Topological Edge states

Introduction: Topological Insulator has drawn a lot of attention recently in condensed matter physics. It describes the phase of matter in a different way, and gives us a new perspective toward materials. What is Topological Insulator(TI)? It is a material with bulk bandgap. However, at the surface of TI, it has edge states that propagate like a metal. We can imagine this like plastic tube wrapped with a aluminum foil around the tube. And it is proposed that it might be a potential candidate as fault tolerant quantum computation because the edge states are protected by time reversal symmetry. Protected state means that it is robust against impurity or imperfections in the crystal. The first 3D TI was observed in semiconducting alloy \(Bi_{1-x}Sb_x\) with angle resolved photoemission spectroscopy(ARPES) (Hsieh 2008). The reason to choose Bisumth is due to its strong spin orbit interaction which is essential to see TI edge states. But the \(Bi_{1-x}Sb_x\) surface states are complicated, so it invokes other materials such as \(Bi_2Se_3\) (Xia 2009).

Integer Quantum Hall Effect: To discuss TI surface edge states, we begin with the very similar cousin of TI edge states, which is quantum hall effect(QHE), or more accurately integer quantum hall effect. First observation of quantum hall effect was conducted in 1980 in MOSFET system in the low temperature and high magnetic field environment(Klitzing 1980). The astonishing experiment result is that the conductivity(or resistivity) of the system has plateau with increasing magnetic field as following.

Quantum Hall Effect dependence on magnetic field.(Avron 2003)

The integer QHE has a deep connection with the topology and berry phase. We can calculate the current in the following equation. Consider 2D time dependent system with translation symmetry in x and y directions. \[\textit{I}_x\sim\langle{u_k}|\frac{\partial{H(k(t))}}{\partial{k_x}}|u_k\rangle =\partial_x\textit{E}_k-\langle{\partial_x{u_k}}|\textit{H(k)}|u_k\rangle-\langle{u_k}|\textit{H(k)}|\partial_x{u_k}\rangle\] where \(\partial_x=\frac{\partial}{\partial{k_x}}\) and \(|u_k\rangle\) is the Bloch state. And with Schrodinger’s equation \[\textit{H(k)}|u_k\rangle=i \frac{\partial}{\partial{t}}|u_k\rangle=i (\dot{k_x}\partial_x+\dot{k_y}\partial_y)|u_k\rangle\] \[\textit{I}_x\sim\partial_x\textit{E}_{k}-b_{z}(k)\dot{k_y}\] \[b_z(k)=i(\langle\partial_x{u_k}|\partial_y{u_k}\rangle-\langle\partial_y{u_k}|\partial_x{u_k}\rangle)\] The \(b_z(k)\) is so called Berry curvature. When we apply the electric field in the y-axis, we have \(\dot(k_y)\sim-\frac{eE_y}{\hbar}\). Therefore, the hall conductivity can be approximated as \(\sigma_{xy}\sim{b_z(k)}\).



This “\(v\)” number is called TKNN number or Chern number. Now we know that the conductivity is proportional to Chern number. And it can be proven that each occupied Landaul level contributes to one Chern number. The energy eigenvalue of Landaul level is of the form \[\epsilon_n=(n+\frac{1}{2})\hbar\omega_c\] where \(\omega_c=\frac{eB}{m}\). We are ready to explain why increasing B would increase the conductivity(or reduce the resistivity). When we increase the B field, the separation between each Landaul level starts to increase. And the system’s Fermi energy does not change. Therefore, as the B increases, the total number of occupied Landual levels decreases. Then the conductivity decreases( or resistivity increases as Fig.1). The complete derivation is in (Takahashi 2015).