Author: Kristof Lebecki
Company: IF PAN

Micromagnetic modeling – periodic boundary conditions

One can speak about new trends not only with regard to fashion. The same applies to sciences. Recently it is popular to use a term “nano”. Beside a nanocar, which is not yet sold commercially (maybe it is too small? ☺), we speak about nano-physics, nano-objects, and nano-magnetism as well. The word “nano” is a shortcut for nanometer, which is so small that a human hair has a diameter of roughly 100,000 nanometers. Because of historical reasons, instead of “nanomagnetism” physicists use rather the word “micromagnetism” that was introduced in early sixties (the nano-trend was not present at that time). 

Magnetically active nanoobjects are easy to produce. One example of a device where micromagnetism has an important application is the hard disk, present nowadays in almost every computer. In the hard disk, the information is stored in small magnetized islands, and it is transmitted by a small writing head. Although it is easy to produce micromagnetic nanoobjects, it is much more difficult to observe them. Often specialized microscopes are needed for this purpose. The theory of micromagnetism is complicated, as well. Because of these reasons, numerical micromagnetism is recently widely used. In this approach the behavior of the ferromagnetic sample is modeled using specialized computer programs. Such simulations help to design future devices, as modeling is often cheaper and faster than building a real test device and checking its functionality. Another, often used approach is to perform a complicated experiment and to explain its results using micromagnetic simulations. No wonder, that roughly in every second publication related to micromagnetism the simulations are mentioned.

Last term present in my PhD thesis title is probably most difficult to explain. To proceed, I will use a comparison with a fairy tale structure. The introduction is already well known: “Over the hills and far away”. For my purpose the standard ending will be more important: “They lived happily over after”. This short sentence gives an impression that the heroes of the story will happily continue their live. In some sense the episode is ended, but there will be maybe next interesting adventures in their future live. The same trick is used in TV serials: every episode is ended in such a way that it is easy to continue it. In some sense all the episodes (in physics: parts) are similar (physics: exactly the same) and as a result the whole serial resembles an endless story (physics: it is infinite).

Similar idea was introduced in physics by Max Born. Sometimes we can evaluate only a small part of a system. For instance, we are currently not able to simulate the behavior of the whole hard disk, we can model only, say, one-millionth part of it. The question now arises, what is outside this simulated part? Nothing, it means this part is alone in the universe? But the magnetic interactions are active over long distances! The magnetic needle in the compass reacts to the Earth’s core, although they are thousands of miles away. The more so, different areas of the hard disk interact with each other. Instead of evaluating our part as being alone in the universe, Max Born made following suggestion: imagine that our selected part (episode) has everywhere same neighbors: to the left and to the right. We end up with a situation where only one part is considered, but due to Born’s periodic boundary conditions infinite set of parts is evaluated (the whole serial). This concept allows evaluation of a small part (we call it: a simulation window) of the system, while the geometrical shape (we call it: the aspect ratio) of the whole system is retained in our analyze.

PS Text similar to the shown above was presented on my English exam. The main reproof concerned allegedly inappropriate language register. I mean, it was not enough scientific. Because of that, I have prepared following scientific terms, which you can use to “enrich” it. Just take them and drop several times in random places. Certainly after this procedure, it will look more sophisticated and scientific. Have fun ☺
Discretization mesh, demagnetization tensor, anisotropy constant, roundoff errors, distance cut-off, prolate ellipsoid, quadrupoles, one dimensional, double-precision, exchange length, finite element method, nucleation field, magnetization saturation, magnetic permeability.

K. M. Lebecki, 2008