Electric conductivity of the line-centered honeycomb lattice

Highlights

• This work is a careful theoretical analysis on the optical conductivity of a line-centered honeycomb lattice.

• We demonstrate that the flat band can lead to several unconventional behaviors of the system conductivity when it is broken by the disorder.

• We find that the low-energy conductivity might increase with the disorder strength and decrease with the chemical potential energy, these two features are very contrary to the physics intuition.

Abstract

The line-centered honeycomb (LCH) lattice has a single-valley Dirac band intersected by a dispersionless flat band at the band center. We investigate in this work the low-energy conductivity of the LCH lattice by considering the short-range impurities and focus on the role of the flat band when it is broken by disorders. Based on the self-consistent Born approximation and real space Kernel Polynomial methods, we showed that the system has a zero-energy conductivity like Graphene but it varies with the disorder strength. The broken flat band due to the disorder effect will contribute to the electron transport and this delocalization effect can lead to the exotic behaviors of conductivity in the low-energy regime as it slightly increases with the disorder strength and decreases with the chemical potential energy of the system.

Introduction

Since the superconductivity was discovered in the twisted bilayer Graphene [1], [2] where the nearly flat band was believed to play a key role, the study interest in the topological flat band systems have been ignited and a huge amount of investigations have been implemented in the last several years [3], [4], [5], [6], [7], [8], [9], [10], [11]. Similar to the linear energy-dispersion Dirac band, the flat band is also quite exotic because the electrons have a zero group velocity. The divergent density of states of the flat band together with the Coulomb interaction may give rise to some peculiar quantum phases like ferromagnetism [12], [13], [14], [15], high-temperature superconductivity [16], [17], zero-field fractional quantum Hall effect [18], [19], Bose–Einstein condensation [20], Wigner crystallization [21], and etc.

The Lieb, Kagome, and Dice lattices [22], [23], [24], [25], [26] were among the early proposed flat band systems and have been intensively studied. In a generic flat band lattice model, the Dirac band often exists due to the fact that the flat band is originated in a system with a chiral sublattice symmetry and an imbalance among the number of sublattice sites. The chiral symmetry would lead to the linear Dirac bands while the extra imbalance atoms can result in the formation of the flat band, which intersects generally the Dirac band at the Dirac point. The Dice lattice [26] possesses a honeycomb lattice similar to the Graphene lattice but an extra hub atom sits in the center of each original hexagon, so a flat band is born to cross the two Dirac bands at $K$ and $K^{\prime }$ points of the Brillouin Zone. Similarly, the line-centered rectangle and line-centered Graphene lattices are also the typical examples possessing a flat band [27].

Recently, Lee et al. [28] found that the material $1$T-TaS${}_2$ with the charge density wave phase [29] can be mapped into a honeycomb lattice with some extra atoms on each edge (bond) of the hexagon, and display multiple flat bands intersecting Dirac-cone bands. The number of the flat band number ($m$) is the extra atom number on each edge (bond) of the hexagon, e.g., the $m=1$ case represents the line-centered honeycomb lattice (LCH) as shown in Fig. 1(a). The LCH has also been employed to describe another flat-band material [30], the synthesized metal–organic framework, namely, bis(iminothiolato)nickel monolayer [31]. Different from the Graphene lattice, the LCH lattice has a peculiar band structure as it has a single-valley Dirac cone together with a flat band at the band center.

Besides those induced novel quantum phases, the flat band can also indirectly affect the transport properties of the intersecting Dirac band, although it may not contribute to the transport directly due to zero group velocity. For instance, the Dirac electrons in both the Lieb and Dice lattice models [32], [33], [34], [35] have been shown to display the so-called super Klein tunneling behavior because of existence of the flat band. The conductivity of the flat band systems have also been studied intensively but the role of the flat band is not so much unanimously agreed upon. Several works [36], [37], [38], [39] were dedicated to discussion whether the zero-energy conductivity (ZEC) in the Dice lattice exists or not. For example, Vigh et al. [37] argued that in the Dice lattice, the interband transition between the flat and adjacent propagating (Dirac) bands could lead to a divergent ZEC that increases inversely with the disorder strength. While Louvet et al. [36] studied the same Dice model by calculating the transmission and concluded a zero ZEC. Two of authors of the present work studied the latter case [39] and suggested that the disorder effect could recover a finite ZEC in the same system.

In this work, we study the single-valley Dirac band conductivity of the LCH system and focus on the effect of the flat band when it is broken by impurities. Based on the self-consistent Born approximation (SCBA) and the real-space Kernel Polynomial method (KPM), we show that the system has a nonconstant ZEC varying with the disorder strength. The broken flat band due to disorders will contribute to the electron transport and lead to the exotic behaviors of conductivity in the low-energy regime as it slightly increases with disorder and decreases with the chemical potential energy.

This work is organized as follows. In Section 2, we introduce the LCH lattice model and present the derivation of the low-energy conductivity based on the SCBA method. In Section 3, the numerical results of the conductivity are given based on the real-space KPM. A brief conclusion is drawn in the last section.