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version, currently submit to Physical Review B

Ferromagnetism and possible heavy fermion behavior in single crystals of NdOs4Sb12

P.-C. Ho, W. M. Yuhasz, N. P. Butch, N. A. Frederick, T. A. Sayles, J. R. Jeffries, and M. B. MapleDepartment of Physics and Institute for Pure and Applied Physical Sciences,

University of California, San Diego, La Jolla, CA 92093-0360, U.S.A.

J. B. Betts and A. H. LacerdaNational High Magnetic Field Laboratory/LANL, Los Alamos, NM 87545

P. RoglInstitut fuer Physikalische Chemie, Universitaet Wien, A-1090 Wien, Währingerstr. 42, Austria

G. GiesterInstitut fuer Mineralogie und Kristallographie, Universitaet Wien, A-1090 Wien, Althanstr. 14, Austria

(Dated: July 02, 2005)

Single crystals of the filled-skutterudite compound NdOs4Sb12 have been investigated by meansof electrical resistivity, magnetization, and specific heat measurements. The NdOs4Sb12 crystalshave the LaFe4P12-type cubic structure with a lattice parameter of 9.3 Å. Possible heavy-fermionbehavior is inferred from specific heat measurements, which reveal a large electronic specific heatcoefficient γ ≈ 520mJ/mol-K2, corresponding to an effective mass m∗ ≈ 98 me. Features relatedto a ferromagnetic transition at ∼ 0.9K can be observed in electrical resistivity, magnetization andspecific heat. Conventional Arrott-plot analysis indicates that NdOs4Sb12 conforms to mean-fieldferromagnetism.

PACS numbers: 75.40.Cx, 71.27.+a, 71.70.Jp, 71.70.Ch

I. INTRODUCTION

The filled skutterudite compounds have the chemicalformula MT4X12, where M is an alkali (Na or K), al-kaline earth (Ca, Sr, Ba), lanthanide or actinide atom;T is a transition-metal atom (Fe, Ru, Os); and X is apnictogen atom (P, As, Sb). The compounds crystal-lize in a LaFe4P12-type structure with the space groupIm3̄.1 Due to the strong hybridization between f- and con-duction electrons and their unique crystal structure, thelanthanide- and actinide-based filled skutterudite mate-rials display a wide range of strongly-correlated-electronphenomena, such as BCS-like superconductivity (e.g.,PrRu4Sb12),

2,3 heavy fermion behavior (e.g., PrFe4P12),4

heavy fermion superconductivity (e.g., PrOs4Sb12),5,6

ferromagnetism (e.g., PrFe4Sb12),7 metal-insulator tran-

sitions (e.g., PrRu4P12),8 Kondo-insulator behavior (e.g.,

UFe4P12 and CeFe4P12),9 valence fluctuation behavior

(e.g., YbFe4Sb12),10,11 and non-Fermi-liquid behavior

(e.g., CeRu4Sb12).12,13

Previous studies have shown ferromagnetic order-ing in Nd-based filled skutterudites. Measurements ofNdFe4Sb12 revealed ferromagnetic order below 16.5Kwith a Nd ordered moment of 2.04µB and a Fecollinear moment of 0.27µB.

14 Lower ferromagnetictransition temperatures were found in NdFe4P12 at2K,15 NdRu4P12 at 1.6K,

16 NdRu4Sb12 at 1.3K,2,3 and

NdOs4Sb12 at 0.8K.16 Since NdOs4Sb12 displays ferro-

magnetism with a low Curie temperature and its neigh-boring compound PrOs4Sb12 shows heavy fermion be-havior and unconventional superconductivity,6,17,18,19,20

possibly involving triplet spin pairing of electrons, there

is a strong likelihood that PrOs4Sb12 is near a ferromag-netic quantum critical point. Thus, by thoroughly char-acterizing the physical properties of NdOs4Sb12, a deeperinsight into the unusual behavior of PrOs4Sb12 may beattained. Earlier studies of the compound NdOs4Sb12only reported the results of structural refinement21 andthe value of the Curie temperature.16 In this report, wepresent a new and detailed investigation of NdOs4Sb12single crystals, including measurements of X-ray diffrac-tion, electrical resistivity, magnetization, and specificheat. We also discuss possible heavy fermion behaviorin this Nd-based filled skutterudite compound.

II. EXPERIMENTAL DETAILS

NdOs4Sb12 single crystals were grown in a molten Sbflux as described previously,17 using high purity Nd, Os(3.5N), and Sb (6N). X-ray powder diffraction measure-ments were performed with a Rigaku D/MAX B x-raymachine on a powder prepared by grinding several singlecrystals, which indicated single phase NdOs4Sb12 with aminor impurity peak of Sb (. 10%). The crystals hada LaFe4P12-type BCC structure,

1 with a lattice param-eter a = 9.30 Å. Two single crystals of similar dimensionwere selected for single crystal X-ray diffraction measure-ments. The data were collected on a four-circle NoniusKappa diffractometer at 296K using Mo Kα radiation(λ = 0.071073nm). No absorption corrections were nec-essary because of the rather regular crystal shape andsmall dimensions of the investigated crystals. The struc-ture was refined with the SHELXS-97 program.22

http://arxiv.org/abs/cond-mat/0412713v3

2

Electrical resistivity ρ(T,H) was measured using thestandard 4-wire technique in a Quantum Design PPMSsystem and in a 3He-4He dilution refrigerator in fieldsup to 8T. The low temperature (0.02K - 2.6K) andhigh-field (8T - 18T) ρ(T,H) measurements were per-formed in the National High Magnetic Field Laboratoryat Los Alamos National Laboratory. The electrical cur-rent applied to the sample was perpendicular to the ap-plied magnetic field, which was along the [001] direction,in all ρ(T,H) measurements. Measurements of ρ(T, P )were made under hydrostatic pressures up to 28 kbar ina beryllium-copper piston-cylinder clamp23 and a 4Hecryostat. The pressure was determined inductively fromthe pressure-dependent superconducting transition of aPb manometer.

DC magnetic susceptibility from 2K to 300K was mea-sured in a Quantum Design MPMS SQUID magnetome-ter. The magnetizationM(H,T ) measurements were car-ried out in a 3He Faraday magnetometer with a gradientfield of∼ 0.05 - 0.1T/cm in external fields up to 5.5T andat temperatures between 0.4K and 2K. For the Faradaymagnetometer measurements, several single crystals (to-tal mass of 21.3mg) were combined in a mosaic fashionand measurements were performed with magnetic fieldapplied along the [001] axis. Specific heat C(T ) of multi-ple single crystals (total mass of 42.15mg) was measuredbetween 0.5K and 70K in a 3He calorimeter using a semi-adiabatic heat-pulse technique.

III. RESULTS AND DISCUSSION

A. Single crystal structural refinement

Structural refinement was performed on X-ray diffrac-tion data collected from single crystals of NdOs4Sb12;the results are listed in Table I. The thermal displace-ment parameters Uii of the Nd atoms are isotropic andhave large values compared with the Uii for the Os andSb atoms, a common feature in the filled skutterudites.The Nd sites are fully occupied, which is not always thecase for filled skutterudites, such as Pr0.73Fe4Sb12 andEu0.95Fe4Sb12.

7,24 If the NdOs4Sb12 crystal is consideredto be a simple Debye solid with the Nd atoms behavinglike Einstein oscillators, the thermal displacement andthe Einstein temperature ΘE are related by

U =~2

2mNdkBΘE

coth

(

ΘE2T

)

, (1)

where mNd

is the atomic mass of Nd. For NdOs4Sb12,ΘE is estimated as ∼ 45K, which is close to the valuesfound for thallium-filled antimony skutterudites such asTl0.22Co4Sb12, Tl0.5Co3.5Fe0.5Sb12, Tl0.8Co3FeSb12, andTl0.8Co4Sb11Sn.

25,26

0

1

2

3

4

5

6

7

0 5 10

M/H @ 500 Oe

χ dc

(cm

3 /m

ol)

T (K)

NdOs4Sb12

Λ = 1.39 Nd-mol/cm3

xLLW = - 0.1, W = 3.9

Γ8(2) (421 K)

Γ8(1) (179 K)Γ6 (0 K)

xLLW = - 0.4, W = -7.2

Γ6 (596 K)

Γ8(1) (218 K)

Γ8(2) (0 K)

100

200

0 100 200 300

H/M @ 500 Oe(xLLW = -0.1, W = 3.9)

(xLLW = -0.4, W = -7.2)

χ dc-

1 (m

ol/c

m3 )

T (K)

Fig. 1 Ho, P.-C., et al.Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

(a)

(b)

FIG. 1: (a) DC magnetic susceptibility χdc vs temperatureT from 0K to 10K measured at 500Oe. Fits of χdc(T ) toa CEF model in which the ground state is either Γ6 (dashed

line) or Γ(2)8 (solid line). Below 10K, these two lines overlap.

(b) χ−1dc vs T from 0K to 300K. Fits of χdc(T ) and χ−1dc (T ) to

a CEF model in which the ground state is either Γ6 (dashed

line) or Γ(2)8 (solid line).

B. Magnetic Properties

The dc magnetic susceptibility χdc(T ) of NdOs4Sb12was measured at 500Oe and is displayed in Fig. 1(a).The χ−1dc (T ) data (Fig. 1(b)) exhibit different slopes at

high and low temperatures. The linear slope of χ−1dc (T )between 65K and 300K yields a Curie constant (CCW≡ (NAµ

2eff)/(3kB)) ∼ 1.85 cm

3K/mol, a negative Curie-Wiess temperature (ΘCW) ∼ -43K, and a effective mo-ment µeff ∼ 3.84µB, which is close to the Nd

3+ free-ion value of 3.62µB. The Curie-Weiss fit to the low-temperature range of χ−1dc (T ) (Fig. 2(c)) gives CCW ∼0.69 cm3K/mol, a positive ΘCW ∼ 1K, and a value ofµeff ∼ 2.35µB. The curvature in χ

−1dc (T ) is due to the

influence of the crystalline electric field (CEF), and thepositive ΘCW from the low-temperature fit indicates fer-romagnetic order developing below 1K.Although the crystal structure of NdOs4Sb12 has tetra-

hedral symmetry (Th),27 this is only a slight deviation

from cubic symmetry (Oh). Thus, to simplify the CEFanalysis, only Oh symmetry was considered. In an ionic(localized) model with cubic symmetry, the Nd3+ ten-fold degenerate J = 9/2 Hund’s rule ground state multi-

plet splits into a Γ6 doublet and two Γ8 (Γ(1)8 ,Γ

(2)8 ) quar-

tet states. In the treatment of Lea, Leask, and Wolf,28

these energy levels and their corresponding wave func-tions in cubic Oh symmetry can be parameterized by thevariables xLLW and W , where xLLW is the ratio of the

3

TABLE I: Single crystal structural data measured at T = 296K for NdOs4Sb12. The crystal structure is LaFe4P12-type withthe space group Im3̄ (No. 204). The range of X-ray scattering angle is 2◦ < 2θ < 80◦.

NdOs4Sb12

Crystal size 64 × 78 × 84 µm3 Lattice parameter a [Å] 9.3075(2) Density ρ [g/cm3] 9.745

Reflection in refinements 473 ≤ 4 σ(F0) of 482 Number of variables 11 R2

F=

∑

|F 20− F 2

c|/

∑

F 20

0.0186Goodness of fit 1.255

Nd in 2a (0, 0, 0); Thermal displacement [Å2] Interatomic distances [Å]

Occupancy 1.00(1) Nd: U11 = U22 = U33 0.0482(5) Nd - 12 Sb 3.4831

Os in 8c (1/4, 1/4, 1/4); Thermal displacement [Å2] Interatomic distances [Å]

Occupancy 1.00(1) Os: U11 = U22 = U33 0.0025(1) Os - 6 Sb 2.6239

Sb in 24g (0, y, z); y: 0.15597(3) Thermal displacement [Å2] Interatomic distances [Å]

z: 0.34017(3) Sb: U11 0.0026(1) Sb - 1 Sb 2.9033Occupancy 1.00(1) U22 0.0044(1) - 1 Sb 2.9752

U33 0.0065(1) - 2 Os 2.6239- 1 Nd 3.4831

fourth- and sixth-order terms of the angular momentumoperators and W is an overall energy scale. The CEFcontribution to the molar magnetic susceptibility can bedetermined from the expression29

χCEF(T ) = NAg2Jµ2B

∑

i

|〈i|Jz|i〉|2

kBTpi − 2

∑

i,j( 6=i)

|〈i|Jz |j〉|2

Ei − Ejpi

,

(2)whereNA is Avogadro’s number, gJ is the Landé g-factor,µB is the Bohr magneton, pi = e

−Ei/(kBT )/Z is the ther-mal population probability (Z is the partition function),and the Ei’s are the energies of the multiplets. However,the occurrence of ferromagnetic order in NdOs4Sb12 re-quires the presence of a molecular field constant Λ toaccount for the effective field, with χ−1 = χ−1CEF − Λ.The low-temperature value of µeff (2.35 µB) indicates

that the ground state of Nd3+ is either Γ6 or Γ(2)8 . A

wide range of xLLW values with a Γ6 ground state can fitthe χ(T ) data reasonably well (−1 ≤ xLLW ≤ 0.1). For aΓ8 ground state, a good fit to the χ(T ) data is only foundfor xLLW ≈ −0.4. All the fits imply that the splitting be-tween the ground and first excited states is greater than120K. When compared with the ρ(T ) data (to be dis-cussed later), the best fits are (I) xLLW = −0.1, W = 3.9,and (II) xLLW = −0.4, W = −7.2, corresponding to

the following level schemes: (I) Γ6 (0K), Γ(1)8 (180K),

Γ(2)8 (420K) and (II) Γ

(2)8 (0K), Γ

(1)8 (220K), Γ6 (600K),

with Λ = 1.39 Nd-mol/cm3 and an effective ground statemoment ∼ 2.31µB for both schemes. These fits are dis-played in Figs. 1(a) and (b). The Heisenberg interactionstrength between a Nd ion and its eight nearest neighborsis estimated to be 0.522K from Λ.Isothermal magnetization measurements M(H), dis-

played in Fig. 2(a), were made above and below the Curietemperature (ΘC). In the paramagnetic state, the M(H)isotherms between 2K and 5K can be fit well by a Bril-louin function with effective J = 2.7 that includes a tem-perature shift of 1K due to ferromagnetism. The M(H)isotherms in the vicinity of ΘC were used to constructa conventional Arrott plot consisting of M2 vs H/Misotherms, shown in Fig. 2(b). The isotherms in the Ar-rott plot are linear and parallel in the vicinity of ΘC forH ≤ 1T, indicating that NdOs4Sb12 is a mean-field fer-

0

0.5

1

1.5

2

0 1 2 3 4 5

0.4 K0.9 K1.0 K1.2 K1.4 K2.0 K3.0 K

M (

µB/f.u

.)

H (T)

Msat

= 1.73 µB

0

1

2

3

4

5

0 1 2 3 4

χdc

-1

inverse initial χ from Arrott plot

H/M

(m

ol/cm

3)

T(K)

ΘCW

~ 1 K

µeff

~ 2.35 µB

(a)

-6

-4

-2

0

2

4

0 0.5 1 1.5 2 2.5

M2 a

xis

inte

rcept

(µB/f

.u.)

2

T (K)

ΘC ~ 0.93 K

1.76 µB/f.u.

Msp

~ 0.7 µB

(c)

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3

NdOs4Sb

12

1.6 K1.5 K1.4 K1.3 K1.1 K1.0 K0.9 K0.6 K0.5 K0.4 K

M2 (

µB/f

.u.)

2

H/M (T/µB)

(b)

Fig.2 Ho, P.-C., et al.Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

(d)

FIG. 2: (a) Magnetization M vs magnetic field H isothermsand the saturation magnetization Msat determined from Mvs H isotherms below the Curie temperature ΘC. (b) M

2

vs H/M isotherms (Arrott plot) for NdOs4Sb12. (c) Inverseinitial magnetic susceptibility (open circles) determined fromM2 = 0 intercepts of M2 vs H/M isotherms, compared tothe χ−1dc (T ) data (pluses). The line is a Curie-Weiss fit. (d)H/M = 0 intercepts of M2 vs H/M isotherms plotted versusT . Positive values correspond to the spontaneous magnetiza-tion Msp.

romagnet. The intercepts of the linear fits to the (H/M)axis (≡ the inverse initial magnetic susceptibility) agreewell with the low-temperature χ−1dc (T ) data (Fig. 2(c)).The intercepts of the linear fit to the M2 axis are shownin Fig. 2(d), where zero identifies ΘC = 0.93K. BelowΘC, the intercept of the M

2-axis corresponds to thesquare of the spontaneous magnetization (M2sp), whichlevels off and results in a small value ofMsp ∼ 0.7µB/f.u..However, a linear extrapolation of the negative M2-axisintercept back to zero temperature yields a much largervalue of ∼ 1.76 µB/f.u., which is comparable to the satu-ration magnetization Msat of ∼ 1.73 µB/f.u. determined

4

10

12

14

16

18

0 1 2

8 T

10 T

12 T

14 T

16 T

18 T

ρ (µΩ

-cm

)

T (K)

NdOs4Sb12

1

2

3

4

5

0 1 2 3 4 5 6

T d (K

)

H(T)

(b)

0

100

200

300

400

0 100 200 300

ρ (µΩ

-cm

)

T (K)

0 T

9 T

Fig.3 Ho, P.-C., et al.: Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs4Sb

12

10

12

14

16

18

0 1 2

ρ (µΩ

-cm

)

T (K)

NdOs4Sb12

1 T

0.5 T

0 T

1.5T

2 T

3T

4 T

6 T

(a)

(c) (d)

FIG. 3: (a) Low-temperature electrical resistivity ρ vs tem-perature T for magnetic fields H between 0T and 6T forNdOs4Sb12. The solid lines are guides to the eye. (b) Low-temperature electrical resistivity ρ vs T for magnetic fieldsH between 8T and 18T for NdOs4Sb12. The solid lines areguides to the eye. (c) Position of the shoulder Td in ρ(T )(which corresponds to ΘC) vs H , with vertical bars indicat-ing the width of the transition at Td as described in the text.(d) High-T resistivity ρ vs T at H = 0T, 0.5 T, 1T, 2T, 4T,6T, 8T, 9T.

directly from the M(H) data of NdOs4Sb12 (Fig. 2(a)).The value of Msat is also consistent with the saturationmagnetization found in NdFe4P12 (1.72µB/f.u.).

15

C. Electrical Resistivity

Low-temperature electrical resistivity ρ(T ) data forNdOs4Sb12 in various magnetic fields from 0T to 18Tare shown in Figs. 3(a) and (b). The zero-field residualresistivity ratio RRR ≡ ρ(300K)/ρ(0.02K) of ∼ 45 indi-cates that the single crystal studied is of good metallurgi-cal quality (Fig. 3(a) and (d)). A shoulder occurs in thezero-field ρ(T ) curve at ∼ 1.2K, below which ρ(T ) hasa sharp drop, indicating the development of an orderedstate. The temperature Td at which this drop occurs isdefined as the intercept of two lines, one of which is alinear fit to the data above the transition while the otheris a linear fit to the data below the transition. The up-per and lower limits of the transition are defined as thetemperatures midway between Td and the temperaturesat which the data deviate from the linear fits. Shown inFig. 3(c) is the field dependence of Td, which increaseswith increasing field up to 6T, and is no longer observedin the ρ(T ) data at higher fields. The temperature Td cor-relates with the onset of ferromagnetic order at 0.9K de-termined from magnetization and specific heat measure-

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3

ρ (µ

Ω-c

m)

T (K)

0 T

8 T

2 T

4 T

14 T

18 T

10

12

14

16

0 4 8 12 16

ρ 0 (

µΩ-c

m)

H (T)

(a)

0

1

2

3

4

5

0 4 8 12 16

∆ sp

w (

K)

H (T)

(c)

3

3.5

4

4.5

0 4 8 12 16

n

H (T)

(b)

Fig.4 Ho, P.-C., et al.Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

(d)

FIG. 4: (a) Low-T electrical resistivity ρ vs temperature Twith power-law (dashed line) and spin-wave (solid line) fits atvarious magnetic fields for NdOs4Sb12. (b) Residual resistiv-ity ρ0 vs H . (c) Exponent n of the power-law fit vs H . (d)Energy gap ∆spw from the spin-wave fit vs H .

ments. Displayed in Fig. 3(d) are the high-temperatureρ(T , 0T ≤ H ≤ 9T) data for NdOs4Sb12, which show aslight increase in ρ(T ) with increasing field.In order to analyze the behavior of the resistivity be-

low ΘC, the ρ(T ) data were fit with a power-law of theform ρ(T ) = ρ0 + BT

n. The fitting curves are plot-ted as dashed lines in Fig. 4(a). The residual resistivityρ0 increases with increasing field and has a linear H-dependence above 8T (Fig. 4(b)). Between 0T and 18T,the exponent n varies from 3 to 4, which indicates thatNdOs4Sb12 exhibits neither typical Fermi-liquid (n ∼ 2)nor typical non-Fermi-liquid (n < 2) behavior (Fig. 4(c)).Since ferromagnetic ordering occurs below ΘC, electron-spin wave scattering was considered with the form30

ρ(T ) = ρ0 +AT

∆spw

(

1 + 2T

∆spw

)

exp

(

−∆spwT

)

, (3)

where ∆spw is the spin wave energy gap, which may re-sult either from magnetic anisotropy or from broken sym-metry due to presence of a CEF. This formula describesthe ρ(T,H) data well, and the fitting curves are shown assolid lines in Fig. 4(a). As determined from these fits, thespin-wave energy gap ∆spw is ∼ 0.75K at 0T, increasesapproximately linearly to ∼ 4.5K as the field increasesto 12T, and then drops to ∼ 3.4K at 18T (Fig. 4(d)).Figure 5(a) displays the zero-field ρ(T ) data, which

have a slight negative curvature at ∼ 130K that maybe related to scattering from the CEF levels. In orderto analyze the CEF contribution to ρ(T ) at 0T, it isnecessary to subtract a lattice contribution ρlat(T ) andan impurity contribution ρimp (∼ 9.4µΩ-cm) from theresistivity data. Usually, ρlat(T ) is estimated from anisostructural nonmagnetic reference compound; in the

5

0

100

200

300

400

0 100 200 300

ρestimated ρlat+ρimp

ρ (µΩ

-cm

)

T (K)

NdOs4Sb12

0

40

80

120

160

0 100 200 300

ρ-ρlat-ρimp(xLLW = -0.1, W = 3.9, x = 0.45)

(xLLW = -0.4, W = -7.2, x = 0.1)

∆ρ (µ

Ω-cm

)

T (K)

(a)

(b)

Fig. 5 Ho, P.-C., et al.Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

FIG. 5: (a) Zero-field resistivity ρ and the estimated ρlat+ρimpvs temperature T for NdOs4Sb12, where ρimp ∼ 9.4 µΩ-cm.(b) Temperature dependence of the incremental resistivity∆ρ = ρ− ρlat − ρimp and CEF fits for two different ground

states: Γ6 (dashed line) and Γ(2)8 (solid line), where x/(1− x)

is the ratio of s-f exchange to aspherical Coulomb scatteringfor NdOs4Sb12.

case of NdOs4Sb12, the resistivity of LaOs4Sb12, whichhas an empty 4-f shell, was used. However, above 100K,ρ(T ) of LaOs4Sb12 exhibits a significant negative cur-vature that is common in La-based compounds such asLaAl2.

31,32 This curvature is generally less pronouncedin Sc-, Y-, and Lu-based compounds, which have com-pletely empty or filled 4f-electron shells. However, thecompounds ScOs4Sb12, YOs4Sb12 and LuOs4Sb12 havenot yet been synthesized. Above 100K, ρlat(T ) was es-timated from SmOs4Sb12, as ρ(T ) of SmOs4Sb12 has anapproximately linear-T dependence between 100K and300K.33 The s-f exchange scattering effect in ρ(T ) ofSmOs4Sb12 only appears below 80K due to the small en-ergy splitting (∼ 35K) between the ground and the firstexcited states in that compound. Thus it is reasonable toassume that ρ(T ) of SmOs4Sb12 for 100K ≤ T ≤ 300Kis due to electron-phonon scattering and use it as anestimate of the high-temperature portion of ρlat(T ) forNdOs4Sb12.The incremental resistivity ∆ρ(T ) = ρ(T )− ρlat(T )−

ρimp (Fig. 5(b)) is best described by two energy-levelschemes that are consistent with the CEF analysis ofχ(T ) discussed previously. During the CEF analysisof ∆ρ(T ), it was also found that s-f exchange scatter-ing alone could not entirely account for ∆ρ(T ); other-wise, ρimp would always be negative, which is unphysical.Thus, the effect of aspherical Coulomb scattering34,35,36

was also considered, with the ratio between the s-f ex-change scattering and the aspherical Coulomb scattering

1

2

3

4

5

0 4 8 12 16

ρE

x+

Asph (

arb

itra

ry u

nit)

H(T)

0.02 K

40 K

20 K

15 K

4.5 K

2.75 K

0.90.35

1.7 K

10 K

(xLLW = -0.1, W = 3.9, x = 0.45)

(b)

10

20

30

0 4 8 12 16

ρ (µ

Ω-c

m)

H (T)

15 K

0.02 K & 0.35 K

4.5 K

2.75 K

0.91.7

10 K

20 K

NdOs4Sb12

(a)

1

2

3

4

5

0 4 8 12 16

ρE

x+

Asph (

arb

itra

ry u

nit)

H(T)

0.02 K

40 K

20 K15 K

4.5 K2.75 K

0.9 K0.35 K

1.7 K

10 K

(xLLW = -0.4, W = -7.2, x = 0.1)

(c)

Fig. 6 Ho, P.-C., et al.Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

FIG. 6: (a) Isotherms of electrical resistivity ρ vs magneticfield H for NdOs4Sb12. (b) and (c) Calculated ρ vs Hisotherms including the effects of s-f exchange and aspheri-cal Coulomb scattering using the CEF parameters determinedfrom zero-field data.

defined as x : (1-x). In Fig. 5(b), the fit curves represent-ing the two best-fit energy schemes are plotted in com-parison with ∆ρ(T ), which is described well by both fitsabove 40K. The departure of ∆ρ(T ) from the fits below40K may be due to a reduction of the electron scattering,resulting from the development of the coherent heavy-fermion ground state in NdOs4Sb12 as the temperatureis lowered. The ρ(H) isotherms are displayed in Fig. 6(a)and qualitatively agree with the ρ(H) isotherms gener-ated using the cubic-CEF parameters determined fromthe zero-field fits to ∆ρ(T ) (Figs. 6(b) and (c)).Measurements of ρ(T ) were performed from 1 K to 300

K under nearly hydrostatic pressure P between 0 kbarand 28 kbar (Fig. 7). In Fig. 7 it can clearly be seenthat the pressure-induced change in the high-T electricalresistivity is much more pronounced than the resistivitychange induced by high field (Fig. 3(c)), in the pressureand temperature ranges of this investigation. However,at low temperatures, the value of Td is more stronglyinfluenced by an increase in field H (Fig. 3(b)) than byvariation of P (Fig. 7 inset (b)).

D. Specific Heat

Specific heat divided by temperature C/T vs T datafor NdOs4Sb12 are shown in Fig. 8(a). The data reveal apronounced peak at ∼ 0.8K that correlates well withthe magnetic ordering temperature ΘC inferred fromthe shoulder of ρ(T ), the divergence of χdc(T ), and theArrott-plot analysis. No obvious Schottky anomaly as-sociated with CEF energy splitting was observed in theC(T ) data below 20K. This is consistent with the fact

6

0

100

200

300

400

0 100 200 300

NdOs4Sb

12

0 kbar6 kbar8.7 kbar14.5 kbar

20 kbar

22.4 kbar24.5 kbar28.4 kbar

ρ (

µΩ

-cm

)

T (K)

P

10

12

14

16

18

20

1 1.5 2 2.5 3 3.5 4

20 kbar22.4 kbar24.5 kbar28.4 kbar

0 & 6 kbar

8.7 kbar

14.5 kbar

1.2

1.6

2

0 5 10 15 20 25 30

Td (

K)

P (kbar)

(a)

(b)

Fig. 7 Ho, P.-C., et al. Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

FIG. 7: Electrical resistivity ρ vs temperature T forNdOs4Sb12 at various pressures P up to 28 kbar. Inset (a):Expanded view of the resistive ferromagnetic transitions. In-set (b): Temperature of the drop in ρ(T ) due to the onsetof ferromagnetism Td (approximately corresponding to theCurie temperature ΘC) vs P with vertical bars indicating thewidth of the transition.

that all the fits to the dc magnetic susceptibility dataimply that the CEF splitting between the ground andfirst excited states is greater than 120K. The Schottkyanomaly due to 120K CEF splitting would exhibit apeak at ∼ 40K and make a negligible contribution toC(T ) 20K. Such a peak would also be difficult to resolveagainst the large lattice background.

In Fig. 8(a), the specific heat of LaOs4Sb12 is dis-played in comparison with that of NdOs4Sb12, wherethe electronic specific heat coefficient γ and the Debyetemperature ΘD of LaOs4Sb12 are ∼ 36mJ/mol-K

2 and∼ 280K, respectively. After the specific heat of nonmag-netic LaOs4Sb12 (an estimate of the lattice heat capac-ity of NdOs4Sb12) is subtracted from the specific heatof NdOs4Sb12, the difference divided by temperatureδC/T is plotted against T 2 and shown in the inset ofFig. 8(a). The value of γ for NdOs4Sb12 estimated fromthe δC/T vs T 2 plot ranges from ∼ 436mJ/mol-K2 to∼ 530mJ/mol-K2. Below 13K, δC/T vs T 2 is not con-stant, which could be due to a difference between the ac-tual lattice heat capacities of NdOs4Sb12 and LaOs4Sb12.Nevertheless, the curvature in C/T vs T 2 of NdOs4Sb12(inset of Fig. 8(b)) and the magnetic transition occurringat ∼ 1K cause difficulties in the analysis of the data us-ing the typical formula C/T = γ + βT 2. Since we havestrong evidence against the possibility of a CEF Schot-tky contribution from the analysis of the χ(T ) data, thecurvature in C/T vs T 2 is possibly due to either the rat-

0

10

20

30

40

0 5 10 15 20

C (

J/m

ol-K

)

T(K)

NdOs4Sb

12

LaOs4Sb

12

400

600

800

1000

0 100 200 300 400

δC/T

(m

J/m

ol-

K2)

T2 (K

2)

0

10

20

30

40

0 5 10 15 20

C (

J/m

ol-K

)

T(K)

NdOs4Sb

12

Cel + C

latγ ~ 520 mJ/mol-K2

ΘD ~ 255 K, Θ

E ~ 39 K, r ~ 0.48

Fig. 8 Ho, P.-C., et al.Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

(a)

(b)

400

800

1200

1600

0 100 200 300 400

C/T

(m

J/m

ol-

K2)

T2 (K

2)

NdOs4Sb

12

FIG. 8: (a) Zero-field C vs T for NdOs4Sb12 and LaOs4Sb12below 20K. Inset: δC/T vs T 2 below 20K, where δC ≡C(NdOs4Sb12) - C(LaOs4Sb12). The extrapolated values ofthe two dashed lines at 0K set the lower and upper limitsof γ(NdOs4Sb12) - γ(LaOs4Sb12). (b) A comparison betweenC of NdOs4Sb12 and a fit of Cel + Clat, where Cel = γTand Clat is composed of CDeb and CEin as described in thetext. The electronic specific heat coefficient γ, the Debyetemperature (ΘD), the Einstein temperature (ΘE),and thecoupling constant r for the Einstein oscillator are estimatedas 520mJ/mol-K2, 255K, 39K and 0.48 respectively. Inset:C/T vs T 2 for NdOs4Sb12 below 20K for NdOs4Sb12.

tling motion of the Nd atoms or a narrow peak in thedensity of electronic states near Fermi energy, such asa Kondo resonance.37 However, we do not consider theapplication of the resonance level model (RLM)37 to beappropriate, because the magnetization data above ΘCare fit well by a Brillouin function and no obvious Kondoeffect is observed in the ρ(T ) data of NdOs4Sb12. Inthe previous specific heat studies of the Tl0.22Co4Sb12filled skutterudite compound, Sales et al.25 found thatthe difference in heat capacity between Tl0.22Co4Sb12and the unfilled skutterudite compound Co4Sb12 can beaccurately described by a quantized oscillator (Einsteinmodel) with an Einstein temperature ΘE of 55K. Sincethe X-ray structural refinement at room temperature forNdOs4Sb12 indicates a small ΘE ∼ 45K associated withthe Nd atoms, it can be assumed that the Nd atoms par-tially act like Einstein oscillators with a mixing ratio r,and the lattice contribution to the specific heat can be

7

0

4

8

12

0

4

8

12

0 4 8 12 16 20

∆C

(J/m

ol-

K) S (J

/mo

l-K)

T(K)

Rln4

Rln2

∆C = C - Cel - C

lat

Smag

NdOs4Sb

12ΘC

0.1

1

10

0.1 1

∆C

(J/m

ol-K

)

T(K)

∆C ~ T2.3

NdOs4Sb

12

∆C ~ T1.5

exp(-∆spw

/T)∆

spw ~ 0.54 K

Fig. 9 Ho, P.-C., et al.Weak ferromagnetism and heavy fermion behavior in single crystals of NdOs

4Sb

12

(a)

(b)

FIG. 9: (a) Incremental specific heat ∆C (∆C ≡ C − Cel −Clat) (left axis) and the magnetic entropy Smag (right axis)vs temperature T . (b) Logarithmic plot of the power-law fit(dotted line) and the anisotropic-spin-wave fit (dashed line) tothe incremental specific heat ∆C(T ) after the electronic andlattice contributions have been removed (in a very limitedmeasuring range below ΘC).

expressed as Clat = CEin + CDeb,

CEin = r

[

3R(ΘE/T )

2e(ΘE/T)

[e(ΘE/T) − 1]2

]

, (4)

CDeb = (17− r) ·12π4

5R

(

T

ΘD

)3

, (5)

where R is the universal gas constant, ΘD representsthe Debye temperature, and r ≤ 1. The effective De-bye contribution is (17 − r)/r times bigger than that ofthe Einstein-like Nd rattling motion due to the partici-pation of the rest of the atoms in the unit cell. Below20K, a least squares fit of Cel + Clat to the C(T ) datawas performed, where Cel = γT is the electronic specificheat, from which estimated values of γ,ΘD,ΘE, and rwere derived. The fitting curve is plotted in Fig. 8(b)along with the C(T ) data of NdOs4Sb12 as a compari-son. The values obtained for γ,ΘD, ΘE, and r are es-timated as 520mJ/mol-K2, 255K, 39K, and 0.48, re-spectively. The value of ΘD is close to the Debye tem-perature of LaOs4Sb12, the value of ΘE is comparableto that determined from the X-ray data, and the valueof γ is close to that estimated from the simple subtrac-tion of the LaOs4Sb12 specific heat data, suggesting thatNdOs4Sb12 is possibly a heavy fermion compound.

The temperature dependence of the magnetic entropySmag was derived from the integration of ∆C/T vs T andis shown in Fig. 9(a), where ∆C ≡ C − Cel − Clat. Themagnetic entropy (Smag) in NdOs4Sb12 reaches 0.69R (≈Rln2) at 0.85K and levels off at a value of ∼ 1.14R(≈ Rln3). The magnetic entropy reaches ∼ 74% of itsfull value at ΘC, and a noticeable magnetic contribu-tion persists up to ∼ 3K, revealing the existence of mag-netic fluctuations above ΘC. It has been argued forthe antiferromagnetic system Gd1−xYxNi2Si2 that mag-netic fluctuations can contribute to C(T ) at tempera-tures up to 5 times the Néel temperature.38 Thus, wecannot completely rule out the possibility that the shortrange magnetic correlations near the ferromagnetic tran-sition temperature regime give rise to enhancement ofthe specific heat of NdOs4Sb12.

38,39,40 However, it is un-likely that they would account for a large fraction ofthe enhanced specific heat, due to the following argu-ments: (i) The paramagnetic-state M(H) isotherm data(2 - 5K) scale well with a Brillouin function. The M(H)data at 2K and 3K, and the χ−1dc (T ) data along withthe initial χ−1 from the Arrott Plot analysis do notshow obvious evidence of magnetic fluctuations above2K (displayed in Fig. 2 (a) and (b)). (ii) The valueestimated for γ (520mJ/mol-K2) is within the upperand lower limits of γ (436 - 530mJ/mol-K2) estimatedfrom the analysis done by comparing the specific heatof NdOs4Sb12 with that of the nonmagnetic compoundLaOs4Sb12 suggesting that our analysis is justified. (iii)The lower limit (4K) of the fitting range used to deter-mine γ from the formula C = Cel+Clat is four times theCurie temperature, which is safely higher than the tem-perature at which the calculated Smag saturates. (iv)Heavy fermion behavior has been found in the neighbor-ing compounds PrOs4Sb12 (γ ∼ 600 mJ/mol-K

2)41 andSmOs4Sb12 (γ ≈ 880 mJ/mol-K

2);33 earlier studies ofthe related compound NdFe4Sb12 also reported possibleevidence for an enhanced electron mass.15,42 Consider-ing all of these points, it seems likely that NdOs4Sb12displays heavy fermion behavior.

Since Smag lies between Rln2 (= 0.69R) and Rln4 (=1.39R), it is difficult to determine conclusively whether

the Γ6 doublet or Γ(2)8 quartet is the Nd

3+ ground state.If the Γ6 doublet is the ground state, then the extra en-tropy may result from another degree of freedom, such asa tunnelling mode or off-center mode of Nd3+ ion in an

Sb-icosahedron cage.43 If the Γ(2)8 quartet is the ground

state, the smaller entropy may be due to an overesti-mated lattice contribution to the specific heat or transferof entropy to the conduction electron system.

Figure 9(b) displays the incremental specific heat∆C(T ) after the electronic and lattice contributions havebeen removed. Even though the range of the ∆C(T ) databelow ΘC is very limited, the data were fit with spin-wave formulas ∆C(T ) ∝ T n for magnetically isotropicmetals and ∆C(T ) ∝ T 3/2 exp(−∆spw/T ) for magneti-cally anisotropic metals.44 From the first formula, thevalue of the exponent n ≈ 2.3 is higher than the value of

8

1.5 expected from a spin wave in an isotropic metal. Thespin-wave energy gap ∆spw determined from the secondformula is ∼ 0.54K, consistent with the value of 0.75Kdetermined from the zero-field resistivity data.

IV. SUMMARY

We have performed X-ray diffraction, electrical re-sistivity, magnetization, and specific heat studies ofNdOs4Sb12 single crystals, which exhibit interestingstrongly-correlated-electron behavior. X-ray experi-ments have revealed full occupancy of Nd sites and alarge mean square displacement of Nd ions in NdOs4Sb12.The compound NdOs4Sb12 exhibits mean-field-type fer-romagnetism with ΘC ∼ 0.9K. The value of γ es-timated from the analysis of the specific heat islarge, ∼ 520mJ/mol-K2 (m∗ ∼ 98me), indicating thatNdOs4Sb12 is a possible heavy fermion compound. Acubic CEF analysis suggests two best-fit energy level

schemes: (I) Γ6 (0K), Γ(1)8 (180K), Γ

(2)8 (420K); (II)

Γ(2)8 (0K), Γ

(1)8 (220K), Γ6 (600K), both with a molecu-

lar field parameter Λ = 1.39 Nd-mol/cm3. The electricalresistivity data indicate that both s-f exchange and as-pherical Coulomb scattering are present in NdOs4Sb12.

Low-temperature electrical resistivity data suggest thepossible existence of spin-wave excitations below ΘC.The uncertainty in the CEF-energy-level scheme groundstate and the possible existence of spin-wave excitationsmay be resolved by further neutron scattering experi-ments.

Acknowledgments

We thank Dr. C. Capan for technical support atNHMFL at Los Alamos and Prof. D. P. Arovas at UCSDfor helpful discussions. We also thank S. K. Kim and A.P. Thrall for assistance in sample preparation. Researchat UCSD was supported by the U. S. Department of En-ergy under Grant No. DE-FG02-04ER46105, the U. S.National Science Foundation under Grant No. DMR-0335173, and the National Nuclear Security Administra-tion under the Stewardship Science Academic Alliancesprogram through DOE Research Grant No. DE-FG52-03NA00068. The work at the NHMFL Pulsed Field Fa-cility (Los Alamos National Laboratory) was performedunder the auspices of the NSF, the State of Florida andthe US Department of Energy.

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