Sanchayita Mondal
email: sanchayita.mondal87@gmail.com

Krishanu Ghosh

R Ranganathan

Eric Alleno

Chandan Mazumdar
email: chandan.mazumdar@saha.ac.in

Thermoelectric properties of Ru 2 TiGe Heusler phase

We report a study of structural, magnetic, heat capacity and thermoelectric properties of a Ru-based Heusler alloy, Ru 2 TiGe. The magnetic measurements reveal that at higher temperatures, diamagnetic and Pauli paramagnetic contributions dominate the magnetic behaviour whereas, at lower temperatures (T ≤ 20 K), superparamagnetic interaction among clusters is observed. The developement of such magnetism, however, does not exhibit any noticeable influence in thermoelectric properties. Though the temperature dependence of resistivity exhibits a metal-like nature, the large value of Seebeck coefficient leads to an appreciable power factor of the order of 0.54 mW/mK 2 at 300 K. Large power factor as well as low thermal conductivity results in a value of zT = 0.025 at 390 K for Ru 2 TiGe that is orders of magnitude higher than that of the other pure Heusler alloys and point towards its high potential for practical thermoelectric applications.

Introduction

Power generation and refrigeration using thermoelectric devices are attracting more and more attention from researchers on a global scale due to their environment-friendly technology as well as long-term maintenance-free operation. The efficiency of a thermoelectric device in recovering waste heat depends on the figure of merit, zT (zT = S 2 T ρκ , where S is the thermopower or Seebeck coefficient, ρ and κ are the electrical resistivity and thermal conductivity, respectively), of the thermoelectric material used in the device. However, most of the materials commercially available for use in thermoelectric devices, are known to be rather toxic and expensive as well as not efficient enough, which limit their practical applications [START_REF] Goldsmid | Thermoelectric refrigeration[END_REF][START_REF] Dughaish | Lead telluride as a thermoelectric material for thermoelectric power generation[END_REF]. Therefore, to optimize the performance of a thermoelectric device, it is crucial to identify new thermoelectric materials with large zT. Recently, compounds like skutterudites [START_REF] Nolas | The effect of rare-earth filling on the lattice thermal conductivity of skutterudites[END_REF], clathrates [START_REF] Kleinke | New bulk materials for thermoelectric power generation: clathrates and complex antimonides[END_REF], and Heusler alloys families [START_REF] Huang | Enhancement of thermoelectric properties at room temperature through isoelectronic ruthenium-substitution in n-type halfheusler vfesb compounds[END_REF][START_REF] Culp | Effect of substitutions on the thermoelectric figure of merit of half-Heusler phases at 800 C[END_REF][START_REF] Nishino | Thermoelectric properties of the pseudogap Fe 2 VAl system[END_REF], etc. have been investigated for their potential as new thermoelectric materials.

Among the different families of compounds, Heusler alloys have aroused great interest for the possibility of doping/substituting each of its constituting elements in order to optimize their physical properties [START_REF] Graf | Simple rules for the understanding of heusler compounds[END_REF].

Heusler alloys, X 2 YZ (X /Y are primarily transition metals, rare earth metals or actinides, Z = a p-block element), crystallize in the cubic structure L2 1 having the space group: Fm 3m. The L2 1 structure consists of four interpenetrating facecentered cubic (fcc) sublattices where the X -atoms occupy the positions ( 1 ) and (0 0 0), respectively [START_REF] Graf | Simple rules for the understanding of heusler compounds[END_REF][START_REF] Heusler | Kristallstruktur und ferromagnetismus der mangan-aluminium-kupferlegierungen[END_REF]. Interestingly, many different properties of Heusler alloys can be uniquely controlled by their valence electron counts (VEC). Heusler compounds having VEC 24 have shown promise for thermoelectric applications due to their semiconducting or semi-metallic ground state. Starting with Fe 2 VAl [START_REF] Nishino | Semiconductorlike behavior of electrical resistivity in Heusler-type Fe 2 VAl compound[END_REF], many new Heusler alloys (Fe 2 VGa, Fe 2 TiSn etc. [START_REF] Ślebarski | Electronic structure and thermodynamic properties of Fe 2 VGa[END_REF][START_REF] Ślebarski | Electronic structure and thermodynamic properties of the Heusler alloys Fe 2 Ti 1-x VxSn[END_REF]) having VEC 24 have been discovered and studied so far in search of large zT. The main challenge one usually faces while identifying a good thermoelectric material is to decrease the thermal conductivity and improve the power factor (PF = S 2 /ρ) at the same time. It was found that in case of Fe 2 VAl, the lattice thermal conductivity could be significantly reduced by doping/substitution with a heavier element [START_REF] Mikami | Thermoelectric properties of tungstensubstituted Heusler Fe 2 VAl alloy[END_REF]. By extending this idea, one would therefore be tempted to use heavier elements to identify and synthesize new Heusler materials, maintaining the number of valence electrons at 24. As Ru is isoelectronic to Fe and has a larger atomic mass, the existence of some Ru-based Heusler alloys having a VEC 24 have been reported recently. Among those alloys, a semi-metallic ground state has been observed in Ru 2 NbAl [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF], Ru 2 NbGa [START_REF] Kuo | Ru 2 NbGa: A Heusler-type compound with semimetallic characteristics[END_REF] and Ru 2 TaAl [START_REF] Tseng | Semimetallic behavior in Heusler-type Ru 2 TaAl and thermoelectric performance improved by off-stoichiometry[END_REF] while Ru 2 VZ (Z = Al, Ga) [START_REF] Mondal | Structural and transport properties of two new Heusler type Ru 2 VAl and Ru 2 VGa compounds[END_REF][START_REF] Mondal | Ru 2 VAl and Ru 2 VGa: Two new heusler-type compounds[END_REF][START_REF] Ramachandran | Thermoelectric properties of Heusler-type Ru 2 VAl 1-x Gax alloys[END_REF] shows metallic behaviour. Although having semi-metallic ground state, only Ru 2 NbAl among the pristine compounds showed an appreciable value of zT at 300 K (zT 300K = 0.0052) [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF]. Even though most Ru-based compounds exhibit relatively low thermal conductivity compared to that of Fe-based Heusler alloys, the difficulties in achieving high zT lie in their low values of thermopower. Among these Ru-based alloys, a recent theoretical study predicted that Ru 2 TiGe would be a potentially good thermoelectric material [START_REF] Krishnaveni | Band gap engineering in ruthenium-based Heusler alloys for thermoelectric applications[END_REF] because it presents the possibility of having a high thermopower as well as a semi-metallic ground state, which are necessary for practical use. In comparison to Fe 2 VAl, Ru 2 TiGe contains two heavier elements, viz., Ru and Ge, while the atomic weights of V and Ti are almost comparable.

In addition to their potential as good thermoelectric material, Heusler alloys are also known to exhibit many intriguing magnetic properties.

Interestingly, the total magnetic moment (M ) per unit cell of a Heusler material can be estimated using the Slater-Pauling rule: M (µ B ) = |VEC-24| [START_REF] Galanakis | Slater-Pauling behavior and origin of the half-metallicity of the full-Heusler alloys[END_REF]. Thus, Heusler alloys with VEC 24 should be non-magnetic with zero total magnetic moment per unit cell. Several materials with VEC 24 are indeed found to be non-magnetic [START_REF] Nishino | Semiconductorlike behavior of electrical resistivity in Heusler-type Fe 2 VAl compound[END_REF][START_REF] Lue | Superparamagnetism and magnetic defects in Fe 2 VAl and Fe 2 VGa[END_REF][START_REF] Lue | NMR and Mössbauer study of spin dynamics and electronic structure of Fe 2+x V 1-x Al and Fe 2 VGa[END_REF]. Nevertheless, close inspections of experimental measurements of the magnetic properties of these alloys reveal that many of the reported compounds could even be classified as marginally magnetic instead, and in some cases the reported deviation in magnetic behavior have also been explained differently by different investigators. For example, while one study reported cluster glass behavior for Fe 2 VAl [START_REF] Vasundhara | Evidence for cluster glass behavior in Fe 2 VAl Heusler alloys[END_REF], other studies revealed the presence of superparamagnetism (SPM) or even ferromagnetism [START_REF] Feng | Physical properties of heusler-like Fe 2 VAl[END_REF][START_REF] Gao | Significant off-stoichiometry effect leading to the N-type conduction and ferromagnetic properties in titanium doped Fe 2 VAl thin films[END_REF]. SPM is also observed in other Heusler alloys based on Fe and Ru having VEC 24, namely Fe 2 VGa, Ru 2 NbAl, etc. [START_REF] Ślebarski | Electronic structure and thermodynamic properties of Fe 2 VGa[END_REF][START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF]. Some other compounds of this class, e.g., Fe 2 TiSn, Ru 2 NbAl etc. [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF][START_REF] Ślebarski | Weak ferromagnetism induced by atomic disorder in Fe 2 TiSn[END_REF] also exhibit ferromagnetic interactions. It is generally accepted that most of these various magnetic behaviors are due to the antisite type structural defects that usually develop during the synthesis and annealing process. Thus, a thorough study of the magnetic properties of Ru 2 TiGe could also help us to understand the microstructural aspects which in turn can influence the thermoelectric properties.

In the present work, we therefore report a detailed study on Ru 2 TiGe through structural, magnetic, heat capacity as well as thermoelectric measurements.

It should be noted here that although none of the constituting elements usually considered to be magnetic, our study reveals the presence of superparamagnetically interacting aggregates in Ru 2 TiGe, most likely originating from antiste defects detected in the material. Thermoelectric measurements suggest that despite a metal-like ground state, a relatively large thermopower can be observed in this compound, leading to a zT value of about 0.025 at 390 K, the highest among undoped Heusler alloys yet reported.

Experimental methods

A polycrystalline Ru 2 TiGe ingot was synthesized by arc melting technique under flowing Ar atmosphere.

Stoichiometric amounts of the constituent elements Ru (>99.9%), Ti (>99.99%) and Ge (>99.9999%) were melted several times to achieve homogeneity. The weight loss in this process was observed to be less than 0.5%. The as-cast ingot was then wrapped in Ta-foil and annealed at 1273 K for 48 hours in a vacuumsealed quartz tube followed by quenching in icewater. The sample was annealed again at 1223 K for 12 hours after cleaning its surface and following the same quenching procedure. The ingot was then cut in appropriate shapes and polished and again annealed for 2 hours at 1173 K following the similar procedure to eliminate the surface strain that might have been generated due to the mechanical stress in the process of scraping, cutting and polishing, as some Heusler alloys are observed to be highly sensible to such coldwork [START_REF] Maier | Order-disorder transitions in the Fe 2 VAl Heusler alloy[END_REF][START_REF] Schaf | Extended and local effects of cold work in Heusler alloys[END_REF]

Results and Discussion

Structural details

The XRD pattern of Ru 2 TiGe taken at 300 K is shown in Fig. 1. All the diffraction lines except a peak with negligible intensity (∼ 1% of most intense peak) at 2θ ∼ 37 • , can be indexed considering the L2 1 crystal structure (Fig. 1). The lattice parameter estimated to be 6.034(1) Å, matches well with the earlier reported value of 6.035 Å [START_REF] Yin | Standard enthalpies of formation of selected Ru 2 YZ Heusler compounds[END_REF]. The extra peak at 2θ ∼ 37 • , appeared due to the presence of a negligible amount of unreacted Ru, and have earlier been reported in many other Ru-based Heusler compounds [START_REF] Kuo | Ru 2 NbGa: A Heusler-type compound with semimetallic characteristics[END_REF][START_REF] Yin | Standard enthalpies of formation of selected Ru 2 YZ Heusler compounds[END_REF]. Another secondary phase was detected by EPMA (white areas in the inset of Fig. 1), where localized analysis found Ru 1.06 Ge 0.94 as an apparent composition, most likely corresponding to the RuGe-Ru eutectic reported in the binary Ru-Ge phase diagram [32]. However, the Bragg peaks associated with the RuGe phase can not be seen in the XRD pattern of Ru 2 TiGe. One of the possible reason for the visual absence of the Bragg peaks of RuGe is that the compound forms in cubic B2-type crystal structure, whose lattice parameter (a = 3.011 Å) is nearly half of that of Ru 2 TiGe (a = 6.034 Å that can be expressed as 2×3.017 Å). As a result, the Bragg peaks from both the compound could overlap with each other, and can not be resolved separately. The presence of such small fraction of Ru as a secondary phase in these materials has been argued to have no influence in their transport and magnetic properties at all [START_REF] Kuo | Ru 2 NbGa: A Heusler-type compound with semimetallic characteristics[END_REF][START_REF] Yin | Standard enthalpies of formation of selected Ru 2 YZ Heusler compounds[END_REF]. Similarly, we argue here that a minute fraction of non-magnetic and metallic RuGe [START_REF] Sakai | New phases of binary compounds: CsCl-type RuGe and RuSn[END_REF] does not affect the physical properties of Ru 2 TiGe either. It may also be noted here that the signature of structural antisite disorder is frequently observed in the Heusler compounds in perturbation to the ordered L2 1 phase. When Y and Z atoms exchange their respective positions and distribute evenly, the B2 type disordered state is resulted. The occurrence of B2 phase is generally reflected in the XRD pattern by diminished intensity of (111) diffraction line. In case of Ru 2 TiGe forming in the ordered L2 1 crystal structure, the intensity of (111) peak is calculated to be 2 % of the most intense line. In the present diffraction pattern, the measured intensity of the said diffraction line is ∼1.9 % suggesting the sample is primarily forming in the ordered L2 1 phase. The elemental composition of the matrix, derived from the EPMA measurements is Ru 1.91(1) Ti 1.05(3) Ge 1.03 [START_REF] Nolas | The effect of rare-earth filling on the lattice thermal conductivity of skutterudites[END_REF] . This deviation from the stoichiometry is in agreement with the occurrence of Ru and RuGe secondary phases and also indicates the presence of Ti Ru and Ge Ru antisite defects where some of the Ti (Y ) and Ge (Z ) atoms occupies Ru (X ) sites.

The XRD patterns collected at different temperatures between 300 K to 12 K do not show any detectable change in primary diffraction pattern-type indicating that the crystal structure remains unaltered down to 12 K, the lowest temperature achievable in our diffractometer (Fig. 2). The lattice parameter gradually decreases with decreasing temperature similar to that observed in most of the materials. The unit-cell volume of Ru 2 TiGe is plotted as a function of temperature and fitted (Fig. 2, inset) using the following equation [START_REF] Sayetat | An easy method for the determination of Debye temperature from thermal expansion analyses[END_REF]

V (T ) = γ G U (T )/K 0 + V 0 , (1)
where V 0 represents the unit-cell volume at T = 0 K, K 0 is the bulk modulus, and γ G is the Grüneisen parameter. U(T), the internal elastic energy can be described by considering the Debye approximation as

U (T ) = 9N K B T T Θ D 3 Θ D T 0 x 3 e x -1 dx (2)
where N is the number of atoms per unit cell and x = hω/k B T. From the fit of V(T) curve of Ru 2 TiGe, the Debye temperature Θ D and the Grüneisen parameter γ G are estimated to be 400 K and 1.6, respectively. The theoretically calculated value of K 0 =228 GPa [START_REF] Iftikhar | Computational Study of Ru 2 TiZ (Z= Si, Ge, Sn) for Structural, Mechanical and Vibrational Properties[END_REF] is used for the estimation of γ G .

Magnetic properties 3.2.1. Magnetic susceptibility

The temperature dependence of magnetic susceptibility [χ(T )] behaviour of Ru 2 TiGe has been investigated applying external magnetic field (H ) of strengths 10 kOe and 70 kOe. The susceptibility values measured under zero field cooled (ZFC) and field cooled (FC) protocols at 10 kOe do not show any thermoremanence behaviour. The values of χ(T ) at 10 kOe found to be very small though remain positive throughout the whole temperature region (Fig. 3, top). In contrast, the susceptibility values measured at 70 kOe become negative (Fig. 3, bottom) in the intermediate temperature range (100-200 K) suggesting the presence of diamagnetism in Ru 2 TiGe. However, a minor upturn could be seen in the χ(T ) curves for both the fields at higher temperatures, indicating Pauli paramagnetic (PPM) behaviour.

Experimentally observed χ(T ) curves can be well explained considering a modified Curie-Weiss equation [START_REF] Blanchard | Anomalous thermal expansion in orthorhombic perovskite SrIrO 3 : Interplay between spin-orbit coupling and the crystal lattice[END_REF], expressed as where the first term is the well known Curie-Weiss expression, χ 0 is temperature independent diamagnetic (χ dia ) and/or Pauli paramagnetic (χ P P M ) contributions and αT 2 is the higher order term in the expanded Pauli paramagnetic equation which is generally neglected in the zeroth order approximation [START_REF] Blanchard | Anomalous thermal expansion in orthorhombic perovskite SrIrO 3 : Interplay between spin-orbit coupling and the crystal lattice[END_REF].

χ(T ) = C T -θ p + χ 0 + αT 2 , (3)
Using Eq. 3, the ZFC susceptibilities for both 10 kOe and 70 kOe applied field have been fitted and the fit parameters are shown in Table 2. The values of χ 0 for both the fields are found to be negative suggesting the predominant magnetic contribution is diamagnetic in Ru 2 TiGe. It may be noted that the temperature independent contribution, χ 0 , was reported to be positive in the earlier preliminary magnetic susceptibility analysis for the same compound measured under 10 kOe field [START_REF] Mizusaki | Metal-insulator-metal transition in Ti substituted antiferromagentic Ru 2 MnGe Heusler alloy[END_REF]. However, it may be pointed out that the analysis method appears to be quite different, as the earlier report might not have considered the higher order αT 2 term. As the earlier report did not present any detailed analysis, except mentioning it to be paramagnetic, it is not possible to compare the magnetic properties presented in this work with that reported earlier [START_REF] Mizusaki | Metal-insulator-metal transition in Ti substituted antiferromagentic Ru 2 MnGe Heusler alloy[END_REF]. The finer details could be different due to their synthesis condition and resulting differences in inherent

H χ 0 µ ef f θ p α (kOe) (emu/mol-Oe) (µ B ) (K) (emu/mol-Oe-T 2 ) 10
-0.75×10 -6 0.058 -1.32 1.93×10 -10 70 -6.32×10 -6 0.069 -11.85 0.78×10 -10 microstructures. In the present study, the magnetic susceptibility measured at 70 kOe, however resulted in a higher absolute value of χ 0 , revealing that effect of Pauli paramagnetic contribution has been suppressed on applying higher field. Our estimated values of χ 0 are the resultant of these two components, viz., positive Pauli paramagnetic and negative diamagnetic (χ 0 = χ P P M -χ dia ).

Apart from these two contributions, very small values of effective paramagnetic moment (µ ef f ) and paramagnetic Curie temperature (θ p ) point towards the presence of another additional short range magnetic component, effect of which is evident in the relatively sharp upturn in χ-T curves at low temperatures below ∼ 25 K (Fig. 3). Such additional magnetic contribution may have been originated from the discernible deviation of stoichiometry that leads to small percentage of Ti Ru and Ge Ru antisite defects in this compound. It should be pointed out here that appearance of magnetic contribution from these defects is although rare, yet similar short range magnetization have earlier been reported in quite a few compounds e.g., Ru 2 NbAl [14], SrRuO 3 [START_REF] Cao | Thermal, magnetic, and transport properties of singlecrystal Sr 1-x Cax RuO 3 (0≤ x≤1. 0)[END_REF] etc., where none of the constituent elements are generally considered to be magnetic in nature. Effect of those defects and disorders in Ru 2 TiGe are further explored through isothermal magnetization measurements described below.

Isothermal magnetization

In order to further understand the magnetic behaviour of Ru 2 TiGe, the isothermal magnetisation data have been collected at different temperatures as a function of externally applied magnetic field. The magnetic isothermal [M(H)] curve, taken at 300 K [ Fig 3,inset(a)], increases initially upon increasing magnetic field up to 10 kOe after which the slope of the curve changes and becomes negative throughout the remaining magnetic field range scanned (10 < H ≤ 70 kOe). Such magnetic behaviour could be a result of simultaneous presence of two different type of magnetic contributions; one having positive magnetization and other with a negative component. Generally, negative magnetization has its origin in the diamagnetic contribution which varies linearly with increasing applied magnetic field, yielding a negative slope. To explain the nonlinear variation of the isothermal magnetization, the positive component of magnetisation must be non-linear. If different magnetic interactions contribute towards this positive magnetisation, at least one such contribution should be nonlinear, in order to maintain the non-linearity of overall magnetisation. The positive contribution in this case increases rapidly with increasing applied field up to 10 kOe and then must have approached a saturating tendency suggesting that the positive contribution either could be of ferromagnetic type or a combination of paramagnetic and ferromagnetic interactions. In contrast, the isothermal magnetization at 2 K increases slowly with applied field and shows a approach-to-saturation-like behaviour [ Fig 3,inset(b)]. In both the isotherms, taken at 300 K and 2 K, no hysteresis behaviour were observed. Since, signature of any ferromagnetic long range order could not be detected in the susceptibility behaviour of Ru 2 TiGe, the presence of a subtle ferromagnetic contribution in the magnetic isotherm taken only at 300 K could possibly be generated from an impurity phase, present beyond the detection limit of both the XRD as well as EPMA analysis performed on this material. Elemental Fe could be a possible candidate for the impurity phase as it also saturates near 10 kOe though its saturation magnetisation, 2.2 µ B , is much higher than the observed value (∼ 3×10 -5 µ B at 10 kOe) at 300 K in Ru 2 TiGe. Considering the whole positive contribution to the M(H) curve at 300 K is the manifestation of Fe impurity in Ru 2 TiGe, the amount of Fe present in the system is estimated to be 0.0015 % or 15 ppm (parts per million). To avoid such percentage of impurity, one need to prepare sample with all the starting elements having purity >99.9985%. Although for the synthesis of Ru 2 TiGe, we have used Ti and Ge having purity >99.9985%, the purity of Ru was limited to 99.9%. Thus, 99.9% pure Ru available to us might easily be the source of the minuscule Fe impurity discussed above. Such a small impurity would also remain beyond the detection limit of measurements like XRD, EPMA, etc. employed in this study. However, it may be noted here that aforesaid impurity has a visible influence mostly at higher temperatures, where the positive paramagnetic contribution of Ru 2 TiGe is quite low and the diamagnetic contribution dominates. In addition, this diamagnetic contribution also has a very weak dependence (negative dM/dH ) of magnetic field. On the other hand, the magnetization of the FM impurity has a rather large positive dM/dH in the low field region, but the magnetization saturates as the magnetic field increases. As a result, the effect of FM impurity, though of very tiny fraction, has an overwhelming influence in the low field region. Once the FM contribution to isothermal magnetization saturates at higher field, the negative dM/dH of the diamagnetic contribution starts to manifest its domination. In contrast, at 2 K, the inherent magnetization value is quite large, increases as the magnetic field is raised and reaches toward a magnetization value ∼ 4×10 -4 µ B at 70 kOe, which is one order of magnitude higher than that of the FM impurity. Such larger magnetization values suppress the manifestation of magnetic impurity effect at the low temperature isothermal magnetization result. Similar presence of elemental magnetic impurity is also reported earlier in literature [START_REF] Pandey | Magnetic behavior of binary intermetallic compound YPd 3[END_REF]. Such type of impurity, however, have very little effects in their transport properties. It may be pointed out here that the effect of such FM impurity does not have any significant influence on the χ(T ) data taken at 10 kOe and 70 kOe in the temperature range 2-300 K, presented in the previous section. The measured temperature range is well below the FM ordering temperature of Fe (1043 K) and both the applied magnetic fields (10 kOe and 70 kOe) falls within the magnetic saturation region of the isothermal magnetization characteristics of the Fe impurity. Thus the minute presence of the said Fe impurity can only contribute a very small positive temperature independent term (∼10 -6 emu/mol-Oe) to the χ(T ) curves. The effect of removal of such constant term from each χ(T ) curves only results in a bit higher χ 0 values than that given in Table 2.

To check the slope of the M(H) curve at intermediate temperatures, the data have also been taken at 20 K, 15 K, 10 K, and 5 K (Fig 4). It is observed that for all the temperatures the magnetic isotherms are nonlinear and show positive slopes. Isothermal magnetization data at various temperature for Ru 2 TiGe. The solid lines represent the fit of the data using Eq. 4.

The nonlinearity increases as the temperature decreases. Since the susceptibility behaviour negate the possibility of the presence of any long range ordering in this compound, the deviation from the linear nature of the magnetic isotherms (expected in purely paramagnetic system) indicate towards the presence of a short range interaction. Generally, such S-shaped anhysteretic behaviour is observed in the materials having short range ferromagnetic interactions or a superparamagnetic (SPM) state.

To find the origin of the approach to such saturation-like behaviour, we have analyzed all the M(H) curves below 20 K considering the presence of SPM state in this material.

To properly assign SPM to any system, two different conditions need to be simultaneously satisfied. Firstly, Sshaped anhysteretic curves should be described by a Langevin function and secondly, all the magnetic isotherms in the temperature range where the superparamagnetism is manifested, should overlap in a universal curve when plotted as a function of H/T [START_REF] Hurd | Varieties of magnetic order in solids[END_REF]. We found that all the M(H) curves below 20 K can be well-explained using the Langevin function [START_REF] Vasundhara | Evidence for cluster glass behavior in Fe 2 VAl Heusler alloys[END_REF][START_REF] Feng | Physical properties of heusler-like Fe 2 VAl[END_REF], given by

T µ M s M s /µ N (K) (µ B ) (×10 -4 µ B /f.u.) (×10 -5 /f.u.) (×10
M (H) = M S L(x) (4)
where x = µH k B T , M S is the saturation magnetization, µ represents the average magnetic moment per cluster and L(x) = coth(x) -1/x, is the Langevin function. The parameters extracted from the fits are listed in Table 3. It can be observed that for all the temperature in the region 20 K to 2 K, the magnetic moment per SPM cluster shows a little variation around ∼6 µ B . The saturation magnetization and the number of SPM cluster per mole are estimated to be ∼ 4×10 -4 µ B /f.u. and ∼ 10 20 /mole and are found to be close enough in the temperature range from 2-20 K. In order to check the other condition, all the magnetic isotherms below 20 K are plotted as a function of H/T (Fig. 5). Interestingly, it can be seen that they all indeed follow a single universal curve, confirming the presence of SPM in Ru 2 TiGe below 20 K. It can be concluded from the analysis of the isothermal magnetization of Ru 2 TiGe that the superparamagnetically interacting clusters develop below 20 K, whereas a competition between diamagnetic and ferromagnetic contributions is apparent at the higher temperatures.

A similar or comparable magnetic feature is not restricted to Ru 2 TiGe, but also found in quite a few other Heusler alloys including Ru 2 NbAl [14], Fe 2 VAl [START_REF] Feng | Physical properties of heusler-like Fe 2 VAl[END_REF], etc. having VEC 24. In case of Fe 2 VAl, it was shown that the local perturbation caused by the Fe V and Fe Al -type antisite disorders result in the development of unpaired d -electron and the observed magnetism in that material is the direct outcome of those unpaired electrons belonging to the region containing antisite defects [START_REF] Graf | Simple rules for the understanding of heusler compounds[END_REF][START_REF] Bilc | Electronic and thermoelectric properties of Fe 2 VAl: the role of defects and disorder[END_REF]. As the theoretical calculation was carried out considering the atomic disorders within the basic L2 1 -cubic structure, this particular crystal structure certainly play a major role in such magnetism [START_REF] Graf | Simple rules for the understanding of heusler compounds[END_REF][START_REF] Bilc | Electronic and thermoelectric properties of Fe 2 VAl: the role of defects and disorder[END_REF]. However, the development of such unpaired electron need not be restricted to any particular chemical composition, as long as the compound is forming with transition metal elements. For Ru 2 NbAl too, the magnetism was also argued to have originated from Ru N b and Al N btype local disorders [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF]. In the present case of Ru 2 TiGe, development of Ti Ru and Ge Ru -types of antisite disorders have also been inferred from the XRD as well as EPMA measurements. The superparamagnetic behaviour in this material thus also likely to have originated due to the formation of small magnetic clusters around those defects.

Heat capacity

To ensure the veracity of the magnetization measurements, another independent experimental technique, the heat capacity [C(T)] has been carried out in the temperature range 2-300 K for Ru 2 TiGe in absence of any external magnetic field (Fig. 6). The C(T) curve starts to increase as the temperature raises and does not show any signature of long range magnetic order supporting the results inferred from the magnetic measurements.

Generally, the observed heat capacity behaviour can be explained considering the Debye-Sommerfeld equation [START_REF] Kittel | Introduction to solid state physics[END_REF], given by,

C(T ) = γ S T +9nR T Θ D 3 Θ D T 0 x 4 e x (e x -1) 2 dx (5)
where γ S T is the electronic specific heat and the second term represents the lattice/phonon contribution to the heat capacity. The Sommerfeld coefficient γ

S = 1 3 π 2 D(E F )k B 2
, where D(E F ) is the density of states at the Fermi level, E F and n represents the number of atoms per formula unit (for Ru 2 TiGe: n = 4), Θ D is the Debye temperature and x = hω/k B T. This above discussed Debye-Sommerfeld model could reproduce well the heat capacity data in the temperature range 35-300 K.

The value of Θ D and γ S estimated from the fit are 407 K and 7.6 mJ/mol-K 2 respectively. The value of Θ D is found similar to that obtained from the fit of lattice volume as a function of temperature (Sec. 3.1). A close inspection at the lower temperatures however reveals that the fit is not as good as it is in the higher temperatures and this is also reflected in the relatively high estimated value of γ S comparing standard metallic system [START_REF] Kittel | Introduction to solid state physics[END_REF]. This value of γ S , obtained from the fitting of heat capacity data to Eq. 5, is not proper as γ S is generally a low temperature phenomenon [START_REF] Kittel | Introduction to solid state physics[END_REF] and thus a proper analysis of heat capacity data at lower temperature is crucial.

In the low temperature region Eq. 5 can be simplified as

C(T ) = γ S T + βT 3 + δT 5 + • • • • • • (6)
where β, δ are the coefficients. Below ∼ Θ D /50, δT 5 and other higher order terms could be neglected and the heat capacity behavior in the representation of C/T vs. T 2 is expected to show a linear dependance. The Debye temperature can be estimated from the slope (β) of such straight line: β = (12π 4 /5)N k B /Θ D 3 . It may be mentioned here that in some related off-stoichometric Heusler alloys [START_REF] Lonchakov | Examination of the specific features of the electron density of states of weakly nonstoichiometric Fe-V-Al alloys through the analysis of low-temperature heat capacity[END_REF][START_REF] Okulov | Semiconductor-like behavior of electric transport in Fe-V-Al-based metallic alloys and their uncommon magnetic properties[END_REF], the C/T vs. T 2 representation are known to deviate from linearity, although it remain lienar in Ru 2 TiGe as expected in standard materials (Fig. 6, inset). The value of γ S and β are found to be 0.2 mJ/mol-K 2 and 0.067 mJ/mol-K 4 respectively. The value of Θ D , calculated using obtained β is 487 K, found to be higher than that estimated from the whole temperature range fit. It is primarily because Θ D s, extracted using Debye-Sommerfeld equation, are known to have a small temperature dependence [START_REF]Specific Heats at Low Temperatures[END_REF][START_REF] Goetsch | Structural, thermal, magnetic, and electronic transport properties of the LaNi 2 (Ge 1-x Px) 2 system[END_REF]. In the Debye model, it was assumed that only low-frequency modes of lattice vibrations i.e., acoustic modes contributes to the lattice specific heat for the large temperature region although this assumption is actually valid only at low temperatures. Therefore, at very low temperature, Θ D should be temperature independent. As the temperature increases, not only the acoustic modes, but optical modes of lattice vibrations also found to contribute to the lattice specific heat that in turn decreases the value of Θ D at higher temperature. Generally, the variations of Θ D remain within 10% around their mean value for most of the elements, though some exceptions, for instance, 20% variations have also been observed in case of zinc, cadmium etc. [START_REF]Specific Heats at Low Temperatures[END_REF][START_REF] Goetsch | Structural, thermal, magnetic, and electronic transport properties of the LaNi 2 (Ge 1-x Px) 2 system[END_REF]. For general use, the value of Θ D (Θ D = 407 K for Ru 2 TiGe), estimated from fit of the C(T) data in the temperature range 2-300 K should have better acceptability as it is averaged over the entire temperature region. The low but non-zero value of γ S in Ru 2 TiGe reveals that only a small number of free carriers are available for the electric conduction for this material, yielding a bad metal-like/semimetallic ground state.

Transport properties

Electrical resistivity

The electrical resistivity behaviour as a function of temperature [ρ(T )] of Ru 2 TiGe has been studied in the temperature range 2-390 K, shown in Fig. 7. The ρ(T ) curve exhibits a positive temperature coefficient of resistivity (TCR) over a large range of temperatures from 40 K to 390 K, suggesting metallic nature, similar to that found earlier on another experimental observation [START_REF] Mizusaki | Metal-insulator-metal transition in Ti substituted antiferromagentic Ru 2 MnGe Heusler alloy[END_REF]. However, the experimental result found to be in contravention to the theoretical investigation of band structure calculations of Ru 2 TiGe [START_REF] Krishnaveni | Band gap engineering in ruthenium-based Heusler alloys for thermoelectric applications[END_REF][START_REF] Hamri | Ru 2 Ti 1-x FexGe: Novel candidate for the spintronic applications[END_REF] that had predicted the possibility of having a pseudo-gap ∼200 meV, which would have manifested in a semiconductorlike temperature dependence of resistivity. This deviation of ρ(T ) could be attributed to the structural antisite defects and disorders, noticed in this compound. The excess Ti-d and Ge-p bands originated from the Ti Ru and Ge Ru types of antisite defects are likely to introduce additional states in pseudo-gap region leading to a metal-like ρ(T ) behaviour, as experimentally observed in Ru 2 TiGe. The magnitude of ρ at 390 K is 443 µΩ-cm whereas at 2 K it is found to be 362 µΩ-cm giving a residual resistivity ratio (RRR = ρ 390K /ρ 2K ) value of ∼ 1.22. Despite having a metal-like resistivity, where resistance decreases with decreasing temperature, low value of RRR suggests Ru 2 TiGe to be a bad metal. A close inspection at the lower temperatures reveals that the TCR changes its sign from positive to negative at around 40 K and remain negative down to lowest measured temperature. It may be noted here that in systems having high resistances, sharp upturn at low temperature may arise from different quantum interference effects such as weaklocalization, electron-electron interactions etc. [START_REF] Anderson | Possible explanation of nonlinear conductivity in thin-film metal wires[END_REF][START_REF] Altshuler | Electron-Electron Interaction In Disordered Conductors[END_REF][START_REF] Bergmann | Physical interpretation of weak localization: A time-of-flight experiment with conduction electrons[END_REF]. Though ρ(T ) curve of Ru 2 TiGe exhibits negative TCR below 40 K, the value of ρ changes at a very slow rate. Thus, the above mentioned processes may not be the primary basis of the negative TCR in Ru 2 TiGe. We have discussed earlier that the structural analysis of Ru 2 TiGe suggests small presence of Ti Ru and Ge Ru antisite defects. Such structural defects are often argued to be responsible for the localization of charge carriers [START_REF] Kelekar | Epitaxial growth of the Heusler alloy Co 2 Cr 1-x FexAl[END_REF]. Mott's variable range hopping (VRH) [START_REF] Mott | Conduction in glasses containing transition metal ions[END_REF] of electrons between exponentially localized states is a mechanism that could explain both the negative TCR and high value of ρ in such systems at low temperature. In this mechanism, the conduction is expected to propagate via electrons, hopping between localized sites that are energetically closed but not necessarily close in space. The conduction behaviour of VRH in three dimensional systems can be expressed as

ρ(T ) = ρ 0 exp T 0 T 1/4 (7)
where ρ 0 represents the residual resistivity and T 0 , the activation temperature, depends on the localization length (ξ) as ξ -3 [START_REF] Mott | Conduction in glasses containing transition metal ions[END_REF]. The resistivity data of Ru 2 TiGe has been plotted as ln(ρ) vs. T -1/4 below 40 K (Fig. 7, Inset) and fitted using Eq. 7. In spite of having an excellent fit, the value of T 0 is found to be extremely low ∼7×10 -7 K than that of the values reported in the literature [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF][START_REF] Vasundhara | Low-temperature electrical transport in Heusler-type Fe 2 V(AlSi) alloys[END_REF] thus also ruling out the possibility of any significant role of VRH mechanism in this material.

On the other hand, it may be noted here that the magnetic measurements discussed earlier suggest the development of superparamagnetically interacting clusters in the low temperature region in this material. In many non-magnetic systems containing a minute magnetic impurity, such resistivity minimum at low temperatures is found to be originated from the Kondo effect [START_REF] Okulov | Semiconductor-like behavior of electric transport in Fe-V-Al-based metallic alloys and their uncommon magnetic properties[END_REF][START_REF] Kondo | Resistance minimum in dilute magnetic alloys[END_REF][START_REF] Lonchakov | New manifestations of a pseudogap state and electron spin scattering in the low-temperature thermal properties of nearstoichiometric iron-vanadium-aluminum alloys[END_REF]. Recently, in some strongly correlated manganites as well as ferromagnetic metals, the Kondo effect is observed [START_REF] Vasundhara | Electronic transport in Heusler-type Fe 2 VA 1-x Mx alloys (M = B, In, Si)[END_REF][START_REF] Zhang | Kondo-like transport and its correlation with the spinglass phase in perovskite manganites[END_REF][START_REF] Pasupathy | The Kondo effect in the presence of ferromagnetism[END_REF]. According to Kondo's theory, the resistivity below the minimum increases following the relation: ρ Kondo = ρ 0 -ρ s ln T , where ρ 0 is the residual resistivity, the second term is the contribution from the interaction between the conduction electrons and the magnetic spins and ρ s represents the strength of the Kondo spin scattering [START_REF] Kondo | Resistance minimum in dilute magnetic alloys[END_REF]. Therefore, it appears that the negative TCR in Ru 2 TiGe could be attributed to the formation of those magnetic clusters, pointing towards the Kondo-scattering mechanism. The ρ(T ) behaviour above 40 K is almost linear and can be described using the Bloch-Grüneisen model [START_REF] Blatt | Physics of electronic conduction in solids[END_REF] that considers the scattering of conduction electrons by the acoustic lattice vibrations. The temperature-dependent electrical resistivity of a metal according to the Bloch-Grüneisen model is given by

ρ BG (T ) = ρ ΘD T Θ D 5 Θ D T 0 x 5 (e x -1)(1 -e -x ) dx (8)
where, ρ ΘD is the value of resistivity at Debye temperature. In order to fit the ρ(T ) curve for the entire temperature region, the following equation has been used combining both Kondo effect and the Bloch-Grüneisen formula

ρ(T ) = ρ 0 -ρ s ln T + ρ BG (T ) (9)
The ρ(T ) curve for Ru 2 TiGe fitted very well using the above equation (Fig. 7). The parameters extracted from the fit are Θ D = 508 K, ρ ΘD = 110 µΩ-cm, ρ 0 = 361 µΩ-cm and ρ s = 1.1 µΩ-cm. It may be noted here that the value of ρ s , estimated above is relatively smaller in comparison to that reported in many magnetically ordered or glassy systems [START_REF] Zhang | Kondo-like transport and its correlation with the spinglass phase in perovskite manganites[END_REF][START_REF] Huang | Effect of cu-doping on the magnetic and electrical transport properties of three-quarter heusler alloy zrco1. 5sn[END_REF], although even smaller values of ρ s are also reported in some other Heusler compounds, e.g., Fe 2 VAl 1-x B x [START_REF] Vasundhara | Electronic transport in Heusler-type Fe 2 VA 1-x Mx alloys (M = B, In, Si)[END_REF]. It may nevertheless be pointed out that although the magnitude of ρ s , estimated for Ru 2 TiGe, appears to be quite reasonable, the absence of any signature of Kondo scattering in the heat capacity data at low temperature cast a serious doubt on the possibility of Kondo scattering in this compound. Thus, the negative TCR, observed at the lower temperatures could also be a manifestation of the semiconductor nature of conductivity, as expected in VEC 24 Heusler alloys. It is important to note here that the values of Θ D , estimated from the resistivity measurement, found to be higher than that of the values extracted from the lattice thermal expansion and heat capacity data. We have already discussed that both longitudinal as well as transverse acoustic lattice vibrations were considered in the Debye model to calculate lattice specific heat theoretically. In contrast, only longitudinal phonons have been considered in the theory to develop Bloch-Grüneisen formula [START_REF]Specific Heats at Low Temperatures[END_REF]. As the basic assumptions behind the Bloch-Grüneisen model and the Debye theory are quite different, consequently, the Debye temperature evaluated from electrical resistivity measurement differs from the Θ D , obtained by the heat capacity data. However, the Θ D (400 K), evaluated from lattice thermal expansion is found to be close enough to that of the heat capacity. It is important to note that the thermal expansion parameter (Θ D ) exhibit a Debye behaviour in the entire temperature region as the main contributions is provided by the phonons to thermal expansion. The coefficient of thermal expansion, α, is related to the heat capacity C and bulk modulus K 0 by α = (1/V )(∂V /∂T ) P = γ G C /K 0 V, where γ G is Grüneisen parameter [START_REF] Sayetat | An easy method for the determination of Debye temperature from thermal expansion analyses[END_REF].

Generally, γ G and K 0 have very weak dependence on temperature, the plot of α as a function of temperature essentially follows a Debye behavior. Because of these, the value of Θ D estimated from the analysis of different data often found to differ in different analysis [START_REF]Specific Heats at Low Temperatures[END_REF][START_REF] Goetsch | Structural, thermal, magnetic, and electronic transport properties of the LaNi 2 (Ge 1-x Px) 2 system[END_REF]. Among all three Θ D values of Ru 2 TiGe, Θ D = 407 K estimated from the heat capacity data is considered as the best representative of Θ D in this system since the approximations considered in the Debye model, used to explain the temperature variation of heat capacity are most inclusive in nature.

Seebeck coefficient

In order to explore the thermoelectric properties of Ru 2 TiGe, the temperature dependence of Seebeck coefficient [S(T)] has been investigated in the temperature range 2-390 K, as shown in Fig. 8. The value of Seebeck coefficient at 300 K is 47 µV/K, found to be higher than that of other Heusler alloys having VEC 24 like Fe 2 VAl (∼35 µV/K), Fe 2 VGa (∼30 µV/K), Ru 2 NbAl (∼22 µV/K) and Ru 2 NbGa (∼20 µV/K) [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF][START_REF] Vasundhara | Low-temperature electrical transport in Heusler-type Fe 2 V(AlSi) alloys[END_REF][START_REF] Nishino | Effect of off-stoichiometry on the transport properties of the Heusler-type Fe 2 VAl compound[END_REF][START_REF] Lue | Thermoelectric properties of the semimetallic Heusler compounds Fe 2-x V 1+x M (M = Al, Ga)[END_REF][START_REF] Nishino | Thermal and transport properties of the Heusler-type Fe 2 VAl 1-x Gex (0≤x≤0.20) alloys: Effect of doping on lattice thermal conductivity, electrical resistivity, and Seebeck coefficient[END_REF][START_REF] Lue | Off-stoichiometric effect on the transport and pseudogap characteristics of Fe 2 VGa[END_REF][START_REF] Lue | Effects of Ge substitution on the thermoelectric properties and pseudogap characteristics of Fe 2 VGa[END_REF]. The magnitude of the S(T) remains positive throughout the entire temperature region examined suggesting that the majority carriers for the thermoelectric transport must be holes in Ru 2 TiGe. As the temperature raises, S(T) increases gradually and after exhibiting a humplike shape with a maximum at T ≈ 50 K, it varies almost linearly above 120 K and attains a value of 60 µV/K at 390 K. The S(T) onset of linear variations does not coincide with the onset of monotonous increase of the resistivity at around 80 K (Fig. 7). The maximum in S(T) rather matches with the lattice thermal conductivity maximum at 50 K (Fig. 9) discussed later. The appearance of the hump-like shape in S(T) curve of Ru 2 TiGe can therefore be ascribed to the phonon drag effect, similar to that reported earlier in Fe 2 TiSn [START_REF] Lue | Thermal and transport properties of the Heusler-type compounds Fe 2-x Ti 1+x Sn[END_REF]. However, the linear nature of S(T) curve at higher temperatures is in sync with observed metallic electric conduction behaviour of Ru 2 TiGe.

Hall coefficient

The Hall coefficients (R H ) were measured for Ru 2 TiGe in the temperature range 5-300 K for externally applied magnetic field of 50 kOe in order to check the sign of the Seebeck coefficient already discussed. The Hall resistivity increases with rising magnetic field almost with a constant slope for all the investigating temperatures and the value of R H has been calculated using that constant slope. The magnitude of R H remains positive and almost temperature independent for the entire temperature regime (Fig. 8, Inset:(a)) indicating the dominance of hole-type carrier for thermoelectric transport in conformity with the positive sign of S(T) observed in this compound (Fig. 8). The value of observed R H is found to be ∼ 10 -8 m 3 /C, which is 100-1000 times larger than that of a conventional metals and nearly close to the elemental semimetals such as Sb. Similar values of R H are also reported previously in the isoelectronic Heusler compounds Fe 2 VAl (∼2 ×10 -8 ) [START_REF] Nishino | Effect of off-stoichiometry on the transport properties of the Heusler-type Fe 2 VAl compound[END_REF] and Fe 2 VGa (∼1 ×10 -8 ) [START_REF] Fukuhara | Hall effect in heusler alloys fe2+ x v1-x al and fe2+ x v1-x ga[END_REF]. Assuming the presence of only one type of carrier i.e. holes, the hole concentrations(n) are calculated ∼3.3 × 10 20 cm -3 (Fig. 8, Inset:(b)) using the relation n = 1/eR H , where e is the charge of electron. Such carrier concentration is one order of magnitude smaller than that of the metallic Fe 2 TiSn (∼10 21 ) [START_REF] Pallecchi | Thermoelectric properties of chemically substituted full-heusler f e 2 tis n 1-x s b x (x= 0, 0.1, and 0.2) compounds[END_REF] and subsequently 100-1000 times lower than a conventional metal. Having similar carrier concentration (∼10 20 ), isoelectronic Fe 2 VAl and Fe 2 VGa exhibit non-metallic electrical resistivity [START_REF] Nishino | Effect of off-stoichiometry on the transport properties of the Heusler-type Fe 2 VAl compound[END_REF][START_REF] Fukuhara | Hall effect in heusler alloys fe2+ x v1-x al and fe2+ x v1-x ga[END_REF]. In case of Ru 2 TiGe, such small numbers of carriers are responsible for its low residual resistivity ratio and large thermopower that are indeed beneficial for achiving large zT.

Thermal conductivity

To further evaluate the thermoelectric performance of Ru 2 TiGe, thermal conductivity (κ) was measured in the temperature range 2-390 K (Fig. 9). At low temperatures, κ starts to increase rapidly with temperature and shows a sharp peak at ∼50 K. Above 50 K, the value of κ decreases and attains 12.4 W/m-K at 390 K (Fig. 9). For ordinary metals and semimetals, the total thermal conductivity is a sum of lattice (κ L ) and electronic (κ e ) contributions. The electronic thermal conductivity can be extracted using the Wiedemann-Franz law κ e ρ/T = L 0 , where ρ represents the measured dc electric resistivity and L 0 = 2.45×10 -8 WΩK -2 is the Lorenz number. The lattice thermal conductivity κ L can then be evaluated by subtracting κ e from the observed κ. The estimated κ e is found to be only a small contribution to κ (Fig. 9), indicating that the thermal conductivity of this material is essentially originated from κ L (Fig. 9). At low temperatures, κ L increases with temperature and a sharp maximum appears around 50 K because of the reduction in thermal scattering at low temperatures. The height of this peak in κ L commonly represents the degree of crystallographic order in a compound and for a structurally disordered system this peak in κ L remains absent [START_REF] Kuo | Ru 2 NbGa: A Heusler-type compound with semimetallic characteristics[END_REF][START_REF] Lue | Thermoelectric properties of quaternary Heusler alloys Fe 2 VAl 1-x Six[END_REF]. Thus, the sharp peak in κ L for the present compound agrees with its ordered L2 1 structure. The κ L value of Ru 2 TiGe at 300 K is estimated to be 11.2 W/m-K (Fig. 9), which is however lower than those in Fe 2 VAl (∼ 28 W/m-K) and Fe 2 VGa (∼ 17 W/m-K) [START_REF] Lue | Thermoelectric properties of the semimetallic Heusler compounds Fe 2-x V 1+x M (M = Al, Ga)[END_REF][START_REF] Nishino | Thermal and transport properties of the Heusler-type Fe 2 VAl 1-x Gex (0≤x≤0.20) alloys: Effect of doping on lattice thermal conductivity, electrical resistivity, and Seebeck coefficient[END_REF][START_REF] Lue | Off-stoichiometric effect on the transport and pseudogap characteristics of Fe 2 VGa[END_REF][START_REF] Lue | Effects of Ge substitution on the thermoelectric properties and pseudogap characteristics of Fe 2 VGa[END_REF]. This reduction may be attributed to smaller Debye temperature and speed of sound caused by substituting Fe by heavier Ru atom. Despite having a metal-like ground state, appreciably large value of thermopower as well as low κ in Ru 2 TiGe increase the possibility of large zT in this compound.

In order to understand the influence of various phonon scattering mechanism on the lattice thermal conductivity of Ru 2 TiGe, the κ L (T ) data was analyzed using the Debye-Callaway model [START_REF] Callaway | Model for lattice thermal conductivity at low temperatures[END_REF][START_REF] Walker | Phonon scattering by point defects[END_REF]. According to this model, the lattice thermal conductivity as a function of temperature is described by the following equation:

κ L (T ) = k B 2π 2 v k B T h 3 θ D T 0
x 4 e x τ P -1 (e x -1) 2 dx (10) where x = hω k B T , ω and θ D are the phonon frequency and the Debye temperature, respectively, and v represents the average phonon velocity that is approximately equal to the speed of sound in the studied material. The phonon scattering relaxation time, τ P , is considered as the sum of different phonon scattering mechanism, defined as

τ P -1 = v L +Aω 4 +Bω 2 T e -θ D /3T +Cω+Dω 2 (11)
where L is the characteristic length defining the sample size and A, B, C and D are the free A(s 3 ) B(s/K) C D(s) 5.7×10 -43 5.5×10 -18 3.6×10 -4 3.3×10 -17 fitting parameters. The terms in the right of Eq. 11 are the scattering rates of phonon by the sample boundaries (SB), the point or mass defects (PD), the phonons (PP), the grain boundaries (GB) and the electrons (EP), respectively [START_REF] Callaway | Model for lattice thermal conductivity at low temperatures[END_REF][START_REF] Wang | Thermal conductivity of nanocrystalline silicon: importance of grain size and frequencydependent mean free paths[END_REF][START_REF] Tritt | Thermal conductivity: theory, properties, and applications[END_REF]. The frequency dependent expression (Cω) for the phonon scattering rate by the grain boundaries has been included in Eq. 11 since κ L shows a T 2 variation (Fig. 10) at very low temperature in Ru 2 TiGe. Such T 2 variation of κ L for silicon below ∼70 K had previously been explained while deriving the above mentioned frequency dependent term Cω [START_REF] Wang | Thermal conductivity of nanocrystalline silicon: importance of grain size and frequencydependent mean free paths[END_REF]. A metal-like ground state and the hole concentration of ∼ 3.3 ×10 20 cm -3 ) in Ru 2 TiGe compel to adapt the term Dω 2 considering the phonon scattering rate by electron [START_REF] Tritt | Thermal conductivity: theory, properties, and applications[END_REF]. To fit the κ L (T), the value of θ D = 407 K estimated from the specific heat data is utilized and v = 3200 ms -1 is taken as the average speed of sound. Here, L = 2 mm was held as a fixed parameter giving its weak influence on the results.

A good agreement between the fitted curve and the measured values can be seen in Fig. 10. The deviation at the high temperature region most likely arises from radiation artifacts on the measurement. The fitted parameters are presented in Table 4. The relative weight of each scattering process on the lattice thermal conductivity can be better understood by replacing τ P -1 in Eq. 10 by the individual scattering rate, yielding a theoretical limit imposed by each process on the lattice thermal conductivity. It can be interpreted from Fig. 10 that scattering by the sample boundaries and by the mass or point defects play a minor role in the κ L of Ru 2 TiGe. At low temperature, grain boundary scattering dominates while electron and phonon scattering play a major role at intermediate and high temperature regions, respectively.

Figure of merit

Using all three measured thermoelectric parameters i.e., Seebeck coefficient [S(T)], electrical resistivity [ρ(T )] and thermal conductivity [κ(T )], the value of the power factor (PF=S 2 /ρ) and the dimensionless figure of merit (zT =S 2 T /ρκ) are estimated and illustrated in Fig. 11. It should be mentioned here that the developement of superparamagnetic interaction below 20 K does not exhibit any noticeable influence in the power factor and figure of merit, as neither Seebeck coefficient nor thermal conductivity exhibit any discernible change in this temperature range. The small upturn in resistivity below 40 K is also likely to be due to the semiconducting nature. Additionally, since the practical application demands an improved zT value in the high temperature region (T ≥ 300 K), the effect of magnetism in the low temperature region may not have any practical significance. In the high temperature region, although Ru 2 TiGe exhibits metal-like electric conduction, its comparably large Seebeck coefficient leads to achieve higher value of power factor (0.54 mW/mK 2 at 300 K) for this material than that of the existing Fe-based and other Ru-based stoichiometric Heusler alloys. For example, the room temperature PF values of 0.12 and 0.16 mW/mK 2 have been reported earlier in literature for Ru 2 NbAl [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF] and Fe 2 VAl [START_REF] Lue | Thermoelectric properties of quaternary Heusler alloys Fe 2 VAl 1-x Six[END_REF], respectively. With the rise of temperature, the PF curve after exhibiting a hump-like shape at around 40 K resulting from the phonon drag effect already observed in the S(T) curve, increases and reaches at a value of 0.84 mW/mK 2 at 390 K. Such PF value for present compound is of the same order of magnitude to Fe 2 VAl-based and Fe 2 VGa-based Heusler alloys that are suitably doped for the improvisation of their zT values [START_REF] Vasundhara | Electronic transport in Heusler-type Fe 2 VA 1-x Mx alloys (M = B, In, Si)[END_REF][START_REF] Wei | Thermoelectric properties optimization of Fe 2 VGa by tuning electronic density of states via titanium doping[END_REF][START_REF] Renard | Thermoelectric properties of the Heusler-type Fe 2 VTaxAl 1-x alloys[END_REF]. Similar to power factor, the estimated value of zT at 300 K (zT 300K = 0.012) is also higher than that of the most investigated non-doped thermoelectric material Fe 2 VAl. The zT of Ru 2 TiGe even exceeds the zT of Ru 2 NbAl for which largest value of zT (0.0052 at 300 K) was obtained among all reported pristine Heusler compounds [START_REF] Mondal | Ferromagnetically correlated clusters in semimetallic Ru 2 NbAl Heusler alloy and its thermoelectric properties[END_REF]. Temperature dependance of zT exhibit a increasing trend and yields almost two times higher value of 0.025 at 390 K than at 300 K. Nonetheless, it is still one order of magnitude smaller than that of the state-of-the-art thermoelectric material at 300 K, Bi 2 Te 3 which displays zT = 0.8 at 300 K. Enhancement in zT values are earlier reported by suitable doping/substitution in proper site of the pristine Heusler alloys viz. Fe 2 VAl, Ru 2 NbGa, Ru 2 TaAl etc. [START_REF] Tseng | Semimetallic behavior in Heusler-type Ru 2 TaAl and thermoelectric performance improved by off-stoichiometry[END_REF][START_REF] Renard | Thermoelectric properties of the Heusler-type Fe 2 VTaxAl 1-x alloys[END_REF][START_REF] Garmroudi | Large thermoelectric power factors by opening the band gap in semimetallic Heusler alloys[END_REF][START_REF] Bag | Thermoelectric properties of chemically substituted Heusler-type Ru 2-x Nb 1+x Ga and Ru 2 NbGa 1-x Mx (M = In, Ge, and Sn) alloys[END_REF]. It may be noted here that a 20-times increased zT value (0.036) at 300 K was reported in Fe 2 VAl 0.9 Si 0.1 [START_REF] Lue | Thermoelectric properties of quaternary Heusler alloys Fe 2 VAl 1-x Six[END_REF] than in non-doped Fe 2 VAl and by replacing heavier Ge atom in place of Si in Fe 2 VAl 0.9 Si 0.1 , a total 72-fold increased zT value of 0.13 at 300 K was achieved in Fe 2 VAl 0.9 Ge 0.1 [START_REF] Nishino | Thermal and transport properties of the Heusler-type Fe 2 VAl 1-x Gex (0≤x≤0.20) alloys: Effect of doping on lattice thermal conductivity, electrical resistivity, and Seebeck coefficient[END_REF].

Interestingly, being only a pristine compound, Ru 2 TiGe exhibits such a large value of zT at room temperature, further improvement in zT for Ru 2 TiGe is highly anticipated by following similar strategy through doping to alter the metal-like ground state to a semiconductor-like one by adjusting the charge carrier concentration and substitution to reduce the lattice thermal conductivity.

Conclusion

An elaborate study on a Ru-based Heusler alloy, Ru 2 TiGe, is performed through structural, magnetic, heat capacity and thermoelectric properties measurements. The magnetic properties at higher temperatures are dominated by temperature independent diamagnetic and Pauli paramagnetic contributions. In contrast to the observed magnetic behaviour at higher temperatures, superparamagnetic interaction among clusters is observed at lower temperatures below 20 K, despite the fact that none of the constituent atoms in Ru 2 TiGe is usually known to order magnetically in nature.

This short range magnetic interaction may have its origin in the small off-stoichiometry, noticed in otherwise structurally ordered (in L2 1 phase) Ru 2 TiGe. Larger Hall coefficient (R H ∼ 10 -8 m 3 /C), smaller Sommerfeld coefficient (γ S = 0.2 mJ/mol-K 2 ) as well as lower RRR (ρ 390K /ρ 2K =1.22) than a conventional metal categorize Ru 2 TiGe as a bad metal or a semi-metal. In spite of having a metallike resistivity, a large value of Seebeck coefficient in this compound results in a larger power factor of the order of 0.54 mW/mK 2 at 300 K than that of the other reported non-doped Heusler alloys. Large power factor as well as low thermal conductivity which is the result of employing heavy constituent elements like Ru and Ge in Ru 2 TiGe, give rise to a superior value of zT = 0.012 at 300 K in comparison to those reported in other pure Heusler alloys and point towards its high potential for practical thermoelectric applications. Adopting the strategies reported in literature for achieving many-fold increment of zT in Fe 2 VAl through appropriate elemental substitution, similar great enhancement in the thermoelectric performances of Ru 2 TiGe is also highly anticipated.

  while Y and Z atoms are located at the positions (

Figure 1 :

 1 Figure 1: Powdered X-ray diffraction pattern of Ru 2 TiGe, measured at room temperature & indexed considering the L2 1 crystal structure. Presence of minor amount of unreacted Ru is marked by * symbol. Inset: Back scattered image of Ru 2 TiGe.

  . The sample homogeneity and chemical composition of the annealed sample were evaluated by using the Wavelength Dispersive Spectroscopy based Electron Probe Micro-Analysis (WDS-EPMA) technique [Model: SX 100, M/s Cameca, France]. The powdered X-ray diffraction (XRD) technique was adapted to check the single phase nature of Ru 2 TiGe at 300 K using Cu K α radiation in a powder diffractometer equipped with a rotating anode X-ray source at 9 kW [Model: TTREX III, M/s Rigaku Corp., Japan]. The XRD measurements were further carried out down to 12 K in order to look for any probable structural distortion. The Rietveld refinement technique by using FULLPROF software [30] is used to analyze the XRD patterns. Thermal transport [ρ(T ), S (T ), κ(T )], heat capacity [C ], Hall coefficient [R H ] and magnetic [M (T, H )] properties were measured using commercial set ups [Models: PPMS Evercool-II and SQUID-VSM, M/s Quantum Design Inc., USA]. For transport measurement we have used a long rectangular shaped sample piece.

Figure 2 :

 2 Figure 2: Low temperature XRD pattern of Ru 2 TiGe down to 12 K; Inset: Temperature dependence of unit-cell volume of Ru 2 TiGe. Solid line represents a fit to Eq. 1.

Figure 3 :

 3 Figure 3: Temperature dependence of magnetic susceptibility of Ru 2 TiGe measured at (top) H = 10 kOe and (bottom) 70 kOe under ZFC configuration along with a fit to Eq. 3; Insets: Isothermal magnetization at (a) 300 K and (b) 2 K of the same sample.

  Figure 4:Isothermal magnetization data at various temperature for Ru 2 TiGe. The solid lines represent the fit of the data using Eq. 4.

Figure 5 :

 5 Figure 5: The SPM state follows universal curve at low temperature isotherm for Ru 2 TiGe.

Figure 6 :

 6 Figure6: Specific heat as a function of temperature of Ru 2 TiGe. Solid line represents the fit to Eq. 5; Inset: C/T vs T 2 plot at low temperatures along with the fit to Eq. 6.

Figure 7 :

 7 Figure 7: Resistivity as a function of temperature for Ru 2 TiGe with a fit to Eq. 9; Inset: Plot of ln(ρ) vs. T -1/4 below 40 K.

/ c m 3 )Figure 8 :

 38 Figure 8: Temperature dependence of Seebeck coefficient of Ru 2 TiGe. Insets: (a) Hall coefficient and (b) hole concentration as a function of temperature

Figure 9 :

 9 Figure 9: Temperature variations of the total thermal conductivity κ, lattice thermal conductivity κ L , and electronic thermal conductivity κe for Ru 2 TiGe as a function of temperature.

Figure 10 :

 10 Figure 10: Temperature variations of lattice thermal conductivity κ L for Ru 2 TiGe. The open circles are the experimental data; the solid line is the fit to Eq. 10; the dashed curves are the theoretical limits on the lattice thermal conductivity imposed by different phonon scattering processes.

Figure 11 :

 11 Figure 11: zT and power factor as a function of temperature for Ru 2 TiGe.

Table 1 : Crystallographic parameters of Ru 2 TiGe obtained from the Rietveld refinement.

 1

	Ru 2 TiGe
	Structure	Cubic L2 1
	Space group	Fm 3m, No. 225
	Lattice parameter	6.034(1)	Å
	Element	Occupancy
	Ge (4a : 0,0,0)	0.986
	Ti (4b : 1 2 , 1 2 , 1 2 ) Ru (8c : 1 4 , 1 4 , 1 4 )	1.000 1.944
	R p	3.61 %
	R wp	5.15 %
	Volume	219.69(2)
	Density	9.79 g/cm 3

Table 2 :

 2 Parameters extracted from the fit of ZFC χ(T ) curves for H = 10 kOe and 70 kOe to Eq. 3 for Ru 2 TiGe.

Table 3 :

 3 The parameters, µ and Ms, obtained from fitting of isothermal magnetisation at different temperatures, T, using Eq. 4 are presented here. Number of SPM cluster per f.u. (Ms/µ) and per mole (N = Ms µ ×6.023×10 23 ) are also given.

Table 4 :

 4 Parameters of the Debye-Callaway model fitted to the lattice thermal conductivity data in Ru 2 TiGe. See main text for their definition.


Thibault Marsan

Yann Landon

Bruno Watier

À la recherche d'un critère de performance pour les athlètes d'escrime fauteuil

Introduction

L'escrime fauteuil est un sport de duel pratiqué sur un fauteuil spécifique et avec les mêmes armes que les athlètes valides. Ce sport est devenu depuis 2020 un sport paralympique mais les critères de performance permettant aux athlètes d'évaluer leurs niveaux manquent. Les critères de performance trouvés pour les athlètes valides étaient la vitesse de la fente et de changement de direction (Chen et al., 2017). Aucun de ces paramètres n'a été relié à la performance en escrime fauteuil. Après discussions avec les entraîneurs, les paramètres correspondants pour l'escrime fauteuil seraient la vitesse d'arme et leur capacité à se déplacer rapidement dans leur fauteuil, celui-ci restant fixe sur le sol. L'objectif de cette étude était alors de déterminer des critères de performance, en comparant ces paramètres avec des paramètres anthropométriques, le classement et les années de pratique. L'hypothèse faite était que la vitesse d'arme ainsi que la capacité à se déplacer rapidement dans le fauteuil corrèlerait avec ces paramètres.

Méthodologie

Onze para-escrimeurs de l'équipe nationale française de para-escrime étaient volontaires pour cette étude. Leur âge moyen ± écart-type, leur poids et leur année de pratique étaient de 30,2 ± 11,3 ans, 66,8 ± 9,8 kg et 18,5 ± 8,6 ans. Tous les participants avaient leur propre équipement d'escrime fauteuil. Cette étude a été autorisée par EudraCT / CRDI n° 2020-A02919-30. Deux participants se faisaient face. Un participant devait réaliser cinq attaques à vitesse maximale, pendant que l'autre était immobile. Ensuite, pour évaluer la capacité de déplacement dans le fauteuil, un l'athlète devait réaliser une attaque suivie d'une retraite, pour esquiver l'attaque de l'athlète adverse. Les positions des marqueurs réfléchissants ont été capturées à l'aide d'un système de capture de mouvement optoélectronique à 22 caméras VICON fonctionnant à 200 Hz. La trajectoire du marqueur sur la lame a été dérivée pour calculer la vitesse de l'arme. Le mouvement du milieu des marqueurs sur les épaules des athlètes représentait le mouvement d'avance et de recul. La corrélation entre la vitesse de l'arme, et la vitesse de recul et la taille, la masse, le classement mondial, et le nombre d'années de pratique a été évaluée.

Résultats et discussionLa seule corrélation significative était entre l'amplitude de mouvement du torse et la masse des athlètes qui manient une épée (r=0,82, p-value=0,02). Ce résultat était obtenu pour les athlètes maniant une épée, et représentait l'effectif le plus faible avec six athlètes. Aucune autre corrélation n'était toutefois significative L'hypothèse initiale n'a donc pas pu être vérifiée. Cela peut être dû au faible nombre de sujets impliqués dans l'étude mais il est aussi possible que cela soit lié aux différences de pathologies entre les athlètes. En effet, bien que répartis dans des trois catégories, les onze athlètes avaient onze handicaps différents.