Inelastic magnetic scattering effect on local density of states of topological insulators

Magnetic ions such as Fe, Mn, and Co with localized spins may be adsorbed on the surface of topological insulators such as Bi${}_2$Se${}_3$. They form scattering centers for the helical surface states which have a Dirac cone dispersion as long as the local spins are disordered. However, the local density of states (LDOS) may be severely modified by the formation of bound states. Commonly, only elastic scattering due to normal and exchange potentials of the adatom is assumed. Magnetization measurements show, however, that considerable magnetic single-ion anisotropies exist which lead to a splitting of the local impurity spin states, resulting in a singlet ground state. Therefore inelastic scattering processes of helical Dirac electrons become possible, as described by a dynamical local self-energy of second order in the exchange interaction. The self energy influences bound-state formation and leads to significant new anomalies in the LDOS at low energies and low temperatures, which we calculate within the $T$-matrix approach. We propose that they may be used for spectroscopy of local impurity spin states by appropriate tuning of the chemical potential and magnetic field.

Figure 1

(left) Energy level scheme of localized spin states split by the anisotropy term in Eq. (1) with $\Delta =K_0$ and ${\Delta }^{\prime }=4K_0$. (right) Second-order self-energy for Dirac electrons (solid line) due to scattering from local spin states $\alpha ,\beta$ (dashed line).

Figure 2

(a) Real part of conduction electron Green's function ${\overline{G}}_0^{\tau }{\left(\omega \right)}^{\prime }$ from analytical (solid line) and numerical (dashed line) calculation. The latter is performed with a soft cutoff of width $\gamma /E_c=0.05$ at $E_c$. (b) Corresponding imaginary parts ${\overline{G}}_0^{\tau }{\left(\omega \right)}^{\prime \prime }$. The DOS of Dirac half cones $(\tau =\pm 1)$ is given by $N_{\tau }\left(\omega \right)/\left(D_0{E}_c\right)=-\frac{1}{\pi }{\overline{G}}_0^{\tau }{\left(\omega \right)}^{\prime \prime }$.

Figure 3

Real and imaginary parts of self-energy ${\overline{\Sigma }}_2\left(\omega \right)$ at $T=0$ for $\mu /E_c=0.2$; $\Delta /E_c=0.1$ and $({\lambda }_0,\lambda )=(0,1)$. The singular points given in Eq. (27) at high energies close to the surface band edge ($H_1,H_2$) and low energies close to the Dirac point $(L_i,i=1-3)$ are indicated. The solid line corresponds to the exact result using Eqs. (23) and (24) with a sharp cutoff at $E_c$, and the dashed line corresponds to numerical integration of Eq. (20) with a soft cutoff $\gamma /E_c=0.05$ and imaginary part $\eta =0.5\times {10}^{-3}$.

Figure 4

LDOS for $\mu =0$, $\Delta =0$, and various scattering strengths $({\lambda }_0,\lambda )$. Unperturbed $D\left(\omega \right)=D_0\left|\omega \right|$ is shown as the dotted line. The numerical evaluation with $\eta /E_c=0.5\times {10}^{-3}$ is used.

Figure 5

LDOS at $T=0$ for $\mu =0$ and scattering strengths $({\lambda }_0,\lambda )=(0,1)$ for various $\Delta$. For $\Delta /E_c=0.2$ the thick line is from numerical calculations $(\eta /E_c=0.5\times {10}^{-3}),$ and the thin line is from exact analytical results. Unperturbed $D\left(\omega \right)=D_0\left|\omega \right|$ is shown as the dotted line.

Figure 6

LDOS at $T=0$ for $\Delta /E_c=0.1$, scattering strengths $({\lambda }_0,\lambda )=(0,1)$, and various $\mu$. In the bottom panel the thick line represents numerical results $(\eta /E_c=0.5\times {10}^{-3})$, and thin line represents the exact analytical results. Arrows indicate the position of $L_{2,3}$ singularities in the self-energy due to local spin excitations.

Figure 7

Contour plot of LDOS with ${\lambda }_0=0,\lambda (T=0)=1$. (a) In the $\omega$–$\Delta$ plane with chemical potential at $\mu /E_c=0.1$. Increasing $\Delta$ leads to separation and increasing prominence of bound-state peaks with asymmetric intensity for nonzero $\mu$. (b) In the $\omega$–$T$ plane for the symmetric case $\mu =0$ and $\Delta /E_c=0.1$.

Figure 8

Temperature dependence of LDOS (numerical calculations for $\Delta /E_c=0.1$ with $\eta /E_c=0.5\times {10}^{-3})$. (top) $\mu /E_c=0$. (bottom) $\mu /E_c=0.2$.

Figure 9

Magnetic field effect on LDOS for easy-plane and easy-axis anisotropy $K_0$ of impurity spin S. (a) $K_0>0$. Two transitions to excited impurity states $|\pm 1\rangle$ with split energies ${\Delta }_{\pm }=\Delta \pm \delta$ lead to field-induced splitting of the LDOS inelastic peak ($T=0$, $\mu /E_c=0.15,\Delta /E_c=0.1$).Thick and thin solid lines correspond to numerical and analytical results, respectively. The zero-field case is shown as a dashed line for comparison. (b) $K_0<0$. Only one inelastic transition to the $|+1\rangle$ excited state is present with shifted energy ${\tilde{\Delta }}_{+}=\tilde{\Delta }+\delta$ appearing in the LDOS spectrum. The dashed line is as before.