Thermal expansion of skutteruditesG. Rogl,1,2,4 L. Zhang,1,2,4 P. Rogl,1,a� A. Grytsiv,1 M. Falmbigl,1 D. Rajs,2 M. Kriegisch,2

H. Müller,2 E. Bauer,2 J. Koppensteiner,3 W. Schranz,3 M. Zehetbauer,4 Z. Henkie,5

and M. B. Maple6

1Institute of Physical Chemistry, University of Vienna, Währingerstr. 42, A-1090 Wien, Austria2Institute of Solid State Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040 Wien,Austria3Nonlinear Physics Group, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria4Group Physics of Nanostructured Materials, University of Vienna, Boltzmanngasse 5, A-1090 Wien,Austria5Institute of Low Temperature and Structure Research, Polish Academy of Science, PL-50-950 Wroclaw,Poland6Department of Physics, University of California, San Diego, La Jolla, California 92093, USA

�Received 24 July 2009; accepted 5 December 2009; published online 18 February 2010�

The current paper gives an overview of the newly obtained thermal expansion coefficients ofskutterudites as well as those so far available in literature. Thermal expansion was determined forCoSb3, Pt4Sn4.4Sb7.6, for As- and Ge-based skutterudites as well as for various high-ZT skutterudites�micro- and nanostructured� with didymium �DD� and mischmetal �Mm� as filler atoms inframeworks of �Fe1−xCox�4Sb12 and �Fe1−xNix�4Sb12, and for double and triple-filled skutteruditessuch as Ca0.07Ba0.23Co3.95Ni0.05Sb12 and Sr0.025Ba0.075Yb0.1Co4Sb12. For low temperatures, acapacitance dilatometer was used �4–300 K�, whereas for temperatures 300�T�750 K, a dynamicmechanical analyzer was employed. For a set of Ge-, P-, and Sb-based skutterudites, latticeparameters of single crystals, measured at three different temperatures, were used to derive thethermal expansion coefficient. The semiclassical model of Mukherjee �Phys. Rev. Lett. 76, 1876�1996�� has been successfully used to quantitatively describe the thermal expansion coefficient interms of Einstein and Debye temperatures, which compare well with the corresponding results fromspecific heat, electrical resistivity, or temperature dependent x-ray measurements. © 2010 AmericanInstitute of Physics. �doi:10.1063/1.3284088�

I. INTRODUCTION

Thermoelectric generators directly convert heat flow intoelectrical power. Energy conversion efficiency of thermo-electric materials is a function of the dimensionless thermo-electric figure of merit ZT=S2T / ����, where S is the See-beck coefficient, T is the temperature, � is the electricalresistivity, and � is the thermal conductivity. With thermo-electric energy conversion efficiencies of more than 10%�ZT�1�, skutterudites have been considered as suitable ther-moelectric generator materials for an application range 300–700 K. For a flawless long-term and cyclic temperature per-formance of thermoelectric devices, it is essential thatthermal expansion coefficients of p- and n-legs as well as ofcontacting materials are chosen as similar as possible. Al-ready in the 1990s in the Jet Propulsion Laboratory not onlytransport behavior of skutterudites but also related problemssuch as thermal expansion were investigated; however, atthat time thermal expansion coefficients were only reportedfor CoSb3,1,2 RhSb3,1 and IrSb3.1,3,4 From our comprehensiveliterature search in two major electronic libraries, chemicalabstracts service �CAS� and INSPEC, scanning entries up to2009, it became obvious that only a few research groupshave dealt with thermal expansion �see data and references in

Table II�. Besides these directly accessible data on thermalexpansion, all data in literature were collected, which al-lowed us to extract thermal expansion coefficients. There-fore, the aim of the present work is threefold: �i� to supplynew data on thermal expansion from a series of high ZT p-and n-type skutterudites, �ii� to extract thermal expansionfrom experimental data in literature, where expansion coef-ficients have not yet been evaluated by the authors, and�iii� a general discussion of all thermal expansion dataavailable covering antimony-, phosphorous- arsenic-, andgermanium-based skutterudites. It is interesting to note thatour literature search did not reveal any expansion data onarsenide skutterudites. The experimental work presentedherein is concerned with skutterudites �micro- and nanostruc-tured� where didymium �DD� �4.76% Pr and 95.2% Nd� andmischmetal �Mm� �50.8% Ce, 28.1% La, 16.1% Nd,and 5.0% Pr� act as filler atoms in the frame-works of �Fe1−xCox�4Sb12 and �Fe1−xNix�4Sb12. From theseseries of samples, we selected those with a ZT�1�DD0.68Fe3.2Ni0.8Sb12 and DD0.76Fe3.4Ni0.6Sb12�, includingnanostructured �ball milled �BM�� as well as micro-structured materials. These were DD0.44Fe2.1Co1.9Sb12

and DD0.44Fe2.1Co1.9Sb12 BM, Mm0.76Fe4Sb12 andMm0.70Fe4Sb12 BM, and DD and Mm-samples with the samenominal composition �DDFe4Sb12 BM and MmFe4Sb12

BM�.5,6 We compare the thermal expansion of the above-mentioned samples not only with multifilled n-type

a�Author to whom correspondence should be addressed. Tel.: �43-1-4277-52456. FAX: �43-1-4277-95245. Electronic mail:peter.franz.rogl@univie.ac.at.

JOURNAL OF APPLIED PHYSICS 107, 043507 �2010�

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skutterudites, Ca0.07Ba0.23Co3.95Ni0.05Sb12 �Ref. 7� andSr0.025Ba0.075Yb0.1Co4Sb12,

8 which both have a ZT�1,CoSb3, and Pt4Sn4.4Sb7.6 but also with values reported in theliterature.9–15 The semiclassical model of Mukherjee16 willbe used to quantitatively evaluate the thermal expansion andto derive Einstein and Debye temperatures for all thosesamples where lattice parameters or dilatometric data areavailable as a function of temperature over a larger tempera-ture range starting from 4.2 K.

II. EXPERIMENTAL DETAILS

All DD and Mm samples, CoSb3, Pt4Sn4.4Sb7.6 as well asCa0.07Ba0.23Co3.95Ni0.05Sb12, and Sr0.025Ba0.075Yb0.1Co4Sb12,were prepared via an optimized melting reaction technique.The solids obtained were ground into fine powders in a WCmortar or BM and in both cases hot pressed in an argonatmosphere at 600 °C under a pressure of 50 MPa. For fur-ther details, see Refs. 5–8. Pt4Sn4.4Sb7.6 was prepared in theform of cold compacted sintered pellets.

Lattice constants for polycrystalline powders were ob-tained at room temperature from Guinier x-ray diffractiondata, applying Cu K�1 radiation and employing a leastsquares evaluation with the program STRUKTUR.17 Chemicalcomposition and microstructure were determined by electronprobe microanalysis �EPMA�. Filling levels were obtainedfrom combined evaluation of EPMA and Rietveld x-ray pat-tern refinements. Lattice parameters of skutterudite singlecrystals were obtained at three different temperatures �300,200, and 100 K, N2 cooling the crystal� from an Enraf Non-ius Kappa charge-coupled device instrument with monochro-matic Mo K� radiation under a flow of equilibrated nitrogengas from a cryostat. LaFe4As12 and PrFe4As12 are singlecrystals, grown from elements with purities �99.9% by us-ing a molten metal flux method at high temperature and pres-sure. Details on growth, structural, and physical propertiesare reported elsewhere.18–20

The thermal expansion from 4.2 to 300 K was measuredin a miniature capacitance dilatometer,21 using the tilted plateprinciple.22,23 For this measurement, the sample is placed in ahole of the lower ringlike capacitance plate made of silver,which is separated from the silver upper capacitor plate bytwo needle bearings. All DD samples, Mm0.76Fe4Sb12,Mm0.70Fe3CoSb12 BM, Ca0.07Ba0.23Co3.95Ni0.05Sb12,Sr0.025Ba0.075Yb0.1Co4Sb12, and CoSb3, Pt4Sn4.4Sb7.6, andboth Ge- and As-based samples were measured with this lowtemperature capacitance dilatometer. For the measurement ofthe thermal expansion at a temperature range from 300 to700 K, a dynamic mechanical analyzer DMA7 �Perkin ElmerInc.� was employed. The sample is positioned in a parallelplate mode with a quartz rod on top of the sample and dataare gained using the thermodilatometric analysis �TDA�.TDA is often referred to as zero force thermomechanicalanalysis. With this method the change in the dimension of asample is measured while subjected to a temperature changewithout using any force. The length of the sample is mea-sured via the movement of the quartz rod. This movement isdetected using electromagnetic inductive coupling. The ab-solute length l and the length change �l are acquired with a

resolution of 10 nm,24 for further details see also Refs.25–27. All Mm samples, except Mm0.70Fe3CoSb12 BM,DD0.08Fe2Ni2Sb12, and Ca0.07Ba0.23Co3.95Ni0.05Sb12, weremeasured with this method.24 Porosity was obtained fromdensity measurements in distilled water, using Archimedes’principle, and the calculation of the x-ray density dX

= �ZM� / �NV�, where M is the molar mass, Z is the numberof formula units per cell, N is Avogadro’s number, and V isthe volume of the unit cell. Resistivities of the DD alloys inthe temperature range from 4.2 to 300 K were measuredusing a dc four-point technique. The resistivity curves ofDD0.68Fe4Sb12, DD0.76Fe3.4Ni0.6Sb12, DD0.44Fe2.1Co1.9Sb12,and DD0.44Fe2.1Co1.9Sb12 BM showed metallic behavior andtherefore could be fitted well with the Bloch–Grüneisenfunction yielding also the Debye temperature; fordetails see Ref. 5. The same technique was used forCa0.07Ba0.23Co3.95Ni0.05Sb12.

7

Time of flight of sound pulse measurements were per-formed on cylinders with a frequency of 10 MHz using ahome made equipment27 to provide the data for longitudinal�vl� and shear �transversal� �vs� sound velocities.

III. RESULTS AND DISCUSSION

Table I summarizes the thermoelectric properties of se-lected DD and Mm samples,5,6 Ca0.07Ba0.23Co3.95Ni0.05Sb12

�Ref. 7� and Sr0.025Ba0.075Yb0.1Co4Sb12,8 published recently

and the porosity of all these alloys.All samples are single phase except Mm0.70Fe4Sb12

�80 at. %�, which contains also FeSb2, Sb, and Mm2O3. Inall cases the Rietveld refinement showed an ordered atomarrangement with respect to the atom site distribution amongDD/Mm, �Fe/Co, Fe/Ni�, and Sb sublattices. DD and Mmcontents agree well with the data obtained from EPMAwhich applies also for the Ba and Ca contents and Co/Nicontribution in Ca0.07Ba0.23Co3.95Ni0.05Sb12 and the Sr, Ba,and Yb contents of Sr0.025Ba0.075Yb0.1Co4Sb12.

Figures 1 and 2 show the thermal expansion �l / l of theaforementioned skutterudites as a function of temperature. InFig. 1, the data from the low temperature measurements aredisplayed, and those from the high temperature measure-ments in Fig. 2.

From Fig. 1, it is obvious that �l / l0 of all measuredskutterudites decreases linearly in temperature from roomtemperature to about 150 K, whereas for temperatures below150 K, a nonlinear behavior is evident. The thermal expan-sion coefficient � follows from a temperature derivative ofthe length change, i.e.,

� = � ��l

�T� 1

l0. �1�

� was calculated in the temperature range from about 150 to300 K. The thermal expansion coefficients derived in thepresent article together with data available in the literatureare listed in Table II. Although the thermal expansion for allDD samples �9.4510−6 K−1���11.3010−6 K−1� isslightly lower than for the Mm samples �9.9710−6 K−1

���12.4210−6 K−1�, the difference within DD as wellas within Mm skutterudites is not high, which applies also

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for the difference between the n-type and the p-type DDsamples. As expected, the difference in � between nanostruc-tured �BM� and microstructured samples and of samples withlower or higher porosity is very small, as can be seen whencomparing the graphs of Mm0.76Fe4Sb12 and Mm0.70Fe4Sb12

BM in Fig. 2 as well as DD0.44Fe2.1Co1.9Sb12 andDD0.44Fe2.1Co1.9Sb12 BM in Fig. 3 or the values of � of thesesamples as well as for Mm0.70Fe3CoSb12 BM, measuredat low temperatures, and Mm0.70Fe3CoSb12, measuredat high temperatures �see Table II�. However, Figs. 1 and 2also show that both n-type multifilled skutterudites,Ca0.07Ba0.23Co3.95Ni0.05Sb12 ��=9.1410−6 K−1� andSr0.025Ba0.075Yb0.1Co4Sb12 ��=8.3510−6 K−1�, have a sig-nificantly smaller thermal expansion. These observationslead to the conclusion that the grain size does not influencethe thermal expansion, but filler atoms do have some influ-ence on thermal expansion. Figure 4 shows that in DD orMm skutterudites, an increasing Fe-content �and a simulta-neously increasing filler-content� enlarges the thermal expan-sion. This is also the case for other pairs of samples; e.g.,La0.743Fe2.74Co1.26Sb12 ��=9.0810−6 K−1� �Ref. 15� and

La0.9Fe4Sb12 ��=11.6910−6 K−1�, CexCo4Sb12, 0�x�0.1 ��=810−6 K−1� �Ref. 14� and Ce0.9Fe4Sb12 ��=13.9310−6 K−1�, or Ca0.07Ba0.23Co3.95Ni0.05Sb12 ��=9.1410−6 K−1� and CaFe4Sb12 ��=10.910−6 K−1�.10

For CoSb3 two rather different thermal expansion coeffi-cients were found in literature: �=13.510−6 K−1 �no de-tails given, Ref. 1� and �=6.3610−6 K−1 �from singlecrystal in a range from 300 to 930 K, Ref. 2�. Therefore,thermal expansion was remeasured for a CoSb3 sample �BMand hot pressed� revealing a coefficient �=9.110−6 K−1

�120–220 K�. This value fits well to the dependency of DDand Mm alloys shown in Fig. 4.

The lattice parameter of a cubic material at varying tem-peratures is in relationship to the thermal expansion coeffi-cient gained from TDA or capacitance data via the relation

� =

a2 − a1

a1

�T, �2�

where ax is the lattice parameter at the temperature x. Formost of our calculations, we used the lattice parameter a2

FIG. 1. �Color online� Temperature dependent thermal expansion �l / l0 ofvarious skutterudites for 4.2 K�T�300 K.

FIG. 2. �Color online� Temperature dependent thermal expansion �l / l0 ofvarious skutterudites for 300 K�T�700 K.

TABLE I. Physical properties at 297 K �ZT also at 700 K� of selected skutterudites.

Skutterudite TypePorosity

�%��

�� cm�S

�V /K��

�mW/cm K� ZT �297 K� ZT �700 K� Ref.

Ca0.07Ba0.23Co3.95Ni0.05Sb12 n 5 317 �109 53 0.18 1.10 7Sr0.025Ba0.075Yb0.1Co4Sb12 BM n 0.6 295 �170 32 0.46 1.28 8DD0.08Fe2Ni2Sb12 n 1.4 3730 �70 21 0.02 0.24 5DD0.76Fe3.4Ni0.6Sb12 p 6.3 690 126 19 0.34 1.05 5DD0.68Fe3.2Ni0.8Sb12 p 2.3 552 104 22 0.26 0.93 5DD0.44Fe2.1Co1.9Sb12 p 4.1 780 105 19 0.20 0.44 5DD0.44Fe2.1Co1.9Sb12 BM p 0.2 900 91 15 0.18 0.43 5DD0.86Fe4Sb12 BM p 0.6 370 74 32 0.13 0.86 a

Mm0.76Fe4Sb12 p 5.1 449 77 26 0.15 0.61 6Mm0.70Fe4Sb12�not single phased� BM p 3.3 408 79 21 0.22 0.75 6Mm0.20Fe2.5Ni1.5Sb12 p 1.5 ¯ ¯ ¯ ¯ ¯ 6Mm0.70Fe3CoSb12 BM p 1.8 785 113 15 0.33 1.16 6Mm0.68Fe3CoSb12 BM p 1.2 650 103 17 0.28 1.09 6Mm0.05FeCo3Sb12 p 1.9 ¯ ¯ ¯ ¯ ¯ 6

aThis work, prepared as described in Ref. 5.

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TABLE II. Lattice parameters and thermal expansion coefficients of Ge-, P-, As-, and Sb-based skutterudites.

Lattice parameter a, RT �nm� Ref. �10−6 �K−1� T �K� Method Ref.

Ge-based skutteruditesBaPt4Ge12 0.86928�2� a 10.2 160–245 DMb a

UPt4Ge12 0.85835�3� a 9.15 160–245 DM a

P-based skutteruditesLaRu4P12 0.80605�2� 28 4.7 10–300 SCLPc a

PrRu4P12 0.80493 a 5.64 10–300 SCLP a

GdRu4P12 0.80375 29 5.4 150–300 LPd a

LaOs4P12 0.80932�3� a 5.1 100–300 LP a

CeOs4P12 0.80751�3� a 5.08 100–300 SCLP a

PrOs4P12 0.80813�2� a 4.77 100–300 SCLP a

NdOs4P12 0.80790�2� a 5.8 100–300 SCLP a

SmOs4P12 0.80731�2� a 5.39 100–300 SCLP a

As-based skutteruditesPrFe4As12 0.8310�2� 18 10.3 160–250 DM a

LaFe4As12 0.83273�2� 20 9.25 160–250 DM a

Sb-based skutteruditesCoSb3 0.903484�2� a 9.1 120–220 DM a

0.90345�3� 2 13.5 ¯ ¯ 16.36 300–930 ¯ 2

RhSb3 12.7 ¯ ¯ 1IrSb3 0.92503�3� 9 6.89 RT LP a

0.92503�3� 3 6.69 300–673 LP 30.92533 1 7.96 ¯ ¯ 1

8 ¯ ¯ 4NaFe4Sb12 0.91759�3� 10 11 100–300 LP 10CaFe4Sb12 0.9171�4� 33 10.9 150–300 LP a

CaxCo4Sb12 0.9052 11 9 100–300 LP 11Ru0.5Pd0.5Sb3 0.9298 12 9.09 ¯ ¯ 12Tl0.22Co4Sb12 0.9056 13 7.4 180–300 NDe 13TlCo3FeSb12 0.9112 13 9.5 180–300 ND 13Tl0.5Co4Sb11.5Sn0.5 0.9082 13 9.5 180–300 ND 13La0.9Fe4Sb12 0.91503 a 11.69 100–300 SCLP a

Ce0.9Fe4Sb12 0.91406�3� a 12.7 100–300 LP a

CexCo4Sb12�0�x�0.1� ¯ ¯ 8 300 ¯ 14PrFe4Sb12 0.91290 a 11.21 125–300 LP a

Nd0.85Fe4Sb12 0.91412�2� a 12.50 100–300 SCLP a

Eu0.93Fe4Sb12 0.91725�2� a 10.59 100–296 SCLP a

Yb0.95Fe4Sb12 0.91586�8� 34 12 110–295 LP a

YbxCo4Sb12 0.9048 11 8.17 100–300 LP 11DD0.86Fe4Sb12 BM 0.91357�2� a 11.26 160–245 DM a

DD0.68Fe3NiSb12 0.91208�4� a 12.09 160–245 DM a

DD0.76Fe3.4Ni0.6Sb12 0.91219�3� a 11.29 160–245 DM a

DD0.08Fe2Ni2Sb12 0.90927�3� a 9.81 160–245 DM a

9.82 300–600DD0.44Fe2.1Co1.9Sb12 0.90920�4� a 9.51 160–245 DM a

DD0.44Fe2.1Co1.9Sb12 BM 0.90878�3� a 9.45 160–245 DM a

Mm0.76Fe4Sb12 0.91370�5� a 13.43 160–250 DM a

11.94 300–500Mm0.70Fe4Sb12 BM 0.91384�3� a 12.42 300–500 DM a

Mm0.20Fe2.5Ni1.5Sb12 0.91009�1� a 10.78 300–500 DM a

Mm0.70Fe3CoSb12 BM 0.91165�3� a 11.33 160–245 DM a

Mm0.68Fe3CoSb12 0.91167�3� a 11.43 300–500 DM a

Mm0.05FeCo3Sb12 0.90624�3� a 9.97 300–500 DM a

Ca0.07Ba0.23Co0.95Ni0.5Sb12 0.90665�3� a 9.19 160–280 DM a

10.16 300–650Sr0.025Ba0.075Yb0.1Co4Sb12 0.90617�4� a 8.85 160–245 DM a

La0.743Fe2.74Co1.26Sb12 0.90971�3� 15 9 ND 158.9 100–300 LP a

Pt4Sn4.4Sb7.6 0.93304�2� a 6.94 130–230 DM a

aThis work.bDM dilatometer measurements.cSCLP calculated from single crystal lattice parameter.dLP calculated from lattice parameter.eND data from neutron diffraction.

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gained from precise x-ray diffraction data on polycrystallineor single crystal specimens.

Figure 5 gives an overview of the temperature dependentlattice parameters of all measured DD and Mm samples andshows that the slopes do not differ very much. Figures 5 and6 demonstrate that for DD0.08Fe2Ni2Sb12, Mm0.70Fe4Sb12,and Ca0.07Ba0.23Co3.95Ni0.05Sb12, the measurements at lowand high temperatures, with different equipment used, fitwell together. Also the values calculated for two differentmeasurement ranges fit within the measurable accuracy.While in the case of DD0.08Fe2Ni2Sb12, the values for thethermal expansion ��160–240�=9.8110−6 K−1 for low and��300–600�=9.8210−6 K−1 for high temperatures are equal,for MmFe4Sb12 and Ca0.07Ba0.23Co3.95Ni0.05Sb12 �see Figs. 5and 6�, the difference in these temperature ranges is smallerthan 110−6 K−1.

Figures 7 show temperature dependent lattice parametersfor various Ge-, P-, As- and Sb-based skutterudites. For thecalculation of the thermal expansion coefficient �, either our

data or literature data of lattice parameters were used �seealso Table II�. Slack3 derived the thermal expansion coeffi-cient �=6.6910−6 K−1 for IrSb3 from x-ray data in thetemperature range from 300 to 673 K in good agreementwith our calculation, �=6.8910−6 K−1, using the latticeparameters of Kjekshus9 �see Fig. 7�a��. Both values arelower than the thermal expansion �=810−6 K−1 found byKjekshus earlier.4 In Fig. 7�a�, the data for the DD and Mmskutterudites based on Sb are compared with the correspond-ing La, Ce, Pr, Nd, Eu, and Yb skutterudites. The almostidentical slopes, i.e., the expansion coefficients �, againshow that � is almost insensitive to the filler elements asalready concluded above. All thermal expansion coefficientsof the P-based skutterudites were calculated from lattice pa-rameters obtained from single crystal measurements. It isremarkable that all thermal expansion coefficients of P-basedskutterudites MOs4P12 �M=La, Ce, Pr, Nd, Sm�, in a range�4.8–5.8�10−6 K−1, are only half as large as those of Sb-based skutterudites where � ranges from 7.510−6 to 1410−6 K−1. Similarly, data of LaRu4P12 �Ref. 28� with �=4.710−6 K−1, PrRu4P12 with �=5.6410−6 K−1, andGdRu4P12 �Ref. 29� with �=5.410−6 K−1 are also in thisrange of relatively low thermal expansion, indicating stron-

FIG. 3. �Color online� Temperature dependent thermal expansion �left axis�and calculated lattice parameter �right axis� versus temperature for nano-structured �BM� and microstructured DD0.44Fe2.1Co1.9Sb12.

FIG. 4. �Color online� Thermal expansion coefficient � vs DD and Mmcontents �a� and versus Fe content �b�. The solid line is a guide to the eyes.

FIG. 5. �Color online� Temperature dependent lattice parameters a of DDand Mm compounds.

FIG. 6. �Color online� Thermal expansion �left axis� and lattice parameter�right axis� as function of temperature for Ca0.07Ba0.23Co3.95Ni0.05Sb12 withDebye temperature D and Einstein temperature E gained from the fit.

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ger covalent bonding in the P framework than in the Sbframework. Figure 7�c� shows a good agreement between thelattice parameters of the Ge-based skutterudites gained fromsingle crystal measurements and those extracted from ther-mal expansion measurements. The thermal expansion coeffi-cients are in the range of Sb-based skutterudites. Both ars-enide samples �Fig. 7�d��, PrFe4As12 ��=10.310−6 K−1�and LaFe4As12 ��=9.2510−6 K−1�, show a smaller ther-mal expansion coefficient than their Sb-based counterpartsPrFe4Sb12 ��=11.2110−6 K−1� and LaFe4Sb12 ��=11.6910−6 K−1�. The inset of Fig. 8 shows the low temperaturethermal expansion of PrFe4Sb12 below 25 K. In agreementwith measurements of electrical resistivity, specific heat,elastic constants, and magnetization,18,19 a magnetic phasetransition at TC=18 K becomes obvious also from �l / l0 ver-sus T. At T�=12 K, a second phase transition �found fromsusceptibility, magnetization, and specific heatmeasurements19� is evident from a change in the slope of�l / l0 versus T.

To analyze the thermal expansion as a function of tem-perature in the full temperature range, we followed a semi-classical treatment by Mukherjee �details are described inRef. 16� taking into account three- and four-phonon interac-tions, considering an anharmonic potential, and using boththe Debye model for the acoustic phonons and the Einsteinapproximation for the optical modes. The length change�l / l�T0� is given by

�l

l�T0�=

�xT − �xT0

x0�xT =

�

2T2 +

3g

4c2 ��-G�2 − F�3� ,

� = �3

p�3kBT� T

D�3�

0

D/T z3dz

ez − 1+ �p − 3

p� kB E

e D/T − 1� ,

�3�

where � is the electronic contribution to the average latticedisplacement, D is the Debye temperature, E is the Einsteintemperature, and p is the average number of phononbranches actually excited over the temperature range. G, F, c,and g are further material dependent constants. The Debyeand Einstein temperatures, D and E were obtained fromleast squares fits of Eq. �3� to the experimental data.Fits were performed for all skutterudites measured inthe low temperature range and exemplary graphs areshown for Ca0.07Ba0.23Co3.95Ni0.05Sb12 �Fig. 6�, and forDD0.76Fe3.4Ni0.6Sb12 �Fig. 9�, the two Ge-based �Fig. 10�,and the two As-based �Fig. 8� skutterudites. Debye tempera-tures are in very good agreement with those gained from fitsto resistivity data5,7 and in the case of the BaPt4Ge12 andPrFe4As12 in good agreement with calculations from heatcapacity.30,19 For UPt4Ge12 the value for the Einstein tem-perature is almost the same as the one gained from theatomic displacement parameters.

FIG. 7. �Color online� �a� Temperature dependent lattice parameters of various Sb-based skutterudites. �b� Temperature dependent lattice parameters of variousP-based skutterudites. �c� Temperature dependent lattice parameters of Ge-based skutterudites with data from single crystal measurements. �d� Temperaturedependent lattice parameters of As-based skutterudites.

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The Debye temperatures for Mm0.70Fe4Sb12 andMm0.68Fe3CoSb12 were calculated using the values of themeasured longitudinal �vl� and shear �vs� sound velocitiesemploying Anderson‘s expressions:31

D =h

kB�3nN�

4�M�1/3

vm and vm = �1

3 2

vs3 +

1

vl3��−1/3

,

�4�

where h is Planck’s constant, kB is Boltzmann’s constant, Nis Avogadro’s number, � is the density, M is the molecularweight, and n is the number of atoms in the molecule. ForMm0.70Fe4Sb12, vs=2.70105 cm /s, vl=4.10105 cm /s,and therefore vm=2.96105 cm /s; for Mm0.68Fe3CoSb12,vS=2.72105 cm /s, vl=4.43105 cm /s, and thereforevm=3.00105 cm /s. With these values, D=306 K and D=313 K, respectively, could be calculated. Both values fitnicely to the values gained via least squares fits of Eq. �3�.For Sr0.025Ba0.075Yb0.1Co4Sb12, vS=2.79105 cm /s and vl

=4.65105 cm /s were measured yielding a calculated vm

=3.08105 cm /s and D=327 K.

The compounds UPt4Ge12, ReyOs4P12 �RE=La, Ce, Pr,Nd�, and REyFe4Sb12 �Re=La, Ce, Nd, Eu� can be consid-ered as simple Debye solids with the rare earth atoms behav-ing like Einstein oscillators. The Einstein temperatures E,ii

and the thermal displacements are related by

Uii =�2

2mkB E,iicoth� E,ii

2T� , �5�

where m is the atomic mass of the rattling atoms. From thelinear slope �Uii /�T in Fig. 11 �high temperature approxi-mation for h��2kBT�, the force constants Kii

= �kB�T� /�Uii, the frequency of vibrations �ii

=1 /2��Kii /m�1/2, and finally the Einstein temperature E,ii

= �h·�ii� /kB can be extracted. For further details, see Ref. 32.Table III gives an overlook over all these Debye and

Einstein temperature data. It is seen that the results of thiswork are in good agreement with those from the literature;e.g., compare the data for LaFe4Sb12 although the methods ofmeasurements and of calculations vary strongly. Debye andEinstein temperatures are of importance to define the vibra-

FIG. 8. �Color online� Temperature dependent thermal expansion ofLaFe4As12 and PrFe4As12. The solid line is a least squares fit according toEq. �3� for LaFe4As12, and the dashed line for PrFe4As12. The inset showsthe thermal expansion below 25 K for PrFe4As12, marked with arrows are T�

and TC.

FIG. 9. �Color online� Temperature dependent thermal expansion �left axis�and lattice parameter �right axis� curve of DD0.76Fe3.4Ni0.6Sb12. The solidline is a least squares fit according to Eq. �3�.

FIG. 10. �Color online� Temperature dependent thermal expansion ofBaPt4Ge12 and UPt4Ge12 with solid line and dashed line as least squares fitaccording to Eq. �3�.

FIG. 11. �Color online� Thermal displacements vs temperature for variousGe, P-, and Sb-based skutterudites.

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TABLE III. Debye � D� and Einstein � E� temperatures of selected skutterudites.

D

�K� Method Ref. E

�K� Method Ref.

Ge-based skutteruditesBaPt4Ge12 260 FTE a 82 FTE a

247 Cpb 30

UPt4Ge12 260 FTE a 62 FTE a

59 ADPc a

P-based skutteruditesLaOs4P12 125 ADP a

CeOs4P12 122.8 ADP a

PrOs4P12 117.6 ADP a

NdOs4P12 100.2 ADP a

As-based skutteruditesPrFe4As12 356 Cp 18 88 FTE a

360 FTE a

LaFe4As12 470 FTE a 98 FTE a

Sb-based skutteruditesCoSb3 314 FTE a

307 SVd 2306 SV 35321 SV 36325 Cp 37

La0.9Fe4Sb12 260 Cpa 85 ADP a

LaFe4Sb12 260 Cp 38248 Cp 39299 ADP 40 79 ADP 40331 ADP 13 70 ADP 36298 Cp�0.5–5 K� 37242 Cp�6.5–10� 37

Ce0.9Fe4Sb12 79 ADP a

CeFe4Sb12 86 EXAe 41Ce0.85Fe4Sb12 65 ¯ 42Nd0.85Fe4Sb12 72.8 ADP a

Eu0.93Fe4Sb12 82 ADP a

DD0.86Fe4Sb12 BM 210 FTEf a 98 FTE a

DD0.68Fe3NiSb12 240 FTE a 93 FTE a

235 FRg 5DD0.76Fe3.4Ni0.6Sb12 254 FTE a 75 FTE a

227 FR 5DD0.08Fe2Ni2Sb12 225 FTE a 97 FTE a

DD0.44Fe2.1Co1.9Sb12 270 FTE a 95 FTE a

267 FRDD0.44Fe2.1Co1.9Sb12 BM 265 FTE a 98 FTE a

198 FRMm0.76Fe4Sb12 312 FTE a a

Mm0.70Fe4Sb12 BM 306 SVd a a

Mm0.70Fe3CoSb12 BM 319 FTE a 92 FTE a

Mm0.68Fe3CoSb12 BM 313 SV a a

Ca0.07Ba0.23Co3.95Ni0.5Sb12 205 FTE a 51 FTE a

206 FR 7Sr0.025Ba0.075Yb0.1Co4Sb12 327 SV a a

Pt4Sn4.4Sb7.6 420 FTE a¯

a

aThis work.bCp heat capacity measurements.cADP calculated from temperature dependent atomic displacement parameters.dSV gained from sound velocity.eEXA extended x-ray absorption fine structure measurements.fFTE gained from thermal expansion fit.gFR gained from resistivity fit.

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tional spectra of thermoelectric materials where scattering ofheat carrying phonons on rattling modes helps to reduce bulkthermal conductivity as a key route to improve the thermo-electric figure of merit.

Despite the fact that there is no real physical explanationfor a relation between the figure of merit ZT and the thermalexpansion coefficient �, one could get the impression fromFig. 12 that a higher ZT is related to a higher � or vice versa.

Although thermal expansion coefficients are not avail-able for all corresponding members of P-, Ge-, As-, and Sb-based skutterudite families, a general trend emerges fromTable II, i.e., big values for Sb-based, middle sized values forAs- and Ge-based and smallest values for P-based skutteru-dites, documenting the increasing covalent bonding strengthwithin the framework cages �see Fig. 13�.

IV. CONCLUSIONS

This paper presents a comprehensive evaluation of ther-mal expansion data on skutterudites combining new mea-surements with all data hitherto available in the literature.Thermal expansion coefficients � for Sb-based skutteruditeswere found to be about double the size of P-based skutteru-dites. Although the differences in thermal expansion withinDDy�Fe1−xCox�4Sb12 and within Mmy�Fe1−xCox�4Sb12

samples are negligible, an increasing amount of fillers and ofFe content increases the thermal expansion. The influence ofthe filling level in combination with the Fe content on ther-mal expansion may be the reason that all Co-based n-typeskutterudites investigated have a smaller thermal expansioncoefficient than our Fe-based p-type skutterudites. The grainsize does not affect the thermal expansion. Fits of the ther-mal expansion measurements by means of the semiclassicaltreatment of Mukherjee proved well, by achieving Debyeand Einstein temperatures very close to those from resistiv-ity, sound velocity, or specific heat measurements. The Ein-stein temperatures derived are consistent with low frequencymodes of the filler atoms �rattling modes�, which reduce ther-mal conductivity via scattering of heat carrying phonons.

ACKNOWLEDGMENTS

Support by the University of Vienna within the IC Ex-perimental Materials Science “Bulk Nanostructured Materi-als” and the Austrian FWF, Grant No. P19284-N20 is grate-fully acknowledged. This work was partially supported bythe Austrian Science Foundation FWF, Grant No. S10406-N16. Z.H., G.R., and P.R. are grateful to the OEAD for abilateral exchange program Austria-Poland, Grant No. PL-06/2009.

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