Electronic and Optical Properties of Rare Earth Oxides: *Ab Initio* Calculation ()

Sezen Horoz^{1}, Sevket Simsek^{2}, Selami Palaz^{3}, Amirullah M. Mamedov^{4,5*}

^{1}Institute of Natural Sciences, Cukurova University, Adana, Turkey.

^{2}Department of Material Science and Engineering, Hakkari University, Hakkari, Turkey.

^{3}Department of Physics, Faculty of Science and Letters, Harran University, Sanliurfa, Turkey.

^{4}Nanotechnology Research Center (NANOTAM), Bilkent University, Ankara, Turkey.

^{5}International Scientific Center, Baku State University, Baku, Azerbaijan.

**DOI: **10.4236/wjcmp.2015.52011
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In this work, we have investigated the electronic and optical properties
of the technologically important rare earth oxide compounds—X_{2}O_{3} (X: Gd, Tb) using the density functional theory within the GGA. The band
structure of X_{2}O_{3} have been calculated along high
symmetry directions in the first brillouin zone. The real and imaginary parts
of dilectric functions and the other optical responses such as energy-loss
function, the effective number of valence electrons and the effective optical
dielectric constants of the rare earth sesquioxides (Gd_{2}O_{3} and Tb_{2}O_{3}) were calculated.

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Horoz, S. , Simsek, S. , Palaz, S. and Mamedov, A. (2015) Electronic and Optical Properties of Rare Earth Oxides: *Ab Initio* Calculation. *World Journal of Condensed Matter Physics*, **5**, 78-85. doi: 10.4236/wjcmp.2015.52011.

1. Introduction

X_{2}O_{3} (X:Gd, Tb) are the interesting materials from both fundamental and industrial perspectives and have a wide range of applications. They are thermodynamically stable, making them useful for corrosion resistive coating [1] -[5] . Additionally, their high refractive indices lead to applications in optics, such as antireflection coatings, switches, filters and modulators [1] [4] . The most recent interest of them is due to their high dielectric constants and electrical stability, making them good candidates for a new class of gate oxides in metal-oxide semiconductor field-effect transistors [1] . In addition, many properties of rare-earth sesquioxides are determined by their semicore f-levels. While being mainly localized on the rare-earth atoms and usually not participating in bonding and electronic conduction, f-shell electrons are available for optical transition and can establish strong magnetic order [1] . So far as we know, no ab initio general potential calculation of the optical properties of the rare-earth sesquioxides has been reported. The main purpose of this work is to provide some additional information to the existing features of Gd_{2}O_{3} and Tb_{2}O_{3} by using density functional theory. Therefore, in this work, we have investigated the electronic and optical properties of Gd_{2}O_{3} and Tb_{2}O_{3} compounds.

2. Method of Calculation

In the present paper, all calculations have been carried out using the ab-initio total-energy and molecular-dy- namics program VASP (Vienna ab-initio simulation program) developed at the Faculty of Physics of the University of Vienna [6] - [9] within the density functional theory (DFT) [10] . The exchange-correlation energy function is treated within the GGA (generalized gradient approximation) by the density functional of Perdew et al. [11] . We get a good convergence using a Monkhorst-Pack [12] mesh grid for the total-energy calculation with a cutoff energy of 510 eV for both compunds. The electronic iterations convergence is using the Normal (blocked Davidson) algorithm and reciprocal space projection operators. These values were found to be sufficient for studying the electronic and optical properties of X_{2}O_{3} crystals.

3. Results and Discussion

3.1. Structural and Electronic Properties

In the first step of our calculations, we have carried out the equilibrium lattice constants of Gd_{2}O_{3}, and Tb_{2}O_{3} by minimizing the ratio of the total energy of the crystal to its volume using the experimental data [13] [15] . We have compared the present results for lattice parameters of X_{2}O_{3} with previous experimental values [13] - [29] and are given in Table 1. These results are within the accuracy range of calculations based on density functional theory.

Table 1. The calculatedequilibriumlattice parameters and direct band gaps together with the available experimental values for Gd_{2}O_{3} and Tb_{2}O_{3}.

The investigation of electronic band structure for understanding the electronic and optical properties of X_{2}O_{3 }is very useful. The band structures of the X_{2}O_{3} were calculated using GGA. The electronic band structures were calculated along the special lines connecting the high-symmetry points Г, H, N, and P for X_{2}O_{3} in the k-space. The electronic band structure of Gd_{2}O_{3}, and Tb_{2}O_{3} along the high symmetry directions have been calculated by using the equilibrium lattice constants and are given in Figure 1 and Figure 2.

As can be seen in Figure 1, the Gd_{2}O_{3} compound has a direct band gap semiconductor with the value 3.86 eV (in Г-high symmetry point). The band gap with the value 3.82 eV of Tb_{2}O_{3 }compound has the same character of that of Gd_{2}O_{3} (Figure 2). The band gap values obtained for X_{2}O_{3} are good agreement with the earlier theoretical resuts, but is less than the estimated experimental results [1] [3] [5] . In these figures (Figure 1 and Figure 2), the lowest valance bands that occur between 0 and −3.5 eV (72 energy states) are dominated by O 2p states while the valence bands that occur between −14 eV and −16.5 eV (24 energy states) are dominated by Gd 6s and Tb 6s states. The lowest occupied valance bands are essentially dominated by O 2s (−19 eV and −21.5 eV and include 48 energy states).

3.2. Optical Properties

It is well known that the effect of the electric field vector, , of the incoming light is to polarize the material. At the level of a linear response, this polarization can be calculated using the following relation [29] :

Figure 1. The calculated electronic band structure and Density of State for Gd_{2}O_{3}.

Figure 2. The calculated electronic band structure Density of State for Tb_{2}O_{3}.

(1)

where is the linear optical susceptibility tensor and it is given by [30]

(2)

where denote energy bands, is the Fermi occupation factor, is the normalization volume. are the frequency differences, is the energy of band n at wave vector. The are the matrix elements of the position operator [30] .

As can be seen in Equation (2), the dielectric function and the imaginary part of, is given by

(3)

The real part of, , can be obtained by using the Kramers-Kroning transformation [30] . Be-

cause the Kohn-Sham equations determine the ground state properties, the unoccupied conduction bands as calculated, have no physical significance.

The known sum rules [30] can be used to determine some quantitative parameters, particularly the effective number of the valence electrons per unit cell, as well as the effective optical dielectric constant, which make a contribution to the optical constants of a crystal at the energy. One can obtain an estimate of the distribution of oscillator strengths for both intraband and interband transitions by computing the defined according to

(4)

where is the density of atoms in a crystal, e and m are the charge and mass of the electron, respectively, and is the effective number of electrons contributing to optical transitions below an energy of.

Further information on the role of the core and semi-core bands may be obtained by computing the contribution that the various bands make to the static dielectric constant,. According to the Kramers-Kronig relations, one has

(5)

One can therefore define an “effective” dielectric constant, that represents a different mean of the interband transitions from that represented by the sum rule, Equation (5), according to the relation

(6)

The physical meaning of is quite clear: is the effective optical dielectric constant governed by the

interband transitions in the energy range from zero to, i.e. by the polarization of the electron shells.

We first calculated the real and imaginary parts of the linear dielectric function of the Gd_{2}O_{3}, and Tb_{2}O_{3} compounds (Figure 3 and Figure 4). In order to calculate the optical response by using the calculated band structure, we have chosen a photon energy range of 0 - 65 eV and have seen that a 0 - 40 eV photon energy range is sufficient for most optical functions. We first calculated the real and imaginary parts of linear dielectric function of the Gd_{2}O_{3} and Tb_{2}O_{3} compounds (Figure 4 and Figure 5). All the Gd_{2}O_{3} and Tb_{2}O_{3} compounds

Figure 3. The real and imaginary parts of the linear dielectric function and Electron energy-loss spectrum of Gd_{2}O_{3}.

Figure 4. The real and imaginary parts of the linear dielectric function and Electron energy-loss spectrum of Tb_{2}O_{3}.

studied so far have ε_{1} are equal to zero in the energy region between 9 eV and 40 eV for decreasing and increasing of (see, Table 2). Also, values of versus photon energy have main peaks in

the energy region 4 eV and 30 eV. Some of the principal features and singularities of the for both investigated compounds are shown in Table 3.

The peaks of the correspond to the optical transitions from the valence band to the conduction band and are in agreement with the previous results. The maximum peak values of for Gd_{2}O_{3} and Tb_{2}O_{3} are around 9.31 eV and 9.49 eV, respectively.

The corresponding energy-loss functions, , were calculated using Equation (7) and are also presented in Figure 3 and Figure 4. The describes the energy loss of fast electrons traversing the material. The sharp maxima in the energy loss function are associated with the existence of plasma oscillations [30] . The curves of L have a maximum near 35 eV (Gd_{2}O_{3}) and 38 eV (Tb_{2}O_{3}).

(7)

The calculated effective number of valence electrons is given in Figure 5(a). The effective number of valence electron per unit cell, up to 5 eV is zero (below the band gap) then reaches saturation values at

Figure 5. The calculated (a) effective number of electrons participating in the interband transitions and (b) effective optical dielectric constant.

Table 2. The energy values at the zero point of real part of dielectric function for Gd_{2}O_{3} and Tb_{2}O_{3}.

Table 3. The maximum peak values of the imaginary part of the dielectric function for Gd_{2}O_{3} and Tb_{2}O_{3}.

about 30 eV (Gd_{2}O_{3}) and 35 eV (Tb_{2}O_{3}). This means that deep-lying valence orbitals participate in the interband transitiond as well (see Figure 1 and Figure 2). The effective optical dielectric constant, , is shown in Figure 5(b).

The curves of can be arbitrarily divided into two parts. The first part is characterized by a rapid growth of and extends up to 12 eV. The second part shows a smoother and slower growth of and reaches a saturation values at about 30 eV(Gd_{2}O_{3}) and 35 eV (Tb_{2}O_{3}). This means that the largest contribution to is made by transitions corresponding to the bands at ~5 eV and ~12 eV.

4. Conclusion

In the present work, we have made a detailed investigation of the electronic, and frequency-dependent linear optical properties of the X_{2}O_{3} (X: Gd and Tb) crystals using the density functional methods. The result of the structural optimization implemented using the GGA are in good agreement with the experimental and theoretical results. We have examined photon-energy dependent dielectric functions, some optical properties such as the energy-loss function, the effective number of valance electrons and the effective optical dielectric constants for both materials.

Acknowledgements

This work is supported by the projects DPT-HAMIT, DPT-FOTON, NATO-SET-193 and TUBITAK under Project Nos., 113E331, 109A015, 109E301.

NOTES

^{*}Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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