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Investigation of high figure of merit in semiconductor XHfGe (X = Ni and Pd) half-Heusler alloys: Ab-initio study
Computational Condensed Matter 21 (2019) e00407
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Investigation of high figure of merit in semiconductor XHfGe (X ¼ Ni and Pd) half-Heusler alloys: Ab-initio study F. Bendahma a, *, M. Mana b, S. Terkhi a, S. Cherid a, B. Bestani c, S. Bentata a, d a
Technology and Solids Properties Laboratory, Abdelhamid Ibn Badis University, 27000, Mostaganem, Algeria Abdelhamid Ibn Badis University, 27000, Mostaganem, Algeria c Structure, Elaboration and Application of Molecular Materials Laboratory, AbdelhamidIbnBadis University, 27000, Mostaganem, Algeria d Mustapha Stambouli University of Mascara, 29000, Mascara, Algeria b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 April 2019 Received in revised form 24 May 2019 Accepted 25 May 2019
Using the full potential Linearized augmented Plane Wave (FP-LAPW) method to the density functional theory (DFT) as part of the Generalized Gradient Approximation (GGA), we investigated the structural, electronic, elastic, thermodynamic, and thermoelectric properties of ternary XHfGe (X ¼ Ni and Pd) HalfHeusler alloys. The calculation shows that the materials studied are non-magnetic (NM) with semiconductor nature, elastically stable and anisotropic. In the quasi-harmonic model of Debye, the volume V, the bulk modulus B, the thermal capacity Cv, the entropy S and the temperature of Debye qD are also studied. The semi-classical Boltzmann theory as implemented in the BoltzTraP code is applied to study the thermoelectric (TE) properties. The extremely high figure of merit ZT ~1.14 and 0.95 at 1100 K have been achieved for NiHfGe and PdHfGe, respectively, to make these systems promising candidates for TE applications. Published by Elsevier B.V.
Keywords: Half-heusler Generalized gradient approximation (GGA) Semiconductor behavior Thermoelectric applications
1. Introduction In general, Heusler alloys have drawn growing attention during the last few decades in the scientific community [1]. In particular, half Heusler (HH) with general formula XYZ exhibit a big interest for their properties such as shape memory, thermoelectric, ferromagnetic, spin polarization effects and superconductivity [2e10], that could lead to new mechanical and electronic devices. These properties are due to many substitutions of transition metal, a noble metal, or rare - earth elements at both the crystallographic sites X, Y and (IIIeV) elements at the Z sites [11e13].Thermoelectric materials (TEMs) become interesting because of their ability to directly convert heat into electricity [14e16]. This physical phenomenon motivated the researchers to find better materials having a high thermoelectric figure of merit (ZT) approaching ~1, including the three important parameters describing thermoelectric materials which are: Electric conductivity (s/t), Seebeck coefficient (S), and thermal conductivity (k/t) [17,18]. Recently, (HH) alloys with a valence electron count of 18 belong to the semiconducting materials have been extensively studied as potential high temperature
* Corresponding author. E-mail address: [email protected] (F. Bendahma). https://doi.org/10.1016/j.cocom.2019.e00407 2352-2143/Published by Elsevier B.V.
TEMs. This is mainly to their thermal stability, non-toxic components and cost effectiveness and their narrow band gaps including the density of states sharp slope near the Fermi level. Among them, MNiSn (M ¼ Ti, Zr, Hf) and MCoSb (M ¼ Ti, Zr, Hf) half Heusler compounds exhibit semiconductor behavior and have narrow indirect band gap (e.g., Eg ¼ 0.95eV for TiCoSb, Eg ¼ 0.42 eV for TiNiSn and Eg ¼ 0.5 eV for ZrNiSn) [19e22]. So, thermoelectric power generators (TEPGs) better performance, a material having a narrow band gap become a key solution [23]. In this case and in the absence of experimental and theoretical data, our efforts have been focused on the ab-initio study in terms of structural, electronic, elastic, thermodynamic and thermoelectric properties of two half Heusler compounds NiHfGe and PdHfGe. The arrangement of this paper is as follows: A theoretical background is presented in section 2. Results and discussion are presented in section 3. A summary of the results is given in section 4.
2. Computational details In this paper, we have employed the full potential linearized augmented plane-wave (FPLAPW) method to the density functional theory (DFT) implemented with the Wien2k package [24]. The generalized gradient approximation proposed by
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PerdeweBurkeeErnzerhof (GGA-PBE96) [25] was used for exchanges and correlation. The integration of the full Brillouin zone is performed with 12 12 12 k-points mesh. The plane wave cutoff parameter RMT K max was chosen to be 8 where RMT is the smallest radius of the muffin-tin spheres and Kmax gives the magnitude of the largest k-vector in the plane wave expansion. These Half-Heusler compounds XHfGe (X ¼ Ni, Pd) crystallize in the cubic MgAgAs (C1b) structure with the F-43 m space group. In this structure with corresponding Wyckoff positions: X, Hf and Ge atoms occupy 4a (0.25, 0.25, and 0.25), 4b (0, 0, and 0) and 4c (0.5, 0.5, and 0.5) sites, respectively. To study the thermodynamic properties of the cubic XHfGe (X ¼ Ni,Pd) Half-Heusler compounds we have applied the quasi-harmonic Debye model as implemented in the Gibbs program [26].The Boltztrap code which is based on Boltzmann transport theory [27] was used to analysis the transport properties. We need a large number of k-points for the thermoelectric properties and we used 100000k-points in the irreducible wedge of the Brillouin zone, to ensure the convergence of the transport properties. 3. Results and discussions 3.1. Structural properties The total energy-volume dependence estimated by fitting Birch Murnaghan's equation of state is shown in Fig. 1 [28]. It is clear from these plots that for both Half-Heusler alloys (NiHfGe and PdHfGe), with their lower computed total energies, that the nonmagnetic (NM) state is more favorable than the corresponding ferromagnetic (FM) or antiferromagnetic (AFM) configuration in the cubic structure. Table 1 summarizes most important parameters obtained at static equilibrium corresponding to (0K and 0 GPa) such as lattice constant a0, bulk modulus B, its first pressure derivative B’, the volume V0 and minimum total energy (E0) of cubic NiHfGe and PdHfGe Half-Heusler compounds. As it can be observed from Table 1, the ɑ0 values of XHfGe compounds increase in the following sequence: ɑ0 (NiHfGe) ˂ ɑ0 (PdHfGe). As Hf and Ge atoms are the same in the two compounds, this result can be explained by considering the atomic radii of Ni and Pd: R(Ni) ¼ 1.35 Å and R(Pd) ¼ 1.40 Å,i.e. the lattice constant increases with increasing atomic size of the X element in XHfGe compounds. Whereas, the B values decrease in the following sequence: B (NiHfGe) > B (PdHfGe), i.e. in inverse sequence to ɑ0, in agreement with the well-known relationship between B and the unit cell volume: B f V1 0 .
Table 1 Refined crystallographic parameters: lattice constant a0 (bohr), unit cell volume V (bohr3), bulk modulus B (GPa), its pressure derivative B’ and ground state energies E0 (Ry) of cubic NiHfGe and PdHfGe Half-Heusler compounds. Compound
a0
V0
B
B0
E0
NiHfGe PdHfGe
11.09 11.62
343.02 394.11
145.35 137.33
4.48 4.22
37435.672092 44488.094061
3.2. Electronic properties The electronic band structures of XHfGe (X ¼ Ni, Pd) compounds are calculated at the theoretical lattice constant along highsymmetry directions in the first Brillouin zone, as shown in Fig. 2. Using GGA approximation, the valence band maximum (VBM) is 0 eV at the G point but the conduction band minimum (CBM) is 0.66eV and 0.55 eV at the X point for NiHfGe and PdHfGe, respectively. This confirms that NiHfGe and PdHfGe have an indirect band gap equal to 0.66 and 0.55 eV, respectively. For better understanding the electronic properties of the studied half Heuslers and to explain the contribution of different states in the band structures, both total density of states (TDOS) and partial densities of states (PDOS) of XHfGe (X ¼ Ni, Pd) were evaluated respectively as shown in Fig. 3. The Fermi level EF is set at 0 eV. The states Ni (4s2, 3d8), Pd (4d10), Hf (6s2, 5d2), and Ge (4s2,4p2) are treated as valence electrons. In Fig. 3, it can be seen that the region between 4 and 0 eV, is mainly due to (3d Ni/4d Pd) states for NiHfGe and PdHfGe, respectively and a small contribution of 5d states of Hf and 4p of Ge. Furthermore, the region between 0 and 4 eV presents a strong contribution of 5d Hf state and a weak one of (3d Ni/4d Pd) atom for both considered materials. Consequently, NiHfGe and PdHfGe have a semi-conductor behavior and their density of states are dominated by the (Ni/Pd)-d, Hf-d (d-e g, d-t 2 g) and Ge-p states. 3.3. 3. Elastic properties Even, Half-Heusler alloys are known to be remarkably stable and strong materials. The elastic constants of the studied compounds were checked in this study using Stress-strain method in order to confirm their stability. By doing so, only three independent elastic constants Cij, namely C11, C12 and C44 for a cubic system are needed to evaluate all the other elastic moduli. Based on the parameter values presented in Table 2, we remark that the calculated bulk modulus from the elastic constants (B ¼ (1/ 3) (C11 þ 2 C12)) and from Birch Murnaghan equation of state EOS fits are almost identical, which gives more credence to our obtained
Fig. 1. Total energy as a function of unit cell volume forNiHfGe and PdHfGe with ferromagnetic (FM), anti-ferromagnetic (AFM) and non-magnetic (NM) states.
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Fig. 2. The calculated band structures for NiHfGe and PdHfGe by using GGA for non-spin polarized state.
Fig. 3. Total and partial density of states for NiHfGe and PdHfGe with GGA approximation.
Table 2 Calculated elastic constants Cij, bulk modulus B (GPa), anisotropy factor A, shear modulus G (GPa), Young's modulus E (GPa) and the Poisson's ratio n of cubic NiHfGe and PdHfGe Half-Heusler compounds. Compound
C11
C12
C44
B
A
G
E
B/G
n
NiHfGe PdHfGe
243.24 179.16
97.15 117.36
104.78 90.78
145.85 137.96
1.43 2.94
90.71 53.87
225.4 143.0
1.61 2.56
0.22 0.35
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results (see Table 1). In addition, other important macroscopic parameters such as the anisotropy factor A, the shear modulus G, the Poisson's ratio n and the Young's modulus E have also been investigated in this study using the Voigt-Reuss-Hill approximations for the cubic structures [29e31]:
critical one. Another important characteristic measuring the resistance offered by an elastic body to deformation, that is to say the solid stiffness measurements estimated by Young's modulus (E) have also been investigated. The very large E values of 225.4 and 143.0 GPa for NiHfGe and PdHfGe, respectively are in fact an indication that the considered half-heusler are stiffer.
BV ¼ BR ¼ ðC11 þ 2C12 Þ=3
(1)
GV ¼ ðC11 C12 þ 3C44 Þ=5
(2)
3.4. 4.Thermodynamic properties
GR ¼ 5C44 ðC11 C12 Þ=½4C44 þ 3ðC11 C12 Þ
(3)
A ¼ 2C44 =ðC11 C12 Þ
(4)
G ¼ ðGv þ GR Þ=2
(5)
Thermodynamic properties of NiHfGe and PdHfGe compounds have also been computed for supplementary useful information. So, thermal properties determination in the range of 0e1200 K and pressure effect in the range of 0e20 GPa were investigated for this purpose. Firstly, Fig. 4 shows the volume-temperature relationship at different pressures for both NiHfGe and PdHfGe compounds, in which, the volume (V) increases with increasing temperature with very slow rate. Secondly, at a given temperature, there is a decrease in volume with increasing pressure. At initial state (0K and 0 GPa), the calculated volumes for NiHfGe and PdHfGe are 343.8 and 394.13 bohr3 respectively. These volume values confirm those obtained in the structural properties, which are 343.02 and 393.11 bohr3 for NiHfGe and PdHfGe, respectively. We notice that during compression, the volume reduction of NiHfGe is less than the PdHfGe one because of the bulk modulus of the first compound is higher. In Fig. 5 are shown the effects of pressure and temperature on the bulk modulus (B). The bulk modulus value is almost decreasing linearly with increasing temperature. This decrease in bulk modulus values is an indication that XHfGe (X ¼ Ni, Pd) fcc crystal structure is entering a new bonding regime and becoming softer. At 0K and 0 GPa conditions, the obtained bulk modulus values for NiHfGe and PdHfGe are 145.72 and 137.29 GPa respectively which are in good agreement with the previously determined ones using the structural and elastic properties. Furthermore, at ambient temperature and 0 GPa, the high bulk modulus values of about 140.01 and 135.38 GPa corresponding to NiHfGe and PdHfGe is respectively; confirm that these later ternary Heuslers present a strong hardness and an important compressibility rate [35e37]. We can see from heat capacity Cv versus temperature plots at
On the other hand, equations (6) and (7) were used to compute the Young's modulus E and Poisson's ratio n
E ¼ 9BG=ð3B þ GÞ
(6)
n ¼ ð3B 2GÞ=½2ð3B þ GÞ
(7)
It is noticed that for a cubic crystal, requirements for the compounds mechanical stability conditions should be (C11 C 12> 0); (C11 þ 2 C12 > 0; C44 > 0 and C12 1.75, a material is considered to be ductile; otherwise, it will be fragile and hard. In this study, the B/G ratio was equal to 1.61 and 2.56 for NiHfGe and PdHfGe respectively; indicating then, that the first material is in fact fragile, while the second one is ductile. Poisson's ratio n can also be used for estimating the ductility of a material. According to the Frantsevich rule [34], if n > 1/3, the material is considered to be ductile, otherwise it is considered to be brittle. Based on the obtained results for n values presented in Table 2, we can say that NiHfGe and PdHfGe are brittle and ductile, respectively and are also anisotropic since their A values are greater than the
Fig. 4. Volume of unit cell parameters versus temperature for NiHfGe and PdHfGeat different pressures.
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Fig. 5. Temperature dependence of the bulk modulus at different pressures for NiHfGe and PdHfGe.
different pressures presented in Fig. 6 that the pressure has no effect on Cv.Whereas increasing the temperature three important regions were identified: In the first region when T 300K, the increase in temperature, will increase Cv rapidly for both materials, according to Debye law (since Cv is proportional to T3) [38]. At 300 K and 0 GPa conditions, Cv value approaches approximately 68.26 and 68.41 Jmol1K1for NiHfGe and PdHfGe, respectively. In the second region when 300˂ T ˂800 K, Cv increases slowly with increasing temperature; this is mainly due to the atomic vibrations. In the third region when T 800K, Cv increases slowly with temperature. At temperature limit (T ¼ 800 K), no more rate increase is observed and its value approaches approximately 73.42 and 73.84 J mol1 K1 for NiHfGe and PdHfGe, respectively. Furthermore, it should be noted that the heat capacity
was quite close to the DulongePetit classical limit, (75 Jmol1K1), a common phenomena in all solids [39]. The most important parameter in the quasi harmonic Debye model approach is the Debye temperature qD, which relates different physical properties of solids, such as specific heat, elastic constants and melting temperature [40]. Fig. 7 shows the dependence of the Debye temperature qD and temperature at different pressures. Obtained values of 371.24 and 346.17K at 0 GPa and 300K for NiHfGe and PdHfGe compounds respectively indicate that PdHfGe has a lower thermal conductivity than NiHfGe at these conditions. In addition, with the pressure increase from 0 to 20 GPa at the given temperature, there is an increase in calculated qD values. However, at a constant pressure, qD decreases linearly with increasing temperature. Material's entropy S knowledge provides an essential insight on its vibration properties, which lead to important consequences for the performance of
Fig. 6. Variation of the heat capacity Cvwith temperature at different pressures forNiHfGe and PdHfGe.
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Fig. 7. Debye temperature versus temperature of NiHfGe and PdHfGe at different pressures.
many dispositives as heat engines, refrigerators, and heat pumps. Effects of temperature on entropy at fixed pressures are shown in Fig. 8. It is observed that the stable equilibrium state corresponds to the starting curves values (S ¼ 0) at 0 K and 0 GPa, then, the entropy will increase exponentially with temperature above 0 K. In addition, the entropy obtained values of 88.41 and 90.94 Jmol1K1for NiHfGe and PdHfGe, respectively, indicate that PdHfGe is less ordered than NiHfGe at ambient and 0 GPa conditions. At temperatures exceeding 700K the quantity ‘entropy’ becomes very high. This is due mainly to both, the increase of the vibration entropy of a material with increasing temperature and its decrease with increasing cohesive energy. 3.5. 5.Thermoelectric properties Once the structural stability of the two materials realized, by studying the elastic and thermodynamic properties, their thermoelectric ones must be checked in order to have a clear idea and
more information about these HH alloys. The evolution of electrical conductivity data (s/t) per relaxation time (t z 1015 s) within a temperature ranging from 0 to 1100 K is represented in Fig. 9. In which two different temperature ranges are obtained from the curves: In the range of (300 ˂ T ˂ 600 K), results obtained in this study show that the top electric conductivity is 5.54 1019 and 6.98 1019(Ums)1 for NiHfGe and PdHfGe, respectively, at the ambient temperature (300K) meaning that PdHfGe is a good electrical conductor than NiHfGe at ambient. Beyond this temperature value, the electric conductivity decreases as temperature increases for the two compounds. In the range of (600 T 1100 K), there is slow increase in the electric conductivity with temperature. This temperature dependence is essentially due to both the intrinsically narrow band gaps between 0.66 eV (NiHfGe) and 0.55 eV (PdHfGe), rising then s/t to large values, and the sharp increase in the density of states around the Fermi level. The Seebeck coefficient (S) or thermoelectric power is related to the fact that electrons are both carriers of electricity and heat. For a good thermoelectric device, a
Fig. 8. Variation of entropy with temperature at different pressures of NiHfGe andPdHfGe.
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Fig. 9. Electrical conductivity versus temperature of NiHfGe and PdHfGe.
Fig. 11. Thermal conductivity versus temperature of NiHfGe and PdHfGe.
high Seebeck coefficient is required. In Fig. 10 are shown the Seebeck coefficient for NiHfGe and PdHfGe compounds as a function of temperature. We observe clearly that the highest value of Seebeck coefficient for both compounds is achieved at high temperatures (Seebeck coefficient around 150 mV/K for NiHfGe and 138 mV/K for PdHfGe at 1100 K). Furthermore, both the valence band maximum and conduction band minimum are at low asymmetry points of the Brillouin zone, presenting then, that the present electronic properties performance of the studied materials giving rise to their high Seebeck coefficients [41], which is an important factor for the efficient behavior of thermoelectric devices. This means that these HH compounds semiconductor behaviors are good thermoelectric devices. The thermal conductivity (K) of a material is a measure of its ability to energy transfer as heat. It is defined as: K¼ Ke þ Kl, where the Ke and Kl are respectively the electrical and lattice vibrations of total thermal conductivity K [42]. In this paper research, the BoltzTraP code used which neglects the lattice vibrations part (Kl) compared to the electronic part (Ke). According to Wiedemanne Franz law, the electronic thermal conductivity (Ke) is defined as follows: Ke ¼ LsT, (L is the Lorentz number ¼ 2.44 108 W.U. K2for free electrons). Plots of the thermal conductivity versus temperature for both compounds are presented in Fig. 11. At room
temperature, the thermal conductivities are 1.73 1014 and 1.16 1014 W/m K s for NiHfGe and PdHfGe, respectively. According to these calculated values, the thermal conductivities at ambient are lower than those corresponding to (T) exceeding 300 K. These low values in (K) are due to the atomic contribution to phonons and their states densities (DOS). At T > 300 K, the thermal conductivity increase for both materials is related to the force of attraction between the atoms that causes the lattice vibrations. Also, it's noted that, the lower thermal conductivity at the ambient is proportional to the higher electrical one obtained previously. According to WiedemanneFranz law which states that a good thermoelectric material requires a low thermal conductivity with high electrical conductivity, our result is in good agreement with this law [43]. Knowing that the thermoelectric energy TE conversion efficiency depends on the transport coefficients of the constituent materials through the figure of merit ZT ¼ S2s T/K, where S, s, K, and T are the Seebeck coefficient, electrical conductivity, thermal conductivity, and absolute working temperature, respectively [44]. A material defined by a ZT 1 represents the best elements for the thermoelectric devices [45,46]. In this context, evolutions of ZT versus temperature for NiHfGe and PdHfGe are represented in Fig. 12. According to the quasi-linear profile, an increase of ZT for the two
Fig. 10. Variation of the Seebeck coefficient as a function of temperature of NiHfGe and PdHfGe.
Fig. 12. Figure of merit ZT versus temperature of NiHfGe and PdHfGe.
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compounds is observed with temperature increase, which is in good agreement with the Seebeck coefficient behavior found previously. The obtained results show that the figure of merit's maximum value is 1.14 and 0.95 at 1100 K for NiHfGe and PdHfGe, respectively. These ZT values are definitely higher to those reported in the literature for HH compounds such as ZrNiSn(0.3) [47], Ta doped ZrNiSn(0.75) [48], TaIrSn(0.61) [49] and TaRhSn(0.55) [50]. 4. Conclusion
[12]
[13]
[14] [15] [16]
In summary, we have performed ab-initio study, within FPLAPW method to evaluate the structural, electronic, elastic, thermodynamic and thermoelectric properties of XHfGe (X ¼ Ni, Pd) HH compounds with 18 valence electrons. First, the total energy calculations show that the nonmagnetic (NM) state is the more favorable configuration in the cubic MgAgAs (C1b) structure for both materials. Subsequently, the electronic structure and the density of states (DOS) around the Fermi level confirm that the studied materials are of semiconductor character. Afterwards, the elastic constants Cij obey the cubic stability conditions and indicate that NiHfGe and PdHfGe are brittle and ductile, respectively. In addition, using the quasi-harmonic Debye model, the volume V, bulk modulus B, heat capacity Cv, Debye temperature qD and entropy S have been calculated successfully and give more credence for our results, which prove their belonging to a new class of thermodynamically stable HH semiconductors. The BoltzTraP code within the Wien2k program was also used for thermoelectric properties investigated. Our results show that the high figure of merit ZT of ~1.14 and 0.95 at 1100 K has been obtained for NiHfGe and PdHfGe, respectively. Finally, we specify that our initial predictions which demonstrate that NiHfGe and PdHfGe have great potential for high temperature power generation (HTPG) and will lead to fundamental advances in the design of future materials for thermoelectric applications are open to experimental verifications. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.cocom.2019.e00407.
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Computational Condensed Matter journal homepage: http://ees.elsevier.com/cocom/default.asp
Investigation of high figure of merit in semiconductor XHfGe (X ¼ Ni and Pd) half-Heusler alloys: Ab-initio study F. Bendahma a, *, M. Mana b, S. Terkhi a, S. Cherid a, B. Bestani c, S. Bentata a, d a
Technology and Solids Properties Laboratory, Abdelhamid Ibn Badis University, 27000, Mostaganem, Algeria Abdelhamid Ibn Badis University, 27000, Mostaganem, Algeria c Structure, Elaboration and Application of Molecular Materials Laboratory, AbdelhamidIbnBadis University, 27000, Mostaganem, Algeria d Mustapha Stambouli University of Mascara, 29000, Mascara, Algeria b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 April 2019 Received in revised form 24 May 2019 Accepted 25 May 2019
Using the full potential Linearized augmented Plane Wave (FP-LAPW) method to the density functional theory (DFT) as part of the Generalized Gradient Approximation (GGA), we investigated the structural, electronic, elastic, thermodynamic, and thermoelectric properties of ternary XHfGe (X ¼ Ni and Pd) HalfHeusler alloys. The calculation shows that the materials studied are non-magnetic (NM) with semiconductor nature, elastically stable and anisotropic. In the quasi-harmonic model of Debye, the volume V, the bulk modulus B, the thermal capacity Cv, the entropy S and the temperature of Debye qD are also studied. The semi-classical Boltzmann theory as implemented in the BoltzTraP code is applied to study the thermoelectric (TE) properties. The extremely high figure of merit ZT ~1.14 and 0.95 at 1100 K have been achieved for NiHfGe and PdHfGe, respectively, to make these systems promising candidates for TE applications. Published by Elsevier B.V.
Keywords: Half-heusler Generalized gradient approximation (GGA) Semiconductor behavior Thermoelectric applications
1. Introduction In general, Heusler alloys have drawn growing attention during the last few decades in the scientific community [1]. In particular, half Heusler (HH) with general formula XYZ exhibit a big interest for their properties such as shape memory, thermoelectric, ferromagnetic, spin polarization effects and superconductivity [2e10], that could lead to new mechanical and electronic devices. These properties are due to many substitutions of transition metal, a noble metal, or rare - earth elements at both the crystallographic sites X, Y and (IIIeV) elements at the Z sites [11e13].Thermoelectric materials (TEMs) become interesting because of their ability to directly convert heat into electricity [14e16]. This physical phenomenon motivated the researchers to find better materials having a high thermoelectric figure of merit (ZT) approaching ~1, including the three important parameters describing thermoelectric materials which are: Electric conductivity (s/t), Seebeck coefficient (S), and thermal conductivity (k/t) [17,18]. Recently, (HH) alloys with a valence electron count of 18 belong to the semiconducting materials have been extensively studied as potential high temperature
* Corresponding author. E-mail address: [email protected] (F. Bendahma). https://doi.org/10.1016/j.cocom.2019.e00407 2352-2143/Published by Elsevier B.V.
TEMs. This is mainly to their thermal stability, non-toxic components and cost effectiveness and their narrow band gaps including the density of states sharp slope near the Fermi level. Among them, MNiSn (M ¼ Ti, Zr, Hf) and MCoSb (M ¼ Ti, Zr, Hf) half Heusler compounds exhibit semiconductor behavior and have narrow indirect band gap (e.g., Eg ¼ 0.95eV for TiCoSb, Eg ¼ 0.42 eV for TiNiSn and Eg ¼ 0.5 eV for ZrNiSn) [19e22]. So, thermoelectric power generators (TEPGs) better performance, a material having a narrow band gap become a key solution [23]. In this case and in the absence of experimental and theoretical data, our efforts have been focused on the ab-initio study in terms of structural, electronic, elastic, thermodynamic and thermoelectric properties of two half Heusler compounds NiHfGe and PdHfGe. The arrangement of this paper is as follows: A theoretical background is presented in section 2. Results and discussion are presented in section 3. A summary of the results is given in section 4.
2. Computational details In this paper, we have employed the full potential linearized augmented plane-wave (FPLAPW) method to the density functional theory (DFT) implemented with the Wien2k package [24]. The generalized gradient approximation proposed by
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PerdeweBurkeeErnzerhof (GGA-PBE96) [25] was used for exchanges and correlation. The integration of the full Brillouin zone is performed with 12 12 12 k-points mesh. The plane wave cutoff parameter RMT K max was chosen to be 8 where RMT is the smallest radius of the muffin-tin spheres and Kmax gives the magnitude of the largest k-vector in the plane wave expansion. These Half-Heusler compounds XHfGe (X ¼ Ni, Pd) crystallize in the cubic MgAgAs (C1b) structure with the F-43 m space group. In this structure with corresponding Wyckoff positions: X, Hf and Ge atoms occupy 4a (0.25, 0.25, and 0.25), 4b (0, 0, and 0) and 4c (0.5, 0.5, and 0.5) sites, respectively. To study the thermodynamic properties of the cubic XHfGe (X ¼ Ni,Pd) Half-Heusler compounds we have applied the quasi-harmonic Debye model as implemented in the Gibbs program [26].The Boltztrap code which is based on Boltzmann transport theory [27] was used to analysis the transport properties. We need a large number of k-points for the thermoelectric properties and we used 100000k-points in the irreducible wedge of the Brillouin zone, to ensure the convergence of the transport properties. 3. Results and discussions 3.1. Structural properties The total energy-volume dependence estimated by fitting Birch Murnaghan's equation of state is shown in Fig. 1 [28]. It is clear from these plots that for both Half-Heusler alloys (NiHfGe and PdHfGe), with their lower computed total energies, that the nonmagnetic (NM) state is more favorable than the corresponding ferromagnetic (FM) or antiferromagnetic (AFM) configuration in the cubic structure. Table 1 summarizes most important parameters obtained at static equilibrium corresponding to (0K and 0 GPa) such as lattice constant a0, bulk modulus B, its first pressure derivative B’, the volume V0 and minimum total energy (E0) of cubic NiHfGe and PdHfGe Half-Heusler compounds. As it can be observed from Table 1, the ɑ0 values of XHfGe compounds increase in the following sequence: ɑ0 (NiHfGe) ˂ ɑ0 (PdHfGe). As Hf and Ge atoms are the same in the two compounds, this result can be explained by considering the atomic radii of Ni and Pd: R(Ni) ¼ 1.35 Å and R(Pd) ¼ 1.40 Å,i.e. the lattice constant increases with increasing atomic size of the X element in XHfGe compounds. Whereas, the B values decrease in the following sequence: B (NiHfGe) > B (PdHfGe), i.e. in inverse sequence to ɑ0, in agreement with the well-known relationship between B and the unit cell volume: B f V1 0 .
Table 1 Refined crystallographic parameters: lattice constant a0 (bohr), unit cell volume V (bohr3), bulk modulus B (GPa), its pressure derivative B’ and ground state energies E0 (Ry) of cubic NiHfGe and PdHfGe Half-Heusler compounds. Compound
a0
V0
B
B0
E0
NiHfGe PdHfGe
11.09 11.62
343.02 394.11
145.35 137.33
4.48 4.22
37435.672092 44488.094061
3.2. Electronic properties The electronic band structures of XHfGe (X ¼ Ni, Pd) compounds are calculated at the theoretical lattice constant along highsymmetry directions in the first Brillouin zone, as shown in Fig. 2. Using GGA approximation, the valence band maximum (VBM) is 0 eV at the G point but the conduction band minimum (CBM) is 0.66eV and 0.55 eV at the X point for NiHfGe and PdHfGe, respectively. This confirms that NiHfGe and PdHfGe have an indirect band gap equal to 0.66 and 0.55 eV, respectively. For better understanding the electronic properties of the studied half Heuslers and to explain the contribution of different states in the band structures, both total density of states (TDOS) and partial densities of states (PDOS) of XHfGe (X ¼ Ni, Pd) were evaluated respectively as shown in Fig. 3. The Fermi level EF is set at 0 eV. The states Ni (4s2, 3d8), Pd (4d10), Hf (6s2, 5d2), and Ge (4s2,4p2) are treated as valence electrons. In Fig. 3, it can be seen that the region between 4 and 0 eV, is mainly due to (3d Ni/4d Pd) states for NiHfGe and PdHfGe, respectively and a small contribution of 5d states of Hf and 4p of Ge. Furthermore, the region between 0 and 4 eV presents a strong contribution of 5d Hf state and a weak one of (3d Ni/4d Pd) atom for both considered materials. Consequently, NiHfGe and PdHfGe have a semi-conductor behavior and their density of states are dominated by the (Ni/Pd)-d, Hf-d (d-e g, d-t 2 g) and Ge-p states. 3.3. 3. Elastic properties Even, Half-Heusler alloys are known to be remarkably stable and strong materials. The elastic constants of the studied compounds were checked in this study using Stress-strain method in order to confirm their stability. By doing so, only three independent elastic constants Cij, namely C11, C12 and C44 for a cubic system are needed to evaluate all the other elastic moduli. Based on the parameter values presented in Table 2, we remark that the calculated bulk modulus from the elastic constants (B ¼ (1/ 3) (C11 þ 2 C12)) and from Birch Murnaghan equation of state EOS fits are almost identical, which gives more credence to our obtained
Fig. 1. Total energy as a function of unit cell volume forNiHfGe and PdHfGe with ferromagnetic (FM), anti-ferromagnetic (AFM) and non-magnetic (NM) states.
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3
Fig. 2. The calculated band structures for NiHfGe and PdHfGe by using GGA for non-spin polarized state.
Fig. 3. Total and partial density of states for NiHfGe and PdHfGe with GGA approximation.
Table 2 Calculated elastic constants Cij, bulk modulus B (GPa), anisotropy factor A, shear modulus G (GPa), Young's modulus E (GPa) and the Poisson's ratio n of cubic NiHfGe and PdHfGe Half-Heusler compounds. Compound
C11
C12
C44
B
A
G
E
B/G
n
NiHfGe PdHfGe
243.24 179.16
97.15 117.36
104.78 90.78
145.85 137.96
1.43 2.94
90.71 53.87
225.4 143.0
1.61 2.56
0.22 0.35
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results (see Table 1). In addition, other important macroscopic parameters such as the anisotropy factor A, the shear modulus G, the Poisson's ratio n and the Young's modulus E have also been investigated in this study using the Voigt-Reuss-Hill approximations for the cubic structures [29e31]:
critical one. Another important characteristic measuring the resistance offered by an elastic body to deformation, that is to say the solid stiffness measurements estimated by Young's modulus (E) have also been investigated. The very large E values of 225.4 and 143.0 GPa for NiHfGe and PdHfGe, respectively are in fact an indication that the considered half-heusler are stiffer.
BV ¼ BR ¼ ðC11 þ 2C12 Þ=3
(1)
GV ¼ ðC11 C12 þ 3C44 Þ=5
(2)
3.4. 4.Thermodynamic properties
GR ¼ 5C44 ðC11 C12 Þ=½4C44 þ 3ðC11 C12 Þ
(3)
A ¼ 2C44 =ðC11 C12 Þ
(4)
G ¼ ðGv þ GR Þ=2
(5)
Thermodynamic properties of NiHfGe and PdHfGe compounds have also been computed for supplementary useful information. So, thermal properties determination in the range of 0e1200 K and pressure effect in the range of 0e20 GPa were investigated for this purpose. Firstly, Fig. 4 shows the volume-temperature relationship at different pressures for both NiHfGe and PdHfGe compounds, in which, the volume (V) increases with increasing temperature with very slow rate. Secondly, at a given temperature, there is a decrease in volume with increasing pressure. At initial state (0K and 0 GPa), the calculated volumes for NiHfGe and PdHfGe are 343.8 and 394.13 bohr3 respectively. These volume values confirm those obtained in the structural properties, which are 343.02 and 393.11 bohr3 for NiHfGe and PdHfGe, respectively. We notice that during compression, the volume reduction of NiHfGe is less than the PdHfGe one because of the bulk modulus of the first compound is higher. In Fig. 5 are shown the effects of pressure and temperature on the bulk modulus (B). The bulk modulus value is almost decreasing linearly with increasing temperature. This decrease in bulk modulus values is an indication that XHfGe (X ¼ Ni, Pd) fcc crystal structure is entering a new bonding regime and becoming softer. At 0K and 0 GPa conditions, the obtained bulk modulus values for NiHfGe and PdHfGe are 145.72 and 137.29 GPa respectively which are in good agreement with the previously determined ones using the structural and elastic properties. Furthermore, at ambient temperature and 0 GPa, the high bulk modulus values of about 140.01 and 135.38 GPa corresponding to NiHfGe and PdHfGe is respectively; confirm that these later ternary Heuslers present a strong hardness and an important compressibility rate [35e37]. We can see from heat capacity Cv versus temperature plots at
On the other hand, equations (6) and (7) were used to compute the Young's modulus E and Poisson's ratio n
E ¼ 9BG=ð3B þ GÞ
(6)
n ¼ ð3B 2GÞ=½2ð3B þ GÞ
(7)
It is noticed that for a cubic crystal, requirements for the compounds mechanical stability conditions should be (C11 C 12> 0); (C11 þ 2 C12 > 0; C44 > 0 and C12 1.75, a material is considered to be ductile; otherwise, it will be fragile and hard. In this study, the B/G ratio was equal to 1.61 and 2.56 for NiHfGe and PdHfGe respectively; indicating then, that the first material is in fact fragile, while the second one is ductile. Poisson's ratio n can also be used for estimating the ductility of a material. According to the Frantsevich rule [34], if n > 1/3, the material is considered to be ductile, otherwise it is considered to be brittle. Based on the obtained results for n values presented in Table 2, we can say that NiHfGe and PdHfGe are brittle and ductile, respectively and are also anisotropic since their A values are greater than the
Fig. 4. Volume of unit cell parameters versus temperature for NiHfGe and PdHfGeat different pressures.
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5
Fig. 5. Temperature dependence of the bulk modulus at different pressures for NiHfGe and PdHfGe.
different pressures presented in Fig. 6 that the pressure has no effect on Cv.Whereas increasing the temperature three important regions were identified: In the first region when T 300K, the increase in temperature, will increase Cv rapidly for both materials, according to Debye law (since Cv is proportional to T3) [38]. At 300 K and 0 GPa conditions, Cv value approaches approximately 68.26 and 68.41 Jmol1K1for NiHfGe and PdHfGe, respectively. In the second region when 300˂ T ˂800 K, Cv increases slowly with increasing temperature; this is mainly due to the atomic vibrations. In the third region when T 800K, Cv increases slowly with temperature. At temperature limit (T ¼ 800 K), no more rate increase is observed and its value approaches approximately 73.42 and 73.84 J mol1 K1 for NiHfGe and PdHfGe, respectively. Furthermore, it should be noted that the heat capacity
was quite close to the DulongePetit classical limit, (75 Jmol1K1), a common phenomena in all solids [39]. The most important parameter in the quasi harmonic Debye model approach is the Debye temperature qD, which relates different physical properties of solids, such as specific heat, elastic constants and melting temperature [40]. Fig. 7 shows the dependence of the Debye temperature qD and temperature at different pressures. Obtained values of 371.24 and 346.17K at 0 GPa and 300K for NiHfGe and PdHfGe compounds respectively indicate that PdHfGe has a lower thermal conductivity than NiHfGe at these conditions. In addition, with the pressure increase from 0 to 20 GPa at the given temperature, there is an increase in calculated qD values. However, at a constant pressure, qD decreases linearly with increasing temperature. Material's entropy S knowledge provides an essential insight on its vibration properties, which lead to important consequences for the performance of
Fig. 6. Variation of the heat capacity Cvwith temperature at different pressures forNiHfGe and PdHfGe.
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Fig. 7. Debye temperature versus temperature of NiHfGe and PdHfGe at different pressures.
many dispositives as heat engines, refrigerators, and heat pumps. Effects of temperature on entropy at fixed pressures are shown in Fig. 8. It is observed that the stable equilibrium state corresponds to the starting curves values (S ¼ 0) at 0 K and 0 GPa, then, the entropy will increase exponentially with temperature above 0 K. In addition, the entropy obtained values of 88.41 and 90.94 Jmol1K1for NiHfGe and PdHfGe, respectively, indicate that PdHfGe is less ordered than NiHfGe at ambient and 0 GPa conditions. At temperatures exceeding 700K the quantity ‘entropy’ becomes very high. This is due mainly to both, the increase of the vibration entropy of a material with increasing temperature and its decrease with increasing cohesive energy. 3.5. 5.Thermoelectric properties Once the structural stability of the two materials realized, by studying the elastic and thermodynamic properties, their thermoelectric ones must be checked in order to have a clear idea and
more information about these HH alloys. The evolution of electrical conductivity data (s/t) per relaxation time (t z 1015 s) within a temperature ranging from 0 to 1100 K is represented in Fig. 9. In which two different temperature ranges are obtained from the curves: In the range of (300 ˂ T ˂ 600 K), results obtained in this study show that the top electric conductivity is 5.54 1019 and 6.98 1019(Ums)1 for NiHfGe and PdHfGe, respectively, at the ambient temperature (300K) meaning that PdHfGe is a good electrical conductor than NiHfGe at ambient. Beyond this temperature value, the electric conductivity decreases as temperature increases for the two compounds. In the range of (600 T 1100 K), there is slow increase in the electric conductivity with temperature. This temperature dependence is essentially due to both the intrinsically narrow band gaps between 0.66 eV (NiHfGe) and 0.55 eV (PdHfGe), rising then s/t to large values, and the sharp increase in the density of states around the Fermi level. The Seebeck coefficient (S) or thermoelectric power is related to the fact that electrons are both carriers of electricity and heat. For a good thermoelectric device, a
Fig. 8. Variation of entropy with temperature at different pressures of NiHfGe andPdHfGe.
F. Bendahma et al. / Computational Condensed Matter 21 (2019) e00407
7
Fig. 9. Electrical conductivity versus temperature of NiHfGe and PdHfGe.
Fig. 11. Thermal conductivity versus temperature of NiHfGe and PdHfGe.
high Seebeck coefficient is required. In Fig. 10 are shown the Seebeck coefficient for NiHfGe and PdHfGe compounds as a function of temperature. We observe clearly that the highest value of Seebeck coefficient for both compounds is achieved at high temperatures (Seebeck coefficient around 150 mV/K for NiHfGe and 138 mV/K for PdHfGe at 1100 K). Furthermore, both the valence band maximum and conduction band minimum are at low asymmetry points of the Brillouin zone, presenting then, that the present electronic properties performance of the studied materials giving rise to their high Seebeck coefficients [41], which is an important factor for the efficient behavior of thermoelectric devices. This means that these HH compounds semiconductor behaviors are good thermoelectric devices. The thermal conductivity (K) of a material is a measure of its ability to energy transfer as heat. It is defined as: K¼ Ke þ Kl, where the Ke and Kl are respectively the electrical and lattice vibrations of total thermal conductivity K [42]. In this paper research, the BoltzTraP code used which neglects the lattice vibrations part (Kl) compared to the electronic part (Ke). According to Wiedemanne Franz law, the electronic thermal conductivity (Ke) is defined as follows: Ke ¼ LsT, (L is the Lorentz number ¼ 2.44 108 W.U. K2for free electrons). Plots of the thermal conductivity versus temperature for both compounds are presented in Fig. 11. At room
temperature, the thermal conductivities are 1.73 1014 and 1.16 1014 W/m K s for NiHfGe and PdHfGe, respectively. According to these calculated values, the thermal conductivities at ambient are lower than those corresponding to (T) exceeding 300 K. These low values in (K) are due to the atomic contribution to phonons and their states densities (DOS). At T > 300 K, the thermal conductivity increase for both materials is related to the force of attraction between the atoms that causes the lattice vibrations. Also, it's noted that, the lower thermal conductivity at the ambient is proportional to the higher electrical one obtained previously. According to WiedemanneFranz law which states that a good thermoelectric material requires a low thermal conductivity with high electrical conductivity, our result is in good agreement with this law [43]. Knowing that the thermoelectric energy TE conversion efficiency depends on the transport coefficients of the constituent materials through the figure of merit ZT ¼ S2s T/K, where S, s, K, and T are the Seebeck coefficient, electrical conductivity, thermal conductivity, and absolute working temperature, respectively [44]. A material defined by a ZT 1 represents the best elements for the thermoelectric devices [45,46]. In this context, evolutions of ZT versus temperature for NiHfGe and PdHfGe are represented in Fig. 12. According to the quasi-linear profile, an increase of ZT for the two
Fig. 10. Variation of the Seebeck coefficient as a function of temperature of NiHfGe and PdHfGe.
Fig. 12. Figure of merit ZT versus temperature of NiHfGe and PdHfGe.
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compounds is observed with temperature increase, which is in good agreement with the Seebeck coefficient behavior found previously. The obtained results show that the figure of merit's maximum value is 1.14 and 0.95 at 1100 K for NiHfGe and PdHfGe, respectively. These ZT values are definitely higher to those reported in the literature for HH compounds such as ZrNiSn(0.3) [47], Ta doped ZrNiSn(0.75) [48], TaIrSn(0.61) [49] and TaRhSn(0.55) [50]. 4. Conclusion
[12]
[13]
[14] [15] [16]
In summary, we have performed ab-initio study, within FPLAPW method to evaluate the structural, electronic, elastic, thermodynamic and thermoelectric properties of XHfGe (X ¼ Ni, Pd) HH compounds with 18 valence electrons. First, the total energy calculations show that the nonmagnetic (NM) state is the more favorable configuration in the cubic MgAgAs (C1b) structure for both materials. Subsequently, the electronic structure and the density of states (DOS) around the Fermi level confirm that the studied materials are of semiconductor character. Afterwards, the elastic constants Cij obey the cubic stability conditions and indicate that NiHfGe and PdHfGe are brittle and ductile, respectively. In addition, using the quasi-harmonic Debye model, the volume V, bulk modulus B, heat capacity Cv, Debye temperature qD and entropy S have been calculated successfully and give more credence for our results, which prove their belonging to a new class of thermodynamically stable HH semiconductors. The BoltzTraP code within the Wien2k program was also used for thermoelectric properties investigated. Our results show that the high figure of merit ZT of ~1.14 and 0.95 at 1100 K has been obtained for NiHfGe and PdHfGe, respectively. Finally, we specify that our initial predictions which demonstrate that NiHfGe and PdHfGe have great potential for high temperature power generation (HTPG) and will lead to fundamental advances in the design of future materials for thermoelectric applications are open to experimental verifications. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.cocom.2019.e00407.
[17] [18]
[19] [20]
[21] [22] [23] [24]
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