1. Introduction

In 1940s, Hippel et al. and Wul & Goldman discovered ferroelectric behavior in BaTiO₃ compounds ^[1,2]. After discovery of ferroelectric effect in BaTiO₃ compound, ferroelectric materials offer a variety of applications in technological areas such as ultrasonic transducers, piezoelectric transformers and acoustic scanners, etc. ^[3].

Classifying ferroelectric materials according to their structure, we end up with four groups: a) the tungsten-bronze group, b) pyrochlores, c) the bismuth layer-structure group, and d) perovskites. The families, including lead element, represent most of the ferroelectric ceramics manufactured in the world ^[4]. Unfortunately, the lead-based ferroelectric materials are hazardous for environmental and they need to be replaced with lead-free compounds. The bismuth-based compounds are lead-free compounds ^[5,6]. In recent years, especially perovskite-type Bi compounds have received more attention. Researchers have intensively investigated and analyzed bismuth-based compounds due to their extraordinary properties such as lead-free piezoelectric (BiXO₃ (X = Al, Sc, Fe, …) and modified version) ^[7,8,9,10], photocatalysts (BiFeO₃ and Ga doped version of this compound) ^[11,12] and multiferroics (BiXO₃ (X = Mn, Co, Fe, Cr)) ^{[11,12,13,14,15,16]}. There are some Bi-perovskites compounds (BiXO₃ (X = Ti, Cu, Rh, …)) that has been devoted limited studies in literature ^[17,18].

BiRhO₃, having similar electronic analogue to BiCoO₃ compound, is one of almost untouched compounds among the BiXO₃ (X = Mn, Co, Fe, …) materials ^[18,19]. In literature, there are limited studies related to BiRhO₃ or modified version of compound ^{[18,19,20,21,22]}. Due to this reason, information about its structure and dielectric properties is narrow. Therefore, electronic, optical, and structural properties of BiRhO₃ are still under cover; hence the compound needs further investigation. The aim of the paper is to obtain electronic, structural and optical properties of BiRhO₃ and to give some theoretical information about compound for further investigations.

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2. Computational Methods

The calculation described in this paper was performed using Linearized Augmented Plane Wave (LAPW) ^[23] implemented into Density Functional Theory (DFT). Pseudopotential calculations of total energy, electronic band structure and optical properties were obtained using the WIEN2k packet program ^[24]. We used 25 hartree for cut-off energy and an appropriate set of k points in chosen grid (21 × 29 × 20) in the first Brillouin zone (BZ) to compute the density, potential and wave function. Practical applications of DFT require an approximate exchange-correlation energy. To choose the best approximate exchange-correlation energy for true exchange-correlation functional, four different potentials implemented in WIEN2k program were used. These potentials are namely Local Spin Density Approximation (LSDA), Generalized Gradient Approximation (GGA) using the Perdew-Burke-Ernzerhof ^[25], WC-GGA ^[26] and PBEsol-GGA ^[27]. In the calculation, −8.5 Ry energy value was used to separate between valence and core states.

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3. Results and Discussion

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3.1. Structural Properties

The space group of orthorhombic BiRhO₃, non-magnetic material, is Pnma (62). In unit cell, there are 20 atoms. The location of each atom in the unit cell is given in Table 1 and Figure 1. BiRhO₃ is perovskite-type compound crystallizing in an orthorhombic system with a = 5.8089 Å ^[28], b = 7.7720 Å ^[15], c = 5.3510 Å ^[28]. The atomic position and space group information about the BiRhO3 compound in unit cell were taken from ref ^[19]. Atomic electronic configuration of compound is Bi [Xe] 4f¹⁴5d¹⁰6s²6p³, Rh [Kr] 4d⁸5s¹ and O [He] 2s²2p⁴. Before starting to calculate physical properties of compound, we implemented the spin-orbit coupling effect on our calculation because 4d electrons have spin-orbit coupling.

As mentioned above, in Wien2k software, there are four type potentials to be used as an exchange-correlation potential. Therefore, we ran volume optimization calculation for each potential to obtain the best theoretical lattice parameters, which are the closest to experimental value and gave the minimum Energy value (Figure 2). For this aim, we used Birch-Murnaghan equation of state ^[29,30] to fit volume versus energy graph to obtain the theoretical values of the lattice constant depending on selected potential. All obtained lattice values are inserted in Table 2. In addition, we also obtained bulk modulus, derivative bulk modulus and ground state volume for BiRhO₃ and all values were inserted in Table 2.

It is clear from Table 2 that WC-GGA potential gives the closet value of the calculated lattice constant to experimental one. After this point, we used the WC-GGA potential for the rest of the calculations instead of exchange-correlation potential. In Figure 3, we also graphed pressure versus volume for four potentials. As seen from Figure 3, zero pressure can be achieved where the minimum volume of the compound was obtained.

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3.2. Electronic Properties

The calculated band structure of BiRhO₃ was given in Figure 4. The Fermi energy level was set to originate. We can see that, there are bands at −17 eV and −11 eV called deep valence band of the compound. To determine forbidden energy bandgap value between the conduction band and valence band to classify compound, y axis was enlarged (Figure 5). As seen from Figure 5, BiRhO₃ has an energy band gap 0.302 eV. This result confirms the result of Yi et al. ^[19]. According to this band gap value, BiRhO₃ compound is classified as a semiconductor. Furthermore, while the top of the valence band is located at X high symmetry point, the bottom of the conduction band is located at Γ high symmetry point in the BZ. As a result, BiRhO₃ has indirect transition. This kind of transition includes both photon and phonon for momentum conservation.

Also, we mentioned that we did include the spin-orbit effect to calculation due to the d orbital of Rh element. We also performed without spin-orbit calculation. Comparing these two calculations, we could not see any significant differences.

The electronic density of states (DOS) and the atom resolved partial density of states (PDOS) of BiRhO₃ at 0 GPa were also calculated and shown in Figure 6. As seen from Figure 6a, there is a deep valley which is the border line between the bonding state and antibonding states.

All atoms contribute to not only the conduction band, but also the valence band (Figure 6a). This contribution mainly comes from Rh and O atoms comparing with Bi atom. Main contribution to conduction and valence bands comes from Rh atom. Analysing more in detail, the valence band is constructed by Rh-d and O-p orbitals, regardless of where O atoms sit in the unit cell (Figure 6c–d). The conduction band is mixed by Bi-p, O-p and Rh-d orbitals. Moreover, s orbital of Bi atom contributes to core region.

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3.3. Optical Properties

Next, we will discuss the optical properties of BiRhO₃ compound. The linear response of a system due to an external electromagnetic field can be described by the complex dielectric function $\varepsilon \left(\omega \right) = {\varepsilon _1}\left(\omega \right) + i{\varepsilon _2}\left(\omega \right)$. In this equation, ${\varepsilon _1}\left(\omega \right)$ and ${\varepsilon _2}\left(\omega \right)$ are real and imaginary parts of the complex dielectric function, respectively. To calculate dielectric function of compound, we have to consider the material structure. Calculating dielectric function, independent from selected axes, are sufficient in cubic crystal structure because ${\varepsilon _x}\left(\omega \right)$, ${\varepsilon _y}\left(\omega \right)$ and ${\varepsilon _z}\left(\omega \right)$ are equal to each other. In contrary, if compound has non-cubic crystal structure, we have to pay attention axes direction to calculate dielectric function.

Since BiRhO₃ compound has orthorhombic structure, all dielectric components, dependence on axes, were calculated. The sufficient photon energy range was chosen for the optical response of BiRhO₃. The real, ${\varepsilon _1}\left(\omega \right)$, and imaginary part, ${\varepsilon _2}\left(\omega \right)$, of dielectric constant as a function of photon energy were plotted and shown in Figure 7 and Figure 8. We only considered the case of the incident radiation with the linear polarization.

As seen in Figure 7, the static values of the real part of dielectric constant depending on axes, $\varepsilon _1^{xx}\left(0 \right)$, $\varepsilon _1^{yy}\left(0 \right)$ and $\varepsilon _1^{zz}\left(0 \right)$ were found 26.3, 23.6 and 22.7 (small window), respectively. In addition, ε₁(ω) has negative values at the photon energy regions between 4.69–9.075, 13.32–16.04 and 27.36–28.01 eV. In these regions, ε₁(ω) has negative value and the incident electromagnetic waves are entirely reflected. There is no transition after 28.01 eV energy values since ε₁(ω) gets closer to zero.

The imaginary part of the dielectric function is connected to the energy band structure. Careful analyzing imaginary part of the dielectric function, it was observed that absorption starts before 0.3 eV (small window in Figure 8). In addition, there is a dramatic increase at 0.3 corresponding to the energy band gap where the transition occurs from the valence band to the conduction band. The maximum peak of the imaginary part of dielectric constant located at 1.102 eV is responsible for strong optical transitions.

Figure 9 shows the refractive index according to obtained crystal axes. It is common knowledge refractive index define how light propagates through the material. Due to the structure of the compound, the axes dependence static index of the sample was found as an ${n_x}\left(0 \right) = 5.14$, ${n_y}\left(0 \right) = 4.86$ and ${n_z}\left(0 \right) = 4.77$ (small window).

Finally, Figures 10 and 11 represent three optical constants namely reflectivity R(ω), extinction coefficient k(ω) and energy loss L(ω). They are calculated as functions of incident radiation up to 60 eV. According to our intensive and detailed search, we could not come cross and optical constant of BiRhO₃ compound obtained from experimental or other theoretical studies. So they should be understood as a theoretical prediction.

While the reflectivity (Figure 10) is less than 50% for energy range 0 to 3 eV along y and z-axes and more than 50% along the x axis. More than 50% of the incoming electromagnetic beam through 5–10 eV was reflected back at y axis (small window). In this region, photon penetrates through compounds along x and z axes comparing with the y axis.

The local minimum of extinction coefficient k(ω) plot in Figure 11 corresponds to the zero of ε₁(ω). The function L(ω) also shown in Figure 11 describes the energy loss of a fast electron traversing the material. It is a well-known fact that the prominent peak in $L\left(\omega \right)$ spectra presents the characteristic associated with plasma resonance and corresponding frequency is the so-called plasma frequency ${\omega _p}$. As seen from Figure 10, sharp peak is associated with plasma oscillation, which correspond negative values of the real part of the dielectric constant. The peak of the L(ω) calculated by the WIEN2k code is located around 28 eV for all axes.

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4. Conclusion

In literature, there are so limited studies related BiRhO₃ compound. Hence, we reported electronic, structural and optical properties of BiRhO₃. WC-GGA implemented in Wien2K code was used for exchanged and correlation interaction. The obtained relaxed lattice constant is consistent with the experimental result. According to our results, BiRhO₃ is a semiconductor and has indirect band gap transition from X to Γ high symmetry point in the Brillioun zone with the value of 0.3 eV. In addition, we also plot DOS and PDOS. We also calculated energy dependent optical constant of the compound. Unfortunately, we do not come cross with any optical data for compound, so we could not make any comparison.

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Acknowledgements

This work was supported by the Unit of Scientific Research Projects of Yuzuncu Yil University under project No. 2015-FED-YL348. The authors also would like to special thanks to Dr. Mustafa TEKPINAR for his critical reading. Also grammer was checked with Gingerly grammer program.

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Conflict of Interest

All authors declare no conflicts of interest in this paper.

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