Multiferroic Properties of Rare-Earth-Doped VOCl₂ Monolayers: A First-Principles Study

Abstract

The structural and electronic properties of the rare-earth-modified VOCl₂ monolayer (V_1−xX_xOCl₂, where X = Nd, Sm and Eu) are explored, by using the density functional theory calculations. In particular, the influence of the rare-earth (X) concentration on the physical properties is investigated for $x=0.166$, 0.083, and 0.062. The lattice parameters for all the optimized structures reveal an increase, while the crystal structure changes from rectangular (with Pmm2 space-group) to oblique for the $x=0.166$ concentration, preserving the original space-group for the other compositions. The structural analyses also revealed moderate changes in the VO₂Cl₄ distortions, after the inclusion of the rare-earth elements. On the other hand, the electronic properties have shown that the substitution of V by the Nd, Sm and Eu cations also preserves the semiconductor behavior of the studied system. The obtained results for the density of state reveal a non-zero total magnetization and show that the inclusion of the X cations promotes a transition from the antiferromagnetic to the ferrimagnetic state in the V_1−xX_xOCl₂ compositions. Furthermore, the modern theory of polarization reveals the ferroelectric character for the pure and modified system. These results show that the controlled substitution at the V-site with rare-earth elements simultaneously modifies the structural, electronic, magnetic and multiferroic properties of the VOCl₂ system, offering promising potential of the studied system for application in 2D-based materials and electronic devices with enhanced multifunctional properties.

1. Introduction

Two-dimensional (2D) materials with multiferroic response have achieved significant relevance in recent years due to the coexistence (and even coupling) of the electrical and magnetic orders in the same phase, at the nanometric scale. This characteristic enables their application in advanced electronic devices, such as non-volatile memories, high-sensitivity sensors, and spintronics-related components [1,2,3,4]. Since the discovery of graphene, the world’s thinnest and strongest material (one-atom-thick layer of carbon ions arranged in a hexagonal lattice), with a similar density to carbon fiber and lighter than aluminum [5], the exploration of 2D materials has extraordinarily contributed to the understanding of quantum phenomena associated with the composition and dimensions of materials. In this context, the detection of ferroic properties in low-dimensional materials has led to a considerable increase in the interest from the scientific community to continuously explore their intrinsic responses, thus generating a large number of theoretical and experimental studies related to the 2D materials’ properties. For instance, the two-dimensional NbSX₂ (X = Cl, Br, and I) family exhibits both ferroelectric and antiferromagnetic properties [6], and the TiOX₂ compounds present only the ferroelectric phenomenon. On the other hand, it has been also reported that vanadium oxyhalide (VOX₂, where X = F, Cl and Br) monolayers exhibit improved piezoelectric response, being even higher than that observed in some bulk piezoelectric materials, such as w-AlN or w-GaN [7]. Other studies carried out in NbOCl₂ monolayers, by using the density functional theory (DFT) and many-body quantum perturbation theory, also have shown that adding holes (p-type doping) to NbOCl₂ can induce significant electronic and magnetic phase transitions, turning the semiconductor into a bipolar magnetic semiconductor (or even a half-metal), driven by the emergence of flat bands, which is crucial for potential spintronic applications [8]. Furthermore, as recently reported, the formation of Van der Waals (vdW) heterostructures has become an effective way to obtain tunable physical and chemical properties in two-dimensional (2D) materials, where the VOCl₂/PtTe₂ heterostructure stands out, having a ferromagnetic metal in its ground state with a Curie temperature around 111 K, and the ferromagnetic–antiferromagnetic transition can be induced by applying biaxial compressive strains [9].

From the fundamental point of view, the search for intrinsic magnetism down to the 2D limit has been severely restricted to the ability to reliably exfoliate large air-stable nanosheets. It is known that the chemical exfoliation method offers many advantages, including a high adaptability degree. In particular, transition-metal oxyhalides have been revealed to be ideal candidates for chemical exfoliation due to their large interlayer spacing and the wide variety of interesting magnetic properties [10]. In this way, the vanadium oxychloride dioxide (VOCl₂) monolayer, which is a two-dimensional (2D) material with a thickness of a few nanometers that can be easily obtained by exfoliation from the bulk material, has emerged as a prototype system within the new generation of 2D multiferroic materials. Because of the coexistence of the ferroelectricity and ferromagnetism phenomena at low temperatures, as well as exhibiting a high intrinsic 2D spontaneous electric polarization in the plane (312 pC/m) and stable antiferromagnetism with a Néel’s temperature around 177 K, it is revealed as a promising material for high-density multi-state data storage [11]. The ferroelectricity and magnetism in the VOCl₂ monolayer can be independently tuned via tensile strain along different directions in the crystallographic plane. From the applicability point of view, this material offers an opportunity to design 2D multiferroic-based nanoelectronic devices, having the flexibility to increase (or decrease) the electrical polarization without affecting the monolayer’s magnetic properties, and vice-versa [12]. Studies reported in the literature show that, by using Green’s function method, the antisymmetric magnetoelectric interaction may be responsible for the spin reorientation transition without a change in the magnetic moments’ ordering, so that by changing the sign of the exchange magnetic interaction, a ferromagnetic ordering is obtained without observing a spin reorientation [13]. Calculations based on the density functional theory (DFT) have also shown that the ferroelectricity and the magnetism originate from the same V cation, where the partially filled d-orbitals are perpendicular to the spontaneous polarization that produces the ferroelectric response, indicating a violation of the conventional $d^0$ rule and enabling the combination of polarizations and magnetoelectric coupling [11,14]. The DFT theory and PBE functionals underestimate the energy bandgap, in particular, for systems composed of d- and f-electrons, which affects the electronic and functional interpretation of the material. This effect may be more relevant in 2D monolayers, such as VOCl₂, which indeed is corrected by using the Hubbard (U) potentials to ensure the robustness and reliability of the results and conclusions. It is worth pointing out that a quantitatively accurate prediction of $E_g$ would require higher-level approaches such as hybrid functionals, or many-body perturbation theory, as in fact reported by Heyd et al. [15].

In materials engineering, the doping process by substitutional modifications is recognized as an effective way to modify the electronic structure, magnetic exchange energy, and ferroelectric polarization [16]. The controlled incorporation of magnetic (or rare-earth) impurities into 2D ferroelectric materials can lead to new magnetoelectric coupling regimes, and even induce additional multiferroic phases [3]. In this context, the partial substitution of V⁴⁺ by rare-earth cations (such as Nd, Sm and Eu) in VOCl₂ could promote 3d-4f hybrid states and change the intrinsic ferroelectric character, thus generating local distortions that modify the density of states and the total magnetization of the system. Previous studies on three-dimensional ferroelectric perovskites, such as BaTiO₃ and BiFeO₃, have shown that the introduction of rare-earth (RE) elements improves the thermal stability and the dielectric response [17,18,19], suggesting that a similar effect could be reproduced in 2D monolayers, where electronic correlations and structural anisotropy are more pronounced. Therefore, RE-doped VOCl₂ is presented as an ideal platform for investigating the interaction between localized 4f and 3d transition metal orbitals, contributing to the development of multifunctional materials for application in magnetoelectric sensors, flexible electronics and advanced spintronic devices [3,20]. In this work, we use first-principles study based on the density functional theory (DFT) [21,22], with the Perdew–Burke–Ernzerhof (PBE) approximation for the exchange and correlation energy functional, for a detailed investigation on the structural, electronic, magnetic and multiferroic properties of RE-modified VOCl₂ monolayers (RE = Nd, Sm, and Eu). In particular, the effect of the partial substitution of the V cation on the band structure, local density of states, spontaneous polarization and magnetization is carefully explored. The obtained results could contribute to the establishment of a solid theoretical framework for the understanding of the multiferroic behavior in 2D systems with controlled doping, offering alternative strategies for the design of multifunctional materials for emerging applications in spintronics and next-generation nanoelectronics.

2. Methodology and Computational Procedures

First-principles calculations, based on density functional theory (DFT), implemented in the QUANTUM ESPRESSO software have been used to investigate the structural, electronic, magnetic and multiferroic properties of the studied system [23,24,25]. The electron–nucleus interaction was described using the PAW pseudopotentials [26], and the Generalized Gradient Approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) functional [27] was used for the description of the exchange and correlation energy. In order to successfully describe the 3d- and 4f-electrons of V and the RE elements, respectively, the DFT+U approximation is also taken into account, where the Hubbard potentials (U), obtained from the linear response approach [28,29,30] implemented in the hp.x code [31], are found to be around 1.0 eV (for the V-3d electrons), and around 4.6, 6.4 and 6.9 eV for the 4f-electrons of Nd, Sm and Eu, respectively. In the VOCl₂, the Hubbard parameter (U) for V-3d has a direct influence on both the energy bandgap and magnetic order, so that when U increases, the energy bandgap also increases. In this way, the U value for the V-3d electrons has been selected in such a way that the calculated energy bandgap for the VOCl₂ (AFM-3) is consistent with the literature reports [11]. In the case of the 4f orbitals, the selected U values allow them to be delocalized to form hybridizations with the O-2p and V-3d states. The observed magnetic-ordering trends are driven primarily by the RE f-electron contribution and exchange splitting, whereas moderate variations of U around the linear-response value mainly shift the relative position of localized d/f bands. It is worth pointing out that the U values used for the 3d-electrons of V are within the range of magnitudes reported for the VOCl₂ compound, and the obtained results for the energy bandgap, magnetization, and polarization are consistent with the reported ones the literature [11,12], whereas those used for the 4f-electrons of the rare-earth metals are comparable to those reported in the literature [19,32,33]. The Nd, Sm and Eu cations have been selected here as representative rare-earth dopants due to their systematically varying ionic radii and $4f$ electron configurations, thus enabling the investigation of chemical pressure (that is, size effect) and f-electron effects on multiferroicity. The chosen concentrations correspond to dilute and moderate substitution levels within a periodic supercell, allowing the identification of robust trends without constructing a full-composition phase diagram.

The model used to describe the VOCl₂ monolayer with Pmm2 space-group and antiferromagnetic (AFM-3) behavior [11,12] is represented in Figure 1a by the $2\times 2$ supercell with 2a and 2b basis vectors, where a and b are the basis vectors of the $1\times 1$ VOCl₂ unit-cell represented by the red dashed-line rectangle. For the description of the antiferromagnetism in Figure 1, the V sublattice with spin-up electrons is represented by red spheres, while the V sublattice with spin-down electrons is represented by blue spheres; the O and Cl ions are represented by purple and green spheres, respectively. The VOCl₂ (AFM-3) supercell is represented in Figure 1a projected on the crystalline plane formed by the a and b basis vectors (z-axis). At the same time, Figure 1 (b and c) show the VOCl₂ monolayer projected on the crystalline plane formed by the a and c (y-axis) and b and c (x-axis) basis vectors, respectively. It is important to note that the VOCl₂ monolayers with AFM-3 magnetic configuration present more stability and the lowest energy with respect to other VOCl₂ monolayer magnetic configurations [11]. In order to analyze the properties of the V_1−xX_xOCl₂ modified system (where X = Nd, Sm, Eu), we use supercells whose initial configuration is constructed according to the doping content, represented in Figure 2 for $x=0.166$, $0.083$, and $0.062$ concentrations by $2\times 3$ (a), $3\times 4$ (b), and $4\times 4$ (c) supercells, respectively, where the substitutional ion is represented by the sky-blue sphere. In order to ensure that the obtained results correspond to the VOCl₂ monolayer, the interlayer separation in the z-direction has been considered to be $c=30$ Å for all the studied compositions. The selected rare-earth atoms (X = Nd, Sm and Eu) substitute the V-sites within a periodic supercell. Due to the periodic boundary conditions, the models represent ordered substitution at the specified concentration rather than a random alloy configuration.

For the simulation of the structural, electronic and optical properties of the pure and modified VOCl₂ system, the considered valence band of the material is formed by 13 valence electrons for V ($3s^{2}3p^{6}3d^{3}4s^2$), 6 valence electrons for O ($2s^{2}2p^4$) and 7 valence electrons for Cl ($3s^{2}3p^5$). Similarly, for the modified systems, the valence bands include 24 electrons of Nd ($4d^{10}5s^{2}5p^{6}4f^{3}5d^{1}6s^2$), 26 electrons of Sm ($4d^{10}5s^{2}5p^{6}4f^{5}5d^{1}6s^2$) and 27 electrons of Eu ($4d^{10}5s^{2}5p^{6}4f^{6}5d^{1}6s^2$). Before all the simulations were performed, an optimization process of the unit-cells and atomic positions was carried out using the BFGS algorithm [34,35], with a convergence criterion for total energy of ${10}^{-8}$ Ry, considering as a starting point the unit-cell represented in Figure 2 for the modified systems. The cutoff energy was set at 50 Ry and 580 Ry for the plane waves and the charge density, respectively, which ensures the convergence of the structural and electronic properties [11,12]. The integration in the reciprocal space was performed using $12\times 12\times 1$k-mesh points according to the Monkhorst–Pack scheme [36]. The lattice parameters were extracted by fitting E(V) to the Birch–Murnaghan equation of state [37]. After the optimization process, we performed the self-consistent (scf) solution of the Kohn–Sham equations, in order to obtain the Kohn–Sham wave functions and energy levels. The band structure was constructed using the Kohn–Sham energy levels and the Γ–X–R–Y–Γ–R path in the first Brillouin zone. The total magnetization was calculated as the spin-up and spin-down electronic density of state difference. For the characterization of the ferroelectric behavior, we first evaluate the energy gap of the material, which is an important parameter that allows us to use the modern electric polarization theory to determine the spontaneous electric polarization in the plane of the monolayer, in terms of the Berry phases of the Kohn–Sham wave functions associated with the occupied states of the valence band [38,39]. The thickness of the VOCl₂ monolayer ($3.34$ Å) was used to transform the volumetric spontaneous polarization into two-dimensional spontaneous polarization.

3. Results and Discussion

3.1. Structural Properties

In order to study the structural properties of the VOCl₂ (AFM-3) system, with Pmm2 space-group, we use the $2\times 2$ supercell shown in Figure 1a. After the optimization process of the crystal structure, the lattice parameters for the $1\times 1$ unit-cell were obtained using the Birch–Murnaghan equation of state [37] and the obtained results are summarized in

Table 1. Calculated formation energy and structural parameters for the pure VOCl₂ and modified V_1−xX_xOCl₂ systems.

System	Supercell	${\mathit{E}}_{\mathit{f}}$ (eV)	a (Å)	b (Å)	$\mathit{\alpha }$ (°)	Area (Å)2
VOCl2	1 × 1	−9.227	3.743	3.205	90.000	11.996
V0.834Nd0.166OCl2	2 × 3	−10.992	7.554	11.157	90.177	84.285
V0.834Sm0.166OCl2	2 × 3	−12.392	7.515	11.098	90.069	83.414
V0.834Eu0.166OCl2	2 × 3	−11.695	7.505	11.083	90.179	83.190
V0.917Nd0.083OCl2	3 × 4	−13.488	11.606	13.801	89.999	160.194
V0.917Sm0.083OCl2	3 × 4	−14.993	11.548	13.762	89.999	158.937
V0.917Eu0.083OCl2	3 × 4	−14.344	11.534	13.756	89.999	158.664
V0.938Nd0.062OCl2	4 × 4	−17.149	15.340	13.713	90.000	210.371
V0.938Sm0.062OCl2	4 × 4	−18.660	15.301	13.683	89.999	209.388
V0.938Eu0.062OCl2	4 × 4	−18.008	15.278	13.668	90.000	208.840

, which appear to be in agreement with the theoretical and experimental results reported in the literature [11,12,40]. For the modified V_1−xX_xOCl₂ system (with X = Nd, Sm, and Eu), the initial crystalline structure is represented in Figure 2, where a V(↑) atom has been substituted by a rare-earth ion (represented by sky-blue sphere). In this case, after the crystal structure optimization process, considering the energy minimization, noticeable changes are observed in the obtained lattice parameters, being also dependent on the doping concentration (see Table 1). For instance, for the V_0.834X_0.166OCl₂ composition it can be seen that a increases by $0.908$% ($0.034$ Å), $0.387$% ($0.015$ Å) and $0.254$% ($0.010$ Å), for the Nd, Sm, and Eu RE elements, respectively, whereas b increases by $16.037$% ($0.514$ Å), $15.424$% ($0.494$ Å) and $15.268$% ($0.489$ Å), respectively for Nd, Sm, and Eu. Furthermore, the unit-cell angle ($\alpha$) increases by $0.158$%, $0.196$%, and $0.197$%, respectively, for the Nd, Sm, and Eu RE elements. On the other hand, for the $x=0.083$ concentration, it was observed that the lattice parameters a and b also increase, with variation for a around $3.357$%, $2.841$% and $2.716$%, and that for b around $7.652$%, $7.348$% and $7.301$%, for Nd, Sm, and Eu, respectively. However, no change was observed in the crystal angle for this composition. For the highest doping concentration ($x=0.062$), both lattice parameters (a and b) increase, with the variation in a being around $2.458$%, $2.197$% and $2.044$% and that in b being around $6.966$%, $6.732$%, and $6.615$%, for the Nd, Sm, and Eu elements, respectively. For this composition, the observed changes in $\alpha$ are also negligible. The observed behavior in the lattice parameters with increasing both the doping concentration and RE element correlates with the ionic radii of the substitutional cations as well as the chemical bonds between nearest neighbors.

The thermodynamic stability of the optimized crystal structures of the modified V_1−xX_xOCl₂ systems is given by the formation energies defined by

$$E_f=E\left({\mathrm{V}}_{1-x}{\mathrm{X}}_x{\mathrm{OCl}}_2\right)-E\left({\mathrm{VOCl}}_2\right)-u_{\mathrm{X}}+u_{\mathrm{V}}$$

where E(V_1−xX_xOCl₂) is the total energy of the modified system, E(VOCl₂) is the total energy of the VOCl₂ supercell equivalent to the V_1−xX_xOCl₂ system, and $u_{\mathrm{X}}$ and $u_{\mathrm{V}}$ parameters are the chemical potentials of the doping cation (X) and the V atom, respectively [41,42,43]. The calculated values for $E_f$ are reported in Table 1, where a formation energy value of around $-9.227$ eV for the pure VOCl₂ system has been obtained, which guarantees the stability of the crystalline structure and the charge redistribution. For the modified V_1−xX_xOCl₂ systems, the formation energy values are lower than those obtained for the pure VOCl₂ system and decrease as the doping concentration decreases. That is, the formation energy for the V_0.938X_0.062OCl₂ (with X = Nd, Sm, and Eu) composition is lower than that for the V_0.917X_0.083OCl₂ system, which in turn is lower than that for the V_0.834X_0.166OCl₂ system. The V_0.938X_0.062OCl₂ composition described using $4\times 4$ supercells has the highest thermodynamic stability. These results reveal that at lower concentrations of the substitutional RE cations, the probability for the formation of the crystal structure is higher, in particular for the Sm element.

3.2. Electronic Properties of the VOCl₂ (AFM-3) Monolayer

The VOCl₂ monolayer in its antiferromagnetic ground state (AFM-3), as represented in Figure 2, behaves as an indirect semiconductor, according to previous studies on 2D vanadium oxyhalides, that exhibit magnetic order and ferroelectricity outside the $d^0$ rule [11,12,14,44]. Figure 3 shows the band structure (a) and density of states (b) for the VOCl₂ system. As observed, the obtained band structure (see Figure 3a), by using the Kohn–Sham energy levels and represented using the high-symmetry Γ-X-R-Y-Γ-R trajectory in the first Brillouin zone, confirms the semiconductor behavior with calculated indirect bandgap ($E_g$) around $1.62$ eV, where the highest occupied states have energy level at the R point and the unoccupied state with minimal energy at the Γ point. In addition, the local density of states (LDOS) represented in Figure 3b reveals that the contribution of the 3d-states of V is the most important for the unoccupied states with the lowest energy in the conduction band. Similarly, below the Fermi energy level, we observe that the contribution of the 3d-electrons of V is very important for the formation of valence band states with energies ($E-E_F$) around $-1.0$ eV, where the 3d-states exhibit mainly hybridizations with the 2p-electrons of Cl. The contribution of the 2p-electrons of O is very important for the stability of the VOCl₂ crystal structure, which can be confirmed for $E-E_F$ energies below $-2.0$ eV. On the other hand, results of Figure 3b clearly indicate that for all the $E-E_F$ energy values, the LDOS for spin-up electrons is exactly the same to that for spin-down electrons. This result justifies the zero total magnetization, which is consistent with the two V sublattices with up and down electrons considered in Figure 1, as a model to describe the AFM-3 behavior of the VOCl₂ system [11,12].

The ferroelectric behavior of the VOCl₂ system is analyzed by considering the spontaneous polarization (P_s) in the plane as the order parameter, which is calculated within the King-Smith and Venderbilt (KSV) modern theory of the electric polarization. This theory expresses the spontaneous polarization in terms of the Berry phases of the occupied Kohn–Sham states within the DFT formalism. The calculated value for the 2D spontaneous polarization was found to be around 309 pC/m, in agreement with the reported results in the literature [11,12,13,14]. From the microscopic point of view, P_s arises from the off-center displacement of the V cation in the VO₂Cl₄ octahedron and from spatial distortions produced by the pseudo-Jahn–Teller mechanism, which breaks the inversion center for a polar ground state [45]. That is, such a displacement creates an electric dipole moment within the unit-cell, which aligns with neighboring dipoles to produce a net spontaneous polarization that can be reversed by applying an external electric field. The presence of unoccupied d-orbitals in the V⁴⁺ ion allows for this structural distortion (known as the $d^0$ rule mechanism), which is a fundamental condition for the material to be ferroelectric [17]. In this context, the observed polarization can be associated with a lattice instability and metal–ligand hybridization and/or asymmetric bonding, which can also occur in non-$d^0$ systems, particularly in low-dimensional lattices where competing instabilities (magnetic order, orbital effects, and anisotropic bonding) can coexist [16].

3.3. Electronic Properties for the V_1−xX_xOCl₂ (with X = Nd, Sm, Eu)

Considering the initial models used for the V_1−xX_xOCl₂ system (as shown in Figure 2), and the structural parameters reported in Table 1, we performed the self-consistent calculation of the Kohn–Sham energy levels and wave functions, whose results were used for the construction of the band structures and local density of states represented in Figure 4. Figure 4a describes the fat-bands for the V_0.834Nd_0.166OCl₂ composition, including the spin-up projected density-of-states (PDOS) contribution of the Nd(up) orbitals, corresponding to the occupied states with $E-E_F$ energies around $-0.8$ eV for the 3d-states of V. The conduction band states around $0.2$ eV correspond to the hybridization of the 3d-states of V and the 2p unoccupied states of Cl. The 4f occupied states’ contribution for Nd to the valence band is observed in orange color for the $E-E_F$ energy around $-1.6$ eV, and around $2.0$ eV in the conduction band. Figure 4b shows the band structure for spin-down electrons of the V_0.834Nd_0.166OCl₂ system, where the contribution of the Nd-4f down states has been included, which is negligible compared to the other atoms contributions. In this band structure, the occupied states for the $E-E_F$ energy around $-0.2$ and $-0.8$ eV correspond to the 3d spin-down electrons of V, and the conduction band states for energy around $0.3$ eV correspond to the 3d unoccupied states of V. The semiconductor behavior of the V_0.834Nd_0.166OCl₂ system is guaranteed with the calculated indirect bandgap, reaching an energy value around $0.38$ eV, with occupied states showing higher energy in the valence band near to the R point and unoccupied states showing lower energy in the conduction band near the Γ point in the first Brillouin zone. The local density of states (LDOS) for each atom and the density of states (DOS) represented in Figure 4c confirm the results obtained by the band structures of the V_0.834Nd_0.166OCl₂ composition, shown in Figure 4a,b, and also confirm the calculated total magnetization (M_T) value around $1.28$ ${\mu }_B$/cell, which is explained by the difference between the spin-up and spin-down electron density of states of the modified system. This total magnetization and the calculated absolute magnetization of $7.39$ ${\mu }_B$/cell evidence the imbalance in the number of spin-up electrons and spin-down electrons, modifying the antiferromagnetic behavior, mainly due to the 4f electrons of Nd presence, transforming the AFM-3 behavior of VOCl₂ to Ferrimagnetic behavior in the V_0.834Nd_0.166OCl₂ system. The V, O and Cl atoms maintain an important contribution to the valence band formation of $E-E_F$ energies between $-1.8$ and $-6.0$ eV, with the contribution of the spin-up 4f-states of Nd being determinant for $E-E_F$ energies between $-1.8$ and $-2.8$ eV.

Figure 4d depicts the fat-bands for the V_0.834Sm_0.166OCl₂ composition, including the PDOS of the 4f-states of spin-up Sm orbitals, corresponding to the occupied states with $E-E_F$ energies around $-0.8$ eV for the 3d-states of V. The conduction band states observed around $0.2$ eV correspond to the hybridization of the 3d-states of V and the 2p unoccupied states of Cl, similar to the V_0.834Nd_0.166OCl₂ composition. The 4f unoccupied spin-up states of Sm are important for the conduction band formation, as shown by the energy levels in yellow colors for the $E-E_F$ energies around $0.7$ and $1.0$ eV. Figure 4e shows the band structure for spin-down electrons of the V_0.834Sm_0.166OCl₂ system, where the PDOS of 4f down-states is negligible, when compared to the contributions of the other atoms. In this band structure the occupied states for the $E-E_F$ energy around $-0.2$ and $-0.8$ eV correspond to the 3d spin-down electrons of V, and the conduction band states for energy around $0.7$ eV correspond to the 3d unoccupied states of V. The calculated indirect bandgap between the R and Γ point in the first Brillouin zone of $0.37$ eV reveals the semiconductor behavior of the V_0.834Sm_0.166OCl₂ composition. The LDOS per atom and the DOS shown in Figure 4f confirm the band structure results, with a total magnetization ($M_T$) value of $3.31$ ${\mu }_B$/cell, which, together with the obtained values for the absolute magnetization around $9.38$ ${\mu }_B$/cell, reveals the ferrimagnetic behavior of the V_0.834Sm_0.166OCl₂ composition, mainly due to the presence of the 4f-electrons of Sm. For energy values between $-2.0$ and $-6.0$ eV the V, O and Cl maintain their important contribution for the formation of the valence band.

The spin-up band structure for the V_0.834Eu_0.166OCl₂ composition is shown in Figure 4g, including the PDOS of the 4f spin-up states of Eu (shown by yellow color), which, in contrast to the 4f spin-up electrons of Nd and Sm, constitutes the valence band close to the Fermi level, forming hybridizations with the 3d spin-down electrons of V. The occupied states with $E-E_F$ energies around $-1.0$ eV correspond to the 4f-electrons of Eu. In the conduction band, for $E-E_F$ energies around $0.4$ eV, the unoccupied 3d and 2p states of V and Cl, respectively, hybridize. Figure 4h shows the spin-down band structure for the V_0.834Eu_0.166OCl₂ composition, where the PDOS of 4f down-states is negligible. On the other hand, for $E-E_F$ energy values around $0.2$ and $0.4$ eV, the 3d spin-down unoccupied states of V are important. In addition, the calculated indirect bandgap of $0.10$ eV reveals the semiconductor behavior of the V_0.834Eu_0.166OCl₂ system. The LDOS per atom and the DOS shown in Figure 4i also confirm the results for the band structures, and the total magnetization (M_T) around $5.65$ ${\mu }_B$/cell, with the absolute magnetization of $10.74$ ${\mu }_B$/cell, reveals the ferrimagnetic behavior of the V_0.834Eu_0.166OCl₂ composition, due to the influence of the 4f-electrons of Eu. For energy values between $-1.5$ eV and $-5.5$ eV the V, O and Cl also maintain their important contribution for the formation of the valence band.

The initial configuration to study the V_1−xX_xOCl₂ (with X = Nd, Sm and Eu) system for $x=0.083$ concentration is represented in Figure 2b, and the results of the band structure, LDOS and DOS, obtained using the Kohn–Sham energies and wave functions, is represented in Figure 5. The band structure for the V_0.917Nd_0.083OCl₂ composition around the Fermi energy, described by $E-E_F=0$ eV, is shown in Figure 5a, including the PDOS of the Nd-4f spin-up states. The occupied states with $E-E_F$ energies between $-0.2$ and $-0.4$ eV correspond to the hybridization of the 3d-states of V, while those with energies between $-0.5$ and $-0.8$ eV correspond to the 3p-states of Cl. The 4f spin-up states of Nd, represented by orange color, also have an important contribution. For $E-E_F$ energies below $-0.9$ eV up to $-5.8$ eV, the Nd-4f, V-3d, O-2p and Cl-3p spin-up states exhibit chemical bonds, as described in violet color. The conduction band for energies above $1.5$ eV corresponds to the V-3d spin-up states. Figure 5b shows the spin-down band structure for the V_0.917Nd_0.083OCl₂ composition, with Nd-4f spin-down states, which is negligible compared to the other atoms’ contributions. The occupied states for energies around $-0.2$ and $-0.4$ eV correspond to the V-3d down-spin electrons, whereas the states above $1.5$ eV are associated with V-3d unoccupied states. The direct bandgap of $1.59$ eV reveals the semiconductor behavior of the V_0.917Nd_0.083OCl₂ system, calculated at the Γ point in the first Brillouin zone. The LDOS per atom and the DOS shown in Figure 5c confirm the electronic properties observed in Figure 5a,b for this composition. The obtained values for the total magnetization and the absolute magnetization around $1.94$ ${\mu }_B$/cell and $16.02$ ${\mu }_B$/cell, respectively, reveal the structural change from an antiferromagnetic to a ferrimagnetic behavior for the V_0.917Nd_0.083OCl₂ system.

Figure 5d represents the spin-up band structure for the V_0.917Sm_0.083OCl₂ composition, with 4f spin-up PDOS of Sm. The occupied states for $E-E_F$ energies between $-0.2$ and $-0.4$ eV correspond to the V-3d and Cl-3p hybridized states. On the other hand, the Nd-4f spin-up states are negligible between −0.2 and −4.5 eV, but have an important contribution to the valence bands between −4.5 and −5.5 eV. For energies between −1.0 and −6.0 eV, the spin-up states of V, O and Cl constitute the valence band. Around 1.9 eV, the Nd-4f spin-up states (represented by yellow lines) also have important contributions to the conduction band. Figure 5e shows the spin-down band structure of the V_0.917Sm_0.083OCl₂ system, where the Sm-4f spin-down PDOS is negligible. As can be seen, the occupied states for $E-E_F$ energies between −0.2 and −0.4 eV correspond to the 3d spin-down electrons of V, while the observed ones in the conduction band above 1.5 eV correspond to unoccupied 3d states of V. The calculated direct bandgap at the Γ point was found to be around 1.57 eV and reveals the semiconductor behavior of the V_0.917Sm_0.083OCl₂ composition. The LDOS per atom and DOS shown in Figure 5f confirm the results of the band structures, where the observed value for the total magnetization (M_T) around $3.97$ ${\mu }_B$/cell, together with the absolute magnetization obtained around $18.05$ ${\mu }_B$/cell, indicates the ferrimagnetic behavior of the V_0.917Sm_0.083OCl₂ composition, promoted by the 4f electrons of Sm.

The spin-up band structure with Eu-4f spin-up PDOS for the V_0.917Eu_0.083OCl₂ composition is represented in Figure 5g, where the occupied states located between $-0.2$ and $-0.4$ eV correspond to the 3d-states of V, with the 4f spin-up states of Eu being negligible. As observed, yellow lines around $0.7$ eV in the conduction band represent the Eu-4f unoccupied spin-up states. For energy values above $1.4$ eV, the unoccupied states associated with V, O and Cl are important to form the conduction band. The spin-down band structure shown in Figure 5h reveals that the Eu-4f spin-down states have a negligible contribution for the formation of the band structure near the Fermi energy level. However, it is noticed that the V-3d spin-down occupied states have important contributions for the formation of the valence band between $-0.2$ and $-0.4$ eV, as well as for the formation of the conduction band above $1.4$ eV. The indirect bandgap of $0.83$ eV is calculated between the R and Γ points in the first Brillouin zone, revealing the semiconductor behavior of the V_0.917Eu_0.083OCl₂ composition. The LDOS per atom and the DOS shown in Figure 5i confirm the results of the band structures. At the same time, the total magnetization value of $4.96$ ${\mu }_B$/cell, together with the absolute magnetization of $19.06$ ${\mu }_B$/cell, reveals the ferrimagnetic behavior of the V_0.917Eu_0.083OCl₂ system, promoted by the presence of the Eu-4f electrons.

In order to study the properties of the V_1−xX_xOCl₂ system, for $x=0.062$, represented by the initial model of Figure 2c, we present in Figure 6 the results of the band structures, LDOS and DOS, where the spin-up band structure for the V_0.938Nd_0.062OCl₂ system with Nd-4f spin-up PDOS is shown in Figure 6a. The valence band below the Fermi level ($E-E_F=0$ eV) up to $-0.4$ eV is formed by V and Cl spin-up states. For energies around −0.7 eV the Nd-4f spin-up states (represented by orange lines) have remarkable contributions. It can be observed that from $-0.9$ up to $-2.0$ eV there is a formation of Cl-dominant states, followed by the contributions of O, Nd (violet lines) and V. In the conduction band, above $1.5$ eV, the 3d-states of V are predominant. Figure 6e shows the spin-down band structure for the V_0.938Nd_0.062OCl₂ system, where the contribution of the Nd-4f spin-down states is almost zero. The energy bandgap of $1.56$ eV, calculated by the energy difference between the highest occupied state (around the X point) and the lowest unoccupied state (Γ point), reveals the semiconductor behavior of the V_0.938Nd_0.062OCl₂ system. The LDOS per atom and the DOS shown in Figure 6c confirm the results shown in Figure 6a,b. The total magnetization value of $2.01$ ${\mu }_B$/cell and the obtained absolute magnetization around $20.93$ ${\mu }_B$/cell reveal the change from antiferromagnetic to ferrimagnetic behavior of the modified system, induced by the presence of the 4f-electrons of Nd. The V, O, and Cl have important contributions to the valence band between $-0.7$ and $-5.5$ eV, with a predominance of the Cl states.

Figure 6d represents the spin-up band structure near the Fermi energy for the V_0.938Sm_0.062OCl₂ composition, where the valence bands with energies between $-0.1$ and $-0.4$ eV correspond mainly to the 3d-states of V, while the contribution of the spin-up states of Nd is negligible. In contrast, for the unoccupied states of the conduction band, around $1.4$ eV, the 4f states of Sm have a considerable contribution, although lower than the contribution of V. Figure 6e shows the spin-down band structure of the V_0.938Sm_0.062OCl₂ system, where the contribution of the Sm-4f spin-down states is negligible. The semiconductor behavior of the V_0.938Sm_0.062OCl₂ system is revealed by the presence of an indirect bandgap around $1.53$ eV, with the highest energy occupied states in the valence band (near to the X point) and the lowest energy unoccupied states in the conduction band at the Γ point of the first Brillouin zone. The LDOS per atom and the DOS shown in Figure 6f reinforce the results shown by the band structures. The total magnetization of $4.04$ ${\mu }_B$/cell and the absolute magnetization value of $22.99$ ${\mu }_B$/cell reveal the ferrimagnetic behavior of the V_0.938Sm_0.062OCl₂ system. For energies between $-5.2$ and $-6.0$ eV (in the valence band) and between $1.8$ and $2.2$ eV (in the conduction band), the V, O, and Cl atoms also have significant contribution, with the Cl states being more important, followed by the O, V and Sm-4f states, respectively.

The spin-up band structure of the V_0.938Eu_0.062OCl₂ composition, shown in Figure 6g with Eu-4f spin-up PDOS, reveals that the valence band for energies between $-0.1$ and $-0.4$ eV is formed mainly by V and Cl states, with predominant V states. Yellow lines around $0.6$ eV in the conduction band represent the Eu-4f unoccupied spin-up states. Above $1.3$ eV, the unoccupied states are associated with the V, O and Cl states. Figure 6h shows the band structure for the down-spin electrons of the V_0.938Eu_0.062OCl₂ system, where the contribution of the Eu-4f spin-down states are negligible. The semiconductor behavior for this composition is revealed by the calculated indirect bandgap around $0.78$ eV, with the highest energy occupied states in the valence band (near the X point) and the lowest energy unoccupied states in the conduction band, near the Y point in the first Brillouin zone. The LDOS per atom and DOS shown in Figure 6i confirm the results shown by the band structures. The total magnetization value of $5.05$ ${\mu }_B$/cell and the absolute magnetization found around $23.99$ ${\mu }_B$/cell reveal the ferrimagnetic behavior of the V_0.938Eu_0.062OCl₂ system. The Eu-4f spin-up states have important contributions in the valence band for energies between $-5.7$ and $-6.0$ eV. The

Table 2. Electronic and magnetic properties of the pure VOCl₂ and V_1−xX_xOCl₂ (with X = Nd, Sm and Eu) monolayers.

System	${\mathit{E}}_{\mathit{g}}$ (eV)	${\mathit{P}}_{\mathit{s}}$ (pC/m)	${\widehat{\mathit{\mu }}}_{{\mathit{P}}_{\mathit{s}}}$	${\mathit{M}}_{\mathit{T}}$ (${\mathit{\mu }}_{\mathit{B}}$/Cell)	${\mathit{M}}_{\mathbf{abs}}$ (${\mathit{\mu }}_{\mathit{B}}$/Cell)
VOCl2	1.62	308.77	(−1.00,0.00)	0.00	5.06
V0.834Nd0.166OCl2	0.38	296.36	(−0.98,−0.17)	1.28	7.39
V0.834Sm0.166OCl2	0.37	333.91	(0.97,0.24)	3.31	9.38
V0.834Eu0.166OCl2	0.10	341.65	(0.98,−0.19)	5.65	10.74
V0.917Nd0.083OCl2	1.59	275.72	(0.99,0.00)	1.94	16.02
V0.917Sm0.083OCl2	1.57	290.89	(0.98,0.19)	3.97	18.05
V0.917Eu0.083OCl2	0.83	269.80	(0.99,0.09)	4.96	19.06
V0.938Nd0.062OCl2	1.56	273.41	(0.99,−0.12)	2.01	20.93
V0.938Sm0.062OCl2	1.53	268.02	(0.98,0.17)	4.04	22.99
V0.938Eu0.062OCl2	0.78	243.41	(−0.98,−0.19)	5.05	23.99

summarizes all total magnetization $M_T$ and absolute magnetization $M_{abs}$, calculated using the Projected Density of States.

Table 3. Magnetic moment of the rare-earth (RE) elements and the associated vanadium (V) in the V_1−xX_xOCl₂ monolayers.

M(${\mathit{\mu }}_{\mathit{B}}$/Cell)	V1−xNdxOCl2			V1−xSmxOCl2			V1−xEuxOCl2
	0.166	0.083	0.062	0.166	0.083	0.062	0.166	0.083	0.062
RE	2.817	2.783	2.770	4.839	4.801	4.771	6.462	5.807	5.762
V	−1.145	−0.702	−0.655	−1.133	−0.690	−1.133	−0.794	−0.703	−0.645
RE + V	1.672	2.081	2.115	3.709	4.111	3.638	5.668	5.104	5.117

summarizes the magnetic moment contribution of the rare-earth (RE) cations and the magnetic moment (M) associated with vanadium (V); the obtained results reinforce the idea that the magnetic behavior of the modified systems is mainly due to the presence of the doping elements. That is, while the RE dopant carries the dominant positive moment, the V sublattice develops an antiparallel (negative) net moment, which is consistent with ferrimagnetism.

3.4. Ferroelectric Properties

For the analysis of the ferroelectric behavior in the studied compositions, we consider that, according to the above discussed results, the band structure and the density of states revealed semiconductor behavior, which allows us to use the modern theory of electric polarization [38,39]. This approach expresses the spontaneous polarization using the Berry phases of occupied Kohn–Sham wave functions and an adiabatic pathway connecting a centrosymmetric and polar state, where the 2D polarization is calculated as the product of the 3D polarization and the monolayer thickness. The calculated spontaneous polarization for VOCl₂ (AFM-3) was found to be around 308.77 pC/m and can be associated with the Berry phases of the wave functions and the spatial symmetry break by the V ions’ displacement in $-0.229\widehat{x}$ Å from their centrosymmetric position. In the modified systems, the observed changes in the spontaneous polarization are mainly related to the symmetry changes promoted by the rare-earth ion displacement, with respect to the V host ion’s position. For instance, in the V_0.834Nd_0.166OCl₂ composition the spontaneous polarization decreases to 296.36 pC/m, due to the Nd displacement of $0.39\widehat{x}+0.16\widehat{y}$ Å, the near-oxygen average displacement of $0.07\widehat{x}+0.26\widehat{y}$ Å, and the near-Cl ion displacement at $0.12\widehat{x}+0.28\widehat{y}$ Å. On the other hand, the spontaneous polarization value of 333.91 pC/m for the V_0.834Sm_0.166OCl₂ system is generated by the Sm displacement of $0.36\widehat{x}+0.15\widehat{y}$ Å and two near-V neighbors’ displacement at $0.12\widehat{x}+0.17\widehat{y}$ Å, in addition to the O and Cl near-neighbor displacements. At the same time, the polarization value of 341.65 pC/m obtained for the V_0.834Eu_0.166OCl₂ system is generated by the Eu displacement of $0.36\widehat{x}+0.15\widehat{y}$ Å, a V near-neighbor displacement of $0.22\widehat{x}+0.19\widehat{y}$ Å, a near-oxygen displacement of $0.27\widehat{x}+0.34\widehat{y}$ Å and near-Cl displacement of $0.17\widehat{x}+0.23\widehat{y}$ Å. Also, it was found that the observed changes in the spontaneous polarization for all the other V_1−xSm_xOCl₂ studied systems are related to the displacement of the dopant ions and their nearest neighbors. The results are summarized in Table 2, and can be verified using the effective Born charges. For all the cases, the polarization direction is located in the a-b plane of the crystal structure and is reported in pC/m (2D units). The calculated spontaneous polarization in the z-direction (perpendicular to the a-b plane) is negligible for all the cases. As can be seen, for the V_0.834Sm_0.166OCl₂ and V_0.834Eu_0.166OCl₂ systems, the 2D spontaneous polarization increases, when compared with the pure VOCl₂ system, by 8.14% and 10.65% respectively, whereas for all the other cases lower 2D spontaneous polarization values have been found. Regarding the dependence of spontaneous electrical polarization $\left(P_s\right)$ on the concentration of rare-earth ions, it can be observed that, for the V_1−xX_xOCl₂ studied systems, the calculated spontaneous polarization is lower for $x=0.062$ and shows an intermediate value for $x=0.083$ in all the cases. On the other hand, $P_s$ reaches the highest value for $x=0.166$; the $\Delta P_s=P_s\left(0.166\right)-P_s\left(0.062\right)$ values’ difference is around 22.95, 65.89 and 98.24 pC/m for X = Nd, Sm, and Eu, respectively. These results indicate a higher and smaller change in $\Delta P_s$ in the presence of Eu and Nd, respectively. It is worthwhile to point out that the direction of the polarization in the V_1−xX_xOCl₂ compound changes along the a-b plane, from the –x direction for the pure VOCl₂ system to approximately the +x direction for the doped compositions, except in the case of the V_0.938Eu_0.062OCl₂ system, where the deviation angle is small. According to the data reported in Table 2, the more noticeable change in the direction of the polarization occurs for the V_0.917Sm_0.166OCl₂ composition.

The observed changes in the spontaneous polarization can be explained by the electron charge density distortions caused by the presence of the rare-earth ions, as well as pseudo-Jahn–Teller distortions. Figure 7 shows the charge density on the a-b plane for the V_1−xNd_xOCl₂ system for different concentrations (x), where the highest displacement ($0.16\widehat{x}+0.42\widehat{y}$ Å) of the Nd ions is found for $x=0.166$, as observed in Figure 7a. It is important to emphasize that an identical configuration is observed in the charge density distribution for the V_1−xSm_xOCl₂ and V_1−xEu_xOCl₂ systems, except for a small increase in the intensity of the charge density at the positions of the Sm and Eu dopants.

The observed trends are consistent with previous reports on two-dimensional VOX₂ (X = Cl, Br, I) multiferroics, where ferroelectricity and magnetism originate from the same transition-metal cation [11,14]. Similar to rare-earth-doped bulk multiferroics such as BiFeO₃, dopant-induced lattice distortions modulate the electric polarization, whereas changes in the $p-d$ hybridization govern bandgap evolution [18]. Within the modern Berry-phase framework, the spontaneous polarization is governed by both ionic displacements and electronic charge redistribution [38]. However, rare-earth substitution modifies local V-centered coordination, changing the bond lengths and angles and, in some cases, inducing Jahn–Teller-like distortions [39]. In this context, such V-centered distortions (including possible Jahn–Teller-like local symmetry lowering) can modify V-O/Cl bond lengths, thereby changing both the ionic contribution to the polarization and the degree of $p-d$ hybridization. These effects directly influence the magnitude of the polar distortion and the associated charge transfer, explaining the observed change of the polarization with the dopant type (X = Nd, Sm and Eu) and concentration (x = 0.166, 0.083 and 0.062) [38,39].

From the experimental point of view, rare-earth substitution is a standard route in solid-state chemistry for tuning ferroic responses (as widely demonstrated in bulk multiferroics); the predicted energetic favorability suggests plausible thermodynamic accessibility [18]. On the other hand, layered/van der Waals-type 2D materials can often be exfoliated from bulk precursors, or even grown from advanced thin-film growth routes, which include the Molecular Beam Epitaxy (MBE) and Chemical Vapor Deposition (CVD) techniques, depending on chemistry. Furthermore, according to the literature reports, a pristine VOCl₂ monolayer has been proposed to be synthesized as mechanically strippable from a layered bulk parent, suggesting that exfoliation-based routes may be viable [11]. While direct experimental reports on RE-doped VOCl₂ monolayers are (to the best of our knowledge) still lacking, chemical exfoliation has been demonstrated for the related layered oxychloride VOCl₂ down to the monolayer limit [10], thus supporting the practical accessibility of vanadium oxyhalide monolayers. On the other hand, rare-earth (RE) substitution could, in principle, be pursued during bulk crystal growth (followed by exfoliation), or via post-synthetic routes (e.g., ion-exchange/intercalation-assisted strategies), depending on the phase stability. In the present study, the VOCl₂ system belongs to the VOX₂ family where both magnetism and ferroelectricity can originate from the same transition-metal cation, enabling large polarization with strong magnetoelectric coupling, as indeed recently reported for VOCl₂-related monolayers [11,14]. However, compared to other 2D multiferroics (e.g., NbOI₂-based 2D materials) [14,46,47], it can be seen that the studied VOCl₂-based system is distinctive in both magnetism and ferroelectricity responses, originating from the same V cation, offering enhanced tunability and potentially stronger magnetoelectric coupling.

4. Conclusions

The inclusion of rare-earth ions (Nd, Sm, Eu) as V substitutional modifiers into the VOCl₂ (AFM-3) monolayer promotes considerable and correlated changes in the structural, electronic and multiferroic properties. It was found that the rare-earth doping induces local distortions in the crystal structure, thus affecting the space-group and the lattice parameters, which modulates the energy bandgap, the intrinsic ferroelectricity and the magnetic behavior of the system. As a consequence, a structural transformation from the antiferromagnetic behavior of the VOCl₂ monolayer to the ferrimagnetic behavior in the V_1−xX_xOCl₂ monolayers is observed, preserving the multiferroicity of the system, which is an important property for applications in spintronics. On the other hand, it was also observed that the area of the V_1−xX_xOCl₂ unit-cell changes directly with the variation in the ionic radius of X (Nd > Sm > Eu) as well as with the doping concentration (x). The VOCl₂ monolayer crystal structure changes from a rectangular crystal structure, with Pmm2 space-group, to an oblique-type crystal structure for the V_1−xX_xOCl₂ monolayers. These observed changes are more noticeable for $x=0.166$, while in the other compositions the rectangular crystalline structure is preserved. The strong distortion observed in the VO₂Cl₄ octahedra, but negligible in the XO₂Cl₄ octahedra, is attributed to the (pseudo-) Jahn–Teller effect. The substitution of V with Nd and Sm reduces the bandgap of the VOCl₂ system, with the highest variation occurring at $x=0.166$. However, at $x=0.062$, the values approach those of the pure VOCl₂ system. The ferroelectricity (characterized by the spontaneous polarization, P_s) of VOCl₂ increases for $x=0.166$, while it decreases for all the other concentrations. Furthermore, the substitution of V with Nd/Sm/Eu in VOCl₂ changes the spin-up/down DOS symmetry of the pure system, giving the pure VOCl₂ monolayer a total magnetization M_T = 0. Conversely, all the modified compositions show M_T ≠ 0, due to the presence of the rare-earth $f-$electrons, which have preferential (up) alignment. M_T ≠ 0, along with a high absolute magnetization value observed in the modified systems, indicates a transition from the initial AFM-3 state to a ferrimagnetic state of the modified VOCl₂.