Spin-charge locking and tunneling into a helical metal

We derive a kinetic equation for the electrons moving on the surface of a threedimensional topological insulator. Due to the helical nature of the excitations backward scattering is suppressed in the collision integral, and the spin dynamics is entirely constrained by that of the charge. We further analyze the tunneling between the helical metal and a conventional metal or ferromagnet. We find that the tunnel resistance strongly depends on the angle between the magnetization in the ferromagnet and the current in the helical metal. A nonmagnetic layer on top of the helical metal amplifies the current-induced spin polarization. editor’s choice Copyright c © EPLA, 2011 Topological insulators [1,2] have recently attracted considerable interest, especially after their experimental discovery in two [3] and three dimensions [4–9]. While insulating in the bulk, such materials possess gapless helical edge states whose existence depends on —and is protected by— time-reversal invariance [10–16]. This makes the latter robust against time-reversal symmetric perturbations (such as impurity scattering) and at the same time very sensitive to time-reversal breaking ones (such as magnetic fields). When the topological insulator is a three-dimensional system, the gapless excitations are confined to its surface and form a two-dimensional conductor which presents novel and interesting properties, see for example ref. [17] for a recent summary. In particular, Burkov and Hawthorn [18] considered the problem of spin-charge coupled transport on a helical metal and derived diffusion equations for charge and spin. They predicted a distinctive magnetoresistance effect when the helical metal is placed between a ferromagnet and a normal metal. In this paper we extend their work in several ways. We first derive a kinetic equation which is valid even beyond the diffusive regime. In this latter regime we obtain a diffusion equation which agrees with that of Burkov and Hawthorn as far as the charge component is concerned. On the other hand for the spin density we find a different behaviour, namely the spin dynamics is constrained to follow the charge one. (a)E-mail: [email protected] Secondly, we consider the effect of bringing the helical metal in contact with a ferromagnet and discuss its unconventional magnetoresistance. For the simplest case the effective Hamiltonian describing the surface states of a topological insulator has the form [2,19] H = vFk× ez ·σ, (1) where the parameter vF is the velocity of the gapless excitations, ez is a unit vector perpendicular to the surface, k is the two-dimensional momentum operator, σ are the Pauli matrices, and units of measure such that = 1 have been used. The eigenstates ofH form two bands with linear dispersion, ± =±vF k, and we will assume in the following that the Fermi energy is located deep enough in the upper band for states in the lower band to remain fully occupied and thus not relevant for the dynamics of the system. Because of the helical nature of the excitations, such a surface conductor is called a helical metal and presents novel and interesting properties. For instance the velocity operator is given by ẋ= vFez ×σ, so that the particle current becomes j= 2vFez × s, s being the spin polarization. This means that the particle current is entirely constrained by the spin density or that, vice versa, the in-plane components of the spin density are constrained by the particle current. Such a constraint will also become apparent later in the kinetic equation for a disordered helical metal. The