Nowadays Half-Heusler alloys are known as the best materials for different applications including optoelectronics, electrical, thermoelectric and piezo-electric applications (Masuri et al., 2016). Ternary form of half-Heusler alloys is XYZ and of full-Heusler alloys X_2YZ (Hülsen et al., 2009; Lee et al., 2005; Galanakis and Mavropoulos, 2007), having four fcc lattices and usually crystallize in L2_1 and C1_b structure. Where, X and Y represents the most and intermediate electropositive element respect-ively, and Z denotes the most electronegative element. High spin polarization, half-metallic property, and sustain aggressive environment are possessed by Half-Heusler alloys (Andrea et al., 2014; Bhat et al., 2015; Zhang et al., 2014). In the Half-metallic (HM) materials one spin state is metal and the other behaves like semiconducting or insulating. Because of their potential applications making spin electronics devices like spin diode, spin field effect tube, spin throttle valve and spin filter, it plays an important role in materials science (Katsnelson et al., 2008; Li et al., 2004).

The first Heusler compound (NiMnSb) was discovered and observed that there was a gap at the Fermi energy for the minority spin channel (usually spin-down), but not for the majority spin channel (usually spin-up). It gives rise to 100 % spin-polarization in one of the spin channels at the Fermi level due to the existence of the band gap. In the half metals, it produces a fully spin-polarized current which helps in maximizing the efficiency of magneto-electronic devices. Like transition metal pnictides and metal-chalcogenides (Ge and Zhang, 2009; and Guo and Liu, 2011), different types of perovskites (Zhu and Yan, 2009), oxides (Hongliang, 2007; Dho et al., 2010), magnetic semiconductors (Kronik et al., 2002; Naefa and Rahman, 2020; Noor et al., 2011), and all types of Heusler alloys (Nour-mohammadi and Abolhasani, 2010; Wang and Wei, 2011) have shown the properties half-metals and this  have an important applications in the field of spintronics (Jimbo et al., 1994). 

Having outstanding applications and uses in several sectors, Half-Heusler compounds are attracting researchers attention. Recently, various physical properties are investi-gated of these types of compounds. For example, the electronic and physical properties of PtBiZ (Z = Fe, Mn, Ni, Co) are investigated in and of PdMnBi in (Huang et al., 2014). Electronic and magnetic properties of several half-Heusler compounds are investigated and discussed in (Guan-Nan and Ying-Jiu, 2009; Rozale et al., 2013). 

Optoelectronic properties of RbSrZ (Z = C, Si, Ge) are examined in (Ahmad et al., 2013), of CoCrZ (Z = Al, Ga) in (Missoum et al., 2014), and of NiTiX and CoVX (X = Sb, Sn) in (Ameri et al., 2014). Half-Heusler compounds RhCrZ (Si, Ge) are investigated in (Mehmood et al., 2017). Though a number of half-Heusler compounds have been predicted to be half-metallic using DFT based calculations (Kandpal et al., 2006), a comprehensive analysis of the structural, mechanical, electronic, and optical properties of the half-Heusler family is necessary, because it is not sure how many half-metallic half-Heuslers that can be imagined, are stable. Thus a systematic study of the structural stability of the half-Heusler (C1b) family should provide guidance for future experiments. In this article, we have investigated the structural, electronic, elastic, and optical properties of the half-Heusler alloys RuVAs at zero and different pressure up to 40 GPa. Since only the structural and magnetic properties of compound RuVAs have been studied in (Ma et al., 2017), therefore we are interested to study the detail physical properties of this compound at 0-40 GPa. Here we have tried to show the variation of different physical properties of RuVAs under different pressure.

2. Computational details

First-principles calculations have been carried out using the DFT-based CASTEP computer program with the generalized gradient approximation (GGA) and the PBE exchange correlation functional was chosen for this calculation (Clark et al., 2005; Rahman et al., 2016; Perdew et al., 2008). In this calculation V-3s^2 3p^6 3d^3 4s^2, As-4s^2 4p^3, Ru-4s^2 4p^6 4d^7 5s^1 are used as valence electrons. In a plane-wave basis set the energy cut-off of 400 eV are used for electromagnetic wave function expansion and 12×12×12 k-points are used for the sampling of Brillouin zone for all calculations. 

In the geometry optimization, the convergence parameters were set as follows: (i) 1.0×10-5eV/atom is set for the total energy difference (ii) 0.001 Å is set for the maximum ionic displacement (iii) 0.03 eV/Å is set for the maximum ionic force (iv) 0.05 Gpa is set for the maximum stress. With space group F43m (space gr. No. 216) the Wyckoff positions for Ru, V and As are (0.25, 0.25, 0.25), (0.50, 0.50, 0.50) and (0, 0, 0) as shown in Fig 1. The bulk modulus can be obtained from the elastic constants and the elastic stiffness constants of RuVAs are obtained using the stress–strain method (Fan et al., 2006).

3.1. Structural properties of RuVAs

The half-Heusler RuVAs compound has cubic structure with the space group F4-3m (216) and the equilibrium lattice parameter has a value of 5.76 Å which is similar to the calculated value. The optimized lattice parameter and the calculated unit cell volume at zero to 40 GPa pressure are tabulated in Table 1. Minimizing total energy the lattice constants and atomic positions have been optimized with a function of normal stress which is shown in Fig 1.

Table 1:The calculated equilibrium lattice parameters a0 (Å), unit cell volume V0 (Å3), normalized lattice parameter and volume of RuVAs at zero to 40 GPa pressure.

To study the effect of pressure on the crystal structure of RuVAs, we have studied the variation of lattice parameter and unit cell volume under pressure (0 to 40 Gpa) which is shown in Fig 2(a). From Fig 2(a), it can be noticed that the ratio a/a_0 and normalized volume V/V_0 decreases with the increase of pressure, where a_0 and V_0 represent the equilibrium lattice Parameter and volume at zero pressure, respectively. Due to the reducing of the atomic distance with increasing pressure, the volume decrease. The pressure-volume curves of RuVAs intermetallic from the calculation are shown in Fig 2(b), where the pressure-volume data are fitted to a third order Birch-Murnaghan equation of state (EOS).

3.2. Elastic properties of RuVAs

Many different fundamental solid-state properties are closely related with elastic properties of a material (Fu et al., 2008). By the study of elastic constants, the stability of a material, hardness, ductility, brittleness, elasticity, anisotropy nature and other mechanical phenomena are described (Wang et al., 2011). The elastic constants are determined by employing stress strain role belongs to Hooks law. In Table 2 our investigated results of elastic constants are included with available theoretical data. Due to the different calculations method a little bit deviation is observed between the two theoretical values (zero pressure). 

For cubic crystal, the mechanically stability the single elastic constants should convince the Born stability criteria which are following:

C11 >0, C44>0, C11–C12>0 

and C11 + 2C12 >0                                     (1)

From Table 2, we observed that the calculated values of elastic constants are positive and satisfied the above restriction which suggests that the RuVAs is mechanically stable up to 40 GPa. There are no other theoretical data available in literature for the elastic constants of RuVAs under pressure and so we are not able to compare our results with other data. 

From Fig 3(a) we observed that the pressure dependence of elastic constants C_ijincluding C11, C12 and C44 increase with applied pressure except C_44 because of the lattice parameters become smaller with increasing pressure. The variation of C44 which demonstrates that the cubic structure is unstable phase and at high pressure the phase transition may happen. For a cubic crystal, while the elasticity in length is characterized by C11, C12 and C44 represent the elasticity in shape (Prikhodko et al., 2002). According to Bouhemadou et al. (2015) a transverse strain forms the change in shape with a constant volume where longitudinal strain changes the C11.

We can be seen from Table 2 that C11 is larger than C44 which means that the alloy presents a stronger resistance to the unidirectional than resistance to the pure shear deformation. We have noticed form Fig 3(a) that C11 is more sensitive to pressure than C12 and C44 indicating the shear constant is more sensitive than the Youngs modulus. For approximating the brittle or ductile nature of materials an important criterion is Cauchys pressure, which is given by (C12−C44). This parameter specified the metallic and nonmetallic feature of material. The positive and negative Cauchy pressure not only prefers the metallic nature and nonmetallic nature respectively (Liu et al., 2012) but also the negative Cauchy pressure indicates the brittle nature and positive Cauchy pressure indicates the ductile nature of material. From our investigation we observed that the Cauchy pressure is positive and increases with increasing pressure confirming the metallic character of RuVAs and this metallic behavior becomes stronger with pressure. For positive value of Cauchy pressure our compound shows the ductile manner. For understand the mechanical behavior of materials the polycrystalline elastic parameter such as bulk   modulus B, shear modulus G, Youngs modulus E and Poissons ratio ν, anisotropy factor A are very vital parameters. Therefore we investigate the variations of these parameters under pressure. We have employed the Voigt-Reuss–Hill (VRH) averaging scheme to estimate these parameters. For cubic crystal, B and G can be expressed as follows (Wu et al., 2007):

〖  B〗_v=B_R=(C_11+2C_12)/3                                (2)

G_v=  ((C_11-C_12+ 3C_44 ))/5                            (3)

G_R=  (5C_44 (C_11-C_12 ))/[4C_44+3(C_11-C_12 )]                        (4)

The average value of B and G according to Hill can be expressed:

B=  1/2 (B_R+ B_v )                                                   (5)

G=  1/2 (G_v+ G_R )                                           (6)

The Youngs modulus (E) and Poissons ratio (ν) can be determined by using the following relations:

E=  9GB/(3B+G)                                             (7)

ν=  (3B-2G)/(2(3B+G))                                        (8)

The calculated values of B, G, Eat different pressures up to 40 GPa are tabulated in Table 3 and plotted in Fig 3(b). The resistance to volume changes and shape changes are measured by bulk modulus, B and shear modulus G respectively. On the other hand, the material stiffness is measured by Youngs modulus E. The harder of the material is ensured by the large value of Youngs modulus. From Fig 3(b) we have observed that with variation of pressure the value of bulk modulus increases which indicate that the stronger the capacity of the resist deformation is increased with increasing pressure. The value of B and G increases with pressure up to 20 Gpa and then decreases. It happens because the anisotropy factor also increases up to 20 GPa and then decreases.

Table 3: The calculated bulk modulus B (GPa), shear modulus G (GPa), Youngs modulus E (GPa), B/G values, Poissons ratio ν and anisotropy factor A of  RuVAscompound under hydrostatic pressure.

The calculated value of A, ν and B/G are shown in Fig 4. The Pugh ratio, B/G also provides information about the ductile/brittle nature of any material. If B/G>1.75 then the material behaves ductile nature and the material will show brittle nature for B/G<1.75. According our results we observed that the values of B/G at different pressure are larger than 1.75 and increases with pressure. So, we can say that with increasing pressure the ductile nature of half-Heusler alloy RuVAs is increases. Poissons ratio parameter revealed the bonding and stability nature against shear of material and if ν ranges between 0.25 and 0.50 (Cao et al., 2013), the inter atomic force in a material is central. The materials flexibility is big with rising of Poissons ratio. Since the Poissons ratio are increased with increasing pressure indicating the increase of plasticity with pressure. 

The small value of Poissons ratio for covalent bond is (ν = 0.1) and for ionic bond ν = 0.25. From Table 3 we see that the value of ν are in the range of 0.39-0.40, which indicating the existing of central force in RuVAs from 0 to 40 GPa. In a solid material the degree of anisotropy is denotes by the Zener anisotropy factor, A. For isotropic material ‘A will be unity whereas values smaller or greater than unity reflect the degree of elastic anisotropy (Evecen, 2017). We have calculated the values of ‘A utilizing the following expression which are included in Table 3.

A=〖2C〗_44/((C_11-C_12 ) )                                            (9)

From Fig 4 we can say that the anisotropy behavior of this material is increased up to 20 GPa then it decreases with the increase of pressure.

3.3. Electronic properties (band structure and density of states) 

To provide valuable data about their experimental modelling and eventual utility in fabricating many type energy devices, the knowledge of the electronic structure is essential (Amudhavalli et al., 2018). To determine the electronic nature of RuVAs the Density of states (DOS) and band structures are calculated within the GGA scheme under pressure up to 40 GPa. The band structures in the energy range -8 to 10 eV are seen in Fig 5 and however, energy range remains same with increasing applied pressure. From Fig 5, we can be seen that the conduction band and valance band are not overlapping and the top of the valence band occurs at the L point and bottom of the conduction band occurs at X point respectively which implying that the RuVAs compound shows the half-metallic behavior. The band gap of RuVAs compound is 0.159 eV at 0 GPa which is good agreement with the other theoretical values 0.24 (Ma et al., 2017), 0.19 (Shi et al., 2017), 0.09 (Chibani et al., 2018). 

From Fig 5, it can be found that without any influence of pressure our compound shows the direct band gap material and it also confirm that after applying external hydrostatic pressure its direct band gap changed into indirect small band gap of  0.073eV and 0.029 eV at 20 Gpa and 30 Gpa respectively. The band gap almost decreases linearly with the increasing pressure and this trend agrees well with the values obtained by Wu et al. (2015). The band gap almost decreases linearly with the increasing pressure as shown in Fig 7. It is clear from Fig 5 that when the pressure is applied up to 40GPa the conduction bands shift downwards and valance bands shift upwards and then the band-gap decreases. We can be noticed from Fig 5 there is no band gap at Fermi level at pressure 40 Gpa and then it shows the metallic behavior. So, we say that phase transition occur at 40 GPa of RuVAs compound.

For understanding the bonding the DOS is also an important factor of a material (Duan et al., 2011). The DOS in the energy range -20 to 20 eV for RuVAs under zero pressure are depicted in the Fig 6(a). According to Fig 6 (a) we can see that the major contributions come from As-p and the d states of Ruthenium (Ru) and Vanadium (V). In the case of Ru, at higher energy level the DOS profile is relatively low and flat due to fully occupied d orbits of Ru. Besides, As-p formed the top of the valence band and it keeps at the Fermi level, while the bottom of the conduct band formed by V-d moves downwards, for this reason the band gap decreases. The underlying physical mechanism may be that when the applied pressure increases the distance between As and V shortens and thats why the outer orbitals overlap and then enhances the covalency between As and V, reducing the energy gap between the bonding and anti-bonding orbitals of V-As bonds and ultimately the decrease of the band gap occur. And it is consistent with our investigated band structure.

To elucidate the nature of the Total density of state in the energy range -6 to 10 eV of RuVAs at varying pressures up to 40 GPa were calculated and presented in Fig 6(b). From Fig 6(b) it can be seen that with the increase in external applied pressure, the numbers of new peaks are created and started to appear in the overlap energy states. However, energy range remains same with increasing applied pressure and the lowest energy level almost remain unaffected because of no pressure effect on the lowest energy level. At 0 Gpa the peak is minimum in conduction band, other peak started to seen in conduction band with the rising of pressure (10-40 Gpa). Also, there are lowest energy states in valence band from 0 to 40 Gpa. From Fig 5 we noticed the position of valence band and conduction band almost same at the pressure of 20 Gpa and 30 GPa respectively. At Fermi level the decreases or increases of DOSs correspond to the hardening or softening of the materials, respectively (Yan and Chen, 2014). From our investigation we can be seen that the peak of total DOS at Fermi level are decreasing with increasing pressure which indicate that RuVAs compound become harder with increasing pressure. And it is consistent with the investigation of the elastic properties. Near EF the main peak is right shifted from valence bands to EF and this analysis ensured that the delocalization of RuVAs becomes stronger gradually under pressure (Mao et al., 2014).

3.4. Optical Properties 

In this section, we will discuss the study of optical properties of RuVAs compound under various pressures for understanding the electronic system. For calculating the optical properties of RuVAs, we used the frequency-dependent complex dielectric function which is given by, ε(ω)= ε_1 (ω)+〖iε〗_2 (ω) is used.The imaginary part ε_2 (ω)can be written through momentum matrix elements and by considering the transition between the filled and unfilled electronic state by the equation (Accelrys, 2010); 

〖  ε〗_2 (ω)=(2e^2 π)/(Ωε_0 ) ∑_(k,v ,c)▒〖|ѱ_k^c |u.r|ѱ_k^v |〗^2  δ(E_k^c-E_k^v-E)      (10)

Where, u  is indicating as the polarization vector of the incident electric field, ω denotes frequency of light, e is denoted as the electronic charge and ѱ_k^c and ѱ_k^v are the conduction and valence band wave functions at k, respectively. According to the Kramers-Kronig transform, the real part ε_1 (ω)can be derived from the imaginary part. All other optical constants such as loss-function, refractive index, absorption spectrum, conductivity and reflectivity are calculated using the equations given in (Hadi et al., 2016). For photon energies up to 60 eV the optical functions of these compounds under pressure are calculated. For all calculations the value of Gaussian smearing is 0.5 eV.

To understand the polarizability and energy loss of a material, dielectric function is used as a drastic parameter where the electromagnetic wave propagates through it. The real and imaginary dielectric functions at different pressure are shown in Fig 8(a), and (b) respectively. From analysis Fig 8(a), we can see that when with increasing pressure then dielectric constant also increases and from this it can be deduced that flux density also increases for RuVAs. We also noticed that the lower energy limit ε_1 (ω)  of dielectric constant is ε_1 (0) and peak is gradually disappearing above Fermi level and below the Fermi level getting sharp gradually. For different pressure the value of static dielectric constant changes from 33.66 to 35.66 at 0 to 40 Gpa.  The imaginary part of ε_2 (ω) shows major peak at visible and minor peak at ultraviolet regions. The absorption and conductivity occurred at the non-zero potion of the imaginary part of the dielectric constant

The calculated optical reflectivity spectra as a function of photon energy of RuVAs are shown in Fig 9(a). It is observed that the reflectivity is in the range of 0.61-0.63 in the infrared region for 0-40 GPa and the value drops rapidly in the visible region (1.8-3.0 eV) and then again increases in ultraviolet region. In our study the highest peaks is between 4.4 to 5.9 eV photon energy ranges of RuVAs compound where the conductivity, loss function fall rapidly and absorption quality is good. To avoid solar heating, the high value of reflectivity in the high energy region can be used as a good coating material. It is so interesting to notice that with the increase of pressure reflectivity also increases. Fig 9(b) shows that the optical absorption co-efficient of RuVAs compound under pressure. The absorption coefficient provides data about the most solar energy conversion efficiency which shows how far of a specific light energy penetrate into the material up to absorption (Roknuzzaman and Islam, 2013). 

From 0.2 eV photon energy the absorption co-efficient start for RuVAs compound at zero pressure which notifies the small semiconductor behaviors and the corresponding peaks lie at 2.4 eV, 4.0 eV, 4.5 eV, 6.1 eV, 6.6 eV from 0-40 GPa. Different peaks are showed in the spectra at the ultraviolet region and there is no absorption from 33.54 eV to 36.96 eV photon energy for (0 - 40) Gpa. At 40 GPa the absorption co-efficient start from zero photon energy because the phase transition occurred in this pressure which ensured from the electronic properties. Due to decrease in band gap the absorption edge changed with respect to pressure. In our study, we notice that RuVAs possess good absorption coefficient in the high energy region and the highest peaks are located at 40 eV. 

The optical conductivity of RuVAs as a function of photon energy is plotted in Fig 9(c). Due to the result of absorbing photons the photoconductivity of RuVAs increases (Sun et al., 2006). 

The loss function is a significant parameter and its explained the energy loss of a fast electron traversing the material (Parvin et al., 2015). The peaks are related to the plasma resonance in the loss function graph and which is associated frequency is called the plasma frequency ???????? and it is shown in Fig 10(a). In the energy range of about 25.5-28.8 eV, a few number of large peaks in the energy loss spectra is present. At these high energy range, the real part and imaginary part of the dielectric function is small and correspond to the trailing edge of R(ω) (Saniz et al., 2006). With the rising of pressure from 0-30 GPa the loss function is decreasing and with pressure 40 GPa the loss function is increasing because phase transition occurred here. Therefore from energy loss spectrum, the peaks of plasmon energy lie at 26.5 and 28.8 eV respectively, at 0 and 40 GPa pressures, which show that the plasmon energy curve also shifts towards the higher energy side with rising pressure. How much light is refracted/bent when entering into material is determined by the refractive index and it is a dimensional quantity (Russell,  2003). The refractive index (real part) at different pressure is shown in Fig 10(b). Within the range up to 40 GPa the refractive index values varies from 5.81 to 5.95. From the figure of refractive index we notice that the value of static refractive index is high in the infrared region then highly decrease in the visible region and slowly reduced in the ultraviolet region. The almost similar patterns of the refractive index are observed at various pressures.

In this work, we have investigated the structural, elastic, electronic and optical properties of the half-Heusler alloy RuVAs under various pressures up to 40 GPa by performing first-principles calculations. The structural properties including cell volume and the normalized structural parameters a/a0 and V/V0 decrease with the increase of pressure.  The pressure effects on elastic modulus including bulk modulus B, shear modulus G and Youngs modulus E show that all the parameters are increases with the increase of pressure. The ductility of these RuVAs compounds have been analyzed using the Pughs rule and the Cauchy pressure and Pughs ratio ensured that the compound RuVAs shows ductile behavior. The half-Heusler alloy RuVAs is mechanically stable within the pressure range from 0 GPa to 40 GPa and the material becomes unstable under the pressure more than 40 GPa.  The compound shows the half-metallic nature with the small indirect band gaps which are revealed from electronic band gap and DOS analysis. The compound RuVAs shows semi-metallic nature up to 30 GPa but it shows totally metallic nature at 40 GPa, which ensures the occurring of phase transition of RuVAs compound at 40 GPa. The optical properties including dielectric constant, optical absorption, optical conductivity, refractive index, reflectivity and loss function shows the increasing behavior with the increase of pressure. We tried our best to discuss the optical properties of RuVAs under pressure which might be the key reason to enhance the near/mid infrared light activity.

We would like to thank Department of Physics, Pabna University of Science and Technology, Bangladesh for the condensed matter lab support.

We have no conflict of interest about this article.