OLCAO based Electronic Structure Calculations and Optical ...

CHINESE JOURNAL OF PHYSICS

VOL. 52, NO. 6

December 2014

OLCAO based Electronic Structure Calculations and Optical Properties of Ni3 Ga Intermetallic Alloy System Altaf Hussain,∗ M. Nasir Rasool, Salman Mehmood, and M. Shakil Department of Physics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan (Received June 17, 2013; Revised May 30, 2014) The first-principles orthogonalized linear combination of atomic orbitals (OLCAO) method has been employed to study the electronic structure and optical properties of the Ni3 Ga intermetallic alloy system. To see the effects of ordering, we have simulated the ordered (L12 ) and disorder models. The calculated optical conductivity of the ordered phase is rich in structures and in reasonable agreement with the experimental data, while the conductivity spectrum of the disordered phase shows a single broadened peak. It is found that the average charge transfer from Ga to Ni in the ordered phase was greater (0.376 electrons per atom) as compared to the disordered phase (0.234 electrons per atom). The localization index (LI) calculations show that in case of the ordered phase there were localized states present below as well as above the Fermi-level. Localization of states is also present in the disordered phase, but both the models have shown different behavior. Other results on the band structure, density of states, and dielectric functions have also been discussed and described in detail. DOI: 10.6122/CJP.20140722

PACS numbers: 73.61.At, 78.30.Er, 78.30.Ly

I. INTRODUCTION Intermetallic alloys show a series of very promising characteristics [1, 2]; binary intermetallic compounds that contain a transition metal and a 3rd group metal display interesting mechanical, electronic, magnetic, and optical properties [3–8]. One of the candidates is Ni3 Ga, which crystallizes in the simple cubic Cu3 Au (L12 ) structure. Ni3 Ga is an exchange enhanced paramagnetic material [6, 9]. The unique properties exhibited by this intermetallic compound are due, in part, to the nature of its atomic/crystal and electronic structure [3]. The intermetallic compound Ni3 Ga has been a subject of intensive study both theoretically as well as experimentally. Experimental studies for Ni3 Ga include those of the magnetic properties, de Hass-van Alphen effect, elastic constants, and thermal properties [6, 10–13]. The electronic structure and optical properties of this system have been studied extensively by various experimental and theoretical methods. Pong et al. [14] investigated the charge transfer and p-Ni d hybridization effects by X-ray absorption near edge structure (XANES) measurements. Hsu et al. [4] used high-resolution X-ray photoemission spectroscopy (XPS) and bremsstrahlung isochromat spectroscopy (BIS) to characterize the electronic structure of this alloy system. Theoretically, the electronic structure calculations have been performed by using the linear-muffin-tin-orbital (LMTO) method in the local∗

Electronic address: [email protected]

http://PSROC.phys.ntu.edu.tw/cjp

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c 2014 THE PHYSICAL SOCIETY ⃝ OF THE REPUBLIC OF CHINA

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spin-density (LSD) approximation [6]. The dielectric functions and optical properties of polycrystalline Ni3 Ga were measured by the spectroscopic ellipsometry technique in the energy range of 0.6 to 4 eV [5]. Due to the diverse nature of the results presented in the literature [3–6, 14], it is obvious that more comprehensive investigations should be performed in this direction. The concepts of localization index (LI) and effective charge (Q∗ ) will really add important information to the present understanding of the alloy system. Therefore, in this work we have performed theoretical calculations on both the ordered (L12 ) as well as disordered (theoretically simulated) phases of the Ni3 Ga intermetallic alloy system by using the first-principles orthogonalized linear combination of atomic orbitals (OLCAO) method [15, 16]. The OLCAO method uses localized atomic orbitals as the basis set for wave function expansion. It is very efficient and versatile for electronic, optical, and spectroscopic calculations. In Section II, the method of calculation is described. In Section III, the results of the electronic structures and optical properties are explained and discussed. Section IV contains some conclusions drawn from the obtained results.

II. METHOD OF CALCULATIONS To calculate the electronic structure and optical properties of the ordered (L12 ) and disordered phases of the Ni3 Ga intermetallic alloy system, we used the first-principles based orthogonalized linear combination of atomic orbitals (OLCAO) method [15, 16]. OLCAO is based on the local density approximation (LDA) of density functional theory (DFT) [17, 18]. This method has been demonstrated as highly accurate and efficient when dealing with materials with complex structures for both crystalline [19, 20] and non-crystalline systems [21, 22]. In the OLCAO method, the solid state wave functions are expanded in atomic orbitals which consist of Gaussian type orbitals (GTOs) and spherical harmonics appropriate for the angular momentum quantum number. The concept of using a linear combination of atomic orbitals (LCAO), or the tight-binding method, in the band theory of solids was originally introduced by Bloch in 1928 [23]. The introduction of orthogonalization to the core technique in the real space formalism makes the method particularly appealing for complex systems including amorphous or disordered solids. The essential key to the success of this method is the ability to calculate the multicenter integrals accurately and efficiently. The crystal potential is usually calculated from a superposition of free-atom charge densities using the local-exchange approximation. Self-consistency can be achieved by an iterative process [24, 25]. We used the full basis set in these calculations for Ni and Ga. In OLCAO the full basis set for Ni includes the core-orbitals (e.g., 1s, 2s, 3s, 2p, 3p), the occupied valence-orbitals (e.g., 4s, 3d), and additional empty-orbitals (e.g., 5s, 4p, 5p, 4d), while the full basis set for Ga includes the core-orbitals (1s, 2s, 3s, 2p, 3p), the occupied valence-orbitals (4s, 4p, 3d), and additional empty-orbitals (5s, 5p, 4d). A full basis set provides additional variational freedom to the Bloch function and to get higher accuracy. For the primitive unit cell and supercell (consisting on 256-atoms) calculations of the ordered phase of Ni3 Ga intermetallic alloy system, we used 120 and 4 olcao k-points

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respectively, while to perform calculations on a theoretically simulated disordered phase of the alloy system we used 27 olcao k-points. From the self-consistent energy eigen-values and wave functions, many useful physical quantities may be obtained. The most fundamental one is the band structure, or the E(k) dispersion relation along the symmetry lines of the Brillouin zone (BZ). For metallic systems, a Fermi surface can be constructed which is the locus of the k-points in the BZ satisfying E(k) = EF , the Fermi energy. The density of states (DOS) can be resolved into partial components called partial DOS (PDOS) according to different atoms γ, different orbitals i, and even different spins in the spin-polarized calculations. This is most commonly done by the Mulliken population analysis [26]. This procedure amounts to dividing each n (k) according to the electron state ψn (k, r) of energy En (k) into fractional charges qi,γ relations: ∫ ∑ n qi,γ (k) = |ψn (k, r)|2 dr = 1, (1) i,γ

n qi,γ (k) =

∑

∗

Ani,γ · Anj,δ · Siγ,jδ (k).

(2)

j,δ

When summed over the BZ and over all bands n, the PDOS can be projected out according to the atomic specification δ and orbital specification i. The effective charge Q∗α , is defined as the calculated valence electrons associated with an atom α in the crystal. Q∗α provides information about charge transfer (gain or loss of electronic charge from the neutral atom). This number is always positive and should not be confused with the valence charge state or other forms of the effective charge description. Q∗α calculated here is based on the Mullikan scheme [26] of the molecular orbital theory. An effective charge Q∗α for each atom α can be defined from Eq. (1) as ∑∑∑ ∗n n Q∗α = Ciα Cjβ Siα,jβ . (3) i

n,occ j,β

n are the eigenvector coefficients and S In Eq. 3, Cjβ iα,jβ are the overlap integrals between the ith orbital of the αth atom and jth orbital of the βth atom. A quantitative number for the effective charge is extremely useful in providing physical insight for the interpretation of experimental data, and is considered to be one of the strong points of the OLCAO method [15, 16]. The localization index (LI) is a calculation related to the approximate measure of the spread of the electron charge around each atom and is particularly useful for identifying the degree of disorder in amorphous systems. The LI for the state n with energy En is defined as [N ]1/2 ∑ L (En ) = (ρnα )2 , (4) α=1

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where ρnα is the fractional electron charge assigned to the αth atom according to Mulliken’s prescription [26] and calculated from the resulting eigenfunctions. Since the normalization N ∑ condition implies ρnα = 1, L(E) = 1/N for a completely delocalized state and L(E) = 1 α=1

for a completely localized one. For amorphous systems the localization of states is typically encountered near band edges, in accordance with the Anderson model [27]. Further, the localization of states has some of its distribution near the valence band edge, which can be used to estimate the mobility edge [28]. These concepts have been discussed previously in greater detail with regard to the OLCAO method [15, 16, 24, 25, 29]. The interband optical conductivity is another important physical observable which can be directly compared with experimental optical data. In the OLCAO approach, this comparison is conducted as follows: The real part of the interband optical conductivity σl (E) is calculated in the random phase approximation (RPA) by using the KuboGreenwood (K-G) formula [30]: σl (E) =

2πe2 Ω mEΩ (2π)3 ∫ ∑ 2 − → |⟨ψn (k, r)| P |ψl (k, r)⟩| × fl (k) [1 − fn (k)] δ (En (k) − El (k) − E) × dk n,l

(5) where E = ¯hω is the photon energy, f (k) is the Fermi distribution function, and l labels an occupied state and n an unoccupied state. The integration over the BZ for Eq. (5) utilizes the linear analytic tetrahedron (LAT) method for the density of states (DOS) calculation.

III. RESULTS AND DISCUSSIONS III-1. Basic Structures The ordered phase of Ni3 Ga intermetallic alloy is a cubic structure like Cu3 Au with Ni atoms at the face centered positions and Ga atoms at the corners of the cube, as shown in Fig. 1 (a). The disordered phase of the alloy system has been simulated by randomly placing the Ni and Ga atoms at the fcc lattice as shown in Fig. 1 (b). The Strukturbericht designation for Ni3 Ga is L12 with space group Pmm. In this work we used 56 k-points (12 × 12 × 12) in self-consistent field (SCF) calculations and 120 k-points (16 × 16 × 16) for OLCAO calculations of the primitive cell in ordered Ni3 Ga. On the other hand for the ordered supercell phase only 1 SCF k-point (2 × 2 × 2) and 4 OLCAO k-points (4 × 4 × 4) as well as for disordered phase of the Ni3 Ga, 1 SCF k-point (1 × 1 × 1) and 27 OLCAO k-points (3 × 3 × 3) were used.

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FIG. 1: Unit cell of the Ni3 Ga intermetallic alloy system. (a) Ordered phase: Ni atoms (filled circles) are on face centers and Ga atoms (circles filled with horizontal bricks) are at corners, (b) Disordered phase of the alloy system: here the Ni and Ga atoms are randomly distributed.

III-2. Electronic Structure III-2-1. Band Structure

The band structure calculation gives the most direct comparison with experiments. The band structure is the plot of the energy eigenvalues of the system as a function of reciprocal space k-points. The k-points were chosen specifically within the irreducible portion of the Brillion zone. These Brillion zones are defined by highly symmetric special k-points and are dependent on the Bravias lattice of the given system [6]. The classification for the material as an insulator or a metal and the determination of the energy gap are the most basic electronic structure measurements. Fig. 2 shows the OLCAO-LDA band structure of the Ni3 Ga intermetallic alloy along the high symmetry directions in the energy range −12 eV to 8 eV. The energy eigenvalues are plotted along the y-axis and reciprocal space k-points (or wave vector) along the x-axis. The Fermi level (EF ) has been set at 0 eV energy value and is represented by the dotted line in the plot. The band structure shows that the valence and conduction bands are overlapped. This overlapping of bands indicates that Ni3 Ga carries conducting features. We have observed that six bands 19–24 cut the Fermi level. The band structure is in good agreement with Hayden et al. [6], except in the number of bands cutting the Fermi level. These calculations are self-consistent and relativistic but without spin-orbit interactions. The wave functions associated with bands 19–24, which cross the Fermi level found their roots from the d orbitals of the Ni atoms. As far as the Ga is concerned, it contributes little to the band structure of the system. III-2-2. Density of States

The number of electronic states per unit energy range is called the density of states (DOS). The total density of states (TDOS) represents what percentage of electrons in the system exist at what energies. Generally for crystalline systems, the TDOS can be observed to have a number of different peak structures at various points both above and below the highest occupied energy state, which is known as the Fermi level in metals.

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FIG. 2: Band structure of the ordered Ni3 Ga intermetallic alloy system in Au3 Cu structure along high-symmetry lines. A dotted line has been added at the Fermi level (EF = 0 eV).

When TDOS are broken into the partial density of states (PDOS), these peaks can usually be seen to come mainly from some particular orbitals and atomic types or a combination of just a few atomic types. The PDOS show which orbitals and atoms lead to prominent bands in the band structure diagrams, in particular it can depict the orbitals and atoms leading to the top of the occupied states and the bottom of the unoccupied states. On opposite sides of the highest occupied energy level, the difference in energy between various peaks can also assist in identifying which orbitals and atoms lead to specific structures of the optical spectra in the valence band. Further, the relationship of the DOS with bonding gives particular results. The TDOS of the valence band can also be compared directly with experimental measurements, such as X-ray emission spectroscopy (XES) and X-ray photoemission spectroscopy (XPS). In both cases, the resolution of the calculations is higher compared to experiment, which has the obvious limitation that not every physical phenomenon can be explained in the calculation. Fig. 3 is the plot of the TDOS and atom resolved PDOS for the ordered phase of Ni3 Ga using primitive cell and super cell calculations. Here in these calculations the Fermi level (EF ) is set at the 0 eV energy value. We see that the TDOS spectra were dominated by Ni-3d states. The contribution from Ga atoms is very small. In case of the primitive cell the major peak in the TDOS spectrum is at −0.69 eV below the Fermi level (EF ). In the PDOS plots, the peaks for Ni and Ga atoms were observed at −0.69 and −14.71 eV below

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the Fermi level (EF ), respectively. The TDOS and PDOS results calculated for the ordered supercell of Ni3 Ga are very close to that of the primitive cell. In the ordered supercell the major peak in the TDOS appears at −0.69 eV and the major peak positions for Ni and Ga in the PDOS were observed at −0.69 and −14.68 eV below the Fermi level (EF ), respectively. Fig. 4 shows the plot of the TDOS and atom resolved PDOS spectra of the disordered phase of Ni3 Ga alloy. The electronic structure calculations for the disordered phase are not similar to those of the ordered phase. It was observed that the disordered phase of Ni3 Ga exhibits fewer structures, as compared to that of the ordered phase of Ni3 Ga. The Fermi level (EF ) is again set at the zero energy value. The main peak in the TDOS plot was observed at −0.89 eV. The contribution from Ga is again very small, as was observed in the ordered phase. In short the disordered spectrum is more or less the broadened version of the ordered supercell spectrum. The results of the DOS at the Fermi level (EF ) for Ni3 Ga are summarized in Table I.

FIG. 3: Total and partial DOS for the ordered Ni3 Ga intermetallic alloy system using primitive unit cell (a) and supercell (b) calculations. The Fermi level (EF ) is at 0 eV.

III-2-3. Localization Index (LI)

The localization index (LI) is a calculation related to the approximate measure of the spread of the electronic charge around each atom and is particularly useful for identifying

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FIG. 4: Total and partial DOS for the disordered Ni3 Ga intermetallic alloy system. The Fermi level (EF ) is at 0 eV.

TABLE I: Calculated results for the DOS at the Fermi level (EF ) both for the ordered and disordered phases of the Ni3 Ga intermetallic alloy system. Model Used

N (EF )

Ordered primitive cell

6.8758

Ordered supercell

6.8926

Disordered supercell

5.0799

the degree of disorder in amorphous systems. Due to the implication of the normalization condition, L(E) = 1/N for a completely delocalized state and L(E) = 1 for a completely localized one. Moreover, the localization of states near the valence band edge has been employed to determine the mobility edge [28]. Fig. 5 shows the calculated localization index (LI) of the ordered and disordered phases using the supercell according to Equation (4) in the energy range of −10 eV to +10 eV. The generally accepted notion regarding electron localization in a disordered solid such as an amorphous semiconductor is that the states at the band edges are localized and those at the center of the bands are delocalized. In metallic systems, this concept is less clear,

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because the mobility of electrons in metals is sure (i.e., totally delocalized), although s- or p-orbitals are usually less localized than d-orbitals. In the present work we have focused on the electron states within a few eV of the Fermi level, where Ni-3d states dominate. In the ordered crystalline phase, the d-orbitals of Ni with specific interaction with Ga atoms form symmetry related bands. Some of them were relatively more localized. The localization index in the ordered supercell increases rapidly for the states from −3.81 eV including the Fermi level (EF = 0 eV) up to 0.23 eV. The states above 0.23 eV were almost delocalized. The localization index of the states below −3.81 eV increases gradually. The trend was different in the case of the disordered cell. The LI of the states below −4.42 eV gradually increases. The LI of the states between 3.14 eV and 1 eV shows some different behavior. The states above 1 eV were completely delocalized in the disordered phase of Ni3 Ga.

FIG. 5: Localization index (LI) plots of the states of the ordered (a) and disordered (b) Ni3 Ga intermetallic alloy system.

III-2-4. Effective Charge

The important Q∗ data calculated in this work is summarized in Table II. The primitive as well as the supercell approach has been used to calculate this data. In this work the Q∗ value of 256 atoms in total (for the disordered phase only) are plotted, 192 of Ni and 64 of Ga. This effective charge calculation depicts much of the microscopic information about the electronic structure of the material. The Q∗ value for Ni ranges from 9.962 to 10.233 electrons/atom while that of Ga remains constant at 12.633 electrons/atom in the primitive cell calculation of the ordered phase of Ni3 Ga. Our calculated results show that Ni atoms have the tendency to gain charge and Ga atoms have charge losing behavior. The average Q∗ values for Ni and Ga are 10.123 electrons/atom and 12.633 electrons/atom, respectively. The average Q∗ values indicate a considerable charge transfer from Ga to Ni. Ni atoms gain a net charge of 0.123 electrons/atom on average, while the Ga atoms lose an average of 0.367 electrons/atom in primitive cell calculations. The Q∗ calculation for the ordered supercell shows that the net charge transfer to Ni atoms has increased. Ni atoms have gained 0.125 electrons/atom on the average, while the net charge lost by Ga is increased as

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well, that is, it loses 0.376 electrons/atom, on the average. The Q∗ value for the ordered supercell phase ranges from 10.122 to 10.131 electrons/atom for Ni and that of Ga remains constant at 12.624 electrons/atom. The calculated average Q∗ values are 10.125 and 12.624 electrons/atom for Ni and Ga, respectively. TABLE II: Calculated results of the electronic structure of disordered and ordered Ni1−x Gax (x = 0.25) (256-atom model). Average effective charges (electrons) ∗

∗

Charge transfer (electrons)

Q (Ni)

Q (Ga)

∆Q∗ (Ni)

∆Q∗ (Ga)

Disordered Ni3 Ga

10.078

12.766

0.078

−0.234

Ordered Ni3 Ga

10.123

12.624

0.123

−0.376

The Q∗ values for the disordered phase of Ni3 Ga intermetallic alloy system were calculated and shown in Table II. The associated charge transfer from one atom to the other is also listed in Table II. The Q∗ value in the disordered supercell ranges from 9.699 to 10.259 electrons/atom for Ni and 12.742 to 12.817 electrons/atom for Ga. The average values of Q∗ are 10.078 and 12.766 electrons/atom for Ni and Ga, respectively. In the case of the disordered phase Ni gains 0.078 electrons/atom and Ga loses 0.234 electrons/atom. To make it more clear, the results have been plotted in Fig. 6.

FIG. 6: Effective charges (Q∗ ) of Ni and Ga atoms in a 256-atom disordered Ni3 Ga alloy model. Top row represents Ga atoms and lower row Ni atoms.

III-3. Optical Properties III-3-1. Dielectric function

The imaginary part (ε2 ) of the dielectric function of both the ordered and disordered phases of the Ni3 Ga intermetallic alloy system using the supercell approach has been cal-

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culated in the energy range 0.6 – 3.5 eV and plotted in Fig. 7 (a) and (b), respectively. Our calculated results (solid lines) are compared to that of the experimental results by Hsu et al. [5]. Three main peaks were identified in the data obtained through our calculations. In the start there is a deep valley like structure from 0.60 – 0.74 eV with tip at point D at 0.62 eV in the plot. The first peak A was observed at 0.75 eV, peak C at 1.62 eV, and the largest peak B was found between A and C at 1.19 eV. There is an acute decrease in ε2 along with the right arm of peak B. Some small bumps were also seen after peak C in these calculations. No peak like structure was seen in the experimental data [5]. The curve shows non-interesting behavior above 3 eV. However the general trend of the calculated and experimentally measured curve is nearly the same and consistent. In the case of the disordered phase, the experimentally measured curve matches very well with the theoretically calculated one, as can be seen in Fig. 7 (b). Our calculated curve shows absolute values just above the experimental curve. This difference in the absolute values may be due to the rough surface of the sample used in the experiment.

FIG. 7: The experimental (dotted from Ref. 5) and our calculated imaginary part of the dielectric function for ordered (a) and disordered (b) Ni3 Ga.

III-3-2. Optical Conductivity (σ)

The calculated optical conductivity of the ordered phase of Ni3 Ga using the supercell technique is shown in Fig. 8 (a). Here in this plot the calculated results are compared with the experimental data of Hsu et al. [5]. The comparison shows that the experimentally obtained curve has much larger amplitude. The calculated optical conductivity spectrum has three prominent peaks A, B, and C located at 1.20 eV, 1.64 eV, and 3.95 eV respectively. Among these A has the largest amplitude and is the most prominent peak. There is also a shoulder S on the left of peak A at 0.78 eV. The peak A was observed to be shifted slightly towards the high energy value and has a small amplitude compared to that of the experimentally observed peak [5]. The peaks B and C are not as prominent as that of peak A but their presence cannot be ignored. These are wider and are lying on the right of the main peak A. There are also some small peak like features E and F present between B and

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C at 2.22 eV and 2.73 eV. A deep valley like structure D is also present at 0.31 eV. This feature has not been observed in experimental data due to the limited range. Moreover the calculated spectrum is more rich in structures than the experimental [5]. However the general trend in the calculated and measured spectra appears to be consistent.

FIG. 8: Optical conductivity spectra for the ordered (a) and disordered (b) Ni3 Ga intermetallic alloy system. In (a) the top curve represents the experimental data from Ref. [5].

The part of the spectrum near the low energy end (∼ 0 eV) shows a sharp increase in conductivity. This may be due to the intraband transitions in metals. In the present work intraband transitions were calculated using the sufficiently large supercell technique. The wave functions for these transitions were calculated according to the Drude model (σ(ω) = (σo /1 + iωτ )), where τ is the relaxation time. It was also observed that the interband transitions in Ni3 Ga start at about 0.32 eV. The optical conductivity for the disordered phase of Ni3 Ga is shown in Fig. 8 (b). It has a single broad peak α at 4.85 eV, starting from 3.70 eV and ending gradually, and a very weak feature β at 1.90 eV. At energy close to 0 eV, the optical conductivity increases rapidly, showing the metallic conductivity as in the crystalline case and in all metallic systems. Such types of results may be obtained only if we use a sufficiently large supercell in these calculations. The comparison between the optical conductivity of the ordered phase of Ni3 Ga (Fig. 8 (a)) with that of the disordered phase (Fig. 8 (b)) is of great interest. The spectra in the ordered phase are rich in structures, as discussed earlier. In the disordered phase the conductivity spectrum is very simple and different, having a single broad peak α and a slight peak like feature at β. This difference in the disordered phase can be accounted for, because the inter-atomic pairs (Ni-Ni, Ni-Al, Al-Al) were distributed randomly, that gave different values of the LI (localization index). This is the reason for different electron states in both the occupied and unoccupied case, and is responsible for the very different optical conductivities.

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IV. CONCLUDING REMARKS In this study, computations on the electronic structures and optical properties of Ni3 Ga binary intermetallic alloy system in both the ordered and disordered phases have been performed. The calculations are based on the first-principles OLCAO method using DFT. The band structure calculations revealed the conducting nature of the alloy. It was observed from the band structure that the valence and conduction bands are overlapped, that confirms the conducting behavior of the system. The density of states of the primitive as well as supercell for the ordered and disordered phases were calculated. The DOS results indicated that both phases are different in nature. The total DOS was found to be dominated by Ni-3d states and the contribution from Ga atoms is very small in both phases. The peak points at −0.69 eV for both the primitive and supercell in the ordered phase and −0.89 eV for the disordered phase. We concluded that the electronic structure calculations for the disordered phase are similar to that of the ordered phase, but the disordered phase of Ni3 Ga exhibits fewer structures as compared to the ordered phase. The localization index (LI) calculations show that in the ordered supercell LI increases rapidly, for states from −3.81 eV including the Fermi level up to 0.23 eV. The states above 0.23 eV are almost delocalized. While in the disordered cell states above 1 eV are completely delocalized and the LI of the states below −4.42 eV gradually increases. It was concluded from these values that the behavior of the ordered and disordered phases is different for LI. In the present work the effective charge (Q∗ ) values of 256 atoms, 192 of Ni, and 64 of Ga were also calculated. The Q∗ value ranges from 9.962–10.233 electrons/atom for Ni and for Ga, it remains constant at 12.633 electrons/atom in the primitive cell. By considering the average Q∗ values, it was concluded that Ni atoms gain a net charge 0.123 and Ga loses 0.367 electrons/atom in the primitive cell. For the ordered supercell average Q∗ values for Ni and Ga are 10.125 and 10.624 electrons/atom, respectively. Ni atoms gain 0.125 electrons/atom in the ordered supercell. In the case of the disordered cell average Q∗ values calculated are 10.078 and 12.766 for Ni and Ga, respectively, showing that Ni gains 0.078 and Ga loses 0.234 electrons/atom. The Q∗ calculations indicated that there is a charge transfer in Ni3 Ga alloy from Ga to Ni. These calculations revealed that Ga atoms show a charge losing property and Ni atoms a charge gaining property. It was also taken into attention that law of conservation of charge remains valid in these calculations. The results for the imaginary part of the dielectric function (ε2 ) of the ordered and disordered phases of Ni3 Ga were compared with the experimental data. The obtained results are found to be very close to that of the experimental values. Our calculated curve exhibits absolute values just above the experimental curve in Ref. [5], however both curves show a similar trend and are consistent. Finally, the calculations were performed for the optical conductivity (σ). The optical conductivity curve for the ordered phase of Ni3 Ga shows three prominent peaks at 1.20 eV, 1.64 eV, and 3.95 eV. The largest peak observed among these is at 1.20 eV, which is similar to the experimental data, but our calculated peak has a small amplitude and was found to be shifted towards high energy values. We concluded that the calculated spectra of the ordered phase of Ni3 Ga are rich in structures in comparison to the experimental spectrum. In the case of the disordered phase a single peak at 4.85 eV was observed.

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By comparing the spectra we concluded that the spectrum of the ordered phase is rich in structures as compared to the disordered phase, while the disordered phase has a single prominent peak. This difference is mainly due to the random distribution of inter-atomic pairs (Ni-Ni, Ni-Al, Al-Al).

Acknowledgement The authors gratefully acknowledge Prof. Dr. Wai-Yim Ching for providing access to computational facilities at the Department of Physics, University of Missouri-Kansas City (UMKC).

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