There is a widespread belief that quantum confinement induced by an external electrostatic potential is not possible for charge carriers in a pristine borderless graphene sheet. This belief is based on the perfectly valid statement that in graphene an electron striking a potential barrier at normal incidence penetrates this barrier without reflection (this is the graphene analogy of the Klein paradox. well-known for ultra- relativistic quantum particles). The Klein paradox together with the absence of an energy gap in the graphene spectrum leads to the conclusion that the states in any electrostatically-defined quantum well in graphene are "leaky" as there is always an extended state with the same energy in the continuum outside the well. There is, however, an important feature in the density of states associated with the two-dimensional nature and linear dispersion of charged carriers in graphene. Namely, the density of states vanishes at zero energy (corresponding to the Dirac point in graphene spectrum). Therefore, zero-energy states can not "leak" outside of the quantum well and are totally confined. Zero-energy states were ignored in previous theoretical studies of electronic states in graphene quantum wells, because these studies dealt with sharply terminated potentials (square wells) which do not support zero-energy states: the two-component structure of the electron wavefunction requires simultaneous termination of the wavefunction and its derivative at the well's edges which is not possible. The situation changes drastically when the confinement potential decays smoothly, which is precisely what happens in realistic graphene structures.
This PhD project will be devoted to the study of zero-energy confined states in smooth non-singular potentials vanishing at large distances. We have recently found such states for a model potential which supports exact analytical solutions (see http://link.aps.org/doi/10.1103/PhysRevB.81.245431). In particular, we have shown that there is a threshold value of the product of the potential strength to its effective width for which the first confined state appears, in striking contrast to the non-relativistic case. The future work will be focused on realistic potentials created by top gates. We plan to develop a universal numerical technique which would allow finding the number of zero-energy confined modes in a smooth graphene waveguide with an arbitrary potential profile. We will also study the influence of disorder on the mode propagation in such a channel taking into account both short- and long-range scatterers. Waveguides created by narrow magnetic stripes or areas of controlled strain and edge states will be considered as well. The proposed research will also be relevant for understanding the formation of charged carrier puddles in disordered graphene and will be ultimately aimed at understanding the nature of the non-vanishing minimal conductivity in graphene.
The project is inspired by the pioneering experimental work on top-gated graphene structures carried out by Prof. Savchenko's group in Exeter. We expect our results to be helpful for explaining experimental data and designing optimal structures for creating graphene-based transistors with high on/off current ratios.