Electronic Structure Calculation of Cr³⁺ and Fe³⁺ in Phosphor Host Materials Based on Relaxed Structures by Molecular Dynamics Simulation

Abstract

The electronic structures of the luminescent center ions Cr³⁺ and Fe³⁺ in the deep red phosphors LiAl₅O₈:Cr³⁺, α-Al₂O₃:Cr³⁺, and γ-LiAlO₂:Fe³⁺ were calculated by the DV-Xα method, in which the local distortion induced by the replacement of Al³⁺ sites in the host crystals by the luminescent center ions was reproduced by classical molecular dynamics (MD) simulation. The MD simulations based on classical dynamics allowed for the handling of more than 1000 atoms for the lattice relaxation calculations, which was advantageous to simulate situations in which a small number of foreign atoms (ions) were dispersed in the host lattice as in phosphors, even when typical periodic boundary conditions were applied. The relaxed lattices obtained after MD indicated that the coordination polyhedra around Cr³⁺ and Fe³⁺ expanded in accordance with the size difference between the luminescent center ions and Al³⁺ in the host crystals. The overall profiles of the partial density of states (p-DOSs) of the isolated Cr³⁺ and Fe³⁺ 3d orbitals were not significantly affected by the lattice relaxation, whereas the widths of the energy splitting of the 3d orbitals were reduced. The electronic structure calculations for Fe–Fe pairs in γ-LiAlO₂ showed that the antiferromagnetic interactions with antiparallel electron spins between the Fe³⁺ ions were preferred, especially when the Fe–Fe pair was on the first-nearest neighboring cation sites.

1. Introduction

Phosphors are materials that emit luminescence, usually in the visible range, when they are stimulated by high-energy radiation such as X-rays or ultraviolet (UV) rays, or by electron beams. Inorganic phosphors have been practically used in fluorescent lamps and cathode-ray tubes, and they are increasingly being applied to lighting applications using white LED lamps. Yttrium aluminum garnet (YAG) doped with Ce³⁺ is a conventional yellow phosphor excited by blue light emitted from LED chips, and SiAlON and CaAlSiN₃ doped with Eu²⁺ have recently been developed as phosphors that emit orange to red luminescence in white LED lamps [1,2,3,4,5,6,7,8].

Recently, we have focused on the 3d transition metals Cr³⁺, Mn⁴⁺, and Fe³⁺ as luminescent centers [9,10] since the emission exists in the deep red region, as in Mg₂TiO₄:Mn⁴⁺, α-Al₂O₃:Cr³⁺, LiAl₅O₈:Fe³⁺, and γ-LiAlO₂:Fe³⁺. Such deep red emission is difficult to achieve with rare earth phosphors, and their relatively sharp emission peaks are advantageous for improving color rendering in lighting applications.

This paper proposes a computational approach that combines classical MD simulations and molecular orbital (MO) calculations using the DV-Xα method in order to effectively investigate the electronic structures of luminescent ions in phosphors. The advantages of this approach are as follows.

(1)

MD reproduces the local lattice distortion induced by substitution of foreign ions (in this case, luminescent center ions) prior to the electronic structure calculations. For example, γ-LiAlO₂:Fe³⁺ is a deep-red phosphor [9,11] in which Fe³⁺ replaces Al³⁺ in the host lattice. The ionic radii of Fe³⁺ and Al³⁺ in four-hold coordination are 0.49 Å and 0.39 Å, respectively [12], with the larger Fe³⁺ ion pushing the ligand oxygens away and expanding the coordination polyhedron. Such local distortion affects the electronic structure of the central Fe³⁺ according to the conventional crystal field theory.

MD can readily give a relaxed arrangement of the constituent atoms, even when two or more foreign ions or vacancies are introduced in a host lattice.

(2)

With the development of computational techniques, ab-initio MDs such as CASTEP [13], VASP [14,15,16,17], and FMO-MD [18] have become available over the decades. In contrast to these ab-initio MDs, classical MDs describe atomic interactions between the constituent atoms by simple two-body potentials, allowing a larger number of atoms to be handled in the simulation with reasonable accuracy. It is suitable for simulating a dilute situation in which luminescent center ions are randomly dispersed in the lattice of the phosphor host crystal. Unexpected periodicity in the arrangement of luminescent center ions in the calculation cell, due to periodic boundary conditions often applied in MD simulations, can be avoided, as it can occur when the cell is constructed with a limited number of atoms.

A similar approach has been reported for glass materials to investigate the electronic structures of glass under strong electric fields [19], in which MD was effectively used to simulate the atomic arrangement, since structural analysis such as the X-ray diffraction technique is not applicable to glass. Quite limited research has attempted to reveal the effects of local distortion using a combination of classical MD and MO techniques in dilute systems such as phosphors. This work is a pioneering study applying this approach to crystalline materials, focusing on LiAl₅O₈:Cr³⁺, α-Al₂O₃:Cr³⁺, and γ-LiAlO₂:Fe³⁺ deep red phosphors.

Figure S1 shows the crystal structures of the host materials. LiAl₅O₈ has a spinel-type structure [20], with tetrahedral and octahedral sites for the cations, and Cr³⁺ is considered to replace Al³⁺ in the octahedral sites. Corundum-type α-Al₂O₃ has a crystal structure in which two AlO₆ octahedra share one face to form an Al₂O₉ dimer, providing octahedral sites for Cr³⁺ [21]. The crystal structure of γ-LiAlO₂ contains tetrahedra of AlO₄ and LiO₄ [22], offering tetrahedral sites for the luminescent center ion Fe³⁺. Figure S2 shows the emission and excitation spectra of these deep red phosphors. Cr³⁺ shows sharp line spectra at 716 nm for LiAl₅O₈ and 694 nm for α-Al₂O₃, with two excitation bands around 420 nm and 570 nm due to the d-d transitions. Fe³⁺ in γ-LiAlO₂ shows a rather broad emission peak from 640 to 900 nm, with the peak top around 740 nm. The main excitation band of Fe³⁺ in the UV region around 240 nm is due to the charge transfer transition; the smaller excitation bands at 390 nm and 450 nm are due to the d-d transitions.

The objective of this work is to demonstrate the effectiveness and potential of our combined MD and MO approach to elucidate the effect of lattice relaxation on the electronic structures of the luminescent center ions in the host lattices. This approach will lead to a better understanding of phosphor materials and to help in developing materials with better properties.

2. Methods

2.1. Molecular Dynamics (MD) Simulation

MD simulations were carried out using the MXDORTO code [23]. The interaction between the constituent atoms was described by the Born–Mayer–Huggins type pair potential (Equation (1)).

$$U_{ij}=\frac{Z_i{Z}_j{e}^2}{r_{ij}}{+A}_{ij}exp\left(-\frac{r_{ij}}{{\rho }_{ij}}\right)$$

(1)

In Equation (1), U_ij is the potential of a pair of atoms i and j, and z_i and z_j are the oxidation numbers of ions i and j, respectively. The constituent atoms were assumed to be fully ionized; Al, Cr, and Fe were assumed to be +3, Li +1, and O −2. e is the electron charge, and r_ij is the atomic distance between ions i and j. The first term represents the electronic interaction between ions i and j, and A_ij and ρ_ij in the second term are potential parameters that determine the short-range repulsive interaction. These potential parameters were refined to reproduce the reference crystal structures of α-Al₂O₃ [21], Li₂O [24], α-LiAlO₂ [25], γ-LiAlO₂ [22], LiAl₅O₈ [20], Cr₂O₃ [26], LiCrO₂ [27], Fe₂O₃ [28], and LiFe₅O₈ [29] with agreement within 2% of the lattice parameters in the NPT ensemble (N: number of atoms, P: pressure, and T: temperature) at 300 K (Table S1). The potential parameters A_ij and ρ_ij are listed in

Table 1. Potential parameters, A_ij and ρ_ij, used in this work. Short-range repulsion was not considered for the cation pairs except Li–Li.

Atom Pair	Aij (eV)	ρij (Å)
O–O	1997.28	0.2810
Al–O	1716.41	0.2810
Li–O	960.0	0.2690
Li–Li	98.9357	0.2994
Cr–O	1156.68	0.3131
Fe–O	1200.68	0.3151

. Table S1 shows the coincidence between the simulated and reference crystal structures with respect to the lattice parameters and the mean square displacement (MSD) of the constituent atoms in MD. The MSD was comparable to the magnitude of typical thermal vibrations. It should be noted that a single set of the potential parameters for each atom pair reproduced the reference crystal structures properly, even though both octahedral and tetrahedral coordination sites exist for Li⁺, Al³⁺, and Fe³⁺ (only octahedral for Cr³⁺) in the reference crystals.

The initial atomic arrangement of each host crystal was obtained from the crystal structure reported in the literature [20,21,22]. The orthogonal MD cells were constructed by repeating the crystallographic unit cells with x, y, and z axes of 23–26 Å and contained 1512, 1800, and 1600 atoms for LiAl₅O₈, α-Al₂O₃, and γ-LiAlO₂, respectively. The periodic boundary conditions were applied to the calculations. Cr³⁺ and Fe³⁺ ions were introduced in the MD cells by substituting Al³⁺ sites according to the following conditions.

• To investigate the effect of lattice relaxation for the isolated luminescent center ion, one Cr³⁺ ion was introduced to replace the Al³⁺ on the octahedral site in LiAl₅O₈ and α-Al₂O₃; in γ-LiAlO₂, the Al³⁺ on the tetrahedral site was replaced by one Fe³⁺ ion.
• To investigate the interaction between Fe–Fe pairs in γ-LiAlO₂, two Fe³⁺ ions were placed at the first-, second-, and third-nearest neighboring Al³⁺ positions.

The initial structures were relaxed for 1000 steps at 300 K with a time step of 2 fs in the NVT ensemble (V: volume), and for 3000 steps in the NPT ensemble. The temperature of the systems was then increased to 1500 K at a ramp rate of 0.1 K/step in the NPT ensemble and maintained at 1500 K for 100,000 steps. The systems were cooled to 300 K at a ramp rate of −0.1 K/step and relaxed for 31,000 steps at 300 K. The relaxed positions of the constituent atoms were obtained by averaging the atomic coordinates of the last 1000 steps.

The coordination environments around the cations were discussed using the frequency distribution (FD) corresponding to the number of focusing atoms surrounding the central atom in the thin shell at a distance R with a width of ΔR (here 0.01 Å).

2.2. Molecular Orbital (MO) Calculations

The electronic structures of the luminescent center ions of Cr³⁺ and Fe³⁺ were calculated by the DV-Xα method using the DVSCAT code [30,31], where the relativistic effects were not taken into account. The calculation clusters for MO were extracted from the relaxed structurers obtained by the MD simulations. The clusters were arranged with Cr³⁺ or Fe³⁺ at the centers, with Al and Li octahedra or tetrahedra surrounding the central CrO₆ or FeO₄. The clusters used in the MO calculations are shown in

Table 2. Clusters for molecular orbital calculations.

Luminescent Center	Host Crystal	Calculation Cluster
Cr3+ (octahedral)	LiAl5O8	(Li2Al10CrO38)41−
	α-Al2O3	(Al13CrO45)48−
Fe3+ (tetrahedral)	γ-LiAlO2	(Li9Al6FeO32)34− for isolated Fe3+ (Li9Al5Fe2O32)34− for Fe3+–Fe3+ interaction at the first- and second-nearest neighbors(Li9Al6Fe2O34)35− for Fe3+–Fe3+ interaction at the third-nearest neighbor

and Figures S3 and S4.

The calculation clusters were embedded in the Madelung potential generated by the Evjen method [32], and atoms outside the clusters were treated as point charges placed at the positions obtained by MD. The spin polarization was taken into account in the MO calculations, and both Cr³⁺ and Fe³⁺ were started in the high-spin states.

3. Results and Discussion

3.1. Electronic Structures of Isolated Cr³⁺

Figure 1 shows the averaged FDs of O atoms around Al in LiAl₅O₈:Cr³⁺ (a) and α-Al₂O₃:Cr³⁺ (b). Spinel-type LiAl₅O₈ contains tetrahedral and octahedral sites for Al³⁺. The crystallographic data of LiAl₅O₈ [20] provide 1.779 Å × 3, 1.835 Å × 1 for the A–O distances at the tetrahedral site and 1.856 Å × 2, 1.905 Å × 2, and 1.940 Å × 2 for the octahedral site. The corundum structure of α-Al₂O₃ consists of AlO₆ octahedra forming face-sharing Al₂O₉ dimers (Figure S1), with Al³⁺ at the octahedral off-center position and the Al–O distances of 1.857 Å × 3 and 1.970 Å × 3 [21].

The lattice relaxation by MD resulted in bimodal Al–O distribution with peak tops at around 1.78 Å and 1.91 Å for LiAl₅O₈:Cr³⁺ (a) and 1.80 Å and 2.00 Å for α-Al₂O₃:Cr³⁺ (b), as shown Figure 1. The red arrows in Figure 1 indicate the Cr–O distances obtained by MD: 1.84 Å × 2 (Cr–O₁), 1.96 Å × 2 (Cr–O₂), and 1.99 Å × 2 (Cr–O₃) for LiAl₅O₈:Cr³⁺ (a), and 1.85 Å × 3 (Cr–O₁) and 2.07 Å × 3 (Cr–O₂) for α-Al₂O₃:Cr³⁺ (b). The arrows are on the longer sides of the Al–O peaks, indicating that the larger Cr³⁺ ions expanded the CrO₆ octahedron compared to AlO₆. The MD results showed that the Cr³⁺ in the AlCrO₉ dimer in α-Al₂O₃:Cr³⁺ was displaced by 0.06 Å toward the octahedral void compared to the position of Al³⁺ in the Al₂O₉ dimer, which was consistent with ~0.1 Å estimated in the crystal field calculations for α-Al₂O₃:Cr³⁺ [33].

Figure 2 shows the partial density of states (p-DOSs) of Cr 3d and O 2p orbitals in LiAl₅O₈:Cr³⁺ (a) and α-Al₂O₃:Cr³⁺ (b), calculated based on the lattice relaxed by MD (solid line) and the unrelaxed lattice without MD (dotted line), where the HOMO levels are set at 0 eV. Figure 2 shows that the 3d orbitals of Cr³⁺ were divided into two groups, and that Cr³⁺ preferred the high-spin state regardless of the host material or lattice relaxation. These basic features were consistent with the considerations based on the conventional crystal field theory for Cr³⁺ in the octahedral symmetry.

The CrO₆ octahedra were not in the ideal octahedral symmetry (O_h) even without relaxation, but for ease of understanding, the conventional notations t_2g and e_g are used to represent the split d orbitals. The HOMO level is in the t_2g orbitals, reflecting the d³ configuration of Cr³⁺. Hybridization of Cr 3d and O 2p was found to be limited, with the O 2p and Cr 3d contributions observed separately at −10 to −4 eV and −1 to 6 eV, respectively.

The lattice relaxation by MD did not significantly affect the overall profiles of the p-DOSs, but the d-orbital splitting between t_2g and e_g orbitals of Cr 3d decreased from 2.52 eV to 2.30 eV in LiAl₅O₈ and from 2.37 eV to 2.01 eV in α-Al₂O₃ due to the lattice relaxation, reflecting a reduction in the ligand field strength due to the expansion of the coordination polyhedra.

3.2. Electronic Structures of Fe³⁺

3.2.1. Isolated Fe³⁺

Figure 3 shows the FD of Al–O in γ-LiAlO₂:Fe³⁺ obtained by MD. The averaged Fe–O distance is on the longer side of the Al–O peak, indicating that the FeO₄ tetrahedron expanded due to the size difference between Fe³⁺ and Al³⁺.

Figure 4 plots the p-DOSs of Fe 3d and O 2p orbitals for the lattices with MD (solid line) and without MD (dotted line), with the HOMO levels set at 0 eV. Fe³⁺ in the d⁵ configuration converged to the high-spin state with two groups of the d orbitals. Although the FeO₄ tetrahedron was not in the ideal tetrahedral symmetry (T_d), the conventional notations t₂ and e are used as discussed for Cr³⁺ above. The HOMO level is in the t₂ orbital. The lattice relaxation by MD did not significantly affect the overall profile of the p-DOS as in the case of Cr³⁺ (Figure 4), but it did reduce the energy splitting between the t₂ and e orbitals from 1.22 eV (dotted line) to 1.17 eV (solid line).

The MO calculations show that the Fe 3d orbitals were highly hybridized with O 2p, which was more pronounced for the up-spin electrons. From −8 to −3 eV, the large contribution of Fe 3d was recognized along with O 2p, while from −2 to 1 eV, the contribution of O 2p was observed along with Fe 3d.

3.2.2. Fe³⁺–Fe³⁺ Interaction

To investigate the magnetic interactions between the Fe–Fe pairs, the additional Fe³⁺ was placed on the first-, second-, and third-nearest neighboring Al³⁺ sites from the central Fe³⁺ (Figure S4). Figure S4 shows the clusters for the MO calculations for the Fe–Fe pairs.

Figure 5 shows the FD of Al–Al. The arrows on the figure indicate the distances of Fe–Fe pairs on the first-, second-, and third-nearest neighboring Al³⁺ sites.

Compared to the Al–Al distances, the distance between the first-nearest Fe–Fe pair was elongated, and the distances between the second- and third-nearest Fe–Fe pairs were almost the same as the corresponding Al–Al distance. It indicated that the effect of the size difference between Al³⁺ and Fe³⁺ was probably counteracted by the connection angles between the point-sharing Al(Fe)O₄ tetrahedra.

Figure 6 shows the p-DOSs of the Fe 3d and O 2p orbitals for the first-nearest Fe–Fe pair. The second- and third-nearest Fe–Fe pairs yielded the p-DOSs with substantially the same features as in Figure 6. The initial ferromagnetic and antiferromagnetic configurations converged to their respective configurations. To compare the relative stability of the systems with the different magnetic configurations, the HOMO levels of the p-DOSs were not set at 0 eV; the HOMO levels were indicated by the dashed lines.

Table 3. Difference in the HOMO energies between antiferromagnetic and ferromagnetic configurations in the Fe–Fe pairs.

	HOMO Energy (eV)		Energy Difference (eV)
	Antiferromagnetic	Ferromagnetic	Anti.−Ferro.
Nearest	0.27	0.76	−0.49
Second nearest	0.30	0.32	−0.02
Third nearest	0.18	0.23	−0.05

compares the HOMO energies obtained for the different magnetic configurations. It should be noted here that the comparison should be limited to within the same calculation cluster, since the calculated energy is also influenced by the cluster type itself. Table 3 shows that the antiferromagnetic interaction was favored by 0.49 eV, 0.02 eV, and 0.05 eV for the Fe–Fe pairs at the first-, second-, and third-nearest neighboring positions, and the stabilizing effect of the antiferromagnetic interaction was particularly pronounced for the first nearest Fe–Fe pair. Such magnetic interaction was considered to be related to the well-known “superexchange interaction” often observed in compounds containing magnetically active cations [34,35,36]. Fe³⁺–O²⁻–Fe³⁺ in Figure S4a was about 130°, intermediate between 90° and 180°. The magnetic interaction between Fe³⁺ occupying tetrahedral sites have rarely been reported in the literature, whereas for Fe³⁺ in octahedral sites, the antiferromagnetic interaction have been reported, for example, in spinel-type ZnFe₂O₄ and colundum-type Fe₂O₃ with the Neel temperatures of ~10 K and 950 K, respectively.

4. Conclusions

This work demonstrated the MO calculations of the electronic structures of the luminescent center ions Cr³⁺ and Fe³⁺ introduced in the host lattices LiAl₅O₈, α-Al₂O₃, and γ-LiAlO₂. The local distortions induced by replacing the smaller Al sites were reproduced by the MD simulations based on the classical dynamics. For the isolated Cr³⁺ and Fe³⁺ in these host crystals, the MD simulations confirmed the expansion of the coordination polyhedra around these cations. The MO calculations showed that these luminescent center ions preferred the high-spin states, and that the lattice relaxation by MD reduced the energy splitting of the d orbitals between t_2g and e_g for Cr³⁺ on the octahedral site and t₂ and e for Fe³⁺ on the tetrahedral site. Without the lattice relaxation by MD, the energy splitting was overestimated by about 10% for LiAl₅O₈:Cr³⁺, 20% for α-Al₂O₃:Cr³⁺, and 4% for γ-LiAlO₂:Fe³⁺. The MD simulations also indicated that the paired Fe³⁺ ions in γ-LiAlO₂:Fe³⁺ shifted away from each other, which was more pronounced for the first-nearest neighboring pair. The Fe–Fe pairs preferred the antiferromagnetic interactions, and the degree of the stabilization of the antiparallel orientation of the d⁵ electrons was found to be maximal at about 0.5 eV for the first-neighboring Fe–Fe pair.