Unified Framework for Charge-Spin Interconversion in Spin-Orbit Materials

Materials with spin-orbit coupling are of great interest for various spintronics applications due to the efficient electrical generation and detection of spin-polarized electrons. Over the past decade, many materials have been studied, including topological insulators, transition metals, Kondo insulators, semimetals, semiconductors, and oxides; however, there is no unifying physical framework for understanding the physics and therefore designing a material system and devices with the desired properties. We present a model that binds together the experimental data observed on the wide variety of materials in a unified manner. We show that in a material with a given spin-momentum locking, the density of states plays a crucial role in determining the charge-spin interconversion efficiency, and a simple inverse relationship can be obtained. Remarkably, experimental data obtained over the last decade on many different materials closely follow such an inverse relationship. We further deduce two figure of merits of great current interest: the spin-orbit-torque (SOT) efficiency (for the direct effect) and the inverse Rashba-Edelstein effect length (for the inverse effect), which statistically show good agreement with the existing experimental data on wide varieties of materials. Particularly, we identify a scaling law for the SOT efficiency with respect to the carrier concentration in the sample, which agrees with existing data. Such an agreement is intriguing since our transport model includes only Fermi surface contributions and fundamentally different from the conventional views of the SOT efficiency that includes contributions from all the occupied states.

Figure 1

(a) Three-terminal-based structure with a spin-orbit material. (b) Circuit representation for charge-to-spin conversion in a spin-orbit material.

Figure 2

(a) Setup to measure charge-current-induced spin voltage in a spin-orbit material using a ferromagnetic contact with magnetization $m$. (b) Charge-spin interconversion resistance, $\mathrm{\Delta }{R}_S$, in diverse classes of materials as a function of channel number of modes (M_T), which is related to the material density of states near the Fermi energy. Experimental data points include topological insulators [$({\mathrm{Bi}}_{0.5}{\mathrm{Sb}}_{0.5}{)}_2{\mathrm{Te}}_3$ [4], (${\mathrm{Bi}}_{0.53}{\mathrm{Sb}}_{0.47}{)}_2{\mathrm{Te}}_3$ [2], ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ [3, 4, 30], ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ [5], and ${\mathrm{Bi}\mathrm{Sb}\mathrm{Te}\mathrm{Se}}_2$ [7] ], transition metals (Ta [31], $\mathrm{Pt}$ [31, 32], W [31], and Ir [31]), metallic interface ($\mathrm{Cu}$$|$$\mathrm{Bi}$ [33]), narrow bandgap semiconductor ($\mathrm{In}\mathrm{As}$ [8, 9]), topological Kondo insulator (${\mathrm{Sm}\mathrm{B}}_6$ [34]), and semimetals (${\mathrm{WTe}}_2$ [19] and Cd₃As₂ [44]). Solid black line represents the theoretical trend.

Figure 3

(a) Structure for SOT-related experiments. (b) SOT efficiency ${\zeta }_{\mathrm{SOT}}$ in various metallic interfaces and comparison with Eq. (18).

Figure 4

(a) Resistivity and Hall carrier concentration of bare ${\mathrm{Sr}\mathrm{Ir}\mathrm{O}}_3$ as a function of thickness, taken from Ref. [22]. (b) Spin-orbit torque (SOT) efficiency ${\zeta }_{\mathrm{SOT}}$ in ${\mathrm{Sr}\mathrm{Ir}\mathrm{O}}_3$ calculated using Eq. (19) with carrier concentration in Fig. 4, and comparison with experiments in Ref. [23].

Figure 5

(a) Structure for experiments on inverse effects. (b) Inverse Rashba-Edelstein effect length in diverse classes of materials including $\mathrm{Ag}|\mathrm{Bi}$ [78], $\mathrm{Cu}|\mathrm{Bi}$ [33], $\mathrm{Ag}|{\mathrm{Bi}}_2{\mathrm{O}}_3$ [84], $\mathrm{Cu}|{\mathrm{Bi}}_2{\mathrm{O}}_3$ [84], $\mathrm{Au}|{\mathrm{Bi}}_2{\mathrm{O}}_3$ [84], $\mathrm{Al}|{\mathrm{Bi}}_2{\mathrm{O}}_3$ [84], ${\mathrm{La}\mathrm{Al}\mathrm{O}}_3|{\mathrm{Sr}\mathrm{Ti}\mathrm{O}}_3$ ($\mathrm{LAO}|\mathrm{STO}$) [20], ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ [87], Fe/Ge(111) [85], and ${\mathrm{Mo}\mathrm{S}}_2$ [86, 97]. The solid line represents Eq. (32).

Figure 6

Scattering mechanisms considered in the formalism among the four groups of electronic states, classified according to the spin-polarization index (up or down) and the sign of the group velocity (positive or negative).