Table Of ContentTypeset with jpsj2.cls <ver.1.2> Letter
Anomalous Metal-Insulator Transition in Filled Skutterudite CeOs Sb
4 12
Yoshiki Imai, Keitaro Sakurazawa and Tetsuro Saso
Department of Physics, Saitama University, Saitama, 338-8570
(Received )
6
Anomalous metal-insulator transition observed in filled skutterudite CeOs4Sb12 is investi-
0
gated by constructingtheeffective tight-bindingmodel with theCoulomb repulsion between f
0
electrons. By usingthemean field approximation, magnetic susceptibilities are calculated and
2
the phase diagram is obtained. When the band structure has a semimetallic character with
n
small electron and hole pocketsat Γ andH points, a spin densitywave transition with theor-
a
dering vector Q=(1,0,0) occurs dueto the nesting property of the Fermi surfaces. Magnetic
J
field enhances thisphase in accord with theexperiments.
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KEYWORDS: filled-skutteruditecompounds, CeOs4Sb12, field induced metal-insulator transition
]
l
e
-
r Filled-skutterudite compounds RM X (R: rare body-centeredcubic(BCC)lattice.Theforbitalsofrare-
4 12
t
s earth, M: Fe, Ru, and Os, and X: P, As, and Sb) with earthatomsstronglyhybridizewiththeporbitalsinX12
.
t theuniquebody-centeredcubicstructureshaveattracted clustersclosetheFermilevel.TheX12 clusterhastheTh
a
muchinterestduetowidevarietyofphysicalphenomena, symmetry, but its effect on J=5/2 f1 state in Ce is the
m
e.g.magneticandquadrupoleordering,Kondoinsulating same as O .
h
d- behavior, unconventional superconductivity, etc.1,2 In order to investigate the properties of Ce-
n Recently, anomalous phase transition from paramag- skutterudites, we previously proposeda simplified tight-
o netic metal (PM) to spin density wave(SDW) state was bindingbandmodelforCeRu Sb consistingofthe sin-
4 12
c
found atT =0.8Kin CeOs Sb .Surprisingly,the phase glevalencebandandthesevenfstateswiththespin-orbit
[ 4 12
boundary is shifted to higher temperatures by applying interaction and under the cubic crystalline field.11 The
1 the externalmagnetic field.2,3 Such increaseofthe tran- former is representedby the top-mostone ofthe valence
v
sition temperature have been often observed in the sys- bandsmadeupofthep-orbitalsonX clusters,andhas
2 12
tems with quadrupole-ordering,4 but the magnetic mea- theA symmetrywhichhybridizeswiththef-stateinthe
6 u
0 surement5 suggests that the crystal-field (CF) ground cluster center. The Coulomb interaction was introduced
1 state of f electron is Γ which has no quadrupole mo- to allthe bandequallytomakethe useofthe dynamical
7
0 ment, and the specific heat measurement2 shows only mean-field approximation possible.
6
a small entropy change at the transition, excluding lo- Here, we simplify the model further to investigate the
0
cal moment ordering but suggesting an SDW or CDW magnetic properties of CeOs Sb . The present model
/ 4 12
t transitions with an opening of a gap over a small por- consists of a single f band (representing the J=5/2, Γ
a 7
m tion of the Fermi surfaces of itinerant electronic states. ground states) and a single valence band (the above-
The neutron scattering experiment6 was done but the mentioned A band), hybridizing with each other. Then
- u
d CF was not clearly observed. The NQR measurements7 Hamiltonian is given by
n
claims the first-order transition since the hysterisis was
o observed, but the specific heat data2 clearly show peaks H = ǫckc†k,σck,σ+ ǫfkfk†,σfk,σ
:c at the transition indicating the second order transition Xk,σ Xk,σ
v
with some short range order.
Xi CeOs4Sb12 shows semiconducting behavior in the re- + (cid:18)Vc†k,σfk,σ+h.c(cid:19)+U nfi↑nfi↓
r sistivityexperimentwithasmallchargegap(Eg ∼10K) Xk,σ X
a at low temperatures.5 The overall structure of the op-
+ B σ nc +βnf (1)
tical conductivity and its temperature dependence show ext (cid:18) k,σ k,σ(cid:19)
similar behaviors to those of CeRu Sb which exhibits Xk,σ
4 12
a pseudo-gap and the mid-infrared peak with strong where c† (f† ) creates a conduction (f) electron with
temperature-dependence.8,9 Although the band struc- the momk,eσntukm,σ k and pseudo-spin σ (representing Γ
7
ture of CeOs4Sb12 obtained by the band calculation is doublet), and nf =f† f with the site index i. We
quite similar to that of CeRu Sb ,10 CeOs Sb shows iσ i,σ i,σ
4 12 4 12 have assumed the BCC tight-binding band with near-
semiconductingbehavior,whereasCeRu4Sb12 ismetallic est and next-nearest hopping for conduction band, ǫc =
k
in experiment.5
αctk, where tk = cos(kxa/2)cos(kya/2)cos(kza/2) +
In this letter, we construct the effective model with
α [cos(k a)+cos(k a)+cos(k a)].Thelatticeconstanta
2 x y z
the characteristic feature of the filled skutterudite com-
issetequaltounityinthefollowing.α andα determine
c 2
pound CeOs Sb and investigate the anomalous metal-
4 12 the hopping amplitude. The f state is assumed to have
insulator transition observed in this material. thedispersionǫfk =Ef+αftk,namelythedispersivepart
The basic crystal structure of filled skutterudites is isproportionaltoǫc butwithasmallcoefficientα .This
k f
that the X clusters, filled by R in the center, form the
12 would be a reasonable choice since both X and R oc-
12
2 J.Phys.Soc.Jpn. Letter Online-JournalSubcommittee
6
5
E
F
y 00 4
g
r q)
ne 0(
E 3
c
2
-1
1
G N P G H P N H G N P G H P N H
Fig. 1. The band structure for the non-interacting system with-
outtheexternalfieldwhereαc=1.0,αf=0.03,α2=0.07,Ef=0.3, Fig. 2. Staticirreduciblesusceptibilityforvariousmomentawith-
andV=0.15.EF representstheFermilevel. out the external field at T = 0.002 where χ0(q) = χ0↑↓(q) =
χ0 (q)=χ0 (q).
↑↑ ↓↓
cupythesameBCCsites.V isthehybridizationbetween
conductionandfelectrons.Itsk-dependenceisneglected. RPA overestimates the transition temperature, we use
Thesechoiceofourmodelbandimmenselyfacilitatesthe the small value of the Coulomb interactions here, which
calculations for the strong correlation, which will be re- is considered as the renormalized one at the low-energy
ported in a separate paper.12 Actually, of course, the regime in the rare-earthcompounds. Furthermore, since
f-dispersion is created through the X12 clusters, so that theCoulombinteractionactsonlyonfelectrons,ordered
it must be self-consistently determined with ǫck and the states will be mainly determined by the localized f elec-
hybridization Vk. U is the Coulomb repulsion between f trons, so that only the f-f component of the correlation
electrons. It is well-known that ǫck and ǫfk have perfect function is considered, for simplicity. It is essential for
nesting characters for α2 = 0, V=0, and at half-filling. the following theory to calculate both the parallel and
The last term in eq. (1) represents the Zeeman energy, perpendicular elements to the externalfield of the static
whereBext denotesthe externalmagneticfieldalongthe susceptibility, which are given by
z direction and β is coefficient of effective magnetic mo-
ment, which corresponds to gJhJzfi=6/7×5/6=5/7 for χx(q)=χy(q)= β2 χ0↑↓(q) , (2)
Ce. 2 1−Uχ0↑↓(q)
WithouttheCoulombinteraction,thediagonalizeden-
ergy dispersions are obtained as Ek± =[ǫck+ǫfk±{(ǫck− χz(q)= β2χ0↑↑(q)+χ0↓↓(q)+2Uχ0↑↑(q)χ0↓↓(q)(3,)
ǫf)2 + 4V2}1/2]/2, which are the so-called hybridized 4 1−U2χ0↑↑(q)χ0↓↓(q)
k
bands, but since ǫf has also a finite dispersion, the two whereχ0 isthebandsusceptibility,whichisdefinedby
k σσ′
bands can have finite overlapif α is not too small.
f −T
Consideringtheempiricaltrendofthelatticeconstant χ0σσ′(q)= N gσf(k+q,iωn)gσf′(k,iωn),(4)
andtheenergygapofCe-skutterudites,CeOs4Sb12 must kX,ωn
be a semimetal.2 Our assumed band structure is shown
1
inFig.1,whichhasa semi-metallic character:the topof gσf(k,iωn)= V2 .(5)
lower band (Γ point) and the bottom of upper band (H iω +µ−ǫf −
n kσ iω +µ−ǫc
point) slightly touch the Fermi level, respectively. The n kσ
overlapping of the bands is about 0.001α . T is the temperature and N is the number of sites. µ is
c
Although our band structure is quite simplified, Fig. the chemical potential and ω (=(2n+1)πT) represents
n
1 captures the low-energy structure of the band calcula- theoddMatsubarafrequency.ǫc andǫf aretheenergy
kσ kσ
tion for CeOs4Sb1210 rather well. In the published band dispersion with the Zeemanterms. Note that we use the
calculation,10 one of the band made of Γ8 states has a following relation that χ↓↑(q) is always equal to χ↑↓(q)
minimumatΓpointandalmosttoucheswiththevalence within the RPA to derive χx.
bandthere,butitis neglectedhere.Also,onlythe single We numerically calculate the above-mentioned equa-
valence band is taken into account and the lower bands tions. The summations are efficiently carried out by us-
are completely neglected. These may affect the quanti- ing the fastFourier transformwith N =128×128×128
tative results but at least the qualitative feature may pointsintheksummationand4096Matsubarafrequen-
remain unchanged. The existence of Γ8 band may be a cies in the ωn summation. Note that hopping amplitude
cause of the lower temperature anomaly.7 α is taken to be unity for simplicity.
c
In order to investigate a possibility of a magnetic To discuss the instability to ordered states, the band
transition and an anomalous magnetic field effect, the susceptibility, χ0(q) ≡ χ0 (q) = χ0 (q) = χ0 (q) for
↑↑ ↓↓ ↑↓
random phase approximation (RPA) is employed. Since B =0, is shown in Fig. 2. The largest value appears
ext
J.Phys.Soc.Jpn. Letter Online-JournalSubcommittee 3
c x )
T
) 3 c z d (10
q el
z( 0.1 c fi
xc(q), Bext magneti 5
ct 2 0 1 2
s
e Temperature (K)
g
r
a
L
1
0 0.02 0.04 0.06 0.08 0.1 0 0.002 0.004 0.006 0.008 0.01
B T
ext
Fig. 4. TheT-B Phasediagramof themetal-SDW transitionat
Fig. 3. Thelargestsusceptibilities as afunction of the magnetic
(fiχelzd(qa)t)T. =0.002andU=0.0.Solid(dashed)linerepresentsχx(q) USug=aw0a.2r.aTetheal.i2nset shows the experimental result obtained by
(fluctuation-exchange) method was used.14
at H point q=(1,0,0) in unit of 2π/a. Since the Fermi
Ingeneral,theresponsefunctionsarestronglyaffected
surfaces of electrons and holes are quite small spheres,
bytheformofFermisurface.Thepresentbandstructure
the peak of χ0(q) appears at q = (1,0,0) at low tem-
at B = 0 is close to the perfect nesting, so that the
peratures, which corresponds to the difference between ext
change of magnetic susceptibilities becomes quite large
thetopoflowerband(Γpoint)andthebottomofupper
for the external field. Therefore, it is possible that when
band(Hpoint)inFig.1.Recently,the resultofthe neu-
the weak external field is applied, the nesting condition
tronscatteringforCeOs Sb atB =0foundthatthe
4 12 ext
isratherimprovedbecauseoneofthespinbandisshifted
nesting vector corresponds to q = (1,0,0).13 Therefore
by the field and approaches the nesting condition.
our obtained result of the susceptibilities is consistent
These results indicate that the presence of the ex-
with the experiment at B =0.
ext
ternal field enhances instability for ordered states. In
The basic lattice structure of CeOs Sb is BCC. For
4 12
the 3-dimensional system, ordered states can occur at
the single band Hubbard model at half-filling with only
a finite temperature. Namely, the anomalous metal-
the nearest neighbor hopping, there exists the perfect
insulatortransitionmayoccurinthepresentmodelwith
nestingatthewavevectorq=(1,0,0).Inthatcase,itis
appropriate value of the Coulomb repulsion. Figure 4
knownthattheinfinitesimallysmallU causesthemetal-
shows the temperature-external field phase diagram of
SDW transitionat T=0.Although the perfect nesting is
metal-SDW transition. Note that the instability occurs
lost in the present model due to the hybridization and
not in χz(q=(1,0,0)) but in χx(q=(1,0,0)) (and χy),
the next nearest neighbor hoppings, the Fermi surfaces
which indicates the appearanceofthe magnetic moment
are in the vicinity of the nesting. Therefore, when the
component perpendicular to the field. It can be checked
externalfieldisabsent,themetal-SDWtransitionoccurs
by an experiment in a future. In the small external field
at U ∼ 0.2 at T=0.002 within the RPA in the present
c region,the phase transitiontemperature T is enhanced
model. N
withincreasingB ,sothattheslopeofphaseboundary
Next, let us discuss effects due to the external mag- ext
becomes positive. Thus the anomalous phase transition
netic field. The field dependence of the largest values of
observed in CeOs Sb can be qualitatively reproduced.
susceptibilities is shown in Fig. 3. With increasing the 4 12
Noteherethatinordertocalculatemagneticsuscepti-
magnetic field alongthe z direction,χx is firstenhanced
bilities,we employthe RPA,whichis insufficientto take
at small field region. As the external field is increased
the effect of quantum fluctuations into account. Quan-
further, χx is suppressed. The vector q which gives the
tum fluctuations particularly become important in the
largest values of susceptibilities does not change within
vicinity of quantum critical points where antiferromag-
B < 0.2. In contrast to the behavior of χx, χz is
ext netic correlations are enhanced.14 However, correlation
first suppressed. This behavior is reasonable since when
effects are essential in the weak external field region. By
the weak external field is applied, the spin-spin correla-
taking quantum fluctuations into accounte.g.via FLEX
tion functions along the external field are generally sup-
approximation,14themetal-SDWtransitiontemperature
pressed. Furthermore, with increasing the external field
is reduced andthe transitionboundary as a whole shifts
further,maximum value of χz is enhanced. Inthis range
to the low temperature region. In particular, the shift
offield,thevectorqwhichgivesthemaximumofthesus-
becomes conspicuous at the weak external field region.
ceptibilitiesisgraduallychanged.Bothofsusceptibilities
However, overall structure of the phase boundary ob-
are suppressed under larger external field (not shown).
tained here may not qualitatively changed.
These behaviors and the mechanisms are similar to our
Furthermore, we comment the multi-band effects of f
previous theory for CeRhIn where more precise FLEX
5
orbitals. In our model, we neglect the degeneracy of f
4 J.Phys.Soc.Jpn. Letter Online-JournalSubcommittee
orbitals, for simplicity. However, in the published band New Quantum Phenomena Realized in the Filled Skut-
calculation,one of the bands made fromthe 4f Γ states terudite Structure”, No. 16037204. A part of computa-
8
has a minimum at Γ point, though slightly above E , tions was done at the Supercomputer Center at the In-
F
which is neglected in our model. However, we consider stitute for Solid State Physics, University of Tokyo.
thatthe nesting propertybetweenthe topofthe valence
band at Γ and the bottom of the Γ band at H is the
7
mainsourceoftheSDWtransitionandanomalousphase 1) E.D.Bauer, N.A.Frederick, P.-C.Ho, V.S. Zapf and M.B.
diagram. Therefore qualitative properties of CeOs Sb Maple:Phys.Rev.B65(2002)R100506.
4 12
2) H.Sugawara, S.Osaki,M.Kobayashi, T.Namiki,S.R.Saha,
can be reproduced by using the present simple model.
Y.Aoki,andH.Sato: Phys.Rev.B71(2005) 125127.
Asimilaranomalousmagneticfieldeffectisalsoinves-
3) T.Namiki,Y.Aoki,H.SugawaraandH.Sato:Acta.Phys.Pol.
tigatedfortheKondoinsulator,15,16inwhichtheordered B34(2003)1161.
phase appears at finite field since the perpendicular el- 4) A.Takase, K.Kojima,T.Komatsubara and T.Kasuya: Solid
ement of spin-spin correlations to the external field is StateCommun.36(1980)461.
5) E.D.Bauer,A.Slebarski,E.J.Freeman,C.SirventandM.B.
enhanced.
Maple:J.Phys.Condens.Matter 13(2001) 4495.
Insummary,weinvestigatetheanomalousphasetran-
6) C.Yang,M.Kohgi,K.Iwasa,H.SugawaraandH.Sato:J.Phy.
sition observed in CeOs4Sb12. By constructing the ef- Soc.Jpn.74(2005)2862.
fective tight-binding model with the Coulomb interac- 7) M. Yogi, H. Kotegawa, G. Zheng, Y. Kitaoka, S. Ohsaki, H.
tion between f electrons, whose band structure is in the SugawaraandHSato: J.Phy.Soc.Jpn.74(2005) 1950.
8) S. V. Dordevic, D. N. Basov, N. R. Dilley, E. D. Bauer and
vicinity of the nesting, the metal-SDW phase boundary
M.B.Maple:Phys.Rev.Lett. 86(2001)684.
is determined within the mean field approximation.The
9) M. Matsunami, H.Okamura, T. Nanba, H. Sugawara and H.
nesting condition of Fermi surfaces is improved when Sato: J.Phy.Soc.Jpn.72(2003) 2722.
the weak external field is applied, so that the slope of 10) H.Harimaand K.Takegahara: J.Phys. Condens. Matter 15
metal-SDW phase boundary becomes positive and the (2003) S2081,andprivatecommunication.
11) T.MutouandT.Saso:J.Phy.Soc.Jpn.73(2004)2900.
anomalous transition can be reproduced. A small mo-
12) T.Saso:submittedtoJ.Phys.Soc.Jpn.
ment perpendicular to the applied magnetic field should
13) K.Iwasa:privatecommunications.
beinduced,whichcanbecheckedbyafutureexperiment. 14) K.Sakurazawa,H.KontaniandT.Saso:J.Phy.Soc.Jpn.74
WewouldliketothankProfessorsM.Kohgi,K.Iwasa, (2005) 271.
H.Sugawara,H.Harima,K.TakegaharaandH.Kontani 15) I.Milat,F.AssaadandM.Sigrist:Eur.Phys.J.B38(2004)
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forusefulinformationsandilluminatingdiscussions.This
16) T.Ohashi,A.Koga,S.SugaandN.Kawakami:Phys.Rev.B
workwas supportedby The Grant-in-Aidfromthe Min-
70(2004) 245104.
istry of Education, Science and Culture: “Evolution of
