The ternary system Au–Ba–Si: Clathrate solution, electronic structure, physical properties, phase equilibria and crystal structures

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Acta Materialia 60 (2012) 2324–2336 www.elsevier.com/locate/actamat

The ternary system Au–Ba–Si: Clathrate solution, electronic structure, physical properties, phase equilibria and crystal structures I. Zeiringer a, MingXing Chen a, A. Grytsiv a, E. Bauer b, R. Podloucky a, H. Effenberger c, P. Rogl a,⇑ b

a Institute of Physical Chemistry, University of Vienna, Wa¨hringerstrasse 42, A-1090 Wien, Austria Institute of Solid State Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria c Institute of Mineralogy and Crystallography, University of Vienna, Althanstrasse 14, A-1090 Wien, Austria

Received 14 September 2011; received in revised form 21 December 2011; accepted 22 December 2011 Available online 1 March 2012

Abstract We report on (i) the phase relations at 800 °C within the ternary system Au–Ba–Si up to 33.3 at.% Ba, (ii) on the crystallographic data of new ternary compounds, (iii) on details of the clathrate type I solid solution, (iv) on electrical and thermal transport measurements for Ba8Au5.1Si40.9 supported by (v) density functional theory calculations. The clathrate type I solid solution Ba8AuxSi46x at 800 °C extends from Ba8Au4Si42 (a = 1.039 nm) to Ba8Au6Si40 (a = 1.042 nm). The cubic primitive symmetry (space group Pm 3n) was confirmed by Xray powder diffraction in the whole homogeneity region. The lattice parameters of the solid solution show an almost linear increase with increasing gold content and site preferences from X-ray refinement confirm that gold atoms preferably occupy the 6d site in random mixture with Si atoms. The phase equilibria at 800 °C are characterized by seven ternary phases in the investigated region up to 33.3 at.% Ba. The homogeneity range has been established for Ba(Au1xSix)2 (AlB2 type, extending from BaAu0.4Si1.6 to BaAu0.9Si1.1). BaAu2+xSi2x (unknown structure type) exhibits a very small homogeneity range (x = 0.6–0.7) and two other ternary phases exist at about 22 at.% Ba, 52 at.% Au and 28 at.% Si and 20 at.% Ba, 58 at.% Au and 22 at.% Si (structure types for both unknown). The crystal structures of two further novel phases in the gold-rich part have been determined from single crystal X-ray data: BaAu3+xSi1x of BaAu3Ge type (x = 0–0.3, space group P4/nmm, x = 0: a = 0.6488(2), c = 0.5305(2) nm) and BaAu5xSi2+x (x = 0–0.2, own structure type, space group Pnma, x = 0: a = 0.8935(2), b = 0.6939(2), c = 1.0363(2) nm). The proximity of Ba8AuxSi46x to a metal to insulator transition is corroborated by density functional theory electronic structure calculations. A gap in the electronic density of states, located near the Fermi energy, gives rise to distinct features of the temperature-dependent electrical resistivity and Seebeck effect. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Crystal structure; Density functional; Transport properties; Electronic structure; X-ray diffraction

1. Introduction Since the discovery of superconductivity in Na- and Badoped silicon clathrates of type I (Na,Ba)xSi46 [1,2], the clathrate solution Ba8AuxSi46x has also been intensively studied. Details concerning different clathrate types and crystal structures can be found in Nolas et al. [3] and Rogl [4]. Single crystal X-ray refinements proved isotypism with ⇑ Corresponding author.

E-mail address: [email protected] (P. Rogl).

the clathrate type I structure for three alloys Ba8AuxSi46x, x = 5.4, 5.9 [5] and 6.0 [6]. In contrast to the X-ray single crystal structure determination by Cordier [6] suggesting full atom order with Au atoms in the 6d sites, the X-ray analysis by Jaussaud et al. [5] revealed a lower gold content within the 6d sites, but a small random substitution of Au for Si within the 24k sites. (Wyckoff sites refer to the setting of the crystal structure standardized with the program Structure Tidy. Note that according to this standardization Wyckoff sites 6c and 6d interchange.) Both groups of authors [5,6] agreed on a defect-free Si–Au framework.

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.12.040

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The maximum superconducting fraction in the clathrate solution Ba8AuxSi46x (0.2 6 x 6 6) prepared at 800 °C and 3 GPa was found for x = 1 [7]. The thermoelectric properties, measured for two compounds with x = 5.4 and 5.9, yielded either n-type or p-type behaviour, respectively, but the Seebeck coefficients were rather small for both alloys [5]. 29Si magic angle spinning nuclear magnetic resonance (MAS NMR) spectroscopy indicated a strong contribution of the Si 3s orbital to the band at the Fermi level [5]. Investigations of the electronic structure revealed a metallic character for Ba8Au6Si40, and transmission electron microscopy (TEM) studies confirmed a simple type I unit cell (space group Pm 3n) [8]. Vibrational dynamics and electronic structure have been analysed for Ba8AuxSi46x (x = 1–6) by means of first principles calculations assisted by Raman spectroscopy [9]. It was demonstrated that Au substitution within the Si framework of “Ba8Si46” significantly reduced the electron density at the Fermi level, leaving behind a poor metal BaAu6Si40 [9]. High pressure Raman spectroscopy experiments (up to 27 GPa) have been carried out on Ba8Au6Si40 [10] in order to find an isostructural phase transition similar to Ba8Si46 under high pressure. Besides the type I clathrate, the high temperature and high pressure synthesis of BaAu2Si2 (ThCr2Si2 type, a = 0.45331(6), c = 1.0297(1) nm) has been reported [11]. The aims of the present work were many-fold: (i) to provide consistent information on the phase relations in the isothermal region at 800 °C for the area with Ba contents up to 33.33 at.%, (ii) to establish details of the homogeneity regions of binary and ternary compounds in combination with atomic ordering in the corresponding crystal structures, (iii) to reveal temperature-dependent thermoelectric data for the compound Ba8Au5.1Si40.9 as part of the type I homogeneity region, and, finally, (iv) to understand the correlation between structure and physical properties in combination with density functional theory (DFT) calculations.

in distilled water. In comparison with the calculated X-ray density dx, the derived relative density (d/dx) was >96%. Details of the various techniques of characterization, (i) scanning electron microscopy (SEM) with electron probe microanalysis (EPMA) in a Zeiss Supra 55 VP operated at 20 keV and 60 lA using energy-dispersive X-ray analysis (EDX) for the quantitative analysis, (ii) X-ray powder diffraction (XRD) and single crystal diffraction (XSCD) to determine crystal structure and (iii) determination of the physical properties, have been reported in previous papers [12,13].

2. Experimental details

4. Results and discussion

Samples (about 1 g each) were prepared by arc melting mixtures of the pure elements (99.9 mass% Ba, 99.9 mass% Au and 99.999 mass% Si, all from Alfa Aesar) under an argon atmosphere without significant weight loss (<1 mass%). The arc-melted buttons were vacuum sealed in quartz tubes and annealed at 800 °C for 1 week. One sample (of about 1 cm3) with nominal composition Ba8Au5.1Si40.9 was prepared in a different way for physical properties measurements (thermopower, resistivity and thermal conductivity): five alloys of 1–2 g were argon arc melted from the elements. From these reguli powders were obtained via high energy ball milling in a Vario Planetenmu¨hle Pulverisette 4 and were then compacted under argon in a hot press (HP W 200/250-2200-200-Ks, FCT Systems GmbH) at 800 °C at a pressure of 56 MPa. To determine the relative percentage density (dr) we used Archimedes’ principle for the density d, which we measured on a cylindrical sample

4.1. The clathrate I solid solution Ba8AuxSi46x

3. Computational details DFT calculations for the Ba8AuxSi46x compounds were carried out by applying the Vienna Ab Initio Simulation Package (VASP) [14,15]. The generalized gradient approximation (GGA) is used for the exchange correlation functional as parameterized by Perdew et al. [16], and the pseudo-potential is treated within the framework of Blo¨chl’s projector augmented wave method [17,18]. The valence states configuration for the construction of the pseudo-potentials included 5s25p66s2 states for Ba, 5d106s1 states for Au, and 3s23p2 states for Si. In all calculations a 5  5  5 Monkhorst grid was constructed for sampling over the Brillouin zone. Transport properties such as the Seebeck coefficient (S) and the electric resistivity (q) were derived within the semi-classical Boltzmann transport theory. More details about the actual calculations are given in Zeiringer et al. [19]. Within Boltzmann’s transport theory the effects of electron–electron and electron–phonon scattering are merged into the relaxation time s. Because a first principles calculation of s is not feasible for systems with a large number of atoms per unit cell, it was considered as an empirical parameter by fitting to one experimental value at a given temperature, as described below.

The solubility range of the ternary type I clathrate at 800 °C was studied by XRD and EPMA on a series of samples with nominal composition Ba8AuxSi46x (x = 3,4,5,6). Table 1 shows a comparison of the EPMA data with the results of Rietveld refinement of the X-ray spectra. In all cases the spectra could be fully indexed on the basis of a cubic clathrate type I lattice (space group Pm3n) with minor amounts of (Si) in some cases. As no binary Si-based type I clathrate exists at normal pressure the ternary homogeneity region is limited at 800 °C and was found to extend from 7.4 to 11.3 at.% Au (i.e. 4 6 x 6 6.0). The lattice parameters for the clathrate series increase linearly with increasing gold content and are in perfect agreement with single crystal data reported by Cordier et al. (a = 1.0422(2) nm for Ba8Au6Si40 [6]) and single crystal data of Jaussaud et al.

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Table 1 Results from EPMA and Rietveld refinement for type I clathrates Ba8AuxSi46x. Nominal composition Ba8Au3Si43a Ba8Au5Si41 Ba8Au6Si40 a

EPMA (at.%) Au

Ba

Si

7.4 10.0 11.3

14.2 14.6 15.1

78.4 75.4 73.6

Accepted composition

Lattice parameter (nm)

Si2 in 16i (x, x, x)

Si3 in 24k (0, y, z)

Ba8Au4Si42 Ba8Au5.1Si40.9 Ba8Au6Si40

1.03963(4) 1.04130(2) 1.04191(2)

0.1825(2) 0.1834(1) 0.1852(1)

0.1195(2), 0.3079(2) 0.1164(1), 0.3047(1) Au/Si 0.1168(1), 0.3063(1)

Three phase sample containing j1 + (Si) + BaSi2.

Fig. 1. Lattice parameters versus gold content for the alloys Ba8AuxSi46x, quenched from 800 °C, including literature data [5,6,8,31]. The dashed line is a guide for the eye.

(a = 1.0414(1) nm for Ba8Au5.43Si40.57 and a = 1.0419(1) nm for Ba8Au5.89Si40.11 [5]) (Fig. 1). Our refinement of the XRD spectra unambiguously located the Ba atoms in sites 2a (0, 0, 0) and 6c (1=4 , 0, ½), Au and Si atoms were found to randomly share the 6d site (1=4 , ½, 0), and silicon atoms occupied the sites 16i (x, x, x) and 24k (0, y, z). Cordier et al. [6] reported that Au preferentially occupies the 6d site, but our XRD refinements are in good agreement with the results of Jaussaud et al. [5], who reported for two single crystals that minor amounts of Au also occupied the 24k site. No vacancies were encountered. Substitution of silicon by gold atoms in the crystal lattice over the limited homogeneity range leads to Au-dependent atom parameters for xSi in 16i (x, x, x) and ySi, zSi for Si3 atoms in 24k (0, y, z). The results obtained for the clathrate I series from Rietveld refinement and EPMA are summarized in Table 1. 4.2. Phase equilibria in the system Au–Ba–Si for 0–33.3 at.% Ba The partial isothermal section at 800 °C, evaluated by XRD and EPMA on about 35 alloys, is shown in Fig. 2. Information concerning the binary phase diagrams Ba–Si and Au–Si is taken from Massalski [20] and a summary of all the phases appearing at 800 °C is given in Table 2. At low gold contents s1 Ba8AuxSi46x is in equilibrium with

Fig. 2. Partial isothermal section for the ternary system Au–Ba–Si at 800 °C.

silicon and BaSi2 (see Fig. 3a) and with BaSi2 and s2 Ba(Au1xSix)2 (AlB2 type) (see Fig. 3c). s2 exhibits a quite large homogeneity region, extending from BaAu0.4 Si1.6 (a = 0.4193(2), c = 0.5032(2) nm) to BaAu0.9Si1.1 (a = 0.4353(1), c = 0.4853(1) nm), in all cases, with c/a  1.1. The Rietveld refinement for BaAu0.9Si1.1 as the main phase is shown in Fig. 4. Besides BaAu0.9Si1.1, small amounts of BaAu2 (<2%) and traces of s7 BaAu3+xSi1x (as seen by EPMA, but no additional peaks were present in the XRD spectra) can be found in the sample. A SEM picture of a sample (20.8 at.% Ba, 56.2 at.% Au, 23 at.% Si) representing this three-phase equilibrium is shown in Fig. 3d. BaAu2 (c/a  0.88, AlB2 type) solves about 7.5 at.% Si at 800 °C. Both compounds with 33.3 at.% Ba, BaAu2 and s2, oxidize in air. Five further new ternary phases were encountered in the Au–Ba–Si system at 800 °C. s3 Ba2Au4+xSi3x (formula derived from EPMA measurements) exhibits a homogeneity range of only a few atomic per cent of gold at 800 °C, with hitherto unknown structure type. At the gold-rich end of the solubility range of s3 a narrow three-phase equilibrium exists among s3, s4 BaAu2+xSi2x (structure unknown) and s6 BaAu3xSi1+x also with an unknown structure type. Fig. 3e shows the three-phase equilibrium between (Si), s4 and s5 BaAu5xSi2+x (own structure type), which has a small homogeneity region (x = 0–0.2) at 800 °C. Binary BaAu5, s5 and s7 were found to be in equilibrium with the

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Table 2 Crystallographic data on the solid phases of the ternary system Au–Ba–Si at 800 °C. Space group

Structure type

Lattice parameter (nm)

Fd 3m Fm3n Pnma Cmcm P6/mmm P6/mmm Pm3n P6/mmm oI tP Pnma Unknown P4/nmm

Cdiam Cu BaSi2 TlI AlB2 CaCu5 K4Ge23x AlB2 Unknown Unknown BaAu5Si2 Unknown BaAu3Ge

0.54303, 0.5429(2) 0.40773(1) 0.8942(4) 0.5042(5) 0.4804 0.5690 1.0396(3), 1.0419(1) 0.4193(2), 0.4305(1) 0.65097(4) 1.053 0.8935(2)

0.82267(5) 1.053 0.6939(2)

0.5032(2), 0.490(1) 2.38516(13) 0.845 1.0365(2)

0.6488(2), 0.6509(1)

–

0.5305(2), 0.5289(1)

a (Si) (Au) BaSi2 BaSi BaAu2 BaAu5 s1 Ba8AuxSi46x s2 BaAuxSi2x s3 Ba2Au4+xSi3x s4 BaAu2+xSi2x s5 BaAu5xSi2+x s6 BaAu3xSi1+x s7 BaAu3+xSi1x a

Ref. b

0.6733(3) 1.197(1)

c

1.1555(5) 0.4142(2) 0.4119 0.4542

[20]a [20] [20] [20] [29] [30] x = 4, x = 6a x = 0.4, x = 1a x = 0a x = 0a x = 0a a

x = 0, x = 0.3a

This work.

liquid at 800 °C. The crystal structure of s7 BaAu3+xSi1x (BaAu3Ge type) has been determined from single crystal analysis and its homogeneity region extends from x = 0 up to x = 3. The crystal structures of the new ternary phases s5 and s7 will be discussed in detail in the next section, while those of s3, s4 and s6 are part of an ongoing investigation. The results from EPMA and X-ray phase analyses are summarized in Table 3 and SEM pictures representing all the three-phase equilibria are shown in Fig. 3. 4.3. Crystal structure of the new ternary compounds s5 and s7 4.3.1. s7 BaAu3+xSi1x (BaAu3Ge type) The crystal structure of BaAu3+xSi1x (x = 0) was solved employing direct methods and was found to be isotypic with the structure type of BaAu3Ge, which has recently been reported [21] (space group P4/nmm, a = 0.6448(2), c = 0.5305(2) nm). The lattice of this structure type is fully ordered with Ba atoms occupying the crystallographic site 2b (3=4 , 1=4 , ½), Au atoms in site 4d (0, 0, 0) and 2c (1=4 , 1=4 , z) and Si atoms in site 2c (Table 4). Although the Au–Si distances are rather short (0.25 nm), full occupancy in all Wyckoff positions was proven during the refinement. Characteristic polyhedra in this structure are a square pyramid around the silicon atoms in 2c (Fig. 5a), a distorted “rectangle” around the gold atoms Au1 in 4d (Fig. 5c) and a 16 atom coordination polyhedron around the Ba atoms (Fig. 5b). The polyhedra around the Au2 atoms are very similar to those of the isotypic compound BaAu3Ge, with the exception of one additional Si atom among the quite short Au–Si distances (all interatomic distances can be found in Supplementary information). The structure can be described as a tetragonal derivative of the CuAu3 type and, therefore, is even more closely related to the tetragonal CePt3B type (space group P4mm). A detailed description is given in Zeiringer et al. [21]. As a small range of homogeneity (0 6 x 6 0.3) exists at 800 °C we also analysed a sample at the Au-rich end of the solid solution via Rietveld refinement revealing

mixed occupancy by Au and Si of site 2c (i.e. Au atoms enter the Si site 2c for concentrations x > 0). 4.3.2. s5 BaAu5xSi2+x (own type) The crystal structure of s5 BaAu5xSi2+x (x = 0) was solved by direct methods and refined to R = 0.029 with ˚ 3, yieldresidual electron densities smaller than ±4.5 e A ing a completely ordered atomic arrangement (see Table 5). The crystal structure of BaAu5Si2 adopts the orthorhombic primitive space group Pnma as the highest possible crystal symmetry (a = 0.8935(2), b = 0.6939(2), c = 1.0365(2) nm), revealing Ba, Au and Si atoms in the crystallographic site 4c (x, 1=4 , z) and Au atoms in site 8d (x, y, z). Refinement of the occupancy of all atomic sites and anisotropic atomic displacement parameters show full occupation in all Wyckoff positions (see Table 5). Interatomic distances can be found in Supplementary information. The structure in three-dimensional view is shown in Fig. 6a, with an emphasis on the Ba and Au3 polyhedra. Each Ba atom is surrounded by 12 gold atoms and these Ba polyhedra are interconnected, sharing rectangular faces to form chains along the direction. The Au3 atoms are surrounded by three Si and six Au atoms and share their triangular faces to build up chains in the same direction. The whole structure can be described by alternating layering of these two types of chains. The other coordination polyhedra of this structure type are shown in Fig. 6b–e. 4.4. DFT results 4.4.1. Structure and energetics The lattice parameters for the cubic clathrate type I are derived from fully relaxed VASP calculations, from which the enthalpies of formation DH in terms of differences in total energies at zero pressure are calculated. Three types of systems were studied, namely Si46, Ba8Si46, and Ba8AuxSi46x for x = 1 and 6. From experimentally derived X-ray data it was concluded that the Au atoms are

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Fig. 3. SEM (backscatter detector) images of Au–Ba–Si alloys annealed at 800 °C revealing phase equilibria. Composition of the phases a–l refer to Table 3 (s1, clathrate I; s2, Ba(Au1xSix)2; s3, Ba2Au4+xSi3x; s4, BaAu2+xSi2x; s5, BaAu5xSi2+x; s6, BaAu3xSi1+x; s7, BaAu3+xSi1x).

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Ba8AuxGe46x as a function of the concentration x behaves very similarly to the Ba–Au–Si case, but shifted down by a constant, i.e. the difference in DH of the binary phases Ba8Si46 and Ba8Ge46. In general, for all the studied cases it appears that the formation enthalpy is lowered with increasing x, i.e. replacement of cage atoms by Ag or Au.

Fig. 4. Rietveld refinement for s2 Ba(Au1xSix)2 (AlB2 type, x = 0.55) as the main phase with less than 3% BaAu2.

randomly distributed on Si 6d sites. It should be noted that for the DFT calculations the doping atoms are placed on specific sites within the unit cell, which, in general, implies changes in lattice symmetry and cell shape. However, it turns out that the deviations from a cubic shape are rather small. Therefore, for consistency with the experimental results a cubic unit cell was enforced for all calculations. More details are given in Zeiringer et al. [19]. The calculated structural parameters are listed in Table 6. Viewing the experimental data for the lattice parameter in Fig. 7 one can see that the DFT derived values are larger by about 1%, showing the same trend of increase for increasing x as the experiment. The (rather small) difference between first principles theory and the experimental results is basically due to approximation of the exchange correlation functional. For the Ag–Ge-based clathrates [19] similar deviations were found, but there the first principles lattice parameters were smaller by about 1%. It should be noted that for the Ag–Ge-based clathrate a local density approximation (LDA) type parameterization of the exchange correlation functional was chosen, whereas for the Si-based compounds a GGA parameterization was selected. The different choices were made in order to account for the different sizes, i.e. atomic numbers, of Si and Ge. According to Fig. 7 the enthalpy of formation decreases (i.e. bonding is enhanced) with increasing Au content, very similar to the results for Ba8AgxGe46x [19], which for the sake of completeness are also shown. Interestingly, the enthalpy cost of forming the cage structure of Si46 or Ge46 in comparison with the diamond ground state structure is very similar, but the bonding energy gain (i.e. lowering of DH) for Ba8Ge46 is significantly stronger than for Ba8Si46, which is partially attributed to the different choices of exchange correlation functionals. Because of a recent study on the type I clathrate Ba8Au5.3Ge40.7 [22] we also calculated the formation energies of Ba8AuxGe46x compounds, as shown in Fig. 7. As mentioned above in our first principles calculations, varying x resulted in all structural parameters being fully relaxed, in contrast to the electronic structure calculations in Zhang et al. [22]. Most interestingly, the formation enthalpy of

4.4.2. Densities of states Electronic densities of states (DOS) for Ba8AuxSi46x are shown in Fig. 8, in which significant changes induced by doping can be observed. The artificial cage crystal Si46 is a semiconductor with a gap of 1.14 eV. It should be noted that applying the same type of DFT calculation to Si in its diamond ground state structure results in a gap of about 0.5–0.6 eV, reflecting the problems of standard DFT calculations in deriving correct gap sizes. Therefore, the sizeable DFT gap for the Si cage solid indicates that the experimental gap, if it could be measured, would be about 2 eV, because calculated DFT gap sizes are often smaller by a factor of 2 than the measured ones. On filling the voids in the Si46 cage with Ba the gap is maintained, but shrinks in size due to hybridization between the Ba and Si states. The gap decreases significantly on adding Au, which implies hybridization between the dopants and Si framework. For Ba8Au5Si41 there are two peaks in the DOS around the Fermi energy, which is located at a satellite of the higher energy peak. Doping with up to six Au atoms lowers the magnitude of the higher energy peak for Ba8Au6Si40 in and moves the Fermi energy to the center of the peak at a lower energy. The small difference in the shape of the peaks for the DOS between these two compounds may be due to the difference in crystal symmetry in the calculations. However, there is also a significant similarity between Ba8Au5Si41 and Ba8Au6Si40 in that the peaks around the Fermi energy remain rather unchanged, as well as a pseudogap between them being retained. The similarity in DOS features (i.e. of the electronic structure) justifies the rigid band approximation when variations in atomic composition (i.e. variation in the number of electrons) is handled by only varying the position of the Fermi energy. Of course, the change in the number of valence electrons must be sufficiently small. 4.5. Physical properties of Ba8Au5.1Si40.9 4.5.1. Temperature-dependent thermal conductivity The temperature-dependent thermal conductivity of Ba8Au5.1Si40.9 is shown in Fig. 9, including an error in the measurement of about 10%. The “glass-like” temperature dependency of k(T) without a peak at very low temperatures is related to the scattering of both electrons and heat-carrying phonons by impurities, vacancies, grain boundaries and static defects. The overall values are rather small, with lattice thermal conductivity (kph) being the dominant contribution to the overall thermal conductivity (Fig. 9), k ¼ kph þ ke :

ð1Þ

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Table 3 EPMA and X-ray phase analysis data for ternary three phase alloys (quenched from 800 °C) in the system Au–Ba–Si (a–l correspond to the SEM images in Fig. 3). Nominal composition (at.%) Au

X-ray phase analysis

Ba

Si

Structure type

Composition EPMA (at.%) Au

Ba

Si

(a)

5.6

14.8

79.6

(Si) s1 BaSi2

Cdiam K4Ge23x BaSi2

0.0 8.3 1.0

0.0 14.2 31.5

100.0 77.5 67.5

(b)

22.0

23.2

54.8

s1 s2 s3

K4Ge23x AlB2 –

11.5 17.8 43.3

15.6 33.6 21.7

72.9 48.6 35.0

(c)

8.0

26.0

66.0

s2 s1 BaSi2

AlB2 K4Ge23x BaSi2

12.7 8.6 0.0

32.3 14.9 32.2

55.0 76.5 67.8

(d)

57.0

20.0

23.0

s2 s7 BaAu2

AlB2 BaAu3Ge AlB2

32.3 60.3 62.0

31.8 19.2 31.3

35.9 20.5 6.7

(e)

23.5

5.9

70.6

(Si) s5 s4

Cdiam BaAu5Si2 Unknown

0.0 60.0 54.5

0.0 12.5 17.1

100.0 27.5 28.4

(f)

43.0

20.0

37.0

s1 s3 s4

K4Ge23x Unknown Unknown

11.3 43.4 52.0

15.5 22.0 19.7

72.2 34.6 28.3

(g)

47.0

18.5

34.5

(Si) s1 s4

Cdiam K4Ge23x Unknown

0.0 11.3 51.9

0.0 15.8 20.0

100.0 72.9 28.1

(h)

49.5

22.3

28.2

s2 s3 s6

AlB2 Unknown Unknown

19.6 47.5 58.9

33.3 21.4 20.2

47.1 31.1 20.9

(i)

52.3

20.7

27.0

s3 s4 s6

Unknown Unknown Unknown

46.9 53.5 58.7

22.1 20.0 20.8

31.0 26.5 20.5

(j)

70.0

20.0

10.0

BaAu5 BaAu2 s7

CaCu5 AlB2 BaAu3Ge

83.5 67.2 65.0

16.5 32.8 19.0

0.0 0.0 16.0

(k)

76.0

14.0

10.0

BaAu5 s7 Liquid

CaCu5 BaAu3Ge –

84.9 65.7 73.9

15.1 19.1 11.7

0.0 15.2 14.4

(l)

85.0

9.7

5.3

BaAu5 Liquid Au

CaCu5 – (Cu)

83.6 75.7 100

16.4 12.8 0.0

0.0 11.5 0.0

s1, clathrate I; s2, Ba(Au1xSix)2; s3, Ba2Au4+xSi3x; s4, BaAu2+xSi2x; s5, BaAu5xSi2+x; s6, BaAu3xSi1+x; s7, BaAu3+xSi1x.

The electronic contribution to the total thermal conductivity (ke, Fig. 9) can be derived from the electrical resistivity (Fig. 10) using the Wiedemann–Franz law (L0 = 2.45  108 W O K2). This is, in general, valid only for free electron systems, but is often also used for clathrates and other complex materials. The temperature dependency of the lattice thermal conductivity can be modelled according to Callaway [23] as Z sc x 4 e x kph ¼ CT 3 dx þ FT 3 ð2Þ 2 x ðe  1Þ

1 1 1 1 s1 C ¼ sB þ sD þ sU þ sE

where a radiation loss term (FT3) is added. The overall relaxation time for phonon scattering (sc) is expressed as the sum of various scattering processes

The Debye temperature (hD = 290 K) has been taken from a recent publication [25] dealing with the thermal expansion of several Si- and Ge-based clathrate compounds.

ð3Þ

B corresponds to scattering by boundaries, D to point defect scattering, U to “Umklapp” processes and E to scattering by electrons. A least squares fit to the data according to Eq. (2) is shown in Fig. 9. The minimum thermal conductivity in Fig. 9 (dark pink dotted line) has been calculated using a formula derived by Cahill and Pohl [24]  13 2 2 Z T =hD 3n k B T x3 ex kmin ¼ dx ð4Þ 2 4p hhD 0 ðex  1Þ

I. Zeiringer et al. / Acta Materialia 60 (2012) 2324–2336 Table 4 X-ray single crystal data for s7 BaAu3Si at RT, standardized with the program Structure Tidy [32] (Mo Ka radiation, 2° 6 2H 6 70°, x scans, scan width 2°, 150 s per frame, anisotropic displacement parameters in [102 nm2]). Parameter

s7-BaAu3Si

Space group Composition from EPMA (at.%) Formula from EPMA Formula from refinement a, c (nm) SC a, c (nm) Ge standard labs (mm1) V (nm3) qx (g cm3) Reflections in refinement Number P of variables P R2F ¼ jF 20  F 2c j= F 20 RInt wR2 GOF Extinction (Zachariasen) ˚ 3) (max, min) Residual density (e A

P4/nmm Au60.1Ba21.0Si18.9 BaAu3Si BaAu3Si 0.6488(2), 0.5305(2) 0.6488(1), 0.5299(1) 106.99 0.2233 11.25 290 P 4r(F0) of 311 16 0.023 0.042 0.062 1.324 0.0049(5) 4.85, 2.85

Atom parameters Ba in 2b (3=4 , 1=4 , ½); occupancy U11, U22 = U33 Au1 in 4d (0, 0, 0); occupancy U11, U22 = U33 Au2 in 2c (1=4 , 1=4 , z); occupancy z U11 = U22, U33 Si in 2c (1=4 , 1=4 , z); occupancy z U11 = U22, U33

1.009(7) 0.0113(3), 0.0202(5) 0.993(9) 0.0235(2), 0.0148(3) 0.997(5) 0.6613 (1) 0.0134(2), 0.0158(3) 0.995(6) 0.2012(6) 0.0039(6), 0.0049(5)

The Debye temperature resulting from the Callaway fit is 270 K, in quite good agreement with the published value from the thermal expansion fit. The number of atoms per

Fig. 5. Crystal structure of s7 BaAu3Si (BaAu3Ge type). (a) Threedimensional view of the unit cell with anisotropic displacement parameters from single crystal refinement and the corresponding coordination polyhedra.

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unit cell volume n = 4.8  1028 m3. The measured thermal conductivity is about three times larger than the minimum thermal conductivity. 4.5.2. Electrical resistivity The temperature dependency of the electrical resistivity in the temperature range between 4.5 and 1000 K is shown in Fig. 10, including literature data for Ba8Au5.4Si40.6 and Ba8Au5.9Si40.1 [5] in the low temperature range. The estimated error of our measurements is only about 2%. Ba8Au5.4Si40.6 is a semiconductor and Ba8Au5.9Si40.1 shows a simple metal-like behaviour in the temperature range up to 300 K, with an about six times larger resistivity compared with Ba8Au5.1Si40.9. The electrical resistivity of Ba8Au5.1Si40.9 is characterized by a distinct minimum at very low temperatures and a broad maximum at high temperatures. Such a complex behaviour can never be described by the standard Bloch– Gru¨neisen equation. We adopted a model [26] containing Table 5 X-ray single crystal data for s5-BaAu5Si2 at RT, standardized with the program Structure Tidy [32] (Mo Ka radiation, 2° 6 2H 6 70°, x scans, scan width 2°, 150 s per frame, anisotropic displacement parameters in [102 nm2]). Parameter

s5 BaAu5Si2

Space group Composition from EPMA (at.%) Formula from EPMA Formula from refinement a, b, c (nm) SC a, b, c (nm) Ge standard labs (mm1) V (nm3) qx (g cm3) Reflections in refinement Number P of variables P R2F ¼ jF 20  F 2c j= F 20 RInt wR2 GOF Extinction (Zachariasen) ˚ 3) (max, min) Residual density (e A

Pnma Au63.5Ba12.9Si23.6 BaAu5Si2 BaAu5Si2 0.8935(2), 0.6939(2), 1.0365(2) 0.8929(2), 0.6932(1), 1.0364(1) 119.97 0.6427 12.18 1355 P 4r(F0) of 1512 44 0.029 0.065 0.066 1.131 0.00080(6) 4.44, 4.08

Atom parameters Au1 in 8d (x, y, z); occupancy x, y, z U11, U22, U33 Au2 in 8d (x, y, z); occupancy x, y, z U11, U22, U33 Au3 in 4c (x, 1=4 , z); occupancy x, z U11, U22, U33 Ba in 4c (x, 1=4 , z); occupancy x, z U11, U22, U33 Si1 in 4c (x, 1=4 , z); occupancy x, z U11, U22, U33 Si2 in 4c (x, 1=4 , z); occupancy x, z U11, U22, U33

1.011(4) 0.09011(3), 0.04489(4), 0.60857(2) 0.0102(2), 0.0122(1), 0.0139(1) 1.004(4) 0.09919(3), 0.03388(4), 0.16865(3) 0.0118(1), 0.0131(1), 0.0184(1) 1.002(4) 0.33198(5), 0.31754(4) 0.0090(2); 0.0163(2); 0.0154(2) 1.00(1) 0.22133(8), 0.89391(5) 0.0122(3), 0.0137(3), 0.0134(2) 1.09(2) 0.0613(4), 0.3804(2) 0.008(1), 0.009(1), 0.015(1) 1.03(2) 0.3238(3), 0.5516(3) 0.009(1), 0.011(1), 0.015(1)

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Fig. 6. Crystal structure of s5 BaAu5Si2 (own structure type). (a) Three-dimensional view (2 unit cells in the x-direction) with anisotropic displacement parameters from single crystal refinement and polyhedra around the Ba and Au3 atoms. (b–e) Coordination polyhedra around (b) Au1, (c) Au2, (d) Si2 and (e) Si1.

Table 6 Lattice constant and formation energy per atom (DH) for various compositions of type I clathrates; as derived from DFT. Compound

Lattice parameter (nm)

Si2 in 16i (x, x, x)

Si3 in 24k (0, y, z)

DH (kJ mol1)

Si46 Ba8Si46 Ba8Au1Si45 Ba8Au2Si44 Ba8Au3Si43 Ba8Au4Si42 Ba8Au5Si41 Ba8Au6Si40

1.025 1.038 1.042 1.044 1.047 1.048 1.049 1.052

0.181 0.182 0.183 0.185 0.183 0.183 0.183 0.184

0.116, 0.118, 0.119, 0.119, 0.119, 0.118, 0.119, 0.117,

6.3 10.2 13.1 16.7 20.0 23.2 25.7 27.1

Fig. 7. Calculated formation energy DH (filled red circles) and lattice constants a (blue diamond) of Ba8AuxSi46x as a function of Au doping. Experimental lattice constants are shown as green crosses. The figure also shows the results for the unfilled cage solid Si46. For comparison the energies of formation of Ba8AgxGe46x [19] are also inserted (empty triangles). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.304 0.303 0.306 0.307 0.303 0.304 0.301 0.304

Fig. 8. Density of states (DOS) of Ba8AuxSi46x. The Fermi energy has been shifted to zero.

the Bloch–Gru¨neisen formula to account for the electron– phonon interaction, but allowing for a temperature-dependent charge carrier density n(T). A Mott–Jones term, AT3,

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resistivity beyond the maximum implies an increase in the charge carrier density caused by thermal excitations across the band gap, which according to the fit is 0.45 eV.

Fig. 9. Measured thermal conductivity (red circles), electronic contribution (black solid line) and phonon part (dashed-dotted-dotted black line) and total thermal conductivity (grey squares). The blue dashed line is the least squares fit described in the text, the green dashed-dotted line is the radiation loss and the dark pink dotted line is the minimum thermal conductivity as explained in the text. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

was added to specify scattering of electrons in narrow states (in general d-like) near to the Fermi energy [26],  5 Z T z5 ð5Þ qph ¼ R z hD ðe  1Þð1  ez Þ and qðT Þ ¼

q0 n0 þ qph AT 3 þ : nðT Þ nðT Þ

ð6Þ

In Eqs. (5) and (6) q0 is the residual resistivity (T ! 0), n0 is the residual charge carrier density, R is the electron–phonon interaction constant and hD is the Debye temperature. The high temperature maximum can be described quite well with this model (see fit in Fig. 10). The decrease in

Fig. 10. Temperature dependency of the electrical resistivity (q) and literature data from Jaussaud et al. [5]. The blue solid line is the least squares fit explained in the text. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4.5.3. Thermopower, Hall effect and dimensionless figure of merit The thermopower of Ba8Au5.1Si40.9 has been measured in the temperature range 4.5–1000 K. The measured results, including the experimental error of about 5%, are shown in Fig. 11a, including the literature data [5] for Ba8Au5.4Si40.6 and Ba8Au5.9Si40.1. The thermopower values for Ba8Au5.4Si40.6 are very small and negative over the whole temperature range (see Fig. 11, Smax(300 K)  10 lV K1), whereas the thermopower values of Ba8Au5.9Si40.1 are slightly larger and positive (Smax(300 K)  35 lV K1), indicating hole-dominated transport. Ba8Au5.1Si40.9 is also an n-type material, in which electrons are the main charge carriers, but its thermopower value at 300 K is about six times that of Ba8Au5.4Si40.6. For temperatures well below 500 K S(T) behaves almost linearly before reaching a maximum at higher temperatures. Using Mott’s formula [27], 1 p2 k 2B 2m T ð7Þ 3 eh2 ð3np2 Þ23 the charge carrier density n can be calculated (m is the mass of the carriers and e is the respective charge). Eq. (7) is valid for systems without significant electronic correlations and leads to n = 1.65  1020 cm3. At high temperatures, however, charge carriers are excited across the gap in the electronic density of states, hence n(T) increases. As a consequence (compare Eq. (6)) S(T) starts to decrease. Note that the maximum in S(T) coincides with the maximum in q(T), suggesting the same physical origin. From the maximum in S(T) we estimated that the energy gap Eg  Smax  2e  Tmax = 0.12 eV [28], where Tmax is the absolute temperature at which Smax appears and e is the elementary charge. The energy gap derived from the resistivity fit is somewhat higher, but of the same order of magnitude. In order to confirm the charge carrier concentration obtained from Eq. (7), Hall measurements were carried out in the temperature range 4.5–350 K (9 and 5 T) in a Physical Property Measurement System (PPMS). The charge carrier concentration (electrons) was calculated from the Hall coefficient as n = 1/eRH (Fig. 12b) and has been found to be in perfect agreement with the estimation from the slope of the S(T) curve. The negative values of the Hall coefficients (Fig. 12a) are also in good agreement with the Seebeck coefficient. From the Hall coefficient and the electrical resistivity we also calculated the Hall mobility l = RH/q, which is about 12 cm2 V1 s1 at room temperature (Fig. 12c). The calculated temperature dependency of the figure of merit can be seen in Fig. 11b. As a high ZT (>1) requires a high value of thermopower, good electrical conductivity and low thermal conductivity, the maximum ZT value of S d ðT < HD Þ ¼

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Fig. 11. (a) Temperature dependency of the thermopower (S) and literature data from Jaussaud et al. [5]. (b) The calculated temperature dependency of the dimensionless figure of merit ZT for Ba8Au5.1Si40.9.

Fig. 12. (a) Hall coefficient (RH), (b) charge carrier concentration (n) and (c) Hall mobility (l) for Ba8Au5.1Si40.9.

0.09 at 800 °C is currently too low for any thermoelectric applications. The low ZT is a result of the high electrical resistivity and small thermopower value. 4.5.4. DFT derived transport properties For small variations in the dopant (i.e. variation Dn of the number of valence electrons) the rigid band approximation was employed according to Z 1 N¼ gðEÞf ðT ; lÞdE ð8Þ 1

to calculate the chemical potential l (i.e. the Fermi energy) for N = N0 + Dn electrons. The number N0 represents the number of valence electrons of the chosen reference compound with its DOS g(E). In the Ba–Au–Si clathrates Ba, Au and Si contribute 2, 1 and 4 valence electrons to the system, respectively. To compare the calculations with the most interesting experimental sample composition Ba8Au5.1Si40.9,

the composition Ba8Au5Si41 was chosen as the reference. Consequently, Dn = 0.3, 1.2 and 2.7, namely hole doping, correspond to Ba8Au5.1Si40.9, Ba8Au5.4Si40.6 and Ba8Au5.9Si40.1, respectively. Fig. 13 (left side) depicts the calculated temperature-dependent Seebeck coefficient for the three compounds, which demonstrate that the Seebeck coefficient is very sensitive to doping. For Ba8Au5.1Si40.9 a negative S(T) with a minimum around 600 K is obtained. S(T) changes sign when the Au concentration is increased from 5.1 to 5.4 at low temperature and becomes negative as the temperature is increased. One should also note that the sign of S(T) for Ba8Au5.4Si40.6 at low temperature is different from the experimental measurement, as shown in Fig. 11a, the reason for which will be discussed below. Ba8Au5.9Si40.1 has a positive S(T) with a maximum around 700 K. Fig. 13 (right side) shows the thermopower and Seebeck coefficient S(T) for Ba8AuxSi46x with variations in x close to the chemical composition of the experimental sample Ba8Au5.1Si40.9. One can see that the calculated S(T) for Ba8Au5.1Si40.9 is comparable with the experiment results in terms of both the sign and position of the minimum. The Seebeck coefficient for Ba8AuxSi46x (at least at low temperatures) can be understood by Mott’s formula, valid for very low temperatures, in which the sign and magnitude of the Seebeck coefficient is determined by the energy derivdgðEÞ ative of DOS g(E) at the Fermi energy, i.e.  gðEÞdE jE F , which results in an opposite sign of the Seebeck coefficient and slope of the DOS at EF. One can also see that a small j give rise to a large jgðEÞjEF together with a large dgðEÞ dðEÞ EF Seebeck coefficient, for which the Fermi energy should be as close as possible to the gap. As the temperature increases the Fermi energy moves closer to or further away from the gap, accompanied by an increase or decrease in S(T). According to the electron counting rule formulated in our early work for doping Ba-filled clathrates [19] 5.33 Au atoms per unit cell are required to place the Fermi energy in the gap. According to this rule the Fermi energy of Ba8Au5.1Si40.9 is somewhere between that for Ba8Au5Si41 and the gap. From Fig. 8 one derives a positive derivative

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Fig. 13. DFT derived Seebeck coefficient of Ba8AuxSi46x.

was chosen as the relaxation time in order to fit the calculated resistivity of Ba8Ag5.1Ge40.9 to the corresponding experimental value at the high temperature maximum, indicating that the characteristic peak shifts towards higher temperatures with increasing x. From the discussion above one can see that the Fermi energies for all three compounds are above the gap and move closer to it as x increases gradually from 5.07 to 5.13. Because of that, an increasing critical temperature is required to exit states below the gap for transport. Our calculations indicate that the thermoelectric properties and the conductivity of Ba8AuxSi46x alloys can be well described by the first principles approach used in our work. 5. Conclusions Fig. 14. DFT derived electrical resistivity of Ba8AuxSi46x for x = 5.07, 5.10 and 5.13. dgðEÞ j dE EF

and, consequently, a negative Seebeck coefficient, in accord with the experimental results. Likewise, the Fermi energy of Ba8Au5.4Si40.6 is below but very close to the j , which produces a gap where one finds a negative dgðEÞ dE EF positive S(T). It should be noted that rather small uncertainties in the stoichiometry, which are inherent in the experiment, result in significant fluctuations in the number of valence electrons, which leads to a substantial variation in the Seebeck coefficient. Thus the difference in S(T) between the calculations and experimental results can be attributed to uncertainties and defects in the experiments. The maximum electrical resistivity is also closely related to the position of the Fermi energy. In a recent work on Ba8AgxGe46x it was found that the peak of electrical resistivity arises due to placement of the Fermi energy close to a gap and the temperature dependence of the energy derivative of the Fermi function f(E) [19]. More specifically, if the Fermi energy is very close to the gap but at a higher energy, the states below the gap will involve electronic transport above a certain temperature due to the temperature broadening of f(E). As the Fermi energy shifts closer to or further away from the gap, lower or higher temperatures are required to exit states below the gap. Fig. 14 shows the calculated electrical resistivity for three compounds with x = 5.07, 5.10 and 5.13, respectively. s = 7.96  1014 s

We have derived the phase relations at 800 °C within the ternary system Au–Ba–Si up to 33.3 at.% Ba. Equilibria are characterized by a clathrate type I solid solution, Ba8AuxSi46x, extending at 800 °C from Ba8Au4Si42 (a = 1.039 nm) to Ba8Au6Si40 (a = 1.042 nm). Cubic primitive symmetry (space group Pm3n) was confirmed for the entire clathrate solution by X-ray powder diffraction. Consistent with a monotonous almost linear rise in the lattice parameters with increasing gold content, atomic site preferences from X-ray refinement indicate that gold atoms preferably occupy the 6d site in random mixture with Si atoms. The phase equilibria at 800 °C reveal a total of seven ternary phases in the investigated region up to 33.3 at.% barium. The homogeneity range has been established for Ba(Au1xSix)2 (AlB2 type, extending from BaAu0.4Si1.6 to BaAu0.9Si1.1). BaAu2+xSi2x (unknown structure type) exhibits a very small homogeneity range (x = 0.6–0.7) and two other ternary phases of unknown structure exist at about 22 at.% Ba, 52 at.% Au and 28 at.% Si and 20 at.% Ba, 58 at.% Au and 22 at.% Si. The crystal structures of two further novel phases in the gold-rich part have been determined from single crystal X-ray data: BaAu3+xSi1x of BaAu3Ge type (space group P4/nmm, x = 0: a = 0.6488(2), c = 0.5305(2) nm) and BaAu5xSi2+x (own structure type, space group Pnma, x = 0: a = 0.8935(2), b = 0.6939(2), c = 1.0363(2) nm). The proximity of Ba8AuxSi46x to a metal–insulator transition is

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corroborated by DFT electronic structure calculations. A gap in the electronic density of states, located near the Fermi energy, gives rise to distinct features of the temperature-dependent electrical resistivity and Seebeck effect. Acknowledgements This work was supported by the Austrian FWF Project “P22295” and the FWF funded Science Colleague “Computational Materials Science”. Computations were done on the Vienna Scientific Cluster (VSC). All EPMA measurements were carried out in the Faculty Centre for Nanostructure Research. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.actamat.2011.12.040. References [1] Kawai H, Horie HO, Yamanaka S, Ishikawa M. Phys Rev Lett 1995;74:1427. [2] Yamanaka S, Enishi E, Fukuoka H, Yasukawa M. Inorg Chem 2000;39:56. [3] Nolas GS, Slack GA, Schujman SB. Semiconductors and semimetals, vol. 69. New York: Academic Press; 2000. p. 255. [4] Rogl P. Thermoelectrics handbook: macro to nano, vol. 2. London: Taylor and Francis; 2006. p. 32. [5] Jaussaud N, Gravereau P, Pechev S, Chevalier B, Menetrier M, Dordor P, et al. CR Chim 2005;8:39. [6] Cordier G, Woll P. J Less-Common Met 1991;169:291. [7] Herrmann RFW, Tanigaki K, Kuroshima S, Suematsu H. Chem Phys Lett 1998;283:29. [8] Herrmann RFW, Tanigaki K, Kawaguchi T, Kuroshima S, Zhou O. Phys Rev B 1998;60:13245.

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