Massive Dirac Fermions and Hofstadter Butterfly in a van der Waals Heterostructure
The remarkable transport properties of graphene, such as the high electron mobility, make it a promising material for electronics. However, unlike semiconductors such as silicon, graphene's electronic structure lacks a band gap, and a transistor made out of graphene would not have an “off” state. Ben Hunt and his colleagues modulated the electronic properties of graphene by building a heterostructure consisting of a graphene flake resting on hexagonal boron nitride (hBN), which has the same honeycomb structure as graphene, but consists of alternating boron and nitrogen atoms instead of carbons. The natural mismatch between the graphene and hBN lattices led to a moire pattern with a large wavelength, causing the opening of a band gap, the formation of an elusive fractional quantum Hall state, and, at high magnetic fields, a fractal phenomenon in the electronic structure called the Hofstadter butterfly.
Ben Hunt, PhD and his former team at MIT
One of the most striking properties of graphene, a single-atom-thick layer of carbon, is that the electrons behave as if they have no mass. They move at a constant velocity, regardless of their energy, much like photons, the more familiar massless particles of light. Special relativity tells us that a minimum energy E = 2m0c2 is required to create a particle and antiparticle of rest mass m0 (c is the speed of light; the 2 occurs because two particles are created). Because photons have no rest mass, a pair of photons can be created with energies all the way down to zero energy. In a solid, the band gap energy Eg = 2m0v2 is the energy required to create an electron and hole (particle and antiparticle), where m0 is the effective mass and v is the Fermi velocity (typically less than the speed of light by a factor of several hundred). Thus, mass and band gap are intimately related; no mass equates to no band gap, and until now that was the end of the story in graphene. Ben Hunt and his colleagues showed that electrons in graphene can gain a mass under the right circumstances in their paper in Science.
The massless property of graphene's electrons is due to the symmetry of the lattice: The simplest repeat unit, the unit cell, has two identical carbon atoms (panel A in the figure). There are thus two zero-energy states: one in which the electron resides on atom A, the other in which the electron resides on atom B. Both the electron and hole states exist at exactly zero energy, hence zero band gap and zero mass (panel B). But what happens if the two atoms in the unit cell are not identical? An extreme case is hexagonal boron nitride (hBN)—it too has a hexagonal lattice structure analogous to that of graphene, but with one boron atom and one nitrogen atom in the unit cell. Here, the electrons do have a mass and a large (>5 eV) band gap, but too large to be technologically useful.
Mass and band gap
(A) Graphene has two atoms in its unit cell, labeled A and B.
(B) If A and B are identical, then all the electronic states exist equally on both atoms (denoted by magenta in dispersion relation).
(C) When the energy of atom A is raised relative to atom B, electron states primarily on atom A (red in dispersion relation) have higher energy than electron states primarily on atom B (blue in dispersion relation) and a band gap Eg is opened. If we examine electron states that reside primarily on atom A (red in dispersion relation), we find a positive curvature of the energy versus momentum relation (red dashed curve) and thus positive mass for these states.
(D) When the energy of atom A is lowered relative to atom B, states on atom A have an energy versus momentum relation with negative curvature (red dashed curve) and thus negative mass.
The team showed that hBN in contact with graphene can slightly alter the potential felt at atom A versus atom B, and this effect is large enough that the graphene electrons develop a mass, and hence a band gap. In this case the band gap is much smaller, about 30 meV, but large enough to alter graphene's properties in technologically interesting ways.
One curious aspect of the sort of mass and band gap that can be created for massless particles such as electrons in graphene is that the mass can be positive or negative. Consider then an arrangement of graphene on hBN that slightly raises the energy of an electron on atom A relative to atom B such that it has a positive mass (see the figure, panel C). Then likewise an arrangement that raises the energy of atom B should have a negative electron mass (see the figure, panel D). These two versions of graphene-with-a-mass behave alike and would be indistinguishable in an optical spectroscopy experiment. But an interesting thing happens when they come into contact: An electron traveling from a positive-mass region to a negative-mass region must go through an intermediate region where its mass once again becomes zero. This intermediate massless region is gapless and therefore metallic. The presence of such metallic modes at the boundaries between semiconducting regions of opposite-sign mass is a hallmark of a topological phase. “Massive” graphene displays much the same physics as topological insulators.
Even if the graphene layer is perfectly aligned in angle to the underlying hBN lattice, a small difference in the atomic spacing in the two materials means that they go in and out of registry over ∼14 nm. Hunt et al. use this property in their massive graphene on hBN to examine the Hofstadter butterfly, as have two other groups—one in bilayer graphene on hBN, another in graphene encapsulated above and below by hBN. The effect results from the interplay of the length scale determined by the repeating overlap of graphene and hBN atoms with another length: the size of electron orbits in a magnetic field. The conductivity of such a sample as a function of these two parameters exhibits a fractal pattern whose butterfly shape gave its name to the phenomenon.
The observation of a controllable mass and band gap in graphene leads to a new way to engineer its properties. Confining graphene electrons is notoriously difficult; for example, theory tells us that massless electrons will leak out through even the highest barrier. However, if the mass in graphene can be controlled, electrons can be confined to massless regions in graphene by surrounding them with massive regions. This would allow the patterning of quantum dots, wires, and other mesoscopic structures. Intriguingly, generating adjacent areas of opposite-sign mass in graphene will create one-dimensional conductors along the boundary. These wires would be protected against backscattering and could carry currents without dissipation. The ability to control both the magnitude and sign of the mass of charge carriers is fundamentally new and is likely to spawn unanticipated new phenomena.
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