Modeling all-electrical detection of the inverse Edelstein effect by spin-polarized tunneling in a topological-insulator/ferromagnetic-metal heterostructure

Abstract

The spin-momentum locking of the surface states in a three dimensional topological insulator (TI) allows a charge current on the surface of the TI induced by an applied spin current onto the surface, which is known as the inverse Edelstein effect (IEE), that could be achieved either by injecting pure spin current by spin-pumping from a ferromagnetic metal (FM) layer or by injecting spin-polarized charge current by direct tunneling of electrons from the FM to the TI. Here, we present a theory of the observed IEE effect in a TI-FM heterostructure for the spin-polarized tunneling experiments. If an electrical current is passed from the FM to the surface of the TI, because of density of states polarization of the FM, an effective imbalance of spin-polarized electrons occurs on the surface of the TI. Due to the spin-momentum helical locking of the surface states in the TI, a difference of transverse charge accumulation appears on the TI surface in a direction orthogonal to the direction of the magnetization of the FM, which is measured as a voltage difference. Here, we derive the two-dimensional transport equations of electrons on the surface of a diffusive TI, coupled to a FM, starting from quantum kinetic equation, and analytically solve the equations for a rectangular geometry to calculate the voltage difference.

Theory of spin detection on the surface of diffusive topological insulators by means of ferromagnets: Establishing Onsager reciprocity and the importance of tunnel contact

“…These two equations, Eqs. (30) and 31, are consistent with those we obtained in 2D transport in our previous work 34 and what was obtained from a semi-classical drift-diffusion model on the TI surface with tunneling calculated from the Golden Rule of scattering between the FM majority/minority electrons and the TI surface states 16 . Following the literature 16,24 , we define the charge electrochemical potential µ c in the FM as µ c ≡ (µ ↑ + µ ↓ )/2 and the spin electrochemical potential µ s in the FM as µ s ≡ (µ ↑ −µ ↓ ).…”

“…The 180 0 rotation symmetry is also present in Eqs. (34) and (35), which remain unchanged after letting…”

“…The derivation of Eq. (1) is given in detail in the appendix following our previous work 34 . However, in our previous work, to obtain the continuity equation of the charge density and the charge current density on the TI surface, we solved the quasi-classical Green's function g of the TI surface states assuming projection of g on the conduction band of the Hamiltonian, i.e., g = g 0 (p, ε)[σ 0 + (p ×ẑ) • σ σ σ], and expanding the angular dependence of g in the zeroth and the first harmonics, i.e.…”

“…, where we obtain the momentum relaxation time τ tr to be τ tr = 2τ p because of the spin-momentum locking of the TI surface states 24,34,36 (note that τ s = 2τ p = τ tr ). We refer to τ tr as "the transport relaxation time" consistent with Schwab et al 24 .…”

“…The electrochemical potential µ + and µ − on the TI surface with respect to the electrochemical potential µ 0 c on the FM will be determined in terms of the current density on the TI surface after solving Eqs. (34) and (35) with relevant boundary conditions on j ±…”