Twisted bilayer graphene creates new type of molecular bond

The atoms appear locally to be isolated from one another, but because of non-local quantum correlations the structure remains molecular. The result was obtained in a theoretical study conducted at São Paulo State University.

By José Tadeu Arantes  |  Agência FAPESP – “Twistronics” is a variant of electronics that describes what happens when two almost two-dimensional graphene sheets are mechanically stacked and misaligned by twisting one of them on the other. When the twist reaches a specific angle, known as the “magic” angle and corresponding to about 1.1 degrees, the twisted bilayer graphene (TBG) exhibits competing insulator and superconductor behaviors, which are normally antagonistic.

The phenomenon was obtained experimentally by a group of researchers led by Pablo Jarillo-Herrero in the Physics Department of the Massachusetts Institute of Technology (MIT) in the United States and is described in an article published in the journal Nature in 2018.

Instead of the insulating and superconducting states, a new study, this time theoretical, obtained a metallic state, also known to be possible. The study was supported by FAPESP and conducted by a team led by Professor Antonio Seridonio in the Physics and Chemistry Department of São Paulo State University (DFQ-UNESP) at Ilha Solteira, Brazil. The results are published in the journal 2D Materials.

“In the insulator state, technically called the ‘Mott insulating state’, electron-electron interaction is repulsive and dominant. Repulsion hinders electron transport by the system, and this characterizes the condition of electrical insulation. In the superconductor state, however, we observed counterintuitive behavior in the shape of attraction between electrons. This generated new charge carriers called ‘Cooper pairs’, each consisting of two electrons that flow together through the material without energy dissipation, which is the definition of superconductivity,” Seridonio explained.

“In the metallic state analyzed in our study, the TBG didn’t exhibit a gap between the valence band [fully populated by tightly bound electrons] and the conduction band [containing free electrons and empty spaces]. However, because the material has a periodic atomic number, its discrete energy spectra overlap and give rise to continuous spectra with infinitely close energies.” 

To characterize this metallic state more fully, the study considered the adsorption of impurities in the moiré superlattice created by the relative movement of the two layers. In textiles, moiré patterned fabric has a rippled or wavy look. In physics, moiré is the geometrical interference pattern that results when two networks are superimposed but slightly offset or rotated. The term superlattice is used in this case because the interference pattern has a much larger scale than a graphene network [see figure A in the infographic below].

“What we found was a new type of molecular covalent bond. The usual covalent bond is one where atoms of the molecule share electrons owing to overlapping atomic orbits. This is drastically modified by TBG ‘magic’ when an electrical field is applied to the system. The field breaks the inversion symmetry of the Dirac cones, enabling an atomically frustrated molecular state to emerge. In this state the atoms look locally as if they were isolated from one another but remain molecular in nature because of non-local quantum correlations mediated by the moiré superlattice,” Seridonio said.

A parenthesis is needed here to explain the term “Dirac cones”. Named for the British physicist Paul Dirac (1902-84), who made fundamental contributions to the development of quantum mechanics and quantum electrodynamics, these hourglass-like conical surfaces describe the electronic configurations of certain materials, such as graphene and others, at specific energy levels. The upper and lower halves of the hourglass represent cones that correspond to the conduction and valence bands respectively. The bands meet only at a central point called a Dirac point.

“For a better understanding of the role played by Dirac cones and electrical fields in the appearance of the molecular bonds we found in our study, it’s worth using an analogy. Imagine that the Dirac cones are the two halves of an hourglass with their vertices joined at a single point. The point is a zero-energy state as small as a grain of sand but initially empty owing to the absence of an external electrical field. We call this grain-less empty point a ‘pseudogap’ or ‘Dirac point’,” Seridonio said.

“The top cone, which is upside-down, plays the role of a conduction band and is balanced out by the vertex of the bottom cone, which emulates the valence band. The bottom cone is filled with sand to just below the ‘pseudogap’ and represents the ideal number of electrons for the system. Because the TBG comprises two layers of graphene, there are two hourglasses in this state [figure B]. In the ‘magic angle’, it’s as if the hourglasses were perfectly flat since the slope of the cones becomes zero. The electrical field closes the ‘pseudogap’ by filling it with sand, and the flattening of the cones compresses all levels of molecular energy at this point, which admits only a grain of sand and corresponds to the zero-energy mode [total flattening of the cones reduces their graphical representation to a horizontal line; figure C shows a transition mode in which the cones appear flat but not completely].”

As long as the cones are flat, he added, the electronic configuration does not exit the zero-energy level. The entire molecular spectrum is reduced to a single quantum state of zero energy and becomes robust in this sense. The molecule “tries” to split up into independent atoms but fails because the energy levels are tightly “squeezed” by the collapse of the Dirac cones, and the molecule, therefore, becomes “atomically frustrated”.

“This type of configuration is called a zero-energy mode. If a gradual increase in the size of the electrical field can’t remove this mode, and it remains stuck forever at zero, it becomes immune to outside disturbances and is considered ‘robust’. Robustness is one of the main requirements for successful topological quantum computing,” Seridonio said.

The study led by Seridonio was part of the PhD research of William Nobuhiro Mizobata, first author of the article and a graduate student in materials science at DFQ-UNESP. PhD candidates José Eduardo Cardoso Sanches and Willian Carvalho da Silva also took part, as well as master’s candidate Mathaus Penha, undergraduate Carlos Alberto Batista Carvalho, and professor Valdeci Pereira Mariano de Souza (UNESP Rio Claro) and Marcos Sérgio Figueira da Silva (Fluminense Federal University, UFF).

The article “Atomic frustration-based twistronics” is at: iopscience.iop.org/article/10.1088/2053-1583/ac277f. A preprint is also available from arXiv at: arxiv.org/abs/2110.04909.