Blog entry

Where does ferromagnetism come from?

By Josep Ingla-Aynes, University of Groningen

We are all familiar with magnets, we use them every day for different purposes (Our PC does it to store information in the hard drive, the souvenirs that stay atached in the fridge...). In this entry I am going to explain the reason why some solids show such magnetic behavior.

Net magnetic moments in solids are the result of a combination between the effect of the Pauli prin- ciple and the electrostatic interactions between electrons. But let’s start from the beginning:
Solids are assemblies of atoms that contain large amounts of electrons (In the order of 1032 per cubic meter) and these electrons have their own intrinsic magnetic moment: the so called spin (Jose Lado’s entry in this blog). One could think that these magnetic moments alone can explain the magnetism in materials but the interaction between them is very weak (in the order of μeV if they are separated 0.1 nm). Not enough to give ferromagnetic order at temperatures higher than 10 mK (-273 ◦C ). A stronger interaction affecting electrons comes from the fact that they possess an intrinsic electric charge (the fa- mous e = −1.602 × 10−19C) that leads to mutual repulsion between them due to Coulomb interaction. This interaction alone cannot affect the spin configuration of the electrons but, together with the Pauli principle, it can give rise to magnetic ordering.

In order to explain this effect I will use a simplified model: Let’s assume we have an atom with a closed shell of electrons and 2 unpaired ones; In this particular case we can determine the magnetic state of the atom just by studying the arrangements of the unpaired electrons (this can be done because a closed shell of electrons has zero net magnetic moment). These unpaired electrons have some freedom to select their states; they can choose their spin and have few orbitals they can occupy (Since we are studying confined electrons their energies are quantized).

 

Figure 1: Spin configurations in a 2 electron system. Since we are dealing with identical particles the 2 states highlighted are not distinguishable. From them 2 new states can be defined as linear combinations of both states: One symmetric (+) and the other antisymmetric (-). The symmetric configurations are the ones in the first row and the (+) linear combination of the 2nd rew and have spin 1. The antisymmetric one (-) has spin zero.

In this system, electrons have 4 possible configurations (Fig. 1) and they are very close together (they belong to the same atom and orbital). Hence, their wave functions have a big overlapping and interact with each other as identical Fermions. This means that they have to follow the Pauli principle and the overall state has to be antisymmetric. This constriction couples the spatial distribution of the electrons with their spin configuration and, since electrons have electrostatic charges, different spin symmetries have different electrostatic energies. This energy difference is called ’echange interaction’ and is the re- sponsible for magnetism in solids.


Table 1: Allowed symmetry configurations according to the Pauli principle in a 2 Fermion system

If this exhcange interaction is bigger than the thermal energy of the electrons we can assure that our system will have magnetic behavior. This condition is more intuitive than it looks like: A finite tem- perature gives the electrons a kinetic energy proportional to this temperature (Ec = kT where k is the Boltzman constant and T the absolute temperature in Kelvins). If the kinetic energy is higher than the echange interaction then the electrons can occupy both, symmetric and antisymmetric spin states in an indistingushable way, because they have the energy to jump between them. This is called the Stoner criterium and the full theory is called Stoner ferromagnetism.

Of course, in real solids we have more than 2 electrons and everything gets more tricky (The crystal structure plays a crucial role, different elements have a different amount of valence electrons...) but the spin splitting of the energy is still caused by echange interactions and has to be explained in terms of electrostatic interaction and Pauli principle.