Ab Initio Studies of the Structural, Electronic, and

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The structural, electronic, optical and elastic properties of the potassium .... LDA. W. Figure 1. The calculated band structure of K2SiF6. The GGA- and LDA-.
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Journal of The Electrochemical Society, 159 (6) J212-J216 (2012) 0013-4651/2012/159(6)/J212/5/$28.00 © The Electrochemical Society

Ab Initio Studies of the Structural, Electronic, and Optical Properties of K2 SiF6 Single Crystals at Ambient and Elevated Hydrostatic Pressure Mikhail G. Brika,z and Alok M. Srivastavab,∗ a Institute b GE

of Physics, University of Tartu, Tartu 51014, Estonia Global Research, Niskayuna, New York 12309, USA

The structural, electronic, optical and elastic properties of the potassium hexafluorosilicate (K2 SiF6 ) lattice have been evaluated by the Density Functional Theory (DFT)-based ab initio calculating techniques. The first estimations of the bandgap, elastic constants, and Debye temperature are reported. Further, the dependence of these properties of the K2 SiF6 lattice on pressure is evaluated and discussed. © 2012 The Electrochemical Society. [DOI: 10.1149/2.071206jes] All rights reserved. Manuscript submitted January 6, 2012; revised manuscript received February 29, 2012. Published April 5, 2012.

Crystal Structure and Methods of Calculations Potassium hexafluorosilicate K2 SiF6 crystallizes in a cubic crystal structure, space group Fm3m (No. 225), lattice constant 8.134 Å, with four formula unit per unit cell.3 The K+ ions are 12-fold coordinated by the fluorine ions with the K - F distance of 2.897 Å. The Si4+ ions are surrounded by six fluorine ions at the distance of 1.683 Å. For the ab initio calculations we have used the Cambridge Serial Total Energy Package (CASTEP) module4 of the Materials Studio 4.0 package; both the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof functional5 and the Ceperley-AlderPerdew-Zunger parameterization6, 7 in the local density approximation (LDA) were applied to treat the exchange-correlation effects. The ultrasoft pseudopotentials were employed for a description of interaction between the ionic cores and valence electrons. The electronic configurations used for the calculations were as follows: 3s2 3p6 4s1 for K, 3s2 3p2 for Si, and 2s2 2p5 for F. The convergence parameters were set as follows: energy tolerance 5 × 10−6 eV/atom, force tolerance 0.01 eV/Å; stress tolerance 0.02 GPa, and maximum displacement 5 × 10−4 Å. The Monkhorst-Pack k-points grid sampling was set as 10 × 10 × 10, and the plane-waves cut off energy (which determines the number of the plane waves in the basis set) was 450 eV. The calculations have been performed for a primitive cell. Structural, Electronic, Optical and Elastic Properties of K2 SiF6 at Ambient Pressure Optimization of the crystal lattice constants with the aforementioned calculating settings has lead to the following lattice constants (in Å): 8.1684 (GGA) and 7.7026 (LDA). The difference between these values and experimental data (8.134 Å 3 ) is +0.42% and −5.3%, ∗ z

Electrochemical Society Active Member. E-mail: [email protected]

respectively. As usually happens, the LDA/GGA lattice constants are somewhat under-/overestimated, respectively, with respect to the experimental data. The calculated Si-F distance is 1.68821 Å (GGA) and 1.67644 Å (LDA). The calculated K-F distance is 2.90957 Å (GGA) and 2.73467 Å (LDA). For both distances, the GGA-calculated results are closer to the experimental values. The calculated band structure of K2 SiF6 is shown in Fig. 1. The calculated direct bandgap is 7.588 eV (GGA) and 8.092 eV (LDA), which allows to classify this compound as a wide band-gap dielectric. The only mentioning of the experimental bandgap for this host is given in Ref. 1 as 5.6 eV, but this seems to be underestimated, especially if we take into account the usual underestimation of the bandgap by the ab initio calculations. The calculated density of states (DOS) diagrams that are shown in Fig. 2 allow for the following assignment of the electronic bands: a wide - about 7 eV - conduction band (CB) consists mainly of the Si 3s and 3p states, to which the K 4s states also add a minor contribution. The highest dispersion of the CB electronic states is realized around the Brillouin zone center. The valence band (VB) has the width of

GGA LDA

16 14 12 10

Energy, eV

The synthesis and optical properties of K2 SiF6 :Mn4+ phosphor have been recently reported in the archival literature.1 In this study, a detailed analysis of the absorption and emission spectra of the Mn4+ ion was performed; it was shown that this material is capable of supporting efficient red emission from the Mn4+ activator ions. We have recently presented further theoretical studies on this phosphor pertaining to the crystal field analysis of the Mn4+ energy levels.2 However, up to now there are no studies on the calculations of the band structure and elastic properties of this material. The motivation of our study is to fill this gap in the knowledge of the K2 SiF6 material. This is accomplished via the first principles calculations of the structural, electronic, optical and elastic properties of pure K2 SiF6 . All obtained results are compared (where available) with the corresponding experimental data.

8 6 0 -2 -4 -6

W

L

G

X

W K

Figure 1. The calculated band structure of K2 SiF6 . The GGA- and LDAcalculated electronic bands are shown by the solid and dotted lines, respectively. The Fermi level as set at zero. The coordinates of the special points of the Brillouin zone are (in terms of the reciprocal lattice unit vectors): W(1/2, 1/4, 3/4); L(1/2, 1/2, 1/2); G(0,0,0); X(1/2, 0, 1/2); K(3/8, 3/8, 3/4).

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Journal of The Electrochemical Society, 159 (6) J212-J216 (2012)

20 10

140000

s states p states

K

J213

(a)

GGA LDA

120000

1

Absorption, cm-1

Density of states, electrons/eV

0 s states p states

Si

0 2

s states p states

F

100000 80000 60000 40000

0

20000

20

0

10 0 -30

0 -25

-20

-15

-10

-5

0

5

10

15

5

10

15

20

25

30

35

Energy, eV

20

Energy, eV 2.5 Figure 2. The calculated DOS diagrams for K2 SiF6 . From top to bottom: the K, Si, F partial DOS and total DOS.

(b)

about 6 eV and consists of several narrow sub-bands, which exhibit almost no dispersion at all. The dominating contribution to the VB comes from the F ions 2p states. The Si 3s and 3p states, which are localized at the bottom of the VB, produce minor contributions. The Si and F states are hybridized, since they participate in formation of the Si - F chemical bond. The F 2s states are spread from −22 to −19 eV (the Si 3s, 3p states contributions to these bands are also noticed). The K 3s and 3p states are peaked at −22 eV and −9 eV, respectively. The calculated absorption spectrum of K2 SiF6 is shown in Fig. 3. Both GGA- and LDA-calculated spectra are very similar; the only difference is that the intensity of the calculated peaks is somewhat higher in the LDA-calculated one. The calculated dielectric function ε (Fig. 3) exhibits a similar behavior as well: the LDA-calculated values are greater for both real and imaginary parts of ε. The estimated value of the refractive index (determined as the square root of the real part of the dielectric function at zeroth energy) is 1.44/1.34 for the LDA/GGA calculations, respectively. The GGA estimation of the refractive index is in excellent agreement with the experimental value of 1.34.8 Figure 4 shows certain variances in the electron density difference distribution in the space between the Si - F and K - F ions. The K ions are bonded to the F ions by the ionic bonds, whereas the high degree of covalency of the Si - F bonds is quite noticeable. This conclusion will be also confirmed in the next section by the calculated effective charges of all ions. The calculated elastic constants for K2 SiF6 (which, to the best of the authors’ knowledge, are given here for the first time) are collected in Table I. All the LDA-calculated elastic constants are greater than the GGA-ones, which is a common trend for such kind of calculations. The obtained values of the elastic constants and unit cell volume can be used for calculations of the sound velocity and Debye temperature θ D :9    h 3n N A ρ 1/3 θD = vm , [1] k 4π M

Dielectric function

2.0

Re(ε), LDA Im(ε), LDA Re(ε), GGA Im(ε), GGA

1.5 1.0 0.5 0.0 0

5

10

15

20

25

30

Energy, eV Figure 3. Comparison of the calculated absorption spectra (a) and dielectric function (b) for K2 SiF6 obtained by the LDA and GGA calculations.

where h and k are the Planck’s and Boltzmann’s constants, respectively, NA is the Avogadro’s number, ρ is the crystal density, M is the molecular weight, n denotes the number of atoms per one formula unit (9 for K2 SiF6 ), vm is the averaged sound velocity expressed in terms of the longitudinal vl and transverse vt sound velocities as −1/3   1 2 1 + . [2] vm = 3 v3t vl3 The longitudinal vl and transverse vt sound velocities can be estimated as10   3B + 4G G , vt = , [3] vl = 3ρ ρ R where B is the bulk modulus and G = G V +G is the isotropic shear 2 modulus, which is obtained by averaging the Voigt’s shear modulus

Table I. Calculated elastic constants Cij , bulk modulus B, Young modulus E (all in GPa), elastic compliance constants Sij (in GPa−1 ), and non-dimensional Poisson ratio v for K2 SiF6 . C11 GGA LDA

31.90 59.49

C12

C44

S11

S12

S44

B

E

v

9.28 33.67

15.10 22.07

0.03608 0.02844

−0.00813 −0.01028

0.06624 0.04531

16.82 42.28

27.71 35.16

0.2254 0.3614

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Journal of The Electrochemical Society, 159 (6) J212-J216 (2012)

Figure 4. The cross-sections of the electron density difference in K2 SiF6 in the planes containing only the Si and F ions (top) and K, Si, F ions (bottom).

GV (an upper limit for G values) and the Reuss’s shear modulus GR (a lower limit for G values). The analytical expressions for GV and GR are as follows: 5 4 3 = + . GR C11 − C12 C44

[4]

Table II summarizes the results of the sound velocities and Debye temperature calculations for K2 SiF6 . From the measurements of the Mn4+ emission line widths in K2 SiF6 as the function of temperature, the authors of Ref. 1 estimated the mean phonon energy in this host (which is proportional to the Debye temperature) as 400 K. Our results give a little bit lower values for θ D . We also note that the GGA-calculated density of K2 SiF6 is in excellent agreement with the experimental value of 2665 kg/m3 .8

1.0 Relative volume change, V/V0

C11 − C12 + 3C44 , GV = 5

B=34.72+/-1.15 GPa B'=4.99+/-0.26

0.9

0.8

0.7

B=21.79+/-0.40 GPa B'=4.47+/-0.10 0

Effects of Pressure on the Structural, Electronic and Optical Properties of K2 SiF6 One of the most obvious pressure effects is decrease of volume with increasing pressure. Figure 5 shows the variation of the relative volume change V/V0 with pressure P (V0 stands for the unit cell volume at ambient pressure). The calculated results were fitted to the

GGA LDA

4

8

12

16

20

Pressure, GPa Figure 5. The calculated volume - pressure dependence for K2 SiF6 (symbols) and fits to the Murnaghan equation of state (dashed lines). The parameters of the fit are given in the figure.

Table II. Calculated density, shear moduli, sound velocities and Debye temperature for K2 SiF6 .

This work, GGA This work, LDA

ρ, kg/m3

GV , GPa

GR , GPa

G, GPa

vl , m/s

vt , m/s

vm , m/s

θD , K

2684 3200

13.58 18.41

13.32 17.19

13.45 17.80

3599 4542

2239 2358

2467 2639

298 338

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Journal of The Electrochemical Society, 159 (6) J212-J216 (2012)

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200000

Eg, GGA

GGA 160000

Eg, LDA

Absorption, cm-1

Band gap Eg, eV

10

9

8

0 4 8 12 16 20

120000

80000

40000 7 0

4

8

12

16

0

20

5

Pressure, GPa

10

20

25

Energy, eV 240000

8.0

LDA 200000

0 4 8 12 16 20

7.0

Absorption, cm-1

7.5

Distance, A

15

Lattice constant, GGA Lattice constant, LDA Si-F distance, GGA Si-F distance, LDA

1.70

160000 120000 80000 40000

1.65

0 5

0

4

8

12

16

10

20

15

20

25

Energy, eV

Pressure, GPa Figure 6. The calculated dependencies of the bandgap (a) and lattice constants and Si - F distances (b) in K2 SiF6 as the functions of pressure. The calculated values are given by symbols, the second power (top figure) and linear (bottom figure) fits are shown by the dashed lines. See text for the further details and fitting equations.

Murnaghan equation of state  1  B  − B V = 1+ P , V0 B

[5]

where B and B = dB/dP are the bulk modulus and its pressure derivative, respectively.11 The values of the bulk modulus B that are obtained from this fit are consistent with those reported in Table I. In addition, the pressure derivative B of the bulk modulus has been estimated as well; it is about 4.5–5.0 and lies within the typical limits reported for different solids. The pressure dependence of the calculated bandgap is shown in Fig. 6. The bandgap grows up non-linearly with pressure, and it is best of all approximated by the following second order polynomials: E g = 7.6033 + 0.1819P − 0.0036P 2 (GGA) and E g = 8.1143 + 0.1394P − 0.0024P 2 (LDA). In both equations the pressure P is measured in GPa, and the bandgap value is obtained in eV. The pressure dependence of the lattice constant a (Fig. 6) is linear. It decreases with pressure as a = 8.04571 − 0.0448P (GGA) and a = 7.6322 − 0.0320P (LDA). Here again the pressure P is expressed in GPa and the lattice constant in Å. It is interesting to note that the

Figure 7. The calculated absorption spectra at different pressures (in GPa, see the legend) for K2 SiF6 .

Si - F distance is also decreasing linearly with pressure (Fig. 6), but at considerably lower rate (about 30–40 times smaller than the pressure coefficient of the lattice constant). The linear fit equations for the Si - F distance are d = 1.68528 − 0.00123P Å (GGA) and d = 1.67549 − 0.0011P Å (LDA) with P in GPa. Figure 7 shows the calculated absorption spectra of K2 SiF6 at different pressures, from 0 to 20 GPa. Apart from the blueshift of the calculated spectra, intensity of all absorption peaks is strongly enhanced. Increase of the absorption intensity can be explained by a stronger overlap and mixture of the Si and F wave functions. Changes in the electron density distribution, induced by the applied hydrostatic pressure, can be also revealed by the analysis of the calculated effective Mulliken charges,12 collected in Table III. The

Table III. Calculated Mulliken charges (GGA/LDA results) for all ions in K2 SiF6 as a function of external pressure. Pressure, GPa 0 4 8 12 16 20

K

Si

F

0.97/1.02 1.02/1.08 1.07/1.14 1.11/1.18 1.16/1.22 1.19/1.26

2.11/2.05 2.07/2.02 2.05/1.99 2.02/1.96 2.00/1.94 1.98/1.92

−0.67/−0.68 −0.69/−0.70 −0.70/−0.71 −0.71/−0.72 −0.72/−0.73 −0.73/−0.74

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Journal of The Electrochemical Society, 159 (6) J212-J216 (2012)

positive charge on the K ions is increased with pressure, which means a shift of the electron density from the potassium ions to the silicon and fluorine ions, whose positive and negative charges are decreasing and increasing, respectively.

Conclusions The first ab initio analysis of the structural, electronic, optical, and elastic properties of potassium hexafluorosilicate K2 SiF6 host material has been performed and presented in this paper. Effects of the hydrostatic pressure on the electronic and optical properties were revealed by calculations of the optimized crystal structure at different pressures. Some of the obtained parameters - like the refractive index, density, and Debye temperature - have been compared to the available experimental data and indirect estimations from previous publications with good agreement demonstrated. The remaining calculated parameters like the bandgap and its pressure coefficient, elastic constants and pressure coefficients of the Si - F chemical bond still remain as theoretical estimations, which should be compared with the results of any future experiments on the K2 SiF6 material.

Acknowledgment The authors thank Prof. U. Lille (Tallinn University of Technology) for giving an opportunity to use the Materials Studio package. M. G. Brik also thanks the support from European Union through the European Regional Development Fund (Centre of Excellence “Mesosystems: Theory and Applications”, TK114). References 1. T. Takahashi and S. Adachi, J. Electrochem. Soc., 155, E183 (2008). 2. M. G. Brik and A. M. Srivastava, J. Lumin. (in press, doi:10.1016/j.jlumin.2011.08.047). 3. J. H. Loehlin, Acta Cryst. C, 40, 570 (1984). 4. M. D. Segall, P. J. D. Lindan, M. J. Probert, C. J. Pickard, P. J. Hasnip, S. J. Clark, and M. C. Payne, J. Phys.: Condens. Matter, 14, 2717 (2002). 5. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett., 77, 3865 (1996). 6. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett., 45, 566 (1980). 7. J. P. Perdew and A. Zunger, Phys. Rev. B, 23, 5048 (1981). 8. http://webmineral.com/data/Hieratite.shtml. 9. O. L. Anderson, J. Phys. Chem. Solids, 24, 909 (1963). 10. E. Schreiber, O. L. Anderson, and N. Soga, Elastic Constants and Their Measurements, McGraw-Hill, New York, 1973. 11. F. D. Murnaghan, Proc. Natl. Acad. Sci., 30, 244 (1944). 12. R. S. Mulliken, J. Chem. Phys., 23, 1833 (1955).

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