Vol. 115 (2009) ACTA PHYSICA POLONICA A No. 1 Proceedings of the European Conference \Physics of Magnetism" (PM'08), Pozna¶n2008 Kondo Lattice and Magnetic Properties in Strongly Correlated Electron Systems B. Coqblin¤ Laboratoire de Physique des Solides (UMR 8502), CNRS, Universit¶eParis-Sud 91405 Orsay, France We review here the magnetic properties of some strongly correlated electron systems and especially cerium and uranium compounds. The normal Kondo lattice model with a localized S = 1=2 spin can account for the Kondo-magnetism competition observed in cerium or ytterbium systems, while the underscreened Kondo lattice model with a localized S = 1 spin can well account for the ferromagnetism | Kondo coexistence observed in some uranium compounds such as UTe. Then, we discuss the spin glass-Kondo competition and present the resulting phase diagrams showing spin glass, Kondo and magnetically ordered phases observed in disordered heavy fermion cerium alloys such as CeNixCu1¡x alloys. PACS numbers: 71.27.+a, 72.15.Qm, 75.10.Nr, 75.20.Hr 1. Introduction tion [7] or also the \Kondo collapse" model [8, 9]; in this model, the e®ective Kondo temperature is taken to be It is well known that rare-earth metals are magnetic much larger in the ® phase than in the γ phase and the with a magnetic moment corresponding to the con¯g- e®ective valence equal to the number of conduction elec- uration 4f n with integer n values, except the \anoma- trons remains relatively small of order 3.3 in the ® phase. lous" metals, cerium, europium and ytterbium metals. Thus, in spite of di®erences in the theoretical explana- An anomalous behavior has been also observed in some tions, the phase diagram of cerium can be considered as alloys and compounds containing these three rare-earths, a very good example of the di®erence between the mag- but also praseodymium, samarium and thulium. On the netic case with an almost integer number of 4f electrons other hand, magnetism occurs only for curium in the N and the \intermediate valence" case with an interme- middle of the series of the actinide metals, but many f diate N value. Moreover, the number of 4f electrons is uranium, neptunium, plutonium and americium systems f often not an integer and varies with the external pres- are known to be magnetic or to present an anomalous sure or the matrix concentration in many systems, like behavior [1{3]. Eu(Pd Au ) Si alloys [10] or YbInCu [11]. Cerium systems, which present several di®erent behav- 1¡x x 2 2 4 iors, have been extensively studied. First of all, the phase On the other hand, Kondo [12] explains in 1964 the diagram of cerium metal is very interesting. At room decrease in the magnetic resistivity in logT and, there- temperature and normal pressure, cerium metal is in the fore, the minimum of the total electrical resistivity, by a γ phase which is almost trivalent with a number of 4f perturbation calculation performed on the spin exchange electrons very close to 1 and is magnetic with a magne- Hamiltonian given by JKscSf in the case of an antifer- 1 tization corresponding to the 4f con¯guration. When romagnetic coupling with JK > 0. The \Kondo e®ect" pressure is applied at room temperature, cerium under- has been observed in many magnetic dilute alloys such as goes a ¯rst-order transition from this γ phase to the ® CuFe, LaCe, AuYb, and also in many compounds with phase which is no more magnetic and has an \interme- cerium or ytterbium or with other anomalous rare-earths diate valence" with a smaller number of 4f electrons [4]. such as PrSn3 or TmS. Then, at very low temperatures, The ¯rst theoretical explanation was based on the \pro- the single Kondo impurity has a heavy fermion behavior, motion" model and starts from a treatment of the spin characterized by very large values of the electronic spe- and orbitally degenerate Anderson Hamiltonian, which ci¯c heat constant γ and of the coe±cient of the T 2 law of gives a transition from a magnetic trivalent phase to a the electrical resistivity and the localized magnetic mo- non-magnetic intermediate valence one [5]. Indeed, there ment Sf = 1=2 is completely screened by the conduction have been several other theoretical explanations of the electron spins sc = 1=2 at T = 0. phase diagram of cerium, based on a magnetic transition The Kondo behavior corresponds to a number of f elec- within the promotion model [6] or on the Mott transi- trons close to an integer and it is well known that we can separate at ¯rst sight the anomalous behaviors observed in rare-earth or actinide strongly correlated systems in either \intermediate valence" or \Kondo" ones. Here, ¤ e-mail: [email protected] (13) 14 B. Coqblin c we will review only the Kondo systems and we will dis- For ½JK values smaller than the value ½JK of the cuss essentially the case of a Kondo lattice in strongly quantum critical point Q, cerium compounds are mag- correlated electron systems. netically ordered with a small heavy fermion character Thus, we will review here briefly three main recent the- and with relatively small ordering temperatures, typi- c oretical results on the normal Kondo lattice model with cally of order 5{10 K. For ½JK values larger than ½JK, localized spins S = 1=2 which can describe the competi- cerium compounds are non-magnetic and can have a very tion between the Kondo e®ect and the magnetic ordering large heavy fermion character with a γ typically of order in cerium compounds [13, 14], then on the underscreened 1 J/(mol K2) [1]. Kondo lattice (UKL) model with localized spins S = 1 The Kondo lattice model has been studied in detail which can be applied to the Kondo-ferromagnetism co- within the mean-¯eld approximation [14]. We have in- existence observed in some uranium compounds [15] and troduced a Kondo lattice model with both an intrasite ¯nally on the competition between the Kondo e®ect and Kondo exchange interaction and an intersite magnetic the spin glass in disordered cerium alloys [16]. exchange interaction between neighboring localized spins and the resulting Hamiltonian is, therefore, given by 2. The Kondo lattice model X X X c f H = "kn + E0n + JK si ¢ Si In the regular Kondo problem, the spin S = 1=2 of the kσ iσ f kσ iσ i localized 4f electron (or hole) corresponding to the 4f 1 X 13 (or 4f ) con¯guration is completely screened at very low +JH Si ¢ Sj; (1) temperatures by the spin sc = 1=2 of the conduction elec- hiji trons. It is well known that, in the regular Kondo lattice, where "k is the energy of the conduction band, JK (> 0) there is a competition between the Kondo e®ect and the is the Kondo coupling between the localized spin Si and magnetic ordering arising from the Ruderman{Kittel{ the spin si of a conduction electron at the same site and Kasuya{Yosida (RKKY) interaction between rare-earth JH is the interaction between nearest-neighboring local- atoms at di®erent lattice sites, which has been well de- ized spins. We consider here the case of choosing JH to be scribed by the Doniach diagram [17]. This diagram, positive, implying that intersite interactions are antifer- shown in Fig. 1, gives the variation of the ordering tem- romagnetic, as it is the case of most cerium compounds. Then, in all these methods used to describe both the Kondo e®ect and either short-range magnetic correla- tions or a given magnetic order (which could be antifer- romagnetic, ferromagnetic or spin glass . ), we use two types of operators, describing the Kondo correlations or the magnetic correlations. In our ¯rst study [13], we have used a mean-¯eld approximation with two correlators, ¸, describing the intrasite Kondo correlation, and ¡ , repre- senting an intersite correlation between two neighboring moments + + ¸ = hciσfiσi; ¡ = hfiσfjσi: (2) Here, we summarize briefly the magnetism-Kondo competition and we will discuss here the problem of Fig. 1. The Doniach diagram giving di®erent temper- \exhaustion" principle which has been introduced by atures versus ½JK. TK0, TN0 and TN design respectively the single-impurity Kondo temperature, the correspond- Nozi`eres [21]. Detailed calculations are presented in ing N¶eeltemperature and the N¶eeltemperature for the Refs. [13, 14]. In the mean ¯eld approximation, the lattice. Kondo temperature TK and the correlation temperature Tcor are de¯ned as the temperatures at which, respec- perature and of the Kondo temperature with increas- tively, ¸ and ¡ become zero. We have studied the exis- ing JK. The resulting ordering temperature TN of the tence of the Kondo phase (with ¸ di®erent from zero) and lattice is increasing initially with increasing ½JK, then of the short-range magnetic phase (with ¡ di®erent from passes through a maximum and tends to zero at a critical zero), as a function of the two parameters JK and JH , but c value ½JK corresponding to a \quantum critical point" also of the number nc of conduction electrons. Figure 2 (QCP). Such a behavior of TN has been experimentally shows the e®ect of the di®erent parameters: JK tends observed with increasing pressure in many cerium com- to favor the Kondo e®ect, while JH tends to favor mag- pounds, such as CeAl2 [18] or CeRh2Si2 [19]. A similar netism and is opposite to the Kondo e®ect. Moreover, the e®ect has been observed in YbCu2Si2 [20] or in other yt- Kondo temperature and the Kondo phase decrease when terbium compounds, where the N¶eeltemperature starts nc decreases, as clearly shown in Fig.