Table Of ContentTheory of the thermoelectricity of intermetallic compounds with Ce or Yb ions
V. Zlati´c1 and R. Monnier2
1Institute of Physics, Bijeniˇcka cesta 46, P. O. Box 304, 10001 Zagreb, Croatia and
2ETH H¨onggerberg, Laboratorium fu¨r Festk¨orperphysik, 8093 Zu¨rich, Switzerland
ThethermoelectricpropertiesofintermetalliccompoundswithCeorYbionsareexplainedbythe
5
single-impurityAndersonmodelwhichtakesintoaccountthecrystal-fieldsplittingofthe4fground-
0
state multiplet, and assumes a strong Coulomb repulsion which restricts the numberof f electrons
0
2 or f holes to nf ≤ 1 for Ce and nhfole ≤1 for Yb ions. Using the non-crossing approximation and
imposing thecharge neutrality constraint on the local scattering problem at each temperature and
n
pressure, the excitation spectrum and the transport coefficients of the model are obtained. The
a
thermopower calculated in such a way exhibits all the characteristic features observed in Ce and
J
Yb intermetallics. Calculating the effect of pressure on various characteristic energy scales of the
1 model, we obtain the (T,p) phase diagram which agrees with the experimental data on CeRu2Si2,
2
CeCu2Si2,CePd2Si2, and similar compounds. The evolution of thethermopower and the electrical
resistance as a function of temperature, pressure or doping is explained in terms of the crossovers
]
l betweenvariousfixedpointsofthemodelandtheredistributionofthesingle-particlespectralweight
e within theFermi window.
-
r
t PACSnumbers: 75.30.Mb,72.15.Jf,62.50.+p,75.30.Kz,
s
.
t
a
I. INTRODUCTION not resolved at all.8,16,17,18
m
- The clue to these various types of behavior
d The thermoelectric power, S(T), of intermetallic com- comes from the high-pressure2,11,19,20,21 and doping
n pounds with Cerium and Ytterbium ions exhibits some studies,1,14,16,24,25,26,27,28 which show that the ther-
o
characteristic features which allow the classification of mopower of Cerium compounds changes continuously
c
[ thesecompounds. intoseveraldistinctgroups.1,2,3 Inthe fromtype (a) to type (d). A typicalexample is provided
case of Cerium ions, the thermopower of the compounds by the S(T) of CeRu2Ge2, which is plotted in Fig.1 as a
1
belonging to the first group (type (a) systems) has a function of temperature, for various pressures.11 At am-
v
deep negative minimum at low temperatures1,2,4,5,6 and bient pressure, CeRu Ge is a type (a) system with a
9 2 2
1 a high-temperature maximum, typically between 100 K magnetic ground state and negative thermopower below
5 and 300 K. At the maximum, S(T) could be either posi- 300 K. An increase of pressure leads to a thermopower
1 tiveornegative,asshowninFig. 1. Atverylowtempera- withasmallpositivepeakatlowtemperaturesandanen-
0 tures,thetype(a)systemsordermagneticallyorbecome hancedpeak at high temperatures. A further increaseof
5
superconducting. The compounds of the second group pressureenhancesbothpeaks,shiftsthelow-temperature
0
(type(b)systems)haveanegativelow-temperaturemin- peak towards the high-temperature one, and makes the
/
t imumanda positivehigh-temperaturemaximumbut, in thermopoweratintermediatetemperatureslessnegative.
a
m addition,thethermopowershowsasmallerpositivepeak For large enough pressure, the sign-change does not oc-
atlowesttemperatures.1,7,8,9,10Thissecondpeakissome- curatallandforveryhighpressurethelow-temperature
-
d times concealed by a low-temperature phase transition; peak merges with the high-temperature one, and trans-
n forexample,inCeCu Si itbecomesvisibleonlyinanex- forms into a shoulder (see inset to Fig. 1). The high-
2 2
o ternalmagneticfieldwhichsuppressesthesuperconduct- temperaturepeakgrowscontinuouslybutitspositionre-
c ing transition,4 and in CeRu Ge it shows up when the mains more or less constant, as S(T) changes from type
: 2 2
v external pressure suppresses the magnetic transition.11 (a) to (c). Eventually, for pressures above 10 GPa, the
i The experimental evidence is now accumulating that S(T) assumes the (d) shape. Here, the initial slope of
X
the initial slope of the thermopower S(T)/T is positive S(T)decreasesandthepositionofthemaximumshiftsto
r
a for this class of (heavy fermion) materials, provided the higher temperatures, but its magnitude does not change
measurementsareperformedatlowenoughtemperature as pressure increases. Similar behavior is also seen in
and with sufficient accuracy.6,7,8,12 In the third group thehigh-pressuredataof,CeCu Si ,21CeCu Ge ,20,22or
2 2 2 2
(type (c) systems), the low-temperature peak is well CePd Si .23 As regards doping, the substitutions which
2 2
pronounced and shifted towards the high-temperature reducethevolumeandmakeCeionslessmagnetic,trans-
peak. The main difference with respect to the type form S(T) from type (a) to type (b),24 from (b) to
(b) systems is that the sign-change of S(T) does not (c),14,16 or from (a) to(c),25,26,27 while the substitutions
occur.13,14,15,16 Finally, in some cases (type (d) systems) which expand the volume and make the Ce more mag-
the thermopower growsmonotonicallytowardsthe high- netic, transform the thermopower from, say, type (d) to
temperature maximum, and the low-temperature struc- type (c) or from type (c) to type (b).28 This variationof
ture appears only as a shoulder on a broad peak, or is shape is an indication that the local environment plays
2
80 or doping30,33 stabilizes the magnetic 4f13 configuration
p (GPa) CeRu Ge of Yb ions, and transforms S(T) from, say, type (b) to
2 2
8.0 S ^ c type (a), from (c) to (b), or from (c) to (a).
9.6
40 The experimental results show that 4f systems
60 10.4 with similar thermopowers exhibit similarities in other
15.6
K) thermodynamic35 and transport37,38,39 properties, and
T V/
K 20mS ( T thereisanobviouscorrelationbetweentheshapeofS(T)
S and the magnetic character of the 4f ions. The ther-
)
K 40 p (GPa) mopower measurements provide a simple and sensitive
/
V 0 7.0 tool for characterizing the magnetic state of a 4f ion
m( in a given metallic matrix: the shape of S(T) changes
S 3 10 100 300 from the (a)-type in the case of magnetic Ce (Yb) ions
T (K)
20 5.7 withstablef1(4f13)configurationtothe(d)-typefornon-
magnetic Ce (Yb) ions which fluctuate between the 4f1
(4f13) and 4f0 (4f14) configurations.
3.4
We explain the thermoelectric properties of Ce and
0 Yb ions in terms of a single-impurity Anderson model
which takes into account the splitting of the 4f states
0 by the crystalline electric field (CF), and assumes an in-
0.9 finitely large f–f Coulomb repulsion, which restricts the
2.1 number of f electrons or f holes to n ≤ 1 for Ce and
-20 f
10 100 300 nhfole ≤ 1 for Yb. We assume that pressure changes the
couplingandtherelativeoccupationofthefandconduc-
T (K)
tion states, and impose the charge-neutrality constraint
onthe localscattering problemat eachtemperature and
FIG.1: Temperaturedependenceofthethermoelectricpower pressure. The total charge conservationprovides a mini-
S(T) of CeRu2Ge2 for various pressures. TK and TS label malself-consistencyconditionforapoor-man’streatment
thecentreof broad, pressure-inducedmaxima, related to the of pressure effects in stoichiometric compounds. The ex-
Kondo effect and the crystalline electric field, respectively.
citation spectrum of such a model in the vicinity of var-
TheinsetshowsS(T)dataofCeRu2Ge2 inthenon-magnetic
ious fixed points, the crossoversinduced by temperature
phase.
andpressure,andthe correspondingeffects onS(T),the
numberoffparticles,n (T),andtheelectricalresistance,
f
ρ(T), are calculated by the non-crossing approximation
animportantrolein determining the magneticcharacter (NCA).Thedescriptionofthestoichiometriccompounds
of Ce and Yb ions. Even at high temperatures, where in terms of an impurity model is certainly inadequate at
each4fionisanindependentscatterer,thethermopower lowtemperatureswherethefelectronsbecomecoherent.
of a sample with a high concentration of 4f ions cannot The errors due to such an approximation and the low-
be obtained by rescaling the low-concentrationdata. temperature errorsinherent in the NCA calculations are
discussed in detail at the end of Sec. II.
The Ytterbium intermetallics can be classified using
the mirror-image analogy with Cerium systems. This Our paper extends the long-standing theory of
holds because the Yb ions fluctuate between 4f13 and Coqblin,41,42 which described the high-temperature
4f14, while the Ce ions fluctuate between 4f1 and 4f0 properties of Ce and Yb intermetallics by the Coqblin-
configurations, and the dynamics of a single f hole and Schrieffer (CS) model with CF splitting, and its more
a single f electron is the same. A well-defined local recentversion3 whichimprovedthe low-temperaturecal-
moment leads in Yb systems to the type (a) behav- culationsbyrescalingthecouplingconstants. Thesepre-
ior, such that the thermopower has a negative minimum vioustheoriesexplainedthemainfeaturesofthetemper-
at high temperatures and a positive maximum at low ature dependence of the thermopower3,42 and the mag-
temperatures;29,30 the size of the minimum is about the netic susceptibility35 but couldnotdescribe the pressure
same as the size of the maximum. The thermopower of effects, because the CS model neglects charge fluctua-
(b)-type Yb systems31,32,33 mirrors the (b)-type Ce sys- tions. Furthermore,theapproximationsusedtosolvethe
tems. Here, one finds two negative minima separated effective high- and low-temperature models cease to be
by a small positive maximum. The type (c) Yb systems valid at temperatures at which S(T) changessign,3 such
have a nonmonotonic thermopower with a large (nega- that the shape of S(T) between the two maxima (min-
tive)minimumathightemperaturesandasmalleroneat ima) in Ce (Yb) systems could only be inferred from an
low temperatures, but there is no sign-change.30,31,32,33 interpolation.
Finally,thethermopowerwithasinglenegativepeakcen- Here, we consider both the local spin and charge fluc-
teredaround100K18,30,31,34 mirrorsthe type (d)behav- tuations, and provide the full description of the im-
ior of Ce systems. The reduction of volume by pressure purity problem at various pressures and temperatures,
3
from above the CF temperature to below the Kondo of electrons between 4f and conduction states. In the
temperature, including the intermediate regime where absence of mixing, the conduction band is described by
S(T) changes sign. Our results explain the shapes a semielliptical density of states, N(ǫ), centered at E0
c
(a) to (d) of the thermopower, which are found in and of half-width W, and the unrenormalized f states
the systems like CeAl ,9 CeRu Ge ,11,40 CeCu Si ,21 are represented by a set of delta functions at E0 and
3 2 2 2 2 f
CeCu2Ge2,20,22 or CePd2Si2,23 and the ’mirror-image’ Efi = Ef0 +∆i. The conduction states and the f states
shapes found in systems like YbNiSn, YnInAu2, YbSi haveacommonchemicalpotential,whichistakenasthe
or YbCu2Si2.30 We also explain the pressure data like originoftheenergyaxis. Thepropertiesofthemodelde-
on CeRu2Ge211 or CeCu2Si2,21 and the chemical pres- pend inanessentialwayonthe CF splittings andonthe
suredataonCexLa1−xPdSn27 andCexLa1−xRu2Si215 or coupling constant g = Γ/π|Ef|, where Γ = πV2N(Ec0)
YbCu2Si2.33 measures the coupling strength between the f electrons
The paper is organized as follows. In Sec. II we in- and the conduction band, E = N−1N Ei/N, and
f i=0 i f
troduce the model, discuss its limitations, and describe N = N−1N is the total degePneracy. We assume
the method of solution. In Sec. III we provide the re- i=0 i
sults for the transport coefficients of Ce- and Yb-based Ec0 > 0P, Ef < 0, and Γ,∆i ≪ |Ef| ≃ W, i.e., g ≪ 1.
Since a single f hole is dynamically equivalentto a single
intermetallics. In Sec. IV we discuss the effects of tem-
f electron, we obtain the results for Yb ions by perform-
perature and pressure on the spectral function, analyze
ing the model calculations for a 8-fold degenerate f hole
the fixed-point behavior, and relate the shapes of the
subject to the appropriate CF.
thermopowertothepropertiesofelementaryexcitations.
Sec. V, gives the summary and the conclusions. Theg ≪1limitoftheAndersonmodeliscontrolledby
several fixed points which are well understood.43 In the
case of a Ce ion with two CF levels split by ∆, the fixed
II. THEORETICAL DESCRIPTION
point analysis can be summarized as follows. At small
coupling, such that Γ < ∆ ≪ |E |, we find n (T) ≃ 1
f f
We model the intermetallic compounds by taking as andthe modelexhibits the Kondoeffect. Thatis,allthe
many Ce or Yb ions per unit cell as required by the physical properties depend only on the Kondo tempera-
structure, but assuming that the scattering of conduc- ture, T0, which is uniquely determined by g, ∆, and the
tionelectronsonagiven4fiondoesnotdependonother degeneracies of the CF states (for the NCA definition of
4f ions, except through the modification of the chemi- T0 see Ref. 43 and Sec.IV below). The low-temperature
cal potential. In other words, we consider an effective behavior is characterized by the Fermi liquid (FL) fixed
impurity model which treats the 4f states as scattering point, which describes a singlet formed by an antifer-
resonancesratherthanBlochstatesbuttakeintoaccount romagnetically coupled f electron and conduction elec-
thechargetransferduetothelocalscatteringandadjust tron. An increase of temperature breaks the singlet and
the chemical potential, µ, so as to maintain the overall gives rise, at about T0, to a transition to the local mo-
charge neutrality of the compound. Such a description ment(LM)fixedpoint,whichdescribesaCF-splitfstate
of the lattice problem applies at temperatures at which weaklycoupledtotheconductionband. Theeffectivede-
the mean free path of the conduction electrons is short generacyof this f state is defined by the lowestCF level.
andthe scatteringis incoherent. We considermainlythe For T > ∆, there is a further crossover to another LM
Ce intermetallics and present only a few preliminary re- fixed point, which describes the scattering of conduction
sults for Yb intermetallics. The Cerium ions are allowed electrons on a fully degenerate local moment. At higher
to fluctuate between the 4f0 and 4f1 configurations by coupling, such that Γ ≃ 2∆ ≪ |E |, the f charge is re-
f
exchanging electrons with the conduction band; the (av- duced to 0.8 < n (T) < 0.95, and the impurity still
f
erage) energy difference between the two configurations behaves as a Kondo ion, but the Kondo scale is much
is |E | and the hopping is characterized by the matrix higher than in the case Γ < ∆ ≪ |E |. The two LM
f f
element V. The 4f2 configurationis excluded, i.e., anin- regimes are now close together and the crossover from
finitely strong Coulomb repulsion U between f electrons the FL to the LM regime occurs at temperatures which
is assumed. The 4f1 configuration is represented by N are comparable to ∆. At very high coupling, such that
crystal field levels: there are N −1 excited states sepa- 2∆≤Γ≪|E | andn (T)≤0.8,the f ions appear to be
f f
rated from the CF ground level by energies ∆ ≪ |E |, non-magnetic at all accessible temperatures due to the
i f
where i = 1,...,N −1. The local symmetry is taken mixing of the 4f0 and 4f1 configurations. In this valence
into account by specifying the respective degeneracies of fluctuating (VF) regime, the behavior is non-universal
these levels, N . Thus, the low-energy excitations of Ce andchangesslightly,whenthecalculationsareperformed
i
intermetallicsaremodeledbyaneffectivesingle-impurity fordifferentsetsofparameters. Awayfromn ≃1,more
f
Anderson Hamiltonian,43 than one energy scale is needed to fully characterize the
model. Other CF schemes pertinent to Ce and Yb ions
HA =Hband+Himp+Hmix, (1) in a different environment are characterized by similar
fixed points.
where H describes the conduction band, H de-
band imp
scribes the CF states, and H describes the transfer Our calculations show that the functional form of the
mix
4
response functions changes atthe crossoverand that the
g ≪1limit ofthe Andersonmodelcaptures allthe main
1 L
features of the experimental results on Ce and Yb inter- S =− 12, (3)
|e|T L
metallics. To explain the pressure effects, which changes 11
thethermopowerofCeandYbsystemsinoppositeways,
where L and L are given by the static limits of
11 12
we assume that the exchange coupling g increases in Ce
the current-current and current-heat current correlation
and diminishes in Yb compounds, as pressure increases.
functions, respectively. In the absence of nonresonant
This difference arises because Ce fluctuates between 4f0
scattering the vertex corrections vanish and the trans-
and 4f1, while Yb fluctuates between 4f14 and 4f13 con- port integrals can be written as,43,45,48
figurations,sothatthepressure-inducedreductionofthe
∞
numberofelectronsinthef-shellmakesCeionslessmag- σ df(ω)
L = 0 dω − τ(ω)ωi+j−2, (4)
netic and Yb ions more magnetic. ij e2 Z−∞ (cid:18) dω (cid:19)
In Ce intermetallics, there is a substantial overlap be-
tween the f wave functions of Ce and those of the neigh- where σ0 is material-specific constant, f(ω) = 1/[1 +
boring atoms, and we associate the pressure-induced exp(ω/kBT)] is the Fermi function, 1/τ(ω) is the
increase of g ≃ Γ/|E | with an enhancement of the conduction-electron scattering rate,43
f
hybridization Γ. This enhances the Kondo tempera-
1
ture, and pushes the system from the Kondo to the =cNπV2A(ω), (5)
τ(ω)
VF limit. In stoichiometric compounds, the pressure-
induced reduction of nf is accompanied by the increase A(ω) = ∓1Im G (ω ± i0+) is the f-electron spectral
of n , because the total charge of a given compound, π f
c function, G (z)isthe Green’sfunction, andcis the con-
f
n =n +n ,isconstant. Theconservationofparticles
tot c f centration of f ions. Eqs.(4) and (5) show clearly that
is enforced by adjusting µ, and since all the energies are
the sign and the magnitude of S(T) are determined by
measured with respect to µ, this amounts to shifting E
f thespectralweightwithintheFermiwindow(Fwindow),
and E by some amount δµ(T,Γ(p)). Thus, we describe
c i.e., by the shape of A(ω) for |ω| ≤ 2k T. The sign of
B
the pressure effects for a given Ce compound by chang-
S(T)ispositiveiftheFwindowshowsmorestatesabove
ing Γ and keeping E −E constant. The changesin the
c f thanbelow the chemicalpotential,andis negativeinthe
band-width and the CF splitting are neglected. opposite case. The difficult part is to find G (ω±i0+)
f
In Yb intermetallics, the f states are more localized and, here, we solve this problem by the NCA, following
than in Ce systems, and we assume that the decrease in closely Refs.43,47, where all the technical details can be
the radius of the 4f shell as it loses charge at elevated found. The main difference with respect to these NCA
pressure is sufficient to compensate for the increase in calculationsis that we take c=1 andenforce the overall
hybridizationbroughtaboutbythereductioninunitcell charge neutrality.
volume. The reduction of g ≃ Γ/|Ef| in Yb compounds A detailed comparison with the experimental data
is achieved through an enhancement of the hole binding shows that the transport coefficients obtained from the
energyEf asthe neighbouringions getcloserto the rare single-impurity Anderson model have all the hallmarks
earth,whileΓremainsessentiallyconstant. Thisreduces ofthe experiments,butdiscrepanciesappearatlowtem-
the Kondo temperature, and drives the system towards peratures. Thisindicatesthelimitationsofourapproach
the Kondolimit. Since Γ istreatedasa material-specific which should be considered before presenting the NCA
constant,wemodelthepressureeffectsinYbsystemsby results.
shifting Ef and solving for Ec, so as to preserve nhtootle. There are two main causes for the breakdown of the
ThisprocedureshiftsEf andEcbydifferentamountsand single-impurity model and the NCA calculations. First,
makesthe separationEc−Ef pressuredependent. How- at temperatures much below T0 the NCA spectral func-
ever, when temperature is changedat constant pressure, tion develops anunphysical spike,such that the resistiv-
the charge neutrality is enforced in the same way as for ity and the thermopower become artificially enhanced.
Ce compounds, by shifting the chemical potential with- This error becomes particularly severe at high pressure,
outchangingtheseparationEc−Ef. Thechangesinthe because the characteristicscaleT0 increasesveryrapidly
bandwidth and the CF splitting are neglected. Describ- withΓandthenon-analyticNCAspikeappearsatrather
ing the pressure effects in such a way, we can calculate high temperatures. The unphysical enhancement of the
the response functions of the model for any value of the low-frequency part of A(ω) reduces the integral for L ,
11
external parameters, and study the transitions between which is strongly underestimated at low-temperature.
various fixed points. The integral L is less affected by this pathology, be-
12
The electrical resistivity and the thermopower of the cause it has an additional ω factor which removes the
single-impurity Anderson model are obtained from the states within the Fermi window. Thus, the overallshape
usual expressions,44 of S(T) ≃ LNCA/LNCA seems to be qualitatively cor-
12 11
rect, even though the low-temperature part of the curve
1 has an unphysical enhancement. These difficulties are
ρmag = e2L , (2) wellknown43 andrelativelyeasytoresolveintheKondo
11
5
limit,wherethemodelhasauniquelowenergyscale,T . 100
0
We can find T in the LM regime, where the NCA is re- G (meV)
0
liable, and infer the low-temperature behavior from the
120
universalpowerlawswhichholdinthe FLregime. Com-
bining the NCA results and FL theory, we can discuss
the experimental data at all temperatures at which the 110
single-ion approximation holds.
50
Amoreseriousproblemisthat, instoichiometriccom-
100
pounds, the f electrons become coherent at low enough
temperatures. This leads to a magnetic transition in
)
CeRu Ge ,theformationofaheavyFLinCeRu Si ,su- K
2 2 2 2 / 90
V
perconductivity in CeCu2Si2, or some more complicated
ground states. At high pressure, the coherence sets in m (
S
at very high temperatures, as revealed by low values of 80
0
the electricalresistance. Theonsetofcoherence(likethe
NCA pathology) has its main impact on the low-energy
70
states, giving Llattice ≫ Limpurity, so that the impurity
11 11 60
result badly overestimates the low-temperature electri-
cal resistance. However, the considerations for the pe-
riodic Anderson model48, or other models with on-site (a)
correlation49, show that the integral for L also con-
12 -50
tains an additional ω factor which reduces the contri-
10 100 800
bution of the low-energy coherent states to L , like in
12
the single impurity case. Thus, our results for S(T) pro- T (K)
videaqualitativedescriptionoftheexperimentaldataat
temperatures well below the onset of coherence, but the FIG. 2: Thermopower of Ce ions calculated by the NCA for
calculated values of S(T) around T are overestimated. the CF splitting ∆ = 0.07 eV is plotted as a function of
0
Wementionalsothattheanalysisofthedopingeffects temperature for several values of the hybridization strength
in terms of ”chemicalpressure”is not complete, because Γ<2∆,asindicatedinthefigure. Thebottomcurve,Γ=0.06
eV,corresponds to ambient pressure.
doping might give rise to a charge transfer or change
the character of the ground state, and that the mirror-
image analogybetween Ce and Yb systems holds for the
resonant scattering but may be lost in the presence of for Γ = 0. (In the absence of coupling we have n = 1
f
any additional scattering channel. Despite these draw- and obtain n by integrating the unperturbed density of
c
backs,theNCAsolutionoftheAndersonmodelprovides states.) At high temperatures, the f-state is almost de-
asurprisinglyaccuratedescriptionforalargebodyofex- coupledfromtheconductionband,therenormalizationof
perimental data above the magnetic or superconducting the parameters is small, and the numerics converge very
transitiontemperature,andatlowtomoderatepressure. fast. We thenreducethe temperature,find the new shift
δµ(T,Γ) ensuring the charge conservation, and calculate
the response functions for the resulting values of E and
c
III. THERMOELECTRIC POWER RESULTS E . This process is continued until the NCA equations
f
breakdownatT ≪T . Tomodelthesamesystematdif-
0
In this section we present the NCA results for the ferent pressure, we change Γ, find again δµ(T∆,Γ), and
transportcoefficients, describe the thermopowerand the repeat the same procedure as at ambient pressure for
electrical resistance due to Ce ions in some detail, show T < T∆. Note, ntot and Ec −Ef are conserved at all
the results for the f occupation, present a few prelim- temperatures and pressures.
inary results for the thermopower of Yb, and compare As a numerical example we consider a semielliptical
our results with the experimental data. conduction band of half-width W = 4 eV, centered at
Our strategy for Ce intermetallics is to illustrate the E0 = 0.7 eV, and a f state split into a doublet and a
c
behavior of one particular compound as a function of quartet46 by the CF with ∆ = 0.07 eV. We take n =
tot
temperatureatvariouspressures. Thecompoundischar- n +n =5.6301 electrons per ion (0.9383 electrons for
c f
acterized at ambient pressure and high temperature by each one of N ’effective spin’ channels). The transport
an initial parameter set {W,E0,N,Γ,E0,∆ ,N }, where coefficients are calculated for the hybridization strength
c f i i
E0 > 0 and E0 < 0 are measured with respect to µ changing from 0.06 eV to 0.20 eV, i.e., for Γ varying
c f
and the high-temperature limit is defined by tempera- from Γ < ∆ to Γ > 2∆. The single-particle excitation
ture kBT∆ =∆N−1. For given values of Γ and Ec0−Ef0, spectra corresponding to these parameters are discussed
we start the calculations at T = T and find δµ(T ,Γ) in Sec. IV.
∆ ∆
suchthatthetotalchargeisthesameastheoneobtained UsingtheprocedureoutlinedaboveweobtainforS(T)
6
120 1
120
)
K
80V/ 0.9
40mS (max
80 n(T)f0.8
0
) 50 100 150 200
K G (meV) 0.7
/
V
m(
S G (eV)
0.6
40 0 200 400 600
0.13
T [K]
FIG. 4: f-electron number, n , calculated by the NCA for
f
the CF splitting ∆ = 0.07 eV is plotted as a function of
temperature for several values of the hybridisation strength
0.2 (b)
Γ. For theuppermost curveΓ=0.06 eV and then it increases
in steps of 0.02 eV. Atthe bottom curveΓ=0.20 eV.
0
10 100 800
T (K)
inwhichS(T)is positive. Suchabehavior,whichis typ-
ical of Kondo systems with small T , is in a qualitative
0
FIG.3: ThermopowercalculatedbytheNCAfortheCFsplit- agreementwiththethermopowerofthetype(a)systems
ting ∆=0.07 eV is plotted as a function of temperature for described in Sec.I, and with the data on CeRu Ge 11
2 2
severallargevaluesofthehybridisationstrengthΓ,increasing at low pressures (below 4 GPa) and above the ordering
from 0.13 eV to0.2 eV.The bottom curve,Γ=0.20 eV,isfor
temperature, as shown in Fig.1.
the highest pressure. The inset shows the high-temperature
For ∆ < Γ < 2∆, we have T <150 K and n ≥ 0.8,
maximum of S(T),plotted as a function of Γ. 0 f
and still obtain S(T) with the two maxima. But S
max
is now positive, the value of S is enhanced, the temper-
0
ature interval in which S(T) < 0 is reduced, and S(T)
theresultsshowninFigs.2and 3. Thecalculatedcurves at the minimum is less negative than for smaller Γ. As
exhibitalltheshapes(a)to(d)foundintheexperimentas we increase Γ (by increasing pressure), the sign-change
andgiveS(T)oftherightmagnitude,exceptatlowtem- of S(T) is removed, S and S are further enhanced,
0 max
peratures where the calculatedpeak is too largewith re- but T is not changed. The two peaks are coming closer
S
specttotheexperimentaldata. Asdiscussedalready,the together and are merging eventually. These features are
reasonforthisdiscrepancyisthattheNCAoverestimates typicalof Kondosystems with moderate T , say T ≥10
0 0
the Fermi-level scattering rate for T ≪ T0, and that we K, and are in a qualitative agreement with the data on
neglected the coherent scattering, which sets in at tem- the type (b) and (c) systems mentioned in Sec. I. They
peratures of the order of T0. Thus, our low-temperature are also shown by the CeRu2Ge2 data11 at intermediate
result for L11 is artificially reduced, which makes S(T) pressures,(see the curvesinFig.1 forpressureabove3.4
too large. The sign and the topology of the S(T) curves GPa.)
do not seem to be affected by this error.
At Γ ≈ 2∆, the system enters the VF regime, S(T)
For Γ ≤ ∆, we have T < 5 K and n ≥ 0.95, and acquiresasinglemaximumatT ,withashoulderonthe
0 f S
obtain S(T) with two well separated peaks, as shown low-temperatureside. Eventhisshouldervanishes,when
by the Γ=60 and Γ=70 meV curves in Fig. 2. The Γisincreasedfurther,asshowninFig.3. ForΓ>2∆the
high-temperature peak is centered at T ≃ T /2 and thermopoweris ofthe type (d), witha single peakwhich
S ∆
for our choice of parameters S = S(T ) < 0. The ismuchsteeperonthehigh-thanonthelow-temperature
max S
low-temperature maximum is at about T ≪ T and side. This peak shifts to higher temperatures with in-
0 S
S = S(T ) > 0. The thermopower between the two creasing Γ (pressure) and S saturates (see inset to
0 0 max
maxima is mainly negative. Since most of the type (a) Fig. 3); the initial slope of S(T) decreases continuously
and(b) systemsordermagneticallyorbecome supercon- with Γ. Such a behavior is in a qualitative agreement
ductingaboveT ,thelow-temperaturepeakisnotshown with the thermopowerdata onvalence fluctuators16,17,18
0
inFig.2 forΓ≤100meV.AsmallincreaseofΓ(due to, and with the high-pressure data on CeRu Ge 11 (cf. in-
2 2
say, an increase of pressure) reduces n (see Fig. 4), en- set to Fig. 1) and CeCu Si 5. However, a large dis-
f 2 2
hancesT andS ,andexpandsthetemperaturerange crepancy appearsbetween theory and experiment at low
0 max
7
5
temperatures, because A(ω) is overestimated for ω ≃ 0,
whichmakestheNCAcurveslargerthanthe experimen-
tal ones. A possible correctionof the initial S(T)/T val-
G (meV)
ues is discussed below.
The f-electron number n , calculated for the param- 4 100
f
eters used in Figs. 2 and 3, is plotted in Fig. 4 as a
functionoftemperature. Theoveralltemperaturedepen-
90
dence is rather slow, but two different types of behavior
can still be seen. For Γ<2∆≪−Ef, we find that nf is s) 3
nearly independent of temperature and close to 1, which t
ni
is typical of Kondo systems.36 For Γ>2∆, we find that u 80
n is less than 0.8 and nearly constant at low tempera- .
f b
ttuhreerse bisutanatinacbroeuatseTfo≃llo∆w/ed3kbBy(2th7e1 sKatfuorrat∆io=n0.a0t7heiVgh) (ar 2
g
temperatures. Considered as a function of Γ (pressure), ma 70
nf(Γ) shows different behavior at high and low temper- r
atures. At high temperatures nf decreases uniformly as 60
Γ increases. At low temperatures n does not change 1
f
much for Γ ≪ ∆ and Γ ≫ ∆, but drops rapidly around
Γ≃ 2∆, indicating the crossover from the Kondo to the
VF regime.
The electrical resistivity, ρ (T), obtained for small 0
mag
and intermediate values of Γ, is shown in Fig. 5. The 10 100 800
interesting feature is the high-temperature maximum,
T (K)
which appears for Γ < ∆, and correlates very well
with the maximum in S(T). For temperatures below
the maximum, ρ (T) drops to a minimum and then FIG. 5: Electrical resistivity vs. temperature calculated by
mag
the NCA for the CF splitting ∆ = 0.07 eV and for several
rises logarithmically as T is approached. This mini-
0
values of the hybridization strength Γ, as indicated in the
mum and the subsequent low-temperature upturn are of
figure.
a purely electronic origin and appear in systems with
small T and large CF splitting. In these systems, one
0
can follow the evolution of the two peaks in ρ (T),
mag the initial parameter set {W,E0,N,Γ,E0,∆ ,N }. At
and observe the disappearance of the minimum with the c f i i
application of pressure. For example, in CeRu2Ge2,11 lowertemperaturesweshiftEcandEf withrespecttoEc0
CePd2Ge2,37 CePd2Si2,38 or CeCu5Au39 the minimum and Ef0 by δµ, so as to conserve nhtootle, and calculate the
becomes more shallow, transforms into a shoulder, and responsefunctionsforthisnewparameterset;thisproce-
vanishes at high enough pressure. As discussed already, dure does not change Ec−Ef. At a higher pressure, we
the NCA overestimates the low-temperature scattering startagain at T =T∆, change Ef so as to increase nhfole
anddistortstherelativemagnitude ofthe high-andlow- andfindE whichconservesnhole. SinceΓisnotchanged
c tot
temperature peaks. In addition, the single-ion approx- by this procedure,we now haveE −E 6=E0−E0. For
c f c f
imation always gives ρ (T) which saturates at low temperatures below T , the properties of the systemare
mag ∆
temperature andcannotexplainthe low-temperaturere- calculatedbythesameprocedureasatambientpressure,
duction of ρ (T), which is seen in stoichiometric com- i.e.,nhole isconservedbyshiftingE andE bythesame
mag tot c f
poundsbelowtheonsetofcoherence. Theelectricalresis- amount δµ.
tance of Ce-based Kondo systems at very high pressure, Taking W = 4 eV, E0 = 1.0 eV, Γ=0.08 eV, E0 =
c f
andtheambientpressuredataofvalencefluctuators,can- −0.8 eV, and three excited CF doublets at ∆ =0.02 eV,
1
notbe describedby the NCAsolution. Inthese systems, ∆ =0.04 eV, ∆ =0.08 eV, respectively, we find n =
2 3 tot
the scattering on f ions remains coherent up to rather 6.444 at T = T . The S(T) obtained for E ranging
∆ f
high temperatures and the NCA solution is valid only from E = −0.6 eV to E = −1.0 eV, is shown in Fig.
f f
above the high-temperature maximum and is not physi- 6. Werecall,thatanincreaseofpressuremakesE more
f
cally relevant. negative. For E = −0.6 eV, the thermopower shows a
f
ToillustratethesituationinYbintermetallics,weper- deep minimum, typical of Yb ions in the VF state. For
form a generic calculationfor a semielliptical (hole) con- E = −0.8 eV, S(T) develops a small maximum, which
f
ductionbandandfourCFdoublets. TheNCAequations separates the high-temperature minimum at T and the
S
are solved for a single f hole (N = 8). Following the low-temperature minimum at T . By making E more
0 f
procedure outlined in Sec. II, we start the calculations negativeweshiftT tolowervalues,asshowninFig. 6by
0
atambientpressureandtemperatureT ,suchthatthe f the curves obtained for E = −0.8 eV, E = −0.85 eV,
∆ f f
stateisalmostfree,andcalculatenhole =nhole+nholefor and E = −0.9 eV, respectively. The low-temperature
tot c f f
8
values, which go from 32 µV/K2 for Γ=60 meV to 1
µV/K2 for Γ=130 meV and 0.1 µV/K2 for Γ=200 meV,
0 are in the range reported recently for various Ce-based
heavyfermionsandvalencefluctuators(seeTable1inref.
12). The initial slope of S(T) decreases as we move from
K) the Kondo to the VF limit, in agreement with pressure
V/ experiments.11 At higher temperatures, the non-linear
-50
mS( corrections reduce S(T) and give rise to a maximum at
T . These non-linear corrections are non-universal, and
0
alargeslopedoesnotnecessarytranslateintolargeS(T)
at T . The corresponding calculations for the Yb curves
0
-100 plotted in Fig. 6 show that an application of pressure
enhances S(T)/T.
0 200 400 600 The initial slope of S(T) can also be related to the γ
T [K]
coefficient of the specific heat, with the important result
that the ratio S(T)/γT = (2π/|e|N )cot(πn /N ) is in-
FIG. 6: Thermopower dueto Yb ions, obtained by the NCA 0 f 0
dependent of Z. This expression and Eq.(6) are valid
for four CF doublets, is plotted as a function of temperature
for various values of E0, i.e. for various pressures. Starting in the single-impurity regime, and it is not obvious that
f
fromtheuppermostcurve,weshowS(T)for−E0=1.00;0.95; they wouldproducethe correctresultsforstoichiometric
f
0.90; 0.85, 0.80, 0.70, and 0.60 eV, respectively. The bottom compounds. However, the characteristic energy scales of
curve(E =-0.6 eV) corresponds to thelowest pressure. a coherent FL (inferred from the experimental data on
f
the initial slope of the thermopower, or the specific heat
coefficient) do not seem to be much different from the
rangeinwhichS(T)isnegativeshrinkswithpressure,in single-ion scale T of the LM regime (inferred, say, from
0
agreementwiththeexperimentaldata.30However,forEf the peakinthe thermopowerorthe Curie-Weisstemper-
much below µ, the NCA calculations break down before ature).
thisminimumisreached. AsregardsthevalueofS(T)at
the maximum, it is negative at first but it becomes posi-
tive as pressure increases, i.e., the thermopower changes IV. DISCUSSION OF SPECTRAL PROPERTIES
from (c)-type to (b)-type. At very high pressures, such
that the nhole ≃ 1, the thermopower is dominated at
tot In this section we present the NCA results for A(ω),
low temperatures by a large positive peak and at high
study the low-energy spectral features in the vicinity of
temperaturesbyanegativeminimum,whichistypicalof
various fixed points, discuss the changes induced by the
Yb-based systems with a small Kondo scale. The shape
crossovers, and explain the behavior of S(T) in terms
of S(T) is directly related to the magnetic character of
of the redistribution of spectral weight within the Fermi
Yb ions and our calculations explain the qualitative fea-
turesofthethermopowerofYbAu ,YbAu ,29 andYbSi, window (F window). Only the Ce case is consideredand
2 3
YnNi Si ,30 which are of the (a)-type, of Yb Ir Al ,32 itisassumedthatpressuregivesrisetoanincreaseofthe
2 2 2 3 9
YbAuCu ,31 andYbNiSn,30 whichareofthe(b)-type,of hybridization width Γ. The results obtained for Γ < ∆
4
YbPdCu ,31andYb Rh Al ,32whichareofthe(c)-type, are shown in Fig. 7, where A(ω) is plotted as a function
4 2 3 9
andofYbAgCu ,YbPd Cu ,31andYbInAu 30whichare of frequency for several temperatures. At high tempera-
4 2 2 2
(d)-type. The pressure effects in YbSi,30 and the chemi- tures, T ≃T∆, the spectralfunction has a broadcharge-
calpressureeffectsinYbCu Si 33arealsoinaqualitative excitationpeaksomewhataboveEf andanarrowerreso-
2 2
nanceofhalf-width∆,centeredbelowµ. Thislow-energy
agreement with our results.
resonanceisamany-bodyeffectdue tothe hybridization
An estimate of the low-temperature properties of the
of the conduction states with the 4f states and is typi-
single-impurity model canbe obtained by combining the
cal of the exchange scattering on the full multiplet. In
NCAresultswiththeuniversalFLlaws. TheSommerfeld
expansion of transport coefficients in Eq.(4) gives45 this temperature range, the F window shows more spec-
tral weight below than above µ (see the middle panel in
ΓS(T) π2k 2π Fig. 7) and S(T) < 0. The magnetic susceptibility43 is
B
lim = sin( n )Z, (6)
f Curie-Weiss like, with a very small Curie-Weiss temper-
T→0 kBT 3|e| N0
ature and a Curie constant which is close to the free Ce
where π2k /3|e|=283.5 µV/K and Z is the enhance- ion value. The maximum of S(T) at about T ≃ T /2
B S ∆
ment factor defined by the Fermi-level derivative of the is here negative, S < 0, but a slight increase of Γ
max
f-electron self-energy, Z = [1−∂Σ/∂ω] . Γ/Z is wouldmake S positive. All these features are typical
ω=T=0 max
related to the Kondo scale by a factor of order 1. Set- oftheLMfixedpointcorrespondingtoafullydegenerate
ting T =Γ/Z,andusingfor n (T =0)the NCAresults f-state. At lower temperatures, T < T , the CF splits
0 f S
of Fig. 4, we can estimate from Eq.(6) the initial slope the many-body resonance into two peaks. The larger
of the curves plotted in Figs. 2 and 3. The obtained one grows below µ and the smaller one above µ (see the
9
middle panel in Fig. 7). This asymmetry is enhanced
1.5
as Γ is reduced, which is typical of the Anderson model
withCFsplittings;43 theincreaseofthelow-energyspec-
tral weight below µ gives rise to a large negative ther-
mopower. A further reduction of temperature leads, for
1 T=2 K T ≪ T , to a rapid growth of an additional peak very
T=41 K ∆
A(w) T=209 K closetoµ,suchthatA(ω)acquiresthreepronouncedlow-
f T=670 K
energy peaks (see Fig. 7). (The physical origin of these
many-body resonances is explained in detail in Ref.43)
0.5 The peak centered at ω ≪ ∆ is the Kondo resonance
0
and its appearance below T ≤ 40 K marks the onset of
the LM regime, which is due to scattering of conduction
electronsonthe lowestCF level. The two CFpeaks cen-
0 tered at about ω ±∆ are outside the F window, and
-1.4 -0.7 0 0.7 0
w [eV] do not affect the low-temperature transport and ther-
modynamics. Once the Kondo peak peak appears, the
1.5
F window shows more spectral weight above than be-
low µ and S(T) is positive, which is just the opposite to
what one finds for T ≥ T . The center of the Kondo
∆
resonance saturates at low temperatures at the energy
1
ω >0, whichprovidesthe NCA definition ofthe Kondo
0
A(w) scale, kBT0 = ω0. In the symmetric Anderson model
f
T is related to the width of the Kondo resonance but
0
in the highly asymmetric case we are dealing with here,
0.5 the currentdefinitionismoreappropriate. The compari-
son with numerical renormalization group (NRG) calcu-
lations50 shows that ω gives a reliable estimate of the
0
KondotemperatureevenforadoublydegenerateAnder-
0 sonmodel,andwe assumethatthe NCAdefinitionofT
-0.14 -0.07 0 0.07 0.14 0
w [eV] provides the correct Kondo scale of the CF-split single-
impurity Anderson model as well. Because the Kondo
1.5
resonance is asymmetric with respect to the ω = 0 line
and has more states above than below µ, the reduction
of temperature enhances S(T) unltil it reaches, at T ,
0
the maximum value S . A further temperature reduc-
0
1
tionleavesthe topofthe Kondoresonanceoutside the F
A(w) window, and the thermopower drops. However,most Ce
f
and Yb system with very small T have a phase transi-
0
tionaboveT ,andtodiscussthenormal-stateproperties
0
0.5 of (a)-type systems it is sufficient to consider the NCA
solution for T ≥T .
0
An increase of the coupling to Γ>∆ has a drastic ef-
fectonA(ω), asillustratedin Fig.8, whereA(ω)is plot-
0
-0.001 0 0.001 tedforΓ=0.12eV.The charge-excitationpeakistrans-
w [eV]
formed into a broad background (see the upper panel in
Fig. 8) and the only prominent feature at T ≃ T is
∆
FIG.7: SpectralfunctionA(ω),calculatedforthehybridiza-
the low-energyresonanceofhalf-width∆centeredabove
tion strength Γ=0.06 eV and the CF splitting ∆=0.07 eV,
µ. This low-energy resonance is due to the exchange
plotted as a function of frequency for several temperatures.
scattering of conduction electrons on the full CF mul-
The solid, dashed, dashed-dotted and dotted curves corre-
spondtoT =2,41,209,and670K,respectively. Thecharge- tiplet, which gives rise to the maximum of S(T) in the
excitation peak is visible in the upper panel. The middle LM regime. The F window (see lower panel in Fig. 8)
panel shows the evolution of the CF and Kondo resonances shows more spectral weight above than below µ, so that
with temperature. For T ≤ ∆, the many-body resonance of S(T) > 0. The reduction of temperature below T re-
S
half-width ∆ is centered well below µ. The F window has moves some spectral weight above µ and brings addi-
more states below than above µ and S(T) < 0. The lower tional spectral weight below µ, which reduces S(T) and
panelshowsthepositionoftheKondoresonanceaboveµ. Its leads to a minimum.51 A further reduction of tempera-
center defines T0 =1 K. For T ≤ T0 the F window has more ture leads to the rapid growth of the Kondo peak at ω ,
states above than below µ and S(T)>0. 0
and the CF peak at ω +∆, but the negative CF peak
0
10
7
T=2 K
6 T=41 K
T-209 K
T=670 K 6
5
4
A(w) A(w)
f f
3
3
2
1
0 0
-2 -1 0 1 2 -0.1 0 0.1 0.2
w [eV] w [eV]
FIG.9: Spectral function A(ω),calculated for thehybridiza-
tion strength Γ=0.140 eV and theCF splitting ∆=0.07 eV,
6
plotted as a function of frequency for several temperatures.
The solid, dashed, dashed-dotted and dotted curves corre-
spond to T=2, 41, 209, and 670 K, respectively. For Γ>∆,
theKondo resonance is reduced to a small humpabove the f
4
level.
A(w)
f
However,onceT isobtainedfromtheNCAcalculations,
0
2
the low-temperature transport can be inferred from the
universal power laws which hold in the FL regime, as
discussed in the previous section.
A further increase of Γ shifts the Kondo and the CF
0
-0.14 -0.07 0 0.07 0.14 peakstohigherenergies,andchangestheirrelativespec-
w [eV]
tral weight, as shown in Fig. 9, where the low-frequency
part of A(ω) is shown for Γ = 2∆. The Kondo scale
FIG.8: SpectralfunctionA(ω),calculatedforthehybridiza-
is still defined by the center of the Kondo peak, even
tion strength Γ=0.12 eV and the CF splitting ∆=0.07 eV,
though it is now reduced to a hump on the low-energy
plotted as a function of frequency for several temperatures.
The solid, dashed, dashed-dotted and dotted curves corre- side of a large peak centered at ω0+∆. The unphysical
spond to T =2, 41, 209, and 670 K, respectively. The upper NCA spike at ω =0 can be seen at lowesttemperatures.
panel shows the overall features. The two many body reso- Thethermopoweris positiveatalltemperaturesandhas
nances are resolved but the lower CF peak and the charge- only a shoulder below T . A quantitative comparison
S
excitation peakareabsent. Thelowerpanelshowstheevolu- between T , defined by the position of the Kondo reso-
0
tion of low-energy resonances with temperature. For Γ>∆, nance, and the position of the Kondo anomaly in S(T)
there is more spectral weight above than below µ at all tem-
becomes difficult.
peratures and S(T) is always positive.
Finally, for Γ> 2∆, we find A(ω) with a single broad
peak centred at E˜ > 0, as shown in Fig. 10. The CF
f
excitations are now absent, which is typical for the An-
does not develop. That is, an increase of pressure re- derson model in the vicinity of the VF fixed point. The
moves the lower CF peak, and shifts the Kondo and the relevant energy scale at low temperature is defined as
upperCFpeaktohigherenergies,withoutchangingtheir kBT0 = E˜f, and shows an almost linear dependence on
separation∆. TheFwindowshowsmorespectralweight Γ. The thermopoweris alwayspositive andgrowsmono-
above than below µ, so that S(T) is positive and grows tonicallyfromsmallvaluesatlowtemperaturestowardsa
as temperature is lowered. The maximum S0 is reached high-temperaturemaximumatTS. Theunphysicalspike
at T when the Kondo resonance is fully developed. The at ω = 0 appears at higher temperatures and is more
0
characteristic energy scale is defined again by the posi- pronounced than for small Γ, as illustrated in Fig. 10.
tion of the Kondo peak, k T = ω , which can now be The initial slope of S(T) obtained from the NCA result
B 0 0
quite large. For T ≤T the F window becomes narrower fortransportcoefficientsisverymuchoverestimatedwith
0
thantheKondoresonanceandS(T)dropsbelowS . For respect to the FL result based on Eq.(6). An increase of
0
T ≪T0, where the FL behavior is expected,43 the NCA temperature above TS modifies the excitation spectrum
gives A(ω) with an unphysical spike at µ, which makes on an energy scale of the order of E˜f and reduces S(T).
ρ (T) and S(T) much larger than the exact result. ThevaluesofE˜ andT donotseemtoberelatedinany
mag f S
