Superparamagnetism - The Néel Relaxation in The Absence of Magnetic Field

The 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 and is given by the following Néel-Arrhenius equation:

,

where:

• is thus the average length of time that it takes for the nanoparticle’s magnetization to randomly flip as a result of thermal fluctuations.
• 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 10−9–10−10 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.
• k_B 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.