Néel relaxation theory

Superparamagnetism occurs in ferromagnetic and ferrimagnetic 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. This is what people are working on in the subfield of superparamagnetism call “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 magnetic easy axis of the nanoparticle. 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 and is given by the Néel-Arrhenius equation:[1]

${\displaystyle \tau _{\text{N}}=\tau _{0}\exp \left({\frac {KV}{k_{\text{B}}T}}\right)}$,

where KV is the height of the energy barrier, a product of the magnetic anisotropy energy density K and volume V; k_B is the Boltzmann constant, T the temperature and their product the thermal energy; and τ₀ is a length of time, characteristic of the material, called the attempt time or attempt period (its reciprocal is called the attempt frequency). Typical values for τ₀ are between 10⁻⁹ and 10⁻¹⁰ seconds.

The Néel relaxation time can be anywhere from a few nanoseconds to years or much longer. In particular, it is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles.

Suppose that the magnetization of a single superparamagnetic nanoparticle is measured over a time τ_m. If this time is much greater than the relaxation time τ_N, the nanoparticle magnetization will flip several times during the measurement. In zero field, the measured magnetization will average to zero. If τ_m ≪ τ_N, the magnetization will not flip during the measurement, so the measured magnetization will be equal to the initial magnetization. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will be blocked in its initial state. The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time. A transition between superparamagnetism and the blocked state occurs when τ_m = τ_N. In several experiments, the measurement time is kept constant but the temperature is varied, so the transition between superparamagnetism and blocked state is a function of the temperature. The temperature for which τ_m = τ_N is called the blocking temperature:

${\displaystyle T_{\text{B}}={\frac {KV}{k_{\text{B}}\ln \left(\tau _{\text{m}}/\tau _{0}\right)}}}$

For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25.