Superparamagnetism

Superparamagnetism

Superparamagnetism is a form of magnetism, which appear in small ferromagnetic or ferrimagnetic nanoparticles.

In small enough 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 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 the one of paramagnets.

Description of 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.

This is what people working in the field of superparamagnetism call the "macro-spin approximation".

In this state, there is a finite probability for the magnetization (the giant moment) of the nanoparticle 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 following Néel-Arrhenius equation:

,

where:

τ_N is thus the average length of time that it takes for the nanoparticle magnetization to randomly flip as a result of thermal fluctuations.

τ₀ 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 magnetic anisotropy and V its volume. KV can be thought of as the energy barrier associated with the magnetization moving from its initial "easy axis" direction, through a "hard axis", ending at another easy axis.

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 a function of the exponential of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles.

Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us call τ_m the measurement time. If τ_m >> τ_N, the nanoparticle magnetization will flip several times during the measurement so the magnetization measured will be zero. If τ_m << τ_N, its magnetization will not flip during the measurement so the magnetization measured will be the magnetic moment carried by the nanoparticle. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will appear to be ferromagnetic. The state of the nanoparticle (superparamagnetic or ferromagnetic) depends on the measurement time. A transition between superparamagnetism and ferromagnetism 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 ferromagnetism is seen as a function of the temperature. The temperature for which τ_m = τ_N is called the blocking temperature because, below this temperature, the magnetization is seen "blocked" on the time scale of the measurement.

Superparamagnetic nanoparticles in the presence of a magnetic field

When an external magnetic field is applied to superparamagnetic nanoparticles, they tend to align along the magnetic field, leading to a net magnetization. In the case of an assembly of N identical nanoparticles with randomly oriented easy axis, the hysteresis loop is a Langevin function: , with . µ is the magnetic moment carried by one nanoparticle and H is the magnetic field. The initial slope of the M(H) function is the magnetic susceptibility of the nanoparticle χ, so that . It can be seen from this equation that large nanoparticles have a larger µ and so a larger susceptibility. This explain why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment.
When the magnetic field is removed, the clusters will not randomize their direction immediately, but rather it will take some length of time to do so. Larger clusters tend hold their magnetization for much longer.

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/τ_N, there will be a different magnetic response than when the frequency is much lower than 1/τ_N, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization.^[1] 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).

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. Current hard disk technology with longitudinal recording has an estimated limit of 100 to 200 Gbit/in², though this estimate is constantly changing.

One suggested technique to further extend recording densities on hard disks is to use perpendicular recording rather than the conventional longitudinal recording. This changes the geometry of the disk and alters the strength of the superparamagnetic effect.^[3][4] Perpendicular recording is predicted to allow information densities of up to around 1 Tbit/in² (1024 Gbit/in²).

Another technique in development is the use of HAMR drives, which use materials that are stable at much smaller sizes. But, they require heating before the magnetic orientation of a bit can be changed.

Applications of superparamagnetism

General Applications

Ferrofluid: tunable viscosity

Data analysis: superparamagnetic clustering (SPC) and its extension global SPC (gSPC)

Biomedical applications

Imaging: Contrast agents in Magnetic Resonance Imaging (MRI)

Magnetic separation: cell-, DNA-, protein- separation, RNA fishing

Treatments: targeted drug delivery, magnetic hyperthermia, magnetofection