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Dirac cones for a mean-field model of graphene

Jean Cazalis

Abstract

In this article, we show that, in the dissociation regime and under a non-degeneracy assumption, the reduced Hartree-Fock theory of graphene presents Dirac points at the vertices of the first Brillouin zone and that the Fermi level is exactly at the coincidence point of the cones. For this purpose, we first consider a general Schrödinger operator $H=-Δ+V_L$ acting on $L^2(\mathbb{R}^2)$ with a potential $V_L$ which is assumed to be periodic with respect to some lattice with length scale $L$. Under some assumptions which covers periodic reduced Hartree-Fock theory, we show that, in the limit $L\to\infty$, the low-lying spectral bands of $H_L$ are given to leading order by the tight-binding model. For the hexagonal lattice of graphene, the latter presents singularities at the vertices of the Brillouin zone. In addition, the shape of the Bloch bands is so that the Fermi level is exactly on the cones.

Dirac cones for a mean-field model of graphene

Abstract

In this article, we show that, in the dissociation regime and under a non-degeneracy assumption, the reduced Hartree-Fock theory of graphene presents Dirac points at the vertices of the first Brillouin zone and that the Fermi level is exactly at the coincidence point of the cones. For this purpose, we first consider a general Schrödinger operator acting on with a potential which is assumed to be periodic with respect to some lattice with length scale . Under some assumptions which covers periodic reduced Hartree-Fock theory, we show that, in the limit , the low-lying spectral bands of are given to leading order by the tight-binding model. For the hexagonal lattice of graphene, the latter presents singularities at the vertices of the Brillouin zone. In addition, the shape of the Bloch bands is so that the Fermi level is exactly on the cones.
Paper Structure (47 sections, 35 theorems, 298 equations, 4 figures)

This paper contains 47 sections, 35 theorems, 298 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Lattices
  4. A class of periodic operators on LL
  5. Periodic operator
  6. Reference operator
  7. Effective mono-atomic operators
  8. Convergence to the tight-binding model
  9. The honeycomb lattice case
  10. Application: the periodic reduced Hartree-Fock model
  11. Proof of Theorem \ref{['theo: feshbach-schur']}
  12. Strategy of proof
  13. Notation
  14. A regularity result
  15. Kato's inequalities
  16. ...and 32 more sections

Key Result

Theorem 3

Let $\delta\in\mathopen{(}0 \mathclose{}\mathpunct{},1/2 \mathclose{)}$ be the parameter introduced in the definition eq: definition_chi of the cut-off function $\chi$. Under Assumptions hypo_0--hypo_4, for all $\epsilon\in\mathopen{(}0 \mathclose{}\mathpunct{},\delta \mathclose{)}$ and for all for $L$ large enough and where the $O$ is independent from $\mathbf{k}$. Here $d_1>d_0$ denotes the

Figures (4)

  • Figure 1: Examples of lattices $\mathscr{L}^\mathbf{R}$ with different parameters.
  • Figure 2: The Wallace model. The dispersion relation is invariant with respect to the rotation by $\pi/3$ about the origin. We observe conical singularities, Dirac points, at the vertices of the Brillouin zone $\Gamma^*$.
  • Figure 3: The honeycomb lattice $\mathscr{L}^H$.
  • Figure 4: Numerical estimations by DFTK software of the fifth first Bloch eigenvalues of $H_L^\mathrm{MF}$ along the section $(\Gamma, \mathrm{K},\mathrm{M},\Gamma)$ joining the special symmetry points, see Figure \ref{['fig:triangular_lattice_BZ']}. The Fermi level for $q=2$ is shown in black. For $L=0$, Figure \ref{['fig:L=0']} shows the dispersion relation of $-\Delta$ on the lattice of fixed size $2\cdot\mathscr{L}^H$.

Theorems & Definitions (68)

  • Remark 1
  • Remark 2
  • Theorem 3: Convergence to the tight-binding model
  • Remark 4
  • Theorem 5: Dirac points
  • Theorem 6
  • Remark 7: Assumptions on $V^\mathrm{pp}$
  • Corollary 8: Dirac points in rHF for the honeycomb lattice
  • Proposition 9
  • Lemma 10
  • ...and 58 more