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Thermodynamic study of the Ge–Mn–Si system
Journal of Alloys and Compounds 632 (2015) 10–16
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Thermodynamic study of the Ge–Mn–Si system Alexandre Berche ⇑, Elodie Théron-Ruiz, Jean-Claude Tédenac, Philippe Jund ICGM – Université Montpellier II, UMR-CNRS 5253, Pl. E. Bataillon CC1506, Montpellier 34095, France
a r t i c l e
i n f o
Article history: Received 3 October 2014 Received in revised form 8 December 2014 Accepted 12 January 2015 Available online 19 January 2015 Keywords: DFT calculations Solid solution Calphad method
a b s t r a c t 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). Moreover it appears that the figure-of-merit can be improved by germanium doping. The optimization of the elaboration of such alloys needs the knowledge of the ternary Ge–Mn–Si system. The solid–solid equilibrium in the ternary system seems to be highly driven by the D88 phase an isostructural form of Mn5Si3 and Mn5Ge3. However, the existence of a complete solid solution between these two compounds is not clearly established in the literature. First principles calculations are performed to clarify the mixing enthalpies in this potential solid solution. The predicted phase separation at low temperature is then experimentally confirmed by metallographic examinations on 3 annealed samples. On the basis of these new data, a thermodynamic description of the ternary system is performed using the Calphad method. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction 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). Moreover it appears that the figure-of-merit can be improved by germanium doping [1]. The optimization of the elaboration of such alloys needs the knowledge of the ternary Ge–Mn–Si system and of its constitutive binaries. The aim of this work is to obtain a better description of the ternary system and more especially in the vicinity of the HMS phase. In previous studies, Ge–Mn [2] and Mn–Si [3,4] systems have been studied and described using the Calphad method. Coupled with the database on the Ge–Si system assessed by Olesinski [5], several isothermal sections have been drawn using the Calphad method. From these calculations, it appears that the equilibriums involving the HMS phase are guided by the stability of the potential solid solution linking Mn5Ge3–Mn5Si3 in the D88 structure. This hexagonal structure has a P63/mcm space group (number 193), the Mn atoms are on 4d (1/3, 2/3, 0) and 6g (0.2358, 0, 1/4) positions whereas Ge and Si are located on another 6g (0.5992, 0, 1/4) position according to Aronsson [6].
⇑ Corresponding author at: ICGM–MESO–Université Montpellier II, UMR-CNRS 5253, Pl. E. Bataillon CC1506, Montpellier, France. Tel.: +33 (0)4 67 14 93 52; fax: +33 (0)4 67 14 42 90. E-mail address: [email protected] (A. Berche). http://dx.doi.org/10.1016/j.jallcom.2015.01.072 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.
In the literature, two authors report studies on this solid solution. Kieu Van Con [7] studied several alloys with xMn = 0.625 on the silicon side and according to his analyses, two phases are observed for xGe = 0.15, however no additional details (the annealing temperature for example) are available. In contradiction to this first result, Kappel [8] reinvestigated this section by XRD analyses on 12 alloys melted in an induction furnace and annealed at 1123 K during 4 or 7 days depending on the germanium content. A continuous solid solution is observed at this temperature. Since this D88 phase is the key phase of the ternary system, the potential solid solution is studied using DFT and experimental analyses in this paper. In the first part, theoretical and experimental details are given, in the second part, results concerning the D88 phase are given, in the third part, the Calphad model of the ternary system is described and finally conclusions are given in the last part.
2. Theoretical and experimental details In this work, three different procedures are employed: a theoretical one within the density functional theory (DFT), an experimental one consisting of synthesis, X-ray diffraction (XRD) and metallography and the Calphad method for modeling the phase diagram. Such methods are very classical and only the specific points are detailed in this part. The DFT calculations are performed using the Vienna ab initio simulation package (VASP) [9,10] using the projector augmented waves (PAW) technique within the generalized gradient approximation (GGA). The Perdew–Burke–Ernzerhof parameterization (PBE) is applied. Standard versions of the PAW potentials for Ge (3d54s24p2), Mn (3p64s23d5) and Si (3s23p2) are used. The first Brillouin zone is
A. Berche et al. / Journal of Alloys and Compounds 632 (2015) 10–16
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integrated using Monkhorst–Pack k-point meshes. The plane-wave energy cutoff and the reciprocal space meshes (k-points) are increased to achieve total energies with a precision better than 0.5 meV/atom. The selected cutoff energy is of 500 eV and the number of k-points goes from 15 to 22 depending on the cell. The calculations are performed using the ‘‘accurate’’ precision setting in the VASP input file to avoid wrap-around errors. The ISMEAR tag is taken equal to 5 (tetrahedron method with Blöchl corrections). The total energy of each structure is minimized by relaxing the volume of the cell and the ionic positions starting from the experimental parameters. Two kinds of cells are taken into consideration. The first one consists in the primitive cell (containing 16 atoms). In a second step a larger cell (2 1 1 containing 32 atoms) is considered. This second case will allow checking if the primitive cell is big enough to allow total relaxation of the atomic positions in the solid solutions. The synthesized alloys are prepared as follows. The pure Ge (Alfa Aesar, 99.9999%), Mn (Alfa Aesar, 99.9%) and Si (Alfa Aesar, 99.9999%) are first crushed, weighted under stoichiometric ratios and then mechanically compacted. The obtained tablet is then melted 3 times in an arc-melting furnace under argon atmosphere to obtain a homogeneous sample. The as-cast samples are finally annealed 15 or 30 d at 900 K in quartz tubes under vacuum prior to be quenched in water. The samples are analyzed by X-ray diffraction (XRD) and Scanning Electron Microscope (SEM). Powder diffraction patterns are recorded in the [20–100°] 2h range, with a step size of 0.0170° and a step time of 400 s on a Philips Expert diffractometer (XPert Pro) equipped with a copper Ka1 anticathode. To avoid the fluorescence phenomenon, a monochromator is added between the sample and the detector. The pattern matching was done using the Powdercell software [11]. In the Calphad assessment, the liquid phase and the solid solutions based on the allotropic forms of Mn, Ge and Si are described with a substitution model using the Redlich–Kister formalism. The liquid is described as (Ge, Mn, Mn5Ge3, and Si) since an associate model is taken in the Ge–Mn system. The variation to the stoichiometry is modeled for a few phases (MnGe2, Mn5Ge2, MnGe3, R-Mn6Si and v-Mn9Si2). For these phases, anti-sites are assumed to be the main defects. More details are available in [2,4].
3. Results on the D88 phase
3.1. DFT calculations In a first step, the potential stability of the D88 solid solution is investigated using DFT calculations. For this, calculations are performed on a primitive cell composed of 16 atoms where the 6 g (0.5992, 0, 1/4) positions are occupied by Ge, Si or a mix of Ge and Si. Since the 6 g positions are equivalent, 7 crystals are generated. For each composition, several configurations have been tested. The configuration with the lowest total energy is selected in each case. It appears that for each composition, the more stable state corresponds to a configuration where the substitute atoms are as far as possible in the structure. The cell parameters and the mean magnetic moment of the Mn atoms are calculated for each atomic position. As can be seen in Fig. 1, the cell parameters vary linearly with the silicon content in good agreement with the experimental results of Kieu Von Con [7] and Kappel [8]. The same behavior is noticed for the magnetic moments of manganese on the 4a and 6g sites. The magnetic moments calculated for Mn5Si3 are in good agreement with previous calculations of Arras [12] and in an acceptable agreement with the measurements of Forsyth [13]. The mixing enthalpy in the D88 potential solid solution at 0 K is calculated using Eq. (1).
Dm H0 K ¼ EðMn0:625 Ge0:375ð1xÞ Si0:375x Þ ð1 xÞ EðMn0:625 Ge0:375 Þ x EðMn0:625 Si0:375 Þ
ð1Þ
where x is given by Eq. (2),
x ¼ NSi =N6g
ð2Þ
where NSi is the number of Si atoms on the 6g positions in the cell and N6g is the total number of 6g positions in the cell. For a single cell, N6g = 6 and for a double cell, N6g = 12.
Fig. 1. physical properties of the D88 solid solution: (a) a cell parameter; (b) c cell parameter; (c) mean magnetic moment of the Mn atoms.
E(Mn0.625Ge0.375(1x)Si0.375x) is the calculated energy (for one atom) of a particular case of the D88 solid solution, and E(Mn0.625
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A. Berche et al. / Journal of Alloys and Compounds 632 (2015) 10–16
Fig. 2. Mixing enthalpy in the D88 solid solution.
Ge0.375), E(Mn0.625Si0.375) are the calculated energies of the crystals of respectively Mn5Ge3 and Mn5Si3 in the D88 structure given for one atom. DmH0K is then given in eV/atom or in J/mol of atom. As one can see in Fig. 2, the calculated mixing enthalpy in the D88 phase at 0 K is positive. Moreover, in this case, the size of the cell (primitive or larger) does not modify the results which is consistent with our previous published work [14]. A positive mixing enthalpy is consistent with the existence of a possible phase separation at low temperature in agreement with the results of Kieu Von Con [7]. However, since the maximum of the mixing enthalpy curve is not very high (1285 J/mol of atom), it could be possible that an entropy term induces the existence of a continuous solid solution at higher temperature as it was shown by Kappel [8]. To check this point, some experiments must be performed. 3.2. Experimental verification Three alloys have been synthesized between Mn5Ge3 and Mn5Si3. The compositions are given in Table 1. According to Kappel [8], at 1123 K, a homogenization time of 4–7 days is necessary depending on the silicon content of the samples. In a trial, the alloys are annealed at 900 K for 2 weeks. All the alloys are mainly composed of two phases (Fig. 3). According to EDX analyses, the Mn content difference is low in the two phases. The Si content is higher in the white phase. According to these measurements and depending on the sample, the dark phase contains between 2 and 7 at.% of Si, whereas the white phase contains between 12 and 30 at.% Si. The composition of the white phase is obtained with a better accuracy in the sample with the highest silicon content (Fig. 3c). At the light of this observation, the samples show the presence of two phases, crystallographically similar but with slight differences in the compositions, which is
Fig. 3. Metallographic picture of the Mn0.6250Ge1xSix alloys annealed 2 weeks at 900 K with xSi equal to: (a) 0.0940; (b) 0.1875; (c) 0.2810.
Table 1 Experimental data measured on the samples annealed at 900 K during 2 weeks: global initial composition in Si, cell parameters and EDX composition of the two phases contained in each sample. xSi
Cell parameter (Å) – XRD Phase 1
0.0940 0.1875 0.2810
Composition – EDX/XRD Phase 2
Phase 1/dark
Phase 2/white
a
c
a
c
xSi
xSi
7.1721(1) 7.1520(8) 7.203(56)
5.0123(3) 4.9995(6) 5.021(77)
7.096(01) 6.9792(7) 6.9887(3)
4.939(30) 4.8673(9) 4.8696(7)
0.042/0.040 0.020/0.070 (0.067/0.020)
(0.121/0.160) 0.170/0.300 0.300/0.290
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Fig. 4. Diffractograms of the Mn0.6250Ge1xSix alloys annealed 2 weeks at 900 K with xSi equal to: 0.0940; 0.1875 and 0.2810.
Fig. 5. Diffractograms of the Mn0.6250Ge0.1875Si0.1875 as-cast, annealed 2 and 4 weeks at 900 K.
significant of a phase separation between two solid-solutions (Mn5(Ge, Si)3 a and Mn5(Ge, Si)3 b). By combining the cell parameters with the Vegard’s law measured by Kappel (Fig. 1a and b), the composition of each phase is estimated (Table 1). The so-obtained values are 4 ± 3 at.% Si and 30 ± 3 at.% Si for respectively the dark and white phases. Such values are in good agreement with the EDX analyses. At the light of these results, we evidence at low temperature a phase separation in the D88 phase as previously suggested by Kieu Von Con [7]. However, regarding to both XRD and SEM analyses, we suggest that the thermodynamic equilibrium is not reached after 2 weeks of annealing at 900 K. We observe gray shades in the sample containing 18.75 at.% Si (Fig. 3b) which can indicate composition changes in the phase. Instead of a D88 phase (labeled Phase 3 in Fig. 4) with intermediate cell parameters, we note that a continuum in the peaks is observed and probably due to a continuous change of composition. It shows that in our conditions, the kinetic reaction is slow and that after two weeks, the reaction is not complete. Nevertheless, assuming a medium composition of the alloys, the cell parameters of the ‘‘Phase 3’’ can be estimated as follows: a = 7.11 Å and c = 4.97 Å . To check this point, this sample was annealed 2 more weeks at 900 K. The diffractograms are compared for the as-cast sample and the two annealed samples after 2 and 4 weeks. Results are presented in Fig. 5. The phenomenon is temperature dependent and not time dependent. Indeed, the difference between the as-cast and annealed
Fig. 6. Assessed phase diagram in the: (a) Ge–Mn system [2]; (b) Ge–Si system [5]; (c) Mn–Si system [4].
samples is not obvious as it is shown in Fig. 5. The intensity of the diffraction peaks of the Phase 3 seems to go down with annealing time, but the intensity of some of the peaks of the Phases 1 and 2
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Fig. 7. Assessed sections of the ternary system: (a) isothermal section at 500 K; (b) isothermal section at 900 K; (c) isothermal section at 1173 K, blue circles correspond to the compositions of the diamond phase measured by Zhou [15]; (d) isothermal section at 1350 K; (e) isopleth section between Mn5Ge3 and Mn5Si3; (f) zoom of the isopleth section with xMn = 0.6363. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
are also reduced after a longer annealing process. This seems to indicate that annealing will lead to a phase separation. Such results are consistent with the observations of Kappel who observed that the thermodynamic equilibrium was reached after 1 week but for an annealing temperature 200 K higher. Such a slow kinetic process may indicate that the temperature and the compositions are very narrow close to the consolute point.
We can also notice the presence of 2 unmatched peaks on the diffractograms (Figs. 4 and 5). These peaks may correspond to another phase which is observed because of a variation to the nominal composition due to the vaporization of germanium during the synthesis of the samples. Since the intensity of the D88 phase is very low, these peaks could correspond to a very low quantity of a phase which exhibits more intense peaks (such as Mn3Ge for
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Table 2 Models and parameters of the phases (this work), assessed parameters are in bold type, other parameters are taken from [2,4,5]. Phase Model
Parameters (J/mol)
Liquid (Ge, Mn5Ge3, Mn, Si)1
0
D88 (Mn)0.625(Ge, Si)0.375
L(Mn5Ge3, Si) = 25,000
0
L(Mn:Ge, Si) = 5000 L(Mn:Ge, Si) = 1500
2
MnSix (Mn)0.146(Mn, VA)0.220 (Ge, Mn, Si)0.244(Ge, Si)0.390
0 SER 0 SER bcc diam G(MnSix, Mn:Mn:Si:Ge) 0.39 ⁄ 0HSER + 0.244 ⁄ Gdiam Ge 0.366 ⁄ HMn 0.244 ⁄ HSi = 11,000 0.5 ⁄ T + 0.366 ⁄ GMn + 0.39 ⁄ GGe Si 0 SER 0 SER bcc diam G(MnSix, Mn:Mn:Ge:Si) 0.244 ⁄ 0HSER + 0.39 ⁄ Gdiam Ge 0.366 ⁄ HMn 0.39 ⁄ HSi = 11,000 0.5 ⁄ T + 0.366 ⁄ GMn + 0.244 ⁄ GGe Si 0 SER bcc diam G(MnSix, Mn:Mn:Ge:Ge) 0.634 ⁄ 0HSER Ge 0.366 ⁄ HMn = 0.366 ⁄ GMn + 0.634 ⁄ GGe 0 SER bcc diam G(MnSix, Mn:VA:Ge:Ge) 0.634 ⁄ 0HSER Ge 0.146 ⁄ HMn = 0.146 ⁄ GMn + 0.634 ⁄ GGe 0 SER bcc diam G(MnSix, Mn:Mn:Mn:Ge) 0.39 ⁄ 0HSER Ge 0.61⁄ HMn = 0.61 ⁄ GMn + 0.39 ⁄ GGe 0 SER bcc diam G(MnSix, Mn:VA:Mn:Ge) 0.39 ⁄ 0HSER Ge 0.39 ⁄ HMn = 0.39 ⁄ GMn + 0.39 ⁄ GGe 0 L(Mn:Mn:Si:Ge, Si) = 5000 0 L(Mn:Mn:Ge, Si:Si) = 5000 0 L(Mn:Mn:Ge, Si:Ge, Si) = 7000
B20_MnSi (Mn)0.5(Ge, Si)0.5
0 SER bcc diam G(B20_MnSi, Mn:Ge) 0.5 ⁄ 0HSER GE 0.5 ⁄ HMn = 8000 – 0.5 ⁄ T + 0.5 ⁄ GMn + 0.5 ⁄ GSi L(B20_MnSi, Mn:Ge, Si) = 8000
Diamond (Ge, Mn, Si)1
0
L(Ge, Mn) = 22,000 L(Ge, Mn, Si) = 140,000
0
example). We can notice that after 4 weeks, the intensity of these unidentified peaks is practically included in the baseline. As a conclusion, regarding the mixing enthalpies calculated and the experimental results, it seems that a phase separation exists in the D88 phase at low temperature. However for temperatures close to 900 K, a phase separation transforms the complete solid solution in a mixture of Mn5Ge3 and Mn5Si3. 4. Calphad assessment of the Ge–Mn–Si system In this work, the three binary systems are accepted from the literature [2,4,5] as shown in Fig. 6. The sub-lattice models of the allotropic forms of pure manganese are consistent between Ge–Mn and Mn–Si. Moreover, Mn5Ge3 and Mn5Si3 have the same crystallographic structure (D88 as previously mentioned). As a consequence, the Mn5(Ge, Si)3 solid solution will be named D88 in the next parts of this paper. In Fig. 7, different isothermal sections (Fig. 7a–d) and isopleth sections (Fig. 7e and f) of the Ge–Mn–Si system are drawn. The interaction parameters in the D88 phase are modeled from the mixing enthalpies calculated in Paragraph 3.1. To reproduce this mixing enthalpy, two interaction parameters (0L and 2L) are necessary in the model (Table 2). The as-obtained curve is in very good agreement with the DFT calculations as shown in Fig. 2. Using this model, the isopleth section with xMn = 0.625 is drawn in Fig. 7e. The assessed section is consistent with the existence of the phase separation observed in this work (Paragraph 3.2) at 900 K (see Fig. 7b) and with the continuous solid solution at 1123 K (Fig. 7c) as seen by Kappel [8]. Regarding this agreement, no temperature contribution is added. This study should be completed by an experimental work aiming to determinate the temperature where the phase separation ends. Moreover, it would be interesting to check some of the 3-phase regions such as those surrounding the D88 phase. More particularly, the compositions of the phases involved would have to be compared to those calculated with the present database. Zhou [15] synthesized alloys in the (MnSix + diamond) region by induction furnace. In this study, the as-cast samples were crushed and the obtained powders were sintered at 1173 K. The compositions of the phases observed in the samples are measured by EDX. All the diamond phases contain manganese (blue circles in Fig. 7c corresponds to the compositions). Such results are
consistent with the assessed Mn–Si phase diagram (Fig. 6c). A ternary interaction term in the diamond phase is in consequence added to take into account this solubility. Moreover, according to Aoyama [16], the solubility of Ge in the MnSix phase is measured at 1 at.% Ge by EPMA and ICP on samples obtained by the Bridgman method. Zhou [15] also indicates similar solubilities by EDX measurements. However, according to previous results, one-phased alloys are obtained in MnSix phases containing 2 at.% of Ge. This last value is selected for the present assessment as shown in Fig. 7f. Finally a ternary parameter in the liquid phase between the associate Mn5Ge3 and the silicon is added to correct the too strong influence of the associate model of the Ge–Mn liquid phase. In agreement with previous experimental work, the HMS phase can be doped with around 2 atomic percent of Ge. For higher concentrations, MnSi and a Ge–Si phase will be formed in the alloy (Fig. 7). Using the database, solidification calculations were performed using the Scheil–Gulliver model. For a non-doped HMS phase, the peritectic decomposition of the phase leads to an as-cast alloy that may contain small fractions of MnSi and pure silicon. For a 2 at.% Ge doped HMS, MnSi and a germanium rich Ge–Si alloy should be observed in an as-cast alloy. Using a direct melting method, HMS alloys have to be chemically homogenized by an annealing step at a temperature below 1150 K for pure HMS or below1040 K for doped HMS. An alternative synthesis method could be ball milling. However, it is well-known that with this method, the phases with the highest enthalpy of formation will be formed in the first stages of the synthesis. Regarding the enthalpies of formation, the most stable phases are MnSi and MnSix. A ternary mixing of MnSi, HMS and Si (or Ge–Si) has to be expected. 5. Conclusions This work provides the first ternary assessment of the Ge–Mn– Si system. Based on pertinent DFT calculations and measurements, a thermodynamic description of the system is given. Of course, this first version will have to be improved with additional experimentations such as metallographic examination of the 3-phases regions (to confirm the phase equilibrium) or thermal analyses to fix the liquidus curves.
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The database will be useful in the next step of this work which will consist in synthesizing various doped MnSix samples for measuring their thermoelectric properties. 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). We would like to thanks Dominique Granier of the Institut Charles Gerhard and Claude Gril respectively for the XRD and SEM analyses. We are also thankful to Antony Lluch who worked on this system during his Master internship. 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.jallcom.2015.01. 072. References [1] Unpublished results from the Renoter research consortium,. [2] A. Berche, J.C. Tédenac, P. Jund, Thermodynamic modeling of the germaniummanganese system, Intermetallics 47 (2014) 24–30.
[3] A. Berche, J.C. Tédenac, P. Jund, First-principles determination oft he enthalpy of formation of Mn–Si system, Solid State Commun. 188 (2014) 49–52. [4] A. Berche, E. Ruiz-Théron, J.C. Tedenac, R.M. Ayral, F. Rouessac, P. Jund, Thermodynamic description of the Mn–Si system: an experimental and calculation work, J. Alloys Comp. 615 (2014) 693–702 [5] R.W. Olesinski, G.J. Abbaschian, The Ge–Si (germanium–silicon) system, Bull. Alloy Phase Diagrams 5 (2) (1984) 180–183. [6] B. Aronsson, A note on the compositions and crystal structures of MnB2, Mn3Si, Mn5Si3 and FeSi2, Acta Chem. Scand. 14 (1960) 1414–1418. [7] M. Kieu Von Con, Etude des silicogermaniures de manganèse cristallisant dans la structure D88, C.R. Acad. Sci. Paris 260 (groupe 6) (1965) 111–113. [8] G. Kappel, G. Fischer, A. Jaéglé, Magnetic investigation of the system Mn5Ge3– Mn5Si3, Phys. Status Solidi A 34 (1976) 691–696. [9] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1996) 15–50. [10] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169– 11186. [11] W. Kraus, G. Nolze, Program PowderCell 2.4, FIMRT, Berlin, Germany, 2000. [12] E. Arras, D. Caliste, T. Deutsch, F. Lançon, P. Pochet, Phase diagram, structure, and magnetic properties of the Ge–Mn system: a first-principles study, Phys. Rev. B 83 (2011) 174103. [13] J.B. Forsyth, P.J. Brown, The spatial distribution of magnetisation density in Mn5Ge3, J. Phys. Condens. Matter 2 (1990) 2713–2720. [14] A. Berche, J.C. Tédenac, P. Jund, Thermodynamic study of the Cr–Mn–Si system, Calphad. (submitted for publication). [15] A.J. Zhou, X.B. Zhao, T.J. Zhu, S.H. Yang, T. Dasgupta, C. Stiewe, R. Hassdorf, E. Mueller, Microstructure and thermoelectric properties of SiGe-added higher manganese silicides, Mater. Chem. Phys. 124 (2010) 1001–1005. [16] I. Aoyama, M.I. Fedorov, V.K. Zaitsev, F.Y. Solomkin, I.S. Eremin, A.Y. Samunin, M. Mukoujima, S. Sano, T. Tuji, Effects of Ge doping on micromorphology of MnSi in MnSi1:7 and on their thermoelectric transport properties, Jpn. J. Appl. Phys. 44 (12) (2005) 8562–8570.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Thermodynamic study of the Ge–Mn–Si system Alexandre Berche ⇑, Elodie Théron-Ruiz, Jean-Claude Tédenac, Philippe Jund ICGM – Université Montpellier II, UMR-CNRS 5253, Pl. E. Bataillon CC1506, Montpellier 34095, France
a r t i c l e
i n f o
Article history: Received 3 October 2014 Received in revised form 8 December 2014 Accepted 12 January 2015 Available online 19 January 2015 Keywords: DFT calculations Solid solution Calphad method
a b s t r a c t 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). Moreover it appears that the figure-of-merit can be improved by germanium doping. The optimization of the elaboration of such alloys needs the knowledge of the ternary Ge–Mn–Si system. The solid–solid equilibrium in the ternary system seems to be highly driven by the D88 phase an isostructural form of Mn5Si3 and Mn5Ge3. However, the existence of a complete solid solution between these two compounds is not clearly established in the literature. First principles calculations are performed to clarify the mixing enthalpies in this potential solid solution. The predicted phase separation at low temperature is then experimentally confirmed by metallographic examinations on 3 annealed samples. On the basis of these new data, a thermodynamic description of the ternary system is performed using the Calphad method. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction 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). Moreover it appears that the figure-of-merit can be improved by germanium doping [1]. The optimization of the elaboration of such alloys needs the knowledge of the ternary Ge–Mn–Si system and of its constitutive binaries. The aim of this work is to obtain a better description of the ternary system and more especially in the vicinity of the HMS phase. In previous studies, Ge–Mn [2] and Mn–Si [3,4] systems have been studied and described using the Calphad method. Coupled with the database on the Ge–Si system assessed by Olesinski [5], several isothermal sections have been drawn using the Calphad method. From these calculations, it appears that the equilibriums involving the HMS phase are guided by the stability of the potential solid solution linking Mn5Ge3–Mn5Si3 in the D88 structure. This hexagonal structure has a P63/mcm space group (number 193), the Mn atoms are on 4d (1/3, 2/3, 0) and 6g (0.2358, 0, 1/4) positions whereas Ge and Si are located on another 6g (0.5992, 0, 1/4) position according to Aronsson [6].
⇑ Corresponding author at: ICGM–MESO–Université Montpellier II, UMR-CNRS 5253, Pl. E. Bataillon CC1506, Montpellier, France. Tel.: +33 (0)4 67 14 93 52; fax: +33 (0)4 67 14 42 90. E-mail address: [email protected] (A. Berche). http://dx.doi.org/10.1016/j.jallcom.2015.01.072 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.
In the literature, two authors report studies on this solid solution. Kieu Van Con [7] studied several alloys with xMn = 0.625 on the silicon side and according to his analyses, two phases are observed for xGe = 0.15, however no additional details (the annealing temperature for example) are available. In contradiction to this first result, Kappel [8] reinvestigated this section by XRD analyses on 12 alloys melted in an induction furnace and annealed at 1123 K during 4 or 7 days depending on the germanium content. A continuous solid solution is observed at this temperature. Since this D88 phase is the key phase of the ternary system, the potential solid solution is studied using DFT and experimental analyses in this paper. In the first part, theoretical and experimental details are given, in the second part, results concerning the D88 phase are given, in the third part, the Calphad model of the ternary system is described and finally conclusions are given in the last part.
2. Theoretical and experimental details In this work, three different procedures are employed: a theoretical one within the density functional theory (DFT), an experimental one consisting of synthesis, X-ray diffraction (XRD) and metallography and the Calphad method for modeling the phase diagram. Such methods are very classical and only the specific points are detailed in this part. The DFT calculations are performed using the Vienna ab initio simulation package (VASP) [9,10] using the projector augmented waves (PAW) technique within the generalized gradient approximation (GGA). The Perdew–Burke–Ernzerhof parameterization (PBE) is applied. Standard versions of the PAW potentials for Ge (3d54s24p2), Mn (3p64s23d5) and Si (3s23p2) are used. The first Brillouin zone is
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integrated using Monkhorst–Pack k-point meshes. The plane-wave energy cutoff and the reciprocal space meshes (k-points) are increased to achieve total energies with a precision better than 0.5 meV/atom. The selected cutoff energy is of 500 eV and the number of k-points goes from 15 to 22 depending on the cell. The calculations are performed using the ‘‘accurate’’ precision setting in the VASP input file to avoid wrap-around errors. The ISMEAR tag is taken equal to 5 (tetrahedron method with Blöchl corrections). The total energy of each structure is minimized by relaxing the volume of the cell and the ionic positions starting from the experimental parameters. Two kinds of cells are taken into consideration. The first one consists in the primitive cell (containing 16 atoms). In a second step a larger cell (2 1 1 containing 32 atoms) is considered. This second case will allow checking if the primitive cell is big enough to allow total relaxation of the atomic positions in the solid solutions. The synthesized alloys are prepared as follows. The pure Ge (Alfa Aesar, 99.9999%), Mn (Alfa Aesar, 99.9%) and Si (Alfa Aesar, 99.9999%) are first crushed, weighted under stoichiometric ratios and then mechanically compacted. The obtained tablet is then melted 3 times in an arc-melting furnace under argon atmosphere to obtain a homogeneous sample. The as-cast samples are finally annealed 15 or 30 d at 900 K in quartz tubes under vacuum prior to be quenched in water. The samples are analyzed by X-ray diffraction (XRD) and Scanning Electron Microscope (SEM). Powder diffraction patterns are recorded in the [20–100°] 2h range, with a step size of 0.0170° and a step time of 400 s on a Philips Expert diffractometer (XPert Pro) equipped with a copper Ka1 anticathode. To avoid the fluorescence phenomenon, a monochromator is added between the sample and the detector. The pattern matching was done using the Powdercell software [11]. In the Calphad assessment, the liquid phase and the solid solutions based on the allotropic forms of Mn, Ge and Si are described with a substitution model using the Redlich–Kister formalism. The liquid is described as (Ge, Mn, Mn5Ge3, and Si) since an associate model is taken in the Ge–Mn system. The variation to the stoichiometry is modeled for a few phases (MnGe2, Mn5Ge2, MnGe3, R-Mn6Si and v-Mn9Si2). For these phases, anti-sites are assumed to be the main defects. More details are available in [2,4].
3. Results on the D88 phase
3.1. DFT calculations In a first step, the potential stability of the D88 solid solution is investigated using DFT calculations. For this, calculations are performed on a primitive cell composed of 16 atoms where the 6 g (0.5992, 0, 1/4) positions are occupied by Ge, Si or a mix of Ge and Si. Since the 6 g positions are equivalent, 7 crystals are generated. For each composition, several configurations have been tested. The configuration with the lowest total energy is selected in each case. It appears that for each composition, the more stable state corresponds to a configuration where the substitute atoms are as far as possible in the structure. The cell parameters and the mean magnetic moment of the Mn atoms are calculated for each atomic position. As can be seen in Fig. 1, the cell parameters vary linearly with the silicon content in good agreement with the experimental results of Kieu Von Con [7] and Kappel [8]. The same behavior is noticed for the magnetic moments of manganese on the 4a and 6g sites. The magnetic moments calculated for Mn5Si3 are in good agreement with previous calculations of Arras [12] and in an acceptable agreement with the measurements of Forsyth [13]. The mixing enthalpy in the D88 potential solid solution at 0 K is calculated using Eq. (1).
Dm H0 K ¼ EðMn0:625 Ge0:375ð1xÞ Si0:375x Þ ð1 xÞ EðMn0:625 Ge0:375 Þ x EðMn0:625 Si0:375 Þ
ð1Þ
where x is given by Eq. (2),
x ¼ NSi =N6g
ð2Þ
where NSi is the number of Si atoms on the 6g positions in the cell and N6g is the total number of 6g positions in the cell. For a single cell, N6g = 6 and for a double cell, N6g = 12.
Fig. 1. physical properties of the D88 solid solution: (a) a cell parameter; (b) c cell parameter; (c) mean magnetic moment of the Mn atoms.
E(Mn0.625Ge0.375(1x)Si0.375x) is the calculated energy (for one atom) of a particular case of the D88 solid solution, and E(Mn0.625
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Fig. 2. Mixing enthalpy in the D88 solid solution.
Ge0.375), E(Mn0.625Si0.375) are the calculated energies of the crystals of respectively Mn5Ge3 and Mn5Si3 in the D88 structure given for one atom. DmH0K is then given in eV/atom or in J/mol of atom. As one can see in Fig. 2, the calculated mixing enthalpy in the D88 phase at 0 K is positive. Moreover, in this case, the size of the cell (primitive or larger) does not modify the results which is consistent with our previous published work [14]. A positive mixing enthalpy is consistent with the existence of a possible phase separation at low temperature in agreement with the results of Kieu Von Con [7]. However, since the maximum of the mixing enthalpy curve is not very high (1285 J/mol of atom), it could be possible that an entropy term induces the existence of a continuous solid solution at higher temperature as it was shown by Kappel [8]. To check this point, some experiments must be performed. 3.2. Experimental verification Three alloys have been synthesized between Mn5Ge3 and Mn5Si3. The compositions are given in Table 1. According to Kappel [8], at 1123 K, a homogenization time of 4–7 days is necessary depending on the silicon content of the samples. In a trial, the alloys are annealed at 900 K for 2 weeks. All the alloys are mainly composed of two phases (Fig. 3). According to EDX analyses, the Mn content difference is low in the two phases. The Si content is higher in the white phase. According to these measurements and depending on the sample, the dark phase contains between 2 and 7 at.% of Si, whereas the white phase contains between 12 and 30 at.% Si. The composition of the white phase is obtained with a better accuracy in the sample with the highest silicon content (Fig. 3c). At the light of this observation, the samples show the presence of two phases, crystallographically similar but with slight differences in the compositions, which is
Fig. 3. Metallographic picture of the Mn0.6250Ge1xSix alloys annealed 2 weeks at 900 K with xSi equal to: (a) 0.0940; (b) 0.1875; (c) 0.2810.
Table 1 Experimental data measured on the samples annealed at 900 K during 2 weeks: global initial composition in Si, cell parameters and EDX composition of the two phases contained in each sample. xSi
Cell parameter (Å) – XRD Phase 1
0.0940 0.1875 0.2810
Composition – EDX/XRD Phase 2
Phase 1/dark
Phase 2/white
a
c
a
c
xSi
xSi
7.1721(1) 7.1520(8) 7.203(56)
5.0123(3) 4.9995(6) 5.021(77)
7.096(01) 6.9792(7) 6.9887(3)
4.939(30) 4.8673(9) 4.8696(7)
0.042/0.040 0.020/0.070 (0.067/0.020)
(0.121/0.160) 0.170/0.300 0.300/0.290
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Fig. 4. Diffractograms of the Mn0.6250Ge1xSix alloys annealed 2 weeks at 900 K with xSi equal to: 0.0940; 0.1875 and 0.2810.
Fig. 5. Diffractograms of the Mn0.6250Ge0.1875Si0.1875 as-cast, annealed 2 and 4 weeks at 900 K.
significant of a phase separation between two solid-solutions (Mn5(Ge, Si)3 a and Mn5(Ge, Si)3 b). By combining the cell parameters with the Vegard’s law measured by Kappel (Fig. 1a and b), the composition of each phase is estimated (Table 1). The so-obtained values are 4 ± 3 at.% Si and 30 ± 3 at.% Si for respectively the dark and white phases. Such values are in good agreement with the EDX analyses. At the light of these results, we evidence at low temperature a phase separation in the D88 phase as previously suggested by Kieu Von Con [7]. However, regarding to both XRD and SEM analyses, we suggest that the thermodynamic equilibrium is not reached after 2 weeks of annealing at 900 K. We observe gray shades in the sample containing 18.75 at.% Si (Fig. 3b) which can indicate composition changes in the phase. Instead of a D88 phase (labeled Phase 3 in Fig. 4) with intermediate cell parameters, we note that a continuum in the peaks is observed and probably due to a continuous change of composition. It shows that in our conditions, the kinetic reaction is slow and that after two weeks, the reaction is not complete. Nevertheless, assuming a medium composition of the alloys, the cell parameters of the ‘‘Phase 3’’ can be estimated as follows: a = 7.11 Å and c = 4.97 Å . To check this point, this sample was annealed 2 more weeks at 900 K. The diffractograms are compared for the as-cast sample and the two annealed samples after 2 and 4 weeks. Results are presented in Fig. 5. The phenomenon is temperature dependent and not time dependent. Indeed, the difference between the as-cast and annealed
Fig. 6. Assessed phase diagram in the: (a) Ge–Mn system [2]; (b) Ge–Si system [5]; (c) Mn–Si system [4].
samples is not obvious as it is shown in Fig. 5. The intensity of the diffraction peaks of the Phase 3 seems to go down with annealing time, but the intensity of some of the peaks of the Phases 1 and 2
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Fig. 7. Assessed sections of the ternary system: (a) isothermal section at 500 K; (b) isothermal section at 900 K; (c) isothermal section at 1173 K, blue circles correspond to the compositions of the diamond phase measured by Zhou [15]; (d) isothermal section at 1350 K; (e) isopleth section between Mn5Ge3 and Mn5Si3; (f) zoom of the isopleth section with xMn = 0.6363. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
are also reduced after a longer annealing process. This seems to indicate that annealing will lead to a phase separation. Such results are consistent with the observations of Kappel who observed that the thermodynamic equilibrium was reached after 1 week but for an annealing temperature 200 K higher. Such a slow kinetic process may indicate that the temperature and the compositions are very narrow close to the consolute point.
We can also notice the presence of 2 unmatched peaks on the diffractograms (Figs. 4 and 5). These peaks may correspond to another phase which is observed because of a variation to the nominal composition due to the vaporization of germanium during the synthesis of the samples. Since the intensity of the D88 phase is very low, these peaks could correspond to a very low quantity of a phase which exhibits more intense peaks (such as Mn3Ge for
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Table 2 Models and parameters of the phases (this work), assessed parameters are in bold type, other parameters are taken from [2,4,5]. Phase Model
Parameters (J/mol)
Liquid (Ge, Mn5Ge3, Mn, Si)1
0
D88 (Mn)0.625(Ge, Si)0.375
L(Mn5Ge3, Si) = 25,000
0
L(Mn:Ge, Si) = 5000 L(Mn:Ge, Si) = 1500
2
MnSix (Mn)0.146(Mn, VA)0.220 (Ge, Mn, Si)0.244(Ge, Si)0.390
0 SER 0 SER bcc diam G(MnSix, Mn:Mn:Si:Ge) 0.39 ⁄ 0HSER + 0.244 ⁄ Gdiam Ge 0.366 ⁄ HMn 0.244 ⁄ HSi = 11,000 0.5 ⁄ T + 0.366 ⁄ GMn + 0.39 ⁄ GGe Si 0 SER 0 SER bcc diam G(MnSix, Mn:Mn:Ge:Si) 0.244 ⁄ 0HSER + 0.39 ⁄ Gdiam Ge 0.366 ⁄ HMn 0.39 ⁄ HSi = 11,000 0.5 ⁄ T + 0.366 ⁄ GMn + 0.244 ⁄ GGe Si 0 SER bcc diam G(MnSix, Mn:Mn:Ge:Ge) 0.634 ⁄ 0HSER Ge 0.366 ⁄ HMn = 0.366 ⁄ GMn + 0.634 ⁄ GGe 0 SER bcc diam G(MnSix, Mn:VA:Ge:Ge) 0.634 ⁄ 0HSER Ge 0.146 ⁄ HMn = 0.146 ⁄ GMn + 0.634 ⁄ GGe 0 SER bcc diam G(MnSix, Mn:Mn:Mn:Ge) 0.39 ⁄ 0HSER Ge 0.61⁄ HMn = 0.61 ⁄ GMn + 0.39 ⁄ GGe 0 SER bcc diam G(MnSix, Mn:VA:Mn:Ge) 0.39 ⁄ 0HSER Ge 0.39 ⁄ HMn = 0.39 ⁄ GMn + 0.39 ⁄ GGe 0 L(Mn:Mn:Si:Ge, Si) = 5000 0 L(Mn:Mn:Ge, Si:Si) = 5000 0 L(Mn:Mn:Ge, Si:Ge, Si) = 7000
B20_MnSi (Mn)0.5(Ge, Si)0.5
0 SER bcc diam G(B20_MnSi, Mn:Ge) 0.5 ⁄ 0HSER GE 0.5 ⁄ HMn = 8000 – 0.5 ⁄ T + 0.5 ⁄ GMn + 0.5 ⁄ GSi L(B20_MnSi, Mn:Ge, Si) = 8000
Diamond (Ge, Mn, Si)1
0
L(Ge, Mn) = 22,000 L(Ge, Mn, Si) = 140,000
0
example). We can notice that after 4 weeks, the intensity of these unidentified peaks is practically included in the baseline. As a conclusion, regarding the mixing enthalpies calculated and the experimental results, it seems that a phase separation exists in the D88 phase at low temperature. However for temperatures close to 900 K, a phase separation transforms the complete solid solution in a mixture of Mn5Ge3 and Mn5Si3. 4. Calphad assessment of the Ge–Mn–Si system In this work, the three binary systems are accepted from the literature [2,4,5] as shown in Fig. 6. The sub-lattice models of the allotropic forms of pure manganese are consistent between Ge–Mn and Mn–Si. Moreover, Mn5Ge3 and Mn5Si3 have the same crystallographic structure (D88 as previously mentioned). As a consequence, the Mn5(Ge, Si)3 solid solution will be named D88 in the next parts of this paper. In Fig. 7, different isothermal sections (Fig. 7a–d) and isopleth sections (Fig. 7e and f) of the Ge–Mn–Si system are drawn. The interaction parameters in the D88 phase are modeled from the mixing enthalpies calculated in Paragraph 3.1. To reproduce this mixing enthalpy, two interaction parameters (0L and 2L) are necessary in the model (Table 2). The as-obtained curve is in very good agreement with the DFT calculations as shown in Fig. 2. Using this model, the isopleth section with xMn = 0.625 is drawn in Fig. 7e. The assessed section is consistent with the existence of the phase separation observed in this work (Paragraph 3.2) at 900 K (see Fig. 7b) and with the continuous solid solution at 1123 K (Fig. 7c) as seen by Kappel [8]. Regarding this agreement, no temperature contribution is added. This study should be completed by an experimental work aiming to determinate the temperature where the phase separation ends. Moreover, it would be interesting to check some of the 3-phase regions such as those surrounding the D88 phase. More particularly, the compositions of the phases involved would have to be compared to those calculated with the present database. Zhou [15] synthesized alloys in the (MnSix + diamond) region by induction furnace. In this study, the as-cast samples were crushed and the obtained powders were sintered at 1173 K. The compositions of the phases observed in the samples are measured by EDX. All the diamond phases contain manganese (blue circles in Fig. 7c corresponds to the compositions). Such results are
consistent with the assessed Mn–Si phase diagram (Fig. 6c). A ternary interaction term in the diamond phase is in consequence added to take into account this solubility. Moreover, according to Aoyama [16], the solubility of Ge in the MnSix phase is measured at 1 at.% Ge by EPMA and ICP on samples obtained by the Bridgman method. Zhou [15] also indicates similar solubilities by EDX measurements. However, according to previous results, one-phased alloys are obtained in MnSix phases containing 2 at.% of Ge. This last value is selected for the present assessment as shown in Fig. 7f. Finally a ternary parameter in the liquid phase between the associate Mn5Ge3 and the silicon is added to correct the too strong influence of the associate model of the Ge–Mn liquid phase. In agreement with previous experimental work, the HMS phase can be doped with around 2 atomic percent of Ge. For higher concentrations, MnSi and a Ge–Si phase will be formed in the alloy (Fig. 7). Using the database, solidification calculations were performed using the Scheil–Gulliver model. For a non-doped HMS phase, the peritectic decomposition of the phase leads to an as-cast alloy that may contain small fractions of MnSi and pure silicon. For a 2 at.% Ge doped HMS, MnSi and a germanium rich Ge–Si alloy should be observed in an as-cast alloy. Using a direct melting method, HMS alloys have to be chemically homogenized by an annealing step at a temperature below 1150 K for pure HMS or below1040 K for doped HMS. An alternative synthesis method could be ball milling. However, it is well-known that with this method, the phases with the highest enthalpy of formation will be formed in the first stages of the synthesis. Regarding the enthalpies of formation, the most stable phases are MnSi and MnSix. A ternary mixing of MnSi, HMS and Si (or Ge–Si) has to be expected. 5. Conclusions This work provides the first ternary assessment of the Ge–Mn– Si system. Based on pertinent DFT calculations and measurements, a thermodynamic description of the system is given. Of course, this first version will have to be improved with additional experimentations such as metallographic examination of the 3-phases regions (to confirm the phase equilibrium) or thermal analyses to fix the liquidus curves.
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The database will be useful in the next step of this work which will consist in synthesizing various doped MnSix samples for measuring their thermoelectric properties. 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). We would like to thanks Dominique Granier of the Institut Charles Gerhard and Claude Gril respectively for the XRD and SEM analyses. We are also thankful to Antony Lluch who worked on this system during his Master internship. 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.jallcom.2015.01. 072. References [1] Unpublished results from the Renoter research consortium,
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