Proving Nontrivial Topology of Pure Bismuth by Quantum Confinement

The topology of pure Bi is controversial because of its very small ($\sim 10\mathrm{meV}$) band gap. Here we perform high-resolution angle-resolved photoelectron spectroscopy measurements systematically on 14–202 bilayer Bi films. Using high-quality films, we succeed in observing quantized bulk bands with energy separations down to $\sim 10\mathrm{meV}$. Detailed analyses on the phase shift of the confined wave functions precisely determine the surface and bulk electronic structures, which unambiguously show nontrivial topology. The present results not only prove the fundamental property of Bi but also introduce a capability of the quantum-confinement approach.

Figure 1

Schematic representation of (a) the bulk and surface Brillouin zone of Bi crystal in the [111] direction and (b) the Fermi surface. (c) Near-$E_F$ structure of the bulk projections at $\overline{\mathrm{M}}$. (d)–(g) Possible band structures along the $\overline{\mathrm{\Gamma }}\overline{\mathrm{M}}$ direction on the Bi(111) surface. The blue and red lines indicate the two spin-splitting surface bands, SS1 and SS2, respectively.

Figure 2

(a),(b) The Fermi surface and the band structures measured along the $\overline{\mathrm{\Gamma }}\overline{\mathrm{M}}$ direction in a 14 BL Bi(111) film at $h\nu =21\mathrm{eV}$. Solid lines in (b) indicate bulk projections calculated by a tight-binding method [21]. (c) Band structures obtained by the first-principles calculations for a 14 BL Bi slab. (d) Plane-averaged electron densities within the film calculated at the four $k$ points marked in (c). (e),(f) Possible band assignments in an ultrathin Bi film. Gray areas illustrate positions of the VB maximum (VBM) and CB minimum (CBM).

Figure 3

(a) Wide-range band structures measured along the $\overline{\mathrm{\Gamma }}\overline{\mathrm{M}}$ direction in 14, 18, and 79 BL Bi(111) films at $h\nu =21\mathrm{eV}$. The colored images were produced using a curvature method for better visualization [49]. (b) Near-$E_{\mathit{F}}$ band structures measured at $h\nu =8.437\mathrm{eV}$ inside the red box in (a). The thickness is systematically increased from 14 to 202 BL.

Figure 4

(a) EDCs extracted at $\overline{\mathrm{M}}$ ($k=0.8{\AA }^{-1}$). The triangles show peak positions fitted by Lorentzian functions (inset). (b) $E\text{−}{k}_{\perp }$ dispersion experimentally obtained using Eq. (2). The solid line represents a linear fit. (c) $E\text{−}{k}_{\perp }$ dispersion obtained from a tight-binding calculation [21]. (d) Total phase shifts experimentally derived using Eq. (1). (e) A plot of the $N\text{−}E$ relation in QWSs (a structure plot). Solid lines are drawn using Eq. (3).

Figure 5

(a) EDCs at $\overline{\mathrm{M}}$ magnified around $E_F$. (b) Evolutions of peak positions extracted in (a) and those of VBM ($n=1$ QWS in VB) against an inverse thickness $1/N$. The meanings of the solid lines are discussed in the text. (c) Schematic representation of the evolution in electronic structures of Bi films approaching the bulk limit.