Dirac quantum well engineering on the surface of a topological insulator

We investigate a quantum well that consists of a thin topological insulator sandwiched between two trivial insulators. More specifically, we consider smooth interfaces between these different types of materials such that the interfaces host not only the chiral interface states, whose existence is dictated by the bulk-edge correspondence, but also massive Volkov-Pankratov states. We investigate possible hybridization between these interface states as a function of the width of the topological material and of the characteristic interface size. Most saliently, we find a strong qualitative difference between an extremely weak effect on the chiral interface states and a more common hybridization of the massive Volkov-Pankratov states that can be easily understood in terms of quantum tunneling in the framework of the model of a (Dirac) quantum well we introduce here.

Figure 1

(a) Schematic of a single Dirac QW. (b) Schematic of a double Dirac QW. The spatially varying gap $\mathrm{\Delta }\left(z\right)$ switches from positive sign to negative when one passes from the blue area (trivial phase) to the red area (topological phase) and the gap is vanishing somewhere in-between. $l$, $l_1$, and $l_2$ characterize the smoothness of domain wall between phases.

Figure 2

(a) Interface profiles described by a spatially varying gap $\mathrm{\Delta }\left(z\right)$ for two values of characteristic interface width $l/\xi =0.9,2.3$. (b) Profiles of Dirac QWs for its corresponding $\mathrm{\Delta }\left(z\right)$ and chirality $\lambda =\pm$. $U_{-}$ is represented by solid lines and $U_{+}$ by dashed lines.

Figure 3

(a) Profile of the spatially varying gap $\mathrm{\Delta }\left(z\right)$ for finite-size system with $l/\xi =1$ and $L/\xi =4$. (b) Profiles of two adjacent Dirac QWs for two chiralities. Two dashed lines, blue and red, indicate the energy level (close to zero) of chiral states of $\lambda =\pm$, respectively.

Figure 4

(a) Reduced energy $\omega$ as a function of smoothness $l$ for the massive VP states $n=1$ to 4 by solving the secular equation (A9) (solid lines). The $l$ dependence of $\omega$ is well described by Eq. (25) (crosses). Inset: zoom-in to show the splitting for the VP state $n=1$. (b) Reduced energy splitting $\mathrm{\Delta }\omega$ as a function of smoothness $l$ for the massive VP states $n=1$ to 4. The distance between two Dirac QWs is set to be $L/\xi =20$. The continuous lines show the splitting obtained from our secular equation (A9) of the double Dirac QW problem, and the crosses indicate the values obtained from the approximate formula (28) based on quantum tunneling between the QWs. The dotted lines show results based on the same formula, where we have used the exact energies for the VP states of a single Dirac QW instead of the approximate ones given in Eq. (25).

Figure 5

Profiles of two infinitely separated Dirac QWs for two chiralities with $l/\xi =3$: (a) for $\lambda =+$ and (b) for $\lambda =-$. The dashed lines, blue and red, indicate the energy level of chiral states and $n=1$ VP states for $\lambda =\pm$, respectively.

Figure 6

Reduced energy $\omega$ as a function of the distance between two Dirac QWs, $L$, for the massive VP states $n=1$ and 2. The results are obtained by solving the secular equation (A9). Inset: the reduced energy splitting $\mathrm{\Delta }\omega$ decays exponentially with increasing $L$, only shown for the massive VP states $n=1$ and 2. The continuous lines indicate the splitting from the secular equation, while the crosses represent the results based on Eq. (28). The dotted lines show results based on the same formula, where we have used the exact energies for the VP states of a single Dirac QW instead of the approximate ones given in Eq. (25).

Figure 7

Top: ${log}_{10}\omega$ vs $l$ for the $n=0$ chiral state and $L/\xi =20$. Bottom: ${log}_{10}\omega$ vs $L$ for the $n=0$ chiral state and $l/\xi =6$. The results from the secular equation (A9) are represented by blue lines, those using Eq. (30) by orange lines, and those using Eq. (32) by green line.