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 in 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.
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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
where:
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
For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25.
Effect of a magnetic field
When an external magnetic field is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. The magnetization curve of the assembly, i.e. the magnetization as a function of the applied field, is a reversible S-shaped increasing function. This function is quite complicated but for some simple cases:
- If all the particles are identical (same energy barrier and same magnetic moment), their easy axes are all oriented parallel to the applied field and the temperature is low enough (TB < T ≲ KV/(10 kB)), then the magnetization of the assembly is
M ( H ) ≈ n μ tanh ( μ 0 H μ k B T ) . - If all the particles are identical and the temperature is high enough (T ≳ KV/kB), then, irrespective of the orientations of the easy axes:
M ( H ) ≈ n μ L ( μ 0 H μ k B T )
In the above equations:
The initial slope of the
The later susceptibility is also valid for all temperatures
It can be seen from these equations that large nanoparticles have a larger µ and so a larger susceptibility. This explains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment.
Time dependence of the magnetization
There is no time-dependence of the magnetization when the nanoparticles are either completely blocked (
where
From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived:
Measurements
A superparamagnetic system can be measured with AC susceptibility measurements, where an applied magnetic field varies in time, and the magnetic response of the system is measured. A superparamagnetic system will show a characteristic frequency dependence: When the frequency is much higher than 1/τN, there will be a different magnetic response than when the frequency is much lower than 1/τN, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization. The precise dependence can be calculated from the Néel-Arrhenius equation, assuming that the neighboring clusters behave independently of one another (if clusters interact, their behavior becomes more complicated). It is also possible to perform magneto-optical AC susceptibility measurements with magneto-optically active superparamagnetic materials such as iron oxide nanoparticles in the visible wavelength range.
Effect on hard drives
Superparamagnetism sets a limit on the storage density of hard disk drives due to the minimum size of particles that can be used. This limit is known as the superparamagnetic limit.