Single-Domain Particles and Superparamagnetism
Sufficiently small ferromagnetic particles possess an important property which makes them attractive for many applications: particles below a certain size are always in a so called single-domain state (Kneller, 1966; Landau and Lifshitz, 1975a). This means that the whole particle volume is occupied by a single magnetic domain - that is, all atomic magnetic moments of such a particle are aligned parallel to each other (homogeneous magnetization state).
The main reasons for the transition to a single-domain state for very small particles are: (1) the demagnetizing energy which favors closed (noncollinear) magnetization configuration decreases when the particle size decreases; whereas (2) the exchange energy of a nonhomogeneous magnetization configuration increases when the size of such a configuration (which cannot be larger than a particle size) decreases. This means that below a certain particle size, the homogeneous (single-
domain) particle magnetization state which has a relatively large demagnetizing energy but very low exchange energy is energetically more favorable than some closed (multi-domain) magnetization configuration with low demagnetizing but large exchange energy.
To estimate the critical size below which a particle should be single-domain (Landau and Lifshitz, 1975a), let us study the size dependence of the energy contributions mentioned above. The demagnetizing energy of a single-domain state can be easily estimated using Eq. (1.60). The demagnetizing field Hdem inside a homogeneously magnetized spherical particle with the saturation magnetization M is Hdem = —■4pM/3 (Kittel, 1986) which, according to Eq. (1.60), leads to the demagnetizing energy of the order Edem @ M2 V, where V is the particle volume. The exchange energy density of a nonhomogeneous magnetization configuration inside a particle if large (@ M) magnetization changes occur at the length scale of the particle size a is eexch @ aM2/a2 (see Eq. 1.50), so that the total exchange energy is Eexch @ eexchV @ (aM2/a2)V (and only large magnetization rotations leading to closed magnetization configurations can provide substantial decrease of the demagnetizing energy).
The particle ''prefers'' a single-domain state if the corresponding demagnetizing energy is less than the exchange energy of a closed magnetization state inside a particle: Edem < Eexch, or M2V < (aM2/a2)V. This means that the particle is in a single-domain state if its size is less than acr @ v'a. For materials with a large magnetic anisotropy constant K, one should also take into account the anisotropy energy of a nonhomogeneous magnetization configuration which leads to the estimate acr @ y/aK/M2. The critical sizes for common ferromagnetic materials such as Fe or Ni are @10 ... 100 nm.
The energy calculation for a single-domain particle can be greatly simplified. First, its magnetization configuration can be described by a single vector m of its total dipole magnetic moment the magnitude of which is simply proportional to the particle volume: m = MsV (Ms is the saturation magnetization of the particle material). The exchange energy (see Eq. 1.50) for the homogeneous magnetization configuration is Eexch = 0. Further, for a spherical particle its demagnetizing energy does not depend on the moment orientation and hence can be omitted as any constant in the energy expression. Thus, the particle energy can be written as a sum of its magnetic moment energy in the external field H0 (see Eq. 1.37) and its magnetic anisotropy energy (see, e.g., Eq. 1.53):
Here, we have used the unit vector m along the particle magnetic moment and the unit vector n along the particle anisotropy axis.
The system of uniaxial single-domain particles each of which possesses the energy (Eq. 1.66) is known as a Stoner-Wohlfarth model (Stoner and Wohlfarth, 1948), and is widely used in fine magnetic particle theory due to its (apparently) simple properties. One of its most important features is the existence of a magnetization hysteresis in a collection of such particles (see the corresponding explana tion in Section 126.96.36.199). Due to the simple energy expression (Eq. 1.66) for a single particle, many magnetic characteristics of the noninteracting Stoner-Wohlfarth model such as initial susceptibility, permanent magnetization and coercive force can be computed either analytically or by very transparent numerical calculations (Kneller, 1966).
Another interesting property of fine particle systems is that, above a certain temperature Tsp, such a system behaves like a paramagnetic body, although Tsp is still much lower than the Curie point Tc for the corresponding ferromagnetic material. To understand this behavior (Kneller, 1966), let us consider a system of uniaxial particles with the energies described by Eq. (1.66).
In the absence of an external field the magnetic moment of each particle has two equivalent equilibrium positions (states) along two opposite directions of the aniso-tropy axis. These states are separated by the energy barrier, with height equal to the maximal possible anisotropy energy: E^^ = KV. If the system temperature is sufficiently low such that the thermal activation energy kT is much less than this barrier height, then the magnetic moment of each particle will (almost) forever stay in one of these two states depending on the previous system history (i.e., in which direction a strong external field was applied, say, several years ago). However, for sufficiently high temperatures T > Tsp, thermal transitions between the two equilibrium states may occur on the observable time scale so that after some time each moment can be found with equal probabilities in one of these two states.
For such temperatures the total magnetic moment of a fine particle system in the absence of an external field is zero (as for paramagnetic and diamagnetic substances), because each moment can be oriented with equal probabilities in two opposite directions. When a small external field is applied, then the moment orientation along that direction of the anisotropy axis that has the smallest angle with this field is preferred and the system demonstrates a net average magnetization along the applied field (as usual paramagnets do). However, the magnetic susceptibility (which characterizes the system response to the applied field) for such a system of fine ferromagnetic particles is about 104 ... 106 times larger than for usual paramagnetic materials because the moment of small particles which now play the role of single atoms (molecules) of a paramagnet is much larger than any atomic or molecular magnetic moment. For this reason, the behavior of a fine particle system for T > Tsp is known as superparamagnetism.
To estimate the temperature of this superparamagnetic transition Tsp (which is also known as a blocking temperature Tbl), we are reminded that, according to the Arrhenius law, for a system with the temperature T the average transition time between two states separated by an energy barrier AE is ttr @ t0 exp(AE/kT), k being the Boltzmann constant, AE @ KV (see above). The prefactor t0 should be measured experimentally, and for magnetic phenomena under study is about t0 @ 10~9 s. To observe the superparamagnetic behavior, the observation time tobs should be at least of the same order of magnitude as ttr, which leads to the relationship tobs b t0 exp(AE/kT). Hence, for the given observation time tobs the blocking temperature can be estimated as Tsp @ KV/ln(tobs/t0). The corresponding value, for example for iron particles of size @10 nm and an observation time tobs = 1 is TSp @ 102 K.
In real fine particle systems the transition to a superparamagnetic state with increasing temperature occurs gradually due to the always-present particle size and shape distribution. These distributions lead to a spread of the energy barrier heights, and this results in different transition temperatures for different particles.
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