Tunable Magnetic Anisotropy from Higher-Harmonics Exchange Scattering on the Surface of a Topological Insulator

We show that higher-harmonics exchange scattering from a magnetic adatom on the surface of a three dimensional topological insulator leads to a magnetic anisotropy whose magnitude and sign may be tuned by adjusting the chemical potential of the helical surface band. As the chemical potential moves from the Dirac point towards the surface band edge, the surface normal is found to change from a magnetic easy to a hard axis. Hexagonal warping is shown to diminish the region with easy axis anisotropy, and to suppress the anisotropy altogether. This indirect contribution can be comparable in magnitude to the intrinsic term arising from crystal field splitting and atomic spin-orbit coupling, and its tunability with the chemical potential makes the two contributions experimentally discernible, and endows this source of anisotropy with potentially interesting magnetic functionality.

Figure 1

(a) The conical TSS band with an indication of the opposite helicity of positive- and negative-energy states, and an interband scattering process from $\mathit{k}$ to ${\mathit{k}}^{\prime }$. (b) Illustration of $s$-wave (orange, dotted) and $p$-wave (purple-green, striped) scattering amplitudes in Eq. (6) and their relation to the scattering angle for the interband scattering process shown in panel (a). (c) Anisotropy energy resulting from two constant amplitudes $2j^1=j^0=10\mathrm{eV}{\AA }^2$ and using estimated values for pristine ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ surface states with $v_F\approx 3000\mathrm{meV}\AA$, and half bandwidth $D\approx 400\mathrm{meV}$ (where the built-in chemical potential lies around 300 meV) [11, 12]. The thick black curve shows the perturbative result in Eq. (12). Curves in red to magenta (and smaller dashing) correspond to stronger warping, with $\lambda =\{0,1,2,5,10,20\}\times 10^5{\mathrm{meV}}^2{\AA }^3$, calculated by numerical $k$-space diagonalization.

Figure 2

(a) Plot of the maximum value $j_{kk}^l$ of the 6 lowest harmonics of the isotropic step-function impurity potential ${\mathcal{J}}_0(r)$, with range $a$. (b) Double-logarithmic plot of the corresponding longitudinal magnetic anisotropy energy at $\mu =0$ as a function of impurity range $a$ for a number of different amplitudes ${\mathcal{J}}_0$. Curves in red to magenta (and smaller dashing) correspond to larger values of ${\mathcal{J}}_0$. The range is measured in units of the smallest available wavelength of the TSS Dirac band, $k_D^{-1}=v_F/D=7.5\AA$. Thin gray lines show the power law $0.77(W^3/v_F^4)(a^2{\mathcal{J}}_0{)}^2(a{k}_D{)}^6$, matching the curves very well for $a{k}_D\lesssim 1$.

Figure 3

Plots of the longitudinal magnetic anisotropy energy for an isotropic step-function impurity potential ${\mathcal{J}}_0(r)$, with varying ranges $a$ and fixed amplitudes ${\mathcal{J}}_0=1\mathrm{eV}$ (upper panel) and ${\mathcal{J}}_0=10\mathrm{eV}$ (lower panel). Curves in red to magenta (and smaller dashing) correspond to larger values of $a$, as listed in the insets.