CHINESE JOURNAL OF PHYSICS

VOL. 38, NO. 1

FEBRUARY 2000

Energy Barrier, Coercivity and Blocking Temperature Variation of Fine-Particle Systems Jing Ju Lu and Huei Li Huang Department of Physics, National Taiwan University, Taipei, Taiwan 106, R.O.C. (Received July 2, 1999)

Coercivity and energy barrier height variation of a dipole interacting fine particle system has been evaluated by means of the fixed step Monte Carlo simulation supplemented by a perturbation calculation. The dipole interaction field can be decomposed into a longitudinal and a transverse field component. The superposition of the two field component leads to local maxima at ¯ = 0 and ¼=2 and a global minimum of coercivity at ¯ » 60± . Thus, the coercivity of a random dipole interacting system generally becomes lower compared to that of a noninteracting system since the probability of finding the bonding angle of the system smaller than 30± is less than 13%. The asymmetric fanning mode is shown to be favored in comparison with the other modes in making the ("#) ) (##) transition if ¯ is neither zero nor ¼=2. Whenever the bonding angle is close to ¼=2, the system may exhibit a two-step switching behavior causing the system to broaden its switching field distribution. Finally, the fieldcooled and zero-field-cooled magnetization relaxation behavior and its blocking temperature distribution can be well accounted for. PACS. 75.60.Jp – Fine-particle systems. PACS. 75.60.Nt – Temperature-hysteresis effect. PACS. 76.60.Es – Relaxation effect.

I. Introduction High density recording calls for the magnetic medium to consist of single domain particles homogeneous in particle size, isolated and well oriented along the track direction so as to reduce particle size fluctuations and interparticle-interaction-induced particle noise. The particles need to be small enough in linear dimension to assure high bit density, while at the same time the product of the particle’s anisotropy constant and volume, KV, should be large enough to ensure high thermal stability [1-3]. In light of this, it is necessary to ask the question how serious and to what extent the ubiquitous interparticle interaction, dipolar interaction in the case of particulate media, has to do with these problems. It is important to carefully delineate whether the interparticle interaction enhances or depresses the energy barrier for magnetization relaxation and its corresponding coercivity for magnetization reversal and what specific media may satisfy the future technology need. Effect of fluctuations in the relevant physical parameters such as particles size, anisotropy constant and misorientation of the easy axis with respect to the curvilinear vector of the pulsed recording head field has been discussed previously in the context of writing transition noise [3]. Energy barrier, coercivity and blocking temperature variations and the related relaxation 0

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phenomena for fine magnetic particles has also been extensively studied and reviewed [4]. Energy barrier for thermal relaxation as a function of the bonding angle (between the line connecting two particles and the applied field direction, i.e., the z-axis.) for well oriented dipole interacting particles has been studied to some extent [5-7] in which the energy barrier was shown to increase monotonically at zero bonding angle whilst the reverse is true when the bonding angle is ¼=2. Recently, Mossbauer spectroscopic analysis and magnetic measurements of noninteracting (i.e., for particles far apart) and interacting magnetic fine particles such as has raised various questions regarding the dynamics of the moments and the effects of interaction [8-9]. In effect, there exists conflicting interpretations of the observation data as regard the interparticle interaction effect on blocking temperature, or equivalently, energy barrier, variation of magnetic particles [8-12]. For example, in the analysis of the blocking temperature TB in iron grains [9] it was deduced that the energy barrier increases with the interparticle interaction, whilst an opposite conclusion has been reached in analyzing the weakly interacting ° ¡ F e2 O3 particles [11-12] suggesting that the effective dipolar interaction decreases with decreasing interparticle separation. In this communication a thorough study in this connection is presented aiming to clarify the confusing dependence of the energy barrier, coercivity and blocking temperature variation for thermal relaxation of a dipole interacting fine particle system which is potentially applicable to a typical recording media. The present analysis is based upon Monte Carlo simulation [13; 14] in combination with and argumented by the perturbation calculation. Due comments will be given on blocking temperature variation of a fine particle system. The present perturbation approach causes less than 8% relative error compared to an exact nu- merical calculation if particles are < reasonably apart such that the packing fraction of the media is ¼ 40% (see text) as elaborated below. II. Perturbation calculation Consider a pair of well oriented dipole interacting single domain particles of volume V1 = V0 v1 and V2 = V0 v2 , where v1 < v2 , each with the easy axis of the particles oriented along the z-axis. The configuration of the system is shown in Fig. 1. Energy of the system is E(µ1 ; µ2 ) = KV0 [v1 sin2 µ1 + v2 sin2 µ2 ¡ 2h(v1 cos µ1 + v2 cos µ2 ) +2½v1 v2 fcos(µ1 ¡ µ2 ) ¡ 3 cos(µ1 ¡ ¯) cos(µ2 ¡ ¯)g];

(1)

= E (0) (µ1 ; µ2 ) + ²(µ1 ; µ2 ) where E (0) (µ1 ; µ2 ) is the energy of the noninteracting system, ²(µ1 ; µ2 ) is the dipolar term, K is the anisotropy constant, µi is the angle between the iÄth magnetization vector with the z-axis, MS is the saturation magnetization h = H=HK , HK = 2K=MS and ½ = MS2 V0 =2r3 K is the dipolar interaction strength, and ¯ is the bonding angle between the line connecting two centers and the z-axis (we take 00 · ¯ · 90± ). In general, energy barrier for thermal relaxation will be shifted in the presence of dipolar interaction and has to be evaluated numerically. If the dipolar interaction strength ½ is weak, one may use perturbation method to evaluate the shift of the energy minima or saddle points of energy surface E (0) (µ1 ; µ2 ). We have p (2) EBi = KV0 [vi (1 ¡ h)2 ¡ 2½vi vj f3 sin ¯ cos ¯(1 ¡ h) § (1 ¡ 3 cos2 ¯) 1 ¡ h2 ]:

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0 = KV (1¡ If the dipole field is weak, the energy barrier for thermal relaxation is simply p EB h) p . The corresponding coercivity for magnetization reversal is simply hC = 1 ¡ EBC =KV = 1¡ T =TB where TB is the blocking temperature. Thus, at T = 0, the coercivity field is h = 1, or H = HK ; at T = TB no field is required to effect the magnetization reversal. For a homogenuous system, the general expression for energy barrier up to the second order in perturbation expansion is 2

p EB = (1 ¡ h)2 + 2hl (1 ¡ h) ¡ 2h? 1 ¡ h2 KV · ¸ 2h 1 + h + h2 2 2 hk h? + h? hk ¡ p 1 ¡ h2 1 ¡ h2

(3)

where hk = ¡½(1 ¡ 3 cos ¯ sin ¯); h? = 3½ cos ¯ sin ¯ represents, respectively, the longitudinal and transverse component of the dipole field. Note that Eq. (3) reduces to EB =KV = 1 + 2hk ¡ 2h? + [h2k + h2? ] if h = 0. As it turns out, comparison of the result of the energy barrier variation between purturbation and numerical evaluation is excellent if it is by the second order perturbation calculation, remains quite good by the linear order perturbation calculation provided if the dipolar < 0:05. Figure 2 shows such comparison between the linear perturbation interaction strength is ½» and numerical calculation. Energy barriers for magnetization relaxation with the bonding angle ¯ = 0± and ¼=2 only, and ½ < 1=3 has been evaluated previously, but with an error [5]. The relevant correct expression for ""=)"# transition can be re-stated and summarized as follows. p EB = [(1 + ½)(1 ¨ 3½) + h2 ¡ 4h + 2h (1 ¡ ½)(1 ¨ 3½)] KV

(4)

where the upper sign is good for ¯ = 0± , the lower sign for ¯ = ¼=2. For ¯ = 0± , the magnetization reverses by means of an asymmetric fanning rotation, (±µ1 =±µ2 < 0, ±µ1 6= ¾µ2 ). For ¯ = ¼=2, the mode of the magnetization reversal is by means of the asymmetric coherent rotation, (±µ1 =±µ2 > 0; ±µ1 6= ±µ2 ). Equation (4) differs from that given in Ref. [5]. However, the two expressions are not quantitatively dissimilar very much from each other. According to Eq. (4) EB is anisotropic with respect to ¯. It increases monotonically with ½ along the ¯ = 0 direction whilst decreases with ½ >h , E turns along the ¯ = ¼=2 direction. When the external field exceeds the critical field h» c B negative and the magnetization reverses instantaneously. (0) In the absence of the dipole term, the minima and saddle points of the system Em (µ1 ; µ2 ) (0) (0) are given by the conditions @E (µ1 ; µ2 )[email protected]µ1 = 0, @E (µ1 ; µ2 )[email protected]µ2 = 0. If the dipole term is present, the extreme positions are shifted and satisfy the relation (0) Em (µ1 + ¢µ1 ; µ2 + ¢µ2 ; ½) ¼ Em (µ1 ; µ2 ) + ½ ¢ @²(µ1 ; µ2 )[email protected]½;

(5)

where @²(µ1 ; µ2 )[email protected]½ = 2KV fcos(µ1 ¡ µ2 ) ¡ 3 cos(µ1 ¡ ¯) cos(µ2 ¡ ¯)g. The shift of the energy minima or saddle points from the corresponding noninteracting case can be obtained easily. For example,

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FIG. 1. Configuration of a dipole interacting system.

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FIG. 2. Variation of the energy barrier EB versus the dipole interaction strength ½ for ¯ = 0± and ¼=2. EB is anisotropic with respect to ¯ and varies with the dipole interaction strength ½. When the external field exceeds a certain critical value the magnetization reverses instantaneously as EB turns negative.

¾Em (0; 0)=2½KV » = 2½(1 ¡ 3 cos2 ¯) » = ±Em (0; 0)=2½KV

(5a)

p ±Em (0; § cos¡1 (h)) » = (h ¡ 3 cos ¯[h cos ¯ § sin ¯ 1 ¡ h2 ]) 2½KV

(5b)

Equation (5b) corresponds to the saddle points for Em (0; 0) ) Em (0; ¼) transition. Note that during the transition there are two saddle points, of which ±Em (0; + cos¡1 (h)) is lower than ±Em (0; ¡ cos¡1 (h)). This means that during the transitions, µ1 and µ2 reverse in the opposite direction, ±µ1 =±µ2 < 0 suggesting that an asymmetric fanning mode costs less energy compared to a coherent rotation mode. Thus, one will not be in the position to detect the difference in energy between these two modes if the consideration is apriori limited only to the cases where ¯ = 0 or ¯ = ¼=2 only such that sin ¯ = 0 or cos ¯ = 0 identically, as was done in previously [5]. The net magnetic moment for E(0; ¼) configuration is zero when h = 0. It begins to tilt toward the – z- direction when a small – h field, however weak, was applied. The energy barrier and coercivity corresponding to the Em (0; 0) ) Em (0; ¼) transition is p EB ' (1 ¡ h)2 ¡ 2½[(1 ¡ h)(1 ¡ 3 cos2 ¯) + 3 cos ¯ sin ¯ 1 ¡ h2 ]; (6) KV

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q p p 2 hC » 1 ¡ T =TB ¡ ½[(1 ¡ 3 cos ¯) + 3 cos ¯ sin ¯ 2 TB =T ¡ 1];

(7)

q ³ p p ¼´ hC ¯ = = 2 ¡ (1 ¡ ½)(1 + 3½) ¡ 4 + 8½ ¡ 4 (1 ¡ ½)(1 + 3½) + T =TB : 2

(9)

so that the corresponding analytic expressions for coercivity for ¯ = 0 and ¼=2 is, respectively, q p p hC (¯ = 0) = 2 ¡ (1 ¡ ½)(1 ¡ 3½) ¡ 4 ¡ 8½ ¡ 4 (1 ¡ ½)(1 ¡ 3½) + T=TB (8)

III. Monte carlo simulation Monte Carlo simulation has been successfully applied to study superparamagnetism of small magnetic particles [7; 13-14]. The method has been proved to be useful whenever a very short time interval and or very high frequency sweep rate is involved in the simulation of the system. In the calculation to follow, the Monte Carlo sampling method [14] was employed to simulate magnetization relaxation of a dipole interacting fine particle system in which the state of the system in phase space is assumed to jump randomly from (µ1i ; µ2i ) state to (µ1f ; µ2f ) state. The jumping probability p is defined as µ µ ¶¶ ¡¢Ei!f p = min 1; exp ; (10) kB T where ¢Ei!f = E(µ1i ; µ2i ) ¡ E(µ1f ; µ2f ) so that the probability is p = exp(¡¢Ei!f =kB T ) whenever E(µ1i ; µ2i ) > E(µ1f ; µ2f ). With reference to Ref. [14] a constraint to each Monte Carlo step (MCS) was imposed in our simulation scheme such that each random step is finite and is limited by a fixed maximum step. The characteristic frequency of the system is on the order of f0 = e25 Hz, or e25 MCS steps per second, and a total of 7200 MCS steps in one period were chosen in the simulation. This is equivalent to a disk drive under a 10 MHz field sweep rate. Each MCS step as prescribed is dependent upon not only the barrier height but also the breadth of the energy barrier, that is, dependent upon a three dimensional character of a barrier. By contrast, a perturbation calculation takes care only of the barrier height. Under the circumstance, a perturbation calculation has an inherent shortcoming compared to the MCS simulation. Due comments will be given below. The hysteresis loops and coercivities reported hereunder correspond to an averaged value of the simulation results from a collection of 2000 pairs of the interacting dipoles. IV. Results and discussions IV-1. Energy barrier and coercivity variation As remarked earlier, recent literature for interacting fine particle system suggested that the particle interaction may cause either enhancement [9] or depression [11-12] of blocking temperature, or equivalently the energy barrier, compared to the noninteracting values. It is therefore of

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particular interest to examine the situation closely. From Eqs. (6)-(7) it is apparent that the situation is sensitively dependent upon the critical bonding angle ¯C if the relevant material parameters of the system remain unchanged. From Eq. (6) the critical bonding angle can be expressed as " ! !# Ãr Ãr 1 1 ¡ h 1 ¡ h ¯C = ¼ ¡ cos¡1 ¡ cos¡1 : (11) 2 18 2 Thus, EB becomes enhanced due to the presence of the longitudinal component of the dipole field if ¯ < ¯C . From Eq. (11) it is clear that the cone of the bonding angle gets narrower with increasing external field. At h = 0 the critical bonding angle is ¯C = 24:6± . At h = 0:6 it reduces to ¯C = 20:7± . As the external field increases the critical bonding angle decreases correspondingly. For example, as h = 0 ) 0:8, we have ¯C = 24:6± ) 10:7± . A plot of the energy barrier and coercivity variation as a function of the coupling strength ½ based on Eqs. (6)-(7) for h = 0 and 0.6 for several values of ¯ are shown in Fig. 3 (top). The corresponding analytic results based on Eqs. (4) and (8)-(9) are also shown (solid lines) for comparison. The solid lines with a positive (negative) slope corresponds to the case where ¯ = 0 (¼=2). The perturbation results agree reasonably well with the analytical ones to within less than 5% when the dipolar coupling strength ½ · 0:05. Energy barrier changes substantially with the bonding angle as well as the external field. At ¯ = 30± , EB is nearly flat with respect to ½ when h = 0, or nearly unchanged from that of the noninteracting case. At larger field values, EB decreases fairly quickly and falls below that of the noninteracting value faster than the case for ¯ = 90± . By similar procedures, coercivity is found to be enhanced when ¯ < ¯C , depressed if ¯ > ¯C , due to dipole interaction. For an example, at EB =KV = T =TB = 0:5, coercivity becomes enhanced if ¯ < ¯C = 24:0± , depressed if the ¯ > ¯C . A plot of the coercivity variation versus the dipolar coupling strength ½ under thermal relaxation for several ¯ values are depicted in Fig. 3 (bottom). The exact analytic results are shown in solid lines for comparison. If the barrier height is lowered, say, EB =KV = 1 ) 0:2, the ¯C¢ angles will be correspondingly reduced, from 29:3± ) 18:6± . This is equivalent to lower the temperature or increase the field sweep rate as shown elsewhere [15]. Figure 4(a) shows comparison of the energy barrier variation based on the perturbation calculation and that obtained by numerical solution. Again, the energy barrier generally becomes enhanced when fB falls into a small cone of angle » 15± ¡ 30± . The larger the external field the smaller is the cone angle. Both energy barrier and coercivity become depressed if the bonding angle is larger. Both energy barrier EB (¯) and coercivity hc (¯) have a local maxima at ¯ = 0 and ¼=2, and a global minimum near ¯ » 60± at all temperatures. The global minimum of the coercivity and energy barrier is generally lower than the corresponding noninteracting values. Since, for a 3D random system, the probability of finding the bonding angle less than < ¯C = 30± is about ¼ 13%, the average coercivity and energy barrier of a dipole interacting system is likely to be lower compfared to that of the noninteracting system. Thus, by either lowering the temperature, increasing the field sweep rate or making the product of the KV value larger [15] the critical bonding angle of the system will be lowered so that both coercivity and energy barrier of the system can be further reduced in an actual laboratory measurements. The energy barrier variation depends upon the relative orientation of the moments, either the transition of the type ("; ") ) ("; #) or its reverse transition ("; #) ) ("; "). The former

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FIG. 3. Comparison of the energy barrier and coereivity variation as a function of the dipole interaction strength ½ under thermal relaxation effect. Marked: perturbation results; Solid lines: analytic results.

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FIG. 4. (a) Comparison of the energy barrier variation versus bonding angle for several ½ values. Circles and squares: numerical results; dashed lines: perturbation calculations. Both energy barrier and coercivity become enhanced whenever ¯ is less than ¯C ¼ 15± » 30± , depressed if ¯ is larger than ¯C , and have a global minimum near ¯ » 60± at all temperature. The variation depends slightly upon the presence of an external field. (b) Dependence of the energy barrier variation upon the relative orientation of the moments.

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corresponds exactly to the situation described above. For the latter, the global minimum of the energy barrier occurs at ¯ » 30± instead. Figure 4(b) depicts the energy barrier variation of such a situation. IV-2. Monte carlo simulation Comparison of the coercivity variation as a function of the dipole interaction strength ½ for ¯ = 0± and ¼=2 obtained by MCS simulation (heavy dots), analytic calculation (solid lines) based on Eqs. (8), (9) and perturbation calculation (dashed lines) is shown in Fig. 5. The results from MCS simulation and analytic ones agree quitw well for all the range of ½ values calculated. The perturbation results appear acceptable provided if ½ is small. In MCS simulations we input as parameters KV=kB T = 45, f = 10 MHz and EBC =KV = 0:0886. The first parameter is roughly equivalent to a half-life of » 10 years, typical of the present day magnetic recording media. The parameter EBC =KV = 0:0886 was chosen in order to make the three curves shown in Fig. 5 coincide with each other at ½ = 0. This value is obtained by first solving Eq. (7) for hc via MCS simulation by putting ½ = 0 and substituting the balance back into Eq. (4). Numerically, EBC =KV = 0:0886 is equivalent to EBC ¼ 4kB T which corresponds to a half-life of » 1/7000 of a period in tracing out a hysteresis loop. The MCS simulation result agrees excellently with the analytic ones for the dipole interaction strength falling in the range ½ = (0; 0:2) calculated. For dipole interacting ellipsoidal particles such as ° ¡ F e2 O3 whose anisotropy energy is dominated by shape anisotropy the interaction strength ½ » 0:8 corresponds fairly close to a packing fraction of the present day particulate recording media. Thus, the present MCS simulation argumented with perturbation calculation can be usefully exploited to simulate a dipole interacting particulate magnetic recording media. Comparison of the coercivity variation versus bonding angle obtained perturbation calculation (top figure) and MCS simulation (bottom figure) is shown in Fig. 6. The input parameters remain the same as given in Fig. 5. There appears to have an appreciable relative deviation between the two set of curves near ¯ » = 60± . A possible source of difference in mechanism between the two approaches which may have been responsible for the deviation is as follows. First, each random step in MCS simulation, both the barrier height and barrier breadth of any three dimensional character can be sensed and taken into account automatically. Not so in perturbation

FIG. 5. Comparison of the coercivity variation versus the coupling strength ½. Dots: MCS simulation; solid lines: exact numerical result; dashed lines: perturbation results.

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FIG. 6. Comparison of the coercivity variation based on perturbation calculation (top figure) and MCS simulation (bottom figure). The input parameters remain the same as in Fig. 5.

calculation in which one takes note of existence of the barrier height only while usually disregarding the ‘barrier breadth’ nearly completely. Second, the reason for the set of three curves to intersect at one point (top figure) is mostly due to the first order perturbation approximation perse. Reminded that during the perturbation calculation of coercivity the energy barrier is usually treated approximately as constant. This is not quite correct as has been elaborated [15]. In fact, any small (transverse) perturbation will always result in a corresponding change in the barrier height, hence, coercivity. Nevertheless, general character of the two set of curves bears good resemblance to each other. IV-3. Packing fraction and dipole interaction For ellipsoidal magnetic particles such as ° ¡ F e2 O3 where shape anisotropy energy dominates, one can establish the relationship between magnetic anisotropy energy, dipole coupling strength ½ and packing fraction, P , through the relation V ¼¼ 3 = r

µ

MS2 v 2K r3

¶

2( 12 (NA ¡ NC )MS ) 2K 2K = ½ = ½ = ½(NA ¡ NC ) MS2 m2S MS2

(12)

where (NA ¡NC ) is the demagnetization factor for the ellipsoidal particles. If the aspect ratio of the ellipsoidal particle is c=a = 5, then (NA ¡NC ) = 5:23, so that the packing fraction is P ¼ 5:23½. For dipole interaction strength ½ = 0:05 » 0:08, we have, approximately, P » = 0:26 » 0:42 which is a realistic figure for a typical particulate magnetic recording media. This suggests that the perturbation calculation is reasonable for the dipole interacting particulate recording media. One

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additional comment is due. In order to carry out MCS simulation for magnetization reversal for a dipole interacting system, it is essential that one has a good sound prior knowledge of the extreme of the entire energy surfaces as to how and through what passages the moment may reverse. In this respect, the present treatment is well guided and supplemented by the perturbation calculation. IV-4. Dipole field contribution To explain why dipole field is responsible for enhancing and depressing the coercivity and energy barrier height of a particulate system one takes note of the following. Dipole interaction ~ ef f = ¡r¡3 (M ~ ¡ 3^ ~ ¢ r^) may be decomposed into two parts: a longitudinal component field H r (M Hk and a transverse one H? . The longitudinal component Hl is portional to » ½(1 ¡ cos2 ¯) whose contribution to coercivity decreases with increasing ¯, turns negative when ¯ · 54:7± . p 2 The effective transverse component H? is proportional to » ¡½ cos ¯ sin ¯ which (is usually neglected in the mean field calculation) has a negative contribution to hc (¯) for all bonding angles, and has a minimum at ¯ = 45± . The superposition of Hl and H? contributions leads to the local maxima at ¯ = 0 and ¼=2 and a global minimum of hc (¯) at ¯ » 60± . The former results in coercivity enhancement when the bonding angles is less than a certain critical bonding angle ¯C ¼ 15± » 30± , and coercivity depression due to the presence a global minimum of hc (¯) at ¯ » 60± , as schematically depicted in Fig. 7. From Fig. 7 we notice that coercivity can be enhanced if ¯ ¯C = 15± » 30± , depressed whenever ¯ > ¯C . Depression is most appreciable when ¯ is in the range of ¼ 50± » 70± . IV-5. Bonding angles versus hysteresis loops As discussed earlier related to Eqs. (4) and (5) the dipole interacting system has three minimum energy states, (1)("; "); (2)("; #) or (#; ") and (3) (#; #) (four if the particles are not identical) and two saddle points corresponding to the transitions (1) ) (2) and (2) ) (3). The corresponding barrier height is p EB =KV » (13) = (1 ¡ h)2 ¨ 2½[(1 ¡ h)(1 ¡ 3 cos2 ¯) § 3 cos ¯ sin ¯ 1 ¡ h2

where the upper [lower] sign is for (1) ) (2) [(2) ) (3)] transition. If the barrier height for (1) ("") ) (2) ("#) transition is higher than the corresponding transition (2) ("#) ) (3) (##), there exists no intermediate state at (2) ("#), instead, it makes a direct jump from (1) ("") in transition to ) (3) (##). On the other hand, if the barrier height for (1) ) (2) transition is lower than (2) ) (3) transition, one intermediate state at (2) ("#) will exist so that magnetic moment may stay for an indefinite period before jumping to (3) (##) upon application of a large enough field to effect magnetic reversal. The existence of whether one-or two-stage transition in the magnetization reversal depends upon relative barrier height between the two transitions (1) ) (2) and (2) ) (3). In fact it is largely dependent upon the sign of the term (1 ¡ 3 cos2 ¯). This term becomes negative if ¯ < 54:7± so that the energy barrier for the transition (1) ) (2) is higher than that for (2) ) (3) transition and there exists no intermediate transition state. The reverse is true if ¯ > 54:7± , the barrier height for (2) ) (3) becomes higher and a two-stage transition results. Physically, this corresponds to the fact that the state (2) ("#) is more stable than either of the state (1) ("") or (3) (##) when the bonding angle ¯ is greater than 54:7± .

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FIG. 7. Relative contribution of the longitudinal and transverse component of the effective dipolar field which contributes to the shift in coercivity variation. Superposition of the two contributions results in the coercivity minimum at ¯ » 60± . A sample average of the coercivity is thus depressed compared to that of the noninteracting system.

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FIG. 8. MCS simulation of the hysteresis behavior for several bonding angles and for a 2D random system. The system undergoes a two-stage magnetization reversal processes whenever ¯ is, or close to, ¼=2.

A summary of the MCS simulation results on hysteretic behavior of a pair of dipole interacting SDPs for various bonding angles and dipole coupling strengths are shown in Figs. 8 and 9. As shown in Fig. 8, the coercivity is larger at small bonding angles than at larger bonding angles. Note that the system may undergo a two-stage magnetization transition processes when accompanied by a non-zero dipole coupling strength whenever the bonding angle is close to > 60± as depicted in Fig. 9. The system ¯ = ¼=2. The system begins to show sign of it when ¯ » may begin to switch at a lower field value, say hc1 , resulting in a net negative magnetization until at h = hc2 < hc1 when the second or the last stage of the magnetization reversal takes place. As a consequence, the hysteretic behavior of this dipole interacting system, when the media is thin enough so that the system is independent of the azimuthal angle of the magnetization, the initiation of the magnetization transition may take place at a smaller field value, and by virtue of the two-step switching, the slope of the hysteresis curve maybe further reduced, resulting in a wider switching field distribution. Thus, jdM=dHj becomes smaller overall, hence lower in coercivity compared to that of the noninteracting case. The dashed line in Fig. 8 depicts the hysteresis behavior for a 2D random system, whilst the solid lines corresponds to that of the noninteracting case. Similar features for a 2D random system are reproduced in Fig. 9 (top) for several ½ values as a parameter. The behavior is similar to that of a dipole interacting thin film

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FIG. 9. MCS simulation results of hysteresis behavior for several ¯’s and for a 2D random system where the loops are obtained by averaging over the ¯¡ values. Slope of the magnetization curves of the latter is appreciably smaller, leading to a wider switching field distribution and lower coercivity.

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FIG. 10. (a) Behavior of the MZF C and MF C magnetization variation of a dipole interacting system with a log normal volume distribution f (V ) with ½ = 0:04, ½V = 04, (b) Derivative of (MZF C MF C )=MS with respect to temperature giving the blocking temperature distribution of the system.

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media. A final remark is in order. The coercivity decreases steadily with increasing coupling strength except when the bonding angle is close to zero when the corresponding energy barrier becomes enhanced. IV-6. Thermal relaxation and blocking temperature distribution The present calculation maybe successfully extended to discuss the zero-field-cooled, MZF C , field-cooled, MF C , and net magnetization variation, (MZF C ¡ MF C ), of the system based on the simple Langevin function and Arhennius relation. Langevin function is defined as L (HV =T ) = [exp(HV MS =kB T ) ¡ exp(¡HV MS =kB T )] = [exp(HV MS =kB T ) + exp(¡HV MS =kB T )] in which the field H is an effective field which is assumed to be small compared to the anisotropy field (say, H = 0:01 HK ). The blocking temperature distribution of the system TB (V ) corresponding to a system with an arbitrary log normal volume distribution f(V ) = exp[¡(ln(V=V0 ))2 =2¾V2 ] where ¾V is the half-width of the distribution can be easily obtained [16]. Figure 10(a) shows the ZFC and FC magnetization relaxation behavior the system with a continuous volume distribution f (V ) under various field values at ½ = 0:04 and ¾V = 0:4, and Fig. 10(b) depicts the derivative of (MZF C - MF C )=MS with respect to temperature which effectively represents the blocking temperature distribution of the system. Note that Fig. 10(a) bears good resemblance to the ZFC and FC magnetization relaxation curves observed [4; 9] and that the absolute magnetization value depends on the level of applied field, half width of the volume distribution ¾V and dipole interaction strength ½. Further, the blocking temperature distribution TB (V ) nearly exactly duplicates the weighted volume distribution of the system V f(V ). V. Summary Coercivity and energy barrier variation of a fine dipole interacting system has been discussed in terms of a fixed step Monte Carlo simulation supplemented by perturbation scheme. Dipole interaction field can be decomposed into a longitudinal and a transverse component. The superposition of the two component field leads to the local maxima at ¯ = 0 and ¼=2 and a global minimum in coercivity at ¯ » 60± . Coercivity of a random dipole interacting system thus becomes lower compared to that of a noninteracting system since the probability of finding the bonding angle of the system smaller than 30± is less than 13%. The asymmetric fanning mode is favored compared to other modes in making ("#) ) (##) transition if ¯ 6= 0, ¼=2. In addition, whenever the bonding angle ¯ ! ¼=2, the system may exhibit a two-step switching behavior causing the system to broaden its switching field distribution. Finally, the field-cooled and zero-field-cooled magnetization relaxation behavior and its blocking temperature distribution can be well accounted for based on the simple Langevin function formulation. Acknowledgment We gratefully acknowledge the financial help through a research grant from NSC of the ROC under: NSC-88-2216- E- 002-021 and -022. References [ 1 ] H. N. Bertram, Theory of magnetic recording, (Cambrige Univ. Cambrige, 1994), Chap. 12. [ 2 ] Pu-Ling Lu and S. H. Charap. IEEE Trans. Magn. 30, 4230, 1994.

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[ 3 ] Borlian Chen and H. L. Huang, J. Magn. Magn. Mats., 171, 218 (1997). [ 4 ] J. L. Dormann, D. Fiorani, E. Tronc, Adv. in Chem. Phys., CXVIII, (pp. 283-493); eds. I. Prigogine and S. A. Rice. (John Wiley, 1997). [ 5 ] W. Chen, S. Zhang and H. N. Bertram, J. Appl. Phys. 71, 5579, (1992). [ 6 ] I. Klik, J. S. Yang and C. R. Chang, J. Appl. Phys. 76, 6493 (1994); see also I. Klik and C. R. Chang, Phys. Rev B, 52, 3540 (1995). [ 7 ] H. L. Huang et al., J. Appl. Phys. 85, 5558 (1999). [ 8 ] L. Dormann, L. Bessaia, and D. Fiorani, J. Phys. C: Solid State Phys. 21, 2015 (1988). [ 9 ] P. Prene´ , E. Tronc, J-P. Jolivert, J. Livage, R. Cherkaoui, M. Nogue´ s, J. L. Dormann, D. Fiorani, IEEE Trans. magn 29, 2658 (1993). [10] S. Mørup and E. Tronc, Phys. Rev. Lett. 72, 3278 (1994). [11] L. Dormann et al., Phys. Rev. B53, 14291 (1996). [12] F. Hansen and S. Mørup, J. Magn. Magn. Mats. 184, 262 (1998). [13] K. Binder and D. W. Heerman, Monte Carlo Simulationin Statistical Physics, (Spriner, Berlin 1980). [14] D. A. Dimitrov and G. M. Wysin, Phys. Rev. B54, 9237 (1996). [15] J. J. Lu and H. L. Huang, J. Appl. Phys. 77, 3323, 1995. [16] J. J. Lu and H. L. Huang, ISAMT’99, paper BD-5 (invited), May, 1999

VOL. 38, NO. 1

FEBRUARY 2000

Energy Barrier, Coercivity and Blocking Temperature Variation of Fine-Particle Systems Jing Ju Lu and Huei Li Huang Department of Physics, National Taiwan University, Taipei, Taiwan 106, R.O.C. (Received July 2, 1999)

Coercivity and energy barrier height variation of a dipole interacting fine particle system has been evaluated by means of the fixed step Monte Carlo simulation supplemented by a perturbation calculation. The dipole interaction field can be decomposed into a longitudinal and a transverse field component. The superposition of the two field component leads to local maxima at ¯ = 0 and ¼=2 and a global minimum of coercivity at ¯ » 60± . Thus, the coercivity of a random dipole interacting system generally becomes lower compared to that of a noninteracting system since the probability of finding the bonding angle of the system smaller than 30± is less than 13%. The asymmetric fanning mode is shown to be favored in comparison with the other modes in making the ("#) ) (##) transition if ¯ is neither zero nor ¼=2. Whenever the bonding angle is close to ¼=2, the system may exhibit a two-step switching behavior causing the system to broaden its switching field distribution. Finally, the fieldcooled and zero-field-cooled magnetization relaxation behavior and its blocking temperature distribution can be well accounted for. PACS. 75.60.Jp – Fine-particle systems. PACS. 75.60.Nt – Temperature-hysteresis effect. PACS. 76.60.Es – Relaxation effect.

I. Introduction High density recording calls for the magnetic medium to consist of single domain particles homogeneous in particle size, isolated and well oriented along the track direction so as to reduce particle size fluctuations and interparticle-interaction-induced particle noise. The particles need to be small enough in linear dimension to assure high bit density, while at the same time the product of the particle’s anisotropy constant and volume, KV, should be large enough to ensure high thermal stability [1-3]. In light of this, it is necessary to ask the question how serious and to what extent the ubiquitous interparticle interaction, dipolar interaction in the case of particulate media, has to do with these problems. It is important to carefully delineate whether the interparticle interaction enhances or depresses the energy barrier for magnetization relaxation and its corresponding coercivity for magnetization reversal and what specific media may satisfy the future technology need. Effect of fluctuations in the relevant physical parameters such as particles size, anisotropy constant and misorientation of the easy axis with respect to the curvilinear vector of the pulsed recording head field has been discussed previously in the context of writing transition noise [3]. Energy barrier, coercivity and blocking temperature variations and the related relaxation 0

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phenomena for fine magnetic particles has also been extensively studied and reviewed [4]. Energy barrier for thermal relaxation as a function of the bonding angle (between the line connecting two particles and the applied field direction, i.e., the z-axis.) for well oriented dipole interacting particles has been studied to some extent [5-7] in which the energy barrier was shown to increase monotonically at zero bonding angle whilst the reverse is true when the bonding angle is ¼=2. Recently, Mossbauer spectroscopic analysis and magnetic measurements of noninteracting (i.e., for particles far apart) and interacting magnetic fine particles such as has raised various questions regarding the dynamics of the moments and the effects of interaction [8-9]. In effect, there exists conflicting interpretations of the observation data as regard the interparticle interaction effect on blocking temperature, or equivalently, energy barrier, variation of magnetic particles [8-12]. For example, in the analysis of the blocking temperature TB in iron grains [9] it was deduced that the energy barrier increases with the interparticle interaction, whilst an opposite conclusion has been reached in analyzing the weakly interacting ° ¡ F e2 O3 particles [11-12] suggesting that the effective dipolar interaction decreases with decreasing interparticle separation. In this communication a thorough study in this connection is presented aiming to clarify the confusing dependence of the energy barrier, coercivity and blocking temperature variation for thermal relaxation of a dipole interacting fine particle system which is potentially applicable to a typical recording media. The present analysis is based upon Monte Carlo simulation [13; 14] in combination with and argumented by the perturbation calculation. Due comments will be given on blocking temperature variation of a fine particle system. The present perturbation approach causes less than 8% relative error compared to an exact nu- merical calculation if particles are < reasonably apart such that the packing fraction of the media is ¼ 40% (see text) as elaborated below. II. Perturbation calculation Consider a pair of well oriented dipole interacting single domain particles of volume V1 = V0 v1 and V2 = V0 v2 , where v1 < v2 , each with the easy axis of the particles oriented along the z-axis. The configuration of the system is shown in Fig. 1. Energy of the system is E(µ1 ; µ2 ) = KV0 [v1 sin2 µ1 + v2 sin2 µ2 ¡ 2h(v1 cos µ1 + v2 cos µ2 ) +2½v1 v2 fcos(µ1 ¡ µ2 ) ¡ 3 cos(µ1 ¡ ¯) cos(µ2 ¡ ¯)g];

(1)

= E (0) (µ1 ; µ2 ) + ²(µ1 ; µ2 ) where E (0) (µ1 ; µ2 ) is the energy of the noninteracting system, ²(µ1 ; µ2 ) is the dipolar term, K is the anisotropy constant, µi is the angle between the iÄth magnetization vector with the z-axis, MS is the saturation magnetization h = H=HK , HK = 2K=MS and ½ = MS2 V0 =2r3 K is the dipolar interaction strength, and ¯ is the bonding angle between the line connecting two centers and the z-axis (we take 00 · ¯ · 90± ). In general, energy barrier for thermal relaxation will be shifted in the presence of dipolar interaction and has to be evaluated numerically. If the dipolar interaction strength ½ is weak, one may use perturbation method to evaluate the shift of the energy minima or saddle points of energy surface E (0) (µ1 ; µ2 ). We have p (2) EBi = KV0 [vi (1 ¡ h)2 ¡ 2½vi vj f3 sin ¯ cos ¯(1 ¡ h) § (1 ¡ 3 cos2 ¯) 1 ¡ h2 ]:

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0 = KV (1¡ If the dipole field is weak, the energy barrier for thermal relaxation is simply p EB h) p . The corresponding coercivity for magnetization reversal is simply hC = 1 ¡ EBC =KV = 1¡ T =TB where TB is the blocking temperature. Thus, at T = 0, the coercivity field is h = 1, or H = HK ; at T = TB no field is required to effect the magnetization reversal. For a homogenuous system, the general expression for energy barrier up to the second order in perturbation expansion is 2

p EB = (1 ¡ h)2 + 2hl (1 ¡ h) ¡ 2h? 1 ¡ h2 KV · ¸ 2h 1 + h + h2 2 2 hk h? + h? hk ¡ p 1 ¡ h2 1 ¡ h2

(3)

where hk = ¡½(1 ¡ 3 cos ¯ sin ¯); h? = 3½ cos ¯ sin ¯ represents, respectively, the longitudinal and transverse component of the dipole field. Note that Eq. (3) reduces to EB =KV = 1 + 2hk ¡ 2h? + [h2k + h2? ] if h = 0. As it turns out, comparison of the result of the energy barrier variation between purturbation and numerical evaluation is excellent if it is by the second order perturbation calculation, remains quite good by the linear order perturbation calculation provided if the dipolar < 0:05. Figure 2 shows such comparison between the linear perturbation interaction strength is ½» and numerical calculation. Energy barriers for magnetization relaxation with the bonding angle ¯ = 0± and ¼=2 only, and ½ < 1=3 has been evaluated previously, but with an error [5]. The relevant correct expression for ""=)"# transition can be re-stated and summarized as follows. p EB = [(1 + ½)(1 ¨ 3½) + h2 ¡ 4h + 2h (1 ¡ ½)(1 ¨ 3½)] KV

(4)

where the upper sign is good for ¯ = 0± , the lower sign for ¯ = ¼=2. For ¯ = 0± , the magnetization reverses by means of an asymmetric fanning rotation, (±µ1 =±µ2 < 0, ±µ1 6= ¾µ2 ). For ¯ = ¼=2, the mode of the magnetization reversal is by means of the asymmetric coherent rotation, (±µ1 =±µ2 > 0; ±µ1 6= ±µ2 ). Equation (4) differs from that given in Ref. [5]. However, the two expressions are not quantitatively dissimilar very much from each other. According to Eq. (4) EB is anisotropic with respect to ¯. It increases monotonically with ½ along the ¯ = 0 direction whilst decreases with ½ >h , E turns along the ¯ = ¼=2 direction. When the external field exceeds the critical field h» c B negative and the magnetization reverses instantaneously. (0) In the absence of the dipole term, the minima and saddle points of the system Em (µ1 ; µ2 ) (0) (0) are given by the conditions @E (µ1 ; µ2 )[email protected]µ1 = 0, @E (µ1 ; µ2 )[email protected]µ2 = 0. If the dipole term is present, the extreme positions are shifted and satisfy the relation (0) Em (µ1 + ¢µ1 ; µ2 + ¢µ2 ; ½) ¼ Em (µ1 ; µ2 ) + ½ ¢ @²(µ1 ; µ2 )[email protected]½;

(5)

where @²(µ1 ; µ2 )[email protected]½ = 2KV fcos(µ1 ¡ µ2 ) ¡ 3 cos(µ1 ¡ ¯) cos(µ2 ¡ ¯)g. The shift of the energy minima or saddle points from the corresponding noninteracting case can be obtained easily. For example,

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FIG. 1. Configuration of a dipole interacting system.

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FIG. 2. Variation of the energy barrier EB versus the dipole interaction strength ½ for ¯ = 0± and ¼=2. EB is anisotropic with respect to ¯ and varies with the dipole interaction strength ½. When the external field exceeds a certain critical value the magnetization reverses instantaneously as EB turns negative.

¾Em (0; 0)=2½KV » = 2½(1 ¡ 3 cos2 ¯) » = ±Em (0; 0)=2½KV

(5a)

p ±Em (0; § cos¡1 (h)) » = (h ¡ 3 cos ¯[h cos ¯ § sin ¯ 1 ¡ h2 ]) 2½KV

(5b)

Equation (5b) corresponds to the saddle points for Em (0; 0) ) Em (0; ¼) transition. Note that during the transition there are two saddle points, of which ±Em (0; + cos¡1 (h)) is lower than ±Em (0; ¡ cos¡1 (h)). This means that during the transitions, µ1 and µ2 reverse in the opposite direction, ±µ1 =±µ2 < 0 suggesting that an asymmetric fanning mode costs less energy compared to a coherent rotation mode. Thus, one will not be in the position to detect the difference in energy between these two modes if the consideration is apriori limited only to the cases where ¯ = 0 or ¯ = ¼=2 only such that sin ¯ = 0 or cos ¯ = 0 identically, as was done in previously [5]. The net magnetic moment for E(0; ¼) configuration is zero when h = 0. It begins to tilt toward the – z- direction when a small – h field, however weak, was applied. The energy barrier and coercivity corresponding to the Em (0; 0) ) Em (0; ¼) transition is p EB ' (1 ¡ h)2 ¡ 2½[(1 ¡ h)(1 ¡ 3 cos2 ¯) + 3 cos ¯ sin ¯ 1 ¡ h2 ]; (6) KV

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q p p 2 hC » 1 ¡ T =TB ¡ ½[(1 ¡ 3 cos ¯) + 3 cos ¯ sin ¯ 2 TB =T ¡ 1];

(7)

q ³ p p ¼´ hC ¯ = = 2 ¡ (1 ¡ ½)(1 + 3½) ¡ 4 + 8½ ¡ 4 (1 ¡ ½)(1 + 3½) + T =TB : 2

(9)

so that the corresponding analytic expressions for coercivity for ¯ = 0 and ¼=2 is, respectively, q p p hC (¯ = 0) = 2 ¡ (1 ¡ ½)(1 ¡ 3½) ¡ 4 ¡ 8½ ¡ 4 (1 ¡ ½)(1 ¡ 3½) + T=TB (8)

III. Monte carlo simulation Monte Carlo simulation has been successfully applied to study superparamagnetism of small magnetic particles [7; 13-14]. The method has been proved to be useful whenever a very short time interval and or very high frequency sweep rate is involved in the simulation of the system. In the calculation to follow, the Monte Carlo sampling method [14] was employed to simulate magnetization relaxation of a dipole interacting fine particle system in which the state of the system in phase space is assumed to jump randomly from (µ1i ; µ2i ) state to (µ1f ; µ2f ) state. The jumping probability p is defined as µ µ ¶¶ ¡¢Ei!f p = min 1; exp ; (10) kB T where ¢Ei!f = E(µ1i ; µ2i ) ¡ E(µ1f ; µ2f ) so that the probability is p = exp(¡¢Ei!f =kB T ) whenever E(µ1i ; µ2i ) > E(µ1f ; µ2f ). With reference to Ref. [14] a constraint to each Monte Carlo step (MCS) was imposed in our simulation scheme such that each random step is finite and is limited by a fixed maximum step. The characteristic frequency of the system is on the order of f0 = e25 Hz, or e25 MCS steps per second, and a total of 7200 MCS steps in one period were chosen in the simulation. This is equivalent to a disk drive under a 10 MHz field sweep rate. Each MCS step as prescribed is dependent upon not only the barrier height but also the breadth of the energy barrier, that is, dependent upon a three dimensional character of a barrier. By contrast, a perturbation calculation takes care only of the barrier height. Under the circumstance, a perturbation calculation has an inherent shortcoming compared to the MCS simulation. Due comments will be given below. The hysteresis loops and coercivities reported hereunder correspond to an averaged value of the simulation results from a collection of 2000 pairs of the interacting dipoles. IV. Results and discussions IV-1. Energy barrier and coercivity variation As remarked earlier, recent literature for interacting fine particle system suggested that the particle interaction may cause either enhancement [9] or depression [11-12] of blocking temperature, or equivalently the energy barrier, compared to the noninteracting values. It is therefore of

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particular interest to examine the situation closely. From Eqs. (6)-(7) it is apparent that the situation is sensitively dependent upon the critical bonding angle ¯C if the relevant material parameters of the system remain unchanged. From Eq. (6) the critical bonding angle can be expressed as " ! !# Ãr Ãr 1 1 ¡ h 1 ¡ h ¯C = ¼ ¡ cos¡1 ¡ cos¡1 : (11) 2 18 2 Thus, EB becomes enhanced due to the presence of the longitudinal component of the dipole field if ¯ < ¯C . From Eq. (11) it is clear that the cone of the bonding angle gets narrower with increasing external field. At h = 0 the critical bonding angle is ¯C = 24:6± . At h = 0:6 it reduces to ¯C = 20:7± . As the external field increases the critical bonding angle decreases correspondingly. For example, as h = 0 ) 0:8, we have ¯C = 24:6± ) 10:7± . A plot of the energy barrier and coercivity variation as a function of the coupling strength ½ based on Eqs. (6)-(7) for h = 0 and 0.6 for several values of ¯ are shown in Fig. 3 (top). The corresponding analytic results based on Eqs. (4) and (8)-(9) are also shown (solid lines) for comparison. The solid lines with a positive (negative) slope corresponds to the case where ¯ = 0 (¼=2). The perturbation results agree reasonably well with the analytical ones to within less than 5% when the dipolar coupling strength ½ · 0:05. Energy barrier changes substantially with the bonding angle as well as the external field. At ¯ = 30± , EB is nearly flat with respect to ½ when h = 0, or nearly unchanged from that of the noninteracting case. At larger field values, EB decreases fairly quickly and falls below that of the noninteracting value faster than the case for ¯ = 90± . By similar procedures, coercivity is found to be enhanced when ¯ < ¯C , depressed if ¯ > ¯C , due to dipole interaction. For an example, at EB =KV = T =TB = 0:5, coercivity becomes enhanced if ¯ < ¯C = 24:0± , depressed if the ¯ > ¯C . A plot of the coercivity variation versus the dipolar coupling strength ½ under thermal relaxation for several ¯ values are depicted in Fig. 3 (bottom). The exact analytic results are shown in solid lines for comparison. If the barrier height is lowered, say, EB =KV = 1 ) 0:2, the ¯C¢ angles will be correspondingly reduced, from 29:3± ) 18:6± . This is equivalent to lower the temperature or increase the field sweep rate as shown elsewhere [15]. Figure 4(a) shows comparison of the energy barrier variation based on the perturbation calculation and that obtained by numerical solution. Again, the energy barrier generally becomes enhanced when fB falls into a small cone of angle » 15± ¡ 30± . The larger the external field the smaller is the cone angle. Both energy barrier and coercivity become depressed if the bonding angle is larger. Both energy barrier EB (¯) and coercivity hc (¯) have a local maxima at ¯ = 0 and ¼=2, and a global minimum near ¯ » 60± at all temperatures. The global minimum of the coercivity and energy barrier is generally lower than the corresponding noninteracting values. Since, for a 3D random system, the probability of finding the bonding angle less than < ¯C = 30± is about ¼ 13%, the average coercivity and energy barrier of a dipole interacting system is likely to be lower compfared to that of the noninteracting system. Thus, by either lowering the temperature, increasing the field sweep rate or making the product of the KV value larger [15] the critical bonding angle of the system will be lowered so that both coercivity and energy barrier of the system can be further reduced in an actual laboratory measurements. The energy barrier variation depends upon the relative orientation of the moments, either the transition of the type ("; ") ) ("; #) or its reverse transition ("; #) ) ("; "). The former

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FIG. 3. Comparison of the energy barrier and coereivity variation as a function of the dipole interaction strength ½ under thermal relaxation effect. Marked: perturbation results; Solid lines: analytic results.

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FIG. 4. (a) Comparison of the energy barrier variation versus bonding angle for several ½ values. Circles and squares: numerical results; dashed lines: perturbation calculations. Both energy barrier and coercivity become enhanced whenever ¯ is less than ¯C ¼ 15± » 30± , depressed if ¯ is larger than ¯C , and have a global minimum near ¯ » 60± at all temperature. The variation depends slightly upon the presence of an external field. (b) Dependence of the energy barrier variation upon the relative orientation of the moments.

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corresponds exactly to the situation described above. For the latter, the global minimum of the energy barrier occurs at ¯ » 30± instead. Figure 4(b) depicts the energy barrier variation of such a situation. IV-2. Monte carlo simulation Comparison of the coercivity variation as a function of the dipole interaction strength ½ for ¯ = 0± and ¼=2 obtained by MCS simulation (heavy dots), analytic calculation (solid lines) based on Eqs. (8), (9) and perturbation calculation (dashed lines) is shown in Fig. 5. The results from MCS simulation and analytic ones agree quitw well for all the range of ½ values calculated. The perturbation results appear acceptable provided if ½ is small. In MCS simulations we input as parameters KV=kB T = 45, f = 10 MHz and EBC =KV = 0:0886. The first parameter is roughly equivalent to a half-life of » 10 years, typical of the present day magnetic recording media. The parameter EBC =KV = 0:0886 was chosen in order to make the three curves shown in Fig. 5 coincide with each other at ½ = 0. This value is obtained by first solving Eq. (7) for hc via MCS simulation by putting ½ = 0 and substituting the balance back into Eq. (4). Numerically, EBC =KV = 0:0886 is equivalent to EBC ¼ 4kB T which corresponds to a half-life of » 1/7000 of a period in tracing out a hysteresis loop. The MCS simulation result agrees excellently with the analytic ones for the dipole interaction strength falling in the range ½ = (0; 0:2) calculated. For dipole interacting ellipsoidal particles such as ° ¡ F e2 O3 whose anisotropy energy is dominated by shape anisotropy the interaction strength ½ » 0:8 corresponds fairly close to a packing fraction of the present day particulate recording media. Thus, the present MCS simulation argumented with perturbation calculation can be usefully exploited to simulate a dipole interacting particulate magnetic recording media. Comparison of the coercivity variation versus bonding angle obtained perturbation calculation (top figure) and MCS simulation (bottom figure) is shown in Fig. 6. The input parameters remain the same as given in Fig. 5. There appears to have an appreciable relative deviation between the two set of curves near ¯ » = 60± . A possible source of difference in mechanism between the two approaches which may have been responsible for the deviation is as follows. First, each random step in MCS simulation, both the barrier height and barrier breadth of any three dimensional character can be sensed and taken into account automatically. Not so in perturbation

FIG. 5. Comparison of the coercivity variation versus the coupling strength ½. Dots: MCS simulation; solid lines: exact numerical result; dashed lines: perturbation results.

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FIG. 6. Comparison of the coercivity variation based on perturbation calculation (top figure) and MCS simulation (bottom figure). The input parameters remain the same as in Fig. 5.

calculation in which one takes note of existence of the barrier height only while usually disregarding the ‘barrier breadth’ nearly completely. Second, the reason for the set of three curves to intersect at one point (top figure) is mostly due to the first order perturbation approximation perse. Reminded that during the perturbation calculation of coercivity the energy barrier is usually treated approximately as constant. This is not quite correct as has been elaborated [15]. In fact, any small (transverse) perturbation will always result in a corresponding change in the barrier height, hence, coercivity. Nevertheless, general character of the two set of curves bears good resemblance to each other. IV-3. Packing fraction and dipole interaction For ellipsoidal magnetic particles such as ° ¡ F e2 O3 where shape anisotropy energy dominates, one can establish the relationship between magnetic anisotropy energy, dipole coupling strength ½ and packing fraction, P , through the relation V ¼¼ 3 = r

µ

MS2 v 2K r3

¶

2( 12 (NA ¡ NC )MS ) 2K 2K = ½ = ½ = ½(NA ¡ NC ) MS2 m2S MS2

(12)

where (NA ¡NC ) is the demagnetization factor for the ellipsoidal particles. If the aspect ratio of the ellipsoidal particle is c=a = 5, then (NA ¡NC ) = 5:23, so that the packing fraction is P ¼ 5:23½. For dipole interaction strength ½ = 0:05 » 0:08, we have, approximately, P » = 0:26 » 0:42 which is a realistic figure for a typical particulate magnetic recording media. This suggests that the perturbation calculation is reasonable for the dipole interacting particulate recording media. One

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additional comment is due. In order to carry out MCS simulation for magnetization reversal for a dipole interacting system, it is essential that one has a good sound prior knowledge of the extreme of the entire energy surfaces as to how and through what passages the moment may reverse. In this respect, the present treatment is well guided and supplemented by the perturbation calculation. IV-4. Dipole field contribution To explain why dipole field is responsible for enhancing and depressing the coercivity and energy barrier height of a particulate system one takes note of the following. Dipole interaction ~ ef f = ¡r¡3 (M ~ ¡ 3^ ~ ¢ r^) may be decomposed into two parts: a longitudinal component field H r (M Hk and a transverse one H? . The longitudinal component Hl is portional to » ½(1 ¡ cos2 ¯) whose contribution to coercivity decreases with increasing ¯, turns negative when ¯ · 54:7± . p 2 The effective transverse component H? is proportional to » ¡½ cos ¯ sin ¯ which (is usually neglected in the mean field calculation) has a negative contribution to hc (¯) for all bonding angles, and has a minimum at ¯ = 45± . The superposition of Hl and H? contributions leads to the local maxima at ¯ = 0 and ¼=2 and a global minimum of hc (¯) at ¯ » 60± . The former results in coercivity enhancement when the bonding angles is less than a certain critical bonding angle ¯C ¼ 15± » 30± , and coercivity depression due to the presence a global minimum of hc (¯) at ¯ » 60± , as schematically depicted in Fig. 7. From Fig. 7 we notice that coercivity can be enhanced if ¯ ¯C = 15± » 30± , depressed whenever ¯ > ¯C . Depression is most appreciable when ¯ is in the range of ¼ 50± » 70± . IV-5. Bonding angles versus hysteresis loops As discussed earlier related to Eqs. (4) and (5) the dipole interacting system has three minimum energy states, (1)("; "); (2)("; #) or (#; ") and (3) (#; #) (four if the particles are not identical) and two saddle points corresponding to the transitions (1) ) (2) and (2) ) (3). The corresponding barrier height is p EB =KV » (13) = (1 ¡ h)2 ¨ 2½[(1 ¡ h)(1 ¡ 3 cos2 ¯) § 3 cos ¯ sin ¯ 1 ¡ h2

where the upper [lower] sign is for (1) ) (2) [(2) ) (3)] transition. If the barrier height for (1) ("") ) (2) ("#) transition is higher than the corresponding transition (2) ("#) ) (3) (##), there exists no intermediate state at (2) ("#), instead, it makes a direct jump from (1) ("") in transition to ) (3) (##). On the other hand, if the barrier height for (1) ) (2) transition is lower than (2) ) (3) transition, one intermediate state at (2) ("#) will exist so that magnetic moment may stay for an indefinite period before jumping to (3) (##) upon application of a large enough field to effect magnetic reversal. The existence of whether one-or two-stage transition in the magnetization reversal depends upon relative barrier height between the two transitions (1) ) (2) and (2) ) (3). In fact it is largely dependent upon the sign of the term (1 ¡ 3 cos2 ¯). This term becomes negative if ¯ < 54:7± so that the energy barrier for the transition (1) ) (2) is higher than that for (2) ) (3) transition and there exists no intermediate transition state. The reverse is true if ¯ > 54:7± , the barrier height for (2) ) (3) becomes higher and a two-stage transition results. Physically, this corresponds to the fact that the state (2) ("#) is more stable than either of the state (1) ("") or (3) (##) when the bonding angle ¯ is greater than 54:7± .

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FIG. 7. Relative contribution of the longitudinal and transverse component of the effective dipolar field which contributes to the shift in coercivity variation. Superposition of the two contributions results in the coercivity minimum at ¯ » 60± . A sample average of the coercivity is thus depressed compared to that of the noninteracting system.

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FIG. 8. MCS simulation of the hysteresis behavior for several bonding angles and for a 2D random system. The system undergoes a two-stage magnetization reversal processes whenever ¯ is, or close to, ¼=2.

A summary of the MCS simulation results on hysteretic behavior of a pair of dipole interacting SDPs for various bonding angles and dipole coupling strengths are shown in Figs. 8 and 9. As shown in Fig. 8, the coercivity is larger at small bonding angles than at larger bonding angles. Note that the system may undergo a two-stage magnetization transition processes when accompanied by a non-zero dipole coupling strength whenever the bonding angle is close to > 60± as depicted in Fig. 9. The system ¯ = ¼=2. The system begins to show sign of it when ¯ » may begin to switch at a lower field value, say hc1 , resulting in a net negative magnetization until at h = hc2 < hc1 when the second or the last stage of the magnetization reversal takes place. As a consequence, the hysteretic behavior of this dipole interacting system, when the media is thin enough so that the system is independent of the azimuthal angle of the magnetization, the initiation of the magnetization transition may take place at a smaller field value, and by virtue of the two-step switching, the slope of the hysteresis curve maybe further reduced, resulting in a wider switching field distribution. Thus, jdM=dHj becomes smaller overall, hence lower in coercivity compared to that of the noninteracting case. The dashed line in Fig. 8 depicts the hysteresis behavior for a 2D random system, whilst the solid lines corresponds to that of the noninteracting case. Similar features for a 2D random system are reproduced in Fig. 9 (top) for several ½ values as a parameter. The behavior is similar to that of a dipole interacting thin film

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FIG. 9. MCS simulation results of hysteresis behavior for several ¯’s and for a 2D random system where the loops are obtained by averaging over the ¯¡ values. Slope of the magnetization curves of the latter is appreciably smaller, leading to a wider switching field distribution and lower coercivity.

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FIG. 10. (a) Behavior of the MZF C and MF C magnetization variation of a dipole interacting system with a log normal volume distribution f (V ) with ½ = 0:04, ½V = 04, (b) Derivative of (MZF C MF C )=MS with respect to temperature giving the blocking temperature distribution of the system.

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media. A final remark is in order. The coercivity decreases steadily with increasing coupling strength except when the bonding angle is close to zero when the corresponding energy barrier becomes enhanced. IV-6. Thermal relaxation and blocking temperature distribution The present calculation maybe successfully extended to discuss the zero-field-cooled, MZF C , field-cooled, MF C , and net magnetization variation, (MZF C ¡ MF C ), of the system based on the simple Langevin function and Arhennius relation. Langevin function is defined as L (HV =T ) = [exp(HV MS =kB T ) ¡ exp(¡HV MS =kB T )] = [exp(HV MS =kB T ) + exp(¡HV MS =kB T )] in which the field H is an effective field which is assumed to be small compared to the anisotropy field (say, H = 0:01 HK ). The blocking temperature distribution of the system TB (V ) corresponding to a system with an arbitrary log normal volume distribution f(V ) = exp[¡(ln(V=V0 ))2 =2¾V2 ] where ¾V is the half-width of the distribution can be easily obtained [16]. Figure 10(a) shows the ZFC and FC magnetization relaxation behavior the system with a continuous volume distribution f (V ) under various field values at ½ = 0:04 and ¾V = 0:4, and Fig. 10(b) depicts the derivative of (MZF C - MF C )=MS with respect to temperature which effectively represents the blocking temperature distribution of the system. Note that Fig. 10(a) bears good resemblance to the ZFC and FC magnetization relaxation curves observed [4; 9] and that the absolute magnetization value depends on the level of applied field, half width of the volume distribution ¾V and dipole interaction strength ½. Further, the blocking temperature distribution TB (V ) nearly exactly duplicates the weighted volume distribution of the system V f(V ). V. Summary Coercivity and energy barrier variation of a fine dipole interacting system has been discussed in terms of a fixed step Monte Carlo simulation supplemented by perturbation scheme. Dipole interaction field can be decomposed into a longitudinal and a transverse component. The superposition of the two component field leads to the local maxima at ¯ = 0 and ¼=2 and a global minimum in coercivity at ¯ » 60± . Coercivity of a random dipole interacting system thus becomes lower compared to that of a noninteracting system since the probability of finding the bonding angle of the system smaller than 30± is less than 13%. The asymmetric fanning mode is favored compared to other modes in making ("#) ) (##) transition if ¯ 6= 0, ¼=2. In addition, whenever the bonding angle ¯ ! ¼=2, the system may exhibit a two-step switching behavior causing the system to broaden its switching field distribution. Finally, the field-cooled and zero-field-cooled magnetization relaxation behavior and its blocking temperature distribution can be well accounted for based on the simple Langevin function formulation. Acknowledgment We gratefully acknowledge the financial help through a research grant from NSC of the ROC under: NSC-88-2216- E- 002-021 and -022. References [ 1 ] H. N. Bertram, Theory of magnetic recording, (Cambrige Univ. Cambrige, 1994), Chap. 12. [ 2 ] Pu-Ling Lu and S. H. Charap. IEEE Trans. Magn. 30, 4230, 1994.

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[ 3 ] Borlian Chen and H. L. Huang, J. Magn. Magn. Mats., 171, 218 (1997). [ 4 ] J. L. Dormann, D. Fiorani, E. Tronc, Adv. in Chem. Phys., CXVIII, (pp. 283-493); eds. I. Prigogine and S. A. Rice. (John Wiley, 1997). [ 5 ] W. Chen, S. Zhang and H. N. Bertram, J. Appl. Phys. 71, 5579, (1992). [ 6 ] I. Klik, J. S. Yang and C. R. Chang, J. Appl. Phys. 76, 6493 (1994); see also I. Klik and C. R. Chang, Phys. Rev B, 52, 3540 (1995). [ 7 ] H. L. Huang et al., J. Appl. Phys. 85, 5558 (1999). [ 8 ] L. Dormann, L. Bessaia, and D. Fiorani, J. Phys. C: Solid State Phys. 21, 2015 (1988). [ 9 ] P. Prene´ , E. Tronc, J-P. Jolivert, J. Livage, R. Cherkaoui, M. Nogue´ s, J. L. Dormann, D. Fiorani, IEEE Trans. magn 29, 2658 (1993). [10] S. Mørup and E. Tronc, Phys. Rev. Lett. 72, 3278 (1994). [11] L. Dormann et al., Phys. Rev. B53, 14291 (1996). [12] F. Hansen and S. Mørup, J. Magn. Magn. Mats. 184, 262 (1998). [13] K. Binder and D. W. Heerman, Monte Carlo Simulationin Statistical Physics, (Spriner, Berlin 1980). [14] D. A. Dimitrov and G. M. Wysin, Phys. Rev. B54, 9237 (1996). [15] J. J. Lu and H. L. Huang, J. Appl. Phys. 77, 3323, 1995. [16] J. J. Lu and H. L. Huang, ISAMT’99, paper BD-5 (invited), May, 1999