Structural, electronic, elastic and thermodynamic properties of XFe4P12 (X = Tb and Dy) filled skutterudite using FP-LMTO method

Accepted Manuscript Title: Structural, electronic, elastic and thermodynamic properties of XFe4 P12 (X = Tb and Dy) filled skutterudite using FP-LMTO method Author: Samir Mustapha Laoufi Amina Touia Mohammed Ameri Ibrahim Ameri Fatima Boufadi Keltouma Boudia Amel Slamani Fadila Belkharroubi Y. Al-Douri PII: DOI: Reference:

S0030-4026(16)30308-4 http://dx.doi.org/doi:10.1016/j.ijleo.2016.04.034 IJLEO 57517

To appear in: Received date: Accepted date:

12-2-2016 12-4-2016

Please cite this article as: {http://dx.doi.org/ 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.

Structural, Electronic, Elastic and Thermodynamic Properties of XFe4P12 (X=Tb and Dy) Filled Skutterudite using FP-LMTO Method. Samir Mustapha Laoufi1, Amina Touia1, Mohammed Ameri1,a,*, Ibrahim Ameri2, Fatima Boufadi1, Keltouma Boudia1, Amel Slamani1, Fadila Belkharroubi1, Y. Al-Douri3. 1

Laboratoire Physico-Chimie des Matériaux Avancés (LPCMA), Université Djilali Liabès de Sidi BelAbbès, Sidi Bel-Abbès, 22000, Algérie. 2 Djillali Liabes University, Faculty of Exact Sciences, Department of Physics , PO Box 089, Sidi Bel Abbes, Algeria 22000. 3 Institute of Nono Electronic Engineering, University Malaysia Perlis, 01000 Kangar, Perlis, Malaysia.

Abstract The physical properties of TbFe4P12 and DyFe4P12 ternary alloys have been described using the full potential linear muffin-tin orbital’s (FP-LMTO) method within local density approximation (LDA) and local spin-density approximation (LSDA). The equilibrium lattice constants of both alloys are in good agreement with experimental data. The electronic band structure and density of state depict that the alloys as conductors. The computed elastic constants Cij, the bulk modulus B, anisotropy factor A, shear modulus G ,Young’s modulus E, Poisson’s ratio ν and B/G rations of XFe4P12 (X=Tb,Dy) at different pressures using LDA are found. The sound velocities and Debye temperature are too anticipated from elastic constants. Eventually, the fluctuations of the primitive cell volume, expansion coefficient α, bulk modulus B, heat capacity ( and

), Debye temperature θD, and entropy S, at pressure 0-50

GPa and temperature between 0-3000 K are obtained.

Keywords: FP-LMTO; Alloy; Structural properties; Thermodynamic properties. *

Corresponding author. E-mail: [email protected] ; [email protected] Web site: ahttp://univ-sba.dz/lpcma/ , Tel: +213772534447 1

1. Introduction The ternary alloys with the general formula LnT4X12 (Ln = lighter lanthanide, T = Fe, Ru and Os, X = P, As and Sb) crystallize at skutterudite type structure (cubic, space group: Im-3) [1,2] (see Fig. 1). These skutterudites with lighter lanthanides were frequently prepared by flux methods [1,2]. However, we have prepared many skutterudite compounds with lighter lanthanides at high temperatures and high pressures [3]. These materials show interesting physical properties at low temperatures. Superconducting [4,5], semiconducting [6], metalinsulator transition [7,8], magnetic [9], heavy fermions [10], intermediate valence [11,12], and non-Fermi liquid behavior [13] have been observed in these materials. Further, skutterudite compounds exhibit remarkable thermoelectric properties [14, 15]. The potential of materials for thermoelectric applications [16,17] is determined by the figure of merit, Z = S2·σ/κ where S is the Seebeck coefficient, σ is the electrical conductivity, and κ is the thermal conductivity. Better thermoelectric properties are determined by a combination of high mobility (for higher σ with reasonable carrier concentrations), higher band masses (to obtain higher values of S), and lower lattice thermal conductivities, κ = κe + κ1 which include both electronic and phonon contributions, respectively. The physical properties of skutterudite compounds were reviewed by Sales [18].

We have prepared new filled skutterudites with heavy lanthanides (including Y), LnT4P12 (Ln= heavy lanthanide, T = Fe, Ru and Os) at high temperatures and pressures [19–20]. The new filled skutterudite TbFe4P12 has been prepared at around 4 GPa and 1050 °C. Powder Xray diffraction of TbT4P12 (T = Fe and Ru) has been studied with synchrotron radiation at ambient pressure [21]. The electrical and magnetic properties of several new filled skutterudites with heavy lanthanide have been studied at low temperatures; DyFe4P12 [22] and TbFe4P12 [21] exhibit a ferromagnetic ordering around 10 K. These materials could be

2

potential in multistage thermoelectric devices as it has been shown that they have ZT>1.0, but their properties are not well known. Recent findings on the physical-chemical and transport properties of skutterudite materials are reviewed. Results obtained on binary compounds showed that their lattice thermal conductivity is too high to achieve high thermoelectric figures of merit. Several approaches have been recently proposed to decrease the lattice thermal conductivity of skutterudite materials. They are described and illustrated by recent experimental results. It is shown that low thermal conductivity can be achieved and high ZT values are possible for skutterudites [23] .We have already refined the crystal structures of some filled skutterudites on powder X-ray diffraction data with Rietveld methods [6,12]. In this work, the structural, electronic, elastic and thermodynamic properties using the full potential method linear muffin-tin orbital (FP-LMTO) for different values of pressure and temperature are investigated. The arrangement of this paper is as complies: The FP-LMTO computational details in section 2 is depicted and section 3 presents the results and discussion for structural, electronic, elastic and thermodynamic properties for TbFe4P12 and DyFe4P12 compounds. Eventually, a conclusion is given in section 4.

2. Computational method Computations depicted here, were performed using the FP-LMTO method within The density-functional theory (DFT) [24-25] and implemented in the Lmtart computer code in this method, the space is divided in two parts: an interstitial regions (IR) and non-overlapping (MT) spheres centered on the atomic sites. In the IR regions, the Fourier series represent the basic function. In side on MT spheres, the basic set is treated as a linear combination of radial functions times spherical harmonics. The charge density and the potential are represented inside the muffin-tin sphere radius (MTS) by spherical harmonics up to lmax=6. The exchangecorrelation (XC) effects are treated with the local density approximation (LDA) using the scheme developed by Perdew and Wang [26-27] and the local spin density approximation 3

(LSDA) [28]. The k-integrations over the Brillouin zone (BZ) is carried out up to (6,6,6) grid in the irreducible Brillouin zone, using the tetrahedron method [29] the values of the sphere radii, energy cut-off and the number of PLWs for TbFe4P12 and DyFe4P12 are listed in Table 1.

2.1. Elastic properties The elastic constants of materials describe its response to an applied stress or, conversely, the stress required to maintain a given deformation. Both stress and strain have three tensile and three shear components, giving six components in total. The linear elastic constants form 6×6 symmetric matrix, having 27 different components, such that and strains,

=

for small stresses, ,

[30]. Any symmetry presents in the structure may make some of these

components equals each other’s and may be fixed at some of these components equaled and others may be fixed at zero. Thus, cubic crystal has only three different symmetry elements (C11, C12 and C44), each one represents three equal elastic constants (C11=C22= C33, C12= C23= C31, C44= C55= C66). A single strain with non-zero first and fourth components, yielding a very efficient method for the calculation of elastic constants for the cubic system. Thus for the calculation of elastic constants C11, C12 and c44 we have used the Mehl method [31-32] of imposing the conservation of volume of the sample under the pressure effect. To calculate the difference of elastic modulus, C11-C22, we apply the following orthorhombic strain tensor.   0   0   0 0 

where

   0  1   1 1   2  0

(1)

is the applied stress, varies between 5×10-4 and 2.5×10-3. The application of this

strain affects the total energy: E ( )  E ( )  E (0)  6(C11  C12 )V0 2  [ 4 ]

4

(2)

where E (0) is the energy of the system to the initial state (without stress) and V0 is the volume of unit cell. Moreover, for an isotropic cubic crystal, the bulk modulus is related to Cij constants according to the equation [33-34]:

B  (C11  2C12 ) / 3

(3)

For determining coefficient C44, the following monoclinic strain tensor is used.  1    2  0 

 2 1 0

    0  4  (4   2 )  0

(4)

This changes the total energy to: 1 E ( )  E ( )  E (0)  C44V0 2  [ 4 ] 2

(5)

Combining equations (3) and (4), one can easily determine the two elastic constants C11 and C12, while the third elastic constant, of equation (6), is deduced by C44 from the elastic constants. We obtain the anisotropy parameter A (for an isotropic crystal, A equals 1, while another value greater or less than 1 means that it is an anisotropic crystal). Poisson’s ratio ν characterizes the tension of the solid perpendicular to the direction of the force applied, Young’s modulus E measures the resistance change from solid to its length expression and the shear modulus G measures the interior of the solid with planes parallel to the later which is represented by the following equations:

A

2C 44 C11  C12

(6)



3B  E 6B

(7)

E

9 BG 3B  G

(9)

5

G

C11  C12  3C 44 5

(10)

where B is the bulk modulus given by equation (3). From these results, we find that the stability criteria [35-36]; C11-C12>0, C11>0, C44>0, (C11+2C12)>0 and C12
Table 2 lists the elastic constant (C11, C12,and C44), bulk modulus B, shear modulus G and Young’s modulus E, Poisson’s ratio ν, the anisotropic parameter A and B/G rations of XFe4 P12(X=Tb,Dy) compounds under a broad of pressures (0-50 GPa) at 0 K using

LDA

approximation. To investigate the elastic constants, it is seen that the elastic constants C11, C12, C44 and bulk modulus B, increase monotonically with the applied pressure (see Fig. 2). The results from, G and E of XFe4 P12 (X=Tb,Dy) compounds at various pressures are also calculated and the results are depicted in Table 2, the results indicate that the values of G and E increase with increasing pressure. A typical Poisson’s ratio value for covalent materials is about 0.1, whereas that for ionic materials is about 0.25 [37]. In this work, Poisson ratio ν of XFe4 P12 (X=Tb,Dy) alloys at various pressures are around 0.1. This indicates that this compound is highly covalent. One of the most important property of crystalline solids is the elastic anisotropy ratio which is defined as A = 2C44/ (C11-C12). This property has an important implication in engineering field since it is highly correlated with the possibility to introduce micro crack in materials [38]. Essentially, all known crystals are elastically anisotropic. For isotropic crystals «A» equals 1.0 while any value smaller or larger than 1.0 indicates anisotropy. The magnitude of a deviation from 1.0 measures the degree of elastic anisotropy possessed by the crystal. The obtained anisotropic «A» values vary from 0.8684 to 1.3534, indicates that this compound is anisotropy. According to the empirical formula of Pugh of ductility [39], which states that the critical value which separates ductile and brittle 6

materials is around 1.75; if B/G >1.75, the material behaves in a ductile manner; otherwise the material behaves a brittle manner. The computed values of ratio B/G vary from 0.9065 to 1.0712 at various pressures, indicates that this compound is classified as brittle material. Eventually; The elastic constant C11, C12, C44 (in GPa ), bulk modulus B (in GPa),Young’s and shear modulus E,G (in GPa), Poisson’s ratio ν, the anisotropic parameter A and B/G values are similar for two compounds . The elastic constants give important information about the binding characteristic between adjacent atomic planes, anisotropic character of binding structural stability [40]. The requirement of mechanical stability criteria in this skutterudites structure leads to following restrictions on the elastic constants C11-C12>0, C11>0, C44>0, (C11+2C12)>0 and C12
2.2. Debye temperature Debye temperature calculated from elastic constants is the same as that determined from specific measurement. One of the standard methods to calculate the Debye temperature θD, is from the elastic constant data; also θD could be obtained from the average sound velocity,

,

by the following relation [41]:

=

(11)

where h is Plank’s constant, kB is Boltzmann’s constant and

is the atomic volume. The

average sound velocity in the polycrystalline material is given by [42]:

= where

(12) and

are the longitudinal and transverse sound velocities obtained using the shear

modulus G and the bulk modulus B from Navier’s equation [43]: 7

=

(13)

=

(14)

The obtained sound velocities and Debye temperature as well as the density for XFe4P12 (X=Tb,Dy) alloys at different pressures using LDA approximation are given in Table 3. The ensues show that the values of νl, νt, νm and θD increase with increasing pressure. We have found at θD (TbFe4P12) > θD(DyFe4P12), the decrease of the Debye temperature in the sequence (Tb→Dy) can be attributed to the progressive decreasing of average sound velocity νm in the same sequence. Up to date, there are no experimental and theoretical available data for comparison.

3. Results and discussion 3.1. Structural properties The skutterudite structure is characterized by two non-equivalent atomic positions y and z, which are not fixed by symmetry. The y and z positions have been optimized by minimizing the total energy with keeping the volume fixed at the experimentally observed value [44]. The optimized values of y and z are used to calculate the total energy at different unit-cell volume. The atomic positions in the ternary skutterudites TbFe4P12 and DyFe4P12 are listed in Table 4. Figure 3 indicates the computed total energy of TbFe4P12 and DyFe4P12 as a function of unit cell volume has been calculated for the local spin density approximation LSDA and local density approximation LDA. The results show that the total energy in the LSDA is lower than that of the LDA for both compounds. The curve of the total energy versus unit cell volume is fitted to Murnaghan’s equation of state (EOS) to define the ground state properties like the equilibrium lattice constant a0, the bulk modulus B0 and its pressure derivative B′ [45]. The calculated lattice constant a, bulk modulus B0 and its pressure derivative B’0 of TbFe4P12 and 8

DyFe4P12 alloys at 0 GPa and 0 K are listed in Table 5. It is seen that, the lattice constant values are similar for two compounds, which can be easily explained by considering the atomic radii and it indicates the lattice constant of TbFe4P12 and DyFe4P12 are underestimated compared to the experimental data. This underestimation is generally attributed to LSDA and LDA. The obtained lattice constants for both compounds are in good agreement with experimental data. Our calculated lattice constants for TbFe4P12 and DyFe4P12 compounds using LDA and LSDA are 1.98% and 1.92% for LDA and 0.80% and 0.97% for LSDA, respectively smaller than corresponding experimental values. Up to date, no experimental or theoretical available data for the bulk modulus and its pressure derivatives are available in the literature. Finally, the equilibrium lattice constants and bulk modulus of TbFe4P12 almost agree with those of DyFe4P12.

3.2. Electronic properties Figure 4 shows the energy band structures for XFe4P12 (X=Tb,Dy) alloys at the equilibrium lattice constant within LSDA approximation. This is drawn along the symmetry in the first Brillouin zone. These spectra are calculated for spin-up and spin-down within the LSDA; as can be seen, the conduction-band and valence-band states overlap and they have no forbidden energy gap at the Fermi level in both spin up and spin down. Therefore, these compounds have a metallic behavior within LSDA. In the first structure (a) of TbFe4P12, the valence band is essentially dominated by the Tb-4f, Fe-3d and Tb-5d below the Fermi energy with minor contributions from the others states of the elements forming the compound. The region including the Fermi energy arises predominately from the Fe-3d and Tb-5d. The part just beyond Fermi level energy is basically dominated by Tb-5d, 4f states. In The second structure (b) of DyFe4P12, the valence band is essentially dominated by the Fe-3d, P-3P and Tb-5d below the Fermi energy with minor contributions from the others states of the elements

9

forming the compound. The region including the Fermi energy arises predominately from the Fe-3d and Tb-5d, 4f. The part just beyond Fermi level energy is basically dominated by Tb4f, 5d states. To stand further information electronic states, which form the band structures, we have too computed the total and the partial densities of states (DOS) of these compounds. These are exhibited in Fig. 5 for XFe4P12 (X= Tb, Dy) compounds. In The first structure (a) of TbFe4P12 the valence bands have a character of hybridized Tb-4f, Fe-3d and Tb-5d states. To below Fermi level energy the valence band is primarily due to the P-3p states with a few contributions of Tb-6s. The region around the Fermi level is essentially dominated by Fe-3 and Tb-4f states with an admixture of P-3p, 2s states. The part just beyond Fermi level energy is basically dominated by Fe-3d and Tb-5d, 4f states, with small contributions from P-3d, 3p states. While, in the second structure (b) of DyFe4P12, the valence bands have a character of hybridized Fe-3d states. Below Fermi level, the valence band is primarily due to the P-3p and Dy-6s, 5p states with a few contributions of Dy-4f. The region around the Fermi level is essentially dominated by Fe-3d, Tb-5d states with an admixture of, Tb-4f, Dy-5p-5d states and P-3p states. The part beyond Fermi level is basically dominated by Fe-3d, Dy-6s and P-3p states, with small contributions from, Fe-4s and P-3d states. However, the total and partial densities of states (DOS) of TbFe4P12 are different to those of DyFe4P12. The spin interaction has significant influence on the band structures because the split of f-d band show the ferromagnetic nature of this type of compounds.

3.3. Thermodynamic properties The thermal ensues in this work was done inside the quasi-harmonic Debye model used in Gibbs program [46-47], which permits to value the Debye temperature. To find the Gibbs free energy G*(V, P, T) [48]: G*(V, P, T)=E(V) +PV+

[θ(V); T]

(15)

10

For a solid depicted by an energy-volume (E-V) relationship in the static approximation, was expressed out and fitted with a numerical EOS in prescribe to find its structural parameters at P = 0 and T= 0, and then derive the macroscopic properties as function of P and T from standard thermodynamic relations [49], where E (V) is the total energy for XFe4P12 (X=Tb,Dy), PV is the hydrostatic pressure condition, θ (V) is the Debye temperature, and is the Helmholtz free energy which can be written as [50-51]:  D  9    vib D ,T   n B  D  3ln 1  e T   D  D        8

16)

where n is the number of atoms per formula unit, D (θ/T) is the Debye integral. The Debye temperature θ is meant as [51]:

D =



1  B 6 2 nV 2   



1

3

f  

(17)

BS 

where M is the molecular mass per unit cell and BS is the adiabatic bulk modulus, which is approximately given by static compressibility [46]:

BS  B V   V

d 2 E V 

18)

dV 2

f(σ) is given by eq. (19) [52-51]: f (σ)=

(19)

where σ is Poisson ratio. Therefore, the non-equilibrium Gibbs function G*(V, P, T) as a function of (V, P, T) can be minimized with respect to volume V as:

 G V ,,     0  V   P ,T

(20)

The thermal equation-of-state (EOS) V (P, T) can be obtained by solving eq. (20). The isothermal bulk modulus BT is given by [46]: 11

BT(P,T)=V

(21)

The thermodynamic quantities, like heat capacities CV at stable volume, CP [53] at stable pressure, entropy S, and internal energies have been calculated by applying the following relations [46]:

   CV  3nK B  4 D  D  T 

 3 D       T   (22)   D  e T  1 

CP= Cv (1+αγT)

(23)

S= nK [4D (θ/T) -3ln (1-e-θ/T)]

(24)

U= nkT

(25)

where α is the thermal expansion coefficient and γ is the Grüneissen parameters which are given by following equations [46]:



 CV BT T

(26)

   dIn  DV  dInV

(27)

In this work, the prediction of thermal properties at pressure 0-50 GPa and temperature between 0-3000 K are obtained for XFe4P12 (X=Tb,Dy) alloys using LDA approximation. The fluctuation of the volume is shown in Fig. 6. It is indicates that the volume increases nearly almost linearly with increasing temperature, but the rate is more important for temperature above 100 K. For a given temperature, the primitive cell volume decreases with increasing pressure. These results are due to the fact that the effect of increasing pressure of the material is the same as the decreasing temperature of the material. The calculated equilibrium primitive cell volume V at zero pressure and room temperature are 800.5812 Å and 765.1687 Å for TbFe4P12 and DyFe4P12, respectively. Figure. 7 shows the bulk modulus 12

variation against temperature at a donated pressure. It is seen that the bulk modulus is almost constant from 0 to 100 K and decrements additively with raising temperature from T > 100 K.

The compressibility increments with raising temperature at a donated pressure and decrements with pressure at a donated temperature. These effects are owing to raising pressure on the material is the same as the diminishing temperature of the material. The fluctuation of the heat capacities CV against temperature at pressure 0-50 GPa is depicted in Fig. 8. We clearly see that when T <1000 K, the heat capacity CV is a function of the temperature and pressure. If the temperature is constant, the heat capacity CV decrements with pressure. We also note that for high temperature, the heat capacity CV does not depend on the temperature and tends to the limit of Dulong-Petit [52], that bears the value of 422.49 (J .mol-1.K-1). The fluctuation of the heat capacity CP at pressure 0-50 GPa and temperature between 0-3000 K for XFe4P12 (X=Tb,Dy) alloys is depicted in Fig. 9. At a given temperature, we can see the variation of the heat capacity Cp is alike to those of Cv exactly at low temperatures but at high temperature, Cp behaves other wise of Cv, it does not tend to a constant value. The result of pressure on the heat capacity at constant pressure Cp is alike to that of Cv. Figure 10 indicates the variations of the volume expansion coefficient α as a function of temperature. At a given pressure, we can see that α varies for low temperatures and gradually tends to increase linearly with increasing temperature, with a growth rate that is low, which leads to say that the dependence of α with temperature is lower for higher temperatures. At zero pressure, the variation of the thermal expansion coefficient with temperature is more important. For a given temperature, the thermal expansion coefficient decreases with increasing pressure. Figure 11 shows the variation of the entropy S with the temperature and pressure For XFe4P12 (X=Tb,Dy) alloys. For a given pressure, the entropy S increases strongly with increasing temperature and decreases with increasing pressure at a given temperature. When the temperature increases,

13

the main vibration contributions to the entropy increases and therefore the entropy increases with the temperature. The fluctuations of the primitive cell volume, expansion coefficient α, bulk modulus B, heat capacity ( and

), Debye temperature θD, and entropy S at pressure 0-

50 GPa and temperature between 0-3000 K of TbFe4P12 almost agree with those of DyFe4P12.

4. Conclusion Using the FP-LMTO approach based on the DFT within LDA and LSDA, we investigated the physical properties of filled skutterudite XFe4P12 (X=Tb,Dy) alloys. The main conclusions can be summarized as follows:

 The equilibrium lattice constants and bulk modulus of TbFe4P12 almost agree with those of DyFe4P12.  The computed spin-polarized band structure for XFe4P12 (X=Tb,Dy) alloys exhibit metallic character.  The total and the partial densities of states (DOS) of TbFe4P12 are different to those of DyFe4P12.  The elastic constant C11, C12, C44 (in GPa ), bulk modulus B (in GPa), Young’s and shear modulus E,G (in GPa), Poisson’s ratio ν, the anisotropic parameter A and B/G values are similar for two alloys.  The calculated Poisson ratio ν of XFe4P12 (X=Tb,Dy) at various pressures are around 0.1. This indicates that this alloys is highly covalent.  By analyzing the B/G ratio, we conclude that XFe4P12 (X=Tb,Dy) classify as brittle material.  The anisotropy factor suggests that XFe4P12(X=Tb,Dy) compound exhibits anisotropic elasticity.

14

 The sound velocities (vl,vt and vm) and the Debye temperature of TbFe4P12 are larger than that of DyFe4P12.  The fluctuations of the primitive cell volume, expansion coefficient α, bulk modulus B, heat capacity ( and

), Debye temperature θD, and entropy S, values are similar

for both alloys.  There are no experimental and other theoretical data for comparison, so we consider the present results as a prediction study for the first time, which still awaits an experimental confirmation.

Acknowledgments Y.A. would like to thank University Malaysia Perlis for grant No. 9007-00185 and TWASItaly for the full support of his visit to JUST-Jordan under a TWAS-UNESCO Associateship.

15

References [1] W. Jeitschko, D. Braun "LaFe4P12 with filled CoAs3-type structure and isotypic lanthanoid-transition metal polyphosphides" Acta Crystallogr. B 33 (1977) 3401-3409 [2] I. Shirotani, J. Hayashi, T. Adachi, C. Sekine,T. Kawakami, T. Nakanishi, H. Takahashi, J. Tang,A. Matsushita, T. Matsumoto, "Metal to insulator transition of filled skutterudite PrRu4P12 at low temperatures and high pressures" Physica B 322 (2002) 408-415 [3] I. Shirotani,"Superconductivity of Ternary Metal Compounds Prepared at High Pressures" Bulletin of the Chemical Society of Japan Vol. 76 (2003) 1291-1307 [4] I. Shirotani, T. Uchiumi, K. Ohno, C. Sekine,Y. Nakazawa, K. Kanoda, S. Todo, T. Yagi, "Superconductivity of filled skutterudites LaRu4As12 and PrRu4As12 "Phys. Rev.B 56 (1997) 7866-7876 [5] A. Miyake, K. Shimizu, C. Sekine, K. Kihou, I. Shirotani," Pressure-induced superconductivity in filled skutterudite PrRu4P12"J. Phys. Soc. Jpn. 73 (2004) 2370-2377 [6] I. Shirotani, T. Uchiumi, C. Sekine, M. Hori,S. Kimura, N. Hamaya, J. Solid State Chem." Electrical and Magnetic Properties of La1-xCexRu4P12 and CeOs4P12 with the Skutterudite Structure" 142 (1999) 146-155 [7] C. Sekine, T. Uchiumi, I. Shirotani, T. Yagi, "Metal-Insulator Transition in PrRu4P12 with Skutterudite Structure" Phys. Rev.Lett. 79 (1997) 3218-3225 [8] K. Matsuhira, Y. Doi, M. Wakeshima, Y. Hinatsu,H. Amitsuka, Y. Shimaya, R. Giri, C. Sekine, I. Shirotani"Specific Heat Study on Sm-based Filled Skutterudite Phosphides SmT4P12 (T =Fe, Ru and Os) " J. Phys. Soc. Jpn. 74 (2005) 1030-1035 [9] M. E. Danebrock, B. Christroph, H. Evers,W. Jeitschko," Magnetic properties of alkaline earth and lanthanoid iron antimonides AFe4Sb12 (A = Ca, Sr, Ba, La--Nd, Sm, Eu) with the LaFe4P12 structure" J. Phys. Chem. Solids 57 (1996) 381-390 [10] E. D. Bauer, A. Slebarski, E. J. Freeman, C. Siirvent,M. B. Maple," Kondo insulating behaviour in the filled skutterudite compound CeOs4Sb12" J.Phys.:Condens. Matter 13 (2001) 4495-4505 [11] N. R. Dilley, E. J. Freeman, E. D. Bauer, M.B. Maple"Intermediate valence in the filled skutterudite compound YbFe4Sb12", Phys. Rev. B 58 (1998) 6287-6299 [12] I. Shirotani, J. Hayashi, K. Takeda, R. Nakada,Y. Ohishi," X-ray study with synchrotron radiation for filled skutterudite YbFe4P12 at ambient and high pressures" Physica B, 382 (2006) 8-15

16

[13] N. Takeda, M. Ishikawa,"The effect of La substitution and magnetic field on non-Fermiliquid behaviour in CeRu4Sb12" J. Phys.: Condens. Matter 13 (2001) 5971-5988 [14] B. C. Sales, D. Mandrus, R. K. Williams," Filled Skutterudite Antimonides: A New Class of Thermoelectric Materials" Science 272 (1996) 1325-1333 [15] C. Sekine, K. Akita, N. Yanase, I. Shirotani, I. Inagawa,C. H. Lee," Thermoelectric Properties of CeRu4P12 and CeOs4P12 with Filled Skutterudite-Type Structure " Jpn. J. Appl. Phys. 40 (2001) 3326-3333 [16] T. Caillat, A. Borshchevsky, J.P. Fleurial, Proceedings of the 11th International Conference on thermoelectrics, Arlington, USA, 7–9 October, 1992, p. 3; G.J. Snyder, T. Caillat, Using The Compatibility Factor To Design High Efficiency Segmented Thermoelectric Generators Materials Research Society Symposium Proceedings 793 (2003) 37-45 [17] K. Nouneh, Ali H. Reshak, S. Auluck , I.V. Kityk ,R. Viennois , S. Benet, S. Charar "Band energy and thermoelectricity of filled skutterudites LaFe4Sb12 and CeFe4Sb12" Journal of Alloys and Compounds 437 (2007) 39–46 [18] B. C. Sales, Handbook on the Physics and Chemistryof Rare Earths, edited by K. A. Gschneidner (Jr.), J.-C. G. Bunzli, V. K. Pecharsky (Elsevier Science, 2003) 33, p. 1. [19] I. Shirotani, Y. Shimaya, K. Kihou, C. Sekine, T. Yagi," Systematic high-pressure synthesis of new filled skutterudites with heavy lanthanide, LnFe4P12 (Ln=heavy lanthanide, including Y" J. Solid State Chem. 174 (2003) 32-45 [20] I. Shirotani, N. Araseki, Y. Shimaya, R. Nakata, K. Kihou,C. Sekine, T. Yagi," Electrical and magnetic properties of new filled skutterudites LnFe4P12 (Ln = Ho, Er, Tm and Yb) and YRu4P12 with heavy lanthanide (including Y) prepared at high pressure" J. Phys.: Condens. Matter 17 (2005) 4383-4395 [21] Ichimin Shirotania, Keiki Takedaa, Chihiro Sekinea, Junichi Hayashia, Ryusuke Nakadaa, Kunihiro Kihoua, Yasuo Ohishib, and Takehiko Yagic "High-pressure Synthesis and Structural, Electrical and Magnetic Properties of a New Filled Skutterudite TbFe4P12 " Z. Naturforsch. 61b (2006) 1471 – 1476 [22] I. Shirotani, Y. Shimaya, K. Kihou, C. Sekine, T. Yagi," Systematic high-pressure synthesis of new filled skutterudites with heavy lanthanide, LnFe4P12 (Ln=heavy lanthanide, including Y" J. Solid State Chem. 174 (2003) 32-45 [23]: Caillat, T. Fleurial, J.P. ; Borshchevsky, A. « Skutterudites for thermoelectric applications » 10.1109/ICT.1996.553265.

17

[24] S. Savrasov, D. Savrasov, Full-potential linear-muffin-tin-orbital method for calculating total energies and forces, Phys. Rev. B 46 (1992) 12181–12195 [25] S.Y. Savrasov, Linear-response theory and lattice dynamics: a muffin-tin-orbital approach, Phys. Rev. B 54 (1996) 16470–16486 [26] J.P.Perdew,«density-functional approximation for the correlation energy of the inhomogeneous electron gas». Phys. Rev. B 33 (1986) 8822-8824 [27] J.P.Perdew,Y.Wang,«Accurate and simple analytic representation of electron-gaz correlation energy».Phys. Rev. B 45 (1992) 13244–13249 [28] J.P. Perdew, Y. Wang. « Pair-distribution function and its coupling-constant average for the spin-polarized electron gas ».Phys. Rev. B 46 (1992) 12947-12955 [29] Blochl P,jepsen O ,Andersen OK.«improved tetrahedron method for Brillouin-zone integrations». Phys. Rev B. 49 (1994) 16223-16233 [30] Ashcroft NW,mermin ND.«solid state physics.philadelphia,PA:saunders college» 1976. [31] Mehl MJ.Osburn JE Paraconstantantopoulos DA .Klein BM. «Structural properties of ordered high-melting-temperature intermetallic alloys from first-principles total-energy calculations».Phys. Rev B 41 (1990) 10311-10323 [32]Rached H,Rchad D , Khenata R,Reshak AH ,Rabah M.«First-principles calculations of structural, elastic and electronic properties of Ni2mNz(Z=Al,Ga and In ) heusler alloys».Phys Stat Sol (b) 246 (2009) 1580-1586 [33] Schreiber E,Anderson OL , Soga N. «Elastic constants and their measurement. New York: Mc Graw-Hill» 1973. [34] Nakanishi Y,Yamaguchi T,Hazama H,Goto T.Matsuda TD ,Sugawara H.Sato H.Yoshizawa M.«Elastic properties of filled skutterudite compounds».J Phys Soc Jpn. 71 (2002) 249-172 [35] Born M, «On the stability of crystal lattices».Proc Cambridge Philos Soc. 36 (1940) 160172 [36] Born M,Huang K,In:C larendon,editor.«Dynamical theory of crystal lattices». OxfordUniversity Press; 1956. [37] Haines J,Leger JM, Bocquillon G.«Synthesis and design of super hard materials.Annu» Rev Mater Res. 31 (2001) 1-23 [38] V. Tvergaard and J. W. Hutchinson, «Microcracking in Ceramics Induced by Thermal Expansion or Elastic Anisotropy» J. Am. Ceram. Soc. 71 (1988) 157-166 [39] P ugh SF. «Predicted studies of semiconductors ».Philos Mag. 45 (1954) 823-843

18

[40]S. Benalia, M.Hachemaoui, D.Rached, , R.Khenata, , N.Bettahar, M.Benyahia “FPLMTO calculations of elastic and electronic properties of the filled skutterudite CeRu4P12” Journal of Physics and Chemistry of Solids 70 (2009) 622–626 [41] Wachter P, Filzmoser M,Rebizant J .Electronic and elastic properties of the light actinide tellurides.Physica B. 93 (2001) 199.223 [42] Voigt W.Semiconductors and semimetals. «Lehrbuch der K ristall-physik». Leipzing: Taubner; 1929. [43] Schreiber E, Anderson OL,Soga N.«Elastic constants and their measurements». New York : Mc Graw-Hill; 1973. [44]:M. Ameri , K.Boudia , R.Khenata , B.Bouhafs , A.Rais , S.Bin Omran , B. Abidri , Y.AlDouri “Structural, elastic, electronic and thermodynamic properties of the filled skutterudite CeOs4Sb12 determined by density functional theory” Materials Science in Semiconductor Processing 16 (2013) 1508–1516 [45] Murnaghan FD.«the compressibility of media under extreme pressures».Proc Natl Acad Sci USA. 30 (1944) 244-247 [46] M.A. Blanco, E. Francisco, V. Luaña, Comput.« GIBBS: isothermal-isobaric thermodynamics of solids from energy curves using a quasi-harmonic Debye model» Phys. Commun. 158 (2004) 57-66 [47] F. Peng, H.Z. Fu, X.D. Yang, « Ab initio study of phase transition and thermodynamic properties of PtN» Physica B 403 (2008) 2851-2859 [48]:A. Bouhemadou, Y. Al-Douri, R. Khenata, K. Haddadi «Structural, elastic, electronic, optical and thermal properties of c-SiGe2N4 »The European Physical Journal B 71 (2009) 185-194 [49]: A. Bouhemadou, K. Haddadi “Structural, elastic, electronic and thermal properties of the cubic perovskite-type BaSnO3 “Solid State Sciences 12 (2010) 630–636 [50] M.A. Blanco, A. Martín Pendás, E. Francisco, J.M. Recio, R. Franco. « Thermodynamical properties of solids from microscopic theory: applications to MgF2 and Al2O3» J. Mol. Struct. (Theochem) 368 (1996) 245-259 [51] E. Francisco, M.A. Blanco, G. Sanjurjo, « Atomistic simulation of SrF2 polymorphs » Phys. Rev. B 63 (2001) 094107-094115 [52] E. Francisco, J.M. Recio, M.A. Blanco, A. Martín Pendás, A. Costales. « Quantummechanical study of thermodynamic and bonding properties of MgF2» J. Phys.Chem. 102 (1998) 1595-1604

19

[53] N Chouit, S Amara Korba, M Slimani, H Meradji, S Ghemid, and R Khenata “Firstprinciples study of the structural, electronic and thermal properties of CaLiF3” Physica Scripta 55 (2013) 257-269

20

Table 1. The number of plane wave (NPW), energy cut-off (in Ryd) and the MTS (in a.u), used in our calculations. NLW LDA Ecut-off (Ryd) LDA MTS ( a.u.)

Tb

LDA

3.415

TbFe4P12

DyFe4P12

34412

34412

122.3970 Fe 2.038

P

Dy

2.121

3.415

122.3817 Fe 2.039

P 2.122

Table 2. Calculated elastic constant c11,c12,c44 (in GPa ),bulk modulus B (in GPa),Young’s and shear modulus E,G(in GPa),Poisson’s ratio ν ,the anisotropic parameter A and B/G rations with LDA at different pressures for XFe4P12 (X=Tb,Dy). TbFe4P12 P(GPa)

c11

c12

c44

B

G

E

ν

A

B/G

0 10 20 30 40 50

522.2429 573.6014 630.0745 680.5807 734.9227 784.1502

39.4565 64.8641 89.7600 115.5780 139.9733 164.8748

210.4228 253.0234 296.3106 335.6153 376.4784 419.0729

201.9795 235.1000 269.6700 304.3400 338.9900 373.6000

222.8100 253.5615 285.8493 314.3697 344.8769 375.2980

488.7234 559.2900 633.8405 701.5518 772.6188 843.4635

0.0967 0.1033 0.1086 0.1158 0.1201 0.1237

0.8717 0.9947 1.0968 1.1880 1.2655 1.3534

0.9065 0.9271 0.9433 0.9680 0.9829 0.9954

DyFe4P12 P(GPa)

c11

c12

c44

B

G

E

ν

A

B/G

0 10 20 30 40 50

523.3084 575.7322 630.5807 680.5807 731.9392 784.1502

45.6000 69.6767 94.6084 119.5418 145.9136 170.3676

207.4320 241.4224 276.7314 311.3079 345.1518 379.728

205.1082 238.4200 272.7300 307.0100 341.3100 375.5800

220.0009 246.0645 273.2333 298.9925 324.2962 350.5933

486.1768 549.2426 614.4915 677.1541 738.8744 802.1769

0.1049 0.1160 0.1244 0.1303 0.1391 0.1440

0.8684 0.9541 1.0326 1.1097 1.1779 1.2373

0.9323 0.9689 0.9981 1.0268 1.0524 1.0712

21

Table 3. Longitudinal, transverse and average sound velocities (νl,νt,νm) respectively, (in ms-1), and Debye temperature(θD, in K) with LDA at different pressures for the XFe4P12 (X=Tb,Dy).

P(GPa) 0 10 20 30 40 50

-1

νl(ms ) 9711.06400 10407.2600 11092.1200 11692.5500 12286.1600 12851.2500 -1

P(GPa)

νl(ms )

0 10 20 30 40 50

9675.839 10315.3300 10938.6700 11512.800 12055.0400 12583.5908

TbFe4P12 νt(ms-1) νm(ms-1) 6488.7100 6922.0020 7349.5140 7707.4450 8072.7620 8421.2910

7091.3530 7569.3750 8040.7700 8437.859 8841.3390 9226.1330

DyFe4P12 νt(ms-1) νm(ms-1) 6428.2560 6798.381 7163.8720 7493.9560 7804.6230 8114.8930

7030.4690 7442.8150 7849.0970 8216.8830 8563.0860 8907.6640

θD(K) 438.6259 468.1932 497.3507 521.9121 546.8688 570.6697

θD(K) 435.3193 460.8513 486.0079 508.7808 530.2173 551.5533

Table 4. The atomic positions in the ternary skutterudite XFe4P12(X=Tb, Dy). Atom X=Tb,Dy Fe P

Atomic positions (0, 0,0) (1/4,-1/4,1/4) (1/4,1/4,-1/4) (-z, 0, y) (o,-y,-z) (y, z, o) (0,-y, z) (y,-z, 0) (-z, 0,-y)

(-1/4,1/4,1/4) (0, y, z) (0, y,-z) (z, 0, y)

(1/4,1/4,1/4) (z, 0,-y) (-y,-z, 0) (-y, z, 0)

Table 5. Lattice constant a0 (in Å),pnicogen atom-free internal parameters y and z, bulk modulus B (in GPa), its pressure derivative B’and minimum energy at equilibrium E0(in Ryd) ( experimental data for comparison). TbFe4P12 Present LDA LSDA Exp.[13] DyFe4P12 Present LDA LSDA Exp.[13]

a0

y

z

B

B’

E0

7.638 7.729 7.792a

0.353 0.353

0.1501 0.1501

201.579 263.038

3.472 3.8898

-41771.6007 -41771.9624

7.639 7.713 7.7891a

0.353 0.353

0.1501 0.1501

205.10824 281.12327

3.438 3.005

-42669.9124 -42669.9869

a

Ref [19]

22

Figure captions Figure 1: Crystal structure of filled skutterudite XFe4P12(X=Tb,Dy) Figure 2: The pressure dependence of the elastic constants Cij and the bulk modulus B of XFe4P12 (X=Tb,Dy). Figure 3: Total energy versus volume curves for XFe4P12(X=Tb, Dy) within LSDA and LDA. Figure 4: The band structure of XFe4P12(X=Tb,Dy) at the equilibrium structure using LSDA approximation. Figure 5: The total and partial densities of states of XFe4P12(X=Tb,Dy): at the equilibrium structure using LSDA approximation. Figure 6: The variation of the primitive cell volume as a function of temperature for XFe4P12 (X=Tb,Dy).at different pressures. Figure 7: The variation of the bulk modulus B as a function of temperature for XFe4P12 (X=Tb,Dy).at different pressures. Figure 8: The variation of the heat capacity Cv as a function of temperature for XFe4P12 (X=Tb,Dy):at different pressures. Figure 9: The variation of the heat capacity CP as a function of temperature for XFe4P12 (X=Tb,Dy) :at different pressures. Figure 10: The Variation of the thermal expansion coefficient as a function of temperature for XFe4P12(X=Tb,Dy).at different pressures. Figure 11: The Variation of entropy as a function of temperature for XFe4P12(X=Tb,Dy) at different pressures.

23

Figure 1

24

Figure 2

25

Figure 3

26

Figure 4

27

Figure 5

28

Figure 6

29

Figure 7

30

Figure 8

31

Figure 9

32

Figure 10

33

Figure 11

34