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The role of electronic and crystal structure in the effect of volumetric band convergence
Computational Materials Science 153 (2018) 141–145
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The role of electronic and crystal structure in the effect of volumetric band convergence
T
⁎
Vladislav Pelenitsyna, , Pavel Korotaeva,b a b
Center for Applied and Fundamental Research, Dukhov Research Institute for Automatics, 127055, Sushchevskaya 22, Moscow, Russia Material Modeling and Development Laboratory, NUST “MISiS”, 119991 Leninskiy pr. 4, Moscow, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermoelectric materials Electronic structure Volumetric bands convergence Transport properties
The effect of rearrangement of the electronic band structure by changing the volume of a crystalline cell was investigated by means of first-principles calculation. The corresponding effect on the thermoelectric power factor was studied in Boltzmann approximation. The separate role of electronic structure of constituent elements and crystal structure was investigated. As the example, cubic Mg2Si and orthorhombic Sr2Sn were considered, and vice versa, hypothetical orthorhombic Mg2Si and cubic Sr2Sn. We found that in all studied cases there is rearrangement of electronic bands, mostly conduction ones, with the change of volume of the cell. In all cases it may leads to the increase of power factor.
1. Introduction Electrical energy is the most practical form of energy. Therefore, the search for effective converters of thermal, mechanical and other forms of energy into electrical is an important task. Much attention is paid to direct methods of converting thermal energy into electrical energy, which include, in particular, solid-state thermoelectric materials [1]. Such converters have a number of advantages: the absence of moving parts, reliability, compactness. The operating principle of the thermoelectric material is based on the Seebeck effect. The effectiveness of such a material is determined by a dimensionless quantity ZT, figure of S2σ merit: ZT = k + k T , where S is the Seebeck coefficient, σ is the conel
latt
ductivity, kel is the electronic thermal conductivity, and klatt is the lattice thermal conductivity. To optimize ZT, usually one try to increase the numerator S 2σ , which is called the power factor (PF). Another approach is to lower the denominator kel + klatt , focusing on the lattice thermal conductivity [2]. Of course, all these quantities are interrelated, which makes the optimization task difficult [3]. An increase in the PF can be achieved by optimizing the electronic band structure (BS) of the material. Optimization is achieved, usually, in two ways: increasing the degeneracy of the electronic bands near Fermi level and creating gradients in the energy density of electronic states. In the first case, one try to achieve convergence of the electronic levels near the Fermi level [4–6]. In the second approach, one try to find doping, that create sharp changes in the density of electronic states [7,8]. Let us consider the first of these ways of increasing efficiency: the ⁎
Corresponding author. E-mail address: [email protected] (V. Pelenitsyn).
https://doi.org/10.1016/j.commatsci.2018.06.038 Received 9 April 2018; Received in revised form 20 June 2018; Accepted 23 June 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
rearrangement of the BS. To do this is to change the cell volume, for example, by alloying. It has been shown experimentally that Mg2Si1 − x Snx compounds have a higher PF than the Mg2Si and Mg2Sn compounds separately [9]. Also Mg2Si1 − x Snx studied theoretically, where an increase in the power factor was observed [10,11]. Indeed, in the work [4], the change in transport properties was studied with a change in the cell volume. It was shown that an increase in cell volume leads to an increase in the PF. In particular, magnesium compounds Mg2X (X = Si, Ge) were considered, which proved to be ones of the best in terms of the PF gain [12]. Lets focus on what role does the crystal structure and electronic structure of the valence shells of elements play in the realization of the effect. Specifically, if one change the lattice type or replace atoms in the given compound by the atoms from the same group, how BS and PF behave with the cell volume change? It is of practical importance, because the thermal conductivity of compounds with heavy atoms generally will be lower, which may lead to an improvement in the thermoelectric figure of merit. As an example we focus on the A2B type compounds of the elements II and IVa groups. To study the mentioned problem, the cubic Mg2Si (Fm3m) and orthorhombic Sr2Sn (Pnma) compounds were considered. To clarify the contribution of the crystal lattice, hypothetical structures of cubic Sr2Sn (Fm3m) and orthorhombic Mg2Si (Pnma) were studied. 2. Method of calculation The calculations were carried out using the PAW package VASP
Computational Materials Science 153 (2018) 141–145
V. Pelenitsyn, P. Korotaev
is well-studied [4,20]. Our results are very close, so we present BS and PF in the supplementary material. With the increase of volume of the cell, two lowest bands at point X converge (Fig. S1), leading to the increase in PF (Fig. S2a–b).
Table 1 Table of parameters used in calculation. Compound Mg2Si
Structure
k-points gridsa
Ecut, eV
cubic
14 × 14 × 14 (44 × 44 × 44) 16 × 12 × 12 (32 × 24 × 24) 16 × 12 × 12 (28 × 21 × 21) 14 × 14 × 14 (44 × 44 × 44)
340
orth. Sr2Sn
orth. cubic
380
3.2. Sr2Sn in the orthorhombic structure of Pnma
350
The band structures of orthorhombic Sr2Sn is shown in Fig. 1a. As the cell volume increases, the C1 and C2 conduction bands converge, as well as C1 and C3. There are also changes in the position of other conduction bands. The valence band, however, remains almost unchanged. The energy difference between these bands as function of the relative cell volume, is presented in Fig. 1b. The most significant convergence of the bands is observed near the minimum of the conduction band, as well as in the cubic Mg2Si (Fig. 1b). The PF as function of the chemical potential at a fixed temperature is presented in Fig. 2a–b. As the cell volume increases, the maximum value of the PF increases. The peak value of the PF is shown in Table 2. The largest effect is observed for the conduction bands, which is associated with a significant change of their position. It should be noted, that at given value of the chemical potential (relate to the concentration of carriers), the PF can even decrease, for example, for the p-type. This suggests that it is necessary to optimize not only the electronic structure, but also the carrier concentration, to get a maximum benefit. The obtained result illustrates that the BS rearrangement effect is observed near the edge of the conduction band not only in the cubic structure of magnesium compounds, but also in the orthorhombic structure of Sr2Sn and also leads to an increase in the PF.
475
a Top for self-consistent calculation, bottom for calculation on a densified grid.
5.4.1 [13,14]. The cubic structure of Mg2Si (Fm3m, No 225) was specified by a primitive unit cell containing three atoms with lattice parameter a = 6.338 Å [15]. To determine the structure of Sr2Sn (Pnma, No 62), the generators of this group were used [16] and the parameters: a = 5.378 Å, b = 8.402 Å, c = 10.078 Å, x = 0.6561, z = 0.0723, obtained in the experimental and theoretically works [18,17]. The Sr2Sn unit cell contains 12 atoms. The hypothetical cubic structure Sr2Sn (Fm3m, No 225) same as stable Mg2Si was specified by a primitive unit cell containing three atoms. The lattice parameter a = 7.985 Å was obtained by the optimization of the structure. Then, by analogy, the hypothetical orthorhombic structure Mg2Si (Pnma, No 62) containing 12 atoms was considered. The lattice parameters are obtained from the optimization of the structure: a = 4.378 Å, b = 7.353 Å, c = 14.144 Å. For self-consistent calculation, we used a Γ -centered grids of k-points and a cutoff energy of the basis of plane waves Ecut shown in Table 1. To obtain transport coefficients semiclassical Boltzmann theory was used, as implemented in BoltzTraP code [19]. Densified grids of kpoints were used to calculate transport properties shown in Table 1. 2 Note that we calculate the reduced power factor S σ , when τ is the reτ laxation time. In general, the relaxation time depend on temperature, chemical potential and crystal structure which could change the shape of the curves of the obtained dependencies. We assume that the relaxation time qualitatively behaves closely for all crystal structures of the same material. In the follow by PF we mean the reduced power factor.
3.3. Sr2Sn in the cubic structure Fm3m Taking into account the previous result, one can make the assumption that the effect, observed near the conduction band minimum, is determined mainly by the electronic structure. In A2B type stable compounds of the elements II and IVa groups there are two types of lattice: cubic and orthorhombic. It is interesting to see how the effect under investigation will behave in the considered compound if one change the lattice type. To do this, we first consider the hypothetical cubic structure of Sr2Sn of the space group Fm3m. The calculated enthalpy of formation of this structure is Hform(cubic Sr2Sn) = −0.6 eV/ atom, for the orthorhombic structure Hform(orth. Sr2Sn) = −0.63 eV/ atom. The difference from the ground state is Δ Hform = 0.03 eV/atom. The calculated band structures are shown in Fig. 3a. It can be seen that near the minimum of the conduction band, the convergence of the bands (C3 and C2 bands at the point X) is not as pronounced as in the previous case. However, band C1 at the point Γ significantly changes its
3. Results and discussion 3.1. Mg2Si in the cubic structure of Fm3m The effect of band convergence in cubic Mg2Si with volume change
Fig. 1. (a) Band structure of orthorhombic Sr2Sn for different cell volumes: experimental and increased by 9 vol.% of the experimental. In the figure, C1 and C2 is the first and second lowest conduction bands at the point Γ , respectively. C3 is the lowest conduction band at the point U; (b) dependence of the energy difference between the bands indicated on the band structure, on the relative volume of the cell. Also, for comparison, the dependence of the energy difference between of two lowest conduction bands at the point X (Fig. S1) of the cubic Mg2Si is presented. 142
Computational Materials Science 153 (2018) 141–145
V. Pelenitsyn, P. Korotaev
Fig. 2. Power factor as function of chemical potential at a fixed temperature (300 K and 700 K on the left and right, respectively) for: (a)–(b) Sr2Sn with an orthorhombic lattice; (c)–(d) Sr2Sn with a hypothetical cubic lattice; (e)–(f) Mg2Si with a hypothetical orthorhombic lattice. The boundaries of the chemical potential correspond to the approximate carrier concentration n= ± 10 22 carr./cm3 .
structure is Hform(orth. Mg2Si) = 0.29 eV/atom, enthalpy of the ground state is Hform(cubic Mg2Si) = −0.15 eV/atom. The difference is Δ Hform(Mg2Si) = 0.44 eV/atom. The calculated band structures are shown in Fig. 4a. It can be seen that the effect is not directly observed near the edge of the conduction band at the point Γ , but the bands still change their position. The band C3 at the point S became closer to the band C1 at the point Γ . In the valence band, the positions of the energy levels also vary, but generally far from the maximum of the valence band. Dependence of the energy difference between the bands indicated on the BS is shown in Fig. 4b. The PF as function of the chemical potential is shown in Fig. 2e–f. In this case, the change of PF is clear. For the n-type the increase of PF can be obtained with an optimal choice of the carrier concentration (Table 2). This result shows that the effect for the considered compounds does not depend on the type of lattice. The change in the position of the bands is present in all the structures studied (Figs. 1, 3, 4).
position. The structure of the valence band remains practically unchanged. Also, unlike the cubic Mg2Si, the maximum of the valence band is at the point X, and not in Γ . Dependence of the energy difference between the bands indicated on the BS is shown in Fig. 3b. The PF as function of the chemical potential is shown in Fig. 2c–d. One can see that an increase in the cell volume also leads to an increase in the maximum of the PF, but unlike Sr2Sn in orthorhombic structure (Fig. 1) and Mg2Si in the cubic structure (Fig. S2a–b) is not so significant (Table 2). As a result the convergence of energy levels is also present in the hypothetical cubic structure of Sr2Sn and can lead to an increase in the PF with an optimal choice of the carrier concentration. However, this increase is small. 3.4. Mg2Si in the orthorhombic structure Pnma By analogy, the hypothetical orthorhombic Mg2Si structure of the space group Pnma was studied. The enthalpy of the formation of this
143
Computational Materials Science 153 (2018) 141–145
V. Pelenitsyn, P. Korotaev
Sr2Sn and Mg2Si with cubic and orthorhombic lattices, respectively, are considered. It is established that the convergence of electronic bands with respect to energy occurs with an increase in the cell volume in all the cases considered. For the orthorhombic Sr2Sn the largest convergence of bands in energy was Δ Emax = 0.47 eV, for the cubic Sr2Sn Δ Emax = 0.38 eV and for the orthorhombic Mg2Si Δ Emax = 0.42 eV. Transport properties were also analyzed, namely the reduced power factor of these structures. It is shown that, in generally, the maximum value of the reduced power factor increases with increasing volume of the crystalline cell. For the orthorhombic Sr2Sn and Mg2Si the maximum value increased by 24% and 10%, respectively, for the cubic Sr2Sn by 14%. However, for a certain fixed carrier concentration, the power factor may decrease with a change in the cell volume. Thus, for the considered materials, the presence of the band structure rearrangement effect does not depend on the type of the lattice. However, it cannot be said for sure what has a largest impact on the studied effect: the electronic structure of atoms or the lattice type. Therefore, the study of thermoelectric materials, in particular the compounds of elements of groups II and IVa, it is necessary to take into account the effect of rearrangement of the band structure. It can lead to both an insignificant and significant increase in the power factor, which, in turn, can have a positive effect on thermoelectric efficiency.
Table 2 Table of the maximum power factor value. The table shows the peak of PF and the corresponding values of the change in cell volume, carrier concentration and chemical potential. Also, for comparison, these values are for an equilibrium volume. Structure
orth. Sr2Sn
T, K
300 700
cubic Sr2Sn
300 700
cubic Mg2Si
300 700
orth. Mg2Si
300 700
V/V0
S2σ , τ
1014
μW cm K2 s
n, 1021
carr . cm3
μ , eV
1.09 1.00 1.09 1.00
21.38 17.25 52.72 44.37
1.3 0.7 −0.5 1.1
−0.4 −0.33 0.36 −0.32
1.09 1.00 1.09 1.00
34.43 30.12 58.79 57.07
0.8 6.5 2.6 1.9
−0.29 −0.26 −0.29 −0.25
1.09 1.00 1.09 1.00
28.09 23.94 94.47 83
−0.14 0.15 −0.5 −1
0.21 −0.14 0.28 0.45
1.00 1.06 1.00
47.12 119.25 108.29
−1.8 −1.6 −1.9
0.72 0.66 0.71
4. Conclusion
Data availability
In this article, we considered the effect of rearrangement of the band structure with cell volume change in A2B type compounds of the elements of the II and IVa groups. The compounds Mg2Si with cubic lattice, Sr2Sn with an orthorhombic lattice and hypothetical structures of
The raw and processed data required to reproduce these findings are available to download from Mendeley Data (DOI: http://dx.doi.org/10. 17632/psgzt9zp4h.1).
Fig. 3. (a) The band structure of Sr2Sn with a hypothetical cubic lattice for different cell volumes: optimal and increased by 9 vol.% of the optimal. The figure shows the different bands; (b) dependence of the energy difference between the bands indicated on the band structure, on the relative volume of the cell.
Fig. 4. (a) The band structure of Mg2Si with a hypothetical orthorhombic lattice for different cell volumes: optimal and increased by 9 vol.% of the optimal. The figure shows the different bands; (b) dependence of the energy difference between the bands indicated on the band structure, on the relative volume of the cell. 144
Computational Materials Science 153 (2018) 141–145
V. Pelenitsyn, P. Korotaev
Acknowledgments
[7] J.P. Heremans, D. Rowe (Ed.), Materials, Preparation and Characterization in Thermoelectrics, CRC Press, Taylor & Francis, FL, Boca Raton, 2006. [8] E. Flage-Larsen, O.M. Lovvik, D. Rowe (Ed.), Materials, Preparation and Characterization in Thermoelectrics, CRC Press, Taylor & Francis, FL, Boca Raton, 2006. [9] V.K. Zaitsev, M.I. Fedorov, E.A. Gurieva, I.S. Eremin, P.P. Konstantinov, A.Y. Samunin, M.V. Vedernikov, Phys. Rev. B74 (2006) 045207. [10] J.J. Pulikkotil, D.J. Singh, S. Auluck, M. Saravanan, D.K. Misra, A. Dhar, R.C. Budhani, Phys. Rev. B 86 (2012) 155204. [11] T. Dasgupta, C. Stiewe, J.D. Boor, E. Mller, Phys. Status Solidi (A) 211 (2014) 1250–1254. [12] J. Sun, D.J. Singh, Phys. Rev. Appl. 5 (2016) 024006. [13] G. Kresse, J. Furthmller, Phys. Rev. B54 (1996) 11169. [14] G. Kresse, D. Joubert, Phys. Rev. B59 (1999) 1758. [15] M. Pekguleryuz, K. Kainer, A. Kaya, Fundamentals of Magnesium Alloy Metallurgy, Woodhead Publishing Series in Metals and Surface Engineering, Elsevier Science, 2013. [16] T. Hahn, Space-group symmetry, International Tables for Crystallography (Published for the International Union of Crystallography by D. Reidel Pub. Co., 1987). [17] A. Palenzona, M. Pani, J. Alloy. Compd. 384 (2004) 227–230. [18] A. Widera, H. Schfer, J. Less Common Met. 77 (1981) 29. [19] G.K.H. Madsen, D.J. Singh, Comput. Phys. Commun. 175 (2006) 67. [20] W. Liu, X. Tan, K. Yin, H. Liu, X. Tang, J. Shi, Q. Zhang, C. Uher, Phys. Rev. Lett. 108 (2012) 166601.
P.K. thank Ministry of Education and Science of Russian Federation Grant No. K2-2017-080 supported in the framework of Increase Competitiveness Program of NUST “MISIS” implemented by a governmental decree dated 16 March 2013, No. 211. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.commatsci.2018.06. 038. References [1] [2] [3] [4] [5] [6]
D.M. Rowe, CRC Handbook of Thermoelectrics, CRC Press, Taylor & Francis, 1995. G.J. Snyder, E.S. Toberer, Nat. Mater. 7 (2008) 105. G. Mahan, Solid State Phys. 51 (1997) 81–157. S. Bhattacharya, G.K.H. Madsen, Phys. Rev. B 92 (2015) 085205. Y. Pei, H. Wang, G.J. Snyder, Adv. Mater. 24 (2012) 6125. S. Bhattacharya, R. Chmielowski, G. Dennler, G.K.H. Madsen, J. Mater. Chem. A 4 (2016) 11086.
145
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The role of electronic and crystal structure in the effect of volumetric band convergence
T
⁎
Vladislav Pelenitsyna, , Pavel Korotaeva,b a b
Center for Applied and Fundamental Research, Dukhov Research Institute for Automatics, 127055, Sushchevskaya 22, Moscow, Russia Material Modeling and Development Laboratory, NUST “MISiS”, 119991 Leninskiy pr. 4, Moscow, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermoelectric materials Electronic structure Volumetric bands convergence Transport properties
The effect of rearrangement of the electronic band structure by changing the volume of a crystalline cell was investigated by means of first-principles calculation. The corresponding effect on the thermoelectric power factor was studied in Boltzmann approximation. The separate role of electronic structure of constituent elements and crystal structure was investigated. As the example, cubic Mg2Si and orthorhombic Sr2Sn were considered, and vice versa, hypothetical orthorhombic Mg2Si and cubic Sr2Sn. We found that in all studied cases there is rearrangement of electronic bands, mostly conduction ones, with the change of volume of the cell. In all cases it may leads to the increase of power factor.
1. Introduction Electrical energy is the most practical form of energy. Therefore, the search for effective converters of thermal, mechanical and other forms of energy into electrical is an important task. Much attention is paid to direct methods of converting thermal energy into electrical energy, which include, in particular, solid-state thermoelectric materials [1]. Such converters have a number of advantages: the absence of moving parts, reliability, compactness. The operating principle of the thermoelectric material is based on the Seebeck effect. The effectiveness of such a material is determined by a dimensionless quantity ZT, figure of S2σ merit: ZT = k + k T , where S is the Seebeck coefficient, σ is the conel
latt
ductivity, kel is the electronic thermal conductivity, and klatt is the lattice thermal conductivity. To optimize ZT, usually one try to increase the numerator S 2σ , which is called the power factor (PF). Another approach is to lower the denominator kel + klatt , focusing on the lattice thermal conductivity [2]. Of course, all these quantities are interrelated, which makes the optimization task difficult [3]. An increase in the PF can be achieved by optimizing the electronic band structure (BS) of the material. Optimization is achieved, usually, in two ways: increasing the degeneracy of the electronic bands near Fermi level and creating gradients in the energy density of electronic states. In the first case, one try to achieve convergence of the electronic levels near the Fermi level [4–6]. In the second approach, one try to find doping, that create sharp changes in the density of electronic states [7,8]. Let us consider the first of these ways of increasing efficiency: the ⁎
Corresponding author. E-mail address: [email protected] (V. Pelenitsyn).
https://doi.org/10.1016/j.commatsci.2018.06.038 Received 9 April 2018; Received in revised form 20 June 2018; Accepted 23 June 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
rearrangement of the BS. To do this is to change the cell volume, for example, by alloying. It has been shown experimentally that Mg2Si1 − x Snx compounds have a higher PF than the Mg2Si and Mg2Sn compounds separately [9]. Also Mg2Si1 − x Snx studied theoretically, where an increase in the power factor was observed [10,11]. Indeed, in the work [4], the change in transport properties was studied with a change in the cell volume. It was shown that an increase in cell volume leads to an increase in the PF. In particular, magnesium compounds Mg2X (X = Si, Ge) were considered, which proved to be ones of the best in terms of the PF gain [12]. Lets focus on what role does the crystal structure and electronic structure of the valence shells of elements play in the realization of the effect. Specifically, if one change the lattice type or replace atoms in the given compound by the atoms from the same group, how BS and PF behave with the cell volume change? It is of practical importance, because the thermal conductivity of compounds with heavy atoms generally will be lower, which may lead to an improvement in the thermoelectric figure of merit. As an example we focus on the A2B type compounds of the elements II and IVa groups. To study the mentioned problem, the cubic Mg2Si (Fm3m) and orthorhombic Sr2Sn (Pnma) compounds were considered. To clarify the contribution of the crystal lattice, hypothetical structures of cubic Sr2Sn (Fm3m) and orthorhombic Mg2Si (Pnma) were studied. 2. Method of calculation The calculations were carried out using the PAW package VASP
Computational Materials Science 153 (2018) 141–145
V. Pelenitsyn, P. Korotaev
is well-studied [4,20]. Our results are very close, so we present BS and PF in the supplementary material. With the increase of volume of the cell, two lowest bands at point X converge (Fig. S1), leading to the increase in PF (Fig. S2a–b).
Table 1 Table of parameters used in calculation. Compound Mg2Si
Structure
k-points gridsa
Ecut, eV
cubic
14 × 14 × 14 (44 × 44 × 44) 16 × 12 × 12 (32 × 24 × 24) 16 × 12 × 12 (28 × 21 × 21) 14 × 14 × 14 (44 × 44 × 44)
340
orth. Sr2Sn
orth. cubic
380
3.2. Sr2Sn in the orthorhombic structure of Pnma
350
The band structures of orthorhombic Sr2Sn is shown in Fig. 1a. As the cell volume increases, the C1 and C2 conduction bands converge, as well as C1 and C3. There are also changes in the position of other conduction bands. The valence band, however, remains almost unchanged. The energy difference between these bands as function of the relative cell volume, is presented in Fig. 1b. The most significant convergence of the bands is observed near the minimum of the conduction band, as well as in the cubic Mg2Si (Fig. 1b). The PF as function of the chemical potential at a fixed temperature is presented in Fig. 2a–b. As the cell volume increases, the maximum value of the PF increases. The peak value of the PF is shown in Table 2. The largest effect is observed for the conduction bands, which is associated with a significant change of their position. It should be noted, that at given value of the chemical potential (relate to the concentration of carriers), the PF can even decrease, for example, for the p-type. This suggests that it is necessary to optimize not only the electronic structure, but also the carrier concentration, to get a maximum benefit. The obtained result illustrates that the BS rearrangement effect is observed near the edge of the conduction band not only in the cubic structure of magnesium compounds, but also in the orthorhombic structure of Sr2Sn and also leads to an increase in the PF.
475
a Top for self-consistent calculation, bottom for calculation on a densified grid.
5.4.1 [13,14]. The cubic structure of Mg2Si (Fm3m, No 225) was specified by a primitive unit cell containing three atoms with lattice parameter a = 6.338 Å [15]. To determine the structure of Sr2Sn (Pnma, No 62), the generators of this group were used [16] and the parameters: a = 5.378 Å, b = 8.402 Å, c = 10.078 Å, x = 0.6561, z = 0.0723, obtained in the experimental and theoretically works [18,17]. The Sr2Sn unit cell contains 12 atoms. The hypothetical cubic structure Sr2Sn (Fm3m, No 225) same as stable Mg2Si was specified by a primitive unit cell containing three atoms. The lattice parameter a = 7.985 Å was obtained by the optimization of the structure. Then, by analogy, the hypothetical orthorhombic structure Mg2Si (Pnma, No 62) containing 12 atoms was considered. The lattice parameters are obtained from the optimization of the structure: a = 4.378 Å, b = 7.353 Å, c = 14.144 Å. For self-consistent calculation, we used a Γ -centered grids of k-points and a cutoff energy of the basis of plane waves Ecut shown in Table 1. To obtain transport coefficients semiclassical Boltzmann theory was used, as implemented in BoltzTraP code [19]. Densified grids of kpoints were used to calculate transport properties shown in Table 1. 2 Note that we calculate the reduced power factor S σ , when τ is the reτ laxation time. In general, the relaxation time depend on temperature, chemical potential and crystal structure which could change the shape of the curves of the obtained dependencies. We assume that the relaxation time qualitatively behaves closely for all crystal structures of the same material. In the follow by PF we mean the reduced power factor.
3.3. Sr2Sn in the cubic structure Fm3m Taking into account the previous result, one can make the assumption that the effect, observed near the conduction band minimum, is determined mainly by the electronic structure. In A2B type stable compounds of the elements II and IVa groups there are two types of lattice: cubic and orthorhombic. It is interesting to see how the effect under investigation will behave in the considered compound if one change the lattice type. To do this, we first consider the hypothetical cubic structure of Sr2Sn of the space group Fm3m. The calculated enthalpy of formation of this structure is Hform(cubic Sr2Sn) = −0.6 eV/ atom, for the orthorhombic structure Hform(orth. Sr2Sn) = −0.63 eV/ atom. The difference from the ground state is Δ Hform = 0.03 eV/atom. The calculated band structures are shown in Fig. 3a. It can be seen that near the minimum of the conduction band, the convergence of the bands (C3 and C2 bands at the point X) is not as pronounced as in the previous case. However, band C1 at the point Γ significantly changes its
3. Results and discussion 3.1. Mg2Si in the cubic structure of Fm3m The effect of band convergence in cubic Mg2Si with volume change
Fig. 1. (a) Band structure of orthorhombic Sr2Sn for different cell volumes: experimental and increased by 9 vol.% of the experimental. In the figure, C1 and C2 is the first and second lowest conduction bands at the point Γ , respectively. C3 is the lowest conduction band at the point U; (b) dependence of the energy difference between the bands indicated on the band structure, on the relative volume of the cell. Also, for comparison, the dependence of the energy difference between of two lowest conduction bands at the point X (Fig. S1) of the cubic Mg2Si is presented. 142
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Fig. 2. Power factor as function of chemical potential at a fixed temperature (300 K and 700 K on the left and right, respectively) for: (a)–(b) Sr2Sn with an orthorhombic lattice; (c)–(d) Sr2Sn with a hypothetical cubic lattice; (e)–(f) Mg2Si with a hypothetical orthorhombic lattice. The boundaries of the chemical potential correspond to the approximate carrier concentration n= ± 10 22 carr./cm3 .
structure is Hform(orth. Mg2Si) = 0.29 eV/atom, enthalpy of the ground state is Hform(cubic Mg2Si) = −0.15 eV/atom. The difference is Δ Hform(Mg2Si) = 0.44 eV/atom. The calculated band structures are shown in Fig. 4a. It can be seen that the effect is not directly observed near the edge of the conduction band at the point Γ , but the bands still change their position. The band C3 at the point S became closer to the band C1 at the point Γ . In the valence band, the positions of the energy levels also vary, but generally far from the maximum of the valence band. Dependence of the energy difference between the bands indicated on the BS is shown in Fig. 4b. The PF as function of the chemical potential is shown in Fig. 2e–f. In this case, the change of PF is clear. For the n-type the increase of PF can be obtained with an optimal choice of the carrier concentration (Table 2). This result shows that the effect for the considered compounds does not depend on the type of lattice. The change in the position of the bands is present in all the structures studied (Figs. 1, 3, 4).
position. The structure of the valence band remains practically unchanged. Also, unlike the cubic Mg2Si, the maximum of the valence band is at the point X, and not in Γ . Dependence of the energy difference between the bands indicated on the BS is shown in Fig. 3b. The PF as function of the chemical potential is shown in Fig. 2c–d. One can see that an increase in the cell volume also leads to an increase in the maximum of the PF, but unlike Sr2Sn in orthorhombic structure (Fig. 1) and Mg2Si in the cubic structure (Fig. S2a–b) is not so significant (Table 2). As a result the convergence of energy levels is also present in the hypothetical cubic structure of Sr2Sn and can lead to an increase in the PF with an optimal choice of the carrier concentration. However, this increase is small. 3.4. Mg2Si in the orthorhombic structure Pnma By analogy, the hypothetical orthorhombic Mg2Si structure of the space group Pnma was studied. The enthalpy of the formation of this
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Sr2Sn and Mg2Si with cubic and orthorhombic lattices, respectively, are considered. It is established that the convergence of electronic bands with respect to energy occurs with an increase in the cell volume in all the cases considered. For the orthorhombic Sr2Sn the largest convergence of bands in energy was Δ Emax = 0.47 eV, for the cubic Sr2Sn Δ Emax = 0.38 eV and for the orthorhombic Mg2Si Δ Emax = 0.42 eV. Transport properties were also analyzed, namely the reduced power factor of these structures. It is shown that, in generally, the maximum value of the reduced power factor increases with increasing volume of the crystalline cell. For the orthorhombic Sr2Sn and Mg2Si the maximum value increased by 24% and 10%, respectively, for the cubic Sr2Sn by 14%. However, for a certain fixed carrier concentration, the power factor may decrease with a change in the cell volume. Thus, for the considered materials, the presence of the band structure rearrangement effect does not depend on the type of the lattice. However, it cannot be said for sure what has a largest impact on the studied effect: the electronic structure of atoms or the lattice type. Therefore, the study of thermoelectric materials, in particular the compounds of elements of groups II and IVa, it is necessary to take into account the effect of rearrangement of the band structure. It can lead to both an insignificant and significant increase in the power factor, which, in turn, can have a positive effect on thermoelectric efficiency.
Table 2 Table of the maximum power factor value. The table shows the peak of PF and the corresponding values of the change in cell volume, carrier concentration and chemical potential. Also, for comparison, these values are for an equilibrium volume. Structure
orth. Sr2Sn
T, K
300 700
cubic Sr2Sn
300 700
cubic Mg2Si
300 700
orth. Mg2Si
300 700
V/V0
S2σ , τ
1014
μW cm K2 s
n, 1021
carr . cm3
μ , eV
1.09 1.00 1.09 1.00
21.38 17.25 52.72 44.37
1.3 0.7 −0.5 1.1
−0.4 −0.33 0.36 −0.32
1.09 1.00 1.09 1.00
34.43 30.12 58.79 57.07
0.8 6.5 2.6 1.9
−0.29 −0.26 −0.29 −0.25
1.09 1.00 1.09 1.00
28.09 23.94 94.47 83
−0.14 0.15 −0.5 −1
0.21 −0.14 0.28 0.45
1.00 1.06 1.00
47.12 119.25 108.29
−1.8 −1.6 −1.9
0.72 0.66 0.71
4. Conclusion
Data availability
In this article, we considered the effect of rearrangement of the band structure with cell volume change in A2B type compounds of the elements of the II and IVa groups. The compounds Mg2Si with cubic lattice, Sr2Sn with an orthorhombic lattice and hypothetical structures of
The raw and processed data required to reproduce these findings are available to download from Mendeley Data (DOI: http://dx.doi.org/10. 17632/psgzt9zp4h.1).
Fig. 3. (a) The band structure of Sr2Sn with a hypothetical cubic lattice for different cell volumes: optimal and increased by 9 vol.% of the optimal. The figure shows the different bands; (b) dependence of the energy difference between the bands indicated on the band structure, on the relative volume of the cell.
Fig. 4. (a) The band structure of Mg2Si with a hypothetical orthorhombic lattice for different cell volumes: optimal and increased by 9 vol.% of the optimal. The figure shows the different bands; (b) dependence of the energy difference between the bands indicated on the band structure, on the relative volume of the cell. 144
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V. Pelenitsyn, P. Korotaev
Acknowledgments
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