AC susceptibility and hysteresis in Ising magnets

ELSEVIER

J H Journalof magnetism ~ i and magnetic J H materials Journal of Magnetism and Magnetic Materials 136 (1994) L29-L32

Letter to the Editor

AC susceptibility and hysteresis in Ising magnets Muktish Acharyya *, Bikas K. Chakrabarti 1 Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Calcutta-64, India Received 21 March 1994

Abstract

We study here, in Ising magnets in presence of a sinusoidally varying external field, the nature of ac susceptibility arising out of the thermodynamic relaxation dynamics. The signature of the accompanying hysteresis and the dynamic phase transition are investigated in the susceptibility behaviour. These studies are made by using mean field equation of motion and the Monte Carlo method for a three dimensional Ising system. It is found that A/' (and X') has a prominent peak (dip) as the dynamic transition phase boundary is crossed.

AC susceptibility measurements are now commonly used to determine magnetic and superconducting properties of materials [ 1,2]. Typically, in an electrically conducting (magnetic or superconducting) material, if the external magnetic field (h) is periodically varied (say sinusoidally) in time (t), then the induced eddy current [ 3 ], in turn, produces a response magnetisation m(t) given by a complex susceptibility (X) having both the real or in phase (X') and the imaginary or loss or out-of-phase (X") parts. The m-h plot also gives the hysteresis loops with finite area (loss; typically very small) as the A/' is nonvanishing. The temperature and frequency variation of the hysteresis loss or of A/' (coming from the temperature variation of the conductivity of the sample [3]) are then studied [2] to locate the transition point of the (electrical) conductivity of the (superconducting) samples. There are also comparatively older reports [ 1 ] on such properties of A/' (and X' ) in magnetic materials.

* Corresponding author. t e-mail:[email protected] 0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved

SSDIO304-8853(94)OO316-J

The above mentioned out-of-phase part (A/') of the susceptibility and the consequent hysteresis loss (in conducting magnets) arise due to the delayed response (magnetisation m(t) =moCOS(tot-8); )C = (mo/ ho)sin(8)) to the external field variation ( h ( t ) = hocos (tot)). This loss originates from the nature of the induced eddy currents [3], or from the laws of electrodynamics. Recently, there has been considerable study on the origin of hysteresis (in magnetic insulators), arising out of the relaxationally delayed thermodynamic response [ 4-9 ]. Although these studies explore the nature of thermodynamic hysteresis (usually with loss much larger compared to the above eddy loss), arising out of the relaxational delay [7] in the magnetisation m(t) in response to periodically varying applied magnetic field, they do not address directly the question of complex susceptibility for such thermodynamic response. Most of these studies investigate the low frequency (power law) variation of hysteresis loss in various magnetic models, like the n-vector model in the n ~ o~limit [5], of Ising model using Monte Carlo dynamics [6], or in real magnetic samples [9]. Acharyya and Chakrabarti [ 7] have studied the time variation properties of magnetisation (m(t)) in response to

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M. Acharyya, B.K. Chakrabarti /Journal of Magnetism and Magnetic Materials 136 (1994) L29-L32

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t Fig. 1. (a) Timevariationof response (magnetisation)functionand the (perturbing) field variation in the mean field study. The inset shows the corresponding Lissajous plot. (b) Time variation of response (magnetisation)functionand 1heperturbingfieldvariation in the Monte Carlo study (of d= 3 Ising system). The inset shows the correspondingLissajousplot. the sinusoidal applied field (,~(t) = hocos(wt) ) for wide range of frequency (w) and amplitude (ho) of the perturbation and of temperature (T) of Ising magnets in dimensions d = 1 to 4, using the Monte Carlo dynamics and using the mean field dynamics. They observed that the (thermodynamic) response function is always periodic (although not necessarily sinusoidal, as in the case of electrodynamic response) with the same frequency as that of the perturbing field, and gets delayed in general due to relaxation. The hysteresis loop is then identified as the Lissajous plot of the response function versus the pev:urbing field. The delay in the response gives rise to the nonvanishing width of the hysteresis loops. The equalitly in the frequencies (of h(t) and m(t) ) gives rise to the quadratic equation for the Lissajous figures, giving the double valuedness and stability of magnetisation in the hysteresis loop. Acharyya and Chakrabarti also fitted [7] the observed variation of the loop area (A = ~m dh) to a scaling form

a ~ hgT -Bg( og/h~T ~) with g ( x ) ~ x ~ e -cx~ for Monte Carlo results and

or Lorentzian, in the mean field case. Although the form of the scaling function (and the frequency dependence, in particular its Lorentzian form in the mean field case) was found to be quite robust, the exponents a,/3, y, 6 and e were found not to be so stable but assume constant values (e.g., c~-1.0_+0.02, /3=0.75_+0.05, y-~1.25_+0.08, 6 = 1.55 + 0.05 and e = 0.45 for d = 3 Ising system Monte Carlo results and ~ = 1,/3= 1/2, y = 1/2, 6 - - 0 for the mean field case) for a reasonable wide range of (high) temperature (T) and (low) amplitude (ho) values [ 7 ]. These scaling forms reduce to the previously observed [5,6] power law in the low frequency and amplitude limit. Another important study, in such systems, have been regarding of dynamic phase transition: The order parameter q = ~m dt makes a transition (from q = 0 for large ho and T values to q 4:0 for low ho and T) across the dynamic phase boundary line in the ho-Tplane (the boundary is in general frequency dependent). Such dynamic phase transition has been observed and the phase boundary line has been determined in both the mean field [7] and the Monte Carlo dynamics [5,7] studies. As mentioned before, these studies have not, so far, been extended to investigate the ac susceptibility and its thermodynamic variation in such magnetic insulators (as in the case of conducting magnetic systems). We show here the thermodynamic variations of the ac susceptibility of an Ising system, using numerical solution of the mean field equation of motion for the dynamics and using the Monte Carlo dynamics for a three dimensional system. The properties of X' and ~', in particular their temperature and frequency variations, are obtained here for an Ising system, using Monte Carlo dynamics (in three dimensions) and also solving the mean field equation of motion numerically (and analytically in the linearised limit). These behaviour are also compared with those for conducting magnets [ 1-3]. It is found that X' has a sharp dip and )C has a peak at the temperature Td(ho, w) at which the dynamic phase transition occurs (q 4:0 for T < Tj(ho, w) and q = 0 above). X' has another broad peak (also X"; but very weak) at a higher temperature which is identified as the signature of the transition point of the original zero field (ho = 0) order- disorder (mo = 0 to mo~ 0) phase transition. We thus find a new method to detect the dynamic phase transition [7,10] in such systems, from the study of temperature variations of)(" (and X' ) for fixed ho and w.

1

L E T T E R TO THE E D I T O R

M. Acharyya, B.K. Chakrabarti / Journal of Magnetism and Magnetic Materials 136 (1994) L29-L32

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The equation for the dynamics of the magnetisation (mo) of a magnet in presence of a sinusoidally varying magnetic field h(t) = hocos(tot), may be written in the mean field approximation (cf. [ 11 ] ) as dm . , r m + hocos(tot)] -~ j, r--~-t= - m + tann k

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where ~-is the microscopic relaxation time (the Boltzmann constant and the spin-spin interaction strength is taken to be unity). We have solved the equation numerically using fourth order Runge-Kutta method to get m(t). A typical solution of m(t) is shown in Fig. l ( a ) (the Fig. 1 (b) shows a similar response from the Monte Carlo study described below), where its time variation is compared with that of the field h(t). It is found that the response m(t) is always periodic with the same period as that of the external field h(t), although m(t) is not necessarily sinusoidal (except in the limit ho ~ 0 and T>> 1, where linearisation of Eq. (1) is possible). The response however, generally gets delayed compared to h(t). We can thus express m(t) as P ( h ( t - reff) ) where P induces the same periodicity as that of h(t) and %ff is the effective delay due to relax-

ation (determined in general by ~-and T, ho, to etc.). Solving m(t) for h(t), one gets the quadratic equation (because of same periodicity in m and h) for the Lissajous plots or the hysteresis loops [7]. Instead of solving for the hysteresis loops (Lissajous plots), we study here the complex susceptibility X=X ' +ix", with X '= (mo/ho)cos(toreff) and x " = (mo/ho)sin(to%fD, where mo is the amplitude of m(t). Numerical solutions (of Eq. (1) ) for X' and x" are shown in Fig. 2(a) where we also show the variation of the response amplitude mo and order parameter (q = Sm dt) for the dynamic phase transition. In Fig. 2(b), the same are shown for Monte Carlo dynamics [ 7] of d = 3 (simple cubic lattice of size 203, averaging over 30 Monte Carlo seed values) Ising system. One can easily identify the high temperature peak in X' (and also weak appearance in x") with the high temperature decay of magnetisation amplitude (mo) and the second (low temperature) prominent peak in x" (and the dip in X') with the dynamic transition (q changing from zero to nonzero value). In fact, the high temperature peak in X' (and its weak presence in x") can be formally identified, at least in the mean field case, as the signature

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M. Acharyya, B.K. Chakrabarti / Journal of Magnetism and Magnetic Materials 136 (1994) L29-L32

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of the original order-disorder transition for ho = 0. In the limit of high T (T > 1) and small ho (ho ~ 0), one can linearise Eq. (1) as dm em + hocos ( tot).. 1 "r~-t = T , ~= 1-~.

field case. These observations for frequency dependence of X' and )(" are also very much comparable to those observed in conducting magnets [ 1 ]. Our study on the temperature variation of the complex susceptibility of an Ising model in a periodically varying external field shows that a low temperature prominent peak in ~ ' (and dip in X') occurs (at Td) as one crosses the dynamic phase boundary (q 4:0 for T < Ta(ho,to) and q = 0 for T> Ta). The experimental measurements of the behaviour of)(" in magnetic insulators would thus be able to detect the intriguing dynamic phase transition (and determine the phase boundary), which has so far been only observed in theoretical studies [ 10,7].

Acknowledgements We are grateful to R. Ranganathan for many informative discussions.

References

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The solution can be written as [7] m ( t ) = moCOS(tot-6), where mo=ho(e2+to2Ta)l/2/T and 6 = sin- 1( toT/( e"2+ toaz2) 1/2). One can then express the complex susceptibility as X' = (mo/ho)cos (6) and ){'= (mo/ho)sin(6). Employing the above expression for mo and 6, one finds peaks in X" and X' near T = 1 in the to --* 0 limit. The low temperature peaks (or the dip) in )¢" and X' can not be obtained from such linearised theory as the occurrence of dynamic transition (q 4: 0) already invalidates the linearisation approximation (the sinusoidal solution for m(t) is no longer appropriate). We have also studied the frequency variations of X' and ,~' both from the mean field equation of motion (1) and using the Monte Carlo method (for d = 3 Ising system). The results are shown in Fig. 3. The linearised mean field theory, as discussed above, can easily account for such (observed) variations in the mean

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