Soluciones tecnológicas durante el calado

1.5.1 Magnetic hysteresis in a single domain nanoparticle

Figure 1.4 The orientation of magnetization vector Ms and external magnetic field H with respect

to the easy axis. (a) magnetic nanorod (b) magnetic nanosphere.

Consider a single domain magnetic nanoparticle with a uniaxial magnetic anisotropy. It can be a magnetic nanorod with high aspect ratio or a spherical nanoparticle with uniaxial crystalline anisotropy (for example: cobalt). As shown in Figure 1.4, we can define the direction of the magnetization vector Ms and external magnetic field H with respect to the easy axis by introducing angles θ and φ respectively in both cases.

Keeping only the second order term, the energy density can be written as:

2 0 s sin cos( ) E K M H V      (1.14)

where E is the total energy of the particle, V is the volume of the particle, K is the anisotropy coefficient and μ0 is the magnetic permeability of vacuum. First term on the right hand side corresponds to the uniaxial anisotropic energy and shows how strong the spins are bonded to the easy axis. The second term is the magnetostatic energy between the particle and external magnetic field.

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Figure 1.5 (a) The total energy density as a function of orientation of magnetization θ for different external magnetic fields H. (b) Correspondence of the energy minimums to the points on the hysteresis loop.

Figure 1.5 (a) shows an example of the calculated energy as a function of magnetization orientation θ under five different external magnetic fields H=±1.2K/μ0Ms, ±0.6K/μ0Ms, 0. In all cases, the magnetic fields are in the same direction: φ=π/3. The energy minimums correspond to possible equilibrium orientations of the magnetization vector. In the real experiment, the direction of the magnetic field φ is fixed and only the field magnitude H is varied. The instrument can only measure the component of

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magnetization along the field direction which is Mscos(φ-θmin). θmin corresponds to an energy minimum at a particular H. At H=±1.2K/μ0Ms, there is only one energy minimum and magnetization vector will always follow that direction. At H =0, ±0.6K/μ0Ms, there are two energy minimums, equilibrium direction of magnetization vector depends on the history of application of the external magnetic field. If H is decreasing from + (1.20.60-0.6-1.2), θmin varies in the following order: θ1, θ2, θ3, θ4, θ5 (Figure 1.5

(a)). These equilibrium positions sit on the upper branch of the hysteresis loop (Figure 1.5 (b)). If H is increasing from - (-1.2-0.600.61.2), θmin doesn’t follow the same path but varies in a different order: θ5, θ6, θ7, θ8, θ1 (Figure 1.5 (a)). These points sit on

the lower branch of the hysteresis loop(Figure 1.5 (b)). The presence of the multiple equilibrium positions is the origin of hysteresis for a single domain ferromagnetic nanoparticle.

Scanning the magnitude of external magnetic field H from + to - and picking the correct energy minimum for each H, one can construct the hysteresis loop for a particular angle φ. Figure 1.6(a) shows a series of hysteresis loops calculated using the developed algorithm for the external fields with different directions φ. When external magnetic field H is parallel to the easy axis (φ=0), the hysteresis loop takes on a rectangular shape. In the other limit, when H is perpendicular to the easy axis (φ=π/2), no hysteresis can be observed. As a result, the magnetization M(H,φ) is a function of both the magnitude of external magnetic field H and its orientation φ.

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Figure 1.6 (a) Hysteresis loop for a single domain nanoparticle with uniaxial anisotropy. Shape of the hysteresis loop varies as the direction (φ) of external magnetic field H changes. (b) The hysteresis loop for an assembly of single domain nanoparticles with randomly oriented easy axes. (c) The experimental hysteresis loop for a powder of nickel nanorods (see details in chapter III ).

Experimentally, we usually deal with an assembly of magnetic nanoparticles. We consider them as the single domain nanoparticles and assume their easy axes to be randomly oriented. To construct the theoretical hysteresis loop for this case, one should scan φ from 0 to π/2 and obtain a series of hysteresis loops M(H,φ) for different φ (Figure 1.6(a)). Then the average hysteresis loop for the assembly of nanoparticles is interpreted as: / 2 s min 0 / 2 0 cos[ ( ) ]sin ( ) sin M H d M H d          





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The average magnetization is calculated in 3-D space with the orientation distribution function f(φ, φa)=1/4π and the average over azimuth angle φa is not considered due to the uniaxial symmetry. The calculated averaged hysteresis loop is shown in Figure 1.6(b).

There are usually three parameters characterizing the hysteresis loop: Saturation magnetization Ms, Remanence magnetization Mr and Coercive force Hc. These parameters are defined from the hysteresis loop shown in Figure 1.6(b) and (c). Figure 1.6(c) is the experimental hysteresis loop for the powders composed of nickel nanorods.

Saturation magnetization Ms, as defined in section 1.2, is the maximum magnetization achievable by the material. For a single domain nanoparticle, the magnitude of the magnetization vector is always Ms: magnetization vector only rotates under the external magnetic field.

Remanence magnetization Mr is also named as the spontaneous magnetization. It is the magnetization remained in the material when the external magnetic field is removed. It is only proportional to the saturation magnetization Ms. At H=0, the magnetization vector will follow the direction of the easy axis (θmin=0,π). As a result, the remanence magnetization for a particular φ is written as: Mr=Mscosφ. Substituting it into eq.(1.15), the remanence magnetization for an assembly of single domain nanoparticles with randomly oriented easy axes is interpreted as:

/ 2 s 0 s r / 2 0 cos sin 2 sin M d _M M d        





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Thus, for an assembly of randomly orientated single domain nanoparticles, remanence magnetization Mr is always half the saturation magnetization Ms.

Coercive force Hc is the magnetic field needed to demagnetize the ferromagnetic material. The external magnetic field in Figure 1.6 is normalized by the term K/μ0Ms, meaning that the coercive force will be proportional to this term. According to the numerical results, for an assembly of single domain nanoparticles, the coercive force

Hc≈0.96 K/μ0Ms.

The model of an assembly of randomly orientated nanoparticles is very attractive: it provides a simple method for the estimation of uniaxial coefficient K by measuring the saturation magnetization Ms and coercive force Hc experimentally.

It should be noted that the calculation above assumed a coherent rotation of magnetization vector i.e. all the spins rotate in unison. In reality, different modes of magnetization reversal are possible.

Figure 1.7 Different modes of magnetization reversal (a) coherent rotation (b) curling (c) buckling (d) fanning (e) domino effects[8].

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Figure 1.7 is taken from Ref[8] and shows different modes of magnetization reversal. The idea behind these modes is to decrease the magnetostatic energy of the single domain nanoparticles without creating new domains. For all the modes shown in Figure 1.7 (b-e), the calculated coercive force will be lower than that of the coherent rotation case Figure 1.7 (a) [8].

1.5.2 Superparamagnetic nanoparticle

Temperature is always an important factor in magnetism. For any ferromagnetic material, there’s a Curie temperature Tc, above which the material will become paramagnetic. For a single domain nanoparticle, if there are no thermal fluctuations, the spins will be frozen at the easy axis. For a spherical nanoparticle without shape anisotropy, the strength of the spin bonding the easy axis is characterized by magnetocrystalline energy Ea. This energy Ea is proportional to the particle volume V. As the particle size decreases, at some critical size the thermal energy kBT will be able to

overcome the energy barrier Ea~kBT to flip the spins. If the observation time τm is much greater than the characteristic flipping time τ, the observed magnetization will be zero and material behaves as paramagnetic. However, in this case, the magnetic susceptibility (χ=M/H) is very high hence the material is called superparamagnetic. The superparamagnetic nanoparticles do not exhibit any hysteresis i.e. Mr=Hc=0.

To quantify the effects of thermal fluctuations, we consider a particle with a uniaxial symmetry. The anisotropy energy Ea is the first term on the right hand side of eq.(1.14), Ea=KVsin2θ.

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Figure 1.8 The uniaxial anisotropy energy as a function of the spin orientation θ.

Figure 1.8 shows that, KV is the energy barrier for the spins sitting along the easy axis (θ=0) to jump through the hard direction (θ=π/2) to reach the opposite direction (θ=π) of the same easy axis. Following the Neel-Arrhenius equation, we can estimate the Neel relaxation time τ[9]: 0 B exp(KV) k T   (1.17)

τ0 is the characteristic time scale for a single jump over the energy barrier KV. It is a material parameter and has the typical value 10-9 to 10-10 second[10]. The Neel relaxation time τ characterizes the time needed for a successful jump over the energy barrier. If the measurement time τm is much greater than τ, the nanoparticle behaves as a superparamagentic nanoparticle because the spin will flip many times during the measurement and the measured average spontaneous magnetization will be zero. On the other hand, if τm<<τ , the spin wouldn’t flip during the experiment and the material behaves as ferromagnetic. Assuming that τm = τ, one can define the blocking temperature

22 B m B 0 ln( ) KV T k    (1.18)

At the blocking temperature TB, the measurement time τm equals to the characteristic jumping time τ. Eq.(1.18) defines the transition of material’s behavior from superparamgentic to ferromagnetic. Below this temperature TB, τm<τ, the flip of spin is blocked and the material behaves as ferromagnetic. Above TB, τm>τ, the flip of spin is allowed, hence the material behaves as superparamagnetic.

We can also calculate the critical size for a nanoparticle to be superparamagnetic. Assume that the nanoparticle has a spherical shape, V=πD3/6. Substituting this volume in eq. (1.18) and solving for D, one can define the critical size Ds of a nanoparticle, below which the particle is expected to behave as superparamagnetic at temperature T0.

1/3 B 0 m s 0 [6 ln( )k T ] D K     (1.19)

Equations (1.13) and (1.19) show that there are two critical sizes for magnetic nanoparticles. One particle size distinguishes a multi-domain structure from a single domain one. The other critical size sets the boundary between ferromagnetic and superparamagnetic behavior of nanoparticles. Figure 1.9 is taken from Ref[11] showing these critical particle sizes for different materials estimated from this theory.

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Figure 1.9 Superparamagnetic, single domain and multidomain regions for spherical magnetic nanoparticles. The shaded region corresponds to superparamagentism. The black bar represents the ferromagnetic single domain nanoparticle. The multi-domain particles sit to the right of the black bar[11] .