Universality in heavy-fermion systems with general degeneracy

0.14 MB·English

by  N. Tsujii

Tsujii et al
Universality in heavy-fermion systems with general degeneracy
N. Tsujii1 , H. Kontani2, and K. Yoshimura3
1National Institute for Materials Science, Sengen 1-2-1, Tsukuba, 305-0047, Japan
2Graduate School of Science, Nagoya University,
Fro-cho, Chikusa-ku, Nagoya City 464-8602, Japan and
3Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
(Dated: February 2, 2008)
5
0 We discuss the relation between the T2-coefficient of electrical resistivity A and the T-linear
0 specific-heat coefficient γ for heavy-fermion systems with general N, where N is the degeneracy
2 of quasi-particles. A set of experimental data reveals that the Kadowaki-Woods relation; A/γ2 =
1×10−5µΩ(Kmol/mJ)2,collapsesremarkablyforlarge-N systems,althoughthisrelationhasbeen
n
a regardedtobecommonlyapplicabletotheFermi-liquids. Instead,basedontheFermi-liquidtheory
J we propose a new relation; A˜/γ˜2 = 1×10−5 with A˜ = A/21N(N −1) and γ˜ = γ/21N(N −1).
This new relation exhibits an excellent agreement with the data for whole the range of degenerate
1
heavy-fermions.
1

] PACSnumbers: 71.10.Ay,71.27.+a,75.30.Mb
l
e
-
r The Fermi-liquid theory [1] is the most fundamen- liquid theory has suggested [10] that the values of
st tal one to understand the electronic state of metal- A/γ2, so far considered to be unique and independent
t. lic systems. This theory has achieved a great success on materials, do depend on the number of degeneracy of
a
in describing not only the electronic properties of nor- quasi-particles N. For isolated atoms, N is defined as
m
mal metals but also unusual properties of the strongly- N = 2J + 1 with J the total angular momentum. In
- correlated electron systems [2, 3] like f-electron based solids, N can vary due to the competition between the
d
n heavy-fermion compounds [2, 3, 4, 5] and d-electron crystal-field splitting ∆ and the Kondo temperature TK.
o based intermetallics and oxides [2, 6]. In this theory, ForTK <∆,thelow-temperaturepropertiesarebasically
c theeffectofelectron-electroninteractionsareinvolvedin explainedby N = 2 (S = 1/2)Kondo model, since most
[ the effective mass of quasi-particles, m∗. This enables ofthe degeneracyare lostdue to the large ∆ [5, 11, 12].

1 a very simple representation of physical properties: the For TK > ∆, on the contrary, the crystal field splittings
v electronic specific heat C and the electrical resistivity ρ are covered by the large Kondo effect, and the degener-
7 are described as C = γT and ρ = AT2 with γ ∝ m∗ acy are almost preserved down to low temperatures. In
3 and A∝m∗2. Such a temperature dependence has been this case, the theory [10] gives the failed universality of
2 actuallyobservedinnumerouskindofmetals. Moreover, A/γ2.
1 this description implies that the ratio A/γ2 does not de- In this paper, we make a quantitative comparison of
0 pend on m∗, resulting in the universal value of A/γ2. the experimental data in ref.[8] and several recent works
5
0 In fact, it has been revealed that many f-electron based with the theoretical results of ref.[10]. The results dis-
/ systems show the universal behavior; A/γ2 = 1.0×10−5 playabeautifulagreementbetweenexperimentsandthe-
at µΩcm(Kmol/mJ)2 [7], called as the Kadowaki-Woods ory. Furthermore, we propose an advanced relation for
m (KW) relation. The KW-relation has therefore been ac- A and γ based on these results. Using A˜ and γ˜, the
- ceptedasthemostessentialrelationshowingthevalidity values of A and γ normalized by 1N(N −1), we show
2
d of the Fermi-liquid theory. thatthesetwovaluesofheavy-fermionsystemswithgen-
on Recently, however, significant and systematic devia- eral N are related by a very simple formula; A˜/γ˜2 =
c tions from the relation have been observed in many 1×10−5 µΩcm(Kmol/mJ)2. This new relation, namely,
: heavy-fermion compounds, in spite that they appar- the ‘grand-KW-relation’, will be an important waymark
v
i entlyshowFermi-liquidbehavioratlowtemperature [8]. for the research of strongly-correlated electron systems
X This class of compounds includes Yb-based compounds with degeneracies, and remarkably extends the validity
r like YbCu5, YbAgCu4, YbCuAl, YbNi2Ge2, YbInCu4, of the Fermi-liquid theory.
a
YbAl3, and Ce-based compounds like CeNi9Si4 [9] and At first, we briefly describe the theoretical results of
CeSn3. Notably, the deviations in these systems are al- ref. [10]. For the case of strong-coupling limit where
most ’universal’; A/γ2 ≈ 0.4×10−6µΩcm(Kmol/mJ)2. m∗/m ≫ 1 (m∗ and m being the mass of heavy quasi-
This systematic and large deviationcannotbe explained particles and free electrons, respectively), the orbitally-
by specific characters of materials, like carrier density, degenerate periodic Anderson (ODPA) model gives [13]:
band structure, anisotropy, etc. Instead, there seems to
existacommonphysicalorigin. The originofthis devia- hk2 3π6
A = B N(N −1)Γ2 (0,0)ρ4(0), (1)
tion therefore rises an important issue for the generality e2 2k4a3 loc f
F
of the Fermi-liquid theory.
π2
Very recently, a theoretical work based on the Fermi- γ = NAkB2 N(N −1)Γloc(0,0)ρ2f(0). (2)
6
2
where h is the Plank constant, kB the Boltzmann con-
stant, kF the Fermi momentum, and NA the Avogadro 102
number. In addition, a is the unit-cell length, and N = 2 UBe13
ρf(0) the density of states per f-orbit at the Fermi en- A/γ2 = 1×10-5 CCeCeAu6l
ergy. Γloc(0,0) represents the effective interaction be- µΩcm(K mol/mJ)2 3
YbRhSi
tween quasi-particles. Note that A and γ given in eq.(1) CeCuSi 2 2
and (2) are not simply proportional to N(N − 1), be- 101 N ≅ 8 2 2
cause Γloc(0,0) also depends on N. The value A/γ2 is N ≅ 6
then deduced as [14]: N ≅ 4 YbNi2B2C UPt3
N ≅ 2 YbRh2Si2 N=4
γA2 = e2kBh2NA2 · 9(3nπ4/23)a−31/3 12N(N1 −1) 100 5f Ce(RHu =2S 6iT2)CeB6 N=6
≈ 121N×(N10−−51) µΩcm(Kmol/mJ)2. (3) 2Ωcm/K)10-1 UPt2 UUUASPl2nt3 YbCuAl Y SbSmCmFuOe5s4P4S12b12
ForthecaseofN =2,thisformulagivestheKW-relation. µA ( Yb2Co3Ga9 Eu(Pt0.8Ni0.2)2Si2
ForgeneralN,thisgivesasetofuniversalrelations. This CePd3 UIn3 YbCu4.5Ag0.5
is shown in Figure 1 as the solid lines for N = 2, 4, 6, UGa3 Eu(Pt Ni )Si
and 8. -2 CeNi 0.75 0.252 2
10
In the figure, experimental data are also plotted af-
YbCuAg
YbInAu 4
ter ref.[7, 8, 9, 15, 16, 17, 18]. At first, one can see that 2
CeNiSi
9 4
manyheavy-fermionsystemssuchasCeCu6 orCeCu2Si2 YbNiGe
agree with the KW-relation; i.e., the theoretical predic- 2 2
tion for N = 2. This is consistent with the situation 10-3 CeSn3 N=8
TK < ∆, which results in the low degeneracy of N = 2 YbYAblInCu4 A/γ2 = 0.36×10-6
[19]. Moreover, it is clear that many Yb- and Ce-based YbAl2 3 µΩcm(mol K/mJ)2
systems,whichhavedeviatedfromtheKW-relation,well
agree with the theoretical predictions for N = 6-8. 10-4
It should also be noted that the A/γ2 of Eu- and Sm- 10 2 3 4 56 100 2 3 4 56 1000 2 3
based compounds agrees very well with the line for N γ (mJ/mol K2)
= 8 and N = 4, respectively. These Eu-compounds are
considered to be intermediate-valent between Eu2+(S =
7/2) and Eu3+(J = 0) [16]. The Fermi-liquid state of
FIG. 1: T2-coefficient of electrical resistivity A, vs. T-
themishenceconsideredtobeemergedoutofthedegen-
linear coefficient of specific heat γ of heavy-fermion systems
eracy N = 2S +1 = 8. For the two Sm-based systems,
with various degeneracy. Experimental data are taken from
the value of N = 4 has been assumed, since the lowest
ref.[7, 8, 9, 16, 17, 18]. The black line corresponds to
CEFlevelsareconsideredtobeaquartet[17,18]. These
the Kadowaki-Woods relation [7]. Other solid lines are the
quantitativeagreementsofA/γ2 withrespectivetheoret-
prediction from the orbitally-degenerate periodic-Anderson
ical lines evidences that A/γ2 of heavy-fermion systems model [10]. Colors of the symbols represent the degeneracy
arenotspecifictomaterials,butarecommonlyscaledby N probable for the systems; black, yellow, blue, and red in-
degeneracy. dicate N = 2, 4, 6 and 8, respectively. The value of N of
Inthefollowing,wegoforwardtounifytheserelations U-compoundsare not determined.
into a single relation. If the value of N is determined
experimentally, we candefine the normalizedcoefficients
A˜ and γ˜ from the eq.(1) and (2) as: the eq.(4) scales A˜ and γ˜ universally for a wide range of
materials. This fact shows the validity of the theoretical
A γ approachusingthe ODPAmodel,andextends the valid-
A˜= , γ˜ = .
1N(N −1) 1N(N −1) ity of the Fermi-liquid theory. We would like to stress
2 2
thatournew relationhas the sameformjust asthe orig-
Then A˜/γ˜2 is obtained from the eq.(3) as: inal KW-relation. The formula (4) may hence be called
as ‘grand-KW-relationfor general degeneracy’.
A˜/γ˜2 ≈1×10−5 µΩcm(Kmol/mJ)2. (4) Here it would be interesting to discuss to what extent
this rule holds when system is reachedtoward the quan-
ThisformuladoesnotincludeanyN-dependence. Hence, tum critical point (QCP). Even in the vicinity of the
this should be applicable to arbitrary-N systems. QCP,the Fermi-liquidstate isrealizedatsufficientlylow
In Figure 2, we plot A˜ and γ˜ of f-electron based sys- temperatures below a characteristic temperature (Tcoh,
tems. For uranium compounds, we have tentatively as- in literatures) as far as the system is in the magneti-
sumedN = 2,whichisdiscussedlater. One canseethat cally disorderedside. In this case, our theory yields that
3
stabilizetheFermi-liquidstate,becauseN-dependenceof
102 TK (∝ e−1/ρNJK) will be much prominent than that of
UBe13 CeAl3 TN(∝ N2JRKKY). A large mass-enhancement is realized
Ã/γ~ 2 = 1 × 10-5 YbRh2Si2 (0T) CeCu6 with relatively higher TK when N >2.
101 µΩcm(K mol/mJ)2 CeCu2Si2 theThfoerrmeushlaou(l3d),,otfhceoruartsieo,Aex/iγst2eaxscewpetlilonass.tAhastisosfeAe˜n/γ˜in2
UPt depend on the carrier concentration n, wave number at
3
Ce
100 Sm CeB6 YCbeRRhu2SSii2 (6T) athree eFxetrrmemi eenlyerdgiyffekrFen,tanfrdomsotoynp.icaIfl oonnees,ofA˜t/hγ˜e2secavnaludees-
2K ) 10-1 EYUub UInUPUt2AUlS2nU3PtSmYbONs4iS2Bb122C2 2 veroeqnial.ea(tt4eci)oar,nentmh[s7oae]ur.ekgaThtbhhClaiyste.PdCSdiuse3cPcrhwdeep3alanlsnahcegoyxrwaermseesspauwlleltiatsihrsfgrtCeohmedePeodtvrh3iia.egtiilnIoananrlgFKefirgWdo.me2--,
m / UGa33 SmFe4P12 generacy,N = 6 for CePd3 [25]. It should be noted that
Ω c 10-2 YbCuAl YbCu5 CePd3hasverysmallcarrier-concentration(0.3electrons
µÃ ( Yb2CCoe3PGda9 CeNiYbCu4E.5uA(Pgt0.5Ni )Si apserprf.oup.)or[t2i6o]n.alTthoenA−/4/γ32fvroamlueeqis.(3fo)uannddtaolsdoefpreonmdootnhenr
10-3 3 CeNi9Si4 0.8 0.22 2 theoretical studies [2, 20, 21, 27]. Taking this into con-
YbInAu YbCu4Ag Eu(Pt0.75Ni0.25)2Si2 sideration,thedeviationofCePd3fromtheuniversalline
2 isreasonable. SimilardeviationisreportedfortheKondo
YbNiGe
10-4 2 2 semiconductor CeNiSn [28]. Anomalously large A value
CeSn (54 µΩcm/K2) compared to its γ (40 mJ/mol K2) has
3
been attributed to its extremely low carrier concentra-
YbInCu
10-5 YbAl YbAl3 4 tion [28].
2 The compounds CeNi(N = 6) and YbCuAl(N = 8)
also show slight deviations, possibly due to the error in
10-6 the N estimations. For other exceptions, YbInAu2 and
10-1 100 101 102 103 104 Yb2Co3Ga9, we have no explanation for the origin of
γ~ ( mJ / mol K2) deviation. Other causes such as multi-Fermi-surface ef-
fect [9] may have to be considered. In addition, strong
anisotropyofthe Fermisurfacecancausedeviationfrom
the universal relation [2, 29]. This effect would be in
FIG.2: TheplotofA˜andγ˜ofheavy-fermionsystems. A˜and general more prominent in d-electron systems [30, 31].
γ˜arethedividedvaluesofAandγby 1N(N−1),respectively. For U-based compounds, its degeneracy has been the
2
N of the U-based compoundsis tentatively assumed to be 2. subject of arguments. If the 5f-electrons are well lo-
The dotted line represents the grand KW-relation (4) given calized, N can be determined experimentally, as in the
in thetext;A˜/γ˜2 = 1×10−5 µΩ(K mol/mJ)2. case of UPd3 [32]. In most of U-compounds, however,
itisconsideredthatthe5f-electronshavemore-itinerant
character than 4f, since 5f-orbitals are spacially more
the value of A˜/γ˜2 defined at the low temperature limit expanded. The definitionofN in U-compounds is there-
(T → 0), follows the relation (4) even in the vicinity of foreambiguous. Here,onecanseeinFig.1andFig.2that
QCP.Thisisbecauseourtheoryisderivedforthelimitof those U-compounds well agree with the theoretical pre-
T →0. Itshouldalsobenoticeabletopointthatthethe- dictionforN =2. Thiscanleadustothepossibilitythat
oreticalcalculationsofA/γ2forN =2basedonthespin- the orbital degree of freedom is quenched and only the
fluctuation theory show that the ratio is approximately spin degree of freedom participates in the Fermi-liquid
independent of the distance from the QCP [20, 21]. In state in these 5f-systems, similar to transition metals.
fact, A/γ2 of YbRh2Si2 and CeInCo5, both of which are Although the estimation of N from the A/γ2 plot is not
consideredtobeinthevicinityofQCP,arealmostacon- conclusive, this plot may serve as a hint to discuss the
stant as external magnetic field is varied [22, 23]. Mean- puzzling 5f-electrons.
while, forthe caseof‘very’close to the QCP,whereTcoh Inaddition,wenotethatthegrand-KW-relationisalso
is quite low, a deviation from the universality may be powerfultodescribethepressuredependentpropertiesof
observed, as is suggested theoretically [20] and experi- heavy-fermion systems. In CeCu2Ge2 (or YbNi2Ge2), it
mentallyonYbRh2(Si,Ge)2 [24]. Thisdeviationhowever issuggestedthatthevalueofA/γ2 reduces(orincreases)
seems to occur in an extremely narrow condition where about 25 times at high pressures [33, 34] probably due
the Fermi-liquid description is probably not valid. Ex- to the change of N by pressures. In our plot of A˜and γ˜,
ceptforsuchextremecases,thegrand-KWrelationisone these crossover would be described on the single scaling
ofthecommonbehaviorofFermiliquids,eveninsystems without breaking the universality. This situation may
closetotheQCP.Notethatthef-orbitaldegeneracywill be hence ideal for the continuity principle of the Landau
4
Fermi-liquid theory [35]. Current interests in strongly- quanta,like anomalousHalleffect [36],etc. The analysis
correlatedelectronsystemsareextendedtotheorbitally- using the ODPA modelwillbe henceforthindispensable.
degenerate cases. Hence, the grand-KW-relation will be
oneofthemostfundamentalrelationinFermi-liquidsys- Authors acknowledge K. Yamada, A. Mitsuda, Y.
tems. Wealsocommentthattheeffectofthedegeneracy Aoki, G. Kido and H. Kitazawa for fruitful discussions
must be taken into considerationin many other physical and comments.
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includeonly one rare-earth ion. hencemuchsmallerthaneq.(4).Here,notethatthethe-
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specificheatusingtheimpurity-Kondomodels.Although case ofthesemetals. Forsuchweak-couplingcases, devi-
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