-
arX
iv:c
ond-
mat
/041
2713
v3 [
cond
-mat
.str
-el]
4 J
ul 2
005
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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