Unconventional Fractional Quantum Hall Sequence in Graphene
Authors: Ben Feldman and Amir Yacoby
Affiliation: Department of Physics, Harvard University, USA
Graphene, a two-dimensional sheet of carbon that is one atom thick, has attracted considerable interest due to its unique and potentially useful physical properties. Like other two-dimensional materials, application of a perpendicular magnetic field leads to the formation of a sequence of flat energy bands called Landau levels (LLs). At high magnetic fields or when samples are very clean, interactions among electrons become important and produce additional energy gaps, even when the LLs are only partially filled. This phenomenon is known as the fractional quantum Hall effect (FQHE), and it leads to striking physical consequences such as excitations with a fraction of an electron charge [1-3].
Graphene provides an especially rich platform to study the FQHE. The low dielectric constant and unique band structure lead to FQH states with energy gaps that are larger than in GaAs at the same field. Moreover, charge carriers in graphene have an overall fourfold degeneracy that arises from their spin and valley degrees of freedom. This means that graphene can support FQH states that have no analogues in more conventional systems. Suspending samples above the substrate or depositing them on boron nitride minimizes disorder, and the FQHE was recently observed in such devices at all multiples of filling factor ν = 1/3 up to 13/3, except at ν = 5/3 [4-7]. The absence of a state at ν = 5/3 might result from low-lying excitations associated with the underlying symmetries in graphene, but alternate scenarios associated with disorder could not be ruled out in prior studies.
Figure 1: Picture of the scanning Single-Electron Transistor (SET) microscope setup
To further explore this behavior, we used a scanning single-electron transistor (SET) to probe a suspended graphene flake . The SET is a unique local probe that is particularly non-invasive. It measures the presence of energy gaps in the electronic spectrum with sensitivity that no other technique can provide and therefore is very well adapted to the exploration of the FQHE. Moreover, we are able to study small regions of a graphene flake, and these local measurements reveal a dramatic improvement in sample quality relative to prior studies of larger-scale areas.
Our measurements show that electron-electron interactions in graphene produce different types and patterns of electronic states from what has been observed in more conventional materials. Although we observe the standard sequence of FQH states between ν = 0 and 1, states only occur at even-numerator fractions between ν = 1 and 2. This suggests that both spin and valley degeneracy are lifted below ν = 1, but one symmetry remains between ν = 1 and 2. The pattern of states that we observe and their corresponding energy gaps indicate an intriguing interplay between electron-electron interactions and the underlying symmetries of graphene.
Figure 2: Schematic of the measurement setup. The scanning single-electron transistor is held about 100 nm above a suspended graphene flake, and it measures the energy cost of adding additional electrons to the system.
Moreover, the scanning technique allows us to study variations in behavior as a function of position. Although all regions of the graphene flake show qualitatively similar behavior, local doping shifts the gate voltage required to observe each FQH state. Global measurements such as transport studies therefore require an especially homogenous sample to observe the delicate effects associated with interactions among electrons, whereas using a local probe allows us to observe especially clean regions and therefore observe more of the intrinsic physics.
Figure 3: Inverse compressibility of graphene as a function of carrier density and magnetic field. Incompressible behavior, which indicates the presence of an energy gap, is labeled at integer and certain fractional filling factors.
In the future, we are interested in continuing to explore the unusual FQH in graphene. In particular, we hope to better understand how the electrons are ordered in the various FQH states. We are also interested in learning more about the FQHE at higher filling factors and in related materials such as bilayer graphene.
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