Two-dimensional ferromagnetic-ferroelectric multiferroics in violation   of the d0 rule

Hengxin Tan; Menglei Li; Haitao Liu; Zhirong Liu; Yuanchang Li; and; Wenhui Duan

arXiv:1905.03545·cond-mat.mtrl-sci·June 5, 2019

Two-dimensional ferromagnetic-ferroelectric multiferroics in violation of the d0 rule

Hengxin Tan, Menglei Li, Haitao Liu, Zhirong Liu, Yuanchang Li, and, Wenhui Duan

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TL;DR

This paper introduces a new class of monolayer VOX2 multiferroics that combine ferroelectricity and magnetism from the same V cation, violating the d0 rule and enabling large polarization with strong magneto-electric coupling.

Contribution

It reports monolayer VOX2 as a novel multiferroic material where ferroelectricity and magnetism originate from the same atom, violating the traditional d0 rule.

Findings

01

VOX2 monolayers exhibit both ferroelectricity and magnetism from V cations.

02

The d-orbital orientation allows ferroelectricity without hindrance.

03

Large polarization and strong magneto-electric coupling are achieved.

Abstract

Contribution of d-electron to ferroelectricity of type-II multiferroics causes strong magneto-electric coupling and distinguishes them from the conventional type-I multiferroics. However, their therein polarization is too small because the ferroelectricity is merely a derivative from the magnetic order. Here we report a new class of multiferroic materials, monolayer VOX2 (X = Cl, Br, and I), which combine the advantages of type-I and type-II multiferroics. Both ferroelectricity and magnetism arise from the same V cation, where the filled d-orbital is perpendicular to an a priori ferroelectric polarization and thus poses no hindrance to ferroelectricity, indicating a violation of the usual d0 rule. This makes the combination of large polarizations and strong magneto-electric coupling possible. Our findings not only add new ferromagnetic-ferroelectric multiferroics, but also point to a…

Tables1

Table 1. Table 1: The lattice constants a 𝑎 a , b 𝑏 b (Å) and spontaneous polarization P 𝑃 P (10 -10 C/m) of the VO X 2 subscript 𝑋 2 X_{2} under the ground state magnetic configuration (GMC). Note that the polarizations are slightly sensitive to the long-range magnetic order, as can be seen in Table S1 of the Supplemental Material SM .

	GMC	$a$	$b$	$P$
VOCl₂	AFM3	3.815	3.380	3.0
VOBr₂	AFM1	3.799	3.585	2.7
VOI₂	FM	3.810	3.956	2.3

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Contribution of $d$ -electron to ferroelectricity of type-II multiferroics causes strong magneto-electric coupling and distinguishes them from the conventional type-I multiferroics. However, their therein polarization is too small because the ferroelectricity is merely a derivative from the magnetic order. Here we report a new class of multiferroic materials, monolayer VO $X_{2}$ ( $X$ = Cl, Br, and I), which combine the advantages of type-I and type-II multiferroics. Both ferroelectricity and magnetism arise from the same V cation, where the filled $d$ -orbital is perpendicular to an a priori ferroelectric polarization and thus poses no hindrance to ferroelectricity, indicating a violation of the usual $d^{0}$ rule. This makes the combination of large polarizations and strong magneto-electric coupling possible. Our findings not only add new ferromagnetic-ferroelectric multiferroics, but also point to a unique mechanism to engineer multiferroics.

Two-dimensional ferromagnetic-ferroelectric multiferroics in violation of the $d^{0}$ rule

Hengxin Tan

State Key Laboratory of Low-Dimensional Quantum Physics and Collaborative Innovation Center of Quantum Matter, Department of Physics, Tsinghua University, Beijing 100084, China

Menglei Li

[email protected]

Department of Physics, Capital Normal University, Beijing 100084, China

Haitao Liu

Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China

Zhirong Liu

[email protected]

Center for Nanochemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China

Yuanchang Li

[email protected]

Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China

Wenhui Duan

State Key Laboratory of Low-Dimensional Quantum Physics and Collaborative Innovation Center of Quantum Matter, Department of Physics, Tsinghua University, Beijing 100084, China

Institute for Advanced Study, Tsinghua University, Beijing 100084, China

Multiferroics possessing both magnetic and ferroelectric ordersSchmid (1994); Eerenstein et al. (2006) are of great importance because interactions between the magnetic and electric polarizations lead to multifarious physical effects and potential applications. Of particular interest is the ferromagnetic-ferroelectric case for new device paradigm based on four logic states Spaldin and Fiebig (2005); Gajek et al. (2007); Scott (2007). Unfortunately, ferromagnetic-ferroelectric multiferroics are rare in nature and their design was proved unexpectedly difficult Spaldin and Fiebig (2005). In recent years, the boom of multiferroic research has been reignited, one key factor of which is the discovery of fascinating properties in thin film systems Dawber et al. (2005); Ma et al. (2011); Martin and Rappe (2016), e.g., enlarged polarizations Wang et al. (2003); Choi et al. (2004); Ramesh and Spaldin (2007); Dong et al. (2015). The search for multiferroics in the two-dimensional (2D) limit is further stimulated after the observation of stable ferroelectricity with enhanced transition temperature in monolayer SnTe Chang et al. (2016), and robust ferromagnetism in ultra-thin transition-metal compounds of Cr2Ge2Te6 Gong et al. (2017) and CrI3 Huang et al. (2017). However, up to now, only few candidates of 2D multiferroics were proposed, including magnetic-ferroelectric Hf2VC2F2 Zhang et al. (2018), ferromagnetic-antiferroelectric monolayer transition-metal phosphorus chalcogenides Qi et al. (2018) and ferromagnetic-ferroelectric electron-doped CrBr3 Huang et al. (2018).

Generally, the magnetism originates from the partially occupied $d$ -orbitals of transition-metal cations, which, however, is believed to suppress the occurrence of ferroelectricity. This is well known as the $d^{0}$ rule in multiferroics Hill (2000). In this regard, the ferroelectric mechanism lies at the center of the multiferroic study. Several alternative ferroelectric mechanisms have been reported Fiebig et al. (2016) to circumvent this contraindication, such as the 6 $s$ lone-pair activity in BiFeO3 Wang et al. (2003) and PbVO3 Shpanchenko et al. (2004), the charge ordering in the mixed valency systems of LuFe2O4 Ikeda et al. (2005) and Fe3O4 Alexe et al. (2009), the geometric distortion in YMnO3 Van Aken et al. (2004) and BaMnF4 Zhou et al. (2015), and the spin driven symmetry breaking in TbMnO3 Kimura1 et al. (2003); Xiang et al. (2008). The first three mechanisms give rise to the so-called type-I multiferroics Khomskii (2009) where ferroelectricity and magnetism have independent origins, resulting in large polarizations and high transition temperatures but very weak magneto-electric coupling. The last one corresponds to the type-II multiferroics Khomskii (2009) where the ferroelectricity is a derivative of magnetism, resulting in small polarizations and low transition temperatures despite of the favorable strong magneto-electric coupling. Ideal multiferroics should combine the advantages of the type-I and type-II multiferroics. This hightlights the uniqueness of the single-phase one-cation multiferroics where the cation would be undoubtedly responsible for both ferroelectricity and magnetism. Occurrence of such multiferroics of type-I is particularly unusual because it means some yet unrevealed interplay beyond the $d^{0}$ rule between $d$ -electrons and ferroelectricity. Whereas, such occurrence is somewhat trivial in the type-II multiferroics because the ferroelectric polarization is only a consequence of the magnetic order, still compatible with the $d^{0}$ rule. As a matter of fact, only copper oxides/oxyhalides Kimura et al. (2008); Zhao et al. (2016); Seki et al. (2010); Zhao et al. (2012) are reported so far to be single-phase one-cation multiferroics, which, however, all belong to the type-II class.

In this work, based on the first-principles calculations, we identify three single-phase one-cation type-I multiferroic monolayers, including the ferromagnetic-ferroelectric VOI2 and antiferromagnetic-ferroelectric VOCl2 and VOBr2. Most prominently, ferroelectric properties are directly related to the V4+ cation although it possesses a $d_{xy}$ electron, which thus violates the conventional $d^{0}$ rule. The origin of the coexistence rather than mutual suppression of two ferroic orders is that the occupied $d_{xy}$ -orbital locates in a plane perpendicular to an a priori polarization and thus hardly suppresses the ferroelectricity. Such characteristics make it possible to achieve the advantages of type I and type II simultaneously, i.e., large polarizations and strong magneto-electric coupling. It also points to new multiferroic physics beyond the $d^{0}$ rule, as well as new strategies of designing novel multiferroic materials. Note that bulk vanadium oxide dihalides with layered structures have been studied in experiments previously Seifert and Uebach (1981); Hillebrecht et al. (1997). The nature of van der Waals interlayer interaction makes the synthesis of 2D monolayer highly probable.

All electronic and magnetic properties are calculated within the framework of density functional theory as implemented in the Vienna $Ab$ - $initio$ Simulation Package (vasp) Kresse and Furthmüller (1996a); Kresse and Furthmüller (1996b) with the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional Heyd et al. (2003) based on the geometric structures optimized with the Perdew-Burke-Ernzerhof (PBE) Perdew et al. (1996) functional. Noteworthy, the HSE approach has been successfully applied in describing different metal/insulator phases of strongly correlated vanadium oxides such as VO2 Eyert (2011) and V2O3 Guo et al. (2014). The electron-ion interaction is mimicked by the projected augmented wave method Blöchl (1994) with an energy cutoff of 600 eV. The $k$ -point grids of 10 $\times$ 10 $\times$ 1 and 6 $\times$ 6 $\times$ 1 are employed to sample the Brillouin zone for the unit cell and 2 $\times$ 2 $\times$ 1 supercell, respectively. A criterion of 0.005 eV/Å is used for the Hellman-Feynman forces during the structural relaxation. A vacuum layer of at least 15 Å is added to avoid the spurious interaction between monolayer and its periodic images. The polarization is calculated by the berry phase method King-Smith and Vanderbilt (1993).

We start with the orthorhombic monolayer VO $X_{2}$ ( $X$ = Cl, Br, and I) having inversion symmetry ( $Pmmm$ space group), where two V-O (four V- $X$ ) bonds are of equal length with an O-V- $X$ bond angle of 90∘, as shown in Fig. 1(a) for VOI2 and the Supplemental Material SM for VOCl2 and VOBr2. However, this paraelectric phase is unstable at zero temperature as manifested by the existence of imaginary frequency in the phonon spectrum [See Fig. 1(b)], indicating a tendency of spontaneous structural distortion. The imaginary band along the $\Gamma$ -Y is a polar mode corresponding to the relative movement of atoms along the V-O chain. This consequently breaks the inversion symmetry and leads to a ferroelectric polarization.

As it turns out, the distorted structure with lower $Pmm2$ symmetry removes the imaginary phonon mode [See the right panels of Fig. 1]. Now two V-O bonds becomes unequal, e.g., 2.14 versus 1.67 Å for the VOI2. It is worth noting that the off-center displacement of V in bulk VOCl2 has been observed by scanning tunnel microscope and atomic force microscope Hillebrecht et al. (1997). Although four V- $X$ bonds remain equivalent, the O-V- $X$ bond angle deviates from 90∘, resulting in the separation of positive and negative charge centers. This leads to polarizations of 2.8, 2.6 and 2.3 $\times$ 10*-10* C/m for VOCl2, VOBr2 and VOI2 respectively [See Fig. S2 (a) in the Supplemental Material SM for the polarization along the switching path of VOCl2 as an example], which are comparable to that of the group IV monochalcogenides Fei et al. (2016). Note that the polarization weakly depends on the magnetic structures which will be studied later, and the ones corresponding to the ground state are 3.0, 2.7 and 2.3 $\times$ 10*-10* C/m for VOCl2, VOBr2 and VOI2, as summarized in Table I. Given their monolayer thickness of 3.3, 3.6 and 3.8 Å, the equivalent bulk polarizations are 91, 75 and 61 $\mu$ C/cm2 which are at the same level as those of the traditional perovskite ferroelectrics of PbTiO3 and BiFeO3 Sun et al. (2016), and an order of magnitude larger than those of improper ferroelectrics Cheong and Mostovoy (2007) of the hexagonal rare-earth ferrites and manganites Xu et al. (2014); Tan et al. (2016). Symmetry-lowering leads to an energy decreasing of 207, 177 and 105 meV per cation for Cl, Br and I systems. The total energy along the polarization switching path has the shape of a double well potential which is the character of ferroelectrics, as shown in Fig. S2(b) for VOCl2 as an example. Such depths of the double well potential are similar to those of typical ferroelectrics such as PbTiO3, BiFeO3 and YMnO3 Bilc et al. (2008); Zhang et al. (2017), strongly implying the high stability of the ferroelectric phase. We also calculated the intrinsic hysteresis loop of VOCl2 [Fig. S2 (c)] by the approach proposed in Ref. Sai et al., 2002, where the determined ferroelectric coercive field is comparable to that of BaTiO3.

All three systems are spin-polarized with a total magnetic moment of 1 $\mu_{B}$ , mainly coming from the V cation. We considered four magnetic configurations as illustrated in Fig. 2(a) to explore the long-range magnetic order. The results are summarized in Table I (More details are found in Table S1 of the Supplemental Material SM ). A ferromagnetic ground state is achieved for the VOI2, while different antiferromagnetic ones, namely, checkerboard (AFM3) and stripe (AFM1), are found for the VOCl2 and VOBr2, respectively. Significantly, VOI2 is the only 2D material that possesses inherent ferromagnetism and ferroelectricity, so far as we know.

Actually, these three systems are so alike except for the radii of different halogen atoms. This is reflected by the large difference between lattice constant $b$ as well as similar $a$ (See Table I). To explain their different magnetic orders, we schematically illustrate the exchange paths along V- $X$ chains in Fig. 2(b). There are two kinds of exchange interactions. One is a direct interaction between the local moments on adjacent V which favors an anti-parallel spin alignment. The other is the halogen-mediated super-exchange which favors a parallel spin alignment in terms of the Goodenough-Kanamori rule Goodenough (1955, 1958); Kanamori (1959). Their competition determines the magnetic order. The larger the halogen radius, the larger the V-V separation along the $b$ direction, leading to the weaker direct exchange. In contrast, the electronic structures shown in Fig. 3 reveal an increased $p$ - $d$ hybridization between V and halogen with larger radius, which implies a strengthened super-exchange from the VOCl2 to VOI2. For VOCl2 and VOBr2, the direct exchange dominates owing to the relatively small halogen radius, leading to a favourable antiferromagnetic order along the V- $X$ chains. When the direct exchange decreases to be less significant than the super-exchange, a ferromagnetic coupling becomes energetically stable, corresponding to the case of VOI2.

We now turn to the central observation of this paper to examine how the ferroelectricity survives in such a non- $d^{0}$ system. To reveal the role played by the very $d$ -electron in VO $X_{2}$ where V has a $d^{1}$ configuration due to its +4 valency, we first consider an analogue satisfying the $d^{0}$ requirement, TiO $X_{2}$ . Similar spontaneous symmetry-lowering is observed in TiO $X_{2}$ to result in ferroelectricity. An example is shown in Fig. 4(a) for TiOI2. Detailed geometries, phonon spectra and electronic structures of TiO $X_{2}$ are provided in the Supplemental Material SM . Due to the $d^{0}$ nature of the Ti systems, the origin of their ferroelectricity is the same as that of the traditional perovskite BaTiO3 Megaw (1952); Zhong et al. (1995), i.e., an effect driven by the phonon mode softening associated with the covalent bonding between Ti and anions. Off-center displacements of Ti endow TiOCl2, TiOBr2 and TiOI2 with polarizations of 2.3, 2.1 and 1.9 $\times$ 10*-10* C/m, which are 0.7, 0.6 and 0.4 $\times$ 10*-10* C/m smaller than their V counterparts (values for ground states). The difference between V and Ti lies at an extra $d$ -electron in V. In this regard, the occupation of $d$ -orbital in VO $X_{2}$ enhances the ferroelectric polarization instead of inhibiting it, surprisingly opposed to the $d^{0}$ rule.

It is the configuration of the occupied $d$ -orbital that plays a central role for the survival of the polarization in VO $X_{2}$ . Under the local octahedral field established by oxygen and halogen, the cation $d$ -orbitals split into four subgroups as revealed by our calculations: the $d_{xy}$ singlet, the $d_{xz}$ and $d_{yz}$ doublet, the $d_{x^{2}-y^{2}}$ singlet and the $d_{z^{2}}$ singlet. This is schematically shown in the lower panels of Fig. 4. In TiO $X_{2}$ , no $d$ -orbital is occupied, so $d$ -orbital does not cause any hindrance to the off-center displacements of Ti, in consistence with the $d^{0}$ rule. In VO $X_{2}$ , on the other hand, the lowest $d_{xy}$ singlet is occupied. However, $d_{xy}$ is distributed in the plane perpendicular to the V-O chain, as shown by the corresponding charge density plot in Fig. 4(b). Within the Slater-Koster approximation, the coupling between $d_{xy}$ and the O $p$ orbitals is zero Slater and Koster (1954); Li et al. (2015). Therefore, $d_{xy}$ does not cause any hindrance when V moves along the V-O chain. In contrast, $d_{xz}$ and $d_{yz}$ have nonzero coupling with the O $p$ orbitals Slater and Koster (1954); Li et al. (2015), so one may expect them to cause some hindrance to the polarization if occupied. As a result, the absolute energy (relative to the vacuum level) of $d_{xy}$ is less sensitive to the movement of V along the V-O chain than that of $d_{xz}$ / $d_{yz}$ . Indeed, the projected density of states show an almost unchanged $d_{xy}$ in the ferroelectric and paraelectric phases for VOI2, and the energy shift of $d_{xy}$ is smaller than that of $d_{xz}$ / $d_{yz}$ for VOCl2 and VOBr2 (see Fig. S4 of the Supplemental Material SM ). If one more $d$ -electron is added, occupation of $d_{xz}/d_{yz}$ will occur, thus causing the suppression of polarization. This is confirmed in the analogous CrOI2 monolayer where the off-center displacement of cation is almost completely removed, as shown by the O-Cr-I bond angle close to 90∘ in Fig. 4(c). Interestingly, CrOI2 is a half metal (see Fig. S8 of the Supplemental Material SM for the band structures).

In fact, the $d_{xy}$ electron also directly contributes to the polarization of VO $X_{2}$ . This is readily understood provided that we equivalently consider VO $X_{2}$ as TiO $X_{2}$ plus an extra $d_{xy}$ electron. Although $d_{xy}$ does not couple with the O $p$ orbitals, it couples with the halogen $p$ orbitals. Owing to the existence of an a priori ferroelectric distortion like TiO $X_{2}$ , the hybridization between $d_{xy}$ and $X$ $p$ would cause further separation of the positive V nucleus and the negative $d_{xy}$ orbital, therefore contributing to an extra polarization. This can be quantitatively evaluated by estimating the average effective charge $\bar{q}$ of cation based on the relation, $Pab=\bar{q}\Delta x$ , where $P$ , $a$ and $b$ are the polarization and lattice constants of a unit cell and $\Delta x$ is the shift distance of cation from the inversion center. $\bar{q}$ of V in VOI2 is determined to be 9.2 $e$ , while $\bar{q}$ of Ti in TiOI2 is 8.6 $e$ . Therefore, the filled $d_{xy}$ orbital contributes about 0.6 $e$ to $\bar{q}$ . We also calculate their average Born effective charges and the results are 9.5 $e$ and 8.7 $e$ , respectively, for V and Ti (see Fig. S7 of the Supplemental Material SM for details). Difference of 0.8 $e$ agrees well with the estimation, confirming the $d_{xy}$ contribution. It is worth noting that the $d$ -electron alone should not cause ferroelectric polarization, but it can enhance the ferroelectricity when parasitizing on an a priori distortion. This opens a new route to engineer multiferroicity via the controlled magnetic doping in an intrinsic ferroelectric material.

Two characteristics distinguish VO $X_{2}$ from other multiferroics. On the one hand, the existence of conventional polarization endows it with the feature of the type-I multiferroic materials. On the other hand, both magnetic and ferroelectric orders arise from the same V cation, and there is even a parasitic polarization from $d$ -electron, which presents a feature intrinsic to the type-II multiferroics. Such a duality makes it highly possible to combine large polarization with strong magneto-electric coupling.

Finally, we tentatively evaluate the magneto-electric coupling by monitoring the magnetic response to the variation of ferroelectric polarization. We move the V atom away from the ground state along the V-O chain to change the polarization monotonously. Figure 5 shows the corresponding energy dependence of the four magnetic configurations on the polarization with ferromagnetic phase as the energy reference. Although the ground state is always ferromagnetic for VOI2, we interestingly observe a crossover of the long-range magnetic order in VOCl2 and VOBr2 during the change of polarizations. That is, the long-range magnetic order changes from AFM3 to AFM1 for VOCl2 while from AFM1 to FM for VOBr2 with the decrease of the polarization. Besides, the energy differences between the different magnetic states vary remarkably with the polarization, reflecting the influence of polarization on the magnetic property. Such behaviors may be clues of sizable magneto-electric coupling.

In summary, we have predicted three monolayer multiferroic materials in violation of the $d^{0}$ rule. VOI2 shows the fascinating ferromagnetism and ferroelectricity while VOCl2 and VOBr2 show the antiferromagnetism and ferroelectricity. We unravel a novel mechanism for multiferroics beyond the $d^{0}$ rule, i.e., the filled $d$ -orbital of V is perpendicular to the a priori polarization and contributes a parasitic polarization, so as to realize the coexistence of $d$ -electron ferroelectricity and magnetism. Our works thus offer a new way on searching for and engineering practical multiferroic materials.

Acknowledgements.

This work was supported by the Basic Science Center Project of National Natural Science Foundation of China (Grant No. 51788104), the Ministry of Science and Technology of China (Grant No. 2016YFA0301001), the National Natural Science Foundation of China (Grant No. 11674071, No. 21773002, No. 11704038, and No. 11874089), the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (Grant No. KF201712), the Beijing Advanced Innovation Center for Future Chip (ICFC), and the Beijing Institute of Technology Research Fund Program for Young Scholars.

$Note$ $added$ : After we submitted the manuscript, we became aware of the work Ai et al. (2019) which predicted the coexistence of ferroelectricity and antiferromagnetism in monolayer VOCl2 and VOBr2.

Bibliography59

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Two-dimensional ferromagnetic-ferroelectric multiferroics in violation of the d0d^{0}d0 rule

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Two-dimensional ferromagnetic-ferroelectric multiferroics in violation of the $d^{0}$ rule