Comparative Study of the Compressibility of M₃V₂O₈ (M = Cd, Zn, Mg, Ni) Orthovanadates

Abstract

We report herein a theoretical study of the high-pressure compressibility of Cd₃V₂O₈, Zn₃V₂O₈, Mg₃V₂O₈, and Ni₃V₂O₈. For Cd₃V₂O₈, we also present a study of its structural stability. Computer simulations were performed by means of first-principles methods using the CRYSTAL program. In Cd₃V₂O₈, we found a previously unreported polymorph which is thermodynamically more stable than the already known polymorph. We also determined the compressibility of all compounds and evaluated the different contributions of polyhedral units to compressibility. We found that the studied vanadates have an anisotropic response to compression and that the change in volume is basically determined by the compressibility of the divalent-cation coordination polyhedra. A systematic discussion of the bulk modulus of M₃V₂O₈ orthovanadates will also be included.

1. Introduction

In modern society, every day the need for less contaminated and more energy efficient materials is more evident. Complex vanadium materials have received a great deal of attention for developing green technologies. Nowadays, they are being considered as electrodes for rechargeable batteries and as photocatalytic materials [1,2,3,4,5,6]. Orthovanadates, with formula M₃V₂O₈, being M a divalent atom, are also interesting because of their optical, dielectric, and magnetic properties [7,8,9,10]. On top of that, these materials have received a great deal of attention owing to their luminescence properties [11], which make them promising materials as laser-host matrices and for developing light-emitting diodes [12]. In addition, M₃V₂O₈ vanadates have attracted attention because of their self-activated photoluminescence [13]. Interestingly, the colors of their photoluminescence cover almost the whole visible-light region from blue to yellow.

From the fundamental point of view, the high-pressure (HP) behavior of M₃V₂O₈ vanadates has also been the focus of recent studies [14,15,16]. X-ray diffraction experiments have shown that Zn₃V₂O₈ [14] and Ni₃V₂O₈ [15] remain stable in the low-pressure phase up to 15 GPa. In Mn₃V₂O₈ [16], the onset of a structural phase transition has been located beyond 10 GPa. On the other hand, the bulk moduli of M₃V₂O₈ vanadates have been determined to cover a range of values from 100 GPa to 140 GPa. However, a systematic understanding of this physical parameter has not been stablished yet. On top of that, there are materials such as Mg₃V₂O₈ and Cd₃V₂O₈ which have never been studied under high-pressure conditions.

Here, we report a study of the compressibility of Zn₃V₂O₈, Ni₃V₂O₈, Mg₃V₂O₈, and Cd₃V₂O₈ and of the HP structural stability of Cd₃V₂O₈. To achieve our goals, we have performed density-functional theory (DFT) calculations. This method has been proven to provide an efficient framework to study the behavior under pressure of materials including ternary oxides [17,18,19,20]. The obtained results have been validated by a comparison with experiments. We have found that the volume change under HP is mainly due to the volume decrease in MO₆ octahedral units. We also found evidence supporting the existence in Cd₃V₂O₈ of a previously unreported polymorph.

2. Computational Details

First-principles total-energy calculations were performed within the periodic DFT framework using the CRYSTAL14 program package [21]. The exchange correlation functionals were described using three different approximations: Becke-Lee–Yang–Parr (B3LYP) [22,23] and Heyd–Scuseria–Ernzerhof (HSE06) [24] hybrid functionals, and Perdew–Burke–Ernzerhof (PBE) standard functionals [25]. The electron basis sets employed to describe Cd, Ni, Zn, Mg, V, and O atoms were taken from the Crystal website [26]. Using the Pack–Monkhorst/Gilat shrinking factors IS = ISP = 4 we have controlled the diagonalization of the Fock matrix. This procedure was carried out using adequate fine grids of irreducible k-points in the reciprocal space. The TOLINTEG parameters, controlling the tolerance factors for the Coulomb and exchange integrals, were set to 10⁻¹⁴ to assure a convergence in total energy that is more accurate than 10⁻⁷ Hartree. Relative numerical errors in the determination of unit-cell parameters are smaller than 10⁻⁴. The percent of Fock/Kohn-Sham matrices mixing was set to 40. The choice of the exchange−correlation functional is of critical importance as it has a significant influence on the description of physical properties [27]. Based upon our previous work in related Zn₂V₂O₇ [28] and Cd₂V₂O₇ [29], and on the fact that in the present calculations we found that B3LYP better describes the structures of all studied compounds at ambient pressure, we employed only B3LYP functionals for present high-pressure studies. Since computer simulations have been performed with the intention of comparing the results among the studied materials at room-temperature, in which Ni₃V₂O₈ is paramagnetic, calculations for this compound were performed only for the paramagnetic configuration. To consider long-range Van der Waals interactions, we included them in calculations using the semi-empirical contributions proposed by Grimme [30], which are especially useful to give an accurate description for metal oxides. All the calculated structures are dynamically stable, having real frequencies over the full Brillouin Zone. From total-energy calculations, the pressure (P)–volume (V) relationship was calculated from the energy (E) versus volume (V) curves. The enthalpy (H) of different polymorphs was determined using H = E + P × V. Crystal structures were represented using VESTA [31], which was also used to calculate polyhedral volumes.

3. Results and Discussion

3.1. Crystal Structure

The crystal structures of Zn₃V₂O₈, Mg₃V₂O₈, and Ni₃V₂O₈ are isomorphic. Their prototypic structure (also shared by Co₃V₂O₈ and Mn₃V₂O₈) is described by the orthorhombic space group Cmca and has four formula units per unit-cell [32,33,34]. This orthorhombic structure is represented in Figure 1a. It is formed by edge-sharing MO₆ (M = Mg, Ni, Zn) octahedra forming two-dimensional Kagome-type staircases which are separated by VO₄ isolated tetrahedra which corner-shared oxygen atoms with the MnO₆ octahedra.

The crystal structure of Cd₃V₂O₈ is also orthorhombic. A schematic representation of it is shown in Figure 1b. The structure has been described in the literature by space group Pnma and has two formula units per unit cell [35]. However, in this description, there is one 4c Wyckoff position occupied by Cd atoms with an occupation factor equal to 1/2. We found that if the structure is described instead by orthorhombic space group Pmc2₁ (a subgroup of Pnma) the 4c Wyckoff position of Cd splits in to one 2a and one 2b Wyckoff positions. Assuming that Cd atoms are at 2a sites and 2b sites are vacant, an equivalent crystal structure is obtained. However, in this case, all atomic positions have a structural occupation factor equal to 1. Both descriptions give identical X-ray diffraction patterns and can be used to describe the structure of Cd₃V₂O₈. Nevertheless, the second one (space group Pmc2₁) is the most convenient for calculations. Consequently, when describing Cd₃V₂O₈ in this work, we will use space group Pmc2₁. As shown in Figure 1b, the structure of cadmium orthovanadate is composed by CdO₆ octahedra that share edges forming chains in the [100] direction separated by a chain of alternating CdO₄ and VO₄ tetrahedral units.

In our study, we have also considered a second possible polymorph for Cd₃V₂O₈ which we assumed was isomorphic to Zn₃V₂O₈, Mg₃V₂O₈, and Ni₃V₂O₈. We will name it α-Cd₃V₂O₈ (to use the same name used in the literature for the isomorphic vanadates), and we will name the polymorph previously reported in the literature (described by space group Pmc2₁) as β-Cd₃V₂O₈. Interestingly, our calculations found that α-Cd₃V₂O₈ has a lower enthalpy than β-Cd₃V₂O₈ from ambient pressure up to 14 GPa (see Figure 2), with the enthalpy difference increasing as pressure increases. This result means that α-Cd₃V₂O₈ is thermodynamically more stable. This means that it might be synthesized at ambient conditions using a different synthesis route than that used by Ben Yahia et al. [35] or the β-α phase transition could be driven by relatively low pressures at room temperature. The fact that the volume per formula unit of α-Cd₃V₂O₈ (168.1 ų) is 6% smaller than the volume per formula unit of β-Cd₃V₂O₈ (178.5 ų) is consistent with the second hypothesis. The calculated atomic positions of α-Cd₃V₂O₈ are given in

Table 1. Calculated atomic positions of α-Cd₃V₂O₈.

Atom	Site	x	y	z
Cd1	4a	0	0	0
Cd2	8e	0.25	0.1554	0.25
V1	8f	0	0.3902	0.1247
O1	8f	0	0.2784	0.2491
O2	8g	0	0.0121	0.247
O3	16g	0.2134	0.3815	0.9847

. Its unit-cell parameters can be found in

Table 2. Unit-cell parameters and volume of different orthovanadates at zero pressure calculated with B3LYP functionals. They are compared with experiments [32,33,34]. The last column is the relative difference between calculations and experiments (ε) in %.

α-Cd3V2O8	B3LYP
a (Å)	6.5837
b (Å)	11.9604
c (Å)	8.5473
V (Å3)	673.046
β-Cd3V2O8	B3LYP	Exp. [34]	ε(%)
a (Å)	6.8744	6.9882	−1.6
b (Å)	5.3107	5.3251	−0.3
c (Å)	9.7796	9.8133	−0.3
V (Å3)	357.032	365.18	−2.2
Zn3V2O8	B3LYP	Exp. [32]	ε(%)
a (Å)	6.0332	6.088	−0.9
b (Å)	11.4247	11.489	−0.6
c (Å)	8.1709	8.280	−1.3
V (Å3)	563.200	579.145	−2.8
Mg3V2O8	B3LYP	Exp. [33]	ε(%)
a (Å)	5.9061	6.053	−2.4
b (Å)	11.2516	11.442	−1.7
c (Å)	8.1216	8.33	−2.5
V (Å3)	539.705	576.923	−6.4
Ni3V2O8	B3LYP	Exp. [34]	ε(%)
a (Å)	5.685	5.936	−4.2
b (Å)	11.4508	11.42	−0.3
c (Å)	8.1093	8.24	−1.5
V (Å3)	527.903	558.582	−5.5

.

The unit-cell parameters of the crystal structures optimized for all studied compounds at zero pressure using B3LYP functionals are provided in Table 2. In the table, they are also compared with experiments when available. The differences between calculated parameters and experiments are comparable to the typical relative differences between calculations and experiments in complex oxides when using B3LYP functionals [36]. A similar underestimation of the calculated volume has been obtained for Zn₂V₂O₇ [28] and Cd₂V₂O₇ [29] using B3LYP functionals [32,33,34].

In

Table 3. Volume of studied compounds calculated with HSE06 and PBE functionals. The relative difference between calculations and experiments (ε) is also given. Results from the Materials Project (MP) are also included.

Compound	VHSE06 (Å3)	VPBE (Å3)	εHSE06(%)	εPBE(%)	VMP (Å3)	εMP(%)
α-Cd3V2O8	656.413	669.242
β-Cd3V2O8	348.866	355.061	−4.5	−2.8	384.85	5.4
Zn3V2O8	549.367	559.630	−5.1	−3.4	609.12	5.2
Mg3V2O8	530.461	541.487	−8.1	−6.1	599.51	3.9
Ni3V2O8	503.762	504.320	−9.8	−8.7	579.38	3.7

, we have summarized the optimized unit-cell volume at zero pressure obtained using HSE06 and PBE functionals. The comparison of results in Table 2 and Table 3 supports that the B3LYP functionals describe orthovanadates more accurately than the other two functionals we have tested. If our results are compared with results reported in the Materials Project [37], which have been obtained using VASP [38,39] and PBE functionals, it can be also seen that B3LYP implemented in CRYSTAL14 gives the most precise description of the studied compounds. In particular, the VASP calculations systematically overestimate the volume at zero pressure.

3.2. Pressure–Volume Equation of State and Compressibility

Now we will comment the evolution of the unit-cell parameters of all compounds. In Figure 3, we represent the pressure dependence of unit-cell parameters. From these results, we have determined the linear compressibility coefficients ($k_l=-\frac{1}{l}\frac{\partial l}{\partial P}$), which are summarized in

Table 4. Calculated linear compressibility coefficients of different compounds compared with experiments [14,15].

	$k_a=2.73\left(4\right){10}^{-3}$ GPa−1
α-Cd3V2O8	$k_b=1.76\left(6\right){10}^{-3}$ GPa−1
	$k_c=2.27\left(11\right){10}^{-3}$ GPa−1
	$k_a=4.16\left(4\right){10}^{-3}$ GPa−1
β-Cd3V2O8	$k_b=1.88\left(1\right){10}^{-3}$ GPa−1
	$k_c=1.56\left(3\right){10}^{-3}$ GPa−1
	$k_a=2.31\left(3\right){10}^{-3}$ GPa−1	$k_a=2.9\left(1\right){10}^{-3}$ GPa−1 [14]
Zn3V2O8	$k_b=1.59\left(2\right){10}^{-3}$ GPa−1	$k_b=1.9\left(1\right){10}^{-3}$ GPa−1
	$k_c=2.24\left(3\right){10}^{-3}$ GPa−1	$k_c=2.7\left(1\right){10}^{-3}$ GPa−1
	$k_a=2.59\left(1\right){10}^{-3}$ GPa−1
Mg3V2O8	$k_b=1.59\left(1\right){10}^{-3}$ GPa−1
	$k_c=2.37\left(1\right){10}^{-3}$ GPa−1
	$k_a=2.89\left(51\right){10}^{-3}$ GPa−1	$k_a=2.7\left(1\right){10}^{-3}$ GPa−1 [15]
Ni3V2O8	$k_b=1.46\left(4\right){10}^{-3}$ GPa−1	$k_b=1.79\left(6\right){10}^{-3}$ GPa−1
	$k_c=2.33\left(3\right){10}^{-3}$ GPa−1	$k_c=2.33\left(5\right){10}^{-3}$ GPa−1

. Notice that all compounds described by space group Cmca show a similar anisotropy. In the four compounds, we found that $k_a$ > $k_c$ > $k_b$. In the case of Ni₃V₂O₈ and Zn₃V₂O₈, the results from calculations agree with experiments [14,15]. In the case of β-Cd₃V₂O₈, the response to compression is also anisotropic, being also the material which is considerably more compressible along the a-axis than along the perpendicular directions.

In the case of the four isomorphic compounds (those described by space group Cmca), the observed anisotropy is related to the layered nature of the Kagome-type structure. For these materials, it can be assumed that pressure-induced changes in the crystal structure are mainly determined by changes in the octahedral units [40,41]. This is because the VO₄ tetrahedron, due to the strong V 3d-O 2p hybridization [40], behaves as a rigid unit and changes little under compression [41]. As can be seen in Figure 1a, the crystal structure of these orthovanadates contains two-dimensional Kagome ladders of compressible MO₆ octahedral units. These octahedral chains form compressible layers that lie perpendicular to the b-axis of the structure. These layers are separated by the less compressible VO₄ tetrahedra, which are arranged to form pillars between them, reducing the compressibility along the b-axis and making it the least compressible axis. A similar picture can be used to explain the anisotropy of β-Cd₃V₂O₈. In this polymorph, there are linear chains of CdO₆ octahedra aligned along the a-axis. The layers are connected by corner-sharing VO₄ tetrahedra. These characteristics make the a-axis the easiest direction of compression.

The calculated pressure dependence of the unit-cell volume is shown in Figure 4. We have fitted these results using a third-order Birch–Murnaghan equation of state [42] for each of the studied compounds. From the fits, we have obtained the zero-pressure unit-cell volume [V₀], bulk modulus [K₀], and its pressure derivative [K₀’]. We summarize the values obtained for these parameters in

Table 5. The unit-cell volume (Å3), bulk modulus (GPa), and bulk modulus pressure derivative at ambient pressure determined using a third-order Birch–Murnaghan EOS.

Phase	${\mathit{V}}_0\left({\mathbf{\AA }}^3\right)$	$\mathit{K}{\left(\mathbf{G}\mathbf{P}\mathbf{a}\right)}_{ }$	${{\mathit{K}}_0^{’}}_{ }$
α-Cd3V2O8	673.0	112	5.0
β-Cd3V2O8	357.0	92	4.0
Zn3V2O8	563.2	136	5.4
Mg3V2O8	539.7	146	4.4
Ni3V2O8	527.9	171	4.5

. The fittings have been carried out using the program Eosfit [43]. We have found that there is a tendency to a decrease in K₀ as V₀ increases. This is a consequence of the increase in the packing efficiency as the volume is reduced. In addition, we have noticed that in the cases of Ni₃V₂O₈ and Zn₃V₂O₈ (the two compounds for which experimental results are available), our results systematically underestimate the experimental compressibility (see Figure 4), which leads to an overestimation of the bulk modulus by approximately 20 GPa [14,15]. This is consistent with the fact that our calculations underestimate V₀. Notice that both parameters are correlated and an underestimated V₀ will always lead to an overestimated B₀ [44,45]. Both facts can be a consequence of an overestimation of the interactions between atoms, which can cause over-hybridization of the M–O bonds because of a self-interaction error [46].

To further understand the changes in the unit-cell volume of the studied compounds under compression, we have analysed changes in polyhedral volumes with pressure. In Figure 5, we report relative changes in the unit-cell volume and relative changes in the polyhedral volumes for each compound. In the figure, it can be seen that, in all compounds, most of the volume change is accounted for by the compression of the MO₆ octahedra. This observation supports the hypothesis we used to explain the anisotropy of the compressibility. In particular, in Mg₃V₂O₈ and Zn₃V₂O₈, it is quite notorious that the relative compressibility of MgO₆ and ZnO₆ is comparable to that of the respective compounds. In the other three cases, the contribution of the volume change in VO₄ is less negligible than in Mg₃V₂O₈ and Zn₃V₂O₈.

On the other hand, our findings also agree with the conclusions extracted by Hazen et al. [47] from the analysis of multiple ternary oxides. These authors established that the bulk modulus of ternary oxides is determined by the compressibility of the largest polyhedra. They proposed that the bulk modulus is expected to have a linear dependence with the average volume of the larger cation polyhedron multiplied by the cation formal charge. In our cases, this means that K₀ is proportional to Z/d³, where Z = 2 is the formal charge of divalent M cation, and d is the average M-O bond distance within the MO₆ octahedron. In Figure 6, we plot our calculated K₀ versus Z/d³. In the figure, we also include available experimental results [15,16]. Results from Cu₃V₂O₈ have been excluded, because it has a very different crystal structure where part of the Cu atoms are in highly unusual square-planar geometry [48]. As a consequence, this compound is extremely unstable under HP, decomposing into 3CuO + V₂O₅. In Figure 6, it can be seen that in spite of the offset between experiments and calculations, both sets of results follow the expected linear dependence, confirming that the changes induced in the structure by pressure are mainly due to changes in MO₆ octahedral units.

In addition to the bulk modulus, the hardness, a measure of the resistance to localized plastic deformation, is a very important physical parameter for technological applications, in particular for using M₃V₂O₈ as laser-host materials. To the best of our knowledge, their hardness is unkown. We have estimated it from the calculated bulk moduli using the relation proposed by She et al. [49]. We have obtained that the hardness goes from 12(2) GPa in Cd₃V₂O₈ to 20(2) GPa in Ni₃V₂O₈. These values are similar to the values previosuly reported for zircon-type orthovanadates [50].

4. Conclusions

In this work, we have studied and compared the compressibility behavior under pressure of four different metal orthovanadates using density-functional theory calculations. We have determined that B3LYP functionals describe better than HSE06 and PBE functionals the crystal structure of Cd₃V₂O₈, Zn₃V₂O₈, Mg₃V₂O₈, and Ni₃V₂O₈. In Cd₃V₂O₈, we have also discovered a new polymorph which is thermodynamically more favorable than the previously known polymorph. In addition, we have determined the pressure dependence of unit-cell parameters. The compressibility of the different compounds is anisotropic. For the different materials, the anisotropy has been discussed in detail, being the principal compression axis determined. We have also determined a pressure–volume equation of states from which the bulk moduli have been obtained. This magnitude has been related to changes in the volume of the divalent cation coordination octahedra. The comparison of the compounds allowed us to verify that their bulk modulus can be explained using the model proposed by Hazen et al. for ternary oxides.