Magnetoresistance in Ferromagnetically Coupled Three-Dimensional Topological Insulator Strips

In this work, we calculate the eigenstates and dispersion relations for the 2-D surface states of 3-D topological insulator (TI) planar nanostrips of finite width coupled to a proximate insulating ferromagnetic layer. The magnetization of the ferromagnetic layer in the transverse, longitudinal and out-of-plane directions have differing effects on the dispersion relations. We also calculated the transmission through a short central TI strip sandwiched between TI source and drain leads of semi-infinite length. We investigate the transport through the setup when the magnetization of the ferromagnetic layer on top of the central region differs from that of the leads. We found that when the lead magnetization is in the longitudinal direction, i.e., along the current direction, the transmission is maximal for both the parallel and anti-parallel alignment of the lead and central magnetizations. However, for the case where the lead magnetization is in the transverse direction, the transmission is maximal for parallel alignment and minimal for the antiparallel alignment.