Signatures of Wigner molecule formation in interacting Dirac fermion quantum dots

Abstract

We study N interacting massless Dirac fermions confined in a two-dimensional quantum dot. Physical realizations of this problem include a graphene monolayer and the surface state of a strong topological insulator. We consider both a magnetic confinement and an infinite mass confinement. The ground state energy is computed as a function of the effective interaction parameter α from the Hartree-Fock approximation and, alternatively, by employing the Müller exchange functional. For N = 2, we compare those approximations to exact diagonalization results. The Hartree-Fock energies are highly accurate for the most relevant interaction range α 2, but the Müller functional leads to an unphysical instability when α 0.756. Up to 20 particles were studied using HartreeFock calculations. Wigner molecule formation was observed for strong but realistic interactions, accompanied by a rich peak structure in the addition energy spectrum.

Electrostatic energy of polygonal charge distributions

“…Formation of polygonal patterns is observed in both experimental measurements and numerical studies of such diverse systems as electrons in quantum dots at the classical limit (the Wigner crystals) [2][3][4][5][6][7], ions in dusty plasmas [8][9][10], triboelectrically charged macroscopic objects [11], and vertices in mesoscopic superconducting disks [12,13]. These patterns persist upon inclusion of small-magnitude quantum and finite-temperature effects [2,6,15].…”

“…However, 2D assemblies of equicharged particles interacting with external potentials of cylindrical symmetry almost invariably involve either patterns of polygons inscribed on concentric rings or fragments of triangular lattices, the latter being more prevalent in species composed of larger numbers of particles [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Formation of polygonal patterns is observed in both experimental measurements and numerical studies of such diverse systems as electrons in quantum dots at the classical limit (the Wigner crystals) [2][3][4][5][6][7], ions in dusty plasmas [8][9][10], triboelectrically charged macroscopic objects [11], and vertices in mesoscopic superconducting disks [12,13]. These patterns persist upon inclusion of small-magnitude quantum and finite-temperature effects [2,6,15].…”

Confined electronic states and their modulations in graphene nanorings

“…The Eq. (8) shows that the upper component of the wave function with l = 0 is nonzero at r = 0. It feels more compression from the inner Dirac gap.…”

“…The nanostructures such as nanoribbons, 2,11 quantum dots, 2,5,[8][9][10][11][12][13][14] and quantum rings (QRs) 2,5-7,11 based on graphene have attracted much attention. It has been illustrated that the confinement of Dirac fermions at a nanometer scale is not trivial due to Klein's paradox, 15,16 which seriously limits graphene's potential application for building electronic devices, because confining carriers are crucial for applications.…”

Electrostatic self-energies of discrete charge distributions on Jordan curves

“…The combination of Coulombic interparticle interactions and two-dimensional confining potentials of cylindrical symmetry usually produces assemblies of particles positioned on either vertices of polygons inscribed on concentric rings or nodes of triangular lattice [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Formation of such patterns, which is observed both in experimental settings and numerical simulations, occurs in systems ranging from electrons in quantum dots [2][3][4][5][6][7] to ions in dusty plasmas [8][9][10] and triboelectrically charged objects [11].…”