Size-dependent structural and electronic properties of Bi(111) ultrathin nanofilms from first principles

Few layer bismuth nanofilms with (111) orientation have shown striking electronic properties, especially as building blocks of novel two-dimensional heterostructures. In this paper we present state-of-the-art first principles calculations, based on both density functional theory and maximally localized Wannier functions, that encompass electronic and structural properties of free-standing Bi(111) nanofilms. We accurately evaluate both the in-plane lattice constant and, by including the van der Waals interaction between bismuth bilayers, the intra/interlayer distances. Interestingly and somehow unexpectedly, the in-plane lattice constant is predicted to shrink by about 5% going from the thickest investigated nanofilm $\left(\sim 80\AA \right)$ to single bilayer Bi(111), entailing a thickness dependent lattice mismatch in complex heterostructures involving ultrathin Bi(111). Moreover, quantum confinement effects, that would be expected to rule the electronic structure at this size range, compete with surface states that appear close to and across the Fermi level. The implication is that not only all but the thinnest films have a metallic band structure but also that such surface states might play a role in either the formation of interfaces with other materials or for sensing applications. Finally, the calculated electronic structure compares extremely well with ARPES measurements.

Figure 1

Atomistic model of Bi(111) nanofilms and surface: the example of the 6 BL nanofilm (the top and side views are shown in the top and bottom panels, respectively). The shadowed region highlights the surface unit cell.

Figure 2

The optimized surface unit cell parameter as a function of the number of Bi BLs (empty circles). The thick solid line shows an exponential fit of the calculated data, whereas the thin solid line, at $a\sim 4.57\AA$, is its asymptotic value for large nanofilm thickness. Experimental data from Ref. [13] are included (stars), together with error bars. The inset shows the percentage variations $\mathrm{\Delta }a$ of the calculated values with respect to the experimental ones. The single BL nanofilm has been assigned zero thickness.

Figure 3

(left) A Bi(111) surface model showing the topmost atomic BLs and the definition of the $d_{ij}$ distances. (right) The variation of intra- $(d_{12}$, top panel) and inter-BL $(d_{23},d_{45},d_{67}$, bottom panel) distances (as defined in the left panel), as a function of the number of BLs.

Figure 4

The percent variation of the $d_{12}$ (top panel) and $d_{23}$ (bottom panel) distances (as defined in the left panel of Fig. 3), as a function of the number of BLs. The variation is computed with respect to the calculated bulk values, $d_{12b}$ and $d_{23b}$, respectively. The experimental results (as extrapolated at 0 K) of Ref. [29] are evidenced by the colored rectangle and can be compared with the thickest film calculation.

Figure 5

Weighted band structure (analogous of the weighted spectral function introduced in Ref. [28]) of selected Bi(111)-$n\mathrm{BL}$ nanofilms with different thickness along the $\mathrm{\Gamma }\text{−}M$ segment in the first BZ of the 2D hexagonal lattice. The size of the dots is proportional to the projection of the corresponding Kohn-Sham eigenfunction on the surface MLWFs. The Fermi level is set at $E_F=0$ eV and is represented by the dashed line.

Figure 6

(left) The Bi(111)-10BL nanofilm band structure of Fig. 5 (solid lines and circles) compared with the ARPES measurements of Ref. [8] (shaded regions). (right) A closeup of the band dispersion along the $\mathrm{\Gamma }\text{−}M$ line around the Fermi energy and close to $\mathrm{\Gamma }$ (experimental data are reported as shaded regions where different colors distinguish the highest but different intensities of the photoemission results). The Fermi level is set at $E_F=0$ eV. No relative shift of the theoretical data with respect to the experimental ones has been applied.

Figure 7

Hole (purple) and electron (yellow) pockets in the Fermi surface for the Bi(111)-20BL nanofilm are evidenced in the band structure along the $\mathrm{\Gamma }\text{−}M$ direction. The Fermi level is set at $E_F=0$.

Figure 8

The Fermi surface for the Bi(111)-6BL (top panel), Bi(111)-10BL (middle panel), and Bi(111)-20BL nanofilms. The Fermi level is set at $E_F=0$ and depicted in blue. Yellow (purple) regions correspond to states below (above) $E_F$. The color bar labels are in eV.