We derive analytical solutions for the zero-energy states of degenerate shell obtained as a singular eigenvalue problem found in tight-binding ͑TB͒ Hamiltonian of triangular graphene quantum dots with zigzag edges. These analytical solutions are in agreement with previous TB and density-functional theory results for small graphene triangles and extend to arbitrary size. We also generalize these solutions to trapezoidal structure which allow us to study bowtie graphene devices.

We show that the ground state and magnetization of the macroscopically degenerate shell of electronic states in triangular gated graphene quantum dots depends on the filling fraction of the shell. The effect of degeneracy, finite size, and electron-electron interactions are treated nonperturbatively using a combination of density functional theory, tight-binding, Hartree-Fock and configuration interaction methods. We show that electronic correlations play a crucial role in determining the nature of the ground state as a function of filling fraction of the degenerate shell at the Fermi level. We find that the half-filled charge neutral shell leads to full spin polarization but this magnetic moment can be completely destroyed by adding a single electron.

We present a theory of electronic properties of gated triangular graphene quantum dots with zigzag edges as a function of size and carrier density. We focus on electronic correlations, spin and geometrical effects using a combination of atomistic tight-binding, Hartree-Fock and configuration interaction methods (TB+HF+CI) including long range Coulomb interactions. The single particle energy spectrum of triangular dots with zigzag edges exhibits a degenerate shell at the Fermi level with a degeneracy N edge proportional to the edge size. We determine the effect of the electronelectron interactions on the ground state, the total spin and the excitation spectrum as a function of a shell filling and the degeneracy of the shell using TB+HF+CI for N edge < 12 and approximate CI method for N edge ≥ 12. For a half-filled neutral shell we find spin polarized ground state for structures up to N = 500 atoms in agreement with previous ab initio and mean-field calculations, and in agreement with Lieb's theorem for a Hubbard model on a bipartite lattice. Adding a single electron leads to the complete spin depolarization for N edge ≤ 9. For larger structures, the spin depolarization is shown to occur at different filling factors. Away from half-fillings excess electrons(holes) are shown to form Wigner-like spin polarized triangular molecules corresponding to large gaps in the excitation spectrum. The validity of conclusions is assessed by a comparison of results obtained from different levels of approximations. While for the charge neutral system all methods give qualitatively similar results, away from the charge neutrality an inclusion of all Coulomb scattering terms is necessary to produce results presented here.

Properties of the 'electron gas'-in which conduction electrons interact by means of Coulomb forces but ionic potentials are neglected-change dramatically depending on the balance between kinetic energy and Coulomb repulsion. The limits are well understood 1 . For very weak interactions (high density), the system behaves as a Fermi liquid, with delocalized electrons. In contrast, in the strongly interacting limit (low density), the electrons localize and order into a Wigner crystal phase. The physics at intermediate densities, however, remains a subject of fundamental research 2-8 . Here, we study the intermediate-density electron gas confined to a circular disc, where the degree of confinement can be tuned to control the density. Using accurate quantum Monte Carlo techniques 9 , we show that the electron-electron correlation induced by an increase of the interaction first smoothly causes rings, and then angular modulation, without any signature of a sharp transition in this density range. This suggests that inhomogeneities in a confined system, which exist even without interactions, are significantly enhanced by correlations.Quantum dots 10 -a nanoscale island containing a puddle of electrons-provide a highly tunable and simple setting to study the effects of large Coulomb interaction. They introduce level quantization and quantum interference in a controlled way, and can, in principle, be made in the very-low-density regime, where correlation effects are strong 11 . In addition, there are natural parallels between quantum dots and other confined systems of interacting particles, such as cold atoms in traps.Therefore, we consider a model quantum dot consisting of electrons moving in a two-dimensional (2D) plane, with kinetic energy (−(1/2) i ∇ with n being the density of electrons. For our confined system in which n(r) varies, we define r s in the same way using the mean densityn ≡ n 2 (r)dr/N. We have studied this system up to N = 20 electrons. The spring constant ω makes the oscillator potential narrow (for large ω) or shallow (for small ω); it thereby tunes the average density of electrons between high and low values, thus controlling r s . For example, for N = 20, varying ω between 3 and 0.0075 changes r s from 0.4 to 17.7. The radius of the dot grows significantly as r s increases, in an approximately linear fashion (see Fig. 1).In the bulk 2D electron gas, numerical work suggests a transition from a Fermi-liquid state to a Wigner crystal around r c s ≈ 30-35 (refs 2-4,8). On the other hand, experiments on the 2D electron gas (which include, of course, disorder and residual effects of the ions) show more-complex behaviour, including evidence for a metal-insulator transition 5 . Circular quantum dots have been studied previously using a variety of methods, yielding a largely inconclusive scenario. Many studies 12-14 have used density functional theory or the HartreeFock method. These typically predict charge or spin-density-wave order even for modest r s (unless the symmetry is restored after the fact 14 ), ...

We study the development of electron-electron correlations in circular quantum dots as the density is decreased. We consider a wide range of both electron number, N ≤ 20, and electron gas parameter, rs < ∼ 18, using the diffusion quantum Monte Carlo technique. Features associated with correlation appear to develop very differently in quantum dots than in bulk. The main reason is that translational symmetry is necessarily broken in a dot, leading to density modulation and inhomogeneity. Electron-electron interactions act to enhance this modulation ultimately leading to localization. This process appears to be completely smooth and occurs over a wide range of density. Thus there is a broad regime of "incipient" Wigner crystallization in these quantum dots. Our specific conclusions are: (i) The density develops sharp rings while the pair density shows both radial and angular inhomogeneity. (ii) The spin of the ground state is consistent with Hund's (first) rule throughout our entire range of rs for all 4 ≤ N ≤ 20. (iii) The addition energy curve first becomes smoother as interactions strengthen -the mesoscopic fluctuations are damped by correlation -and then starts to show features characteristic of the classical addition energy. (iv) Localization effects are stronger for a smaller number of electrons. (v) Finally, the gap to certain spin excitations becomes small at the strong interaction (large rs) side of our regime.