Study on Local-Structure Symmetrization of K₂XF₆ Crystals Doped with Mn⁴⁺ Ions by First-Principles Calculations

Abstract

The crystals of Mn⁴⁺-activated fluorides, such as those of the hexafluorometallate family, are widely known for their luminescence properties. The most commonly reported red phosphors are A₂XF₆: Mn⁴⁺ and BXF₆: Mn⁴⁺ fluorides, where A represents alkali metal ions such as Li, Na, K, Rb, Cs; X=Ti, Si, Ge, Zr, Sn, B = Ba and Zn; and X = Si, Ge, Zr, Sn, and Ti. Their performance is heavily influenced by the local structure around dopant ions. Many well-known research organizations have focused their attention on this area in recent years. However, there has been no report on the effect of local structural symmetrization on the luminescence properties of red phosphors. The purpose of this research was to investigate the effect of local structural symmetrization on the polytypes of K₂XF₆ crystals, namely O_h-K₂MnF₆, C_3v-K₂MnF₆, O_h-K₂SiF₆, C_3v-K₂SiF₆, D_3d-K₂GeF₆, and C_3v-K₂GeF₆. These crystal formations yielded seven-atom model clusters. Discrete Variational Xα (DV-Xα) and Discrete Variational Multi Electron (DVME) were the first principles methods used to compute the Molecular orbital energies, multiplet energy levels, and Coulomb integrals of these compounds. The multiplet energies of Mn⁴⁺ doped K₂XF₆ crystals were qualitatively reproduced by taking lattice relaxation, Configuration Dependent Correction (CDC), and Correlation Correction (CC) into account. The ⁴A_2g→⁴T_2g (⁴F) and ⁴A_2g→⁴T_1g (⁴F) energies increased when the Mn-F bond length decreased, but the ²E_g → ⁴A_2g energy decreased. Because of the low symmetry, the magnitude of the Coulomb integral became smaller. As a result, the decreasing trend in the R-line energy could be attributed to a decreased electron–electron repulsion.

1. Introduction

Incandescent and fluorescent lighting sources have been rapidly replaced by White Light Emitting Diodes (WLEDs) in homes, offices, and public areas. They are made of a blue LED chip with a yellow phosphor. WLED is the most energy-efficient conversion source compared to previously existing lighting sources, yet it creates pseudo-white light due to a lack of red emissions. Rather than employing all basic colors of LED chips, mixing blue LED chips with yellow and red phosphors is simpler and less expensive. The blue LED chips are typically made of InGaN [1], whereas the yellow phosphor components are composed of Y₃Al₅O₁₂: Ce³⁺ [2]. The high-performance red phosphors are Eu²⁺ doped nitrides [3,4,5,6,7,8,9]. Unfortunately, red phosphors are expensive due to scarcity and challenging synthesis conditions, such as extreme temperatures and nitrogen pressure. Finding novel red phosphor materials that are appropriate for WLED is currently challenging. Significant performance factors for white light that are used in general lighting include high Quantum Efficiency (QE > 70%), resistance to thermal quenching (preferably > 80% of the luminescence intensity should be sustained at 450 K), and strong color quality, which includes a low Correlated Color Temperature (CCT) of 3000 K and a high Color Rendering Index (CRI > 70).

The most commonly reported red phosphors are fluoride-based, such as A₂XF₆: Mn⁴⁺ and BXF₆: Mn⁴⁺, where A represents alkali metal ions, such as Li, Na, K, Rb, Cs; X = Ti, Si, Ge, Zr, Sn), B = Ba and Zn and X = Si, Ge, Zr, Sn, and Ti. K₂SiF₆: Mn⁴⁺, KNa₂SiF₆: Mn⁴⁺, and K₂TiF₆: Mn⁴⁺, in particular, have shown good potential for WLED as a red phosphor under blue LED chip stimulation. The first red Mn⁴⁺-doped fluoride phosphor, K₂SiF₆: Mn⁴⁺, was published in 1973 [10]. K₂SiF₆ is one of the most promising hexafluoride hosts, with a slightly higher Luminous Efficacy of Radiation (LER) upon Mn⁴⁺ doping than K₂TiF₆ and a 30% higher External Quantum Efficiency (EQE) than KnaSiF₆: Mn⁴⁺ [11]. Mn⁴⁺, when doped in K₂SiF₆ or K₂TiF₆ as a red phosphor, yields WLEDs with warm-white CCTs ~3000 K and good CRIs ~90, as demonstrated by Setlur et al. [12]. The d–d transitions in Mn⁴⁺ cause the particular red emission line detected in K₂SiF₆: Mn⁴⁺ to be approximately 630 nm (15,873 cm⁻¹ or 1.97 eV) [13]. Nevertheless, the chemical and thermal stability problems and safety hazards of K₂SiF₆ and K₂TiF₆ doped with Mn⁴⁺ have been reported.

The aforementioned red-phosphor performance is highly dependent on local structure. Numerous research teams have concentrated on the modification and enhancement of phosphor luminescence properties through the alteration of the local crystal structure. The “Cation-Size-Mismatch effect”, “Neighboring-Cation Substitution effect”, and “Nanosegregation and Neighbor-Cation Control effect”, among other new luminescence mechanisms, were reported by Liu’s group in Ce³⁺ and Eu²⁺-doped (oxy)nitrides based on the variation in the local crystal structure [14,15,16,17]. Ram’s team also showed that slight modifications to the local structure of phosphor systems such as La₃xCexSi₆N₁₁, SrxBa₂xSiO₄: Eu²⁺, etc., could lead to appreciable gains in luminescence performance [18,19]. Cheetham’s team discovered that local crystal structural deformation accounted for a significant spectrum change from blue to yellow light from Ca₂SiO₄: Ce [20].

The Ligand Field Theory (LFT) has been frequently used to successfully evaluate the multiplet energy levels and optical spectra of Transition Metal (TM) ions in crystals [21]. However, it is an empirical method in which the measured spectrum is used to determine the Racah parameters and crystal field splitting. Watanabe and Kamimura produced the first non-empirical forecast in the late 1980s [22,23] using a combination of the local density approximation (LDA) and LFT. On the other hand, a number of teams, including Daul et al. [24], Wissing et al. [25], and Oliveira et al. [26,27], have also performed first-principle calculations based on the Density Functional Theory (DFT). However, obtaining the many-electron wave functions proved unfeasible. During the previous ten years, Ogasawara’s team created the Discrete Variational Multi-Electron (DVME) approach [28]: a non-empirical first-principles many-electron calculation technique. It uses both a Configuration Interaction (CI) computation and DFT. DVME consists of two phases. To begin, one-electron Molecular Orbital (MO) calculations are performed using the Discrete Variational Xα (DV-Xα) method. The CI method is then used to perform many-electron computations, which is the main stage of the DVME approach. It has been shown that DVME is a powerful tool for estimating absorption spectra, energy levels, transition energies, etc., without the use of any empirical parameters.

Up until recently, there has been no study on the influence of local structure on symmetry (switching from a high-symmetry to a low-symmetry configuration) or on the luminous qualities of red phosphors. Therefore, the goal of this research was to investigate the effect of local structural symmetrization on the polytypes of K₂XF₆ crystals, namely O_h−-K₂MnF₆, C_3v-K₂MnF₆, O_h-K₂SiF₆, C_3v-K₂SiF₆, D_3d-K₂GeF₆, and C_3v-K₂GeF₆. The DVME method was used to calculate their multiplet energy levels.

2. Materials and Methods

Polytypes of various K₂XF₆ crystals were used to create seven-atom model clusters. The cubic K₂MnF₆ ICSD #47213 had a = 8.221 lattice parameter, a space group $Fm\overline{3}m$, and O_h symmetry [29]. The lattice parameters of hexagonal K₂MnF₆ ICSD #60417 were a = 5.719 Å and c = 9.330 Å, with space group $P63mc$ and C_3v Symmetry [30]. The cubic K₂SiF₆ ICSD #2940 had a= 8.134 lattice parameters, a space group $Fm\overline{3}m$, and O_h symmetry [31]. The lattice parameters of hexagonal K₂SiF₆ ICSD #158483 were a = 5.6461 Å and c = 9.2322 Å, with the space group $P63mc$ and C_3v Symmetry [32]. The lattice parameters of rhombohedral K₂GeF₆ ICSD #24026 were a = 5.63 and c = 4.66, with the space group $P\overline{3}m1$ and D_3d Symmetry [33]. The lattice parameters of hexagonal K₂GeF₆ ICSD #30310 were a = 5.71 Å and c = 9.27 Å, with the space group $P63mc$ and C_3v Symmetry [34]. The computations were performed using O_h, D_3d, and C_3v symmetry for clusters built from K₂XF₆ (X = Mn, Si, or Ge) and crystals with cubic, rhombohedral, and hexagonal structures, respectively. Figure 1a–c depicts the various types of crystal structures of the materials under consideration, including namely cubic, rhombohedral, and hexagonal structures. Figure 1d–f were model clusters made up of seven atoms, one X⁴⁺ ion surrounded by 6 F⁻. Here, we adopted the results of the Mn K-edge Extended X-ray Absorption Fine Structure (EXAFS) measurement of K₂XF₆ (X = Si, or Ge): Mn⁴⁺ [35]. The Mn-F bond lengths for K₂SiF₆: Mn⁴⁺ and K₂GeF₆: Mn⁴⁺ were 1.807 and 1.810 Å, respectively. The one-electron calculations utilizing the DV-Xα method were then carried out [36,37,38]. The DVME approach was used to account for the many-electron effects [28]. The energy corrections such as Configuration Dependent Correction (CDC) and Correlation Correction (CC) were also considered. Racah parameters were used to calculate the Coulomb integrals as well. These methods’ specific steps are described in Reference [35].

3. Results

3.1. Bond Lengths

The Mn-F bond lengths of O_h-K₂MnF₆, C_3v-K₂MnF₆, O_h-K₂SiF₆, C_3v-K₂SiF₆, D_3d-K₂GeF₆, and C_3v-K₂GeF₆. are shown in

Table 1. Mn-F Bond lengths (Å) of O_h-K₂MnF₆, C_3v-K₂MnF₆, O_h-K₂SiF₆, C_3v-K₂SiF₆, D_3d-K₂GeF₆, and C_3v-K₂GeF₆ doped with Mn⁴⁺.

K2XF6 Crystals	d1	d2	d3	d4	d5	d6
Without relaxation
Oh-K2MnF6	2.005920	2.005920	2.005920	2.005920	2.005920	2.005920
C3v-K2MnF6	1.785311	1.785311	1.785314	1.792269	1.792271	1.792271
Oh-K2SiF6: Mn4+	1.682920	1.682920	1.682920	1.682920	1.682920	1.682920
C3v-K2SiF6: Mn4+	1.680538	1.680542	1.680542	1.688463	1.688463	1.688461
D3d-K2GeF6: Mn4+	1.770284	1.770280	1.770280	1.770284	1.770284	1.770284
C3v-K2GeF6: Mn4+	1.777360	1.777357	1.777357	1.805570	1.805570	1.805564
With relaxation
Oh-K2MnF6	2.005920	2.005920	2.005920	2.005920	2.005920	2.005920
C3v-K2MnF6	1.785311	1.785311	1.785314	1.792269	1.792271	1.792271
Oh-K2SiF6: Mn4+	1.807000	1.807000	1.807000	1.807000	1.807000	1.807000
C3v-K2SiF6: Mn4+	1.802744	1.802753	1.802753	1.811248	1.811248	1.811247
D3d-K2GeF6: Mn4+	1.810000	1.809997	1.809997	1.810000	1.810000	1.810000
C3v-K2GeF6: Mn4+	1.795749	1.795750	1.795750	1.824250	1.824250	1.824244

. All six bond lengths are represented by letters d1, d2, d3, d4, d5, and d6, respectively. When the lattice relaxation effect was not used, the lengths of the Mn-F bonds dropped from O_h-K₂MnF₆ to C_3v-K₂MnF₆. This was similar to the trend for O_h-K₂SiF₆: Mn⁴⁺ to C_3v-K₂SiF₆: Mn⁴⁺. On the other hand, the trend for D_3d-K₂GeF₆: Mn⁴⁺ to C_3v-K₂GeF₆: Mn⁴⁺ was reversed. When the lattice relaxation effect was used, however, the Mn-F bond lengths decreased in all situations.

3.2. Molecular Orbital Energies

Figure 2 depicts the molecular orbital energies of O_h-K₂MnF₆, C_3v-K₂MnF₆, O_h-K₂SiF₆, C_3v-K₂SiF₆, D_3d-K₂GeF₆, and C_3v-K₂GeF₆. The Valence Band (VB) is represented by black solid lines. The Conduction Band (CB) is shown by the black dashed lines. The impurity levels are represented as t_2g and e_g, with solid red and dashed blue lines, respectively. The tops of the VBs were set to zero. For O_h-K₂MnF₆ and C_3v-K₂MnF₆, the crystal field splitting (10Dq, defined as the differential energy between t_2g and e_g levels) was estimated to be 1.79 and 2.68 eV, respectively. Without accounting for the lattice relaxation effect, the 10Dq of O_h-K₂SiF₆ and C_3v-K₂SiF₆ were estimated to be 3.52 and 3.44 eV, respectively. They fell to 2.63 and 2.53 eV when the lattice relation effect was taken into account. In the case of D_3d-K₂GeF₆ and C_3v-K₂GeF₆, the 10Dq was determined to be 2.76 and 2.72 eV, respectively. After accounting for the lattice relaxation effect, they fell to 2.52 and 2.61 eV, respectively.

3.3. Multiplet Energy Levels

Since the d–d transitions of K₂XF₆: Mn⁴⁺ was prohibited by the parity selection rule, the transition probabilities could not be determined. As a result, this report is restricted to energy levels. We estimated the doublet states ²E,.²T_2, and ²T_1, as well as the quartet states ⁴T₂ and ⁴T_1a. The absorption transitions start from the ground ⁴A₂ state to ⁴T₂ and ⁴T_1a states which often appeared as wide bands and were referred to as the U- and Y-band, respectively. On the other hand, the emission transition started from the ²E state to the ground ⁴A₂ state, which generally appeared as a sharp line and was referred to as R-line.

The pure K₂MnF₆ and K₂SiF₆: Mn⁴⁺ computed multiplet energy diagrams with O_h and C_3v symmetry are shown in Figure 3. A few adjustments, including CDC, CC, and lattice relaxation, were also assessed. Figure 3 demonstrates that quite often, the doublet states decreased when reduced symmetry was employed. Furthermore, CDC-CC correction had a smaller impact on O_h-K₂MnF₆ than it did on C_3v-K₂MnF₆, suggesting that C_3v-K₂MnF₆ benefited more from correlation correction. On the other hand, the quartet states increased for pure K₂MnF₆ from O_h to C_3v while they dropped for K₂SiF₆: Mn⁴⁺ in the same order. This was expected because the Mn-F bond length, which varied widely depending on the material, primarily affected the quartet states.

The predicted multiplet energy diagrams of K₂GeF₆: Mn⁴⁺ with D_3d and C_3v symmetry are shown in Figure 4. CDC, CC, and lattice relaxation were also evaluated, similar to Figure 3. These findings showed that the average doublet state values for the two clusters were remarkably similar. Low symmetry was also found to have an impact on multiplet splitting. While the splitting of the ⁴T₂ state decreased, it increased for the ⁴T_1a, ²T_2, and ²T₁ states.

3.4. Coulomb Integrals

The Coulomb integrals of pure K₂MnF₆, K₂SiF₆: Mn⁴⁺, and K₂GeF₆: Mn⁴⁺ are shown in

Table 2. Using the MnF₆²⁻ model clusters with O_h, D_3d, and C_3v symmetry, Coulomb integral (eV) for the pure TM−3d atomic orbitals ($J_{\mathrm{AO}}$) and the molecular orbitals ($J_{\mathrm{MO}}$) were calculated. The adjustments were contrasted, including those with and without lattice relaxation. The orbital deformation parameter ($\lambda$) and the correlation correction factor (c) were multiplied by $J_{\mathrm{AO}}$ to calculate the effective Coulomb integrals ($J_{\mathrm{eff}}$).

Compound	K2MnF6		K2MnF6: Mn4+		K2SiF6: Mn4+ Relaxed		K2GeF6: Mn4+		K2GeF6: Mn4+ Relaxed
Symmetry	Oh	C3v	Oh	C3v	Oh	C3v	D3d	C3v	D3d	C3v
$J_{\mathrm{AO}}$	23.92	24.30	24.36	24.12	24.21	23.88	24.35	24.29	24.31	24.26
$J_{\mathrm{MO}\left(t2g\right)}$	19.19	19.96	20.40	19.30	19.91	18.35	20.07	19.95	19.91	19.88
$J_{\mathrm{MO}\left(eg\right)}$	16.50	18.43	19.08	18.91	18.09	17.93	18.66	18.32	18.38	18.18
${\lambda }_{\left(t2g\right)}$	0.80	0.82	0.84	0.80	0.82	0.77	0.82	0.82	0.82	0.82
${\lambda }_{\left(eg\right)}$	0.69	0.76	0.78	0.78	0.75	0.75	0.77	0.75	0.76	0.75
c factor	1.01	0.84	0.78	0.77	0.85	0.84	0.83	0.85	0.86	0.86
$c{\lambda }_{\left(t2g\right)}$	0.81	0.69	0.65	0.61	0.70	0.64	0.69	0.69	0.70	0.70
$c{\lambda }_{\left(t2g\right)}$	0.70	0.64	0.61	0.60	0.64	0.63	0.64	0.64	0.65	0.64
$J_{\mathrm{eff}\left(t2g\right)}$	19.42	16.81	15.86	14.77	16.94	15.39	16.69	16.87	17.07	17.05
$J_{\mathrm{eff}\left(eg\right)}$	16.70	15.52	14.83	14.47	15.39	15.04	15.51	15.49	15.75	15.60

. When low symmetry was used, the effective Coulomb integrals $J_{\mathrm{eff}}$ estimated by $c\lambda J_{\mathrm{AO}}$ almost always decreased. Although the $J_{\mathrm{eff}\left(t2g\right)}$ of K₂GeF₆: Mn⁴⁺ without lattice relaxation was greater than that of D_3d-K₂GeF₆: Mn⁴⁺, its tendency improved when lattice relaxation was considered. These findings suggest that reduced symmetry resulted in a smaller Coulomb integral. As a result, the decreasing trend of R-line energy could be attributed to a decreased electron–electron repulsion.

4. Discussion

LEDs are now used in a variety of commonplace applications, including display backlights for smartphones, tablets, and televisions, as well as warm-white LEDs for energy-efficient lighting. A portion of the blue light from the LED chip is converted into white light by color-converting luminescent materials. This is accomplished by using doped wide-bandgap materials, also referred to as phosphors or Colloidal Quantum Dots (QDs). The color quality of white LEDs is improved when red-emitting phosphors are added when compared to the prototype’s arrangement of a blue LED and a yellow Y₃Al₅O₁₂: Ce³⁺. These luminous materials have an incredibly high luminescence efficiency, especially at room temperature and above, due to the involvement and stimulation of thermal phonons.

A rare-earth ion such as Eu²⁺ and Ce³⁺ or a transition metal such as Mn²⁺ and Mn⁴⁺ are doped into an inorganic host material to create phosphor materials. Rare-earth ions are frequently used in conventional LED phosphors, with the main component being Y₃Al₅O₁₂: Ce³⁺ (YAG: Ce) [39]. By altering the host compound’s composition, the dopant Eu²⁺ can change the emission spectrum, for example, from a green emission in SrGa₂S₄: Eu²⁺ [3,4] and SrSi₂O₂N₂: Eu²⁺ [5] to a red emission in Sr₂Si₅N₈: Eu²⁺ [5,6], (Ca, Sr)S: Eu²⁺ [3,7], CaAlSiN₃: Eu²⁺ [8] and Sr[LiAl₃N₄]: Eu²⁺ [9]. A current trend toward creating non-rare-earth element LED phosphors is the result of environmental challenges, including the scarcity of rare-earth materials.

The requirements for a phosphor to be suitable for LED applications were described by Smet et al. [39] in detail. According to the Color Quality Scale (CQS) [40] or the CRI [41], the resulting white light source had a high color rendering. This was significant for illumination. For display applications to produce a broad color spectrum or high color purity, saturated colors were necessary. The lower the filtering losses, the better the phosphors’ emission spectrum fits the color filters. Second, a phosphor must have a high LER, which is a metric for the average eye sensitivity of the spectrum, measured in lm/W) and a high Internal Quantum Efficiency (IQE), which is stable at high temperatures. Third, there needs to be significant blue light absorption, which raises the EQE. A phosphor can only be considered a serious candidate for LED applications when all four requirements are satisfied simultaneously.

The Mn⁴⁺ emission center prefers to remain in the octahedral or modified octahedral position of the host due to the large ligand-field stabilizing energy of Mn⁴⁺ in the six-fold coordination. In the initial LFT simulation, only the octahedral (O_h) crystal field was taken into account [42]. The doubly degenerate e_g level had +6Dq more energy than the fivefold degenerate 3d level, and the triply degenerate t_2g level had −4Dq more energy. The intensity of the crystal field, Dq, changed based on the ion-crystal combination, and as a result, the crystal field splitting was 10Dq. Through the use of the electron–electron repulsion parameters A, B, and C, also referred to as Racah parameters, the impact of covalency could also be taken into account in this situation. The crystal field strength Dq, the Racah parameters B and C, and the multiplet energy levels E_i could thus be used to explain them. The recognized Tanabe Sugano diagrams [21,43], which depict E_i/B as functions of Dq/B for a fixed value of C/B, describe the energy levels of all d^N systems in an octahedral crystal field as functions of Dq. The spectral characteristics of red phosphor materials were also described using the absorption and emission spectra. There were certain doublet states, such as ²E, ²T₁, etc., and some quartet states, such as ⁴A₂, ⁴T₂, ⁴T_1a, ⁴T_1b, etc., since Mn⁴⁺-doped compounds contained three electrons filling ten degenerate 3d orbitals (3d³). The energy was lowest in the ground state (⁴A₂). The transitions from ⁴A₂ to ⁴T₂ (U-band) and ⁴T_1a (Y-band) were utilized for absorption, while the transition from ²E to ⁴A₂ (R-line) was employed for emission when used as red phosphor materials.

Previously, we studied the potential of oxide and fluoride materials for red phosphor materials in WLED by DV-Xα and DVME methods. The investigation included lattice relaxation, orbital energy, multiplet energy, absorption spectra, energy correction, pressure dependence, and emitted light utilizing CIE 1931 color space [35,44,45,46,47,48,49,50,51,52]. Because of the length of the Mn-F bond, the quartet state energies typically had a strong relationship with crystal field splitting. On the other hand, doublet state energies strongly depended on the correlation correction. The computational conditions to reproduce those optical properties depend on the material itself.

The study of low-structure symmetrization was required when understanding the properties of novel phosphor materials. According to our findings, a low structure had a substantial effect on multiplet structures, which could affect their performance as phosphor materials. Table 1 indicates that considering the lattice relaxation effect caused by the Mn⁴⁺, substitution resulted in a considerable shift in the bond lengths. The crystal field splittings were approximated using the one-electron DV-Xα approach, as shown in Figure 2. The 10Dq crystal field splitting tendency was caused by the lengths of the Mn-F bonds. Furthermore, the multiplet energies in Figure 3 and Figure 4 were determined using the many-electron DVME approach. The splitting of the corresponding multiplet energy levels was visible in lower structure symmetrization.

5. Conclusions

By taking into consideration lattice relaxation, CDC, and CC, the multiplet energies of Mn⁴⁺ doped K₂XF₆ crystals were qualitatively estimated. The fluoride compounds exhibited here were suitable materials to be used as red phosphors for white LEDs since the Mn⁴⁺ impurities in these hosts were emitted at approximately 620−630 nm (the proper spectral range to obtain a “warm” white light from the LEDs). In addition, they appeared to be thermally stable since they are known as commercial red phosphors (especially K₂SiF₆: Mn⁴⁺). We found that the Mn-F bond length dropped, yet the U- and Y-band energies increased. By contrasting various fluoride crystal symmetry polytypes, the impact of lower symmetry was explored. The Mn-F bond length decreased, while the absorption energies of ⁴A_2g→⁴T_2g (⁴F) and ⁴A_2g→⁴T_1g (⁴F) increased, yet the ²E_g → ⁴A_2g emission energy decreased. A reduced Coulomb integral result was produced when symmetry was low. It followed that a decrease in electron–electron repulsion was the cause of the declining trend in R-line energy. The CC factor c dominated the R-line energy, while the Mn-F bond length and crystal field splitting was principally responsible for the U- and Y-band energies. We discovered that low symmetry decreased the attraction between the electrons.