Computer Simulation of the Incorporation of V²⁺, V³⁺, V⁴⁺, V⁵⁺ and Mo³⁺, Mo⁴⁺, Mo⁵⁺, Mo⁶⁺ Dopants in LiNbO₃

Abstract

The doping of LiNbO₃ with V²⁺, V³⁺, V⁴⁺ and V⁵⁺ as well as Mo³⁺, Mo⁴⁺, Mo⁵⁺ and Mo⁶⁺ ions is of interest in enhancing its photorefractive properties. In this paper, possible incorporation mechanisms for these ions in LiNbO₃ are modelled, using a new set of interaction potentials fitted to the oxides VO, V₂O₃, VO₂, V₂O₅ and to LiMoO₂, Li₂MoO₃, LiMoO₃, Li₂MoO₄.

1. Introduction

Ferroelectric lithium niobate is a material that has been extensively studied because of its many technological applications, including optical integrated circuits, electro-optical modulators, optical memories, acoustic filters, high-frequency beam deflectors, frequency converters and holographic volume storage [1,2,3,4,5,6,7,8,9], for which holographic volume storage performance is very important [10,11,12,13,14,15]. This paper looks at the doping of LiNbO₃ with vanadium and molybdenum ions in different charge states, with the aim of predicting the optimum location of dopants, and charge compensation mechanisms where needed.

Previous work on vanadium and molybdenum doped lithium niobate has included experimental studies of how its photorefractive properties are enhanced by doping with molybdenum ions [16,17] where it is suggested that the Mo⁶⁺ ion dopes at the Nb⁵⁺ site. Another study looks at LiNbO₃ co-doped with Mg and V, concluding that some of the vanadium dopes at the Nb site in the 5+ charge state, but that V⁴⁺_Li, V³⁺_Li and V²⁺_Li defects are also observed [18]. Finally, another recent publication [19] has looked at the photorefractive response of Zn and Mo co-doped LiNbO₃ in the visible region, and concluded that the presence of Mo⁶⁺ ions helps promote fast response and multi-wavelength holographic storage, which is attributed to their occupation of regular niobium sites in the lattice.

In a Density Functional Theory (DFT) study [20], vanadium doping was modelled, and it was concluded that vanadium substitutes at the Li⁺ site as V⁴⁺, but that it dopes at the Nb site as a neutral defect as the Fermi level is increased. In another DFT study [21], molybdenum doping was modelled and it was concluded that the most stable configuration involves doping at the Nb⁵⁺ site, in agreement with the previously mentioned experimental studies [16,17]. It is noted that in the DFT studies, predictions were made on the basis of defect formation energies, as opposed to the solution energy approach adopted in this paper.

This paper presents a computer modelling study of V²⁺, V³⁺, V⁴⁺ and V⁵⁺ as well as Mo³⁺, Mo⁴⁺, Mo⁵⁺ and Mo⁶⁺ doping in LiNbO₃ using interatomic potentials. Such calculations enable predictions to be made of the sites occupied by dopant ions, and the form of charge compensation adopted, if needed. These calculations provide information about how the defects behave in the material, and how they influence its properties in the applications mentioned previously. It follows a series of papers by the authors on LiNbO₃ doped with a range of ions [22,23,24,25,26,27].

2. Materials and Methods

2.1. Interatomic Potentials

The interatomic potentials used in this work consist of Buckingham potentials, supplemented by an electrostatic term, as given below:

$$V(r_{ij})=\frac{q_i{q}_j}{r_{ij}}+A_{ij}\exp \left(\frac{-r_{ij}}{{\rho }_{ij}}\right)-C_{ij}{r}_{ij}{}^{-6}$$

(1)

This expression shows that for each pair of ions it is necessary to determine three parameters: A_ij, ρ_ij and C_ij, which are constants for each interaction, q_i, q_j represent the charges of the ions i and j, and r_ij is the interatomic distance. The parameters are determined by empirical fitting, and formal charges are used for q_i and q_j. The procedure by which potentials were obtained for LiNbO₃ is explained in the work of Jackson and Valério [22], and the derivation of the potentials for the vanadium and molybdenum dopants is described in Section 3.1 below. The potentials for LiNbO₃ have been the subject of recent studies on the doping of the structure with rare earth ions [23,24], doping with Sc, Cr, Fe and In [25], metal co-doping [26] and doping with Hf [27]. These papers show that modelling can predict the energetically optimal locations of the dopant ions and calculate the energy involved in the doping process. This paper extends this procedure to the study of V²⁺, V³⁺, V⁴⁺ and V⁵⁺ as well as Mo³⁺, Mo⁴⁺, Mo⁵⁺ and Mo⁶⁺ doped lithium niobate, with the aim of establishing the optimal doping site and charge compensation scheme for both sets of ions.

2.2. Defect Formation Energies

The calculation of defect formation energies is carried out using the Mott–Littleton approximation [28], in which the crystal is divided into two regions: region I, which contains the defect, and region II, which extends from the edge of region I to infinity. In region I, the positions of the ions are adjusted until the resulting force is zero. The radius of region I is selected such that the forces in region II are relatively weak and the relaxation can be treated according to the harmonic response to the defect (a dielectric continuum). An interfacial region IIa is introduced to treat interactions between region I and region II.

3. Results and Discussion

3.1. Derivation of Interatomic Potential Parameters

It was necessary to derive potential parameters for the dopant oxide structures: VO, V₂O_3, VO₂ and V₂O₅ as well as LiMoO₂, Li₂MoO₃, Li₃MoO₄ and Li₂MoO₄. For V²⁺-O²⁻, V³⁺-O²⁻, V⁴⁺-O²⁻ and V⁵⁺-O²⁻ as well as Mo³⁺-Li⁺, Mo⁴⁺-Li⁺, Mo⁵⁺-Li⁺, Mo⁶⁺-Li⁺, Mo³⁺-O²⁻, Mo⁴⁺-O²⁻, Mo⁵⁺-O²⁻ and Mo⁶⁺-O²⁻ interactions, a new set of potentials was derived empirically by fitting to the observed structures as shown in

Table 1. Interionic potentials obtained from a fit to the VO, V₂O₃, VO₂, V₂O₅, LiMoO₂, Li₂MoO₃, Li₃MoO₄ and Li₂MoO₄ structures.

Interaction	Aij(eV)	ρij(Å)	Cij(Å6 eV)
Licore-Oshell	950.0	0.2610	0.0
Vcore-Oshell	293.240087	0.475181	0.0
Mocore-Licore	573.532325	0.369602	0.0
Mocore-O2−shell	3003.79	0.3474	0.0
Mocore-Ocore	600.263736	0.328558	0.0
O2−shell-O2−shell	22764.0	0.1490	27.88
Harmonic	k(eV Å2)	ro(Å)
Vcore-Ocore	46.997833	1.942956
Mocore-Ocore	385.638986	2.073074
Species			Y(e)
Mocore			3.0 4.0 5.0 6.0
Vcore			2.0 3.0 4.0 5.0
Ocore			0.9
Oshell			−2.9
Spring			k(Å−2 eV)
Ocore-Ooore			70.0

. The O²⁻-O²⁻ potential was obtained by Sanders et al. [29] and uses the shell model for O [30], which is a representation of ionic polarisability, in which each ion is represented by a core and a shell, coupled by a harmonic spring, and the Li-O potential was taken from [22]. In all cases, the dopant-oxide potentials were obtained by fitting to parent oxide structures.

Table 2. Comparison of calculated (calc.) and experimental (expt.) lattice parameters.

Oxide	Lattice Parameter	Exp.	Calc. (0 K)	Δ%	Calc. (293 K)	Δ%
VO	a(Å) = b(Å) = c(Å)	4.067800	4.108237	0.99	4.10683	0.98
V2O3	a(Å) = b(Å) = c(Å)	9.393000	9.304757	0.90	9.346331	0.94
VO2	a (Å) = b(Å)	4.556100	4.569483	0.20	4.566212	0.22
	c(Å)	2.859800	2.866421	0.23	2.857861	0.07
V2O5	a(Å)	11.971900	11.99652	0.20	12.01247	0.33
	b(Å)	4.701700	4.722561	0.44	4.660343	0.88
	c(Å)	5.325300	5.355671	0.57	5.371149	0.86
Lithium Molybdates	Lattice Parameter	Exp.	Calc. (0 K)	Δ%	Calc. (293 K)	Δ%
LiMoO2	a(Å) = b(Å)	2.866300	2.880528	0.50	2.887246	0.73
	c(Å)	15.474300	15.409390	0.42	15.595024	0.78
Li2MoO3	a(Å) = b(Å)	2.878000	2.854443	0.82	2.859809	0.63
	c(Å)	14.91190	15.002886	0.61	15.04632	0.90
Li3MoO4	a(Å) = b(Å) = c(Å)	4.1389	4.107762	0.75	4.106941	0.77
Li2MoO4	a(Å) = b(Å)	14.330000	14.301305	0.20	14.384501	0.38
	c(Å)	9.584	9.492067	0.96	9.632413	0.96

compares experimental and calculated structures of VO [31], V₂O₃ [32]_, VO₂ [33] and V₂O₅ [34] oxides as well as LiMoO₂ [35], Li₂MoO₃ [36], Li₃MoO₄ [37] and Li₂MoO₄ [38] lithium molybdate structures, using the potentials in Table 1. It is seen that the experimental and calculated lattice parameters differ by less than 1%, confirming that the potentials can be used in further simulations of defect properties. The calculations were carried at 0 K (the default for the modelling code and used in most other theoretical studies) and at 293 K for comparison with room temperature results. In this way, we can see how the structure and energies vary with temperature.

3.2. Defect Calculations

In this section, calculated energies for dopant ions in LiNbO₃ are reported. The divalent, trivalent, tetravalent, pentavalent and hexavalent dopants can substitute at Li and Nb sites in the LiNbO₃ matrix with charge compensation taking place in a number of ways. The proposed schemes described in the following subsections are written as solid state reactions using the Kroger–Vink notation [39]. This notation appears in the tables in Section 3.2.1, Section 3.2.2, Section 3.2.3, Section 3.2.4 and Section 3.2.5 where the dot/bullet (·) means a net positive charge and the dash/prime (′) means a net negative charge.

3.2.1. Divalent Dopants

The substitution of the divalent dopant V²⁺ in the Li⁺ and Nb⁵⁺ host sites requires a charge-compensating defect, which can involve Li and Nb vacancies, Nb_Li anti-sites, interstitial oxygen, self-compensation and oxygen vacancies. The modes of substitution considered for divalent cations are shown in

Table 3. Types of defects considered due to M = V²⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i) ${\mathrm{MO}+2 \mathrm{Li}}_{\mathrm{Li}} \to { \mathrm{M}}_{\mathrm{Li}}^{•}{+\mathrm{V}}_{\mathrm{Li}}^{\prime }{+\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii) ${5\mathrm{MO}+5\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}}\to {5\mathrm{M}}_{\mathrm{Li}}^{•}{+\mathrm{V}}_{\mathrm{Nb}}^{″‴}+{2.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii) ${2\mathrm{MO}+2\mathrm{Li}}_{\mathrm{Li}}\to {2\mathrm{M}}_{\mathrm{Li}}^{•}{+\mathrm{O}}_{\mathrm{i}}^{″}{+\mathrm{Li}}_2\mathrm{O}$
Li+ and Nb5+	Self-Compensation	(iv) $4{\mathrm{MO}+3 \mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to { 3\mathrm{M}}_{\mathrm{Li}}^{•}{+\mathrm{M}}_{\mathrm{Nb}}^{‴}+ {1.5\mathrm{Li}}_2\mathrm{O}+{0.5 \mathrm{Nb}}_2{\mathrm{O}}_5$
Nb5+	Lithium Vacancies and Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(v) ${\mathrm{MO}+2\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{M}}_{\mathrm{Nb}}^{‴}{+\mathrm{V}}_{\mathrm{Li}}^{\prime }+{ \mathrm{Nb}}_{\mathrm{Li}}^{•••}{+\mathrm{Li}}_2\mathrm{O}$
	Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(vi) $4{\mathrm{MO}+3\mathrm{Li}}_{\mathrm{Li}}{+4\mathrm{Nb}}_{\mathrm{Nb}} \to 4{\mathrm{M}}_{\mathrm{Nb}}^{‴}{+3\mathrm{Nb}}_{\mathrm{Li}}^{••••}{+\mathrm{Li}}_2{\mathrm{O}+\mathrm{LiNbO}}_3$
		(vii) ${4\mathrm{MO}+3\mathrm{Li}}_{\mathrm{Li}}{+4\mathrm{Nb}}_{\mathrm{Nb}}\to {4\mathrm{M}}_{\mathrm{Nb}}^{‴}{+3\mathrm{Nb}}_{\mathrm{Li}}^{••••}+{1.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Vacancies	(viii) ${2\mathrm{MO}+2\mathrm{Nb}}_{\mathrm{Nb}}{+3\mathrm{O}}_{\mathrm{O}} \to 2{\mathrm{M}}_{\mathrm{Nb}}^{‴}{+3\mathrm{V}}_{\mathrm{O}}^{••}{+\mathrm{Nb}}_2{\mathrm{O}}_5$

.

The solution energies for the divalent (V²⁺) dopant with different charge-compensating mechanisms were evaluated and plotted as a function of the reaction schemes. Based on the lowest energy value, it seems that the incorporation of a divalent (V²⁺) ion is energetically favourable at the lithium and niobium sites, taking into account the first in relation to the c axis. In schemes (i) and (iv), the energy difference in eV is small at both temperatures in the first neighbours, indicating that it can be incorporated at the lithium site compensated by a lithium vacancy as well as by self-compensation as shown in Figure 1. This can be attributed to the similarity between the ionic radius of V²⁺, which is 0.79 Å, and those of the Li⁺ site, which varies between 0.59 and 0.74 Å, and the Nb⁵⁺ site, which varies between 0.32 and 0.71 Å [40].

3.2.2. Trivalent Dopants

As with the divalent ion V²⁺, the trivalent V³⁺ and Mo³⁺ dopants can be incorporated at the lithium and niobium sites in the LiNbO₃ matrix through various schemes as shown in

Table 4. Types of defects considered due to V³⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i)$0.5{\mathrm{M}}_2{\mathrm{O}}_3{+3\mathrm{Li}}_{\mathrm{Li}}\to { \mathrm{M}}_{\mathrm{Li}}^{••}{+2\mathrm{V}}_{\mathrm{Li}}^{\prime }+{1.5\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii)$2.5{\mathrm{M}}_2{\mathrm{O}}_3{+5\mathrm{Li}}_{\mathrm{Li}}{+2\mathrm{Nb}}_{\mathrm{Nb}} \to { 5\mathrm{M}}_{\mathrm{Li}}^{••}{+2\mathrm{V}}_{\mathrm{Nb}}^{″‴}+{2.5\mathrm{Li}}_2{\mathrm{O}+\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii)${0.5\mathrm{M}}_2{\mathrm{O}}_3{+\mathrm{Li}}_{\mathrm{Li}} \to { \mathrm{M}}_{\mathrm{Li}}^{••}{+\mathrm{O}}_{\mathrm{i}}^{″}+{0.5\mathrm{Li}}_2\mathrm{O}$
Li+ and Nb5+	Self-Compensation	(iv)${\mathrm{M}}_2{\mathrm{O}}_3{+\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{M}}_{\mathrm{Li}}^{••}{+\mathrm{M}}_{\mathrm{Nb}}^{″}+ {0.5\mathrm{Li}}_2\mathrm{O}+{0.5 \mathrm{Nb}}_2{\mathrm{O}}_5$
Nb5+	Oxygen Vacancies	(v)$0.5{\mathrm{M}}_2{\mathrm{O}}_3{+\mathrm{Nb}}_{\mathrm{Nb}}{+\mathrm{O}}_{\mathrm{O}}\to { \mathrm{M}}_{\mathrm{Nb}}^{″}{+\mathrm{V}}_{\mathrm{O}}^{••}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(vi)${\mathrm{M}}_2{\mathrm{O}}_3{+\mathrm{Li}}_{\mathrm{Li}}{+2\mathrm{Nb}}_{\mathrm{Nb}} \to { 2\mathrm{M}}_{\mathrm{Nb}}^{″}{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}{+\mathrm{LiNbO}}_3$
	Lithium Vacancies and Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(vii)${0.5\mathrm{M}}_2{\mathrm{O}}_3{+3\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to {\mathrm{M}}_{\mathrm{Nb}}^{″}+{ 2\mathrm{V}}_{\mathrm{Li}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}+{1.5\mathrm{Li}}_2\mathrm{O}$

and

Table 5. Types of defects considered due to Mo³⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i)${\mathrm{LiMoO}}_2{+3\mathrm{Li}}_{\mathrm{Li}} \to { \mathrm{Mo}}_{\mathrm{Li}}^{••}{+2\mathrm{V}}_{\mathrm{Li}}^{\prime }{+2\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii)$5{\mathrm{LiMoO}}_2{+5\mathrm{Li}}_{\mathrm{Li}}{+2\mathrm{Nb}}_{\mathrm{Nb}} \to 5{\mathrm{Mo}}_{\mathrm{Li}}^{••}{+2\mathrm{V}}_{\mathrm{Nb}}^{″‴}{+5\mathrm{Li}}_2{\mathrm{O}+\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii)${\mathrm{LiMoO}}_2{+\mathrm{Li}}_{\mathrm{Li}} \to {\mathrm{Mo}}_{\mathrm{Li}}^{••}{+\mathrm{O}}_{\mathrm{i}}^{″}{+\mathrm{Li}}_2\mathrm{O}$
Li+ and Nb5+	Self-Compensation	(iv)${2\mathrm{LiMoO}}_2{+2\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to {\mathrm{Mo}}_{\mathrm{Li}}^{••}{+\mathrm{Mo}}_{\mathrm{Nb}}^{″}+{1.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
Nb5+	Oxygen Vacancies	(v)${\mathrm{LiMoO}}_2{+\mathrm{Nb}}_{\mathrm{Nb}}+{ \mathrm{O}}_{\mathrm{O}}\to {\mathrm{Mo}}_{\mathrm{Nb}}^{″}{+\mathrm{V}}_{\mathrm{O}}^{••}+{0.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
Nb5+	Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(vi)$2{\mathrm{LiMoO}}_2{+\mathrm{Li}}_{\mathrm{Li}}{+2\mathrm{Nb}}_{\mathrm{Nb}}\to 2{\mathrm{Mo}}_{\mathrm{Nb}}^{″}{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}+{1.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
Nb5+	Lithium Vacancies and Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(vii)${\mathrm{LiMoO}}_2{+3\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to {\mathrm{Mo}}_{\mathrm{Nb}}^{″}+2{\mathrm{V}}_{\mathrm{Li}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}{+2\mathrm{Li}}_2\mathrm{O}$

. When these ions are substituted at Li and Nb sites, the extra positive charge can, as noted earlier, be compensated by the creation of vacancies, interstitials, anti-site defects or self-compensation.

According to Figure 2 and Figure 3 for the first and second neighbours with respect to the c axis, the trivalent V³⁺ and Mo³⁺ ions prefer to occupy both the Li and Nb sites according to scheme (iv) which is also observed in other trivalent ions [23,24,25]. This can be attributed to the similarity between the ionic radius of V³⁺ which is 0.64 Å and Mo³⁺ which is 0.67 Å [40] and that of Li⁺ and Nb^{5 +}. The ionic radius of Li⁺ varies between 0.59 Å and 0.74 Å and Nb⁵⁺ varies from 0.32 Å to 0. 66 Å [40]. All these ionic radii are in relation to the coordination sphere with oxygen atoms.

3.2.3. Tetravalent Dopants

Like other divalent and trivalent cations, tetravalent V⁴⁺ and M⁴⁺ dopant ions can also substitute at either the Li⁺ or Nb⁵⁺ sites. When these ions substitute at the Li⁺ and Nb⁵⁺ site charge compensation is required, and various schemes involving vacancies, interstitials, anti-sites and self-compensation are adopted, as shown in

Table 6. Types of defects considered due to M = V⁴⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i)${\mathrm{MO}}_2{+4\mathrm{Li}}_{\mathrm{Li}}\to {\mathrm{M}}_{\mathrm{Li}}^{•••}{+3\mathrm{V}}_{\mathrm{Li}}^{\prime }+{ 2\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii)${5\mathrm{MO}}_2{+5\mathrm{Li}}_{\mathrm{Li}}{+3\mathrm{Nb}}_{\mathrm{Nb}}\to 5{\mathrm{M}}_{\mathrm{Li}}^{•••}{+3\mathrm{V}}_{\mathrm{Nb}}^{″‴}+ {2.5\mathrm{Li}}_2\mathrm{O}+{1.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii)${2\mathrm{MO}}_2{+2\mathrm{Li}}_{\mathrm{Li}} \to { 2\mathrm{M}}_{\mathrm{Li}}^{•••}{+3\mathrm{O}}_{\mathrm{i}}^{″}{+\mathrm{Li}}_2\mathrm{O}$
Li+ and Nb5+	Self-Compensation	(iv)${4\mathrm{MO}}_2{+\mathrm{Li}}_{\mathrm{Li}}{+3\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{M}}_{\mathrm{Li}}^{•••}{+3\mathrm{M}}_{\mathrm{Nb}}^{\prime }+ {0.5\mathrm{Li}}_2\mathrm{O}+{1.5 \mathrm{Nb}}_2{\mathrm{O}}_5$
Nb5+	Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(v)${4\mathrm{MO}}_2{+\mathrm{Li}}_{\mathrm{Li}}{+4\mathrm{Nb}}_{\mathrm{Nb}} \to { 4\mathrm{M}}_{\mathrm{Nb}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}+{0.5\mathrm{Li}}_2\mathrm{O}+{1.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Lithium Vacancies and Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(vi)${\mathrm{MO}}_2{+4\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{M}}_{\mathrm{Nb}}^{\prime }{+3\mathrm{V}}_{\mathrm{Li}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}+{ 2\mathrm{Li}}_2\mathrm{O}$
		(vii)${2\mathrm{MO}}_2{+3\mathrm{Li}}_{\mathrm{Li}}{+2\mathrm{Nb}}_{\mathrm{Nb}}\to {2\mathrm{M}}_{\mathrm{Nb}}^{\prime }{+2\mathrm{V}}_{\mathrm{Li}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}{+\mathrm{Li}}_2{\mathrm{O}+\mathrm{LiNbO}}_3$
		(viii)${3\mathrm{MO}}_2{+2\mathrm{Li}}_{\mathrm{Li}}{+3\mathrm{Nb}}_{\mathrm{Nb}}\to {3\mathrm{M}}_{\mathrm{Nb}}^{\prime }{+\mathrm{V}}_{\mathrm{Li}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}{+\mathrm{LiNbO}}_3$
	Oxygen Vacancies	(ix)${2\mathrm{MO}}_2{+2\mathrm{Nb}}_{\mathrm{Nb}}{+\mathrm{O}}_{\mathrm{O}}\to { 2\mathrm{M}}_{\mathrm{Nb}}^{\prime }{+\mathrm{V}}_{\mathrm{O}}^{••}{+\mathrm{Nb}}_2{\mathrm{O}}_5$

and

Table 7. Types of defects considered due to M=Mo⁴⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i)${\mathrm{Li}}_2{\mathrm{MoO}}_3{+4\mathrm{Li}}_{\mathrm{Li}}\to {\mathrm{Mo}}_{\mathrm{Li}}^{•••}{+3\mathrm{V}}_{\mathrm{Li}}^{\prime }+{ 3\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii)${5\mathrm{Li}}_2{\mathrm{MoO}}_3{+5\mathrm{Li}}_{\mathrm{Li}}{+3\mathrm{Nb}}_{\mathrm{Nb}}\to 5{\mathrm{Mo}}_{\mathrm{Li}}^{•••}{+3\mathrm{V}}_{\mathrm{Nb}}^{″‴}+ {7.5\mathrm{Li}}_2\mathrm{O}+{1.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii)${2\mathrm{Li}}_2{\mathrm{MoO}}_3{+2\mathrm{Li}}_{\mathrm{Li}} \to { 2\mathrm{Mo}}_{\mathrm{Li}}^{•••}{+3\mathrm{O}}_{\mathrm{i}}^{″}{+3\mathrm{Li}}_2\mathrm{O}$
Li+ and Nb5+	Self-Compensation	(iv)${4\mathrm{Li}}_2{\mathrm{MoO}}_3{+\mathrm{Li}}_{\mathrm{Li}}{+3\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{Mo}}_{\mathrm{Li}}^{•••}{+3\mathrm{Mo}}_{\mathrm{Nb}}^{\prime }+ {4.5\mathrm{Li}}_2\mathrm{O}+{1.5 \mathrm{Nb}}_2{\mathrm{O}}_5$
Nb5+	Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(v)${4\mathrm{Li}}_2{\mathrm{MoO}}_3{+\mathrm{Li}}_{\mathrm{Li}}{+4\mathrm{Nb}}_{\mathrm{Nb}} \to { 4\mathrm{Mo}}_{\mathrm{Nb}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}+{4.5\mathrm{Li}}_2\mathrm{O}+{1.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Lithium Vacancies and Anti-site (${\mathrm{Nb}}_{\mathrm{Li}}^{••••}$)	(vi)${\mathrm{Li}}_2{\mathrm{MoO}}_3{+4\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{Mo}}_{\mathrm{Nb}}^{\prime }{+3\mathrm{V}}_{\mathrm{Li}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}+{ 3\mathrm{Li}}_2\mathrm{O}$
		(vii)${2\mathrm{Li}}_2{\mathrm{MoO}}_3{+3\mathrm{Li}}_{\mathrm{Li}}{+2\mathrm{Nb}}_{\mathrm{Nb}}\to {2\mathrm{Mo}}_{\mathrm{Nb}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}{+2\mathrm{V}}_{\mathrm{Li}}^{\prime }{+3\mathrm{Li}}_2{\mathrm{O}+\mathrm{LiNbO}}_3$
		(viii)${3\mathrm{Li}}_2{\mathrm{MoO}}_3{+2\mathrm{Li}}_{\mathrm{Li}}{+3\mathrm{Nb}}_{\mathrm{Nb}}\to {3\mathrm{Mo}}_{\mathrm{Nb}}^{\prime }{+\mathrm{Nb}}_{\mathrm{Li}}^{••••}{+\mathrm{V}}_{\mathrm{Li}}^{\prime }{+3\mathrm{Li}}_2{\mathrm{O}+2\mathrm{LiNbO}}_3$
	Oxygen Vacancies	(ix)${2\mathrm{Li}}_2{\mathrm{MoO}}_3{+2\mathrm{Nb}}_{\mathrm{Nb}}{+\mathrm{O}}_{\mathrm{O}}\to { 2\mathrm{Mo}}_{\mathrm{Nb}}^{\prime }{+\mathrm{V}}_{\mathrm{O}}^{••}{+2\mathrm{Li}}_2{\mathrm{o}+\mathrm{Nb}}_2{\mathrm{O}}_5$

.

The results obtained from these calculations are given in Figure 4 and Figure 5. By inspecting these figures, it can be seen that the tetravalent cation V⁴⁺ prefers to be incorporated at the Li⁺ and Nb⁵⁺ sites through scheme (iv), while the Mo⁴⁺ ion prefers to be incorporated at the niobium site compensated by an oxygen vacancy according to scheme (ix). Similar to the divalent and trivalent dopants, this preference is related to the proximity with the ionic radii of Li⁺ and Nb⁵⁺.

3.2.4. Pentavalent Dopants

For the pentavalent dopants V⁵⁺ and Mo⁵⁺, no charge compensation is required for the substitution at the Nb⁵⁺ host site, but it is required when the substitution is at the Li⁺ host site, as shown in

Table 8. Types of defects considered due to M = V⁵⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i)${0.5\mathrm{M}}_2{\mathrm{O}}_5{+5\mathrm{Li}}_{\mathrm{Li}}\to {\mathrm{M}}_{\mathrm{Li}}^{••••}{+4\mathrm{V}}_{\mathrm{Li}}^{\prime }+ {2.5\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii)${2.5\mathrm{M}}_2{\mathrm{O}}_5{+5\mathrm{Li}}_{\mathrm{Li}}{+5\mathrm{Nb}}_{\mathrm{Nb}}\to 5{\mathrm{M}}_{\mathrm{Li}}^{••••}{+4\mathrm{V}}_{\mathrm{Nb}}^{″‴}+ {2.5\mathrm{Li}}_2{\mathrm{O}+2\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii)${0.5\mathrm{M}}_2{\mathrm{O}}_5{+\mathrm{Li}}_{\mathrm{Li}} \to { \mathrm{M}}_{\mathrm{Li}}^{••••}{+2\mathrm{O}}_{\mathrm{i}}^{″}+{0.5\mathrm{Li}}_2\mathrm{O}$
Nb5+	No Charge Compensation	(iv)${0.5\mathrm{M}}_2{\mathrm{O}}_5{+\mathrm{Nb}}_{\mathrm{Nb}}\to { \mathrm{M}}_{\mathrm{Nb}}^{}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$

and

Table 9. Types of defects considered due to Mo⁵⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i)${\mathrm{Li}}_3{\mathrm{MoO}}_4{+5\mathrm{Li}}_{\mathrm{Li}}\to {\mathrm{Mo}}_{\mathrm{Li}}^{••••}{+4\mathrm{V}}_{\mathrm{Li}}^{\prime }+{ 4\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii)${5\mathrm{Li}}_3{\mathrm{MoO}}_4{+5\mathrm{Li}}_{\mathrm{Li}}{+4\mathrm{Nb}}_{\mathrm{Nb}}\to 5{\mathrm{Mo}}_{\mathrm{Li}}^{••••}{+4\mathrm{V}}_{\mathrm{Nb}}^{″‴}+{ 10\mathrm{Li}}_2{\mathrm{O}+2\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii)${\mathrm{Li}}_3{\mathrm{MoO}}_4{+\mathrm{Li}}_{\mathrm{Li}} \to { \mathrm{Mo}}_{\mathrm{Li}}^{••••}{+2\mathrm{O}}_{\mathrm{i}}^{″}{+2\mathrm{Li}}_2\mathrm{O}$
Nb5+	No Charge Compensation	(iv)${\mathrm{Li}}_3{\mathrm{MoO}}_4{+\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{Mo}}_{\mathrm{Nb}}^{}+{1.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$

.

The solution energies for the pentavalent (V⁵⁺) and (Mo⁵⁺) dopants with different charge compensation mechanisms were evaluated and plotted as a function of the reaction scheme. Based on the lowest energy value, it seems that the incorporation of pentavalent (V⁵⁺) and (Mo⁵⁺) ions at an Nb site is energetically more favourable than at an Li site, according to scheme (iv) as shown in Figure 6 and Figure 7 at temperatures 0 K and 293 K. This can be attributed to the similarity between the charge of the V⁵⁺ and Mo⁵⁺ ions and the Nb⁵⁺ host, which can contribute to a small deformation in the lattice and consequently a lower solution energy. Experimental results by Kong et al. [17] and Tian et al. [16] show that substitution occurs at the Nb⁵⁺ site.

3.2.5. Hexavalent Dopants

For the hexavalent dopant Mo⁶⁺, as with the pentavalent ions, there is no self-compensation mechanism and charge compensation schemes are possible when replacing Li and Nb in the LiNbO₃ matrix as shown in

Table 10. Types of defects considered due to Mo⁶⁺ incorporation in LiNbO₃.

Site	Charge Compensation	Reaction
Li+	Lithium Vacancies	(i)${\mathrm{Li}}_2{\mathrm{MoO}}_4{+6\mathrm{Li}}_{\mathrm{Li}} \to { \mathrm{Mo}}_{\mathrm{Li}}^{•••••}{+5\mathrm{V}}_{\mathrm{Li}}^{\prime }{+4\mathrm{Li}}_2\mathrm{O}$
	Niobium Vacancies	(ii)${\mathrm{Li}}_2{\mathrm{MoO}}_4{+\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}}\to {\mathrm{Mo}}_{\mathrm{Li}}^{•••••}{+\mathrm{V}}_{\mathrm{Nb}}^{″‴}+{1.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(iii)${2\mathrm{Li}}_2{\mathrm{MoO}}_4{+2\mathrm{Li}}_{\mathrm{Li}}\to 2{\mathrm{Mo}}_{\mathrm{Li}}^{•••••}{+5\mathrm{O}}_i^{″}{+3\mathrm{Li}}_2\mathrm{O}$
Nb5+	Lithium Vacancies	(iv)${\mathrm{Li}}_2{\mathrm{MoO}}_4{+\mathrm{Li}}_{\mathrm{Li}}{+\mathrm{Nb}}_{\mathrm{Nb}} \to { \mathrm{Mo}}_{\mathrm{Nb}}^{•}{+\mathrm{V}}_{\mathrm{Li}}^{\prime }+{1.5\mathrm{Li}}_2\mathrm{O}+{0.5\mathrm{Nb}}_2{\mathrm{O}}_5$
	Niobium Vacancies	(v)${5\mathrm{Li}}_2{\mathrm{MoO}}_4{+6\mathrm{Nb}}_{\mathrm{Nb}}\to 5{\mathrm{Mo}}_{\mathrm{Nb}}^{•}{+\mathrm{V}}_{\mathrm{Nb}}^{″‴}{+5\mathrm{Li}}_2{\mathrm{O}+3\mathrm{Nb}}_2{\mathrm{O}}_5$
	Oxygen Interstitial	(vi)${2\mathrm{Li}}_2{\mathrm{MoO}}_4{+2\mathrm{Nb}}_{\mathrm{Nb}}\to {2\mathrm{Mo}}_{\mathrm{Nb}}^{•}{+\mathrm{O}}_{\mathrm{i}}^{″}{+2\mathrm{Li}}_2{\mathrm{O}+\mathrm{Nb}}_2{\mathrm{O}}_5$

.

The solution energies for the hexavalent (Mo⁶⁺) dopants with different charge-compensation mechanisms were evaluated and plotted as a function of the reaction scheme. Based on the lowest energy value, it seems that the incorporation of hexavalent (Mo⁶⁺) ions at an Nb site is energetically more favourable than at an Li site, according to scheme (iv) as shown in Figure 8 at temperatures 0 K and 293 K. This can be attributed to the similarity between the ionic radii of Mo⁶⁺ ions and the Nb⁵⁺ host site (0.32–0.71 Å) [40]. The ionic radii of Mo⁶⁺, taking into account the coordination number, vary between 0.42 and 0.67 Å [40], and the small difference between the Mo⁶⁺ dopant ions and Nb⁵⁺ ions can contribute to a small deformation in the lattice and consequently a lower solution energy. This result reveals that global trends of dopant solution energies are controlled by the combination of dopant ion size [40] and its electrostatic interactions, demonstrating that there is a relation between the energetically preferred site and the types of defect mechanisms involved in the doping process. Experimental results from Kong et al. [17] and Zhu et al. [41] show that substitution occurs at the Nb⁵⁺ site.

In all cases, the energy involved in doping was obtained by calculating the solution energy, which includes all terms of the thermodynamic cycle involved in the solution process. For example, the solution energy, E_sol, corresponding to the incorporation of V²⁺ at the Li⁺ site (second equation in Table 3) is given by:

$${\mathrm{E}}_{\mathrm{Sol}}{=\mathrm{E}}_{\mathrm{Def}}{(5\mathrm{M}}_{\mathrm{Li}}^{•}{+\mathrm{V}}_{\mathrm{Nb}}^{″‴})+{2.5\mathrm{E}}_{\mathrm{Latt}}{(\mathrm{Li}}_2\mathrm{O})+{0.5\mathrm{E}}_{\mathrm{Latt}}{(\mathrm{Nb}}_2{\mathrm{O}}_5)-{5\mathrm{E}}_{\mathrm{Latt}}\left(\mathrm{MO}\right)$$

(2)

where the E_latt and E_Def terms are lattice energies and defect energy.

All energies were normalised by the number of dopants, i.e., the solution energy is divided by the number of dopants involved. For example, for scheme (ii) of Table 3, the energy must be divided by five, since five lithium sites are occupied. This is done because the number of dopants varies for each mechanism. Lattice energies, E_latt, required to calculate the solution energies are given in

Table 11. Lattice energies used in the solution energy calculations (eV).

Compound	Lattice Energy	Lattice Energy
	0 K	293 K
LiNbO3	−174.45	−174.66
Li2O	−33.16	−32.92
Nb2O5	−314.37	−313.39
VO	−22.06	−22.07
V2O3	−124.37	−124.39
VO2	−111.54	−111.57
V2O5	−315.65	−274.18
LiMoO2	−98.07	−97.09
Li2MoO3	−150.38	−149.10
Li3MoO4	−181.28	−178.88
Li2MoO4	−234.06	−234.12

.

3.2.6. Summary of Results for Vanadium and Molybdenum Dopants in LiNbO₃

In this sub-section, the results presented in the last five subsections are summarised.

Divalent dopants: the calculations predict that, for V²⁺, self-compensation (simultaneous doping at lithium and niobium sites) and doping at the lithium site with lithium vacancy compensation are most likely. It is noted that V²⁺_Li defects have been observed experimentally [18].

Trivalent dopants: both V³⁺ and Mo³⁺ ions are predicted to self-compensate. Experimental data from [18] support V³⁺ doping at the lithium site, as with V²⁺.

Tetravalent dopants: here, different behaviour is predicted for vanadium and molybdenum. V⁴⁺ is predicted to self-compensate, while Mo⁴⁺ is predicted to occupy a niobium site with oxygen vacancy charge compensation. Again, [18] suggests that V⁴⁺ can dope at a lithium site.

Pentavalent dopants: both V⁵⁺ and Mo⁵⁺ are predicted to dope at the niobium site (no charge compensation is needed), agreeing with experimental results [16,17].

Hexavalent dopants: Mo⁶⁺ is predicted to dope at the niobium site, with charge compensation by lithium vacancy formation. The occupation of the niobium site is supported by experimental data [16,17,19].

4. Conclusions

This paper has presented a computational study of VO, V₂O₃, VO₂ and V₂O₅ as well as LiMoO₂, Li₂MoO₃, Li₃MoO₄ and Li₂MoO₄ structures doped into LiNbO₃. New interatomic potential parameters for VO, V₂O_3, VO₂ and V₂O₅ as well as LiMoO₂, Li₂MoO₃, Li₃MoO₄ and Li₂MoO₄ have been developed. It was found that divalent (V²⁺), trivalent (V³⁺, Mo³⁺) and tetravalent (V⁴⁺) ions are more favourably incorporated at the Li and Nb sites through the self-compensation mechanism. The tetravalent (Mo⁴⁺) ion is more favourably incorporated at the niobium site, compensated by an oxygen vacancy. The pentavalent ions (V⁵⁺, Mo⁵⁺) and hexavalent (Mo⁶⁺) ion are more favourably incorporated at the Nb site, and the lowest energy schemes involve, respectively, no charge compensation, and for the Mo⁶⁺ ion, charge compensation with lithium vacancy. This is shown to be consistent with some experimental data, although future calculations involving finite V⁵⁺ and Mo⁶⁺ concentrations will be carried out to investigate this further.

Finally, to summarise, in this paper we have looked in detail at vanadium and molybdenum dopants in various charge states in LiNbO₃, and through the use of solution energies, identified the energetically favoured sites and charge compensation mechanisms, while comparing the results with available experimental and theoretical work in this field.