Influence of Isostatic Pressure on the Elastic and Electronic Properties of K₂SiF₆:Mn⁴⁺

Abstract

Isostatic pressure effects on the elastic and electronic properties of non-doped and Mn⁴⁺-doped K₂SiF₆ (KSF) have been investigated by first-principles calculations within density functional theory (DFT). Bulk modulus was obtained by the Murnaghan’s equation of states (EOS) using the relationship between volume and pressures at pressures between 0 and 40 GPa, and elastic constants were calculated by the stress–strain relationship giving small distortions at each pressure point. The other elastic parameters such as shear modulus, sound velocity and Debye temperature, which can be obtained from the elastic constants, were also estimated. The influence of external isostatic pressure on the electronic properties, such as crystal field strength 10Dq and emission energy of ²E → ⁴A² transition (E_em), of KSF:Mn⁴⁺ was also studied. The results suggest that 10Dq and E_em linearly increase and decrease, respectively, with increasing pressure.

1. Introduction

Mn⁴⁺-doped phosphors as red photon generating systems have been extensively studied over the past decades due to their great promise for use in LED devices [1,2,3,4,5,6,7]. Among a large number of Mn⁴⁺ activated red-emitting phosphors, K₂SiF₆:Mn⁴⁺ (KSF:Mn⁴⁺) has been developed as generators of red photons in phosphor-converted white LEDs (pc-LEDs). The sharp-line emission of KSF:Mn⁴⁺ peaks at about 630 nm, where the human eye sensitivity to red light is still quite high. With minimum emission beyond 650 nm, the emission spectrum of KSF:Mn⁴⁺ is suitable for supplying the red photons necessary to produce white light with high efficacy (lumens per watt) and color rendering index (CRI) in pc-LEDs. Given its commercial importance, detailed analyses of the spectroscopic properties, such as absorption and emission spectra, electronic and geometric properties of KSF:Mn⁴⁺ were performed [8,9,10] and recently this red phosphor was commercialized [11].

The optical properties of the Mn⁴⁺ ions (with the 3d³ electron configuration) are strongly influenced by the composition and crystal structure of the host lattice. In the Tanabe–Sugano diagram for the d³ electron configuration, the “Mn⁴⁺-ligand” bonding covalence defines the emission energy of the ²E → ⁴A₂ spin-forbidden transition and the strength of the octahedral crystal field denoted as 10Dq. It was determined by DFT calculations that weak Mn⁴⁺–ligand hybridization generally leads to higher Mn⁴⁺ emission energies [12]. The hybridization between the Mn⁴⁺ ions and ligands is related to the Mn⁴⁺–ligand distances. The application of pressure is one of the state parameters that can change the peak energy and 10Dq by reducing the inter-atomic distances. Studies of the Mn⁴⁺ optical properties under external pressure have been reported in the archival literature [13,14,15]. The goal of this study was to quantitatively explore by first-principles calculations within density functional theory, the electronic, elastic and optical properties of K₂SiF₆ in the absence and presence of the activator ion (Mn⁴⁺).

2. Computational Method

All the density functional calculations in this paper were performed by the plane-wave basis projector augmented wave package, VASP [16], using the generalized gradient approximation proposed by Perdew, Burke and Ernzerhof (GGA-PBE) [17] to express electron–electron correlation. The space group of KSF is Fm-3m with a lattice constant of 8.134 Å [18], in which the Si⁴⁺ ions are surrounded by six F⁻ ions and the local site symmetry is described by the O_h point group. The Mn⁴⁺-doped KSF models were constructed by replacing one Si⁴⁺ ion by one Mn⁴⁺ ion in the unit cell of KSF, which includes 36 atoms corresponding to four formula units. After careful convergence tests with respect to the plane wave cut-off energy, 900 eV was selected for the energy cut-off in all the calculations. The Monkhorst–Pack k-points grid sampling [19] was set as 4 × 4 × 4. The structural parameters of non-doped KSF and KSF:Mn⁴⁺ were optimized by allowing relaxations of the lattice constants and internal atomic positions under isostatic external pressures between 0 and 40 GPa by a 5 GPa step.

3. Results and Discussion

The calculated and experimental lattice constants of non-doped KSF and KSF:Mn⁴⁺ are summarized in

Table 1. Comparison of lattice constants, a, and bond lengths of Si–F and Mn–F in the non-doped and Mn⁴⁺-doped K₂SiF₆, respectively, between experiments and calculations.

System		Calc. (Å)	Exp. (Å)
K2SiF6	a	8.336	8.134 a
	Si-F	1.720	1.683 a
K2SiF6:Mn4+	a	8.357
	Mn-F	1.835	1.807 b

^a Ref. [18]. ^b Ref. [20].

together with the calculated Si-F and Mn-F bond lengths. The experimental lattice constant of non-doped KSF, a = 8.134 Å [18], is reproduced well by the current calculations, a = 8.336 Å, within a typical overestimation due to the GGA-PBE functional use. It is noted here that a slight volume expansion occurs after inclusion of the Mn⁴⁺ ions into KSF host. The calculated lattice constant of KSF:Mn⁴⁺ is 8.357 Å, which is larger than that of KSF 8.336 Å. This expansion can be simply explained by the difference in ionic radii of Si⁴⁺ (0.40 Å) and Mn⁴⁺ (0.53 Å). The calculated Mn⁴⁺–F^– bond-length is also larger than the calculated Si⁴⁺–F^– bond-length by 6.69%, which supports the earlier experimental result of +7.43% [20].

The calculated pressure dependence of the relative volume change V/V₀ for non-doped KSF is plotted in Figure 1a, which is fitted to the Murnaghan’s equation of state (EOS) [21] expressed by:

$$V/V_0={\left(1+P\frac{B^{\prime }}{B}\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$B^{\prime }$}\right.}$$

(1)

where V and V₀ are volumes at pressure P and ambient pressure, respectively, and B and B’ are bulk modulus and its pressure derivative, respectively. This fitting yields that B and B’ are 20.01 GPa and 4.68, respectively. Our results agree well with the earlier study [22] using GGA-PBE, B = 21.79 GPa and B’ = 4.47, although a slight difference appears between these two from a difference in calculating methods, i.e., projector augmented wave and pseudopotential methods in the current and earlier [22] studies, respectively. To the best of our knowledge, no experimental bulk modulus for KSF has been reported yet. For Mn⁴⁺-doped KSF, V/V₀ is also plotted in Figure 1b, which is also fitted to the Murnaghan’s EOS. The fitting yields B = 19.84 GPa and B’ = 4.68, which implies that inclusion of Mn⁴⁺ into KSF leads to a decrease of the bulk modulus.

The elastic constants of non-doped and Mn⁴⁺-doped KSF were calculated using a stress-strain method [23] implemented in VASP, which are summarized in

Table 2. Pressure dependence of the elastic constants C_ij (all in GPa) for the non-doped and Mn⁴⁺- doped K₂SiF₆.

System	Pressure	C11	C12	C44
K2SiF6	0	28.2	11.3	14.3
	5	55.6	35.2	20.4
	10	77.1	52.7	30.5
	15	100.1	70.5	36.6
	20	119.0	84.9	46.0
	25	137.5	99.2	52.5
	30	157.4	113.3	60.1
	35	176.7	127.9	68.0
	40	195.7	141.5	75.8
K2SiF6:Mn4+	0	23.7	9.0	11.0
	5	45.8	28.1	20.1
	10	62.9	40.3	28.4
	15	90.3	62.5	36.4
	20	112.0	78.3	45.0
	25	133.0	93.9	53.5
	30	154.4	109.4	61.9
	35	175.0	123.7	70.5
	40	195.3	137.4	79.2

and plotted in Figure 2 as a function of pressure. Here the optimized structure at each pressure was used to calculate the elastic constants. Three irreducible elastic constants for the cubic structure, i.e., C₁₁, C₁₂ and C₄₄, at zero-pressure were reported in the earlier study [22]; the values are 31.90, 9.28 and 15.10 GPa, respectively. Those in the current study show similar values of 28.2, 11.3 and 14.3 GPa for C₁₁, C₁₂ and C₄₄, respectively. The difference between these two studies is derived from the difference in calculating conditions as discussed above for the bulk modulus. The calculated C₁₁ and C₁₂ values for the non-doped KSF are larger than those of KSF:Mn⁴⁺ at all pressures calculated in this work. On the other hand, the calculated C₄₄ of non-doped KSF is larger than that of KSF:Mn⁴⁺ at zero pressure, the difference between these two becoming smaller as pressure increases, and finally C₄₄ of KSF:Mn⁴⁺ becomes larger than that of non-doped KSF between 20 and 25 GPa. It can be noted that all the elastic constants of both non-doped and KSF:Mn⁴⁺ increase with increase of pressure, in which C₁₁ and C₁₂ increase more rapidly than C₄₄ as shown in Figure 3.

Using these calculated elastic constants, sound velocities can be obtained. The mean sound velocity v_m is expressed [24] in terms of the longitudinal sound velocity v_l and the transverse one v_t as:

$$v_m={\left[\frac{1}{3}\left(\frac{2}{v_t^3}+\frac{1}{v_l^3}\right)\right]}^{-1/3}$$

(2)

in which v_l and v_t are calculated [25] by:

$$v_l=\sqrt{\frac{3B+4G}{3\rho }}\mathrm{and}{v}_t=\sqrt{\frac{G}{\rho }}$$

(3)

Here $G=\left(G_V+G_R\right)/2$ is the isotropic shear modulus, in which $G_V=\left(C_{11}-C_{12}+3C_{44}\right)/5$ is the Voight’s shear modulus (an upper limit for G), and $G_R=5\left(C_{11}-C_{12}\right)C_{44}/\left[4C_{44}+3\left(C_{11}-C_{12}\right)\right]$ is the Reuss’s shear modulus (a lower limit for G). Debye temperature, Θ_D, is calculated using the following equation [25]:

$${\Theta }_D=\frac{h}{k}{\left[\frac{3n}{4\pi }\left(\frac{N_A\rho }{M}\right)\right]}^{1/3}{v}_m$$

(4)

where h and k are the Planck’s and Boltzmann’s constants, respectively, N_A is the Avogadro’s number, ρ is the density, M is the molecular weight, and n denotes the number of atoms per formula unit (here nine for KSF). The calculated shear modulus, G, sound velocities, v_m, v_t and v_l, and Debye temperature, Θ_D, as a function of pressure for both non-doped and Mn⁴⁺-doped KSF are summarized in

Table 3. Calculated shear moduli, sound velocities and Debye temperatures for the non-doped and Mn⁴⁺-doped K₂SiF₆.

System	P, GPa	G, GPa	vt, m/s	vl, m/s	vm, m/s	ΘD, K
K2SiF6	0	11.59	2146.70	3590.80	2375.61	280
	5	15.44	2278.49	4589.10	2557.08	319
	10	21.15	2547.46	5227.87	2861.95	368
	15	25.48	2708.88	5738.69	3048.37	400
	20	30.91	2909.98	6138.43	3273.96	437
	25	35.06	3037.20	6463.19	3418.59	462
	30	40.26	3199.14	6796.39	3600.57	493
	35	45.13	3331.08	7089.15	3749.39	519
	40	50.24	3467.45	7364.17	3902.49	545
K2SiF6:Mn4+	0	9.36	1900.48	3193.65	2104.06	248
	5	14.49	2174.34	4174.21	2432.98	303
	10	19.65	2419.06	4696.07	2708.78	348
	15	24.72	2628.23	5410.22	2953.19	388
	20	30.38	2842.32	5881.25	3194.64	426
	25	35.74	3020.83	6283.09	3396.21	459
	30	41.28	3191.09	6653.71	3588.09	491
	35	47.08	3361.47	6990.49	3779.15	522
	40	52.97	3518.48	7289.33	3954.87	551

and plotted in Figure 3a–c for shear moduli, sound velocities, and Debye temperature, respectively. The G, v_m, v_t, v_l and Θ_D values for both systems increase with pressure. Inclusion of Mn⁴⁺ into KSF leads to a decrease of G, v_m, v_t, v_l and Θ_D at zero pressure. Each of the differences in those parameters between non-doped and Mn⁴⁺-doped KSF becomes smaller with increase of pressure. Calculated G of KSF:Mn⁴⁺ becomes larger than that of non-doped KSF at pressures between 20 and 25 GPa as in the case of C₄₄, while v_m, v_l and Θ_D of KSF:Mn⁴⁺ becomes larger than that of the non-doped one between 30 and 35 GPa.

Calculated electronic densities of states (DOSs) of non-doped and Mn⁴⁺-doped KSF at 0 GPa are shown in Figure 4. As illustrated in this figure, new orbitals associated with the Mn⁴⁺ 3d orbitals appear in the band gap due to a doping of Mn⁴⁺ ion. Current results are consistent with our previous report [26], although the previous one was undertaken with larger super cells expanded by 2 × 1 × 1. The Mn⁴⁺ ion in KSF is surrounded by six F⁻ ions in a crystal environment of cubic symmetry. Hence, the 3d ground-state splits into the triply and doubly degenerated t_2g and e_g orbitals, respectively. The3d states of Mn⁴⁺ hybridize with the F⁻ 2p states in both t_2g and e_g orbitals as shown in Figure 4b. The Mn⁴⁺ ion has three 3d electrons, which fully occupy t_2g up-spin state.

To discuss the influence of pressure on the electronic structure, calculated DOSs of KSF:Mn⁴⁺ are compared among those at different pressures between 0 and 40 GPa, which are shown in Figure 5. The energies of the top of each state originating from 3d orbital of Mn⁴⁺ ion, i.e., up-spin and down-spin states of t_2g and e_g, are plotted in Figure 6. As the top of the occupied band was set to zero in Figure 5, the tops of all t_2g up-spin states were located at zero. On the other hand, down-spin t_2g states shift to the lower energy side, while up- and down-spin e_g states shift to the higher energy side with increasing pressure. Calculated 10Dq, which denotes the crystal field strength and is defined as a difference in energy between up-spin t_2g and e_g states, at different pressures is summarized in

Table 4. Comparison of the experimental and calculated ²E → ⁴A₂ emission transition energy and 10Dq for the Mn⁴⁺-doped K₂SiF₆ at different pressures.

	Pressure (GPa)	Calc. (eV)	Exp. (eV)
10Dq	0	2.76	2.74 a
	5	2.83
	10	2.87
	15	2.90
	20	2.93
	25	2.96
	30	3.00
	35	3.03
	40	3.06
2E → 4A2	0	2.74	1.99 b
	5	2.72
	10	2.70
	15	2.68
	20	2.66
	25	2.64
	30	2.63
	35	2.62
	40	2.60

^a Ref. [27]. ^b Ref. [9].

and plotted in Figure 7. Calculated 10Dq value at zero pressure of 2.76 eV agrees well with the experimental value of 2.74 eV [27], although our current calculations were carried out by GGA-PBE without the effect of strong electron correlation for Mn 3d electrons. As shown in these tables and figures, 10Dq increases linearly as pressure increases. Least square fitting to the pressure dependence of 10Dq yields 10Dq = 0.00346 P + 2.734. In addition, from a geometrical point of view, 10Dq values are plotted in Figure 8 as a function of bond length between Mn⁴⁺ and sounding F⁻ ions. As shown in this figure, a very good linear relation can be found between 10Dq and Mn-F bond length. The least-square fitting result is 10Dq = −6.37 d_Mn-F + 14.46. This result implies that 10Dq can be estimated by this equation for other Mn⁴⁺-doped fluorides, in which the doped Mn⁴⁺ ion is located at the O_h symmetry point indicated by six F⁻ ions. This result is consistent with our previous study on A₂SiF₆ [26], where A = K, Rb and Cs, in which 10Dq decreases with increasing Mn⁴⁺−F⁻ bond-length.

The red emission from Mn⁴⁺ ion is assigned to the transition from the first excited state ²E to the ground state ⁴A₂ of the Mn⁴⁺ 3d state. Electron configuration of the ⁴A₂ state of the Mn⁴⁺ ion is all three electrons on up-spin t_2g while that of ²E is two electrons on up-spin t_2g and one on down-spin t_2g states. Here, the transition energy from ²E to ⁴A₂, E_em, is calculated by an energy gap between the up-spin and down-spin t_2g states within a one electron approximation to avoid complicated calculations considering multielectron effects to obtain total electronic energy difference between the ²E and ⁴A₂ states. The calculated E_em value at zero pressure is 2.74 eV, which is larger than the experimental value of 1.99 eV [9]. This overestimation is derived from the method to obtain E_em. However, the change in E_em can be discussed qualitatively as in our earlier report on KSF:Mn⁴⁺ [26]. As the d and f electrons are localized, consideration of a strong electron–electron correlation effect, such as the DFT + U method we have undertaken previously [26], may lead to a better reproduction of E_em than the current calculations with GGA-PBE. In addition, the calculated E_em at zero pressure, i.e., 2.74 eV, is larger than that in the previous study on KSF:Mn⁴⁺ [26]. Here, a 1 × 1 × 1 unit cell was used, whereas a 2 × 1 × 1 super cell was employed in our previous study [26], which yield a difference in calculated E_em at zero pressure. E_em as a function of pressure is summarized in Table 4 and plotted in Figure 7. Contrary to the change in 10Dq, E_em decreases almost linearly, E_em = −0.00346 P + 2.734, with increasing pressure. The calculated E_em as a function of Mn⁴⁺–F⁻ bond length is also plotted in Figure 8 for KSF:Mn⁴⁺. It can be seen that the ²E → ⁴A₂ transition energy has very good linear relation with Mn–F bond length, E_em = 3.11 d_Mn-F – 2.97.

4. Conclusions

First-principles calculations have been carried out for the non-doped and Mn⁴⁺-doped K₂SiF₆ to study the influence of isostatic pressure on the geometric structure, elastic and electronic properties of K₂SiF₆:Mn⁴⁺ red phosphor. In particular, the pressure effect on the elastic properties such as elastic constants, shear modulus, sound velocity and Debye temperature between non-doped and Mn⁴⁺-doped systems was discussed. Pressure dependence of two important electronic parameters, i.e., 10Dq and the emission energy of the ²E → ⁴A₂ transition, of KSF:Mn⁴⁺ have been investigated, which yield good linear relationships both between 10Dq and external pressure and between emission energy of ²E → ⁴A₂ and pressure. It can be noted that the current analysis of the pressure effect on 10Dq and emission energy of the ²E → ⁴A₂ transition provided empirical formulae to obtain these values as a function of bond-length between Mn⁴⁺ and surrounding F⁻ ions, which can be used for the estimation of these important parameters for other Mn-doped fluorides.