In physics, it is already well-known that electrons behave very differently in three dimensions (cube), two dimensions (sheet) or one dimension (wire). These behaviours give rise to different possibilities for technological applications and electronic systems. But what happens if electrons live in 1.58 dimensions – and what does it actually mean to live in 1.58 dimensions? Theoretical and experimental physicists at Utrecht University investigated that in a new study that will be published in Nature Physics on 12 November.

It may be difficult to imagine 1.58 dimensions, but the idea is more familiar to you than you think at first glance. Non-integer dimensions, such as 1.58, can be found in fractal structures, such as your lungs. A fractal is a self-similar structure that scales in a different way than normal objects: if you zoom in, you will see the same structure again. For example, a small piece of Romanesco broccoli typically looks similar to the whole head of broccoli. In electronics, fractals are used a lot in antennas for their properties of receiving and transmitting signals in a large frequency range.

## Electronic nanoscale fractals

A relatively new topic in fractals is the quantum behaviour that emerges if you zoom in all the way to the scale of electrons. Using a quantum simulator, Utrecht physicists Sander Kempkes and Marlou Slot were able to build such a fractal out of electrons. The researchers made a ‘muffin tin’ in which the electrons would confine to a fractal shape, by placing carbon monoxide molecules in just the right shape on a copper background with a scanning tunneling microscope. The resulting triangular fractal shape in which the electrons were confined is called a Sierpiński triangle, which has a fractal dimension of 1.58. The researchers observed that the electrons in the triangle actually behave as if they live in 1.58 dimensions.