~\ref{subfig:feti}.  %as shown schematically in Fig.~\ref{subfig:feti}. 
The two polarization states in these ferroelectric topological insulators could be used, for example, to create spin-selected collimated electron beams~\cite{rappe_feti} or to control the spin texture around the Dirac cones of the surface states~\cite{picozzi_topo_hyperferroelectrics}. Several materials have been proposed to exhibit these properties: superlattices of the ferroelectric material GeTe and the topological insulator Sb$_2$Te$_3$ exhibit electric field control of topological order~\cite{murakami_ferro_topo_superlattice}; strained LiZnSb and CsPbI$_3$ are candidate ferroelectric topological insulators~\cite{narayan_abc_feti,rappe_feti}; and ferroelectricity and topological order coexist in some $ABC$ hyperferroelectrics, which could be candidates for thin-film applications~\cite{picozzi_topo_hyperferroelectrics}. 

%Liu and co-workers have shown that an external electric field can induce a band inversion in phosphorene driving the system into a topological insulator~\cite{efield_ti_zunger},

% Statement of what we do: antiferroelectric topological insulator
In this work we explore the possibility of controlling topological order in antiferroelectric materials, which exhibit an antipolar ground state and two polar states that can be accessed using an electric field, as shown schematically in Fig.~\ref{subfig:afti}. In ferroelectric materials, the two polar states are related by inversion symmetry, imposing the same topological character on both. Antiferroelectrics are more flexible, as inversion symmetry also dictates that the two polar states have the same topological character, but the antipolar state could belong to a different topological class. %As a consequence, using antiferroelectrics would provide the same potential applications as using ferroelectrics, with the addition that topological phase transitions could also be controlled using an electric field. 
We thus define an antiferroelectric topological insulator (AFTI) as an antiferroelectric material in which at least one of the states is a topological insulator. There can be three different types of AFTIs, depending on which states are topological insulators:
(i) a type-I AFTI with an antipolar normal state and polar topological states, (ii) a type-II AFTI with an antipolar topological state and polar normal states, and (iii) a type-III AFTI with antipolar and polar topological states.
Electric fields could be used to drive topological phase transitions in AFTIs of types I and II. The schematic of Fig.~\ref{subfig:afti} shows an example of an AFTI of type I.
%

\begin{figure}[b]
\centering
\subfloat[][]{\includegraphics[scale=0.30]{./feti.pdf}\label{subfig:feti}}
\hspace{0.3cm}
\subfloat[][]{\includegraphics[scale=0.30]{./afti.pdf}\label{subfig:afti}}
\caption{Schematic representation of (a) a ferroelectric topological insulator and (b) an antiferroelectric topological insulator with an antipolar normal insulator (NI) state and two polar topological insulator (TI) states, referred to as type-I antiferroelectric topological insulator in the text.}
\label{fig:afti}
\end{figure}


% Description of system class and justification
We study potential AFTIs in the recently proposed $ABC$ family of materials, which includes ferroelectrics~\cite{abc_ferroelectrics}, antiferroelectrics~\cite{abc_antiferroelectrics}, and hyperferroelectrics~\cite{abc_hyperferroelectrics}. 
The antiferroelectric $ABC$ compounds have a nonpolar reference structure of hexagonal $P6_3/mmc$ space group with energy-lowering distortions to an antipolar orthorhombic $Pnma$ structure and to a polar hexagonal $P6_3mc$ structure. The $Pnma$ antipolar structure is the ground state, and there is a small energy barrier to the $P6_3mc$ polar structure~\cite{abc_antiferroelectrics}.
 We adopt a convention in which the $B$ and $C$ atoms form the hexagonal layers of the $P6_3/mmc$ structure, and the $A$ atoms are arranged in the stuffing sites in between. 
%Bennett and co-workers have proposed a range of compounds in which the $Pnma$ antipolar structure has the lowest energy, with a small energy barrier to the $P6_3mc$ polar structure, which suggests the possibility of using these materials as antiferroelectrics~\cite{abc_antiferroelectrics}. 
The large number of possible $ABC$ compounds with these characteristics make them an excellent platform to search for the coexistence of antiferroelectricity and other types of order. 

% Search strategy and target compounds
Topological insulators exhibit band inversion driven by the spin-orbit interaction. Since the strength of the spin-orbit interaction increases with the atomic number, we focus the search for topological materials in $ABC$ compounds to those containing heavy elements. Furthermore, the energy scale associated with spin-orbit coupling (of up to about $1$~eV) needs to be comparable to the band gap in order to induce a band inversion, and therefore we focus on materials with band gaps below $1$~eV. We use these criteria to search over the $ABC$ antiferroelectrics reported in Ref.~\cite{abc_antiferroelectrics}, and we indentify three compounds, LiMgBi, NaMgBi, and KMgBi, that are promising candidates as AFTIs. We refer collectively to these three compounds as $A$MgBi (for $A=$Li, Na, Mg). 

\paragraph{First-principles calculations.}
% DFT structural calculations
Our calculations are based on density functional theory (DFT) as implemented in the {\sc vasp} package~\cite{vasp1,vasp2,vasp3,vasp4}. We use the local density approximation to the exchange correlation functional~\cite{PhysRevLett.45.566,PhysRevB.23.5048} and the projector augmented wave method~\cite{paw_original,paw_us_relation} with an energy cut-off of $300$~eV and a $\mathbf{k}$-point grid size of $8\times8\times8$ to sample the Brillouin zone (BZ). All calculations include spin-orbit coupling. The lattice vectors and internal coordinates are optimized until the forces on all atoms are smaller than $0.001$~eV/\AA\@, and the pressure is within $0.01$~GPa of the target value when applying hydrostatic pressure. In-plane epitaxial strain is enforced by constraining the in-plane lattice constants and allowing the out-of-plane lattice constant and the internal coordinates to fully relax. Strains are reported using the hexagonal $P6_3mc$ equilibrium structure as reference, with in-plane lattice constants equal to $4.68$~\AA\@, $4.86$~\AA\@, and $5.05$~\AA\@ for LiMgBi, NaMgBi, and KMgBi, respectively. The $Pnma$ equilibrium structure differs from any epitaxially strained $Pnma$ structure because the imposition of a hexagonal substrate forces the two in-plane lattice vectors of the orthorhombic structure to have the exact ratio $\sqrt{3}$, a condition that is not exactly obeyed in the unstrained structure.

% DFT band gap and Z2 calculations
We make a precise determination of the band gaps by Wannier interpolation of the band structure~\cite{wannier_rmp} using the {\sc wannier90} package~\cite{wannier90_2008,wannier90_2014}. We use the iterative approach of Ref.~\cite{liu_noncentrosymm_weyl}, initially sampling the electronic BZ with a coarse $\mathbf{k}$ grid (typically of size $40\times40\times40$), then determining the $\mathbf{k}_0$-point having the smallest gap, and then searching over a denser $\mathbf{k}$-point grid centered around the point $\mathbf{k}_0$. This procedure is iterated until convergence is reached. The topological $Z_2$ index for the insulating phases is determined by following the adiabatic pumping of Wannier charge centers over the BZ~\cite{z2_index_calculation} as implemented in the {\sc z2pack} code~\cite{z2pack}. This approach is appropriate for systems without inversion symmetry. 


% Ambient conditions
\begin{table}[b]
  \setlength{\tabcolsep}{2pt} % Default is 6pt
  \caption{Energy difference $\Delta E$ between the $P6_3mc$ structure and the $Pnma$ structure in meV per formula unit (f.u.), and the band gaps $E_{\mathrm{g}}^{P6_3mc}$ and $E_{\mathrm{g}}^{Pnma}$ of the $P6_3mc$ and the $Pnma$ structures in meV. All results correspond to the equilibrium structures.}
  \label{tab:ambient}
  \begin{ruledtabular}
  \begin{tabular}{c|ccc}
            &  $\Delta E$~(meV/f.u.)   &   $E_{\mathrm{g}}^{P6_3mc}$~(meV)   & $E_{\mathrm{g}}^{Pnma}$~(meV)   \\
  \hline
  LiMgBi &  $59$  & $66$  & $12$  \\
  NaMgBi &  $122$ & $85$  & $22$  \\
  KMgBi  &  $239$ & $96$  & $16$  \\
\end{tabular}
\end{ruledtabular}
\end{table}

\paragraph{Equilibrium structures.}
The $Pnma$ structure is lower in energy compared to the $P6_3mc$ structure in all three $A$MgBi materials, as shown in Table~\ref{tab:ambient}. All three materials in both the $P6_3mc$ and $Pnma$ structures are strong topological insulators with topological indices $(1;000)$. The minimum band gaps, shown in Table~\ref{tab:ambient}, occur near the $\Gamma$ point, with states dominated by bismuth and magnesium. These results indicate that the $A$MgBi compounds are candidate AFTIs of type III in which the the antipolar and both polar states have the same $Z_2$ indices. Finally, we note that a polymorph of LiMgBi, namely MgLiBi in which the Li and Mg atoms are exchanged, is more stable than LiMgBi in our calculations.

\begin{figure}
\centering
\includegraphics[width=1.00\textwidth]{./gap_lmb_strain.pdf}
%\includegraphics[width=0.45\textwidth]{./gap_lmb_strain.pdf}
\caption{Band gap of the LiMgBi compound of polar $P6_3mc$ (top) and antipolar $Pnma$ (bottom) structures as a function of epitaxial strain, together with their topological character. The $P6_3mc$ struture exhibitis regions in which it is a topological insulator (TI), a Weyl semimetal (WSM), and a normal insulator (NI), whereas the $Pnma$ structure exhibits metallic and normal insulator phases.}
\label{fig:gap_lmb}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=1.00\textwidth]{./bands.pdf}
%\includegraphics[width=0.45\textwidth]{./bands.pdf}
\caption{Band structure of LiMgBi in the polar $P6_3mc$ structure at strains of $-2.86\%$ (left) corresponding to a normal insulator, and $+0.34\%$ (right) corresponding to a topological insulator.}
\label{fig:bands}
\end{figure}


\paragraph{Epitaxial strain.} 
% Epitaxial strain -- LiMgBi
We further investigate how the relative energies between the $Pnma$ and $P6_3mc$ structures, as well as their topological character, can be tuned by means of epitaxial strain. We find that the unstrained $P6_3mc$ structure of all three compounds corresponds to a topological insulator, and compressive epitaxial strain drives them through a topological phase transition to a normal insulator via an intermediate Weyl semimetal phase, as shown in Fig.~\ref{fig:gap_lmb} for LiMgBi. The intermediate semimetallic phase is a universal feature of a topological phase transition in materials without inversion symmetry~\cite{murakami_weyl_semimetal,liu_noncentrosymm_weyl}. In Fig.~\ref{fig:bands} we show the band structure of polar $P6_3mc$ LiMgBi around the $\Gamma$ point for epitaxial strains of $-2.86\%$, corresponding to a normal insulator, and $+0.34\%$, corresponding to a topological insulator, with gaps of $93$ and $94$~meV respectively. The bands exhibit Rashba splitting, the strength of which can also be tuned by means of epitaxial strain. The projection on the bismuth orbitals demonstrates the band inversion associated with the topological phase transition. 

In LiMgBi, the $Pnma$ epitaxial structure is a normal insulator under compressive strain, and becomes a metal at about $0.34\%$ tensile strain (Fig.~\ref{fig:gap_lmb}). The fact that the epitaxial LiMgBi $Pnma$ structure is either a normal insulator or a metal, whereas the corresponding unstrained structure is a topological insulator, derives from the fixed $\sqrt{3}$ ratio that the two in-plane orthorhombic axes obey under epitaxial strain. We have performed band gap and $Z_2$ index calculations of several structures linearly interpolating between the epitaxially strained and unstrained $Pnma$ structures, confirming that there is a gap closure and a topological phase transition from the unstrained topological insulator to the epitaxial normal insulator. 

% LMB -- AFTI
The band gap results presented in Fig.~\ref{fig:gap_lmb} indicate that epitaxial LiMgBi is a candidate AFTI of type I with the polar states corresponding to a topological insulator, and the antipolar state to a normal insulator, the situation depicted in Fig~\ref{subfig:afti}. Furthermore, under compressive strain, the energy difference between the $Pnma$ and the $P6_3mc$ structures decreases, promoting the possibility that an electric field can be used to switch between the polar and antipolar states. Finally, we note that the MgLiBi polymorph is more stable than LiMgBi for compressive strains below $0.20\%$, so that the potential synthesis of LiMgBi as an AFTI should focus in the epitaxial range from $-0.20\%$ to $+0.34\%$.

% Epitaxial strain -- NMB and KMB
In NaMgBi and KMgBi, the $Pnma$ structure under epitaxial strain does not exhibit a normal insulator regime, and therefore
an antipolar normal state cannot coexist with a polar topological state
in these materials.
Instead, in NaMgBi a polar normal insulator coexists with an antipolar topological insulator at strains between $-4.72\%$ and $-4.00\%$, which corresponds to a type-II AFTI. However, the band gap for both antipolar and polar states in that region is below $20$~meV, which may make experimental realization difficult. KMgBi under strain behaves analogously to NaMgBi, and the polar normal insulator coexists with the antipolar topological insulator at strains between $-4.95\%$ and $-3.96\%$, but again the band gaps are small. Additional details of NaMgBi and KMgBi under strain are provided in the Supplemental Material.

\begin{figure}
\centering
\includegraphics[width=1.00\textwidth]{./gap_kmb_pressure.pdf}
%\includegraphics[width=0.45\textwidth]{./gap_kmb_pressure.pdf}
\caption{Band gap of the KMgBi compound of polar $P6_3mc$ (top) and antipolar $Pnma$ (bottom) structures as a function of hydrostatic pressure, together with their topological character. The $P6_3mc$ struture exhibitis a topological insulator (TI) regime and a metallic regime, whereas the $Pnma$ structure exhibits topological insulator and normal insulator (NI) phases.}
\label{fig:gap_kmb}
\end{figure}


\paragraph{Hydrostatic pressure.}
% Hydrostatic pressure -- KMB
Under hydrostatic pressure, the $P6_3mc$ structure of NaMgBi and KMgBi undergoes a transition from a topological insulator into a metal, as shown in Fig.~\ref{fig:gap_kmb} for KMgBi. The $Pnma$ structure of NaMgBi and KMgBi undergoes a topological phase transition from a topological to a normal insulator, as shown for KMgBi in Fig.~\ref{fig:gap_kmb}. LiMgBi under hydrostatic pressure is higher in energy compared to the polymorph MgLiBi.

% KMB -- AFTI
The band gap results of Fig.~\ref{fig:gap_kmb} suggest that KMgBi under hydrostatic pressure is also a candidate AFTI of type I (Fig.~\ref{subfig:afti}). In particular, this functional phase should be observable at pressure between $0.89$~GPa and $2.50$~GPa. 
% Hydrostatic pressure - LMB and NMB
NaMgBi has a similar phase diagram to that of KMgBi, but there is no pressure range in which the normal antipolar and topological polar phases coexist, as shown in the Supplemental Material. 

\begin{table*}[b]
  \setlength{\tabcolsep}{2pt} % Default is 6pt
  \caption{Candidate AFTIs from the $A$MgBi family of compounds. Type I refers to normal antipolar and topological polar states, type II to topological antipolar and normal polar states, and type III to topological antipolar and polar states.}
  \label{tab:afti_summary}
  \begin{ruledtabular}
  \begin{tabular}{c|ccc}
            &  Equilibrium  &   Epitaxial strain   & Hydrostatic pressure   \\
  \hline
  LiMgBi &  --       & Type I ($-0.20$\% to $+0.34$\%)   & --  \\
  NaMgBi &  Type III & Type II ($-4.72$\% to $-4.00$\%)  & --  \\
  KMgBi  &  Type III & Type II ($-4.95$\% to $-3.96$\%)  & Type I ($0.89$~GPa to $2.50$~GPa)  \\
\end{tabular}
\end{ruledtabular}
\end{table*}

\paragraph{Discussion.}
% Discussion
Overall, our first-principles calculations indicate that the $A$MgBi compounds are promising candidates as AFTIs, as summarized in Table~\ref{tab:afti_summary}. Of these, the most promising candidates are LiMgBi under epitaxial strain in the range from $-0.20\%$ to $+0.34\%$, and KMgBi under hydrostatic pressure between $0.89$~GPa and $2.50$~GPa, which are AFTIs of type I with polar topological states and an antipolar normal state as depicted in Fig.~\ref{subfig:afti}. In these materials, an electric field could be used to induced a topological phase transition. This possibility raises interesting questions for future research, such as the precise switching mechanism in AFTIs (as well as in ferroelectric topological insulators). Switching typically occurs via the motion of domain walls, but in an AFTI of type I or II these domain walls separate phases of distinct topological order.  As a result they are necessarily metallic, which may influence the switching dynamics in unusual and interesting ways.


% Conclusions
\paragraph{Conclusions.}
We propose a new type of functional material, an antiferroelectric topological insulator, in which an electric field can be used to induce topological phase transitions and to control topological order. Using first-principles methods, we show that the orthorhombic $ABC$ compounds provide a rich playground in which to explore the existence of antiferroelectric topological insulators. In particular, the compounds $A$MgBi ($A=$Li, Na, K) exhibits an interesting interplay between antiferroelectric and topological properties as a function of both epitaxial strain and hydrostatic pressure, with LiMgBi under strain and KMgBi under pressure appearing as promising candidates for the experimental realization of an antiferroelectric topological insulator.

% Future work
%Further manipulation of the $A$MgBi compounds, such as the construction of superlattices, could lead to additional candidate antiferroelectric topological insulators. 



\acknowledgments

This work was funded by NSF grant DMR-1408838. B.M.~thanks Robinson
College, Cambridge, and the Cambridge Philosophical Society for a Henslow 
Research Fellowship.



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\bibliography{ti}

\end{document}
