Famous Einstein's memo E = m c^2 is an "ultraheavy" limiting case of a more general mass-energy equivalence, generalized to Dirac equation in case of electrons. Another limiting case is E = c p, denoting "ultrarelativistic" quasiparticles (travelling at effective speed of light c). In our recent paper, we show how to construct stable ultraheavy and ultrarelativistic Dirac quasiparticles at the same energy scale.
Graphene is an atomically-thin crystal of carbon. Since its discovery in 2004, the material became a surprisingly vivid toy for theoretical physics due to its chemical simplicity, and yet the beautiful analogy of Quantum Electrodynamics (QED) "in the pencil": electronic excitations (quasiparticles) in a graphene sheet are accurately described by relativistic Dirac equation, with a different effective "speed of light". In other words, quasiparticles in graphene looks like if they were ultrarelativistic: they have a linear spectrum E = ħck, and are described by spinor, two-component wave functions corresponding to pseudospin, constructed from two graphene sublattices. This analogy is very robust, leading to a number of predictions on the basis of Dirac equations, for the first time confirmed experimentally (e.g. relativistic Landau levels with QED degeneracy). There was also a hope to dope graphene further to induce superconductivity in the monolayer; but it was not succeeded for the single layer alone. Thus the search is focused on producing dramatically different electronic structures on the basis of graphene, to reach different scenarios of superconductivity. One ultimate goal would be to have both "ultraheavy" and "ultrarelativistic" quasiparticles at the same energy scale, which will allow band hybridization and thus promote enhanced electronic mobility on the background of electronic localization.
When three graphene layers of graphene are rotated in a certain configuration [top, left], the electronic structure features a coexistence of flat (ultraheavy) band piercing steep (ultrarelativistic) bands [bottom, right]. Figure from [Carr2020].
To achieve these conditions, we decided to use several layers of graphene. It is known that one can reach the situation with the flat bands it twisted bilayer graphene; to construct both the flat bands piercing the steep bands we decided to add an extra layer. The resulting material looks like a sandwich: it is a three-layer system, in which the "bread" layers are perfectly aligned, while the middle one is slightly rotated. Rotating the the middle layer creates the long-periodic moiré pattern, which modifies the way the electrons interact between the layers. At rotation angle of 1.5 degree, this system shows a robust coexistence of flat and steep electronic bands at the same energy scale (see Figure), precisely the situation we were looking for. Importantly, this phenomenon of the flat bands piercing steep bands is robust in the graphene sandwich: this configuration is protected against accidental layer shifts by a large barrier of 20 meV/nm^2. This barrier is a consequence of complicated atomic relaxations in the sandwich: the atoms tend to relax to the most beneficial positions, slightly different from their positions in the isolated monolayer graphene. It is remarkable that the lattice relaxation mechanism, which is usually seen as a hindrance for experiments, plays a significant role in stabilization of the flat band/steep band scenario, and thus coexistence of ultraheavy and ultrarelativistic Dirac quasiparticles at the same energy scale.
We hope that our construction will become a fruitful platform for further exploration of Dirac physics in condensed matter experiments, and a possible setup for testing "flat band/steep band" scenario for unconventional superconductivity.
Further reading: S. Carr, C. Li, Z. Zhu, E. Kaxiras, S. Sachdev, and A. Kruchkov, Ultraheavy and Ultrarelativistic Dirac Quasiparticles in Sandwiched Graphenes, Nano Letters 20 (2020).