MAGNETIC POLARON IN FERROMAGNETIC CRYSTAL

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MAGNETIC POLARON IN FERROMAGNETIC CRYSTAL

Yu. Izyumov, F. Kassan-Ogly, M. Medvedev

To cite this version:

Yu. Izyumov, F. Kassan-Ogly, M. Medvedev. MAGNETIC POLARON IN FERRO- MAGNETIC CRYSTAL. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1076-C1-1078.

�10.1051/jphyscol:19711387�. �jpa-00214426�

JOURNAL DE PHYSIQUE

Coiloque C 1, supplkment au no 2-3, Tome 32, Fbvrier-Mars 1971, page C I - 1076

MAGNETIC POLARON IN FERROMAGNETIC CRYSTAL

Yu. A. IZYUMOV, F. A. KASSAN-OGLY, M. V. MEDVEDEV Institute for Metal Physics Sverdlovsk, U. S. S. R.

RhurnC.

- Les polarons magnktiques apparaissent dans un cristal ferromagnbtique non mCtalllque, dans lequel le signe de l'integrale d'echange s-d est negatif. 11s representent la superposition de deux Ctats. Dans un de ces Ctats, I'Clec- tron se deplace dans une structure magnCtique totalement ordonnk avec une orientation antiparallkle a son propre spin.

L'autre etat reprksente I'Clectron se deplagant dans le cristal et accompagne d'une onde de spin. Les energies des Ctats polarons sont obtenues comme les pbles de l'amplitude de diffusion d'un Clectron par une onde de spin, calcule dans une approximation de gaz, correspondant aux diagrammes de type en Cchelle. L'equation de dispersion qui determine les energies de polaron est valable pour des valeurs arbitraires de I'intkgrale d'echange s-d.

Abstract. -

In nonmetal ferromagnetic crystal with the negative sign of s-d exchange integral there appear themagne- tic polarons. They represent the superposition of two states in one of which electron moves in completely ordered magnetic structure with antiparallel orientation of its spin. The other state representates the electron moving in the crystal and being accompanied

by

a spin-wave.

The energies of polaron states are obtained as the poles of scattering amplitude of an electron on a spin-wave, having been calculated in the gas approximation corresponding to the diagrams of the ladder type.

The dispersion equation determining polaron energies is valid at arbitrary value of s-d exchange integral.

A conduction electron moving in nonmetal magneto-

ordered crystal deforms the magnetic structure of the crystal. This deformation can translate together with an electron along the crystal and this quasiparticle is called magnetic polaron [I].

The s-d exchange interaction is the mechanism of this coupling of an electron and magnetic system of the crystal. Magnetic polaron have been theoretically investigated in the papers [2, 31 at T

= 00

and in the papers [4, 51 at nonzero temperatures. As it was marked in

[2]

the results of calculation of polaron effective mass being obtained in [4] did not pass on to the results of [2] at zero temperature. The insensitive- ness of the main results to the sign of s-d exchange integral is the other obstacle of the approach of

[4]

and

[ S ]

essentially using the mathematical apparatus of the theory of the usual small-size polaron in the pro- blem of magnetic polaron.

We develop a method which allows to describe magnetic polaron at low temperatures ( T

4

T,) and gives the agreable results with

[2]

at

T = 00.

For magnetic polaron may be treated as the bound state of an electron and spin-waves it is necessary to investigate the scattering amplitude of an electron on a spin-wave.

Let us consider ferromagnetic crystal containing conduction electrons of a small concentration. The Hamiltonian of the system may be written in the repre- sentation of the secondary quantization as

-

A 2 (sSm)oo*

^{d o}

am,, (1)

moo'

Let us pass from spin-operators to boson operators of spin deviations in

(1) :

and introduce the

temperature

Green's functions [6]

of electrons and spin-waves

:

Here Zmo(z)

=

eJer a,,, e-Jez , etc.

For the calculation of Green's functions it is possible to develop diagram technique in which the dynamic part of s-d exchange

+(2~)"a,:, a",? b*+(ail a,",-a:, amt)b,+

b,)

(4) is considered as perturbation. Green's functions (3):of zero approximation in impulse representation are such as follow

:

where p

=

(p, up)-4-vector

; E;

and

E~

are the energies of noninterating conduction electron and a spin-wave

:

\ A

where the first term describes the Bloch's energy of

conduction electrons, the second term - the exchange

el =

B 2 eipA ~ f ;

=

EL - A S .

spin-spin energy and the third one - the s-d exchange (7)

A

interaction. Here a;o is an operator of birth of an Green's functions of electron with spin

f

and 1 will

be

electron in the site m with the spin

^o ; S, is the operator

drawn by solid lines with a light and a dark arrows of atomic spin

;

s is the operator of electron spin

;

accordingly and the Green's function of spin-waves

A

runs over the neatest neighbours.

will

be drawn by a wavy line.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711387

MAGNETIC POLARON IN FERROMAGNETIC CRYSTAL C 1

-

1077

Hence it is easy to see that four following types of axis the pole of scattering amplitude responds to the elementary vertex parts take place (Fig. 1). energy of two-particle states of the system.

It is not difficult to show that a t arbitrary negative values of s-d exchange parameter A the equa-

l $1 ^- tion (10) has an isolated root for any K provided the

(~$9 according energy is always less than

^EB 1

corresponding to the state of electron without polaron effect being taken into account.

FIG. 1.

On figure

3

there is shown the spectrum structure

The first two graphs describe the electron scattering with the overturn of spin and the latter two-the scatter- ingqwithout spin overturning.

Consider a scattering amplitude of an electron on a spin-wave

:

(Here 4-impulses are denoted by ciphers).

For electron and spin-wave concentrations are supposed to be small it is possible to use the gas appro- ximation. The graph equations taking into account the graphs of only the ladder type are shown on figure 2.

After summarizing over the frequencies o, in these expressions we obtain the expression for the amplitude answering the gas approximation :

F(12

;

34)

=

-

-

1

- AS

-

A (

2 i ( c o l + c o 2 1 - ~ l + 2

A 1

- _{2 N} --C

p

i(m1 + ^{02) -} &I-,

^{- E ~ 1-I. + ~Y(9)}

The poles E

=

i(o, + m2) of the amplitude are determined from the equation

:

I---

AS

A

1

=

0 (10)

~ - 8 b 2

N E-&(K,Z)+~-E(K,Z)-~

in which there are introduced the total and the relative impulses of scattering particles :

K = 1 4 - 2 ;

q = i ( l - 2 ) .

After an analytical continuation of E onto the real

FIG. 3.

-

Spectrum of the two-particle exitations as a function of wave-vector k along the direction 1111

1.

Calculated for para- meters values :

I

A/B

I

⁼ 0,l (for S = 1) A / B = 8 (for curves

1 and 1') ; A / B = 2 (for curves 2 and 2').

calculated for the simple cubic lattice at two sets of parameters values as a function of wavevector K along the direction [I 1 I]. The solid lines correspond to the genuine eigenstates of the system and the discrete ones - to the states without polaron effect being taken into account. i. e. to the energies clc.

1

It should be marked that due to polaron effect the average value of an electron spin z-projection turns out to be less than its maximum value $. This magnitude had been investigated thoroughly in the paper [3].

At A > 0 the bound state neither appears for elec- tron with spin t nor for electron with spin 3. betow the energy band.

The essential role of s-d exchange interaction sign

may be understood from the simple physical considera-

tions. Consider the completely ordered ferromagnetic

crystal at T

=

00. If A > 0 then electron spin orientates

along the spontaneous moment of the system. If

otherwise

A

< 0 then obviously the antiparallel

orientation turns out to be advantageous and the total

spin of the system is St,,

=

NS - 4. But another one

C 1

-

1078 YU. A. IZYUMOV, F. A. KASSAN-OGLY, M. V. MEDVEDEV

state when in ferromagnetic system a single spin and coincided with the equation (10) which determines deviation takes place and an electron spin is orientated the poles of scattering amplitude.

along the spontaneous moment belongs to the same It is rather obvious for spin

s =

3 that considered value of the total spin. Hence the eigenstate of the states respond to the stable polaron but at other values system at A <

0

should be the superposition of the of S it is not excluded that the bound state of several two these states. The exact equation for eigen energies spin-waves having the lower energy in comparison of these states had been obtained in the papers [2,

31

with the above mentioned can appear.

References

[l]

DE

^GENNES

(P. G.),

Phys. Rev., 1960,118, 141. [5]

ZALITCHICHIS (A.

L.),

IRCHIN (Yu. P.),

Solid State [2]

METHFESSEL

(S.),

MATTIS

(D.

C.), Hundbuch der

Physics, 1968, 10, 1974.

Physik. XYIII/l,

1968.

[3]

IZYUMOV (Yu. A.), MEDVEDEV (M.

V.), JETP., 1970, ^{ABRIKOSOV (*'}

A')' P')7

DZYALOSHINS-

59, 553. ^KIJ

(I.

E.),

Metody kvantovoi teorii polya

v

[4]

WOLFRAM

(T.),

CALLAWAY

(J.), Phys. Rev., 1962,

statisticheskoi fizike. Moskva, Fizmatgiz.,

1962.

127, 1605.