Optoelectronic properties of RuCrX (X = Si, Sn, Ge) Half Heusler alloys: a DFT study

The aim of this study was to investigate the structural, electronic, optical, and thermal properties of optoelectronic Half Heusler Alloys, RuCrX (X = Si, Sn, Ge). The characterizations of these Half Heusler Alloys, RuCrX (X = Si, Sn, Ge) have been performed using Density Functional Theory (DFT) through first-principles calculations with the aid of WIEN2K code. The Generalized Gradient Approximation (GGA) was utilized as an exchange-correlation function in WIEN2K-Package to optimize the structures.. To obtain the necessary observational quality and desired properties, Full Potential Linearized Augmented Plane Wave (FPLAPW) was applied. The calculation of the lattice constants and band gaps was crucial to determine suitable materials for specific optoelectronic applications. This study also emphasized the complex dielectric function and elastic properties leading to the imaginary part of the dielectric functions showed that compounds were optically metallic and transparent with ductile properties. Also, the optical spectra and band structure transitions were studied in detail. Hence, the study predicted that Half Heusler Alloys, RuCrX (X = Si, Sn, Ge), have the potential for applications in optoelectronic devices.


Introduction
The half-metallicity concept arises due to an unusual electronic structure of materials. A half-metal behaves differently for electrons of opposite spins. It is a half-metal for electrons of a particular spin while an insulator or semiconductor for electrons of opposite spin [1]. Half-metals have attracted much interest in recent years, mainly due to their applications in spintronics. The band gap in one of the spin channels gives rise to 100% spin polarization at the Fermi level. This produces a fully spin-polarized current in the half metals [2]. Such current helps in maximizing the efficiency of magneto-electronic devices. It is evident that some materials like iron (Fe), cobalt (Co), and Nickel (Ni) are in abundant use naturally and present ferromagnetic behavior [3]. It is, in general, the observation that these materials show attraction to external magnetic fields, and they can keep their magnetic properties intact in the absence of external magnetic fields. The beginning of the 20th century shows a stark revolution in ferromagnetism. It was analyzed that Copper (Cu), Manganese (Mn), and Aluminum (Al) composition execute magnetic nature at room temperature, although these elements are not naturally magnetic [4]. A German genius Heusler showed a deep inclination towards ferromagnetic elements and their remarkable properties, revealing that the Cu-Mn-Al composition shows ferromagnetic only owing to the heat treatments and compositions. It was considered that only the solid solution of Cu(Mn) 3 Al was responsible for its ferromagnetic. It was only revealed in 1928-1929 that Cu 2 MnAl showed ferromagnetism because of its' BCC structure [5]. The credit for studying the first Heusler single crystal of Cu 2 MnAl and tracing the exact atomic Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
positions of individual atoms in this complex structure goes to Potter. Cu 2 MnAl was famous as Heusler Alloy at that time. Afterward, all the alloys having a similar composition and nature were categorized as Heusler Alloys. The weird diversity in properties such that anti-magnetism and tremendous magnetization caused the prominence of Heusler Alloys [6]. Optical and electronic properties of the half-metallic full Heusler compounds Co 2 CrZ(Z = Ga, Ge, As) are investigated [7]. Until the end of the 20th century, many Heusler Alloys and their derivatives had been explored [8]. It happened in 1903 that Heusler reported that Cu-Mn alloy becomes a material when an sp element (In, Al, Bi, and Sn) is added to it. The alloy contains no ferromagnetic element [1,6]. It was predicted that Heusler Alloys show half-metallic nature. In the case of Heusler Alloys, high spin polarization, and large curie temperatures were confirmed by experimental investigation [9]. Optical properties like dielectric function, refractive index, reflectivity, conductivity, and absorption coefficient are calculated and discussed [10]. In the case of Heusler Alloys, there is flexibility for the partial or the complete alternative substitutions of one or more than one of the constituent elements [11]. New materials with more physical properties, such as see-beck coefficient, heat capacity, thermal conductivity, optical and electrical resistivity, can be manufactured using the Heusler Alloys' flexible nature [12]. Heusler alloys consist of a large family of more than 1000 materials with exotic properties like high tunability, strong spin-orbit coupling, half-metallicity, mechanical and thermal stability, and interchange of symmetry [5,13]. These instinctive properties lead to tremendous applications in spintronics, thermoelectric, piezoelectric semiconductors, optoelectronics, and topological insulators [14]. These are pollution-free materials and getting special attention from green-energy scientists because of the drastic climate change and global energy crisis [15].
The current study is organized into four sections. The detail of these four sections is as under: • Section 1 explains the background and introduction of desired materials.
• Section 2 describes the computational details of WIEN2ksoftware and its' application in this research.
• Section 3 presents the research outcomes and discussion in graphical and tabular representations.
• Section 4 presents concluding remarks and future recommendations to continue the research further.

Computational methods
Recent progress in DFT (density functional theory) makes it possible to predict accurate and reliable optoelectronic properties based on a dynamical lattice model [1,16]. The input data for the first principles calculations of optoelectronic properties is entirely based on the structure of the crystal and the number of atoms within the crystal. So structures are optimized in WIEN2k-package with the Full Potential Linearized Augmented Plane Wave (FP-LAPW) method. PBE-GGA is used as an exchange and correlation function [10].
The numbers of plane waves are limited by R MT × K max = 7.0. The cut-off energy to separate core and valence states is 6.0 Ry for three materials. Self-Consistent Field (SCF) calculations are accomplished with 120 k-gen based on a mesh of (1000+) k-points in the irreducible Brillouin zone scheme. The convergence criteria are fixed to 10 -3 e for charges and 10 -5 Ry for energy. The structural stability is checked by the Birch-Murnaghan equation of state. These compounds are used for FCC lattices Monkhorst-Pack mesh which yields k-points at an irreducible wedge of integrated Brillouin zone [17]. The convergence threshold means self-consistency is achieved for each cycle when energy changes are less than a specific given value. The present study selects it to be 10 -6 Ry. The optical properties have been estimated by observing optical conductivity, absorption coefficient, energy-loss-functional, dielectric constant, reflectivity, refractive index, and extinction coefficient. Optical conductivity (σ) defines the connection between current density and the electric field at particular frequencies [4,12]. The absorption coefficient (α) determines the fraction of incident electromagnetic radiation absorbed per unit mass or unit thickness [5,18]. Energy loss functional (L) shows the Table 1. Calculated values of Lattice Constants, Bulk Modulus, volume, and total energy of RuCrX (X = Si, Sn, Ge).

Alloy
Other work 25,26 Bulk modulus B 0 (Gpa) loss of energy inside the material and the cost of energy for fast-moving electrons [3]. The dielectric constant (ε) represents the ratio of the permittivity of a material to the permittivity of free space [9,11]. Reflectivity (R) predicts the bouncing back of radiations from the material surface [18]. The refractive index (n) elaborates on the ratio of the speed of light in a vacuum to the speed of light in a medium [19]. The extinction coefficient (K) denotes light absorption measurement in the given material [6,13,20]. After the convergence of the Self-Consisted field cycle, it becomes possible to calculate different material properties, i.e. DOS, El. Dens, and band structure. The features of all the properties are in the Tasks menu. Electron density tells about the occupancy of electrons per unit volume for specific material [15,21]. The density of state describes the number of states that are available to be occupied by the system at each level of energy [22]. Band structure determines the nature of the material for this purpose, a very large strength of k-points (2000+) in order to initialize and perform the SCF cycle is used.

Structural properties
The HHAs RuCrX (X = Si, Sn, Ge) crystallize in cubic C1 b prototype structure. Each unit cell contains four formula units with Ru: 4a (0, 0, 0), Cr: 4b (1/2, 1/2, 1/2), and X at 4c (1/4, 1/4, 1/4) Wyckoff positions, respectively [1,20]. So, the schematic structure of these HHAs allocated by XCryDen, are shown as shown in figure 1. The optimization in each structure is done using DFT embodied in the WIEN2k package. The optimized graph for each alloy is shown in figures 2, 3 and 4, which show that energy is a function of volume. The optimized parameters are listed in table 1.

Optical properties 3.2.1. Optical conductivity
Optical conductivity is used to explore materials' allowed interband optical transition [21,23]. It comprises two parts Real and Imaginary Part [10,24]. These parts give their values by showing peak and valley distribution, as shown in figures 5 and 6.
Real Part: The graph shows the 1st real peak value of optical conductivity at approx. 1eV. On increasing energy to 8eV, it shows unstable and decreasing nature. So, no real peak value is predicted in this region. At a little above 8eV, an abrupt increase in optical conductivity is noticed. At 9eV, 10eV, and 11eV, the second real  peak values of optical conductivity for the given materials are obtained, respectively [25,26]. A sudden decrease occurs after the above values, and a minute random increase occurs until 16eV.
Imaginary Part: Optical Conductivity has -ve values too as shown by the Imaginary Part. The graph shows a tendency to shift from zero to negative values of optical conductivity at the initial stage. The first imaginary peak of optical conductivity is negative at 0 eV. The second one is also negative at approximately 6.2eV while the third, fourth and fifth peak values of optical conductivity for RuCrX(X = Si, Sn, Ge) are positive and their corresponding values of energies are about 1.5eV, 6eV and 11eV, respectively.

Absorption coefficient
The key condition for the absorption of electromagnetic radiation occurs only when the energy of the band gap of the given specimen is equal to the energy of the photons of the incident electromagnetic radiation [26]. As the graph shows, the absorption coefficient for zero energy will be zero. The graphical illustration of absorption   co-efficient and energy relationship shows the variations in absorption coefficient for RuCrSi, RuCrSn, and RuCrGe, respectively. At 0eV, the absorption coefficient for all three materials indicates similar behavior showing zero absorption coefficient following the standard theory. Hence, the first peak values for RuCrSi, RuCrSn, and RuCrGe are obtained at 1.25, 0.75, and 1.0eV, respectively. At the same time, the second peak values for these respective materials are found at 6.25, 4.8, and 5.5eV [23]. Finally, the third peak values for the above respective materials are 11.25, 9.0, and 10.25eV, respectively, as depicted in figure 7.

Energy loss functional
The mathematical interpretation of energy loss is represented by equation (1).
Graphically, it is observed that the energy loss is minimal at low energy between 0eV to 1eV, but a quick increase is noticed when an increase in energy. The first energy loss peak value is observed separately at about 2.5eV, 1.5eV, and 2.0eV for RuCrSi, RuCrSn, and RuCrGe. The second energy loss peak value is noticed at the same energy level, i.e., 7.5eV for RuCrSi and RuCrGe, but the same peak value is obtained at approximately 6.0eV for RuCrSn. Finally, the third peak values appear at 12.0eV for all these materials, as presented in figure 8 [2].

Dielectric constant
The mathematical relation which is used to express the dielectric constant ε is followed by equation (2).
Where ε1 (ω) is the real part of the dielectric constant, which describes the electric polarization of materials, ε2 (ω) is the imaginary part of the dielectric constant, which explains the electric absorption [24,27]. Real Part: In the graph of the real part of the dielectric constant the peak values for RuCrSi, RuCrSn and RuCrGe are observed at 0eV. After a small increase in energy up to 1eV the Real part of the dielectric constant decreases. Then it increases gradually till 2.5eV and further, it tends to be uniform for greater energies till about 14eV as shown in figure 9.
Imaginary Part: The graphical representation shows that at 0eV imaginary part of the dielectric constant of RuCrSi, RuCrSn and RuCrGe executes some value of approximately 30eV but it shows its peak value at about 0.5eV then decreases to 2.5eV, and then at higher energy levels, it shows fluctuating nature in figure 10.

Reflectivity
The bouncing back of radiations from the surface of the material is known as reflectivity. It is used to study the nature of the surfaces of various materials. The following graph represents the Reflectivity factor of RuCrX (X = Si, Sn, Ge). It describes that at 0eV, the reflectivity is at maximum peak value [17,25]. The reflectivity trends are cleared in figure 11.
When increasing a small amount of energy up to 1eV, the reflectivity fluctuates, then reflectivity decreases rapidly up to 2.5eV. After the increase in energy from 2.5eV, the reflectivity changes irregularly in an increasing way. So, the second peak values for the reflectivity of RuCrSi, RuCrSn, and RuCrGe are noted at 6.0, 4.8, and 5.5eV, respectively, and third peak values for these materials are at 11.25, 9.8, and 11.0eV, respectively.

Refractive index
In optical physics, a dimensionless constant is used to express the ratio between the speed of light in a vacuum and the speed of light in some medium. It is called the refractive index. It can be denoted by (n) [17,20]. The graphical representation of the refractive index describes that at 0eV, the materials RuCrX(X = Si, Sn, Ge) show their maximum refractive index. Then with the increase in energy up to 2.0eV, the refractive in-dices decrease rapidly. Furthermore, by increasing energy, they do not show their second apex values and show irregular changes, as depicted in figure 12.

Extinction coefficient (K)
The measurement of light absorption in any material is defined by a specific coefficient called the Extinction Coefficient. Its symbolic representation is κ. The relationship between the extinction coefficient and absorption coefficient of the material is followed by equation (3).
Where α is the absorption coefficient, and κ is the extinction coefficient [9,24]. This relation shows that if the absorption coefficient increases, the extinction coefficient increases consequently. It is clear from figure 13 that at 0eV, the materials RuCrSi, RuCrSn, and RuCrGe show some extinction coefficient. By an increase of a small amount in energy, even as little as 0.75eV, the value extinction coefficient reaches its maximum peak value, but by increasing energy up to 1eV, extinction coefficient κ decreases abnormally. The extinction coefficient shows erratic behavior in the last region, where the energy values exceed 2.5eV.

Thermal properties
The thermodynamic properties can be predicted by the Debye model quasi-harmonic implemented Gibbs code in which Gibbs function G * (V, P, T), heat capacity C v [28], and thermal expansion coefficient. The fermi energy E F and the transport energy E T for these alloys are enlisted in table 2:

Debye temperature
In the quasi-harmonic Debye model [29], the non-equilibrium Gibbs free energy of a solid is given by equation (4).  Where, E(V ) is the total energy per unit cell of the material, θD(V ) is the Debye temperature, and A Vib is the vibrational Helmholtz free energy. For the RuCrX (X = Si, Sn, Ge), the behavior of Debye temperature between (0-500) K is shown in figure 14. Therefore, it is evident that none of these materials show zero Debye temperature at 0 K. The general expression for Debye temperature can be computed as: Where, M is the molecular mass per formula unit, and B is the static Bulk Modulus. Figure 15 explains the variation of this bulk modulus at the corresponding temperatures.

Heat capacity
The lattice vibration properties can be accessed through the heat capacity of a material. The heat capacity CV depends on temperature as C V is proportional to T 3 [6,26]. Therefore, the heat capacity at constant volume, C V , was calculated as a function of temperature ranges from 0K to 500K, as shown in figure 16. Obviously, the C V curve increases.

Thermal expansion coefficient
The thermal expansion coefficient α has an important theoretical and experimental significance and is also essential for predicting the thermodynamic equation of state. Figure 17 presents the effect of the temperature and the thermal expansion coefficient α. It is shown that between 0K to 500K, from 0K to 100K, alpha values increase linearly, and from 100K to 500K, it increases exponentially.

Electronic properties 3.4.1. Electron density
In the execution menu, the obtained graph of electron density in figure 18 is by performing a few easy steps using the calculations performed by the SCF cycle. There are two possibilities for output: a 3D plot or a 2D plot called contour. For specific matters, electron occupancy can be explained by it [16]. The variation of ED is represented by a contour map which comprises that the ED reduces from the center as it moves far from the center. Figures 14(a)-(c) denote two-dimensional ED contour maps for RuCrSi, RuCrSn and RuCrGe, respectively, for the plane (110). It shows the bonding nature between the elements of the material. The observed results show ionic bonding between Ru and X(X = Si, Sn, Ge), while weak covalent bonding exists between Ru and Cr [17,23].

Energy band gap
The nature of the material, whether it is an insulator, a conductor, or a semiconductor, can be described by the EBG. To get accurate and precise results about the calculation of band structure, First have to perform a very large number of symmetry operations and, for this purpose, take an enormous strength of k-points, i.e. (2000+) to initialize and perform SCF-Cycle. It is apparent in figure 22 that there is no energy band gap in RuCrSi, RuCrSn, and RuCrGe. It means clearly indicates that these alloys are metallic [3, 10, 21].

Conclusion
A comprehensive analysis was conducted on the optoelectronic properties of RuCrX (X = Si, Sn, Ge) with the assistance of DFT calculations in WIEN2K Codes and the results were in line with standard theoretical studies. The discussion focused on various characteristics including optical conductivity, absorption coefficient, energy loss function, dielectric constant, reflectivity, refractive index, and extinction coefficient as practical tools to measure theoptical properties. Additionally, the optimization curves were used to showcase the typical characteristics such as ground state energy and lattice constant. After careful analysis, the study findings were found to be consistent with the established theory. The research output determined that the studied Half Heusler Alloys of RuCrX (X=Si, Sn, Ge) are pure materials as they do not exhibit magnetic properties individually but do show magnetic nature when combined. The lattice constants a 0 (A) of RuCrSi, RuCrSn and RuCrGe were measured to be 5.64, 5.76 and 5.97, respectively, and the Bulk Modulus B 0 (Gpa) for these materials were found to be 203.26, 185.64 and 159.85 respectively. The study also accurately determined the band gaps of semiconductorswith an orbitalindependent exchange-correlation potential, whichdepends solely on APW in the PBE-GGA. The temperature was found to affect the fundamental properties, such as the thermal expansion coefficient α, which increases linearly from 0K to 100K and then exponentially from 100K to 500K. The research used various intrinsic properties, including lattice parameters, energy band gaps, and static optical properties to identifyinnovative alloys for optoelectronic devices. The calculated values of Fermi Energy and Transport Energy showed that RuCrSi had the highest value, followed by RuCrGe, and RuCrSn has the least value. The study also observed a decrease in Dybye temperature and Bulk modulus and an increase in specific heat capacity and thermal expansion coefficient from 0K to 500K.