Pairs of Ln(III) dopant ions in crystalline solid luminophores: an ab initio computational study

JOURNAL OF RARE EARTHS, Vol. 34, No. 8, Aug. 2016, P. 820

Pairs of Ln(III) dopant ions in crystalline solid luminophores: an ab initio computational study A. Shyichuk1, G. Meinrath2, S. Lis1,* (1. Department of Rare Earths, Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznań, Poland; 2. Radioactivity Environment Remediation (RER) Consultants, Fuchsbauerweg 50, 94036 Passau, Germany) Received 8 January 2016; revised 8 April 2016

Abstract: Formation of dopant ions clusters in solid (glass) luminophores may affect efficiency of non-radiative energy transfer processes between dopant (photoactivator) ions via shortening of the effective distance between them. This study was based on the assumption that the distance between the dopant ions affects the energy of crystal volume at proximity. According to this idea, semi-empirical and ab initio density functional theory (DFT) calculations were performed on various supercells of YVO4:Eu3+ as a model system. It was noted that a shorter Eu–Eu distance resulted in lower total energy of the system, compared to an analogous structure with distant Eu3+ ions. As lower energy configurations are preferred, the observed phenomenon was considered to be related to dopant ions clusters formation. Additionally, the values of energies obtained from DFT calculations were used to estimate the percentage of dopant ions occurring as pairs, for different dopant concentrations. The estimation agreed quite well with the available literature data. Keywords: crystalline; luminophore; lanthanides; vanadate; DFT; dopant clustering; rare earths

Lanthanide cations, Ln3+, due to their special photoluminescence properties[1] have been widely used as luminescence activators in various organic and inorganic materials[2,3]. Potential applications Ln3+-based luminophores[3] include lasers[4], bioimaging[5–7], telecommunication[8] and light sources[9,10]. Usually, small amounts of Ln3+ ions are added to the materials (matrices) that are not luminescent otherwise. The characteristics of a Ln3+-based luminophore such as absorption and emission intensities, photoefficiency, luminescence lifetime and quenching processes depend to a large extent on the matrix material. One of the important characteristics of the luminophores are Ln-Ln energy transfer processes. Both advantageous (sensitization) and disadvantageous (concentration quenching) processes depend on the energy transfer rates. The latter depend on the distance between the interacting ions[11]. If the distance is known, the energy transfer rates can be calculated. However, a problem of finding the effective Ln-Ln distance arises[12]. On the one hand, there are numerous crystallographic positions suitable for dopant ions and the exact locations of the dopant ions are not known. This results in numerous Ln–Ln distances, each of them characterized by some probability of occurrence. A simple idea is assuming a simple random (equiprobable) distribution of dopant ions among the available crystallographic sites. Then, struc-

ture data can be used to calculate the interatomic distances and obtain the effective energy transfer rates[13]. On the other hand, what if the dopant ions interact somehow and tend to locate close to each other? If the dopant ions tend to distribute inhomogeneously in the matrix compound and occur in clusters, the resulting effective distances would be shorter than expected. The energy transfer rates would increase respectively[14–18]. For instance, it was shown that both repulsive and attractive interactions between the dopant ions can affect a macroscopic property of a luminophore, a photoluminescence decay rate[19]. Although the phenomenon of dopant clustering is known[17,18,20–24] (mostly for glass materials), it is not commonly studied in the research concerning Ln3+ photoluminescence in solids. There are experimental evidences showing that the dopant ions may occur as pairs in neighboring sites more often than it is suggested by a simple statistical distribution[25–27]. Sometimes, dopant ions might separate into another phase[28,29]. However, the agglomeration of dopant ions was scarcely investigated theoretically. Kokou and coworkers used molecular dynamics to analyze Eu3+-doped silicate glasses emphasizing several factors important for dopant ions cluster formation[17]. Muñoz-García and Seijo have indicated that Ce3+ and La3+ dopant ions in Y3Al5O12 (YAG) tend to occur as close as possible[30].

Foundation item: Project supported by the National Science Centre (NCN, Poland) (Grant DEC-2012/06/M/ST5/00325, Etiuda Project 2013/08/T/ ST5/00490) * Corresponding author: S. Lis (E-mail: [email protected]; Tel.: +48 61 829 1679) DOI: 10.1016/S1002-0721(16)60100-X

A. Shyichuk et al., Pairs of Ln(III) dopant ions in crystalline solid luminophores: an ab initio computational study

Please note that we referred to the chemical clusters, as defined by Auzel and Goldner[18] (i.e. dopant ions at the distance of several 10–1 nm). The goal of our paper was applying semi-empirical and ab initio calculations in order to investigate how does the distance between the dopant ions affect the energy of crystal fragment, a supercell, as well as correlate the results to available experimental data. We consider the reason of dopant ion clusters formation as a preference of an Eu3+ ion to occupy a crystallographic site at proximity of another Eu3+ ion, and this preference is (presumably) driven by total energy of a crystal fragment. The total energy, in turn, is affected by interaction between the two ions, which is, likely, a weak Eu–Eu bond formation involving d and f electrons. The lower energy corresponds to a higher thermodynamic stability of a system. If a more compact location of dopant ions in the structure resulted in an energy decrease (in respect to the structure with homogeneous location of dopant ions), the agglomeration of dopant ions should be considered as possible. Please note, that we did not assume any interstitial or other defects. We have simply substituted selected ions of the matrix material with the ions of dopant. Such substitution can be made at both neighboring or distant positions. However, we did not analyze, how any of the studied structures can be formed. Tetragonal yttrium vanadate, t-YVO4, was selected as a model system. It is a known host for Ln3+ activators[31–33], studied in form of crystals[33,34] and nanopowders[35,36]. The structure of the material is relatively simple and symmetric, with only one kind of crystallographic site of Y3+ available, which makes it easy to work with. Finally, this material is characterized by the numerous different Y–Y distances. Consequently, more data points for the plot of energy as a function of distances between the dopant ions can be provided. Quantum-mechanical computations using semi-empirical and ab initio DFT methods were performed on the selected material. Several Eu3+-doped structures (supercells) based on t-YVO4 unit cell were analyzed in order to investigate if the distances between Eu3+ dopant ions in YVO4 had any effect on the electronic energy of the system. The electronic energies of structures with mutually proximate or distant dopant ions were calculated.

detailed description of calculation is given in the next section). The pre-optimized unit cell was replicated along principal axes in order to form the supercells. The supercells were marked as i×j×k, where i, j and k indicate, how many times a unit cell was replicated by the respective axes x, y and z. Cubic 2×2×2 supercell (named “C”, 8 unit cells), a long rod-like 1×8×1 supercell (L8, 8 unit cells) as well as smaller 1×1×3 (L3z, 3 unit cells), 1×3×1 (L3y, 3 unit cells) 3×1×1 (L3x, 3 unit cells) were considered. Please note, that in all of the supercells periodic boundary conditions (PBC) were applied. That is, the calculations were performed as if the supercells were replicated infinitely along principal axes. Due to the PBC, the dopant ions in the L8, L3x, L3y and L3z supercells are proximate to their reflections along x and z (L8, L3y), x and y (L3z) or y and z (L3x) axes. Consequently, the distance between the dopant ions in question is, in fact, a distance between dopant ion planes. This is, on the one hand, a considerable drawback of the model, which is a result of compromise between size of the model system and calculation quality. On the other hand, the distance between planes still reflects the general idea of this paper. There were only two Eu3+ ions introduced into each supercell. Such approach results in only one Eu–Eu distance. Larger amount of dopant ions would have risen the question of how to take multiple distances into account. The study followed the scheme provided in Table 1, with gradual increase of plane wave cutoff energy. Table 1 List of the performed computations, by supercell kind and level of theory Supercell, number of unit cells

Method

Range of the Eu–Eu

Level of theory/

distances/nm

plane wave cutoff

C, 8

Sparkle/PM6

0.4–1.0

Semi-empirical

C, 8

GGA/PBE

0.4–1.0

DFT, 430 eV

L8, 8

GGA/PBE

0.4–2.5

DFT, 550 eV

L3x, L3y, L3z, 3

GGA/PBE

0.4–1.2

DFT, 600 eV

1 Experimental 1.1 Details of the model: the supercells The experimental[37] crystal structure of tetragonal YVO4 was obtained from the Crystallography Open Database (COD)[38,39]. The unit cell of the material is characterized by I41/amd:2 space group. The database unit cell geometry has been optimized using CASTEP (the

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Fig. 1 Supercells of YVO4 in question

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JOURNAL OF RARE EARTHS, Vol. 34, No. 8, Aug. 2016

1.2 Computations A preliminary study was performed using semi-empirical Sparkle/PM6 method. The method is the only semi-empirical approach with parametrization of Y, V and O (PM6), as well as Eu (Sparkle). The C supercell was composed using the unmodified geometry from the database. Only cell parameters of the structures were optimized. Fractional coordinates of atoms and lattice angles remained unchanged. MOPAC2012 code with default settings was used to calculate the total energies, while we used an original code to perform a simple steepest-descent relaxation of the lattice parameters. As the computations were a preliminary research, the results shall be described only briefly. In the DFT[40,41] computations, the relaxation was done using plane-wave DFT CASTEP[42,43] code, used with the Materials Studio package (the provided GUI was used to set up the calculations). The unit cell was pre-optimized using CASTEP, with the settings listed in Table 2. Both atom positions and lattice parameters were optimized while the original unit cell space group and symmetry restrictions were preserved. Next, the supercells were composed using the pre-optimized unit cell. In the following calculations, only lattice vectors were relaxed. Fractional coordinates and lattice angles were constrained. A variant of BFGS geometry optimization method from CASTEP[44] with fixed basis quality was used. General gradient approximation (GGA) PerdewBurke-Ertzenhoff (PBE)[45] exchange-correlation functional was used. This particular functional is rather commonly used for solid-state calculations (including Ln3+ ions with 4f as valence electrons[46–50]). Also, this functional is ab initio based. The systems were treated as spin-polarized and open-shell (“fix occupancy” feature was disabled). The initial number of unpaired electrons in the system was 12, provided by the two Eu3+. All of the computations were using Pulay mixing in the self-consistent field achievement (SCF) procedure with convergence window of 3. Other parameters of the SCF and BFGS procedures are given in Table 2. The computations were performed using ultrasoft effective core potentials in the reciprocal space representation. The configurations of valence electrons on atoms were: O 2s2 2p4,

V 3s2 3p6 3d3 4s2, Y 4s2 4p6 4d1 5s2, Eu 4f7 5s2 5p6 6s2. The finite basis set correction was performed automatically via single point energy calculations (preceding the geometry optimization) with the cutoff energies of E–10 eV, E–5 eV and E, where E (eV) is the plane wave cutoff energy of the calculation. Since the crystal radii of Eu3+ and Y3+ are close (eight-fold coordination, 0.1206 and 0.1159 nm, respectively)[51], no distortions at the proximity of dopant ions were modeled. In other words, selected Y atoms were substituted with Eu and no further changes (except for the lattice vector lengths) were introduced.

2 Results and discussion One of the reasons for selecting YVO4 as the model system, was a rich set of distances between the Y3+ positions available in this material. The effect of mutual location of dopant ions on energy transfer rates between them is only significant if the distances between the ions are relatively short. As the Ln–Ln energy transfer rates decay very fast with the donor-acceptor distance[11], those become very small at about 1–2 nm, depending on a particular material and the interacting ions. Even in a relatively small 2×2×2 (C) supercell, there are 11 possible Y–Y distances. This offers a possibility to get more data points in the total energy as a function of Ln–Ln distance plot. 2.1 Semi-empirical computations For the computations with Sparkle/PM6 method, one hundred input structures were prepared. In each of them, two Eu3+ ions were introduced into randomly selected Y3+ positions, replacing the original ions. After the lattice vectors optimization was finished, the total energies from every modeled structure were obtained. In MOPAC2012, the total energy is a sum of one-center (single-electron electron-nuclei and two-electron electron-electron) and two-center (resonance, exchange, electron-electron repulsion, electron-nuclei attraction and nuclei-nuclei repulsion) terms[52]. This total energies were plotted as a function of the Eu–Eu distance in the respective structures. A simple exponential function, y=y0+A·e–(x–x0)/τ, was fitted to the data points in Fig. 2. The distance between

Table 2 Parameters and thresholds of the DFT calculations SCF parameters

Plane wave Supercell

energy

Energy conv. Charge mixing

Spin mixing

Energy/

Force/

(eV/atom)

(eV/nm)

nm

GPa

15

1×10–5

0.3

1×10–4

0.05

2.0

15

–5

1×10

0.3

1×10–4

0.05

0.5

15

1×10–5

0.3

1×10–4

0.05

15

–6

0.1

–5

0.02

cutoff/eV

threshold/eV

amplitude

amplitude

340

1×10–6

0.5

2.0

C

430

–6

1×10

0.5

L8

550

1×10–6

0.5

600

–7

Unit cell

L3x, L3y, L3z

5×10

0.5

BFGS convergence thresholds Charge and spin

2.0

density mixing g-vectors/nm–1

5×10

Displacement/ Stress/

5×10

A. Shyichuk et al., Pairs of Ln(III) dopant ions in crystalline solid luminophores: an ab initio computational study

Fig. 2 A plot of total energy of YVO4:Eu3+ C supercell (computed with Sparkle/PM6) as a function of distance between the dopant ions; the numbers of the overlapping data points are indicated

the dopant ions was x, while the energy was y. An apparent decrease of the total energy with the decrease of the Eu–Eu distance can be noticed (Fig. 2). It can be concluded at this point, that more dense packing of the dopant ions within the matrix material results in decrease of the system total energy. Consequently, the total energy might be a factor promoting the tendency of dopant ions to agglomerate, that is, to occupy neighboring positions in the crystal. Most of the data points in Fig. 2 are overlapping. That is, different input structures with different particular positions occupied by the Eu3+ ions are likely to have the same energy provided the same Eu–Eu distance. On the one hand, the result of this semi-empirical modeling is quite promising. It agrees well with the idea of the study. On the other hand, the level of theory of semi-empirical computations is too low for this kind of study. The Sparkle model represents the Ln3+ cations as point charges. Next, the electron correlation is missing in PM6. We have performed several additional PM6-only computations on YVO4:La3+ and YVO4:Lu3+ supercells in order to exclude the effect of the Sparkle model. Both La and Lu are parametrized as 3d metals in the PM6 model. Again, a strong dependence of the total energies on the Ln–Ln distance was found. In the case of YVO4:La3+ the trend was similar to the one of YVO4:Eu3+ (i.e., shorter Ln–Ln distance corresponded to lower total energy). However, in the case of YVO4:Lu3+ the trend was opposite. Summarizing, due to the inconsistencies mentioned above, the results of Sparkle/PM6 modeling were considered only as preliminary. The next step was to repeat the study using ab initio DFT. 2.2 DFT calculations The first step of the DFT study was a simple repetition of the Sparkle/PM6 calculations, with two exceptions.

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The amount of input structures was reduced to 50, while the dopant ions were not placed at the supercell wall. (Computations with dopant ions at supercell wall were failing repeatedly.) The BFGS final energies from the computations were collected. In CASTEP, total energy is derived accordingly to Kohn-Sham equations, using functional of electron density. The functional includes parts corresponding to effective (Kohn-Sham) potential, electron kinetic energy, electron Coulomb (Hartree) repulsion and electron exchange-correlation[42]. The obtained plot of the total energy vs. Eu–Eu distance (Fig. 3) reveals a similar trend to the one of the Sparkle/PM6 results (Fig. 2). Similarly to the previous figure, an exponential fit was applied to the data points. A conclusion can be made that shorter Eu–Eu distance corresponds to lower energy of the system. The next step was to clarify the dependence shown in Fig. 3. The idea was to create a system in which no random substitution was required. Here, linear supercells were used. First, critical distance of the studied interaction was analyzed. For this purpose, an untypical 1×8×1 (L8) supercell was used. In the L8 supercell, there are 32 positions of Y, of which 16 are located at the supercell wall. Another 16 form a zig-zag line through the middle of the supercell. This latter positions were used for the substitution, that started with the two close-most positions at the supercell center. The next step used the next two positions to each side. Finally, the two furthermost positions were substituted, resulting in total of 8 structures, where distances between the Eu3+ ions varied between 0.393 and 2.522 nm (optimized values). The data points in Fig 4 are presented in the described order. It is to be noted that in the first four cases, the Eu–Eu line segment is located inside the supercell, while in the next four structures, the shortest Eu–Eu line segment pierces through the smallest face of the supercell to its reflection at periodical boundary. Thus, the Eu–Eu distance increases and then decreases.

Fig. 3 A plot of the energies of YVO4:Eu3+ C supercell (computed with DFT) as a function of distance between the dopant ions; the numbers of the overlapping data points are indicated

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Fig. 4 A plot of the energies of YVO4:Eu3+ L8 supercell (computed with DFT) as a function of distance between the dopant ions; note that the first four and the second four data points correspond to different structures

The first conclusion is that the periodic boundary conditions (PBC) did not significantly affect the analyzed phenomenon: there is no much difference in the total energy values between the structures with the Eu–Eu line segment crossing the supercell wall (marked with ′ in Fig. 4) and the structures with Eu–Eu line segment inside the supercell. Next, the interaction of interest is relatively short-ranged: the values of energy for proximate location of Eu3+ ions (0.393 nm) are clearly smaller than those corresponding to distant location (1.09 nm and more). The latter data points fluctuate slightly. The next step was aimed to obtain more data points in the 0.4–1.0 nm range of the Eu–Eu distance. For this purpose, smaller supercells of three unit cells of YVO4 were used, i.e. L3z, L3y and L3x supercells (Fig. 1). In the case of these small supercells, one of the Eu3+ ions was located in the Y3+ site at the smallest face of the supercells, while one of the other Y3+ sites was substituted by the second Eu3+ ion (sites in the supercell wall were excluded). The results of computations are presented in Fig. 5. The data

points can be fitted with the exponential function with high values of R2. In this case, a tendency of exponential decrease of the energy with the Eu–Eu distance is very clearly seen. The energy difference between the systems with furthermost location of the dopant ions and the systems with proximate location of them is about 0.08 eV in the case of the C supercell, 0.03 eV in the case of L8 supercell and about 0.1 eV in the cases of the L3z, L3y and L3x supercells. Note, that these values are orders of magnitude higher, then the SCF convergence threshold. The computations resulted in lattice vectors values that are similar but not identical. In the case of the C and L8 supercells, the values did not show any systematic dependence on the Eu–Eu distance and were rather randomly varied. In the case of the L3z, L3y and L3x supercells, the lengths of a and b lattice vectors increase slightly with increase of the Eu–Eu distance, while the c lattice vector decreases a little. However, the dependencies are not quite clear and the data points are rather scattered. The data on the variations of the lattice vectors are collected in Table 3. We have made an attempt to predict the probability, for a given supercell, of the two dopant ions to be found at neighboring positions, using the calculated energies and Boltzmann distribution. Let us assume there are n ambiguous states, each i-th state characterized by energy Ei. Then, at given temperature T, the probability p(i) of occurrence of particular i-th state in population is given by (k is Boltzmann constant): exp( − Ei / kT ) p (i ) = n (1)

∑ exp(− E

j

/ kT )

j =1

We have defined a “state” as a supercell in which the two Eu3+ ions are located at the distance li and characterized by a total energy Ei. That is, a particular li corresponded to a particular Ei, for a given supercell. We assume that analyzing a cubic 8-unit-cell piece of a real YVO4 doped with 6.25 mol.% Eu3+ (corresponding to the C supercell), one could find the two Eu3+ ions at the distance li with the p(i) probability. The E=f(l) functions were derived using exponential fits of the data points in Figs. 3 and 5. The lines in the graphs represent the functions. Functions in the followTable 3 Variation of the lattice vector lengths Lattice vector a Supercell

3+

Fig. 5 Plots of the final energies of YVO4:Eu L3z, L3y and L3x supercells (computed with DFT) as a function of distance between the dopant ions

Value/ nm

±/pm

±/%

Lattice vector b Value/ nm

±/pm ±/%

Lattice vector c Value/ nm

±/pm

±/% 0.008

C

1.4777

0.08

0.006 1.4777

0.08 0.006 1.2723

0.10

L8

0.7161

0.06

0.009 5.7525

0.42 0.007 0.6386

0.04

0.006

L3z

0.7233

0.11

0.015 0.7233

0.10 0.014 1.8972

0.27

0.014

L3y

0.7233

0.03

0.005 2.1698

0.05 0.002 0.6324

0.02

0.003

L3x

2.1698

0.10

0.004 0.7233

0.05 0.007 0.6324

0.03

0.005

A. Shyichuk et al., Pairs of Ln(III) dopant ions in crystalline solid luminophores: an ab initio computational study

ing Eqs. (2–5) correspond to the C, L3x, L3y and L3z supercells, respectively (l in nm): E=–130418.15507–3.09714 exp (–l/0.103941) (2) E=–60528.29326–0.95902 exp (–l/0.168692) (3) E=–60528.29715–1.09723 exp (–l/0.156607) (4) E=–60528.26052–4.48266 exp (–l/0.117837) (5) Please note that for some i, there are ni geometries (ni>1) with the same li and different particular positions of the dopant ions. That is, there are ni states characterized by the same Ei and li. Although calculations show that the energies are not exactly the same for such structures, the differences were neglected. It can be clearly seen in Figs. 3–5 that the distance between the ions causes much larger effect on the energy. The Eq. (1) must be extended in order to take ni into account: p (i ) =

ni exp( − Ei / kT )

(6)

n

∑n

j

exp( − E j / kT )

j =1

The values of ni were acquired from the crystal structure of YVO4. Finally, as the energies are quite large, it is more convenient to compute e–ΔE/kT (instead of e–E/kT), which was done in the actual calculations using a Python script:

⎛ n ⎞ ⎜ ∑ n j exp( − E j / kT ) ⎟ j =1 ⎟ p (i ) = ni ⎜ ⎜ exp( − Ei / kT ) ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ n ⎛ −E + E = ni ⎜ ∑ n j exp ⎜ j i ⎝ kT ⎝ j =1

−1

⎞⎞ ⎟⎟ ⎠⎠

825

rial, namely YVO4:Eu3+, was used as a test case. The model was based on the assumption that different arrangements of the dopant ions in the Y positions of the YVO4 lattice would result in different electronic energies of the system. Both semi-empirical and DFT quantum-chemical computational approaches were used to test this idea. It was found out that the decrease of Eu–Eu distance results in apparent decease of system's total energy. As far as the lower-energy configurations are preferred and the shorter distance corresponds to the lower energy, the observed phenomenon was considered one of the reasons of dopant ions pairs and larger clusters formation. Finally, using the calculated energies, an estimation of the probability of occurrence of configurations with distant or proximate dopant ions was made, on the basis of Boltzmann distribution. Such estimation correlated quite well with the experimental data available in literature. Acknowledgements: The computations were performed using the resources of Poznań Supercomputing and Networking Centre (Grant No. 175) and Wrocław Centre of Networking and Supercomputing (Grant No. 300). The latter provided an access to Materials Studio software.

References: (7) −1

According to the calculations, for the C supercell (or, for YVO4:6.25 mol.%Eu3+), the probability of the ions to occur at the shortest possible distance (0.3891 nm) is about 40%. In case of the small supercells (e.g., YVO4:16.667 mol.%Eu3+), the probability of the dopant ions to be found at the shortest distance between them is about 77%. Assuming that the probability of pairing depends linearly on the amount of dopant, even at doping rate below 1%, about 10% to 20% of dopant ions should occur in pairs. According to Ref. [25], the abundance of pairs in YVO4:0.58 mol.%Nd3+ is about 15%. Thus, our very crude approximation results in reasonable agreement with the experimentally observed data and might serve at least as a good starting point in further computational studies concerning dopant agglomeration and clustering in lanthanide-doped materials.

3 Conclusions A theoretical model based on supercell representation of a crystal was used to analyze the effects of distance between the dopant ions on electronic energy of the system (a considered cause of dopant ions agglomeration in Ln3+-based luminophores). Particular luminescent mate-

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