Spin and orbital magnetic response on the surface of a topological insulator

Coupling of the spin and orbital degrees of freedom on the surface of a strong three-dimensional insulator, on the one hand, and textured magnetic configuration in an adjacent ferromagnetic film, on the other, is studied using a combination of transport and thermodynamic considerations. Expressing exchange coupling between the localized magnetic moments and Dirac electrons in terms of the electrons' out-of-plane orbital and spin magnetizations, we relate the thermodynamic properties of a general ferromagnetic spin texture to the physics in the zeroth Landau level. Persistent currents carried by Dirac electrons endow the magnetic texture with a Dzyaloshinski-Moriya interaction, which exhibits a universal scaling form as a function of electron temperature, chemical potential, and the time-reversal symmetry breaking gap. In addition, the orbital motion of electrons establishes a direct magnetoelectric coupling between the unscreened electric field and local magnetic order, which furnishes complex long-ranged interactions within the magnetic film.

Figure 1

Schematic of the low-energy description, which captures contributions to the spin, $M_s$, and orbital, $M_o$, components of the magnetization that stem from the Dirac electrons.

Figure 2

Orbital magnetization, Eq. (9), is shown by black curves: Solid, dashed, and dotted corresponding to temperatures $k_{B}T/{\Delta }_0=0.01$, 1, and 10, respectively, setting $\zeta =0$. The grey curves (with the dotted one essentially overlapping with the black curve) show derivatives ${\chi }_{\Delta }$, Eq. (17). The shaded areas are the derivatives ${\chi }_{\mu }$, Eq. (18). Note that ${\chi }_{\mu }$ vanishes in the extreme limits of both $T\to 0$ and $T\to \infty$ for $|\mu /{\Delta }_0|>1$ [3].