Publish Date: March 6, 2024

A Twist in the Saga of Fractals

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Moir\’{e} means watered textile. This is created when two layers of fabrics are pressed again each other to create a wavy colour pattern, and has been known in silk industry for more than a century. Essentially this moir\'{e} pattern is an interference pattern that emerges when once two or more lattices are overlaid on each other but with a relative twist or displacement between them. They manifest in a whole range of fields spanning textiles, architecture, optics etc. Material world become fascinated in such pattern a little more than a  decade back when it was understood that if a two dimensional crystal such as graphene (discovered in 2005 and Nobel prize on Physics was awarded to it in 2011), was placed one atop another and relatively twisted, it changes its properties drastically and can possibly even make it superconducting under ambient condition. The field which is now popularly known as “Twistronics” ( Electronics of twisted electrons), registered an explosive growth with the experimental confirmation of the above idea in 2018 by an experimental group in MIT. The idea was generalised to a large class of twisted layered materials which form a special class of so called van der Waals structures, now also known as moir\'{e} materials.

Now in a recent work published in Physical Review B published by American Physical Society, and selected as “Editor’s suggestions” (, investigators at the Physics Department of Indian Institute of Technology Delhi, uncovers an unexpected manifestation of fractal geometry in the electronic properties of such moir\’{e} materials. The ubiquity of fractals in nature has long captivated researchers, as these self-similar patterns appear across diverse physical systems, such as flower petals, or coast-line. The researchers demonstrate that when two-dimensional crystals such as graphene or HBN (hexagonal boron-nitride) are multiply stacked and then twisted by a specific angle, the resulting super-moiré pattern exhibits electronic fractality. The researchers name this new class of electronic fractals as moiré fractals, and derive an elegant formula relating its fractal dimension to the number of electronic bands in a given energy window. Their findings have significant implications for controlling the conductive properties of these novel van der Waals materials and provides a powerful tool for engineering designer quantum materials. They also find parallels between moiré fractals and those arising in transportation and commerce, as described by the central place theory of economic geography, again underlying the ubiquity of such fractal structures.

Sankalpa Ghosh is funded by a MATRICS grant from DST SERB, Govt. of India and Deepanshu Aggarwal is funded by a fellowship by UGC, Govt. of India.

Prof. Sankalpa Ghosh and Prof. Rohit Narula, whose group published this work (the lead author is a Ph.D. student Mr. Deepanshu Agarwal) commented that if the existence of moir\’{e} fractals is experimentally confirmed, it will help to create interesting class of quantum materials with tailored properties that may play role in future semiconductor technology. They pointed out that a number of experimental groups in India and abroad already showed interest in their work.