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arXiv:1510.05913 · Physical review. B./Physical review. B

Wigner crystallization at graphene edges

Alev Devrim Güçlü

Published Jan 12, 2016• 4 citations
arXivHTMLPDF

Abstract

Using many-body configuration interaction techniques, we show that Wigner crystallization occurs at the zigzag edges of graphene at surprisingly high electronic densities up to $0.8\phantom{\rule{0.28em}{0ex}}{\text{nm}}^{\ensuremath{-}1}$. In contrast with one-dimensional electron gas, the flatband structure of the edge states makes the system interaction dominated, facilitating electronic localization. The resulting Wigner crystal manifests itself in pair-correlation functions, and evolves smoothly as the edge electron density is lowered. We also show that the crystallization affects the magnetization of the edges. While the edges are fully polarized when the system is charge neutral (i.e., high density), above the critical density, the spin-spin correlations between neighboring electrons go through a smooth transition from antiferromagnetic to magnetic coupling as the electronic density is lowered.

DOI
10.1103/physrevb.93.045114

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Using many-body configuration interaction techniques, we show that Wigner crystallization occurs at the zigzag edges of graphene at surprisingly high electronic densities up to 0.8nm\ensuremath−10.8\phantom{\rule{0.28em}{0ex}}{\text{nm}}^{\ensuremath{-}1}0.8nm\ensuremath−1. In contrast with one-dimensional electron gas, the flatband structure of the edge states makes the system interaction dominated, facilitating electronic localization. The resulting Wigner crystal manifests itself in pair-correlation functions, and evolves smoothly as the edge electron density is lowered. We also show that the crystallization affects the magnetization of the edges. While the edges are fully polarized when the system is charge neutral (i.e., high density), above the critical density, the spin-spin correlations between neighboring electrons go through a smooth transition from antiferromagnetic to magnetic coupling as the electronic density is lowered.
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Abstract. The physics of interacting quantum wires has attracted a lot of attention recently. When the density of electrons in the wire is very low, the strong repulsion between electrons leads to the formation of a Wigner crystal. We review the rich spin and orbital properties of the Wigner crystal, both in the one-dimensional and quasi-one-dimensional regime. In the one-dimensional Wigner crystal the electron spins form an antiferromagnetic Heisenberg chain with exponentially small exchange coupling. In the presence of leads the resulting inhomogeneity of the electron density causes a violation of spin-charge separation. As a consequence the spin degrees of freedom affect the conductance of the wire. Upon increasing the electron density, the Wigner crystal starts deviating from the strictly one-dimensional geometry, forming a zigzag structure instead. Spin interactions in this regime are dominated by ring exchanges, and the phase diagram of the resulting zigzag spin chain has a number of unpolarized phases as well as regions of complete and partial spin polarization. Finally we address the orbital properties in the vicinity of the transition from a onedimensional to a quasi-one-dimensional state. Due to the locking between chains in the zigzag Wigner crystal, only one gapless mode exists. Manifestations of Wigner crystal physics at weak interactions are explored by studying the fate of the additional gapped low-energy mode as a function of interaction strength. PACS numbers: 71.10.Pm2 1.
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We consider interacting electrons in a quantum wire in the case of a shallow confining potential and low electron density. In a certain range of densities, the electrons form a two-row (zigzag) Wigner crystal whose spin properties are determined by nearest and next-nearest neighbor exchange as well as by three- and four-particle ring exchange processes. The phase diagram of the resulting zigzag spin chain has regions of complete spin polarization and partial spin polarization in addition to a number of unpolarized phases, including antiferromagnetism and dimer order as well as a novel phase generated by the four-particle ring exchange.
In this thesis we study two different one dimensional systems. The first project is on the transverse dynamical structure factors of the XXZ spin chain and the second project is on magnetism of zigzag edges of graphene nano-ribbons. In chapter 2, we apply field theory methods, first developed to study x-ray edge singularities, to interacting one-dimensional systems in order to include band curvature effects and study edge singularities at arbitrary momentum. We point out that spin chains with uniform Dzyaloshinskii-Moriya interactions provide an opportunity to test these theories since these interactions may be exactly eliminated by a gauge transformation that shifts the momentum. However, this requires an extension of these x-ray edge methods to the transverse spectral function of the XXZ spin chain in a magnetic field. In chapter 3, by considering the Hubbard model in the weak coupling limit, U << t, for bearded as well as zigzag edges, we argue for existence of magnetic edges. We first present an argument based on Lieb's theorem. Then, projecting the Hubbard interactions onto the flat edge band, we prove that the resulting one-dimensional model has a fully polarized ferromagnetic ground state. We also study excitons and the effects of second neighbor hopping as well as a potential energy term acting on the edge only, proposing a simple and possibly exact phase diagram with the magnetic moment varying smoothly to zero. Finally, we consider corrections of second order in U, arising from integrating out the gapless bulk Dirac excitations.