EXCITATIONS ÉLÉMENTAIRES DANS LES FERROMAGNÉTIQUES, DIFFUSION DES NEUTRONSELEMENTARY EXCITATIONS IN ITINERANT ELECTRON FERROMAGNETS

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EXCITATIONS ÉLÉMENTAIRES DANS LES FERROMAGNÉTIQUES, DIFFUSION DES NEUTRONSELEMENTARY EXCITATIONS IN

ITINERANT ELECTRON FERROMAGNETS

T. Izuyama

To cite this version:

T. Izuyama. EXCITATIONS ÉLÉMENTAIRES DANS LES FERROMAGNÉTIQUES, DIFFUSION DES NEUTRONSELEMENTARY EXCITATIONS IN ITINERANT ELECTRON FERROMAG- NETS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-809-C1-811. �10.1051/jphyscol:19711285�.

�jpa-00214314�

EXCI TA TIONS E L ~ M E N TAIRES D A NS L ES FERROMA GNETIQUES, DIFFUSION DES NEUTRONS

ELEMENTARY EXCITATIONS

IN ITINERANT ELECTRON FERROMAGNET S

by T. IZUYAMA

Institute of Physics, College of General Education, University of Tokyo, Komaba, Tokyo, Japan

and

Service de Physique Thtorique, Centre d7Etudes Nucltaires de Saclay BP no 2, 91, Gif-sur-Yvette, France

RBsumb.

On montre d'une f a ~ o n rigoureuse qu'un ferromagnktique non relativiste ayant une symktrie d'inversion et un ktat fondamental

peine dkgknkrk, contient toujours un, et seulement un, mode de magnon acoustique et son spectre

est aigu. On obtient l'expression la plus gknkrale pour

On dkveloppe une approche gknQale aux dynamiques de spin des liquides ferromagnktiques de Fermi

basses tempkratures. On obtient les dependances du moment et de la temp6rature sur l'amortissement des ondes de spin froides.

Abstract. -

It is shown rigorously that a non-relativistic ferromagnet with inversion symmetry and quasi-non- degenerate ground state contains always a single and only single acoustic magnon mode and its spectrum is

which is sharp. The most general expression for

is obtained. A general approach to the spin dynamics of ferromagnetic Fermi liquids at low temperatures is developed. The momentum and temperature dependences of the damping of cold spin waves are obtained.

I. Introduction.

The purpose of this paper is to give a general theoretical approach to the spin dyna- mics in the itinerant electron ferromagnets at low tem- peratures without resorting to the approximations such as R. P. A. or higher R. P. A. that may easily be criticized. The approach is based, however, on the assumption that the ferromagnets are described by the Landau picture of Fermi liquid [I]. The most basic difference between the ferromagnetic Fermi liquid considered here and the normal paramagnetic one is that we have now magnons in addition to the indivi- dual-particle-like excitations as the low-lying excita- tion modes.

As a lemma, it is shown firstly that any non-relati- vistic ferromagnet should have a sharp or well-defined magnon mode with the excitation spectrum w

~q~

for small wave numbers. The proof is quite general and has been performed rigorously. We have to assume, however, that there is no such acoustic mode as that with vanishing contribution to the transverse spin- spin correlation function S+ -(q, w) in the limit of small q. Then, according to Landau, the low-lying excited states are uniquely labelled by ( nPk, } and { vq ), where n,,,

[the number of the quasi-electrons with the crystal momentum k and spin a in the p-th energy band] and v,

[the number of the magnons with crystal momentum q]. For the sake of simplicity, all the electrons that may give rise to the low-lying excitation are assumed to have the same direction of spin, say, along the positive z-axis a t T

0

Then, as it turns out, the individual electron excita- tions accompanying the spin fiips show an energy gap [2] in their excitation energies and hence such excitations may completely be ignored for the analysis of low temperature behaviour of the system. There- fore, the labelling of the low-lying excited states by { npk ) and { v, } does not cause any doubt of over- counting the excited states (from now on npk

n,,?).

11. General theorems on the existence of the magnon mode 131. - Suppose we have a ferromagne- tism in a non-relativistic system with inversion sym- metry. The ground state

is assumed t o be non- degenerate except for the spin degeneracy due to the various eigenvalues of S,. Then, introducing the n-th moment of the transverse excitations by

-

-

-

- m

with

M,(q) = C eiq.'j

, X

hamiltonian ,

j

and a = c/qY (0 <

< 2 and

> 0), we get rigorously for q

< >,

^Dq2 and < ^{w 2} >,

where

In the above, the Pauli spin matrices oj are chosen so that

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711285

C 1

-

for

S , I Y o > = M I Y O >

A

and

= q/q. Therefore, it has been proved rigorously that there exists a single and only single magnon mode with the frequency spectrum o

Dq2 (for small q) and also the mode is well-defined, because

If all electron spins are up, the exchange stiffness becomes

where

111. Ferromagnetic fermi liquid.

POSTULATE (1)

I The one magnon state > A,

I Yo>.

Define A's so that

[A,, A,]

0

POSTULATE (2)

I The one pair state (one excited electron + one hole >

&, ^{Z p k I Yo} >. The exis- tence of the unitary transformation :

where apkt refers to the state with the crystal momen- tum k but does not necessarily to an eigenstate of the single-particle energy in the non-interacting limit.

The Fermi surface is supposed to have a well-defi- ned meaning

a",, I Yo >

0 if

is outside the F. S.

iifi, I Yo >

0 if inside .

POSTULATE (3) : Define A, so that

or more specifically,

< Y, I Aq[IX, At,] I ^Yp >

C I ( P . ~ ) + C, q2 (for any state

with the crystal momentum

This is a physical postulate so that A: can have a real meaning of a magnon creation not only in the ground state but also in any excited states.

Let us construct an over complete set by

where k i stands for p i k i .

s thus defined are not necessarily the eigenfunctions of X . Consider

Let us bring X step by step until it operates directly on 1 Yo >, leaving the commutators [X, A;], [X, i:],

etc ... Next we bring, for instance, [X, At] step by step until it operates directly on I Yo >, leaving the com- mutators [[x, At], At], [[X, At], iit], etc.. . This process is repeated. Then, neglecting some kinematical effects, as in the case of Dyson'r theory of spin waves

we obtain the following Hamiltonian

where

=

< ^y6 I a",; ... 2,; A,; ... A,;,

x

L...

...[ E, A:,] ... ALm] auk] ... a":,] I Yo > .

The neglect of the kinematical effect, which seems to keep the exact nature of the present approach, enables us to regard

I {

}) as the complete orthonormal set in an ideal Hilbert space just like in the Dyson theory of spin waves for Heisenberg ferromagnets.

In equilibrium the magnon Green function is obtai- ned uniquely from

P

( q , I;) du < eu' A, e

A: > e

,

0

where

and

C,

271.il//3 ( l = 0, k

k 2 ,... ) .

The latter may be analyzed in the ideal Hilbert space at low temperatures. The Bloch-De Dominicis dia- gram expansion

for F(q, I;,) may be made use of.

At low temperatures the magnon damping is shown to be given mostly by the diagrams illustrated in figure 1, where we have the vertices

FIG. 1. -The diagrams leading to the dominant contribution to the magnon damping at low temperatures. The wavy lines represent the magnon propagation and the solid lines denote the

quasi-electron propagation.

ELEMENTARY EXCITATIONS I N ITINERANT ELECTRON FERROMAGNETS C 1 - 8 1 1

y(r,q + P

r l q , ~ )

b ( ~ . q ) band is spherical ( ~ ( k )

E( I k 1)) the results can be

more explciti

Y ( P , P I ' , Q + P - r )

b(r.q + P

r )

-

1 ( N o b)' kB T '

~ ( k + q - ~ : ~ I k : q ) = a ( k . ~ ) (1l~q)mag-mag

and 6(2 n ) '

( D a i 2 )

/ I m\

y ( k : q I k + q - p : p ) = a ( k . p ) q4 ln2 (y ) ^,

for small p, q and r. Higher powers of the magnon Dq2

momenta have been neglected, because such powers where a. is defined by

No a: with No

the lead to higher temperature powers than those consi- number of lattice points, and

dered here.

-

1' l n ( e x + 1 ) d x Then for cold magnons ( ~ < q kk,

~ the following

(llTq)mag-el -

energy width is obtained

1 2 D

e x + l q2(k, +

SIzq

(1ITq)mag-mag + (sIzq)mag-e~.

+ 105 1 P: Da2 q 6 ,

(l/zq)mag-mag

const. b 2 T q

ln2 (;qT)

_where

is the density of quasi-particle levels at

~ ( k )

The expression for (l/zq)m,g-mag is similar (l/zq),, ,-,,,

const. a2 T'

+ const. aq62. to that for the magnon damping in Ideal Heisenberg If there is no degeneracy (s-band) and if the energy ferromagnets [6].

References

[I]

LANDAU (L. D.),

BLOCH (C.) and

DOMINICIS (C.),

ABRIKOSOV (A. A,) and KHALATNIKOV (I.

1958, 66, 177.

LUTTINGER

and WARD (J. C.),

[2]

IZUYAMA

IZUYAMA

and KUBO

IZUYAMA

to

published

Phys. Rev.

details there.

HARRIS (A.

Rev.,

[4]

DYSON

J.),

Rev.,

KASHCHEEV and KRIVOGLAZ,

3, 1541.