Kuramoto Y.

Kusunose H.

Otsuki J.

English

application/pdf学術論文 (Article)114324 bytesjournal articl

Otsuki, Junya

Kusunose, Hiroaki

Kuramoto, Yoshio

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

The de Haas-van Alphen effect in Kondo systemswith crystalline electric field著者 Otsuki  J, Kusunose  H, Kuramoto  Yjournal orpublication titleICM2006 proceedingsURL http://hdl.handle.net/10097/6427arXiv:cond-mat/0608382 v1   17 Aug 2006Journal of Magnetism and Magnetic Materials 0 (2006) 1–0www.elsevier.com/locate/jmmmThe de Haas-vanAlphen effect in Kondo systemswith crystalline electric fieldJunya Otsuki ∗ , Hiroaki Kusunose , Yoshio KuramotoDepartment of Physics, Tohoku University, Sendai 980-8578Received 12 June 2005; revised 13 June 2005; accepted 14 June 2005AbstractThe effective mass probed by the de Haas-van Alphen oscillations is studied for a model Ce system under magnetic field higherthan the Kondo energy. In the mean-field theory, the mass enhancement per Ce ion in the periodic system is identical with that inthe dilute system. With decreasing magnetic field, the effective mass tends to diverge corresponding to a formation of the Kondoground state. The effective mass can be very different between up and down spins depending on the nature of the 4f wave functions.c© 2006 Elsevier B.V. All rights reserved.PACS: 75.20.Hr; 71.27.+aKeywords: de Haas-van Alphen effect; Kondo effect; cyclotron effective mass; CexLa1−xB6The de Haas-vanAlphen (dHvA) effect is a powerfulmea-sure to observe the quasi-particle behavior of each spin com-ponent separately. In dilute Kondo systems, earlier work[1,2,3,4] concerned about the spin splitting and the Dingletemperature. In more recent dHvA experiments on CeB6,which shows a typical Kondo effect, it has been found thatone of spin components does not contribute to oscillatorymagnetization [5,6]. Furthermore, the cyclotron effectivemass of the observed spin component tends to diverge asapplied magnetic field decreases [7,8]. Since enhancementof the effective mass in CexLa1−xB6 is proportional to Ceconcentration x [7,8,9], the origin of the anomalous massenhancement as well as its spin dependence are ascribed tothe single-site effect. The purpose of this paper is to clar-ify the influence of the Kondo effect on the cyclotron ef-fective mass starting with the dilute limit. The results arecompared with those obtained for periodic systems withSU(N) symmetry [10].The oscillatory partMosc of magnetization per unit vol-ume is given in terms of the cross-sectional area S of theFermi surface, the cyclotron frequency ωc = eH/mc with-out many-body effect, and the self-energy Σσ(iωn) of con-duction electrons with ωn = (2n+ 1)πT the fermion Mat-subara frequency as follows (~ = kB = 1) [3,11]:MoscV=e1/2T(2π3cH)1/2∑σ=±∞∑l=1S∣∣∣∣∂2S∂k2z∣∣∣∣−1/2∗ Corresponding author.Email address: otsuki@cmpt.phys.tohoku.ac.jp (Junya Otsuki).× Re{i exp[2πil(Sc2πeH− φ)∓iπ4]×∞∑n=0exp[2πilωc(iωn − σµBH − Σσ(iωn))]}. (1)The standard Lifshitz-Kosevich formula is recovered by set-ting Σσ(iωn) = 0 in eq. (1).In the dilute systems (x ≪ 1) with completely randomdistribution of impurities, Σσ(iωn) is given by Σσ(iωn) =xtσ(iωn), where tσ(iωn) denotes the impurity t-matrix [12].The t-matrix is diagonal in the crystalline-electric-field(CEF) basis α at the impurity site. In terms of the Blochbasis (k, σ), which describes the cyclotron motion, thet-matrix is given bytσ(ω) =∑αwσ,αtα(ω), (2)where the momentum dependence is neglected. The trans-formation coefficient wσ,α depends strongly on the CEFstates.The component n = 0 gives a dominant contributionnear of the Fermi level in eq. (1), provided 2π2T/~ωc ≫ 1.We expand Σσ(iω0) as follows:Σσ(iω0) ≃ Σσ(0) + iω0∂Σσ(ω)/∂ω|ω=0. (3)Consequently, ReΣσ(0) gives a shift of the dHvA frequency,and ImΣσ(0) yields the relaxation rate or the Dingle tem-perature. The cyclotron mass m is replaced by a spin-dependent effective mass m∗cσ defined in terms of tσ(ω) bym∗cσ/m− 1 = −x∂Retσ(ω)/∂ω|ω=0 ≡ xµσ (4)0304-8853/06/$ - see frontmatter c© 2006 Elsevier B.V. All rights reserved.Junya Otsuki et al. / Journal of Magnetism and Magnetic Materials 0 (2006) 1–0 2In this paper we restrict our consideration to the effectivemass, relegating the spin splitting to another publication.It is well-known that the ground state of theKondo impu-rity system is a local Fermi liquid, provided the number ofinternal degrees of freedom matches the number of screen-ing channels. However the lifetime of conduction electrons,which appears in the dHvA oscillation, can remain finitesince it always refers to the Bloch basis. If the lifetime isshort, the effective mass as defined by eq. (4) can take evena negative value, and is not related to an observable. In sucha case, the impurity specific heat is not determined bym∗cσ,but by the phase shift in the local Fermi liquid. In orderto demonstrate the point in the simplest manner, we takethe N -fold degenerate Anderson model with constant hy-bridization V and infinite Coulomb repulsionU , and use theslave boson mean field approximation (SBMFA). In con-trast with the periodic SU(N) model, however, the Blochstate in our impurity model has only the spin degeneracy.The 4f electron Green function is given byGf,α(ω+) = r2[ω+ − (ǫ˜f − hα) + i∆˜]−1, (5)where r2 is the renormalization factor, hα = gαµBH is theZeeman shift, and we have introduced ω+ = ω + i0 and∆˜ = r2∆ = r2πV 2ρc. We take a constant density of statesρc = 1/2D with 2D being band width. The impurity t-matrix is then given by tα(ω+) = |V |2Gf,α(ω+), and theenhancement factor µα of each channel are given byµα = −∂Retα(ω)∂ω∣∣∣∣ω=0=∆˜πρc(ǫ˜f − hα)2 − ∆˜2[(ǫ˜f − hα)2 + ∆˜2]2. (6)In the high magnetic field regime, |ǫ˜f − hα| ≫ ∆˜, eq. (6)becomesµα ≃∆˜πρc(ǫ˜f − hα)2. (7)In this regime, the enhancement factor determines thephysical effective mass which is relevant to the specificheat.For large-N , renormalized parameters are given interms of the Kondo temperature TK by ǫ˜f = TK and∆˜ = πnfTK/N [13]. In this limit, eq. (7) agrees with thecorresponding quantity obtained for periodic systems [10].Hence under high magnetic field, where the Kondo singletbecomes unstable, there is no difference between periodicand dilute systems in the mass enhancement, except forthe concentration factor. We remark that eq. (7) applieseven in the weak field for the periodic system [10].Figure 1 shows the enhancement µα with gα > 0 forN = 4 in the SBMFA. Since D/TK is of order 104 forCexLa1−xB6, µα is of order 103. In dilute systems, µα be-comes negative around hα ∼ TK. For periodic systems, onthe other hand, µα diverges at hα = TK corresponding tothe Kondo resonance.The negative value of µα in the dilute systems at hα ∼ TKis an outcome of a short lifetime of conduction electrons.In addition to the large relaxation, low magnetic field withsmall energy interval of the Landau levels should make the-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 0  1  2  3  4  5  6µ α T K / Dhα / TKdiluteperiodic 0 20 40 60 0  1  2  3  4  5  6D / µα T KFig. 1. Magnetic field dependence of the enhancement factor µα andµ−1α(inset) of the orbital with gα > 0 for dilute systems, eq. (6),and for periodic systems, eq. (7).dHvA signals almost invisible. Therefore extrapolation ofthe mass enhancement from high field region leads to a di-vergence around hα ∼ TK. The inset of Fig. 1 shows the in-verse effective mass which tends to zero. On the other hand,in periodic systems at hα ∼ TK, the Fermi surface includes4f electrons. Thus the topology should change drasticallyas the field decreases.We now apply the above results to CexLa1−xB6. Spin-dependent enhancement factor µσ is evaluated by µσ =∑αwσ,αµα. Hence the character of each signal is domi-nated by the orbital having the largest value of µα, i.e., themost stable one in the magnetic field. According to a sim-ple calculation with the Γ8 CEF wave function, the lowestorbital under H//[001] includes up and down spins in theproportion of 16/21 to 5/21. Consequently, the conductionband with up spin is more difficult to observe due to theheavier mass and the large relaxation. On the other hand,the down spin is easier to observe, and µσ should behavelike Fig. 1 against magnetic field.We acknowledge valuable discussion with Prof. H. Shibaand Prof. H. Aoki.References[1] H. Miwa, T. Ando and H. Shiba, Solid State Commun. 8 (1970)2161.[2] G. Simpson and B. E. Paton, Phys. Letters 34A (1971) 180.[3] H. Shiba, Prog. Theor. Phys. 50 (1973) 1797; H. Shiba, Phys.cond. Matter 19 (1975) 259.[4] E. W. Fenton, J. Phys. F 6 (1976) 363.[5] W. Joss et al., Phys. Rev. Lett. 59 (1987) 1609.[6] N. Harrison et al., Phys. Rev. Lett. 81 (1998) 870.[7] G. Endo, Ph. D. thesis, Tohoku University (2005).[8] S. Nakamura et al., unpublished.[9] R. G. Goodrich et al., Phys. Rev. Lett. 82 (1999) 3669.[10] A. Wasserman, M. Springford and A. C. Hewson, J. Phys.:Condens. Matter 1 (1989) 2669.[11] S. Engelsberg and G. Simpson, Phys. Rev. B 2 (1970) 1657.[12] H. Shiba, private communication.[13] A. C. Hewson, The Kondo Problem to Heavy Fermions(Cambridge University Press, 1993).

The de Haas-van Alphen effect in Kondo systems with crystalline electric field

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The de Haas-van Alphen effect in Kondo systems with crystalline electric field

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