Frustrated Dipole Order Induces Noncollinear Proper Ferrielectricity in Two Dimensions

TL;DR

This paper predicts a new type of noncollinear ferrielectricity in two-dimensional dioxydihalides, driven by frustrated dipole interactions, leading to unique topological and physical properties.

Contribution

It introduces a theoretical prediction of noncollinear ferrielectricity in 2D materials induced by competing soft modes, a phenomenon rarely observed in electric dipole systems.

Findings

Presence of $ ext{Z}_2 imes ext{Z}_2$ topological domains

Atomic-scale dipole vortices

Negative piezoelectricity observed

Abstract

Within Landau theory, magnetism and polarity are homotopic, displaying a one-to-one correspondence between most physical characteristics. However, despite widely reported noncollinear magnetism, spontaneous noncollinear electric dipole order as ground state is rare. Here a dioxydihalides family is predicted to display noncollinear ferrielectricity, induced by competing ferroelectric and antiferroelectric soft modes. This intrinsic noncollinearity of dipoles generates unique physical properties, such as $\mathbb{Z}_2\times\mathbb{Z}_2$ topological domains, atomic-scale dipole vortices, and negative piezoelectricity.

Full text

The Landau theory of phase transitions provides an elegant common framework for both magnetic and polar systems. The one-to-one correspondence between physical characteristics, such as ordered phases – ferromagnetic (FM) vs ferroelectric (FE) states, antiferromagnetic (AFM) vs antiferroelectric (AFE) states [see Figs. 1(a-b)]–, hysteresis loops, domains, and other properties is well recognized. However, ferrielectric (FiE) systems, with partially compensated collinear dipoles [Figs. 1(c-d)], are rare (except in liquid crystals and in a few solids like hybrid improper ferroelectrics Benedek and Fennie (2011)) Scott et al. (2017), while ferrimagnetic (FiM) materials are fairly common, e.g. Fe3O4.

This incompleteness of dipole orders is even more dramatic with regards to noncollinearity. For magnets, spin noncollinearity has been widely studied Ramirez (1994); Cheong and Mostovoy (2007), leading to exotic magnetism-driven polarization ($P$) Kimura (2007), skyrmions Nagaosa and Tokura (2013), and topological anomalous Hall effect Nagaosa et al. (2010). There are several mechanisms to generate these crucial noncollinear spin orders. For example, in geometrically frustrated systems, such as two-dimensional (2D) triangular lattices, the AFM coupling between nearest-neighbor (NN) spins can generate the $120^{\circ}$ order Ramirez (1994). For other lattices, the exchange frustration, typically involving competition between NN FM ($J_{1}$) and next-nearest-neighbor (NNN) AFM ($J_{2}$) couplings, can generate magnetic cycloid or helical arrangements Cheong and Mostovoy (2007).

By contrast, the electric dipoles within a FE or AFE domain always tend to be parallel or antiparallel, aligned by the dipole-dipole interactions Dawber et al. (2005); Rabe et al. (2007). Although slightly noncollinear dipole orders were proposed in a few FiEs, e.g. BaFe2Se3 and Pb2MnWO6 Dong et al. (2014); Orlandi et al. (2014), their noncollinearities are rigidly fixed by the local crystalline environment and, thus, can be trivially modulated. Noncollinear FiE phases were also predicted for strained BiFeO3 Yang et al. (2012, 2014); Prosandeev et al. (2011), which has attracted much interest while waiting for experimental verification. In addition, dipoles can become noncollinear at some domain walls, as when forming flux-closure domains and even dipole vortices/skymions Jia et al. (2011); Naumov et al. (2004); Nahas et al. (2015); Yadav et al. (2016); Das et al. (2019). However, such noncollinearity is not a primary property of the FE state but driven by electrostatic effects from geometrically-confined boundary conditions.

Inspired by the “frustration” concept from magnets, here a series of 2D materials ($M$O${}_{2}X_{2}$, $M$: group-VI transition metal; $X$: halogen) is studied theoretically, which we predict can host intrinsic noncollinear electric dipole textures spontaneously.

2D FE monolayers (or few-layers) exfoliated from van der Waals (vdW) layered materials are intrinsically superior at the nanoscale as compared with canonical three-dimensional (3D) FE materials Chang et al. (2016); Liu et al. (2016); Ding et al. (2017); Zhou et al. (2017); Wu and Jena (2018). In spite of this potential value, 2D FE materials remain rare Wu and Jena (2018) and their physical properties, such as domain structures, have not been well explored. This Letter demonstrates that these particular 2D noncollinear polar systems can provide an ideal platform to explore exotic polarity and topological domains beyond the standard collinear ferroelectricity, which may be crucial for domain wall nanoelectronics Catalan et al. (2012).

Physical properties.- Starting from the 3D $M$O3 crystal [perovskite-like structure without A-site ions, see Fig. 1(e)], $M$O${}_{2}X_{2}$ can be derived by replacing the apical O2- oxygens by double halide ions $X^{-}$ [Fig. 1(f)], forming a series of vdW layered materials. Further replacement of O2- by $X^{-}$ can lead to quasi-one-dimensional $M$O$X_{4}$ chains and to the zero-dimensional molecular limit $MX_{6}$ [see Figs. S1(a-b) in Supplementary Material (SM) Sup ].

In this dioxydihalides family, WO2Cl2 and MoO2Br2 have been synthesized experimentally Jarchow et al. (1968); Armstrong et al. (2004); Schustereit et al. (2011), while the crystal structures of the corresponding fluorides, iodides, and CrO${}_{2}X_{2}$ are isomeric or unknown. Thus, in the following only WO2Cl2 and MoO2Br2 will be studied using density functional theory (DFT) Sup . Most previous studies of $M$O${}_{2}X_{2}$ were devoted only to their chemical properties, while their physical properties were rarely addressed.

As shown in Fig. 1(g), each $M$O${}_{2}X_{2}$ layer is composed of corner-sharing octahedra Schustereit et al. (2011). Contrary to most 2D materials which display compact honeycomb or triangular atomic arrangements, the “square” lattice of $M$O${}_{2}X_{2}$ is spatially loose, which is advantageous for polar distortions. Due to the weak vdW interactions, there are two stacking modes for $M$O${}_{2}X_{2}$ layers observed in experiments: the A-B stacking $\alpha$-type (Fig. 1(f)] and the A-A stacking $\beta$-type [Fig. S1 in SM Sup ), corresponding to the space groups $Bb$ (No. $9$) and $Pb2_{1}m$ (No. $26$), respectively Schustereit et al. (2011); Jarchow et al. (1968). The optimal distances ($d_{0}$’s) between adjacent layers are shorter in the $\alpha$ phases (see Table S1 in SM Sup ), which are lower in energy than the $\beta$ phases by $70$ meV/W and $77$ meV/Mo.

The cleavage energies were calculated to analyze whether it is possible to exfoliate $M$O${}_{2}X_{2}$ monolayers, as shown in Fig. 1(h). For the $\alpha$ phase, the cleavage energies are $0.22$ J/m2 and $0.18$ J/m2 for MoO2Br2 and WO2Cl2, respectively. For comparison, the cleavage energy for graphite is $0.325$ J/m2 theoretically Mounet et al. (2018) and $0.37$ J/m2 experimentally Wang et al. (2015). Thus, the exfoliation of a monolayer, or few-layers, from bulk $M$O${}_{2}X_{2}$ should be feasible experimentally. Furthermore, our molecular dynamic simulation confirms the thermal stabilities of these monolayers at $300$ K and $400$ K (Fig. S2 in SM Sup ).

Additional physical properties of $M$O${}_{2}X_{2}$ monolayers and bulk forms are summarized in SM (Fig. S3 and Table. S2) Sup .

Noncollinear dipole order.- The paraelectric (PE) structure of $M$O${}_{2}X_{2}$ is shown in Fig. 2(a), where all $M$ ions are restored to the central positions of the O${}_{4}X_{2}$ octahedra. In its phonon spectrum there are two imaginary-frequency branches, which will lead to spontaneous distortions. The symmetric $B_{1g}$ (i.e. $X_{2}^{+}$) phonon mode at $X$ (and $Y$) leads to AFE-type distortions [Fig. 2(b)]. The double-degenerate $E_{u}$ (i.e. $\Gamma_{5}^{-}$) phonon mode at $\Gamma$ leads to FE-type distortions [Fig. 2(c)]. Both these distortions lower the symmetry from tetragonal to orthorhombic.

The most striking physical result is that the cooperation of these two distortion modes, which resemble the exchange frustration in magnets, leads to a net FiE structure [Fig. 2(d)], which is dynamically stable according to its phonon spectrum. In this FiE state, the AFE and FE ordering directions are orthogonal, along the $a$ and $b$ axes, respectively. If the vdW $d_{0}$ is used as the thickness of a monolayer, the calculated net $P$’s along the $b$ axis are slightly larger than their bulk values (see Table S2 in SM Sup ), themselves only a little larger than those of BaTiO3 ($\sim 20-25$ $\mu$C/cm2).

The $d^{0}$ rule (i.e. the formation of coordination bonds between $M$’s empty $d$ orbitals and O’s $2p$ orbitals) should be the driving force for the polar distortions. This was confirmed by the Bader charge calculation Henkelman et al. (2006). As shown in Table S3 in SM Sup , a small portion of valence electrons “leak” from O2- to $M^{6+}$, accompanying the FiE distortions, which can also be visualized in Fig. 2(e). Although the $d^{0}$ rule is well known for FE/AFE perovskites, such as BaTiO3 and PbZrO3, interestingly none of the previously studied 2D FE systems belongs to this $d^{0}$ category Wu and Jena (2018), since none have perovskite-like structures.

The novel FiE state unveiled here is exotic. Due to the frustration between FE and AFE modes, here the local dipole moments are noncollinear, with a canting angle $\sim 129^{\circ}$ for WO2Cl2 ($\sim 128^{\circ}$ for MoO2Br2) between NN dipoles along the $b$ axis. This unexpected noncollinearity leads to unique physics, as discussed below.

Domain & domain walls.- The DFT energy contour as a function of distortion modes for WO2Cl2 is shown in Fig. 3(a) and its Landau energy fitting can be found in SM Sup . For selected orthorhombic axes, four degenerate wells exist (characterized by the $\pm Q_{1}$ and $\pm Q_{2}$ of the $X_{2}^{+}$ and $\Gamma_{5}^{-}$ modes), corresponding to the four domains: A+, A-, B+, and B-. An intuitive conclusion arising from Fig. 3(a) is that the favoured domain walls are between A+/A-, A+/B+, B+/B-, and A-/B-, while domain walls between A+/B- or A-/B+ are highly energetic, thus unfavourable. A similar situation occurs in hexagonal manganites Choi et al. (2010); Artyukhin et al. (2014), whose energy contour shows a Mexican-hat sixfold symmetry and, thus, prefers the well-known $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ topological domain patterns Choi et al. (2010); Artyukhin et al. (2014). Contrary to WO2Cl2, the collaborative modes in hexagonal manganites are the FE distortion and trimerization of the Mn-sublattice, not the AFE mode.

The possible FE and AFE domain walls in WO2Cl2 are shown in Fig. 3(b-c). Because the head-to-head and tail-to-tail charged domain walls are highly energetically unfavorable, we consider only the shoulder-by-shoulder charge-neutral domain walls. More details can be found in Sup . According to the DFT structural relaxations, these domain walls are atomically sharp, leading to distinct domains. The domain wall energies for FE and AFE domain walls are $1.6$ meV/bond and $5.0$ meV/bond for WO2Cl2 ($11.5$ meV/bond and $8.5$ meV/bond for MoO2Br2), respectively.

As a consequence, it is natural to expect $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ topological domain patterns in $M$O${}_{2}X_{2}$, as sketched in Fig. 3(d). If crystalline twinning and high-energetic charged domains can be excluded, the domain structure will be quite regular: the FE domain walls can only propagate along the $b$ axis while the AFE domain walls can only propagate along the $a$ axis, forming perpendicular cross points, namely atomic-scale $\mathbb{Z}_{4}$ antiphase vortices/antivortices. More details of domains and domain walls can be found in the SM Sup .

In addition, atomic-scale dipole vortices/antivortices form at the FE domain walls, which is different to the dipole vortices in PbTiO3/SrTiO3 superlattices and BiTiO3 nanostructures, whose size scale is much larger ($\sim 3-5$ nm) Naumov et al. (2004); Nahas et al. (2015); Yadav et al. (2016); Das et al. (2019). Our dipole vortices/antivortices are also different from those in the predicted $P2_{1}2_{1}2_{1}$ phase of strained BiFeO3, whose atomic-scale dipole vortices/antivortices form a closely packed array with a fixed position in the whole crystal Prosandeev et al. (2011), while ours exist only at some domain walls and thus are movable. The study of a dipole vortex, in correspondence to a magnetic vortex (or skyrmion) in magnetism, is an emerging topic in the field of polar materials. FiE $M$O${}_{2}X_{2}$ can provide a superior playground for the dipole vortex due to its intrinsic non-collinearity.

Negative piezoelectricity.- For applications of FE materials, the switching of $P$’s are important physical properties. Interestingly, the switching paths for $M$O${}_{2}X_{2}$ are nontrivial. As sketched in Fig. 4(a), the $180^{\circ}$ reversal of the net $P$ does not require the $180^{\circ}$ flip of local dipoles, different from the plain FE cases. Instead, the collaborative $\sim 50^{\circ}$ rotations of local dipoles are enough, another unique property of the unveiled noncollinear ferrielectricity. As shown in Fig. 4(b), the AFE state can act as the nonpolar intermediate state, leading to a moderate energy barrier for $P$ reversal. Comparing with their bulk values, the energy barriers do not change much, namely from $77$ meV/f.u. (bulk) to $79$ meV/f.u. (monolayer) for WO2Cl2.

Another interesting issue is the $90^{\circ}$ switching of $P$, which corresponds to the interchange of crystalline axes $a$ and $b$. Two possible two-step paths are proposed to achieve this flop. As shown in Fig. 4(c), in each step half of the dipoles rotate while the other half remain fixed. Interestingly, the middle states just correspond to two kinds of domain walls, as sketched in Fig. 3(d). The corresponding switching energy barriers are plotted in Fig. 4(d), suggesting a slightly lower barrier for path I.

An interesting property of this $90^{\circ}$ flop of $P$ is the resulting negative piezoelectricity due to noncollinear ferrielectricity. Starting from the tetragonal PE state, the elongated axis of the orthorhombic FiE state is perpendicular to the direction of net $P$, i.e. $a>b$ for $P||b$, which is unusual as compared with most ferroelectrics. This anomalous behavior is due to the noncollinearity of local dipoles, since the $a$-axis component of each local dipole is larger than the $b$-axis component. Then, the $90^{\circ}$ flop of $P$ will lead to a negative piezoelectric coefficient $d_{33}$ (see Figs. 4(e-f)), at least for the partial intermediate process during the $90^{\circ}$ flop. In recent years, the existence of exotic negative piezoelectricity was predicted in special materials Liu and Cohen (2017), but experimentally has only been observed in the organic FE polymer poly(vinylidene fluoride) (PVDF) Katsouras et al. (2015) and CuInP2S6 You et al. (2019).

Our calculations predict that the monolayer dioxydihalides $M$O${}_{2}X_{2}$’s are promising 2D polar materials with exotic noncollinear ferrielectricity. The $d^{0}$ rule, which works in 3D ferroelectric perovskites but has not been found in 2D ferroelectrics before, is the driving force for the polar distortions. Noncollinear ferrielectric order persisting up to room temperature are expected. More importantly, the frustration between the FE and AFE modes generates an intrinsically noncollinear dipole texture, which leads to unique physics in the novel FiE state unveiled here, such as $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ antiphase domain vortices, atomic-level dipole vortices at domain walls, negative piezoelectricity, and others. The key idea introduced here –the frustration of phonon instabilities – provide one more route to pursuit a variety of noncollinear dipole orders, such as cycloid dipole textures with particular chiralities or even dipole-based skyrmions. As a consequence of this noncollinearity, exotic physics is expected to emerge.