在石墨烯上由閘極觸發之一維拓樸能態之原由

Abstract

本篇探討由閘極在石墨烯中觸發之ㄧ維拓樸能態之原由。我們考慮在石墨烯中加入閘極偏壓，可觸發ㄧ種ㄧ維拓樸能態。對比於Tamm態在半導體表面中，此ㄧ維拓樸能態沒有最小偏壓的觸發限制其物理來源與之不同。我們可以由以下兩個方向來證明此一維拓樸能態之原由: 1.我們發現此ㄧ維拓樸能態的形成必須滿足由閘極電位引發的贋自旋轉動，當電位開啟時才會形成一轉軸與贋自旋所在之平面垂直的轉動矩陣並在所有波向量中滿足且通過Dirac點。2.在我們的系統中實空間贋自旋方向角在周期內隨波向量的演進中在跨越Dirac點時會出現不連續之奇異點。我們知道奇異點的出現是拓樸能態的特點。基於以上證明，我們認為由閘極在石墨烯中觸發之ㄧ維能態是拓樸能態。

In this work, we consider induced localized states that are found out of a long metal gate that orients along an armchair atomic chain in a 2D graphene sheet. Specifically, we explore the topological nature of this gate-induced 1D energy branch. In contrast to the Tamm states in semiconductor surfaces, the gate-induced 1D branch does not require a threshold value for the gate-bias voltage V0. Two perspectives are taken to demonstrate the topological nature. First, we show that the boundary condition for the formation of the 1D branch can be cast into a rotation operator D that aims to rotate the'K -valley pseudospinχp-1 the to the K -valley pseudospin χp1 , for a given longitudinal wave vector ky , and at the right energy. We have shown, for a given ky and V0, that the angle between the pseudospins χp1 and χp-1 varies over 2π as the energy scans across the energy gap. Equally importantly, we show that as long as V0 is nonzero, the rotation axis of D is perpendicular to the plane formed by χp1 and χp−1 . This show that the 1D branch boundary condition must be satisfied for any ky and finite V0 values. As such, the branch will touch upon the Dirac point, where ky=0. Second, we look at the real space pseudospin orientation as ky varies over its Brillouin zone. Discontinuous in the pseudospin orientation angle φχ across the Dirac point is obtained. This is a signature of the topological nature of the 1D branch, similar but not exactly the same as that discussed in Hatsugai’s (PRL 2002) [1] and Montambaux’s (PRB 2011) [2]. Base on this two perspectives we have demonstrate the topological nature of the gate-induced 1D branch.