Electron-electron interactions in graphene field-induced quantum dots in a high magnetic field

Abstract

We study the effect of electron-electron interaction in graphene quantum dots defined by an external electrostatic potential and a high magnetic field. To account for the electron-electron interaction, we use the Thomas-Fermi approximation and find that electron screening causes the formation of compressible strips in the potential profile and the electron density. We numerically solve the Dirac equations describing the electron dynamics in quantum dots, and we demonstrate that compressible strips lead to the appearance of plateaus in the electron energies as a function of the magnetic field. Finally, we discuss how our predictions can be observed using the Kelvin probe force microscope measurements.

Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots

“…by simulataneous application of a magniectic field and an electrostatic potential. In the paper [13] such a quantum dot was considered.…”

“…where E is energy level for electrons, B is the strength of the magnetic field perpendicular to the graphene surface, m is the angular momentum, V (r) is the potential that includes electron-electron interactions and define the quantum dot. The potential V (r) that we use was described in detail in the paper [13]. The remaining constants have the values: γ = 646 meV • nm and e/ = 1.52…”

“…The operator H, with the domain X, is self-adjoint, in L 2 (C R ) × L 2 (C R ), when the above scalar product is used [13].…”

“…In the previous paper [13] a numerical discretization of the eigenvalue problem (3), with an equidistant grid {r i } n i=1 , r 1 = 0 and r n = R, was used together with the backward-forward finite difference method to find eigenenergies E and wavefunctions χ 1 (r) and χ 2 (r). To eliminate the singularity of χ 2 (r) at r = 0, it was assumed that χ 2 (0) = 0, which is equivalent to an antisymmetry assumption.…”

“…For example, when designing a new electronic device, the analysis of electron dynamics in the underlying material is required. It is especially important to investigate the possibility of electron localisation and its energy spectrum by solving the eigenvalue problem within the used model [6,9,10,11,13,14,18]. Knowledge of the accuracy in the computed eigenvalues allows us to draw conclusions about which effects to incorporate in our models and which effects that can safely be neglected.…”