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.

We present a microscopic theory of electronic and optical properties of colloidal graphene quantum dots (CGQDs). The single-particle properties are described in the tight-binding model based on the p z carbon orbitals. Electron-electron screened Coulomb direct, exchange, and scattering matrix elements are calculated using Slater p z orbitals. The many-body ground state and excited states are constructed as a linear combination of a finite number of excitations from the Hartree-Fock (HF) ground state (GS) by exact diagonalization techniques. HF ground states corresponding to semiconductor, Mott-insulator, and spin-polarized phases are obtained as a function of the strength of the screened interaction versus the tunneling matrix element. In the semiconducting phase of a triangular CGQD, the top of the valence band and the bottom of the conduction band are found to be degenerate due to rotational symmetry. The singlet and triplet exciton spectra from the HF GS are obtained by solving the Bethe-Salpeter equation. The low-energy exciton spectrum is predicted to consist of two bright-singlet exciton states corresponding to two circular polarizations of light and a lower-energy band of two dark singlets and 12 dark triplets. The robustness of the bright degenerate singlet pair against correlations in the many-body state is demonstrated as well as the breaking of the degeneracy by the lowering of symmetry of the CGQD. The band-gap renormalization, electron-hole attraction, fine structure, oscillator strength, and polarization of the exciton are analyzed as a function of the size, shape, screening, and symmetry of the CGQD. The theoretical results are compared with experimental absorption spectra.

Electronic and magnetic properties of triangular graphene rings potentially fabricated using carbon nanotubes as masks are described as a function of their size and width. The electronic properties of the charge neutral system are calculated using tight-binding method and interactions are treated using the mean-field Hubbard model. We show that for triangular quantum dots with a triangular hole, the magnetic properties are determined by the width of the ring, leading to ferromagnetic corners and antiferrimagnetic sides. The electronic properties of gated graphene quantum rings as a function of additional number of electrons or holes are described by a combination of tight-binding, Hartree-Fock, and configuration interaction methods. The outer edge is found to be maximally spin polarized for almost all filling factors while the evolution of the excitation gap as a function of shell filling shows oscillations as a result of electronic correlations.

We study interaction-induced localization of electrons in an inhomogeneous quasi-one-dimensional system-a wire with two regions, one at low density and the other high. Quantum Monte Carlo techniques are used to treat the strong Coulomb interactions in the low density region, where localization of electrons occurs. The nature of the transition from high to low density depends on the density gradient-if it is steep, a barrier develops between the two regions, causing Coulomb blockade effects. Ferromagnetic spin polarization does not appear for any parameters studied. The picture emerging here is in good agreement with measurements of tunneling between two wires.With the rapid development of nanotechnology over the last decade, experiments have been able to probe strong interaction phenomena in reduced dimensionality systems such as quantum dots, wires, and point contacts. Of particular interest are systems in which the electron density is inhomogeneous. In the low density region of such a system the interaction energy is comparable to the kinetic energy and novel effects occur such as the "0.7 structure" in quantum point contacts or Coulomb blockade effects accompanying localization in a one-dimensional (1D) wire. We perform quantum Monte Carlo calculations of an inhomogeneous, quasi-1D electron system in order to address such effects.The so-called "0.7 structure" [1] remains poorly understood: in a quasi-1D electron gas -a wire, constriction, or quantum point contact (QPC) -decreasing the density causes the conductance G to decrease in integer multiples of 2e 2 /h (one for each transverse mode), except for an extra plateau or shoulder at G ≈ 0.7(2e 2 /h) as the lowest mode is depopulated [2,3,4,5,6,7,8]. Proposed theoretical explanations have been mainly based on three approaches: formation of a bound state leading to a Kondo effect [9,10], spontaneous spin polarization of the low density electrons [2,5,6,7,11,12], and formation of a Wigner crystal [13,14]. An open question is whether the critical features underlying each of these approaches is present in an inhomogeneous quasi-1D system-whether a localized state with Kondo-like correlations, spin polarization, or a Wigner crystal occurs.The formation of a Wigner crystal was investigated directly using tunneling spectroscopy into a quantum wire [15,16,17,18]. Clear evidence of localized electrons was found, accompanied by unexpected single electron phenomena.The general problem of a transition from a liquid to a localized crystal-like phase remains a subject of fundamental research in a variety of bulk and nanoscale systems [19,20,21,22,23,24,25]. Thus, an approach from a quasi-1D point of view is valuable not only for understanding the 0.7 anomaly and the tunneling experiments, but also for bringing a new way of looking at the physics of interaction-induced liquid to crystal transitions.Previous electronic structure calculations investigating inhomogeneous 1D systems have been based on mean field approximations. While some density functional calculations in the ...