Graphene, a 2D honeycomb lattice of carbon, was first isolated in 2004 by a team of researchers at Manchester led by Andre Geim. The material boasts a wealth of fascinating electronic properties, many of which arise from the fact that it is a semiconductor with a zero-energy gap between its valence and conduction bands. Near where the two bands meet, the relationship between the energy and momentum of the electron is described by the Dirac equation and resembles that of a photon. These bands, called Dirac cones, enable electrons to travel through graphene at extremely high speeds, which means that electronic devices, like transistors, made from the material could be faster than any that exist today.

Hexagonal boron nitride (hBN) is not only an excellent substrate for graphene (thanks to the fact that the two materials have very similar lattice constants), it has also allowed researchers to access further “Dirac points” in graphene and observe phenomena like “Hofstadter’s butterfly” at relatively low magnetic fields. These effects occur because of the superlattice (or moiré) potential between graphene and hBN. This potential is the regular pattern created whenever two similar 2D lattices are overlaid.

The commensurate-incommensurate transition

“We have now found a remarkable, new effect as a result of this potential – the commensurate-incommensurate transition,” team member Colin Woods of Manchester told

“Commensurate” literally means “matching in size or proportion” and when applied to this situation means that graphene expands its lattice constant (or size) by 1.8% to match with hBN, he explains. “Incommensurate” means the opposite – not matching. In this state, graphene remains the same size.

The commensurate-incommensurate transition is so-called because the effect depends on the angle between the graphene and hBN lattices. If this angle is small (approximately less than a degree), then the graphene will enter the commensurate state. If the angle is large, then graphene will remain in the incommensurate state. “Although it is extremely difficult to rotate a graphene sheet on a hBN substrate, we have overcome this problem by making many samples at varying angles and testing each one,” said Woods.

The team, which includes researchers from China, the Netherlands, Russia and Japan, says that it observed the commensurate and incommensurate states as a distribution of strain across the surface of a graphene sheet on the hBN. “In the commensurate state, the strain distribution becomes very abrupt,” added Woods. “This is because there must be a network of domain walls (marked yellow in the figure above), also known as solitons in one dimension, between the stretched regions (grey/blue).”

The findings could be a new and exciting way to control and fine-tune the electronic properties of graphene devices, he said. “As well as being of fundamental interest for furthering graphene electronics, our results could also be used to study 2D analogues of the so-called Frenkel-Kontorova model in easily accessible structures. This model describes commensurate transitions in 1D.”

The current work is detailed in Nature Physics doi:10.1038/nphys2954.