A phenomenological approach to study the effect of uniaxial anisotropy on the magnetization of ferromagnetic nanoparticles

Accepted Manuscript A Phenomenological approach to study the effect of uniaxial anisotropy on the magnetization of ferromagnetic nanoparticles N. Sánchez-Marín, A. Cuchillo, M. Knobel, P. Vargas PII: DOI: Reference:

S0304-8853(17)32010-3 https://doi.org/10.1016/j.jmmm.2017.12.077 MAGMA 63546

To appear in:

Journal of Magnetism and Magnetic Materials

Please cite this article as: N. Sánchez-Marín, A. Cuchillo, M. Knobel, P. Vargas, A Phenomenological approach to study the effect of uniaxial anisotropy on the magnetization of ferromagnetic nanoparticles, Journal of Magnetism and Magnetic Materials (2017), doi: https://doi.org/10.1016/j.jmmm.2017.12.077

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A Phenomenological approach to study the effect of uniaxial anisotropy on the magnetization of ferromagnetic nanoparticles N. S´anchez-Mar´ın1,2 , A. Cuchillo3 , M. Knobel4 and P. Vargas1,2 1

4

Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Av Espa˜na 1680, Valpara´ıso, Chile 2 Centro para el Desarrollo de la Nanociencia y la Nanotecnolog´ıa CEDENNA, Santiago, Chile 3 Departamento de F´ısica, Universidad de Atacama, Copiap´ o, Chile Instituto de F´ısica Gleb Wataghin, Universidad Estadual de Campinas(UNICAMP), Caixa Postal 6165, Campinas 13083-970, SP, Brasil

Abstract We study the effect of the uniaxial anisotropy in a system of ideal, noninteracting ferromagnetic nanoparticles by means of a thermodynamical model. We show that the effect of the anisotropy can be easily assimilated in a temperature shift T a∗ , in analogy to what was proposed by Allia et al. [1] in the case of interacting nanomagnets. The phenomenological anisotropic T a∗ parameter can be negative, indicating an antiferromagnetic-like behavior, or positive, indicating a ferromagnetic-like character as seen in the inverse susceptibility behavior as a function of temperature. The study is done considering an easy axis distribution to take into account the anisotropy axis dispersion in real samples (texture). In the case of a volumetric uniform distribution of anisotropy axes, the net effect makes T a∗ to vanish, and the magnetic susceptibility behaves like a conventional superparamagnetic system, whereas in the others a finite value is obtained for T a∗ . When magnetic moment distribution is considered, the effect is to enhance the T a∗ parameter, when the dispersion of the magnetic moments becomes wider. Keywords: Magnetic Anisotropy, Ferromagnetic Nanoparticles, Magnetic Susceptibility, Langevin function.

1

1. Introduction

29 30

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

The interest for studying magnetic nanoparticles, from the 31 fundamental as well as technological point of view, has been 32 continuously growing over the last years, mainly owing to po- 33 tential applications in diverse areas such as hyperthermia, drug 34 delivery and data storage, among others [2, 3]. 35 Experimental studies as well as theoretical models are re- 36 quired in order to precisely control and determine the macro37 scopic properties of such nanosystems. This allows one to prop38 erly design systems with specific functionalities, as required in 39 each different application [4, 5, 6]. In the process of understanding of granular magnetic systems, there are still many re- 40 quirements to be considered and explained by means of models 41 that reconcile the experimental findings with theory. 42 In the early 2000, Allia et al. [1] proposed a phenomeno- 43 logical model where the effects of dipolar interactions among 44 the nanoparticles were taken into account by means of a fic- 45 titious temperature named “phenomenological temperature” or 46 T ∗ . In their work they modified the classical expression of the 47 Langevin function by means of this phenomenological temper- 48 ature, by adding it to the actual temperature and so describ- 49 ing fairly well the magnetic moment of granular systems of 50 Cu90 Co10 . By this simple way, they overcame some contra- 51 dictions and difficulties found in some experimental data, and 52 not explained up to that time by other models [1]. The applica- 53 tion of the T ∗ model to granular solids represents a straightfor- 54 ward way to extend the classical Langevin model. This happens 55 mainly for the following reasons: a) It allows, in very simple 56 Preprint submitted to Journal Name: JMMM

way, from the mathematical viewpoint, to explain the experimental data for these systems; b) it allows to conciliate the discrepancies or contradictions that appear when magnetic moments are infered from magnetic and structural measurements; and c) it allows to draw a phase diagram regarding the different regimes on nanomagnetic systems (BS , single-particle blocked regime; BC , collective blocked regime; ISP, interacting superparamagnetic regime; SP, superparamagnetic regime). The phenomenological temperature model of Allia has been also used in several other magnetic systems [7, 8, 9, 10] and also has been improved using mean field models [11]. However, although the model is appropriate in certain situations and has been extensively applied, its complete validity has been questioned by some authors by means of Monte Carlo simulations [12]. Indeed, at low temperatures the simple description by means of the T ∗ model is not complete, because in the low temperature region there are effects of spatial ordering and the coexistence of anisotropy and dipole-dipole interaction. In any system with non-neglegible anisotropy it must affect the reorientation freedom of the magnetic moments in a different way that a simple, classical Langevin function can reproduce. This occurs because the Langevin function is independent of positional order and it does not consider any anisotropy. On the other hand, it has been reported that the anisotropy generates a region or magnetic state coined “anisotropic superparamagnetism” (ASP) in a temperature region above the blocking temperature [13], in this region the magnetization curves at a given temperature clearly distinguish from the Langevin curves at the December 21, 2017

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

same temperature. Similarly, in the T ∗ model, in the “Interacting superparamagnetic” (ISP) state, the isothermal magnetization curves are clearly distinguished from the Langevin curves. In both cases the magnetization saturates slower than an equivalent Langevin curve. This is a clear indication that when there is a coexistence of interaction and anisotropy in a system, the application of the T ∗ model has to be carefully interpreted. To represent an interacting system by means of a non interacting model, without a knowledge of T ∗ a priori, can induce to the misinterpretation of a non interacting system, leading to spurious results [1]. This is due to the heuristic character of the model. In spite of all these observations, the simplicity of the T ∗ model deserves to be considered and to establish their scope and limitations in an appropriate manner. The purpose of this work is to elucidate the contribution of solely the anisotropy on the Phenomenological Temperature T ∗ presented in the ISP model [1]. In the ISP model, a value of T ∗ is obtained from the experimental data, so in this measure it is not possible to distinguish and separate the contributions due to the anisotropy of the contribution that is due to the interac-103 tion. The value obtained unquestionably considers the contribu-104 tion of both components because the nanoparticles intrinsically105 possess magnetic anisotropy. Since it is not possible to separate106 them experimentally we believe that one way to obtain this contribution is by means of a theoretical study that considers only anisotropy to obtain a value (T ∗ = T a∗ ) associated only to this component and from this knowledge to associate the missing 107 ∗ part to a T int , value associated with the interaction in the case of 108 a real interacting system. A comparison made between the ISP 109 model and the exact thermodynamic model is performed to see if a T ∗ (exact) value can emerge naturally from this comparison, following the same procedure performed in the cited ISP model. In order to be more precise, in the ISP model a modified Langevin is compared with the experimental data and from it T ∗ is obtained, in this work we compare a modified Langevin 110 with numerical (exact) data and from it we obtain T a∗ . 111 112

93

94 95 96 97 98 99 100 101 102

2. Model To model a non interacting array of ferromagnetic nanoparticles, we model first the response of an individual particle to the external field at a given temperature. As we treat with nanopar-113 ticles we suppose that their size is comparable to their exchange114 length, therefore each ferromagnetic nanoparticle can be treated115 as a single macrospin with a given uniaxial anisotropy charac-116 terized by an anisotropy field Ha and a direction uˆ . The energy117 ~ is given by the following118 ~ in an external field H of a macrospin m expression: 119 mHa E = −mH[sin θ sin α sin φ + cos θ cos α] − cos2 θ 2

Figure 1: Schematic view of the geometry. Without loosing generality, the ~ lays in the YZ plane, characterized by the angle α. magnetic field H

applied field, in thermal equilibrium at temperature T , is given by: Z  m π λ m(α) = e 2I1 (Ω) sin2 θ sin α + I0 (Ω) sin 2θ cos α dθ, Z 0 (2) with λ = m cos θ{2H cos α + Ha cos θ}/2kB T , Ω = mH sin α sin θ/kB T , I1 (x) and I0 (x) are modified Bessel functions of first kind [14], are obtained from integration over azimuthal angle, and Z is the partition function, Z=2

π

Z

eλ I0 (Ω) sin θ dθ.

Moreover, by evaluating the partition function, the thermodynamical average of the energy at temperature T and at zero external field (H=0) , i.e. the anisotropy energy, is given by:

Ea = −

mHa 2

"r

  mHa # exp 2k BT 2kB T kB T , q − πmHa mHa mHa erfi 2k BT

Where it is assumed that the z-axis is the anisotropy axis, α120 ~ = H(0, sin α, cos α),121 is the angle defining the magnetic field H θ and φ define the magnetic moment vector of the macrospin as122 ~ = m(sin θ cos φ, sin θ sin φ, cos θ) as shown in Figure 1. The123 m component of the magnetic moment of the macrospin along the124 2

(4)

where erfi(x) denotes the imaginary error function [14]. ∂m The magnetic susceptibility is obtained from χ = ∂H at H = 0. χ=

 m2  εa (2 − 3 sin2 α) + sin2 α , 2kB T

(5)

with εa = −2Ea /mHa . On the other hand, the Interacting Superparamagnetic Model (ISP) proposes a phenomenological temperature T ∗ correcting the real temperature due of the interaction between the nanoparticles [1], and therefore the experimental anhysteretic curve of the magnetization M measured at temperature T is fitted using a Langevin function L(x), so: M = mNL

(1)

(3)

0

mH kB (T + T ∗ )

! (6)

where N is the number of nanoparticles, and T ∗ is a shift in temperature which best fits this relation at low external fields. In our case we are assuming that all nanoparticles have same magnetic moment m. For small external magnetic fields we can rewrite the Eq.(6) as:

M=

155

2

m NH , 3kB (T + T ∗ )

(7)156 157

and the susceptibility

158 159

m2 N χ= 3kB (T + T ∗ )

(8)

160 161 162

then

m2 N T = − T. 3kB χ ∗

163

T a∗ obtained throughout Eq.(10). Here the values of T a∗ that lead to the best fits have negative values. Figure 3 shows the results of the magnetization at 5K, 30K, 100K and 300K, respectively, but now with the magnetic field applied perpendicular to the anisotropy axis. It is worth noting that in these four new cases the values of T a∗ are positive, contrary to what happens when the field is applied parallel to the anisotropy axis. In this last case the fit of the modified Langevin function at low temperatures (see Figure 3a) is insufficient.

(9)

Finally using the expression for χ given by Eq.(5), T ∗ has the following analytical expression. T a∗ = 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139

2 T − T, 3 [εa (2 − 3 sin2 α) + sin2 α]

(10)

corresponding to the correction caused by the anisotropy of the nanoparticle. In the last equation, we have denominated T ∗ as T a∗ to emphasize that this temperature has its origin from purely anisotropy effects, no interactions among nanoparticles, contrary as the original proposal of Ref. [1]. 2.1. Cluster The expression for T a∗ given in Eq.(10) is obtained for a fixed angle between the magnetic field and the anisotropy axis. Considering a large number of magnetic nanoparticles, it is convenient to fix the external magnetic field along the z-axis and then perform an average over the anisotropy axes directions (θ,φ) in the unit circle for a given probability distribution, f (θ, φ) for θ ∈ [0, π] and φ ∈ [0, 2π]. Therefore the magnetization and susceptibility have the following expressions:

M χcluster

=

Z

m(θ) f (θ, φ) sin θdθdφ , ! Z ∂m f (θ, φ) sin θdθdφ . = ∂H H→0

(11) (12)

142

From the last expression and using Eq.(9) we can calculate the new T a∗ for this system of many nanoparticles with a distribution of anisotropy axes.

143

3. Results and Discussion

140 141

144 145 146 147 148 149 150 151 152 153 154

Figure 2: Magnetization vs external magnetic field at different temperatures, T , when the magnetic field is applied parallel to the anisotropy axis. The symbols (filled black dots) depict the exact results, while the blue line shows the Langevin function at this temperature and the red line is the Langevin function at T 0 = T + T a∗ , for the values indicated inside the graphics.

3.1. Single nanoparticle We studied the magnetization, susceptibility and T a∗ for a system of one macrospin of m = 1000µB , and considering an anisotropy field of Ha = 3kOe. Using these values, the Blocking Temperature, T b ≈ mHa /50kB , is of the order of T b = 4K. Figure 2 shows results of the magnetic moment at four different temperatures, 5K, 30K, 100K and 300K, respectively, for a configuration where the external field is applied parallel to164 the anisotropy axis. The figure also shows the corresponding165 Langevin function at the same temperatures, and the corrected166 Langevin function at a four new temperatures T 0 = T + T a∗ , for167 3

Figure 3: Magnetization vs external magnetic field at different temperatures, T , when the magnetic field is applied perpendicular to the anisotropy axis. The symbols (filled black dots) depicts the exact results, while the blue line shows the Langevin function at this temperature and the red line is the Langevin function at T 0 = T + T a∗ , for the values indicated inside the graphics.

Figure 4 shows the results for T a∗ as a function of the temperature T for different angles between the external field and anisotropy axis. From this result one sees that T a∗ is negative for angles α < 54.7 ◦ , and positive for α > 54.7 ◦ . At α ≈ 54.7 ◦

168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186

, T a∗ = 0 and the effect of the anisotropy vanishes. It is in-187 teresting to interpret the effect of the uniaxial anisotropy in the188 behavior of the inverse susceptibility of this system of non inter-189 acting nanoparticles. Figure 5 shows the inverse susceptibility190 curves as a function of temperature. One clearly can observe191 that when α > 54.7 ◦ , i.e. T a∗ > 0 the curve of 1/χ vs T resem-192 bles the one of an antiferromagnetic system. On the contrary,193 for α < 54.7 ◦ the curve behaves like the one expected for a194 ferromagnetic system. This happens because for angles smaller195 than the critical angle (54.7 ◦ ), the anisotropy helps the exter-196 nal field to magnetize the system, equivalently to what occurs197 when a lower temperature is considered (T a∗ negative). Unlike,198 for angles higher than the critical angle, the anisotropy hinders199 the magnetization, in analogy to how a higher temperature hin-200 ders magnetization (T a∗ positive). Note that the critical angle201 (54.7 ◦ ) results because the condition sin2 α = 2/3 in Eq.(5)202 and Eq.(10) cancels the anisotropy contribution. This behavior203 has already been observed in real samples [15], and predicted204 theoretically [16]. 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221

Generally speaking one can note that exact thermodynamic magnetization results are well described by the modified Langevin function with T a∗ (Eq.(10)), except in the perpendicular case at lowest temperature (T = 5 K). For this case the Langevin function largely deviates from the exact thermodynamic results. These deviations in Figure 3a (and later in Figure 8a) are larger at intermediate fields due to anisotropy. To discuss these results we recall the Stoner-Wohlfarth (SW) model, for the same energy model as given in Eq.(1). As we know the SW model is valid a T = 0 K, and therefore it has intrinsic coercivity. In spite of that we can use their general magnetic behavior to see how far deviates from the exact thermodynamic curves at low temperatures. Figure 6a shows the SW curve for the case of external field parallel to the anisotropy axis (green curve), together with the exact thermodynamic results (Eq.(2)) at 10 K (black dots) and also the Langevin function (blue curve) at the same temperature, 10K. Figures 6b and 6c show the same results as before but when the field is applied at 30 and 60 degrees with respect to the anisotropy axis respectively, and Figure 6d shows the result for an applied field perpendicular to the anisotropy axis, here the SW model shows no coercivity. In general we see a very good agreement between the M vs H curves of the SW model and the exact thermodynamic result where they have to coincide. The thermodynamic curve fits very well to the SW model at high values of H and when the field decreases separates from it. The same behaviour is shown respect to the Langevin function. Moreover, at this low temperature the perpendicular exact curve deviates greater from the Langevin than a parallel. As noticed above the anisotropic effects are most noticeable when the applied field is perpendicular to the anisotropy axis. It can be explained by the fact that in the perpendicular case, exact curve is most similar to SW (anisotropic) curve than Langevin (non anisotropic) curve as seen in Figure 6d and hence not fitted by a modified Langevin model.

Figure 4: T a∗ vs Temperature for different angles between external magnetic field and anisotropy axis.

Figure 6: Magnetization curves at 10 K for different models: Exact thermodynamic (black dots), Langevin (blue line) and SW (green line). a) when the field is applied parallel to anisotropy axis, b), c) and d) when the field is applied at 30, 60 and 90 degrees with respect to the anisotropy axis respectively.

Figure 5: Inverse susceptibility vs Temperature for different angles between external magnetic field and anisotropy axis.

4

222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241

Figure 7 shows M vs H curves at different temperatures when the external field is applied perpendicular with respect to the anisotropy axis. The SW model (green line) fits the exact thermodynamic result (black dots), at very low temperatures (T = 0.01K), as shown in Figure 7a. At high temperature, T = 300K (Figure 7d), the Langevin model fits the exact thermodynamic model. In Figure 7c, at intermediate temperature (T = 30K), the exact thermodynamic model does not fit the SW model nor the Langevin model. The separation of the exact thermodynamical model from the SW regime begins, in this case, at the field where H = Ha , see Figure 7b. From these results one can observe that the exact thermodynamic model approaches the SW model at low temperatures, and as the temperature raises, the exact result moves towards the Langevin model. On the other hand, we can also say from the previous results, that the differences of the Langevin curve and249 the exact thermodynamic curve at intermediate fields is solely a250 consequence of the anisotropy. All these features are signatures251 of the anisotropy even for certain cases of random anisotropy252 axes, as will be shown below. 253

a sphere (volumetric case), and assuming a uniform distribution of anisotropy axes, one finds, f (θ, φ) = 1 Consequently the average magnetization is given by: Rπ m(θ) sin(θ)dθ M= 0Rπ sin(θ)dθ 0

(13)

(14)

where m(θ) is given by Eq.(2). According to Eq.(12), by deriving the Eq.(14) with respect to the external field, we obtain the average susceptibility, χvol =

m2 , 3kB T

(15)

i.e. one recovers the Langevin susceptibility of a paramagnetic system at the real temperature T , therefore in this case T a∗ = 0 at all temperatures. We note that this case is similar to the case of a single nanoparticle with anisotropy axis at 54.7 ◦ with respect to the external field, since T a∗ = 0 as shown in Figure 4.

SW model Magnetization at T Langevin at T -3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

1

0

M/Ms

1

T=0.01 K

T=5 K

a)

b)

0

-1 1

0

M/Ms

-1 1

T=30 K

T=300 K

c)

d)

0

-1

-1 -3

-2

-1

0

H/Ha

1

2

3

-3

-2

-1

0

1

2

3

H/Ha

Figure 8: Magnetization vs external magnetic field at T = 5K, 30K, 100K and 300K, respectively, for the case of a sample with an volumetric uniform distribution of the anisotropy axes. The symbols depicts the exact results, the continuous blue line depicts the Langevin function at the same temperature. In all these cases, T a∗ = 0 by Eq.(15), indicating that the anisotropy effect averages out.

Figure 7: Magnetization vs external magnetic field when the magnetic field is applied perpendicular to the anisotropy axis at different temperatures. The black circles shows exact results, the blue line shows the Langevin function, and the green line depict the SW model. a) at T = 0.01K, b) at T = 5K, c) at T = 30K, and d) at T = 300K, here Langevin and the exact results coincide at the figure resolution. 254

242 243 244 245 246 247

248

3.2. Effects of distribution of anisotropy axes: volumetric and planar cases Now, we show results of considering the anisotropy axes distributed in any spatial direction (θ, φ) as discussed in section 2.1. Also we discuss results for a distribution of anisotropy axes in a plane.

3.2.2. Kent distribution for the anisotropy axes. Now, we consider the Kent distribution [17] for the anisotropy axes. This distribution is the analog to a normal distribution on a sphere. It is used to simulate a system that has privileged directions (i.e. texture). The expression for the probability density on the unitary sphere is given by: fk (θ, φ) =

3.2.1. Volumetric uniform distribution of anisotropy axes. We consider first the case of an ensemble of non-interacting255 nanoparticles with random anisotropy axes with uniform distri-256 bution in any direction. By averaging the anisotropy axes over257 5

exp[kγˆ1 · rˆ] exp[β(γˆ2 · rˆ)2 − (γˆ3 · rˆ)2 ] , P 2 j 2 2 j+1/2 I 2π ∞j=0 Γ(Γ(j+1/2) 2 j+1/2 (k) j+1) β ( k )

(16)

where rˆ = (sin θ cos φ, sin θ sin φ, cos θ), γˆ1 is the center of the distribution, γˆ2 and γˆ3 are the major and minor semi-axes respectively (γˆ1 ,γˆ3 ,γˆ3 form an orthonormal set of vectors), the

258 259

parameter k determines the concentration and β determines the271 ellipticity (0 ≤ 2β < k). 272 By using Eq.(12) the susceptibility has the following general273 form, 274  m2  275 χkent = 2εa + (1 − 3εa )h , (17) 276 2kB T with Z Z 2π π

h=

0 260 261 262 263 264 265 266 267

fk (θ, φ) sin3 (θ)dθdφ.

When k tends to infinity, we clearly see that we recover the susceptibility for the cases of a single particle whose anisotropy axis is parallel or perpendicular to the external field respectively. The Figure 10 shows the inverse susceptibility for both cases, but where k is arbitrary i.e. the inverse of the expressions given in Eq.(21) and Eq.(22).

(18)

0

In the limit k → 0 the h function is 2/3. Therefore the Eq.(17), becomes the result of the uniform distribution case already shown (Eq.(15)). In the other limiting case, k → ∞, the expression given in Eq.(17) tends to the susceptibility of a single particle with arbitrary angle between the field and the anisotropy axis, as given in Eq.(5). In the particular cases of circular and centered distributions parallel and perpendicular to the field, shown in Figure 9,

Figure 9: Example of the Kent distribution, for the cases β = 0 (circular) and a) External field parallel to γˆ1 and b) External field perpendicular to γˆ1 .

268 269

the expression for h given in Eq.(18) transforms to: Z π k 2 hk = exp(k cos θ) sin3 θdθ = 2 (k coth(k) − 1). 2 sinh(k) 0 k (19) for the parallel case (Figure 9a), and Z π k 1 + k2 − k coth(k) h⊥ = I0 (k sin θ) sin3 θdθ = . 2 sinh(k) 0 k2 (20) for the perpendicular case (Figure 9b). Therefore the associated susceptibilities are: χk χ⊥

270

= =

m2 [k2 εa + (1 − 3εa )(k coth(k) − 1)] k2 k B T 2 2 m [2k εa + (1 − 3εa )(1 + k2 − k coth(k))] 2k2 kB T

and using Eq.(9) we get:

Figure 10: Inv. susc. as a function of temperature for a circular distribution (β = 0) and different values of k: a) External field parallel to γˆ1 , b) External field perpendicular to γˆ1 . The curves for a single particle are for reference.

(21) 277 278

(22)279 280 281 282

∗ T ak

=

k2 T − T 3[k2 εa + (1 − 3εa )(k coth(k) − 1)]

283

(23)284 285

∗ T a⊥ =

2k2 T − T. 3[2k2 εa + (1 − 3εa )(1 + k2 − k coth(k))]

286

(24)287 288

6

Depending on the concentration, k, the inverse susceptibility curve as a function of temperature transits between the case of uniform distribution of anisotropy axes (k = 0) to the case of a single particle (k → ∞) where the anisotropy axis and the field makes an arbitrary angle, determined by γˆ1 . Now we study three particular cases of elliptic distributions (i.e. textures) that are shown in Figure 11, namely, a) centered ~ with the major semiin direction parallel to the field (ˆγ1 k H) ~ axis in the y-direction (ˆγ2 ⊥ H), b) centered perpendicular to ~ with the major semi-axis in the z-direction the field (ˆγ1 ⊥ H) ~ ~ (ˆγ2 k H), and c) centered perpendicular to the field (ˆγ1 ⊥ H) ~ with the major semi-axis in the x-direction (ˆγ2 ⊥ H).

291 292 293 294 295 296 297 298

Figure 11: Examples of elliptic Kent distribution , a) Parallel case, b) Perpen-299 dicular case where the major semi-axis (γˆ2 ) is in the z-direction and c) perpen-300 dicular case where the major semi-axis (γˆ2 ) is in the x-direction. 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319

320 321 322 323 324 325

whose γˆ 1 is parallel with the external field (shown on Figures 9a and 11a). The inverse susceptibility for elliptical distribution is larger than that of circular distribution. This is because an elliptical distribution has a greater contribution to 1/χ from the anisotropy axes with greater angles with respect to the external field direction. The Figure 12b compares a circular distribution with an elliptical distribution whose γˆ 1 is perpendicular to the external field (shown on Figures 9b, 11b and 11c). Here, we see that the ~ inverse susceptibility of an elliptical distribution where γˆ 2 k H (Fig. 11b) is smaller than that of a circular distribution. This is because the elliptical distribution has a smaller contribution to 1/χ from the anisotropy axes with smaller angles with respect to the external field direction. On other hand, the inverse ~ (Fig. 11c) is greater that of a circususceptibility where γˆ 2 ⊥ H lar distribution. This is because the elliptical distribution has a greater contribution to 1/χ from the anisotropy axes with larger angles with respect to the external field direction. As summary of this section we have shown that the phenomenological temperature T a∗ vanishes when the distribution of anisotropy axes is uniform over the unitary sphere. However any deviation from uniformity will produce a deviation of the magnetic susceptibility from the Curie form, like a single particle with anisotropy axis parallel or perpendicular to the external field. Therefore any system, with a non uniform distribution of anisotropy axes, will show a positive temperature T a∗ when the angle between the majority of the anisotropy axes and the external field is greater than 54.7 ◦ , in the opposite case, T a∗ will be negative. 3.2.3. anisotropy axes with a planar distribution Now, for the planar case, we assume that the anisotropy axes are distributed in the XY-plane, and that the applied field is along the xˆ axis in the plane. It is possible to consider again the Kent distribution, Eq.(16), and adjust it to the planar case and we obtain: exp[kγˆ1 · rˆ] exp[k(γ1x cos φ + γ1y sin φ] = , 2πI0 (k) 2πI0 (k)

fk (φ) = 326 327 328 329 330

Figure 12: Inverse susceptibility vs temperature for the case of: a) Field parallel to γˆ1 , and γ2 is in the y-direction (Fig.11a), where dashed line represents the circular case β = 0 with k = 7, green triangles depict the case of β = 3.49 and k = 7 and continous line the single particle behaviour with its anisotropy axis parallel to the field. b) Field perpendicular to γˆ1 , where dashed line represents the circular case, dark yellow squares show the case of β = 3.49, k = 7 and ~ (Fig.11b), red circles depict the case of β = 3.49, k = 7 and γˆ2 ⊥ H ~ γˆ2 k H (Fig.11c), and continuous line shows the single particle behaviour with easy axis perpendicular to the field. 331 332 289 290

The Figure 12a compares 1/χ vs T for circular distribution333 (β = 0) and elliptical distribution (β , 0) of the anisotropy axes,334 7

(25)

where we did β = 0, and γˆ1 = (γ1x , γ1y ) is the direction in the plane where the distribution is centered. fk (φ) is normalized in [0, 2π] With this planar distribution que can write the susceptibility (Eq.(17)) by evaluating the h function (Eq.(18)) as:

h plan =

Z 0

2π

exp[k(γ1x cos φ + γ1y sin φ] 2 sin φdφ 2πI0 (k) I1 (k) cos(2ψ) = + sin2 ψ I0 (k) k

(26)

where γˆ 1 = (cos ψ, sin ψ). The angle ψ (measured from +x) represents the direction in the XY-plane where the distribution is centered. Therefore we can evaluate analytically the susceptibility for any configuration:

355

χ plan =

" !# 356 m2 I1 (k) cos(2ψ) 2εa + (1 − 3εa ) + sin2 ψ . (27)357 2kB T I0 (k) k 358

335

By using Eq.(9), we get the following expression for T a∗ .

359 360 361

∗ T a,plan

336 337 338

2 = " 3

T 2εa + (1 − 3εa )

362

 I (k) cos(2ψ) 1

I0 (k)

k

+ sin2 ψ

− T. (28)363 # 364

On the same way as in the volumetric distribution, in the limit k → 0, we get an uniform distribution, where the susceptibility is: 365 366

χ plan = 339 340 341 342 343

 m2  εa + 1 , 4kB T

(29)

which is the same as a single particle with anisotropy axis at 45 ◦ with the external field. On the other hand, in the limit k → ∞ (i.e a very narrow distribution in the ψ direction) the susceptibility tends to the case of a single particle with its anisotropy axis at an angle ψ with respect to the external field.

3.3. Distribution of magnetic moments Until now we have considered that all nanoparticles have the same magnetic moment m. However, in order to give a further step in the analysis, it is interesting to discuss the effect of a distribution of magnetic moments in the system. Real samples usually display a lognormal distribution of grain sizes, leading to an equivalent distribution of magnetic moments. Thus, without loosing generality we can assume that the m values have a lognormal distribution [18] characterized with an location parameter m0 , and its scale parameter σ. " # −{ln(m/m0 )}2 1 F(m; m0 , σ) = √ exp . (30) 2σ2 2πmσ This distribution is normalized in the [0, ∞] interval, therefore, Z ∞ M = m(α)F(m; m0 , σ)dm (31) 0 Z ∞ χdist = χF(m; m0 , σ)dm. (32) 0

On the other hand, in the ISP model [1] one has, χdist =

hm2 i 3kB (T + T ∗ )

(33)

we recall that, hm2 i =

Z 0

∞

m2 F(m; m0 , σ)dm = m20 exp(2σ2 )

(34)

which corresponds to the arithmetic average of m2 . Using the equations (32) and (33) one can write ∗ T a,dist =

where ga = 367 368 369

Figure 13: Inverse susceptibility vs temperature for the planar case. Lines are results for single particle behaviour when the angle between field and easy axis370 is: 90 ◦ (black), 45 ◦ (blue in the middle) and 0 ◦ (red). Symbols are results for371 a Kent distribution with values given inside the figure. 372 373 344 345 346 347 348 349 350 351 352 353 354

As summary of this section, and in contrast with the previous374 section, we have found that the phenomenological temperature375 T a∗ does not vanishes when the distribution of anisotropy axes376 is uniform over the unitary circle. In this case T a∗ is negative377 (with a value equal to that obtained for a single particle whose378 easy axis forms 45 ◦ with the external field, see Figures 4 and379 5), whereas in the volumetric case it is zero such as can be seen380 from Eq. (15). However, a non uniform distribution will always381 produce a deviation of the susceptibility from the Curie form382 resulting in a positive or negative value of T a∗ depending on k383 and ψ such as can be seen in Figure 13. 384 8

2 T − T, 3 [ga (2 − 3 sin2 α) + sin2 α] 1 hm2 i

Z

(35)

∞

m2 εa F(m; m0 , σ)dm.

(36)

0

We consider four sets of parameters in the lognormal distribution whose average is always hmi = 1000µB in order to analyze and compare results with the case of a single macrospin. The parameters used are given in the Figure 14. We studied the effect of the magnetic moment distribution in two situations. First, when the applied magnetic field is parallel to the anisotropy axes, and second, when the field is perpendicular to the anisotropy axes of all nanoparticles. As one can see in the Figure 14, as the σ parameter increases, the dispersion becomes larger. On the other hand, Figure 15 shows that the wider the spread σ, the smaller is the slope of 1/χ vs T for both situations (perpendicular and parallel). This is because the larger σ the more nanoparticles exists with a magnetic moment above the average value of 1000µB . Figure 16 shows that the susceptibility growths with the magnetic moment. Therefore, the average of all susceptibilities of the sysP tem χdist = N −1 χi , is larger as long more nanoparticles with larger magnetic moments exists. Consequently 1/χdist will be

385 386 387 388 389 390 391 392 393

smaller than the monodispersion case, where all magnetic moments are equal to m = 1000µB . ∗ We also note that T a∗ (given by Eq.(10)) and T a,dist (given by Eq.(35)) have mathematically the same expression, excepts for ga and εa . In our case ga > εa for the parameters used. Consequently, the larger the value of σ, the larger is the absolute ∗ value of T a,dist , as shown in Figure 17. This result also implies that the larger the value of the magnetic moment, the larger the absolute value of T a∗

∗ Figure 17: T a,dist vs Temperature for different distributions of magnetic moments. We show the cases parallel and perpendicular between external magnetic field and anisotropy axis.

394

4. Conclusions

420

By mean of a thermodynamical model we have investigated the role of the uniaxial magnetic anisotropy on the magnetization and initial susceptibility of a system of non interacting nanoparticles. The effect of the anisotropy can be easily assimilated in a temperature shift T a∗ in analogy as proposed by Allia et al. [1] in the case of interacting nanomagnets. When we consider a distribution of easy axes we found that T a∗ vanishes only for a uniformly weighted random distribution in case of volumetric distribution. For all other cases, T a∗ has a finite value and the anisotropy produces a ferromagnetic-like or antiferromagnetic-like behavior when analyzing its response in the magnetic susceptibility measurements even when there are not interactions between nanoparticles. Explicit relationship between T a∗ and the anisotropy energy is given for the case of a single magnetic moment and for a system with axis distribution centered in some particular directions. The value of T a∗ obtained from this relationship can be added to the actual temperature in the Langevin function in a similar way done by Allia et al. From this study we can assure that a T a∗ value emerges from purely anisotropy effects as expected, that should be considered when purely anisotropic nanoparticle systems are described by mean of Langevin function. In a real system in which anisotropy and dipole-dipole interaction are present, this work can help to disentangle both contributions on T ∗ from the magnetic susceptibility measurements, if the anisotropy axes distribution of the sample is previously known.

421

Acknowlegments

395 396 397

Figure 14: Lognormal distributions corresponding to the four studied cases. In398 399 all cases the average value of the magnetic moment is hmi = 1000µB . 400 401 402 403 404 405 406 407 408 409 410 411 412 413

Figure 15: Inverse susceptibility vs Temperature for different distributions of414 magnetic moments. We show the cases parallel and perpendicular between415 external magnetic field and anisotropy axis. 416 417 418 419

422 423 424 425 426 427

Figure 16: Susceptibilities vs magnetic moments at 150 K for cases where the428 magnetic field is aplied parallel and perpendicular to the anisotropy axes. 429

9

This research was supported by Fondecyt Grants No 1140552 and 1130950, and USM DGIIP grant PI-M-17-3. The program “Proyecto Basal” CEDENNA FB0807. AC acknowledge research funds from DIUDA-22302, MK acknowledges support from FAPESP and CNPq, brazilian agencies. Authors acknowledge to anonymous referees for their valuable discussion, suggestions and comments that have enormously enriched the present work.

430

431 432 433

434 435

Appendix

In this section we summarize the behavior of the phenomenological temperature T a∗ as a function of temperature T , for the different cases studied previously. First we note that T a∗ tends to a constant value at high enough temperature. The limit of Eq.(10) for T → ∞ is

lim

T →∞

436 437 438 439

T a∗

" # mHa 4 2 = 2 sin α − 10kB 3

therefore the ratio T a∗ /T tends to zero at high temperatures. This is in agreement with Ref[1]. The figure 18 show the behavior of T a∗ /T for a single particle with different orientations, α, between the uniaxial anisotropy axis and the magnetic field. Figure 19: Absolute value of T a∗ /T vs. T , for the circular Kent distribution, i.e. β = 0. a) Depict the curves when γˆ1 (the center of the distribution) is parallel to the external field. b) Depict the curves for γˆ1 perpendicular to the external field.

Figure 18: Absolute value of T a∗ /T versus T in logarithmic scale. a) Depict the curves for α < 54.7 ◦ where the anisotropic particle has a ferromagnetic like behavior. b) Depict the curves for α > 54.7 ◦ where the particle shows an antiferromagnetic like behavior. According with Ref[1] T a∗ /T = 1 corresponds to the transition between the SP and ISP regimes, T a∗ /T = 25 indicates the transition between the ISP and collective blocking (Bc ) regimes .

440 441 442

Figure 20: Absolute value of T a∗ /T vs. T , for the elliptical Kent distribution, with k = 7. a) γˆ1 (the center of the distribution) is parallel to the external field and γˆ2 (major semi-axes) perpendicular to the external field. Green curve is the result with β=3.49, blue line is the circular case (β = 0) for comparison. b) γˆ1 perpendicular to the external field. Red curve shows the results with β = 3.49 for γˆ2 parallel to the external field, dark yellow line depicts the results with β = 3.49 for γˆ2 perpendicular to external field and blue line is the circular case (β = 0) for comparison.

Figures 19 to 21 show results for the anisotropic temperature, T a∗ , for a system with anisotropy axes distributed with Kent probability, for different cases. 10

445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472

T a∗ /T

473

Figure 21: Absolute value of versus T , for the planar Kent distribution. 474 a) The curves show the results for three different values of k when the center of ◦ the distribution is parallel to the external field (ψ = 0 ). b) Same as a) but the center of the distribution is perpendicular to the external field (ψ = 90 ◦ ). 443 444

Finally the Figure 22 shows the results for a system with a Lognormal distribution of magnetic moments.

Figure 22: Absolute value of T a∗ /T versus T , for a system with a Lognormal distribution (m0 , σ) of magnetic moments . a) Curves depict results for different (m0 , σ) values. The anisotropy axes are parallel to the field. b) Curves show results for different (m0 , σ) values, with the anisotropy axes perpendicular to the field.

11

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