Multi-Terminal Spin Valve on Channels with Spin-Momentum Locking

It is experimentally established that charge current flowing in a channel with spin-momentum locking such as topological insulator surface states or Rashba interfaces induces a spin voltage, which can be electrically measured with a ferromagnetic contact along the current path. Using this fact in conjunction with Onsager reciprocity arguments, we make the surprising prediction that the anti-parallel resistance of a spin valve can be either larger or smaller than the parallel resistance depending on the direction of spin flow relative to the direction of spin-momentum locking. However, we argue that this remarkable signature of spin-momentum locking can only be observed in multi-terminal measurements. Two-terminal measurements in the linear response regime, will show a single anti-parallel resistance larger than the parallel resistance as commonly observed in channels without spin-orbit coupling. We support this result with detailed numerical calculations based on a semiclassical model that provides insight into the underlying physics.

which results into the following form: for n = 1 to K. Here, K is the number of lattice points, a is the distance between two lattice points, q is the electron charge, {V i } is the terminal voltage in up-down basis at i th contact, [g C ] i is the normalized contact conductance matrix of i th contact. Note that {ϕ} 0 and {ϕ} K+1 are boundary conditions at left and right ends of the channel respectively. The vectors and matrices are given by with g u = (a g u )/(q 2 /h), g d = (a g d )/(q 2 /h), u 1 = r s1 + r N M + t s N M , u 2 = r s2 + r + t s , t s = (a t s )/N , r = (a r)/N , r s1 = (a r s1 )/M , and r s2 = (a r s2 )/N . We use Eq. (I.2) to construct multi-contact based structures as shown in Fig. S1.
FIG. S1. Numerical setup using Eq. (I.2) for multi-contact based structures on channels with spin-momentum locking. The terminals with the subscript L and R are the left and right boundaries of the channel. i th contact (i = 1, 2, 3, and 4) will be a NM contact if gu = g d or a FM contact if gu = g d .

A. Contact Currents
We multiply both sides of Eq. (I.2) with [m] = diag M M N N and sum over all lattice points as representing the sum of all up and down spin polarized currents flowing in/out of the channel. The scattering matrix is given by  with u 1 = r s1 + r + t s , and u 2 = r s2 + r + t s .
The first term which is related to the scattering processes in the channel vanishes for charge current (i.e. 1 1 1 1 N n=1 ([S] {ϕ} n ) = 0) satisfying the charge current conservation law. This term is non-zero for spin current and captures the spin loss in the channel due to spin relaxation. The second term is the up and down spin polarized currents flowing in the channel from the i th external contact, which is the Eq. (12) of the main manuscript. In the discrete lattice space, the total current is the sum over all the lattice points where the contact is connected, given by Note that the contact can be ferromagnetic if g u = g d or non-magnetic if g u = g d . The third and fourth terms are the up/down spin polarized currents at the left and right ends of the channel respectively, which gives Eq. (13) in the main manuscript. Boundary conditions at left and right ends are We can convert the vectors and matrices from up-down to charge-spin basis by multiplying the following matrix: [τ ] = 1 1 1 −1 . (I.6)

B. Boundary Conditions
In all our simulations, we have applied the open circuit boundary condition at the charge terminal and short circuit boundary condition at the spin terminal for the contacts at the left and the right ends of the channel: where i c and v s are terminal charge current and spin voltage respectively. If i th terminal is used to measure an open circuit charge voltage, then the boundary condition at the terminal is given by: On the other hand, if the i th terminal is used for supply, then the boundary condition at the terminal is given by: where v c is the terminal charge voltage, I 0 is the external current source, and V 0 is the external voltage source.