The structural, elastic and thermoelectric properties of Fe2VAl at pressures

Journal of Alloys and Compounds 565 (2013) 22–28

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The structural, elastic and thermoelectric properties of Fe2VAl at pressures Bin Xu a,⇑, Xingfu Li a, Gongqi Yu b, Jing Zhang a,⇑, Shanshan Ma a, Yusheng Wang a, Lin Yi c a

Department of Mathematics and Information Sciences, North China Institute of Water Conservancy and Hydroelectric Power, Zhengzhou 450011, China The Second Artillery Command College, Wuhan 430012, china c Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China b

a r t i c l e

i n f o

Article history: Received 29 September 2012 Received in revised form 20 February 2013 Accepted 24 February 2013 Available online 14 March 2013 Keywords: Heusler alloys First-principles calculation Elastic constants Thermoelectric properties

a b s t r a c t First principles calculations are used to investigate the structural, elastic and thermoelectric properties of Fe2VAl at pressures. The calculated lattice constant, bulk modulus and pressure derivative are very close to those found experimentally and theoretically. The band structures of Fe2VAl are significantly modified near Fermi level with increasing pressures. The total density of states decreases near the Fermi level, whereas increases at the Fermi level with increasing pressures. The elastic constants, shear modulus, Young’s modulus and bulk modulus linearly increase monotonously when pressure is enhanced. It is seen from Table 2 that our results satisfy the stability conditions at different pressures. It is noted that the calculated value of B0/G increases with the pressure increasing and Fe2VAl reveals ductility at high pressure. The Seebeck coefficients S of Fe2VAl rapidly increases for p-doped and n-doped in the low-doping region as the carrier concentration increase. The results show that the calculated electronic conductivity r/s of the undoped Fe2VAl is low; however, as the carrier concentration increase, the calculated electronic conductivity r/s of Fe2VAl rapidly increases for p-doped and n-doped. The findings also show that the power factor S2r/s rapidly increases for n-doped and decreases for p-doped with the increasing pressure, indicating that the n-doped can greatly enhance the electrical transport properties of Fe2VAl compound. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction There has been a renewed interest in the field of thermoelectric driven by the need for the reuse of exhausted heat in these days. So, we have been studying filled-skutterudites, half-Heusler alloys and Heusler alloys to develop a new thermoelectric material or to improve the thermoelectric properties [1–3]. The present study sheds theoretical insight on better understanding of the properties of the Heusler type compounds Fe2VAl. Fe2VAl is one such class of alloys that has been the topic of interest over the past decade because of its rich variety of unusual transport and magnetic properties [4–7]. Semiconductors, semimetals, normal Pauli metals, weak ferromagnets, antiferromagnets, as well as half-metallic ferromagnets exist in this class of materials. Among these alloys, the Fe2VAl compound has been characterized as a nonmagnetic semimetal [5]. A further optical conductivity study on Fe2VAl has confirmed the existence of a pseudogap in the vicinity of the Fermi level [8]. In recent years, there has been considerable interest in the electrical resistivity, specific heat, and thermoelectric properties of Fe2VAl. Many efforts have been made to extend the investigation on (Fe1xCox)2VAl to the low-temperature range by Liu and Morelli

⇑ Corresponding authors. Tel./fax: +86 371 65790758. E-mail address: [email protected] (B. Xu). 0925-8388/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2013.02.160

[9]. Terazawa et al. identified that Ta or W substitution for V is useful for decreasing the lattice thermal conductivity of the Fe2VAl Heusler alloy without greatly affecting the electron transport properties by using first-principles cluster calculations [10]. Furthermore, the effects of W and Sb substitution on thermoelectric properties of the Heusler alloy were evaluated by Fe2V1xWxAl [11] and Fe2VAl1xSbx [12] sintered alloys, respectively. Bilc et al. studied the role of defects and disorder for electronic and thermoelectric properties of Fe2VAl [13]. With the aim of studying the effect of increasing the concentration of vanadium, Kumar et al. calculated the electronic and optical properties for Fe3xVxAl (x = 0–3) compounds [14]. Polycrystalline (Fe1xCox)2VAl (0 6 x 6 0.5) alloys were synthesized through a combined process of arc-melting, annealing and sintering by Lu et al. [15]. The polycrystalline samples of Fe2V1xNbxAl were prepared and the thermoelectric properties were characterized by Kurosaki et al. [16]. They investigated thermoelectric properties from 300 to 850 K. A Fe2VAl sintered alloy with fine microstructure was synthesized to evaluate the effect of grain size on thermoelectric properties by Mikami [17]. Vasundhara et al. investigated the temperature variation of electrical resistivity and Seebeck coefficient of Heuslertype Fe2VAl1xMx alloys (M = B, In, and Si) [18]. Lue et al. revealed the effects of partial substitution of Nb onto the V sites of Fe2VAl by measuring the electrical resistivity, Seebeck coefficient, and thermal conductivity as a function of temperature [19]. The effects of

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B. Xu et al. / Journal of Alloys and Compounds 565 (2013) 22–28

Si substitution on the temperature-dependent electrical resistivity, Seebeck coefficient, as well as thermal conductivity have been investigated in the Heusler-type compound Fe2VAl by Lue et al. [20]. We have studied the magnetic and optical properties of Fe2VAl and Fe3Al also [21]. Nishino et al. reported the thermoelectric properties of the Heusler-type Fe2VAl1xGex alloys with compositions 0 6 x 6 0.20 [22]. The electronic structure of Ti1xScxNiSn half-Heusler systems have calculated by Stopa et al. using the muffin-tin KKR-CPA method as well as the full-potential KKR technique for parent compounds TiNiSn and ScNiSn. The theoretical values of the Seebeck coefficient obtained from the linear extrapolation of S/T to finite temperatures remain in satisfying agreement with experimental data. Their results confirm the electron transport properties of Ti1xScxNiSn [23]. In this paper, the thermoelectric performance for Heusler-type compounds Fe2V1xNbxAl is investigated by first principles calculations based on the full potential linearized augmented plane wave (FPLAPW) method and the semi-classical Boltzmann theory. 2. Computational method

N¼

X ~ v~k ~ v~k~s~k

ð1Þ

~ k

where ~ v~k is the group velocity associated with that state and s~k is the relaxation time. Once it is known, all transport coefficients, necessary to determine ZT, can be directly obtained [36–45]. For calculating the transport tensors, 26  26  26 kmesh was used to calculate eigenenergies by the BOLTZTRAP code [45], which is based on a well tested smoothed Fourier interpolation to obtain an analytical expression of the bands. The original k mesh was interpolated onto a mesh five times as dense as the original [45]. The relaxation time s is inserted as a constant and doping is treated within the rigid band approximation [46]. In addition, the temperature dependence of energy band structure is neglected.

3. Results and discussion 3.1. Electronic structure The crystal structure of Fe2VAl has Fm-3m space group as illustrated in Fig. 1. There are 16 atoms in the unit cell of the Heusler compounds Fe2VAl. The Fe atoms occupy 8c positions (1/4 1/4 1/ 4), the Al atoms occupy the 4a positions (0 0 0) and the V atoms are placed at 4b sites (1/2 1/2 1/2). All atoms have eight nearest

Fig. 1. Crystal structure of Fe2VAl.

-7475.585 -7475.590 -7475.595

Energy (eV)

Our calculations were performed using the FPLAPW method as implemented in the WIEN2K code [24]. The exchange–correlation energy is in the form of Perdew– Burke–Ernzerhof (PBE) [25] with generalized gradient approximations (GGA). We take RmtKmax equal to 8.5 and make the expansion up to l = 10 in the muffin tins (MT) spheres. The use of the full-potential ensures that the calculation is completely independent of the choice of the sphere radii. Nonoverlapping MT sphere radii of 1.8, 2.0 and 2.2 a.u. were used for Fe, V and Al atoms, respectively. The elastic constants have been performed using the Cambridge Serial Total Energy Package (CASTEP) [26], which is based on the density functional theory (DFT) using the plane wave pseudopotential (PWPP) method to describe the electron– electron and the electron–core interaction. The generalized gradient approximation exchange correlation functional in the scheme of Perdew–Bueke–Ernzerhof (GGA– PBE) is employed. The Vanderbilt-type ultrasoft pseudopotentials [27] are employed to generate the pseudopotentials for Mg and H atoms. The kinetic cutoff energy for the plane wave expansion is taken to be 500 eV, which was large enough to obtain good convergence. The states Fe: 3d64s2, V: 3s23p63d3 4s2, Sn: 5s2 5p2 and Al: 3s23p1 were treated as valence states. In the Brillouin zone integrations, 10  10  10 k points were determined according to Monkhorst–Pack scheme [28]. Based on the Broyden Fletcher Goldfarb Shenno (BFGS) [29] minimization technique, the system reached the ground state via self consistent calculation when the total energy is stable within 5  107 eV/atom, and less than 102 eV/Å for the force. The method for the calculation of transport properties of a crystalline solid is based on the semiclassical Boltzmann theory [30] and the rigid band approach. This approach has been successful used in rationalizing and predicting the optimal doping level of known compounds [31–33]. The rigid band approach to conductivity is based on the transport distribution, which is the kernel of all transport coefficients. The transport distribution is defined as [34,35]

-7475.600 -7475.605 -7475.610 -7475.615 -7475.620 280

290

300

310

320

330

340

350

3

Volume (Å ) Fig. 2. Total energy per Fe2VAl versus volume.

neighbors. The V and Al atoms have eight Fe atoms as the nearest neighbors while for Fe there are four V and four Al atoms. We have determined theoretically the lattice constant of Fe2VAl by minimizing the total energy using the first-principles electronic structure calculation within the Fm-3m symmetry. The total energy of the Fe2VAl as a function of the volume per formula unit is shown in Fig. 2. The obtained lattice constant a, the bulk modulus B0, and the pressure derivative of the bulk modulus B0 for Fe2VAl at zero pressure are listed in Table 1. We have fitted our ab initio data to a Birch–Murnaghan equation of state (EOS) and obtained the following results: V0 = 314.5639 Å3, B0 = 216.3 GPa and B0 = 5.1. After geometry optimization, we find that the calculated lattice constant is 5.7129 Å for Fe2VAl, which is very close to those found experimentally and theoretically [47–53]. It is easy to find that the calculated bulk modulus B0 is consistent with other calculated values and the experimental data within 0.85% [48–53], but the calculated pressure derivative B0 is lower than the other calculated value. The excellent agreement strongly supports the choice of pseudopotentials and the GGA approximation for the current study.

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B. Xu et al. / Journal of Alloys and Compounds 565 (2013) 22–28

Table 1 Calculated lattice constants in Å, bulk modulus B0 in GPa, and its pressure derivative B0 of Fe2VAl at the theoretical equilibrium volume. Method

Lattice constant

B0

B0

Theory, this work Expt. Other theory

5.7129 5.76246, 5.75547, 5.76148, 5.7649, 5.71250 5.7650, 5.71251, 5.71252

5.1

216.3

5.3

21250, 220.851, 21252

In this paper all conversions between volume and pressure are based on the two parameter Murnaghan equation [54]

V V0

0.6

1

ð2Þ

5.7

(a)

5.6

20GPa

0.2

40GPa 60GPa

0.0

80GPa 100GPa

-0.2 -0.4 -0.6 -0.8 -1.0 W

K

12 0.6

0GPa 20GPa 40GPa 60GPa 80GPa 100GPa

10 8

0.4 0.2 0.0 -0.2 -0.1 0.0

6

0.1

0.2

4 2

-10

-8

-6

-4

-2

0

2

4

6

8

Energy (eV) 4.0

2.5 2.0 1.5 1.0 0.5 0.0 -1.0 -0.5 0.0 0.5 1.0

0 Gp Fe eg

3.5

Fe t 2g V eg

3.0

V t2g Al s Al p 100 Gp Fe eg

2.5 2.0

Fe t2g V eg

1.5

V t 2g Al s Al p

1.0

5.5

Z W

L

Fig. 4. Band structure of Fe2VAl at different pressures.

DOS

where V0 and B0 are the volume and the bulk modulus per Fe2VAl at zero pressure, respectively; B0 = dB0/dp, which is the bulk modulus pressure derivative. The calculated cell parameter a and cell volume V as a function of pressure is presented in Fig. 3. It is easy to observe that the cell parameter a and cell volume V decrease with the pressure increasing. Fig. 4 presents the results of band structures along symmetry lines at different pressures. The formation of a band gap in Fe2VAl revealed from our calculation is consistent with the previous result calculated by Weht et al., Singh et al. and Bilc et al. at 0 GPa [55,56]. It is clearly seen from Fig. 4 that the band structures of Fe2VAl are significantly modified near Fermi level with increasing pressures. The hole bands along K–C–D are also indicated in Fig. 4. It is shown that the two hole bands consist of one twofold degeneracy. With increasing pressures, the population of the hole carriers above the Fermi energy gradually increases. The calculated energy band gaps at the C point increase with the pressure increasing. The valence band tends to shift toward the high-energy region at the C point and, furthermore, the conduction band shifts toward the higher-energy region also. The valence band crossing with the conduction band at the X point at 40 GPa. This is attributed to band widening by strengthened overlap interaction of the wave function with the shrinking atomic distance under high pressure. Fig. 5 shows total density of states (DOS) at different pressures and partial density of states at 0 and 100 GPa for Fe2VAl. The total DOS decreases near the Fermi level, whereas increases at the Fermi level with increasing pressures in Fig. 5a. It is seen that the valence bands crossings the Fermi level mainly due to Al p orbital and Fe d orbital moving up along C point, and the conduct bands move up due to Al p orbital, Fe d and V d orbital moving down along C point in Fig. 5b. An analysis of partial contributions to the DOS presented in Fig. 5b shows that the top of the valence band is mostly contributed by Fe t2g states, while the bottom of the conduction band by Al p, Fe eg and V eg states.

0GPa

0.4

Energy (eV)

B0 B0

0.8

#

B0

DOS

P¼

"

1.0

0.5

5.4 5.3

0.0

5.2

(b)

180 170

-10

-8

-6

-4

-2

0

2

4

Energy (eV) Fig. 5. Total density of states of Fe2VAl at different pressures (a). Partial density of states of Fe2VAl at 0 GPa and 100 GPa (b).

160 150

3.2. Elastic properties

140 0

10

20

30

40

50

60

70

80

90 100

Pressure (GPa) Fig. 3. Cell parameters a (a), Cell volume V (b) as a function of pressure for Fe2VAl.

The calculated elastic constants of Fe2VAl are presented in Table 2. A simple relationship, which empirically links the plastic properties of materials with their elastic moduli was proposed by Pugh

25

B. Xu et al. / Journal of Alloys and Compounds 565 (2013) 22–28

Table 2 Calculated elastic constants, shear modulus (G), and Young’s modulus (E) all expressed in GPa, Poisson’s ratio (m), Bulk modulus (B0), shear anisotropic factor A and Lame constants (l and k) for Fe2VAl at the theoretical equilibrium volume. P

C11

C12

C44

G

E

m

B0

A

l

k

0 10 20 30 40 50 60 70 80 90 100

404.5 453.7 505.7 557.8 612.9 669.3 714.1 770.6 826.6 864.8 930.0

122.1 140.7 163.8 187.4 215.3 246.1 266.1 298.4 331.3 347.4 391.4

136.1 152.4 171.7 187.3 201.6 215.3 228.1 241.6 253.7 263.4 273.1

138.1 154.0 171.4 186.4 197.6 213.8 226.5 239.4 251.3 261.5 271.6

347.9 387.1 425.8 463.6 501.0 537.0 570.0 604.0 637.1 665.6 698.1

0.204 0.237 0.245 0.251 0.260 0.269 0.272 0.279 0.286 0.287 0.296

216.3 245.0 277.8 310.9 347.8 387.2 415.4 455.8 496.4 519.9 570.9

0.9639 0.974 1.0044 1.0113 1.0141 1.0175 1.0183 1.0233 1.0244 1.0182 1.0141

136.1 152.4 171.7 187.3 201.6 215.3 228.1 241.6 253.7 263.4 273.1

132.3 148.8 162.4 183.3 209.7 238.7 257.9 287.3 319.3 337.9 383.8

600 K G G

Stability criteria

500

400

300

200

0

20

40

60

80

100

Pressure (GPa) Fig. 7. Dependence of stability criteria of Fe2VAl compound with pressure.

120 100

Cauchy pressure, CP 80

GPa

[57]. The shear modulus G represents the resistance to plastic deformation, while the bulk modulus B represents the resistance to fracture. It is shown from Fig. 6 that the elastic constants C11, C12, C44, shear modulus (G), Young’s modulus (E) and bulk modulus B0 all expressed in GPa linearly increase monotonously when pressure is enhanced. Moreover, C44 and G increase very slowly and almost overlap with the elevated pressure. Our calculated elastic constants C11, C12 and C44 obtained from the present calculation are underestimated by 2.7%, 2.6%, and 19.1%, respectively, when compared with the other theoretical value [52]. The calculated shear modulus (G), and Young’s modulus (E), Poisson’s ratio (m), and bulk modulus B are underestimated by 14.0%, 10.2%, 0.8% and 2%, respectively, comparing with the other theoretical values [52]. The deviations in the calculated elastic constants from the experimental values are partly ascribed to temperature and volume effects. Another point of caution is the fact that the calculated values pertain to 0 K, while experiments are performed at room temperature with the pressure increasing, the change ratios of C11, C12, C44, G, E and B0 for Fe2VAl are 525.5%, 269.3%, 137.0%, 133.5 %, 350.2% and 354.6%, respectively. The change of C11 is a little more sensitive to pressure than that of others, and C44 is the most unresponsive one. For an isotropic crystal, the shear anisotropic factor A is equal to 1, while any value smaller or larger than 1 indicates anisotropy. We can see that Fe2VAl is isotropy from Table 2. l and k are Lame constants (l and k) for isotropic material. In Fig. 7, we show the dependence of stability criteria of Fe2VAl compound with pressure. For a cubic crystal under pressure P, the generalized elastic stability criteria [58–60] are: K ¼ 13 ðC 11 þ 2C 12 þ PÞ > 0; G ¼ 12 ðC 11  C 12  2PÞ > 0; G0 ¼ C 44  P > 0 and CS = Our results for elastic constants in Table 2 satisfy these

60 40 20 0

900

C11

800

Elastic moduli (GPa)

0

C12

500 400 300 200 100 0

60

80

100

Fig. 8. Cauchy pressure with pressure.

G E B

600

40

Pressure (GPa)

C44

700

20

20

40

60

80

100

Pressure (GPa) Fig. 6. Calculated pressure dependence of Cij, G, E and B0 for Fe2VAl.

stability conditions at different pressures. Therefore, the cube structure Fe2VAl is mechanical stable at pressure up to 100 GPa. Cauchy pressure, CP  C12C44 is shown in Fig. 8. It is noticeable that the Cauchy pressure, CP  C12C44 is 14.0 for Fe2VAl at 0 GPa, which is higher than the other theoretical value (CP = 45 GPa) [52]. The Cauchy pressure is negative below 30 GPa, however, it is positive above 30 GPa. For metallic bonding the Cauchy pressure is typically positive and for nonmetallic bonding the Cauchy pressure is typically negative. Fe2VAl is typically a covalent character of bonding.

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B. Xu et al. / Journal of Alloys and Compounds 565 (2013) 22–28

4

2.1

n-type 2.0 3

B/G

1.9 2 1.8

p-type

1

S

2

1.7

0GPa 10GPa 20GPa 30GPa 40GPa 50GPa 60GPa 70GPa 80GPa 90GPa 100GPa

1.6 0

20

40

60

80

100

0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

0.2

0.4

0.6

0.8

1.0

Pressure (GPa) Fig. 12. Calculated power factor S2r/s of Fe2VAl as a function of the chemical potential at room temperature under different pressures.

Fig. 9. B/G ratio with pressure for Fe2VAl.

Fe2VAl can be classified as a brittle material. However, the other value of B0/G is 1.37, which is lower than our work [52]. It is noted that the calculated value of B0/G increase with the pressure increasing and Fe2VAl reveals ductility at high pressure. This is the first theoretical prediction of the elastic constants at different pressures. Unfortunately, no experimental and theoretical data of elastic constants are available for our comparison at different pressures, which still awaits experimental confirmation.

60

n-type

0GPa 10GPa 20GPa 30GPa 40GPa 50GPa 60GPa 70GPa 80GPa 90GPa 100GPa

40

/ )

20

S(

0

p-type

-20

3.3. Thermoelectric performance

-40

To assess the thermoelectric performance of Fe2VAl B, we inspect the details of the results on transport properties. In Figs. 10–12, we plotted the chemical potential dependence of electrical transport properties of Fe2VAl. Carrier scattering relaxation time s is treated as a constant in this work [62]. Seebeck coefficients are sindependent, but the electrical conductivities and power factors have to be presented with respect to s. There is no evident correlation between the peaks of Seeback coefficient or Electrical conductivity over relaxation time and the peaks of power factor over relaxation time. The Seebeck coefficients for Fe2VAl are displayed in Fig. 10 at different pressures. In our calculation, the chemical potential l corresponds to the ground state Fermi energy EF in good approximation (l(T = 0) = EF, with the strict definition that the Fermi energy is the energy of the highest occupied state at T = 0). The high chemical potential corresponds to a heavy doping, which may fall outside the rigid-band picture, so we will only focus on those peaks around the Fermi level [41]. The Seebeck coefficient of Fe2VAl is negative for n-doped, while positive for p-doped. It is seen that the calculated Seebeck coefficient S of undoped Fe2VAl is low; however, as the carrier concentration increase, the Seebeck coefficients S of Fe2VAl rapidly increases for p-doped and n-doped in the low-doping region. Our calculations are good agreement with available theoretical results [13]. There is a main peak near the Fermi energy for the Seebeck coefficient at different pressures for n-doped. It is seen that the Seebeck coefficients S decreases with the increasing pressure in the low-doping region, and then the change of the Seebeck coefficients S is small with the increasing pressure in the high-doping region for n-type doping. However, the Seebeck coefficients S always decreases with the increasing pressure for p-type doping. In Fig. 11 we show the calculated electronic conductivity r/s of Fe2VAl. It is seen that the calculated electronic conductivity r/s of the undoped Fe2VAl is low. However, as the carrier concentration increase, the calculated electronic conductivity r/s of Fe2VAl

-60 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 10. Calculated Seebeck coefficient S of Fe2VAl as a function of the chemical potential at room temperature under different pressures.

8

n-type 6

4

0GPa 10GPa 20GPa 30GPa 40GPa 50GPa 60GPa 70GPa 80GPa 90GPa 100GPa

p-type

2

0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 11. Calculated electronic conductivity r/s of Fe2VAl as a function of the chemical potential at room temperature under different pressures.

A high B0/G ratio may then be associated with ductility, whereas a low value would correspond to a more brittle nature. The critical value separating ductile and brittle materials is around 1.75 [61]. The value of B0/G is shown in Fig. 9. In the case of Fe2VAl the value of B/G is 1.566 from our calculated values in Table 2, and therefore

B. Xu et al. / Journal of Alloys and Compounds 565 (2013) 22–28

rapidly increase for p-doped and n-doped in Fig. 11. As the pressures increase, the electronic conductivity r/s rapidly increases. It is interesting that the electronic conductivity r/s of n-doped increases faster than that of p-doped with the increasing pressure, which shows that n-type doping in the Fe2VAl compound may be more favorable than p-type doping. The electronic conductivity r/s is as much as around two times higher at 100 GPa than that at 0 GPa for the n-doped material, which shows that the electronic conductivity can be rapidly enhanced by increasing pressure. The power factor of Fe2VAl sample, S2r/s is plotted in Fig. 12. There is one peak for n-doped; however there are two peaks for p-doped for the calculated power factor S2r/s near the Fermi level at 0 GPa. One peak is low, the other is high. The peak of n-doped is lower than that of p-doped at 0 GPa. The calculated power factor of Fe2VAl rapidly increases for n-doped and p-doped as the carrier concentration increase. However, the power factor S2r/s of ndoped increase faster than that of p-doped with an increase of the carrier concentration. It is seen that the power factor S2r/s rapidly increases for n-doped and decrease for p-doped with the increasing pressure, indicating that the n-doped can greatly enhance the electrical transport properties of Fe2VAl compound. The increase of the power factor S2r/s is mainly due to the enhanced electronic conductivity, but Seebeck coefficient almost does not work. In particular, the maximum value reaches 3.95  1011 W/K2 cm s at the pressure of 100 GPa for n-doped, which is much higher than those of Fe2VAl (1.97  1011 W/K2 cm s) at 0 GPa. This indicates that the thermoelectric properties of Fe2VAl would be improved effectively at high pressure. It can be seen that n-type doping in the Fe2VAl compound may be more suitable for thermoelectric materials than p-type doping.

4. Conclusion First principles calculations are used to investigate the structural, elastic and thermoelectric properties of Fe2VAl at pressures. The calculated lattice constant, bulk modulus and pressure derivative are very close to those found experimentally and theoretically. The cell parameters a and cell volume V decrease with the pressure increasing. The formation of a band gap in Fe2VAl revealed from our calculation is consistent with the previous result. It is clearly seen from Fig. 4 that the band structures of Fe2VAl are significantly modified near Fermi level with increasing pressures. The total Density of states decreases near the Fermi level, whereas increase at the Fermi level with increasing pressures. It is shown from Fig. 6 that the elastic constants C11, C12, C44, shear modulus (G), Young’s modulus (E) and bulk modulus B0 all expressed in GPa linearly increase monotonously when pressure is enhanced. Our results for elastic constants in Table 2 satisfy the stability conditions at different pressures. The Cauchy pressure is negative below 30 GPa, however, it is positive above 30 GPa. It is noted that the calculated value of B0/G increases with the pressure increasing and Fe2VAl reveals ductility at high pressure. The Seebeck coefficients S of Fe2VAl rapidly increases for p-doped and n-doped in the low-doping region as the carrier concentration increase. It is seen that the calculated electronic conductivity r/s of the undoped Fe2VAl is low; however, as the carrier concentration increase, the calculated electronic conductivity r/s of Fe2VAl rapidly increase for p-doped and n-doped. Our calculated results show that the electronic conductivity can be rapidly enhanced by increasing pressure. It is seen that the power factor S2r/s rapidly increases for n-doped and decrease for p-doped with the increasing pressure, indicating that the n-doped can greatly enhance the electrical transport properties of Fe2VAl compound. The increase of the power factor S2r/s is mainly due to the enhanced electronic conductivity, but Seebeck

27

coefficient almost does not work. The results showed that the thermoelectric properties of Fe2VAl would be improved effectively at high pressure. It can be seen that n-type doping in the Fe2VAl compound may be more suitable for thermoelectric materials than p-type doping. Acknowledgments The project is supported by the Science and the National Natural Science Foundation under Grant Nos. 10947162 and 11104072; Technology Research project of Hubei provincial Department of Education under Grant No. D20108202; Basic and Advanced Technology Program of Henan province (Nos. 112300410229, 12A140008), Foundation for University Key Teacher by Henan province (Nos. 2010GGJS-146, 2012GGJS-104). References [1] [2] [3] [4] [5] [6] [7] [8]

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