Ba- ﬁ lled Ni – Sb – Sn based skutterudites with anomalously high lattice thermal conductivity †‡

Novel ﬁ lled skutterudites Ba y Ni 4 Sb 12 − x Sn x ( y max = 0.93) have been prepared by arc melting followed by annealing at 250, 350 and 450 °C up to 30 days in vacuum-sealed quartz vials. Extension of the homogeneity region, solidus temperatures and structural investigations were performed for the skutterudite phase in the ternary Ni – Sn – Sb and in the quaternary Ba – Ni – Sb – Sn systems. Phase equilibria in the Ni – Sn – Sb system at 450 °C were established by means of Electron Probe Microanalysis (EPMA) and X-ray Powder Di ﬀ raction (XPD). With rather small cages Ni 4 (Sb,Sn) 12 , the Ba – Ni – Sn – Sb skutterudite system is perfectly suited to study the in ﬂ uence of ﬁ ller atoms on the phonon thermal conductivity. Single-phase samples with the composition Ni 4 Sb 8.2 Sn 3.8 , Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 were used to measure their physical properties, i.e. temperature dependent electrical resistivity, Seebeck coe ﬃ cient and thermal conductivity. The resistivity data demonstrate a crossover from metallic to semiconducting behaviour. The corresponding gap width was extracted from the maxima in the Seebeck coe ﬃ cient data as a function of temperature. Single crystal X-ray structure analyses at 100, 200 and 300 K revealed the thermal expansion coe ﬃ cients as well as Einstein and Debye temperatures for Ba 0.73 Ni 4 Sb 8.1 Sn 3.9 and Ba 0.95 Ni 4 Sb 6.1 Sn 5.9 . These data were in accordance with the Debye temperatures obtained from the speci ﬁ c heat (4.4 K < T < 140 K) and Mössbauer spectroscopy (10 K < T < 290 K). Rather small atom displacement parameters for the Ba ﬁ ller atoms indicate a severe reduction in the “ rattling behaviour ” consistent with the high levels of lattice thermal conductivity. The elastic moduli, collected from Resonant Ultrasonic Spectroscopy ranged from 100 GPa for Ni 4 Sb 8.2 Sn 3.8 to 116 GPa for Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 . The thermal expansion coe ﬃ cients were 11.8 × 10 − 6 K − 1 for Ni 4 Sb 8.2 Sn 3.8 and 13.8 × 10 − 6 K − 1 for Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 . The room temperature Vickers hardness values vary within the range from 2.6 GPa to 4.7 GPa. Severe plastic deformation via high-pressure torsion was used to introduce nanostructuring; however, the physical properties before and after HPT showed no signi ﬁ cant e ﬀ ect on the materials thermoelectric behaviour.


Introduction
Several classes of thermoelectric materials have been investigated for renewable power generation applications, including tellurides, 1,2 half-Heuslers, 3,4 silicides 5 and skutterudites. Skutterudite-based materials have attracted considerable research interest since a long time 6 because they show a large variety of physical properties. However, for commercial thermoelectric applications, 7 high figures of merit for the p-type as well as for n-type configuration are a precondition. [8][9][10][11] Skutterudites crystallize in the cubic CoAs 3 structure (space group Im3) with the general formula F x T 4 M 12 , where T is a transition metal of the VIII th group located in position 8c ( 1 4 , 1 4 , 1 4 ) and M is a pnictogen, chalcogen or an element of the IV th main-group in the Wyckoff site 24g (0, y, z). These atoms form a cage-like structure with a large icosahedral void at the 2a site (0, 0, 0), which may accommodate various filler atoms F, including alkaline and alkaline earth metals, lanthanoids, actinoids, as well as halogens or in particular cases Y, Hf, 12,13 Pb and Sn. [14][15][16][17] A ternary skutterudite in the system Ni-Sn-Sb was first reported by Grytsiv et al. 18 who defined a wide homogeneity range at 250°C and 350°C by establishing the isothermal sections in the Sn-Sb-NiSb-Ni 3 Sn 2 subsystem at these temperatures. Investigations by Mishra et al. 19 in the Snrich part of the Ni-Sn-Sb phase diagram suggested that the phase equilibria determined by Grytsiv et al. 18 lies within 10°C above or below the declared temperatures.
Numerous reports (see for example ref. 18,[20][21][22][23][24][25] describe mixed occupancies for all three crystallographic sites (24g, 8c, 2a). For Ni-Sn-Sb based skutterudites, the structure with Ni-atoms fully occupying the 8c site seems to be stabilized by a random distribution of Sb and Sn atoms in the 24g position because in the binary systems, Ni-{Sn, P, As, Sb}, only the skutterudite NiP 3 exists as a high temperature modification. 26,27 A special situation occurs for Sn-atoms, which may occupy the 24g site, but may simultaneously enter the 2a site of the same compound reaching filling levels of 0.21 in this position. 18 Ternary and isotypic quaternary skutterudites with Eu and Yb as filler elements have been characterised by their physical properties as well as by Raman and Mössbauer spectroscopy, unambiguously revealing that a small amount of Sn is able to enter the 2a (0, 0, 0) site. 18 The filler atom in 2a in most skutterudites is loosely bound in the large icosahedral cage and is generally believed to decrease the thermal conductivity of the material via rattling modes. 6,28 With Ni and Sn atoms being smaller than Fe, Co and Sb, respectively, the Ni 4 (Sn,Sb) 12 icosahedral cages are smaller in comparison to a large filler atom such as Ba (for details see section 5: the crystal structure and vibration modes of Ba y Ni 4 Sb 12−x Sn x ). Thus, the Ba-Ni-Sn-Sb skutterudite system may become a model system to study the influence of filler atoms on the phonon thermal conductivity.
In addition, not much information is available on the thermal stability of Ni-skutterudites. To the best of our knowledge, the melting temperatures were determined for only two binary skutterudites, CoAs 3 29 and CoSb 3 , 30 and there are no data either on the influence of the filler on the melting points or on the thermal stability of ternary or multi-component skutterudites.
The current article will focus on a series of aspects outlined below: (i) The Ni-Sn-Sb isothermal section at 450°C and the extension of the ternary skutterudite phase Ni 4 Sb 12−x Sn x in comparison with data reported by Grytsiv et al. 18 (ii) The temperature dependent filling levels of Ba y Ni 4 Sb 12−x Sn x .
(iii) The solidus surface for the ternary and quaternary skutterudites, Ni 4 Sb 12−x Sn x and Ba y Ni 4 Sb 12−x Sn x , in dependence of their composition.
(iv) The structure of Ba y Ni 4 Sb 12−x Sn x determined by single crystal X-ray diffraction.
(v) Physical property measurements such as temperature dependent resistivities, thermal conductivities and Seebeck coefficients of three single-phase samples: Ni 4  (viii) A discussion on the thermoelectric behaviour in terms of the electron and phonon mean free paths and the influence of Ba-filler level on the lattice thermal conductivity.

Sample preparation
Starting materials were used in the form of elemental pieces of Ba, Ni, Sb and Sn, all of 99.95 mass% minimum purity. Samples suitable for constitutional analysis were prepared by one of the following optimised melting reactions directly gaining the ternary Ni 4 Sb x Sn 12−x alloys, whereas for the quaternary skutterudites, the filler element was added in a second reaction step: (i) Bulk alloys, each with a total weight of 1-2 g, were synthesized via a Ni 4 Sb x Sn 12−x master alloy by argon arc melting of metal ingots on a water-cooled copper hearth, adding the filler element to the ternary alloy using the same reaction method.
(ii) Samples with the nominal composition Ni 4 Sb x Sn 12−x (1-2 g) were prepared from stoichiometric amounts of high purity Sb and Sn pieces and fine Ni wire. After mixing, the material was sealed into evacuated quartz tubes, heated to 980°C, kept in the liquid state for half an hour prior to quenching the capsules in air. For the quaternary samples, the whole procedure was repeated, adding pieces of Ba to the ternary master alloys. Total weight losses of 1-3 mass% that occurred during sample preparation were attributed to the high vapour pressures of Sb and Ba and were compensated by adding an additional 3-5 mass% of Ba and Sb. All the samples were then sealed in evacuated quartz tubes, annealed at 250, 350 or 450°C for 3 to 30 days for equilibration before quenching in cold water.
Large single-phase samples (6-8 g) in cylindrical form for the study of their physical properties were prepared following either route (i) or (ii); the specimens were further ground to a grain size below at least 100 μm inside an Argon glove box using a WC mortar followed by ball-milling in a Fritsch planetary mill (Pulverisette 4) with WC-balls of 1.6 mm, rotation speed 250 and ratio of −2.5 for 2 h to gain a nanocrystalline powder. These powders were then loaded into 1 cm diameter graphite cartridges for hot-pressing under 1 bar of 5N-argon in a FTC uniaxial hotpress system (HP W 200/250 2200-200-KS).
The Sb-rich specimens, Ni 4 Sb 8.2 Sn 3.8 and Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 , were directly prepared by hot pressing at 450°C, resulting in a densely compacted single-phase material.
For the Sn-rich sample, Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 , a modified preparation method had to be chosen. 5-10 mass% of extra Sb and Sn were added from the very beginning of the synthesis. The ball-milled powder was pre-compacted and annealed under argon at 450°C overnight inside the hotpress, followed by hotpressing at 430°C in order to squeeze out the surplus Sb 2 Sn 3 .
Small single crystals suitable for temperature dependent X-ray structure analysis were obtained from flux techniques: Sn-rich single crystals of Ba 0.95 Ni 4 Sb 6.1 Sn 5.9 were prepared from Sn flux, Sb-rich Ba 0.73 Ni 4 Sb 8.1 Sn 3.9 single crystals from Sb 2 Sn 3 flux by heating the starting materials with compositions 2Ba-12Ni-Sb22.5-Sn63.5 and 2Ba-8Ni-45Sb-45Sn, respectively, to 950°C to reach the liquid state followed by cooling at a rate of 12°C h −1 to 450°C. The alloys were maintained at this temperature for 3 days to reach thermodynamic equilibrium. Subsequently, the samples were removed from the furnace and the Sn-rich sample was treated with hot concentrated HCl acid to dissolve the Sn-rich matrix. For the Sbrich single crystals, the Sb 2 Sn 3 -rich matrix was removed using hot concentrated HNO 3 and then the crystals were washed with cold concentrated HCl.
Sample characterisation X-ray powder diffraction (XPD) data were collected using a Huber Guinier camera with monochromatic Cu K α1 radiation (λ = 0.154056 nm) and an image plate recording system (model G670). Pure Si (a Si = 0.5431065 nm) was used as internal standard to determine the precise lattice parameters via leastsquares fitting to the indexed 2θ values employing the program STRUKTUR. 31 For quantitative Rietveld refinements, we applied the program FULLPROF. 32 The chemical compositions of the different specimens were extracted from electron probe microanalyses (EPMA) using energy-dispersive X-ray (EDX) spectroscopy with an INCA Penta FETx3-Zeiss SUPRA55VP equipment (Oxford Instruments).
After inspection on an AXS-GADDS texture goniometer for quality and crystal symmetry, the X-ray intensity data for the two single crystals were collected on a four-circle Nonius Kappa diffractometer with a charge-coupled device (CCD) area detector and graphite-monochromated Mo K α radiation (λ = 0.07107 nm) at 100, 200 and 300 K under nitrogen gas. The orientation matrix and unit cell parameters were derived using the program DENZO. 33 In addition to the general treatment of absorption effects using the multi-scan technique (SADABS; redundancy of integrated reflections >8), no individual absorption correction was necessary because of the regular shape and small dimensions of the crystals. Employing SHELXS-97 and SHELXL-97 software 34 for the single-crystal X-ray diffraction data, the structures were successfully solved using direct methods and refined.
Using Archimedes' principle and distilled water, the density ρ s of the hot pressed samples was determined and compared to the calculated X-ray densities.
where M is the molar mass, Z is the number of formula units per cell, N is Avogadro's number and V is the volume of the unit cell.

Mössbauer spectroscopy
Mössbauer spectra were obtained on a constant-acceleration spectrometer using a 300 μCi Ca 121 SbO 3 source and a 10 mCi Ca 119m SnO 3 source for 121 Sb-and 119 Sn-Mössbauer spectroscopy, respectively. The velocity was calibrated with α-Fe at room temperature utilizing a 57 Co/Rh source. All Mössbauer spectra were obtained using powder samples and the isomer shifts are with reference to InSb at 10 K and to CaSnO 3 at room temperature. The significant γ-background due to the fluorescence of the source for 119 Sn-Mössbauer spectroscopy was suppressed with a 50 μm Pd foil between the source and detector.

Physical property measurements
The electrical resistivity, Seebeck coefficient and thermal conductivity were measured (4 K < T < 300 K) using a homemade equipment cooled by liquid He (see ref. 35). The Seebeck coefficient and electrical resistivity above room temperature were measured simultaneously with a ULVAC-ZEM 3 (Riko, Japan) apparatus. The thermal diffusivity D t was measured using a flash method (Flashline-3000, ANTER, USA). The thermal conductivity κ was calculated from D t , the specific heat C p and the density ρ s employing the relationship κ = D t ·C p ·ρ s . Hall data were obtained by a Quantum Design physical properties measurement system (PPMS) using a standard six-point method in a magnetic field of 9 T. Specific heat measurements from 2 to 140 K were performed on singlephase samples with masses between 2.5 g and 5 g cooled with liquid He employing an adiabatic step heating technique.

Thermal expansion measurements
The thermal expansion from 4.2 K to 300 K was measured with a miniature capacitance dilatometer 36 using the tilted plate principle. 37,38 For this measurement, the sample was placed in a hole of the lower ring-like silver capacitance plate, which is separated from the silver upper capacitor plate by two needle bearings. For measurement of the thermal expansion between 80 and 420 K, a dynamic mechanical analyzer DMA7 (Perkin Elmer Inc.) was employed. The sample was positioned in a parallel plate mode with a quartz rod on top of the sample and data are gained using the thermo-dilatometric analysis (TDA), see ref. [39][40][41][42].

Elastic property measurements
Resonant ultrasound spectroscopy (RUS) outlined by Migliori et al. 43 was used to determine the elastic properties. For this measurement, the cylindrically shaped samples of 10 mm diameter and masses between 2.5 g and 5 g were mounted "edgeto-edge" between two piezo-transducers and were excited via a network analyser in the 100 to 500 kHz frequency range at room temperature. Macroscopically, the polycrystalline samples are isotropic and the set of eigenfrequencies gained was then fitted providing the values for the Young's modulus E and Poisson's ratio ν.

Hardness measurements
The load-independent Vickers hardness HV was determined using an Anton Paar MHT 4 microhardness tester mounted on a Reichert POLYVAR microscope evaluating all the indentation data using eqn (2.2).
F gives the indenter load and d is defined as with d 1 and d 2 being the resulting diagonal lengths of the indenter, respectively.

Severe plastic deformation (SPD) via high-pressure torsion
The HPT technique is based on the use of a Bridgman anviltype device. A thin disk-shaped sample was subjected to torsional strain in a cavity under high hydrostatic pressure between two anvils at room temperature or at elevated temperatures via induction heating. The shear strain γ is dependent on the number of revolutions n, the radius r, and the thickness t of the specimen in the following way: Therefore, the cylindrical single-phase samples were cut into slices with a thickness (t ) of about ∼1 mm and a diameter of 10 mm. The Sb-rich alloys were processed at 400°C under a hydrostatic pressure of 4 GPa with 1 revolution, whereas the temperature for Sn-rich samples was only 300°C.

DTA measurements
Melting points were determined from DTA measurements recorded on a NETZSCH STA 409 C/CD equipment. Pieces of single-phase samples weighing 500 mg to 600 mg were sealed in evacuated quartz glass crucibles. In general, three heating and three cooling curves were recorded for each sample using a scanning rate of 5 K min −1 .

Crystallite size evaluation
The crystallite size was evaluated from the X-ray diffraction patterns (spectra from Cu Kα 1 radiation) using the MDI JADE 6.0 software (Materials Data Inc., Liverpool, CA). This method yields the crystallite size from the full width at half maximum (FWHM) of a single diffraction peak using the Scherrer formula. 44 Silicon was used as an internal standard for instrumental broadening. The calculations were performed for three well separated reflections, (240), (332) and (422), within a 2θ range from 51°to 58°. For details see ref. 45.

Filling levels, phase equilibria and homogeneity region
Phase equilibria in the ternary system Ni-Sn-Sb at 450°C For the binary boundary systems, we accepted the Ni-Sn phase diagram reported by Schmetterer et al. 46 and the Ni-Sb system reported by Cha et al., 47 whereas the Sn-Sb phase diagram is consistent with a recent reinvestigation reported by Polt et al. 48 Grytsiv et al. reported that the ternary skutterudite phase Sn y Ni 4 Sb 12−x Sn x (τ) exhibits a wide range of homogeneity at 250°C (2.4 ≤ x ≤ 5.6, 0 ≤ y ≤ 0.31) and at 350°C (2.7 ≤ x ≤ 5.0, 0 ≤ y ≤ 0.27). 18 The phase equilibria change drastically by increasing the temperature from 250 to 350°C and the authors 18 suggested that at least two invariant reactions exist in this temperature range. Mishra et al. 19 presented a detailed reaction scheme for the Sn-rich part of the system involving three invariant equilibria in the same temperature range. In order to derive the phase equilibria at 450°C, we annealed several samples at this temperature for at least 30 days. Threephase equilibria (Sb) + NiSb 2 + τ and NiSb 2 + α + τ at 450°C ( Fig. 1a and b) were found to be similar to those observed by Grytsiv et al. at 250 and 350°C. 18 We also observed threephase regions L + (Sb) + τ, L + α + τ, and L + Ni 3 Sn 2 + α ( Fig. 1c-e), which agree well with the Schultz-Scheil diagram derived by Mishra et al. 19 The solid solution α (NiSb-Ni 3 Sn 2 ) was observed in all the as-cast samples and has the widest liquidus field, likely being the most stable phase in the system. Thus, the as-cast alloy 2.7Ni-24.7Sn-72.6Sb shows primary crystallisation of the alpha-phase, which is surrounded by NiSb 2 and the last portion of liquid crystallizes with the composition 40Sn-60Sb. At 450°C, the sample is in solid-liquid state ( Fig. 1c): huge grains of the skutterudite (Ni 4 Sb 9.1 Sn 2.9 ) and antimony (Sb) were grown in equilibrium with the liquid with the composition 0.3Ni-47.75Sn-51.95Sb. Three-phase equilibria with the liquid phase L + Ni 3 Sn 4 + α and L + α + τ are also well confirmed via investigation of the samples 21.5Ni-25.5Sn-53Sb and 35Ni-52Sn-13Sb ( Fig. 1d and e). Between these two phase triangles, a huge two-phase field arises containing an Sn-rich liquid and the α-phase (Fig. 1f ). The latter solid solution, formed by congruent melting compounds, separates the phase diagram into two subsystems for which the phase equilibria may be investigated independently. The equilibria in the Ni-NiSb-Ni 3 Sn 2 subsystem have not been reinvestigated but were introduced after Burkhardt and Schubert (Fig. 2). 49 The composition and lattice parameters of the skutterudite phase τ coexisting in the various three-phase equilibria at 450°C are listed in Table 1. The isothermal section at 450°C is shown in Fig. 2. At 450°C, the homogeneity region of the ternary skutterudite phase τ-Sn y Ni 4 Sb 12−x Sn x extends for 2.4 ≤ x ≤ 3.2. By comparing the compositions of the ternary skutterudites Ni 4 Sb 12−x Sn x as a function of the Sn-content from the homogeneity regions at 250°C and 350°C 18 as well as at 450°C with those after DTA, a good agreement was found (see Fig. 3).
One may see a weak temperature dependence for the Sb-rich end of the skutterudite solid solution that coexists in three-phase solid-state equilibria: (Sb) + NiSb 2 + τ and NiSb 2 + α + τ. However, the Sn-rich side of the homogeneity region in Fig. 1 Microstructure of Ni-Sn-Sb alloys annealed at 450°C. The nominal composition (at% from EPMA) and X-ray phase analyses are given in Table 1. Fig. 2 Isothermal section of the Ni-Sn-Sb system at 450°C. The microstructure of the investigated alloys is given in Fig. 1. The phase equilibria in the (Ni)-NiSb-Ni 3 Sn 2 subsystem are shown by Burkhardt and Schubert. 49 equilibrium with the liquid phase shows a significant temperature dependence. With increasing temperature, the composition of the liquid gets strongly depleted on Sn, resulting in a pronounced phase segregation during the crystallization of the samples. DTA of the single-phase sample Ni 4 Sb 8.2 Sn 3.8 (Fig. 4a) reveals a solidus temperature of 410°C during the first heating (  Table 2); however, the temperature was increased to 453°C after the second and third heating cycles (see Fig. I of the ESI ‡). Further additional thermal effects appear in the temperature range from 415 to 423°C. EPMA of the sample after DTA (Fig. 5a) shows three phases α, β (SnSb) and skutterudite τ with an Sb-enriched composition (Ni 4 Sb 8.9 Sn 3.1 ).
Homogeneity region for the filled skutterudite Ba y Ni 4 Sb 12−x Sn x In general, the filling of the 2a site in the skutterudite structure has a maximal effect on the reduction of the lattice thermal conductivity. 6,28 In order to define the solubility of the electropositive filler (Ba) in Ni 4 Sb 12−x Sn x , two samples with nominal compositions of 1Ba-4Ni-5Sb-7Sn and 1Ba-4Ni-8Sb-4Sn were investigated in the as-cast state and after annealing at 250, 350 and 450°C.
Rietveld refinements for the occupancy in the 2a site in combination with EPMA measurements are shown in Fig. 6. No skutterudites were observed in the as-cast alloys. The skutterudite phase is formed after annealing and appears in the microstructure as the main phase for one of the three temperatures 250, 350 or 450°C (Fig. 7). In all cases, we observed an increase in the filling level with increasing Sn-content reaching y = 0.93 for the Ba filled skutterudite, while Fig. 7 additionally shows a clear dependence of the filling level on the temperature. Furthermore, for the Sn-rich compositions, EPMA reveals two sets of skutterudite compositions in the samples annealed at 350°C and 450°C. Such phase segregation is not likely connected with diffusion, suppressed at these temperatures, because the heat treatment at 250°C yields the formation of a skutterudite with a uniform composition. Considering the interesting behaviour of the Ba filled skutterudite, this compound was selected for detailed investigations at 450°C.
For several quaternary samples, annealed at 450°C, the Ba filling level in dependence of the Sn content was determined by Rietveld refinements combined with microprobe measurements and yielded a large homogeneity region of Ba y Ni 4 Sb 12−x Sn x , as shown in a two-dimensional (2D) projection in Fig. 8. Obviously, the maximal solubility of Ba in the skutterudite (y > 0.93) responds to two cases: (i) an equilibrium with the Sn-rich liquid and (ii) the crystallization of skutterudites with two compositions (Ba 0.94 Ni 4 Sb 6.1 Sn 5.9 and    Ba 0.93 Ni 4 Sb 5.1 Sn 6.9 ) that coexist with Ni 3 Sn 4 . Upon increasing the Sb-content, the solubility decreases to y = 0 in the threephase equilibria with (Sb) and NiSb 2 as well as with α and NiSb 2 . DTA provided further information on the stability of Ba y Ni 4 Sb 12−x Sn x . Three single-phase samples were used: Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 ( Fig. 4c and d) after ball milling and hot pressing, and the compound Ba 0.29 Ni 4 Sb 9.1 Sn 2.9 , which appeared to be a single-phase after annealing at 450°C (Fig. 4b). For these alloys, three heating and three cooling curves were recorded using a scanning rate of 5 K min −1 (see Fig. II to IV of the ESI ‡). In all the first heating curves, only one signal could be detected; however, at lower temperatures, the subsequent heating and cooling curves revealed additional thermal effects associated with the incongruent melting of the alloys (also see Table 2). This was confirmed by the microstructure of the samples after DTA ( Fig. 5b-d). In all cases, we observe the formation of a high melting α-phase with subsequent crystallization of the β-phase and BaSb 3 . A tin-rich liquid was found to form during crystallisation of Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 . Due to incongruent crystallization of Ba y Ni 4 Sb 12−x Sn x , the composition of the skutterudite phase changes similarly to the ternary Ni-Sn-Sb skutterudite (see chapter above). The melting points, T m , for the samples before and after DTA are listed in Table 2 and the compositional dependences of the solidus temperature were compared with data for the Ni-Sb-Sn skutterudite in Fig. 9. The filling of the skutterudite lattice with Ba-atoms clearly results in a significant increase in the melting temperature and this influence is particularly visible for the Sn-rich samples.
Knowledge of the extension of the homogeneity region for Ba y Ni 4 Sb 12−x Sn x as a function of temperature was used to prepare single-phase samples for measurements of their physical properties (Fig. 4a, c, d and Table 3) and to grow single crystals for structural investigations.

Crystal structure and vibration modes of Ba y Ni 4 Sb 12−x Sn x
Rietveld refinements of the X-ray powder data for Ba y Ni 4 Sb 12−x Sn x were fully consistent with the skutterudite structure CoAs 3 in the unfilled case and LaFe 4 Sb 12 in the filled case. The refinements combined with EPMA defined the degree of filling in the 2a site and the Sb/Sn ratio in the 24g site. The compositional data and corresponding lattice parameters for all new phases of Ba y Ni 4 Sb 12−x Sn x are summarized in Table 4. The lattice parameters revealed neither a dependence on the filling level of Ba nor on the Sb/Sn ratio.
For two flux-grown single crystals, Ba y Ni 4 Sb 12−x Sn x (for details see micrographs in Fig. 10), the X-ray diffraction intensities were recorded at three different temperatures (100, 200 and 300 K). The refinement in all cases proved isotypism with the LaFe 4 Sb 12 type (filled skutterudite; space group Im3) with Ni atoms occupying the 8c site, whereas Sb and Sn atoms randomly share the 24g site. The residual density at the 2a site was assigned to Ba atoms. The compositions derived from the structural refinement, namely, Ba 0.73 Ni 4 Sb 8.1 Sn 3.9 and Ba 0.95 Ni 4 Sb 6.1 Sn 5.9 , are in accordance with the microprobe measurements. Final refinements with fixed occupancies (Tables 5 and 6) and anisotropic atom displacement parameters (ADPs) led to a reliability factor R F below 2%. The maximum residual electron density of ∼3 e − Å −3 appears at a distance of 1.5 Å from the centre of the 24g site. This density can be interpreted as a "diffraction ripple" of the Fourier series around the heavy Sb and Sn atoms located at this site. The interatomic distances for Ni and (Sn/Sb) lie within range of values known for CoSb 3 15 and LaFe 4 Sb 12 . 52 Fig. 11 shows the temperature dependence of the ADP for Ba 0.73 Ni 4 Sb 8.1 Sn 3.9 and Ba 0.95 Ni 4 Sb 6.1 Sn 5.9 . Filler atoms in cage compounds such as skutterudites generally exhibit ADPs at RT, which are about three to four times higher than those of the framework atoms. 6 In the case of Ba y Ni 4 Sb 12−x Sn x , this  factor is only slightly above 1.2. Despite their low ADPs, the Ba atoms may be tentatively described as harmonic Einstein oscillators uncoupled from a framework that behaves as a Debye  Compositional dependence of the solidus curve for Ba y Ni 4 Sb 12−x Sn x in comparison to that of Ni 4 Sb 12−x Sn x (for details see Fig. 3). For the melting temperatures, see Table 2.  2)) applied to the framework atoms (Ni, Sb and Sn) revealed only slight differences between θ D for Ni and the random Sn/Sn mixture.

Dalton Transactions Paper
θ E and θ D are listed in Table 7. As the skutterudite framework is built by the Sn and Sb atoms forming octahedra with the Ni atoms in the octahedral centres, one may explain these results as follows: because of its position in an octahedral cage formed by Sb and Sn, the vibration of the strongly bonded Ni atom is not independent from that of the framework and therefore leads to values close to those obtained for Sb/Sn.   By fitting the temperature dependent lattice parameters via a linear function of the form a·x + b (with a = 1.02 × 10 −5 and b = 0.92 in the case of Ba 0.73 Ni 4 Sb 8.1 Sn 3.9 , but a = 1.15 × 10 −5 and b = 0.92 for Ba 0.95 Ni 4 Sb 6.1 Sn 5.9 ), the thermal expansion coefficients for the two compositions could also be determined ( Table 7). All values for α lie within the range discovered for other skutterudites and increase with higher filling levels. 22,23

Mössbauer spectroscopy
In order to access and quantify a potential occupation of the rattling 2a site by Sn, temperature dependent Mössbauer spectroscopy measurements were conducted for the 119 Sn resonance from 10 to 290 K. As potential Sn on the 2a site should be rather weakly bonded, a very low Lamb-Mössbauer factor  (5) 0.00117 (6) 0.00118 (10)   would be expected for the corresponding component, which necessitates the low temperature measurements. The 119 Sn Mössbauer spectra are shown in Fig. 12 and are basically characterized by a broad, slightly asymmetric doublet-like structure. The spectra were fitted with a model consisting of up to three doublets, which represent the different configurations of Sn on the 24g site with zero, one or two Sb next neighbors (NN). As for each Sn atom, the probability of two NN Sn atoms is above 0.1 only for sample Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 , the Mössbauer spectra of the other samples were analyzed using two doublets. Exemplarily, these different components are also shown in Fig. 12 for measurements at 290 K and the Mössbauer parameters of all components at all temperatures are summarized in Table I of the ESI. ‡ With the exception of the Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 sample, which exhibits the highest filling fraction, component I (see Table I of the ESI ‡) consistently shows the biggest quadrupole splitting, which can be related to the more covalent character of the bond 53 in the case of Sb NN. The average isomer shifts and quadrupole splittings are shown in Fig. 13 and reasonably   agree with the previously reported Mössbauer data on Ni 4 Sb 9 Sn 3 . 18 Notably, average isomer shifts and quadrupole splittings in the Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 sample significantly differ from the values obtained for the other compounds (see Fig. 13). Considering the available data, a correlation of Mössbauer parameters with the Ba-or Sn-content is, however, rather ambiguous. As there is no indication for a Mössbauer component with Einstein like behavior and/or significant contribution only at low temperature, which would be the case for Sn on the 2a site, the possibility of Sn on the rattling position is ruled out. The relative absorption area of a Mössbauer spectrum is pro-    portional to the Lamb-Mössbauer factor f LM and thus, the temperature dependence of the former can be used to fit f LM within a Debye model: 54 where E γ is the Mössbauer resonance energy, M is the nuclear mass and θ D is the Debye temperature (sometimes called Mössbauer temperature in this context). For all four skutterudites, the results are displayed in Fig. 14 including the corresponding Debye temperatures. The latter values obtained from Mössbauer spectroscopy are in good agreement with the Debye temperatures based on the X-ray diffraction and heat capacity measurements (see Table 7). In particular, the hardening of the host lattice with increasing filling fraction is reflected in the fitted Debye temperatures and exhibits an almost linear dependency with a slope of about 55(11) K per Ba per f.u.    The 121 Sb-Mössbauer spectra obtained at 10 K are shown in Fig. 15. They are characterized by a single asymmetric peak, which in the case of 121 Sb indicates a quadrupole splitting. All the 121 Sb-spectra could be fitted with a single component. The corresponding Mössbauer parameters are summarized in Table II of the ESI. ‡ Due to the rather large linewidth of the 121 Sb Mössbauer transition, linewidth, quadrupole splitting and the asymmetry term are highly correlated and particularly the asymmetry term cannot be reliably fitted. As fitting with a fixed, vanishing asymmetry term slightly improved the fitting quality when compared to fitting with a fixed, maximal asymmetry term, all the results presented herein were obtained with the vanishing asymmetry term. This aspect and the quadrupole splitting values are in contrast to the reports on CoSb 3 mentioned in ref. 55 and 56, which can, however, be attributed to the reduction of Sb content in the present samples. In any case, the present findings validate the sample quality from the Sb perspective.

Specific heat
To gather further information on the vibrational behavior of the filled and unfilled Ba y Ni 4 Sb 12−x Sn x skutterudites, specific heat measurements in the temperature range from 3 to 140 K were applied to the single-phase specimens Ni 4 Sb 8.2 Sn 3.8 , Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 . The data are displayed in Fig. 16 in the form of a C p /T versus T plot. The specific heat, C p , of simple non-magnetic materials can be expressed as γ is the Sommerfeld coefficient and β is inversely proportional to θ D . N A is Avogadro's number, k B is the Boltzmann factor and n stands for the number of atoms per formula unit. Table 7 summarizes the parameters obtained from the least square fitting below 5 K. Fig. 16 suggests a moderate overall lattice softening with increasing Ba and Sn content. To extract the corresponding Einstein temperatures and Einstein frequencies, the specific heat data were analysed by applying two different methods. The first approach was based on an additive combination of the Debye and Einstein models. It is assumed that the phonon spectrum of a polyatomic compound contains three acoustic branches and 3n-3 optical branches, where the acoustic part of the phonon specific heat can be described by the Debye model (eqn (6.3)) with R being the gas constant and ω D = θ D /T. Herein, the three acoustic branches are taken as one triply degenerated branch. In a similar way, the Einstein model describes the optical branches,  were conducted with a fixed, vanishing asymmetry term. Fig. 16 Temperature-dependent specific heat divided by temperature C p /T for Ni 4  Einstein function was added to describe the increase of phonon modes, with c 3 = 3·y, where y represents the filling level. The heat capacity of the skutterudites follows then from with the sum running over two or three Einstein modes. The least squares fittings obtained according to this model are presented as dashed lines in Fig. 16 and the extracted data for Einstein and Debye temperatures are also listed in Table 7.
The second approach uses the model introduced by Junod et al. 57,58 to gain insight in the complex phonon spectrum. Special functionals of the phonon specific heat can take the form of convolutions of the phonon spectrum F(ω). Therefore, the electronic part of the specific heat is subtracted from C p (T ) (C ph (T ) = C p (T ) − γ·T ) and least squares fitting with two estimated Einstein-modes were applied to the phonon specific heat, as shown in Fig. 17, in the form of C ph /T 3 vs. T. Further details on this method are described for example in ref. 59. In Fig. 17, the corresponding phonon spectra are plotted as solid lines scaled to the right axis. The simple Debye function based on eqn (6.3) is presented as a dotted line using the θ D -data extracted together with the Sommerfeld coefficients. A comparison between filled and unfilled skutterudites shows that the filler element Ba has a strong influence on the Ni-Sn-Sb network. These results are also consistent with the increase in the Debye temperature. All data referring to the specific heat are listed in Table 7. The Debye and Einstein temperatures extracted by the two different methods described above match well with each other for all three samples investigated.

Electronic and thermal transport
As the Ba y Ni 4 Sb 12−x Sn x compound exists within a wide homogeneity region, detailed investigations on the transport properties as a function of the composition were performed.   seems to correlate with an increase of the free carrier concentration, increasing the Sn-content seems to exert the inverse effect. Although the concept introduced by E. Zintl 60,61 is based on simple crystal chemistry, it appropriately describes the changes in ρ(T ) with the composition of Ba y Ni 4 Sb 12−x Sn x alloys on the basis of a simple counting of charge carriers n z , i.e., n z ¼ yÁ2ðBaÞ þ 4Á10ðNiÞ À ð12 À xÞÁ3ðSbÞ À xÁ4ðSnÞ ð7:1Þ

Electrical resistivity
This composition dependent change of carriers can also be seen from the Seebeck coefficients S(T ) of the three samples, as discussed below.
For all the three samples, the temperature dependent electrical resistivity exhibits two different regimes. At low temperatures, a metallic-like behaviour was obtained, which changes to a semiconducting behaviour at higher temperatures, at least up to 700 K. This change cannot be explained by a simple activation-type conductivity mechanism. Thus, a rectangular model for the density of states was considered (2-band model), with a narrow gap lying either slightly above or below the Fermi energy E F , 62 which successfully described resistivities ρ(T ) of various clathrate and skutterudite systems (see for example ref. 18 and 59). While the former case describes holes as primary charge carriers, the latter one accounts for electrons as the dominating carriers. In these models, the unoccupied states above E = E F are available at T = 0 K. Simple metallic conductivity is possible at low temperatures but becomes semiconducting-like once electrons are excited across the energy gap of width E gap . The total number of carriers (electrons and holes) is strongly dependent on the absolute temperature as well as on E gap . On the basis of these assumptions, ρ(T ) can be calculated using where n 0 gives the residual charge carrier density, n(T ) the total charge carrier density, ρ 0 is the residual resistivity and ρ ph accounts for the scattering of electrons on the phonons, taking into account the Bloch Grüneisen law: where < stands for a temperature independent electronphonon interaction constant and A·T 3 represents the Mott-Jones term. 63 Table 7 and can be compared to data obtained using other methods (see below).

Hall data
The Hall effect was studied for all three single-phase samples Ni 4 Fig. 20. The mobility μ was calculated from the relation μ = σ/n·e, with σ being the electrical conductivity, n the charge carrier density and e the carrier charge (in this case electron charge). The temperature independent mobility μ(T ) below 100 K was revealed from detailed analysis, i.e., assuming non-degeneracy for charge carriers and a distribution function given by Maxwell-Boltzmann statistics, usually attributed to neutral impurity scattering in semiconductors. 59 For Ni 4 Sb 8.2 Sn 3.8 and Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 , the decrease in μ with temperature between 100 to 300 K was close to T −1/2 , which suggests alloy scattering as the main scattering process. This effect occurs whenever different atoms in the semiconductor compounds are not arranged periodically in the crystal structure. 64 The electron effective mass m* can be calculated using eqn (7.4), which is usually valid for a single parabolic band at high n and/or low T. 65 Evaluation shows a significant increase in the effective mass with increasing filling level, rising from 3.7m e for Ni 4 Sb 8.2 Sn 3.8 to 7.6m e for Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 . For Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 , a further increase up to a value of 15.2m e was found. In addition to the increase in mobility at higher temperature, this indicates that the assumption of a single parabolic band is not valid. The Pisarenko plot (Fig. 21) shows the Seebeck coefficients at 300 K as a function of charge carrier density. The dashed line indicates the maximum thermoelectric performance for the unfilled sample Ni 4 Sb 8.2 Sn 3.8 at 423 K.

Thermopower
The temperature-dependent Seebeck coefficients, S(T ), for the samples with the composition Ni 4 Sb 8.2 Sn 3.8 , Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 are displayed in Fig. 22a. S(T ) is negative at low temperatures up to at least 400 K, suggesting that the transport phenomena are dominated by electrons as charge carriers, indicating n-type behaviour. This can be explained by the Zintl concept in parallel to the electrical resistivity, as discussed above. Below about 400 K, S(T ) behaves almost linearly before reaching a minimum between 400 and 500 K. The Seebeck coefficient can be understood in terms of Mott's formula: 65 with m* being the effective mass and e the respective charge of the carriers involved. Eqn (7.4) is assumed to be valid for systems without significant electronic correlations. Furthermore, Goldsmid and Sharp 66 showed the possibility for estimating the gap width E gap from the maximum/ minimum of S(T ), i.e., As in the present case, S(T ) is negative over the entire temperature range investigated, the minima data were inserted into eqn (7.5), leading to the results summarized in Table 8. The estimated gap energies E gap are in accordance with those obtained from temperature dependent electrical resistivity and show a significant decrease from Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 to Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 . A higher Sn content in the network (24g site) appears to influence the band structure more than an increased Ba filling level in the icosahedral voids (2a site). Radiation losses in the low-temperature steady-state heat flow measurements were corrected subtracting a Stefan-Boltzmann A·T 3 term. In general, the thermal conductivities are quite high when compared with those of other skutterudites (see for example ref. [8][9][10][11], but they lie in the same range as those obtained by Grytsiv et al. 18 for their Sn filled and unfilled Sn y Ni 4 Sb 12−x Sn x compounds. For a more detailed analysis (shown in Fig. 25), only data between 4 and 300 K were taken into account. The total thermal conductivity can be written as where κ el presents the electronic part and κ ph the phonon part. A number of scattering processes limits both contributions, so that a finite thermal conductivity will result in any case. For simple metals, κ el can be calculated from the temperature dependent electrical resistivity via the Wiedemann Franz law:

Thermal conductivity
where the temperature dependent Lorentz number L 0 (T ) is given by as proposed by Rowe et al. 67 In order to obtain the correct values for L 0 (T ), the reduced Fermi energy, ð7:9Þ   has to be extracted from the measured Seebeck coefficients according to and F n (ξ) being the n th order Fermi integral, ð7:11Þ with k B being the Boltzmann constant, e the electron charge, E F the Fermi energy and s the scattering parameter. Assuming acoustic phonon scattering as the main carrier scattering mechanism, s = −1/2, L 0 (T ) can be calculated by substituting s and ξ in eqn (7.8 Subtracting the κ el term from the measurement data results in κ ph , as shown in Fig. 25a, c and e. For all three samples, the electronic part is quite low but is enhanced at higher temperature. According to the Matthiessen rule, the electronic thermal resistivity, W el of a simple metal can be written in terms of the thermal resistivity, W el,0 caused by electron scattering due to impurities and defects as well as that caused by electrons scattered due to thermally excited phonons, W el,ph : Using the Wilson equation, 68 W el,ph can be expressed as with k F being the wave vector at the Fermi surface, q D the phonon Debye wave vector and A a material constant, which depends on the strength of the electron-phonon interaction, Debye temperature, the effective mass of an electron, the number of unit cells per unit volume, Fermi velocity and the electron wavenumber at the Fermi surface. The integrals J n have the form, In non-metallic systems, the lattice thermal conductivity is the dominant part of the thermal conduction mechanism, which can be described by a model introduced by Callaway. [69][70][71] Accordingly, heat carrying lattice vibrations can be described by with the velocity of sound as derived within the Debye model: where N is the number of atoms per unit volume and ω the phonon frequency. The second integral I 2 in eqn (7.15) can be expressed as Here, τ N −1 , τ D −1 , τ U −1 , τ B −1 and τ E −1 denote the normal 3-phonon scattering process, scattering by low-dimensional lattice defects, Umklapp processes, boundary scattering and scattering of phonons by electrons, respectively. The dashed lines in Fig. 25a, c and e refer to least squares fitting according to eqn (7.6) assuming a combination of the Wilson and Callaway models for κ el and κ ph . The extracted values for the Debye temperatures θ D are listed in Table 7 and show a good agreement with the values gained from various other methods described in this article and compare well with literature data.
The minimum thermal conductivity κ min , presented as shaded area in Fig. 25a, c and Here, N is the number of atoms per unit volume and x is a dimensionless parameter connected to the phonon frequency Fig. 25b, d and f compare the situation for temperatures above 300 K. For the two compounds, Ni 4 Sb 8.2 Sn 3.8 and Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 , κ el is the dominant part of thermal conductivity, whereas κ ph is of the order of κ min . For Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 , both the parts, κ ph as well as κ el , are in the same order in magnitude but 2 to 3 times higher than those found for the previous samples. This may be caused by two effects: (i) substitution of Sb with Sn-atoms in the 24g site is combined with an increase in the thermal conductivity, or more likely, as reported in various articles, 74,75 (ii) the vibrations of the filler atom are not independent from those of the network atoms.

Figure of merit
The dimensionless figure of merit zT characterizes the thermoelectric ability of a single material concerning power generation or cooling. For commercial applications, a zT above 1 is desirable. The temperature dependent figure of merit can be calculated using eqn (1.1) and is shown in Fig. 26a. Although the electrical resistivity for all the three investigated compounds Ni 4 Sb 8.2 Sn 3.8 , Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 is quite low, low absolute values of the Seebeck coefficients S(T ) (<70 μV K −1 ) and high thermal conductivities κ(T ) (>20 mW cm −1 K −1 ) prevent zT from reaching values above 0.11 at RT. As already could be seen from the Pisarenko plot, the highest values for zT are achieved from the unfilled sample Ni 4 Sb 8.2 Sn 3.8 .

Transport properties after highpressure torsion (HPT)
Severe plastic deformation (SPD) via HPT is one technique to significantly increase the performance of thermoelectric materials, [76][77][78] with various effects being responsible for this behaviour. Because of the interdependence of ρ, S and κ el , one strategy to increase zT is to reduce the κ ph by increasing the scattering of heat carrying phonons via various mechanisms, as already discussed in chapter 7 in terms of their corresponding relaxation times τ n (eqn (7.19)). In general, I 2 in eqn (7.18) is negligible for τ N ≫ τ U . A possible route to decrease κ ph therefore is either to reduce the grain size d g , because τ B −1 = v s /d g , and to increase the density of dislocations, as τ D −1 includes the contribution of dislocations.
This term consists of the dislocation core, with N D being the dislocation density, and of the dislocation strain field, Here b represents the Burgers vector of the dislocation. Hence, a smaller grain size and/or higher dislocation density results in enhanced phonon scattering on both electrons as well as on lattice defects and this way decreases κ ph . Furthermore, Hicks and Dresselhaus 79,80 demonstrated that grain sizes, approaching nanometer length scales (favourable <10 nm) are able to influence the Seebeck coefficient. On inspecting Mott's formula (eqn (7.4)), it can be seen that the Seebeck coefficient is primarily dependent on the logarithmic energy derivative of the electronic density of states (DOS) at the Fermi-energy. Therefore, any method that is able to increase the slope at E F for a given number of states N will increase S. HPT uses this concept by the transition from a parabolic elec-tronic DOS curve to a spike-like curve while going from macroscopic bulk structures to nanosized ones. With these observations, a way is opened to influence S independently from ρ and κ el . Based on this knowledge, HPT was performed by applying a hydrostatic pressure of 4 GPa, one revolution and a processing temperature of 400°C on two skutterudite samples, Ni 4 Sb 8.2 Sn 3.8 and Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 , whereas for compound Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 , a temperature of 300°C was used. During the deformation, the sample's geometry remains constant and the application of enhanced hydrostatic pressure enables practically unlimited plastic strain without early failure and crack formation. 81,82 Therefore, with the hydrostatic pressure not only low dimensional defects are created, but also grain boundaries are built up from these defects. [82][83][84][85][86][87] The resulting shear strain corresponds to eqn (2.4); thus, the deformation affects the samples rim more than the centre.  Therefore, the physical properties of the processed specimens always cover the more and less deformed parts. Pieces of all parts of the HPT processed samples were collected and used for X-ray powder analysis. In general, an increase in the half width of the X-ray patterns as well as of the lattice parameters was observed, indicating a reduction of the crystallite size and an increase of the dislocation density. For the Ni 4 Sb 8.2 Sn 3.8 and Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 samples, this behaviour is distinct, whereas for compound Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 , almost no change was visible (Fig. 27). The X-ray peak profile analysis shows a reduction in the grain size for all three samples (Fig. 28), whereas the largest effect was observed for the unfilled sample Ni 4 Sb 8.2 Sn 3.8 (Fig. 28). In the EPMA micrographs before and after HPT (Fig. 29), no changes in the microstructure were observed. Due to microcracks, the density of the materials after HPT processing was lower than before (compare values given in Table 3 and Table 9). Fig. 18b, 22b, 24b, 25b, d and f summarize the effect of HPT treatment on the physical properties, ρ(T ), S(T ) and κ(T ).
The electrical resistivity for all the samples (Fig. 18b) after HPT was higher. While a crossover from metallic to semiconducting behaviour was observed for Ni 4 Sb 8.2 Sn 3.8 and Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 , Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 was semiconducting over the entire temperature range investigated. The data for the Seebeck coefficients after HPT (Fig. 22b) stay in the same range as those before HPT or are even slightly lower, which results in no evident change in the temperature dependent Lorentz number L 0 (T ) (Fig. 24b). The thermal conductivity of the Sb-rich samples ( Fig. 25b and d) seems to stay almost the same. Determining κ el and κ ph as shown in chapter 5 reveals that neither κ el nor κ ph was influenced significantly, resulting in no-evident change in the thermal conductivity κ. For the Sn-rich sample (Fig. 25f ), both components, κ el as well as κ ph , are lowered leading to a decrease in the overall thermal conductivity κ. In contrast to the filled skutterudites without Sb-Sn substitution, all these observations show no significant   change in the figure of merit above room temperature (Fig. 26). Δl/l 0 decreases almost linearly within the temperature range from room temperature to about 150 K, whereas for temperatures below 150 K, nonlinear behaviour was observed. The data gained from the measurements using a dynamic mechanical analyser with the sample first cooled with liquid nitrogen and afterwards heated up or simply heated from room temperature to 400 K show a linear increase with increasing temperature (Fig. 31). The temperature derivative of the length change defines the thermal expansion coefficient α, i.e.

Thermal expansion
α ¼ @Δl @l Á 1 l 0 : ð9:1Þ Applying this model, α was calculated in the temperature range above 150 K up to 300 K (see dashed lines in the insert of Fig. 30) and also from a combination of the low and high temperature measurement data. The corresponding thermal expansion coefficients are listed in Table 7. The values obtained for the current samples lead to a good agreement between the high and low temperature data proven by the combination of the data according to Fig. 31. From Table 7, it follows that the thermal expansion coefficients α for Ba y Ni 4 Sb 12−x Sn x based skutterudites lie within the range obtained for other skutterudites. 22 A higher value of α occurs with a higher Ba filling level, visible at higher temperatures, which can be associated with the rattling behaviour of Ba on the 2a site inside the structural cage. For a cubic material, the lattice parameter varies with the temperature in parallel to the thermal expansion coefficient. Therefore, the lattice para-  meters a n at various temperatures can be derived if a n at a certain temperature is known, by applying eqn (9.2).
The values according to all three specimens were added to Fig. 31 88 was used. This model takes into account threeand four-phonon interactions, considering anharmonic potentials and uses both the Debye model for the acoustic phonons and the Einstein model for the optical modes. The length change Δl/l(T 0 ) is then given by and ε of the form, ð9:5Þ Herein, ξ stands for the electronic contribution to the average lattice displacement, θ D is the Debye temperature, θ E is the Einstein temperature and p is the average number of phonon branches actually exited over the temperature range with G, F, c, and g being material constants. Least squares fitting to the experimental data according to eqn (9.3) were added to Fig. 30 and 31 as dashed lines. The values obtained for θ D and θ E are listed in Table 7 and can be compared with the data gained from fitting to the electrical resistivity, thermal conductivity and specific heat. This comparison shows good agreement among data from the various methods as well as with those available in the literature.

Elastic properties
As the average symmetry of skutterudites is isotropic, the Young's modulus E and Poisson's ratio ν can be gained by fitting the variables of the materials eigenfrequencies obtained from the RUS measurements of all three single-phase samples Ni 4 Sb 8.2 Sn 3.8 , Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 . Based on these data, all other elastic properties were calculated. The shear modulus G and the bulk modulus B can be obtained from For an isotropic material, the elastic constant C 11 is given by The longitudinal (v L ) and the transversal (v T ) sound velocities can be derived from one of the following relations: Herein, ρ s is the samples' density measured via Archimedes' principle. From the knowledge of v L and v T , the materials mean sound velocity v m can be calculated using eqn (10.6).
Anderson's relation 89 yields the Debye temperature θ D from the elastic properties measurements.  Table 10 and are shown in Fig. 32. A comparison with the elastic property data available in the literature (see for example ref. 23) shows that Ba y Ni 4 Sb 12−x Sn x based skutterudites are characterized by a rather low Young's modulus and high Poisson's ratio, continuing this trend for the bulk modulus, longitudinal (v L ), transversal (v T ) and mean sound velocity. The extracted Debye temperatures (see Table 7) lie in the same range as the values gathered using other methods, as listed in the previous chapters. Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 particularly on skutterudite crystallites above 200 nm in size. As the Vickers method is an indentation measurement, the specimen's resistance is determined against deformation due to a constant compression load from a sharp object. Fig. 33 shows the results for the skutterudites in the multiphase samples, while Fig. 34 presents data for the singlephase samples. A comparison with the data available in the literature (see for example ref. 23) shows that the HV of the Ba y Ni 4 Sb 12−x Sn x skutterudites lay perfectly within the range of other skutterudites. From the measurements performed on the single-phase samples, two trends can be derived (Fig. 35 Fig. 34 and 35) shows no measurable change.

Electron and phonon mean free path
To learn more about the transport properties caused by electrons and phonons, their mean free paths were calculated using simple models and then discussed in comparison with the resistivity and thermal conductivity data. Assuming a three-dimensional system with a spherical Fermi surface, 90 the mean free path ℓ el of the electrons can be calculated from the measured resistivity data,  Herein, e is the electronic charge and ħ is the reduced Planck's constant. The Fermi wave vector k F can be approximated by with a as the lattice parameter. Fig. 36  . For all three specimens, ℓ el is five to ten times higher than the lattice parameter a. This confirms the quite low resistivity ρ(T ) found in these alloys. It seems that Ba filling in the 2a site has a positive effect on ℓ el , whereas increasing the Sn content in site 24g decreases ℓ el . From a simple kinetic theory, one can also estimate the phonon mean free path (eqn (12.3)), which is an important parameter for thermal conductivity: 28 In addition to the electronic part κ el , heat transport is conducted by phonons with a group velocity v g and mean free path lengths ℓ ph between the scattering processes. The amount of transported heat is also proportional to the molar heat capacity C v . For the group velocity v g , a common approximation takes the mean sound velocity v m . Taking our data from elastic properties measurements, C v can then be calculated by with the measured heat capacity C p , the thermal expansion coefficient α and the molar volume V m . β is the isothermal compressibility, with β = 1/B and B being the bulk modulus (see our data above). As for a solid C p ≈ C v , the comparison between the calculated values for C v and the measured values for C p (Fig. 16 and 37) confirms that eqn (12.4) holds true in the affected temperature range.
The results for the phonon mean free path lengths in the temperature range from 4 to 110 K are shown in Fig. 38. As one of the many strategies in designing high performance thermoelectric materials is to decrease κ ph by increasing the   As demonstrated in various papers, 74,75 the reason for this may stem from the fact that the vibrational modes of the rattling atom are not entirely independent from the framework vibrations: the small icosahedral voids in Ba y Ni 4 Sb 12−x Sn x squeeze the Ba-filler atoms into tight bonding (for relative cage sizes see chapter 4). It should be recalled here that reduction of the cage volume in isotypic EuFe 4 Sb 12 by external pressure significantly changes the phonon structure increasing the "rattling mode" above the optical mode without avoided crossing. 91

Conclusion
Phase equilibria in the ternary system Ni-Sn-Sb have been determined for the isothermal section at 450°C, revealing a rather large extension of the Ni 4 Sn 12−x Sn x homogeneity region from 2.4 ≤ x ≤ 3.2. A combination of literature data and DTA measurements enabled us to determine the solidus surface for the unfilled Ni 4 Sn 12−x Sn x phase as well as for the Ba-filled Ba y Ni 4 Sn 12−x Sn x skutterudite phase. The Ba-and Sn-concentration dependent homogeneity region of the Ba y Ni 4 Sn 12−x Sn x phase was established. Although the homogeneity region seems to extend into a p-type region, so far only n-type material was obtained.
Single-phase skutterudites samples with the composition Ni 4 Sb 8.2 Sn 3.8 , Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 have been successfully prepared and characterised by their physical and mechanical properties with respect to their formation and crystal structure. Resistivity data were within a crossover from metallic to semiconducting behaviour and were successfully modeled in terms of a temperature-dependent carrier concentration in a rectangular density of states with the Fermi energy slightly below a narrow gap. The corresponding narrow gap widthsextracted from maxima in the Seebeck coefficient data as a function of temperatureare ranged about 0.050 eV. Although the resistivity ρ(T ) of the compounds investigated (Ni 4 Sb 8.2 Sn 3.8 , Ba 0.42 Ni 4 Sb 8.2 Sn 3.8 and Ba 0.92 Ni 4 Sb 6.7 Sn 5.3 ) is quite low, the rather large thermal conductivity κ(T ) prevents the material from reaching attractive figures of merit, zT. The Debye and Einstein temperatures gathered by various methods lie within the same range and are therefore comparable to those of other skutterudites and consistent with high levels of lattice thermal conductivity. As one of the reasons for the low zT, the Ba filler atoms with rather small ADPs (atom displacement factors) at the centres of tight Ni 4 (Sb,Sn) 12 cages are unable to significantly decrease the phonon mean free paths and constitute a severe reduction of the "rattling behaviour".
Severe plastic deformation (SPD) via high-pressure torsion (HPT) was used to introduce nanostructuring; however, the physical properties before and after HPT showed no significant effect on the materials thermoelectric behaviour.