JOURNAL OF APPLIED PHYSICS 107, 043507 共2010兲

Thermal expansion of skutterudites G. Rogl,1,2,4 L. Zhang,1,2,4 P. Rogl,1,a兲 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 1

Institute of Physical Chemistry, University of Vienna, Währingerstr. 42, A-1090 Wien, Austria Institute of Solid State Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria 3 Nonlinear Physics Group, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria 4 Group Physics of Nanostructured Materials, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria 5 Institute of Low Temperature and Structure Research, Polish Academy of Science, PL-50-950 Wroclaw, Poland 6 Department of Physics, University of California, San Diego, La Jolla, California 92093, USA 2

共Received 24 July 2009; accepted 5 December 2009; published online 18 February 2010兲 The current paper gives an overview of the newly obtained thermal expansion coefficients of skutterudites as well as those so far available in literature. Thermal expansion was determined for CoSb3, Pt4Sn4.4Sb7.6, for As- and Ge-based skutterudites as well as for various high-ZT skutterudites 共micro- and nanostructured兲 with didymium 共DD兲 and mischmetal 共Mm兲 as filler atoms in frameworks of 共Fe1−xCox兲4Sb12 and 共Fe1−xNix兲4Sb12, and for double and triple-filled skutterudites such as Ca0.07Ba0.23Co3.95Ni0.05Sb12 and Sr0.025Ba0.075Yb0.1Co4Sb12. For low temperatures, a capacitance dilatometer was used 共4–300 K兲, whereas for temperatures 300⬍ T ⬍ 750 K, a dynamic mechanical analyzer was employed. For a set of Ge-, P-, and Sb-based skutterudites, lattice parameters of single crystals, measured at three different temperatures, were used to derive the thermal expansion coefficient. The semiclassical model of Mukherjee 关Phys. Rev. Lett. 76, 1876 共1996兲兴 has been successfully used to quantitatively describe the thermal expansion coefficient in terms of Einstein and Debye temperatures, which compare well with the corresponding results from specific heat, electrical resistivity, or temperature dependent x-ray measurements. © 2010 American Institute of Physics. 关doi:10.1063/1.3284088兴 I. INTRODUCTION

Thermoelectric generators directly convert heat flow into electrical power. Energy conversion efficiency of thermoelectric materials is a function of the dimensionless thermoelectric figure of merit ZT= S2T / 共␳␭兲, where S is the Seebeck coefficient, T is the temperature, ␳ is the electrical resistivity, and ␭ is the thermal conductivity. With thermoelectric energy conversion efficiencies of more than 10% 共ZTⱖ 1兲, skutterudites have been considered as suitable thermoelectric generator materials for an application range 300– 700 K. For a flawless long-term and cyclic temperature performance of thermoelectric devices, it is essential that thermal expansion coefficients of p- and n-legs as well as of contacting materials are chosen as similar as possible. Already in the 1990s in the Jet Propulsion Laboratory not only transport behavior of skutterudites but also related problems such as thermal expansion were investigated; however, at that time thermal expansion coefficients were only reported for CoSb3,1,2 RhSb3,1 and IrSb3.1,3,4 From our comprehensive literature search in two major electronic libraries, chemical abstracts service 共CAS兲 and INSPEC, scanning entries up to 2009, it became obvious that only a few research groups have dealt with thermal expansion 共see data and references in Author to whom correspondence should be addressed. Tel.: ⫹43-1-427752456. FAX: ⫹43-1-4277-95245. Electronic mail: [email protected].

a兲

0021-8979/2010/107共4兲/043507/10/$30.00

Table II兲. Besides these directly accessible data on thermal expansion, all data in literature were collected, which allowed us to extract thermal expansion coefficients. Therefore, the aim of the present work is threefold: 共i兲 to supply new data on thermal expansion from a series of high ZT pand n-type skutterudites, 共ii兲 to extract thermal expansion from experimental data in literature, where expansion coefficients have not yet been evaluated by the authors, and 共iii兲 a general discussion of all thermal expansion data available covering antimony-, phosphorous- arsenic-, and germanium-based skutterudites. It is interesting to note that our literature search did not reveal any expansion data on arsenide skutterudites. The experimental work presented herein is concerned with skutterudites 共micro- and nanostructured兲 where didymium 共DD兲 共4.76% Pr and 95.2% Nd兲 and mischmetal 共Mm兲 共50.8% Ce, 28.1% La, 16.1% Nd, and 5.0% Pr兲 act as filler atoms in the frameworks of 共Fe1−xCox兲4Sb12 and 共Fe1−xNix兲4Sb12. From these series of samples, we selected those with a ZT⬃ 1 共DD0.68Fe3.2Ni0.8Sb12 and DD0.76Fe3.4Ni0.6Sb12兲, including nanostructured 关ball milled 共BM兲兴 as well as microstructured materials. These were DD0.44Fe2.1Co1.9Sb12 and DD0.44Fe2.1Co1.9Sb12 BM, Mm0.76Fe4Sb12 and Mm0.70Fe4Sb12 BM, and DD and Mm-samples with the same nominal composition 共DDFe4Sb12 BM and MmFe4Sb12 BM兲.5,6 We compare the thermal expansion of the abovementioned samples not only with multifilled n-type

107, 043507-1

© 2010 American Institute of Physics

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-2

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al.

skutterudites, Ca0.07Ba0.23Co3.95Ni0.05Sb12 共Ref. 7兲 and Sr0.025Ba0.075Yb0.1Co4Sb12,8 which both have a ZT⬎ 1, CoSb3, and Pt4Sn4.4Sb7.6 but also with values reported in the literature.9–15 The semiclassical model of Mukherjee16 will be used to quantitatively evaluate the thermal expansion and to derive Einstein and Debye temperatures for all those samples where lattice parameters or dilatometric data are available as a function of temperature over a larger temperature range starting from 4.2 K. II. EXPERIMENTAL DETAILS

All DD and Mm samples, CoSb3, Pt4Sn4.4Sb7.6 as well as Ca0.07Ba0.23Co3.95Ni0.05Sb12, and Sr0.025Ba0.075Yb0.1Co4Sb12, were prepared via an optimized melting reaction technique. The solids obtained were ground into fine powders in a WC mortar or BM and in both cases hot pressed in an argon atmosphere at 600 ° C under a pressure of 50 MPa. For further details, see Refs. 5–8. Pt4Sn4.4Sb7.6 was prepared in the form of cold compacted sintered pellets. Lattice constants for polycrystalline powders were obtained at room temperature from Guinier x-ray diffraction data, applying Cu K␣1 radiation and employing a least squares evaluation with the program STRUKTUR.17 Chemical composition and microstructure were determined by electron probe microanalysis 共EPMA兲. Filling levels were obtained from combined evaluation of EPMA and Rietveld x-ray pattern refinements. Lattice parameters of skutterudite single crystals were obtained at three different temperatures 共300, 200, and 100 K, N2 cooling the crystal兲 from an Enraf Nonius Kappa charge-coupled device instrument with monochromatic Mo K␣ radiation under a flow of equilibrated nitrogen gas from a cryostat. LaFe4As12 and PrFe4As12 are single crystals, grown from elements with purities ⬎99.9% by using a molten metal flux method at high temperature and pressure. Details on growth, structural, and physical properties are reported elsewhere.18–20 The thermal expansion from 4.2 to 300 K was measured in a miniature capacitance dilatometer,21 using the tilted plate principle.22,23 For this measurement, the sample is placed in a hole of the lower ringlike capacitance plate made of silver, which is separated from the silver upper capacitor plate by two needle bearings. All DD samples, Mm0.76Fe4Sb12, Mm0.70Fe3CoSb12 BM, Ca0.07Ba0.23Co3.95Ni0.05Sb12, Sr0.025Ba0.075Yb0.1Co4Sb12, and CoSb3, Pt4Sn4.4Sb7.6, and both Ge- and As-based samples were measured with this low temperature capacitance dilatometer. For the measurement of the thermal expansion at a temperature range from 300 to 700 K, a dynamic mechanical analyzer DMA7 共Perkin Elmer Inc.兲 was employed. The sample is positioned in a parallel plate mode with a quartz rod on top of the sample and data are gained using the thermodilatometric analysis 共TDA兲. TDA is often referred to as zero force thermomechanical analysis. With this method the change in the dimension of a sample is measured while subjected to a temperature change without using any force. The length of the sample is measured via the movement of the quartz rod. This movement is detected using electromagnetic inductive coupling. The absolute 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, were measured with this method.24 Porosity was obtained from density measurements in distilled water, using Archimedes’ principle, and the calculation of the x-ray density dX = 共ZM兲 / 共NV兲, where M is the molar mass, Z is the number of formula units per cell, N is Avogadro’s number, and V is the volume of the unit cell. Resistivities of the DD alloys in the temperature range from 4.2 to 300 K were measured using a dc four-point technique. The resistivity curves of DD0.68Fe4Sb12, DD0.76Fe3.4Ni0.6Sb12, DD0.44Fe2.1Co1.9Sb12, and DD0.44Fe2.1Co1.9Sb12 BM showed metallic behavior and therefore could be fitted well with the Bloch–Grüneisen function yielding also the Debye temperature; for details see Ref. 5. The same technique was used for Ca0.07Ba0.23Co3.95Ni0.05Sb12.7 Time of flight of sound pulse measurements were performed on cylinders with a frequency of 10 MHz using a home made equipment27 to provide the data for longitudinal 共vl兲 and shear 共transversal兲 共vs兲 sound velocities. III. RESULTS AND DISCUSSION

Table I summarizes the thermoelectric properties of selected DD and Mm samples,5,6 Ca0.07Ba0.23Co3.95Ni0.05Sb12 共Ref. 7兲 and Sr0.025Ba0.075Yb0.1Co4Sb12,8 published recently and the porosity of all these alloys. All samples are single phase except Mm0.70Fe4Sb12 共80 at. %兲, which contains also FeSb2, Sb, and Mm2O3. In all cases the Rietveld refinement showed an ordered atom arrangement with respect to the atom site distribution among DD/Mm, 共Fe/Co, Fe/Ni兲, and Sb sublattices. DD and Mm contents agree well with the data obtained from EPMA which applies also for the Ba and Ca contents and Co/Ni contribution 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 the aforementioned skutterudites as a function of temperature. In Fig. 1, the data from the low temperature measurements are displayed, and those from the high temperature measurements in Fig. 2. From Fig. 1, it is obvious that ⌬l / l0 of all measured skutterudites decreases linearly in temperature from room temperature to about 150 K, whereas for temperatures below 150 K, a nonlinear behavior is evident. The thermal expansion coefficient ␣ follows from a temperature derivative of the length change, i.e.,

␣=

冉 冊

⳵ ⌬l 1 . ⳵ T l0

共1兲

␣ was calculated in the temperature range from about 150 to 300 K. The thermal expansion coefficients derived in the present article together with data available in the literature are listed in Table II. Although the thermal expansion for all DD samples 共9.45⫻ 10−6 K−1 ⬍ ␣ ⬍ 11.30⫻ 10−6 K−1兲 is slightly lower than for the Mm samples 共9.97⫻ 10−6 K−1 ⬍ ␣ ⬍ 12.42⫻ 10−6 K−1兲, the difference within DD as well as within Mm skutterudites is not high, which applies also

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-3

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al.

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

Skutterudite Ca0.07Ba0.23Co3.95Ni0.05Sb12 Sr0.025Ba0.075Yb0.1Co4Sb12 DD0.08Fe2Ni2Sb12 DD0.76Fe3.4Ni0.6Sb12 DD0.68Fe3.2Ni0.8Sb12 DD0.44Fe2.1Co1.9Sb12 DD0.44Fe2.1Co1.9Sb12 DD0.86Fe4Sb12 Mm0.76Fe4Sb12 Mm0.70Fe4Sb12共not single phased兲 Mm0.20Fe2.5Ni1.5Sb12 Mm0.70Fe3CoSb12 Mm0.68Fe3CoSb12 Mm0.05FeCo3Sb12

BM

BM BM BM BM BM

Type

Porosity 共%兲

␳ 共␮⍀ cm兲

S 共␮V / K兲

␭ 共mW/cm K兲

ZT 共297 K兲

ZT 共700 K兲

Ref.

n n n p p p p p p p p p p p

5 0.6 1.4 6.3 2.3 4.1 0.2 0.6 5.1 3.3 1.5 1.8 1.2 1.9

317 295 3730 690 552 780 900 370 449 408 ¯ 785 650 ¯

⫺109 ⫺170 ⫺70 126 104 105 91 74 77 79 ¯ 113 103 ¯

53 32 21 19 22 19 15 32 26 21 ¯ 15 17 ¯

0.18 0.46 0.02 0.34 0.26 0.20 0.18 0.13 0.15 0.22 ¯ 0.33 0.28 ¯

1.10 1.28 0.24 1.05 0.93 0.44 0.43 0.86 0.61 0.75 ¯ 1.16 1.09 ¯

7 8 5 5 5 5 5 a

6 6 6 6 6 6

a

This work, prepared as described in Ref. 5.

for the difference between the n-type and the p-type DD samples. As expected, the difference in ␣ between nanostructured 共BM兲 and microstructured samples and of samples with lower or higher porosity is very small, as can be seen when comparing the graphs of Mm0.76Fe4Sb12 and Mm0.70Fe4Sb12 BM in Fig. 2 as well as DD0.44Fe2.1Co1.9Sb12 and DD0.44Fe2.1Co1.9Sb12 BM in Fig. 3 or the values of ␣ of these samples as well as for Mm0.70Fe3CoSb12 BM, measured at low temperatures, and Mm0.70Fe3CoSb12, measured at high temperatures 共see Table II兲. However, Figs. 1 and 2 also show that both n-type multifilled skutterudites, 共␣ = 9.14⫻ 10−6 K−1兲 and Ca0.07Ba0.23Co3.95Ni0.05Sb12 Sr0.025Ba0.075Yb0.1Co4Sb12 共␣ = 8.35⫻ 10−6 K−1兲, have a significantly smaller thermal expansion. These observations lead to the conclusion that the grain size does not influence the thermal expansion, but filler atoms do have some influence on thermal expansion. Figure 4 shows that in DD or Mm skutterudites, an increasing Fe-content 共and a simultaneously increasing filler-content兲 enlarges the thermal expansion. This is also the case for other pairs of samples; e.g., La0.743Fe2.74Co1.26Sb12 共␣ = 9.08⫻ 10−6 K−1兲 共Ref. 15兲 and

La0.9Fe4Sb12 共␣ = 11.69⫻ 10−6 K−1兲, CexCo4Sb12, 0 ⬍ x ⬍ 0.1 共␣ = 8 ⫻ 10−6 K−1兲 共Ref. 14兲 and Ce0.9Fe4Sb12 共␣ = 13.93⫻ 10−6 K−1兲, or Ca0.07Ba0.23Co3.95Ni0.05Sb12 共␣ = 9.14⫻ 10−6 K−1兲 and CaFe4Sb12 共␣ = 10.9⫻ 10−6 K−1兲.10 For CoSb3 two rather different thermal expansion coefficients were found in literature: ␣ = 13.5⫻ 10−6 K−1 共no details given, Ref. 1兲 and ␣ = 6.36⫻ 10−6 K−1 共from single crystal in a range from 300 to 930 K, Ref. 2兲. Therefore, thermal expansion was remeasured for a CoSb3 sample 共BM and hot pressed兲 revealing a coefficient ␣ = 9.1⫻ 10−6 K−1 共120–220 K兲. This value fits well to the dependency of DD and Mm alloys shown in Fig. 4. The lattice parameter of a cubic material at varying temperatures is in relationship to the thermal expansion coefficient gained from TDA or capacitance data via the relation

FIG. 1. 共Color online兲 Temperature dependent thermal expansion ⌬l / l0 of various skutterudites for 4.2 K ⬍ T ⬍ 300 K.

FIG. 2. 共Color online兲 Temperature dependent thermal expansion ⌬l / l0 of various skutterudites for 300 K ⬍ T ⬍ 700 K.

a2 − a1 a1 , ␣= ⌬T

共2兲

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

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-4

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al.

TABLE II. Lattice parameters and thermal expansion coefficients of Ge-, P-, As-, and Sb-based skutterudites. Lattice parameter a, RT 共nm兲

Ref.

␣ ⫻ 10−6 共K−1兲

T 共K兲

Method

10.2 9.15

160–245 160–245

DMb DM

4.7 5.64 5.4 5.1 5.08 4.77 5.8 5.39

10–300 10–300 150–300 100–300 100–300 100–300 100–300 100–300

SCLPc SCLP LPd LP SCLP SCLP SCLP SCLP

10.3 9.25

160–250 160–250

DM DM

a

9.1 13.5 6.36 12.7 6.89 6.69 7.96 8 11 10.9 9 9.09 7.4 9.5 9.5 11.69 12.7 8 11.21 12.50 10.59 12 8.17 11.26 12.09 11.29 9.81 9.82 9.51 9.45 13.43 11.94 12.42 10.78 11.33 11.43 9.97 9.19 10.16 8.85 9 8.9 6.94

120–220 ¯ 300–930 ¯ RT 300–673 ¯ ¯ 100–300 150–300 100–300 ¯ 180–300 180–300 180–300 100–300 100–300 300 125–300 100–300 100–296 110–295 100–300 160–245 160–245 160–245 160–245 300–600 160–245 160–245 160–250 300–500 300–500 300–500 160–245 300–500 300–500 160–280 300–650 160–245

DM ¯ ¯ ¯ LP LP ¯ ¯ LP LP LP ¯ NDe ND ND SCLP LP ¯ LP SCLP SCLP LP LP DM DM DM DM

a

Ref.

Ge-based skutterudites BaPt4Ge12 UPt4Ge12

0.86928共2兲 0.85835共3兲

LaRu4P12 PrRu4P12 GdRu4P12 LaOs4P12 CeOs4P12 PrOs4P12 NdOs4P12 SmOs4P12

0.80605共2兲 0.80493 0.80375 0.80932共3兲 0.80751共3兲 0.80813共2兲 0.80790共2兲 0.80731共2兲

a a

P-based skutterudites 28 a

29 a a a a a

As-based skutterudites 18 20 Sb-based skutterudites

PrFe4As12 LaFe4As12

0.8310共2兲 0.83273共2兲

CoSb3

0.903484共2兲 0.90345共3兲

2

0.92503共3兲 0.92503共3兲 0.92533

9 3 1

NaFe4Sb12 CaFe4Sb12 CaxCo4Sb12 Ru0.5Pd0.5Sb3 Tl0.22Co4Sb12 TlCo3FeSb12 Tl0.5Co4Sb11.5Sn0.5 La0.9Fe4Sb12 Ce0.9Fe4Sb12 CexCo4Sb12共0⬍x⬍0.1兲 PrFe4Sb12 Nd0.85Fe4Sb12 Eu0.93Fe4Sb12 Yb0.95Fe4Sb12 YbxCo4Sb12 DD0.86Fe4Sb12 BM DD0.68Fe3NiSb12 DD0.76Fe3.4Ni0.6Sb12 DD0.08Fe2Ni2Sb12

0.91759共3兲 0.9171共4兲 0.9052 0.9298 0.9056 0.9112 0.9082 0.91503 0.91406共3兲 ¯ 0.91290 0.91412共2兲 0.91725共2兲 0.91586共8兲 0.9048 0.91357共2兲 0.91208共4兲 0.91219共3兲 0.90927共3兲

10 33 11 12 13 13 13

DD0.44Fe2.1Co1.9Sb12 DD0.44Fe2.1Co1.9Sb12 BM Mm0.76Fe4Sb12

0.90920共4兲 0.90878共3兲 0.91370共5兲

a

Mm0.70Fe4Sb12 BM Mm0.20Fe2.5Ni1.5Sb12 Mm0.70Fe3CoSb12 BM Mm0.68Fe3CoSb12 Mm0.05FeCo3Sb12 Ca0.07Ba0.23Co0.95Ni0.5Sb12

0.91384共3兲 0.91009共1兲 0.91165共3兲 0.91167共3兲 0.90624共3兲 0.90665共3兲

a

Sr0.025Ba0.075Yb0.1Co4Sb12 La0.743Fe2.74Co1.26Sb12

0.90617共4兲 0.90971共3兲

a

RhSb3 IrSb3

Pt4Sn4.4Sb7.6

0.93304共2兲

a

a a

¯ a a a

34 11 a a a a

a a

a a a a a

15 a

100–300 130–230

a a

a a a a a a a a

a

1 2 1 a

3 1 4 10 a

11 12 13 13 13 a a

14 a a a a

11 a a a a

DM DM DM

a

DM DM DM DM DM DM

a

DM ND LP DM

a

a a

a a a a a

15 a a

a

This work. DM dilatometer measurements. c SCLP calculated from single crystal lattice parameter. d LP calculated from lattice parameter. e ND data from neutron diffraction. b

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-5

Rogl et al.

J. Appl. Phys. 107, 043507 共2010兲

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

gained from precise x-ray diffraction data on polycrystalline or single crystal specimens. Figure 5 gives an overview of the temperature dependent lattice parameters of all measured DD and Mm samples and shows that the slopes do not differ very much. Figures 5 and 6 demonstrate that for DD0.08Fe2Ni2Sb12, Mm0.70Fe4Sb12, and Ca0.07Ba0.23Co3.95Ni0.05Sb12, the measurements at low and high temperatures, with different equipment used, fit well together. Also the values calculated for two different measurement ranges fit within the measurable accuracy. While in the case of DD0.08Fe2Ni2Sb12, the values for the thermal expansion ␣关160–240兴 = 9.81⫻ 10−6 K−1 for low and ␣关300–600兴 = 9.82⫻ 10−6 K−1 for high temperatures are equal, for MmFe4Sb12 and Ca0.07Ba0.23Co3.95Ni0.05Sb12 共see Figs. 5 and 6兲, the difference in these temperature ranges is smaller than 1 ⫻ 10−6 K−1. Figures 7 show temperature dependent lattice parameters for various Ge-, P-, As- and Sb-based skutterudites. For the calculation of the thermal expansion coefficient ␣, either our

FIG. 4. 共Color online兲 Thermal expansion coefficient ␣ vs DD and Mm contents 共a兲 and versus Fe content 共b兲. The solid line is a guide to the eyes.

FIG. 5. 共Color online兲 Temperature dependent lattice parameters a of DD and Mm compounds.

data or literature data of lattice parameters were used 共see also Table II兲. Slack3 derived the thermal expansion coefficient ␣ = 6.69⫻ 10−6 K−1 for IrSb3 from x-ray data in the temperature range from 300 to 673 K in good agreement with our calculation, ␣ = 6.89⫻ 10−6 K−1, using the lattice parameters of Kjekshus9 关see Fig. 7共a兲兴. Both values are lower than the thermal expansion ␣ = 8 ⫻ 10−6 K−1 found by Kjekshus earlier.4 In Fig. 7共a兲, the data for the DD and Mm skutterudites based on Sb are compared with the corresponding La, Ce, Pr, Nd, Eu, and Yb skutterudites. The almost identical slopes, i.e., the expansion coefficients ␣, again show that ␣ is almost insensitive to the filler elements as already concluded above. All thermal expansion coefficients of the P-based skutterudites were calculated from lattice parameters obtained from single crystal measurements. It is remarkable that all thermal expansion coefficients of P-based skutterudites MOs4P12 共M = La, Ce, Pr, Nd, Sm兲, in a range 共4.8– 5.8兲 ⫻ 10−6 K−1, are only half as large as those of Sbbased skutterudites where ␣ ranges from 7.5⫻ 10−6 to 14 ⫻ 10−6 K−1. Similarly, data of LaRu4P12 共Ref. 28兲 with ␣ = 4.7⫻ 10−6 K−1, PrRu4P12 with ␣ = 5.64⫻ 10−6 K−1, and GdRu4P12 共Ref. 29兲 with ␣ = 5.4⫻ 10−6 K−1 are also in this range of relatively low thermal expansion, indicating stron-

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

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-6

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al.

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

ger covalent bonding in the P framework than in the Sb framework. Figure 7共c兲 shows a good agreement between the lattice parameters of the Ge-based skutterudites gained from single crystal measurements and those extracted from thermal expansion measurements. The thermal expansion coefficients are in the range of Sb-based skutterudites. Both arsenide samples 关Fig. 7共d兲兴, PrFe4As12 共␣ = 10.3⫻ 10−6 K−1兲 and LaFe4As12 共␣ = 9.25⫻ 10−6 K−1兲, show a smaller thermal expansion coefficient than their Sb-based counterparts PrFe4Sb12 共␣ = 11.21⫻ 10−6 K−1兲 and LaFe4Sb12 共␣ = 11.69 ⫻ 10−6 K−1兲. The inset of Fig. 8 shows the low temperature thermal expansion of PrFe4Sb12 below 25 K. In agreement with measurements of electrical resistivity, specific heat, elastic constants, and magnetization,18,19 a magnetic phase transition at TC = 18 K becomes obvious also from ⌬l / l0 versus T. At Tⴱ = 12 K, a second phase transition 共found from susceptibility, magnetization, and specific heat measurements19兲 is evident from a change in the slope of ⌬l / l0 versus T. To analyze the thermal expansion as a function of temperature in the full temperature range, we followed a semiclassical treatment by Mukherjee 共details are described in Ref. 16兲 taking into account three- and four-phonon interactions, considering an anharmonic potential, and using both the Debye model for the acoustic phonons and the Einstein approximation for the optical modes. The length change ⌬l / l共T0兲 is given by

具x典T − 具x典T0 ⌬l = l共T0兲 x0

␧=

3g ␥ 具x典T = T2 + 2 关␧-G␧2 − F␧3兴, 4c 2

再冉 冊 冉 冊 冕 T 3 3kBT p ␪D

3

␪D/T

0

冉 冊

冎

p − 3 k B␪ E z3dz + , z e −1 p e␪D/T − 1 共3兲

where ␥ is the electronic contribution to the average lattice displacement, ␪D is the Debye temperature, ␪E is the Einstein temperature, and p is the average number of phonon branches actually excited over the temperature range. G, F, c, and g are further material dependent constants. The Debye and Einstein temperatures, ␪D and ␪E were obtained from least squares fits of Eq. 共3兲 to the experimental data. Fits were performed for all skutterudites measured in the low temperature range and exemplary graphs are shown for Ca0.07Ba0.23Co3.95Ni0.05Sb12 共Fig. 6兲, and for DD0.76Fe3.4Ni0.6Sb12 共Fig. 9兲, the two Ge-based 共Fig. 10兲, and the two As-based 共Fig. 8兲 skutterudites. Debye temperatures are in very good agreement with those gained from fits to resistivity data5,7 and in the case of the BaPt4Ge12 and PrFe4As12 in good agreement with calculations from heat capacity.30,19 For UPt4Ge12 the value for the Einstein temperature is almost the same as the one gained from the atomic displacement parameters.

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-7

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al.

FIG. 8. 共Color online兲 Temperature dependent thermal expansion of LaFe4As12 and PrFe4As12. The solid line is a least squares fit according to Eq. 共3兲 for LaFe4As12, and the dashed line for PrFe4As12. The inset shows the thermal expansion below 25 K for PrFe4As12, marked with arrows are Tⴱ and TC.

The Debye temperatures for Mm0.70Fe4Sb12 and Mm0.68Fe3CoSb12 were calculated using the values of the measured longitudinal 共vl兲 and shear 共vs兲 sound velocities employing Anderson‘s expressions:31

␪D =

冉 冊

h 3nN␳ kB 4␲M

1/3

vm

and

vm =

冉冋

1 1 2 3 + 3 3 vs vl

册冊

−1/3

, 共4兲

FIG. 10. 共Color online兲 Temperature dependent thermal expansion of BaPt4Ge12 and UPt4Ge12 with solid line and dashed line as least squares fit according to Eq. 共3兲.

The compounds UPt4Ge12, ReyOs4P12 共RE= La, Ce, Pr, Nd兲, and REyFe4Sb12 共Re= La, Ce, Nd, Eu兲 can be considered as simple Debye solids with the rare earth atoms behaving like Einstein oscillators. The Einstein temperatures ␪E,ii and the thermal displacements are related by Uii =

冉 冊

␪E,ii ប2 coth , 2T 2mkB␪E,ii

共5兲

where h is Planck’s constant, kB is Boltzmann’s constant, N is Avogadro’s number, ␳ is the density, M is the molecular weight, and n is the number of atoms in the molecule. For Mm0.70Fe4Sb12, vs = 2.70⫻ 105 cm/ s, vl = 4.10⫻ 105 cm/ s, and therefore vm = 2.96⫻ 105 cm/ s; for Mm0.68Fe3CoSb12, vS = 2.72⫻ 105 cm/ s, vl = 4.43⫻ 105 cm/ s, and therefore vm = 3.00⫻ 105 cm/ s. With these values, ␪D = 306 K and ␪D = 313 K, respectively, could be calculated. Both values fit nicely to the values gained via least squares fits of Eq. 共3兲. For Sr0.025Ba0.075Yb0.1Co4Sb12, vS = 2.79⫻ 105 cm/ s and vl = 4.65⫻ 105 cm/ s were measured yielding a calculated vm = 3.08⫻ 105 cm/ s and ␪D = 327 K.

where m is the atomic mass of the rattling atoms. From the linear slope ⌬Uii / ⌬T in Fig. 11 共high temperature approximation for h␯ Ⰶ 2kBT兲, the force constants Kii = 共kB⌬T兲 / ⌬Uii, the frequency of vibrations ␯ii = 1 / 2␲共Kii / m兲1/2, and finally the Einstein temperature ␪E,ii = 共h · ␯ii兲 / 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 this work are in good agreement with those from the literature; e.g., compare the data for LaFe4Sb12 although the methods of measurements and of calculations vary strongly. Debye and Einstein temperatures are of importance to define the vibra-

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

FIG. 11. 共Color online兲 Thermal displacements vs temperature for various Ge, P-, and Sb-based skutterudites.

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-8

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al.

TABLE III. Debye 共␪D兲 and Einstein 共␪E兲 temperatures of selected skutterudites.

␪D 共K兲 BaPt4Ge12 UPt4Ge12

260 247 260

Method

Ref.

Ge-based skutterudites a FTE b Cp 30 a FTE

␪E 共K兲

Method

Ref.

82

FTE

a

62 59

FTE ADPc

a

125 122.8 117.6 100.2

ADP ADP ADP ADP

a

88

FTE

a

98

FTE

a

85

ADP

a

79 70

ADP ADP

40 36

79 86 65 72.8 82 98 93

ADP EXAe ¯ ADP ADP FTE FTE

75

FTE

a

97 95

FTE FTE

a

98

FTE

a

a

P-based skutterudites LaOs4P12 CeOs4P12 PrOs4P12 NdOs4P12 PrFe4As12 LaFe4As12 CoSb3

La0.9Fe4Sb12 LaFe4Sb12

Ce0.9Fe4Sb12 CeFe4Sb12 Ce0.85Fe4Sb12 Nd0.85Fe4Sb12 Eu0.93Fe4Sb12 DD0.86Fe4Sb12 BM DD0.68Fe3NiSb12 DD0.76Fe3.4Ni0.6Sb12 DD0.08Fe2Ni2Sb12 DD0.44Fe2.1Co1.9Sb12 DD0.44Fe2.1Co1.9Sb12 BM Mm0.76Fe4Sb12 Mm0.70Fe4Sb12 BM Mm0.70Fe3CoSb12 BM Mm0.68Fe3CoSb12 BM Ca0.07Ba0.23Co3.95Ni0.5Sb12 Sr0.025Ba0.075Yb0.1Co4Sb12 Pt4Sn4.4Sb7.6

356 360 470 314 307 306 321 325 260 260 248 299 331 298 242

210 240 235 254 227 225 270 267 265 198 312 306 319 313 205 206 327 420

As-based skutterudites Cp 18 a FTE a FTE Sb-based skutterudites a FTE d SV 2 SV 35 SV 36 37 Cp a Cp Cp 38 39 Cp ADP 40 ADP 13 37 Cp共0.5–5 K兲 37 Cp共6.5–10兲

FTEf FTE FRg FTE FR FTE FTE FR FTE FR FTE SVd FTE SV FTE FR SV FTE

a a

a a a

a

41 42 a a a a

5 a

5 a a

a

a

a

a a

a

a

FTE

a

a

a

a

FTE

a

92 51

7 a

a

a

a

¯

a

This work. Cp heat capacity measurements. c ADP calculated from temperature dependent atomic displacement parameters. d SV gained from sound velocity. e EXA extended x-ray absorption fine structure measurements. f FTE gained from thermal expansion fit. g FR gained from resistivity fit. b

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-9

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al.

FIG. 12. 共Color online兲 ZT in dependency on ␣ for various DD and Mm skutterudites.

tional spectra of thermoelectric materials where scattering of heat carrying phonons on rattling modes helps to reduce bulk thermal conductivity as a key route to improve the thermoelectric figure of merit. Despite the fact that there is no real physical explanation for a relation between the figure of merit ZT and the thermal expansion coefficient ␣, one could get the impression from Fig. 12 that a higher ZT is related to a higher ␣ or vice versa. Although thermal expansion coefficients are not available for all corresponding members of P-, Ge-, As-, and Sbbased skutterudite families, a general trend emerges from Table II, i.e., big values for Sb-based, middle sized values for As- and Ge-based and smallest values for P-based skutterudites, documenting the increasing covalent bonding strength within the framework cages 共see Fig. 13兲.

IV. CONCLUSIONS

This paper presents a comprehensive evaluation of thermal expansion data on skutterudites combining new measurements with all data hitherto available in the literature. Thermal expansion coefficients ␣ for Sb-based skutterudites were found to be about double the size of P-based skutterudites. Although the differences in thermal expansion within DDy共Fe1−xCox兲4Sb12 and within Mmy共Fe1−xCox兲4Sb12 samples are negligible, an increasing amount of fillers and of Fe content increases the thermal expansion. The influence of the filling level in combination with the Fe content on thermal expansion may be the reason that all Co-based n-type skutterudites investigated have a smaller thermal expansion coefficient than our Fe-based p-type skutterudites. The grain size does not affect the thermal expansion. Fits of the thermal expansion measurements by means of the semiclassical treatment of Mukherjee proved well, by achieving Debye and Einstein temperatures very close to those from resistivity, sound velocity, or specific heat measurements. The Einstein temperatures derived are consistent with low frequency modes of the filler atoms 共rattling modes兲, which reduce thermal conductivity via scattering of heat carrying phonons.

FIG. 13. 共Color online兲 Thermal expansion coefficients grouped with respect to various families of skutterudites. ␣m关⫻10−6 共K−1兲兴 is the average value of the thermal expansion coefficients within one family.

ACKNOWLEDGMENTS

Support by the University of Vienna within the IC Experimental Materials Science “Bulk Nanostructured Materials” and the Austrian FWF, Grant No. P19284-N20 is gratefully acknowledged. This work was partially supported by the Austrian Science Foundation FWF, Grant No. S10406N16. Z.H., G.R., and P.R. are grateful to the OEAD for a bilateral exchange program Austria-Poland, Grant No. PL06/2009. 1

T. Caillat, A. Borshchevsky, and J.-P. Fleurial, AIP Conf. Proc. 301, 517 共1994兲. T. Caillat, A. Borshchevsky, and J.-P. Fleurial, J. Appl. Phys. 80, 4442 共1996兲. 3 G. Slack and V. Tsoukala, J. Appl. Phys. 76, 1665 共1994兲. 4 A. Kjekshus, Acta Chem. Scand. 共1947-1973兲 15, 678 共1961兲. 5 G. Rogl, A. Grytsiv, P. Rogl, E. Bauer, and M. Zehetbauer, Intermetallics 18, 57 共2010兲. 6 L. Zhang, A. Grytsiv, M. Kerber, P. Rogl, E. Bauer, M. Zehetbauer, J. Wosik, and G. Nauer, J. Alloys Compd. 481, 106 共2009兲. 7 G. Rogl, A. Grytsiv, E. Bauer, P. Rogl, and M. Zehetbauer, Intermetallics 18, 394 共2010兲. 8 L. Zhang, A. Grytsiv, P. Rogl, E. Bauer, and M. Zehetbauer, J. Phys. D: Appl. Phys. 42, 225405 共2009兲. 9 A. Kjekshus and T. Rakke, Acta Chem. Scand., Ser. A 28a, 99 共1974兲. 10 A. Leithe-Jasper, W. Schnelle, H. Rosner, M. Baenitz, A. Rabis, A. Gippius, E. N. Morozova, H. Borrmann, U. Burkhardt, R. Ramlau, U. Schwarz, J. A. Mydosh, Y. Grin, V. Ksenofontov, and S. Reiman, Phys. Rev. B 70, 214418 共2004兲. 11 V. Da Ros, C. Candolfi, J. Leszczynski, A. Dauscher, B. Lenoir, S. J. Clarke, and R. Smith, Proceedings of the Sixth European Conference on Thermoelectrics, Paris, France, 2008 共unpublished兲. 12 T. Caillat, J. Kulleck, A. Borshchevsky, and J.-P. Fleurial, J. Appl. Phys. 79, 8419 共1996兲. 13 B. C. Sales, B. C. Chakoumakos, and D. Mandrus, Phys. Rev. B 61, 2475 共2000兲. 14 G. Nolas, D. C. Johnson, and D. G. Madrus, Mater. Res. Soc. Symp. Proc. 626, Z10.1.1 共2000兲. 15 B. C. Chakoumakos, B. C. Sales, D. Mandrus, and V. Keppens, Acta Crystallogr., Sect. B: Struct. Sci. 55, 341 共1999兲. 16 G. D. Mukherjee, C. Bansal, and A. Chatterjee, Phys. Rev. Lett. 76, 1876 共1996兲. 17 J. Rodriguez-Carvajal, Abstract of the Satellite Meeting on Powder Diffraction of the 15th Congress, International Union of Crystallography, Talence, France, 1990 共unpublished兲, p. 127. 18 Z. Henkie, M. B. Maple, A. Pietraszko, R. Wawryk, T. Cichorek, R. E. 2

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

043507-10

Baumbach, W. M. Yuhasz, and P. C. Ho, J. Phys. Soc. Jpn. 77, 128 共2008兲. T. A. Sayles, W. M. Yuhasz, J. Paglione, T. Yanagisawa, J. R. Jeffries, M. B. Maple, Z. Henkie, A. Pietraszko, T. Cichorek, R. Wawryk, Y. Nemoto, and T. Goto, Phys. Rev. B 77, 144432 共2008兲. 20 B. Nowak, O. Zogal, A. Pietraszko, R. E. Baumbach, M. B. Maple, and Z. Henkie, Phys. Rev. B 79, 214411 共2009兲. 21 M. Rotter, H. Müller, E. Gratz, M. Dörr, and M. Löwenhaupt, Rev. Sci. Instrum. 69, 2742 共1998兲. 22 G. Brändli and R. Griessen, Cryogenics 13, 299 共1973兲. 23 J. Genossar and M. Steinitz, Rev. Sci. Instrum. 61, 2469 共1990兲. 24 J. Koppensteiner, W. Schranz, and M. R. Puica, Phys. Rev. B 78, 054203 共2008兲. 25 W. Schranz, Phase Transit. 64, 103 共1997兲. 26 W. Schranz and D. Havlik, Phys. Rev. Lett. 73, 2575 共1994兲. 27 R. Wyslouzil, Ph.D. thesis, University of Vienna, 1985. 28 T. Uchiumi, I. Shirotani, C. Sekine, S. Todo, T. Yagi, Y. Nakazawa, and K. Kanoda, J. Phys. Chem. Solids 60, 689 共1999兲. 29 C. Sekine, K. Akahira, T. Takeda, Y. Ohishi, P. Haen, J. Phys.: Conf. Ser. 150, 042179 共2009兲. 30 E. Bauer, A. Grytsiv, X.-Q. Chen, N. Melnychenko-Koblyuk, G. Hilscher, H. Kaldarar, H. Michor, E. Royanian, G. Giester, M. Rotter, R. Podloucky, and P. Rogl, Phys. Rev. Lett. 99, 217001 共2007兲. 31 O. L. Anderson, J. Phys. Chem. Solids 24, 909 共1963兲. 19

J. Appl. Phys. 107, 043507 共2010兲

Rogl et al. 32

B. T. M. Willis and A. W. Pryor, Thermal Vibrations in Crystallography, 共Cambridge University Press, Cambridge, 1975兲. 33 R. Gumeniuk, A. Leithe-Jasper, W. Schnelle, H. Rosner, Y. Grin, and L. Vasylecho, Science Ann. Rep. 1, 845 Hasylab at DESY, Hamburg 共2007兲 hasyweb.desy.de/science/annual_report/part1/contribution/41/20665.pdf. 34 W. Schnelle, A. Leithe-Jasper, M. Schmidt, H. Rosner, H. Borrmann, U. Burkhardt, J. A. Mydosh, and Y. Grin, Phys. Rev. B 72, 020402 共2005兲. 35 A. Borshchevsky, J. P. Fleurial, E. Allevato, and T. Caillat, AIP Conf. Proc. 316, 3 共1995兲. 36 B. C. Sales, D. Mandrus, B. C. Chakoumakos, V. Keppens, and J. R. Thompson, Phys. Rev. B 56, 015081 共1997兲. 37 D. A. Gajewski, N. R. Dilley, E. D. Bauer, E. J. Freeman, R. Chau, M. B. Maple, D. Mandrus, B. C. Sales, and A. H. Lacerda, J. Phys.: Condens. Matter 10, 6973 共1998兲. 38 R. Viennois, I. V. Kityk, F. Terki, S. Charar, A. Muntzer, J. Kasperczyk, D. Ravot, J. C. Tedenac, J. Phys.: Condens. Matter 15, 1585 共2003兲. 39 E. Bauer, St. Berger, A. Galatanu, M. Galli, H. Michor, G. Hilscher, Ch. Paul, B. Ni, M. M. Abd-Elmeguid, V. H. Tran, A. Grytsiv, and P. Rogl, Phys. Rev. B 63, 224414 共2001兲. 40 D. J. Braun and W. Jeitschko, J. Less-Common Met. 72, 147 共1980兲. 41 D. Cao, F. Bridges, C. Chesler, S. Bushart, E. D. Bauer, and M. B. Maple, Phys. Rev. B 70, 094109 共2004兲. 42 D. Berardan, Ph.D. thesis, Université de Paris XII Val de Marne, 2004.

Downloaded 23 Feb 2010 to 131.130.24.23. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp