Superparamagnetism is a form of magnetism which appears in small ferromagnetic or ferrimagnetic nanoparticles. In sufficiently small nanoparticles, magnetization can randomly flip direction under the influence of temperature. The typical time between two flips is called the Néel relaxation time. In the absence of an external magnetic field, when the time used to measure the magnetization of the nanoparticles is much longer than the Néel relaxation time, their magnetization appears to be on average zero; they are said to be in the superparamagnetic state. In this state, an external magnetic field is able to magnetize the nanoparticles, similarly to a paramagnet. However, their magnetic susceptibility is much larger than that of paramagnets.
Néel relaxation in the absence of magnetic field
Normally, any ferromagnetic or ferrimagnetic material undergoes a transition to a paramagnetic state above its Curie temperature. Superparamagnetism is different from this standard transition since it occurs below the Curie temperature of the material. Superparamagnetism occurs in nanoparticles which are single-domain, i.e. composed of a single magnetic domain. This is possible when their diameter is below 3–50 nm, depending on the materials. In this condition, it is considered that the magnetization of the nanoparticles is a single giant magnetic moment, sum of all the individual magnetic moments carried by the atoms of the nanoparticle. Those in the field of superparamagnetism call this "macro-spin approximation". Because of the nanoparticle's magnetic anisotropy, the magnetic moment has usually only two stable orientations antiparallel to each other, separated by an energy barrier. The stable orientations define the nanoparticle's so called "easy axis". At finite temperature, there is a finite probability for the magnetization to flip and reverse its direction. The mean time between two flips is called the Néel relaxation time τ N {\displaystyle \tau _{\text{N}}} and is given by the following Néel–Arrhenius equation:
τ N = τ 0 exp ( K V k B T ) {\displaystyle \tau _{\text{N}}=\tau _{0}\exp \left({\frac {KV}{k_{\text{B}}T}}\right)} , where:
τ N {\displaystyle \tau _{\text{N}}} is thus the average length of time that it takes for the nanoparticle's magnetization to randomly flip as a result of thermal fluctuations;
τ 0 {\displaystyle \tau _{0}} is a length of time, characteristic of the material, called the attempt time or attempt period (its reciprocal is called the attempt frequency); its typical value is between 10−9 and 10−10 second; K is the nanoparticle's magnetic anisotropy energy density and V its volume. KV is therefore the energy barrier associated with the magnetization moving from its initial easy axis direction, through a “hard plane”, to the other easy axis direction; kB is the Boltzmann constant; T is the temperature. This length of time can be anywhere from a few nanoseconds to years or much longer. In particular, it can be seen that the Néel relaxation time is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles.
Blocking temperature Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define τ m {\displaystyle \tau _{\text{m}}} as the measurement time. If τ m ≫ τ N {\displaystyle \tau _{\text{m}}\gg \tau _{\text{N}}} , the nanoparticle magnetization will flip several times during the measurement, then the measured magnetization will average to zero. If τ m ≪ τ N {\displaystyle \tau _{\text{m}}\ll \tau _{\text{N}}} , the magnetization will not flip during the measurement, so the measured magnetization will be what the instantaneous magnetization was at the beginning of the measurement. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will appear to be “blocked” in its initial state. The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time. A transition between superparamagnetism and blocked state occurs when τ m = τ N {\displaystyle \tau _{\text{m}}=\tau _{\text{N}}} . In several experiments, the measurement time is kept constant but the temperature is varied, so the transition between superparamagnetism and blocked state is seen as a function of the temperature. The temperature for which τ m = τ N {\displaystyle \tau _{\text{m}}=\tau _{\text{N}}} is called the blocking temperature:
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