Decay length of surface-state wave functions on Bi(1 1 1).

We calculate the decay length in surface normal direction of the surface-state wave functions on a clean Bi(1 1 1) surface as a function of two-dimensional (2D) wave vector [Formula: see text] along the [Formula: see text] line. For this purpose, we perform a first-principles density functional theory (DFT) calculation for semi-infinite Bi(1 1 1) by employing the surface embedded Green's function technique. The decay length of the two surface bands is found to be  ∼24 Bi bilayers at [Formula: see text], while it remains less than 5 Bi bilayers when [Formula: see text] is away from [Formula: see text] and [Formula: see text]. At [Formula: see text], the degenerate surface bands are split from the upper boundary energy of the projected bulk valence bands only by 5 meV. In spite of this, the decay length of these bands at [Formula: see text] is less than 10 Bi bilayers due to the large effective mass (small curvature) of the highest valence band in the surface normal direction.