Properties of four-component Dirac operators describing graphene quantum dots
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2020
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Abstract
The electronic properties of graphene quantum dots (GQDs) can be described by the two-dimensional Dirac equation with boundary conditions consistent with the tight-binding model on a honeycomb lattice. It is convenient to know which boundary conditions are allowed by elemental physical principles of current conservation.
We consider the four-component two-valley massless Dirac operator on planar domains describing GQDs. We show how to reduce the problem into the study of the two-component Dirac operator. For a large class of boundary conditions, not including the zigzag orientation, we give a proof of their self-adjointness for four-component spinor wave functions in the Sobolev space H¹. In particular, in each case, we find a lower bound to the spectral gap around zero, proportional to the inverse of the square root of the area of the domain and depending only in the mixing angle. We also discuss the boundary conditions conserving (breaking) the electron-hole and time reversal symmetries.
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Tesis (Master of Sciences in Physics)--Pontificia Universidad Católica de Chile, 2020