Influence of the Chemical Pressure on the Magnetic Properties of the Mixed Anion Cuprates Cu₂OX₂ (X = Cl, Br, I)

Abstract

In this study, we theoretically investigate the structural, electronic and magnetic properties of the Cu₂OX₂ (X = Cl, Br, I) compounds. Previous studies reported potential spin-driven ferroelectricity in Cu₂OCl₂, originating from a non-collinear magnetic phase existing below $T_N$∼70 K. However, the nature of this low-temperature magnetic phase is still under debate. Here, we focus on the calculation of J exchange couplings and enhance knowledge in the field by (i) characterizing the low-temperature magnetic order for Cu₂OCl₂ and (ii) evaluating the impact of the chemical pressure on the magnetic interactions, which leads us to consider the two new phases Cu₂OBr₂ and Cu₂OI₂. Our ab initio simulations notably demonstrate the coexistence of strong antiferromagnetic and ferromagnetic interactions, leading to spin frustration. The $T_N$ Néel temperatures were estimated on the basis of a quasi-1D AFM model using the $\mathit{ab}\mathit{initio}$ J couplings. It nicely reproduces the $T_N$ value for Cu₂OCl₂ and allows us to predict an increase of $T_N$ under chemical pressure, with $T_N$ = 120 K for the dynamically stable phase Cu₂OBr₂. This investigation suggests that chemical pressure is an effective key factor to open the door of room-temperature multiferroicity.

1. Introduction

The design of multiferroics (MF) in which magnetic and ferroelectric orders are coupled is one of the hottest current topics in materials science [1]. These compounds appear as a fantastic playground to deeply investigate the origin and the interactions of the related ferroic properties and to design materials with improved or novel properties. Such multi-functional materials are also extremely appealing for the development of spintronic devices. For example, the control of a magnetic order by an electric field is targeted for several applications ranging from magnetic sensors to memory technologies. At the present time, very few MF compounds exhibit such properties at room temperature (RT), which dramatically limits potential industrial interests. In that context, we embarked on the quest for discovering high-temperature range MF materials, which ideally also show strong electric polarization.

Among the prospective strategies to tune MF stability domain(s) is the application of an hydrostatic (physical) pressure [2]. For instance, in 2013 [3], we predicted theoretically that under high pressure values of about 20 GPa, CuO would become MF at RT. This prediction has been confirmed through dielectric constants measurements up to 7 GPa in 2021 [4] and more recently by neutron diffraction up to 18.5 GPa [5]. An alternative route is to apply a chemical pressure through chemical substitutions, leading to create internal constraints. For instance, the two multiferroic compounds CuCl₂ [6] and CuBr₂ [7] are characterized by Néel temperatures of 23.9 K and 73.5 K, respectively. Such an increase of $T_N$ is the direct consequence of the chemical substitution of Cl by Br.

Herein, we propose to study the mixed anion system Cu₂OX₂ (with X = Cl, Br, I). It derives from the CuO compound in which half of the ${\mathrm{O}}^{2-}$ ions have been replaced by two ${\mathrm{X}}^{-}$ ions. Magnetic measurements performed on Cu₂OCl₂ evidenced an antiferromagnetic order with a Néel temperature of ∼70 K [8,9], which was firstly attributed to an all-in-all-out arrangement of magnetic moments carried by Cu²⁺ centers forming a pyrochlore-like lattice [10]. Then, powder neutron diffraction analyses [11] revealed an incommensurate magnetic order, with a propagation vector $\overrightarrow{q}$ = $\left[0.827\left(7\right),0,0\right]$ and concomitant with a ferroelectric phase. It has also been evidenced that the incommensurate magnetic structure induces a spin–phonon coupling similar to the ones observed in perovskite compounds [12]. In 2019, two contradictory neutron-scattering investigations were published. In the first investigation [13], the spin-driven nature of the multiferroicity of Cu₂OCl₂ was demonstrated, leading to the proposition of a cycloidal non-collinear magnetic order with competing magnetic exchange couplings and driven by an inverse Dzyaloshinskii–Moriya mechanism. In the second article [14], the authors claimed the simultaneous existence, at low temperature, of a collinear antiferromagnetic order and an antiferroelectric phase resulting from a Cl→O charge transfer.

In terms of J magnetic exchange values, only one experimental estimation from magnetic susceptibilty [8] has been proposed, one conference’s abstract gave contradicting values deduced from Density Functional Theory (DFT) calculations [15] without providing any details, and one last article [13] focused on the sign of J couplings by studying Wannier functions with DFT. The present study aims to clarify the understanding of the magnetic interactions in Cu₂OCl₂ by doing DFT calculations at ambient pressure but also under chemical pressure, by predicting the magnetic properties of Cu₂OBr₂ and Cu₂OI₂, for the very first time.

2. Materials and Methods

Geometry optimizations on the isostructural Cu₂OX₂ (X = Cl, Br, I) systems were performed within the Density Functional Theory (DFT) framework using the PAW method, as implemented in the VASP code [16,17,18]. Valence electrons were treated using the following electronic configurations for the different species: $3p^{2}4s^{2}3d^9$ for Cu, $3s^{2}3p^5$ for Cl, $4s^{2}4p^5$ for Br, $5s^{2}5p^5$ for I and $2s^{2}2p^4$ for O. The PBE approach was considered for the exchange–correlation functional [19]. To improve the description of the electronic properties for Cu-3d orbitals, a Hubbard effective correction following Dudarev’s scheme [20] was used. We previously showed that $U_{eff}$ = 6.5 eV enables us to recover the experimental volume variation under hydrostatic pressure for the CuO compound [3]. Here, an identical value was chosen. Long-distance van der Waals interactions were taken into account by introducing the DFT-D3 correction proposed by Grimme et al. [21]. Spin-polarized simulations were done with a 5 × 4 × 4 $\mathrm{\Gamma }$-centered k-mesh and a cutoff energy of 550 eV. The dynamical stability of all optimized structures was checked by computing the phonon modes with the finite-differences method for symmetrically non-equivalent displacements on a 2 × 1 × 1 supercell and a 2 × 4 × 4 $\mathrm{\Gamma }$-centered k-mesh.

For each compound, the exchange couplings were estimated based on the optimized structures with the Wien2K code, using DFT within the FP-LAPW approach [22]. The PBE0 on-site functional was chosen for the treatment of the exchange correlation part in the lines of our previous works [3,4]. The $R{K}_{max}$ parameter was set to 7. Muffin-Tin radii were fixed to 1.94, 2.10, 2.20 and 1.71 $a_0$ for Cu, Cl, Br, I and O species, respectively. The first Brillouin zone was sampled with 52–58 irreducible k-points. The magnetic interactions were extracted from a least-squares fitting procedure applied between the DFT energy related to a given collinear magnetic state and the energy of an Ising Hamiltonian used to describe the magnetic structure:

$$E_{Ising}=\sum _{i\ne j}{S}_i·J_{ij}·S_j$$

(1)

where $S_i$ and $S_j$ are the spins carried by the $C{u}_i$ and $C{u}_j$ centers, respectively. The $J_{ij}$ parameter represents the magnetic exchange interactions between $S_i$ and $S_j$. Hereafter, $J>0$ indicates an antiferromagnetic (AFM) coupling, and $J<0$ indicates a ferromagnetic (FM) coupling.

3. Results and Discussion

3.1. Crystal Structure of Cu₂OCl₂

The melanothallite Cu₂OCl₂ crystallizes in the orthorhombic space group $Fddd$ with a = 7.4477 Å, b = 9.5989 Å, c = 9.6888 Å, V = 692.65 ${\mathsf{\AA }}^3$, and Z = 8 [11]. The structure shown in Figure 1a is characterized by one symmetrically non-equivalent Cu²⁺ site located in a strongly distorted octahedral environment due to a Jahn–Teller effect. It can be viewed as a CuO₂Cl₂ square-planar environment (also called plaquette), in which the bond lengths are $d_{Cu-O}$ = 1.943 Å and $d_{Cu-Cl}$ = 2.283 Å. Two longer bonds are formed with chlorine ions in apical positions with $d_{Cu-Cl}$ = 3.123 Å. These CuO₂Cl₂ plaquettes are sharing edges to form ribbons which are inter-connected together by the oxygen ions in a three-dimensional network, which has many common features with the CuO atomic structure. For instance, both Cu₂OCl₂ and CuO [23,24,25] exhibit unusual thermal variations of lattice parameters due to the so-called hinge mechanism [26]. The ribbons are parallel to the $\left[110\right]$ and $\left[\overline{1}10\right]$ directions. As shown in Figure 1b, another vision of the structure is a pyrochlore-like lattice where the tridimensional network is based on O-centered OCu₄ tetrahedra which are sharing corners. Here, the chlorine species are pointing toward the center of cavities.

3.2. Geometry Optimization

Geometry relaxations have been performed at ambient pressure in order to validate our approach by comparing with the experimental data of Cu₂OCl₂ and to predict the atomic structure of Cu₂OBr₂ and Cu₂OI₂. The experimental Cu₂OCl₂ structure reported in Ref. [11] was used as an initial guess for all compounds. The main structural parameters of the optimized geometries are summarized in

Table 1. Optimized structural parameters of the Cu₂OX₂ compounds (X = Cl, Br, I) deduced from DFT+U calculations ($U_{e}ff\left[Cu\left(3d\right)\right]$ = 6.5 eV) and experimental data of Cu₂OCl₂ Ref. [11]. For Cu₂OCl₂, the deviation (in percentage) with respect to the experimental data is given in parentheses.

	Cu2OCl2 (Exp.)	Cu2OCl2	Cu2OBr2	Cu2OI2
a (Å)	7.4477	7.4675 (+0.3)	7.7203	8.1680
b (Å)	9.5989	9.6448 (+0.5)	9.9962	10.5578
c (Å)	9.6888	9.7337 (+0.5)	9.7661	10.1226
V (${\mathsf{\AA }}^3$)	692.65	701.05 (+1.2)	753.69	872.93
$z_X$	0.3241	0.3237 (−0.1)	0.3138	0.3104
Cu-O (Å)	1.943	1.951 (+0.4)	1.996	2.094
Cu-X${}_{eq}$ (Å)	2.283	2.295 (+0.5)	2.408	2.543
Cu-X${}_{ap}$ (Å)	3.123	3.132 (+0.3)	3.214	3.391
Cu-O-Cu (${}^{\circ }$)	102.86	102.82 (0.0)	104.58	105.65
Cu-X${}_{eq}$-Cu (${}^{\circ }$)	83.42	83.25 (−0.2)	81.93	82.01
hinge angle (${}^{\circ }$)	63.20	63.13 (−0.1)	62.42	62.10

, including the experimental data of Cu₂OCl₂.

Regarding Cu₂OCl₂, an excellent agreement is found between the experimental and the theoretical structures. The optimized lattice parameters deviate from the experimental ones by less than 0.5%. Considering the first neighbor’s shell, the Cu-O, Cu-Cl${}_{eq}$ and Cu-Cl${}_{ap}$ bonds of the optimized geometry are 1.95, 2.30 and 3.13 Å, respectively (deviation smaller than 0.5%). Inside a ribbon, the Cu-O-Cu and Cu-Cl${}_{eq}$-Cu bond angles are computed at 102.8${}^{\circ }$ and 85.25${}^{\circ }$ (less than 0.2% deviation), respectively. Finally, the hinge angle that corresponds to the Cu-O-Cu angle between two ribbons is 63.1${}^{\circ }$ (0.1% deviation).

Focusing now on the new Cu₂OBr₂ compound, one may notice that the lattice parameters are larger than the optimized ones of Cu₂OCl₂, as expected from the larger atomic size of bromine. It leads to a = 7.720 Å (3.4% larger compared to Cu₂OCl₂), b = 9.996 Å (3.6% larger), c = 9.766 Å (0.3% larger), and V = 753.69 ${\mathsf{\AA }}^3$ (7.5% larger). This steric effect mainly affects a and b parameters, which are defining the directions of the ribbons. The optimized Cu-O, Cu-Br${}_{eq}$ and Cu-Br${}_{ap}$ bond lengths are 2.00, 2.41 and 3.21 Å, respectively. The angles within a ribbon, i.e., Cu-O-Cu and Cu-Br${}_{eq}$-Cu, are about 2${}^{\circ }$ larger and 1.3${}^{\circ }$ smaller than in Cu₂OCl₂, respectively. The hinge angle appears slightly smaller than in Cu₂OCl₂ at 62.42${}^{\circ }$.

For the second new compound Cu₂OI₂, the lattice parameters are still increasing compared to the two former cases, following the chemical sense directed by the atomic radii ($r_{Cl}<r_{Br}<r_I$). In detail, we found a = 8.168 Å (9.4% larger compared to Cu₂OCl₂), b = 10.558 Å (9.5% larger), c = 10.123 Å (4.0% larger) and V = 872.93 ${\mathsf{\AA }}^3$ (24.5% larger). Once more, the significant variation of volume results from more important changes reported for a and b lattice parameters. Regarding the distorted CuO₂I₄ environment, the optimized Cu-O, Cu-I${}_{eq}$ and Cu-I${}_{ap}$ bond lengths are respectively measured at 2.09, 2.54 and 3.39 Å. Focusing on the ribbons, the Cu-O-Cu angle equals 105.7${}^{\circ }$ (2.3${}^{\circ }$ larger than in Cu₂OCl₂), while the Cu-I${}_{eq}$-Cu angle is reduced to 82.0${}^{\circ }$ (1.2% smaller).

No imaginary phonon frequencies were retrieved for both Cu₂OCl₂ and Cu₂OBr₂ systems. First, this result confirms that the present modeling reproduces properly both the atomic parameters and the dynamical stability of Cu₂OCl₂. Experimentally, the crystals are grown using the chemical vapor transport techniques with a stoichiometric mixture of CuO and CuCl₂ compounds [11]. Second, Cu₂OBr₂ is predicted to be dynamically stable. For Cu₂OI₂, one imaginary phonon frequency was calculated at ∼20i ${\mathrm{cm}}^{-1}$, suggesting that such a phase does not exist. However, Cu₂OI₂ was kept in our investigation as a virtual compound allowing us to probe the chemical pressure effect.

3.3. Electronic Properties

Based on the optimized structures, the electronic properties were investigated using the Wien2K code. The projected density of states (pDOS) of the three Cu₂OX₂ systems are represented in Figure 2. The present PBE0 calculations lead to band gap values of 2.18 eV, 1.85 eV and 0.95 eV for X = Cl, Br and I, respectively, and rather similar pDOS. The valence band (VB) is based on the O-2p, Cl-3p (or Br-4p or I-5p) and Cu-3d states (from −7 to 0 eV with respect to $E_F$). While the top of the VB is mainly composed of O-2p states in Cu₂OCl₂, it is mainly based on Br-4p and I-5p states in Cu₂OBr₂ and Cu₂OI₂, respectively. Such a point is emphasized in Figure 3 by the electronic densities calculated in the energy range from −0.5 to 0 eV (with respect to $E_F$). The magnetically active orbital, i.e., the one carrying the magnetic moment, is the Cu-3$d_{x^2-y^2}$, as expected from the $d^9$ electronic configuration and the square planar environment. The calculated magnetic moment of copper is 0.6–0.7 ${\mathsf{\mu }}_B$ in Cu₂OX₂, which is in good agreement with the available experimental data for Cu₂OCl₂. Indeed, the ordered magnetic moment was found to be 0.64(5) ${\mathsf{\mu }}_B$ and 0.66(2) ${\mathsf{\mu }}_B$ from powder and single-crystal neutron refinements, respectively [13].

3.4. Magnetic Exchange Interactions

Magnetic susceptibiltity measurements on Cu₂OCl₂ provided a Néel temperature $T_N$ of ∼70 K followed by a broad maximum at ∼140 K [8,9]. Many investigations were carried out to determine the nature of the magnetic order at lower temperatures, leading to several propositions such as an all-in-all-out model [10], an incommensurate spin spiral phase [11], an incommensurate spin cycloidal phase [13] and a collinear antiferromagnetic (AFM) phase [14]. However, only one of these experimental investigations proposed magnetic exchange coupling (J) values extracted from magnetic susceptibility fits [8], which are in disagreement with the reported J values estimated from DFT calculations [15]. However, these latter values were found in a conference abstract and no related publication is available, to our knowledge.

Based on the optimized geometries, we have thus calculated the J exchange values using PBE0 on-site functional as we did for CuO [3,4]. Examining the atomic structure leads to defining four magnetic interactions depicted in Figure 4 for both ribbons- and pyrochlore-like lattices. The related $d_{Cu-Cu}$ bond lengths and Cu-O-Cu angles are specified in

Table 2. Geometrical parameters related to the four magnetic couplings for the Cu₂OX₂ compounds (X = Cl, Br, I) deduced from DFT+U relaxation ($U_{e}ff\left[Cu\left(3d\right)\right]$ = 6.5 eV). The experimental data of Cu₂OCl₂ taken from Ref. [11] are given in parentheses.

		${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$	${\mathit{J}}_4$
Cu2OCl2	Cu-Cu (Å)	3.049	3.067	3.426	6.099
		(3.037)	(3.055)	(3.410)	(6.075)
	Cu-O-Cu (${}^{\circ }$)	102.82	103.65	122.82	-
		(102.86)	(103.70)	(122.72)	-
Cu2OBr2	Cu-Cu (Å)	3.158	3.112	3.494	6.315
	Cu-O-Cu (${}^{\circ }$)	104.58	102.47	122.16	-
Cu2OI2	Cu-Cu (Å)	3.337	3.252	3.657	6.674
	Cu-O-Cu (${}^{\circ }$)	105.65	101.87	121.64	-

for Cu₂OCl₂, Cu₂OBr₂ and Cu₂OI₂. More specifically, considering the ribbons-like structure, we targeted two intra-ribbon ($J_1$ and $J_4$) and two inter-ribbon ($J_2$ and $J_3$) couplings. It should be noticed that among these four interactions, only $J_4$ is based on a super-superexchange (SSE) path, i.e., the magnetic moments are mediated through an orbital overlap implying two ligand atoms (namely Cu-L-L-Cu, with L = O, Cl, Br or I), the others being based on a superexchange (SE) path implying only one ligand atom (namely Cu-L-Cu, with L = O, Cl, Br or I).

The J couplings were estimated using the mapping analysis procedure. The strategy consists of (1) determining the DFT energies of a set of magnetic states, (2) defining the related energy expressions in terms of J parameters using an Ising Hamiltonian, and (3) refining the J parameters on top of the DFT results using a least squares method. One may underline that the accuracy of such an approach strongly depends on the size and the quality of the set of magnetic states [27]. In our case, eight collinear magnetic structures were selected to obtain four J values. The results are summarized in

Table 3. Magnetic exchange interactions (in meV) calculated at the PBE0 on-site level for the Cu₂OX₂ compounds (X = Cl, Br, I). The available experimental and theoretical data are given for comparison. $J>0$ indicates an AFM coupling, and $J<0$ indicates an FM coupling.

		${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$	${\mathit{J}}_4$
Cu2OCl2	This work	−14.0	−2.2	19.1	8.7
	Theo. [15]	−15.5	19.0 1		8.6
	Exp. [8]	9.7 2	9.3 2		-
Cu2OBr2	This work	−13.1	1.9	25.2	9.2
Cu2OI2	This work	−8.1	24.2	41.0	15.1

¹ This value has been defined as an interchain coupling without giving more details. It could be alternatively $J_2$, $J_3$ or a combined effective value. ² These two parameters correspond to mean field estimation of $J_{intra-ribbon}=J_1$ and $J_{inter-ribbon}=J_2+J_3$.

.

Before discussing our results, let us on comment the experimental values extracted by Okabe et al. [8] considering the ribbons-like picture. They considered two effective J parameters which were found to be both AFM and very close to each other, i.e., $J_{intra}$ = 9.7 meV and $J_{inter}$ = 9.3 meV, thus leading to strong magnetic frustrations. These mean field values can be compared to our J parameters considering that $J_{intra}=J_1+J_4$ and $J_{inter}=J_2+J_3$. It leads to DFT effective intra- and inter-ribbon couplings, which are respectively FM and AFM, i.e., $J_{intra}$ = −5.3 meV and $J_{inter}$ = 16.8 meV, in disagreement with Okabe et al. [8]. In contrast, in Ref. [15], three DFT J values have been proposed, i.e., two intra-ribbon ($J_1$ = −15.5 meV and $J_4$ = 8.6 meV) and one inter-ribbon ($J_{inter}=J_2+J_3$ = 19.0 meV), which are in good agreement with our values, i.e., $J_1$ = −14.0 meV and $J_4$ = 8.7 meV for the intra-ribbon interactions and $J_2$ = −2.2 meV and $J_3$ = 19.1 meV for the inter-ribbon ones.

In other words, the present calculations evidenced that the ribbons are mainly FM ordered due to the first-neighbor (SE) interaction $J_1$, and the spins are frustrated due to the AFM second-neighbor (SSE) interaction $J_4$, which is 1.6 times smaller than $J_1$ in amplitude. Between the ribbons, the interaction is AFM due to $J_3$ ($J_2$ appears negligible). In the pyrochlore-like lattice, each OCu₄ tetrahedron is globally AFM with two up and two down magnetic centers, which further emphasizes the spin frustration character of the magnetic structure. The resulting most stable collinear magnetic order is depicted in Figure 5.

A similar picture is obtained for Cu₂OBr₂ and Cu₂OI₂ compounds. Regarding the intra-ribbon interactions, one may notice that the FM $J_1$ coupling decreases to −13.1 meV and −8.1 meV, respectively, while the AFM $J_4$ coupling is enhanced to 9.2 meV and 15.1 meV, respectively. For the inter-ribbon interactions, the FM $J_2$ coupling becomes AFM at 1.9 meV for Cu₂OBr₂ and strongly increases up to 24.2 meV for Cu₂OI₂. $J_3$ stays the strongest (AFM) coupling and is increased to 25.2 meV and 41.0 meV, respectively. The larger values for $J_3$ cannot be explained only based on geometrical arguments. Indeed, the related Cu-O bond length and Cu-O-Cu angle, respectively, increases and decreases, when Cl is substituted by Br or I. It is the signature that the halogen element indirectly participates in this interaction by setting the size and the shape of the magnetic orbital due to the nephelauxetic effect, which increases in the series Cl^– < Br^– < I^–, leading to an expansion of the copper 3d orbitals and thus larger interactions. Figure 6 shows that the spin densities of the three compounds are quite similar, with spin-up densities (in yellow) along the FM $J_1$ coupling for both copper and halogen elements involved in this interaction. For oxygen atoms, which are also involved in the AFM couplings ($J_3$ and $J_4$), two lobes are found with one up- and one down-spin densities in yellow and cyan, respectively. These observations have a direct consequence on the magnetic moments carried by the ligands, which are 0.11, 0.10, 0.08 and 0.00 ${\mu }_B$ for Cl, Br, I and O, respectively. Indeed, the magnetic moment of the oxygen is null due to its participation in AFM couplings.

To go further, we focused more deeply on the magnetically active orbital, i.e., Cu-3$d_{x^2-y^2}$, and its interaction with the surrounding ligands. The spin-polarized pDOS of Cu-3$d_{x^2-y^2}$, O-2p and Cl-3p states of one plaquette are given in Figure 7a for Cu₂OCl₂. It appears that the occupied Cu-3$d_{x^2-y^2}$ states are mainly positioned within the energy range from −7 to −5 eV (highlighted by the gray dotted line rectangle). The electronic charge density ($\rho$) and spin density related to this energy window are shown in Figure 7b,c, respectively. First of all, from the analysis of both pDOS and $\rho$, it appears that the Cu-3$d_{x^2-y^2}$ orbital interaction is larger with O-2p than Cl-3p states. It confirms that the magnetic moment of the magnetically active orbital is mainly mediated by oxygen atoms and thus justifies that the largest coupling is along a Cu-O-Cu superexchange path. In addition, the up- and down-pDOS of O-2p states are perfectly symmetric, leading to a net magnetic moment of zero. Similarly, the spin-density around oxygen is constituted of two equivalent lobes with opposite spins (as previously discussed). Looking in more detail, Figure 7c evidenced the pivotal role of oxygen that connects two ribbons (with opposite spins), which are nearly perpendicular. To summarize the results of the present analysis, the magnetically active orbital results from an overlap of Cu-3$d_{x^2-y^2}$ with O-2p and to a less extent with Cl-3p states in Cu₂OCl₂. One may notice that in this energy range, the pDOS of the magnetically active orbital is constituted by two peaks: (1) the lower in energy originating from the Cu-O interaction, while (2) the higher in energy is associated to the Cu-Cl interaction. A similar picture can be retrieved for Cu₂OBr₂ and Cu₂OI₂, except that the first peak (Cu-O interaction) increases and the second peak (Cu-X interaction) decreases (see Figure 2). This observation explains the enhancement of $J_3$ from Cu₂OCl₂ to Cu₂OBr₂ and to Cu₂OI₂.

Interestingly, these three systems can be compared to their parent compound CuO, for which the atomic and magnetic structures are described using different descriptors. While the atomic structure can be viewed as based on ribbons of edge-sharing CuO₄ plaquettes, the magnetic order is mainly governed by AFM spin chains of corner-sharing CuO₄ plaquettes. Here also, it appears that the strongest magnetic coupling, i.e., $J_3$, is at the origin of AFM spin chains of corner-sharing CuO₂X₂ plaquettes, highlighted by a yellow–blue dotted line in Figure 5. The presence of such AFM spin chains can be more clearly seen and understood in the OCu₄-based vision, where these AFM spin chains propagate along tetrahedra.

As we did previously for CuO [3,4], we can thus estimate $T_N$ based on the random phase approximation model of a quasi-1D AFM Heisenberg cubic lattice [28]:

$$J^{\prime }=\frac{T_N}{4c\sqrt{ln\left(\alpha J/T_N\right)+0.5ln\left(ln\left(\alpha J/T_N\right)\right)}}$$

(2)

where $\alpha$ = 2.6 and c = 0.233 are numerical parameters, $J^{\prime }$ is an effective inter-chain coupling and J is the intra-chain coupling. In the present case, half of the spin chains are 90${}^{\circ }$ rotated, leading to half as many inter-chain couplings. The resulting ground-state expression is $E_{GS}\left(model\right)=J+J^{\prime }$, instead of $E_{GS}\left(model\right)=J+2J^{\prime }$ in the CuO model where all chains are oriented in the same direction. Such an expression has to be compared to the collinear ground-state (Figure 5) energy defined as:

$$E_{GS}=J_3-J_1+J_2-J_4$$

(3)

Thus, one may define:

$$J=J_3$$

(4)

$$J^{\prime }=-J_1+J_2-J_4$$

(5)

Using the aforementioned J values and the original c and $\alpha$ parameters, we obtain $T_N$ = 65, 111 and 294 K for Cu₂OCl₂, Cu₂OBr₂ and Cu₂OI₂, respectively (see Figure 8). Experimentally, $T_N\sim$70 K for Cu₂OCl₂. Such a good agreement validates the present magnetic model, which consists of considering AFM spin chains based on $J=J_3$ interacting through an effective coupling defined as $J^{\prime }=-J_1+J_2-J_4$. It should be noticed that if we use the refined value for CuO c = 0.284 [3], we obtain $T_N$ = 77, 131 and 344 K for Cu₂OCl₂, Cu₂OBr₂ and Cu₂OI₂, respectively (see Figure 8). The agreement for Cu₂OCl₂ is still very good with the experiment, and whatever the c value is, we predict a significant increase of $T_N$ when Cl is replaced by either Br or I in the compound. A similar trend has been reported experimentally for CuCl₂ and CuBr₂ [6,7], for which $T_N$ = 23.9 and 77 K, respectively.

4. Conclusions

In summary, we have investigated the magnetic properties of Cu₂OX₂ (X = Cl, Br, I) systems based on first-principles calculations. The atomic structure of Cu₂OX₂ can be viewed either as based on ribbons of CuO₂X₂ edge-sharing plaquettes or as a pyrochlore-like lattice constituted of OCu₄ corner-sharing tetrahedra. The present study shows that the magnetic order is governed by AFM spin chains of CuO₂X₂ corner-sharing plaquettes, which defines an AFM intra-chain coupling $J=J_3$. These spin chains interact with each other through three inter-chain couplings, which leads to an effective interaction $J^{\prime }=-J_1+J_2-J_4$. Using these two parameters J and $J^{\prime }$ in an analytical expression for a quasi-1D AFM Heisenberg system, one may recover the Néel temperature of the known multiferroic compound Cu₂OCl₂ at $T_N$ = 70 K. We also predict that Cu₂OBr₂ could be a stable phase and is expected to have a larger $T_N$ value, which is evaluated to be about 120 K. Using chemical pressure to enhance $T_N$ is thus an effective and appealing technique to design new mixed anion compounds in order to reach RT operating MF. Ongoing investigations in our group consist of combining chemical and physical pressures to reach this goal.