Magnetism and the Ising Model

Some materials have the curious property of being magnetic under normal everyday conditions - for example, they stick to the metallic door of your fridge. Technically speaking, they show a spontaneous magnetisation at room temperature, and are called ferromagnetic, for the Latin name of iron, which is the prototype of a material with these properties. It comes out, the state of being magnetic is a phase, similar to being solid or fluid, and indeed, one can study phase diagrams for magnetic materials. For example, if the temperature of a magnetic chunk of iron is raised above a certain, specific temperature, the magnetisation is lost. This temperature is called the Curie temperature for Marie's husband Pierre, a pioneer of solid-state physics. The Curie temperature of iron is at 1043 K. The appearance (or disappearance) of spontaneous magnetisation at the Curie temperature is not only technologically relevant, it is also very useful for geologists: if ferromagnetic minerals in volcanic lava cool down from red-hot molten rock to below the Curie point, they "freeze in" the orientation of the Earth's magnetic field at that very moment. This allows to reconstruct the orientation and strength of the Earth's magnetic field over history.

One goal of physicists in the early years of the 20th century was to understand how spontaneous magnetisation comes about, and to find a quantitative description of the magnetisation as a function of temperature. To this end, they made simplified assumptions, for example, that atoms behave like miniature compass needles which interact just with their neighbours. One of these models was proposed by the German physicist Wilhelm Lenz in 1920, and then analysed in more detail by his student Ernst Ising - it's the famous Ising model (Ising was born in Cologne, Germany, hence the pronunciation of the name is "eeh-sing", not "eye-sing").

In the Ising model, one assumes that the magnetic moments of atoms can have only two orientations, and that it is energetically favourable if the magnetic moments of neighbouring atoms are oriented in parallel - it costs an energy J to flip one magnetic moment with respect to its neighbour. Then, one applies the rules of statistical mechanics and tries to calculate the magnetisation - the average orientation of the magnetic moments. As it comes out, there is indeed a spontaneous magnetisation below a certain temperature - one of the most elementary examples of spontaneous symmetry breaking. And, even more spectacular from the theorist's point of view, in the special case of a restriction to just two dimensions, Onsager and later Yang (the Yang of parity violation and Yang-Mills theories) could derive an exact formula for the magnetisation M as a function of temperature. It looks pretty complicated,


but the interesting thing is that there is only one free parameter in the formula, the Curie temperature T_C, which depends on the energy J necessary to flip a magnetic moment. Essentially, the magnetisation is 1 at zero temperature (meaning that all magnetic moments point in the same direction), and drops to zero as the eighth root when the temperature approaches the Curie point.

As nice as it may be to have such a formula, it would be interesting to check in an experiment if it is correct. However, there is a drawback: It's valid only in two dimensions, i.e. for planar layers just one atom thick, and it works only for magnetic moments which can only be parallel or antiparallel to one fixed direction.

Fortunately, progress in materials science in the 1990s has made it possible to produce thin ferromagnetic films only a few atomic layers thick, with magnetic moments which show indeed the restricted orientation with respect to an axis as described in the Ising model. So, these films should behave like the Ising model, and one can try to measure the magnetisation as a function of temperature. This is what is shown in this plot by C. Rau, P. Mahavadi, and M. Lu:


Figure taken from C. Rau, P. Mahavadi, and M. Lu: Magnetic order and critical behavior at surfaces of ultrathin Fe(100)p(1×1) films on Pd(100) substrates, J. Appl. Phys. 73 No. 10 (1993) 6757-6759 (DOI: 10.1063/1.352476).

It is, unfortunately, not possible to measure magnetisation directly, so one has to rely on other effects which are directly dependent on magnetisation - in this case, one uses a method called Electron capture spectroscopy (ECS): A beam of ions is shot on the film, the ions capture electrons from the surface, and emit light which can be detected. If the surface is magnetised, the light is polarised, and thus, the polarisation of the emitted light is a measure of magnetisation. This is what is plotted on the vertical axis: the polarisation P, normalised to the polarisation P₀ at low temperatures. For, as it comes out in the experiment, the polarisation - and hence, the magnetisation of the film - is nearly constant at low temperatures, and drops sharply to zero when approaching a specific temperature, to be identified as the Curie temperature T_C. In the figure, normalised polarisation is shown as a function of temperature T, where temperature has been normalised to the Curie temperature. Now, one can compare with the theoretical prediction for the magnetisation of the Ising model as a function of temperature. This is the solid black curve. There are no more free parameters, and, as it comes out, the agreement with experimental data is perfect.

Here is an intriguing circle from experiment to theory back to experiment: Experimental data of ferromagnets measured more than 100 years ago show the appearance of spontaneous magnetisation as temperature drops below the Curie point. Models are constructed to try to understand this, and for a simplified model restricted to two dimensions, an exact formula for the magnetisation can be derived. Finally, real materials show up which correspond to the idealisations and simplifications made in the model, the magnetisation can be measured... and it works!

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This post is part of our 2007 advent calendar A Plottl A Day.