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Tunable topological electronic structures in Sb(111) bilayers: A first-principles study
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View: Figures


Image of FIG. 1.
FIG. 1.

(a) Side view and top view of crystal structure of Sb single bilayer. (b) 2D Brillouin-zones. Specific symmetry points are labeled. (c) Energies of the single bilayer and planar honeycomb structures as a function of stain (lattice constant). The strain is with respect to the lattice constant of crystalline Sb. The vertical distance, d 1, between two Sb atoms as a function of strain (lattice constant) is also included.

Image of FIG. 2.
FIG. 2.

Band structure of one bilayer Sb under (a) 0%, (b) 2.3%, (c) 4.6%, and (d) 6.9% strain. The inset in (c) is the magnified band near the Γ point. The path in Brillouin zones is M-Γ-K-M. The size of the black circles is proportional to the contributions projected on the s-type orbital. The band diagrams are to illustrate (e) the trivial state, (f) the transition point, and (g) the non-trivial state. (h) Plot of band gap as a function of strain (lattice constant) at the Γ point.

Image of FIG. 3.
FIG. 3.

(a) An Sb single bilayer after adding an out-of-plane E-field. Two different colors are used to identify the two kinds of atoms involved to highlight inversion symmetry breaking. Band structures of one bilayer Sb under out-of-plane electric fields (b) 0.84 V/Å, and (c) 0.92 V/Å, and (d) 1.00 V/Å. The red and blue represent counter-clockwise and clockwise spin helicity, respectively. (e) Topological phase transition due to an out-of-plane electric field. (g) Band gaps along M-Γ and Γ-K, respectively, indicated by black and red arrows in (b) are plotted as a function of the electric field. (f) 3D band dispersions around Γ at 0.92 V/Å.

Image of FIG. 4.
FIG. 4.

(a)–(d) are the spin orientations of the four bands (from top to bottom in Fig. 3(f) ) projected on k x and k y . The out-of-plane spin polarization is color-scaled as the background in the 2D plot.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Tunable topological electronic structures in Sb(111) bilayers: A first-principles study