Intrinsic bitunable magnetism/polarity behavior in 2D Janus Cr₂I₃Y₃ (Y = F, Cl, or Br) systems

Abstract

Two-dimensional (2D) magnetoelectric (ME) materials with spontaneous magnetism and polarity have drawn a lot of research interest due to their potentially valuable physics and spintronic applications. One of the obstacles in the current study of 2D ME materials is to achieve flexible magnetism/polarity switches because of a fundamental contradiction between magnetism and polarity. In this work, we propose an intrinsic magnetism/polarity switchable 2D Janus structure Cr₂I₃Y₃ (Y = F, Cl, or Br) with mechanical and thermodynamic stability. Janus Cr₂I₃Y₃ monolayers show indirect-gap semiconductors with ferromagnetic ordering. Intrinsic vertical electric polarity (EP) in Janus Cr₂I₃Y₃ structures is investigated by surface potential difference analysis and dipole calculations. We discover that there is an obvious ME coupling between magnetic anisotropy energy (MAE) and EP. MAE can be obviously tuned by rotating the orientations of spins within yz and zx vertical planes, and the easily distinguishable MAE on vertical planes has strong inherent relevance with vertical EP. Simultaneously, magnetic phase transition and magnetic moments can be manipulated by introducing vertical strain, which directly corresponds to the linear response of the EP strength. Our work provides a promising candidate for realizing the spintronic devices with multiple functionalities.

Introduction

Two-dimensional (2D) material is a significant component of nanomaterials since they could hold better mechanical, optical, transport, and electronic properties than bulk materials^1,2,3,4,5. 2D material has gained increasing research attention involving scientific fields like photoelectric, thermoelectric, detection, biological and medical device, etc.^6,7,8, and is an important candidate for the development of spintronic application due to the pursuit of miniaturization, higher speed and storage density, and lower power consumption beyond three-dimensional systems. The success of the mechanical stripping exfoliation of single-layer graphene¹ in 2004 with excellent physical and chemical properties gives rise to a research boom on 2D materials^{7,8,9,10,11,12,13,14}. However, the development of 2D magnets was lagging behind, which were not realized experimentally and were, for a long time, theoretically challenged by Mermin and Wagner theorem¹⁵, which argues that magnetic order is prohibited in the 2D isotropic Heisenberg model at finite temperature. In 2017, Huang et al. and Gong et al. demonstrated the first atomic thin 2D ferromagnets CrI₃ and Cr₂Ge₂Te₆, and since then, great interest has been stimulated on 2D magnets^16,17,18. Huang et al. used magneto-optical Kerr effect microscopy to verify that the CrI₃ monolayer is an Ising ferromagnet with out-of-plane spin orientation, and its Curie temperature is of 45 K¹⁷. The successful synthesis of 2D CrI₃ and Cr₂Ge₂Te₆ magnet breaks the logjam of Mermin and Wagner theorem and provides a broader and more profound platform for studying magnetism in atomically thin 2D materials.

The urgent demand for miniaturization and multi-functionalization in spintronic devices prompts 2D materials to develop into a stage that should possess multiple and controllable functionalities. 2D magnetoelectric (ME) materials which combine magnetism and electric polarity (EP) in a single 2D phase, play an irreplaceable role due to applicable ME functionalities motivated by potential applications in the areas of multiple-state memories, information storage, and spintronic switches^{6,19,20,21,22,23,24,25}. To achieve 2D ME devices, the ME coupling strength between spin orderings and charge dipoles should be intrinsically strong and finely tunable²⁶. So far, only a small number of 2D systems with intrinsic ME coupling are reported, and most of them are 2D multiferroics. The key obstacle to obtaining 2D ME materials is the interplay between energy-lowering covalent bond formation and energy-raising electronic Coulomb repulsion^27,28. Recently, a few 2D magnetic systems are theoretically demonstrated to be 2D ME systems, examples including: the charged 2D CrBr₃ monolayer that exhibits in-plane multiferroicity due to the combination of orbital and charge ordering as realized by the asymmetric Jahn-Teller distortions of Cr-Br₆ units^29,30,31. ReWCl₆ monolayer with a transition between antiferromagnetic (AFM) and ferromagnetic (FM) phases can be tuned by reversing the in-plane polarization due to the low symmetric C₂ phase³². VOCl₂ is demonstrated to be a 2D multiferroic material due to orbital-charge interactions, and the coexistence of d-electron polarity and magnetism indicates strong ME coupling beyond the d⁰ rule³³. The ME coupling realized by spin and electron polarization can be mostly attributed to the absence of time-reversal symmetry and spatial inversion symmetry. In pursuit of finding ME compounds, a great deal of effort has been devoted to studying the absence of symmetry in 2D materials. 2D Janus structures, due to the structural inversion asymmetry along the normal axis, have demonstrated great potentials in new-generation design in the nanoscale fields including valleytronics, photocatalysis, and spintronics^{13,14,34,35,36,37,38,39,40}. Further development of the Janus system is highly desirable, especially since MoSSe^41,42,43 and WSSe⁴⁴ monolayers have been synthesized successfully by evolving modified chemical vapor deposition with breaking the structural symmetry. Therefore, these facts lead us to speculate that 2D transition-metal based Janus monolayers with ordered spin configuration and out-of-plane EP might hold intrinsic ME coupling to realize bitunable magnetism/polarity behavior, which is of fundamental interest and importance in spintronics.

In this work, we investigate a type of Janus structures Cr₂I₃Y₃ (Y = F, Cl, or Br) with spontaneous magnetism and EP based on first principles calculation. The stability of the Janus structure is confirmed by ab initio molecular dynamics (AIMD) and phonon spectrum calculation. We find that the ground state of Cr₂I₃Y₃ is FM semiconductors with out-of-plane EP. By combining orbital and dipole analysis, we systematically study the bitunable changes between magnetism and EP, and further identify two ME coupling mechanisms: the first is the regulation of polarization through rotation of spin orientation, and the second is the regulation of magnetic moment and ground state through changing electric dipole moment. Furthermore, we confirm that the easily distinguishable magnetic anisotropy energy (MAE) on vertical planes has strong inherent relevance with vertical EP. Besides, the magnetic features controlled by EP are also studied through the modulation of distance ε_d between two different halogen layers. The magnetic moment demonstrates a monotonic decreasing behavior for ε_d, and the magnetic ground state can be reversed from FM to AFM due to the change of EP. We find that the Janus Cr₂I₃Y₃ structure has intrinsic bitunable magnetism/polarity behavior, and its special ME coupling effect will not only promote the future research of ME coupling materials as well as 2D Janus materials but also has great application potential for the preparation of integrated nanoscale devices.

Results and discussion

Geometrical structures and stability

We first analyze the geometrical structure of CrI₃ monolayer¹⁷, which has been successfully synthesized, for better comparison with the following study of CrI₃-like Janus structures. After sufficient structural relaxation without any symmetry constraints, the single-layer structure of pure CrI₃ monolayer shown in Fig. 1a is a sandwich structure with a layer of Cr atoms wrapped in the center by two layers of I atoms, showing a primitive structure of hexagonal lattice with P-31m space group symmetry. Meanwhile, each Cr atom is located at the center of the regular octahedron formed by eight I atoms, and the bond lengths of Cr-I and Cr-Cr are 2.74 Å and 4.04 Å, respectively. The DFT optimized lattice parameters of the CrI₃ monolayer is 7.00 Å and the angle formed by Cr-I-Cr bonds is 95.26°, which agrees well with the previous study¹⁶, indicating that our computational method is reliable. The 2D Janus Cr₂I₃Y₃ structure shown in Fig. 1b is very similar but not identical to that of the pure CrI₃ monolayer. The Janus Cr₂I₃Y₃ sandwich is formed by replacing one layer of I atoms with other halogen atoms Y (Y = F, Cl, or Br) of CrI₃ monolayer. The changes of geometric parameters are listed in Table 1. The layer of Cr atom is no longer at the inversion center, but with a shift due to asymmetry between different up and down halogen layers. Janus Cr₂I₃Y₃ structure is formed in the P31m space group with vertical mirror symmetry, but it lacks horizontal mirror symmetry and inversion center. It can be seen easily that the optimized lattice parameter a of the Janus system in accordance with the order from small to large, which is $a_{Cr_2I_6}$ > $a_{Cr_2I_3Br_3}$ > $a_{Cr_2I_3Cl_3}$ > $a_{Cr_2I_3F_3}$. Meanwhile, the bond lengths (d₂) between Cr and Y atoms are also changed following the upper rule. Such a result can be understood by the radius of the atoms ($R_I$ > $R_{Br}$ > $R_{Cl}$ > $R_F$) and the spatial interaction between different halogen layers. With little change in Cr-I bonds length (d₁) as the radius of Y decreases, Cr-I-Cr bond angle (Θ₁) decreases pronouncedly, indicating that the distance between the magnetic metallic element Cr becomes smaller and the spatial interaction thus be affected. The variation trend of geometric parameters in Table 1 is clearly represented in Supplementary Fig. 1a.

Fig. 1: Structural configurations and stability.

a, b Top and side views of 2D structure isolated Cr₂I₆ monolayer and Janus Cr₂I₃Y₃ (Y = F, Cl, or Br) monolayer. The solid lines indicate the primitive cells. The amethyst, blue, and grey spheres represent the Iodine, Chromium, and Y (Florine, Chlorine, or Bromine) atoms, respectively. c Change of free energies over time in AIMD simulations at 300 K and the snapshots at 0 K and 300 K after a 5000 fs simulation for the Janus Cr₂I₃Br₃ monolayer. d Phonon spectra of the Janus Cr₂I₃Br₃ monolayer.

Table 1 The summarization of structure parameters of 2D Cr₂I₆ and Janus Cr₂I₃Y₃ (Y = F, Cl, or Br) monolayer, corresponding with symbols labeled in Fig. 1b.

Although the most energetically stable structure can be precisely obtained through structural optimization, it may not guarantee the stability of the 2D Cr₂I₃Y₃ Janus structure. Therefore, further analysis in thermal and dynamic stability of Janus structure by AIMD simulations and corresponding phonon spectra are implemented, and details can be seen in Fig. 1c, d. The evolution process of free energy last 5000 fs for Janus Cr₂I₃Y₃ supercell at 300 K during the AIMD simulations. Neither occurrence of major structural distortion nor breaking of chemical bond from two views in Fig. 1c verifies that the morphological structure is dynamically stable at room temperature. Phonon spectra obtained by executing $\surd 3 \times \surd 3$ supercell further strengthens the proof of dynamic stability by showing twenty-one optical and three acoustical phonon branches with the absence of imaginary frequencies throughout the Brillouin zone. In addition, the cohesive energy per atom, ${{\Delta }}E_{{{{\mathrm{coh}}}}}$ defined as ${{\Delta }}E_{{{{\mathrm{coh}}}}} = 1/8 \cdot \left( {E_{Cr_2I_3Y_3} - 2E_{Cr} - 3E_I - 3E_Y} \right)$, where$E_{Cr_2I_3Y_3}$is the total energy of the Cr₂I₃Y₃ monolayers, E_cr, E_I, and E_Y are the energy of isolated Cr, I, and Y (Y = F, Cl, or Br) atoms in a 25 × 25 × 25 ų box, respectively. The negative ΔE_coh as shown in Table 2 suggests that the monolayer is energetically favorable. Our results are in good agreement with previous results⁴⁰. We find that the predicted cohesive energy ΔE_coh of the 2D Janus Cr₂I₃Y₃ are lower than that of the pure Cr₂I₆ monolayer, indicating the stability of the Janus monolayer and the possibility of their synthesis. Due to the streamlining of high-purity CrI₃ and Cr₂Ge₂Te₆ preparation and the successful fabrication of Janus transition-metal dichalcogenides^{16,17,18,41,42,43,44,45}, the experimental preparation of the corresponding Janus structure should also be realizable.

Table 2 Formation energy ΔE_coh of 2D Cr₂I₆, Janus Cr₂I₃Y₃.

Magnetic and electronic properties

To verify the ground magnetic states of Janus Cr₂I₃Y₃ monolayers, the collinear magnetic configurations of FM order and AFM order are considered. Identical calculations on the 2D Cr₂I₆ monolayer are also carried out for the purpose of comparison. For intuitive analysis of spin behaviors of Cr₂I₆ and Cr₂I₃Y₃ monolayers, the spin-resolved band structures are shown in Supplementary Fig. 3. Our results show that the Cr₂I₆ monolayer is a half-semiconductor with FM state, where the conduction band and valence band are spin split with the conduction band (CB) minimum and valence band (VB) maximum possessing the same spin channel as shown in Supplementary Fig. 3a. The indirect band gap E_g is estimated to be 0.81 eV with SOC effect (1.15 eV without SOC effect). Magnetic ground states are determined by evaluating the energy difference between AFM and FM states, and 2D Janus Cr₂I₃Cl₃ and Cr₂I₃Br₃ have the same order of energy difference ΔE as 2D Cr₂I₆ ($\Delta E = 59.99me{{{\mathrm{V}}}}/{{{\mathrm{cell}}}}$), where ΔE = E_AFM − E_FM, which are 51.97 and 40.16 meV/cell, respectively. Cr₂I₃F₃ has a minimum ΔE of 3.56 meV/cell among the Cr₂I₃Y₃ monolayers. Almost all the magnetic moments are contributed by d-electrons of Cr atoms as shown in Table 3. As shown in Supplementary Fig. 3, the 2D Janus Cr₂I₃Y₃ monolayers are also semiconductors with FM states, among which Cr₂I₃Br₃ with E_g of 1.04 eV with SOC effect (1.30 eV without SOC) is a half-semiconductor, while Cr₂I₃Cl₃ and Cr₂I₃F₃ are both bipolar magnetic semiconductors, where the spin channel is opposite at VB maximum and CB minimum, and the band gaps are 0.90 and 0.22 eV with SOC effect (1.20 and 0.54 eV without SOC effect), respectively. The band gaps of Cr₂I₆ and Cr₂I₃Y₃ monolayers with FM and AFM states are both listed in Table 3, and results show that the band gaps of FM states for all the monolayers are smaller than that of AFM states, which are mainly because the competition between FM exchange splitting and band gap.

Table 3 Summarization of energy differences for FM and AFM magnetic configurations for 2D Cr₂I₆ and Janus Cr₂I₃Y₃ monolayers, where the energy of FM ground state is set to be 0 meV/cell.

The atom-resolved band structure with SOC for FM Cr₂I₃Y₃ systems are shown in Fig. 2, it can be clearly observed that all Janus Cr₂I₃Y₃ are semiconductors with the CBs mainly contributed by the transition metal Cr atoms, while the halogen elements I or Y atoms make a major contribution to the VBs. Different I atomic layers make identical contributions to the VBs in CrI₃ monolayer due to centrosymmetry, while for Janus structure Cr₂I₃Y₃, the VBs near the Fermi level are dominated by the I atomic layer instead of the Y atomic layer. Meanwhile, by separating the contribution of Cr atom in a primitive cell for the band near the Fermi level as shown in Supplementary Fig. 2, it is confirmed that the change of Y atom will not affect the contribution of different Cr atoms to each energy level but can change the distribution in CBs. That is, the localization of four degenerated bands near CBs minimum dominated by Cr-d electrons is decreased with the electron negativity of halogen atoms being larger. The reason is that the smaller distance between Cr and halogen atoms introduced by larger electron negativity leads to stronger couplings between Cr-d electrons and dispersive Y-p electrons. We can see that d orbitals of Cr atoms can split into three groups ($a_1 = d_{z^2}$, $e_1 = d_{xy}$ and $d_{x^2 - y^2}$, $e_2 = d_{xz}$ and $d_{yz}$) in Cartesian coordination as shown in Fig. 2 instead of expected octahedral splitting. This can be explained by the mismatch between the Cartesian coordination with the internal coordination of the octahedral crystal field, which makes the projected DOSs of Cr-3d electrons orbitally unresolved. To transfer the d orbitals from trigonal prismatic field to octahedral crystal field ($t_{2g} = d_{xy}$, $d_{xz}$ and $d_{yz}$, $e_g = d_{x^2 - y^2}$ and $d_{z^2}$), a manipulation is executed to align the Cr-I bonds in the octahedrons along the x, y, and z axis. The approximate rotation matrix is determined, and the five d orbitals are reconstructed. Finally, the projected DOSs of Cr-3d orbitals of Cr₂I₆ monolayer, as shown in Fig. 3a, can split into two groups as t_2g and e_g, which is the typical octahedral splitting as expected. For the Janus Cr₂I₃Y₃ monolayer, taking Cr₂I₃Cl₃ in Fig. 3b as an example, the splitting of 3d orbitals of Cr atoms can also obey the rules of splittings under an octahedral crystal field. The above results indicate that Janus structure with noncentrosymmetry induced by different halogen layers will affect the dispersions of Cr-d states and further have an influence on band gaps and magnetic properties, which can therefore be explored further.

Fig. 2: Atom-resolved band structures of Janus monolayers and calculated DOSs.

a–d Atom-resolved band structures of Janus monolayers and corresponding DOSs of (a) Cr₂I₆, (b) Cr₂I₃Br₃, (c) Cr₂I₃Cl₃, and (d) Cr₂I₃F₃. The different proportions of occupancy Cr, I (or upper I), and Y (or lower I) atoms are represented by the green, blue, and red dots in the band scatter diagram, respectively. The black gradient filling in the right part of each diagram represents the total contribution of DOS, while the green, purple, and yellow gradient filling represent the projected contribution of $d_{z^2}$, $d_{yz}/d_{xz}$, $d_{xy}/d_{x^2 - y^2}$ orbitals, respectively.

Fig. 3: Projected DOSs of Cr-3d orbitals by coordination matching.

a, b Projected DOSs of Cr-3d orbitals of (a) Cr₂I₆ and (b) Cr₂I₃Cl₃ without SOC after changing d orbitals from trigonal prismatic field to octahedral crystal field by coordination matching. The insets are schematic of rotation of Cr centered octahedron. Amethyst, blue, and green balls present I, Cr, and Cl atoms, respectively.

As for the octahedral environment coordinated by six halogen anions, Cr³⁺ cation hold a high spin state with three low-lying t_2g orbitals, which indicates the dominant contribution of indirect FM exchange instead of direct AFM exchange⁴⁶. The underlying physical mechanism of the FM ground state in our system is related to the superexchange interaction. Goodenough-Kanamori-Anderson (GKA) rule^46,47,48 points out that if the cation-anion-cation bond angles are of 90° and 180°, two cation spins often favor FM and AFM ordering, respectively. The FM superexchange interaction of Cr atoms through I ions in Cr₂I₆ satisfies the GKA rules since the Cr-I-Cr bond angle (Θ₁ = Θ₂ = 95.26°) is approximately approaching 90°. Similarly, as for Janus Cr₂I₃Y₃ (Y = Br, Cl, or F) structures, the bond angles Θ₂ of Cr-Y-Cr are 100.03°, 105.00°, and 122.61°, respectively, which are also approaching 90° instead of 180°, indicating FM via superexchange results. However, with the atomic numbers of halogen elements increasing, the bond angles Θ₂ of Cr-halogen-Cr are larger, while the bond lengths d₂ of Cr-halogen (Cr-Cr) are decreasing from 2.74 to 2.06 Å (4.04 to 3.61 Å). The results indicate the reduced dominance of FM superexchange interaction and thus explain the decreasing of the energy difference ΔE between AFM and FM states as listed in Table 3.

Further, the intrinsic mechanism of Janus monolayer’s magnetism and the prediction of the associated Curie temperature T_c are briefly discussed. As mentioned above, the magnetic moment of Cr₂I₃Y₃ is mainly contributed by Cr atoms. The contribution of the magnetic moment of Cr atom in Cr₂I₃Y₃ (Y = F, Cl, or Br) is 3.24, 3.23, and 3.27 μ_B, respectively, showing different d-p electron coupling strength between Cr and halogen atoms in different Janus structures. The nearest neighbor Heisenberg model of Cr₂I₆ and Cr₂I₃Y₃ systems in one unit cell (with FM exchange J > 0) should be expressed as

$${{{\mathbf{H}}}} = - J\mathop {\sum}\limits_{ < i,j > } {{{{\mathbf{S}}}}_i \cdot {{{\mathbf{S}}}}_j} = - \frac{1}{2}J\mathop {\sum}\limits_{i,j} {{{{\mathbf{S}}}}_i \cdot {{{\mathbf{S}}}}_j}$$

(1)

where <i, j> represents the number of nearest interactions with the double counting subtracted. There are two Cr atoms in one unit cell and three nearest neighbors for each Cr atom. Thus, the spin model allows to write the energies of the different configurations normalized per unit cell (2 Cr atoms):

$$E_{{{{\mathrm{FM}}}}} = - 3J\left| {{{\mathbf{S}}}} \right|^2$$

(2)

$$E_{{{{\mathrm{AFM}}}}} = 3J\left| {{{\mathbf{S}}}} \right|^2$$

(3)

then the energy difference between FM and AFM magnetic configurations can be written as^30,49:

$$E_{{{{\mathrm{FM - AFM}}}}} = - 6J\left| {{{\mathbf{S}}}} \right|^2$$

(4)

Here, only nearest-neighbor exchange couplings between Cr atoms are considered. This is because previous studies indicated that the nearest neighbor exchange parameters are 12 and 25 times larger than the second and the third ones in CrI₃ systems, respectively, showing robust FM intralayer coupling⁵⁰. The exchange parameter of Cr₂I₆ monolayer for the first nearest neighbors is estimated to be 3.63 meV, and the value is quite consistent with the previous study with $J_{Cr_2I_6} = 3.25m{{{\mathrm{e}}}}V$ of Cr₂I₆ monolayer⁵¹, which indicates the reliability and validity of our method. Thereafter, the corresponding exchange parameters Jof Janus Cr₂I₃Y₃ monolayer ($J_{Cr_2I_3F_3} = 0.23m{{{\mathrm{e}}}}V$, $J_{Cr_2I_3Cl_3} = 2.61m{{{\mathrm{e}}}}V$, $J_{Cr_2I_3Br_3} = 3.30m{{{\mathrm{e}}}}V$) are obtained. And theT_c is obtained by combining the Heisenberg model with the local field theory^16,30,

$$T_{{{\mathrm{C}}}} = - J\frac{{\left| {{{\mathbf{S}}}} \right|}}{{k_B}}$$

(5)

Here, $k_B$ denotes Boltzmann constant while $\left| {{{\mathbf{S}}}} \right|$ is $3/2$. However, the critical temperature in our result (T_C = 77 K) of Cr₂I₆ monolayer estimated by mean-field expression (Eq. 5) is overestimated compared with 45 K obtained experimentally¹⁷. The overestimation can be ascribed to the neglect of magnetic anisotropies in the mean-field theory. To make the result of Curite temperature more reliable, a universal expression that considers magnetic anisotropies and depends on the number of nearest neighbors (N_nm = 3) from the fitting of classical Monte Carlo simulations is taken⁵²:

$$T_{{{\mathrm{C}}}} = \frac{{3.64\left| {{{\mathbf{S}}}} \right|^2J}}{{k_B}}\tanh ^{1/4}\left[ {\frac{6}{{N_{nn}}}\log \left( {1 + \frac{{0.033\left[ {A(2\left| {{{\mathbf{S}}}} \right| - 1) + B\left| {{{\mathbf{S}}}} \right|N_{nn}} \right]}}{{J\left( {(2\left| {{{\mathbf{S}}}} \right| - 1)} \right)}}} \right)} \right]$$

(6)

where parameters A and B satisfy the relation $A = \left( {\Delta E_{{{{\mathrm{FM}}}}} + \Delta E_{{{{\mathrm{AFM}}}}}} \right)/\left( {2S^2} \right)$ and $B = \left( {\Delta E_{{{{\mathrm{FM}}}}} - \Delta E_{{{{\mathrm{AFM}}}}}} \right)/\left( {3\left| {{{\mathbf{S}}}} \right|^2} \right)$ with $\Delta E_{{{{\mathrm{FM}}}}({{{\mathrm{AFM}}}})} = E_{{{{\mathrm{FM}}}}({{{\mathrm{AFM}}}})}^{(x)} - E_{{{{\mathrm{FM}}}}({{{\mathrm{AFM}}}})}^{(z)}$, respectively. N_nm = 3 is the number of nearest neighbors. The calculated Curie temperature of Cr₂I₆ monolayer is 42.8 K, which is in excellent agreement with the experimental value of 45 K. Similarly, the estimated Curie temperatures of Janus Cr₂I₃Y₃ (Y = Br, Cl, or F) monolayers are 22.0, 28.8, and 7.5 K, respectively.

The charge redistribution analysis

The electrostatic potential of the related structures is defined by the following equation:

$$V_{{{{\mathrm{potential}}}}}({{{\mathbf{r}}}}) = V({{{\mathbf{r}}}}) + {\int} {\frac{{{{{\mathrm{n}}}}({{{\mathbf{r}}}}^\prime )}}{{\left| {{{{\mathbf{r}}}} - {{{\mathbf{r}}}}^\prime } \right|}}d{{{\mathbf{r}}}}^\prime }$$

(7)

Where, V(r) is the ionic potential, and the second term is the Hartree potential. The diagram of the electrostatic potential relationship of 2D Janus Cr₂I₃Y₃ is shown in Fig. 4. The plateau represents the vacuum energy level, while the valley represents the change of electrostatic potentials inside the Janus structure. The imbalance of electrostatic potentials for Y and I layers in the Janus structure indicates that the Y layer can accumulate more electrons than the I layer. The accumulation of electrons can be quantitatively given by the Bader charge analysis of the CrI₃ and Janus Cr₂I₃Y₃ monolayer in Fig. 4 for the purpose of clarifying the physical origin of polarization. I atom from different layers gains an average charge of 0.35 e⁻ from the Cr atom in pure CrI₃ monolayer, while the balance is broken by introducing different Y atoms in Janus structure, indicating an internal charge polarization. Through the analysis of the electrostatic potential diagram, it’s clear that there is a potential difference between the I layer and the Y layer induced by the non-uniformity of charge. In our results, the potential differences ΔV of the upper and lower surfaces of Janus Cr₂I₃Y₃ (Y = F, Cl, or Br) monolayer are 1.52 eV, 0.94 eV, 0.54 eV, respectively. The tendency of electrostatic potential difference is quite in accordance with the different electronegativity of halogen layers, that is, the electronegativity of the I (X_I = 2.66) atom and the Y (X_F(3.98) > X_Cl(3.16) > X_Br(2.96)) atom.

Fig. 4: The electrostatic potential and the charge redistribution.

a, d The electrostatic potential of (a) Cr₂I₆, (b) Cr₂I₃Br₃, (c) CrI₃Cl₃ and (d) Cr₂I₃F₃ monolayer. The side views of corresponding structures are inserted in accordance with electrostatic potentials. The electrostatic potential difference ΔV between the upper and lower halogen elements are indicated in red arrows between the dotted extension line and black line of vacuum level, specific values are listed in red as well. Orange arrows and numbers represent the charge redistribution from Cr atom to the surface atom.

Bitunable magnetism/polarity behavior

MAE is an analytical parameter based on magnetic anisotropy that is used to evaluate the ability of magnet systems to resist environmentally thermal fluctuation and the switching effect of spintronic application to maintain or reverse the magnetic ordering^7,30. Previous studies show that SOC originated from halogen atoms makes a dominant contribution to MAE in CrI₃ systems⁴⁹. To figure out the origin of MAE for Cr₂I₆ and Cr₂I₃Y₃, we first study the MAE with a change in the orientation of magnetization M along the x and z axes. We follow the recipe of second-order perturbation theory by Wang et al.⁵³, and the MAE can be expressed as

$$\begin{array}{l} {\Delta E_{{{{\mathrm{MAE}}}}} = E({{{\mathbf{M}}}}_x) - E({{{\mathbf{M}}}}_z)} \\ \qquad\quad\,\,\,\, = \xi ^2\mathop{\displaystyle\sum}\limits_{o^ + ,u^ + } \frac{{\left| {\langle o^ + |\hat L_z|u^ + \rangle } \right|^2 - \left| {\langle o^ + |\hat L_x|u^ + \rangle } \right|^2}}{{E_{u^ + } - E_{o^ + }}} - \xi ^2\mathop{\displaystyle\sum}\limits_{o^ - ,u^ + } \frac{{\left| {\langle o^ - |\hat L_z|u^ + \rangle } \right|^2 - \left| {\langle o^ - |\hat L_x|u^ + \rangle } \right|^2}}{{E_{u^ + } - E_{o^ - }}} \end{array}$$

(8)

where $\xi$ represents the SOC constant, + and − indicate the spin-up and spin-down states, respectively. $E_u - E_o$ represents the energy difference between the unoccupied states u and occupied states o. The angular momentum operators are chosen as $\hat L_z$ ($\hat L_x$) along the out-of-plane (in-plane) axis. In this case, a negative $\Delta E_{{{{\mathrm{MAE}}}}}$ indicates that the spin along the x-axis is an easy axis, whereas the positive value supports the z-axis as the easy axis. Based on the second-order perturbation theory, we can analyze the contributions to MAE from the hybridization between different orbitals of specific element, and the corresponding results of MAE contributions for parent structure Cr₂I₆ and Janus structures Cr₂I₃Y₃ are shown in Fig. 5. The net contributions to MAE provided by Cr-d orbitals are negligible with the values of −0.035, −0.020, −0.118, and −0.089 meV/unit cell for Cr₂I₆, Cr₂I₃Br₃, Cr₂I₃Cl₃, and Cr₂I₃F₃, respectively. The sign and value of final MAE are mainly determined by the contributions of halogen-p orbitals. For Cr₂I₆ and Cr₂I₃Br₃, the net contribution of I-p and Br-p atoms to MAE is positive, indicating that the easy-axis for Cr₂I₆ and Cr₂I₃Br₃ monolayer is the z-axis. For Cr₂I₃Cl₃, the negative contribution of I-p as shown in Fig. 5h can counteract the positive contribution of Cl-p as shown in Fig. 5i, which thoroughly leads to a net negative MAE of −1.23 meV/unit cell with x-axis as an easy axis. For Cr₂I₃F₃, the MAE is mainly dominated by I-p orbitals with hybridization between p_x and p_y, and the contributions of Cr-d and F-p orbitals are tiny, which leads to a net negative MAE of −10.45 meV/unit cell with x-axis as an easy axis. That is to say that the vibration of MAE is dominated by halogen atoms instead of transition metal Cr⁵¹, and the value of $\Delta E_{{{{\mathrm{MAE}}}}}$ declines from positive to negative with different halogen atoms following the sequence of Cr₂I₆ > Cr₂I₃Br₃ > Cr₂I₃Cl₃ > Cr₂I₃F₃.

Fig. 5: Orbital-resolved MAE.

Orbital-resolved MAE of Cr-d orbitals and halogen-p orbitals (I, Cl, Br, and F) in Cr₂I₆ and Janus monolayers. Orbitals of Cr₂I₆ monolayer: a Cr-d orbitals, b up-layer I-p orbitals, c down-layer I-p orbitals; Orbitals of Cr₂I₃Br₃ monolayer: d Cr-d orbitals, e I-p orbitals, f Br-p orbitals; Orbitals of Cr₂I₃Cl₃: g Cr-d orbitals, h I-p orbitals, i Cl-p orbitals; Orbitals of Cr₂I₃F₃ monolayer: j Cr-d orbitals, k I-p orbitals, l F-p orbitals.

In Janus Cr₂I₃Y₃ systems, Y layers can induce the inversion symmetry, which leads to vertical EP. To uncover if there is any relationship between MAE and EP, we put emphasis on MAE with arbitrary spin orientations on xz- and yz-plane and the synergetic evolutions of EP as shown in Fig. 6⁵⁴. For comparison, MAE with spin orientations for different angles within yz- and zx-planes for Cr₂I₆ monolayers is firstly investigated, as shown in Fig. 6a, b, respectively. The reference plane where the 2D plane of the material lies is taken as xy-plane while the other two planes are taken as vertical yz- and zx-plane as illustrated in Supplementary Fig. 1d. As for Cr₂I₆, the smallest MAE value occurs when the spin orientation is along z direction as shown in Fig. 6a, b. Since the inversion symmetry is preserved, there is no intrinsic EP in Cr₂I₆ monolayer with the vibration of dipole being 0 mD. Contrarily, for CrI₃Cl₃, the MAE value of z direction shown in Fig. 6c, e is enhanced to be 1.2 meV, and the smallest MAE value occurs with spin orientations along x/y directions. For spin orientations with different rotation angles in MAE, the corresponding dipole moments regarding EP are also investigated. Taking Cr₂I₃Cl₃ as an example, the dipole moments along the z-axis with respect to spin orientations in yz- and zx-planes are shown in Fig. 6d, f. Noting that parameter ${{{\mathrm{D}}}}_0^{{{{\mathrm{Y}}}} = {{{\mathrm{Cl}}}}}$ (about 0.94 D) in ordinate represents the minimum value of EP. As expected, there is the same period variation between MAE and EP of 2D Janus Cr₂I₃Y₃ related to the spin orientations. Besides, the relation of MAE and EP with spin orientations within vertical planes for Cr₂I₃Br₃ and Cr₂I₃F₃ are shown in Supplementary Fig. 5 and Supplementary Fig. 4 respectively. We can conclude that the variations of the dipole with spin orientations from z-axis to x/y-axis ($D({{{\mathbf{M}}}}_{x/y}) - D({{{\mathbf{M}}}}_z)$) are 0, −2.3, −3.5, and −10.8 mD for Cr₂I₆, Cr₂I₃Br₃, Cr₂I₃Cl₃, and Cr₂I₃F₃, respectively, while the value of $\Delta E_{MAE}$ from z-axis to x/y-axis ($E({{{\mathbf{M}}}}_{x/y}) - E({{{\mathbf{M}}}}_z)$) are 1.79, 0.05, −1.23, and −10.45 meV for Cr₂I₆, Cr₂I₃Br₃, Cr₂I₃Cl₃, and Cr₂I₃F₃, respectively, Thus, we speculate that the relative MAE within the spin orientation perpendicular to the xy-plane (e.g. along z direction) can be triggered by the vertical EP, which originates from vertical inversion symmetry breaking. These facts also indicate that the magnitude of the electric dipole moment of 2D Janus Cr₂I₃Y₃ can be finely tuned by operating spin orientations. Our results show the inherent relationship between magnetism and polarity in the Janus Cr₂I₃Cl₃ structures. Fine-tuned MAE is one of the pivotal parameters for magnetic information storage and modern information data recording and processing. It is of great value to explore the quantitative relationship between MAE and EP in Janus Cr₂I₃Y₃ systems, which can open a new way to achieve bitunable magnetism/polarity behavior for the new generation of spintronic devices.

Fig. 6: The coupling between the EP dipole moment and MAE with spin rotation.

MAE of Cr₂I₆ monolayer and Janus Cr₂I₃Cl₃ with spin orientations for different rotation angles within (a, c) yz-plane, and (b, e) zx-plane, respectively. The corresponding electric dipole moment along the z direction of Janus Cr₂I₃Cl₃ for spin orientations in (d) yz-plane, and (f) zx-plane. The horizontal coordinate-axis refers to rotating angles of spin orientations with an interval of 15° starting from z-axis. Insets in (a, b, c, and e) are MAE corresponding to spin orientations in polar coordinates, and insets in (d and f) are dipole moments along the z direction corresponding to spin orientations in polar coordinates.

Another fact is the converse ME regulation from EP to magnetism, and we demonstrate that the magnetic states can be tuned by changing the strength of EP in Janus Cr₂I₃Y₃ systems. The strength of perpendicular electric dipole in 2D Janus Cr₂I₃Y₃ is mainly derived from the distance between positive and negative charge centers. The electrostatic potentials in Fig. 4 show that the capability of capturing electrons is imbalanced on different halogen layers. Hence, the magnitude of the electric dipole moment in Cr₂I₃Y₃ is related to the spacing distance between I and Y atomic layers, and the vertical variation of the spacing can therefore be straightforwardly characterized by the parameter.ε_d, defined as $\varepsilon _d = (d_z - d)/d \times 100\%$, where d_z represents the tunable perpendicular distance between the layer of I and Y atoms while d represents the special case of d_z for fully optimized equilibrium geometry of Cr₂I₃Y₃. It is worth mentioning that the intralayer spacing of transition metal compound monolayers can be modulated experimentally under high pressure using an energy dispersive synchrotron X-ray diffraction method^55,56. Besides, 2D materials with good mechanical properties can sustain higher critical strains than three-dimensional counterparts^57,58. In-plane strain is one of the general approaches to regulate the structure and therefore the electronic properties of a material. When in-plane strain is applied to normal and auxetic 2D materials, an opposite change is also produced in the longitudinal direction of the material, namely, the material also undergoes a vertical strain⁵⁸. We find that the total energy of the Cr₂I₃Y₃ system which shows a quadratic polynomial with d_z, and the lowest energy point locates at d. Furthermore, we systematically investigate the EP dipole moment, magnetic moment, band gaps, and ground states transitions related to ε_d as shown in Fig. 7. Our study shows that the dipole moment from EP is attributed to misaligned charge centers from different halogen layers and has a direct linear relation with the distance between I and Y layers as shown in Fig. 7a, d, g. In this case, we intend to study how the change of EP can affect the magnetic and electronic properties, and the most direct way to tune EP is to change the distance between I and Y layers. Thus, to make the physical mechanism easier to understand, a large range of vertical strain is applied theoretically to make sure that EP can be linearly tuned within a sizable range. Besides, an external electric field can also tune the vertical EP of 2D materials in an effective way⁵⁹, which can be an alternative approach to study the interplay between EP and MAE and need to be systematically investigated further. Interestingly, a magnetic phase transition from FM to AFM order occurs under proper vertical strain ε_d. Since the magnetic moment is dominated by Cr-3d orbitals and the intralayer spacing between Cr atoms remains the same, we deduce that the magnetic phase transition is totally triggered by perpendicular EP. As a result, the linear descent of the electric dipole moment is accompanied by a decrease in the magnetic moment M_Cr as shown in Fig. 7b, e, h. From the relations of ε_d-P and ε_d-M_Cr, we can deduce a monotonic relation between magnetism and polarity, and reach the conclusion that magnetism features including magnetic moment and magnetic phase can be finely tuned by EP. In addition, the plot of band gap E_g versus ε_d shown in Fig. 7c, f, i indicates that the semiconducting feature of Janus Cr₂I₃Y₃ can be maintained within a large range of ε_d. It is worth mentioning that the direct evidence of the response between magnetism and electric polarity (e.g. ME effect) is the ME susceptibility. Our results show that the simultaneous breaking of space inversion symmetry and time reversal symmetry in Janus Cr₂I₃Y₃ systems may allow them to exhibit the linear response, in which an electric field can induce a magnetization and a magnetization field can induce an EP. This phenomenon can be quantitatively revealed by using Monte Carlo simulations in analysis of the ME susceptibility expressed in terms of spin correlation functions, which needs to be systematically investigated further^60,61.

Fig. 7: Vertical strain effects.

The EP dipole moment with respect to vertical spacing variation ε_d for (a) Cr₂I₃F₃, (d) Cr₂I₃Cl₃, and (g) Cr₂I₃Br₃. The Magnetic moments per Cr atom with respect to vertical spacing variation ε_d for (b) Cr₂I₃F₃, (e) Cr₂I₃Cl₃, and (h) Cr₂I₃Br₃. The band gaps concerning vertical spacing variation ε_d for (c) Cr₂I₃F₃, (f) Cr₂I₃Cl₃, and (i) Cr₂I₃Br₃. The transitions between different phases are labeled differently, e.g., orange, blue, and yellow blocks indicate FM conductor, FM semiconductor, and AFM semiconductor phases, respectively.

In summary, the structural, electric and magnetic properties of Janus Cr₂I₃Y₃ (Y = F, Cl, or Br) system with intrinsic magnetism and EP are studied based on first principles calculation. Cr₂I₃Y₃ are found to be FM semiconductors with vertical EP due to asymmetric halogen atomic layers. The bitunable behaviors between magnetism and EP are systematically investigated. The vertical EP can be periodically tuned by rotating orientation of magnetic moment within out-of-planes of Cr₂I₃Y₃ with FM ground states which indicate that there are direct couplings between EP and MAE. We demonstrate that the magnitude of the out-of-plane MAE is obviously distinguished from the in-plane one. Interestingly, magnetic properties including phase transition of magnetic ground states and magnitude of magnetic moments can be tuned by manually manipulating vertical EP at the same time, which can be achieved by introducing vertical strain ε_d between two halogen atomic layers. We reveal the quantitative relationship between magnetism and EP in Janus Cr₂I₃Y₃ systems, which can open a new way to achieve bitunable magnetism/polarity behavior for the new generation of spintronic devices. In the demand of low-power spintronic memories and logic devices, it is quite desirable to control the magnetic orientations or to turn on and off magnetism by purely electrical approach. For this purpose, one can use the Cr₂I₃Y₃ Janus systems with intrinsic magnetism/polarity behaviors to control the magnetic states by EP through the cross coupling of the two intrinsic orders.

Methods

Density functional theory calculations

Based on density functional theory (DFT), first principles calculations are used for the research of 2D Janus Cr₂I₃Y₃ monolayer. The structure relaxations and the state-energy calculations are performed by the projector augmented wave (PAW) method in the Vienna ab initio simulation package (VASP)^62,63. And the corresponding exchange-correlation function is approximated by the Perdew−Burke−Ernzerhof (PBE) parametrization within the generalized gradient approximation (GGA)⁶⁴. To describe the strongly correlated correction of Cr-3d electrons, GGA+U (U = 3 eV) method⁶⁵ is employed after referencing the previous study on the electronic properties of Cr-based CrX₃ (X = I, Br, Cl)^29,30 and Janus research in CrTeI, CrSeBr³⁶, and the magnetic properties in the parent structure of Cr₂I₆ are able to fit well with experimental findings¹⁷. Besides, by testing the electronic and magnetic properties for Cr₂I₃Y₃ with U of 2.5, 3.0, and 3.5 eV for Cr-3d electrons, the main conclusions on magnetism/polar behaviors will not be affected. Thus, the same U parameter for Cr-3d electrons of the Janus systems is chosen for better comparison. The spin-orbit coupling (SOC) effect is also incorporated into the calculation (if it is not mentioned separately) for the existence of heavy element iodine. The cutoff energy of the plane-wave is set to be 500 eV to ensure the relative accuracy of truncation, while the convergence criteria of the force and the energy are set to be −0.01 eV Å⁻¹ and 10⁻⁶ eV, respectively. k-points mesh is set to be 13 × 13 × 1 in the unit cell of 2D Janus Cr₂I₃Y₃ monolayer’s energy calculation. Namely, the 2D Brillouin zone integration is done with a k-mesh density⁶⁶ of 70/a, where a is the lattice constant in the unit of Å. Calculations of MAE require higher accuracy, and, especially, a denser k mesh to sample the reciprocal space. For MAE calculations, a denser grid of 30 × 30 × 1 is used to sample the Brillouin zone. The MAE is defined as the variation in the total energy as the magnetization takes different orientations. Thus, we calculate MAE in Janus Cr₂I₃Y₃ monolayer which represents the energy difference between different spin orientations in the same structure which can be achieved by tuning the spin component in all directions to distinguish the angle. Consequently, the MAE can be expressed as $\Delta E_{{{{\mathrm{MAE}}}}} = E\left( {{{{\mathbf{M}}}}_i} \right) - E\left( {{{{\mathbf{M}}}}_j} \right)$, where M_i and M_j indicate the magnetization along specific axes i and j, respectively. A vacuum of more than 20 Å is added to the normal axis to avoid interlayer interactions according to the principle of VASP. By setting the center of mass as the reference point, the Berry phase expressions with dipole correction are also used to calculate the dipole of the 2D Janus system.

Band structure and density of states calculations

We obtain the electronic band structure, the density of states (DOS), and magnetic properties with a more accurate location of the Fermi level by using a tetrahedron method with Blöchl corrections in the self-consistency step.

Phonon spectra and Ab initio molecular dynamics simulations

To evaluate the thermal stability of the 2D Janus Cr₂I₃Y₃ structure, free energies as a function of time are obtained by ab initio molecular dynamics (AIMD) simulations, which provide a way of calculating material properties at finite temperatures⁶⁷. And the phonon spectra is calculated by using the DFPT method with phonopy package⁶⁸.