COMPATIBILITY OF THE FRIEDEL-ANDERSON VBS AND IONIC APPROACHES TO THE THEORY OF MAGNETIC IONS IN METALS

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COMPATIBILITY OF THE FRIEDEL-ANDERSON

VBS AND IONIC APPROACHES TO THE THEORY

OF MAGNETIC IONS IN METALS

S. Barnes

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-828

COMPATIBILITY OF THE FRIEDEL"ANDERSON VBS AND IONIC

APPROACHES TO THE THEORY OF MAGNETIC IONS IN METALS

S.E. Barnes

DPMC, Eaole de Physique, Geneva, Switzerland

Résumé.- Une formulation unique reproduit â la fois l'approche ionique et celle de Friedel-Anderson. Le blocage du moment cinétique orbital sans champ cristallin est décrit.

Abstract.- A single formulation reproduces both the ionic and Friedel-Anderson schemes. The quenching of angular momentum without crystal fields is described.

1. INTRODUCTION.- In this contribution I wish to correlate new results with those already published (Barnes /2/) in order to compare the ionic (Hirst

Ik I and Friedel /3/ - Anderson /l/ virtual bound state (vbs) approaches to magnetic ions in metals. I show that while a vbs approach is always valid, the ionic approach is capable of describing nearly all properties in the magnetic limit. An important exception is the impurity density of states. The magnetic-non-magnetic transition is smooth. A sketch of a theory for the covalent quenching of angular moment is given i.e. quenching without crystal fields

2. NEW FORMULATION.- In order to extend the formu-lation to the non-degenerate Anderson model :

H " I \o\a\a + l ednd a+I U J,. nmaVa' +

k o — — — am moym a

+ *•'• + I ("Tr a+ a +hc) (2.1)

ion , *• N ma ka

_kma —

where H! contains all ionic non-Coulombic terms, ion

a set of auxiliary fermion a-„ . •, . and boson b„ . operators 0 . are introduced. These

corres-2n,i n,i _ + _ +

pond to impurity configurations e.g. i=(2 ,1 ,0 ) for an n = 3 electron configuration. An arbitrary operator 0 becomes :

I 0 ^ <ni|0|n'i'> 0n,., (2.2)

ni n1 i'

Explicitly the Hamiltonian becomes :

H = I eka ak c A c+ l(£d+(2n+,)U)al2n+l)ia(2n+.)i+

ka — — — m

^ ^ V ^ L ^ n i - i o n ^ ^ ^ ^

{V, (a+ , . , b .+b+ J_n.,am.)a. + h c ) } (2.3) km n + l i1 n i n + l i ni ka

where i1 = f(i) implies e.g. if i = (2 ,1 ,0 ) and m = 1 then a = + i and i' = (2+,l~, 0+) . Only auxi-liary states with total occupation unity are equi-valent to the original states. Feynman diagrams via Wick's theorem are obtained with an Abriksov pro-jection (see Barnes / 2 / ) . The propagators and ver-tices are illustrated in figure (la).

2n»1i

?V- -iZ—Clfi

-iS—c^'i (la)

ko-

S

^

_ 1 i

^ l

n i 0 a )

Fig. 1

3. IONIC VS FRIEDEL-ANDERSON PICTURES.- Calculations can be performed with either the auxiliary propaga-tors, corresponding to the ionic scheme, or with the true d-electron propagators, corresponding to the vbs scheme.

i) Auxiliary propagators e.g.

t

< TTa( 2 n + l ) i ^ ' a(2ni1)i^°^> c o r r e sPo n d i n8 t o t h e ground configuration are sharp. In figure (2a) is shown, for the non-degenerate Anderson model, the

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D(E) ME)

E= 2in: W2 E.2intO E = i & Fig. 2

spectral density for the two singly occupied, stable, s = f

*,

configurations. The principal pole has a small many body width -47~(pJ,~~)~k~,

Jeff =

v2

[

-

(E~+u)-'

3

.

In contrast the unsta- ble bose propagators have broad poles : figure (2b). Most physical properties e.g. the static and dynamic susceptibilities, the RKKY interactions etc, corres- pond to propagators (normally involving two real

' +

t

d-electrons, e.g. S = a a

₊

) which can be written in terms of auxiliary fermion propagators

alone

.

For example figure (Ib) shows Dyson's equation for the above fermion propagator ; this alone determines the polarisation (nd+-nd+), in the magnetic regime. The bose propagators which occur as intermediate states are evaluated far from their own energy shells and can be viewed as weakly energy dependent (exchange) interaction lines. Thus the ground state properties in the magnetic limit, as manifested through the susceptibility etc, are completely consistent with the ionic model.

ii.2 True d-electron propagators e.g.

<T at ( r ) ,amo(0)> are always broad. These are cal-

T mo

culated with the auxiliary representation

t t

t

a,~(a~+~~,b .+bn+li,ani) i.e. d-electron propagators nl

are given by bubble diagrams. Again for the non-degenerate model, the relevant vertex equations are shown in figure (Ic). For the symmetric case, and with self energies on their energy shells, this leads to the following secular equation :

which has the solutions E = 2i~~J(~/2)2-~2. The schematic density of states are shown in figure (2c). For the non-magnetic limit U+O, E = iA in agreement with Anderson / I /. In the magnetic limit

Anderson, but with double the vbs width. The transition is smooth.

For the magnetic limit, in principle, the static susceptibility could be evaluated with the true d-electron propagator. This gives a trivial identity, as it must, since required in this limit are simple Boltzman not Fermi statistics.

The present formalism is not restricted to the simple ionic ground state. Covalent mixing, without crystal fields, can quench the impurity angular momentum. Figure (Id) shows the equivalent of (Ic) for

3

+

the simplest example Ce

,

4f1. In a free electron model angular momentum is a good quantum number and there is no quenching. This is not so for Bloch states, figure (le), different angular momenta are cou- pled, via adjacent configurations. The coupling energy is therefore of order (A2/u) and

can

overcome the spin-orbit energy 5 well within the magnetic regime.

References

/I/ Anderson, P.M., Phys. Rev. ]24 (1961) 40 /2/ Barnes, S.E., J. Phys. (1976) 115, 1375, and

F7 (1977) 2637

-

/3/ Friedel, J., Nuovo Cim. (Suppl)

1

(1958) 287 /4/ Hirst, L.L., Proc. 1974 Conf. Magnetism and

Magnetic Materials (New York : A/P, 1974).