Potentials-Based Computer Modelling of Lanthanide Doped Mixed Metal Fluorides and Oxides for Optical Applications

1. Introduction

Mixed metal fluorides and oxides, doped with lanthanide ions, have a range of applications in optical devices. Computer modelling, based on the use of interatomic potentials, has been widely used to predict the location of dopant ions and the preferred form of charge compensation (if needed), which can then be used to help in synthesis and characterisation of the doped materials. This article looks at work done by the author and collaborators over several years, and it is hoped that it will be useful reference point for future work in the field.

The article starts with a section on methodology, discussing interatomic potentials and their derivation, lattice energy minimisation for the prediction of structures and the calculation of crystal properties, and the calculation od defect formation energies, and solution energies for dopants in materials.

The materials chosen are all of importance in device applications. The first two materials, BaLiF₃ and BaY₂F₈, are both used in laser applications, and in the case of BaLiF₃, a range of possible doping sites and charge compensation schemes are possible, and the calculations show that the preferred site depends on the specific dopant. The third material, SrAl₂O₄, acts as a phosphor when doped with Dy³⁺ and Eu²⁺ ions, but the experimental papers do not usually discuss the detailed solid state chemistry involved in the doping, making this system a good candidate for modelling studies. The final material, LiNbO₃, has many technological applications including when doped with lanthanide ions, but there is disagreement about the location of dopant ions and charge compensation schemes, again making it an excellent material for modelling studies.

2. Methodology

All calculations were performed using the GULP code [1] which has been widely used in modelling studies of a range of materials. The code can be used to fit interatomic potentials and to carry out energy minimisation and defect calculations, as described in the sub sections that follow this.

2.1. Interatomic Potentials

The computer modelling work presented in this review paper was all performed using interatomic potentials. The Buckingham potential has been employed, augmented with a term to represent the electrostatic interactions between ions, and it has the following form:

V (r) = A exp (-r/ρ) – Cr^-6 + q₁q₂/r

where A, ρ and C are parameters to be obtained by empirical fitting, and q₁, q₂ are charges on the interacting ions with ‘r’ being the interionic distance. The shell model has been employed, usually for anions, to represent ionic polarizability, with the ionic charge being split between a core and shell, connected by a harmonic spring [2].

For each compound described the procedure for derivation of potentials is discussed in that section, but in general, empirical fitting has been used, where potentials are fitted to crystal structures, and (where available) crystal properties including elastic and dielectric constants.

The justification for using potentials-based calculations when quantum mechanical methods are increasingly available is that very detailed calculations can be performed on complex materials in a shorter time than would be possible with, for example, DFT calculations. This is further discussed in the conclusions, where a paper comparing the methods is cited.

2.2. Lattice Energy Minimisation

Lattice energy minimisation can be used to calculate structures corresponding to a minimum in the lattice energy.

The total lattice energy is written as a sum of interactions between ions (described by interatomic potentials):

E_latt = ∑ V (r)

Then, by adjusting the structure (either the lattice parameters or atomic positions or both), the minimum energy structure is obtained. Note that a well derived potential would be expected to reproduce the structure without significant structural adjustment.

2.3. Defect Formation Energies and Solution Energies

The main interest in this paper is in calculating the formation energy of dopants in a given structure, but in addition, intrinsic defect energies (mainly formation energies of vacancies and interstitials) are also important. These have all been calculated by the Mott-Littleton method [3] in which a spherical region (region I) is defined surrounding the defect, in which interactions are summed explicitly, and an outer region (region IIB) which is treated as a dielectric continuum. An interface region (IIA) ensures continuity between the inner and outer regions. There is a detailed discussion of the method, including a useful diagram, in our paper on UO₂ [4].

However, defect formation energies do not, on their own, describe the overall solution process, and to do that, solution energies are defined. For example, consider doping BaY₂F₈ with La³⁺ ions, assuming doping at the Y³⁺ site (although this is considered in more detail in section 3.2). A possible solution process can be written as:

LaF₃ + Y_Y -> La_Y + YF₃

The energy of formation of La³⁺ at the Y³⁺ site is calculated, and then combined with the lattice energies of LaF₃ and YF₃ to give the solution energy for the process:

E_sol = E (La_Y) + E_latt (YF₃) – E_latt (LaF₃)

In the following results sections, the calculation of solution energies is discussed in more detail.

3. Results

Four materials have been chosen from those previously studied. The two mixed metal fluorides, BaLiF₃ and BaY₂F₈ are both used as host materials for lanthanide dopants, and they differ in the preferred doping sites, information that is useful in synthesis of the doped materials. The two mixed metal oxides have different applications. SrAl₂O₄ is used for phosphor applications when doped with lanthanides, but as is noted, the experimental papers are not usually concerned with dopant sites and charge compensation. LiNbO₃ has many applications, but in the case of lanthanide doping there is disagreement on doping sites and charge compensation, and calculations are used to address this issue.

3.1. BaLiF₃

BaLiF₃ is a material which adopts the inverted perovskite structure. It has interesting properties as a potential laser material when doped with both divalent and trivalent cations.

Derivation of a potential for the material was carried out using empirical fitting, and unusually, a range of experimental properties were available to use in the fit, including elastic and dielectric constants. Table 1, see [5], shows calculated properties compared with experimental values, and Table 2 gives the potential parameters. It is seen from Table 1 that very good agreement is obtained between the calculated and experimental values.

Shell model parameters

Y = q (F_shell) = −2:321|e|

k (F_core – F_shell) = 48:40 eV Å^-2

Turning now to doping with the trivalent ions La^3+, Y³⁺ and Nd³⁺, it is noted that these ions can potentially substitute at either the Ba²⁺ site or the Li⁺ site, but that charge compensation will be required. This situation is considered in detail in [6], where the following forms of charge compensation are proposed:

(i) formation of a Li⁺ vacancy.

(ii) formation of a F^- interstitial,

(iii) formation of a Ba²⁺ vacancy,

(iv) Li⁺ substitution at a Ba²⁺ site.

Four possible solution schemes are proposed for solution at each of the Ba²⁺ and Li⁺ sites, and it was found that for La³⁺ and Nd³⁺, the most favourable solution scheme involves substitution at the Ba²⁺ site with compensation by Li⁺ vacancies (where Kroger-Vink notation has been used [7]):

MF₃ + Ba_Ba + Li_Li → M^∙_Ba + V_Li^’ + BaLiF₃

For Y³⁺, the most favourable substitution site is Li⁺, with compensation by Ba²⁺ vacancies:

MF₃ + Ba_Ba + Li_Li → M_Li^∙∙ + V_Ba^’’ + BaLiF₃

The difference in the predicted behaviour of La³⁺ and Nd³⁺ compared with Y³⁺ is rationalised in terms of the mean distances between the dopants and the nearest F^- ion in the corresponding MF₃ structures. These predictions are of use when synthesising doped BaLiF₃, an example of which is co-doping with Yb³⁺ and Er³⁺ for X-ray dosimetry and imaging applications [8], in that a knowledge of the preferential location of dopant ions and any charge compensation can be used to guide the synthesis of the doped materials.

3.2. BaY₂F₈

BaY₂F₈ is a promising material for applications in optical devices, and experimental papers have reported doping with Er, Pr, Tm and Dy [9]. While it might be expected that these ions would dope at the Y³⁺ site, solution energy calculations were carried out to confirm this [10].

As with BaLiF₃, potentials were obtained by empirical fitting, but in this case only structural information was available. Table 3 and Table 4 give, respectively, the structural agreement and the potential parameters themselves. Agreement is within 2%, which is regarded as acceptable when there is limited data to fit to.

Shell model parameters

Y = q (F_shell) = −1:378|e|

k (F_core – F_shell) = 24.36 eV Å^-2

Considering doping with lanthanide ions, three solution schemes were considered:

(i) substitution at the Y3+ site (no charge compensation needed)

MF₃ + Y_Y→M_Y + YF₃

E_sol =−E_latt(MF₃) + E (M_Y) + E_latt(YF₃)

(ii) substitution at the Ba²⁺ site (charge compensation by Ba²⁺ vacancies)

2MF₃ + 3Ba_Ba→ 2M_Ba^∙ +V_Ba^’’ +3BaF₂

E_sol =−2E_latt(MF₃) + 2E (M_Ba^∙) + E (V_Ba^’’)+ 3E_latt(BaF₂)

(iii) substitution at the Ba²⁺ site (charge compensation by Ba²⁺/Y³⁺ substitution)

MF₃ + Ba_Ba + Y_Y → M_Ba^∙ + Ba_Y’ +YF₃

E_sol =−E_latt(MF₃) + E(M_Ba^∙) + E(Ba_Y’) + E_latt(YF₃)

Solution energies for each of these schemes are reported in Table 5, for Pr, Nd, Tb, Dy, Er, and Tm. In the case of Nd and Tb doped BaY₂F₈, they were also the subject of a separate combined experimental and modelling study [11]. It is clear from the table that doping at the Y³⁺ site is energetically preferred in all cases.

In conclusion, for lanthanide doping in BaY₂F₈, solution energy calculations confirm that doping occurs at the Y site, which is important in guiding synthesis of the doped materials.

3.3. SrAl₂O₄

SrAl₂O₄ when doped with Eu²⁺ and Dy³⁺ ions, acts as a phosphor material, and is widely used in applications like emergency signs. In experimental papers (see e.g. [12]), it is widely assumed that doping occurs at the Sr²⁺ site, but there is often no discussion of how the charge is compensated if Dy³⁺ ions also substitute at these sites. This makes doped SrAl₂O₄ an excellent candidate for a computer modelling study [13].

As with the other materials, potentials were needed, but in this case, they were available from the literature [14]. Table 6 compares experimental and calculated lattice parameters for SrAl₂O₄. It is noted that the agreement is not perfect, but these potentials were chosen so that the other relevant oxide phases needed to calculate solution energies could also be calculated.

Now doping by Eu²⁺ and Dy³⁺ ions is considered. In the case of just doping an Eu²⁺ ion at a Sr²⁺ site, the solution scheme is assumed to be:

EuO + Sr_Sr → Eu_Sr + SrO

And the solution energy is calculated from:

E_sol = [E (Eu_Sr) + E_latt (SrO)] – [E_latt (EuO)]

The solution energy is calculated to be 0.06 eV per dopant ion, which suggests doping by Eu²⁺ ions alone is favourable. However, it is now necessary to consider Dy³⁺ ion doping. In terms of ion charges, it might be reasonably assumed that the Dy³⁺ ion might substitute at the Al³⁺ site, with the following solution scheme:

Dy₂O₃ + 2 Al_Al → 2Dy_Al + Al₂O₃

The solution energy for this process is calculated to be 1.72 eV per Dy³⁺ ion, suggesting that doping at this site would be energetically feasible. However, this is not the assumption made in the experimental papers, where doping at the Sr²⁺ site is assumed. If doping by Sr²⁺ ions alone is considered, a possible scheme, with charge compensation by Sr²⁺ vacancies is:

Dy₂O₃ + 2 Sr_Sr → 2Dy^∙_Sr + V_Sr’’+ 3SrO

The solution energy for this process is calculated to be 3.08 eV per Dy³⁺ ion, considerably higher than for solution at the Al³⁺ site, suggesting that doping at this site would be unlikely. The question then is whether co-doping of Eu²⁺ and Dy³⁺ ions at the Sr²⁺ site leads to a lower solution energy. The following solution scheme is considered:

Dy₂O₃ + EuO + 4 Sr_Sr → Eu_Sr + 2Dy^∙_Sr + V_Sr’’+ 4SrO

The solution energy for this process is calculated to be 2.08 eV per dopant ion, which shows that co-doping does reduce the solution energy, and that this solution scheme, involving Sr²⁺ vacancies, can be used to explain Eu²⁺ and Dy³⁺ co-doping in SrAl₂O₄.

The calculations presented on SrAl₂O₄ are useful in that they enable the question of where dopants substitute, and how charge compensation is most optimally achieved, to be answered, which is a point often ignored in the experimental papers.

3.4. LiNbO₃

Lithium niobate, LiNbO₃ is of interest because of its many technological applications [15]. Doping with a range of cations, including lanthanides, is important for device applications, and computer modelling can help determine where dopants are accommodated, and with which form of charge compensation.

A potential was derived by fitting simultaneously to the structures of LiNbO₃, Li₂O and Nb₂O₅ [16] since the oxide phases are important in determining solution energies for various doping schemes. The structure of the ferroelectric phase as determined using this potential is compared with experimental data in Table 7, and it seen that good agreement is obtained. Also discussed in [16] is the calculation of elastic, dielectric and piezoelectric constants, which were not included in the fit, but which compare well with experimental values.

Before considering doping, it is important to consider intrinsic defects, because lithium niobate is lithium deficient (or has niobium in excess), and any model should support this observation. This is considered in detail in [17], and the following conclusions were reached:

(i) The non-stoichiometry is explained by antisite or interstitial niobium compensated by lithium or niobium vacancies.

(ii) Of the various possible schemes, the lowest energy is obtained for antisite niobium compensated by lithium vacancies, as in the following reaction:

5Li_Li + ½ Nb₂O₅ → 4V_Li’ + Nb_Li^∙∙∙∙ + 5/2 Li₂O

Now considering lanthanide dopants, lanthanides can potentially dope at either the Li⁺ or Nb⁵⁺ site, with more than one charge compensation scheme in each case. The following solution schemes were considered:

(i) M₂O₃ + 6Li_Li → 2M_Li ^∙∙ + 4V_Li’ + 3Li₂O

(ii) M₂O₃ + Li_Li + Nb_Nb →M_Li^∙∙ + M_Nb’’ + LiNbO₃

(iii) M₂O₃ + Li_Li + 2Nb_Nb →2M_Nb’’ + Nb_Li^∙∙∙∙ + LiNbO₃

(iv) M₂O₃ + 6Li_Li + 2Nb_Nb →2M_Nb’’ + 4V_Li’ + 2Nb_Li^∙∙∙∙ + 3Li₂O

In Table 8, calculated solution energies for the lanthanide series for each scheme are given.

It is clear that in all cases scheme (ii), which involves doping at both sites (also known as self-compensation) is favoured. This is an intriguing result when compared with experimental data obtained using EXAFS [18], which although applied to Zn dopants, finds very low occupancy of the Nb site. This result is discussed in the context of analysis of the EXAFS data [19]. A similar study, but using a lanthanide dopant, would be of great interest here.

The calculations presented here on lanthanide doped LiNbO₃ show that there are a range of schemes for doping, including dopant sites and charge compensation mechanisms, and that there is still scope for research in this area, given the experimental findings referred to.

4. Discussion and Conclusions

The four materials chosen in this review show the range of research done on lanthanide doping in mixed metal fluorides and oxides. In each case, solution energies are calculated which enable predictions to be made about the energetically optimal location of dopants, and favoured charge compensation schemes. This information can be useful in synthesis of the doped materials, and in characterising them (for example, [11]). The method used, based on interatomic potentials, enables complex systems to be modelled, but increasingly, ab initio methods (often based on Density Functional Theory), are being applied to these materials, and the additional information that they can provide will be of great interest, particularly where electronic properties are concerned. However, a comparison of potentials-based modelling and DFT approaches when applied to Th⁴⁺ incorporation in MgF₂ [[20] points to the difficulty in comparing the results of the two methods.