Tunable unconventional Kondo effect on topological insulator surfaces

We study Kondo physics of a $\text{spin}-\frac{1}{2}$ impurity in electronic matter with strong spin-orbit interaction, which can be realized by depositing magnetic adatoms on the surface of a three-dimensional topological insulator. We show that magnetic properties of topological surface states and the very existence of Kondo screening strongly depend on details of the bulk material, and specifics of surface preparation encoded in time-reversal preserving boundary conditions for electronic wavefunctions. When this tunable Kondo effect occurs, the impurity spin is screened by purely orbital motion of surface electrons. This mechanism gives rise to a transverse magnetic response of the surface metal, and to spin textures that can be used to experimentally probe signatures of a Kondo resonance. Our predictions are particularly relevant for STM measurements in $\mathrm{Pb}\mathrm{Te}$-class crystalline topological insulators, but we also discuss implications for other classes of topological materials.

Figure 1

(a) Geometry of the problem. The TI occupies half-space $z\ge 0$. The unit vector $\mathit{n}$ is an outer normal to the surface. The red arrow represents the impurity spin. (b) Schematic $z$ dependence of the surface-state wave function (6), characterized by two length scales: small $1/\lambda \sim 1/m^{*}v$ (shaded region) and large $1/q\sim v/\mathrm{\Delta }$. The Dirac model (7) is valid at distances $\ge 1/q$. The impurity is located under the surface where the Dirac theory is applicable. The inset shows the dispersion of surface modes. (c) Stability diagram of surface states (10). Thick lines correspond to critical momenta $p_{\perp }^{cr}=\pm (\mathrm{\Delta }/v)\mathrm{ctg}\vartheta$. In the white region, no surface states can exist. In the (light) dark gray area, there are surface modes with (only one, $\tau =-1$) both helicities. (d) Dispersion relation (10) for several values of $\vartheta$. The upper (lower) branches (relative to the point $p_{\perp }=0$) correspond to $\tau =\mp 1$. The Fermi energy is ${\varepsilon }_F=0$.

Figure 2

(a) Helical structure of the surface state (10). Blue arrows indicate the expectation value of the electron spin $\frac{1}{2}\mathbf{\Sigma }$ which points perpendicular to the momentum and has a magnitude $\sim sin\vartheta$. For $\vartheta =\pi$, surface states carry no spin. (b) Schematic illustration of the Kondo interaction $H_K$ in Eq. (12). At weak coupling, one can ignore bulk-surface mixing induced by the impurity and assume that $H_K\approx H_K^{ss}$. The impurity only couples to surface states (blue Dirac cone).

Figure 3

Spin distribution $\langle 1_{p_{\perp }\tau \mu }|\mathit{s}\left({\mathit{x}}_{\perp }\right)|1_{p_{\perp }\tau \mu }\rangle =\langle \mathit{s}\left(r\right)\rangle$, Eq. (15) along radial direction with $\tau =\overline{1}$ and $\vartheta =0.9\pi$. $\langle s^r\rangle \left[\langle s^z\rangle \right]$ is the radial ($z$) component. The blue line is the “total” spin $s^{\mathrm{tot}}=\sqrt{{\langle s^r\rangle }^2+{\langle s^z\rangle }^2}$. Blue arrows show schematic spin distributions at a fixed radius $r$. Thick red dots indicate the impurity location at the origin ${\mathit{x}}_0=(0,0,0)$.

Figure 4

(a) Upper plot: Spin-flip scattering processes leading to the Kondo effect. The red (blue) arrows denote impurity spin at ${\mathit{x}}_{\perp }=0$ [local spin (15) in the conduction band at a distance $|{\mathit{x}}_{\perp }|=r$ from the origin]. The impurity spin is screened by the orbital degrees of freedom (coupled flips of the electron spin and orbital angular momentum $l_z$). The lower plot shows a spiral spin structure (15) along a radial direction ($\rho =p_{\perp }r$) away from the impurity. (b) Same as in panel (a) but for a conventional metal without SOI. Only conduction electrons in the $s$-wave state couple to the impurity and the orbital angular momentum does not participate in the Kondo screening. The spin direction (along an arbitrary direction $\mathit{n}$) does not depend on the radial position.

Figure 5

(a) Schematic plot of the transverse magnetic response on a TI surface. A magnetic field $h$ applied normal to the surface causes a radial electron spin polarization. (b) Same as panel (a), but for the Kondo effect in a usual two-dimensional metal without SOI. There is only longitudinal magnetic response.

Figure 6

Kondo temperature, computed from the Nagaoka-Suhl equation (22), as a function of the BC angle $\vartheta$ [see Eq. (8)]. The arrow shows increasing values of the dimensionless Kondo coupling ${\alpha }_K=J_K{\mathrm{\Delta }}^2/2\pi v^3=0.05$, 0.1, 0.2, 0.4, 0.7, 1.0. Inset: Energies involved in the Hamiltonian (21). The blue color indicates the condensed slave boson ${\chi }_0$.