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Chapter 3

Thermoelectric Properties of Correlated Electron Systems Ln3 Pt4Ge6 and LnPt4Ge12 (Ln = Ce, Pr) and Non-centrosymmetric X2T 12P7 (X=Yb, Hf and T = Fe, Co) B.D. White, M. Janoschek, N. Kanchanavatee, K. Huang, L. Shu, S. Jang, D.Y. Tut ¨ un, ¨ J.J. Hamlin, I.K. Lum, R.E. Baumbach, and M.B. Maple

Abstract We report measurements of the thermoelectric power S and electrical resistivity ρ for correlated electron systems Ln3 Pt4 Ge6 and LnPt4 Ge12 (Ln = Ce, Pr) and X2 T12 P7 (X = Yb, Hf and T = Fe, Co). The thermoelectric power factor S2 /ρ is utilized as a means to assess the potential viability of these materials for thermoelectric applications. S is observed to be sensitive to Ln in Ln3 Pt4 Ge6 and LnPt4 Ge12 with Ce-based compounds providing a much larger S and S2 /ρ than Pr-based materials. The character of S for the Ce-based compounds is consistent with an intermediate Ce valence. In the case of X2 T12 P7 compounds reported herein, it appears that it is possible to tune the magnitude of S more effectively by varying T rather than X; the magnitude of S is significantly larger with T = Fe than when T = Co.

3.1 Introduction The promise of harnessing thermoelectric energy for applications involving power generation and/or refrigeration has long motivated the search for new and better thermoelectric materials. Thermoelectric energy has several potential benefits primarily B.D. White • N. Kanchanavatee • K. Huang • S. Jang • D.Y. Tütün • J.J. Hamlin • I.K. Lum • R.E. Baumbach, • M.B. Maple () Department of Physics, University of California, San Diego, La Jolla, CA 92093-0354, USA e-mail: [email protected] M. Janoschek • R.E. Baumbach Department of Physics, University of California, San Diego, La Jolla, CA 92093-0354, USA Current Address: Los Alamos National Laboratory, Los Alamos, NM 87545, USA L. Shu Department of Physics, University of California, San Diego, La Jolla, CA 92093-0354, USA Current Address: Department of Physics, Fudan University, Shanghai 200438, Peoples Republic of China V. Zlat´ıc and A. Hewson (eds.), New Materials for Thermoelectric Applications: Theory and Experiment, NATO Science for Peace and Security Series B: Physics and Biophysics, DOI 10.1007/978-94-007-4984-9 3, © Springer Science+Business Media Dordrecht 2013

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centered around its environmentally-friendly nature and reliability. For example, it could replace or augment conventional methods for energy generation by recovering energy that would otherwise be lost from sources such as waste heat. This power could be generated by a reliable solid state device without moving parts and used in applications where frequent maintenance is difficult or impossible. Unfortunately, wide-spread use of thermoelectric energy has been inhibited by limitations imposed on device efficiency by the thermoelectric properties of available materials. The suitability of a given material for thermoelectric applications is best characterized by its thermoelectric figure of merit Z=

S2 , ρκ

(3.1)

where S is thermoelectric power, ρ = σ −1 is electrical resistivity (inverse of electrical conductivity), and κ = κe + κph is the total thermal conductivity (sum of electronic and lattice contributions as well as any other relevant contributions). In this form, Z has units of inverse temperature. It is convenient to use a dimensionless figure of merit ZT when making comparisons between different materials, where T is the average temperature of the gradient across the sample. The problem of obtaining more efficient thermoelectric materials therefore reduces to maximizing ZT by any available means (with the goal of obtaining at least ZT > 1, preferably near 300 K). Optimizing ZT involves finding an elusive balance between requirements that are somewhat paradoxical. Conventional metals typically have low ρ , but S is small and κe is large. On the other hand, semiconductors and insulators often have large S and low κe , but ρ is large. Clearly, the search for viable thermoelectric materials should concentrate on unconventional materials or, at the very least, involve strongly perturbing materials via chemical substitution (tuning carrier concentration), applied pressure, nanostructuring, and/or applied magnetic field [1]. In practice, optimizing ZT involves attempting to reduce κph and/or increasing the thermoelectric power factor S2 /ρ to blunt the detrimental effect of κ in lowering ZT [2]. As an example, members of the skutterudite family of compounds crystallize ¯ which contains two large icosahedral voids with the cubic CoAs3 structure (Im3), per unit cell. In filled skutterudites, these voids can be occupied by an undersized rare earth ion with sufficiently weak bonding to the cage as to allow strong thermal displacement of the ion in so-called rattling modes. Slack and Tsoukala hypothesized that these rattling modes strongly scatter heat-carrying phonons and were responsible for the large reduction they observed in κph while studying the thermal transport of skutterudite Ir1−x Rhx Sb3 [3]. By reducing κ , ZT is enhanced, and systematic studies of filling with different elements have demonstrated that an order of magnitude reduction of κph at 300 K (more at low temperature) is possible [4, 5]. When filling the void with a rare earth ion with unfilled f -electron shell, phonon-stimulated transitions between low-lying 4 f electronic energy levels may provide an additional mechanism for phonon scattering, further reducing κph

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[4, 5]. It is worth noting that the concept of rattling modes in filled skutterudites is not universally accepted. For example, results from recent neutron diffraction measurements and ab initio calculations for Fe4 Sb12 -type skutterudites strongly disagree with the rattling ion scenario [6]. However, despite some controversy regarding the origins of reduced κph in filled skutterudites, it is clear that this property is a useful attribute for any potential thermoelectric material. Naturally, even with reduced κ , the magnitudes of S and ρ must be sufficiently large and small, respectively, for ZT > 1. For a given κph , Mahan and Sofo calculated that ZT is optimized when the electronic density of states N(ε ) is described by a Dirac delta function centered near the Fermi energy εF [2]. This scenario may be most closely manifested by materials with f -electrons whose contributions to N(ε ) are typically Lorentzian peaks with narrow width. Mahan and Sofo’s calculations suggest that ZT ∼ 14 may be possible in compounds containing rare earth elements where the distribution of energy carriers is as narrow as possible [2]. Hybridization between localized f -electrons from rare earth ions and the sea of conduction electrons can form a Kondo lattice which, via a renormalized quasiparticle band, introduces a sharp peak in N(ε ) with width ∼TK . We need not necessarily appeal to a Kondo lattice to observe this effect, however. Conventional metals containing magnetic impurities which exhibit the Kondo effect’s trademark minimum in ρ versus temperature (single-ion Kondo effect) have long been known to show anomalously large S (orders of magnitude larger than in clean samples) as a consequence of the formation of virtual bound states [7]. These and other observations have led to a generic “rule of thumb” that enhanced S is typically found in correlated electron systems. These concepts have guided our strategy for obtaining more efficient thermoelectric materials and have led us to study the thermoelectric properties of correlated electron systems LnPt4 Ge12 and Ln3 Pt4 Ge6 (Ln = Ce, Pr) and X2 T12 P7 (X = Yb, Hf and T = Fe, Co). We report measurements of the thermoelectric power S and electrical resistivity ρ and calculations of the thermoelectric power factor S2 /ρ for each system. Calculation of ZT was impossible because thermal conductivity data were unavailable, however, S2 /ρ is a tangible substitute for ZT that we use to assess the potential viability of these systems for thermoelectric applications.

3.2 Experiment Polycrystalline samples of PrPt4 Ge12 and CePt4 Ge12 were synthesized by arc melting high-purity elements in an Ar atmosphere using a Zr getter to minimize oxidation and then post annealing the resulting boules in sealed quartz tubes as reported in Ref. [8]. Single crystals of Pr3 Pt4 Ge6 and Ce3 Pt4 Ge6 were synthesized in a molten in flux as reported in Ref. [9]. Polycrystalline samples of Hf2 T12 P7 with T = Fe, Co and Yb2 Fe12 P7 were prepared by conventional solid state reaction. Single crystals of Yb2 Fe12 P7 were synthesized in a molten Sn flux as reported

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in Ref. [10]. Chemical composition and phase purity were verified by means of powder X-ray diffraction and energy dispersive X-ray spectroscopy measurements. The orientation of Ln3 Pt4 Ge6 (Ln = Ce, Pr) and Yb2 Fe12 P7 single crystals was determined by single-crystal X-ray diffraction. Electrical resistivity ρ measurements were performed using the standard fourwire technique down to ∼1.1 K in a pumped 4 He bucket Dewar. Gold leads were attached to samples by spot-welding or using silver epoxy. The thermoelectric power S was measured between 2 and 300 K in a PPMS Dynacool manufactured by Quantum Design. A static temperature gradient of Δ T /T = 2–5% was applied along the longest physical dimension of each sample (parallel to the a axis for Pr3 Pt4 Ge6 and to the b axis for Ce3 Pt4 Ge6 ) and measured using commercial Cernox 1050 thermometers and a Lakeshore 340 Temperature Controller. Copper leads were attached to the sample with silver epoxy in a two-wire configuration. The DC thermoelectric voltage generated by the sample was measured using a Keithley 2182 Nanovoltmeter and was corrected for a background contribution arising from thermal/compositional asymmetry in the wires running from the sample to the external electronics at room temperature.

3.3 Ge-Based Filled Skutterudites LnPt4 Ge12 (Ln = Ce, Pr) The family of Ge-based filled skutterudites XPt4 Ge12 (X = Sr, Ba, La, Ce, Pr, Nd, Eu, Th, U) was recently synthesized, and several of its members were found to be superconducting [11–13]. Among them is the intriguing case of unconventional superconductivity in PrPt4 Ge12 (Tc = 7.9 K) in which evidence for time-reversal symmetry breaking has been observed [14]. Incorporation of Pt and Ge into the skutterudite structure is unique because the cage-forming elements of skutterudites are conventionally pnictogens such as P, As, or Sb and the transition metal is typically from the Fe or Co groups [11]. Band structure calculations for SrPt4 Ge12 and BaPt4 Ge12 have shown that the Ge 4p electronic states contribute a peak to N(ε ), centered near εF [12], which might help to increase S. However, contributions from unfilled d- or f -electron shells near εF are able to provide a greater number of states in a narrower energy range and are thus more likely to enhance S than p-electron states in the context of Mahan’s and Sofo’s picture. Furthermore, bonding between the alkaline earth ions and Ge cage (which are physically smaller than Sb cages in Sb-based skutterudites) is too strong to allow the ions to rattle [12]. This effectively suppresses one of the principle benefits filled skutterudites have to offer as thermoelectric materials. Fortunately, evidence for rattling modes involving La ions has been observed in heat capacity [11] and nuclear magnetic resonance [15] measurements of LaPt4 Ge12 . Rattling modes have also been observed in CePt4 Ge12 and PrPt4 Ge12 [11,16]. These latter modes may scatter heat-carrying phonons more strongly than those observed in their La-based counterpart because the ratio of the effective Ge-cage size to ionic radius of the filling ion is larger than in LaPt4 Ge12


a

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Fig. 3.1 (a) Thermoelectric power S of CePt4 Ge12 (black points) and PrPt4 Ge12 (red points) versus temperature for temperature range 2–300 K. (b) S of PrPt4 Ge12 versus temperature in applied fields of 0 T (filled red points) and 2.0 T (open red points) highlighting the superconducting transition at Tc ∼ 7.8(3) K. Superconductivity is suppressed by a 2.0 T magnetic field. The inset displays S/T versus T for CePt4 Ge12 . S/T extrapolates to ∼0.63 μ V K−2 in the limit T → 0 K. (c) Electrical resistivity ρ of CePt4 Ge12 (black points) and PrPt4 Ge12 (red points) versus temperature. (d) Thermoelectric power factor S2 /ρ versus temperature for CePt4 Ge12 (black points) and PrPt4 Ge12 (red points). The maximum values are 18.2 µW cm−1 K−2 and 0.5 µW cm−1 K−2 for CePt4 Ge12 and PrPt4 Ge12 , respectively. Lines are guides to the eye

and thermal displacement parameters Ueq obtained at room temperature for Ce and Pr ions are 23% and 26% larger, respectively, than Ueq for La [17]. Of course, all of these arguments depend on the validity of the rattling mode scenario itself. Motivated by potential reduction of κ via rattling modes and enhancement of S through electron correlations associated with f -electron physics, we have studied the thermoelectric properties of PrPt4 Ge12 and CePt4 Ge12 . The thermoelectric power S of CePt4 Ge12 and PrPt4 Ge12 versus temperature is shown in Fig. 3.1a, b. In the case of PrPt4 Ge12 , S ∼ 0 below Tc as highlighted in Fig. 3.1b; the offset is due to an additional thermoelectric voltage in the wires that is a consequence of applying a static thermal gradient across the sample. In an applied magnetic field of H = 2.0 T, superconductivity is suppressed and S = 0 below Tc . S < 0 at low temperature, crossing zero and becoming positive above ∼21 K. A maximum value of S, Smax ∼ 6.3 µV K−1 , is attained near room temperature.

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Such a low magnitude for Smax may be consistent with calculations of the electronic density of states N(ε ) for PrPt4 Ge12 , which show that N(ε ) is relatively flat near εF with the only structure coming from a small peak due to Ge 4p electronic states [11]. S versus temperature for CePt4 Ge12 is also shown in Fig. 3.1a. Its overall character agrees with data reported in Ref. [16]; however, our peak in S has larger magnitude. The data appear to be consistent with the theory of Zlatić and Monnier whereby the distinct behaviors of S in Ce-based intermetallic compounds are classified based on comparisons between the crystal field splitting Δ and hybridization strength Γ energy scales [18]. In “type d” compounds, where Γ > 2Δ , the Ce ions are in the intermediate valence regime (fluctuate between 4 f 1 and 4 f 0 electron configurations). According to calculations, when Γ ≈ 2Δ there should be a single maximum in S at TS ∼ Δ /2kB with a subtle shoulder below TS which disappears as Γ increases [18]. We observe a maximum in S with magnitude Smax 48.8 µV K−1 at TS ∼ 80 K and a shoulder in the vicinity of 25 K. This latter feature suggests that Γ ≈ 2Δ and that CePt4 Ge12 may be near a boundary between valence fluctuating behavior and Kondo physics. Gumeniuk et al. came to the same conclusion in order to explain why their measurements of x-ray absorption near edge structure and fundamental physical properties provided evidence both for intermediate Ce valence and for Kondo physics [16]. S/T versus temperature is plotted in the inset of Fig. 3.1b. In the limit of T → 0 K, S/T 0.63 µV K−2 as indicated by the line which extrapolates to T = 0 K. Using the Sommerfeld coefficient γ = 105 mJ mol−1 K−2 from Ref. [16], we calculate the dimensionless quantity q = (S/T )NA e/γ where NA is Avogadro’s number and e is the charge of an electron which yields q = 0.58. Theory suggests that q = ±1 for a Fermi liquid ground state [19], but our value is closer than q = 0.37 obtained in Ref. [16]. Furthermore, it is likely that the true Fermi liquid regime is at far lower temperature than our lowest S measurement [19] and that q = 1 might yet be obtained by performing measurements at those temperatures. The electrical resistivity ρ of PrPt4 Ge12 and CePt4 Ge12 versus temperature is plotted in Fig. 3.1c. The data are in agreement with other measurements reported in the literature [11, 15, 16]. Using these values of ρ , we have calculated the thermoelectric power factor S2 /ρ and plotted it versus temperature in Fig. 3.1d. The temperature dependence of S2 /ρ is dominated by S, and we see that the maximum S2 /ρ is a few orders of magnitude higher for CePt4 Ge12 than for PrPt4 Ge12 (18 µW cm−1 K−2 compared to 0.5 µW cm−1 K−2 ). This value for CePt4 Ge12 is comparable to other Ce-based filled skutterudites such as CeOs4 As12 (with maximum S2 /ρ ∼ 15 µW cm−1 K−2 ) [20]. The maximum thermoelectric power factor measured in filled skutterudites is ∼55 µW cm−1 K−2 (about a factor of 3 higher than that of CePt4 Ge12 ) [21]. Using thermal conductivity data reported for CePt4 Ge12 [16], we estimate ZT = 0.027 at 75 K (using κ 5 W m−1 K−1 and S2 /ρ = 18.21 µW cm−1 K−2 ). Unfortunately, this upper limit for ZT in CePt4 Ge12 is two orders of magnitude smaller than the goal of ZT > 1. Viability as a thermoelectric material might still be possible via judicious use of chemical substitution, however.


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3.4 Correlated Electron Systems Ln3 Pt4 Ge6 (Ln = Ce, Pr) In our attempts to synthesize single crystals of LnPt4 Ge12 with Ln = Ce and Pr, we unexpectedly obtained single crystals of Ln3 Pt4 Ge6 . The compound Ce3 Pt4 Ge6 crystallizes in a structure with orthorhombic space group Bmmb [22], and since its discovery, only electrical resistivity and magnetization data have been reported [23]. The compounds Ln3 Pt4 Ge6 for Ln = Pr – Dy crystallize in a variant of the Ce3 Pt4 Ge6 structure [24], but the physical properties of Pr3 Pt4 Ge6 are only recently being studied. The compound Y3 Pt4 Ge6 , which crystallizes with a monoclinic variation of the Ce3 Pt4 Ge6 structure, was reported to exhibit weakly-coupled BCS superconductivity below Tc = 2.6 K [25]. Calculations of its density of states N(ε ) show that Pt 5d states contribute the largest number of states near εF [25]. In contrast to the filled skutterudites, Ce and Pr thermal displacement parameters in Ln3 Pt4 Ge6 are comparable to parameters for Pt and Ge ions [11, 24], indicating the absence of anything similar to rattling modes in these systems. Despite this disadvantage, an enhanced thermoelectric power factor might still allow for a high ZT . The thermoelectric power S versus temperature for Ce3 Pt4 Ge6 (∇T b axis) and Pr3 Pt4 Ge6 (∇T a axis) is plotted in Fig. 3.2a, b. As with CePt4 Ge12 and PrPt4 Ge12 ,

a

c

b

d

Fig. 3.2 (a) Thermoelectric power S of Ce3 Pt4 Ge6 (black points) and Pr3 Pt4 Ge6 (red points) versus temperature for 2–300 K. (b) S versus temperature at low temperature. (c) Electrical resistivity ρ of Ce3 Pt4 Ge6 (black points) and Pr3 Pt4 Ge6 (red points) versus temperature. (d) Thermoelectric power factor S2 /ρ is plotted versus temperature, exhibiting maxima of 2.7 µW cm−1 K−2 and 0.4 µW cm−1 K−2 , respectively, for Ce3 Pt4 Ge6 and Pr3 Pt4 Ge6 . Lines are guides to the eye

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the Ce-based sample Ce3 Pt4 Ge6 has a maximum value Smax which is an order of magnitude larger than that of Pr3 Pt4 Ge6 . In Pr3 Pt4 Ge6 , S is relatively featureless and positive throughout the measured temperature range. At low temperature, as seen in Fig. 3.2b, S smoothly approaches zero. In Ce3 Pt4 Ge6 , a broad maximum with Smax 13 µV K−1 is observed near ∼175 K. Invoking the theory of Zlatić and Monnier again [18], we can conclude that the character of S most closely matches “type d” behavior as was true for CePt4 Ge12 . We are able to draw one notable contrast between the two Ce-based compounds, however; evidence for a shoulder in Ce3 Pt4 Ge6 below the maximum centered near TS ∼ 175 K is absent, suggesting that Ce is unambiguously in the intermediate valence regime in Ce3 Pt4 Ge6 . Electrical resistivity ρ versus temperature measurements were performed along the same crystallographic directions as S measurements and are shown in Fig. 3.2c for Ce3 Pt4 Ge6 and Pr3 Pt4 Ge6 . The thermoelectric power factor S2 /ρ is displayed in Fig. 3.2d. S2 /ρ is an order of magnitude larger in Ce3 Pt4 Ge6 than Pr3 Pt4 Ge6 (2.7 µW cm−1 K−2 compared to 0.4 μ W cm−1 K−2 , respectively). We are unable to calculate ZT for these systems; however, even if κ for Ce3 Pt4 Ge6 is an order of magnitude lower than that of CePt4 Ge12 , ZT will be two orders of magnitude lower than ZT ∼ 1. Since there are no rattling modes to reduce κph in Ce3 Pt4 Ge6 , even this scenario is probably overly optimistic. It appears that the Ln3 Pt4 Ge6 compounds are less promising candidates than LnPt4 Ge12 as thermoelectric materials. However, it may still be interesting to study whether better thermoelectric properties might be found along other crystallographic directions in these structurally anisotropic systems.

3.5 Non-centrosymmetric X2 T 12 P7 (X = Yb, Hf and T = Fe, Co) The synthesis of a family of compounds X2 T12 P7 where X = Zr, Ce-Lu and T = Mn, Fe, Co, or Ni was reported by Jeitschko [26], who found them to be isomorphic with the non-centrosymmetric Zr2 Fe12 P7 crystal structure with hexagonal space ¯ In fact, they represent the n = 2 variant of a larger family of nongroup P6. centrosymmetric compounds Lnn(n−1)T(n+1)(n+2)Mn(n+1)+1 where M = P or As [27]. Despite being discovered more than three decades ago, the physical properties of these materials are relatively unexplored. Recent studies have shown that Th2 Fe12 P7 and U2 Fe12 P7 are a Pauli paramagnet and antiferromagnet (TN ∼ 14 K), respectively, each with moderately enhanced γ [28]. The Sm-based compound Sm2 Fe12 P7 was found to be a relatively rare example of a Sm-based heavy fermion material (γ ∼ 450 mJ mol−1 K−2 ) with ferromagnetic order below TC = 6.3 K [29]. The Ybbased compound Yb2 Co12 P7 undergoes ferromagnetic order of its Co moments at TC ∼ 136 K and exhibits evidence for possible order of its Yb moments at TM ∼ 5 K [30]. Finally, Yb2 Fe12 P7 exhibits an unconventional T − H phase diagram wherein a crossover from a magnetically-ordered (TM ∼ 0.9 K at H = 0 T) non-Fermi liquid


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(NFL) phase at low H crosses over to a second NFL phase at higher H [10]. This crossover appears to be decoupled from a putative quantum critical point at H ∼ 1.5 T (where TM extrapolates smoothly to 0 K), indicating an unconventional route to its NFL ground states [10]. To our knowledge, the thermoelectric properties of the “2-12-7s” have never been reported. However, they may make a suitable family of materials in which to search for viable thermoelectric materials. The electronic density of states N(ε ) of Fe2 P, the archetypical n = 1 variant of the Lnn(n−1) T(n+1)(n+2)Mn(n+1)+1 family of compounds, was reported to exhibit a sharp peak at εF associated with its Fe 3d electrons [31]. Though subsequent thermoelectric power measurements obtained a relatively modest maximum Smax ∼ 11 µV K−1 for Fe2 P [32], perhaps incorporating elements with f electrons (such as can be done in X2 T12 P7 ) may enhance the density of states near εF and increase Smax . We are unaware of any band structure calculations for X2 T12 P7 , but the nearly equivalent values for γ in Th2 Fe12 P7 (γ ≈ 95 mJ mol−1 K−2 ) and U2 Fe12 P7 (γ ≈ 100 mJ mol−1 K−2 ) might be evidence that N(εF ) is dominated by their common Fe 3d electronic states [28]. We report measurements of Yb2 Fe12 P7 , Hf2 Fe12 P7 , and Hf2 Co12 P7 , which allows us to compare elements with and without an unfilled f -electron shell (Yb and Hf) and the effect of two distinct transition metals (Fe and Co). The thermoelectric power S of Hf2 T12 P7 (T = Fe, Co) and Yb2 Fe12 P7 are displayed in Fig. 3.3a, b. When the T site is occupied by Fe, S > 0 for the majority of the measured temperature range, in contrast to T = Co where S < 0 is observed. In Hf2 Fe12 P7 , we observe a subtle shoulder at ∼10 K and a change in slope below 5 K (as highlighted in Fig. 3.3b). Near ∼50 K, features appear in both Hf2 Fe12 P7 (dramatic change in slope) and Hf2 Co12 P7 (shallow local minimum) that are expected to have a common origin. Above this temperature, Hf2 Fe12 P7 is approximately linear to 300 K, while Hf2 Co12 P7 exhibits a kink at its ferromagnetic transition near TC ∼ 140 K, becoming linear above TC . In Yb2 Fe12 P7 , there is a minimum in S at ∼28 K and a subtle shoulder near 10 K. The theory of Zlatić and Monnier also classifies distinct types of behaviors observed in Yb-based intermetallic compounds [18]. However, our results for Yb2 Fe12 P7 are inconsistent with the characteristics expected in any of their categories. Strangely, S behaves more like a Ce-based intermetallic system than Yb-based. Further study will be necessary to explain this surprising result, but despite lacking an explanation for this behavior, we are able to observe that the overall magnitude of S is relatively enhanced when T = Fe (Smax 57.5 µV K−1 for Yb2 Fe12 P7 and Smax 50.5 µV K−1 for Hf2 Fe12 P7 ). In contrast, Hf2 Co12 P7 exhibits |Smax | 8.0 µV K−1, probably indicating that the magnitude of S is dominated by contributions from the transition metal. In the context of the Mott formula for S [7], we expect that Fe 3d electrons contribute a larger factor of N −1 (εF )(dN(ε )/dε )|εF than Co 3d electrons contribute. The electrical resistivity ρ versus temperature of Hf2 T12 P7 (T = Fe, Co) and Yb2 Fe12 P7 is plotted in Fig. 3.3c. The data plotted for Yb2 Fe12 P7 were measured on a single crystal sample along the crystallographic c axis and are discussed in detail in Ref. [10]. The properties of Hf2 T12 P7 (T = Fe, Co) including ρ will be the

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a

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Fig. 3.3 (a) Thermoelectric power S of Yb2 Fe12 P7 (black points), Hf2 Co12 P7 (red points), and Hf2 Fe12 P7 (blue points) versus temperature for 2–300 K. Lines are guides to the eye emphasizing the linearity of S in the Hf-based systems at high temperature. (b) S versus temperature at low temperature. (c) Electrical resistivity ρ of Yb2 Fe12 P7 (black points), Hf2 Co12 P7 (red points), and Hf2 Fe12 P7 (blue points) versus temperature. (d) Thermoelectric power factor S2 /ρ versus temperature. The inset highlights the temperature dependence of S2 /ρ for the Hf-based compounds. Maximum values are ∼15.6 µW cm−1 K−2 , 0.6 µW cm−1 K−2 , and 0.56 µW cm−1 K−2 , respectively, for Yb2 Fe12 P7 , Hf2 Co12 P7 , and Hf2 Fe12 P7 . Lines are guides to the eye

subject of a forthcoming manuscript [33]. The thermoelectric power factor S2 /ρ is plotted versus temperature in Fig. 3.3d. The magnitude of S2 /ρ for Hf2 T12 P7 (T = Fe, Co) is much smaller than for Yb2 Fe12 P7 , however, we caution that our results are a rough estimate for Yb2 Fe12 P7 because S and ρ were measured on polycrystalline and single crystalline samples, respectively. Despite this source of uncertainty, we see that S2 /ρ for Yb2 Fe12 P7 is comparable to the values found in many filled skutterudites; however, it is still an order of magnitude smaller than the largest S2 /ρ among Yb-based intermetallic compounds (reported for YbAl3 ) [34]. Interestingly, despite a large disparity in the magnitude of S for the Hf-based compounds, they are compensated by an opposite disparity in ρ such that S2 /ρ is quantitatively similar for each compound as highlighted in the inset of Fig. 3.3d. Further work will be necessary to fully investigate the potential of members of this family of compounds as thermoelectric materials. This will include measuring κ and studying what role, if any, the non-centrosymmetric crystal structure may play


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in thermal transport. Based on observations made from our preliminary results, other systems with general chemical formula R2 Fe12 P7 (R = rare earth element) may be a fruitful place to search for materials with enhanced S. For example, the quasiparticle contribution to N(ε ) near εF in the heavy-fermion compound Sm2 Fe12 P7 might augment S, which is already expected to be enhanced by Fe 3d contributions, such that S is very large at low temperature [29].

3.6 Conclusions Motivated to search for viable thermoelectric materials among correlated electron systems, we have studied the thermoelectric properties of Ln3 Pt4 Ge6 and LnPt4 Ge12 (Ln = Ce, Pr) and X2 T12 P7 (X = Yb, Hf and T = Fe, Co). We have assessed the potential of each system by calculating the thermoelectric power factor S2 /ρ (as a substitute for the thermoelectric figure of merit ZT , which we cannot calculate due to the absence of thermal conductivity data). The magnitude of S is observed to be sensitive to Ln in Ln3 Pt4 Ge6 and LnPt4 Ge12 with Ce-based compounds providing a much larger S and S2 /ρ than Pr-based materials. The character of S in each Ce-based compound exhibits traits that suggest the Ce ions possess an intermediate valence. In the case of X2 T12 P7 systems reported herein, it appears that we are able to tune the magnitude of S more effectively by varying T rather than X; the magnitude of S is significantly larger with T = Fe than when T = Co. The most promising materials among those studied and reported herein are CePt4 Ge12 and Yb2 Fe12 P7 with maximum S2 /ρ values of 18.2 and 15.6 µW cm−1 K−2 , respectively. However, even these systems are expected to fall well short of the goal of obtaining a material with ZT ≥ 1 near 300 K. In the case of CePt4 Ge12 , it would be interesting to determine whether chemical substitution might increase S2 /ρ . Further exploration of X2 Fe12 P7 compounds, focusing especially on materials that show evidence for strong correlations as a result of f -electron physics, may also result in promising candidates for thermoelectric applications. Acknowledgements Sample synthesis and screening for superconductivity were conducted under the auspices of AFOSR-MURI Grant FA9550-09-1-0603. Physical properties measurements were supported by NSF Grant 0802478 and sample characterization (powder X-ray diffraction and energy dispersive X-ray spectroscopy measurements) were supported by the U.S. DOE grant DEFG02-04ER46105. Acquisition of crystal growth equipment used in the synthesis of some of these samples was funded by DOE grant DE-FG02-04ER46178. M. Janoschek acknowledges financial support from the Alexander von Humboldt foundation.

References 1. Snyder GJ, Toberer ES (2008) Complex thermoelectric materials. Nat Mater 7:105 2. Mahan GD, Sofo JO (1996) The best thermoelectric. Proc Natl Acad Sci U S A 93:7436 3. Slack GA, Tsoukala VG (1994) Some properties of semiconducting IrSb3 . J Appl Phys 76:1665

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