4:00 PM - 6:00 PM
△ [14p-P10-19] Graphene Nanoribbon and Quantum Spin Hall Phase
Keywords:Quantum Spin Hall Effect, Graphene
In 2004, Kane and Mele suggested that Quantum Spin Hall (QSH) effect could arise in graphene nanoribbons theoretically. Since this model includes the Rashba term, we cannot distinguish up-spin states from down-spin states. Therefore, we omit the Rashba term in order to distinguish these two states and make sure that helical spin currents emerge at the edge of graphene nanoribbons by estimating the wave functions.
First of all, we make the matrix representation corresponding to the Kane-Mele Hamiltonian without the Rashba term. This Hamiltonian includes the nearest-neighbor-hopping term, spin-orbit-interaction (SOI) term, and staggered-potential term which incorporates the difference of the electrical charges between two sublattices of graphene. The two kinds of Hamiltonian matrices are needed since graphene nanoribbons have two types of edges which are called zigzag-edge and armchair-edge, respectively according to the shapes of the edges.
Secondly, we calculate the eigen-energies and the eigenvalues of the Hamiltonian in zigzag-type of graphene nanoribbons. We obtain the band structure which shows spin splitting between up-spin states and down-spin states. This band structure shows QSH phase or normal insulating phase, depending on the strength of SOI and staggered potential. In this band structure, there are gapless edge states in the case of QSH phase, while the edge states disappear in the case of normal insulating phase. By considering the wave functions, it becomes obvious that the helical spin currents emerge at the both edges of nanoribbons and that the helical spin current at one edge flows in the opposite direction to the other edge.
Finally, we calculate the band structure and consider the QSH phase in armchair-type of graphene nanoribbons which has not been studied so much until now. As a result, we have found that the up-spin states do not change so much from down-spin states in the armchair-type of graphene nanoribbons, compared with the zigzag-type of graphene nanoribbons. Moreover, it is suggested that the diagram of QSH phase in the armchair-type of nanoribbons could be different from that in the zigzag-type of nanoribbons which was given by Kane and Mele. These results indicate that it is important to consider the difference between zigzag-type and armchair-type of graphene nanoribbons.
First of all, we make the matrix representation corresponding to the Kane-Mele Hamiltonian without the Rashba term. This Hamiltonian includes the nearest-neighbor-hopping term, spin-orbit-interaction (SOI) term, and staggered-potential term which incorporates the difference of the electrical charges between two sublattices of graphene. The two kinds of Hamiltonian matrices are needed since graphene nanoribbons have two types of edges which are called zigzag-edge and armchair-edge, respectively according to the shapes of the edges.
Secondly, we calculate the eigen-energies and the eigenvalues of the Hamiltonian in zigzag-type of graphene nanoribbons. We obtain the band structure which shows spin splitting between up-spin states and down-spin states. This band structure shows QSH phase or normal insulating phase, depending on the strength of SOI and staggered potential. In this band structure, there are gapless edge states in the case of QSH phase, while the edge states disappear in the case of normal insulating phase. By considering the wave functions, it becomes obvious that the helical spin currents emerge at the both edges of nanoribbons and that the helical spin current at one edge flows in the opposite direction to the other edge.
Finally, we calculate the band structure and consider the QSH phase in armchair-type of graphene nanoribbons which has not been studied so much until now. As a result, we have found that the up-spin states do not change so much from down-spin states in the armchair-type of graphene nanoribbons, compared with the zigzag-type of graphene nanoribbons. Moreover, it is suggested that the diagram of QSH phase in the armchair-type of nanoribbons could be different from that in the zigzag-type of nanoribbons which was given by Kane and Mele. These results indicate that it is important to consider the difference between zigzag-type and armchair-type of graphene nanoribbons.