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Determination of structural disorder in Heusler-type phases
Computational Materials Science 172 (2020) 109307
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Determination of structural disorder in Heusler-type phases a,⁎
V.V. Romaka , G. Rogl a b c
a,b,c
, A. Grytsiv
a,b,c
, P. Rogl
T
a,c
Institute of Materials Chemistry, University of Vienna, Währingerstrasse 42, A-1090 Wien, Austria Institute of Solid State Physics, TU-Wien, Wiedner Hauptstrasse, 8-10, A-1040 Wien, Austria Christian Doppler Laboratory for Thermoelectricity, Wien, Austria
A R T I C LE I N FO
A B S T R A C T
Keywords: Heusler-type phases DFT calculations Phase diagrams Crystal structure Structural disorder
A number of Heusler-type phases, e.g. TiNi2Sn, TiNiSb, ZrNiSn, TiFeSb, NbCoSb, which, for a long time, were considered to be stoichiometric with point composition, are characterized by homogeneity regions and/or offstoichiometry. Traditional high-throughput DFT calculations aim at the discovery of new fully ordered point composition Heusler-type phases and their potential intrinsic and extrinsic defects. In order to extend the application of DFT methods for determination of structural disorder in half-Heusler and full-Heusler phases we propose a new multi-way approach. It is based on evaluation of the crystal structure model that is used in DFT calculations by getting the best agreement between a set of theoretical and available experimental data (e.g. concentration dependencies of the lattice parameter, homogeneity regions, physical and mechanical properties). This approach allows to determine the model of structural disorder in Heusler-type phases, explain or predict the presence of the homogeneity regions or off-stoichiometry, and reveal the mechanisms of solid-solution formation.
1. Introduction The family of the Heusler-type phases stems from Fritz Heusler who, in 1903, discovered the MnCu2Al ferromagnet [1,2], and later determined its crystal structure [3]. Compounds, the crystal structure of which belongs to MnCu2Al structure type nowadays are called Heusler or full-Heusler (FH) phases. Later, the family of the Heusler-type phases was extended by Nowotny and Sibert [4], who determined the crystal structure of MgAgAs as an ordered ternary variant of the CaF2 type [5]. These structures are closely related to that of the full-Heusler phase; but as only half of the 8c sites are occupied, these phases are called semiHeusler or half-Heusler (HH) phases (Fig. 1). In the HH XYZ compounds the X component occupies the 4a (0 0 0) Wyckoff position, Y – 4c (1/4 1/4 1/4), and Z – 4b (1/2 1/2 1/2). Similar to HH compounds in the FH XY2Z compounds the 4a and 4b site are occupied by X and Z components, respectively, however the Y component occupies the 8c (1/4 1/4 1/4) crystallographic site. The difference between these two structures is that FH phases crystallize in centrosymmetric Fm-3m (#225) space group, while HH phases – in noncentrosymmetric F-43m (#216). This fact excludes the formation of continuous solid solutions between these two structure types. Heusler-type phases exhibit semiconducting properties at different electronic configurations: 18 and 24 valence electrons for HH and FH phases, respectively. Using this simple rule, one may rather reliably ⁎
predict the type of conductivity and various other physical properties of these compounds [6–9]. High symmetry, non-parametric atomic positions, and 12 (in HH phases) or 16 (in FH phases) atoms in the unit cell allow to accurately perform ab initio calculations to explain different properties of existing Heusler-type compounds [10] or to predict the formation of new, not yet discovered, HH phases [11]. As an example, a screening of 2295 ternary atom combinations for the HH symmetry via ab-initio calculations was performed [12] in order to predict compositions that could behave as topological insulators (TI). Recently, both ab-initio calculations and machine learning technique were used to screen for unknown stoichiometric HH phases [13–15]. The main disadvantage of these methods is the treatment of all HH phases as stoichiometric compounds with point composition. Detailed experimental and theoretical studies of Ti-Ni-Sn [16] and Zr-Ni-Sn [17] ternary systems revealed the presence of homogeneity regions in HH and FH phases. An even more complicated case of HH compound was found in Ti-Fe-Sb [18] and NbCo-Sb [19] system where Ti1+xFe1.33-xSb appeared to be off-stoichiometric with homogeneity region. In order to extend the application of DFT calculations on the discovery of homogeneity regions and off-stoichiometry in HH and FH phases we present a new multi-way approach, that allows to determine the model of structural disorder of these phases and to evaluate it by direct comparison with experimental data.
Corresponding author at: Leibniz Institute for Solid State and Materials Research, Helmholtzstr. 20, D-01069 Dresden, Germany. E-mail address: [email protected] (V.V. Romaka).
https://doi.org/10.1016/j.commatsci.2019.109307 Received 31 July 2019; Accepted 25 September 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
Computational Materials Science 172 (2020) 109307
V.V. Romaka, et al.
Fig. 1. Crystal structure of full-Heusler compound with general formula XY2Z and half-Heusler compound with general formula XYZ.
2. Modelling details
characterized by a strictly stoichiometric point composition with no homogeneity region: 1:1:1 – in the case of HHs, or 1:2:1 – in the case of FHs. On the phase diagram they are usually marked with a point/circle (Fig. 2a). Interestingly these phases are reported to have point composition in the whole temperature region of existence. The reason why so many Heusler-type phases were reported as sp is that they were mainly discovered as a series of compounds with different sorts of X, Y, or Z components, and thus the stoichiometric point composition was automatically assigned after the structure type was determined. Only in a few cases the deviation from the stoichiometry of Heusler-type phases was defined in detail. (ii) Heusler-type phases with off-stoichiometric point composition (op) are rather rarely observed in comparison with sp Heuslers, but are gaining more interest due to recent investigations of offstoichiometric HH thermoelectric materials [28,29]. This (op) type of phases still maintains the point composition and is marked as a point/circle on the phase diagrams (Fig. 2b), but deviates from 1:1:1 or 1:2:1 stoichiometry and thus is off-stoichiometric. In most cases the deviation in composition is less than 10 at. % and thus, was usually ignored and the sp type was assigned. In some rare cases a Heusler-type phase that was initially reported as op appeared to be sp. For example, in the Nb-Fe-Sb system the HH NbFeSb compound was reported as op Nb0.9Fe1.2Sb0.9 [30], but recent investigations confirmed it as a sp NbFeSb phase [31]. Usually the off-stoichiometry could be satisfactorily explained using the simple Zintl concept, but in general, the roots of these deviations could be explained from the thermodynamic point of view. (iii) In the Heusler-type phases with stoichiometric homogeneity region (sh) the homogeneity region covers the stoichiometric composition of a phase, but in addition expands toward other directions following different patterns (Fig. 2c), namely expanding towards: a) neighbored binary and ternary phases, independently, whether or not the Heusler-type phase is in a direct equilibrium with any of them; b) towards the pure components (corners of the Gibbs triangle) and along the isoconcentration lines of one or several components. This type of Heusler phases could be established in the case of detailed phase diagram investigations, preferably at different temperatures, and could not be reliably or in principle predicted by the Zintl concept. The category of sh phases comprises HH phases such as TiNi1+xSn [16], ZrNi1+xSn [17,32], HfNi1+xSn [32], formerly known as sp TiNiSn [33,34,35], ZrNiSn [33,34,36], and HfNiSn [33,37], respectively, as well as FH phases TiNi2-xSn [16,38], ZrNi2-xSn [32,17], HfNi2-xSn [32], formerly reported as sp TiNi2Sn [39,35], ZrNi2Sn [33,36], and HfNi2Sn
DFT calculations were carried out using the program packages ELK v4.3.06 [20] and Exciting (carbon) [21] – all-electron full-potential linearized augmented-plane wave (FP-LAPW) codes with exchangecorrelation functionals for solids by Perdew-Burke-Ernzerhoff (PBE) [22] in generalized gradient approximation (GGA). The k-point mesh grid was equal to or higher than 10 × 10 × 10 k-points. The manual optimization of lattice parameters was performed by fitting the universal equation of state [23]. The proper values of the muffin-tin radii were selected automatically at the initial stage of the calculations. Rmin(MT) × {|G + k|} was set to 7, where Rmin(MT) is the minimum muffin-tin radius used in the system. The enthalpy of formation (ΔH) at T = 0 K was calculated as a difference between the total energy of the compound and the energy of the respective amounts of pure elements. For higher temperatures the configurational entropy of mixing was introduced. For intermediate compositions of the compounds the crystal structure models were generated using the supercell package [24]. In order to simulate an alloy with random distribution of atoms, the KKR-CPA-LDA method [25,26] was employed. The energy window that covers conduction band, semi-core and valence states was equal to 21.8 eV. The ground state energy and the distribution of the density of states (DOS) were calculated for each alloy using the experimental values of the lattice parameter. The Brillouin zone integration and density of states calculation were performed on the basis of 1000 kpoints. The crystal structure models were plotted with the VESTA software package [27]. 3. Problems in the Heusler-type systems The challenges that arise during the investigation of the Heuslertype phases can be described as problems of their constitution, structural disorder, and physical properties, however, the proper structural model of Heusler-type phases supplies a key for understanding and explanation of their constitution and physical properties. 3.1. Problem of the constitution of Heusler-type phases From the constitutional point of view all known HH and FH phases can be divided into 4 groups which are characterized by: i) stoichiometric point composition; ii) off-stoichiometric point composition; iii) stoichiometric homogeneity region; iv) off-stoichiometric homogeneity region (Fig. 2). (i) Phases with stoichiometric point composition (sp) are the most frequently observed type of Heusler compounds. These phases are 2
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Fig. 2. Constitutionally derived four groups of Heusler-type phases with stoichiometric point composition (a), off-stoichiometric point composition (b), stoichiometric homogeneity region (c), and off-stoichiometric homogeneity region (d). A red marker shows the actual composition of a phase, whereas a yellow marker indicates the ideal stoichiometric composition.
phase and the second one is limited by the small amount of reliable experimental/theoretical data.
[33,37], respectively. (iv) The last group of Heusler-type phases is characterized by the presence of an off-stoichiometric homogeneity region (oh), which does not cover a stoichiometric composition (Fig. 2d). Similarly, to the sh phases, among the oh Heusler phases various shapes of the homogeneity region could be realized, which includes the expansion of the homogeneity region towards the pure components (corners of the Gibbs triangle) and/or along the isoconcentration lines of one or several components. An excellent example of such an oh phase is HH Ti1+xFe1.33−xSb which was first reported as sp (TiFeSb) [40], then as op (Ti1.27FeSb) [41], later as sh (Ti1+xFeSb) [42], and finally as oh (Ti1+xFe1.33−xSb) [18]. Another example is HH NbCoSb, which was first reported in [43] as sp phase, later as op phase (Nb0.8CoSb) [28], and only recently as oh phase (Nb0.8+xCoSb) [19].
3.2. Problem of the structural disorder in Heusler-type phases The constitution of each Heusler-type phase is determined by the shape of its thermodynamic potential and the thermodynamic potential of all neighboring phases. The thermodynamic potential itself is defined by the crystal structure of a phase in general and by the distribution of atoms among crystallographic sites, in particular. The presence of a homogeneity region and/or off-stoichiometry in Heusler-type phases leads to structural disorder of the initial distribution of atoms i.e. to the formation of vacancies, statistical mixtures of atoms, filling of the vacant crystallographic sites (in the case of HH phases). In the simplest case of structural disorder at least two sorts of atoms could be mixed among their initial crystallographic sites, and thus produce disorder while keeping the sp composition of the phase and the completely occupied crystallographic sites (Fig. 3a). It is a challenging task to determine the presence of such a mixture especially when the occupational factor of a second (mixing) component does not exceed 5%, because the total number of atoms, composition and stoichiometry remains the same, and the sensitivity of the X-ray diffraction (XRD) method and Rietveld refinement is usually not sufficient. In a more complicated case, when the Heusler-type phase is offstoichiometric (op), additionally a vacancy in one or several sites might appear (Fig. 3b), as well as an extra atom of one of the components
The afore mentioned classification of the constitution of Heuslertype phases undercovers their main problem – phases that were initially reported and still are accepted to have sp composition in many cases appear to be op, sh or even oh Heuslers. Despite the fact that the existence/formation of sp and op phases could be predicted by using a simple Zintl concept [44] or by high-throughput transport and defect calculation methods [13,14,15], both techniques are useless (i) for the discovery of the homogeneity regions (sh and oh) of Heusler-type phases, and (ii) the determination of their origin and mechanisms of formation, as the first one lacks a criterion for the homogeneity of a 3
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Fig. 3. Models of atoms distribution in the XYZ half-Heusler phases with: stoichiometric point composition – sp (a); off-stoichiometric point composition – op (b); stoichiometric composition with homogeneity region – sh (c); off-stoichiometric composition with homogeneity region – oh (d).
extra source of electrons or holes, and the semiconductor should be intrinsic or fully compensated with the Fermi level positioned exactly at the middle of the band gap resulting in a very high resistivity and a vanishing Seebeck coefficient. In fact, semiconducting Heusler phases are usually n or p-type with quite high values of Seebeck coefficient. For a deeper insight into the origin of the physical properties ab initio calculations are usually used. The electronic band structure, modelled by ab initio DFT methods using the model of the sp ordered crystal structure is in contradiction to the measured physical properties. The application of a rigid-band model or the Boltzmann transport theory within DFT for the prediction of the electrical transport properties of Heusler-type phases and solid-solutions yields reasonable results only in the case when the bands are not or just slightly affected. In fact, when the Heusler-type phase belong to op, sh, or oh group, the bands are significantly affected (Fig. 4) and for detailed DFT modeling the proper model of the crystal structure is required. Thus, the physical properties of the semiconducting Heusler phases can be explained only in the case of a proper model of their crystal structure. Modeling of anti-site defects in the Ni sublattice of TiNiSn was done in [46], where the authors claimed that this might be considered as a result of a decrease of the state of order of Ni sublattice when the temperature increases.
might occupy a vacant crystallographic site (in the case of HH phases). When a phase is characterized by the homogeneity region, that covers the stoichiometric composition (sh), at least two sorts of atoms can be mixed in one of the crystallographic sites (Fig. 3c); furthermore a vacant site could be gradually filled with an extra atom (in the case of HH phases) or vacancies could gradually appear in one of the crystallographic sites. The most challenging case, however, appears when a phase is characterized by a homogeneity region that does not cover the stoichiometric composition (oh). The crystal structure of such a phase simultaneously reveals a statistical mixture of atoms in one or several sites and vacancies or extra atoms in the occupied or vacant sites, respectively (Fig. 3d). The lack of XRD sensitivity makes it hard or even impossible to reliably refine the composition of the statistical mixture of atoms and vacancies that occupy one crystallographic site or even distinguish the type of species that occupies a crystallographic position if they have close X-ray scattering power. The correct determination of the structural models, that correspond to the homogeneity regions is complicated for both sh and oh Heusler-type phases and creates a real problem for a proper understanding of their properties. The models of structural disorder in Heusler-type phases are not yet completely investigated and a successful solution of this problem requires the combination of both, experimental and theoretical approaches.
3.4. Concepts for modelling of Heusler-type phases The simple Zintl concept, that operates only with the number of valence electrons, is widely used to predict or explain the existence of sp and op Heusler-type phases [44,28,45]. In combination with molecular orbital approach and the crystal field theory it could be used to build a simple band structure model of a Heusler-type phase. This concept can explain the formation of the band gap, the drift of the Fermi level while doping the semiconducting HH phase with n- or p-
3.3. Problem of physical properties in Heusler-type phases Although the semiconducting (intrinsic) properties of some Heuslertype phases can be predicted and explained by using the simple Zintl concept [44,45], this concept fails to explain the n- or p-type of electrical conductivity of Heusler phases, as for the sp phase there is no 4
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The determination of the most favorable intrinsic and extrinsic dopants in Heusler-type phases, especially for the semiconducting phases, is done by using the supercell approach [56,57,58,59]. Within this approach a large supercell is constructed to lower the concentration of the dopant and the energy difference is calculated between the structure without and with the defect. The value, and what is the most important, the sign of this energy determines whether the formation of such a defect is favorable or not. This method is suitable to predict the possible type of defects and compensators in the Heusler-type phases, but lacks the determination of the solubility limits (homogeneity region) of such a doped phase due to the necessity to fit the model of a very diluted dopant. Sometimes, in order to predict the existence of new Heusler-type phases, the researchers focus only on the negative enthalpy of formation as a stability criterion [15]. However, a negative enthalpy of formation for a given sp or op hypothetical Heusler-type phase means only that the product of the chemical reaction has a lower energy than the mixture of the initial components. In consequence it shows that the reaction is in principle possible, but it does not reveal whether or not the phase is realized in a real system, as the presence of other neighboring phases may prevent the formation of this Heusler-type phase. Another argument for the mechanical stability of Heusler-type phase is the absence of imaginary frequencies in the phonon DOS spectrum [60]. However, this criterion can be used only as an additional argument for the existence or the absence of the phase, but not as a criterion of phase formation. All concepts and approaches mentioned above have strong sides as well as some weaknesses, but the extension of their usage, as well as their combination could give another approach for a better understanding and prediction of the different constitutional groups of Heusler-type phases.
Fig. 4. Schematic band structure of an n-type HH phase: modeled as sp phase with the DFT method (a); modeled as electron-doped sp phase with DFT and rigid band model (b); derived from experiment (c). Ec, Ev, and EF correspond to the energy of conduction, valence bands, and Fermi energy, respectively.
type dopants or formation of vacancies in one of the crystallographic sites. However, the concept is unable to either predict the existence or explain the formation of the homogeneity regions of Heusler-type phases, as they depend on the equilibrium with the neighboring phases and the temperature. Another concept is based on the CALPHAD method [47,48,49], which is used to predict and refine the phase diagrams of multinary systems with Heusler-type phases in particular Ti-Ni-Sn [16,50], Hf-NiSn [51], Zr-Ni-Sn [52,17], and (Ti,Zr,Hf)NiSn [53]. This method is rather flexible, as it can use both, experimental and theoretical thermodynamic quantities, e.g. the enthalpy of formation. The CALPHAD approach is usually amplified by the wide usage of DFT calculations, that allow to predict the enthalpy of formation of pure ternary phases and hypothetical solid solutions and model the phase equilibria between the Heusler-type phase and all neighboring phases in a phase diagram. The approach, which is close to the CALPHAD method, is the convex hull method, which relies only on DFT calculations [28,13,18,54]. Within this method the calculations of the enthalpy of formation are performed for a number of intermediate compositions between the desired Heusler-type phase and one or several real/hypothetical phases forming pseudobinary systems (Fig. 5). Then the convex hull is constructed and the stability of the sp Heusler-type phase among the neighboring phases, or the existence of the op Heusler phase is determined. For the modeling of the solid solutions (homogeneity regions) both, the DFT part for the CALPHAD method and the convex hull method, often use SQS approach [55] to generate special quasi-random structures for each intermediate composition of the Heusler phase. These structures, however, do not maintain the symmetry of the initial ordered structure. For each quasi-random structure, the geometry optimization is performed with DFT calculations, which include relaxation of both lattice vectors and atomic coordinates. It is common that after geometry optimization some atoms in such a random structure deviate from their non-parametric positions in the initial Heusler-type structure.
4. Determination of structural disorder in Heusler-type phases To solve the problems described above we propose a new approach based on the determination of structural disorder in multicomponent intermetallic phases through experimental and theoretical evaluation of their thermodynamic stability, crystallographic features, and physical/ mechanical properties. Traditionally the model of the crystal structure of the phase is evaluated at the beginning or at an intermediate stage of a complex research, depending whether the crystal structure prototype is known and is further on used for ab initio DFT modeling of its physical and chemical properties to explain experimental results. The proposed approach contains traditional steps (Fig. 6), but allows at the stage of DFT calculations to further refine the model of the crystal structure of a phase and its stability by modeling thermodynamic, crystallographic, transport, elastic, and energetic parameters with further 3-way data evaluation. Based on the proposed classification of the constitution of Heuslertype phases we can assume that at some higher temperatures sp and oh phases could be potentially transformed into sh and oh phases, respectively, due to the greater impact of the entropy term on the thermodynamic potential thus stabilizing the presence of some structural defects. In order to describe the crystallographic model of the potential homogeneity region the shape of the latter should be determined. As the homogeneity region of the Heusler-type phase in most cases does not exceed 10 at. % from the stoichiometric point composition, two simple patterns can be applied a) for substitutions and b) for interstitials and defects (Fig. 7). The model of the first type of patterns assumes that within two different sublattices of the crystal structure, e.g. half-Heusler A(1+x)BC(1-x) the concentration of one sort of atoms (A) increases by x giving (1 + x) in total, while the other species (C) decreases by the same margin giving (1 − x) in total. All possible permutations of two constituents between their two sublattices, limited by
Fig. 5. The schematic simplified illustration of the convex hull method to determine the stability of a Heusler-type phase (a) and the real thermodynamic potentials for determination of the homogeneity region of the Heusler-type phase (b). 5
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Fig. 6. Schematic presentation of approach for determination of structural disorder in HH and FH phases.
Fig. 8. Application of homogeneity region patterns for DFT examination of ‘TiFeSb’ half-Heusler phase. A yellow marker indicates the ideal stoichiometric composition of TiFeSb, the red marker shows the composition with minimum enthalpy of formation, the green marker indicates the experimentally determined [18] homogeneity region for the half-Heusler phase. Black and red dash-lines represent the potential and actual patterns, respectively, along which the DFT calculations were performed.
Fig. 7. Two types of homogeneity region patterns for Heusler-type phases.
the size of the supercell, can be obtained by using the combinatorial structure-generation approach [24]. The model for the second type of pattern assumes that the concentration of one component constantly changes (increases or decreases), whilst the ratio between all other components remains constant. The increase of concentration of one of the components can be presented as the incorporation of extra atoms into the vacant site (only for HH phases), e.g. AB(Bx)C, or the generation of vacancies in all crystallographic sites except one occupied by the desired component, e.g. (AC)1-xB. Again, all possible permutations of the constituents between possible sublattices can be obtained by using the combinatorial structure-generation approach [24]. The process of structure generation within the patterns of the homogeneity region could be fully automated. An example of application of these patterns for determination of the off-stoichiometry and homogeneity region of Ti1+xFe1.33−xSb is shown in Fig. 8. It should be noted, that the generated crystal structure models (supercells) are presented as a single, doubled, tripled initial cubic cell along each of the basis vectors, despite adopting a potentially different symmetry of the whole supercell. If one tried to optimize the geometry (lattice vectors and atomic coordinates) of such a supercell by using the ‘forces’ (Hellmann-Feynman) method, the non-parametric positions would be interpreted as parametric, and no constraints would be applied to the angles between basis vectors, and their lengths. The
resulting crystal structure would be optimized, but the initial shape of the single cubic cell as well as the initial atomic coordinates will be lost (Fig. 9). Moreover, such an optimization suffers from the large number of vacancies, which makes convergence more difficult. To simplify the optimization procedure, the atomic coordinates should remain constant, and the only parameter that should change is the lattice constant, which should be equally scaled along each direction of the supercell. Varying the scale of the lattice constant in the range of 0.9–1.1 from the optimized sp Heusler-type phase with a step of 0.02 and performing self-consistent ground state DFT calculations for each step, the E(V) dependence could be obtained. It could then be fitted with one of the equations of state (EOS) delivering the optimized volume and total energy of the system. The obtained optimized volume divided by the number of unit cells in the supercell will yield the volume of the averaged cubic cell, and the cube root will yield the lattice constant aopt. At this stage it is possible to define the theoretically determined concentration dependence of the lattice parameter aopt and the boundaries of the homogeneity region. The energetically most 6
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Fig. 9. Initial structure of TiNiSn and relaxed structure of TiNi0.75Cu0.25Sn. Lattice deformation of TiNi0.75Cu0.25Sn is visible in the relaxed structure.
structure model in more complicated systems. For example, the Fe for Co substitution in the isopleth TiCoSb-TiFe1.33Sb [18] could potentially occur in 4c, in 4d, or in both sites. The DFT calculations of the enthalpy of formation revealed that the mechanism of substitution is based on a simultaneous substitution of Fe by Co in the 4c site with decreasing Fe content in the 4d site. The slope of the a(x) dependence for the converged DFT model is in fair agreement with experimental data (Fig. 12a). To the first batch of evaluated data belongs also the density of states (DOS) profile of the intermediate compositions between FH and HH phases. For this, however, one needs to use another DFT method based on the Green’s function method (known as Korringa-Kohn-Rostoker, KKR) with coherent potential approximation (CPA), and not the one with a plane-wave basis set. This method allows to preserve the symmetry of the initial HH-phase while modeling solid solutions towards the FH compound. In recent papers [38,32] it was shown that the filling of the vacant 4d site in HH TiNiSn and ZrNiSn compound with extra Ni atoms does not lead to the gradual change of the DOS profile from the HH to the FH phase (Fig. 13). Instead of this a sharp change in DOS from HH to FH occurs near the composition MNi1.65Sn (M = Ti, Zr). This could help in the identification of the symmetry change from noncentrosymmetric HH phases to centrosymmetric FH compounds. The second batch of evaluated data contains: modelled enthalpy of formation, thermodynamic potential (with configurational entropy term), homogeneity regions of the sh, oh Heusler-type phases, composition of the op Heusler phase and phase equilibria with the
favorable crystal structure model for each composition along each direction of the homogeneity region pattern will define the mechanism of the formation of the homogeneity region. The verification of the correctness of the theoretical data (and as a result, the crystal structure model) is performed by the 3-way evaluation, which includes three independent sets of modelled parameters that are directly compared with experimentally extracted data from the phase diagram, crystal structure, and physical/elastic properties. The first batch of evaluated data contains: modelled composition dependencies of the lattice parameter a of the sh and oh Heusler phases. Despite the fact that absolute values of the experimental and DFT-derived lattice parameter are different and the difference depends on the type of exchange-correlation potential used in the modeling, the slope and curvature of both theoretical and experimental a(x) dependencies should be similar. This is clearly visible in the systems TiNiSn-TiNi2Sn [16] and ZrNiSn-ZrNi2Sn [17] (Fig. 10) where the slope of the experimental a(x) dependencies within the solid solutions for FH TiNi2-xSn and ZrNi2-xSn and HH TiNi1+xSn and ZrNi1+xSn fits with the models of the DFT calculations. Despite both sh FH and HH phases have very similar crystal structures their a(x) slopes are different. Another example for modeling the a(x) dependence is the oh Ti1+xFe1.33-xSb phase [18]. There is a small negative curvature of the a(x) dependence within the homogeneity region which is also observed on the modeled curve (Fig. 11a). The composition dependence of the lattice parameter could also be used as an additional criterion for the correctness of the crystal
Fig. 10. Experimental and modeled composition dependencies of the lattice parameter a in the systems TiNiSn-TiNi2Sn [16] (a) and ZrNiSn-ZrNi2Sn [17] (b). 7
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Fig. 11. Experimental [18] and modeled composition dependencies of the lattice parameter a within the homogeneity region of Ti1+xFe1.33-xSb (a) and calculated enthalpy of formation of the hypothetical TiFe1+xSb and Ti1+xFe1.25-xSb solid solutions (b).
solid solutions HH TiNi1+xSn and FH TiNi2-xSn at 950 and 1100 °C was confirmed experimentally [16] (Fig. 14a) and could be predicted by DFT calculations by adding a configuration entropy of mixing term. For the ZrNiSn-ZrNi2Sn section [17] (Fig. 14b) it was shown that the solubility strongly depends on temperature, which means that the entropy term plays a crucial role in stabilization of the solid solution. In contrast to the systems described above the homogeneity region of the Ti1+xFe1.33-xSb HH phase extends along the isoconcentration line of Sb. The DFT calculations showed the presence of the wide minimum on the ΔHf(x) dependence, which corresponds to the homogeneity region and is stable even at low temperatures (Fig. 11b). For the more complex systems, like 4- or 5-component HH solid solutions the formation of a miscibility gap could be predicted and experimentally confirmed [53,61,62]. The experimentally determined homogeneity region of the Heusler-type phases, standard enthalpies of formation, and other thermodynamic parameters were used for a direct comparison with modeled data and in such a manner it was possible to evaluate the correctness of the crystal structure model. The third batch of evaluated data contains: distribution of the electron and phonon densities of states, type of Heusler phase – semiconductor/metal, type of conductivity, band gap, activation energies from the Fermi level onto the percolation level of the valence or conduction bands, Seebeck coefficient, specific heat, thermal expansion coefficient, Young’s, bulk, and shear modulus, Vicker’s hardness, Poisson number, and their composition dependencies for sh and oh Heusler-type phases. This evaluation becomes extremely important in
neighboring phases. A combination of the enthalpy of mixing within the pattern of the homogeneity region allows to figure out whether offstoichiometry appears for the Heusler-type phase. The optimized total energy for each model of the crystal structure could be used to get the enthalpy of formation. The enthalpy of mixing could also be obtained by E(V) optimization of both end-members of the homogeneity region pattern, and from the convex hull method the solubility could be determined along each pattern direction of the homogeneity region. Additionally, the configuration entropy of mixing (ΔSmix) could be applied to determine the most favorable atomic distribution at each temperature. The model for ΔSmix could be presented as a combination of entropy in every sublattice, e.g. in the hypothetical half-Heusler phase (A1-xCx)B(ByVac1-y)C there is a statistical mixture of A and C components in the A-sublattice (4a site), whereas the vacant 4d site is partially filled by extra By atoms. In this case the configuration entropy of mixing is presented in the following form:
ΔSmix = −R(x lnx + (1 − x )ln(1 − x ) + y lny + (1 − y )ln(1 − y )) For the full-Heusler (A1-xCx)(B1-yVacy)2C:
ΔSmix = −R(x lnx + (1 − x )ln(1 − x ) + 2 × (y lny + (1 − y )ln(1 − y ))) In the case when the enthalpy of mixing shows a positive trend within the hypothetical homogeneity region pattern, the configuration entropy of mixing can be introduced. It was demonstrated [17,54] that a positive ΔHmix at T = 0 K could transform into negative ΔGmix at higher temperatures due to the large impact of the configuration entropy of mixing, producing limited solid solutions. The presence of the
Fig. 12. Experimental [18] and modeled a(x) dependencies for the TiCoSb-TiFe1.33Sb isopleth (a) calculated enthalpy of formation for two models of the solid solution and TiFe1.375Sb (b). 8
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Fig. 13. Calculated DOS profiles for the HH ZrNi(Nix)Sn (a) and FH ZrNi1+xSn (b) hypothetical solid solutions.
Fig. 14. Theoretical evaluation of the solid solution in systems TiNiSn-TiNi2Sn (a), ZrNiSn-ZrNi2Sn (b). Dashed lines correspond to the experimentally determined boundaries of the HH and FH solid solutions [16,17].
5. Summary and conclusions
the case of semiconducting properties of the HH phase. The experimentally determined temperature dependencies of resistivity and Seebeck coefficient yield information on the dominant charge carriers, position of the Fermi level, activation energies from the localized states in the energy gap onto the percolation level of the conduction of valence bands. This information strongly depends on the model of the crystal structure, and thus could be used for its evaluation. For TiNiSb [38] it was shown that the metallic type of conductivity is not explained simply as a feature of the band structure of TiNiSn with the Fermi level located in the conduction band, but with the presence of vacancies in the Ti (4a) site, which induces the closing of the band gap. For ZrCoSb [63] it was shown that within the homogeneity region of the HH phase the semiconducting properties change to metallic. For LuNiSb [29] the modeling of the DOS profiles for different combinations of atoms in the crystal structure allowed to determine the composition and atom distribution that fits the best EPMA data and electrical transport properties. For TiNiSn and ZrNiSn [62] it was shown that even very small amounts of extra Ni atoms in the vacant 4d site could have a dramatic effect on the energy gap, and explain the n-type of conductivity of these HH phases. The results appeared to be in a very good agreement with the band structure model constructed through the analysis of the activation energies on these compounds. Recent experimental and theoretical investigations of mechanical properties of HH phases [31,64,65,54] show good correlation with the crystal structure model and thus can serve as its additional evaluation criterion.
The proposed approach for determination of structural disorder extends the traditional DFT techniques and could be partially or fully automated for prediction of the structural disorder and homogeneity regions in Heusler-type phases. A multi-way approach allows to evaluate the model of the crystal structure, and to get the best agreement between a set of theoretical and available experimental data, starting from the concentration dependencies of the lattice parameter, homogeneity regions, and finishing with electrical transport, magnetic and mechanical properties. Acknowledgements V.V. Romaka and P. Rogl thank the Austrian agency for international mobility and cooperation in education, science and research for support via the Ernst Mach project ICM-2017-06580. References [1] F. Heusler, Über magnetische Manganlegierungen, Verh. Dtsch. Phys. Ges. 5 (1903) 219. [2] F. Heusler, W. Starck, E. Haupt, Magnetisch-chemische Studien, Verh. Dtsch. Phys. Ges. 5 (1903) 220. [3] O. Heusler, Kristallstruktur und Ferromagnetismus der Mangan-AluminiumKupferlegierungen, Adv. Phys. 411 (2) (1934) 155–201.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Determination of structural disorder in Heusler-type phases a,⁎
V.V. Romaka , G. Rogl a b c
a,b,c
, A. Grytsiv
a,b,c
, P. Rogl
T
a,c
Institute of Materials Chemistry, University of Vienna, Währingerstrasse 42, A-1090 Wien, Austria Institute of Solid State Physics, TU-Wien, Wiedner Hauptstrasse, 8-10, A-1040 Wien, Austria Christian Doppler Laboratory for Thermoelectricity, Wien, Austria
A R T I C LE I N FO
A B S T R A C T
Keywords: Heusler-type phases DFT calculations Phase diagrams Crystal structure Structural disorder
A number of Heusler-type phases, e.g. TiNi2Sn, TiNiSb, ZrNiSn, TiFeSb, NbCoSb, which, for a long time, were considered to be stoichiometric with point composition, are characterized by homogeneity regions and/or offstoichiometry. Traditional high-throughput DFT calculations aim at the discovery of new fully ordered point composition Heusler-type phases and their potential intrinsic and extrinsic defects. In order to extend the application of DFT methods for determination of structural disorder in half-Heusler and full-Heusler phases we propose a new multi-way approach. It is based on evaluation of the crystal structure model that is used in DFT calculations by getting the best agreement between a set of theoretical and available experimental data (e.g. concentration dependencies of the lattice parameter, homogeneity regions, physical and mechanical properties). This approach allows to determine the model of structural disorder in Heusler-type phases, explain or predict the presence of the homogeneity regions or off-stoichiometry, and reveal the mechanisms of solid-solution formation.
1. Introduction The family of the Heusler-type phases stems from Fritz Heusler who, in 1903, discovered the MnCu2Al ferromagnet [1,2], and later determined its crystal structure [3]. Compounds, the crystal structure of which belongs to MnCu2Al structure type nowadays are called Heusler or full-Heusler (FH) phases. Later, the family of the Heusler-type phases was extended by Nowotny and Sibert [4], who determined the crystal structure of MgAgAs as an ordered ternary variant of the CaF2 type [5]. These structures are closely related to that of the full-Heusler phase; but as only half of the 8c sites are occupied, these phases are called semiHeusler or half-Heusler (HH) phases (Fig. 1). In the HH XYZ compounds the X component occupies the 4a (0 0 0) Wyckoff position, Y – 4c (1/4 1/4 1/4), and Z – 4b (1/2 1/2 1/2). Similar to HH compounds in the FH XY2Z compounds the 4a and 4b site are occupied by X and Z components, respectively, however the Y component occupies the 8c (1/4 1/4 1/4) crystallographic site. The difference between these two structures is that FH phases crystallize in centrosymmetric Fm-3m (#225) space group, while HH phases – in noncentrosymmetric F-43m (#216). This fact excludes the formation of continuous solid solutions between these two structure types. Heusler-type phases exhibit semiconducting properties at different electronic configurations: 18 and 24 valence electrons for HH and FH phases, respectively. Using this simple rule, one may rather reliably ⁎
predict the type of conductivity and various other physical properties of these compounds [6–9]. High symmetry, non-parametric atomic positions, and 12 (in HH phases) or 16 (in FH phases) atoms in the unit cell allow to accurately perform ab initio calculations to explain different properties of existing Heusler-type compounds [10] or to predict the formation of new, not yet discovered, HH phases [11]. As an example, a screening of 2295 ternary atom combinations for the HH symmetry via ab-initio calculations was performed [12] in order to predict compositions that could behave as topological insulators (TI). Recently, both ab-initio calculations and machine learning technique were used to screen for unknown stoichiometric HH phases [13–15]. The main disadvantage of these methods is the treatment of all HH phases as stoichiometric compounds with point composition. Detailed experimental and theoretical studies of Ti-Ni-Sn [16] and Zr-Ni-Sn [17] ternary systems revealed the presence of homogeneity regions in HH and FH phases. An even more complicated case of HH compound was found in Ti-Fe-Sb [18] and NbCo-Sb [19] system where Ti1+xFe1.33-xSb appeared to be off-stoichiometric with homogeneity region. In order to extend the application of DFT calculations on the discovery of homogeneity regions and off-stoichiometry in HH and FH phases we present a new multi-way approach, that allows to determine the model of structural disorder of these phases and to evaluate it by direct comparison with experimental data.
Corresponding author at: Leibniz Institute for Solid State and Materials Research, Helmholtzstr. 20, D-01069 Dresden, Germany. E-mail address: [email protected] (V.V. Romaka).
https://doi.org/10.1016/j.commatsci.2019.109307 Received 31 July 2019; Accepted 25 September 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
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V.V. Romaka, et al.
Fig. 1. Crystal structure of full-Heusler compound with general formula XY2Z and half-Heusler compound with general formula XYZ.
2. Modelling details
characterized by a strictly stoichiometric point composition with no homogeneity region: 1:1:1 – in the case of HHs, or 1:2:1 – in the case of FHs. On the phase diagram they are usually marked with a point/circle (Fig. 2a). Interestingly these phases are reported to have point composition in the whole temperature region of existence. The reason why so many Heusler-type phases were reported as sp is that they were mainly discovered as a series of compounds with different sorts of X, Y, or Z components, and thus the stoichiometric point composition was automatically assigned after the structure type was determined. Only in a few cases the deviation from the stoichiometry of Heusler-type phases was defined in detail. (ii) Heusler-type phases with off-stoichiometric point composition (op) are rather rarely observed in comparison with sp Heuslers, but are gaining more interest due to recent investigations of offstoichiometric HH thermoelectric materials [28,29]. This (op) type of phases still maintains the point composition and is marked as a point/circle on the phase diagrams (Fig. 2b), but deviates from 1:1:1 or 1:2:1 stoichiometry and thus is off-stoichiometric. In most cases the deviation in composition is less than 10 at. % and thus, was usually ignored and the sp type was assigned. In some rare cases a Heusler-type phase that was initially reported as op appeared to be sp. For example, in the Nb-Fe-Sb system the HH NbFeSb compound was reported as op Nb0.9Fe1.2Sb0.9 [30], but recent investigations confirmed it as a sp NbFeSb phase [31]. Usually the off-stoichiometry could be satisfactorily explained using the simple Zintl concept, but in general, the roots of these deviations could be explained from the thermodynamic point of view. (iii) In the Heusler-type phases with stoichiometric homogeneity region (sh) the homogeneity region covers the stoichiometric composition of a phase, but in addition expands toward other directions following different patterns (Fig. 2c), namely expanding towards: a) neighbored binary and ternary phases, independently, whether or not the Heusler-type phase is in a direct equilibrium with any of them; b) towards the pure components (corners of the Gibbs triangle) and along the isoconcentration lines of one or several components. This type of Heusler phases could be established in the case of detailed phase diagram investigations, preferably at different temperatures, and could not be reliably or in principle predicted by the Zintl concept. The category of sh phases comprises HH phases such as TiNi1+xSn [16], ZrNi1+xSn [17,32], HfNi1+xSn [32], formerly known as sp TiNiSn [33,34,35], ZrNiSn [33,34,36], and HfNiSn [33,37], respectively, as well as FH phases TiNi2-xSn [16,38], ZrNi2-xSn [32,17], HfNi2-xSn [32], formerly reported as sp TiNi2Sn [39,35], ZrNi2Sn [33,36], and HfNi2Sn
DFT calculations were carried out using the program packages ELK v4.3.06 [20] and Exciting (carbon) [21] – all-electron full-potential linearized augmented-plane wave (FP-LAPW) codes with exchangecorrelation functionals for solids by Perdew-Burke-Ernzerhoff (PBE) [22] in generalized gradient approximation (GGA). The k-point mesh grid was equal to or higher than 10 × 10 × 10 k-points. The manual optimization of lattice parameters was performed by fitting the universal equation of state [23]. The proper values of the muffin-tin radii were selected automatically at the initial stage of the calculations. Rmin(MT) × {|G + k|} was set to 7, where Rmin(MT) is the minimum muffin-tin radius used in the system. The enthalpy of formation (ΔH) at T = 0 K was calculated as a difference between the total energy of the compound and the energy of the respective amounts of pure elements. For higher temperatures the configurational entropy of mixing was introduced. For intermediate compositions of the compounds the crystal structure models were generated using the supercell package [24]. In order to simulate an alloy with random distribution of atoms, the KKR-CPA-LDA method [25,26] was employed. The energy window that covers conduction band, semi-core and valence states was equal to 21.8 eV. The ground state energy and the distribution of the density of states (DOS) were calculated for each alloy using the experimental values of the lattice parameter. The Brillouin zone integration and density of states calculation were performed on the basis of 1000 kpoints. The crystal structure models were plotted with the VESTA software package [27]. 3. Problems in the Heusler-type systems The challenges that arise during the investigation of the Heuslertype phases can be described as problems of their constitution, structural disorder, and physical properties, however, the proper structural model of Heusler-type phases supplies a key for understanding and explanation of their constitution and physical properties. 3.1. Problem of the constitution of Heusler-type phases From the constitutional point of view all known HH and FH phases can be divided into 4 groups which are characterized by: i) stoichiometric point composition; ii) off-stoichiometric point composition; iii) stoichiometric homogeneity region; iv) off-stoichiometric homogeneity region (Fig. 2). (i) Phases with stoichiometric point composition (sp) are the most frequently observed type of Heusler compounds. These phases are 2
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Fig. 2. Constitutionally derived four groups of Heusler-type phases with stoichiometric point composition (a), off-stoichiometric point composition (b), stoichiometric homogeneity region (c), and off-stoichiometric homogeneity region (d). A red marker shows the actual composition of a phase, whereas a yellow marker indicates the ideal stoichiometric composition.
phase and the second one is limited by the small amount of reliable experimental/theoretical data.
[33,37], respectively. (iv) The last group of Heusler-type phases is characterized by the presence of an off-stoichiometric homogeneity region (oh), which does not cover a stoichiometric composition (Fig. 2d). Similarly, to the sh phases, among the oh Heusler phases various shapes of the homogeneity region could be realized, which includes the expansion of the homogeneity region towards the pure components (corners of the Gibbs triangle) and/or along the isoconcentration lines of one or several components. An excellent example of such an oh phase is HH Ti1+xFe1.33−xSb which was first reported as sp (TiFeSb) [40], then as op (Ti1.27FeSb) [41], later as sh (Ti1+xFeSb) [42], and finally as oh (Ti1+xFe1.33−xSb) [18]. Another example is HH NbCoSb, which was first reported in [43] as sp phase, later as op phase (Nb0.8CoSb) [28], and only recently as oh phase (Nb0.8+xCoSb) [19].
3.2. Problem of the structural disorder in Heusler-type phases The constitution of each Heusler-type phase is determined by the shape of its thermodynamic potential and the thermodynamic potential of all neighboring phases. The thermodynamic potential itself is defined by the crystal structure of a phase in general and by the distribution of atoms among crystallographic sites, in particular. The presence of a homogeneity region and/or off-stoichiometry in Heusler-type phases leads to structural disorder of the initial distribution of atoms i.e. to the formation of vacancies, statistical mixtures of atoms, filling of the vacant crystallographic sites (in the case of HH phases). In the simplest case of structural disorder at least two sorts of atoms could be mixed among their initial crystallographic sites, and thus produce disorder while keeping the sp composition of the phase and the completely occupied crystallographic sites (Fig. 3a). It is a challenging task to determine the presence of such a mixture especially when the occupational factor of a second (mixing) component does not exceed 5%, because the total number of atoms, composition and stoichiometry remains the same, and the sensitivity of the X-ray diffraction (XRD) method and Rietveld refinement is usually not sufficient. In a more complicated case, when the Heusler-type phase is offstoichiometric (op), additionally a vacancy in one or several sites might appear (Fig. 3b), as well as an extra atom of one of the components
The afore mentioned classification of the constitution of Heuslertype phases undercovers their main problem – phases that were initially reported and still are accepted to have sp composition in many cases appear to be op, sh or even oh Heuslers. Despite the fact that the existence/formation of sp and op phases could be predicted by using a simple Zintl concept [44] or by high-throughput transport and defect calculation methods [13,14,15], both techniques are useless (i) for the discovery of the homogeneity regions (sh and oh) of Heusler-type phases, and (ii) the determination of their origin and mechanisms of formation, as the first one lacks a criterion for the homogeneity of a 3
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Fig. 3. Models of atoms distribution in the XYZ half-Heusler phases with: stoichiometric point composition – sp (a); off-stoichiometric point composition – op (b); stoichiometric composition with homogeneity region – sh (c); off-stoichiometric composition with homogeneity region – oh (d).
extra source of electrons or holes, and the semiconductor should be intrinsic or fully compensated with the Fermi level positioned exactly at the middle of the band gap resulting in a very high resistivity and a vanishing Seebeck coefficient. In fact, semiconducting Heusler phases are usually n or p-type with quite high values of Seebeck coefficient. For a deeper insight into the origin of the physical properties ab initio calculations are usually used. The electronic band structure, modelled by ab initio DFT methods using the model of the sp ordered crystal structure is in contradiction to the measured physical properties. The application of a rigid-band model or the Boltzmann transport theory within DFT for the prediction of the electrical transport properties of Heusler-type phases and solid-solutions yields reasonable results only in the case when the bands are not or just slightly affected. In fact, when the Heusler-type phase belong to op, sh, or oh group, the bands are significantly affected (Fig. 4) and for detailed DFT modeling the proper model of the crystal structure is required. Thus, the physical properties of the semiconducting Heusler phases can be explained only in the case of a proper model of their crystal structure. Modeling of anti-site defects in the Ni sublattice of TiNiSn was done in [46], where the authors claimed that this might be considered as a result of a decrease of the state of order of Ni sublattice when the temperature increases.
might occupy a vacant crystallographic site (in the case of HH phases). When a phase is characterized by the homogeneity region, that covers the stoichiometric composition (sh), at least two sorts of atoms can be mixed in one of the crystallographic sites (Fig. 3c); furthermore a vacant site could be gradually filled with an extra atom (in the case of HH phases) or vacancies could gradually appear in one of the crystallographic sites. The most challenging case, however, appears when a phase is characterized by a homogeneity region that does not cover the stoichiometric composition (oh). The crystal structure of such a phase simultaneously reveals a statistical mixture of atoms in one or several sites and vacancies or extra atoms in the occupied or vacant sites, respectively (Fig. 3d). The lack of XRD sensitivity makes it hard or even impossible to reliably refine the composition of the statistical mixture of atoms and vacancies that occupy one crystallographic site or even distinguish the type of species that occupies a crystallographic position if they have close X-ray scattering power. The correct determination of the structural models, that correspond to the homogeneity regions is complicated for both sh and oh Heusler-type phases and creates a real problem for a proper understanding of their properties. The models of structural disorder in Heusler-type phases are not yet completely investigated and a successful solution of this problem requires the combination of both, experimental and theoretical approaches.
3.4. Concepts for modelling of Heusler-type phases The simple Zintl concept, that operates only with the number of valence electrons, is widely used to predict or explain the existence of sp and op Heusler-type phases [44,28,45]. In combination with molecular orbital approach and the crystal field theory it could be used to build a simple band structure model of a Heusler-type phase. This concept can explain the formation of the band gap, the drift of the Fermi level while doping the semiconducting HH phase with n- or p-
3.3. Problem of physical properties in Heusler-type phases Although the semiconducting (intrinsic) properties of some Heuslertype phases can be predicted and explained by using the simple Zintl concept [44,45], this concept fails to explain the n- or p-type of electrical conductivity of Heusler phases, as for the sp phase there is no 4
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The determination of the most favorable intrinsic and extrinsic dopants in Heusler-type phases, especially for the semiconducting phases, is done by using the supercell approach [56,57,58,59]. Within this approach a large supercell is constructed to lower the concentration of the dopant and the energy difference is calculated between the structure without and with the defect. The value, and what is the most important, the sign of this energy determines whether the formation of such a defect is favorable or not. This method is suitable to predict the possible type of defects and compensators in the Heusler-type phases, but lacks the determination of the solubility limits (homogeneity region) of such a doped phase due to the necessity to fit the model of a very diluted dopant. Sometimes, in order to predict the existence of new Heusler-type phases, the researchers focus only on the negative enthalpy of formation as a stability criterion [15]. However, a negative enthalpy of formation for a given sp or op hypothetical Heusler-type phase means only that the product of the chemical reaction has a lower energy than the mixture of the initial components. In consequence it shows that the reaction is in principle possible, but it does not reveal whether or not the phase is realized in a real system, as the presence of other neighboring phases may prevent the formation of this Heusler-type phase. Another argument for the mechanical stability of Heusler-type phase is the absence of imaginary frequencies in the phonon DOS spectrum [60]. However, this criterion can be used only as an additional argument for the existence or the absence of the phase, but not as a criterion of phase formation. All concepts and approaches mentioned above have strong sides as well as some weaknesses, but the extension of their usage, as well as their combination could give another approach for a better understanding and prediction of the different constitutional groups of Heusler-type phases.
Fig. 4. Schematic band structure of an n-type HH phase: modeled as sp phase with the DFT method (a); modeled as electron-doped sp phase with DFT and rigid band model (b); derived from experiment (c). Ec, Ev, and EF correspond to the energy of conduction, valence bands, and Fermi energy, respectively.
type dopants or formation of vacancies in one of the crystallographic sites. However, the concept is unable to either predict the existence or explain the formation of the homogeneity regions of Heusler-type phases, as they depend on the equilibrium with the neighboring phases and the temperature. Another concept is based on the CALPHAD method [47,48,49], which is used to predict and refine the phase diagrams of multinary systems with Heusler-type phases in particular Ti-Ni-Sn [16,50], Hf-NiSn [51], Zr-Ni-Sn [52,17], and (Ti,Zr,Hf)NiSn [53]. This method is rather flexible, as it can use both, experimental and theoretical thermodynamic quantities, e.g. the enthalpy of formation. The CALPHAD approach is usually amplified by the wide usage of DFT calculations, that allow to predict the enthalpy of formation of pure ternary phases and hypothetical solid solutions and model the phase equilibria between the Heusler-type phase and all neighboring phases in a phase diagram. The approach, which is close to the CALPHAD method, is the convex hull method, which relies only on DFT calculations [28,13,18,54]. Within this method the calculations of the enthalpy of formation are performed for a number of intermediate compositions between the desired Heusler-type phase and one or several real/hypothetical phases forming pseudobinary systems (Fig. 5). Then the convex hull is constructed and the stability of the sp Heusler-type phase among the neighboring phases, or the existence of the op Heusler phase is determined. For the modeling of the solid solutions (homogeneity regions) both, the DFT part for the CALPHAD method and the convex hull method, often use SQS approach [55] to generate special quasi-random structures for each intermediate composition of the Heusler phase. These structures, however, do not maintain the symmetry of the initial ordered structure. For each quasi-random structure, the geometry optimization is performed with DFT calculations, which include relaxation of both lattice vectors and atomic coordinates. It is common that after geometry optimization some atoms in such a random structure deviate from their non-parametric positions in the initial Heusler-type structure.
4. Determination of structural disorder in Heusler-type phases To solve the problems described above we propose a new approach based on the determination of structural disorder in multicomponent intermetallic phases through experimental and theoretical evaluation of their thermodynamic stability, crystallographic features, and physical/ mechanical properties. Traditionally the model of the crystal structure of the phase is evaluated at the beginning or at an intermediate stage of a complex research, depending whether the crystal structure prototype is known and is further on used for ab initio DFT modeling of its physical and chemical properties to explain experimental results. The proposed approach contains traditional steps (Fig. 6), but allows at the stage of DFT calculations to further refine the model of the crystal structure of a phase and its stability by modeling thermodynamic, crystallographic, transport, elastic, and energetic parameters with further 3-way data evaluation. Based on the proposed classification of the constitution of Heuslertype phases we can assume that at some higher temperatures sp and oh phases could be potentially transformed into sh and oh phases, respectively, due to the greater impact of the entropy term on the thermodynamic potential thus stabilizing the presence of some structural defects. In order to describe the crystallographic model of the potential homogeneity region the shape of the latter should be determined. As the homogeneity region of the Heusler-type phase in most cases does not exceed 10 at. % from the stoichiometric point composition, two simple patterns can be applied a) for substitutions and b) for interstitials and defects (Fig. 7). The model of the first type of patterns assumes that within two different sublattices of the crystal structure, e.g. half-Heusler A(1+x)BC(1-x) the concentration of one sort of atoms (A) increases by x giving (1 + x) in total, while the other species (C) decreases by the same margin giving (1 − x) in total. All possible permutations of two constituents between their two sublattices, limited by
Fig. 5. The schematic simplified illustration of the convex hull method to determine the stability of a Heusler-type phase (a) and the real thermodynamic potentials for determination of the homogeneity region of the Heusler-type phase (b). 5
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Fig. 6. Schematic presentation of approach for determination of structural disorder in HH and FH phases.
Fig. 8. Application of homogeneity region patterns for DFT examination of ‘TiFeSb’ half-Heusler phase. A yellow marker indicates the ideal stoichiometric composition of TiFeSb, the red marker shows the composition with minimum enthalpy of formation, the green marker indicates the experimentally determined [18] homogeneity region for the half-Heusler phase. Black and red dash-lines represent the potential and actual patterns, respectively, along which the DFT calculations were performed.
Fig. 7. Two types of homogeneity region patterns for Heusler-type phases.
the size of the supercell, can be obtained by using the combinatorial structure-generation approach [24]. The model for the second type of pattern assumes that the concentration of one component constantly changes (increases or decreases), whilst the ratio between all other components remains constant. The increase of concentration of one of the components can be presented as the incorporation of extra atoms into the vacant site (only for HH phases), e.g. AB(Bx)C, or the generation of vacancies in all crystallographic sites except one occupied by the desired component, e.g. (AC)1-xB. Again, all possible permutations of the constituents between possible sublattices can be obtained by using the combinatorial structure-generation approach [24]. The process of structure generation within the patterns of the homogeneity region could be fully automated. An example of application of these patterns for determination of the off-stoichiometry and homogeneity region of Ti1+xFe1.33−xSb is shown in Fig. 8. It should be noted, that the generated crystal structure models (supercells) are presented as a single, doubled, tripled initial cubic cell along each of the basis vectors, despite adopting a potentially different symmetry of the whole supercell. If one tried to optimize the geometry (lattice vectors and atomic coordinates) of such a supercell by using the ‘forces’ (Hellmann-Feynman) method, the non-parametric positions would be interpreted as parametric, and no constraints would be applied to the angles between basis vectors, and their lengths. The
resulting crystal structure would be optimized, but the initial shape of the single cubic cell as well as the initial atomic coordinates will be lost (Fig. 9). Moreover, such an optimization suffers from the large number of vacancies, which makes convergence more difficult. To simplify the optimization procedure, the atomic coordinates should remain constant, and the only parameter that should change is the lattice constant, which should be equally scaled along each direction of the supercell. Varying the scale of the lattice constant in the range of 0.9–1.1 from the optimized sp Heusler-type phase with a step of 0.02 and performing self-consistent ground state DFT calculations for each step, the E(V) dependence could be obtained. It could then be fitted with one of the equations of state (EOS) delivering the optimized volume and total energy of the system. The obtained optimized volume divided by the number of unit cells in the supercell will yield the volume of the averaged cubic cell, and the cube root will yield the lattice constant aopt. At this stage it is possible to define the theoretically determined concentration dependence of the lattice parameter aopt and the boundaries of the homogeneity region. The energetically most 6
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Fig. 9. Initial structure of TiNiSn and relaxed structure of TiNi0.75Cu0.25Sn. Lattice deformation of TiNi0.75Cu0.25Sn is visible in the relaxed structure.
structure model in more complicated systems. For example, the Fe for Co substitution in the isopleth TiCoSb-TiFe1.33Sb [18] could potentially occur in 4c, in 4d, or in both sites. The DFT calculations of the enthalpy of formation revealed that the mechanism of substitution is based on a simultaneous substitution of Fe by Co in the 4c site with decreasing Fe content in the 4d site. The slope of the a(x) dependence for the converged DFT model is in fair agreement with experimental data (Fig. 12a). To the first batch of evaluated data belongs also the density of states (DOS) profile of the intermediate compositions between FH and HH phases. For this, however, one needs to use another DFT method based on the Green’s function method (known as Korringa-Kohn-Rostoker, KKR) with coherent potential approximation (CPA), and not the one with a plane-wave basis set. This method allows to preserve the symmetry of the initial HH-phase while modeling solid solutions towards the FH compound. In recent papers [38,32] it was shown that the filling of the vacant 4d site in HH TiNiSn and ZrNiSn compound with extra Ni atoms does not lead to the gradual change of the DOS profile from the HH to the FH phase (Fig. 13). Instead of this a sharp change in DOS from HH to FH occurs near the composition MNi1.65Sn (M = Ti, Zr). This could help in the identification of the symmetry change from noncentrosymmetric HH phases to centrosymmetric FH compounds. The second batch of evaluated data contains: modelled enthalpy of formation, thermodynamic potential (with configurational entropy term), homogeneity regions of the sh, oh Heusler-type phases, composition of the op Heusler phase and phase equilibria with the
favorable crystal structure model for each composition along each direction of the homogeneity region pattern will define the mechanism of the formation of the homogeneity region. The verification of the correctness of the theoretical data (and as a result, the crystal structure model) is performed by the 3-way evaluation, which includes three independent sets of modelled parameters that are directly compared with experimentally extracted data from the phase diagram, crystal structure, and physical/elastic properties. The first batch of evaluated data contains: modelled composition dependencies of the lattice parameter a of the sh and oh Heusler phases. Despite the fact that absolute values of the experimental and DFT-derived lattice parameter are different and the difference depends on the type of exchange-correlation potential used in the modeling, the slope and curvature of both theoretical and experimental a(x) dependencies should be similar. This is clearly visible in the systems TiNiSn-TiNi2Sn [16] and ZrNiSn-ZrNi2Sn [17] (Fig. 10) where the slope of the experimental a(x) dependencies within the solid solutions for FH TiNi2-xSn and ZrNi2-xSn and HH TiNi1+xSn and ZrNi1+xSn fits with the models of the DFT calculations. Despite both sh FH and HH phases have very similar crystal structures their a(x) slopes are different. Another example for modeling the a(x) dependence is the oh Ti1+xFe1.33-xSb phase [18]. There is a small negative curvature of the a(x) dependence within the homogeneity region which is also observed on the modeled curve (Fig. 11a). The composition dependence of the lattice parameter could also be used as an additional criterion for the correctness of the crystal
Fig. 10. Experimental and modeled composition dependencies of the lattice parameter a in the systems TiNiSn-TiNi2Sn [16] (a) and ZrNiSn-ZrNi2Sn [17] (b). 7
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Fig. 11. Experimental [18] and modeled composition dependencies of the lattice parameter a within the homogeneity region of Ti1+xFe1.33-xSb (a) and calculated enthalpy of formation of the hypothetical TiFe1+xSb and Ti1+xFe1.25-xSb solid solutions (b).
solid solutions HH TiNi1+xSn and FH TiNi2-xSn at 950 and 1100 °C was confirmed experimentally [16] (Fig. 14a) and could be predicted by DFT calculations by adding a configuration entropy of mixing term. For the ZrNiSn-ZrNi2Sn section [17] (Fig. 14b) it was shown that the solubility strongly depends on temperature, which means that the entropy term plays a crucial role in stabilization of the solid solution. In contrast to the systems described above the homogeneity region of the Ti1+xFe1.33-xSb HH phase extends along the isoconcentration line of Sb. The DFT calculations showed the presence of the wide minimum on the ΔHf(x) dependence, which corresponds to the homogeneity region and is stable even at low temperatures (Fig. 11b). For the more complex systems, like 4- or 5-component HH solid solutions the formation of a miscibility gap could be predicted and experimentally confirmed [53,61,62]. The experimentally determined homogeneity region of the Heusler-type phases, standard enthalpies of formation, and other thermodynamic parameters were used for a direct comparison with modeled data and in such a manner it was possible to evaluate the correctness of the crystal structure model. The third batch of evaluated data contains: distribution of the electron and phonon densities of states, type of Heusler phase – semiconductor/metal, type of conductivity, band gap, activation energies from the Fermi level onto the percolation level of the valence or conduction bands, Seebeck coefficient, specific heat, thermal expansion coefficient, Young’s, bulk, and shear modulus, Vicker’s hardness, Poisson number, and their composition dependencies for sh and oh Heusler-type phases. This evaluation becomes extremely important in
neighboring phases. A combination of the enthalpy of mixing within the pattern of the homogeneity region allows to figure out whether offstoichiometry appears for the Heusler-type phase. The optimized total energy for each model of the crystal structure could be used to get the enthalpy of formation. The enthalpy of mixing could also be obtained by E(V) optimization of both end-members of the homogeneity region pattern, and from the convex hull method the solubility could be determined along each pattern direction of the homogeneity region. Additionally, the configuration entropy of mixing (ΔSmix) could be applied to determine the most favorable atomic distribution at each temperature. The model for ΔSmix could be presented as a combination of entropy in every sublattice, e.g. in the hypothetical half-Heusler phase (A1-xCx)B(ByVac1-y)C there is a statistical mixture of A and C components in the A-sublattice (4a site), whereas the vacant 4d site is partially filled by extra By atoms. In this case the configuration entropy of mixing is presented in the following form:
ΔSmix = −R(x lnx + (1 − x )ln(1 − x ) + y lny + (1 − y )ln(1 − y )) For the full-Heusler (A1-xCx)(B1-yVacy)2C:
ΔSmix = −R(x lnx + (1 − x )ln(1 − x ) + 2 × (y lny + (1 − y )ln(1 − y ))) In the case when the enthalpy of mixing shows a positive trend within the hypothetical homogeneity region pattern, the configuration entropy of mixing can be introduced. It was demonstrated [17,54] that a positive ΔHmix at T = 0 K could transform into negative ΔGmix at higher temperatures due to the large impact of the configuration entropy of mixing, producing limited solid solutions. The presence of the
Fig. 12. Experimental [18] and modeled a(x) dependencies for the TiCoSb-TiFe1.33Sb isopleth (a) calculated enthalpy of formation for two models of the solid solution and TiFe1.375Sb (b). 8
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Fig. 13. Calculated DOS profiles for the HH ZrNi(Nix)Sn (a) and FH ZrNi1+xSn (b) hypothetical solid solutions.
Fig. 14. Theoretical evaluation of the solid solution in systems TiNiSn-TiNi2Sn (a), ZrNiSn-ZrNi2Sn (b). Dashed lines correspond to the experimentally determined boundaries of the HH and FH solid solutions [16,17].
5. Summary and conclusions
the case of semiconducting properties of the HH phase. The experimentally determined temperature dependencies of resistivity and Seebeck coefficient yield information on the dominant charge carriers, position of the Fermi level, activation energies from the localized states in the energy gap onto the percolation level of the conduction of valence bands. This information strongly depends on the model of the crystal structure, and thus could be used for its evaluation. For TiNiSb [38] it was shown that the metallic type of conductivity is not explained simply as a feature of the band structure of TiNiSn with the Fermi level located in the conduction band, but with the presence of vacancies in the Ti (4a) site, which induces the closing of the band gap. For ZrCoSb [63] it was shown that within the homogeneity region of the HH phase the semiconducting properties change to metallic. For LuNiSb [29] the modeling of the DOS profiles for different combinations of atoms in the crystal structure allowed to determine the composition and atom distribution that fits the best EPMA data and electrical transport properties. For TiNiSn and ZrNiSn [62] it was shown that even very small amounts of extra Ni atoms in the vacant 4d site could have a dramatic effect on the energy gap, and explain the n-type of conductivity of these HH phases. The results appeared to be in a very good agreement with the band structure model constructed through the analysis of the activation energies on these compounds. Recent experimental and theoretical investigations of mechanical properties of HH phases [31,64,65,54] show good correlation with the crystal structure model and thus can serve as its additional evaluation criterion.
The proposed approach for determination of structural disorder extends the traditional DFT techniques and could be partially or fully automated for prediction of the structural disorder and homogeneity regions in Heusler-type phases. A multi-way approach allows to evaluate the model of the crystal structure, and to get the best agreement between a set of theoretical and available experimental data, starting from the concentration dependencies of the lattice parameter, homogeneity regions, and finishing with electrical transport, magnetic and mechanical properties. Acknowledgements V.V. Romaka and P. Rogl thank the Austrian agency for international mobility and cooperation in education, science and research for support via the Ernst Mach project ICM-2017-06580. References [1] F. Heusler, Über magnetische Manganlegierungen, Verh. Dtsch. Phys. Ges. 5 (1903) 219. [2] F. Heusler, W. Starck, E. Haupt, Magnetisch-chemische Studien, Verh. Dtsch. Phys. Ges. 5 (1903) 220. [3] O. Heusler, Kristallstruktur und Ferromagnetismus der Mangan-AluminiumKupferlegierungen, Adv. Phys. 411 (2) (1934) 155–201.
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