M. El-Hiloa, J. Al Saeib and R. W Chantrellc

Dipolar Interactions in Superparamagnetic Nano-Granular Magnetic Systems

aDept. of Physics, College of Science, University of Bahrain, P.O. Box 32038, Sakhir, Bahrain

bDept. of Physics, Imperial College London, London, SW7 2AZ c Department of Physics, The University of York,York, YO10 5DD, UK

One of these models represents interactions via a mean statistical field which can be added or subtracted from the applied field [1].

Dipolar interaction effects are complex many-body problems, especially when the particles are poly-dispersed, located randomly,

and have their easy axes randomly oriented.

Many phenomenological models have been suggested to account for the interaction effects in fine particle systems.

Other models, the so-called T* model, in which a fictitious

temperature is introduced in the Langevin response of the superparamagnetic system so that L(H/kT) is replaced by the

modified function L(H/k(T+T*)) [2].

[1] S. Shtrikmann and E.P. Wohlfarth, Phys. Lett. 85 A , 457 (1981).

[2] P. Allia, M. Coisson, M. Knobel, P. Tiberto and F. Vinai, Phys. Rev. B 60, 12 207 (1999).

Introduction

Techniques such as Monte-Carlo simulations (MC) was found to be very useful in examining the effects of dipolar interaction in random

systems.

Using MC simulations, Mao et al [3] have questioned the T* model and concluded that this model does not adequately describe the

magnetization of interacting superparamagnetic assemblies.

The MC simulations of Chantrell et al [4] have shown that magnetization curves are depressed with increasing particle

concentration which was attributed to the formation of flux closure.

In [4], the simulations of the initial susceptibility also predicted an apparent negative ordering temperature. It has been noted that this kind

of order is not indicative of antiferomagnetic ordering.

[3] Z. Mao, D. Chen and Z. He, J. Magn. Magn. Mater. 320 (19), 2335 (2008).

[4] R.W. Chantrell, N. Walmsely, J. Gore and M. Maylin, Phys. Rev. B 63, 24410 (2000).

In a recent MC simulations study [5], the magnetization curves of a nano-granular superparamgentic system are predicted to be always

depressed with increasing particle packing densities.

In [5], the distributions of dipolar interaction fields along the x, y and z directions are predicted to be symmetric and Gaussian in form with

an average very close to zero.

In [5], the reduction in magnetization is attributed to the non-linear response of the magnetization to the applied field, which weights the negative interaction fields more strongly than the positive fields.

In this study, the effect of a Gaussian distribution of longitudinal dipolar fields on the magnetization response of a nano-granular superparmagnetic system in the limit of weak anisotropy is examined, analytically and numerically.

[5] J. Al Saei, M. El-Hilo and R.W. Chantrell, Submitted to J. Appl. Phys. (2011)

The Model

( )a ia i i

B

H HM H L f H dH

k T

For a distribution of longitudinal dipolar fields and in the limit of weak anisotropy, the magnetization of identical superparamagnetic particles is simply described by a Langevin function weighted by the distribution of interaction fields, and is simply expressed as;

L(b) is the Langevin function

2 2/ 2 / 2

.

i HiHi Hi

Hi

f H e is the Gaussian distribution of dipolar fields

with is the spread of dipolar fields

The distribution of longitudinal dipolar fields is assumed to remain invariant with applied field. In real systems, f(Hi) is expected to

becomes narrower at high fields but it wont affect the results since;

2/ 1 1 1

( )B B B B

a s i i i i ia i a a a

k T k T k T k TM H M f H dH H f H dH

H H H H H

Analytical Approach

At low fields, where b1, the Langevin function can be expanded and then the magnetic response of the system can be expressed as:

3 3 5 5

3 5

( ) ( ) 2 ( )....

3 45( ) 945( )a i a i a i

a i iB B B

H H H H H HM H f H dH

k T k T k T

Since f(Hi) is symmetric, then the above equation can be simplified to:

2 2 4 4 3 3 2 2

2 4 3 2

30 20( ) 1 .... 1 .... ....

3 5( ) 315( ) 45( ) 21( )a Hi Hi a Hi

aB B B B B

H HM H

k T k T k T k T k T

Thus every term in the Langevin expansion is depressed as a result of a symmetric Gaussian distribution of longitudinal dipolar fields.

The reduction in magnetization increases when the standard deviation of dipolar longitudinal fields (Hi) is increased.

Thus, this reduction in the magnetization of the system cannot be used to infer that the local interaction field is negative, since both negative

and positive interaction fields are equally probable.

When the magnetic response of the system is linear with applied field i.e. M(Ha)(Ha+Hi), then a symmetric distribution of interaction fields

can have no effect on the magnetization because the weighted average of the interaction field <Hi>=0.

Thus the reduction in magnetization for the system examined is attributed to the non-linear response of the magnetization to the

applied field, which weights the negative interaction fields more strongly than the positive fields.

The Initial Susceptibility2 2 4 4

2 4

0

( / ) 301 ....

3 5( ) 315( )a

s Hi Hii

a B B BH

d M M

dH k T k T k T

In the case where the standard deviation of dipolar fields is small i.e.

Hi<<kBT/, the third term in the above equation becomes negligible,

hence

1 2 22 20

22

( )33 1 ..

55( )

i

HiHiBB

BB

C

T T Tk Tk T

k Tk T

Over a certain small range of temperatures, T0(T) varies slowly with temperature, then

0 0( )i

C C

T T T T T

The predicted negative average ordering temperature is an apparent measure of negative ordering in the system, since

positive and negative fields are equally probable.

Numerical Results

0

0.2

0.4

0.6

0.8

1.0

0 1000 2000 3000 4000

Hi

=0 Langevin}

Hi=300 Oe

Hi

=600 Oe

Hi=900 Oe

Applied Field (Ha)

Re

duce

d M

ag

netiz

atio

n (M

/Ms)

The calculated magnetization curves for a symmetric Gaussian distribution function of longitudinal dipolar fields f(Hi) at different values of the standard deviation Hi=300,

600 and 900 Oe. The solid line represents the Langevin function, the case where Hi0.

Dm=6 nm

The reduction in magnetization is due

to the non-linear response of the

magnetization to the applied field, which weights the negative

interaction fields more strongly than the positive fields.

Numerical Results

0

300

600

900

0 100 200 300

Hi

=30 Oe

Hi=100 Oe

Hi

=200 Oe

Temperature T(K)

Inve

rse

Re

du

ced

In

itia

l Su

sce

ptib

ility

i-1

(O

e)-1

The inverse of the calculated reduced initial susceptibility using a Gaussian distribution function of longitudinal dipolar fields f(Hi) at different values of the standard deviation

Hi=30, 100 and 200 Oe.

when Hi=30 Oe, i.e very small

compared to the range of applied

fields for which the magnetization

is still linear, the variation of i

-1 vs. T

extrapolates to almost zero

ordering temperature

The results show that interaction effects cannot be deduced from changes in the average magnetization of the system.

Conclusions

For a symmetric distribution of longitudinal dipolar fields, the magnetization of a superparamagnetic system is predicted

to be always depressed as the spread of dipolar fields is increased.

The analytical analysis shows that this effect arises from the non-linear response of magnetization, which weights

negative interaction fields more strongly than the positive fields.

Thus the idea of describing dipolar interactions in terms of a mean field that could be added or subtracted from the applied

field in random systems is not justified.

Due to the non-linear effect of magnetization, the temperature variation of the initial susceptibility predicts an

ordering temperature that varies with temperature.

Conclusions

The predicted negative ordering temperature cannot be considered as indicative of anti-ferromagnetic order.

Thus the idea of using an effective temperature (T*) to represent interaction effects does not adequately describe the

magnetization of interacting superparamagnetic random systems.