Electron interactions and ferromagnetism in metals

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Electron interactions and ferromagnetism in metals

D.M. Edwards, E.P. Wohlfarth

To cite this version:

D.M. Edwards, E.P. Wohlfarth. Electron interactions and ferromagnetism in metals. J. Phys. Radium,

1959, 20 (2-3), pp.136-137. �10.1051/jphysrad:01959002002-3013600�. �jpa-00236002�

136

ELECTRON INTERACTIONS AND FERROMAGNETISM IN METALS

By ^D. M. EDWARDS ^{and E. P.} WOHLFARTH,

Departments ^of Mathematics, Queen Mary College ^and Imperial College, London, England.

Résumé. 2014 On propose que les effets des corrélations entre les électrons dans les métaux ferro-

magnétiques, qu’on ^ne saurait oublier dans toute discussion ayant pour base la théorie des élec- trons collectifs, ^soient pris ^en considération dans la théorie des plasmas. On introduit ainsi quelques aspects ^de la théorie des ondes de spin.

Abstract.

It is suggested that the effects of electron correlation in metallic ferromagnetics,

which must be included in discussions based on the collective electron treatment, could be consi- dered from the point ^{of view of} plasma theory, ^{and that some} features of the spin ^{wave treatment}

are thereby introduced.

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM TOME} 20, ^FÉVRIER 1959,

1. Introduction. -The deficiencies ^of the collec- tive electron treatment of ferromagnetism ^{are well} known, ^{the most} important ^one being ^{that corre-}

lation efîects are not properly ^{included. In this}

paper ^a discussion of the interactions of magnetic

carriers is given ^{on the basis of} plasma oscillation

theory. This leads ^to the conclusion that some

aspects ^{of the treatment are conserved but that} spin ^wave characteristics _may also be present ^{as a}

result of the correlation effects. Hence _{many of} the experimental ^facts previously inexplicable ^{on a}

pure collective electron treatment (e.g. neutron ^dif-

fraction [1]) ^{are more} readily understood.

2. The interaction of magnetic ^carriers.

The carriers of ferromagnetism ⁱⁿ the transition metals

are the holes in the d shells. Owing ^{to the reci-} procity ^{of holes and} electrons, ^{which is} justifiable

even for interacting carriers, ^{it is} convenient to consider the carriers ^to be electrons. Two cases

have been considered : Firstly ^a model with a

single ^{energy band} containing ^one electron per atom, ^{the electrons} being subject ^to interactions ;

in the state of complete saturation every ^state in this band is occupied. Secondly ^{a similar model}

but with q electrons per atom, where q ^1.

The first case has already ^{been treated} by

Slater [2]. ^{He found that the states of lowest}

energy with ^a single ^reversed spin ^{have wave func-}

tions describing ^Bloch spin ^{waves with an admix-}

ture describing " polar spin waves " corresponding

to the removal of the electron with reversed spin ^to

a nearest neighbour. ^{These are states} of very

high correlation ; they ^have energies split ^{off from}

the band-like continuum of states with lower corre-

lation which arise from the removal of the electron to a distant atom.

In the second case (q 1) ^the system ^{to be} considered is a crystal containing ^N singly charged ions, ^{a uniform} background ^of negative charge ^of density (1- q) ^e per atom, ^{and n .---} qN ^electrons.

When q « ^{1 the} electronic interaction is unim-

portant ^{so it is} supposed ^here that q - ^{1. Slater’s} configuration interaction approach ^is impracticable owing ^{to the} high degeneracy ^{of the} unperturbed

states. A consequence of this degeneracy ^{is that}

even for complete saturation the lowest states are

part ^{of a band-like} continuum. This problem may be treated on the basis of plasma oscillation theory.

The Coulomb interactions in the system ^ensure

that the n electrons are fairly uniformly ^distri-

buted throughout ^the crystal, ^and plasma ^oscil-

lations occur about such an equilibrium ^distri

bution. An analysis ^{of the} type given by ^Nozières

and Pines [3] ^makes possible, ^{to a} good approxi- mation, ^a separation ^{of the} plasma ^and particle

motions. The particles ^then interact with a

screened Coulomb interaction and their wave

functions are subject ^{to certain} subsidiary ^condi-

tions. The plasma oscillations are normally ⁱⁿ

their ground ^{state so that} they ^{make a constant}

contribution to the energy and need not be consi- dered _any further here. Evidence that the d elec- trons contribute to plasma oscillation in the tran- sition metals is provided by experiments ^{on the}

energy losses of fast electrons in thin films. The Hamiltonian for the particle ^{motion is}

where V(r2) ^{is the} potential ^{energy of the ith} par- ticle in the field of the ions and a uniform negative charge ^of density ^e per atom, ^and

is the short-range part of the Coulomb interaction

(kil - interatomic distance). ^{It is} proposed ^to

treat H,,.,. ^{as a} perturbation ^{and the} unperturbed

wave functions are the eigenfunctions ^{T of the} independent particle HamiltoniAn Ho ^which satisfy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:01959002002-3013600

137

the subsidiary conditions. An approximate ^form

for the subsidiary conditions is

where Pk = £ exp (- ik. Ti), ^the density ^fluc-

i

tuations of the electrons. Since k,-’ - interatomic

distance, equations (2) imply ^that density ^fluc-

tuations can only ^{occur within} regions ^{of this order.}

In addition, ^{since the number} of electrons per ^atom is only of the order 1, the fluctuations must be small even within these regions. Thus the subsi-

diary conditions must ’reduce to a considerable extent the probability ^{of two electons} coming together ^{on the same atom. This is} significant ^for

the calculation of Hs.r. >, ^the expectation ^value

of the short-range interaction, ^{which includes the} exchange energy and determines the " molecular field ". Thus wave functions must be used which

satisfy, ^{at least} approximately, ^the subsidiary ^con-

ditions (2). ^The required ^{functions are linear com-} binations of degenerate ^and nearly degenerate eigenfunctions ^of Ho ^{i.e. linear} combinations of Slater determinants all having nearly ^{the same}

total one-electron _energy. Thus the one-electron energy distribution is nearly ^{the same as in a} simple

band calculation. Owing ^{to the} strong correlation

implied by (2) ^the expression ^for Hs.r. > will

now contain mainly ^{terms like nearest} neighbour exchange integrals involving Wannier functions.

Thus the molecular field will depend mainly ^on

interatomic effects, ^the intra-atomic interactions

being greatly ^reduced. Owing ^{to the} strong ^corre-

lation also the movement of a. reversed spin ^will

have close similarities with the movement of a spin

wave, the increase of .Ss.r. > due to a spin ^rever-

sal depending ^{on the} change ^{in total momentum.}

The coexistence of band-like one-electron energies

and spin-wave-like exchange energies suggests ^that aspects ^{of both the} collective electron treatment and the spin-wave ^{treatment are} present. ^A quan- titative discussion of this problem ^{involves a calcu-}

lation of the ’Y to satisfy (2), leading ^{to a calcu-}

lation of the total energy spectrum ^which may be

applied ^{in a} statistical calculation of temperature effects.

3. Previous work.

(a) ^Friedel [4] ^introduces

the idea ^{of a} highly ^screened Coulomb interaction

but since the subsidiary conditions (2) ^{and hence}

correlation are apparently ignored the only ^contri-

bution to the exchange energy is ^an intra-atomic

one. ( b) ^Marshall [5] ^has applied ^{the ideas of}

Van Vleck [6] ^{on the movement of holes} through

the lattice, ^{states of} high ionisation being ^excluded.

The conclusions of this work may be similar to that described ^here. (c) Herring ^{and Kittel’s} [7] ^dis-

cussion leads ^to results like Slater’s for an insulator and suggests ^that spin ^wave type states may also exist in metals.

REFERENCES

[1] ^ELLIOTT (R. J.), ^Proc. Roy. Soc., 1956, ^A 235, 289.

[2] ^SLATER (J. C.), Phys. Rev., 1937, 52,198.

[3] ^NOZIÈRES (P.) ^{and PINES} (D.), Phys. Rev., 1958, 109,

741.

[4] ^FRIEDEL (J.), ^J. Physique Rad., 1955, 16, ^829.

[5] ^.MARSHALL (W.), ^Nuovo Cimento, Suppl. 3, 1957, 6,

1186.

[6] ^{VAN VLECK} (J. H.), ^{Rev. Mod.} Physics, 1953, 25, 220.

[7] ^HERRING (C.) and KITTEL (C.), Phys. Rev., 1951, 81,

869.

DISCUSSION

Mr. Pratt.

You have pointed ^out that it is Hs,r.

which is the determining ^{factor in} deciding ^whether

on not the ground ^{state is} ferromagnetic. ^But

since there is a Fermi hole’around ^an electron ^so far as other electrons of the same spin ^{are con-} cerned, ^{it must be that} Hs.r. ^{is a} spin dependent quantity. ^{It would then seem that} examination of the ground ^state by Hsjr, ^{must be a rather delicate} question. ^How do you modify ^{the random} phase approximation ^{which I} gather you have made, ^{so as}

to accurately ^describe exchange interactions, ^which

of course are vital to the magnetic ground ^{state ?}

Mr. Edwards.

The screening ^{which reduces the}

interactions to Hs.r. ^{is due to} purely electrostatic ^in- teraction (i.e. ^that giving ^{rise to the coulomb} hole)

and should thus be independent ^of spin. ^{It is} expected ^{that the} exchange interactions are of short range and ^so the exchange energy may be treated

using ^the independent particle Hamiltonian invol-

ving Hs.r..