DYNAMICS OF VALENCE-FLUCTUATING Tm IMPURITIES

A new formalism is presented for describing valence fluctuating Tm impurities. Starting from the selfconsistent perturbation scheme valid at arbitrary temperatures, the present formalism incorporates not only the valence fluctuation between 4f12and 4f13, but also the coupled 4f13 and a conduction electron scattering channel through Tmatrix, the latter yielding a singlet ground state at low temperatures. Contribution of this channel to the f-electron spectral function is also discussed. Tm ion has a unique feature among rare earth elements which show mixed valencies when dissolved in host metals and compounds [I]. It fluctuates between 4f12 (Tm3+)and 4f13 (Tm2+), but both states have nonvanishing total angular momentum (j = 6 for Tm3+ and J= 7/2 for Tm2+), which may be a source that most mixed valent systems containing Tm order magnetically at low temperatures 121. On the other hand, an isolated magnetic ion embedded in a metal can hardly stay magnetic down to absolute zero temperature except some pathological cases [3,4]. Thus the ground state of Tm ion in a metal may be a singlet, as was maintained in preceeding theoretical studies. Yafet et al. [5] was the first to investigate the ground state of Tm impurity by the variational method, and the 1/N studies by Read et aI. [?I followed. Since these studies were limited to the ground state properties, there remains much to be elaborated compared to simpler valence fluctuating systems like Ce ions, where the sophisticated theories have been developped for statics and even dynamics [8]. In the following we develop a theory for valence fluctuating Tm impurity alongsuch scheme in pararell with that for Ce systems. Neglecting the crystal field, the f-part, can be expressed as the sum of the partition function, Zf, three different contributions Here, [j] = (2j + 1) and [J] = (25 + 1) denote the degeneracy of the states. Rj (z) = (z Ej Cj (z))-' and R J (z) = (z EJ C j (z))-l are called as resolvents, describing the f l2 and f l3 sectors, respectively, and C j (z) and C J (z) are the selfenergy functions determined by the following coupled equations (see Figs. l a and b) : Here, I? denotes the mixing matrix element defined by 2 r = BAWA, where A = 5/2 and 712, WA = pVA A (VA = < f1 AX IVl c ' l e ~ ~ > is the single particle mat r ixelement)andB~= [ < J I I f$llj > I 2 / ~ ] [ ~ ] i s t h e geometrical factor including reduced matrix element (B5/2 = 1/56, B7/2 = 3/14). These equations are the same as in [9], where the singlet-forming channel is not taken into account. The last term in equation (5) describes the contribution from the singlet bound state, which consists of f13 and one conduction electron. This state is orthogonal to f12 and f13 channels (the first and the second term in Eq. (5)) since the number of local electrons is different [lo]. RB (z), the resolvent for this bound state channel, is expressed with the T-matrix T,,! (z) for f13 and a conduction electron with incoming and outgoing energies E and E ' , respectively, as RB (z) = p2 Jlo d~ 1: dsl (1 f (E)) (1 f ( E ) ) X 00 RJ (Z E ) x T,,r (z) R j (z E ' ) . (7) Fig. 1. (a) (b) Diagrams for Cj (z) and C J (I). Single and double dash lines denote Rj (a) (4f12) and RJ (a) (4f 13), respectively, and the solid line denotes conduction electrons (holes). (c) Diagram for the T-matrix T,,r (z) (depicted by hatched box). Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888318 C8 704 JOURNAL DE PHYSIQUE The argument z denotes the total incoming energy of composite f13 + c1 state. The T-matrix satisfies the following equation (see Fig. lc) where I? B = P V ~ I<~llf$llj>1~ /[q2 = pv? (391112). It is not easy to solve equation (8) even numerically, but from the analysis in the limiting cases (see the following section) one infers that the T-matrix T,,, (z) may be singular when the total energy of the composite particle, z, coincides with the singlet binding energy, and the dependencies on E and E' may be weak. Therefore, we replace T,,I (z) in r.h.s. of equa tion (8) with its average T (2) = < T,,I (z) >,,I, where the average over E and E' is taken with the weight factor g (&,&I; z) = (1 f (E)) (1 f (E)) RJ (z E) .RJ (z E') . Then, we can solve equation