3.1. Crystal Structure of Zn3V4(PO4)6
Zn
3V
4(PO
4)
6 is three-dimensional with a triclinic space group of
. Its experimental lattice constants were reported to be a = 6.349 Å, b = 7.869 Å, c = 9.324 Å, α = 105.32°, β = 108.66°, and γ = 101.23° [
57]. In this complex structure, phosphorous and vanadium form tetrahedral units and octahedral units with nearest-neighbor oxygen atoms (see
Figure 1a). These two units share their corners and form networks in the crystal. This complex structure was energy-minimized completely to determine the equilibrium lattice constants. The calculated values were in good agreement with the values reported in the experiment showing the efficacy of potential parameters and pseudopotentials (see
Table 2).
The calculated total DOS plot shows that the Zn
3V
4(PO
4)
6 exhibits a metallic characteristic (see
Figure 1b), which is one of the essential conditions for a cathode material.
Figure 1c shows the charge density plot showing the ion–ion interaction in the Zn
3V
4(PO
4)
6.
Bader charges calculated on the atoms in the bulk Zn
3V
4(PO
4)
6 are reported in
Table 3. Based on the formula of the Zn
3V
4(PO
4)
6, ionic charges on the Zn, V, P, and O are +2.00, +3.00, +5.00, and −2.00, respectively. Although the Bader charges of Zn, V, and O are not closer to the values of the full ionic charge model, the bulk Zn
3V
4(PO
4)
6 can still be considered as ionic material.
3.2. Intrinsic Defect Properties
Here, we calculated the Frenkel, Schottky, and anti-site defect formation energy processes. These defects are crucial as they determine the properties of materials. As point defect (vacancy and interstitial) energies are necessary to calculate the defect process energies, vacancy and interstitial formation energies were first calculated. The defect processes are written using Kröger–Vink notations [
58]. In a Schottky defect process, vacancies are introduced in the lattice. A Frenkel defect consists of a vacancy–interstitial pair. The concentration of these two defect processes is temperature-dependent. In general, the concentration of these defects will increase with increasing temperature. The Frenkel defect process controls the ionic conductivity of a material as explained in a previous experimental study on AgI [
59]. An enhancement in the electronic conductivity was explained in the presence of Schottky defects in LaFeO
3 [
60]. Point defect energies were combined to calculate Frenkel, Schottky, and anti-site defect formation energies.
In
Figure 2, calculated defect energies are reported. The Zn-V anti-site cluster exhibits the lowest defect formation energy. In this defect process, a small amount of Zn
2+ ions will be on the V site and vice versa. The defect formation energy difference between the anti-site (isolated) and anti-site (cluster) is defined as the binding energy (−0.18 eV). Exoergic binding shows that isolated defects (
) will aggregate as soon as the isolated defects are present in this material. This defect has been determined in a variety of oxide materials both experimentally and theoretically [
61,
62,
63,
64,
65]. The influence of the cation anti-site defect has been discussed in an experimental study on LiFePO
4 [
66]. The electrochemical performance of LiNiPO
4 upon the Li-Ni anti-site defect has been discussed by Kempaiah et al. [
67]. The cycled structure of Li
2FeSiO
4 exhibited Li-Fe ion mixing and the Li-ion diffusion pathways and activation energies were different to those found in its as-prepared structure [
68]. In a recent study, defect and diffusion properties of spinel-ZnCo
2O
4 were examined using classical simulation [
40]. It was found that the Zn-Co anti-site cluster is the lowest-energy defect with a binding energy of −0.14 eV.
Partial Schottky (ZnO and V2O3) and Zn Frenkel defect energies are closer to each other. The Zn Frenkel is an important defect energy process as it can govern the vacancy-assisted Zn-ion migration in this material. This defect energy is 2.82 eV, much lower than that calculated in ZnCo2O4. The O Frenkel has a defect energy of 3.00 eV. This is higher only by ~0.30 eV than that calculated in the partial Schottky or Zn Frenkel. Other Schottky or Frenkel defects are higher-defect-energy processes and do not form under normal temperatures. In particular, the P Frenkel has the highest defect energy of 10.60 eV.
3.3. Diffusion of Zn2+ Ions
The understanding of Zn
2+ ion diffusion is important as it determines the overall performance of Zn
3V
4(PO
4)
6. Here, we used classical simulation techniques to calculate the energies of activation of local Zn hops and construct diffusion pathways with long range. In a previous classical simulation study [
69], it was reported that Li
+ ion migration in LiFePO
4 is one-dimensional with a curve that was later confirmed in a neutron diffraction study [
70].
Five local Zn hops were found. In the first hop (A), Zn diffuses in the bc plane having an activation energy of 1.05 eV (see
Figure 3). A jump distance of 3.98 Å and its activation energy 1.88 eV were calculated in the second hop. Both activation energies are quite high and, therefore, the diffusion of Zn
2+ would be slow.
Table 4 lists the activation energies of each hoping distance. In particular, longer hop distances (>4.00 Å) have higher activation energies (>3 eV). This means that Zn
2+ ions would diffuse very slowly via those hops. The slow diffusivity of Zn
2+ can be due to several reasons including the high positive charge of 2+, long Zn hop distance, and the crystal structure. Morkhova et al. [
71] recently employed DFT simulations to calculate activation energies of some Zn
2+ ion conductors. The most promising structures were found to be spinel compounds with the chemical formula of ZnM
2O
4 (M = Fe, Co, Cr, and V) and their activation energies ranged between 0.54 eV and 0.68 eV. In the crystal structure of Zn
3S
2O
9, the activation energy of Zn
2+ is high (1.55 eV), although this compound consists of three Zn
2+ ions per formula unit. In a previous classical simulation study of spinel-ZnCo
2O
4, it was reported that the Zn
2+ ions migrate in a linear pathway having an activation energy of 0.71 eV [
40].
An improvement of Zn
2+ ion diffusion can be made by preparing materials at the nanoscale, lowering the Zn-Zn separation. The polymerization intercalation method has also been applied to enhance the rate of diffusion of Zn
2+ ions [
72]. In this method, the electrostatic interaction between Zn
2+ and O
2− ions is weakened to overcome the sluggish diffusion of Zn
2+ ions. A combined experimental and DFT study of ZnCo
2O
4 showed that the oxygen vacancy formation can improve the Zn
2+ ion diffusion by enlarging channels [
73]. A facile hydrothermal method was applied to prepare a composite consisting of Zn
xV
2O
5 and graphene oxide to form a stabilized structure and enhance the diffusion of Zn
2+ ions [
74].
3.4. Solution of Dopants
Substitutional doping of elements is an important process to modify the properties of a material [
75]. The ionic conductivity of zirconia was enhanced by the doping of yttria [
76]. Such an enhancement was shown to associate with the point defects controlling the oxygen diffusion at grain boundaries. Ru doping on the Fe site reduced the Li-Li hop distances and enhanced the Li-ion diffusion in LiFePO
4 [
77]. Here, we considered alkali earth (Mg, Ca, Sr, and Ba), divalent transition metal (Co, Mn, Fe, and Ni), and trivalent (Al, Ga, Gd, In, Sc, Y, and La) dopants to predict candidate dopants that can be tried experimentally. Appropriate lattice energies of dopant oxides and charge-compensating defects were introduced to construct defect reaction equations.
Table 5 reports the potential parameters used for dopant oxides in this study.
First, alkali earth dopants (R = Mg, Ca, Sr, and Ba) were substitutionally doped on the Zn site. The following equation explains the doping process.
Solution energies are reported in
Figure 4a. The Ca
2+ is found to be the most favorable dopant. A negative solution energy (−0.16 eV) is calculated for this dopant. The solution energy calculated for the Mg
2+ is also negative but lower by 0.05 eV than that calculated for the Ca
2+. These dopants are favored partly due to their ionic radii (Mg
2+: 0.89 Å and Ca
2+: 1.12 Å) closely matching with the ionic radius of Zn
2+ (0.88 Å). The solution of Sr
2+ is endoergic with the solution energy of 0.41 eV. The Ba
2+ exhibits a high positive solution energy of 1.73 eV, meaning that this dopant can be doped only at high temperatures. The total DOS plot shows that the Ca-doped Zn
3V
4(PO
4)
6 is still metallic (see
Figure 4b). The states of the Ca lie in the deeper level of the valence band (see
Figure 4c).
Next, divalent transition metal ions (Ni
2+, Co
2+, Fe
2+, and Mn
2+) were considered on the Zn site. The most promising dopant is Fe
2+ and its solution energy is –1.30 eV (see
Figure 5a). The negative solution energy indicates that the Fe
2+ on the Zn site is thermodynamically stable. The second most favorable dopant is Mn
2+. The solution energy for this dopant is negative (−0.08 eV), meaning that this dopant is also promising. Both Ni
2+ and Co
2+ exhibit positive solution energies. The most unfavorable dopant is Ni
2+. The metallic characteristic of Zn
3V
4(PO
4)
6 is retained upon Fe-doping (see
Figure 5b). The states in the Fermi energy level are significantly affected by the d-states of Fe (see
Figure 5c).
Next, trivalent dopants were substituted on the V site. This doping process, as other processes mentioned above, requires no charge-compensating defects as defined by the following equation.
The solution energies reported in
Figure 6a indicate that most of the dopants except Al
3+ exhibit an exoergic solution. The most promising dopant of In
3+ has the solution energy of −1.10 eV. The least promising dopant of Al
3+ exhibits an endothermic solution energy of 0.30 eV. There is an increase in solution energy with increasing ionic radius from Al
3+ to In
3+. Then, there is a gradual drop in the solution energy up to Ga
3+. The solution energy of La
3+ becomes more negative than that calculated for Gd
3+.
The total DOS plot shows that In-doped Zn
3V
4(PO
4)
6 is metallic (see
Figure 6b) and the states occupied at the Fermi energy level are a mixture of s, p, and d states of In (see
Figure 6c).
A deformation charge density plot associated with the most favorable dopants interacting the lattice structure is provided in
Figure 7.
3.5. Synthetic Routes for the Formation of Zn3V4(PO4)6
The synthesis of Zn
3V
4(PO
4)
6 was carried out using a high-temperature roasting method. In this complex method, CH
3COO)
2Zn·2H
2O, NH
4VO
3, and H
2C
2O
4·2H
2O are used as starting materials [
39]. Here, we consider some chemical reactions and calculate the formation energies by optimizing different oxides containing Zn, V, and P. In all cases, the formation of one mole of Zn
3V
4(PO
4)
6 is considered and the reaction energies are exothermic (see
Table 6). This means all five routes are theoretically feasible. In the first reaction, the reaction liberates energy (−6.27 eV) in the form of heat. In this reaction route, binary oxides are taken as reactants. The second reaction has two binary oxides and one ternary oxide. The inclusion of a ternary oxide results in a lower reaction energy of −3.12 eV. In the third reaction, a ternary oxide consisting of Zn, P, and O (Zn
2P
2O
7) is considered as a reactant. This reaction is still exothermic and its reaction energy is lower than those of reactions 2 and 3. In the least feasible reaction, a ternary oxide consisting of high Zn content (Zn
3(PO
4)
2) is used. This reaction is exothermic with a reaction energy of −0.29 eV. Although all the reaction energies are thermodynamically exoergic, the availability, abundance, cost of materials, and ease of experiments should also be considered. In addition, kinetic feasibility is also important as the start of the reaction depends on the activation energy. The kinetic barrier can be dealt with by using appropriate catalysts.