A local mixed valence model

Solid State Communications, Vol. 31, pp. 885—888. Pergamon Press Ltd. 1979. Printed in Great Britain. A LOCAL MIXED VALENCE MODEL P. Schlottmann Institut für Theoretische Physik, Freie Universitat Berlin, 1000 Berlin 33, Arnimallee 3, Germany (Received 20 April 1979 by M Cardona)

We consider a purely electronic model for local valence fluctuations consisting of a spinless localized 41-level interacting with spinless extended Sd-states. We show that (a) the impurity model is related to the anisotropic Kondo problem, (b) a single 4f-level behaves like a resonant level with temperature-dependent width and (c) these results, when extended to a concentrated system of local incoherent mixed valence levels, are qualitatively in agreement with the main experimental features for mixed valence compounds.

IN A METALLIC ENVIRONMENT a group of rare earth ions (Ce, Sm, Eu, Tm) has a non-integer average number of 4f-electrons. In these systems two ionic configur1 have similar energies and ations, i.e. 4f’~and 4f’~~Sd, may coexist in the metal. These intermediate valence ions exist as impurities (Ce in LaAI 2 or Th~Lai_~) and as compounds or metals (SmS, TmS, Ce). Experimentally the thermodynamical properties of intermediate valence compounds are similar to those of isolated mixed valence impurities. The fluctuation rate between the two ionic configurations defines a characteristic energy of the system. For temperatures smaller than this characteristic energy mixed valence systems behave like a Fermi liquid [1). Inelastic neutron scattering shows the presence of a quasi-elastic peak and that the q-dependence of the cross-section is always that of the ionic form factor [2]. l’his leads to the picture of local and incoherent rather than extended and coherent valence fluctuations in compounds. This experimental evidence justifies considering a mixed valence compound as a concentrated system of isolated impurities. Intermediate valence compounds and metals may also show a discontinuous variation of the average number of 4f-electrons with pressure and temperature (SmS, Ce~Th1_~). In this letter we solve a simple model for an isolated 4f-level and extend the results to a concentrated system of non-interacting impurities. Intermediate valence S3(S tems are usually described by localized 4f-levels coupled to itinerant Sd-band states by a hybridization mixing term Vk, and local effective Coulomb interactions. The most simple impurity mixed valence model neglects spins and the orbital degeneracy and is given by H

=

~k ekd~dk+ E0f 7 + ~k Vkf(f~dk+ 47) 885

+ Ufdf~f ~ ~

(1)

kk’

The extension of this model to a “lattice of 4f-levels” is the spinless Falicov and Kimball model [3]. The model has been treated within the Hartree— Fock approximation [4, 5] by considering all possible factorizations including exciton-like correlations between f-holes and band electrons, (d ~f) For a concentrated system of 4f-levels the results depend on whether local uncorrelated or extended coherent valence fluctuations are considered. In the former case ~ = (f~f) may vary discontinuously with E~,(which parametrizes the pressure), while coherent states yield a large exciton condensation which smears possible discontinuities of flf for Vkf ~ 0. The slow motion of the 4f-electrons, however, prevents the formation of a stable molecular field and makes the Hartree—Fock approximation questionable. In this letter we (a) relate the impurity model to a strongly anisotropic Kondo Hamiltonian, (b) give an approximate solution of the impurity problem which contains the solvable limits correctly, and (c) extrapolate the results to a finite concentration of 4f-levels and thereby speculate on a discontinuous transition. 1. RELATION OF THE IMPURITY MODEL TO THE ANISOTROPIC KONDO PROBLEM Two particular limits of our model deserve special attention: (1) If Urd = 0 the Hamiltonian reduces to a resonant level and can be solved exactly. The2,f-level wherehas PFa Lorentzian is the density shape of states with at a width the Fermi of ~ level, = 1rPFand V fl~can be expressed in terms of the digamma function

886

A LOCAL MIXED VALENCE MODEL

n~,= ~-+-~-Im ~(i+~ ~2

E0—e~\ 2~i-T

2irT

(2)

(2) If V = 0 the impurity has no dynamics and the model reduces to a straightforward potential scattering problem [6]. This case also corresponds to the X-ray threshold problem [7]. Here, due to the deep hole created by the X-ray, the initial and final states are not the same and an infinite number of electron-hole excitations are emitted and absorbed near the Fermi level. In the present case “the X-ray spectrum” corresponds to the “exciton formation” response function ((f~d;d ÷1

[PF

—

W

2

+ E0

— CF il~,

(3)

= 1 —(1 6/ir) where 6 = arc tan (rrU,dpF) is the scattering phase shift. This expression diverges for Ufd > 0 and shows that the impurity is unstable with respect to perturbations of the f—d mixing type. Hence, as for instance in the Kondo problem, we have to distinguish between two fundamentally different cases: (a) if V 0, n,. is conserved (= 0 or 1) and has two degrees of freedom (as in the ferromagnetic Kondo case at the fixed point) and (b) if V = 0, fl~if not conserved and the f-level has a finite width as for antiferromagnetic Kondo coupling, where the groundstate is a singlet. It is not possible to relate the two cases by scaling or renormalization groups. Hence, the V = 0 case solved by Hewson and Riseborough [6] is not related to the actual mixed valence problem.

The close relation between the Kondo problem and equation (I) can be shown more rigorously. We perform in inverse order the several asymptotically exact transformations of the Hamiltonian discussed in [8]. In this way H takes the form of the anisotropic Kondo Hamiltonian with spin—flip (J coupling parameters given by 1) and spin—nonflip (Jf1)

Vol.31, No.11

coupling constant [J11~ IJjl, equation (4)],there are some differences with the usual (isotropic) Kondo problem. We give now the theoretical arguments that lead to this approximate treatment and list those results which can be taken over from [81 and [9]. The starting point is the Hamiltoman (1), where the resonant level is treated exactly and the U~interaction as the perturl~,ation.We have shown [91that the invariant coupling associated with the interaction U,d is not renormalized in leading logarithmical order. This cancel. lation is a consequence of the underlying X-ray threshold analogy [71. The resonance width however, ~,

is renormalized by the interaction U,.,~.We discussed two width: (1) Through the hybridization responseresonance function alternative ways to calculate the renormalized (~using renormalization group arguments [9] and (2) convoluting the X-ray spectral correlation function with the 4f-electron propagator [8]. Both procedures yield the same renormalization factor for ~A. The renormalized resonance width, S7(w), is selfconsistently determined by calculating the renormalization factor with the enhanced width f2(~).An approximate expression for ~2(w)is fl(w)

=

~exp (—~{~27TTPF + ~, (~+~ 2irT —

(5) where ~ is the solution of equation (5) at zero temperature and for w = 0 ~o = P~’E 1~Pr1i/(i+~) (6) Since ~ >0 the electron-hole excitations enhance the line width of the resonant level, ~2~> At high ternperatures, T ~‘ ~zo, .~ii,we have f2 = /~(2rrTpF)~ and for c.~= 0 and T~£2~,we obtain ~2(T) 2). In summary, at low temperaf20{ 1 (ri/6)ØrT/fo) tures we have a Fermi liquid behavior as expected for a ~.

—

J 1 = 2V, B =

—

JIIPF = %./2(UfdpF

E0

—

Urd(d~do>.

+ .12

—

I) (4)

Here B plays the role of the external magnetic field and (dd0) is the Sd-electron occupation number at the impurity site. This mapping is valid within the limitations of the long-time approximation [8] only. 2. APPROXIMATE SOLUTION OF THE IMPURITY PROBLEM The analogy to the Kondo problem su~8estswe follow the approximation scheme reported in [8] and [9], which was successful in providing a qualitatively correct Kondo susceptibility for small temperatures and external fields. Because of the strong anistropy of the

Kondo.like impurity with a characteristic energy f1~ and at high temperatures the impurity behaves essentially like a spin S 1/2 in an external magnetic field B. The approximation scheme discussed above interpolates between the resonant level and the X-ray problem;both special cases are exactly included [10]. If we associate a magnetic moment to the f-electron it is possible to define the magnetic susceptibility of the impurity as being proportional to ()~. Since the invariant coupling associated with Ufd is not tenormalized in leading order the susceptibility within the leading logarithmic approximation [8,9] is given by the x(’~’)of the resonance level with the renormalized resonance width ~Z(w= 0, T)

x(~.’)=

—

~:l w +i2ifl F \ 2 ~rc~

2irT

)

Vol. 31, No.11

A LOCAL MIXED VALENCE MODEL (7)

with the valence and since the 4f-level width is of the order of the phonon energies one expects the electron—

0 + U,a(d~do) only shifts the resonance but does not affect the width fZ(T). The static susceptibility is given by (di’ is the trigamma function)

phonon coupling to play a fundamental role in concentrated systems. Thus far, however, no conclusive experimental evidence for the importance of the phonons has been given and the main experimental facts can be explained with purely electronic models. This fact fits

—

~

_____ (~ +~ — + (E — E)J. 2irT /

887

-~

The effective f-level energy E =

x0

=

Re

~,‘(-~-

—

B

=E

+ ~2 IE) /2ir2T 2irT —

(8)

Depending on the values of i~and E, x0 increases or decreases with T for T f20. At high temperatures and forE near the Fermi level the susceptibility behaves like a Curie—Weiss law and forE T ~ ~20like ~Z(T)/~rE~.Within the above approximation scheme the average f-occupation number is given by equation (2) if ~ is replaced by ~2(T) and E0 byE. The dynamical susceptibility x”(w)/c&~has a central peak at c~,= 0 with a width of 2~Z(T)at high temperatures and2othe a width of about ~2(T) forinTaddition ~ ~ For susceptibffity shows bumps E ~ T, ~ at w ±E. They are the consequence of an energy threshold the are excitations: thermalthe bath and the resonance for width too small The to provide energy E necessary to fill or to empty the f-level, hence it must be ‘~

~

supplied by an external source such as photons or neutrons. The physical situation is very similar to a Kondo impurity at low temperatures in a very strong magnetic field [11]. These results agree qualitatively with susceptibility and inelastic neutron scattering measurements for mixed valence compounds [2]. 3. EXTRAPOLATION TO A CONCENTRATED SYSTEM OF IMPURITIES In order to extend the previous results to a many impurity system we assume that the valence fluctuations are mainly local and do not interfere with each other, i.e. we neglect the interactions between the impurities. The chemical potential has to be determined selfconsistently by taking into account the conservation of the total number of electrons. In addition we have to assume that the Fermi level is sufficiently far away from the top and the bottom of the conduction electron band such that the boson approximation for the electronhole excitations, used to obtain equation (5), is justified. Similar generalizations of single site solutions to finite concentrations of impurities have been carried out previously [4, 6, 12]. In the present case the picture of purely local incoherent valence fluctuations is supported by the q-dependence of the neutron scattering crosssection [2] and by a recent sum-rule discussion [13] of ~“(q, w), which shows that its q-dependence is weak. Since the rare earth ionic radii change considerably

into the picture of localofincoherent valence fluctuations, where the interference modes is expected to be destructive. For the sake of simplicity we neglect the lattice dynamics of the system. We assume that there are one f-state and n electrons per unit cell. We also assume that the shape of the conduction electron density of states is constant and not modified by the f-electrons. Measuring the energy from the bottom of the Sd-band the average number of conduction electrons, ~d, is given by U,dnf) (9) where p is the ischemical number off-electrons given bypotential. equation The (2) ifaverage E 0 CF is replaced byE0 + Ufdfld p. Defining 1)we obtain E’ = E0 U,dfl +‘1 (n &X2 Ufd p~ 1 ,~, = + ~2 2irT E’pF +(l 2UfdpFXnf~ 4)). (10) ~

=

= PF(P

—

—

—

—

—

—

—

—

—

~

—

—

2~TP~.

—

As a consequence of the assumed electron-hole symmetry equation (10) is invariant under the simultaneous change of sign of E’ and n~—4. The criterion for the order of the valence transition is now given by dE’1 2ir~T 1 2UfdPF1 = = + f2(T)~ PF

__j

—

(

—

~(I

2

2irT,

j

(11) If ~ <0 the transition is smooth, if ~> 0 the variation of n,. is discontinuous and the critical point is given by ~ = 0. Ajump of n~at zero temperature occurs only if Ufd >

—~—-

2PF

+ 2

The temperature and the resonance width tend to suppress the discontinuity. This result is qualitatively similar to that obtained by da Silva and Falicov [14] in the zero band-width limit. The 4f-electrons contribute to the specific heat with a Schottky-like anomaly since the degrees of free-. dom available at high temperatures are frozen out at low temperatures. The low temperature coefficient ~y, C = 7T, is strongly enhanced if the 41-level is in

888

A LOCAL MIXED VALENCE MODEL

resonance with the Fermi level. In the case of a discontinuous transition the specific heat diverges at the transition due to the large volume changes of the sample [15]. In conclusion, our mixed valence model behaves like a resonant level with temperature dependent resonance width. The (speculative) extension to concentrated systerm of 4f-levels has the properties of a Fermi liquid, in agreement with the main experimental results, i.e. the d.c. susceptibility, the 4f-electron occupancy, neutron scattering and the specific heat. We also stated the conditions for a first order transition within the local mixed valence picture. REFERENCES 1. 2.

C.M. Varma, Valence Instabilities and Related Narrow-Band Phenomena (Edited by R.D. Parks), p. 201. Plenum Press, New York (1977). E. Holland-Moritz, M. Loewenhaupt,W. Schmatz & D. Wohlleben,Thys. Rev. Lett. 38,983 (1977);

3. 4. S. 6. ~ 8. 9. 10. 11. 12. 13. 14. 15.

Vol. 31, No. 11

S.M. Shapiro, J.D. Axe, RJ. Birgeneau, J.M. Lawrence & R.D. Parks, Phys. Rev. B16, 2225 (1977). LM. Falicov & J.C. Kimball, Phys. Rev. Lett. 22, 997 (1969). D.I. Khomskii & A.N. Kocharjan, Solid State Commun. 18, 985 (1976). HJ. Leder, Solid State Commun. 27, 579 (1978). A.C. Hewson P.S.(1977). Riseborough, Solid State Commun. 22, &379 P. Nozières & C.T. de Dominicis, Phys. Rev. 178, 1097 (1969). P. Schlottmann,J. Phys. 39, C6, 1486 (1978). P. Schlottmann, J. Magn. Magn. Mat. 7, 72 (1978). If ~lo = 0 spectrum, in equation (6) we recover the limit X-ray threshold equation (4), in the T-~0. W. Gotze& P. Schlottmann, J. Low Temp. Phys. 16, 87 (1974). D.C. Mattis,Phys. Rev. Lett. 36,483 (1976). C.E.T. Goncalves da Silva, Phys. Rev. Lett. 42, 1305 (1979). C.E.T. Goncalves da Silva & L.M. Falicov, Solid State Commun. 17, 1521 (1975). J.M. Markovics & R.D. Parks, [1], p. 451.