compounds

Journal of Alloys and Compounds 597 (2014) 36–44

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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom

Structural, electronic, optical and thermodynamic properties of cubic REGa3 (RE = Sc or Lu) compounds: Ab initio study G. Murtaza a,⇑, S.K. Gupta b, T. Seddik c, R. Khenata c,⇑, Z.A. Alahmed d, R. Ahmed e, H. Khachai f, P.K. Jha g, S. Bin Omran d a

Materials Modeling Laboratory, Department of Physics, Islamia College Peshawar, Pakistan Department of Physics, Michigan Technological University, Houghton, MI 49931, USA Laboratoire de Physique Quantique et de Modélisation Mathématique, Université de Mascara, 29000 Mascara, Algeria d Department of Physics and Astronomy, College of Science, King Saud University, Riyadh 11451, Saudi Arabia e Department of Physics, Faculty of Science, Universiti Teknologi Malaysia, UTM Skudai, 81310 Johor, Malaysia f Physics Department, Djillali Liabes University of Sidi Bel-Abbes, Algeria g Department of Physics, Maharaja Krishnakumarsinhji Bhavnagar University, Bhavnagar 364001, India b c

a r t i c l e

i n f o

Article history: Received 19 November 2013 Received in revised form 26 January 2014 Accepted 27 January 2014 Available online 4 February 2014 Keywords: FP-APW + lo Electronic properties Optical properties Thermodynamic properties

a b s t r a c t Structural, elastic, optoelectronic and thermodynamic properties of REGa3 (RE = Sc and Lu) compounds have been studied self consistently by employing state of the art full potential (FP) linearized (L) approach of augmented plane wave (APW) plus local orbitals method. Calculations were executed at the level of Perdew–Burke and Ernzerhof (PBE) parameterized generalized gradient approximation (GGA) for exchange correlation functional in addition to modified Becke–Johnson (mBJ) potential. Our obtained results of lattice parameters show reasonable agreement to the previously reported experimental and other theoretical studies. Analysis of the calculated band structure of ScGa3 and LuGa3 compounds demonstrates their metallic character. Moreover, a positive value of calculated Cauchy pressure, in addition to reflecting their ductile nature, endorses their metallic character as well. To understand optical behavior calculations related to the important optical parameters; real and imaginary parts of the dielectric function, reflectivity R(x), refractive index n(x) and electron energy-loss function L(x) have also been performed. In the present work, thermodynamically properties are also investigated by employing lattice vibrations integrated in quasi harmonic Debye model. Obtained results of volume, heat capacity and Debye temperature as a function of temperature for both compounds, at different values of pressure, are found to be consistent. The calculated value of melting temperature for both compounds (ScGa3 and LuGa3) is found to be similar to the experimental data with an underestimation of 5%. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The compounds existing in AuCu3 structure favors the superconductivity [1–3], therefore a large number of rare earth elements based binary compounds REGa3 (where RE: rare earth elements) received great attention in recent years [4–24]. These compounds are also significant due to their simplest crystal structure, Fermi surface behavior, magnetic and transport properties. Matthias was the first who synthesized and characterized the single crystals of REGa3 compounds (RE = Sc, Dy–Tm, Lu), and predicted their cu bic crystalline structure with space group Pm3m – a favorable structure for superconductivity [1]. ⇑ Corresponding authors. Tel.: +92 321 6582416. E-mail addresses: [email protected] (G. Murtaza), [email protected] (R. Khenata). http://dx.doi.org/10.1016/j.jallcom.2014.01.203 0925-8388/Ó 2014 Elsevier B.V. All rights reserved.

Among this family of compounds a few experimental and theoretical studies exist in literature dealing with structural, electronic and transport properties of type I superconducting ScGa3 and LuGa3 [25–31]. Pluzhnikov et al. [28] have determined the geometry of the Fermi surface of the ScGa3 and LuGa3 using the de Hassvan Alphen (dHvA) effect and found a similarity between dHvA spectra for REGa3 (RE = Sc and Lu) compounds. Kletowski et al. [29] investigated the temperature of resistivity for LuGa3 and other REM3 (LaSn3, LaPb3, Luln3 and LaIn3) compounds in the temperature range of 4.2–300 K. Besides this, ab initio band structure calculations have also been reported for LuGa3 within linear muffin-tin orbital method (LMTO) [9]. In another theoretical study a great resemblance in the electronic structures of ScGa3 and LuGa3 has been observed [30]. Also, a theoretical study related to the electronic structure of these compounds has been investigated by Bross [20] using a new version of the modified augmented plane

G. Murtaza et al. / Journal of Alloys and Compounds 597 (2014) 36–44

wave method (MAPW) to describe some experimental features of LuGa3 and its iso-structural compounds with Er, Tm and Yb. More recently, thermodynamic and measurement of the transport properties on single crystal of REGa3 (RE = Sc, Lu) has been studied [31]. Svanidze and Morosan [31] pointed out that ScGa3 and LuGa3 are type-I superconducting compounds at transition temperature in the range of 2.2–2.3 K. In this paper, we focused on the structural, elastic, electronic, optical and thermodynamical properties of single crystals REGa3 (RE = Sc or Lu). To the best of our knowledge, neither experimental nor theoretical data related to optical properties of these compounds are available. Thus, we think worthwhile to perform a systematic investigation on ScGa3 and LuGa3 compounds, in order to provide a comparative and complementary study to the previous theoretical and experimental works of binary REGa3 compounds. We have used in our calculation FP-(L(APW + lo)) method developed within density functional theory (DFT). The rest of the paper is as follows, Section 2 deals with the brief description of the computational detail of this study. In Section 3, details of the obtained results and discussion related to structural, elastic, optoelectronic and thermodynamic properties of ScGa3 and LuGa3 compounds are given. Main conclusions of our present work are summarized in Section 4.

37

and m is the Poisson ratio in the above relation. Minimization of the non-equilibrium Gibbs function G(V; P, T) with respect to the volume V at constant pressure and temperature is attained as:

@G ðV; P; TÞ @V

¼0

ð6Þ

P;T

By solving Eq. (6), one can obtain a relation for V(P,T) i.e. thermal equation of state (EOS). Using Eq. (6) for different thermal properties; isothermal bulk modulus (BT), specific heat capacity at constant volume (CV) and at constant pressure (CP), and thermal expansion coefficient a can be evaluated using relations given as below:

BT ðP; TÞ ¼ V

@ 2 G ðV; P; TÞ @V

! ð7Þ

2 P;T

3H=T C V ¼ 3nkB 4DðH=TÞ H=T 1 e

ð8Þ

C P ¼ C V ð1 þ acTÞ

ð9Þ

a¼

cC V BT V

ð10Þ

where c is used to represent Grüneisen parameter and is calculated by using the following expression:

c¼

d ln HðVÞ d ln V

ð11Þ

2. Computational details Calculations were carried out within FP-L(APW + lo) method [32] in order to solve DFT Kohn Sham equations [33] using WIEN2k computational code [34]. For structural properties, Perdew–Burke and Ernzerhof [35] parameterized generalized gradient approximation (PBE-GGA) was used for the purpose of exchange–correlation energy functional/potential. Whereas for electronic and optical properties we have applied Trans Blaha (TB) modified Becke–Johnson (mBJ) scheme [36]. This scheme improves the accuracy of the band gap energy by correctly predicting d and f states of semiconductors and insulators within semi-local orbital independent exchange–correlation potential [36]. The muffin tin (MT) radii of Sc, Lu and Ga were adopted to be 2.35, 2.42, 2.37 atomic unit, respectively. In the irreducible part of the first Brillouin zone, a mesh of 1000 special k-points was taken into account for both compounds. For the wave function expansion inside the muffin tin (MT) spheres, the maximum value of l quantum number was chosen lmax = 10. Whereas in interstitial region of the unit cell plane wave cut-off of RMTKmax = 12 was used. The charge density is Fourier expanded up to Gmax = 12(Ryd)1/2. The total energy convergence was ensured by varying the plane wave cut-off parameter and the number of k-points. To study thermal effects, quasi-harmonic Debye model realized in Gibbs program [37] has been applied. This model is sufficiently flexible in giving all thermodynamical quantities by incorporating obtained results of energy and volume. We give, here, a brief description of this model [37–43]. In this model non-equilibrium Gibbs function G(V; P, T) is described in the following form:

G ðV; P; TÞ ¼ EðVÞ þ PV þ AVib ðHðVÞ; TÞ

ð1Þ

where E(V) represents total energy/formula unit, PV stands for constant hydrostatic pressure condition, H(V) is used for Debye temperature, AVib is used to present lattice vibration and is expressed as:

AVib ðH; TÞ ¼ nkB T

9H þ 3 lnð1 eH=T Þ DðH=TÞ 8T

rffiffiffiffiffi h 1=3 Bs ½6p2 V 1=2 n f ðrÞ M K

ð3Þ

2

d EðVÞ dV

2

ð4Þ

where f (m) is defined as

f ðmÞ ¼

8 " 91 3 3 #1 =3 < 21 þ m 2 11 þ m 2 3 2 þ : ; 31 2m 31 m

Firstly, total energy calculations are done as a function of increasing and decreasing unit cell volume around equilibrium cell volume (V0) and obtained data for E–V is displayed in Fig. 1 by fitting it into Murnaghan equation of states (EOS) [44]. Fig. 1 indicates that with the increase of unit cell volume for compounds, energy decreases and approaches to its minimum value at a particular volume. This particular volume is called optimized equilibrium volume of the compounds whereas energy corresponding to this volume is termed as ground state energy of the material under study. The obtained results for all equilibrium parameters a0, B0 and B0 at the level of PBE-GGA approximation are listed in Table 1, together with the available experimental and theoretical values. Our computed lattice parameters are in reasonable agreement with the available experimental and other theoretically reported data. However, for the sake of comparison no experimental or theoretical data for these compounds is found in literature and our results are thus predictive. 3.2. Elastic properties

In relation (3), M is for molecular mass, BS is a representative for adiabatic bulk modulus, which is estimated in terms of static compressibility by using following relation:

Bs ffi BðVÞ ¼ V

3.1. Structural properties

ð2Þ

In relation (2), n represents the number of atoms/formula unit, kB is the well known Boltzmann constant and the last term D(H/T) is used to represent Debye integral. Where H is expressed, for anisotropic solid, by the following relation:

H¼

3. Results and discussion

ð5Þ

The knowledge of the mechanical properties like Bulk modulus, Young’s modulus, shears modulus and Poisson’s ratio of a material is crucial to get the right view about its strength. These properties can be derived from the knowledge of the elastic constants of the materials. Moreover, from the practical aspect, the elastic constants of a material describe its linear response to external forces whereas, from a fundamental view point, these constants are connected with a variety of crucial properties of the solids, such as structural stability, interatomic potentials, equation of state, and phonon spectra. In addition to, elastic constants are also associated with thermodynamical properties such as specific heat, thermal expansion, melting point, Debye temperature, and Grüneisen parameter. Therefore, for the characterization of a material, determination of the elastic constants is very essential. In this study, because of the cubic symmetry of ScGa3 and LuGa3 compounds, we only require the calculations of three independent elastic constants

38


When this strain is applied, the initial value of the total energy of the system changes that can be found out by the following equation:

EðeÞ ¼ ðC 11 C 12 Þ6V 0 e2 þ Oðe3 Þ

ð14Þ

where ‘V0’ is the unit cell volume. Lastly, volume-conserving rhombohedral strain tensor is applied in the following form:

0

eB

1 1 1

1

e ¼ @1 1 1C A

!

3

ð15Þ

1 1 1

By the application of this strain, total energy is changed and is expressed in terms of elastic constants as follows:

EðeÞ ¼

Fig. 1. Unit cell volume versus total energy for (a) ScGa3 and (b) LuGa3.

Table 1 The obtained results for all equilibrium parameters a0, B0 and B0 within GGA approximation for ScGa3 and LuGa3, together with the experimental and theoretical values and the elastic constant Cij.

a b c d e f g

ScGa3

Present Expt.

LuGa3

Present Expt. Theory

Ref. Ref. Ref. Ref. Ref. Ref. Ref.

A

B

B0

C11

C12

C44

4.12 4.095a, 4.096b, 4.092c, 4.097d 4.2 4.191e, 4.169f 4.212g

73.14

4.18

116.97

53.16

16.39

68.23

4.23

99.97

54.82

28.68

[21]. [25]. [26]. [27]. [3]. [22]. [20].

C11, C12 and C44. Elastic constants Cij are obtained using Charpin method implemented in WIEN2k code [34]. The following equations are used to find out these three elastic constants and related properties, by following the same steps as performed in some earlier studies [45,46]. First one involves the calculations of the bulk modulus ‘B’ in terms of C11 and C12 elastic constants by the following relation [47]:

B¼

1 ðC 11 þ 2C 12 Þ 3

ð12Þ

e 0 ! e ¼B @0 e 0 0

0

1

0

C A

1 1þe2

1

ð16Þ

The values of calculated elastic constants for REGa3 compounds under study are given in Table 1. C11 measures the unidirectional compression induced along principal crystallographic directions. From Table 1, it can be seen that C11 for both compounds (ScGa3 and LuGa3) is much higher than C44, highlighting that these compounds show a weaker resistance to the pure shear deformation compared to the unidirectional compression resistance. For the mechanical stability of cubic crystals, following conditions regarding elastic constants must be satisfied; C11–C12 > 0, C11 > 0, C44 > 0, C11 + 2C12 > 0, and C12 < B < C11. Our results for elastic constants as given in Table 1 suitably verify above mentioned stability conditions. Pettifor concluded [48] that the Cauchy pressure C12–C44 can describe the angular character of atomic bonding in metals and compounds which in turn is related to the brittle/ductile nature of materials. If the material has directional bonding in conjunction with angular character, gives out the negative value of Cauchy pressure. A larger value of the negative pressure corresponds to added directional character where as the typical positive value of the Cauchy pressure point to the metallic bonding character of the material. In our present case, calculated results of the Cauchy pressure for both ScGa3 and LuGa3 compounds are positive, reflecting ductile nature side by side exposing their metallic bonding character as well. To calculate the elastic anisotropy of these compounds, anisotropy factor as denoted by A is computed by applying relation A = 2C44/(C11–C12) using calculated values of elastic constants as given in Table 1. From the literature it is known that if A = 1 material is absolutely isotropic, whilst less or greater value of A than 1 points to anisotropic nature of the material. Thus scale of elastic anisotropy possessed by a material is measured by the taking the simple difference of the calculated value of A with unity i.e. elastic anisotropy = 1–A. Our calculated values of the elastic anisotropy listed in Table 2 clearly display deviation from 1 for both of the compounds. So ScGa3 and LuGa3 are characterized as anisotropic materials. In order to calculate other important properties (Young’s modulus E, shear modulus G, and Poisson’s ratio m) related to the mechanical strength of these compounds in terms of computed values of the elastic constants Cij = C11, C12, C44, as given in references [49–51], following relations are used:

GV ¼

ðC 11 C 12 þ 3C 44 Þ 5

ð17Þ

GR ¼

5ðC 11 C 12 ÞC 44 4C 44 þ ðC 11 C 12 Þ

ð18Þ

GH ¼

GV þ GR 2

ð19Þ

Second one involves application of the volume-conserving tetragonal strain expressed as:

0

V0 ðC 11 þ 2C 12 þ 4C 44 Þe2 þ Oðe3 Þ 3

ð13Þ

39

G. Murtaza et al. / Journal of Alloys and Compounds 597 (2014) 36–44 Table 2 Calculated anisotropy factor A, Poisson’s ratio m, Young modulus E (in GPa), shear moduli GV, GR and GH (in GPa), melting temperature Tm (K) and GH/B ratio.

a

B

GV

GR

GH

E

m

A

B/G

Tm ± 300

ScGa3

74.43

20.34

22.59

21.47

60.02

0.39

0.51

0.28

LuGa3

69.87

25.88

26.23

26.05

71.80

0.37

1.27

0.37

1244.2 1303a 1143.8 1203a

Experimental data from Ref. [56].

E¼

9BGH 3B þ GH

ð20Þ

3B E 6B

ð21Þ

(a)

and

m¼

The calculated values of the mentioned elastic moduli for polycrystalline ScGa3 and LuGa3 aggregates are listed in Table 2. Ductile/brittle nature of the given material can be known by two factors: Pugh’s index of ductility (G/B) and the Poisson’s ratio (m). The value of G/B ratio, called Pugh’s ratio, identifies material’s brittle/ductile nature [52]. To categorize a material to be ductile/ brittle, critical value of G/B ratio was identified to be 0.57. This ratio in our case is around 0.28 for ScGa3 and 0.37 for LuGa3, indicating that both compounds are ductile in nature. Frantsevich et al. [53] devised a rule on the base of Poisson’s ratio (m) to make a distinction between ductile and brittle materials. With respect to his criterion, the critical value of ‘m’ is 0.26. For brittle materials the value of ‘m’ should be less than 0.26. The values of Poisson’s ratio in our calculations as reported in Table 2 are 0.39 and 0.37 for ScGa3 and LuGa3 respectively, endorsing their ductile nature. Poisson’s ratio also gives information about bonding nature in materials. For different types of bonding, the value of the Poisson’s ratio varies over a range 0.0–0.5. Covalently bonded materials show small value i.e. m = 0.1, for ionic crystals its value is reported around 0.25 [54] whereas its typical value for metal materials is 0.33 [54]. In our case, the m values are around 0.33, signifying metallic bonding of REGa3 compounds. Moreover, using Fine et al. [55] empirical relation, calculations regarding the melting temperature of ScGa3 and LuGa3 by employing the following expression in terms of elastic constants:

T m ¼ ½553 K þ ð5:91 K=GPaÞC 11 300 K

(b)

ð22Þ

Calculated melting temperatures of both ScGa3 and LuGa3 are given in Table 2, along with experimental data [56]. We found that computed results are underestimated by 5% than the experimental data. 3.3. Electronic properties Herein electronic properties of the studied compounds via calculated band structures and densities of states are discussed. The electronic band structure of ScGa3 and LuGa3 compounds are computed using the equilibrium lattice parameters within TB-mBJ approximation. Fig. 2 depicts the computed band structure of the ScGa3 and LuGa3. It is noted there is no energy gap among the bands around the Fermi level. At the Fermi level valence and conduction bands overlap significantly, as a result, both ScGa3 and LuGa3 exhibit a metallic character. This finding is similar to that in REGa3 (RE = Er, Tm, Yb and Lu) single crystals which also exhibit metallic behavior [12,20,31]. As can be seen from Fig. 2, the lowest part of the valence band, situated in between 10.5 and 3.5 eV is mostly dominated by Ga ‘‘s’’ character for both compounds. Besides in LuGa3 band structure (Fig. 2b), the Lu f energy bands are located below EF at about 5 eV with a small width of 1.0 eV. However the

Fig. 2. Band structure within TB-mBJ approximation for (a) ScGa3 and (b) LuGa3.

large part of Sc and Lu ‘‘d’’ states are located just above the Fermi level in the conduction band with a mixture of Ga ‘‘p’’ state with a small amount of Sc/Lu ‘‘s, p’’ states and Ga ‘‘s, d’’ states. Fig. 3 presents the calculated total and partial densities of states (TDOS and PDOS) for ScGa3 and LuGa3 compounds. Following this figure, one can see here also more clearly that for both compounds, the lowest part of the valence band lying in a range of energy 10.5 to 3.5 eV are formed mainly by Ga ‘‘s’’ states, however in the case of LuGa3 these states are hybridized with Lu ‘‘f’’ states in region situated between 5.5 eV and 4.5 eV. Just above these states

40


the region from 3.5 eV up to Fermi level is largely composed by Ga ‘‘p’’ states and Sc/Lu ‘‘d’’ states. We noticed that Ga ‘‘p’’ states hybridize strongly with Sc ‘‘d’’ or Lu ‘‘d’’ states and is occupying bands over an energy range from 3.5 to 10 eV. In a vicinity of the Fermi level EF, the d-states of both Sc and Lu provide the dominant contribution to DOS of ScGa3 and LuGa3. The calculated densities of states N(EF) of ScGa3 and LuGa3 at the Fermi level are 2.06, 1.88 states/eV, respectively. Our computed value of N(EF) for ScGa3 is very close to 2.0 states/eV value obtained by Svanidze and Morosan using FP-LAPW [31]. However for LuGa3, this value is larger than that obtained from LMTO calculation (N(EF) = 0.3 states/eV) [12]. The conduction band consists of Sc or Lu ‘‘d’’ states and Ga ‘‘p’’ states mixed in, with small contribution of Sc or Lu ‘‘s, p’’ states and Ga ‘‘s, d’’ states. 3.4. Optical properties The optical properties of ScGa3 and LuGa3 compounds are determined by the dielectric function e(x) = e1(x) + ie2(x), which is mainly contributed from the electronic structures. Where the real and imaginary parts of dielectric function, e1(x) and e2(x), are calculated by the following relations [57,58]:

Z e X

4p 3m2 x2 2 2

e2 ðxÞ ¼

l;n

2

3

d kjPnl0 j2 d½El ðkÞ En ðkÞ hx 3 BZ ð2pÞ

(a)

and

2

e1 ðxÞ ¼ 1 þ P p

Z 0

1

x0 e2 ðx0 Þ 0 dx x02 x2

ð24Þ

As both the compounds (ScGa3 and LuGa3) under investigations encompass cubic symmetry, hence their linear optical properties require only one dielectric tensor component for their thorough description [59–62]. It is worth to mention here that for the metallic and semimetallic compounds, there are two contributions to e(x), namely intraband and interband transitions according to the following expression [63,64],

e2 ðxÞ ¼ e2inter ðxÞ þ e2intra ðxÞ

ð25Þ

As the compounds under study are metallic, the Drude term [58] (intraband transitions) is included in the calculation of their optical spectra. This term is expressed by,

e2intra ðxÞ ¼

xp s xð1 þ x2 s2 Þ

ð26Þ

where xp is the anisotropic plasma frequency and s is the mean free time between collisions

x2p ¼

ð23Þ

8p X 2 # dðekn Þ 3 kn kn

where #nk is the electron velocity and ðekn Þ is (En(k)–EF).

(b)

Fig. 3. Total and partial DOS for (a) ScGa3 and (b) LuGa3.

ð27Þ


Once the real and imaginary parts of the dielectric function are determined, we can calculate other important functions such as the reflectivity R(x), the refractive index n(x), the electron energy-loss function L(x) and the photoconductivity r(x). Calculations of the real part of the dielectric function along with imaginary part for ScGa3 and LuGa3, performed at the level of TBmBJ, have been displayed in Fig. 4 as a function of incident photon energy up to 25 eV. Value for the spectral broadening was used to be 0.1 eV. Following these figures it can be easily pointed out that these compounds have a different optical spectrum, owing to the fact that the band structures for these compounds are different. This difference is responsible of some changes in the optical transitions result in changing the peak positions and the peak intensity. In these compounds the first optical transitions (thresholds peaks) occur at 0.24 for ScGa3 and for LuGa3 at 0.7 eV. Beyond these transitions the main peaks in spectra are located at around 1.71 eV and 1.45 eV for ScGa3 and LuGa3, respectively. Adjacent to the main peaks there are also peaks positioned at 1.9 eV for ScGa3 and at 4.09 eV for LuGa3. The real dielectric function e1(x) is convoluted from e2(x) by the Kramers–Kronig conversion [58]. From spectra presented in Fig. 4a for the real part e1(x), it is clear from zero frequency limit that the value of the calculated static dielectric constant, e1(0) is 57.11 for ScGa3 and 40.38 for LuGa3, characterizing the dielectric response of these materials to static electric field. The large negative values of e1(x) for both compounds indicate that the two crystals have a Drude-like behavior. Fig. 4c shows the calculated reflectivity R(x) against energy of ScGa3 and LuGa3. Curve indicates that the maximum reflectivity is at energy 11 eV for ScGa3 and 2.5 eV for LuGa3. Abrupt decrease in reflectivity is also noted at 13 eV and 14 eV for ScGa3 and LuGa3 respectively because of collective plasma resonance. e2(x) was used to find out depth of plasma minimum and is measure of the degree of overlapping between inter-band absorption regions [65]. In present work, calculations at the level of TB-mBJ give the value of 0.59 and 0.55 respectively, for the magnitude of the reflectivity coefficient at zero frequency, for ScGa3 and LuGa3. The dispersion curves of refractive index for ScGa3 and LuGa3 presented in Fig. 4b shows that both compounds do not have the same features. It is noted from the figure that both compounds (ScGa3 and LuGa3) in low energy range show a maximum value of refractive indices, whereas their behavior is opposite at the large energy range. In the case of ScGa3 the maximum value is followed by a secondary peak at energy around 1.5 eV. L(x) is an important optical parameter which gives information about the possible interactions inside the material by fast electron. These interactions are not only accountable for interband transitions but also are source of intraband transitions, phonon excitation, inner shell ionizations and plasmon excitations. The main peaks in L(x) spectra presented in Fig. 4d located at about 13 and 14 eV for ScGa3 and LuGa3 respectively, are at an energy coinciding to the rapid decrease in reflectivity R(x) (Fig. 4c) and to the second zero crossing of real part e1(x) (Fig. 4a). The corresponding frequency of these prominent peaks in L(x) are defined as the screened plasma frequency xP [66].

3.5. Thermal properties In our present work, quasi-harmonic Debye model has been applied realized in Gibbs program [37]. From the calculated total energy as function of primitive cell volume (E–V) data at zero pressure and temperature in static approximation, and the standard thermodynamic relations, we have derived the macroscopic properties as a function of pressure and temperature. The thermal properties for both ScGa3 and LuGa3 are determined in

41

Fig. 4. The real and imaginary parts of dielectric function, reflectivity R(x), the refractive index n(x), absorption coefficient n(x) and electron energy-loss function L(x) for ScGa3 and LuGa3.

the temperature range from 0 to 800 K and the pressure effect is studied in the range from 0 to 40 GPa. Fig. 5(a and d) shows the variation of bulk modulus as a function of temperature at different values of pressures. It is noted from the figures that bulk modulus decreases linearly with the increase of temperature for both (ScGa3 and LuGa3) of the compounds, showing that at given pressure, compressibility increases with the temperature rise. Further it is seen that the compressibility decreases with increasing pressure. This indicates that the effect of increasing pressure on the material is similar as that of decreasing temperature. Fig. 5(b and e) displays the relative change in volume as a function of temperature at different fixed values of pressures for ScGa3 as well as LuGa3 compounds. It is observed that in the very low temperature range, relative change in volume is almost negligible. The slope is zero at the starting of the curves however its values rapidly increases in a range of temperature from 50 to 100 K, and the relative changes approaches to linear behavior above 150 K. Conversely, at given temperature, the relative change (V/V0) decreases as the pressure is increased. Specific heat at constant volume (CV) as function of temperature is shown in Fig. 5(c and f) for both (ScGa3 and LuGa3) of the compounds at different values of pressure. It can be seen that in the limit of low temperature, CV follows to Debye T3 power law for both compounds. Whereas in range of high temperature, our calculated results for CV as function of temperature for both compounds

42


approaches to the classical asymptotic limit by following the Dulong–Petit. At zero pressure, and temperature T = 300 K, our calculated value of CV for ScGa3 and LuGa3 are 94.35 and 96.82 J/mol K respectively. Thermal expansion coefficient a was calculated for different pressures and temperatures and plotted in Fig. 6(a and c). It is noted that at P = 0,20 and 40 GPa, a increases exponentially with T at low temperatures and gradually approaches to linear behavior at higher temperature. For P = 40 GPa, the value of a at T = 800 K is a little higher than that at 600 K. It shows that higher temperatures and pressures almost does not affect to a. To describe the thermal properties of solids, Debye temperature (HD) is another essential parameter. Above HD crystals behave classically. It is because the role of thermal vibrations becomes more significant than that of the quantum mechanical effects. Variations in Debye temperature versus temperature, at different fixed values of pressures, are depicted in Fig. 6(b and d). From the figure, it can be seen clearly that HD in a range of temperature from 0 to 100 K approximately remains unaltered and with the further increase in temperature, decreases linearly. At zero pressure and

temperature T = 300 K, the computed HD for ScGa3 compound (332.9 K) is twice smaller than that obtained by Svanidze and Morosan from specific heat and resistivity measurements(HD = 660 K) [31], while for LuGa3 our calculated HD value (263.7 K) is slightly overestimated than measured one (HD = 232 K) [31].

4. Conclusions In this study, a comprehensive study on the structural, elastic, electronic, optical and thermodynamical properties of REGa3 (RE = Sc or Lu) compounds, using FP-L(APW + lo) method at the level of PBE-GGA together with TB-mBJ potential has been reported. We have found that the calculated lattice parameters at equilibrium are in reasonable agreement with the available experimental and theoretical results. The calculated elastic constants are satisfying the mechanical stability conditions. The results of the electronic band structure and density of the states of the ScGa3 and LuGa3 exhibit metallic behavior just like other REGa3 compounds. Additionally, the optical properties of the investigated materials

Fig. 5. The variation of the bulk modulus, normalized volume and the heat capacities CV with temperature at some pressures of LuGa3(left panel) and ScGa3(right panel).


43

Fig. 6. The variation of the thermal expansion coefficient and the Debye temperature with temperature at some pressures of LuGa3(left panel) and ScGa3(right panel).

such as real and imaginary parts of the dielectric function, reflectivity R(x), refractive index n(x) and electron energy-loss function L(x) are calculated and analyzed. By employing the lattice vibrations integrated quasi harmonic Debye model, predicted results of the thermodynamical properties are also presented. At fixed values of different pressure, variations in the relative change in volume, specific heat capacities, thermal expansion and Debye temperature as a function of temperature are successfully achieved. Melting temperatures of both the ScGa3 and LuGa3 compounds were calculated and found that these are underestimated by 5% than the experimental data. The present study would be helpful for future experimental and theoretical explorations.

Acknowledgments SKG acknowledges award of the Fulbright-Nehru Postdoctoral Research Fellowship. S.B.O, Z.A.A and R.K. acknowledge the support by the National Plan for Science, Technology and Innovation under the research project No. 11-NAN1465-02. PKJ also acknowledges the University Grants Commission (UGC) and the Department of Sciences and Technology (DST), Govt. of India.

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