Phase transitions in highly-correlated f-electron systems with fluctuating valence

HAL Id: jpa-00218862

https://hal.archives-ouvertes.fr/jpa-00218862

Submitted on 1 Jan 1979

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Phase transitions in highly-correlated f-electron systems with fluctuating valence

S. Sinha, A. Fedro

To cite this version:

S. Sinha, A. Fedro. Phase transitions in highly-correlated f-electron systems with fluctuating valence.

Journal de Physique Colloques, 1979, 40 (C4), pp.C4-214-C4-217. �10.1051/jphyscol:1979466�. �jpa-

00218862�

JOURNAL DE PHYSIQUE Colloque C4, supplément au n° 4, Tome 40, avril 1979, page C4-214

Phase transitions in highly-correlated f-electron systems with fluctuating valence (*)

S. K. Sinha and A. J. Fedro (t)

Argonne National Laboratory, Argonne, Illinois 60439, U.S.A.

(f) Argonne National Laboratory, Argonne, Illinois 60439, U.S.A. and Northern Illinois University, Dekalb, Illinois 60115, U.S.A.

Résumé. — On présente un formalisme qui permet de traiter les électrons des couches f dans des niveaux multiplets fortement corrélés, avec couplage spin-orbite et également hybrides avec les états de valence ou de conduction. Ceci est obtenu par l'introduction d'opérateurs de transition entre 2 configurations éliminant les autres configurations. On utilise la méthode des opérateurs de projection de Zwanzig-Mori pour obtenir des résultats sur les fonctions de Green électroniques que l'on résout de manière autocohérente. Les solutions autocohérentes à l'ordre le plus bas sont assez voisines de celles de la théorie Hartree-Fock sans restriction.

Les termes d'ordre plus élevé présentent, sous certaines conditions, des singularités de type Kondo. On montre que différentes phases, tant configurationnelles que magnétiques, peuvent exister suivant les valeurs relatives de la largeur de bande des électrons de conduction, de l'énergie d'hybridation et des valeurs des intégrales de Coulomb et d'échange.

Abstract. — A formalism is presented for treating electrons in highly-correlated spin-orbit coupled multiplet levels in f-shells, which are also hybridized with valence or conduction states. This is achieved by introducing operators which cause transitions between the configurations, but project out all other configurations. The Zwanzig-Mori projection operator technique is exploited to derive results for the electron Green's functions, which are then solved self-consistently. The lowest order self-consistent solutions bear a certain resemblance to unrestricted Hartree-Fock theory. Higher order terms exhibit Kondo-like singularities under certain conditions. It is shown that several phases, both configurational and magnetic, are possible depending on the relative values of the conduction electron bandwidth, the hybridization energy and the values of the Coulomb and exchange integrals.

The band theory of metals, in conjunction with the local density functional formalism to account for exchange and correlation effects, has on the whole been very successful in explaining most of the properties of metallic systems, including the light actinide metals and compounds [1, 2]. However, in materials where the f-electrons are more localized and yet are still in the vicinity of the Fermi level, such as the rare earth metals and compounds of the so-called mixed valence type or the heavier actini- des, correlation effects in the f-shells must be treated more carefully for a proper description of the possible phases (magnetic or non-magnetic) which result, and of their temperature dependence. The basic problem with a completely band like descrip- tion is that it allows arbitrary numbers of electrons to hop into f-shells from conduction bands, thus allowing f"

+1

, f"

+2

, ... etc. configurations which actually have drastically reduced probability due to the large correlation energy on the f-shell. Further, it is hard to build in directly the spin-orbit and Hund's Rule interactions on the f-shells which give rise to the familiar multiplet structure in the free ion.

Let us suppose that there are two configurations f"

and f "

_1

between which it is possible for an f-shell to

(*) Work supported partly by U.S.D.O.E.

fluctuate without a large cost in correlation energy.

The approach suggested here is to start with the full J-multiplet structure of these configurations as basis and hybridize the level { J5

a

(f") - EJF~

l) } (the ato- mic f-electron excitation energy) with conduction

band states, while projecting out all the configura- tions f"~

²

, f"

⁺¹

, ... etc. This leads to a self-consistent

energy-band like picture but with some important

differences. For instance, all electrons do not obey Fermi-Dirac statistics as in the band picture. Loca- lized f-electrons obey statistics which becomes Boltzmann-like as the f-level moves away from the Fermi energy. The kinematic restrictions on the filling of the f-shell can also give rise to phase transitions where the f-shell fills continuously or discontinuously as the f-level is lowered relative to the conduction bands (depending on the Coulomb interaction between the conduction electrons and f-electrons) ; or where the f-shells can polarize magnetically (even without the indirect exchange mechanism). As we shall see, the magnetic suscepti- bility also behaves quite differently with tempera- ture from the Pauli susceptibility expected from a simple band model.

Let us assume for simplicity that the ground multiplet state of the atomic f"

_1

configuration is a singlet. Let us label states of atomic f" multiplet by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979466

PHASE TRANSITIONS IN HIGHLY-CORRELATED f-ELECTRON SYSTEMS ^C4-215

a . (These may refer to crystal-field-split states if necessary.) We define operators that create and annihilate an f-electron (or hole) in this configuration at site i, by

c : , ) f ' - ' , i ) = J f n , a , i ) c,,~ If", a , i ) = I f n - ' , i ) but with the property that

thus projecting out all other configurations. The commutation rules for these operators

are

similar to those of other Fermion operators except that

where the quantum average of the operator

QL,

^is

where ( q,, ) is the occupation of orbital y.

We may now write the Hamiltonian a s

where a:" is the usual creation operator for a conduction electron in orbital q with spin u on site i ;

T;ls2

is the conduction electron hopping matrix element,

V y

the hybridization matrix element with the f-level, U,

J

are the (intrasite) Coulomb and exchange integrals, and

We may now calculate the one-electron Green's function, bearing in mind the peculiar commutation rules for the c, c ' operators (eq. (3)). This is done using the Mori-Zwanzig Projection Operator forma- lism. We discuss here, for brevity, the lowest order (Mean-Field) result for the simplified case of an orbitally non-degenerate (spin doublet) f-level hybri- dizing with an orbitally non-degenerate (s-like) band with both spins. Let us label the space-and-time Fourier transforms of the (f-f) Green's function

and that of the (f-s) Green's function ( ( a , ,

,

^{; c:,,(t))) by}

The result is the pair of equations

[ w - a ' , l { l r ! r ( ~ ) - a ' , t ' , A l t ( w ) = ( ~ + , )

= I - ( % ,

>

with a similar pair of equations for down spins, and for the (s-S) and (s-f) Green's functions.

In eqs. (6)

a:", = E [ + U[(n,, ) + ( a , , ) I - J ( n s T ) +

+-

1

C

v , ( m , , > [ l - ( n , , ) I - '

N k

a: = [ I - ( n , , ) I V , + U ( m , ) - (7) - J [ ( m , > + ( m :

)I

a"'

_{k T} = s , ( k ) + UC(nfT )+ ( n , , ) I - J ( n , , )

where s,(k) is the unhybridized s-band energy ; V, the Fourier transform of V::,

and (mk

,

⁾ is the Fourier transform of

-<.:,

^{a,, ) -}

The solutions for the eigenvalues from eq. (6) are

1

,

,)

+

^{4 a}

',ti

'/(I - ( n,,

))I1''

(8) eZ[(a',l\ - a"'

and the projected density of f-states

etc. with similar equations for N,,(w) and f o r down-spins.

Finally the populations ( n,, ) ( n,, ) ( n,, ) ( n,, ) are obtained self-consistently from

( n,, ) = [ I - ( n.,

)I 1

d w ~ , , ( w ) f ( w ) (10) ( n 5 , ) =

1

d w ~ > , ( w ) f ( w )

C4-216 S. K. SINHA A N D A. J. FEDRO

Fig. 1.

-

Populations of upspin (full line) and downspin f-states Fig. 2.

-

Same a s figure 1 but with the hybridization ten times (dashed line) as unhybridized f-level is shifted relative t o the larger (and set equal to the conduction electron hopping matrix s-band. Conduction band hopping M.E. is 0.2 and f-s hopping elements).

M.E. is 0.02 in same units as ^{E , .}

Fig. 3.

-

^{,y versus T for E , = 0.6} and conduction band width and Fig. 4.

-

Same a s figure 3 but with E , = 0.8.

hybridization corresponding to figure 1.

where

f

(o) is the Fermi function. The (

m ,

^{) ( m}

,

⁾

are calculated from

etc. The main physical implications of the above equations are a s follows : (1) [( n , , )

+

( n,

, ^)I

^is

restricted t o be s 1. (2) The hybridization depends self-consistently on the occupation of the f-states.

Thus if ( n , , ) .= 1 , a

',2\

^{= 0,} s o that (s

4

) electrons cannot hop into (f

I

) states. Thus a s ( n , , ⁾ increa- ses, hybridization with (f

&

) states will decrease, the

(f ⁾ states will be lowered relative t o the (f

I

) states, providing a mechanism for cooperative spin polarization of the f-shells if f-level is near the Fermi energy (even in the absence of exchange).

Calculations have been performed for an fcc lat- tice with nearest neighbour (s-s) and (s;f) hopping, f o r various values of the s-band width (

W),

hybridi- zation strength (V). Figure 1 shows the self- consistent solutions for ( n,

,

^{) ( n , , ) as} is lowered relative t o the s-band. One

sees

that the f-shell starts to fill and than polarizes magnetically, becoming non-magnetic again a s E , goes well below the s-band.

(The exchange interaction

J

was set = 0 for this calculation.) Figure 2 shows the behaviour a s

V

is increased. Finally, figures 3 and

4

show the beha-

PHASE TRANSITIONS IN HIGHLY-CORRELATED f-ELECTRON SYSTEMS C4-217 viour of the susceptibility as a function of T higher

T

in figures 3 and 4 is due to thermal sampling (calculated by adding a small magnetic field term to of increasingly f-like hybridized states above the and ~ , ( k ) and computing the induced polarization) Fermi level. At lower temperatures, the main contri- for two different values of E,. At E , = 0.8 (see Fig. 1) bution is from the conduction band states.

the f-shell is only slightly occupied but paramagne-

S.

K. Sinha wishes to thank C.E.C.A.M., Orsay tic. At E , =

0.6

it is more occupied and thus closer to and Prof. C. Moser for their hospitality at a work- a polarization instability. The 1/

T

behaviour of

x

at shop where much of this work was carried out.

References

[I] KOELLING, D . D., J. Physique Colloq. 40 (1979) C4-117.

[2] BROOKS, M. S . S . , J. Physique Colloq. 40 (1979) C4-155.