Proc. Int. Conf. Heavy Electrons (ICHE2010)J. Phys. Soc. Jpn. 80 (2011) SAZwicknaglICHE2010c⃝ 2011 The Physical Society of Japan

Magnetic field-induced anomalies and Kondo effect in lanthanide in heavy-electron systems

Gertrud Zwicknagl

Institut fur Mathematische Physik, Technische Universitat Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig

We investigate the influence of magnetic fields on the heavy quasiparticles in the Kondo lattice systemYbRh2Si2 and in filled skutterudites containing Pr ions. In the Kramers system YbRh2Si2 the Kondosinglets break up in an applied field which leads to a general reduction of the effective mass. The observedanomalous behavior of the specific heat is quantitatively described by the Renormalized Band methodwhere the field-dependent quasi-particle parameters are deduced from Numerical Renormalization Groupcalculations for a single Anderson impurity. In filled skutterudites containing the non-Kramers Pr ions,on the other hand, the heavy quasi-particle mass can be explained by aspherical Coulomb scattering ofconduction electrons off Crystalline Electric Field excitations. The field-induced splitting of the low-lyingtriplet should lead to a strongly field-dependent mass enhancement. It is suggested that a field-inducedKondo effect could occur in the dilute system La1−xPrxOs4Sb12 in close analogy to the Kondo effectfound in integer-spin quantum dots.

KEYWORDS: Heavy electrons, Kondo lattice, filled skutterudites, Crystaline Electric Field effects

1. IntroductionMagnetic fields may strongly affect the electronic proper-

ties of materials containing lanthanide or actinide ions.1) Inthese systems, the relevant energy scales of the electronic sys-tem are strongly reduced due to the strong correlations ofthe partially filled 4f shells (for recent reviews see2, 3) andreferences therein). The small energy scales comparable toa Zeeman energy of ∼6 meV at 50T arise from removinglocal degeneracies by the anisotropy of the crystalline elec-tric field (CEF), from building-up long-range order amongmoments or from forming (local) singlets via the Kondo ef-fect. Important examples for unusual behavior are the field-induced changes of carrier-concentration in CeB6 and Ce-BiPt.4–7) In the heavy fermion system (HFS) CeRu2Si2, adeHaas-vanAlphen (dHvA) frequency changes abruptly atthe metamagnetic transition.8) This reflects changes in theground state from a Fermi liquid with f-derived itinerantquasiparticles and a “large” Fermi surface to a conventionalmetal with polarized local f-moments and, concomitantly, a“small” Fermi surface.9–12) Metamagnetic transitions in Ce-based heavy-fermion compounds are a topic of high currentinterest.13)

The present paper focusses on the evolution of the heavyFermi liquid state with an external magnetic field in thestochiometric compound YbRh2Si2 and in the dilute al-loys La1−xPrxOs4Sb12. Of particular interest is the field-dependence of the heavy effective mass which is reflectedin the coefficient of the linear specific heat at low tempera-tures. We anticipate qualitatively different behavior in thesetwo systems since the free Yb ion has a Kramers-degenerateground state while the degeneracy of the f-shell of the non-Kramers ion Pr is removed by CEF effects.

The temperature vs. magnetic-field phase diagram ofYbRh2Si2 exhibits numerous anomalies.14) This heavyfermion compound which crystallizes in the tetragonalThCr2Si2 structure has been in the focus of interest duringthe past decade because it has emerged as a prototypical sys-tem for investigation of quantum critical phenomena.15) In

its ground state, YbRh2Si2 orders antiferromagnetically be-low the Neel temperature, TN=70 mK.16) By applying a weakmagnetic field of Bc=60mT in the basal plane the magnetic or-der is suppressed17) and the characteristic features of a Fermiliquid are observed. The high effective masses are ascribed toa Kondo effect which removes the Kramers degeneracy of thepartially filled Yb 4f shells. Here we discuss the anomaliesobserved at ∼10T in various thermodynamic and transportproperties.14, 18) We show that the latter result from a com-bination of a coherence effect of the periodic Kondo lattice, i.e., a van-Hove-type peak in the quasiparticle density of states(DOS) and a local many-body effect related to breaking-upthe Kondo singlets.

The heavy fermion behavior of the filled-skutterudite com-pound PrOs4Sb12 is related to the presence of the partiallyfilled 4f shells as can be seen by comparing its linear spe-cific heat coefficient γPr ∼ 350 − 500mJ/molK2 to thatof the isostructural non-f reference system LaOs4Sb12 γLa ∼36mJ/molK2.19, 20) It has been recently shown that the keyto the formation of heavy quasiparticles excitations lies in thecrystalline electric field (CEF) splitting of the J = 4 multipletand that the heavy quasiparticle mass is related to the inelas-tic scattering processes of the conduction electrons.22, 23) Thispicture is supported by the observation of an anti-correlationbetween CEF splitting and effective mass. The concept ofmass enhancement due to CEF excitations has been previ-ously applied to Pr metal using the isotropic dipolar exchangeinteraction Hex.21) In the present communication we suggestto test the underlying hypothesis that the effective mass en-hancement in PrOs4Sb12 is due to aspherical Coulomb scat-tering by investigating the variation with magnetic field ofthe specific heat coefficient in dilute alloys La1−xPrxOs4Sb12with x ≪ 1 where the large Pr Pr distance prevents the forma-tion of magnetic field-induced long-range order. It is an inter-esting speculation whether a Kondo effect can be induced bya magnetic field in a non-Kramers system in close analogy torecent findings in quantum dots.24) We show that a Kondo-likemany-body ground state should exist.

SAZwicknaglICHE2010-1

J. Phys. Soc. Jpn. 80 (2011) SAZwicknaglICHE2010 SAZwicknaglICHE2010-2

The paper is organized as follows: In Section 2 we describethe calculation of the the Renormalized Bands in YbRh2Si2.The variation with magnetic field of the quasiparticle DOSand the Fermi surface in high fields are discussed in Section3. Section 4 is devoted to the model Hamiltonian for the low-energy excitations in filled skutterudites containing Pr. Themass renormalization due to aspherical Coulomb scatteringis derived in Section 5 within (self-consistent) second-orderperturbation theory. In Section 6 we discuss the influence ofan external magnetic field. We show in Section 7 that thedivergence of the effective mass at a critical magnetic fieldmight indicate the formation of a novel Kondo-like many-body ground. A summary is given in Section 8.

2. Renormalized Bands for YbRh2Si2The strongly renormalized heavy quasiparticle bands are

determined by means of the Renormalized Band (RB) schemewhich combines material-specific ab-initio methods and phe-nomenological considerations in the spirit of the Landau the-ory of Fermi liquids. The anatz successfully describes thequasiparticle dispersion in Ce-based HFS.2, 10, 25–29, 31) TheFermi surfaces and effective masses deduced from RB calcu-lation reproduce the Hall effect data of the isostructural andisoelectronic HFS counterpart YbIr2Si2.30) For a detailled de-scription of the method we refer to.9)

We introduce renormalized phase shifts for waves whichhave 4f-symmetry with respect to the rare-earth or actinidesites. Operationally, the renormalization procedure amountsto transforming the f-states of the spin-orbit ground state mul-tiplet at the lanthanide site into the basis of CEF eigenstates|m⟩ and introducing resonance-type phase shifts

ηfm(E) ≃ arctan∆f

E − ϵfm(1)

where the resonance width ∆f reflects the renormalizedquasiparticle mass. The resonance energies ϵfm = ϵf + δmrefer to the centers of gravity of the f-derived quasiparticlebands. Here ϵf denotes the position of the band center corre-sponding to the CEF ground state while the δm are the mea-sured CEF excitation energies. One of the remaining two pa-rameters, ϵf , is determined by imposing the condition thatthe charge distribution is not altered significantly by introduc-ing the renormalization. This makes the RB method a single-parameter scheme. The free parameter, ∆f , is adjusted so asto reproduce the coefficient of the linear specific heat at lowtemperatures.

In the case of Yb-based heavy fermion compounds, wehave to renormalize the 4f j=7/2 channels at the Yb sites. Asthe 4f hole count is slightly less than unity the center of grav-ity ϵf will lie below the Fermi energy. In addition, we have toreverse the hierarchy of the CEF scheme, i. e.,

ϵf < 0 ; ϵfm = ϵf − δm . (2)

In the presence of a magnetic field, the quasiparticles haveto be described by field-dependent parameters for the levelϵfm(h) and the resonance width ∆f (h). The energy h =12geffµBH denotes the (anisotropic) Zeeman splitting of thefree ion CEF ground state which is related to that of an ef-fective spin-1/2-system by introducing anisotropic effectiveg-factors. In the present calculation, we use field-dependent

parameters ϵf (h) and ∆f (h) whose highly non-trivial varia-tion with field is derived from fits to field-dependent quasipar-ticle DOS of the single-impurity Anderson model.32–35) Thelatter are calculated microscopically by means of the Numer-ical Renormalization Group (NRG). This procedure properlyaccounts for the progressive de-renormalization of the quasi-particles with increasing magnetic field and the correlation-enhanced Zeeman splitting. As will be shown below, it allowsfor a quantitative description of the observed field-inducedanomalies as well as for the observed dHvA data in the high-field regime.

The band-structures were obtained by the fully relativis-tic formulation of the linear muffin-tin orbitals (LMTO)method.36–38) The spin-orbit interaction is fully taken into ac-count by solving the Dirac equation. We adopt the atomicsphere approximation (ASA) and include the combined cor-rection term which contains the leading corrections to theASA.36) The calculations are done at the experimental latticeparameters a=b=4.007A, c=9.858A for YbRh2Si2. For detailsof the calculation we refer to.30, 39)

The RB calculations reported here adopt a CEF schemewhich is consistent with susceptibility and the inelasticneutron data.40, 41) The latter indicate that the 4f13 statesin YbRh2Si2 are split into 4 doublets with the energies0-17-25-43 meV. The parameters for the tetragonal CEFare B20=0.5246 meV, B40=0.01195 meV, B60=-0.0004725meV, B44=0.03598 meV, B64=-0.01206 meV.42, 43) The low-energy properties are mainly determined by the CEF groundstate which is a superposition of |j = 7/2; jz = ±5/2⟩ and|j = 7/2; jz = ∓3/2⟩ and which is well separated from theexcited states. We find that the coupling to the conductionstates is rather weak and strongly anisotropic. The resultingg-factors are g∥=0.26 and g⊥=3.79 for magnetic fields paral-lel to the tetragonal axis and in the basal plane.

We use a quasiparticle resonance width of ∆f = 20Kas inferred from specific heat and thermopower measure-ments.14, 44)

3. Magnetic field-dependent quasiparticles in YbRh2Si2The RB calculation yields narrow bands with f-character in

the vicinity of the Fermi energy while the dispersion of thebroad non-f conduction bands remains essentially the same asin the local moment regime. The quasiparticle DOS displayedin Figure 1 is mainly due to the CEF-split 4f-states. The RBcalculations yield a DOS of ∼ 290 states/(eV unit cell) cor-responding to specific heat coefficient ∼ 680mJ/(mole K2).This value should be considered as theoretical zero-field limitof the specific heat coefficient measured in the Fermi liquidstate at small finite magnetic fields. The actual zero-field spe-cific heat data are strongly enhanced by the anomalous fluc-tuations associated with the quantum critical point.

There are three bands intersecting the Fermi energy. Theoverall topology qualitatively agrees with LDA results ofRefs.45–49) The dominant contribution to the quasiparticleDOS and, concomitantly, to the specific heat and magneticsusceptibility comes from the Z-centered hole surface whichhas predominantly f-character. The anisotropy of the CEFground state results in a strongly anisotropic hybridization. Asthe heavy band is rather flat over a wide range of the Brillouinzone we find a very sharp van-Hove type feature in the quasi-

J. Phys. Soc. Jpn. 80 (2011) SAZwicknaglICHE2010 SAZwicknaglICHE2010-3

-6 -4 -2 0 2 4 6(E-EF) (meV)

0

200

400

600

800

1000

Ren

DO

S /(

eV c

ell)

0 5 10 15H (T)

0

100

200

300

400

500

D(0

) (s

tate

s/(e

V u

nit c

ell))

Fig. 1. YbRh2Si2: Zero-field quasiparticle DOS from RB calculation (up-per panel) The low-energy properties are determined by the peak at theFermi energy which exhibits a “coherence gap” and a van-Hove type sin-gularity. Lower panel: Variation with magnetic field in the basal plane ofthe quasiparticle DOS at the Fermi level D(0) derived from the RB calcu-lation (open circles). The field-dependent parameters ϵf (h) and ∆f (h)are determined from fits to NRG calculations. Included for comparisonare predictions assuming single-particle Zeeman splitting (dotted line), de-creasing effective mass plus single-particle Zeeman splitting (dot-dashedline), correlation enhanced Zeeman splitting with zero-field effective mass(dashed line). The DOS deduced from the specific heat coefficient mea-sured at ambient pressure18) is shown for comparison (filled circles).

particle DOS displayed in Figure 1. It is this feature whichleads to the anomalies at ∼10T. The variation with magneticfield of the quasiparticle DOS at the Fermi energy is shownin Figure 1. It describes the experimental specific heat andsusceptibility data.18)

The states forming the “jungle gym” sheet of the Fermi sur-face, on the other hand, are strongly hybridized and conse-quently, less affected by magnetic fields. Several groups re-ported a large extremal orbit corresponding to relatively lightquasiparticles.49, 50) The jungle gym sheet of the RB Fermisurface in Figure 2 is consistent with these findings. In a fieldof ∼15T there is a closed orbit in the 110 plane with areaF=13kT and and effective mass m∗ ∼20m which agrees wellwith the values given by Westerkamp.50) From Figure 2it isobvious that the closed orbit is confined to a relatively narrowrange of magnetic field orientations. It is important to notethat this orbit cannot be assigned to a “small” Fermi surfaceexpected in the local moment regime.

Fig. 2. YbRh2Si2: Multiply-connected sheet in high magnetic field of∼15T in the basal plane. There is a closed orbit with area ∼13kT and m∗ ∼20m in the 110 plane.

4. Model Hamiltonian for La1−xPrxOs4Sb12

The electronic part of the Hamiltonian for the systemLa1−xPrxOs4Sb12 is given by

H = Hel +HCEF +HZ +HAC . (3)

Here Hel contains the conduction band dispersion which maybe described by a n.n.n. tight binding model?) according to

ϵkσ = t cos1

2kx cos

1

2ky cos

1

2kz+t′(cos kx+cos ky+cos kz)

(4)with t=174 meV and t’=-27.84 meV. It corresponds to a singleband originating in Sb-4p states. The transfer integrals t and t’are chosen so as to reproduce the observed linear specific heatcoefficient γ = 36mJ/(mole K2) of the non-f reference com-pound LaOs12Sb12.51) Aside from subtle effects the resultingFermi surface is quite similar to the LDA FS in Ref.52)

The CEF and Zeeman Hamiltonian are

HCEF+HZ =∑i,Γn

EΓ | Γn(i)⟩⟨Γn(i) | +gJµB

∑i

J(i)·H .

(5)The external magnetic field is denoted by H, gJ is the Landefactor and µB is Bohr’s magneton. Furthermore i labels thePr sites, and |Γn⟩ denotes the CEF states with energies EΓn .The data are explained best by a | Γ1⟩ ground state with alow-lying triplet excited state at an energy of ∆ = 8K.53, 54)

The other CEF levels are so high in energy that they can beneglected. The compound has tetrahedral Th site symmetryfor Pr which implies that the triplet state is a superposition oftwo triplets Γ4 and Γ5 of Oh symmetry53, 54)

| Γmt ⟩ =

√1− d2 | Γ5,m⟩+ d | Γ4,m⟩ , m = ±, 0 .

(6)The conduction electrons interact with the CEF energy lev-

els of the Pr3+ ions. We focus here on the aspherical Coulombscattering55)

HAC = g∑i

∑kk′σ

∑αβ

Oiαβfαβ (k,k

′) c†kσck′σei(k−k′)·Ri

(7)where c†kσ(ckσ) are the creation (annihilation) operators forconduction electron with momentum k and spin σ while

J. Phys. Soc. Jpn. 80 (2011) SAZwicknaglICHE2010 SAZwicknaglICHE2010-4

Oiαβ =

√32 (J i

αJiβ +J i

βJiα), αβ = yz, zx, xy denote the three

quadrupole operators with Γ5 symmetry. The remaining Γ3

quadrupole terms are neglected since they do not couple tothe excitations under consideration. The scattering anisotropyis accounted for by the quadrupole form factors

fαβ (k,k′) =

1⟨|k− k′|2

⟩FS

(kα − k′α)(kβ − k′β

)(8)

where the momentum transfer is measured in units of itsFermi surface average. The coupling constant g may in prin-ciple be determined by experiments. A derivation of Eq. (7)may be obtained from Ref.56)

5. Mass renomalization due to aspherical Coulomb scat-tering

The effective mass enhancement due to interactions ofthe conduction electrons follows from the self-energy due toHAC . Neglecting vertex corrections, it is given by

Σ (k, ω) = i∑αβ,n

1

V

∑k

∣∣Λnαβ(k,k

′)∣∣2

∫dω′

2πDn (k− k′, ω + ω′)G (k′, ω′) . (9)

where the momentum dependence of the bare vertexΛnαβ(k,k

′) follows from the quadrupolar Γ5 form factors inEq. (8). Here Dn(k, ω) denotes the boson propagator of CEFexcitations. It is related to the dynamical quadrupolar suscep-tibility of the CEF system. In the present case we will neglecteffective RKKY type interactions between CEF states on dif-ferent sites assuming a local (q- independent) singlet-tripletboson propagator

Dn (q, ω) = Dn(ω) =2δn

δ2n − ω2(10)

with the field dependent singlet-triplet excitation energiesδn(H) = ϵnt (H)− ϵs(H) (n = +,0,-).

The self-energy becomes k-independent. In the wide-band limit, the result for Σ(ω) agrees with that of non-selfconsistent second order perturbation theory. After differ-entiation with respect to ω under the integral and using inte-gration by parts one finally gets

δm∗

m=

m∗

m−1 = − ∂Σ(ω)

∂ω

∣∣∣∣ω=0

= g2N(0)fχQ(H) (11)

with

χQ(H) =∑αβ,n

2 |⟨Γs(H) |Oαβ |Γnt (H)⟩|2

δn(H). (12)

The reference mass m includes band effects and effects ofelectron-phonon coupling. The averaged quadrupolar formfactors f = ⟨|fαβ (k,k′) |2⟩FS is a constant. Further-more χQ(H) in Eq. (12) is the field-dependent static uni-form quadrupolar susceptibility. Explicit expressions for thequadrupolar matrix elements in Eq. (12) are given in.23)

0.0 0.2 0.4 0.6 0.8 1.0H /H

c

0

2

4

6

8

10

12

14

(δm

* / m

) (H

) /

(δm

* /m)

(0) d=0.26 ; δ

n(0)=8K

Fig. 3. Variation with magnetic field of δm∗/m scaled to the zero-fieldvalue. The data are calculated assuming a tetrahedral CEF contribution ofd = 0.26 and a zero-field singlet-triplet gap of δn (0) = 8K which aretypicl values for PrOs4Sb12. The full line is the result of the selfconsistentcalculation for finite band width ϵc = 20δn(0) while the broken line is theϵc → ∞ result. The divergence in the vicinity of the critical magnetic fieldHc ∼ 6.9T is a consequence of the model which assumes dispersionlessundamped CEF excitations.

6. Magnetic field-dependent mass enhancement inLa1−xPrxOs4Sb12

It has been shown that aspherical Coulomb scattering canconsistently and quantitatively explain the zero-field effectivemass enhancement in Pr skutterudites.22, 23) When a magneticfield is applied to the sample the field dependence of the effec-tive mass is completely determined by that of the quadrupolarsusceptibility in Eq. (12). To calculate this quantity we usethe singlet-triplet excitation energies δn(H) and the eigen-states and matrix elements in applied field which were givenby Shiina et al53, 54) in closed form for field applied along cu-bic symmetry directions. The field dependence of δn(H) hasrecently been determined by INS eperiments.57)

The two triplet states |Γ±t ⟩ have a linear Zeeman splitting

independent of the tetrahedral CEF contribution d. When d =0, the energies of |Γs⟩ and |Γ0

t ⟩ will be independent of thefield H. For nonzero tetrahedral contribution these two levelswill repel with increasing field H. For d2 < 0.42 the singletground state level Es and lowest triplet level E+

t cross at acritical field Hc meaning δ+(Hc) = 0. In the case of weaktetrahedral CEF such as realised in Pr1−xLaxOs4Sb12 thelevel repulsion of Γs and Γ0

t is also weak and therefore theΓ+t level crosses the Γs ground state at a critical field Hc in

the dispersionless case.The decrease in the excitation gap for H < Hc and the field

dependence of matrix elements leads to a field dependence ofδm∗/m which is shown in Fig. 3 for d = 0.26. For larger d2

the increase is diminished and eventuallly for d2 > 0.42 thelevel repulsion due to the tetrahedral CEF is strong enoughto lead to an increase in excitation energy and hence to a de-creasing effective mass.

7. Is there a field-induced Kondo effect for non-KramersPr ions?

Let us next turn to the (unphysical) divergent mass renor-malization which is predicted for dispersionless undampedCEF excitations when the triplet level approaches the singlet

J. Phys. Soc. Jpn. 80 (2011) SAZwicknaglICHE2010 SAZwicknaglICHE2010-5

Fig. 4. Electronic density in the Kondo-like many-body ground state. Theorbital degrees of freedom of the Pr 4f states and the conduction electronsare entangled. The density 4f is shown in red while the conduction electrondensity is shown in cyan.

ground state level at the critical field Es(Hc) = E+t (Hc).

As the divergence follows directly from the general analyticstructure of the corresponding electron self energy it persistsalso in the self consistent solution for finite band width.23)

In concentrated systems the singlet-triplet excitations havea pronounced dispersion58)and an applied field of criticalstrength therefore leads to a softening of ωq only in the re-stricted phase space around the AFQ ordering vector Q. Con-sequently the mass renormalization will be finite even at thecritical field hc for AFQ order when ωQ=0.

Motivated by the generic phase diagram of Ce- or Yb-basedHFS we ask whether a heavy fermi liquid state can form bya Kondo-type effect in the vicinity of the critical field, i. e.,whether the local degeneracy degeneracy can be removed bythe coupling to the conduction electrons. We should like tomention that a field-induced Kondo effect has been observedin non-Kramers quantum dots.24)

To address this question we consider a single Pr impurityand simplify the Hamiltonian by projecting onto the conduc-tion electron degrees of freedom which couple to the CEFtransitions. This is achieved by introducing the orthonormallinear combinations |ϵν⟩, ν = 0 and ν = αβ, of Bloch stateswith energy ϵ

|ϵ0⟩ = N0

∑k

δ (ϵ− ϵk) |k⟩

|ϵαβ⟩ = Nαβ

∑k

δ (ϵ− ϵk) kαkβ |k⟩ (13)

where the normalization constants Nν , are expressed in termsof the DOS N(ϵ) and the ϵ-surface average

⟨k2αk

2β

⟩ϵ. We start

from a variational ansatz which has been inspired by Yosida’streatment of the Kondo effect

|Ψ0⟩ = AN(0)∑σ

∫ D

0

dϵ

∑ν=0, αβ

∑n=s,t+

a (ϵν;n) cϵνσ † |FS⟩ |Γn(H)⟩ .(14)

It describes an electron interacting with a Pr ion in the pres-ence of a filled Fermi sea |FS⟩. The creation (annihilation)operators for c†ϵνσ (cϵνσ) refer to conduction electron states

introduced above. The functions a(ϵν;n) are to be determinedvariationally while A is an overall normalization constant andD is the width of the conduction band. The matrix elementsof HAC vary weakly with energy and hence can be evalu-ated at the Fermi energy. Minimizing the energy with respectto the energy-dependent amplitudes a (ϵν;n) yields the self-consistency equation for a (ν;n) = N (0)

∫ 0

−Ddϵa (ϵν;n)

a (ν;n) = N (0) ln

∣∣∣∣E − En

D

∣∣∣∣∑ν′n′

⟨0ν;n|HAC |0ν′;n′⟩ a (ν′;n′)

(15)which has a non-trivial solution provided an eigenvalue of thematrix

K (νn; ν′n′) = (16)

−N (0)√λ (n;E)

∑ν′n′

⟨0ν;n|HAC |0ν′;n′⟩√

λ (n′;E)

with λ(E;n) = ln∣∣∣ DE−En

∣∣∣ is unity. This condition is always

satisfied for an energy E < Es (Hc) = E+t (Hc) since the

hermitian matrix K with vanishing trace has at least one pos-itive eigenvalue.

From these consideration we conclude that a field-inducedKondo effect should exist in the system under consideration.In this state the orbital degrees of freedom of the f- and con-duction states are entangled as can be seen from Figure 4which schematically displays the density distributions of theground state obtained for the present model. A quantitative es-timate of the Kondo scale requires a more refined theoreticaltreatment with more detailled information on the (effective)coupling constant g59)

8. Summary and outlookWe calculated the variation with magnetic field of the

quasiparticle bands in YbRh2Si2 by means of the RB methodwhere the field-dependent quasiparticle parameters extractedfrom a NRG treatment of the single-impurity Anderson modelaccount for the correlation enhanced Zeeman splitting and thereduced effective mass of the CEF ground state doublet. Thecalculated DOS reproduces the observed variation with mag-netic field of the linear specific heat coefficient and the mag-netic susceptibility. The “large” Fermi surface consistentlyexplains the quantum oscillation frequency observed in thehigh-field limit.

For the filled skutterudites Pr1−xLaxOs4Sb12 we havestudied in detail the quasiparticle mass enhancement originat-ing in the aspherical Coulomb scattering of conduction elec-trons from singlet triplet CEF excitations. For small enoughtetrahedral CEF characterised by the parameter d2 ≪ 1 thelowest triplet component crosses the singlet ground state ata critical field Hc. In second order perturbation theory themass enhancement increases with field and becomes singu-lar at Hc. For larger tetrahedral CEF (d2 > 0.42) the excita-tion energy between singlet ground state increases with fieldleading to a decrease of the mass enhancement, similar as hasbeen observed in Pr metal where the mass renormalisation isdue to exchange scattering from a singlet-singlet CEF levelscheme. The singular mass enhancement close to the criticalfield of level crossing is an artefact of the model. Any dis-persion of the singlet-triplet excitations due to effective in-

J. Phys. Soc. Jpn. 80 (2011) SAZwicknaglICHE2010 SAZwicknaglICHE2010-6

tersite quadrupolar interactions will lead to a finite effectivequasiparticle mass. In addition, we showed that a Kondo-typeground state can be induced by applying a magnetic field.

Therefore we propose that the field dependence of the elec-tronic specific heat in mixed crystals of Pr1−xLaxOs4Sb12 issystematically investigated and analysed. It may hold impor-tant clues to the microscopic nature of the heavy-electronstate in PrOs4Sb12.

AcknowledgmentIt is a pleasure to thank P. Fulde, P. Gegenwart, C. Geibel,

N. Oeschler, H. Sato, P. Thalmeier, S. Tornow and T. West-erkamp for numerous helpful discussions and F. B. Andersfor making available the NRG code.

1) Z. Fisk and J. D. Thompson. Physical Phenomena in High MagnetigFields, page 197. Addison-Wesley, 1992.

2) P. Thalmeier and G. Zwicknagl. Handbook on the Physics and Chem-istry of Rare Earth, volume 34, pages 135–287. Elsevier B. V., 2005.

3) Hilbert v. Lohneysen, Achim Rosch, Matthias Vojta, and Peter Wolfle.Rev. Mod. Phys. 79, 1015–1075, 2007.

4) W. Joss, J. M. van Ruitenbeek, G. W. Crabtree, J. L. Tholence, A. P. J.van Deursen, and Z. Fisk. Phys. Rev. Lett., 59(14):1609–1612, 1987.

5) N. Harrison, D. W. Hall, R. G. Goodrich, J. J. Vuillemin, and Z. Fisk.Phys. Rev. Lett., 81(4):870–873, 1998.

6) G. Goll, J. Hagel, H. V. von Lohneysen, T. Pietrus, S. Wanka, J. Wos-nitza, G. Zwicknagl, T. Yoshino, T. Takabatake, and A. G. M. Jansen.Europhys. Lett., 57:233, 2002.

7) N. Kozlova, J. Hagel, M. Doerr, J. Wosnitza, D. Eckert, K.-H.Muller, L. Schultz, I. Opahle, S. Elgazzar, Manuel Richter, G. Goll,H. v. Lohneysen, G. Zwicknagl, T. Yoshino, and T. Takabatake. Phys.Rev. Lett., 95:086403, 2005.

8) H. Aoki, S. Uji, A. K. Albessard, and Y. Onuki. Phys. Rev. Lett., 71,2110, (1993).

9) G. Zwicknagl. Adv. Phys., 41, 203,(1992).10) G. Zwicknagl. Quasiparticles in heavy fermion systems. Physica

Scripta T, 49, 34, (1993).11) C. A. King and G. G. Lonzarich. Physica B, 171, 161, (1991).12) F. S. Tautz, S. R. Julian, G. J. McMullen, and G. G. Lonzarich. Physica

B, 206-207, 29, (1995).13) M Deppe, N Caroca Canales, J G Sereni, and C Geibel. Journal of

Physics: Conference Series, 200, 012026, (2010).14) P Gegenwart, Y Tokiwa, T Westerkamp, F Weickert, J Custers, J Fer-

stl, C Krellner, C Geibel, P Kerschl, K-H Muller, and F Steglich. NewJournal of Physics, 8, 171,(2006).

15) P. Gegenwart, Qimiao Si, and Frank Steglich. Nature Physics, 4, 186,(2008).

16) O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F. M. Grosche,P. Gegenwart, M. Lang, G. Sparn, and F. Steglich. Phys. Rev. Lett., 85,626, (2000).

17) P. Gegenwart, J. Custers, C. Geibel, K. Neumeier, T. Tayama, K. Tenya,O. Trovarelli, and F. Steglich. Phys. Rev. Lett., 89,056402, (2002).

18) Y. Tokiwa, P. Gegenwart, T. Radu, J. Ferstl, G. Sparn, C. Geibel, andF. Steglich. Phys. Rev. Lett., 94, 226402–1, (2005).

19) K. Matsuhira, M. Wakashima, Y. Hinatsu, C. Sekine, I. Shirotani, D.Kikuchi, H. Sugawara, and H. Sato. J. Magn. Magn. Mat., 310, 226(2007).

20) Kenya Tanaka, Takahiro Namiki, Atsushi Imamura, Makoto Ueda,Takashi Saito, Sho Tatsuoka, Ryouichi Miyazaki, Keitaro Kuwahara,Yuji Aoki, and Hideyuki Sato, J. Phys. Soc. Jpn., 78, 063701 (2009).

21) P. Fulde and J. Jensen, Phys. Rev. B 27, 4085 (1983)

22) J. Chang, I. Eremin, P. Thalmeier and P. Fulde, Phys. Rev. B 76,220510(R) (2007)

23) Gertrud Zwicknagl, Peter Thalmeier, and Peter Fulde. Phys. Rev. B, 79,115132, (2009).

24) S. Sasaki, S. DeFranceschi, J. M. Elzermann, W. G. van der Wiel, andL. P. Kouwenhoven, Nature 405, 764 (2000)

25) Gertrud Zwicknagl and Uwe Pulst. Physica B, 186, 895, (1993).26) O. Stockert, E. Faulhaber, G. Zwicknagl, N. Stuesser, H. Jeevan, T. Ci-

chorek, R. Loewenhaupt, C. Geibel, and F. Steglich. Phys. Rev. Lett.,92, 136401, (2004).

27) P.Thalmeier, G. Zwicknagl, O. Stockert, G. Sparn, and F. Steglich.Frontiers in Superconducting Materials, pages 109–182. Springer,2005.

28) G. Zwicknagl. J. Low Temp. Phys., 147, 123, (2007).29) I. Eremin, G. Zwicknagl, P. Thalmeier, and P. Fulde. Phys. Rev. Lett.,

101, 187001, (2008).30) Sven Friedemann, Steffen Wirth, Niels Oeschler, Cornelius Krellner,

Christoph Geibel, Frank Steglich, Sam MaQuilon, Zachary Fisk, SilkePaschen, and Gertrud Zwicknagl. Phys. Rev. B, 82, 035103, (2010).

31) P. Fulde, P. Thalmeier, and G. Zwicknagl. Solid State Physics, vol-ume 60, page 1. Academic Press, (2006).

32) A. C. Hewson, A. Oguri, and D. Meyer. The European Physical JournalB - Condensed Matter and Complex Systems, 40, 177–189, (2004).

33) A. C. Hewson, J. Bauer, and W. Koller. Phys. Rev. B, 73, 045117, (2006).34) J. Bauer and A. C. Hewson. Phys. Rev. B, 76, 035119, (2007).35) Robert Peters, Thomas Pruschke, and Frithjof B. Anders. Phys. Rev. B,

74, 245114, (2006).36) O. K. Andersen. Phys. Rev. B, 12, 3060, (1975).37) H.L. Skriver. The LMTO Method, volume 41 of Springer Series in Solid

State Sciences. Springer-Verlag, Berlin, (1984).38) R.C. Albers, A. M. Boring, and N. E. Christensen. Phys. Rev. B, 33,

8116, (1986).39) G. Zwicknagl, J. Phys.: Condens. Matter 22 to appear 201040) A. Hiess, O. Stockert, M.M. Koza, Z. Hossain, and C. Geibel. Physica

B: Condensed Matter, 378-380, 748–749, (2006).41) O. Stockert, M.M. Koza, J. Ferstl, A.P. Murani, C. Geibel, and

F. Steglich. Physica B: Condensed Matter, 378-380, 157–158, (2006).42) Vilen Zevin and Gertrud Zwicknagl. (2010).43) G. Zwicknagl. In Veljko Zlatic and Alex C. Hewson, editors, Prop-

erties and Applications of Thermoelectric Materials, volume 133 ofNATO Science for Peace and Security Series B: Physics and Biophysics,Berlin, (2009). Springer.

44) U. Kohler, N. Oeschler, F. Steglich, S. Maquilon, and Z. Fisk. Phys.Rev. B, 77, 104412–1, (2008).

45) G. Knebel, R. Boursier, E. Hassinger, G. Lapertot, P. G. Niklowitz,A. Pourret, B. Salce, J. P. Sanchez, I. Sheikin, P. Bonville, H. Harima,and J. Flouquet. J. Phys. Soc. Jpn., 75, 114709, (2006).

46) T. Jeong. Journal of Physics: Condensed Matter, 18, 10529–10529,(2006).

47) P. M. C. Rourke, A. McCollam, G. Lapertot, G. Knebel, J. Flouquet,and S. R. Julian. Phys. Rev. Lett., 101, 237205, (2008).

48) P M C Rourke, A McCollam, G Lapertot, G Knebel, J Flouquet, and S RJulian. Journal of Physics: Conference Series, 150, 042165, (2009).

49) A. B. Sutton, P. M. C. Rourke, V. Taufour, A. McCollam, G. Lapertot,G.and Knebel, J. Flouquet, and S. R. Julian. physica status solidi (b),247, 549–552, (2010).

50) T. Westerkamp. PhD thesis, Technische Universitat Dresden, (2009).51) E. D. Bauer, A. Slebarski, G. J. Freeman, C. Sirvent, and M. P. Maple,

J. Phys.: Condens. Matter 13, 4495 (2001)52) K. Miyake, H. Kohno and H. Harima. J. Phys. Condens. Matter 15, L275

(2003)53) R. Shiina and Y. Aoki, J. Phys. Soc. Jpn. 73, 541 (2004)54) R. Shiina, J. Phys. Soc. Jpn. 73, 2257 (2004)55) P. Fulde, L. Hirst and A. Luther, Z. Phys. 230, 155 (1970)56) H. H. Teitelbaum and P. M. Levy Phys. Rev. B 14, 3058 (1976)57) S. Raymond, K. Kuwahara, K. Kaneko, K. Iwasa, M. Kohgi, A. Hiess, J.

Flouquet, N. Metoki, H. Sugawara, Y. Aoki and H. Sato, (unpublished)58) K. Kuwahara, K. Iwasa, M. Kohgi, K. Kaneko, N. Metoki, S. Raymond,

M.-A. Measson, J. Flouquet, H. Sugawara, Y. Aoki and H. Sato, Phys.Rev. Lett. 95, 107003 (2005)

59) Sabine Tornow and Gertrud Zwicknagl, preprint