Ab-initio Calculations of Electronic and Transport Properties of Calcium Fluoride (CaF2) B. Bohara, L. Franklin, Y. Malozovsky, and D. Bagayoko Department of Mathematics, Physics, and Science and Mathematics Education (MP-SMED) Southern University and A&M College, Baton Rouge, LA, 70813 A Presentation at the American Physical Society(APS) March Meeting Baltimore Convention Center, Baltimore, MD, USA - March 17, 2016

OUTLINE ❖ INTRODUCTION

❖ MOTIVATION ❖ COMPUTATIONAL METHODS

❖ RESULTS & DISCUSSIONS ❖ CONCLUSIONS

❖ SELECTED REFERENCES ❖ ACKNOWLEDGEMENTS

INTRODUCTION ❖Calcium Fluoride (CaF2): a Face centered Cubic crystal with very wide band gap (up to 12 eV). our ab-initio calculation results show that it is an indirect band gap material. ❖ It is used in spectroscopic windows and lenses. It has a high transmission, from 250 nm to 7μm, and low absorption properties. ❖The feature of low absorption and the high damage threshold makes this material useful for excimer laser optics.

MOTIVATIONS ❖ Potential applications of CaF2 and the absence of reliable Density functional Theory (DFT) calculations of its electronic properties. ❖ Electronic properties of large band gap materials are reportedly not well described by DFT. So, we want to see what the application of the BZW-EF method will produce. ❖ Specifically, this work is our attempt to resolve the discrepancies between theoretical and experimental band gap values for CaF2.

Table II. Comparison of band gap values from LDA, GGA, and experimental work with results obtained using our BZW-EF method. Computational methods BZW-EF OLCAO FPLAPW+lo Pseudo potential plane wave LCAO crystal 03 FPLAPW Pseudo potential plane wave Pseudo potentials plane wave FLAPW HF Pseudo potentials plane wave PAW Spectroscopic Ellipsometry (room temperature) Spectroscopy method (bulk, at 900K) High resolution laser calorimetric method (bulk)

Potentials DFT & others LDA(Our work) LDA LDA LDA LDA GGA GGA GGA GGA GGA GGA GGA Reflectance spectra Reflectance spectra Absorption

a (Å) 5.42 5.32 5.30 5.34 5.515 5.50 5.56 5.43 5.52 5.49 5.509

B (GPa) Eg (eV) 82.89 12.98 6.53 103 7.27 103.01 7.43 103 8.72 82.14 7.24 77.5 7.85 7.07 87.53 7.4 93 20.77 79.54 7.39 79 7.3 11.6 11.8 10

COMPUTATIONAL METHOS ❖ The Linear Combination of Atomic Orbitals (LCAO) formalism is utilized. ❖ The eigenvalue equation, Hψ = Eψ , is solved self consistently by taking ❖ Ψ = σ𝑁 𝑖=1 𝑎𝑖 Φ𝑖 , i = 1, N ❖ We use the ab-initio local density approximation (LDA) potential of Ceperley and Alder as parameterized by Vosko, Wilk, and Nusair (VWN).

❖ The distinguishing feature of this work: the implementation of the Bagayoko, Zhao and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF). ❖ BZW-EF method is employed to search for the smallest basis set that leads to the absolute minima of the occupied energies (i.e., the optimal basis set). (see, D. Bagayoko AIP Advances 4, 127104 (2014).

❖ We used successively larger basis set to attain the absolute minima of the occupied energies. ❖ The utilization of Rayleigh theorem ensures the selection of the optimal basis set among the potentially infinite number of possible basis sets.

Table I. Self-consistent calculations for c-CaF2 using the BZW-EF method at low temperature with the experimental lattice constant of 5.44672 Å. Calculation II produces the absolute minimum of occupied energies; its basis set is the optimal basis set.

Cal.

1 2 3 4 5

Calcium Fluorine Valence Band Gap Band Gap Valence electrons valence electrons Wave (X-Γ) eV (at Γ ) eV Functions 3s23p6 4s0 3s23p6 4s04p0 3s23p6 4s04p0 5s0 3s23p6 4s04p0 5s03d0 3s23p6 s04p03d0

2s22p6 2s22p6 2s22p6 2s22p6 2s22p6

13 16 17 22 21

13.049 12.984 12.881 9.664 9.665

13.378 13.314 13.212 9.932 9.933

RESULTS & DISCUSSIONS ❖ These results are to be followed with the ones to be obtained with the best values of the charge densities. ❖ The calculated indirect band gap (X –Γ) of 12.98 eV. ❖ Our calculated bulk modulus (82.9GPa). ❖ Our predicted equilibrium lattice constant of (5.42 Å). ❖ The electron effective masses at the bottom of the conduction band along the Γ-L, Γ-X, and Γ-K directions are 0.60 m0, 0.61 m0, and 0.61 m0, respectively.

FIG. 1.

Band structure of c-CaF2 - at the

experimental lattice constant of 5.44672 Å as

obtained with the optimal basis set of Calculation II. The Fermi energy is set equal to zero. The solid line (—) are the bands from Calculation II and the dashed line (---) are the ones from Calculation III.

FIG. 3. Partial Density of States (pDOS) for cubic Calcium Fluoride (c-CaF2) as derived

from

Calculation II.

the

bands

from

FIG. 4. Total Energy vs Lattice Constant for c- CaF2. The predicted, equilibrium lattice constant is 5.42 Å.

CONCLUSIONS ❖ Our calculated, indirect band gap, 12.98 eV, is among the best computational results. ❖ The calculated bulk modulus (82.89 GPa) is in agreement with the experimental result of (82.0 ± 0.7GPa ). ❖ Our predicted equilibrium lattice constant, 5.42Å, is 0.74% lower than experimental result. ❖ This calculation illustrates that the BZW-EF method is able to produce

accurate, electronic and related properties of semiconductors and insulators using ab-initio LDA and GGA potentials.

ACKNOWLEDGEMENTS The research is funded in part by ❖ The National Science Foundation (NSF) and the Louisiana Board of Regents, through LASiGMA and NSF HRD-1002541 ❖ The US Department of Energy – National, Nuclear Security Administration (NNSA) (Award No. DE-NA0001861 and DENA0002630) ❖ LaSPACE, and LONI-SUBR.

Thank You For Your Attention

ACKNOWLEDGEMENTS The research is funded in part by ❖ The National Science Foundation (NSF) and the Louisiana Board of Regents, through LASiGMA and NSF HRD-1002541 ❖ The US Department of Energy – National, Nuclear Security Administration (NNSA) (Award No. DE-NA0001861 and DENA0002630) ❖ LaSPACE, and LONI-SUBR.

OUTLINE ❖ INTRODUCTION

❖ MOTIVATION ❖ COMPUTATIONAL METHODS

❖ RESULTS & DISCUSSIONS ❖ CONCLUSIONS

❖ SELECTED REFERENCES ❖ ACKNOWLEDGEMENTS

INTRODUCTION ❖Calcium Fluoride (CaF2): a Face centered Cubic crystal with very wide band gap (up to 12 eV). our ab-initio calculation results show that it is an indirect band gap material. ❖ It is used in spectroscopic windows and lenses. It has a high transmission, from 250 nm to 7μm, and low absorption properties. ❖The feature of low absorption and the high damage threshold makes this material useful for excimer laser optics.

MOTIVATIONS ❖ Potential applications of CaF2 and the absence of reliable Density functional Theory (DFT) calculations of its electronic properties. ❖ Electronic properties of large band gap materials are reportedly not well described by DFT. So, we want to see what the application of the BZW-EF method will produce. ❖ Specifically, this work is our attempt to resolve the discrepancies between theoretical and experimental band gap values for CaF2.

Table II. Comparison of band gap values from LDA, GGA, and experimental work with results obtained using our BZW-EF method. Computational methods BZW-EF OLCAO FPLAPW+lo Pseudo potential plane wave LCAO crystal 03 FPLAPW Pseudo potential plane wave Pseudo potentials plane wave FLAPW HF Pseudo potentials plane wave PAW Spectroscopic Ellipsometry (room temperature) Spectroscopy method (bulk, at 900K) High resolution laser calorimetric method (bulk)

Potentials DFT & others LDA(Our work) LDA LDA LDA LDA GGA GGA GGA GGA GGA GGA GGA Reflectance spectra Reflectance spectra Absorption

a (Å) 5.42 5.32 5.30 5.34 5.515 5.50 5.56 5.43 5.52 5.49 5.509

B (GPa) Eg (eV) 82.89 12.98 6.53 103 7.27 103.01 7.43 103 8.72 82.14 7.24 77.5 7.85 7.07 87.53 7.4 93 20.77 79.54 7.39 79 7.3 11.6 11.8 10

COMPUTATIONAL METHOS ❖ The Linear Combination of Atomic Orbitals (LCAO) formalism is utilized. ❖ The eigenvalue equation, Hψ = Eψ , is solved self consistently by taking ❖ Ψ = σ𝑁 𝑖=1 𝑎𝑖 Φ𝑖 , i = 1, N ❖ We use the ab-initio local density approximation (LDA) potential of Ceperley and Alder as parameterized by Vosko, Wilk, and Nusair (VWN).

❖ The distinguishing feature of this work: the implementation of the Bagayoko, Zhao and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF). ❖ BZW-EF method is employed to search for the smallest basis set that leads to the absolute minima of the occupied energies (i.e., the optimal basis set). (see, D. Bagayoko AIP Advances 4, 127104 (2014).

❖ We used successively larger basis set to attain the absolute minima of the occupied energies. ❖ The utilization of Rayleigh theorem ensures the selection of the optimal basis set among the potentially infinite number of possible basis sets.

Table I. Self-consistent calculations for c-CaF2 using the BZW-EF method at low temperature with the experimental lattice constant of 5.44672 Å. Calculation II produces the absolute minimum of occupied energies; its basis set is the optimal basis set.

Cal.

1 2 3 4 5

Calcium Fluorine Valence Band Gap Band Gap Valence electrons valence electrons Wave (X-Γ) eV (at Γ ) eV Functions 3s23p6 4s0 3s23p6 4s04p0 3s23p6 4s04p0 5s0 3s23p6 4s04p0 5s03d0 3s23p6 s04p03d0

2s22p6 2s22p6 2s22p6 2s22p6 2s22p6

13 16 17 22 21

13.049 12.984 12.881 9.664 9.665

13.378 13.314 13.212 9.932 9.933

RESULTS & DISCUSSIONS ❖ These results are to be followed with the ones to be obtained with the best values of the charge densities. ❖ The calculated indirect band gap (X –Γ) of 12.98 eV. ❖ Our calculated bulk modulus (82.9GPa). ❖ Our predicted equilibrium lattice constant of (5.42 Å). ❖ The electron effective masses at the bottom of the conduction band along the Γ-L, Γ-X, and Γ-K directions are 0.60 m0, 0.61 m0, and 0.61 m0, respectively.

FIG. 1.

Band structure of c-CaF2 - at the

experimental lattice constant of 5.44672 Å as

obtained with the optimal basis set of Calculation II. The Fermi energy is set equal to zero. The solid line (—) are the bands from Calculation II and the dashed line (---) are the ones from Calculation III.

FIG. 3. Partial Density of States (pDOS) for cubic Calcium Fluoride (c-CaF2) as derived

from

Calculation II.

the

bands

from

FIG. 4. Total Energy vs Lattice Constant for c- CaF2. The predicted, equilibrium lattice constant is 5.42 Å.

CONCLUSIONS ❖ Our calculated, indirect band gap, 12.98 eV, is among the best computational results. ❖ The calculated bulk modulus (82.89 GPa) is in agreement with the experimental result of (82.0 ± 0.7GPa ). ❖ Our predicted equilibrium lattice constant, 5.42Å, is 0.74% lower than experimental result. ❖ This calculation illustrates that the BZW-EF method is able to produce

accurate, electronic and related properties of semiconductors and insulators using ab-initio LDA and GGA potentials.

ACKNOWLEDGEMENTS The research is funded in part by ❖ The National Science Foundation (NSF) and the Louisiana Board of Regents, through LASiGMA and NSF HRD-1002541 ❖ The US Department of Energy – National, Nuclear Security Administration (NNSA) (Award No. DE-NA0001861 and DENA0002630) ❖ LaSPACE, and LONI-SUBR.

Thank You For Your Attention

ACKNOWLEDGEMENTS The research is funded in part by ❖ The National Science Foundation (NSF) and the Louisiana Board of Regents, through LASiGMA and NSF HRD-1002541 ❖ The US Department of Energy – National, Nuclear Security Administration (NNSA) (Award No. DE-NA0001861 and DENA0002630) ❖ LaSPACE, and LONI-SUBR.