Spin Circuit Model for 2D Channels with Spin-Orbit Coupling

In this paper we present a general theory for an arbitrary 2D channel with “spin momentum locking” due to spin-orbit coupling. It is based on a semiclassical model that classifies all the channel electronic states into four groups based on the sign of the z-component of the spin (up (U), down (D)) and the sign of the x-component of the velocity (+, −). This could be viewed as an extension of the standard spin diffusion model which uses two separate electrochemical potentials for U and D states. Our model uses four: U+, D+, U−, and D−. We use this formulation to develop an equivalent spin circuit that is also benchmarked against a full non-equilibrium Green’s function (NEGF) model. The circuit representation can be used to interpret experiments and estimate important quantities of interest like the charge to spin conversion ratio or the maximum spin current that can be extracted. The model should be applicable to topological insulator surface states with parallel channels as well as to other layered structures with interfacial spin-orbit coupling.

For a given Hamiltonian one can estimate p 0 based on the picture in Fig. 1(b). For example, in the case of TISS (N = 0) it is given by based on a following type of Hamiltonian Heren is an outward normal vector from the surface, ⃗ σ is a vector of the Pauli spin matrices, v 0 is the Fermi velocity. For the channel with Rashba SOC (E ≥ 0), it is given by with the following Hamiltonian The polarization direction of spin in TISS and Rashba is determined byŝ = sign( v 0 )Î ×n withn being an outward normal vector andÎ being a charge current direction so that we have ±ẑ = ±x ×ŷ polarized spin in the structure of Fig. S1.

II. CURRENTS AND VOLTAGES
This section outlines how Eq. (33) in the main manuscript was derived for the 2D channel with spin-orbit coupling (SOC).

A. Charge Current
The semiclassical expression for charge current in the channel is given by . Band diagram 2D channel with Rashba SOC is shown where two Fermi circles having opposite spin polarizations exist with radius k1 and k2 respectively at a given energy EF . Here small arrows represent spin directions and right (left) half circles with solid (dotted) lines have positive (negative) group velocities, vg > 0 (vg < 0). The right and outer half circle corresponds to modes with positive group velocities and up spin (U + with M modes) and the right and inner half circle corresponds to modes with positive group velocities and down spin (D+ with N modes).
where µ eq is the equilibrium electrochemical potential. Under this approximation Eq. (II.1) becomes which gives Eq. (12) and first line of Eq. (33) in the main manuscript, where we defined

B. Spin Voltage
The spin voltage in the channel is given by Subtracting each electrochemical potential by µ eq according to Eq. (II.3) gives second line of Eq. (33) in the main manuscript, noting that v z ≡ẑ · ⃗ v s .

C. Spin Current
The definition of spin current in the channel based on our four electrochemical potentials (see Fig. 1(b) in main manuscript) is straightforward and includes both equilibrium and non-equilibrium conditions.

D. Charge Voltage
The charge voltage in the channel is given by The fourth line of Eq.(33) in the main manuscript is given in terms ofμ as Note that the equilibrium part in charge voltage gives a constant shift which cancels out in our model as we take the difference of charge voltages across the channel.