Classical spins in topological insulators

Following the recent theoretical proposal and experiment on quantum spin Hall effect in HgTe/CdTe quantum wells, we consider a single magnetic impurity localized in the bulk of the system, which we treat as a classical spin. It is shown that there are always localized excited states in the bulk energy gap for arbitrarily strong impurity strength in the inverted region, while the localized excited states vanish for very strong impurity strength in the normal region. Similar conclusion also applies to three-dimensional topological insulators. This distinct difference serves as another criterion for the conventional and topological insulating phases when the time-reversal symmetry is broken, and can be easily experimentally observed through the scanning tunnel microscope and/or angle-resolved photoelectron spectroscopy experiments.

Figure 1

(Color online) $F$ function versus $\omega$ in the bulk-energy gap with several values of $e$ and $c^2=1$. The topological nontrivial regime with $\left|e\right|<2$ is shown in the left column, while the topological trivial regime with $\left|e\right|>2$ is shown in the right column of the figure. $F_e\left(e,\omega \right)$ is plotted as red lines while $F_h\left(e,\omega \right)$ is plotted as cyan (gray) lines. The dotted lines are guidelines for the eyes to indicate the boundary values of the $F$ functions.

Figure 2

(Color online) Energy spectrum obtained by direct diagonalization of Hamiltonian (1) with ${ϵ}_k=0$ as a function of impurity strength in both (a) inverted and (b) normal regimes. The localized states are shown in red (intersecting lines in middle).

Figure 3

(Color online) $F$ function versus $\omega$ with $\omega$ in the bulk energy gap. The $F$ functions are calculated numerically from Eq. (5) with the parameters taken from Ref. 3 for HgTe/CdTe QW at $d=58\text{Å}$ and $d=70\text{Å}$. $F_e\left(\omega \right)$ are shown as red lines while $F_h\left(\omega \right)$ are shown as cyan lines. Inset: the enlarged structures for the nontrivial regime.

Figure 4

Effective impurity concentration $\frac{1}{{\tau }_{s}M}$ as a function of effective impurity strength ${\left(1-{\xi }^2\right)}^2/{\left(1+{\xi }^2\right)}^2$. The shaded area indicates the range of the impurity concentration where the impurity band and the continuum are separated.