The effect of randomly oriented anisotropy on the zero-field-cooled magnetization of a non-interacting magnetic nanoparticle assembly

ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 4032–4038

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The effect of randomly oriented anisotropy on the zero-field-cooled magnetization of a non-interacting magnetic nanoparticle assembly W.X. Fang a,b, Z.H. He a,, D.H. Chen a, Y.Z. Shao a a b

State Key Laboratory of Optoelectronic Materials and Technologies, Institute of condensed matter physics, Sun Yat-sen University, Guangzhou 510275, PR China Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 23 March 2009 Available online 6 August 2009

A model based on localized partition function and master equation was set up to calculate the zerofield-cooled (ZFC) and field-cooled (FC) curves of a non-interacting magnetic nanoparticle assembly with randomly oriented anisotropy. The peak temperature of the ZFC curve corresponds to the highest energy barrier that acts against the unblocking process, and could be described well by an equation covering the heating rate effect. The predicted H2/3 field dependence of the peak temperature is in good agreement with published results. & 2009 Elsevier B.V. All rights reserved.

PACS: 75.75.+a 75.50.Lk 75.90.+w 76.90.+d Keywords: Randomly oriented anisotropy Zero-field-cooled and field-cooled magnetization Field dependence of the peak temperature

1. Introduction One of the features of magnetic nanoparticle assembly [1] is the peak temperature of the zero-field-cooled (ZFC) magnetization curve, which reflects underlying information of energy barriers blocking the flip-over of the nanoparticle moment, and the relaxation time of the assembly. However, it is still tanglesome to separate experimentally and theoretically the factors that affect the peak temperature of the particle assembly, such as the distributions of particle volume, the randomly oriented anisotropy, and the interactions between particles. Among these factors, the distribution of the particle volume V drew most of the attention. The ZFC magnetizations are considered to include two parts [2–17], i.e. Z Vc Z 1 f ðVÞMsp ðH; TÞdV þ f ðVÞMb ðH; TÞdV; ð1Þ Mzfc ðH; TÞ ¼ 0

Vc

where f(V) is the volume distribution, Msp and Mb are the contributions to the magnetization of superparamagnetic and blocked particles, respectively, H the applied magnetic field, and T the temperature. Vc is the critical volume that discriminates between the superparamagnetic particles with volume VoVc and the blocked particles with volume V4Vc. As the temperature  Corresponding author.

E-mail address: [email protected] (Z.H. He). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.07.069

increases from a low enough temperature, more and more particles starting from a blocked state become superparamagnetic. Since the magnetization of the superparamagnetic particle was suppressed with the increase of temperature, a maximum magnetization could be observed at a temperature defined as peak temperature Tp. On the basis of Eq. (1), one can obtain increasing [13] and decreasing [2,8], as well as nonmonotonic [2–4,14] field dependence of Tp by tuning the volume distribution function. Particle interactions can be also considered in the parameter Vc by modifying the relaxation time [15] of the particles. Another factor is the randomly oriented anisotropy, which can be deduced particularly in the non-interacting particle system with a reduced remanence of 0.5 in the magnetic hysteresis loop [18]. There were efforts [2,3] studying the effect of the randomly oriented anisotropy on the ZFC magnetization. For a uniaxial anisotropy, the magnetization has one or two energy minimums dependent on the external magnetic field. A general analytical consideration of the temperature effect was difficult. One of the simplifications was to assume that the temperature only has an influence on the probabilities of the moment locating at the energy minima, but not on the magnetization values at the energy minima [3]. This leads to a relatively rough estimation on the peak position of the resulting ZFC curve. To know the exact dependence of the peak temperature, a detailed calculation with the temperature effect on the magnetization in the energy bin is necessary.

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Due to the existence of the energy barriers, the magnetization properties depend on the heating rate (sometimes called cooling rate when the magnetization is recorded during the cooling process). The heating rate effect on blocking temperature TB was analytically derived [19] in condition of high-energy-barrier by Chantrell and Wohlfarth. They gave [19] ! T0 1 kT_ ; ð2Þ ¼1 ln TB 25 T0

calculated. In Section 3, we present our results and discussion on the FC and ZFC curves and the field dependence of Tp. At last, we conclude.

2. Model description All the particles have an identical volume V, and uniaxial anisotropy (the surface anisotropy could be phenomenologically considered in the effective anisotropy constant [20]. For simplicity, we neglect the surface spin disorder.). One of the easy axes is set as z axis, and the applied magnetic field H in x–z plane at an angle v with respect to the z axis, so that the energy of a particle can be written as E ¼ KV sin2 y  Ms VHðsin y cos j sin v þ cos y cos vÞ;

2.1. The energy barrier In Eq. (3), for a field below the switching field given by Hk(sin2/3 v +cos2/3 v)3/2, there are two energy minima in the plane of the easy axis and the applied magnetic field (j ¼ 0), while for field above the switching field only one minimum exists and no blocking occurs. In the case of randomly oriented anisotropy with v ranging from 0 to p, the switching fields can vary from 0.5Hk to Hk. When the axis is parallel to the applied magnetic field (v ¼ 0, p), the two minima are at ym1 ¼ 0, ym2 ¼ p with energies 7MsVH, and a maximum at ymax ¼ cos1(HMs/2K) with energy KV[1+ (H/Hk)2]. Thus the energy barrier that prevents the particle moment from flipping out of the local minima of E ¼ HMV, has a magnitude of KV(1H/Hk)2. In other words, the height of the energy barrier shows a (1H/Hk)2 dependence, which is com-

H = 20 Oe

39.0 Eb (θm1) Eb (θm2)

38.7

0.0

0.2

0.4

0.6

0.8

H = 1200Oe 50 40 Eb (θm1)

30

Eb (θm2)

20

1.0

0.0

0.2

0.4

v (π)

Eb [unit: (kBT)-1(T=50K)]

0.6

0.8

1.0

v (π)

100

H = 3400Oe

80 Eb (θm1)

60

Eb (θm2)

40 20 0 0.0

0.2

ð3Þ

where y and j are the polar angles of the moment of the particle.

Eb [unit: (kBT)-1 (T = 50K)]

Eb [unit: (kBT)-1 (T = 50K)]

where T0KV(1h2)/25kB, hH/Hk, Hk2K/Ms (anisotropy field), K being the magnetocrystalline anisotropy constant, kB the Boltzmann constant and Ms the saturation magnetization of the nanoparticle, and k a constant in unit of s1. Eq. (2) meant that faster heating resulted in a higher TB. Based on Eq. (2), the heating rate effect on the field-cooled (FC) magnetization could be also derived. Since a large number of adjustable parameters were involved in the derivation and systematic experiments on the heating rate effect were rare, it is so far not clear about the dependence of the heating rate on the peak temperature of ZFC magnetization curve. In this paper, by introducing a localized partition function and solving a master equation numerically, we can study with an improving accuracy the effect of randomly oriented anisotropy on the ZFC and FC magnetization of a non-interacting magnetic nanoparticle assembly, and the dependence of the peak temperature of the ZFC magnetization on the applied field and the heating rate, taking into account the temperature effect on the magnetization in the energy bin. Although only the distribution of easy axes was considered, we reproduced the ZFC and FC curves. (the further study on the relaxation process and the memory effect will be reported elsewhere). We organize our paper as follow. In Section 2, we describe our model and analyze the energy barrier of a particle, then introduce the localized partition function and master equation based on which the magnetization can be

39.3

4033

0.4

0.6

0.8

1.0

v (π) Fig. 1. The energy barriers Eb plotted as functions of v under applied magnetic fields of H ¼ 20 Oe (a), H ¼ 1200 Oe (b), and H ¼ 3400 Oe (c).

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where b denotes 1/kBT. The mean magnetization along the applied magnetic field is calculated by

monly used. In general cases, however, the energy barrier tends to show a (1H/Hk)3/2 dependence [21]. Here, the energy bins ym1, ym2 in the case of va0, p are limited in ranges (0, ymax) and (ymax, p), respectively. The heights of the energy barriers Eb(ym1) and Eb(ym2) for the energy bins ym1, ym2 are defined as the energy differences between the energy maximum and minima along the plane of the easy axis and the applied magnetic field, i.e. Eb(ymi) ¼ E(ymax, j ¼ 0)E(ymi, j ¼ 0). Along this plane, the moment can realize reversal with least energy. The energy barriers are plotted as functions of v at H ¼ 20, 1200 and 3400 Oe in Fig. 1. In all the cases, the energy barrier Eb(ym1) for the orientation v ¼ 0 [equal to Eb(ym2) for v ¼ p] is highest; while the energy barrier Eb(ym2) is lowest for the orientation v ¼ p/4. Increasing the magnetic field leads to the broadening of the energy barrier distribution. At the magnetic field of 3400 Oe, which is higher than the switching fields for v in the range from 0.1p to 0.4p and that from 0.6p to 0.9p, no blocking is found. The characteristic relaxation time t for the moment to flip out of the energy bin is supposed to follow Arrhenius law like t ¼ t0 exp[Eb(H, v)/kBT], where t0 is the inverse of the attempt frequency f0 set as 1010 s1 in our model (f0 is usually in the range 1010–1013 [22]). If a constant observing time is longer than the characteristic relaxation time of the energy bin, the moment is not constrained by the energy barrier. When both the characteristic relaxation times of the energy bins are shorter than the observing time, the particle shows superparamagnetism (SPM).

MðvÞ ¼

:

ð5Þ

The analytical form of M(v) could be obtained only in approximation [24]. In order to obtain the magnetization in broad temperature and magnetic field ranges, numerical method is more direct since no approximation is needed. For blocked particles, the magnetization is contributed by two localized magnetization at the energy bins. Since the precession of the particle moment is very fast compared with the relaxation process between the energy bins, we suppose the mean localized magnetization can be calculated by a localized partition function. For the particle moment blocked in the energy bin with minimum position ymi and energy E(ymi, j ¼ 0), the localized partition function Zbi can be calculated by ZZ 1 Zbi ¼ expðbEi Þsin y dy dj; ð6Þ 4p DEi oEb ðymi Þ

where DEi ¼ Ei(y, j)E(ymi, j ¼ 0), and the continuous integrated region is restricted by DEioEb(ymi). Then the localized magnetization M1(v), M2(v) for the energy bins 1, 2 is given according to Eq. (5). Subsequently, the magnetization relaxation was described in general terms by the master equation for a two-bin system [25]

2.2. Partition function and the magnetization

p_ 1 ¼ p2 =t2  p1 =t1 ;

In superparamagnetic state or a state without energy barrier, the partition function of a single particle is [23] Z 2p Z p 1 expðbEÞsin y dy dj; ð4Þ Z¼ 4p 0 0

ð7Þ

where p1 and p2 are respectively the occupation probabilities of a particle moment localized in bins 1 and 2, and t1 and t2 are correspondingly the characteristic relaxation times of the two bins. The solution of Eq. (8) leads to time-dependent probabilities

H = 1200Oe

H = 20Oe

0.6

0.06

0.624 M/MsV

M/MsV

1 @ ln Z

b @H

FC

0.03

0.4

0.616 30

ZFC

ZFC

40

FC

0.2

SPM

SPM

0.00 60

80

60

100

120 T (K)

T (K)

M (ν)sinν/MsV

M/MsV

H = 1200 Oe; ZFC

0.6

0.798 H = 3400Oe ZFC

0.792

FC

0.4

T = 45K T = 37K T = 34K T = 32K T = 31K T = 25K

0.2

SPM 0.786

0.0 10

20 T (K)

30

0.0

0.2

0.4

0.6

0.8

1.0

ν (π)

Fig. 2. The ZFC, FC and SPM magnetization for the applied magnetic fields of H ¼ 20 Oe (a), H ¼ 1200 Oe (b), H ¼ 3400 Oe (c), and the M spectra of the ZFC process at H ¼ 1200 Oe for different temperatures (d). The inset of (b) shows the details of the regions marked by arrow. In (d), the curves of T ¼ 50 K and T ¼ 45 K show little difference. The particle radius is 2.5 nm and the unit heating duration is 60 s.

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in the form of

     t1  t1 t t i þ ; 1  exp  pi ðtÞ ¼ pi ð0Þ exp  1

t

t

t

ð8Þ

where t1 ¼ t11+t1 2 . Finally, the magnetization of the blocked particle M(v) can be calculated by MðvÞ ¼ M1 ðvÞp1 ðv; tÞ þ M2 ðvÞp2 ðv; tÞ:

ð9Þ

To obtain the entire magnetization M of the system with randomly oriented anisotropy, one just needs to integrate M(v)sin v (called M spectrum) from 0 to p. That is Z 1 p M¼ MðvÞsin v dv: ð10Þ 2 0 The ZFC and FC processes were approximated by a stepwise heating and cooling in a certain temperature step DT. The heating (cooling) rate was related to a unit heating duration z which is defined as the duration to achieve a temperature increment of 1 K. For such a unit heating duration z the calculation time step is related to the temperature step by Dt ¼ DTz. To simulate the ZFC process, p1(v, t ¼ 0) and p2(v, t ¼ 0) were set as 12 and starting from a sufficiently low temperature, then the magnetization within the temperature increment is recorded as a blocked state during the heating process. When the time step Dt is longer than each characteristic relaxation time of the two energy bins, we use the Eqs. (4) and (5) to calculate the reversible magnetization. To simulate the FC process, the magnetization is recorded during the cooling process (and found to be overlap with the record of the heating process at the same rate as cooling) from a high enough temperature. When Dt is shorter than one of the two characteristic relaxation times, the magnetization is recorded as a blocked state, and the initial values of p1(vi) and p2(vi) for Eq. (9) are approximated by g(ym1, j ¼ 0)/[g(ym1, j ¼ 0)+g(ym2, j ¼ 0)]and g(ym2, j ¼ 0)/[g(ym1, j ¼ 0)+g(ym2, j ¼ 0] respectively, where g(y, j)is the probability density of the superparamagnetic particle with the moment at state (y, j) and defined as exp[bE(y, j)]/Z. In our calculation, we always use the parameters of Co as our references, i.e. Ms ¼ 1.78 T, and K ¼ 4.1 105 J/m3.

3. Results and discussion 3.1. ZFC, FC curves and M spectra The ZFC and FC magnetizations of the studied system with a constant unit heating duration z ¼ 60 s for H ¼ 20, 1200, 3400 Oe are shown in Fig. 2. As the temperature increases from a sufficiently low temperature, the obtained ZFC curve could be divided into three regions, i.e. totally blocked region, unblocking region and superparamagnetic region. In the totally blocked region, the transferring of the occupation probability between the two energy bins was extremely slow, so that the magnetization was nearly isotropic. It meant the M spectrum looked like a sinusoidal function (see T ¼ 25 K in Fig. 2d). In the unblocking region, the occupation probability transferred from a shallow energy bin (with a low energy barrier and negative contribution to the magnetization) to a deep energy bin (with a high-energy-barrier and positive contribution to the magnetization). In this region, the ZFC curve climbed up from an unblocking temperature dominated by the lowest energy barrier (at vm ¼ p/4) of the system, to a peak temperature dominated by the highest energy barriers (at v ¼ 0 and v ¼ p/2).

4035

The occurring of unblocking could be recognized with two peaks near vm and (pvm) on the M spectra for different temperatures (a typical example is shown in Fig. 2d for H ¼ 1200 Oe). At 20 Oe applied magnetic field the unblocking occurs at near 61 K, and the ZFC curve peaks at 71.1 K (Fig. 2a). At 1200 Oe, the unblocking process occurs at near 28 K, and the ZFC curve peaks at 45 K (Fig. 2b). At 3400 Oe, the unblocking process begins at a very low temperature which is not shown in the figure, and the ZFC curve peaks at 12 K (Fig. 2c). In the superparamagnetic region (above Tp), the ZFC overlaps with the corresponding FC curve, and could be considered as of SPM or anisotropic SPM [26]. As seen in Fig. 2, ZFC and FC curves bifurcate at the peak of the ZFC curves, and the ZFC curves (in Fig. 2a and b) show a flat bottom below the unblocking temperature. However, if a volume distribution [1,3,9] was involved, the peak and the bifurcation point were not necessary to coincide. Since the bifurcation shifts to higher temperature for larger particles, it can be inferred that for a non-interacting particles assembly with volume distribution, the bifurcation of ZFC and FC curves could be related to the maximum particle volume. In the same way, the temperature where the ZFC magnetization starts to climb up could be related to the minimum volume with the lowest energy barrier in the assembly. Another indication of Fig. 2 is the FC magnetization could be considered as SPM equilibrium state only in the case of a high sufficiently applied magnetic field. The previous controversy about the FC curve is on the part above the peak temperature of ZFC curve. This part was assumed to be exactly of the equilibrium state at the blocking temperature [3,5,11,16] or be of the equilibrium state at the corresponding temperature [27]. In the frame of our model, as seen in Fig. 2a–c, the FC curve tends to be SPM curve with the increase of the magnetic field, supporting the second assumption at a high sufficiently applied magnetic field. The reason is under a high field the contribution of the deep energy bin to the total magnetization is greatly improved for both the FC magnetization and SPM. 3.2. Dependence of the peak temperature Historically, the peak temperature Tp was considered to be the blocking temperature TB. On the one hand, TB was studied on the basis of Brown’s derivation [28] about the characteristic relaxation time t of a single particle with the easy axis parallel to the applied magnetic field under a high-energy-barrier approximation; the relaxation time t follows n ðt=t0 Þ1 ¼ 12Cð1  h2 Þ ð1 þ hÞexp½ðað1 þ hÞ2 Þ o þð1  hÞexp½að1  h2 Þ ; ð11Þ

where a ¼ KV/kBT, and C is constant. By setting a constant t one can study the relation between the blocking temperature and the magnetic field [29,30]. To deal with a real system, the aligned anisotropy case was extended to randomly oriented anisotropy case. El-Hilo suggested Hk in Eq. (11) be replaced by gHk, where g ¼ 1 for aligned anisotropy and g ¼ 0.48 for the randomly oriented anisotropy [31]. On the other hand, TB was phenomenologically given by a simple analytical expression [8,11,14,16,32,33], Eb ðHÞ ; ð12Þ TB ¼ kB lnðtobs =tÞ where tobs is the observing time, Eb(H) the average energy barrier, and equal to KV when H ¼ 0. As the applied field H lower Eb(H), the blocking temperature decreases with the field accordingly.

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Tp0 (1-H/Hk)2

Tp = 71.5-6.61×10-5H2

Tp0 [1-H/(0.48Hk)]1.5

60

Tp0 (1-H/Hk)1.5

70 aligned Tp (K)

Tp (K)

calculation (random) 40

68 Random

20

R = 2.5nm ζ = 60s

66

0 1000

0

2000 H (Oe)

0

3000

Tp (K)

60

aligned Linear fit 160

Linear fit 0

0

80 1000

120

Fig. 5. The field dependences in H2 scale of the peak temperature Tp (data in Fig. 4). A good linear fitting (thick line) is observed at low field for the case of aligned anisotropy.

Tp ðHÞ ¼ Tp0 ð1  hÞ2 ¼

Random

20

100

40

H2/3 (Oe)2/3

20

80

60

80

40

60

Now, let us analyze the field dependence of the peak temperature Tp extracted directly from the ZFC curves. When the field decreases close to zero, Tp has an intercept Tp0, which can be calculated by KV/kB ln (zf0). In the case of randomly oriented anisotropy, the obtained field dependence can be described very well by the following equation (see Fig. 3)

20 40

40

H2 (Oe)2×103

Fig. 3. The field dependences of the peak temperature Tp of the ZFC magnetization for systems with randomly oriented anisotropy. The calculation result (open symbol) is compared with fittings by modifying the exponential indices as 32 (dash dot dot line) and 2 (solid line), and by replacing Hk with 0.48Hk (dash line). The particle radius is 2.5 nm and the unit heating duration is 60 s.

60

20

160 H2/3(Oe)2/3 2000

3000

H (Oe) Fig. 4. The field dependences of the peak temperature Tp of ZFC magnetization for the cases of randomly oriented anisotropy and aligned anisotropy. The H2/3 linear fittings (line in inset) from 350 to 3000 Oe are TRan ¼ 79.0(70.2)0.3H2/3 and TAli ¼ 82.4(70.3)0.3H2/3, respectively. The particle radius is 2.5 nm and the unit heating duration is 60 s.

Based on Eqs. (11) and (12), the dependence of the blocking temperature on the magnetic field could be discussed. There were two methods to fit the relation between the magnetic field and the blocking temperature so far. The first one was to analyze the blocking temperature in the power of H. On the basis of Eq. (11), Wenger and Mydosh demonstrated [30] TBpHd, where d ¼ 2 in the low-field case, and 23 in the high-field case. The second one was to replace the energy barrier Eb(H) in Eq. (12) by KV(1h)a, where a ¼ 2 when the easy axes are aligned along the direction of the applied magnetic field [11,14,16,33], and a ¼ 32 in general [32]. It means that TB ¼ TB0 ð1  hÞa ; where TB0 is defined as KV/kB ln (tobs/t).

ð13Þ

ð1  hÞ2 KV : kB lnðzf0 Þ

ð14Þ

Although the maximum contribution to the magnetization in the unblocking process is v ¼ p/4 orientation (Fig. 2d), the peak temperature is proportional to (1h)2 (solid line in Fig. 3), instead of (1h)3/2 (dash dot dot line in Fig. 3) following the change of energy barrier of p/4 orientation. El-Hilo’s method of replacing Hk by 0.48Hk results in a relative large derivation (dash line in Fig. 3). As the peak temperature indicates the termination of the unblocking process, Eq. (14) implies that the peak of the ZFC curve could correspond to the highest energy barrier (vm ¼ 0, p) that acts against the unblocking process. It is worth here discussing the commonly reported fitting by the power terms of H2 at low field and H2/3 at high field. We plotted the peak temperature in H2/3 scale for both cases of randomly oriented and aligned anisotropy in Fig. 4, and found that in a wide range of the magnetic field between 350 and 3000 Oe the H2/3 linear fitting is good, while H2 fitting (see Fig. 5) is good only at low field (less than 150 Oe) in the case of aligned anisotropy. Fig. 4 indicates the commonly observed H2/3 dependence can be derived from Eq. (14). Meanwhile, the field dependence of the peak temperature in our model is different from that of the blocking temperature derived from Eq. (11) due to the following two facts. (i) In our model the H2/3 linear fitting functions for the randomly oriented anisotropy and aligned anisotropy, TRan ¼ 79.0(70.2)0.3H2/3 and TAli ¼ 82.4(70.3) 0.3H2/3 (Fig. 4) have the nearly equal slopes, while El-Hilo et al. [31] reported the different slopes in both cases. (ii) The slope about the H2 linear fitting of Tp at low field in case of aligned anisotropy is 6.61 105 (in K/Oe2) in our model (Fig. 5); while according to the equation 1(TB/TB0) ¼ ((MsV)2/2KVkBTB0)H2, given by Eq. (11) [30], the slope is 1.1 104 (in K/Oe2) with TB0 replaced by Tp0. The above analyses show that the peak temperature is not necessarily the blocking temperature derived from Eq. (11) by setting a constant relaxation time.

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R = 2.5 nm

system only from the H2/3 dependence of the peak temperature [34]. Now we discuss the effects of the heating rate and particle volume on the peak temperature. Physically, a low heating rate (corresponding to a long unit heating duration in our model) will provide enough time for the magnetic moment to flip over the energy barriers, then to develop to an equilibrium state. Especially, the system will reach an equilibrium magnetization at an infinitely low heating rate. The unit heating duration effect on a system with particle size of 6 nm and under an applied magnetic field of 200 Oe is shown in Fig. 6 (open symbol). The peak temperature decreases with the unit heating duration z as described in Eq. (14) (lines in Fig. 6). The tendency is in agreement with the MC simulation result [35] and the analytical estimation on the blocking temperature [19] like Eq. (2). As the peak temperature decreases sharply at short unit heating duration and much slowly at long (see Fig. 6), low heating rate is preferred to weaken its effect on the peak temperature. As shown in Fig. 7, larger particles lead to a higher peak temperature. The peak temperature for the particle assembly of R ¼ 2 nm can be respectively fitted to TRan ¼ 40.9–0.16H2/3 in a specific range (inset of Fig. 7) for the case of randomly oriented anisotropy and TAli ¼ 43.2–0.16H2/3 (not shown in the figure) for the case of aligned anisotropy, and for R ¼ 3 nm to TRan ¼ 132.8–0.50H2/3 (inset of Fig. 7) and TAli ¼ 138.9–0.52H2/3. As implicated in Eq. (14), the absolute value of the slope of the H2/3 linear fitting increases with the particle size. Although the H2/3 dependence of Tp is not a necessary criterion to distinguish the spin-glass state from a non-interacting magnetic particle assembly, one of the interesting phenomena is the decreasing slope with the particle size could be observed [36] in a NiO nanoparticle system with spin-glass freezing state on the particle surface due to the decrease of the fraction of disorder spin lying on the surface of the particles. It meant that the slope may help to analyse the magnetic state of the magnetic particles.

R = 2 nm

4. Conclusions

Eq. (14) H = 200Oe

Tp (K)

70

calculation H = 200Oe

60

Eq. (14) H = 200Oe

50

calculation H = 1200Oe

40 0

500

1000

1500

ζ (s) Fig. 6. The heating duration z dependence of the peak temperature Tp at applied magnetic fields of 200 and 1200 Oe, respectively, in a randomly oriented anisotropic system with uniform particle radius of 2.5 nm.

120

100

50 R = 3 nm

Tp (K)

80

0 80 160 H2/3(Oe)2/3

0

40

0 0

4037

1000

2000

3000

H (Oe) Fig. 7. The field dependences of the peak temperature Tp for systems with particle size of R ¼ 3 nm, R ¼ 2.5 nm, and R ¼ 2 nm. The open symbols represent the calculation result in our model, while the lines represent Eq. (14). The inset shows the corresponding field dependences in H2/3 scale with linear fitting. The unit heating duration is 60 s.

Supposing that Eq. (14) follows a1H2/3+a2 relation in a specific range, the slope paffiffiffi1 for the best fitting can be derived and 2=3 expressed as ð9 3 2=8ÞðTp0 ðVÞ=Hk Þ. In the next, we verify this expression by comparing it to the slopes calculated based on the published H2/3 fittings. For a Fe3O4 nanoparticle system with physical particle size of 1070.5 nm [33], our expression gives 1.36 in comparison with 1.23 in Ref. [33]; for a simulated Co nanoparticle system with a size of 7 nm [25], our expression gives 0.075 in comparison with 0.078 in Ref. [25]. On the basis of the above comparison, we confirm that for a non-interacting magnetic nanoparticle assembly with uniform particle size and randomly oriented anisotropy, Eq. (14) can describe the underlying field dependence of the peak temperature of ZFC curve and covers the observed H2/3 dependence. Obviously the H2/3 dependence of the peak temperature here shows no relation to the spin-glass state, which is reported to exhibiting H2/3 dependence either. Therefore, it is not correct to determine a spin-glass state of a magnetic

We studied the ZFC and FC processes of a non-interacting magnetic nanoparticle assembly with randomly oriented anisotropy, and the effects of applied magnetic field, heating rate, and particle volume on the peak temperature of the obtained ZFC curves by using the localized partition function and the master equation. With the increase of the temperature, the unblocking is related to the transferring of the occupation probability from the shallow energy bin to the deep energy bin. We conclude that, the peak temperature, located at the bifurcation of the ZFC and FC curves where the unblocking process terminates, corresponds to the highest energy barrier that acts against the unblocking, and is proportional to (1h)2 [Eq. (14)], while the prefactor covers the heating rate effect. Eq. (14) could explain the commonly observed H2/3 field dependence, and the slope of the H2/3 linear fitting contains the physics of the anisotropic field.

Acknowledgement This work was supported by the National Natural Science Foundation (No. 60471023) of China. References [1] D.J. Sellmyer, Y. Liu, D. Shindo, Handbook of Advanced Magnetic Materials, Volume I, Advanced Magnetic Materials: Nanostructural Effects, Tsinghua University Press, 2005.

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