Ab initio lattice dynamics and electron-phonon coupling of Bi(111)

We present a comprehensive ab initio study of structural, electronic, lattice dynamical, and electron-phonon coupling properties of the Bi(111) surface within density functional perturbation theory. Relativistic corrections due to spin-orbit coupling are consistently taken into account. Changes of interatomic couplings are confined mostly to the first two bilayers, resulting in superbulk modes with frequencies higher than the optic bulk spectrum, and in an enhanced density of states at lower frequencies for atoms in the first bilayer. We give results for the momentum-dependent electron-phonon coupling of electronic states belonging to the two surface electronic bands along $\overline{\Gamma M}$ which cross the Fermi energy. For larger momenta, the lower surface band exhibits a moderate electron-phonon coupling of about 0.45, which is larger than the coupling constant of bulk Bi. For momenta close to $\overline{\Gamma }$, states of both surface bands show even stronger couplings because of interband transitions to bulk states near $\overline{\Gamma }$ around the Fermi level. For these cases, the state-dependent Eliashberg functions exhibit pronounced peaks at low energy and strongly deviate in shape from a Debye-type spectrum, indicating that an extraction of the coupling strength from measured electronic self-energies based on this simple model is likely to fail.

Figure 1

The structure of bulk Bi in the rhombohedral representation $(a_r,\alpha$, and $u$ are given in Table 1), where ${\mathbf{a}}_{r1}=b\widehat{y}+c_r\widehat{z},{\mathbf{a}}_{r2}=-b(\sqrt{3}/2\widehat{x}+1/2\widehat{y})+c_r\widehat{z},{\mathbf{a}}_{r2}=b(\sqrt{3}/2\widehat{x}-1/2\widehat{y})+c_r\widehat{z}$; $b=a_r\sqrt{2(1-cos\alpha )/3}$; $c_r=a_r\sqrt{(1+2cos\alpha )/3}$; $d_1=6u{c}_r$; $d_2=3c_r-d_1$. R1 and R2 represent the positions of the two atoms.

Figure 2

The Brillouin zone of the A7 structure of bulk Bi. Figure is taken from Ref. [45].

Figure 3

Calculated phonon dispersion curves of bulk Bi along high-symmetry directions of the Brillouin zone. The black dots correspond to inelastic neutron scattering data taken at 75 K [47]. The point ${\Lambda }^{\prime }$ lies on the trigonal axis with ${\mathbf{k}}_{{\Lambda }^{\prime }}=2/3{\mathbf{k}}_{\mathrm{T}}$.

Figure 4

Side view of a sketch of the Bi(111) structure. A, ${\mathrm{A}}^{\prime }$, B, ${\mathrm{B}}^{\prime }$, C, and ${\mathrm{C}}^{\prime }$ represent the planes defined by each layer. The sticks connecting the balls are used to indicate only first-nearest neighbors (between atoms on planes ${\mathrm{A}}^{\prime }$ and B, ${\mathrm{C}}^{\prime }$ and A, or ${\mathrm{B}}^{\prime }$ and C) to highlight the bilayerlike structure of Bi(111). $d_{ij}$ denotes the vertical spacing between adjacent planes.

Figure 5

The projection of the bulk Brillouin zone of Bi onto the (111) surface. Sketched are the bulk Fermi surfaces (exaggerated in size). Figure is taken from Ref. [10]. The shaded area denotes the mirror plane.

Figure 6

Surface band structure of a Bi(111) slab. Shown are the electronic bands in the vicinity of the Fermi energy for the 12-, 24-, and 36-BL slabs, respectively. The gray areas represent regions of bulk states.

Figure 7

Phonon dispersion curves of a Bi(111) slab of 50 bilayers along the high-symmetry directions of the surface Brillouin zone obtained by slab filling of (a) a 6-bilayer and (b) a 12-bilayer slab. The dots correspond to surface modes whose amplitude weight in the two outermost bilayers is larger than 20% (red) or in the range 10%–20% (black). The light gray lines are bulk modes. In (a), the polarization of the surface modes away from the high-symmetry points is succinctly indicated with SV for shear vertical, L for longitudinal, and SH for shear horizontal or the word “All” if the polarization is not well defined. In (b), symbols correspond to peaks in inelastic helium atom scattering spectra taken at lower temperatures as reported by Tamtögl et al. [22].

Figure 8

Layer-projected phonon density of states (LDOS) obtained for a 12-bilayer Bi(111) slab. The red (full) and blue (dashed) lines show the LDOS of the first and second layers of the outermost bilayer, respectively. The black (dotted) curve corresponds to a central bilayer and represents a bulklike spectrum.

Figure 9

(Top) Thick circles indicate the surface electronic states for which the electron-phonon coupling calculations were done. S1, S2, and S3 denote specific surface states of similar binding energies, which are discussed in the text in more detail. Bulk Bi bands projected onto the (111) plane are indicated by small gray circles. (Bottom) Electron-phonon coupling parameter ${\lambda }_{\mathbf{k}i}$ as a function of electron (hole) momentum along the $\overline{\Gamma M}$ symmetry direction for electronic states in the lower (gray/purple) and higher (black) surface bands.

Figure 10

Electron-phonon coupling parameter ${\lambda }_{\mathbf{k}i}$ as a function of electron energy. The notation is the same as in Fig. 9. The points are connected according to their positions in the $\overline{\Gamma M}$ direction. The dashed line shows the calculated density of electronic states $N\left(E\right)$ as a function of energy.

Figure 11

Average of the absorption and emission spectral functions ${\alpha }^2{F}_{\mathbf{k}i}\left(\omega \right)$, calculated for surface electronic states S1, S2, and S3 (see Fig. 9). The gray areas show the contribution of electronic transitions to the $\overline{\Gamma }$ point region (Fig. 9). The dashed (dotted) line indicates the initial slope of a Debye spectrum assuming a surface (bulk) Debye frequency of 5 meV (10 meV) and scaled by the corresponding coupling constant.