# Optical Properties Of Cubic Perovskite Rbcdf3 Biology Essay

K. Ephraim Babu, A. Veeraiah, N. Murali, D. Tirupathi Swamy, V. Veeraiah*

Modelling and Simulation in Materials Science Laboratory, Department of Physics, Andhra University, Visakhapatnam, Andhra Pradesh, 530003, India.

*Corresponding Author email: v_veeraiah@yahoo.co.in

The first-principles calculation is performed to investigate the energy band structure, density of states (DOS) and optical properties of cubic perovskite RbCdF3, by using density functional theory (DFT) with the generalized gradient approximation (GGA) in WIEN2K package. The calculated lattice constant is in good agreement with the experimental result. The calculated band structure shows an indirect (Γ-M) band gap of 3.201 eV. The contribution of the different bands is analyzed from the total and partial density of states curves. The electronic density plots reveal strong ionic bonding in Rb-F and strong covalent bonding in Cd-F. Calculations of the optical spectra, viz., the dielectric function, optical reflectivity, absorption coefficient, real part of optical conductivity, refractive index, extinction coefficient and electron energy loss, are performed for the energy range 0-30 eV.

PACS numbers: 71.15.Mb, 71.20.Ps, 78.20.-e

## Introduction

Computer modelling investigations have become very important tool to determine and to predict the material properties such as electronic, optical, magnetic, and mechanical etc. The theoretical investigations provide the possibility not only to explain the already known properties of a given material but also to predict what property will be expected for a hypothetical material. It is well known that materials with perovskite structures find applications in many areas of science and technology due to their electro-optic, electro-mechanical and non-linear properties. In the last decade, many experimental and theoretical investigations are devoted to the study of perovskite-type flourides: typically the ABF3 type (A: large cation with different valence and B: transition metal). This class of materials has great potential for a variety of device applications in optical, ferroelectric, antiferromagnetic systems [1-5] due to their wide band gaps. Crystallographic, elastic and Raman scattering investigations of structural phase transitions of RbCdF3 are investigated by Rousseau et al. [6, 7]. RbCdF3 has a cubic perovskite structure at room temperature and undergoes a structural phase transition at 124 K from cubic to tetragonal. When RbCdF3 is doped with some transition metal ions it shows many applications in optically rewritable Bragg grating, holographic storage, passive ultra violet (UV) detection, dosimetry, and tunable laser materials [8-10]. RbCdF3 is the most important perovskite fluoride crystal because it possesses excellent insulator, photoluminescent, ferromagnetic and piezoelectric properties and finds applications in photoelectric devices [11, 12].

The theoretical investigations on RbCdF3 have not yet been done to the best of our knowledge. In the present investigation, we explore the electronic, structural and optical properties of cubic RbCdF3 using the full potential linearizd augmented plane wave (FP-LAPW) method in the density functional theory (DFT) frame work with the generalized gradient approximation (GGA), in WIEN2K package. The lattice constant of RbCdF3 has been reported to be 4.398 Å by Jiang et al. [13]. The crystal structure of RbCdF3 in the cubic phase has been studied experimentally using various techniques [14, 15]. The study of optical properties is aimed at determining the dielectric function, refractive index, absorption coefficient, and energy loss spectroscopy of the RbCdF3 material. Similar investigations of some materials based on the first-principles are reported in the literature [16-18]. The present study is organized as follows: in Section 2, we briefly describe the computational details used in this work. Results and discussions of our study are presented in Section 3. Finally, conclusions and remarks are given in section 4.

## Computational details

Several computational methods are developed to study the electronic structure of materials based on first principles approaches. They are utilizing the full solution of the ground state of the electronic system within the local-density approximation (LDA) or generalized gradient approximation (GGA) to Kohn-Sham density functional theory (DFT) [19, 20]. Many physical and chemical properties of the condensed matter can be predicted accurately using these first principles methods [21, 22]. The crystal structure of RbCdF3 is illustrated in Fig. 1. The lattice parameters are taken to be a = 4.398 Å [13]. The crystal structure belongs to the space group pm3m (221) with Rb at (0.5, 0.5, 0.5), Cd at (0, 0, 0) and F at (0.5 0 0) positions and the origin is chosen to be at (0, 0, 0) [14]. The calculations presented in this work are performed using the FP-LAPW+lo method.

The electronic and optical properties are calculated using the ab initio full-potential linearized augmented-plane-wave (FP-LAPW) method with the WIEN2k code [23] in the generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) exchange–correlation potential [24]. In the FP-LAPW method, the unit cell is divided into two parts: (I) non-overlapping atomic spheres (centred at the atomic sites) and (II) an interstitial region. In this method no shape approximation on either potential or electronic charge density is made. By implementation of this method, which allows the inclusion of local orbitals in the basis, we are improving upon linearization and making possible a consistent treatment of semicore and valence states in one energy window and hence ensuring proper orthogonality. The FP-LAPW method expands the potential in the following form:

( inside sphere ) (1)

and ( outside sphere ) (2)

Where, Ylm () is a linear combination of radial functions times spherical harmonics.

The radii of the atomic muffin-tin (MT) spheres are equal to 2.00 a.u for Rb, 1.80 a.u for Cd, and 1.50 a.u for F. The set of plane waves Kmax is determined from the condition RMT Kmax = 7.0. The cut-off energy, which defines the separation between the core and valency states, is set to -8.0 Ry. The integration over the Brillouin zone is performed by the tetrahedron method, using 56 k-points in the irreducible part of the Brillouin zone (BZ ). The convergence criterion (for the total energy) is equal to 0.0001 Ry. The optoelectronic properties of the compound are calculated using a denser mesh of 3500 k-points in the irreducible Brillion Zone (IBZ).

The dielectric function ε(ω) = ε1(ω) + iε2(ω) is known to describe the optical response of the medium at all photon energies. The imaginary part ε2(ω) is directly related to the electronic band structure of a material and describes the absorptive behaviour. The imaginary part of the dielectric function ε2(ω) is given [25, 26] by

(3)

Where, M is the dipole matrix, i and j are the initial and final states respectively, f i is the Fermi distribution function for the i-th state, and Ei is the energy of electron in the i-th state with crystal wave vector k. The real part ε1(ω) of the dielectric function can be extracted from the imaginary part using the Kramers-Kronig relation in the form [27, 28]:

(4)

Where, P implies the principal value of the integral. The knowledge of both the real and imaginary parts of the dielectric tensor allows the calculation of important optical functions such as the refractive index n(ω), extinction coefficient k(ω) and reflectivity R(ω), using the following expressions:

(5)

(6)

(7)

Other optical parameters like energy loss function L(ω), absorption coefficient α(ω), and frequency dependent optical conductivity σ (ω) are calculated by the following expressions:

(8)

(9)

(10)

where, Wcv is transition probability per unit time.

## Results and discussions

## Structural optimization

The total energy per unit cell of RbCdF3 in the cubic perovskite structure is shown in Fig. 2, as a function of the lattice constant. The volume verses energy is fitted by the Birch-Murnaghan equation of state [29]. From this fit, we can get the equilibrium lattice constant (ao), bulk modlus (B0) and pressure derivative of the bulk modulus (B'). These values are shown in Table 1. Our calculated equilibrium lattice constant is 4.5035 Å, which is 2.39 % large as compared with the experimental value. The overestimation in the equilibrium lattice constant is a common feature with GGA calculations.

## Band structure and density of states

The calculated energy bands along the high symmetry lines in the Brillouin zone and total, partial density of states of RbCdF3 are shown in Figs. 3 and 4 respectively. The zero of energy is chosen to coincide with the valence band maximum (VBM), which occurs at M point, and conduction band minimum (CBM) occurs at the Γ point with indirect band gap of 3.201 eV. Thus, RbCdF3 is an indirect band gap insulator. On the basis of different bands; the total density of states (TDOS) could be grouped into four regions and the contribution of different states in these bands can be seen from the partial density of states (PDOS). The first region around −8.0 eV comprises a narrow band due to the Rb 4p state clearly from fig. 4(b). There is a small contribution of the Rb 4p state at −5.0 eV and −2.0 eV obserbed. In the second region around −5.0 eV majority contribution is due to the Cd 4d state, and minority contributions is due to the F 2p states, which are clearly seen from figs. 4(c) and 4(d). In the third region around −4.0 eV to Fermi energy level, majority contribution is due to F 2p states and minority contribution is due to Cd 4d states as observed from fig. 4(d). From the second and third regions −6.0 eV to Fermi energy level, bands due to Cd 4d and F 2p states are observed. There is hybridization between Cd 4d and F 2p states in this region. The first, second and third regions within the range −8.0 – 0 eV comprise the valence band. The upper part of the valence band is composed of the Rb 3d and F 2p states. The fourth region after the Fermi level is the conduction band. The lower part of this band near the Fermi level is mainly due to the Cd 5s, F 2p states. In the conduction band from 8.0 eV to 20 eV majority contribution is from Rb 3d states. Along with this majority contribution from Rb 3d states, there are minor contributions from Cd 5s states around 5 eV, from F 2p states around 10 eV, from Rb 4p , Cd 4p and F 2p states around 10 to 18 eV, and from the F 2p, Cd 4d, and Cd 4p states at the upper part around 20 eV to the conduction band. The calculated band gap of RbCdF3 is shown in Table 2.

The charge density distributions are shown in Fig. 5. Charge density maps serve as a complementary tool for achieving a proper understanding of the electronic structure of the system being studied. The ionic character of any material can be related to the charge transfer between the cation and anion while covalent character is related to the sharing of the charge among the cation and anion. The covalent behaviour is due to hybridization of Cd 4d with the F 2p states in the valence band near the Fermi Energy level. From the figures it is clear that the highest charge density resides in the immediate vicinity of the nuclei. The near spherical charge distribution around the Rb indicates that the bond between Rb and F is strong ionic, with no charge sharing among the contours of the respective atoms. It can be seen that most of the charge is populated in the Cd-F bond direction, while the maximum charge resides on the Cd and F sites. The corresponding contour maps of the charge density distributions are shown in Fig. 5. Hence, we conclude that there exist a strong ionic bonding in Rb-F and a strong covalent bonding in Cd-F.

## Dielectric and optical Properties

The FP-LAPW is a good theoretical tool for the calculation of the optical properties of a compound. The optical properties give useful information about the internal structure of the RbCdF3 compound. The calculated optical properties of RbCdF3 are shown in Figs. 6 and 7. The imaginary part ε2(ω) and real part ε1(ω) of the dielectric function are shown in fig. 6, as functions of the photon energy in the range 0-30 eV. In the imaginary part ε2(ω), the threshold energy of the dielectric function occurs at E0 = 3.42 eV, which corresponds to the fundamental gap at equilibrium. It is well known that the materials with band gaps larger than 3.1 eV work well in the ultraviolet region of the spectrum [30, 31]. In the Fig. 6(a), there are mainly seven peaks observed at (6.52, 9.78, 12.21, 13.53, 16.96, 18.73 and 20.00 eV). These peaks are identified by the symbols P( 1.78 at 6.52 eV), Q ( 1.20 at 9.78 eV), R ( 1.56 at 12.21 eV), S ( 1.83 at 13.53 eV), T ( 1.93 at 16.96 eV), U ( 3.64 at 18.73 eV), V ( 1.97 at 20.00 eV).

The peak P originates due to the electronic transition from (Cd 4d and F 2p ) of valence band (VB) to Cd 5s of conduction band (CB). The peak Q originates due to the electronic transition from (Cd 4d and F 2p ) of VB to Rb 4d of CB. The peak R originates due to the electronic transition from (Cd 4d and F 2p ) of VB to (Rb 4d and Cd 5p ) of CB. The peaks S and T originate due to the electron transitions from ( Cd 4d and F 2p ) of VB to (Rb 4d and Cd 5p ) of CB. The peaks U and V originate due to the electron transitions from (Cd 4d and F 2p ) of VB to (Rb 4d and Cd 4d ) of CB.

The real part of the dielectric function is also displayed in Fig 6(b). This function ε1(ω) gives us information about the electronic polarizability of a material. The static dielectric constant at zero is obtained as ε1(0) = 2.32. From its zero frequency limit, it starts increasing and reaches the maximum value of 3.09 at 4.58 eV. After 18.06 eV at 1.80 it starts decreasing and goes below 0 in negative scale for the ranges 18.78-19.56 eV and 20.00-21.32 eV. At negative values of real dielectric function ε1(ω), this material shows the metallic behaviour otherwise it is dielectric.

The refractive index and the extinction coefficient are displayed in Figs. 7(a) and 7(b) respectively. When we look at the behaviour of imaginary part of dielectric function ε2(ω) and extinction coefficient k(ω), a similar trend is observed from Figs. 6(a) and 7(b). The extinction coefficient k(ω) reaches the maximum absorption in the medium at 18.84 eV. Frequency dependent refractive index n(ω), reflectivity R(ω), and optical conductivity σ (ω) are also calculated and the salient features of the spectra are presented in Table 3. The static refractive index n(0) is found to have the value 1.523. The refractive index reaches a maximum value of 1.764 at 5.91 eV. The refractive index is greater than one because as photons enter a material they are slowed down by the interaction with electrons. The more photons are slowed down while travelling through a material, the greater the material's refractive index. Generally, any mechanism that increases electron density in a material also increases refractive index. However, refractive index is also closely related to bonding. In general, ionic compounds are having lower values of refractive index than covalent ones. In covalent bonding more electrons are being shared by the ions than in ionic bonding and hence more electrons are distributed through the structure and interact with the incident photons to slow down. The refractive index of the compound starts decreasing from the maximum value and goes below 1 for the ranges 19.00-19.78 eV and 20.00-30.00 eV as mentioned in the table 3. Refractive index less than unity shows that the group velocity (Vg = c / n) of the incident radiation is greater than c. It means that the group velocity shifts to negative domain and the nature of medium changes from linear to non-linear above 20.00 eV energy.

The optical reflectivity R(ω) is displayed in Fig. 7(c). The zero-frequency reflectivity is 4.36 %, which remains almost the same upto 3.20 eV. The small value of reflectivity in the infrared and visible energy range shows that the material is transparent in this range. Thus it can be used as an anti-reflecting coating material in this part of the energy spectrum. The maximum reflectivity value is about 31.21 % which occurs at 19.01 eV. Interestingly, the maximum reflectivity occurs where the real part of dielectric function ε1(ω) goes below zero, as seen from Figs. 6(b) and 7(c). The energy loss function is displayed in Fig. 7(d). The energy loss function L(ω) is an important factor describing the energy loss of a fast electron traversing in a material. The peaks in L (ω) spectra represent the characteristic associated with the plasma resonance. The resonant energy loss is seen at 22.48 eV. The optical conductivity σ(ω) is shown in Fig. 7(e). It starts from 4.32 eV and the maximum value of optical conductivity of the compound is obtained at 18.73 eV with a magnitude of 9135.89 Ω-1 cm-1. Similar features are also observed in absorption coefficient α(ω), in the absorption range upto 30 eV and it is shown in Fig. 7(f). This absorption range predicts the usefulness of the compound for optoelectronic devices.

## 4. Conclusions

In this paper, we have studied the electronic, structural and optical properties of the cubic perovskite RbCdF3 using the FP-LAPW + lo method within the generalized gradient approximation (GGA) in the framework of density functional theory. The lattice constant is found to be in good agreement with the experimental result. It is found that the compound has an indirect band gap of 3.201 eV. The compound exhibits strong ionic bonding in Rb-F, and strong covalent bonding in Cd-F. The optical properties such as dielectric function, reflectivity, absorption coefficient, real part of optical conductivity, refractive index, extinction coefficient and electron energy loss are studied in the energy range 0-30 eV. The above properties of RbCdF3 suggest that it is useful in anti-reflection coatings and optoelectronic devices.