Type-II multiferroic Hf$_{2}$VC$_{2}$F$_{2}$ MXene monolayer with high transition temperature

TL;DR

This paper predicts a new 2D multiferroic material, Hf$_{2}$VC$_{2}$F$_{2}$ MXene, where ferroelectricity is directly driven by its magnetic order, with a transition temperature above room temperature, promising for nanoscale device applications.

Contribution

The study identifies a novel type-II multiferroic 2D MXene monolayer where ferroelectricity arises from magnetism, unlike previous loosely coupled multiferroics.

Findings

Predicted a stable 2D multiferroic MXene with high transition temperature

Demonstrated ferroelectricity directly linked to magnetic order

Opened new avenues for intrinsically strong magnetoelectric 2D materials

Abstract

Achieving multiferroic two-dimensional (2D) materials should enable numerous functionalities in nanoscale devices. Until now, however, predicted 2D multiferroics are very few and with coexisting yet only loosely coupled (type-I) ferroelectricity and magnetism. Here, a type-II multiferroic MXene Hf$_{2}$VC$_{2}$F$_{2}$ monolayer is identified, where ferroelectricity originates directly from its magnetism. The noncollinear Y-type spin order generates a polarization perpendicular to the spin helical plane. Remarkably, the multiferroic transition is estimated to occur above room temperature. Our investigation should open the door to a new branch of 2D materials for pursuit of intrinsically strong magnetoelectricity.

Tables

Table 1. Table 1: The calculated energies ( E 𝐸 E ) of different magnetic structures using the HSE06 functional with SOC. The energy of FM is set as the reference. The corresponding local magnetic moments ( m 𝑚 m ) are also listed.

	NM	FM	G-AFM	UUD	Y-AFM
$E$ (eV/V)	$0.09$	$0$	$-0.12$	$-0.14$	$-0.19$
$m$ (${\mu }_B$/V)		$1.18$	$2.05$	$1.99$	$2.04$

Equations

$$\text{P}=\text{M}\cdot \sum _{<ij>}({\text{S}}_i\times {\text{S}}_j),$$

$$\text{M}=-\left[\begin{array}{ccc}9.571& 0& 0\\ 0& 9.571& 0\\ 0& 0& 1.401\end{array}\right]\times 10^{-3}\mathrm{e}\mathring{\mathrm{A}}.$$

$$H=-J_1\sum _{<ij>}{\text{S}}_i\cdot {\text{S}}_j-J_2\sum _{[kl]}{\text{S}}_k\cdot {\text{S}}_l-A\sum _i(S_i^z{)}^2,$$

Full text

\alsoaffiliation

Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, United States

Two-dimensional (2D) materials have attracted attention since the experimental discovery of graphene 1. Divers properties and functions are discovered in plentiful 2D materials, going beyond the original appeal as new semiconductors. More and more important physical properties existing in three-dimensional (3D) crystals have also been found to appear in the 2D form. For example, 2D superconductivity 2, 3, 4, 5, 6, 2D ferromagnetism 7, 8, 9, as well as 2D ferroelectricity 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, have recently been confirmed in experiments or predicted via calculations.

Ferromagnetism and ferroelectricity, with switchable ferro-vectors, play crucial roles in various devices. Thus, their existence in 2D materials would be very attractive. On one hand, since the first prediction of 2D ferroelectric (FE) hydroxylized graphene in 2013 10, more 2D materials, e.g. 1T-MoS2, In2Se3, 2D materials functionalized with polar groups, etc., have been predicted to be FE 11, 12, 13, 14, 15, 16, 17, 18, 19. Experimentally, in-plane FE polarization ($P$) was observed and manipulated in atom-thick SnTe 20, while out-of-plane FE $P$ was found in few layes CuInP2S6 21. The involved mechanisms are either polar phonon modes or polar functional groups. On the other hand, many 2D ferromagnets have been predicted 8, 9, and recently CrI3 monolayer is experimentally confirmed 7.

The coexisting magnetism and polarization lead to the multiferroicity 22, 23. The coupling between these two ferro-vectors allows the control of charge via magnetic field or the control of spin via electric field. In fact, a few 2D multiferroics were also recently predicted 24, 25, 16, 19, in which the origins of polarization and magnetism are independent of each other (i.e. they are all type-I multiferroics 26). Thus their magnetoelectric coupling is indirect and weak.

To pursue the intrinsically strong magnetoelectricity, a possible route is to design 2D type-II multiferroics (i.e. magnetic ferroelectrics 26), in which the FE $P$ is directly generated and thus fully controlled by magnetic order 22, 23. Although the type-II multiferroics have been extensively studied, these materials have not gone into the zone of 2D materials. Even some type-II multiferroics own layered structures, e.g. CuFeO2 27, Sr3NiTa2O9 28, and Ba3MnNb2O9 29, the inter-layer couplings are via ionic bonds, difficult to be exfoliated.

In this work, monolayer carbides and carbonitrides, i.e. MXenes ($M_{n+1}X_{n}T_{x}$, $M$: early transition metal; $X$: carbon or nitrogen), are predicted as 2D type-II multiferroics. As a new branch of 2D materials, MXenes have been experimentally produced by selectively etching the A-layers from their 3D parent compounds MAX 30, 31, 32. The surface can be easily covered by functional groups (e.g. $T$=F, O, or/and OH), resulting in diversiform chemical and physical properties 33, 5. Recently, an ordered double transition metal MAX with Cr-Al/Mo-Al bonding were synthesized, e.g. Cr2TiAlC2 34 and Mo2TiAlC2 35, in which a Ti-layer is sandwiched between two outer Cr/Mo carbide layers in the $M_{3}AX_{2}$ structure. Then ordered double transition metal carbides $M^{\prime}_{2}MX_{2}$ and $M^{\prime}_{2}M_{2}X_{3}$, e.g. Mo2TiC${}_{2}T_{x}$, Mo2Ti2C${}_{2}T_{x}$ and Cr2TiC${}_{2}T_{x}$, were successfully realized by etching the Al-layers 36. In addition, considering the transition metals involved, many MXenes should be intrinsically magnetic (at least from the theoretical viewpoint), and their magnetism depends on $M$ ($M^{\prime}$) and $T$ 8, 9. For instance, Cr2TiC2F2 and Cr2TiC2(OH)2 are predicted to be antiferromagnetic (AFM), whereas Cr2VC2(OH)2, Cr2VC2F2, and Cr2VC2O2 are ferromagnetic (FM) 9. Herein, derived from experimental Hf3C${}_{2}T_{x}$ monolayer 37, the ordered double transition metal carbides Hf${}_{2}M$C${}_{2}T_{2}$ monolayers (possibly realized via 3D parent Hf${}_{2}M$AlC2) 38 are considered to be a 2D type-II multiferroics. Although both $M$ and $M^{\prime}$ can be magnetic ions, here only the middle $M$ layer is considered to be magnetic.

Candidate 2D MXene. According to the knowledge of type-II multiferroicity, some special frustrated magnetic orders, like noncolliear spiral magnetism or $\uparrow\uparrow\downarrow\downarrow$-type AFM order, may break the space inversion symmetry and thus lead to FE $P$ 22, 23. The in-plane geometry of $M$ ions is triangular, which is inherently frustrated if the nearest-neighbor (NN) exchange is AFM 39, 40, 27, 28. Thus, to find MXenes with NN AFM interaction is the first step. According to the Goodenough-Kanamori rule 41, 42, the ions with half-filled $d$ shell usually lead to strong AFM exchanges. Besides, the half-filled Hubbard bands can lead to insulating, as required for ferroelectricity.

In $M^{\prime}_{2}M$C${}_{2}T_{2}$ MXene monolayer, each $M$C6 forms an octahedron [1(a)]. The crystalline field of octahedron splits $M$’s $d$ orbitals into the low-lying $t_{\rm 2g}$ triplets and the higher-energy $e_{\rm g}$ doublets [1(b)]. To pursuit the half-filled Hubbard bands, high spin V2+ ($3d^{3}$), Nb2+ ($4d^{3}$), Mn4+ ($3d^{3}$), Mn2+ ($3d^{5}$), and Fe3+($3d^{5}$) are possible candidates to play as $M$.

To satisfy aforemention conditions, in the following, Hf2VC2F2, Hf2NbC2F2, Hf2MnC2F2, and Hf2MnC2O2 will be calculated using density functional theory (DFT). Since Fe3+ ($3d^{5}$) can not be obtained if symmetric $T$ layers are used, it is not considered here. In addition, $T$=OH will be not calculated, considering its equal valence (and thus similar physical effects) to F.

DFT results. DFT calculations are performed to verify the multiferroicity of these MXene monolayers. Our calculations find that Hf2VC2F2 is the most possible 2D type-II multiferroic material, while others are unlike due to various reasons (see Supplemental Materials for more details of DFT methods and results 38). The possibility of synthesis for Hf2VC2F2 is also explored, which maybe realized via 3D parent Hf2VAlC2. Based on experimentally produced V3AlC2 32 and Hf3AlC2 43, the hybrid energy for Hf2VAlC2 is about $-90$ meV/cell, implying such mix is more favorable. In addition, the possibility of MAX phase to MXenes for Hf2VAlC2 are also verified by the crystal orbital Hamilton population (COHP) and exfoliation energies calculations 38. Based on the results of COHP, the obtained bond strength between Hf and Al is very weaker than other bond between Hf/V and C in Hf2VAlC2, which is similar to that in Hf3AlC2 and V3AlC2, indicating the iconicity/metallicity characteristic between them. To further examine the progress of exfoliation, exfoliation energies are calculated as E${}_{exfoliation}=-[E_{tot}(MAXphase)-2E_{tot}(MXene)-E_{tot}(Al)]/(4S)$ 44, where Etot(MAX phase), Etot(MXene) and Etot(Al) stand for the total energies of bulk MAX phase, 2D MXene, and most stable bulk Al structure (Fm-3m), respectively. $S=\sqrt{3}a^{2}/2$ is the surface area and a is the lattice parameter of the MAX phase. Due to V3AlC2 was experimentally exfoliated into 2D Mxenes 32, Hf2VAlC2 which ows lower exfoliation energies have a better possbility to be exfoliated into MXenes. In summary, we conclude Hf2VAlC2 is a good candidate of 3D parent phase for the successful exfoliation into 2D Hf2VC2 MXenes. More details can be found in Supplemental Materials 38.

First, various configurations (CG’s) for Hf2VC2F2 are verified. Based on energy comparison and dynamic stability, the AA CG is confirmed to be the most favorable one 38, where F ions stand just above/below the V’s positions [1(a)]. Thus, our following investigation will focus on the AA CG only.

The nominal valences for Hf, V, C, and F are $+4$, $+2$, $-4$, and $-1$, respectively. Then for both C and F, the $2p$-orbitals are fully occupied, while for Hf the $5d$ orbitals are fully empty. In this sense, the magnetism can only come from V, whose $3d$ orbitals own three electrons, as confirmed in the DFT calculation.

The magnetic ground state of Hf2VC2F2 is searched by comparing the energies of various possible magnetic orders, including the nonmagnetic (NM), collinear FM, UUD type ferrimagnetic (stands for the ”up-up-down” ferrimagnetic spin order), stripe AFM (G-AFM), as well as the $120^{\circ}$ noncollinear AFM order (coined as Y-AFM here), as sketched in 1(c-e).

Considering the Hubbard-type correlations and spin-orbit coupling (SOC) for $3d$ and $5d$ orbitals, here a wide parameter space of $U_{\rm eff}$(V) and $U_{\rm eff}$(Hf) are scanned, as shown in  2(a) and Supplemental Materials 38. As expected, the $U_{\rm eff}$(Hf) and SOC only have tiny effects on the magnetism due to Hf’s empty $5d$ orbitals 38. In contrast, with increasing $U_{\rm eff}$(V), the magnetic ground-state undergoes two transitions, from NM to G-AFM first, then finally to Y-AFM. The local magnetic moment of V also depends on $U_{\rm eff}$(V), increasing from [math] to more than $2$ $\mu_{\rm B}$/V ( 2(b)). Accompanying the second magnetic transition, the metal-insulator transition also occurs when $U_{\rm eff}$(V)$>2$ eV.

Due to the lack of experimental result on Hf2VC2F2 monolayer, the HSE06 functional with SOC, are adopted as the benchmark to provide an alternative description 45. As shown in  1, the HSE06 plus SOC calculation predicts that the Y-AFM is the ground state for Hf2VC2F2 monolayer, and the corresponding magnetic moment is in good agreement with the result of $U_{\rm eff}$(V)$=3$ eV and $U_{\rm eff}$(Hf)$=2$ eV, implying this set of parameters is proper. In fact, the same $U_{\rm eff}$ parameters were also adopted in previous studies 8, where only FM and G-AFM were considered. Here, four mostly-possible ones have been considered in DFT calculations. Furthermore, the following Monte Carlo simulation, with no bias of preset magnetic configurations, will be employed to verify the results of DFT. If there’s more stable one, the Monte Carlo simulation should capture it.

The calculated electronic structure of Y-AFM with SOC are shown in  2(c-d). It is clear that Hf2VC2F2 monolayer is a direct-gap semiconductor and the corresponding band gap is about $0.4$ eV with default $U_{\rm eff}$’s. The HSE06 functional calculation leads to very similar electronic structure with a larger band gap ($0.9$ eV) 38. The projection of Bloch states to V’s $d$-orbital is also displayed in  2(c-d). As expected, there are nine occupied bands near the Fermi level mostly contributed by V’s $3d$ orbitals. As expected, the $d^{3}$ configuration of V2+ just occupies the $t_{\rm 2g}$ orbitals in the half-filling manner, while the $e_{\rm g}$ orbitals are above the Fermi level. According to PDOS, there is also moderate $p$-$d$ hybridization, which is a bridge for superexchange interaction.

Since the Y-AFM is a type of helical spin order which breaks inversion symmetry, i.e. clockwise vs counterclockwise [ 3(a,b)], previous studies of triangular-lattice antiferromagnets with a helical spin order have found the magnetism induced FE $P$ 27, 46, 47, 48. Thus it is reasonable to expect the similar multiferroicity in the Hf2VC2F2 monolayer.

For the Y-AFM spin order, the noncollinear spin texture forms a helical plane. It is necessary to know the easy plane/axis first. Our calculation with SOC finds that the out-of-plane $c$ direction is the easy axis. Thus in the ground state, the helical plane should be perpendicular to the monolayer. The energy of $ac$ (or $bc$) plane Y-AFM is lower than that of $ab$ plane Y-AFM by $0.14$ meV/V. Our calculation also finds the rotation symmetry within the monolayer plane. Thus the helical plane can be rotated freely along the $c$-axis, as sketched in  3(c).

The standard Berry phase calculation with SOC gives $1.98\times 10^{-6}$ $\mu$C/m for the Y-AFM state, corresponding to $2700$ $\mu$C/m2 in the 3D unit considering the thickness of monolayer $7.0$ Å. To partition these two contributions, using the high-symmetric crystalline structure, the obtained pure electronic contribution ($P_{e}$) is about $1.95\times 10^{-6}$ $\mu$C/m, very close to the total $P$ with ionic displacements. Therefore, here FE $P$ is almost fully ($\sim 98.5\%$) originated from the bias of electronic cloud while the atomic structure is almost in the high symmetric one. Our calculation also indicates that the direction of $P$ is always perpendicular to the spin helical plane, as sketched in  3(c). And this $P$ can be switched to $-P$, once the chirality of Y-AFM is reversed. For comparison, the higher energy $ab$-plane Y-AFM gives $2.9\times 10^{-7}$ $\mu$C/m, pointing along the $c$-axis.

Although this $P$ is much smaller than those of other 2D FE’s, e.g. for some functionalized 2D materials ($3\times 10^{-5}-1.1\times 10^{-4}$ $\mu$C/m 15), 2D honeycomb binary buckled compounds ($9\times 10^{-7}-1.11\times 10^{-5}$ $\mu$C/m 13), and multiferroic C6N8H organic network ($\sim 4500$ $\mu$C/m2 24), it should be noted that the origin of FE $P$ in Hf2VC2F2 monolayer is conceptually different from other 2D FE’s. In fact, it is common sense that the improper FE $P$’s in the type-II multiferroics are weaker than those typical values of proper FE’s 22, 23. Even though, the origin of ferroelectricity in the type-II multiferroics guarantees the intrinsically strong magnetoelectric coupling, which is rare in other multiferroics.

In fact, $2700$ $\mu$C/m2 is already a very significant value in type-II multiferroics, especially considering the fact that its origin is from the SOC, not exchange striction 22, 23. For reference, the $P$ in polycrystal Ba3MnNb2O9 only reaches $3.45$ $\mu$C/m2 29, and $\sim 600$ $\mu$C/m2 in TbMnO3 49. The relatively large $P$ is probably due to $5d$ Hf ions, which own larger SOC than $3d$ elements. Although Hf’s orbitals do not contribute to magnetism directly, the hybridization between orbitals always exists around the Fermi level, which may enhance the effective SOC. Thus in principle, the macroscopic polarization should be detectable, at least in its corresponding bulk form. In addition, the second-harmonic-generation (SHG) based on nonlinear optical process can also be employed to detect the polarization and its domain without electrodes, as done for TbMnO3 matsubara2015magnetoelectric. Among type-II multiferroics, some polarizations are generated by noncollinear spin order via spin-orbit coupling (SOC), as in our Hf2VC2F2. Since SOC is usually weak especially for 3$d$ electrons, the polarizations in this category are usually much smaller than those in conventional ferroelectrics 23. Furthermore, here the high ratio of $P_{e}$/$P$ is also advantage for ultra-fast switching. Although it’s common sense that type-II multiferroics own high $P_{e}$/$P$ (e.g. $\sim 25\%$ in TbMnO3 50, and $\sim 58\%$ in HoMnO3 51, which were estimated using the same method used here) than proper FE materials, the $P_{e}$/$P$=$98.5\%$ is indeed very high and rather rare.

The origin of ferroelectricity driven by helical spin order is also nontrivial. Although the spin-current model (or the Dzyaloshinskii-Moriya interaction) can explain the origin of ferroelectricity in cycloid spiral magnets 52, 53, its equation $\textbf{e}_{ij}\times(\textbf{S}_{i}\times\textbf{S}_{j})$ gives zero net $P$ for Y-AFM in each triangular unit. Instead, the generalized spin-current model proposed by Xiang et al. 54 can phenomenologically explain the origin of P:

[TABLE]

where the summation is over all NN bonds; S denotes a (normalized) spin vector; M is a $3\times 3$ matrix which can be determined via DFT:

[TABLE]

*MC simulation.*In above DFT calculations, only five magnetic candidates were considered, which could not exclude other possible exotic orders. Thus the unbiased Monte Carlo (MC) simulation is performed to verify the ground state and estimate the transition temperature 38. The Heisenberg spin model is adopted:

[TABLE]

where $J_{1}$ ($J_{2}$) is the exchange interaction between NN (NNN) spin pairs; $A$ is the coefficient for magnetocrystalline anisotropy and $S^{z}$ is the component of spin along the magnetic easy axis. Using the normalized $|\textbf{S}|=1$, these coefficients can be extracted from DFT calculations by comparing the energies of magnetic candidates 38: $J_{1}=-48.1$ meV, $J_{2}=6.7$ meV, and $A=0.14$ meV, respectively. As expected, the NN exchange is strongly AFM, while the NNN is much weaker. The dominant $J_{1}$ leads to the Y-AFM, as confirmed using MC simulation [ 4(a)]. Interestingly, the estimated Néel temperature $T_{\rm N}$ reaches $313$ K, a remarkable high $T_{\rm N}$ above room temperature. The FE $|\textbf{P}|$ just appears below $T_{\rm N}$ [ 4(b)], a character of type-II multiferroicity. A typical MC snapshot of local $\textbf{P}_{i}$’s (of V triangular units) at low temperature is shown in 4(c).

For most 3D type-II multiferroics, the magnetism and ferroelectricity only appear far below room temperature 23. High-temperature type-II multiferroicity is a highly desired property for applications, which is a bottleneck for this category of materials. Till now, in various type-II multiferroics, only a few hexagonal ferrites with very complex crystalline/magnetic structures show magnetoelectricity above room temperature 55. Hf2VC2F2 is another room-temperature type-II multiferroic system, with a much simpler crystalline/magnetic structure.

Physically, its high $T_{\rm N}$ is due to the ideal half-filled $t_{\rm 2g}$ orbitals ($3d^{3}$), which prefers a strong superexchange according to the Goodenough-Kanamori rule 41, 42. The similar case is for various ferrites with Fe3+ ($3d^{5}$) ions, which usually own magnetic orders above room temperature.

As a type-II multiferroic, the induced P can be fully controlled by magnetic fields via the helical plane rotation 38. As shown in  4(d), under an in-plane magnetic field, the energies of $yz$-plane and $xz$-plane Y-AFM (after slight distortions driven by magnetic field) are no longer degenerated. Thus, the helical plane of Y-AFM and its associated P should rotate accompanying the field. Since there’s no intrinsic energy barrier for such a helical plane rotation, this magnetoelectric response should work under small fields.

Last, the MoSe2 substrate is considered to test the possible substrate effect 38. With proximate in-plane lattice constants, the optimized distance between Hf2VC2F2 monolayer and MoSe2 substrate is $\sim 3.5$ Å, indicating the vdW interaction. No charge transfer occurs between Hf2VC2F2 and substrate. The Y-AFM remains the ground state, and the whole system remains insulating. Therefore, the substrate will not change the conclusion of multiferroicity for Hf2VC2F2 monolayer.

Conclusion. The noncollinear $120^{\circ}$ Y-type antiferromagnetic order is predicted to be the ground state in MXene Hf2VC2F2 monolayer, and the estimated Néel point can be above room temperature. More importantly, the inversion symmetry is broken by this particular Y-type antiferromagnetic order, resulting in the improper magnetism-driven ferroelectric polarization. Thus Hf2VC2F2 monolayer is a room-temperature type-II multiferroics, which has intrinsically strong magnetoelectric coupling. The crossover between 2D materials and magnetic ferroelectrics will be a very interesting topic, both fundamentally and to benefit nanoscale devices.

{acknowledgement}

Work was supported by National Natural Science Foundation of China (Grant No. 11674055), Fundamental Research Funds for the Central Universities, Jiangsu Innovation Projects for Graduate Student (Grant No. KYLX16 0116), and National Supercomputer Center in Guangzhou (NSCC-GZ).