User:YohanesPramudya

Introduction

The study of electron transport is at the very heart of condensed matter physics. Band theory explains the physical properties of numerous materials, such as simple insulators, metals, and semiconductors. In metal, the electrons can move freely even at zero temperature because the conduction band is partially filled. On the other hand, insulators and semiconductor have fully filled valence band separated by an energy gap from an empty conduction band.

According to band theory, electrons in a perfectly periodic array of ions experience no collisions at all. However, some transition metal oxides with partially filled $d$ or $f$ bands (predicted to be a metal) are poor conductors or even insulators. The reason for the absence of carrier mobility is electron localization. The first fundamental mechanism of electron localization is the random scattering of mobile electrons caused by impurities or defect (disorder), which is called Anderson localization. The transition from metal into insulator occurs when the mean free path becomes smaller than the De Broglie wavelength. Based on scaling theory of localization by Abraham, Anderson, et.al. [3], the metal-insulator transition (MIT) exists for non-interacting electrons in three dimensions (3D). In addition, the system is an insulator if the Fermi energy is smaller than the characteristic amplitude of the disorder potential. According to this theory, there is no true metallic behavior in two dimensions (2D) and one dimensions (1D) system with non interacting electron.

The second fundamental mechanism in MIT is caused by electron-electron interaction. One of the simplest examples for interaction driven localization is Wigner crystallization. In this system, each electron is confined not by an ionic potential, but due to the formation of a deep potential well produced by repulsion from other electrons embedded in a positive charge background.

Experimental Background

The picture of Wigner crystallization can be realized by ultra low density 2D electron or hole liquid in semiconductor devices. At the lowest density and the absence of disorder, one expects the electrons to freeze into a Wigner crystal, driving the system into an insulator. A Wigner crystal (WC) is believed to be extremely fragile to both quantum and thermal fluctuations [5]. To minimize the potential energy, the electrons form a triangular lattice ($E_{B}=-3.921\epsilon _{0}$) in 2D, and BCC lattice in 3D, where $\epsilon _{0}=e^{2}/a$, and $a$ is the nearest distance between electrons. It does not form a square lattice ($E_{B}=-3.900\epsilon _{0}$) at zero temperature because of unstable shear modes in 2D systems. However, one cannot rule out the possibility that in some range of temperatures the square lattice (or other) is favored over the triangular lattice by entropy considerations [6]. As we increase the density, the kinetic energy becomes sufficiently large, leading to the formation of a metal.

Over the last few years, Tsui and collaborators have searched for more direct evidence for Wigner crystallization, by looking at the cleanest samples at the lowest accessible densities $n\sim 10^{11}cm^{-2}$ in zero magnetic field ($\mu \sim 10^{6}cm^{2}/Vs$ in GaAs, $\mu \sim 10^{4}cm^{2}/Vs$ in Si).

The Jellium Model

Describe the hamiltonian and the high density limit, limit small and big $r_{s}$ number

The Wigner Crystal

The Classical Monte Carlo Approach

• The simplest possible model of classical electron liquid.

• Explanation of Classical Monte Carlo/Metropolis algorithm

Ewald Summation

• Ewald summation to calculate potential from finite size system with periodic boundary condition

Simulation in half filled hypercubic Wigner crystal

Half filled electron system and similarity with anti-ferromagnetic Ising model

Finite size effect and scaling

Finding Tc from order parameter curves

Classical Electron Liquid visualization

Algorithm and visualization of Electron liquid with disorder and electric field (linked to the project with high school students)

Phase diaram of electron liquid

How the phase change with temperature and $r_{s}$.

The Phase Diagram of electron liquid

• References