Superparamagnetism

Effect of a magnetic field

When an external magnetic field H 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:

1. 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 (T_B < T ≲ KV/(10 k_B)), then the magnetization of the assembly is

   ${\displaystyle M(H)\approx n\mu \tanh \left(\frac{{\mu }_{0}H\mu }{k_{\text{B}}T}\right)}$.

2. If all the particles are identical and the temperature is high enough (T ≳ KV/k_B), then, irrespective of the orientations of the easy axes:

   ${\displaystyle M(H)\approx n\mu L\left(\frac{{\mu }_{0}H\mu }{k_{\text{B}}T}\right)}$

In the above equations:

• n is the density of nanoparticles in the sample
• ${\mu }_0$ is the magnetic permeability of vacuum
• $\mu$ is the magnetic moment of a nanoparticle
• $L(x)=\frac{1}{\tanh (x)}-\frac{1}{x}$ is the Langevin function

The initial slope of the ${\displaystyle M(H)}$ function is the magnetic susceptibility of the sample ${\displaystyle \chi }$:

${\displaystyle \chi =\{\begin{array}{ll}{\displaystyle \frac{n{\mu }_0{\mu }^2}{k_{\text{B}}T}}& \text{for the 1st case}\\ {\displaystyle \frac{n{\mu }_0{\mu }^2}{3k_{\text{B}}T}}& \text{for the 2nd case}\end{array}}$

The latter susceptibility is also valid for all temperatures ${\displaystyle T>T_{\text{B}}}$ if the easy axes of the nanoparticles are randomly oriented.

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 (${\displaystyle T\ll T_{\text{B}}}$) or completely superparamagnetic (${\displaystyle T\gg T_{\text{B}}}$). There is, however, a narrow window around ${\displaystyle T_{\text{B}}}$ where the measurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of the susceptibility can be observed. For a randomly oriented sample, the complex susceptibility is:

${\displaystyle \chi (\omega )=\frac{{\chi }_{\text{sp}}+i\omega \tau {\chi }_{\text{b}}}{1+i\omega \tau }}$

where

• $\frac{\omega }{2\pi }$ is the frequency of the applied field
• ${\chi }_{\text{sp}}=\frac{n{\mu }_0{\mu }^2}{3k_{\text{B}}T}$ is the susceptibility in the superparamagnetic state
• ${\chi }_{\text{b}}=\frac{n{\mu }_0{\mu }^2}{3KV}$ is the susceptibility in the blocked state
• $\tau =\frac{{\tau }_{\text{N}}}{2}$ is the relaxation time of the assembly

From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived:

${\displaystyle \tau \frac{\mathrm{d}M}{\mathrm{d}t}+M=\tau {\chi }_{\text{b}}\frac{\mathrm{d}H}{\mathrm{d}t}+{\chi }_{\text{sp}}H}$

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. 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 on areal-density is known as the superparamagnetic limit.

• Older hard disk technology uses longitudinal recording. It has an estimated limit of 100 to 200 Gbit/in².
• Current hard disk technology uses perpendicular recording. As of July 2020 drives with densities of approximately 1 Tbit/in² are available commercially. This is at the limit for conventional magnetic recording that was predicted in 1999.
• Future hard disk technologies currently in development include: heat-assisted magnetic recording (HAMR) and microwave-assisted magnetic recording (MAMR), which use materials that are stable at much smaller sizes. They require localized heating or microwave excitation before the magnetic orientation of a bit can be changed. Bit-patterned recording (BPR) avoids the use of fine-grained media and is another possibility. In addition, magnetic recording technologies based on topological distortions of the magnetization, known as skyrmions, have been proposed.

Applications

General applications

• Ferrofluid: tunable viscosity

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