Topological nonmagnetic impurity states in topological Kondo insulators

We examine the presence of nonmagnetic impurities in a hybridization gap model of a Kondo insulator which has band inversion. The model has been used to predict that ${\mathrm{SmB}}_6$ is a topological insulator. We show that there are two types of nonmagnetic impurity states in a Kondo insulator. The type of states can be categorized as deep impurity states, similar to the impurity states in an ordinary metal, and impurity states which have energies within the hybridization gap. Unlike the deep impurity states which only form if the impurity potential exceeds a critical value of the order of the conduction bandwidth, the in-gap impurity states form for exceptionally small values of the impurity potential comparable to the hybridization gap. This result may explain why Kondo insulators are found to be exceptionally sensitive to impurities. We show that these in-gap states are caused by band inversion and have properties similar to those expected for impurities in a topological insulator.

Figure 1

The unperturbed $f, {\rho }^{0,f}\left(\omega \right)$ (blue), and $d, {\rho }^{0,d}\left(\omega \right)$ (red), hybridized density of states as a function of energy. The $f$-binding energy is $E_{f,\alpha }=W/5$ and hybridization matrix elements $V=W/4$. These values are not representative of ${\mathrm{SmB}}_6$ but were chosen simply for clarity of illustration.

Figure 2

The change in the total density of states $\mathrm{\Delta }{\rho }_T\left(\omega \right)$ due to the presence of the impurity potential. The change in the density of states is shown for the same choice of parameters used in Fig. 1. The value of $\mathrm{\Delta }U$ was chosen as $-W/2$. It is seen that impurity potential removed spectral weight from the edges of the hybridization gap and formed an in-gap impurity state of weight unity [as is described by Eq. (34)]. The $\omega$-integrated total spectral weight $\left(2N{\sum }_{\alpha }{D}_{\alpha }\right)$ is conserved. The in-gap state has the form of a delta function. Although $\mathrm{\Delta }U$ is not sufficiently strong to produce a bound state below the lower hybridized band, it is seen that the impurity potential has moved spectral weight of the lower hybridized band to lower energies, thereby forming a virtual bound state with a Fano antiresonance.

Figure 3

A plot of the real (blue) and imaginary part (red) of the $\underline{k}$-averaged homogeneous $f$-electron Green's function $G_{\underline{k}}^{f,0}\left(\omega \right)$, in units of the inverse $d$-bandwidth $W^{-1}$. The values of $E_f/W$ and $V/W$ were chosen, respectively, as $1/5$ and $1/2$. Bound states occur at energies where $\left(\frac{1}{\mathrm{\Delta }U}\right)$ shown by the horizontal dashed black line intersects with the blue line whenever the imaginary part (shown in red) is zero.

Figure 4

The energy dependence of the local $f$-density of states on the impurity site ${\rho }_{\underline{R}=0}^f\left(\omega \right)$ for three values of $\mathrm{\Delta }U, \mathrm{\Delta }U=-1.5W$ (red), $\mathrm{\Delta }U=-1.0W$ (green), and $\mathrm{\Delta }U=-0.5W$ (blue). The values of $E_f$ and $V$ are the same as in Fig. 2. For all values of $\mathrm{\Delta }U$, there is a bound state within the hybridization gap. The delta function has been given a small width to make it visible. It is seen that the intensity of the in-gap bound state is very small and decreases as $\mathrm{\Delta }{U}^{-2}$ when $\mathrm{\Delta }U$ increases. For $\mathrm{\Delta }U=-1.5W$, a bound state has split off from the bottom of the band and has removed almost all of the spectral weight from the continuous spectra. For $\mathrm{\Delta }U=-0.5W$, an external bound state has not formed, but the on-site impurity potential has shifted spectral weight towards the bottom of the $f$ band producing a virtual bound state.

Figure 5

The local $f$-density of states on the site $R=(1,0,0)$ neighboring the impurity ${\rho }_{(1,0,0)}^f\left(\omega \right)$ as a function of $\omega$, for three values of $\mathrm{\Delta }U, \mathrm{\Delta }U=-1.5W$ (red), $\mathrm{\Delta }U=-1.0W$ (green), and $\mathrm{\Delta }U=-0.5W$ (blue). The values of $E_f$ and $V$ are the same as in Fig. 2. For all values of $\mathrm{\Delta }U$, there is a bound state within the hybridization gap. The delta function has been given a small width to make it visible. The intensity of the in-gap bound state on the nearest neighbor site is seen to decrease as the magnitude of $\mathrm{\Delta }U$ is increased. For $\mathrm{\Delta }U=-1.5W$, a bound state has split off from the bottom of the valence band, however, there is no visible vestige of the external bound state on the neighboring site, indicating that it has a localization length less than the lattice spacing.

Figure 6

The energy dependence of the local $f$-density of states ${\rho }_R^f\left(\omega \right)$ at site $\underline{R}$ from the impurity, for $\mathrm{\Delta }U=-0.5W$. The values of $E_f$ and $V$ are the same as in Fig. 2. For small $\mathrm{\Delta }U$, the local $f$-density of states on the impurity site exhibits a virtual bound state and has a small amplitude of the in-gap bound state. The $f$ weight of the in-gap bound state is primarily located on the sites which are nearest neighbor to the impurity, i.e., $\underline{R}=(1,0,0)$. The delta function has been given a small width to make it visible. The amplitude of the in-gap bound decreases with increasing separation from the impurity site.

Figure 7

The local $d$-density of states on the impurity site $R=(0,0,0)$ for $\mathrm{\Delta }U=-0.5W$ (blue) and for $\mathrm{\Delta }U=-1.0W$ (green) and the bulk $d$-density of states where $|\underline{R}|\to \infty$ (red). The values of $E_f$ and $V$ are the same as in Fig. 3. There is a bound state on the impurity site within the hybridization gap. The delta function has been given a small width to make it visible. The $d$-density of states on the impurity site shows a Fano antiresonance near the energy $E_f+\mathrm{\Delta }U$ where the $f$-density of sates exhibits a virtual bound state.

Figure 8

The energy dependence of the local $d$-density of states on the site $R=(1,0,0)$ neighboring the impurity for $\mathrm{\Delta }U=-0.5W$ (blue) and for $\mathrm{\Delta }U=-1.0W$ (green). The values of $E_f$ and $V$ are the same as in Fig. 3. The delta function representing the in-gap state has been given a small width to make it visible. A residual small Fano antiresonance occurs on the nearest neighbor site around the energy $E_f+\mathrm{\Delta }U$.

Figure 9

(Top panel) The bound state energies for $j=\frac{1}{2}, j=\frac{3}{2}, j=\frac{5}{2}$, ... as a function of the ratio of the radius $a$ to the decay length $\frac{\hslash }{mc}$. (Bottom panel) The radial distribution functions for the upper components ($l=1$ solid blue) and lower components ($l=0$ dashed red) of the $j=\frac{1}{2}$ positive energy bound state for $a=1$ (thick), $a=2$ (medium), and $a=4$ (fine).

Figure 10

The $\omega$ dependence of the real and imaginary parts of the denominator of the $T$ matrix $\mathrm{\Delta }\left(\omega \right)$ in dimensionless units. For the chosen parameters, the $T$ matrix has a pole at an energy inside the hybridization gap and a pole at the energy of a virtual bound state.