EFFECT OF NONMAGNETIC IMPURITIES ON THE THERMODYNAMICS OF A HELIMAGNETIC CHAIN

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EFFECT OF NONMAGNETIC IMPURITIES ON THE

THERMODYNAMICS OF A HELIMAGNETIC

CHAIN

I. Harada, H. Mikeska

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Supplement au no 12, Tome 49, decembre 1988

EFFECT OF NONMAGNETIC IMPURITIES ON THE THERMODYNAMICS OF A HELIMAGNETIC CHAIN

I. Harada and H. J. Mikeska

Institut fur Theoretische Physik, Universitiit Hannover, Appelstrasse 2, 3000 Hannover, F.R.G.

Abstract. - A diluted classical spin chain with competing exchange interactions is studied by the transfer matrix method. It is found that a nonmagnetic impurity perturbs the helical spin structure and leads t o a new local equilibrium spin complex. Numerical results for the correlation functions are discussed in connection with this spin complex.

A nonmagnetic impurity plays a vital role in a helimagnetic chain. It perturbs the delicate balance of the helical spin structure and leads t o the formation of a new local equilibrium spin configuration. (We call it a spin complex.) The purpose of this paper is to study this spin complex in detail and its effect on the thermodynamic properties. We consider a helimagnetic chain described by the following Hamiltonian:

where J1and J2 denote the nearest-neighbor (nn) and

the next-nearest-neighbor (nnn) exchange interaction, respectively, and are assumed to be aptiferromapetic (J1, J 2

>

0)

.

s, is a classical unit vector at the site m,

S

, = (sin 0, cos

a,,

sin 8, sin Qm, cos 0,). We assume the periodic boundary condition, s, = s,+N, where N is the number of sites in the chain.

Following the transfer matrix method for the pure case [I] we first introduce a dual lattice on which a dual spin is defined as a relative angle between adjacent spins on the real lattice. Then the original spin system can be redescribed by a new spin system:

An advantage of this Hamiltonian

H D

over H is that

H D

does not contain nnn interactions, and therefore we adopt

H~

as a basic Hamiltonian for our calculations of thermodynamic quantities.

Now, let us first consider a single nonmagnetic impurity at the zero-th site on the real lattice. The dual spin at the impurity site is defined as a relative angle between nnn spins a t both sides of the impurity (see Fig. 1). It is interesting to note that this dual spin is isolated from its nn spins and is in a magnetic field

-3 -2 -1 0 1 2 3 -c.=--=~----.----~----~---.~>-~=- real .' .. . : : ., .. .. ? . , I . , , lattice . , . . . . . , . , . . , . . : . . . , . _{. . .} ._:: ;_.: . _.. _,... : : . . . . _{. .}_{. .} _{. ;}_{. ,} _{' . .} - + - - - + - - - + ₈ +- - - + - - - + - ' dual -3 -2 -I 0 lattice

Fig. 1. - Real lattice with a single nonmagnetic impurity at the zero-th site and corresponding dual lattice.

not 2J1 but 2J2. Because of this isolation of the impurity spin on the dual lattice all information carried by the transfer matrix is lost a t the impurity site. Fur- thermore, it is apparent that two or more successive impurities break up completely the original magnetic chain. Therefore we can calculate the partition function of the diluted system as a product of the partition functions of independent segments having finite n spins:

In ZN = N x (1

-

x ) ~ ln Zo

+

NP, (z) In Z,, (3) n=2

where x denotes the concentration of nonmagnetic impurities, Z o is the contribution from nnn spins next to an impurity, and Zn is the partition function of the

finite chain with n spins. The probability t o find out a segment with n spins is given by Pn (z) = x2 (1 - x ) ~

PI.

Thus the problem is reduced to calculate Z,, which has been formulated in terms of eigenvalues and eigen- functions of the integral equation for the pure case [I]. We note here that our formulation so far is exact for an arbitrary concentration of nonmagnetic impurities and is applicable also t o the planar model [3].

We adopt in our calculations the parameter value, J z l J1 = 112, which yields in the pure case the helical ground state with the turn angle of f 2 ~ 1 3 . The two angles correspond to opposite chirality, that is, the clockwise and the counter-clockwise turn of spins. It is worth noticing that the material FeMgBOl may be described by our model [4].

First we present in figure 2 the temperature dependence of the nn correlation function Wl and that of the nnn correlation function W2 for the diluted system (x = 0.15). Comparing these to the results for the pure system we find that W2 exhibits a saturation tendency a t low temperatures. This behavior is consistent with the saturation tendency of Wl

+

W2 observed in FeMgBO, [4].

In order to gain more insight into the unusual behavior of W2 a t low temperatures it is instructive to see a spin configuration of a spin complex a t zero temperature. Upper part of figure 3 exhibits angles be-

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C8 - 1408 JOURNAL DE PHYSIQUE

Fig. 2. - Temperature dependence of the nearest-neighbor and the next-nearest-neighbor spin correlation function for the diluted system (x = 0.15)

.

The dashed curves are the results for the pure system.

tween nn spins, 8,,,+1, as a function of the distance from the impurity. Note that the angle at the zero- t h site is an exception and represents the angle between nnn spins located on both sides of the impurity. These nnn spins are coupled by the a n t i f e r r ~ magnetic interaction J 2 without competition. Lower

part of this figure shows the corresponding spin correlation functions: Wl (m

+

112) = cos 6',,,+1 and W2 (m) = cos O,-l,,+l. From this figure we see that

at the center of a spin complex two nnn spins couple antiferromagnetically, a nn spin pair next to an impurity makes an angle of about f 5 ~ / 6 and the nnn pair already recovers almost the bulk value. Note that the size of a spin complex depends on the parameter

52/51. The remarkable features we want to mention

are (1) the very small value of IWz (2)l compared to the rather large value of

I

Wl (1.5)l and (2) four p o s sible spin configurations due to the chiral degeneracy. The feature (1) is apparently an origin of the satur* tion tendency of W2 in the diluted system. The feature (2) implies that the memory of the chirality is lost in the spin complex so that, for instance, the strong peak in the temperature dependent specific heat [I], which is due t o chiral domain walls, is reduced considerably. Consequences of the spin complex appear also in other thermodynamic quantities. These will be reported in a separate paper [5].

In conclusion, the present study based on the .nu- merically exact transfer-matrix method demonstrates an important role of nonmagnetic impurities in a helimagnetic chain. Especially a t low temperatures spins near a nonmagnetic impurity form a new spin complex, which is responsible for characteristic features of the system. In order to obtain reliable values of the

Fig. 3. - Angles between nearest-neighbor spins (upper part) and nearest-neighbor and next-nearest-neighbor spin correlation functions (lower part) at zero temperature as a function of the distance from a nonmagnetic impurity located at the origin of the real lattice. Note that the angles 2 ~ / 3 and 4x13 ( - 2 ~ 1 3 ) represent opposite chirality (see text).

parameters in FeMgBO, we feel that it is necessary to perform analogous calculations for the anisotropic Heisenberg chain.

Acknowledgments

This work has been funded by th.e German Federal Minister for Reseach and Technology (BMFT) under contract number 03-MIlHAN-6.

[I] Harada, I. and Mikeska, H. J., 2. Phys. B 72

(1988) 391.

[2] Harada, I., Suzuki, C. and Tonegawa, T., J. Phys.

Soc. J p n 49 (1980) 942.

[3] Harada, I., J. Phys. Soc. J p n !i3 (1984) 1643. [4] Wiedenmann, I., Burlet, P., Scheuer, H. and Con-

vert, P., Solid State Commun. 38 (1981) 129.

[5] Harada, I. and Mikeska, H. J., to be published in