Figure 4

Ab initio calculations for Sb films and nanoribbon.

(a) Calculated band gap at (E_gΓ) with respect to the strength of spin-orbit coupling (SOC, λ) for 3 and 4 BL infinite Sb(111) films. The SOC strength (λ) is artificially set to partial fractions of the true value of SOC. The calculated band structures for a Sb 4 BL film are depicted for λ of (b) 0 and (c) 1 along the direction. The band inversion between π and π^∗ states (insets) occurs after the reopening of the band gap over λ = 0.8. (d) The calculated band structure of Sb 4 BL when an electric field of 40 meV/Å is applied, The band inversion is maintained with this electric field. (e) Calculated band structure for Sb 4 BL films on Bi₂Te₂Se. The band structure is consistent with that in (d) except for an overall energy shift. The yellow and green shadows in (b,c–e) indicate the trivial and non-trivial band gaps, respectively. In (e), the red and blue dots represent the bands of the top and bottom layers of the Sb film, respectively (see Supplementary Fig. 6f for more details). The step edge structure of a 4 BL zigzag-edged Sb(111) nanoribbon (f ) without [(h) with] the substrate. The top layer has a nanoribbon geometry to generate step edges. The calculated band structure of a 4 BL film with step edges along the direction (g) without [(i) with] the substrate. The band in green dots are localized on the edge atoms [dashed circles in (f ) and (h)]. The edge states bands have two branches dispersing out from the conduction and valance bands and crossing at point to form a topologically non-trivial Dirac band.