1. Superparamagnetic limit, where magnetic particles
begin to reverse their magnetization spontaneously
Energy
Barrier E
Needle-shaped magnetic particles prefer to have their magnetization parallel or anti-parallel to the long axis. That provides two stable magnetizations for storing one bit.
To switch from one orientation to the other requires the so-called demagnetization energy. This energy barrier can be overcome by thermal energy. The thermal flip rate of the magnetization is given by an Arrhenius law, which consists of a Boltzmann factor (the probability for overcoming the energy barrier ΔE) and an attempt frequency . This is usually a vibration frequency of molecules, but here one has to use the much lower Larmor frequency L = eB/4m for a full rotation of an electron spin in the internal magnetic field B of the particle.
To reach a flip rate of less than one in about five years (typical for the lifetime of magnetic hard disk data in a warm place) one needs an energy barrier of more than 40 kBT, which is 1 eV at room temperature. Since the energy barrier is proportional to the number of magnetic atoms in the particle, i.e., to the volume, one has a length^3 power law.
The simple size estimate in Lecture 2, Slide 11 comes to a similar result as this more realistic calculation thanks to a cancellation of errors. We used the condition E = kBT instead of E = 40 kBT , but at the same time greatly underestimated the magnetic energy barrier by using the weak shape anisotropy instead of the stronger crystalline anisotropy that is actually used in hard disks.
Flip Rate  Attempt • exp[-E/kT]
109s 40kBT