Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homogenization of 2D materials in the Thomas-Fermi-von Weizsacker theory

Saad Benjelloun, Salma Lahbabi, Abdelqoddous Moussa

TL;DR

The paper studies homogenization of the Thomas–Fermi–von Weizsäcker (TFW) model for 2D crystals by letting the 2D lattice spacing go to zero, showing that the three-dimensional problem converges to a one-dimensional limit. The limit is described by a 1D energy functional ${\mathcal{E}}_1^{\mu}(\rho)$ with a 1D Hartree interaction $D_1$, and the ground state solves a reduced Euler–Lagrange system with conditions on $u$ and the mean-field potential $\Phi$; the authors also prove convergence properties of the densities and energies. They establish that $\rho_N \to \rho_0$ in $L^1(\Gamma)$ and locally in $L^p(\Gamma)$ for $1 \le p \le 3$, and $\sqrt{\rho_N} \rightharpoonup \sqrt{\rho_0}$ in $H^1_{\rm per}(\Gamma)$, with $I_N \to I_0$, and that $\rho_0$ is invariant in the transverse directions. Numerical simulations corroborate the theoretical homogenization and indicate a rate for the energy decay consistent with the observed convergence, validating the zero-order 1D approximation for the 2D material.

Abstract

We study the homogenization of the Thomas-Fermi-von Weizsacker (TFW) model for 2D materials. It consists in considering 2D-periodic nuclear densities with periods going to zero. We study the behavior of the corresponding ground state electronic densities and ground state energies. The main result is that these three dimensional problems converge to a limit model that is one dimensional. We also illustrate this convergence with numerical simulations and estimate the converging rate for the ground state electronic densities and the ground state energies.

Homogenization of 2D materials in the Thomas-Fermi-von Weizsacker theory

TL;DR

The paper studies homogenization of the Thomas–Fermi–von Weizsäcker (TFW) model for 2D crystals by letting the 2D lattice spacing go to zero, showing that the three-dimensional problem converges to a one-dimensional limit. The limit is described by a 1D energy functional with a 1D Hartree interaction , and the ground state solves a reduced Euler–Lagrange system with conditions on and the mean-field potential ; the authors also prove convergence properties of the densities and energies. They establish that in and locally in for , and in , with , and that is invariant in the transverse directions. Numerical simulations corroborate the theoretical homogenization and indicate a rate for the energy decay consistent with the observed convergence, validating the zero-order 1D approximation for the 2D material.

Abstract

We study the homogenization of the Thomas-Fermi-von Weizsacker (TFW) model for 2D materials. It consists in considering 2D-periodic nuclear densities with periods going to zero. We study the behavior of the corresponding ground state electronic densities and ground state energies. The main result is that these three dimensional problems converge to a limit model that is one dimensional. We also illustrate this convergence with numerical simulations and estimate the converging rate for the ground state electronic densities and the ground state energies.
Paper Structure (7 sections, 13 theorems, 84 equations, 4 figures)

This paper contains 7 sections, 13 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Thomas--Fermi--von Weizsäcker model for finite systems
  3. Thomas--Fermi--von Weizsäcker model for 2D crystals
  4. Homogenization of 2D materials
  5. Hartree interaction
  6. Homogenization procedure: proof of Theorem \ref{['Th-H']}
  7. Numerical illustration

Key Result

Theorem 1.2

Let $m\neq 0$ be a smooth non-negative, ${\mathcal{R}}$-periodic function with compact support w.r.t $x_3$. Then, the minimization problem TFW-per has a unique minimizer $\rho$. Moreover, $u=\sqrt{\rho}$ is the unique solution of the corresponding Euler--Lagrange system where $\lambda \in{\mathbb R}$. In addition, $u\in L^{\infty}(\mathbb{R}^3)$ and $\left| u(x)\right|\leqslant \dfrac{C}{1+\left|

Figures (4)

  • Figure 1: Example of a 2D lattice and its unit cell $\Gamma$.
  • Figure 2: The homogenization process for the nuclear density $m_N$, illustrated at $N=1,4$ and $16$ from left to right.
  • Figure 3: Convergence analysis for $e_N$$e_n$ norms (left) and convergence rates estimation for $e_N$$e_n$ norms and the gradient $L^2$ norm $\left\| \nabla{u_n} -\nabla{u_{0}} \right\|_2$ (right).
  • Figure 4: Convergence analysis for the energies $I_N$ (left) and convergence rate estimation (right).

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 1.2: Ref. BB
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • ...and 17 more