Structural, elastic, electronic and vibrational

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Sep 20, 2017 - charge density map it is observed that both ionic and covalent bond exist. ... Since these compounds have rich physical properties, in recent it has .... crystals as BaRh2P2 and SrIr2As2 superconductors, it has six independent ...

Accepted Manuscript Structural, elastic, electronic and vibrational properties of BaRh2P2 and SrIr2As2 superconductors: A DFT study M.I. Kholil, M.S. Ali, M. Aftabuzzaman PII:

S0925-8388(17)33259-0

DOI:

10.1016/j.jallcom.2017.09.209

Reference:

JALCOM 43265

To appear in:

Journal of Alloys and Compounds

Received Date: 22 July 2017 Accepted Date: 20 September 2017

Please cite this article as: M.I. Kholil, M.S. Ali, M. Aftabuzzaman, Structural, elastic, electronic and vibrational properties of BaRh2P2 and SrIr2As2 superconductors: A DFT study, Journal of Alloys and Compounds (2017), doi: 10.1016/j.jallcom.2017.09.209. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Structural, Elastic, Electronic and Vibrational Properties of BaRh2P2 and SrIr2As2 Superconductors: A DFT Study M. I. Kholil1, M. S. Ali1, * and M. Aftabuzzaman1, 2 1

Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan.

ABSTRACT

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Department of Physics, Pabna University of Science and Technology, Pabna-6600, Bangladesh

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The structural, elastic, electronic, Vickers-Hardness, vibrational, Optical and thermo dynamical properties of potentially technologically important superconductors BaRh2P2 and SrIr2As2 are calculated using density functional theory (DFT) with CASTEP code for the first time. The structural and other physical properties of BaRh2P2 and SrIr2As2 are compared with the results where available and show well accord. The phonon dispersion curve is calculated and the dynamical stability of this compound is investigated. Both the compounds show mechanically stable under the Born stability conditions. For the above condition BaRh2P2 behaves ductile nature and SrIr2As2 indicates the brittle nature. The Mulliken dislocation bonding population and charge density maps show stronger bond between As-Ir compared with Ir-Ir bond. The overall superior conversation reflects that the chemical bonding in BaRh2P2 and SrIr2As2 superconductors can be denominated as an extremely anisotropic connection between ionic, covalent and metallic interactions. At the Fermi level valence band and conduction bands overlapped, so the compounds are metal. From the Fermi surface and charge density map it is observed that both ionic and covalent bond exist. Vickers hardness reveals the both superconductors relatively soft with compare to Diamond. The optical and acoustic modes are observed clearly. We calculate the Helmholtz free energy (F), internal energy (E), entropy (S), and specific heat capacity (Cv) from the phonon density of states. Various optical parameters are also calculated. The reflectance spectrum shows that this compound has the potential to be used as an efficient solar reflector. Debye temperature of BaRh2P2 and SrIr2As2 superconductors are 273.91 K and 341.03 K calculated by using present elastic constants data. The superconducting parameter indicates the phonon-mediated medium coupled BCS superconductors. The acquired results in present investigation could provide a great spur for future studies.

Keywords: Mechanical properties, Bonding analysis and Vickers hardness, Optical properties, Thermodynamic properties and Superconducting properties.

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Corresponding author. E-mail addresses: [email protected] (M. S. Ali)

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ACCEPTED MANUSCRIPT 1. Introduction

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The layered ternary intermetallic compounds with ThCr2Si2-type materials consisting of transition metal elements, which have been fully investigated for their attractive and rich physical phenomena [1-2]. Since these compounds have rich physical properties, in recent it has gained giant interest and great attention of researchers. Body-centered tetragonal ThCr2Si2-type ternary intermetallic compounds also have crucial phenomena on superconductivity and details study of this identical structure may be a smooth path for the inquiry of new superconductor. Already some of iron-free compounds were announced to be superconductors, such as SrNi2As2, LaRu2P2, BaRh2P2, SrPd2Ge2, SrIr2As2, etc., with very low transition temperature [3-6]. The ThCr2Si2-type structure was first predicted by Ban and Sikirica in 1965 [7]. G. Just and P. Paufler describe a complete geometric examination of approximately six hundred compounds with ThCr2Si2 type structure in 1996 [8].

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The parent iron-pnictide superconductor compounds AFe2As2 (A = Ca, Sr, Ba, etc.) appear with a common ThCr2Si2-type crystal structure have been extensively studied due to their intriguing physical properties which mostly emerge from the strongly correlated d electrons, notably for the search of high-Tc superconductors [9-11]. Anand et al. [12] discovered crystals structure of APd2As2 (A = Ca, Sr, Ba) and announced superconductivity for CaPd2As2 and SrPd2As2 with ThCr2Si2-type structure and BaPd2As2 with CeMg2Si2-type structure without bulk superconductivity. Ying and his co-workers were found superconductivity of the ternary lanthanum ruthenium phosphide LaRu2P2 crystallizes in ThCr2Si2 type structure with transition temperature of 4.0 K [13-14]. The polycrystalline sample of ThCr2Si2-type LaRu2As2 synthesized by Guo et al. and predicted bulk superconductivity at 7.8 K [11]. Recently, new Rh and Ir pnictide superconductors BaRh2P2 and SrIr2As2, which are isostructural to (Ba,K)Fe2As2 with Tc = 1.0 and 2.9 K predicted by Wurth et al. and Löhken et al. [15-17].

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There were no first principle studies available in the literature about the details mechanical, thermodynamic and superconducting properties of BaRh2P2 and SrIr2As2 superconductors. Therefore, in this article, we would decide to make a details theoretical investigation of the newly layered pnictides BaRh2P2 and SrIr2As2 superconductors by using the first-principles method base on the density functional theory (DFT) with CASTEP code. Finally, the remaining parts of this article are oriented as follows: A brief description of computational method appears in section 2, Investigated results and its related discussion are shown in section 3 and the summary of this investigation are presented in section 4. 2. Computational Method All the calculations were performed using the Cambridge Serial Total Energy Package (CASTEP) program base on density functional theory (DFT) [18]. The exchange-correlation energy function was described by using the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [19]. The ultrasoft pseudopotentials of Vanderbilt-type [20] were used to describe the Coulomb interactions between valence electrons and ionic core. The valence electron configurations of BaRh2P2 and SrIr2As2 superconductors were 2

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considered P-3s23p3, Rh-4d85s1, Ba-5s25p66s2 and As-4s24p3, Sr-4s24p65s2, Ir-5d76s2 respectively for pseudo atomic calculations. The cut-off energy value for plane-wave basis set was elected as 400.0 eV for BaRh2P2 and as 350.0 eV for SrIr2As2. Monkhorst–Pack schemes [21] were used to evaluate the energy in the irreducible Brillouin zone with 7×7×9 grids for both superconductors BaRh2P2 and SrIr2As2. The Broyden–Fletcher–Goldfarb– Shanno (BFGS) minimizations were used to perform the structural optimizations [22]. For Geometry optimization the convergence tolerance was set as follows: the total energy convergence value is within 5.0×10−6 eV/atom, the maximum Hellmann–Feynaman force is within 0.01 eV/Å; the maximum displacement is within 5.0×10−4 Å; the maximum stress is within 0.02 GPa; and the maximum iterations is within 100. The elastic constants of BaRh2P2 and SrIr2As2 are determined by the stress–strain method [23]. The maximum strain amplitude was elected to 0.003. The criteria for convergence tolerance to calculate the Elastic constants were set to 4.0×10-6 eV/atom for the total energy, 0.01 eV/Å for maximum force and 4.0×10−4 Å for maximum displacement.

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3. Results and discussion 3.1 Structural properties

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The layered pnictides BaRh2P2 and SrIr2As2 superconductors possess the tetragonal ThCr2Si2type crystal structure with space group I4/mmm (No.139) [24]. The Wyckoff positions of both these superconductors are 2a (0, 0, 0) for Ba or Sr, 4d (0, 0.5, 0.25) for Rh or Ir and 4e (0, 0, 0.3593) for P or As [17]. The conventional unit cell and primitive cell of BaRh2P2 and SrIr2As2 superconductors are shown in Fig.1. The optimized equilibrium crystal structure of these compounds is adopted by minimizing the total energy. The optimized lattice parameter and Wyckoff positions are tabulated in Table 1. The optimized equilibrium lattice parameters are well accord with others available experimental values. The evaluated values appears slight default from the experimental values because the temperature dependency of cell parameters and GGA process [25].

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Table 1 Optimized atomic Wyckoff positions, unit cell parameters (a and c, in Å), tetragonal ratio c/a, unit cell volume and bulk modulus of BaRh2P2 and SrIr2As2 superconductors. Atom/Properties

Ba/Sr Rh/Ir P/As a-axis (Å) c-axis (Å) c/a

Wyckoff positions 2a 4d 4e

V0(Å3) B0 (GPa) a

BaRh2P2 Present study

SrIr2As2 Expt.

(0,0,0) (0,0,0)a (0,0.5,0.25) (0,0.5,0.25)a (0,0,0.3514) (0,0,0.3593)a 4.040 3.939b 12.375 12.576b 3.063 3.192b 201.97 195.12b 134.31 137c

Ref.[17]; bRef.[24]; cRef.[26] 3

Present study

Expt.

(0,0,0) (0,0.5,0.25) (0,0,0.3684) 4.099 11.853 2.891 199.15 78.301

(0,0,0)a (0,0.5,0.25)a (0,0,0.3593)a 4.068b 11.794b 2.899b 195.17b -

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Fig.1 Crystal structure of BaRh2P2 and SrIr2As2 superconductors (a) Conventional unit cell and (b) Primitive cell.

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3.2 Elastic properties and Polycrystalline Elastic Modulus

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The elastic properties of any material closely related to the long-wavelength phonon spectrum, in this manner the elastic properties of superconducting material must be investigated [27]. The study of elastic properties of solid provides frequent information about the dynamical properties of crystalline materials. The deep investigation also provides exoteric idea about the nature of bonding in atom. The elastic constants were carrying out from a linear fit of the evaluated stress-strain function according to the Hook’s law [28].For the tetragonal crystals as BaRh2P2 and SrIr2As2 superconductors, it has six independent elastic constants, i.e. C11, C12, C13, C33, C44 and C66 [29]. These constants must be satisfying the well-known stability criteria [30].   0,   0,   0,   0    0,   2  0 1 2     4  0

The evaluated single elastic constants for BaRh2P2 and SrIr2As2 superconductors are listed in Table 2.We are not able to compare our calculated values because no available experimental or other theoretical results. From Table 2, it is evident that the above criteria are satisfied, which indicating that the BaRh2P2 and SrIr2As2 superconductors are mechanically stable.

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ACCEPTED MANUSCRIPT Table 2 The calculated elastic constants Cij (in GPa) of BaRh2P2 and SrIr2As2 superconductors. Compound BaRh2P2 SrIr2As2

C11 147.26 201.59

C12 57.75 -16.38

C13 60.50 26.40

C33 89.79 77.99

C44 26.81 111.89

C66 53.39 94.95

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There are two Cauchy relations C12 = C66 and C33 = C44 appears in the tetragonal crystal [31]. The data from Table 2 definitely shows that the evaluated elastic constants do not fulfill these two Cauchy relations and indicates the non-central forces are highly exigent for BaRh2P2 and SrIr2As2 superconductors by virtue of its covalent character [27].

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For the tetragonal system, the bulk modulus (BVRH), the shear modulus (GVRH), Poisson ratio (ν) and Young’s modulus (E) can be calculated by using the Voigt–Reuss–Hill (VRH) average schemes [32-34]:

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2   =  +  + 2 +  (2) 9 2



(3) 

 =

Where,  = ( +  ) − 2

( + 3 − 3 + 12 + 6 ) (4) 30

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And  =  +  + 2 − 4

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18 6 6 3   + + +  ( −  )   

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 =

! =

! =

(5)

 +  (6) 2

 +  (7) 2

The values of Young’s modulus E and Poisson’s ratio v can be determined by using the equations, #=

9! ! (8) (3! + ! ) 5

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$=

3! # 9 6!

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The subscripts V, R and H in above equations notice to Voigt, Reuss and Hill, respectively. The evaluated values of the bulk modulus (BVRH), shear modulus (GVRH), Young’s modulus (E) and Poisson ratio (ν) are tabulated in Table 3.

Table 3 Calculated bulk modulus B (all in GPa), shear modulus G (all in GPa), Young’s modulus E (all in GPa), the BH/GH ratio and Poisson’s ratio v of BaRh2P2 and SrIr2As2. BR 77.75 55.39

BH 80.08 58.47

GV 35.105 93.396

GR 31.196 75.89

GH 33.150 84.643

BH/GH 2.41 0.69

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Compound BV BaRh2P2 82.42 SrIr2As2 61.55

E 87.39 171.27

v 0.318 0.011

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As can be seen from Table 3 that the bulk modulus for all VHR relatively low ( 1.75) and brittle (< 1.75) nature of materials [34]. For the above condition BaRh2P2 behaves ductile nature and SrIr2As2 indicates the brittle nature. The ductility-brittle manner also appears from Poisson’s ratio. For a material critical value of Poisson’s ratio is 0.33 and above this value the material behaves brittle and lower this value material behaves ductile nature [35]. From table, it is evident that both superconductors behave ductile nature. The calculated both Pugh’s ratio and Poisson’s ratio appears same nature for BaRh2P2, which indicates the accuracy of our present DFT base calculation. The Poisson ratio v for ionic materials is 0.25 [36]. The calculated BaRh2P2 compound assumed to be ionic material.

3.3 Electronic properties

The electronic properties of BaRh2P2 and SrIr2As2 can be achieved by study of the electronic band structure, partial density of state (PDOS) and total density of state (TDOS). The band structure

studied along the high symmetry directions in the Brillouin zone (BZ) as shown in Fig.2. The dotted line considered the Fermi level between the valence band and conduction band at zero energy point with abbreviating few bands and range of total band is -8eV to 8eV. As shown in figure both band of BaRh2P2 and SrIr2As2 are overlapping to each other, for this reason consider the metallic character.

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Fig.2 The electronic band structure for (a) BaRh2P2 and (b) SrIr2As2 crystal structure along the high-symmetry points in the irreducible Brillouin zone.

Fig.3 Total (a) and Partial (b) density of state (DOS) of BaRh2P2 superconductor. The dotted line consider the Fermi level 7

ACCEPTED MANUSCRIPT For more crucial information of electronic structure of BaRh2P2 and SrIr2As2, the total density of states (TDOS) and partial density of states (PDOS) was calculated. The calculated total and partial density of state of BaRh2P2 and SrIr2As2 are shown in Fig.3 and Fig.4 respectively.

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As shown in Fig. 3(a), the contribution appears below the Fermi level at Rh and P state and above Fermi level Ba state only contributed. The calculated density of state of Ba, Rh and P atoms are 0.742 states/eV, 2.267 states/eV and 1.107 states/eV respectively at Fermi level. The Rh atom appears as contributing atom of BaRh2P2 at Fermi level. The calculated total density of state or number of states of BaRh2P2 at the Fermi level N (EF) is 4.128 states eV−1fu−1 (Since formula unit one). From Fig. 3(b), high contribution appears of P-3p and Rh4d states and minor contribution of Rh-5s and P-3s states in valence band. The Ba-6s states merely contributed in conduction band.

Fig.4 Total (a) and Partial (b) density of state (DOS) of SrIr2As2 superconductor. The dotted line consider the Fermi level For Sr, As and Ir atoms the calculated density of state are 0.771 states/eV, 1.185 states/eV and 2.578 states/eV respectively and Ir atom most contributed in SrIr2As2 superconductor and the total density of state of SrIr2As2 is 4.162 states eV-1fu-1 ( Since formula unit one). The overall contribution occurs in Ir-d, As-p and Sr states of Total and Partial section of SrIr2As2.

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ACCEPTED MANUSCRIPT 3.4 Chemical bonding and Fermi surface

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To reveal the nature of chemical bonding and charge transfer in BaRh2P2 and SrIr2As2, we have need to investigate the charge density and charge density difference. That’s why we study the charge density and charge density difference from a deep insight of BaRh2P2 and SrIr2As2 superconductor. The contour of electron charge density and charge density difference of both this superconductors are shown in Fig. 5 and Fig. 6 respectively. A contour scale was shown near to the plot, which reveals the low and high electron densities (in the unit of e/Å3) at blue and red colors.

Fig.5 The total charge density (a) and the charge density difference (b) of BaRh2P2 along the 011 direction.

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As shown in Fig. 5(a), it is explicit that the Rh and As atoms loss electrons while P atoms achieve electrons according to contour scale. So we can say that the charge transfer from Rh and As atoms to P atoms, which reveals the ionic interaction between them. The contour total charge density difference plots represent the exchange of electrons in space [37]. From Fig. 5(b) the covalent behavior appears nears the P and Rh atoms because of overlapping charge distribution and the weak charge transfer appears between P-Ba and Ba-Rh bonds. Similarity appears As-Sr and Ir-Sr bonds in Fig. 6(a). The charge distributions between As-Ir atoms are strong because of interatomic distance. Around Ir atoms spherical shape charge distribution reveals the ionic character of Ir-Ir bonds in SrIr2As2. The ionic character describes the metallic nature of Ir-Ir bonds [38]. As like Fig.5 (b) covalent behaviors appear between Ir-As atoms from Fig. 6(b) and charge distribution between Ir-As and As-Sr atoms are assumed to be same due to similarity distance. The overall superior conversation reflects that the chemical bonding in BaRh2P2 and SrIr2As2 superconductors can be denominated as an extremely anisotropic connection between ionic, covalent and metallic interactions. 9

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Fig.6 The total charge density (a) and the charge density difference (b) of SrIr2As2 along the 011 direction.

Fig.7 Fermi surface of (a) BaRh2P2 and (b) SrIr2As2 superconductor.

To understand further the electronic structure of any metallic compound the Fermi surface (FS) is a crucial quantity in case of electrons close to the Fermi level are mainly responsible for superconductivity [39]. We have plotted the FS for BaRh2P2 and SrIr2As2 superconductors in the Fig 7. The Fermi surface of BaRh2P2 is made of four sheets because four bands are crossing the Fermi level as shown in Fig. 2(a). The four sheets appear as concave at the corner of the Brillouin zone path along the A-M direction in Fig. 7(a). Another sheet appears at the centre as spherical shape shown in Fig. 7(a) parallel to Z-M and Z-R direction and connected with 10

ACCEPTED MANUSCRIPT four corner sheets. Moreover, the Fermi surface of SrIr2As2 is also made of three sheets due to band crossing the Fermi level as shown in Fig. 2(b). The two sheets appears as concave and the center one like as hole nature due to band crossing from conduction band to valence band through both Z and Γ points.

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3.5 Mulliken atomic population and Vickers hardness

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For understanding the bonding nature in crystal systems the details investigation is required about the Mulliken atomic populations [40]. The calculated Mulliken atomic populations of BaRh2P2 and SrIr2As2 superconductor are shown in Table 4. From data of Table 4, it is evident that the bearing charge of P and Rh atoms are negative and Ba atoms positive in case of BaRh2P2 superconductor indicating that the charge transfer from Ba atoms to P and Rh atoms which are equal to 0.45e and 0.03e. For SrIr2As2 superconductor only Ir atoms bearing negative charge and As and Sr atoms positive charge and revealing the amount of 0.51e charge transfer from As and Sr atoms to Ir atoms. In case of BaRh2P2 superconductor the RhBa, P-Ba and Rh-Rh bonds reveal the low value which indicating the ionic bond nature between them, whereas P-Rh, and P-P bonds consider the high value which reveal the increasing level of covalency i.e., highly covalent Character [41] and For SrIr2As2 superconductor the As-Ir, As-As, As-Sr and As-Ir bonds behaves ionic nature and covalent interaction appears in Sr-Ir and Ir-Ir bonds. The calculation of iconicity of any material also is an crucial phenomena for understanding the bonding characteristics which is obtained by using the following equation [42],

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ƒ& = 1 − ' (|*+(*|/* (10)

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Where, PC is the bond overlap population of in a pure covalent crystal and the value of PC is one for covalent crystal and P is the bond overlap population. The value of ƒ& is one and zero represent the ionic and covalent bond nature of crystal. From Table we conclude that for BaRh2P2 the P-P bonds behave purely ionic nature and P-Rh bonds considerably ionic nature because of the value of ƒ& is approximately one which contradicts from Mulliken atomic bond nature. Further investigation is required to overcome the contradiction. Now for SrIr2As2 superconductor Ir-Ir bonds reveal the ionic nature for approximately one value which also contradicts the Mulliken bond nature and only similar covalent nature appears in Sr-Ir bonds.

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ACCEPTED MANUSCRIPT Table 4 Population analysis of BaRh2P2 and SrIr2As2 superconductors. Mulliken Atomic pupulations s 1.88 0.89 1.90

p 3.57 -0.25 6.05

d 0.00 8.39 1.09

Total 5.45 9.03 9.03

Charge -0.45 -0.03 0.97

SrIr2As2

As Ir Sr

1.62 0.64 2.19

3.30 0.88 5.97

0.00 7.99 0.99

4.91 9.51 9.15

0.09 -0.51 0.85

Bond P-Rh P-P Rh-Ba P-Ba P-Rh Rh-Rh As-Ir As-As Sr-Ir As-Sr As-Ir Ir-Ir

Population 0.36 0.01 -0.93 -0.49 0.36 -3.46 -0.01 -3.77 0.62 -1.81 -0.01 0.25

ƒ& 0.83 1 0.83 0.45 0.95

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Species P Rh Ba

Lengths (Å) 2.37804 3.67828 3.69490 4.34836 5.33052 6.18750 2.48458 3.06846 3.57716 4.31680 4.91190 5.85103

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Compounds BaRh2P2

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Hardness evaluates or estimates the ability of a material to negate the plastic deformation and the amount of force per unit area performed inversed to plastic deformation for evaluating the hardness of a material. Hardness also measures the resistance of a material to plastic deformation. The greater hardness reveals the greater resistance to deformation of a material. The Vickers hardness experiment was developed by Robert L. Smith and George E. Sandland at Vickers Ltd in 1921 which experiment as an alternative to the Brinell method to evaluates the hardness of materials [43].The theoretical Vickers hardness (Hv) of crystals obtained from the Mulliken bond population data by using the following relations [44, 45]: 2

45

-. = /0-1 3 

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82′ =

(12)

(; 2 ) ?∑ (13) . 9

@ABCC 7D (14)

Where, 82 is the Mulliken population of the µ-type bond, 82′ is the metallic population of the 2 µ-type bond, $9 is the volume of a bond of type µ, V is the cell volume, @ABCC is the number of free electrons, ; 2 is the bond length of type µ and =9. is the bond number of type v per unit volume.

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ACCEPTED MANUSCRIPT According to the known, in present the Diamond is the hardest material with Vickers hardness in the range 70 to 150 GPa. From Table 5 we see that our present calculated Vickers hardness of BaRh2P2 and SrIr2As2 superconductors are 4.41 GPa and 3.10 GPa respectively. Therefore, we conclude that the both superconductors are relatively soft material compare with Diamond.

; 2 2.37804 3.67828 5.33052 3.57716 5.85103

82 0.36 0.01 0.36 0.62 0.25

82′ 0.0805 0.0805 0.0805 0.0887 0.0887

$9 -1 3.0675 31.926 11.3518 -0.9110 34.5495 0.5641 9.2640 9.621 40.5396 0.249 2

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SrIr2As2

@2 3 2 2 2 2

bond P-Rh P-P P-Rh Sr-Ir Ir-Ir

2

-. 4.41

3.10

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Compounds BaRh2P2

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Table 5 Calculated Mulliken bond overlap population of µ-type bond 82 , bond length ; 2 , 2 2 metallic population82′ , bond volume $9 (Å3) and Vickers hardness of µ-type bond -1 (GPa) and total hardness -. (GPa) of BaRh2P2 and SrIr2As2 superconductors.

3.6 Optical properties

2e π K |ѱNM |O. Q|ѱ.M | R(#MN − #M. − #) (15) ΩεJ M,. ,N

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The optical properties of BaRh2P2 and SrIr2As2 superconductors can be calculated by using the frequency dependent dielectric function ε (ω) = ε1 (ω) + iε2 (ω) and the study of the optical functions of solids provides a crucial information of the electronic structure. The imaginary part, ε2 (ω) of dielectric function obtain from the momentum matrix elements between the filled and the unfilled electronic eigenstates and express by the following equation [46],

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where, u is defined as the polarization of the incident electric field, Ω is the unit cell volume, ω is defined as the frequency of light, e is denoted the charge of electron, ѱNM and ѱ.M are defined as the conduction band wave function and the valence band wave function at K respectively. The real part of dielectric function notify from the Kramers-Kronig transform of imaginary part and also the real part of all the optical function of BaRh2P2 and SrIr2As2 superconductors defined by Eqs. 49 to 54 in ref. [46]. Fig.8 represents the optical functions of BaRh2P2 and SrIr2As2 superconductors and calculated for photon energies up to 50 eV. The value of 0.5 eV Gaussian smearing is used for all optical properties calculations. Reflectivity is a measure of the ability of a surface to reflect radiation. The reflectivity spectra of BaRh2P2 and SrIr2As2 superconductors are shown in Fig. 8(a). From the figure we see that the reflectivity start from the value of 0.70 for BaRh2P2, which is maximum and 0.50 for SrIr2As2 with zero Photon energy. Some peak appears at the infrared and visible region of 13

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BaRh2P2. Maximum reflectivity appears at 0.813 with energy 14.52 eV and one small peak at visible region for SrIr2As2 superconductor. Reflectivity is absent in the ultraviolet region of both these superconductors. The above discussion shows that SrIr2As2 is a good reflector rather than BaRh2P2.

Fig. 8 The optical functions (a) reflectivity, (b) absorption, (c) refractive index, (d) dielectric function, (e) conductivity, and (f) loss function of BaRh2P2 and SrIr2As2 superconductor. The absorption coefficient is a measure of the rate of decrease in the intensity of electromagnetic radiation (as light) as it passes through a given substance before it is absorbed. The evaluated absorption coefficients are presented in Fig. 8(b). The absorption starts with zero values and gradually increases and decrease two times of both superconductors. The absorption quality is good in the infrared and visible regions of both superconductors and low absorption appear at ultraviolet region. Two absorption peaks appears at 2.239x105 (cm-1) and 1.814x105 (cm-1) with energy 9.24 eV and 18.44 eV for 14

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The real and imaginary parts of the refractive indices are shown in Fig. 8(c). The real part of the refractive index denotes the phase velocity and the imaginary part known as extinction coefficient also denotes the amount of absorption loss when the electromagnetic wave (as light) passes through the material. Despite of some variation in heights and positions of peaks, the entire features of our calculated real and imaginary parts of the refractive indices of BaRh2P2and SrIr2As2 superconductors are roughly similar in the whole energy range.

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The real and imaginary parts of the dielectric function for both the superconductors are displayed in Fig. 8(d). The real part of dielectric constant shows one peak at 4.13 eV for BaRh2P2 and two peaks at 3.91eV and 21.10 eV for SrIr2As2 superconductors and the imaginary part shows one peak at 5.06 eV for BaRh2P2 and three peaks at 1.09eV, 5.13eV and 22.33eV for SrIr2As2 superconductors. In ultraviolet energy region the real part of the dielectric function tends to be nearly unity and the imaginary part reaches zero for both the superconductors, which indicates the BaRh2P2 and SrIr2As2 are transparent material as well as optically anisotropic.

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Fig. 8(e) shows the conductivity spectra. The real and imaginary parts start with zero photon energy for both the superconductors, because of no band gap appears which is evident from present band structure as shown in Fig 2(a) and Fig. 2(b). The conductivity spectra show some peaks at infrared and visible region and becomes stable at higher energy region for both parts of these two superconductors.

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The energy loss function represents the energy loss of a fast electron when traversing the material and maximum energy loss appears at that point is defined as the bulk plasma frequency, ωp and the frequency occurs when the ε2(ω) is less than one (ε2 < 1)and ε1 (ω) is equal to the zero point (ε1 = 0) [47,48]. Fig. 8(f) shows the energy loss spectrum of BaRh2P2 and SrIr2As2 superconductor as a function of photon energy. In the energy-loss spectrum we observe that the bulk plasma frequency, ωp for BaRh2P2 and SrIr2As2 superconductor are 22.06 eV and 15.35 eV, respectively. The transparent material appears of both these superconductors when the incident photon frequency is greater than the bulk plasma frequency (i.e., >22.06 eV and >15.35 eV).

3.7 Thermodynamic properties The highest frequency mode of vibration in such a temperature is called Debye temperature. Many physical properties of solid such as melting point, specific heat, thermal expansion etc. influences by the Debye temperature. Debye temperature also describes the high and low temperature regions in a solid. When the temperature of a solid is greater than Debye temperature (T > θD) the vibration mode in every case assumed to be equal with KBT energy 15

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and (T < θD) the vibration mode are at rest. The low temperature vibration mainly arises from acoustic vibration. The Debye temperature (θD) is not an accurately determined parameter. For this reason the various data can be used to evaluate the Debye temperature of a material but we have used to elastic modulus data to determine the Debye temperature (θD) of BaRh2P2 and SrIr2As2 superconductors because of standard methods depending on the elastic moduli [49]. The Debye temperature estimation formula which we have to use through elastic constants, averaged sound velocity (vm), longitudinal sound velocity (vl), and transverse sound velocity (vt) are as follows [50-53]: 

[email protected] =Y Z  ST = W  [ $\ (16) V 4X  

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1 2 1 ( $\ = W   +  [ (17) 3 $] $^ 



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4  + 3 ` (18) $^ = _ Z



$] =   (19) Z

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Where, h is the Planck constant, k is the Boltzmann constant, NA is the Avogadro’s number, ρ is the density, M is the molecular weight and n is the number of atoms in the unit cell of BaRh2P2 and SrIr2As2 superconductors. The calculated values of ρ, vt, vl ,vm and θD are tabulated in Table 6. The calculated Debye temperature (θD) and density (ρ) of BaRh2P2 and SrIr2As2 superconductors are 273.91 K, 341.03 K and 6.58 (gm/cm3), 10.35 (gm/cm3) respectively. We are not able to compare our evaluated data with experimental and other theoretical values of BaRh2P2 and SrIr2As2. For this case we compare others compounds with same structure. In ref. [54] we found the values of vl, vt and vm in the unit of (km/s) but we converted this with (m/s) in case of compare present calculation unit. Hadi, M. A., et al. [54] proved that the material LaRu2As2 behaves lower Debye temperature as like as low thermal conductivity and thermal barrier coating (TBC) material. From Table 6, it is evident that the Debye temperature of SrIr2As2 is low also indicates the low thermal conductivity and behaves as a thermal barrier coating material as compare with LaRu2As2. Table 6 The calculated density ρ (in gm/cm3), transverse (vt), longitudinal (vl), and average sound velocity vm (m/s) and Debye temperature θD (K) of BaRh2P2 and SrIr2As2 superconductors. Compounds ρ (gm/cm3) BaRh2P2 6.58 SrIr2As2 10.35 8.779 LaRu2As2

vl (m/s) 4345.97 4068.58 4371

vt (m/s) 2244.54 2859.73 2676

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vm (m/s) 2512.93 3103.45 2976

θD (K) 273.91 341.03 335

Ref. [This] [This] [54]

ACCEPTED MANUSCRIPT The melting point of a solid is the temperature at which it changes state from solid to liquid at atmospheric pressure. The melting temperature of tetragonal crystal can be evaluated by using the elastic constants Cij from the following equation [55], a\ = 354 + 4.5

2 +  (20) 3

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Where, the unit of a\ is in K and C11 and C33 in Gpa. The calculated values of melting temperature are listed in table 7. We find the melting temperature of BaRh2P2 and SrIr2As2 are 930.46 K and 1075.75 K, respectively and conclude that the melting temperature of SrIr2As2 is higher than that of BaRh2P2 superconductor.

−2/3

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Thermal conductivity is the property of a material to conduct heat. The minimum thermal conductivity is directly related to temperature and when the temperature of a material gradually increased the conductivity of the material then gradually decreased to a certain limit [56]. Many similar expressions have been derived for calculating the minimum thermal conductivity. In present work, the minimum thermal conductivity Kmin can be calculated by using the Clarke expression [57]. The Clarke expression can be denoted as, b\c4 = bd $e 

@ρNA



(21)

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Where, KB is defined as the Boltzmann constant, vm is denoted as the average sound velocity, M is the molecular mass, n is the number of atoms per molecule and NA is defined as the Avogadro’s number. The evaluated values of minimum thermal conductivity of BaRh2P2 and SrIr2As2 superconductors are tabulated in table 7. It is evident from table 7 that the minimum thermal conductivity of both these superconductors relatively same and denotes low thermal conductivity at ambient condition.

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Compounds BaRh2P2

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Table 7 The calculated melting temperature, Tm (K), minimum thermal conductivity, Kmin (in Wm-1K-1) and the Dulong-Petit limit (J/mole.K) of BaRh2P2 and SrIr2As2 superconductors.

SrIr2As2

a\ 930.46

b\c4 0.46

Dulong-Petit limit 124.67

1075.75

0.58

The anharmonic effect of the specific heat capacity CV is suppressed and close to a limit at high temperature, which is generally, refers to as a Dulong-Petit limit [58]. The Dulong-Petit limit of a solid can be evaluated by the following equation [58], [email protected]−8'mnm [email protected]=AKB

(22)

Where, NA represents the Avogadro’s number and KB is the Boltzmann constant. The evaluated value of Dulong-Petit limit of both the superconductors is recorded in table 7.

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pqrs

We were used to the equation [60] o = p

+tu

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Electron-phonon coupling constant is a crucial superconducting parameter for calculate the value of superconducting transition temperature. For evaluating the value of aN the electronphonon coupling constant measures accurately. The method attributed to QUANTUMESPRESSO program [59] can accurately enumerate the value directly. In order to enumerate the electron-phonon coupling constant with significant computational resources a doubledelta function integration over a dense net of electron and phonon vectors k and q vectors is required [54]. This process is further complicated for a crystal system having many atoms per cell like BaRh2P2 and SrIr2As2. – 1 for evaluating the electron-phonon

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coupling constant accurately but the low value appears (o = 0.006 & -0.28) due to the approximately similar values between wCxy and wNz^ for BaRh2P2 superconductor and higher value of wNz^ rather than wCxy for SrIr2As2 superconductor. While we cannot accurately evaluate o, the accuracy of Tc for both the compound under consideration cannot be ensured. In case of consideration of above information the electron-phonon coupling constant can be calculated indirectly by using the McMillan formula [61]. ST  1.45aN λ = (23) S (1 − 0.62μ∗ ) ln  T  − 1.04 1.45aN

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1.04 + |∗ ln 

Where, θD denotes the Debye temperature, µ* stand for coulomb pseudo potential. The coulomb pseudo potential can be obtained from the formula [62] as follows,

=(# ) (24) 1 + =(# )

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| ∗ = 0.26

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Where, =(# ) refers to total density of state at the Fermi level. The value of wNz^ can be evaluated from the density of states (DOS) at the Fermi level [60], wNz^ =

X bd =(# ) (25) 3

The calculated superconducting parameters are recorded in Table 8 with available experimental and other theoretical values. It is evident from Table 8 that the density of state at Fermi level of BaRh2P2 coincides to ref. [65]. The calculated value of wNz^ also coincides to experimental value in ref. [24] and theoretical value in ref. [65] and contradict with ref. [63] and [64] for BaRh2P2 superconductor. Therefore, we conclude that may be some error in ref. [64] and [63] due to calculation. The similarity result appears of λ with ref. [64]. The above discussion shows the reliability of present calculations. The contradiction result also appears between wNz^ and ref. [24] for SrIr2As2 superconductor. More investigation is required for 18

ACCEPTED MANUSCRIPT this contradiction. Moreover, the detailed study of superconducting parameters indicating that both the compounds are phonon-mediated medium coupled BCS superconductors.

Notices This study Exp.[24] [63] Others[64] [65]

= #  4.128 4.18

γ 9.69 9.75 9.2 10.26 9.85

SrIr2As2

This study Exp.[24]

4.162 -

9.77 7.03

4. Conclusion

|∗ 0.20 -

o 0.58 0.43 -

0.20 -

0.68 -

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Compounds BaRh2P2

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Superconducting parameters

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Table 8 The calculated density of states at Fermi level = #  (states eV-1 fu-1), specific heat coefficient γ (mJK-2mol-1), the coulomb pseudo potential | ∗ and electron phonon coupling constant o of BaRh2P2 and SrIr2As2 superconductors.

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In summary, the mechanical properties, bonding and Vickers hardness, vibrational, optical properties, thermodynamics properties and superconducting properties of BaRh2P2 and SrIr2As2 have been investigated by means of Density functional theory. The optimized lattice parameters of BaRh2P2 and SrIr2As2 are in good agreement with available theoretical and experimental values. The band structure of both these compound showed that the materials have metallic character. From our mechanical properties results, we observe that BaRh2P2 and SrIr2As2 are mechanically stable. The values of Pugh’s ratio show that the material BaRh2P2 behaves ductile nature and SrIr2As2 indicates the brittle nature. The bonding properties show the ionic, covalent and metallic nature for both superconductors. The concave and spherical shape sheets appear from the Fermi surface for BaRh2P2 and only hole nature appears for SrIr2As2. The results of hardness reveal that the material BaRh2P2 is harder with compare to SrIr2As2. Maximum reflectivity appears at energy 14.52 eV and SrIr2As2 is a good reflector rather than BaRh2P2. From the absorption curve, it is evident that SrIr2As2 behaves good absorber with compare to BaRh2P2 at energy 10.10 eV and 23.25 eV. The material SrIr2As2 behaves as a thermal barrier coating material as compare with LaRu2As2. The study of melting temperature of SrIr2As2 is higher than that of BaRh2P2 superconductor. We also analyze the minimum thermal conductivity of both these superconductors and denote low thermal conductivity at ambient condition. The evaluated value of Dulong-Petit limit of both the superconductors is 124.67 J/mole.K. The detailed study of superconducting parameters indicating that both the compounds are phonon-mediated medium coupled BCS superconductors.

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Structural, Elastic, Electronic and Vibrational Properties of BaRh2P2 and SrIr2As2 Superconductors: A DFT Study

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M. I. Kholil1, M. S. Ali1, * and M. Aftabuzzaman1, 2 Research Highlights:

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Structural properties, mechanical stability under the study of elastic properties and vibrational properties. Electronic properties, Fermi surface and electronic charge density. Superconducting properties Optical properties Bonding analysis and Vickers hardness.

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* Corresponding author. Tel.: +88 0721 750980; fax: +88 0721 750064. E-mail address: [email protected] (A.K.M.A. Islam)

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