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First-principles determination of the enthalpy of formation of Mn–Si phases
Solid State Communications 188 (2014) 49–52
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
First-principles determination of the enthalpy of formation of Mn–Si phases Alexandre Berche n, Jean-Claude Tédenac, Philippe Jund ICGM-Université Montpellier II, UMR-CNRS 5253, Pl. E. Bataillon CC1506, 34095 Montpellier, France
art ic l e i nf o
a b s t r a c t
Article history: Received 17 December 2013 Received in revised form 10 February 2014 Accepted 17 February 2014 by Ralph Gebauer Available online 26 February 2014
After a careful survey of the literature, the enthalpies of formation of Mn3Si, Mn5Si2, Mn5Si3, MnSi and Mn4Si7 have been calculated at 0 K using the density functional theory. The calculated physical properties are compared to previous calculations and experiments. The enthalpies of formation of the phases are finally discussed. & 2014 Elsevier Ltd. All rights reserved.
Keywords: A. semiconductors D. Thermodynamics properties
1. Introduction Manganese and silicon are especially interesting due to their compatibility with industrialization at a large scale (low-cost production processes, abundant materials) and with the rules of hygiene, health and environment. Moreover, the Mn–Si system contains many intermetallic compounds or phases presenting a real potential for applications. Generating electricity from waste heat by means of “green” thermoelectric generators could reduce the global CO2 footprint of industrial activities. Among the potential materials for thermoelectric applications, Higher Manganese Silicides (HMS) MnSix (with x around 1.75) exhibit interesting figures-of-merit at intermediate temperatures (573–873 K) [1,2]. Other applications of these elements have been studied as the possible use of the spin effects in electronic devices for the transfer, processing, and storage of information [3,4]. Manganese and silicon are also known to improve the mechanical properties of low weight and high strength alloys for automotive and aerospace industries [5]. Depending on the application, physical properties such as the electronic band gap, the magnetic moment or the bulk modulus of the phases are critical data. Moreover, all these properties are evidently sensible to the alloy composition and to the annealing process. As a consequence, optimization of elaboration needs the precise knowledge of the thermodynamic properties and an accurate description of the systems including ternaries and binaries as well. To be as accurate as possible, such descriptions have
n Corresponding author. Present address: ICGM-Université Montpellier II, UMRCNRS 5253, Pl. E. Bataillon CC1506, Montpellier, France Tel.: þ 33 6 31 15 85 93. E-mail address: [email protected] (A. Berche).
http://dx.doi.org/10.1016/j.ssc.2014.02.021 0038-1098 & 2014 Elsevier Ltd. All rights reserved.
to be based on consistent thermodynamic values. Among these data, the enthalpies of formation of intermediate phases are primordial because they represent the reciprocal stability of the phases at low temperature. The paper is organized as follows: in Section 2 a literature survey of the enthalpy of formation is proposed. In Section 3 the computational details are given. In Section 4 the results for the physical properties are given. Finally the relative stability of the phases is discussed in Section 5 and conclusions are given in Section 6.
2. Phase stability and experimental enthalpy of formation in the literature At room temperature and below, 6 phases are mentioned in the literature: Mn, Mn3Si, Mn5Si2, Mn5Si3, MnSi and MnSix. MnSix: the HMS phase whose exact stoichiometry (xE 1.75) and structure are not exactly known. In fact, five structures were suggested for this compound: Mn4Si7, Mn11Si19, Mn15Si26, Mn27Si47 and Mn26Si45 [6]. According to De Ridder [7], even after annealing, several phases are identified in the same sample. De Ridder concludes that all these structures co-exist in the system and constitute a quasicontinuous defect fraction x in the range 0.00 oxo 0.05 for MnSi1.75 x. He also suggests that the energy difference between these phases is very low. Since the crystal structures of these phases correspond to large unit-cells (from 44 to 296 atoms in the structure), only the smallest one (Mn4Si7) is studied in this work. In addition 3 other phases with xSi o0.2 are mentioned in the literature (R-Mn6Si, ν-Mn9Si2 and the γ-phase). Since these phases have random site occupancies, they are not taken into consideration in this work.
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A. Berche et al. / Solid State Communications 188 (2014) 49–52
In the Mn–Si system, even if the enthalpies of formation of the phases have been measured many times [8–19], the results remain very scattered. To compare them more easily, the values are calculated for one mole of atom, whereas they are often given for one mole of formula unit in the original papers. The bomb calorimetry method used by Golutvin [19] seems to be very imprecise regarding the results and their error bars. Moreover, it is possible that the alloys of Golutvin [19] reacted with zirconia crucibles during synthesis. As a consequence, these values are ruled out. The results of Lukashenko [8,15] are in very good agreement with those of Eremenko [12–14], Nowotny [17] and Gertman [9] for Mnrich phases. Moreover, the values of Meschel [10] are in very good agreement with those of Eremenko [12–14] for Mn5Si3 and MnSi. However, the shape of the enthalpy of formation curves varies with the sources: smooth with a minimum around MnSi for Gertman [9], or around Mn5Si3 for Muradov [16] or sharp on MnSi for Eremenko [12–14] and Zaitsev [18]. The global uncertainty both on the value and on the shape of the enthalpy of formation curve implies that more data are necessary. Since the experiments seem to be very difficult and give widespread results, a calculation method is employed in this work using the DFT to try to obtain a correct estimation of the enthalpy of formation of the different Mn–Si phases. Thus, in this paper, the relative stability of the phases is discussed on the basis of the density functional theory (DFT) calculations of the enthalpy of formation of the different phases in the Mn–Si system.
3. Computational details The density functional theory calculations are performed using the Vienna ab initio simulation package (VASP) [20,21] using the projector augmented waves (PAW) technique [22,23] within the generalized gradient approximation (GGA). The Perdew–Burke– Ernzerhof parameterization (PBE) is applied. Standard versions of the PAW potentials for Si and Mn are used. For Si four electronic states are included in the valence shell (3s23p2) whereas 13 electronic states are taken into account for Mn (3p64s23d5). The first Brillouin zone is integrated using Monkhorst–Pack k-point meshes. A Gamma centered k-point grid was used for the hexagonal structure. The plane-wave energy cutoff and the reciprocal space meshes (k-point) are increased to achieve total energies with a precision less than 0.5 meV/atom. The selected cutoff energy varies from 350 to 600 eV and the number of k-points goes from 15 to 364 depending on the crystal structure. The calculations are performed using the “accurate” precision setting in the VASP input file to avoid wrap-around errors. For metallic phases, the ISMEAR tag is taken equal to 1 (1st order Methfessel–Paxton smearing), whereas for semiconductors and half-metals a value of 5 (tetrahedron method with Blöchl corrections) is used. The total energy of each structure is minimized by relaxing the volume of the cell and the ionic positions starting from the experimental parameters. In a second step, the equilibrium volume is slightly compressed or expanded and the total energy is minimized for a given fixed volume. These several total energies are then fitted using the Vinet equation of state [24] which allows calculating the equilibrium volume (V0), the total energy E and the bulk modulus B of the crystal structure. The enthalpy of formation at 0 K of the Mn1 xSix phase (ΔfH0(Mn1 xSix)) is calculated by Δf H 0 ðMn1 x Six Þ ¼ EðMn1 x Six Þ–ð1 xÞEðMnÞ xEðSiÞ
ð1Þ
where E(Mn1 xSix), E(Mn) and E(Si) are the equilibrium firstprinciples calculated total energies of Mn1 xSix, Mn and Si respectively in their stable crystalline structure at 0 K. DFT calculations also allow drawing the density of states (DOS) and the electronic band structure of the phases. In the case of semiconducting materials, the electronic gap is determined from these electronic band structure calculations. For each structure, the calculated DOS are similar to those previously calculated in the literature.
4. Calculated results The physical properties of each phase have been calculated from first-principles calculations and a reasonable agreement with previous calculations and measurements is obtained [25–30]. The calculated cell parameters are not presented in this paper, but their values are less than 3% smaller than the experimental values [31], which is a good agreement. The parameters of the D8m-Mn5Si3 phase are 4% smaller than the measured values of Higgins [32]; however these measurements were performed on nanowires and were never measured on bulk materials. The bulk modulus is calculated and shows a variation with respect to the experimental values of 10–20% for all the phases except for αMn (around 30%). However, for this element, experimental data seems to be very doubtful. The bulk moduli of Mn5Si2 and D8m-Mn5Si3 are calculated at 191 and 176 GPa respectively. The calculated magnetic moments and electronic gap of the semiconducting (Mn4Si7 and Si) or half-metal (MnSi and Mn3Si) of the phases are in good agreement with the previous calculations and measurements. On cooling, Mn5Si3 undergoes a second order phase transition towards an orthorhombic structure at 100 K and towards a tetragonal distorted structure at 62 K according to Gottschilch [33]. In our calculations, the low orthorhombic phase is calculated as ferrimagnetic in contradiction with the previous calculations and experimental works of Zygmund [28]. A metastable paramagnetic form of LT-Mn5Si3 was obtained; however, the magnetic moments (0.27, 0.11 and 0.76) are lower than those calculated by Zygmund. It is worth noting that in this work, DFT calculations were performed on 2 structures which have never been calculated before: Mn5Si2: this low-temperature phase decomposes into Mn3Si and Mn5Si3 around 1123 K according to Senateur [34]. However, after a careful annealing of 100 h at 1023 K, Lukashenko [8] did not observe the presence of this phase. The existence of this phase remains uncertain. This structure could have been stabilized by impurities if it turns out that the starting materials of Shoemaker were less pure than those of Lukashenko (99.999% for Si and 499.8% for Mn). Unfortunately, the purity grade of the materials used by Shoemaker is not known. According to Kwon [27], thin films of Mn5Si2 are non-magnetic. To our knowledge, this is the first time that first-principles calculations are performed on this phase. Its total DOS shows that this phase is metallic. Mn5Si2 is clearly magnetic since its total energy is 53 meV lower than in the non-magnetic form. The magnetic moments are for each site 0.249, 2.383, 2.047, 0.145, 1.883 and 1.79μB. This result is in contradiction with the measurements of Kwon [27] on multiphased thin films. D8m-Mn5Si3: Higgins [32] mentioned the existence of Mn5Si3 nanowires with the D8m structure. As discussed by the author, it is the first time that such a crystalline structure is mentioned for Mn5Si3. The electronic properties of this structure were never calculated. The DFT calculations lead to a ferrimagnetic structure with two magnetic moments of 0.175 and 1.47μB.
A. Berche et al. / Solid State Communications 188 (2014) 49–52
51
Table 1 Crystallographic structure (space group and prototype when available) and calculated total energy at 0 K, enthalpy of formation at 0, 298 and 1363 K for the phases of the Mn–Si system; (a) values extrapolated from the convex hull; (b) values for non-magnetic structures; and (c) from the total energy of the relaxed structures. Space group/prototype
Etot (eV atom 1)
T (K) Reference α-Mn
– [31] I-43m/α-Mn
R-Mn6Si ν-Mn9Si2 Mn3Si
P-3 R-3h Fm-3m/BiF3
Mn5Si2
P41212/Mn5Si2
D88-Mn5Si3
P63/mcm/Mn5Si3
LT-Mn5Si3
Ccmm/Mn5Si3
D8m-Mn5Si3
I4/mcm/W5Si3
MnSi
P213/FeSi
Mn4Si7
P-4c2
Si
Fd-3m/C
0 This work 9.012 8.991(b)(c) – – 8.424 8.321(b)(c) 8.255 8.200(b)(c) 7.998 7.981(b) 8.002 7.981(b) 8.019 7.990(b) 7.669 7.644(b)(c) 7.156 7.156(b)(c) 5.426
ΔfH (kJ mol of atom 1)
0.00 0.00(b) 18.20(a) 22.50(a) 29.79 21.40(b) 25.91 21.71(b) 31.94 31.50(b) 32.34 31.51(b) 33.95 32.35(b) 43.42 42.02(b) 40.99 41.72(b) 0.00
298 [8–10,12–15] 0.00 – 16.25 20.50 26.74 – – – 34.10 – – – – – 39.60 – 34.10 – 0.00
1363 [11] 0.00 – 13.10(a) 15.70(a) 19.00(a) – – – 21.71 – – – – – 20.92 – 16.04 – 0.00
congruent melting phase. Two hypotheses are suggested to explain this phenomenon:
Fig. 1. (Color online) Enthalpy of formation of the phases and convex hull at 0 K by the DFT (this work), 298 K (from the literature); non-magnetic phases are labeled NM.
5. Thermodynamic properties of the phases The enthalpies of formation of the phases at 0 K are calculated from the total energies by using Eq. (1) and are presented in Table 1 and displayed in Fig. 1. The convex hull (Fig. 1) exhibits a smooth shape with a minimum for MnSi. Such a result is in agreement with the experimental results of Nowotny [17], Gertman [9] and Lukashenko [8,15] as detailed in Section 2. Concerning Mn5Si3, the enthalpy of formation was calculated for three crystallographic structures: D88 (2 Mn sites: 4d (⅓; ⅔; 0) and 6g (0.236; 0; ¼) and one Si site on 6g (0.5991; 0; ¼)), D8m (2 Mn sites: 4b (0; ½; ¼) and 16k (0.075; 0.223; 0) and 2 Si sites: 4a (0; 0; ¼) and 8h (0.17; 0.67; 0)) and the low-temperature magnetic distortion (LT) of D88 (3 Mn sites: 4c (0.2637; ½; ¼), 8e (0; 0.33245; 0) and 8g (0.6175; 0.38271; ¼) and 2 Si sites: 4c (0.1002; ½; ¼) and 8g (0.29949; 0.70028; ¼)). The enthalpies of formation of D88 and LT are very close. This is a classical result since the magnetic stabilization of a phase is generally weak. However, it is surprising that the D8m structure is more stable than the D88 and LT ones. Another unexpected result consists in the fact that Mn5Si3 is less stable than Mn3Si þMnSi. This is particularly surprising for a
Mn5Si3 could have been stabilized experimentally by impurities such as boron, carbon or oxygen as it has been demonstrated in D88-Ti5Si3 compounds [35]. Moreover, the implantation of carbon in Mn5Si3 structure has been evidenced by Sürgers [36]. Finally, addition of oxygen or carbon in 2b sites (0; 0; 0) highly stabilizes the D88 structure according to DFT calculations performed by Colinet [37]. Mn5Si3 is less stable than Mn3Si þMnSi at 0 K due to the complex magnetic behavior of the Mn–Si phases. At higher temperatures, the magnetism disappears and the phase Mn5Si3 becomes stable. To check this hypothesis, DFT calculations are performed for non-magnetic structures (Table 1). The enthalpies of formation of Mn-rich compounds are highly modified (Fig. 1), and Mn5Si3 becomes more stable than a mix of Mn3Si þMnSi. To completely explain this point, DFT calculations would be necessary to estimate: the contributions from lattice vibrations and from magnetic excitations to the enthalpy of formation at 0 K; and the energetic contribution of defects and impurities (O, C, B) on the stability of the phases. These two points should be developed in a future work. The enthalpy of formation of Mn5Si2 at 0 K indicates that this phase is less stable than a mix of Mn3Siþ MnSi or Mn3Si þMn5Si3, whatever the crystal structure of Mn5Si3 or the magnetic behavior is. As a consequence, Mn5Si2 is not stable at 0 K. This result is conceivable since the existence of this phase was uncertain regarding the literature data (see Section 4).
6. Conclusions The enthalpies of formation of Mn3Si, Mn5Si2, Mn5Si3, MnSi and Mn4Si7 have been calculated at 0 K using the density functional theory.
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A. Berche et al. / Solid State Communications 188 (2014) 49–52
The Mn5Si3 phase which was supposed to be very stable since it has a congruent melting point is calculated as instable at 0 K. As a consequence, the thermodynamic properties of this phase have to be more carefully studied experimentally. Especially, a great care has to be taken to prepare the phases avoiding the presence of impurities. DFT calculations confirm the non-stability of the Mn5Si2 structure. This study has to be completed by DFT calculation of the enthalpy of formation of the other possible HMS structures: Mn11Si19, Mn15Si26, Mn27Si47 and Mn26Si45. Especially, the most stable defects have to be identified for modeling the solubilities of doping elements in these phases. The calculated enthalpies of formation at 0 K determined in this work will be very useful to improve the thermodynamic assessment of the industrially important Mn–Si systems using the Calphad method. Acknowledgments This work was supported by the ANR (French National Research Agency) project PHIMS (P-type HIgh Manganese Silicides by design for high performance and sustainable thermoelectric materials). Part of the computer simulations has been performed on the computer centers HPC@LR and CINES in Montpellier. The authors thank Dr. Catherine Colinet for fruitful discussions. References [1] A. Zaitsev, in: D.M. Rowe (Ed.), CRC Handbook of Thermoelectrics, 1995. [2] Y.J. Shi, Q.M. Lu, X. Zhang, J.X. Zhang, J. Inorg. Mater. 26 (7) (2011) 691–695. [3] M. Bolduc, C. Awo-Affouda, A. Stollenwerk, M.B. Huang, F.G. Ramos, G. Agnello, V.P. LaBella, Phys. Rev. B 71 (2005) 033302. [4] S. Zhou, K. Potzger, G. Zhang, A. Mücklich, F. Eichhorn, N. Schell, R. Grötzschel, B. Schmidt, W. Skorupa, M. Helm, et al., Phys. Rev. B 75 (2007) 085203. [5] A. Luo, M.O. Pekguleryuz, J. Mater. Sci. 29 (1994) 5259–5271. [6] U. Gottlieb, A. Sulpice, B. Lambert-Andron, O. Laborde, J, Alloys Compd. 361 (2003) 13–18.
[7] R. De Ridder, S. Amelinckx, Mat. Res. Bull. 6 (1971) 1223–1234. [8] G.M. Lukashenko, V.P. Sidorko, B.Y. Kotur, Poroshk. Metall. 8 (1981) 67–70. [9] Y.M. Gertman, P.V. Gel’d, in: Proceedings of the Symposium on Izd-vo Akademii Nauk SSSR, Moscow, 51, 1961. [10] S.V. Meschel, O.J. Kleppa, J. Alloys Compd. 267 (1998) 128–135. [11] L. Rossemyr, T. Rosenqvist, Trans. Met. Soc. AIME 224 (1962) 140–143. [12] V.N. Eremenko, G.M. Lukashenko, V.P. Sidorko, Poroshk. Metall. 5 (1964) 49–50. [13] V.N. Eremenko, G.M. Lukashenko, V.P. Sidorko, Poroshk. Metall. 9 (1965) 765–767. [14] V.N. Eremenko, G.M. Lukashenko, V.P. Sidorko, Poroshk. Metall. 9 (1965) 74–76. [15] G.M. Lukashenko, V.R. Sidorko, Poroshk. Metall. 7 (1982) 67–70. [16] V.G. Muradov, Russ. J. Phys. Chem. 48 (8) (1974) 1274 (1274). [17] H. Nowotny, J. Tomiska, L. Erdelyi, A. Neckel, Monatsh. Chem. 108 (1) (1977) 7–19. [18] A. Zaitsev, M.A. Zemchenko, B.M. Mogutnov, Russ. J. Phys. Chem. 63 (6) (1989) 806–810. [19] Y.M. Golutvin, T.M. Kozlovskaya, E.G. Maslennikova, Russ. J. Phys. Chem. 37 (6) (1963) 723–727. [20] G. Kresse, J. Furthmüller, Comp. Mater. Sci. 6 (1996) 15–50. [21] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186. [22] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758–1775. [23] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953–17979. [24] P. Vinet, J.H. Rose, J. Ferrante, J.R. Smith, J. Phys: Condens. Matter 1 (1989) 1941–1951. [25] D. Hobbs, J. Hafner, D. Spisak, Phys. Rev. B 68 (2003) 014407. [26] T. Jeong, Physica B 407 (2012) 888–890. [27] D. Kwon, H.K. Kim, J.H. Kim, Y.E. Ihm, D. Kim, H. Kim, J.S. Baek, C.S. Kim, K. K. Choo, J. Magn. Magn. Mater. 282 (2004) 240–243. [28] A. Zygmunt, J. Leciejewicz, B. Penc, A. Szytula, A. Jezierski, Acta Phys. Pol. A 113 (4) (2008) 1193–1203. [29] Y. Imai, M. Mukaida, K. Kobayashi, T. Tsunoda, Intermetallics 9 (2001) 261–268. [30] S. Caprara, E. Kulatov, V.V. Tugushev, Eur. Phys. J.B 85 (2012) 149–161. [31] H. Putz, K. Brandenbourg, Pearson's Crystal Data. Crystal Structure Database For Inorganic Compounds. Cd-Rom software version 1.0., 2007. [32] J.M. Higgins, R. Ding, S. Jin, Chem. Mater. 23 (2011) 3848–3853. [33] M. Gottschilch, O. Gourdon, J. Persson, C. De la Cruz, V. Petricek, T. Brueckel, J. Mater. Chem. 22 (2012) 15275–15284. [34] J.P. Senateur, R. Fruchart, C. R. Acad. Sci. Paris 258 (1964) 1524–1525. [35] J.J. Williams, Y.Y. Ye, M.J. Kramer, K.M. Hoa, L. Hong, C.L. Fu, S.K. Malik, Intermetallics 8 (2000) 937–943. [36] C. Sürgers, K. Potzger, T. Strache, W. Möller, G. Fischer, N. Joshi, H.V. Löhneysen, Appl. Phys. Lett. 93 (6) (2008) 062503. [37] C. Colinet, Priv. Commun.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
First-principles determination of the enthalpy of formation of Mn–Si phases Alexandre Berche n, Jean-Claude Tédenac, Philippe Jund ICGM-Université Montpellier II, UMR-CNRS 5253, Pl. E. Bataillon CC1506, 34095 Montpellier, France
art ic l e i nf o
a b s t r a c t
Article history: Received 17 December 2013 Received in revised form 10 February 2014 Accepted 17 February 2014 by Ralph Gebauer Available online 26 February 2014
After a careful survey of the literature, the enthalpies of formation of Mn3Si, Mn5Si2, Mn5Si3, MnSi and Mn4Si7 have been calculated at 0 K using the density functional theory. The calculated physical properties are compared to previous calculations and experiments. The enthalpies of formation of the phases are finally discussed. & 2014 Elsevier Ltd. All rights reserved.
Keywords: A. semiconductors D. Thermodynamics properties
1. Introduction Manganese and silicon are especially interesting due to their compatibility with industrialization at a large scale (low-cost production processes, abundant materials) and with the rules of hygiene, health and environment. Moreover, the Mn–Si system contains many intermetallic compounds or phases presenting a real potential for applications. Generating electricity from waste heat by means of “green” thermoelectric generators could reduce the global CO2 footprint of industrial activities. Among the potential materials for thermoelectric applications, Higher Manganese Silicides (HMS) MnSix (with x around 1.75) exhibit interesting figures-of-merit at intermediate temperatures (573–873 K) [1,2]. Other applications of these elements have been studied as the possible use of the spin effects in electronic devices for the transfer, processing, and storage of information [3,4]. Manganese and silicon are also known to improve the mechanical properties of low weight and high strength alloys for automotive and aerospace industries [5]. Depending on the application, physical properties such as the electronic band gap, the magnetic moment or the bulk modulus of the phases are critical data. Moreover, all these properties are evidently sensible to the alloy composition and to the annealing process. As a consequence, optimization of elaboration needs the precise knowledge of the thermodynamic properties and an accurate description of the systems including ternaries and binaries as well. To be as accurate as possible, such descriptions have
n Corresponding author. Present address: ICGM-Université Montpellier II, UMRCNRS 5253, Pl. E. Bataillon CC1506, Montpellier, France Tel.: þ 33 6 31 15 85 93. E-mail address: [email protected] (A. Berche).
http://dx.doi.org/10.1016/j.ssc.2014.02.021 0038-1098 & 2014 Elsevier Ltd. All rights reserved.
to be based on consistent thermodynamic values. Among these data, the enthalpies of formation of intermediate phases are primordial because they represent the reciprocal stability of the phases at low temperature. The paper is organized as follows: in Section 2 a literature survey of the enthalpy of formation is proposed. In Section 3 the computational details are given. In Section 4 the results for the physical properties are given. Finally the relative stability of the phases is discussed in Section 5 and conclusions are given in Section 6.
2. Phase stability and experimental enthalpy of formation in the literature At room temperature and below, 6 phases are mentioned in the literature: Mn, Mn3Si, Mn5Si2, Mn5Si3, MnSi and MnSix. MnSix: the HMS phase whose exact stoichiometry (xE 1.75) and structure are not exactly known. In fact, five structures were suggested for this compound: Mn4Si7, Mn11Si19, Mn15Si26, Mn27Si47 and Mn26Si45 [6]. According to De Ridder [7], even after annealing, several phases are identified in the same sample. De Ridder concludes that all these structures co-exist in the system and constitute a quasicontinuous defect fraction x in the range 0.00 oxo 0.05 for MnSi1.75 x. He also suggests that the energy difference between these phases is very low. Since the crystal structures of these phases correspond to large unit-cells (from 44 to 296 atoms in the structure), only the smallest one (Mn4Si7) is studied in this work. In addition 3 other phases with xSi o0.2 are mentioned in the literature (R-Mn6Si, ν-Mn9Si2 and the γ-phase). Since these phases have random site occupancies, they are not taken into consideration in this work.
50
A. Berche et al. / Solid State Communications 188 (2014) 49–52
In the Mn–Si system, even if the enthalpies of formation of the phases have been measured many times [8–19], the results remain very scattered. To compare them more easily, the values are calculated for one mole of atom, whereas they are often given for one mole of formula unit in the original papers. The bomb calorimetry method used by Golutvin [19] seems to be very imprecise regarding the results and their error bars. Moreover, it is possible that the alloys of Golutvin [19] reacted with zirconia crucibles during synthesis. As a consequence, these values are ruled out. The results of Lukashenko [8,15] are in very good agreement with those of Eremenko [12–14], Nowotny [17] and Gertman [9] for Mnrich phases. Moreover, the values of Meschel [10] are in very good agreement with those of Eremenko [12–14] for Mn5Si3 and MnSi. However, the shape of the enthalpy of formation curves varies with the sources: smooth with a minimum around MnSi for Gertman [9], or around Mn5Si3 for Muradov [16] or sharp on MnSi for Eremenko [12–14] and Zaitsev [18]. The global uncertainty both on the value and on the shape of the enthalpy of formation curve implies that more data are necessary. Since the experiments seem to be very difficult and give widespread results, a calculation method is employed in this work using the DFT to try to obtain a correct estimation of the enthalpy of formation of the different Mn–Si phases. Thus, in this paper, the relative stability of the phases is discussed on the basis of the density functional theory (DFT) calculations of the enthalpy of formation of the different phases in the Mn–Si system.
3. Computational details The density functional theory calculations are performed using the Vienna ab initio simulation package (VASP) [20,21] using the projector augmented waves (PAW) technique [22,23] within the generalized gradient approximation (GGA). The Perdew–Burke– Ernzerhof parameterization (PBE) is applied. Standard versions of the PAW potentials for Si and Mn are used. For Si four electronic states are included in the valence shell (3s23p2) whereas 13 electronic states are taken into account for Mn (3p64s23d5). The first Brillouin zone is integrated using Monkhorst–Pack k-point meshes. A Gamma centered k-point grid was used for the hexagonal structure. The plane-wave energy cutoff and the reciprocal space meshes (k-point) are increased to achieve total energies with a precision less than 0.5 meV/atom. The selected cutoff energy varies from 350 to 600 eV and the number of k-points goes from 15 to 364 depending on the crystal structure. The calculations are performed using the “accurate” precision setting in the VASP input file to avoid wrap-around errors. For metallic phases, the ISMEAR tag is taken equal to 1 (1st order Methfessel–Paxton smearing), whereas for semiconductors and half-metals a value of 5 (tetrahedron method with Blöchl corrections) is used. The total energy of each structure is minimized by relaxing the volume of the cell and the ionic positions starting from the experimental parameters. In a second step, the equilibrium volume is slightly compressed or expanded and the total energy is minimized for a given fixed volume. These several total energies are then fitted using the Vinet equation of state [24] which allows calculating the equilibrium volume (V0), the total energy E and the bulk modulus B of the crystal structure. The enthalpy of formation at 0 K of the Mn1 xSix phase (ΔfH0(Mn1 xSix)) is calculated by Δf H 0 ðMn1 x Six Þ ¼ EðMn1 x Six Þ–ð1 xÞEðMnÞ xEðSiÞ
ð1Þ
where E(Mn1 xSix), E(Mn) and E(Si) are the equilibrium firstprinciples calculated total energies of Mn1 xSix, Mn and Si respectively in their stable crystalline structure at 0 K. DFT calculations also allow drawing the density of states (DOS) and the electronic band structure of the phases. In the case of semiconducting materials, the electronic gap is determined from these electronic band structure calculations. For each structure, the calculated DOS are similar to those previously calculated in the literature.
4. Calculated results The physical properties of each phase have been calculated from first-principles calculations and a reasonable agreement with previous calculations and measurements is obtained [25–30]. The calculated cell parameters are not presented in this paper, but their values are less than 3% smaller than the experimental values [31], which is a good agreement. The parameters of the D8m-Mn5Si3 phase are 4% smaller than the measured values of Higgins [32]; however these measurements were performed on nanowires and were never measured on bulk materials. The bulk modulus is calculated and shows a variation with respect to the experimental values of 10–20% for all the phases except for αMn (around 30%). However, for this element, experimental data seems to be very doubtful. The bulk moduli of Mn5Si2 and D8m-Mn5Si3 are calculated at 191 and 176 GPa respectively. The calculated magnetic moments and electronic gap of the semiconducting (Mn4Si7 and Si) or half-metal (MnSi and Mn3Si) of the phases are in good agreement with the previous calculations and measurements. On cooling, Mn5Si3 undergoes a second order phase transition towards an orthorhombic structure at 100 K and towards a tetragonal distorted structure at 62 K according to Gottschilch [33]. In our calculations, the low orthorhombic phase is calculated as ferrimagnetic in contradiction with the previous calculations and experimental works of Zygmund [28]. A metastable paramagnetic form of LT-Mn5Si3 was obtained; however, the magnetic moments (0.27, 0.11 and 0.76) are lower than those calculated by Zygmund. It is worth noting that in this work, DFT calculations were performed on 2 structures which have never been calculated before: Mn5Si2: this low-temperature phase decomposes into Mn3Si and Mn5Si3 around 1123 K according to Senateur [34]. However, after a careful annealing of 100 h at 1023 K, Lukashenko [8] did not observe the presence of this phase. The existence of this phase remains uncertain. This structure could have been stabilized by impurities if it turns out that the starting materials of Shoemaker were less pure than those of Lukashenko (99.999% for Si and 499.8% for Mn). Unfortunately, the purity grade of the materials used by Shoemaker is not known. According to Kwon [27], thin films of Mn5Si2 are non-magnetic. To our knowledge, this is the first time that first-principles calculations are performed on this phase. Its total DOS shows that this phase is metallic. Mn5Si2 is clearly magnetic since its total energy is 53 meV lower than in the non-magnetic form. The magnetic moments are for each site 0.249, 2.383, 2.047, 0.145, 1.883 and 1.79μB. This result is in contradiction with the measurements of Kwon [27] on multiphased thin films. D8m-Mn5Si3: Higgins [32] mentioned the existence of Mn5Si3 nanowires with the D8m structure. As discussed by the author, it is the first time that such a crystalline structure is mentioned for Mn5Si3. The electronic properties of this structure were never calculated. The DFT calculations lead to a ferrimagnetic structure with two magnetic moments of 0.175 and 1.47μB.
A. Berche et al. / Solid State Communications 188 (2014) 49–52
51
Table 1 Crystallographic structure (space group and prototype when available) and calculated total energy at 0 K, enthalpy of formation at 0, 298 and 1363 K for the phases of the Mn–Si system; (a) values extrapolated from the convex hull; (b) values for non-magnetic structures; and (c) from the total energy of the relaxed structures. Space group/prototype
Etot (eV atom 1)
T (K) Reference α-Mn
– [31] I-43m/α-Mn
R-Mn6Si ν-Mn9Si2 Mn3Si
P-3 R-3h Fm-3m/BiF3
Mn5Si2
P41212/Mn5Si2
D88-Mn5Si3
P63/mcm/Mn5Si3
LT-Mn5Si3
Ccmm/Mn5Si3
D8m-Mn5Si3
I4/mcm/W5Si3
MnSi
P213/FeSi
Mn4Si7
P-4c2
Si
Fd-3m/C
0 This work 9.012 8.991(b)(c) – – 8.424 8.321(b)(c) 8.255 8.200(b)(c) 7.998 7.981(b) 8.002 7.981(b) 8.019 7.990(b) 7.669 7.644(b)(c) 7.156 7.156(b)(c) 5.426
ΔfH (kJ mol of atom 1)
0.00 0.00(b) 18.20(a) 22.50(a) 29.79 21.40(b) 25.91 21.71(b) 31.94 31.50(b) 32.34 31.51(b) 33.95 32.35(b) 43.42 42.02(b) 40.99 41.72(b) 0.00
298 [8–10,12–15] 0.00 – 16.25 20.50 26.74 – – – 34.10 – – – – – 39.60 – 34.10 – 0.00
1363 [11] 0.00 – 13.10(a) 15.70(a) 19.00(a) – – – 21.71 – – – – – 20.92 – 16.04 – 0.00
congruent melting phase. Two hypotheses are suggested to explain this phenomenon:
Fig. 1. (Color online) Enthalpy of formation of the phases and convex hull at 0 K by the DFT (this work), 298 K (from the literature); non-magnetic phases are labeled NM.
5. Thermodynamic properties of the phases The enthalpies of formation of the phases at 0 K are calculated from the total energies by using Eq. (1) and are presented in Table 1 and displayed in Fig. 1. The convex hull (Fig. 1) exhibits a smooth shape with a minimum for MnSi. Such a result is in agreement with the experimental results of Nowotny [17], Gertman [9] and Lukashenko [8,15] as detailed in Section 2. Concerning Mn5Si3, the enthalpy of formation was calculated for three crystallographic structures: D88 (2 Mn sites: 4d (⅓; ⅔; 0) and 6g (0.236; 0; ¼) and one Si site on 6g (0.5991; 0; ¼)), D8m (2 Mn sites: 4b (0; ½; ¼) and 16k (0.075; 0.223; 0) and 2 Si sites: 4a (0; 0; ¼) and 8h (0.17; 0.67; 0)) and the low-temperature magnetic distortion (LT) of D88 (3 Mn sites: 4c (0.2637; ½; ¼), 8e (0; 0.33245; 0) and 8g (0.6175; 0.38271; ¼) and 2 Si sites: 4c (0.1002; ½; ¼) and 8g (0.29949; 0.70028; ¼)). The enthalpies of formation of D88 and LT are very close. This is a classical result since the magnetic stabilization of a phase is generally weak. However, it is surprising that the D8m structure is more stable than the D88 and LT ones. Another unexpected result consists in the fact that Mn5Si3 is less stable than Mn3Si þMnSi. This is particularly surprising for a
Mn5Si3 could have been stabilized experimentally by impurities such as boron, carbon or oxygen as it has been demonstrated in D88-Ti5Si3 compounds [35]. Moreover, the implantation of carbon in Mn5Si3 structure has been evidenced by Sürgers [36]. Finally, addition of oxygen or carbon in 2b sites (0; 0; 0) highly stabilizes the D88 structure according to DFT calculations performed by Colinet [37]. Mn5Si3 is less stable than Mn3Si þMnSi at 0 K due to the complex magnetic behavior of the Mn–Si phases. At higher temperatures, the magnetism disappears and the phase Mn5Si3 becomes stable. To check this hypothesis, DFT calculations are performed for non-magnetic structures (Table 1). The enthalpies of formation of Mn-rich compounds are highly modified (Fig. 1), and Mn5Si3 becomes more stable than a mix of Mn3Si þMnSi. To completely explain this point, DFT calculations would be necessary to estimate: the contributions from lattice vibrations and from magnetic excitations to the enthalpy of formation at 0 K; and the energetic contribution of defects and impurities (O, C, B) on the stability of the phases. These two points should be developed in a future work. The enthalpy of formation of Mn5Si2 at 0 K indicates that this phase is less stable than a mix of Mn3Siþ MnSi or Mn3Si þMn5Si3, whatever the crystal structure of Mn5Si3 or the magnetic behavior is. As a consequence, Mn5Si2 is not stable at 0 K. This result is conceivable since the existence of this phase was uncertain regarding the literature data (see Section 4).
6. Conclusions The enthalpies of formation of Mn3Si, Mn5Si2, Mn5Si3, MnSi and Mn4Si7 have been calculated at 0 K using the density functional theory.
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A. Berche et al. / Solid State Communications 188 (2014) 49–52
The Mn5Si3 phase which was supposed to be very stable since it has a congruent melting point is calculated as instable at 0 K. As a consequence, the thermodynamic properties of this phase have to be more carefully studied experimentally. Especially, a great care has to be taken to prepare the phases avoiding the presence of impurities. DFT calculations confirm the non-stability of the Mn5Si2 structure. This study has to be completed by DFT calculation of the enthalpy of formation of the other possible HMS structures: Mn11Si19, Mn15Si26, Mn27Si47 and Mn26Si45. Especially, the most stable defects have to be identified for modeling the solubilities of doping elements in these phases. The calculated enthalpies of formation at 0 K determined in this work will be very useful to improve the thermodynamic assessment of the industrially important Mn–Si systems using the Calphad method. Acknowledgments This work was supported by the ANR (French National Research Agency) project PHIMS (P-type HIgh Manganese Silicides by design for high performance and sustainable thermoelectric materials). Part of the computer simulations has been performed on the computer centers HPC@LR and CINES in Montpellier. The authors thank Dr. Catherine Colinet for fruitful discussions. References [1] A. Zaitsev, in: D.M. Rowe (Ed.), CRC Handbook of Thermoelectrics, 1995. [2] Y.J. Shi, Q.M. Lu, X. Zhang, J.X. Zhang, J. Inorg. Mater. 26 (7) (2011) 691–695. [3] M. Bolduc, C. Awo-Affouda, A. Stollenwerk, M.B. Huang, F.G. Ramos, G. Agnello, V.P. LaBella, Phys. Rev. B 71 (2005) 033302. [4] S. Zhou, K. Potzger, G. Zhang, A. Mücklich, F. Eichhorn, N. Schell, R. Grötzschel, B. Schmidt, W. Skorupa, M. Helm, et al., Phys. Rev. B 75 (2007) 085203. [5] A. Luo, M.O. Pekguleryuz, J. Mater. Sci. 29 (1994) 5259–5271. [6] U. Gottlieb, A. Sulpice, B. Lambert-Andron, O. Laborde, J, Alloys Compd. 361 (2003) 13–18.
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