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Theoretical prediction of the structural, elastic, electronic and thermodynamic properties of V3M (M = Si, Ge and Sn) compounds
Superlattices and Microstructures 52 (2012) 697–703
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Theoretical prediction of the structural, elastic, electronic and thermodynamic properties of V3M (M = Si, Ge and Sn) compounds T. Chihi a, M. Fatmi b,c,⇑ a
Laboratory for Elaboration of New Materials and Characterization (LENMC), University of Ferhat Abbas, Setif 19000, Algeria Research Unit on Emerging Materials (RUEM), University of Ferhat Abbas, Setif 19000, Algeria c Laboratory of Physics and Mechanics of Metallic Materials (LP3M), University of Ferhat Abbas, Setif 19000, Algeria b
a r t i c l e
i n f o
Article history: Received 14 May 2012 Accepted 17 June 2012 Available online 23 June 2012 Keywords: Ab initio calculation Electronic structure Structural properties
a b s t r a c t Density functional theory (DFT), is used in our calculations to study the V3M (M = Si, Ge and Sn) compounds, we are found that V3Sn compound is mechanically unstable because of a negative C44 = 19.41 GPa. For each of these compounds considered, the lowest energy structure is found to have the lowest N(Ef) value. Also there is a strong interaction between V and V, the interaction between M (M = Si, Ge, Sn) and V (M and M) is negative, not including Si [Sn]. In phonon density of states PDOS, the element contributions varies from lighter (high frequency) to heaviest (low frequency). Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction An intermetallic compounds, as it applies to solid phases, is a phase which crystallizes with a structure other than those of its components. Intermetallic phases containing more than one element, ordered into different sites in the solid solutions. Intermetallic compounds have been attracted candidates for high temperature structural materials because of their desirable intrinsic properties, generally brittle and high melting point. There have been many investigations carried out on various compounds, but mostly on structures that are ordered from of fcc, bcc and hcp metals.
⇑ Corresponding author at: Research Unit on Emerging Materials (RUEM), University of Ferhat Abbas, Setif 19000, Algeria. Fax: +213 35 63 05 60. E-mail addresses: [email protected] (T. Chihi), [email protected] (M. Fatmi). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.06.009
698
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
Silicides of transition metals have attracted considerable interest due to their desirable properties for application involving high temperatures environment. Among these silicides, the intermetallic compound Cr3Si is a typical example since it has a high melting point (2043 K) [1], good creep resistance, good oxidation resistance, high stiffness, and a comparatively low density. Therefore, though lacking in room temperature toughness [2], it has been recognized as a promising candidate for high temperature structural application in aggressive environment [3], especially in the field of aircraft engine material [4]. Yue et al. [5] was thermodynamically modeled the Sn–V binary system using the CALPHAD approach combined with first-principles calculations. V3Si [6] exhibit a cubic to tetragonal phase transition at low temperature. Delaire [7] studied by the inelastic neutron scattering and firstprinciples computation, the interactions of high-temperature electron–phonon of V3X (X = Si, Ge and Co) compounds. In the present work we are interested in cubic V3M (M = Si, Ge and Sn) compounds. They present a Pm–3n space group. The paper is organized as follows. The computational method is described in Section 2. In Section 3, the results are presented and compared with available experimental and theoretical data. Conclusion is given in Section 4.
2. Computational details Cambridge serial total energy package (CASTEP) [8], a first-principles plane wave pseudo potential (PWPP) method based on the density functional theory (DFT), is used in our calculations. CASTEP uses a plane-wave basis set for the expansion of the single-particle Kohn–Sham wave-functions, and pseudopotentials to describe the computationally expensive electron–ion interaction, in which the exchange-correlation energy by the generalized gradient approximation (GGA) of Perdew et al. is adopted for all elements in our models by adopting the Perdew–Burke–Ernzerhof, known as PBE parameters [9]. The atomic orbitals used in the present calculations are: in which the V (3d3 4s2), Si (3p2 3s2), and Ge (3d10 4p2 4s2) orbitals are treated as valence electrons. According to an ultrasoft condition, the pseudo-wave function which is related to the pseudo potential matches the plane- expanded with beyond cut-off energy. Using high cut-off energy, at the price of spending long computational time, can actually provide accurate results. The cut-off energy for the plane-wave expansion was set at 330 eV and the Brillouin zone sampling was carried out using the 6 6 6 set of Monkhorst–Pack mesh [10]. Atomic positions are relaxed and optimized with a density mixing scheme using the conjugate gradient (CG) method for eigenvalues minimization. The equilibrium lattice parameter is then computed from a structural optimization, using the Broyden–Fletcher–Goldfarb–Shenno (BFGS) minimization technique. This technique provides a fast way of finding out the lowest energy structure out of all the converged structures, with the following thresholds: energy change per atom less than 2 105 eV, residual force less than 0.05 eV/Å, displacement of atoms during the geometry optimization less than 0.002 Å, and maximum stress within 0.1 GPa. The crystal structures are reported (Table 1).
Table 1 Calculated V3Si, V3Ge and V3Sn structures: lattice parameters in Å; volumes per formula unit (f.u.) in Å3; Z is the number of formula units in the unit cell, and the space group and compared with experimental data.
a b c
Space group
a
V
Z
V3Si
Pm–3n
104.52
2
V3Ge
Pm–3n
108.36
2
V3Sn
Pm–3n
4.71 4.71a 4.767 4.760b 4.984 4.953c
123.79
2
Ref. [11]. Ref. [12]. Ref. [5].
699
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
3. Results and discussion 3.1. Structural properties The results for lattice parameters a are reported in Table 1 and compared with experimental and previous theoretical calculations. Our calculated value for lattice parameter a of V3M (M = Si, Ge and Sn) compounds are in excellent agreement with the experimental data [11,12,5] The relative deviation may be attributed to the fact that our calculation corresponds to the perfect bulk material at zero temperature, whereas the experimental sample was synthesized at high temperature. The total energies (stability) of V3M (M = Si, Ge and Sn) compounds increase, the nearest-neighbor distance DV–M and DV–V in these compounds increase and the bulk modulus decrease for M metals in the same column in the periodic table, with increasing atomic number as shown in Table 2. To the best of our knowledge, no experimental data are yet available for those compounds, so that our results can be considered as prediction for future investigations. 3.2. Elastic constants The elastic constants of solids provide a link between mechanical and dynamical behaviours of crystals, and give important information concerning the nature of forces operating in solids. In particular, they provide information on stability and stiffness of materials. It is well known that first-order and second-order derivatives of the potential give forces and elastic constants. Therefore, it is an important issue to check the accuracy of the calculations for forces and elastic constants. Let us recall here that pressure effect upon elastic constants is essential, at least for understanding interatomic interactions, mechanical stability and phase transition mechanism. For a cubic crystal, the generalized elastic stability criteria in terms of elastic constants [13]:
ðC 11 þ 2C 12 Þ=3 > 0 C 44 > 0 ðC 11 þ 2C 12 Þ=2 > 0 These criteria are satisfied in the studied pressure. V3Sn is also mechanically unstable because of a negative C44 = 19.41 GPa Table 3. 3.3. Electronic structures DFT band structure calculations are conducted to understand the electronic structure of different materials. Fig. 1 shows the partial s-, p-, and d-densities of states (DOS) for all V3M (M = Si, Ge and Sn) compounds. We show here only the vicinity of the Fermi energy level. The peaks corresponding to the lowest energy (not shown) are due to the contribution M–s and M–p for all compounds. There are two common features in the DOS profiles for these compounds: (1) V3M (M = Si, Ge and Sn) have similar DOS profiles in the whole energy region. (2) The states near the Fermi level are mainly the V 3d states. The DOS at the Fermi level, N(Ef), for all three compounds, which is often used as a good indicator for the structural stability. For each of these compounds considered here, the lowest energy structure is also found to have the lowest N(Ef) value. The DOS profiles and their N(Ef) values of V3M (M = Si, Ge and Sn) structures are very similar, which explains their tiny energy difference and
Table 2 Distances between nearest-neighbored atoms and energy Fermi level of V3M (M = Si, Ge and Sn) compounds studied in this work.
Ef DV–M DV–V
V3Si
V3Ge
V3Sn
10.900 2.633 2.355
11.238 2.665 2.384
11.370 2.786 2.492
700
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
Table 3 Enthalpies of formation E (eV), zero pressure elastic moduli (Cij) in GPa of cubic V3Si, V3Ge and V3Sn. The polycrystalline bulk moduli (B), shear moduli (G), Young’s moduli (Y), and Poisson’s ratio m were obtained using the Voigt approximation. All the elastic constants are in GPa except for the dimensionless m. Material
E
C11
C12
C44
B
G
B/G
Y
m
V3Si V3Ge V3Sn
12077 12076 12050
252.87 220.51 177.05
155.55 153.55 151.80
44.26 21.23 19.41
187.99 175.87 157.80
46.02 26.13 6.47
4.09 6.730 24.39
134.38 94.45 37.80
0.3809 0.410 0.460
suggests the possibility of the co-existence of both structures. The TDOS at the Fermi level Ef (N(Ef)) for all V3M compounds is listed in Table 2. On the other hand near the Fermi level, the DOS mainly originates from the V–d states, this suggests that our compounds are all conductive, and the d bands of the transition metal play the dominant role in electrical transport. 3.4. Bond orders between atoms Bond order is the overlap population of electrons between atoms, and this is a measure of the strength of the covalent bond between atoms. In Table 4, if the overlap population is positive (+) a bonding-type interaction is operating between atom, whereas if it is negative () an anti-bondingtype interaction is dominant between atoms. It is apparent that the bonding-type interactions are operating between the M (M = Si, Ge, Sn) (3p, 4p and 5p) and the V 3d electrons. Thus, there is a strong interaction between V and V, the interaction between M (M = Si, Ge, Sn) and V [M and M] is negative, not including Si [Sn]. 3.5. Thermodynamic properties The energy of vibration of the crystal lattice or elastic wave is quantized. The quantum of energy of an elastic wave is called a phonon, in analogy to the photon, quantum of electromagnetic energy. The thermal vibration of the crystal is formed thermally excited phonons. The study of phonons is an important part of solid state physics, that play a major role in many of the physical properties of solids, including a material’s thermal and electrical conductivities. Phonon dispersion is the dependence of the frequency x on the wave vector. Show two branches: (i) High frequencies (optic) dues to the molecular vibrations. (ii) Low frequencies (acoustic) who their dependence on the wave vector is linear, which is characteristic of sound waves. Fig. 2 shows the phonon density of states for V3M (M = Si, Ge and Sn) it can be seen that the element contributions varies from lighter (high frequency) to heaviest (low frequency), see Fig. 2. Our calculated phonon density of states is similar to that calculated by Delaire [7] (the scale is different), especially that of the phonon DOS of V3Si is constant with increasing temperature, see Fig. 1 [7], and theoretically from first principles, see Fig. 7 [7]. The look is the same for V3Si, the contribution of the silicon compound appears clearly. For V3Ge the look is almost the same. Knowledge of the heat capacity of a substance is mandatory for many applications. By the standard elastic continuum theory [14] two limiting cases are correctly predicted. At high temperatures, the constant volume heat capacity CV tends to the Petit and Dulong limit [15]. At sufficiently low temperatures, CV is proportional to T3 [15]. At intermediate temperatures, the temperature dependence of CV is governed by the details of vibrations of the atoms and for a long time could only be determined from experiments. The thermal properties are determined in the temperature range from 0 to 1000 K, the investigation on the heat capacity of crystals is an old topic of condensed matter physics with which illustrious names are associated [14,16,17]. Fig. 3 represent the variation of the heat capacity, CV(T), for V3M where M = Si, Ge and Sn indicate a sharp increase up to 400 K which is due to the anharmonic approximation of the Debye model used here, that can only be said to approximate for the lattice contribution to the specific heat, because for metals Fig. 1, the electron contribution to the heat is proportional to T, which at low temperatures dominates the Debye T3 result for lattice vibrations.
Density of states (states/eV)
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
20 15 10 5 0
Total V 3 Si
10
V
s p d
Si
s p
5 0 2 1 0 -20
-10
0
10
20
30
40
50
Energy (eV)
Density of states (states/eV)
18 12
Total V3Ge
6 0 12
V
s p d
Ge
s p
6 0 2
0 -20
-10
0
10
20
30
40
50
Energy (eV)
Density of states (states/eV)
18
Total V 3Sn
12 6 20 15 10 5 0 4 3 2 1 0 -20
-10
0
10
20
30
V
s p d
Sn
s p
40
50
Energy (eV) Fig. 1. Partial state densities for V3M (M = Si, Ge and Sn) structures.
701
702
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703 Table 4 Bond orders in the V3M (M = Si, Ge and Sn) structures.
V–V V–M M–M
V3Si
V3Ge
V3Sn
0.54 0.40 0.37
1.56 2.48 3.02
0.92 1.00 2.05
Density of States
a
V Si
b
V Ge
c V Sn 0
2
4
6
8
10
Frequency (THz) Fig. 2. Partial phonon density of states for V3Si (a), V3Ge (b) and V3Sn (c).
60
Heat capacity (cal/cell K)
Dulong-petit limit 50 40 30
V3Si V3Ge V3Sn
20 10 0 0
200
400
600
800
1000
1200
Temperature (K) Fig. 3. The heat capacity versus temperature for V3 M (M = Si, Ge and Sn).
However, at higher temperature, the anharmonic effect on CV is very close to the Dulong–Petit limit (CV(T) 3R for mono-atomic solids) which in common to all solids at high temperatures.
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
703
4. Summary and conclusion ⁄ V3Sn compound is mechanically unstable because of a negative C44 = 19.41 GPa. ⁄ For each of these compounds considered here, the lowest energy structure is found to have the lowest N(Ef) value. ⁄ There is a strong interaction between V and V, the interaction between M (M = Si, Ge, Sn) and V [M and M] is negative, not including Si [Sn]. ⁄ The element contributions in PDOS varies from lighter (high frequency) to heaviest (low frequency).
References [1] A.B. Gokhale, G.J. Abbaschian, in: Massalski, H. Okamoto, P.R. Subramanian, L. Kaeprzak (Eds.), Binary Alloys Phase Diagrams, vol. 2, American Society for Metals, Metals Park, OH, 1987, pp. 1333. [2] D.L. Anton, D.M. Shah, Wright Research and Development Center, WRDC-TR-90-41222, 1990. [3] C.S. Chang, Dissert. Abstr. Int. (USA) 52 (1992) 206. [4] S.V. Raj, Mater. Sci. Eng. 192 (1995) 583. [5] Q. Yue, Y.Q. Liu, M.Y. Chu, J.Y. Shen, Calphad 33 (2009) 539. [6] J.-E. Jørgensen, S.E. Rasmussen, Acta Cryst. B38 (1982) 346–347. [7] O. Delaire, Appl. Phys. A 99 (2010) 523. [8] M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Haspin, S.J. Clark, M.C. Payne, J. Phys. Condens. Mater. 14 (2002) 2717. [9] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396(E). [10] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [11] H.J.Z. Wallbaum, Metallkunde 31 (1939) 362. [12] H.J. Wallbaum, Naturwissenschaften 32 (1944) 76. [13] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford, 1956. [14] P. Debye, Ann. Phys. 39 (1912) 789. [15] A.T. Petit, P.L. Dulong, Ann. Chim. Phys. 10 (1819) 395. [16] A. Einstein, Ann. Phys. 22 (1907) 180. [17] W. Nernst, A.F. Lindemann, Z. Elektrochem, Angew. Phys. Chem. 17 (1977) 817.
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Theoretical prediction of the structural, elastic, electronic and thermodynamic properties of V3M (M = Si, Ge and Sn) compounds T. Chihi a, M. Fatmi b,c,⇑ a
Laboratory for Elaboration of New Materials and Characterization (LENMC), University of Ferhat Abbas, Setif 19000, Algeria Research Unit on Emerging Materials (RUEM), University of Ferhat Abbas, Setif 19000, Algeria c Laboratory of Physics and Mechanics of Metallic Materials (LP3M), University of Ferhat Abbas, Setif 19000, Algeria b
a r t i c l e
i n f o
Article history: Received 14 May 2012 Accepted 17 June 2012 Available online 23 June 2012 Keywords: Ab initio calculation Electronic structure Structural properties
a b s t r a c t Density functional theory (DFT), is used in our calculations to study the V3M (M = Si, Ge and Sn) compounds, we are found that V3Sn compound is mechanically unstable because of a negative C44 = 19.41 GPa. For each of these compounds considered, the lowest energy structure is found to have the lowest N(Ef) value. Also there is a strong interaction between V and V, the interaction between M (M = Si, Ge, Sn) and V (M and M) is negative, not including Si [Sn]. In phonon density of states PDOS, the element contributions varies from lighter (high frequency) to heaviest (low frequency). Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction An intermetallic compounds, as it applies to solid phases, is a phase which crystallizes with a structure other than those of its components. Intermetallic phases containing more than one element, ordered into different sites in the solid solutions. Intermetallic compounds have been attracted candidates for high temperature structural materials because of their desirable intrinsic properties, generally brittle and high melting point. There have been many investigations carried out on various compounds, but mostly on structures that are ordered from of fcc, bcc and hcp metals.
⇑ Corresponding author at: Research Unit on Emerging Materials (RUEM), University of Ferhat Abbas, Setif 19000, Algeria. Fax: +213 35 63 05 60. E-mail addresses: [email protected] (T. Chihi), [email protected] (M. Fatmi). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.06.009
698
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
Silicides of transition metals have attracted considerable interest due to their desirable properties for application involving high temperatures environment. Among these silicides, the intermetallic compound Cr3Si is a typical example since it has a high melting point (2043 K) [1], good creep resistance, good oxidation resistance, high stiffness, and a comparatively low density. Therefore, though lacking in room temperature toughness [2], it has been recognized as a promising candidate for high temperature structural application in aggressive environment [3], especially in the field of aircraft engine material [4]. Yue et al. [5] was thermodynamically modeled the Sn–V binary system using the CALPHAD approach combined with first-principles calculations. V3Si [6] exhibit a cubic to tetragonal phase transition at low temperature. Delaire [7] studied by the inelastic neutron scattering and firstprinciples computation, the interactions of high-temperature electron–phonon of V3X (X = Si, Ge and Co) compounds. In the present work we are interested in cubic V3M (M = Si, Ge and Sn) compounds. They present a Pm–3n space group. The paper is organized as follows. The computational method is described in Section 2. In Section 3, the results are presented and compared with available experimental and theoretical data. Conclusion is given in Section 4.
2. Computational details Cambridge serial total energy package (CASTEP) [8], a first-principles plane wave pseudo potential (PWPP) method based on the density functional theory (DFT), is used in our calculations. CASTEP uses a plane-wave basis set for the expansion of the single-particle Kohn–Sham wave-functions, and pseudopotentials to describe the computationally expensive electron–ion interaction, in which the exchange-correlation energy by the generalized gradient approximation (GGA) of Perdew et al. is adopted for all elements in our models by adopting the Perdew–Burke–Ernzerhof, known as PBE parameters [9]. The atomic orbitals used in the present calculations are: in which the V (3d3 4s2), Si (3p2 3s2), and Ge (3d10 4p2 4s2) orbitals are treated as valence electrons. According to an ultrasoft condition, the pseudo-wave function which is related to the pseudo potential matches the plane- expanded with beyond cut-off energy. Using high cut-off energy, at the price of spending long computational time, can actually provide accurate results. The cut-off energy for the plane-wave expansion was set at 330 eV and the Brillouin zone sampling was carried out using the 6 6 6 set of Monkhorst–Pack mesh [10]. Atomic positions are relaxed and optimized with a density mixing scheme using the conjugate gradient (CG) method for eigenvalues minimization. The equilibrium lattice parameter is then computed from a structural optimization, using the Broyden–Fletcher–Goldfarb–Shenno (BFGS) minimization technique. This technique provides a fast way of finding out the lowest energy structure out of all the converged structures, with the following thresholds: energy change per atom less than 2 105 eV, residual force less than 0.05 eV/Å, displacement of atoms during the geometry optimization less than 0.002 Å, and maximum stress within 0.1 GPa. The crystal structures are reported (Table 1).
Table 1 Calculated V3Si, V3Ge and V3Sn structures: lattice parameters in Å; volumes per formula unit (f.u.) in Å3; Z is the number of formula units in the unit cell, and the space group and compared with experimental data.
a b c
Space group
a
V
Z
V3Si
Pm–3n
104.52
2
V3Ge
Pm–3n
108.36
2
V3Sn
Pm–3n
4.71 4.71a 4.767 4.760b 4.984 4.953c
123.79
2
Ref. [11]. Ref. [12]. Ref. [5].
699
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
3. Results and discussion 3.1. Structural properties The results for lattice parameters a are reported in Table 1 and compared with experimental and previous theoretical calculations. Our calculated value for lattice parameter a of V3M (M = Si, Ge and Sn) compounds are in excellent agreement with the experimental data [11,12,5] The relative deviation may be attributed to the fact that our calculation corresponds to the perfect bulk material at zero temperature, whereas the experimental sample was synthesized at high temperature. The total energies (stability) of V3M (M = Si, Ge and Sn) compounds increase, the nearest-neighbor distance DV–M and DV–V in these compounds increase and the bulk modulus decrease for M metals in the same column in the periodic table, with increasing atomic number as shown in Table 2. To the best of our knowledge, no experimental data are yet available for those compounds, so that our results can be considered as prediction for future investigations. 3.2. Elastic constants The elastic constants of solids provide a link between mechanical and dynamical behaviours of crystals, and give important information concerning the nature of forces operating in solids. In particular, they provide information on stability and stiffness of materials. It is well known that first-order and second-order derivatives of the potential give forces and elastic constants. Therefore, it is an important issue to check the accuracy of the calculations for forces and elastic constants. Let us recall here that pressure effect upon elastic constants is essential, at least for understanding interatomic interactions, mechanical stability and phase transition mechanism. For a cubic crystal, the generalized elastic stability criteria in terms of elastic constants [13]:
ðC 11 þ 2C 12 Þ=3 > 0 C 44 > 0 ðC 11 þ 2C 12 Þ=2 > 0 These criteria are satisfied in the studied pressure. V3Sn is also mechanically unstable because of a negative C44 = 19.41 GPa Table 3. 3.3. Electronic structures DFT band structure calculations are conducted to understand the electronic structure of different materials. Fig. 1 shows the partial s-, p-, and d-densities of states (DOS) for all V3M (M = Si, Ge and Sn) compounds. We show here only the vicinity of the Fermi energy level. The peaks corresponding to the lowest energy (not shown) are due to the contribution M–s and M–p for all compounds. There are two common features in the DOS profiles for these compounds: (1) V3M (M = Si, Ge and Sn) have similar DOS profiles in the whole energy region. (2) The states near the Fermi level are mainly the V 3d states. The DOS at the Fermi level, N(Ef), for all three compounds, which is often used as a good indicator for the structural stability. For each of these compounds considered here, the lowest energy structure is also found to have the lowest N(Ef) value. The DOS profiles and their N(Ef) values of V3M (M = Si, Ge and Sn) structures are very similar, which explains their tiny energy difference and
Table 2 Distances between nearest-neighbored atoms and energy Fermi level of V3M (M = Si, Ge and Sn) compounds studied in this work.
Ef DV–M DV–V
V3Si
V3Ge
V3Sn
10.900 2.633 2.355
11.238 2.665 2.384
11.370 2.786 2.492
700
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
Table 3 Enthalpies of formation E (eV), zero pressure elastic moduli (Cij) in GPa of cubic V3Si, V3Ge and V3Sn. The polycrystalline bulk moduli (B), shear moduli (G), Young’s moduli (Y), and Poisson’s ratio m were obtained using the Voigt approximation. All the elastic constants are in GPa except for the dimensionless m. Material
E
C11
C12
C44
B
G
B/G
Y
m
V3Si V3Ge V3Sn
12077 12076 12050
252.87 220.51 177.05
155.55 153.55 151.80
44.26 21.23 19.41
187.99 175.87 157.80
46.02 26.13 6.47
4.09 6.730 24.39
134.38 94.45 37.80
0.3809 0.410 0.460
suggests the possibility of the co-existence of both structures. The TDOS at the Fermi level Ef (N(Ef)) for all V3M compounds is listed in Table 2. On the other hand near the Fermi level, the DOS mainly originates from the V–d states, this suggests that our compounds are all conductive, and the d bands of the transition metal play the dominant role in electrical transport. 3.4. Bond orders between atoms Bond order is the overlap population of electrons between atoms, and this is a measure of the strength of the covalent bond between atoms. In Table 4, if the overlap population is positive (+) a bonding-type interaction is operating between atom, whereas if it is negative () an anti-bondingtype interaction is dominant between atoms. It is apparent that the bonding-type interactions are operating between the M (M = Si, Ge, Sn) (3p, 4p and 5p) and the V 3d electrons. Thus, there is a strong interaction between V and V, the interaction between M (M = Si, Ge, Sn) and V [M and M] is negative, not including Si [Sn]. 3.5. Thermodynamic properties The energy of vibration of the crystal lattice or elastic wave is quantized. The quantum of energy of an elastic wave is called a phonon, in analogy to the photon, quantum of electromagnetic energy. The thermal vibration of the crystal is formed thermally excited phonons. The study of phonons is an important part of solid state physics, that play a major role in many of the physical properties of solids, including a material’s thermal and electrical conductivities. Phonon dispersion is the dependence of the frequency x on the wave vector. Show two branches: (i) High frequencies (optic) dues to the molecular vibrations. (ii) Low frequencies (acoustic) who their dependence on the wave vector is linear, which is characteristic of sound waves. Fig. 2 shows the phonon density of states for V3M (M = Si, Ge and Sn) it can be seen that the element contributions varies from lighter (high frequency) to heaviest (low frequency), see Fig. 2. Our calculated phonon density of states is similar to that calculated by Delaire [7] (the scale is different), especially that of the phonon DOS of V3Si is constant with increasing temperature, see Fig. 1 [7], and theoretically from first principles, see Fig. 7 [7]. The look is the same for V3Si, the contribution of the silicon compound appears clearly. For V3Ge the look is almost the same. Knowledge of the heat capacity of a substance is mandatory for many applications. By the standard elastic continuum theory [14] two limiting cases are correctly predicted. At high temperatures, the constant volume heat capacity CV tends to the Petit and Dulong limit [15]. At sufficiently low temperatures, CV is proportional to T3 [15]. At intermediate temperatures, the temperature dependence of CV is governed by the details of vibrations of the atoms and for a long time could only be determined from experiments. The thermal properties are determined in the temperature range from 0 to 1000 K, the investigation on the heat capacity of crystals is an old topic of condensed matter physics with which illustrious names are associated [14,16,17]. Fig. 3 represent the variation of the heat capacity, CV(T), for V3M where M = Si, Ge and Sn indicate a sharp increase up to 400 K which is due to the anharmonic approximation of the Debye model used here, that can only be said to approximate for the lattice contribution to the specific heat, because for metals Fig. 1, the electron contribution to the heat is proportional to T, which at low temperatures dominates the Debye T3 result for lattice vibrations.
Density of states (states/eV)
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
20 15 10 5 0
Total V 3 Si
10
V
s p d
Si
s p
5 0 2 1 0 -20
-10
0
10
20
30
40
50
Energy (eV)
Density of states (states/eV)
18 12
Total V3Ge
6 0 12
V
s p d
Ge
s p
6 0 2
0 -20
-10
0
10
20
30
40
50
Energy (eV)
Density of states (states/eV)
18
Total V 3Sn
12 6 20 15 10 5 0 4 3 2 1 0 -20
-10
0
10
20
30
V
s p d
Sn
s p
40
50
Energy (eV) Fig. 1. Partial state densities for V3M (M = Si, Ge and Sn) structures.
701
702
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703 Table 4 Bond orders in the V3M (M = Si, Ge and Sn) structures.
V–V V–M M–M
V3Si
V3Ge
V3Sn
0.54 0.40 0.37
1.56 2.48 3.02
0.92 1.00 2.05
Density of States
a
V Si
b
V Ge
c V Sn 0
2
4
6
8
10
Frequency (THz) Fig. 2. Partial phonon density of states for V3Si (a), V3Ge (b) and V3Sn (c).
60
Heat capacity (cal/cell K)
Dulong-petit limit 50 40 30
V3Si V3Ge V3Sn
20 10 0 0
200
400
600
800
1000
1200
Temperature (K) Fig. 3. The heat capacity versus temperature for V3 M (M = Si, Ge and Sn).
However, at higher temperature, the anharmonic effect on CV is very close to the Dulong–Petit limit (CV(T) 3R for mono-atomic solids) which in common to all solids at high temperatures.
T. Chihi, M. Fatmi / Superlattices and Microstructures 52 (2012) 697–703
703
4. Summary and conclusion ⁄ V3Sn compound is mechanically unstable because of a negative C44 = 19.41 GPa. ⁄ For each of these compounds considered here, the lowest energy structure is found to have the lowest N(Ef) value. ⁄ There is a strong interaction between V and V, the interaction between M (M = Si, Ge, Sn) and V [M and M] is negative, not including Si [Sn]. ⁄ The element contributions in PDOS varies from lighter (high frequency) to heaviest (low frequency).
References [1] A.B. Gokhale, G.J. Abbaschian, in: Massalski, H. Okamoto, P.R. Subramanian, L. Kaeprzak (Eds.), Binary Alloys Phase Diagrams, vol. 2, American Society for Metals, Metals Park, OH, 1987, pp. 1333. [2] D.L. Anton, D.M. Shah, Wright Research and Development Center, WRDC-TR-90-41222, 1990. [3] C.S. Chang, Dissert. Abstr. Int. (USA) 52 (1992) 206. [4] S.V. Raj, Mater. Sci. Eng. 192 (1995) 583. [5] Q. Yue, Y.Q. Liu, M.Y. Chu, J.Y. Shen, Calphad 33 (2009) 539. [6] J.-E. Jørgensen, S.E. Rasmussen, Acta Cryst. B38 (1982) 346–347. [7] O. Delaire, Appl. Phys. A 99 (2010) 523. [8] M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Haspin, S.J. Clark, M.C. Payne, J. Phys. Condens. Mater. 14 (2002) 2717. [9] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396(E). [10] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [11] H.J.Z. Wallbaum, Metallkunde 31 (1939) 362. [12] H.J. Wallbaum, Naturwissenschaften 32 (1944) 76. [13] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford, 1956. [14] P. Debye, Ann. Phys. 39 (1912) 789. [15] A.T. Petit, P.L. Dulong, Ann. Chim. Phys. 10 (1819) 395. [16] A. Einstein, Ann. Phys. 22 (1907) 180. [17] W. Nernst, A.F. Lindemann, Z. Elektrochem, Angew. Phys. Chem. 17 (1977) 817.