Topological quantum well resonances in metal overlayers

The electronic structure of thin Ca/Sr alloy films epitaxially grown on the topological insulator Bi${}_2$Se${}_3$ is investigated by first-principles calculations. Despite a negligible spin-orbit coupling in Ca/Sr alloys and strict spin degeneracy in freestanding Ca/Sr films, the topological surface states on the pristine Bi${}_2$Se${}_3$ surface are spatially transferred, upon Ca/Sr overlayer growth, into the overlayer to form topological quantum well resonances characterized by a spin-polarized chiral Dirac cone. The physical origin of this effect and the implications regarding applications are discussed.

Figure 1

(a) Schematic transformation of an ordinary insulator into a topological insulator under increasing spin-orbit coupling, which can be realized experimentally by high-$Z$-element substitution. The abbreviations CB and VB denote the conduction and valence bands, respectively. The gap inversion is accompanied by parity reversal of the band edge states. (b) Illustration of the interfacial electronic structure at the junction between a topological insulator and an ordinary insulator. The gap must close at the junction, resulting in a metallic topological interface state, indicated by a red peak. (c) Similar to (b) but with the addition of a metal interlayer between the topological insulator and the ordinary insulator. The topological interface state is expected to stretch out to span the metal film.

Figure 2

(a) Schematic atomic structure of a Bi${}_2$Se${}_3$ film with a six-ML Ca/Sr overlayer. (b) The heights of the top two atomic layers of Bi${}_2$Se${}_3$ and the Ca/Sr atomic layers, referenced to the midpoint of the six-QL Bi${}_2$Se${}_3$ substrate, as a function of overlayer thickness ranging from 0 to 6 MLs.

Figure 3

(a) Band structure of a six-QL freestanding Bi${}_2$Se${}_3$ film. The green and yellow regions indicate the bulk bands and the fundamental gap, respectively. The surface states within the gap form a Dirac cone, $X$, at the zone center. The color indicates the spin polarization ${\sigma }_y$. (b) Band structure of a freestanding six-ML Ca/Sr film. All bands are spin doublets. (c) Band structure of a six-QL Bi${}_2$Se${}_3$ film covered with a six-ML Ca/Sr film on each face. There are two Dirac cones, $Y$ and $Z$. (d)–(f) The corresponding spin-polarized planar charge densities. The dashed and solid vertical lines mark the boundaries of Bi${}_2$Se${}_3$ QLs and Ca/Sr MLs, respectively. Each curve shows the $z$-dependent charge density, and the color filling indicates the $z$-dependent spin polarization ${\sigma }_y\left(z\right)$ of the upper state associated with the Dirac cone slightly offset from the zone center to avoid the spin degeneracy.

Figure 4

(a) Theoretical band structure of a six-QL Bi${}_2$Se${}_3$ slab capped by a two-ML Ca/Sr film on each face. The color shading indicates the bulk band gap as in Fig. 3. (b) The three-dimensional dispersion relations of Dirac cone $Z$ near the zone center. The circles indicate two constant-energy cuts, one above the Dirac point (larger circle), and the other below (smaller circle). (c) The constant-energy contours corresponding to the two cuts. The arrows indicate the calculated directions of spin polarization. The chiral nature of the cone is evident. (d) The three-dimensional charge distribution of the upper branch of Dirac cone $Z$ slightly offset from the zone center.