The half Heusler system Ti 1 + x Fe 1 . 33 − x Sb – TiCoSb with Sb / Sn substitution : phase relations , crystal structures and thermoelectric properties

Investigations of phase relations in the ternary system Ti–Fe–Sb show that the single-phase region of the Heusler phase is significantly shifted from stoichiometric TiFeSb (reported previously in the literature) to the Fe-rich composition TiFe1.33Sb. This compound also exhibits Fe/Ti substitution according to Ti1+xFe1.33−xSb (−0.17 ≤ x ≤ 0.25 at 800 °C). Its stability, crystal symmetry and site preference were established by using X-ray powder techniques and were backed by DFT calculations. The ab initio modeling revealed TiFe1.375Sb to be the most stable composition and established the mechanisms behind Fe/Ti substitution for the region Ti1+xFe1.33−xSb, and of the Fe/Co substitution within the isopleth TiFe1.33Sb– TiCoSb. The calculated residual resistivity of Ti1+xFe1.33−xSb, as well as of the isopleths TiFe1.33Sb–TiCoSb, TiFe0.665Co0.5Sb–TiCoSb0.75Sn0.25 and TiFe0.33Co0.75Sb–TiCoSb0.75Sn0.25, are in a good correlation with the experimental data. From magnetic measurements and Fe Mössbauer spectrometry, a paramagnetic behavior down to 4.2 K was observed for TiFe1.33Sb, with a paramagnetic Curie–Weiss temperature of −8 K and an effective moment of 1.11μB per Fe. Thermoelectric (TE) properties were obtained for the four isopleths Ti1+xFe1.33−xSb, TiFe1.33Sb–TiCoSb, TiFe0.665Co0.5Sb–TiCoSb0.75Sn0.25 and TiFe0.29Co0.78Sb– TiCoSb0.75Sn0.25 by measurements of electrical resistivity (ρ), Seebeck coefficient (S) and thermal conductivity (λ) at temperatures from 300 K to 823 K allowing the calculation of the dimensionless figure of merit (ZT ). Although p-type Ti1+xFe1.33−xSb indicates a semi-conducting behavior for the Fe rich composition (x = −0.133), the conductivity changes to a metallic type with increasing Ti content. The highest ZT = 0.3 at 800 K was found for the composition TiFe1.33Sb. The TE performance also increases with Fe/Co substitution and reaches ZT = 0.42 for TiCo0.5Fe0.665Sb. No further increase of the TE performance was observed for the Sb/Sn substituted compounds within the sections TiFe0.665Co0.5Sb–TiCoSb0.75Sn0.25 and TiFe0.33Co0.75Sb–TiCoSb0.75Sn0.25. However, ZT-values could be enhanced by about 12% via the optimization of the preparation route (ball-mill conditions and heat treatments).

The half Heusler system Ti 1+x Fe 1.33−x Sb-TiCoSb with Sb/Sn substitution: phase relations, crystal structures and thermoelectric properties A. Tavassoli, a,b A. Grytsiv, * a,c,d G. Rogl, a,c,d V. V. Romaka, e H. Michor, c M. Reissner, c E. Bauer, c,d M. Zehetbauer b and P. Rogl a,d Investigations of phase relations in the ternary system Ti-Fe-Sb show that the single-phase region of the Heusler phase is significantly shifted from stoichiometric TiFeSb (reported previously in the literature) to the Fe-rich composition TiFe 1. 33 Sb.This compound also exhibits Fe/Ti substitution according to Ti 1+x Fe 1.33−x Sb (−0.17 ≤ x ≤ 0.25 at 800 °C).Its stability, crystal symmetry and site preference were established by using X-ray powder techniques and were backed by DFT calculations.The ab initio modeling revealed TiFe

Introduction
Thermoelectricity is one of the simplest means of the direct conversion of heat into electricity.Here, the energy conversion efficiency is governed by the dimensionless figure of merit ZT = S 2 σ/(λ e + λ ph ), where S is the Seebeck coefficient, T the absolute temperature, σ the electrical conductivity, and λ e and λ ph the electron and lattice components of the total thermal conductivity λ, respectively.Whilst the Seebeck coefficient of a distinct thermoelectric material increases with decreasing charge carrier concentration (n), the electrical conductivity decreases; thus, an optimum thermoelectric performance of heavily doped semiconductors is obtained for n ∼ 10 19 -10 21 cm −3 .
2][3][4][5][6] HH compounds crystallize in the cubic MgAgAs structure-type, space group F4 ˉ3m, forming three interpenetrating face-centered cubic (fcc) sublattices and one vacant sublattice. 7In view of the Zintl mechanism, HH materials with 18 valence electrons were shown to exhibit semiconducting properties. 8A great advantage in the optimization of the thermoelectric properties of HH compounds is the possibility to individually dope each of these sublattices.The most attractive properties of HH materials for thermoelectrics are their high Seebeck coefficient S up to 300 μV K −1 at room temperature and their high electrical conductivity (100 to 1000 S cm −1 ); however, the drawback of HH alloys is their relatively high thermal conductivity reaching up to 10 W mK −1 at room temperature. 9Many HH compounds were investigated in the past with the aim to improve their thermoelectric properties.So far the highest ZT = 1.5 has been reported by Sakurada and Shutoh 10 for Sb-doped n-type Ti 0.5 Zr 0.25 Hf 0.25 NiSn 0.998 Sb 0.002 at 690 K and by Rogl et al. 11 at a slightly higher temperature of 825 K. Schwall and Balke 12 obtained ZT = 1.2 at 830 K via intrinsic phase separation in Ti 0.5 Zr 0.25 Hf 0.25 NiSn 0.998 Sb 0.002 .More recent attempts have succeeded in obtaining a ZT of ∼1.35 for n-type Ti 0.15 Zr 0.25 Hf 0.65 NiSn 0.995 Sb 0.005 with nano-dispersed ZrO 2 2 and for (Zr 0.40 Hf 0.60 ) 0.00 V 0.01 NiSn 0.998 Sb 0.002 with vanadium resonant states. 13For commercial thermoelectric applications, however, the high priced Hf-metal needs to be avoided.Therefore, it is interesting to note that for the Hf-free grades Ti 0.5 Zr 0.5 NiSn 0.98 Sb 0.02 14 and Ti 0.50 Zr 0.48 Nb 0.02 NiSn 0.98 Sb 0.02 , 11 a ZT = 1.2 was obtained with a thermoelectric efficiency η 300-800 K > 11%.A significant progress was also made for p-type Nb 1−x Hf x FeSb with ZT max ≈ 1.5 at 1200 K (ref.][18] During our study of thermoelectric alloys in the section NbFeSb-TiFeSb, 18 we noticed the absence of transport properties of the parent compound TiFeSb, which also served as the base for Fe/Co and Sb/Sn substituted thermoelectric materials.The Fe/Co substitution in TiFe x Co 1−x Sb results in a significant increase of ZT from 0.05 for TiCoSb 19 to 0.42 for TiFe 0.3 Co 0.7 Sb. 20 These values are close to that reported for Sb/ Sn substituted TiCoSb 0.8 Sn 0.2 (ZT max = 0.5 at 700 K (ref.21)).In addition, a DFT calculation of Ti/Fe substitution in Ti 1−x Fe x CoSb indicated a decrease of ZT from 0.05 for TiCoSb 19 to 0.02 for Ti 0.9 Fe 0.1 CoSb. 22urthermore, we noticed that some controversy concerns the region of existence of the HH phase in the Ti-Fe-Sb system.The HH phase TiFeSb (with symmetry F4 ˉ3m) was first reported by Krypyakevych et al. 23,24 Its symmetry and composition have been confirmed; 25,26 however, a HH phase with composition Ti 1.27 FeSb was reported by Skolozdra et al. 27 with additional Ti atoms occupying the tetrahedral vacancies.From an investigation of the isothermal section of the Ti-Fe-Sb system at 800 °C (mainly based on X-ray diffraction analyses), Melnyk and Tremel 26 determined the homogeneity region for the HH phase as Ti 1+x FeSb (−0.2 ≤ x ≤ 0.27) at 800 °C.In contrast to these findings, weak X-ray reflections but also neutron powder diffraction experiments claimed for TiFeSb a superstructure with 8-fold unit cell (Fm3 ˉm; a = 2a 0 = 1.1898 nm, Cu 4 Mn 3 Bi 4 -type). 28ur preliminary investigations of the phase equilibria in the Ti-Fe-Sb system 29,30 showed that the single phase region of the Heusler phase at 800 °C excludes the stoichiometric composition TiFeSb, which appears to be significantly shifted to a higher Fe-content.For a significant Fe/Ti substitution, the resulting chemical formula is Ti 1+x Fe 1.33−x Sb.The formula TiFe 1.33 Sb is consistent with the composition Ti 30 Fe 40 Sb 30 (in at%; determined with an uncertainty of ∼5 at%) as reported by Naghibolashrafi et al. 31 who, however, labeled this phase as TiFe 1.5 Sb.Based on their Rietveld refinement (R I = 0.1) and selected area electron diffraction (SAED) results, these authors 31 concluded the Heusler symmetry Fm3 ˉm for TiFe 1.5 Sb.Besides this strong discrepancy on the composition of the Ti-Fe-Sb Heusler phase, no reliable data exist on the exact symmetry (Fm3 ˉm or F4 ˉ3m) and atom site preference of this compound.Although new experimental findings located the Heusler phase at a higher Fe-content (TiFe 1+x Sb), recent DFT calculations did not necessarily take them into account.The calculations of Azar et al. 32 for Ti 1+x FeSb were based on the old crystallographic model with additional titanium atoms in the unit cell Ti 1+x FeSb, 26 while Sharma and Kumar treated the stoichiometric Heusler phase TiFe 2 Sb. 33he aforementioned conflicting facts for TiFeSb-based thermoelectrics clearly reveal that both sections TiFe x Co 1−x Sb and Ti 1−x Fe x CoSb were ill chosen demanding a re-optimization of thermoelectric properties along proper isopleths.These arguments prompted us to focus our investigations in this work on several tasks: (i) to clarify the crystal structure and extension of the single-phase region of the Heusler phase in the ternary system Ti-Fe-Sb; and (ii) to re-investigate the homogeneity regions and to study the effects of Fe/Ti, Fe/Co and Sb/Sn substitutions on the TE properties for the Heusler phase in the system Ti-Fe-Co-Sb-Sn (for detailed location of the samples planned and investigated, see Fig. 1).As major means of investigation/alloy characterization we employed X-ray diffraction, electron microprobe analyses and the general techniques for physical property measurements (see the section Experimental details).However, since the X-ray atom scattering powers of the elements Ti, Fe, Co are rather similar and thus might not allow an unambiguous evaluation of atom site distributions in the crystal structures, the present study is assisted by (i) DFT calculations of the electronic structures and in particular their stability, as well as by (ii) Mössbauer and magnetic susceptibility measurements in order to elucidate the details of the Fesublattice.

Sample preparation
Samples with a weight of 1.5 g were prepared by conventional arc melting on a water-cooled copper hearth under an inert gas atmosphere (Ar, 5 N) using appropriate amounts of ingots of elements with purity above 99.9%.In order to compensate the weight losses of antimony during arc melting about 3 wt% Sb were added.Repeated flipping over and re-melting homogenized the reguli.Samples were then sealed in quartz ampoules under an argon pressure of 0.25-0.3bar and were heat treated (HT1) at 800 °C (Ti-Fe-Sb alloys: 2-3 weeks) and 950 °C (Ti-Fe-Co-Sb/Sn: 9-11 days).
In order to obtain bigger samples for the measurement of TE properties, 3-4 reguli were crushed into powders with a grain size below 75 μm and then ball milled in tungsten carbide vessels (volume = 80 ml, Ar filled) in a Fritsch planetary mill (Pulverisette 4) with balls of 10 mm diameter under two conditions: (i) at 200 rpm (main disc) and −500 rpm (vessels) for 2 h (ball milled -BM) or (ii) at 400 rpm (main disc) and −1000 rpm (vessels) for 4 h (high energy ball milled -HEBM).The obtained powder was packed into a graphite die with 10 mm diameter and consolidated to a pellet under 56 MPa employing a uniaxial hot press (HP, FCT, W200/250-2200-200-KS) using 5 N Ar as an inert atmosphere at temperatures from 800 °C to 950 °C (Table 1).In some cases the samples were additionally heat-treated (HT2).The compositions of all samples (from EPMA) are shown in Fig. 1 and specific details for their preparation are listed in Table 1.

Sample characterization (X-ray and micro-structure)
The phase structures of the samples were investigated by X-ray powder diffraction (XPD) collected from a HUBER-Guinier image plate with monochromatic CuK α1 -radiation (λ = 0.154056 nm) and pure Ge (99.9999%) as an internal standard.Precise lattice parameters were calculated by least squares fits to the indexed 2Θ values (calibrated with respect to Ge as an internal standard; a Ge = 0.565791 nm at room temperature) using the program STRUKTUR. 34The microstructure and chemical composition were analyzed by scanning electron microscopy (SEM) and electron probe microanalysis (EPMA) via INCA Penta FETx3 -Zeiss SUPRA™55VP equipment with an EDX detector.Standard deviations on a minimum of ten points measured per phase in equilibrated alloys were less than ±0.2 mass% i.e. never exceeding ±0.5 at% for the accepted composition.For the maximum difference of atomic numbers, the measured deviation was less than 1 at% for the line compound TiSb 2 .Quantitative Rietveld refinement was used to determine the atom site, phase distribution and lattice parameter employing the program FULLPROF. 35,36

Physical property measurements
The Seebeck coefficient and electrical resistivity were measured simultaneously from 300 to 823 K using an ULVAC-RIKO ZEM-3 system.The thermal conductivity was calculated from the thermal diffusivity (D), the heat capacity (C p ) and the density (d ) employing the formula λ = D × C p × d.The thermal diffusivity and the heat capacity were determined using a flash method (ANTER Flashline 3000 unit) in the temperature range of 423-800 K.The density (d A ) was measured by the Archimedes principle in distilled water; the relative density, d R , in percent was calculated by d R = (d A /d X ) 100%; d X is defined as d X-ray = nM/VN A , where n is the number of atoms in the cell, M is the molecular weight, V is the cell volume and N A is Loschmidt's number.Generally, measurement errors for the electrical resistivity and Seebeck measurements are ∼3%, and for thermal conductivity ∼5%. 57Fe Mössbauer measurements were performed at 294 K and at 4.2 K in standard transmission mode using a 57 CoRh source relative to which the values of the center shift are given.Specific heat measurements were performed on a commercial Quantum Design PPMS calorimeter in the temperature range of 0.4-5 K with a 3 He insert and from 2-300 K with a 4 He puck.DC magnetic susceptibility measurements were performed at 0.1 T using a CRYOGENIC SQUID magnetometer in the range of 2-300 K.Additional magnetization data were collected at 1 T, 3 T and 5 T in order to verify the approximate linearity of the field dependent isothermal magnetization.

First-principles calculations
The DFT calculations were carried out using the ELK v2.3.22 package 37 an all-electron full-potential linearized augmented-plane wave (FP-LAPW) code with the Perdew-Burke-Ernzerhof exchange-correlation functional in the generalized gradient approximation (GGA). 38The k-point mesh grid was equal to 10 × 10 × 10 k-points.Prior to the final total energy calculations the lattice parameter a was optimized by using the universal equation of state 39 for the set of 11 values of the lattice parameter with 0.1 nm step in the range of 0.55-0.65 nm.The appropriate values of the muffin-tin radii were selected automatically at the initial stage of the calculations.In general, the enthalpy of formation (ΔH) at T = 0 K for a compound with the general composition Ti a Fe b Co c Sb d Sn e was calculated according to the following formula: package. 40For simulation of alloys with random distribution of atoms the Korringa-Kohn-Rostoker method was employed, 41 which is realized in the SPR-KKR 42 code using the coherent potential approximation (CPA).As exchange-correlation potential a local density approximation (LDA) was used with Vosko-Wilk-Nusair parameterization. 43All alloy systems were treated as magnetic in the relativistic approximation.The ground state calculations were carried out for a 1000 k-points energy mesh referring to the experimentally determined lattice parameters and atom distributions from the FP-LAPW code.
The energy window that covers conduction band, semi-core and valence states was equal to 19 eV.The Brillouin zone integration and density of states (DOS) calculations were performed on 1000 k-points.The linear response calculations (SPR-KKR) of the residual resistivity (vertex corrected) at 0 K were carried out using the Kubo-Greenwood formula for 100 000 k-points in the Brillouin zone.

Results and discussion
3.1.Homogeneity regions, phase relations and crystal structures of the Heusler phase in the system Ti-Fe-Sb Fig. 2 presents the isothermal section of the system Ti-Fe-Sb as reported by Melnyk et al. 26 One can see that the homogeneity region of the HH phase (τ 1 -Ti 1+x FeSb) at 800 °C includes the stoichiometric composition TiFeSb with an extended homogeneity region (−0.2 ≤ x ≤ 0.27).However, our re-investigation of this system shows a significant discrepancy with these data.
The EPMA of the microstructure of the as-cast sample TiFeSb (Fig. 3a) shows primary crystallization of τ 2 with the composition Ti 37.5 Fe 24.5 Sb 38 (at%) and a fine eutectic structure of the composition Ti 21 Fe 60 Sb 19 .After annealing at 950 and 800 °C the sample reveals at both temperatures two phases with compositions TiFe 1.33 Sb and Ti 37.5 Fe 24.5 Sb 38 , respectively.Although XPD clearly shows the presence of a cubic Heusler structure, none of the above mentioned compositions are included in the single phase region of τ 1 as reported by Melnyk et al. 26 SEM, EPMA and XPD for two additional compositions TiFe 1.33 Sb and TiFe 2 Sb in as-cast and annealed states (Fig. 3b and c) unambiguously show τ 1 with the composition TiFe 1.33 Sb.Further investigations of the phase relations of the Heusler phase (Fig. 4) reveal that the homogeneity region for this compound at 800 °C exists with a constant antimony content of about 30 at% Sb but with Ti/Fe substitution characterized by the formula Ti 1+x Fe 1.33−x Sb (−0.17 ≤ x ≤ 0.25).The lattice parameters for this phase (Fig. 5) increase with increasing Ti content in line with the difference between the atomic radii of iron and titanium.The literature data on Ti 1.27 FeSb 27 and Ti 1.25 FeSb 26 fit well to this tendency because these compositions are located near the Ti-rich end of Ti 1+x Fe 1.33−x Sb established in this work (see Fig. 4).The literature data reported for TiFeSb, 24,25,28,44 Ti 0.8 FeSb 26 and TiFe 1.5 Sb 31 do not obey the chemical formula Ti 1+x Fe 1.33−x Sb and therefore they are plotted in Fig. 6 as a function of Ti/(Fe + Ti).One may note that the lattice parameters reported for TiFeSb and Ti 0.8 FeSb drop out from the general tendency and may be explained by incorrect compositions assigned for HH.It should be noted that the lattice parameters for "TiFeSb" 24,25,28,44 are similar to those obtained in this work for a sample with this nominal composition, but the composition of the HH phase in this sample is TiFe 1.33 Sb.Similarly, the lattice parameter reported for Ti 0.8 FeSb 26 corresponds to our Fe-rich end of the solid solution at x = −0.17(Ti 0.83 Fe 1.5 Sb).The significant shift of the location of τ 1 to Fe rich compositions at first glance may raise doubts on the non-centrosymmetric crystal symmetry (F4 ˉ3m) for this compound: τ 1 might crystallize in the centrosymmetric space group (Fm3 ˉm), or even structures with both symmetries may occur within the     2).The values of the lattice parameter a for x = 0.125 and 0.375 were not used as they are systematically higher due to the effect of the doubled unit cell in one direction.It is obvious from Fig. 7 that filling the vacancies at the 4d site with additional Fe atoms is energetically favorable up to x ∼ 0.375, where the minimum of the concentration dependent heat of formation is observed.The minimum is quite sharp, thus excluding the existence of a significant homogeneity region towards the Fe-rich corner of the Ti-Fe-Sb system.This result is in good agreement with the EPMA and XPD data of the Heusler phase composition TiFe 1.33 Sb and earlier DFT modeling of TiFe 1+x Sb reported in ref.31, where the most stable composition was predicted to be TiFe 1.5 Sb for the centrosymmetric symmetry Fm3 ˉm.Moreover, the authors 31 predicted for TiFe 1.5 Sb the R3m symmetry with Fe/vacancy ordering and nonmagnetic semiconducting pro-perties.The lattice parameter a increases with x (Fig. 8) revealing almost linear concentration dependence.A visible change in the slope is observed at x ∼ 0.4, presumably corresponding to a change of bonding between the [Fe] and the [Ti, Sb] sublattices as a result of a symmetry change.
Modeling of the Ti 1+x Fe 1.25−x Sb solid solution is more complex due to several possible crystallographic configurations for the compositions: x = −0.5, −0.25, 0, 0.25, and 0.5 which are listed in Table 2 covering the experimentally established homogeneity region Ti 1+x Fe 1.33−x Sb (−0.17 ≤ x ≤ 0.25 at 800 °C).At the composition Ti 0.5 Fe 1.75 Sb, which corresponds to x ≈ −0.5, the most favorable arrangement of atoms is found for the following configuration: the 4a site is occupied by a mixture of Ti 2 Fe 2 atoms, whereas the 4c and the 4b sites are solely occupied by Fe and Sb atoms, respectively, and the 4d site is partially occupied (25% -1 atom) by additional Fe atoms.An attempt at substituting one Fe atom at the 4d site by Ti as well as three Ti atoms at the 4a site by Fe leads to a less negative heat of formation.A substitution of Fe by Ti at the 4c site and Ti by Fe at the 4a site is energetically unfavorable, yielding a positive heat of formation.Even the introduction of some vacancies at 4a with a three Fe atom configuration for this composition is similar to the previous case, but contains more Ti and less Fe atoms.For TiFe 1.25 Sb (x = 0) only one configuration was tested, corresponding to the fully filled 4a, 4b, and 4c sites with Ti, Sb, and Fe atoms, respectively and the 4d site filled by one additional Fe atom.Two possible configurations were tested for Ti 1.25 FeSb (x = 0.25): in one case additional Ti atoms completely substitute Fe at the 4d site, and in the other configurations they partially substitute Fe in 4c.For the first configuration, the heat of formation appeared to be almost two times more negative than that for the second case.For the composition Ti 1.5 Fe 0.75 Sb (x = 0.5), the most energetically favorable configuration consists of completely filled 4a and 4b sites with Ti and Sb, respectively, an Fe 3 Ti mixture at the 4c site and only one Ti atom in the 4d site.A configuration where the 4c site consists of an Fe 2 Ti 2 mixture, while the 4d site is partially occupied by one additional Fe atom, appeared to be much worse.Additionally, a Ti 1.33 FeSb composition consisting of 4 Ti atoms in 4a, 3 Sb atoms in 4b, and 3 Fe atoms in 4c was tested, but the heat of formation appeared to be less negative than for Ti 1.25 FeSb.The substitution mechanism was established by the selection of the best configuration for each composition.In the x < 0 range of the solid solution, additional Fe atoms substitute Ti atoms at 4a, whilst the 4d site is partially occupied by one Fe atom, and remains unchanged.In the range x > 0, Ti atoms at first completely substitute Fe atoms at the 4d site, and only after that, Fe atoms at the 4c site.The formation of the Ti 1+x Fe 1.25−x Sb solid solution is an exothermal process (Fig. 7) with a broad minimum at the most stable composition (x ≈ 0.15).Unfortunately, the lack of heat-of-formation data for the remaining phases of the Ti-Fe-Sb system does not allow the construction of tangent lines in order to determine the boundaries of the Ti 1+x Fe 1.25−x Sb homogeneity region.With increasing x, the lattice parameter a increases (Fig. 8), but the dependence is nonlinear, which is in a good agreement with the experimental data for the Ti 1+x Fe 1.33−x Sb solid solution (see Fig. 6).
To study the effect of substitution in the Ti 1+x Fe 1.25−x Sb solid solution, the charge distribution as derived by the electron localization function (ELF) was calculated (Fig. 9).In TiFe 1.25 Sb the distribution of ELF around the Ti atoms in 4a is almost spherical.In addition, ELF indicates distinctly different bonding between Fe1 and Sb, compared to that between Fe2 and Sb.Due to charge transfer from Ti to Fe, some localized ELF, shifted to Fe1, is observed between Fe1 and Ti.Such local-ization, however, is absent between Fe2 and Ti.In Ti 0.75 Fe 1.5 Sb, where Ti atoms are partly substituted by additional Fe atoms, the ELF localization between Fe at the 4a site and Fe1 is somewhat lower than that between Fe1 and Ti.The ELF distribution around Fe2 atoms remains the same as in TiFe 1. 25 Sb.With increasing Ti content up to Ti 1.25 FeSb, i.e., a partial occupation of the 4d site by Ti2 atoms, the ELF localization between Fe1 and Ti/Sb increases.The ELF distribution around Ti2 atoms looks almost the same as around Fe2 atoms in TiFe 1.25 Sb.Within the solubility region the bonds between Fe1 and Ti/Sb atoms remain more or less the same, and Ti/Fe a Three closest atoms at the 4c site to the selected atom at the 4d site.
or Fe/Ti substitutions do not have significant effects on the stability of the structure.Physical properties derived from the present DFT calculations are discussed in the following paragraph.The residual resistivity, as well as the distribution of the density of states of Ti 1+x Fe 1.33−x Sb, were modeled by using the KKR-CPA method for several concentrations: x = −0.2,0, 0.1, 0.2, and 0.3 (Table 3).The lowest residual resistivity (the lowest structural disorder) is observed in TiFe 1.33 Sb.It increases in Ti 0.8 Fe 1.53 Sb and Ti 1.1 Fe 1.23 Sb due to the statistical mixtures in both the 4a and 4d sites.However, with an increase of Ti, x > 0.1, the effect of disorder decreases at the 4d site generating a decreasing residual resistivity.This result is in a good agreement with the experimental data.The energy dependent density of states (Fig. 10a) of TiFe 1.33 Sb exhibits a band gap, with the Fermi level located in the valence band.The effective band gap is reduced due to the presence of a localized maximum inside the gap at ∼0.75 eV.This maximum contains an almost equal contribution of Fe1 and Fe2 electronic states and a somewhat lower contribution of Ti and Sb states.At the composition Ti 0.8 Fe 1.53 Sb the DOS distribution (Fig. 10b) reveals a filling of the band gap, mainly by Fe states from the 4c and 4d sites and the Fermi level is located at these states.At a higher Ti content, Ti 1.1 Fe 1.23 Sb (Fig. 10c), two localized maxima are formed by Fe and Ti states inside the band gap, with the Fermi level being located in the valence band.As a consequence, the Seebeck coefficient for x = 0 and x = 0.1 should have a positive sign while the resistivity data could exhibit a metal to insulator like transition with different mechanisms of conductivity.It should be noted that in all calculations no spin polarized DOS was observed, requiring a paramagnetic behavior for the samples studied.This is interesting, as for hypothetically ordered TiFeSb, Tobola et al. 45 obtained a DOS polarization and a total magnetic moment of 0.78µ B .The authors stated that TiFeSb is a metal, but in fact it is a semiconductor with  Table 3 The calculated (ab initio) residual resistivity ρ 0 (μΩ cm), and density of states at the Fermi level N(E F ) (states per eV f.u.) for selected compositions [Ti 1+x Fe

Magnetic susceptibility, Mössbauer spectroscopy and specific heat of TiFe 1.33 Sb
In order to check the magnetic ground state of TiFe 1.33 Sb and to corroborate our DFT results, specific heat (C p ), DC-magnetic susceptibility, magnetization and Mössbauer measurements have been performed from 1.5 to 300 K.The temperature and field dependent magnetization (Fig. 11) reveal a Curie-Weiss type paramagnetic behavior within the whole temperature range, which is superimposed by a small, almost temperature independent ferromagnetic component.An intrinsically paramagnetic state of TiFe 1.33 Sb is corroborated by Mössbauer spectra (see below); accordingly, we attribute the small ferromagnetic component to iron impurities with a molar fraction of less than 0.1% (corresponding to a saturated moment of about 0.003µ B per Fe-atom).Thus, the inverse magnetic sus-  short range or spin-fluctuation-type correlations.A magnetic phase transition is not observed within the measured temperature range.Fig. 13 shows the Mössbauer spectra obtained at 294 K and 4.2 K.Both spectra can be analysed by a superposition of two subspectra, one doublet and one singlet with an intensity ratio of 0.9 : 0.1.Values for the center shift are 0.279 mm s −1 (singlet) and 0.108 mm s −1 (doublet) for the 294 K spectrum, as well as 0.400 mm s −1 (singlet) and 0.220 mm s −1 (doublet) for 4.2 K, respectively.The quadrupole splitting changes from 0.211 mm s −1 to 0.220 mm s −1 , with temperature decreasing from 294 K to 4.2 K.The half-width of 0.18 (294 K) and 0.19 (4.2 K) is approximately 50% larger than for the respective α-Fe calibration spectrum, pointing to the small scatter of the hyperfine parameter between crystallographically identical sites.No sign of magnetic components is found in the temperature range investigated.According to the crystallographic model obtained from XRD-analyses, 4 Fe atoms occupy the 4c sites, and 1.33 Fe atoms are distributed over the 4d sites.This corresponds to an occupation number of 0.33 of Fe on the 4d sites.Assuming a purely random distribution of Fe atoms over the 4d sites, the probability to find no Fe atom in the 4d site around one 4c site is 4.1%.In that case, the electronic surrounding around a 4c site is highly symmetric, with no quadrupole splitting.Therefore, this surrounding should be represented by the singlet in the Mössbauer spectra.For 95.9% of the 4d sites, one or more Fe atoms are in the next 4d shell, thus disturbing the symmetric charge distribution leading to the doublet in the spectrum.The fact that the intensity of the singlet (∼10%) is larger than the expected 4.1% indicates that the distribution of the Fe atoms over the 4d sites is not perfectly random.Micrographs of as-cast samples from these sections indicate that (i) the formation of the HH-phase occurs incongruently, and that (ii) the composition of this compound strongly varies from centre to rim of the grains.Only TiCoSb (stoichiometric composition) was found to be single-phase in the as-cast state.Fig. 14 represents the EPMA results for all measurements Fig. 12 The temperature dependent heat capacity, C p (T )/T, of TiFe 1.33 Sb measured at zero field and at 1 T externally applied magnetic field.  of the HH phase in the as-cast samples TiFe 1.33(1−y) Co y Sb with nominal compositions y = 0.25, 0.5, 0.75 and 1.00.We can observe a gradual increase of Sb-and Ti-contents from 30 at% (at y = 0) to 33.3 at% (y = 1.00).After annealing the samples with y = 0.25 and 0.5 at 950 °C for 9 days, singlephase homogeneous samples were obtained; for a higher cobalt content (y = 0.75), however, annealing results in a still heterogeneous composition similar to that observed in the as-cast state.Almost complete equilibration (Fig. 15) was achieved by annealing the hot-pressed samples produced from fine powders after BM or HEBM (for details see Table 1).These data on the extension of the solid solution are inconsistent an immiscibility gap reported to exist in TiFe 1−y Co y Sb for 0.2 ≤ y ≤ 0.9. 46Unfortunately, these authors did not report the temperature at which this immiscibility gap should exist.The compositional dependence of the lattice parameters for TiFe 1.33(1−y) Co y Sb in Fig. 16 reveals a strong decrease with increasing Co content due to a decrease of the total number of atoms in the unit cell, from 13.33 for TiFe    2).Modeling of TiFe 1−y Co y Sb is quite straightforward as substitutions occur only at the 4c site, keeping the 4d site vacant.With increasing Co concentration, the absolute heat of formation increases almost linearly, while the concentration dependent lattice parameter a decreases and has nonlinear character.The difference in ΔH f between TiFeSb and TiCoSb is quite significant: >19.3 kJ mol −1 .
The presence of atoms at the 4d site makes TiFe 1.25−y Co y Sb more complicated, as several atomic configurations are possible within the same composition.For the composition TiFeCo 0.25 Sb, five configurations were tested (Table 2).The    2).
The KKR-CPA calculations evidence that the density of states at the Fermi energy in TiFe 1.33−y Co y Sb gradually decreases with increasing Co content as it shifts to the band gap (Table 3, Fig. 20), and is comparable to those obtained earlier for TiCoSb. 45,47This should lead to an increase of the resistivity and the Seebeck coefficient up to the compensation point of the semiconductor.Similarly it should lead to an increase of the thermoelectric power factor (PF), but at the same time the structure would become more ordered, which should increase the lattice thermal conductivity.The energy dependent DOS (Fig. 20) indicates that the localized maximum inside the band gap fades away with increasing Co content.It is interesting to note that according to Kaczmarska et al. 47 the Seebeck coefficient at 300 K for TiCoSb ranges from −28 to −40 µV K −1 , whereas Tobola et al. 8 reports +270/−250 µV K −1 .In fact, perfectly ordered TiCoSb should have the Fermi level at the middle of the gap giving zero Seebeck coefficient, as there are no sources of additional electrons or holes.An attempt at clarifying the nature of defects in TiCoSb 48 showed that vacancies in any of the occupied crystallographic sites cause a DOS polarization and appearance of a magnetic moment, which is in contradiction to the experimental findings.A partial substitution of Sb by Sn in TiCoSb 0.75 Sn 0.25 shifts the Fermi level deeper into the valence band (Fig. 21a).In comparison with TiCoSb, the density of states increases, giving a lower resistivity and a lower Seebeck coefficient.A simultaneous substitution of Fe by Co and Sb by Sn in the alloys TiFe 0.43 Co 0.68 Sb 0.88 Sn 0.12 and TiFe 0.17 Co 0.88 Sb 0.88 Sn 0.12 (Fig. 21b and c) leads to the formation of a localized maximum inside the band gap in the first case and its shift to the valence band in the second case.The effective band gap is narrowed, but the density of states decreases in the alloy with the higher Co content, giving rise to a higher resistivity and Seebeck coefficient.In general, the overcompensation of the n-type semiconductor like TiCoSb with Sn should not increase the thermoelectric performance of the system due to the presence of carriers of both signs.Taking the DOS profiles into account, it is clear that the desired Sn content should be less than 0.1 in order to shift the Fermi level to the inflection point, where the Seebeck coefficient should be maximum.In all cases studied, no DOS polarization was observed, causing a paramagnetic state for the selected HH compositions.The  ).The electrical resistivity of the Fe-rich sample (x = −0.133)shows semiconducting behavior, whereas for the two other samples (x = 0.108 and x = 0) a metallic behavior at low temperatures turns to a semiconducting one at about 550 and 650 K, respectively.All Seebeck temperature curves except the one for x = −0.133reveal a pronounced maximum, from which the energy gap (E g ) can be calculated: The results, summarized in Table 1, show that E g changes distinctly from 146 meV (for x = 0.108) to 62 meV (for x = −0.133).Positive S(T ) values indicate that the electronic transport is mostly due to holes as charge carriers.The total thermal conductivity increases with increasing Fe-content, whilst the phonon part shows an opposite behavior.The phonon thermal conductivity λ ph was taken from the difference λ ph = λ − λ e (assuming the validity of the Wiedemann-Franz law, λ e (T ) ∼ L(T )T/ρ(T ) with the Lorenz number L(T ) derived from the measured Seebeck coefficient values and the Fermiintegrals as proposed by D. M. Rowe et al. 50).Consequently the maximum value of ZT = 0.3 at 800 K appears for this series at x = 0. Considering the significant change of the transport properties in the series Ti 1+x Fe 1.33−x Sb, we suggest that the effect of Fe/Ti substitution on the electronic structure is very strong, resulting in a wide ZT range for this compound (0.1 ≤ ZT ≤ 0.3).
The effect of porosity on the mechanical properties of TiFe 1.33 Sb has been reported recently 51 and was investigated for two samples with two different relative densities, d R = 89.9 and d R = 98.0%(see Fig. 22).As expected, the porosity has little effect on the thermopower, but it decreases both the electrical and thermal conductivities.As the effect on the electrical conductivity is stronger than on the thermal conductivity, the figure of merit is lower for the sample with higher porosity.Stadnyk et al. 25 measured resistivity and thermopower for the composition TiFeSb in the temperature range from ∼75 to 375 K.The Seebeck coefficient at room temperature (∼30 μV K −1 ) is about 2-3 times lower than that obtained in this work for Ti 1+x Fe 1.33−x Sb (Fig. 22).On the other hand, the room temperature resistivity (∼6700 μΩ cm) 25 is about 4-8 times higher.In Fig. 23a and b temperature and composition dependent physical properties of TiFe 1.33(1−y) Co y Sb are displayed for y = 0.0, 0.25, 0.5 and 0.75, together with the data from the literature i.e.TiFe 0.5 Co 0.5 Sb and TiFe 0.3 Co 0.7 Sb 20 and TiCoSb. 19The temperature dependent electrical resistivity evidences a change from a metallic (y = 0) to a semiconducting behavior for the cobalt rich compositions (y ≥ 0.5).All electri- cal resistivities collected throughout this work are lower than the values reported by Wu et al.The thermopower of these samples increases with increasing Co content.In parallel to the increasing electrical resistivities, the conductivity type changes from the hole to the electron dominated transport in TiCoSb.The thermopower values of this work are in the same range as those reported in the literature.Concomitantly with the electrical resistivity and thermopower, the maximum of the power factor shifts to higher temperatures with increasing Co content.However, the maximum power factor of TiFe 0.33 Co 0.75 Sb, PF = 1.33 mW mK −2 , is lower than the value of 1.9 mW mK −2 reported for TiFe 0.3 Co 0.7 Sb. 20 The total thermal conductivity for all investigated samples is also lower than the values reported by Wu et al. 20 for the Fe/Co substituted compositions and for TiCoSb. 19The maximum ZT for the Fe/Co substituted compositions investigated is about 0.42 and is higher than that for the end-members TiFe 1.33 Sb (ZT 800 K = 0.3) and TiCoSb (ZT 800 K = 0.05 (ref.19)).Only one significant discrepancy was observed between the ZT values of this investigation and the data reported in the literature for y = 0.5: 20 there, TiFe 0.5 Co 0.5 Sb has a ZT 800 K = 0.2, but   It is known that a significant improvement of the TE performance can be achieved via nano-structuring.Our investigation of p-and n-type skutterudites [52][53][54] shows that the figure of merit can be increased by almost 60% via HEBM of the samples.The effect of BM conditions (BM, HEBM and HEBM ann.) on the TE properties for the three samples with the same chemical composition TiFe 0.29 Co 0.78 Sb is summarized in Fig. 24a and b.All properties change to the better for the HEBM samples in comparison with BM alloys.The most prominent increase (about ∼35%) was observed for the Seebeck coefficient and consequently for the power factor (∼54%).It is worth mentioning that the thermal conductivity of HEBM samples is also increased by ∼50%, which results in a ZT-value close to that obtained for BM samples.Further annealing of HEBM samples at 950 °C for 9 days slightly increases the electrical resistivity but all other TE properties (thermopower, power factor and thermal conductivity) are decreased, resulting in a figure of merit, ZT = 0.44, which is an increase of 12% in comparison with the BM sample.We have to note that the effect of HEBM on TE properties for this HH composition is weaker and different from that observed for skutterudites; 54 therefore, further optimization of the relevant parameters is needed.
As the effect of Sb/Sn substitution is known to be an efficient method for the optimization of the TE performance of HH p-and n-compositions, 21 it was applied to the samples from the sections TiFe 0.665 Co 0.5 Sb-TiCoSb 0.75 Sn 0.25 and TiFe 0.33 Co 0.75 Sb-TiCoSb 0.75 Sn 0.25 as documented in Fig. 25a and b.The temperature dependent electrical resistivity shows that generally all samples are semiconductors; however, annealing the sample TiFe 0.33 Co 0.75 Sb 0.875 Sn 0.125 at 950 °C for 9 days turns it to a metallic behavior at temperatures below 550 K.The compositional dependences of resistivity, Seebeck coefficient and thermal conductivity show rather different tendencies in these two series (Fig. 25b).
Finally, the compositional dependences of ZT for both series are alike, i.e. giving the highest ZT = 0.4 at 825 K for the samples without Sn.Upon annealing, the HEBM sample TiFe 0.33 Co 0.75 Sb 0.875 Sn 0.125 (open triangles in Fig. 25) shows a decrease of thermal conductivity similar to that shown in the previous series (see Fig. 24), resulting in an increase of ZT from 0.3 to 0.4 at 825 K.

Conclusion/summary
In order to investigate the phase relations in the ternary system Ti-Fe-Sb, SEM, EPMA and XPD were applied.The homogeneity region of the Heusler phase is significantly moved from stoichiometric TiFeSb to an Fe-rich composition with an extensive homogeneity region at 800 °C: Ti 1+x Fe 1.33−x Sb (−0.17  the experimental data but did not predict significant gains in thermoelectric performance.
From dc magnetic susceptibility and 57 Fe Mössbauer measurements, a paramagnetic behavior is observed for TiFe 1.33 Sb down to 4.2 K, with a Curie-Weiss temperature of −8 K and an effective moment of 1.11µ B per Fe.
Ternary Ti 1+x Fe 1.33−x Sb exhibits a semi-conducting behavior for the Fe-rich composition (x = −0.133),but the conductivity type changes to metallic with increasing Ti content.The figure of merit (ZT 800 K ) in this solid solution changes from 0.1 to 0.3 at 800 K, and adopts the highest value for the composition TiFe 1.375 Sb to be the most stable composition and established the mechanisms behind Fe/Ti substitution for the region Ti 1+x Fe 1.33−x Sb, and of the Fe/Co substitution within the isopleth TiFe 1.33 Sb-TiCoSb.The calculated residual resistivity of Ti 1+x Fe 1.33−x Sb, as well as of the isopleths TiFe 1.33 Sb-TiCoSb, TiFe 0.665 Co 0.5 Sb-TiCoSb 0.75 Sn 0.25 and TiFe 0.33 Co 0.75 Sb-TiCoSb 0.75 Sn 0.25 , are in a good correlation with the experimental data.From magnetic measurements and 57 Fe Mössbauer spectrometry, a paramagnetic behavior down to 4.2 K was observed for TiFe 1.33 Sb, with a paramagnetic Curie-Weiss temperature of −8 K and an effective moment of 1.11µ B per Fe.Thermoelectric (TE) properties were obtained for the four isopleths Ti 1+x Fe 1.33−x Sb, TiFe 1.33 Sb-TiCoSb, TiFe 0.665 Co 0.5 Sb-TiCoSb 0.75 Sn 0.25 and TiFe 0.29 Co 0.78 Sb-TiCoSb 0.75 Sn 0.25 by measurements of electrical resistivity (ρ), Seebeck coefficient (S) and thermal conductivity (λ) at temperatures from 300 K to 823 K allowing the calculation of the dimensionless figure of merit
where a, b, c, d, and e are the numbers of each type of atom in the crystal lattice of the compound used in calculations; j, k, l, m, and n are the number of atoms in the crystal lattice of Ti, Fe, Co, Sb, and Sn, respectively, used in calculations; and E tot is the total energy of the compound in eV.The distribution of the electron localization function (ϒ) was calculated on a 60 × 60 × 60 point grid and plotted by the VESTA software

Fig. 2
Fig. 2 An isothermal section at 800 °C of the phase diagram Ti-Fe-Sb after ref. 57 based on the original work of Melnyk et al.26The colored filled circles refer to sample compositions (a, b, c) in this investigation.

Fig. 4 A
Fig. 4 A partial isothermal section of the Ti-Fe-Sb system at 800 °C (this work); the vertices of the three-phase equilibria after EPMA; [a] the dashed line indicates the homogeneity region of the HH phase after ref. 26.

Fig. 7
Fig. 7 Heat of formation as a function of composition along the TiFe 1+x Sb and Ti 1+x Fe 1.25−x Sb solid solution.

Fig. 8
Fig. 8 Lattice parameter versus Fe-concentration within the TiFe 1+x Sb and Ti 1+x Fe 1.25−x Sb solid solution.
ceptibility 1/χ(T ) = H/M(T ) (5 T, inset of Fig.11) is analyzed in terms of a modified Curie-Weiss law, χ(T ) = χ 0 + C/(T − Θ p ), where χ 0 relates to a temperature independent component, Θ p is the paramagnetic Curie-Weiss temperature, and C is the Curie constant, which is used to calculate the effective paramagnetic moment, μ eff .The corresponding fit to the experimental data is displayed by a solid line.The temperature independent component χ 0 = 4.3 × 10 −3 emu per mol-Fe is essentially an artifact of the above mentioned saturated moment of iron impurities.The intrinsic susceptibility of TiFe 1.33 Sb is Curie-Weiss paramagnetic with Θ p = −8 K and C = 0.154 emu K per mol-Fe.The latter reveals an effective paramagnetic moment of 1.11µ B per Fe-atom.Fig.12shows the temperature dependent heat capacity C p plotted as C p /T versus T of TiFe 1.33 Sb.A distinct low temperature-increase is obtained indicating an evolution of magnetic

Fig. 10
Fig. 10 The distribution of the total and partial densities of states in TiFe 1.33 Sb (a), Ti 0.8 Fe 1.53 Sb (b) and Ti 1.1 Fe 1.23 Sb (c).The Fermi level is at E = 0 eV.

Fig. 15
Fig. 15 The XRD pattern of the Heusler type.(a) TiFe 1.33(1−y) Co y Sb for various y.For y = 0.75 BM a secondary phase appeared with the composition Ti 28 Fe 30 Sb 42 ; (b) the Rietveld refinement of TiFe 1.33 Sb resulted in R F = 0.026; R I = 0.033.

Fig. 18
Fig. 18 The composition dependence of the heat of formation for the solid solutions TiFe 1−y Co y Sb and TiFe 1.25−y Co y Sb.

Fig. 19
Fig. 19 The composition dependence of the lattice parameter a for the solid solutions TiFe 1−y Co y Sb and TiFe 1.25−y Co y Sb.

Fig. 20
Fig. 20 The distribution of the total and partial densities of states in TiFeCo 0.25 Sb (a), TiFe 0.66 Co 0.5 Sb (b) and TiCoSb (c).The Fermi level is at E = 0 eV.
TiFe 0.665 Co 0.5 Sb of this work has a ZT 800 K = 0.39, showing a difference of almost by a factor of two.Most likely, this discrepancy is caused by the fact that the composition TiFe 0.5 Co 0.5 Sb investigated by Wu20 lies outside of the single-phase region; indeed, we found that TiFe 0.5 Co 0.5 Sb after annealing at 950 °C for 8 days contains about 20% of a secondary phase (τ 2 ).
≤ x ≤ 0.25).A DFT study revealed TiFe 1.375 Sb to be the most stable composition.In Ti 1+x Fe 1.33−x Sb for x < 0, the Ti atoms are substituted by Fe, and for x > 0 the Fe atoms (4d site) by Ti.In the isopleth TiFe 1.33 Sb-TiCoSb, Fe (4c site) is substituted by Co with decreasing Fe content at the 4d site.The DFT calculations of residual resistivity for the isopleths Ti 1+x Fe 1.33−x Sb, TiFe 1.33 Sb-TiCoSb, TiFe 0.665 Co 0.5 Sb-TiCoSb 0.75 Sn 0.25 and TiFe 0.33 Co 0.75 Sb-TiCoSb 0.75 Sn 0.25 are in good agreement with
1.33 Sb.Further optimization of the TE performance was achieved via Fe/Co substitution resulting in a maximum value of ZT = 0.42 (at 825 K) for TiFe 0.665 Co 0.5 Sb.It was shown that dual Fe/Co and Sb/Sn substitutions do not drive an additional increase of ZT, whereas the optimization of the preparation route (by means of ball-milling techniques and heat treatments) increases the ZT-values by about 12%.This journal is © The Royal Society of Chemistry 2018 Dalton Trans., 2018, 47, 879-897 | 895 Open Access Article.Published on 30 November 2017.Downloaded on 18/01/2018 09:56:50.This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.

Table 2
Calculated heat of formation ΔH f (meV per atom), lattice constant a (nm), and bulk modulus B (GPa) for the selected compositions with the half-Heusler structure including the additional 4d crystallographic site 1.33−x ] 1−y Ti y Co y Sb 1−z Sn z