Surface and subsurface phonons of Bi(111) measured with helium atom scattering

The surface phonon dispersion curves of Bi(111) have been measured by inelastic helium atom scattering (HAS) along the two symmetry directions. The complex set of observed dispersion curves, including several branches in the acoustic region, plus a localized and a resonant branch in the optical region, is interpreted by means of calculations based on density functional perturbation theory (DFPT). It is recognized that the upper phonon branches in the acoustic region starting at about $5.3$ meV and $4.3$ meV at zero wave vector correspond to shear-vertical and longitudinal modes localized on the third surface layer (second bilayer), respectively. The HAS ability of detecting subsurface phonons previously observed for Pb(111) multilayers is attributed to the comparatively strong electron-phonon interaction, and confirmed through a DFPT calculation of the phonon-induced surface charge-density oscillations. A comparison of the integrated HAS intensities with those of Pb(111) multilayers measured under similar kinematic conditions allows for an estimation of the electron-phonon mass-enhancement parameter for the Bi(111) surface. An anomaly at the $\overline{\mathrm{M}}$ point, quite sharp at 123 K but very broad at room temperature, is associated with recombination processes of bulk $\overline{\mathrm{M}}$-point pocket electrons with bulk pocket holes at either $\overline{\Gamma }$ or equivalent $\overline{\mathrm{M}}$ points.

Figure 1

(a) Top view of the Bi(111) surface structure together with the two scattering directions. The rhombus (shadowed area) is the surface unit cell with $a=4.538\AA$ (interpolated to 140 K; Ref. 16); atoms of different color are on different planes, as illustrated in panel (c). Panel (b) shows the surface Brillouin zone with the symmetry points, the two reciprocal lattice basis vectors [10] and [01], and the scattering directions in the reciprocal space. The layered structure of Bi(111) is shown in (c) with the bulk interlayer distances at 140 K (Ref. 16); the slab-adapted hexagonal unit cell encompasses three bilayers ($c=11.823\AA$). Although the two hollow positions in the surface unit cell (a) are crystallographically inequivalent, the calculated surface charge density (d) as seen by a He atom impinging vertically with an energy of 17 meV shows equivalent hollow sites as for a single hexagonal layer.

Figure 2

A few time-of-flight spectra converted to the energy transfer ($\Delta E$) scale for different incident angles and about the same incident energy along the $\overline{\Gamma }\overline{\text{K}}$ direction. $\Delta E<0$ $(>0)$ corresponds to a phonon creation (annihilation) process. The scan curves (dot-dashed lines) allow reading the corresponding parallel momentum transfer on the right-hand scale. An example of a Gaussian fit of the spectrum on top of a multiphonon background is shown in the lowest panel (dashed curves). The symbols (squares, triangles, circles, lozenges) on the scan curves mark the main inelastic features and correspond to different phonon dispersion curves in Fig. 4. The crystal was cooled down to $T_S=123\text{K}$ during the measurements.

Figure 3

Same as Fig. 2 for the $\overline{\Gamma }\overline{\mathrm{M}}$ direction and with the crystal at $T_S=103\text{K}$.

Figure 4

Measured surface phonon dispersion relation of Bi(111). The solid blue lines represent a fit of the Rayleigh mode (RW) by a simple force constant model. The black lines show the velocity of the shear transverse sound in bulk bismuth ($v_{t,z}$). The different symbols correspond to the different peaks in the scan curves in Figs. 2 and 3. (a) Sample cooled down to $T_S$. The blue arrow indicates a possible anomaly at the $\overline{\mathrm{M}}$ point. (b) Sample at room temperature.

Figure 5

Dispersion curves of a Bi(111) 12-layer slab from a DFPT calculation without SO coupling. Highlighted branches represent surface-localized modes and resonances for longitudinal (L) (left panel) and shear vertical (SV) (right panel) polarization. Their intensity projected respectively onto the first, second, and third surface layer is given by the gray code.

Figure 6

DFPT phonon dispersion curves of a 20-bilayer slab of Bi(111) obtained with the filling procedure by which a bulk slab of 14 bilayers with its DFPT bulk force constant is sandwiched between the two halves of the present 6-bilayer slab with its actual force constants. The labels indicate the surface branches of sagittal polarization which are distinctly localized outside the surface-projected bulk bands (notations as in Fig. 5).

Figure 7

Comparison of the measured phonon dispersion relation with the DFPT calculations. The calculated surface phonon branches of Fig. 5 which contribute the largest DOS are superimposed onto the experimental data points.

Figure 8

(a) Energy transfer spectrum for helium atoms scattered from Bi(111) along $\overline{\Gamma }\overline{\mathrm{M}}$ with the crystal cooled down to $T_S=123\text{K}$. The spectrum was least-squares fitted by multiple Gaussian peaks. Note that the optical SV1 peak is narrower than those of the acoustic band because the latter are all resonances whereas the optical mode SV1 is localized at the surface. (b)–(d) CDOs calculated with DFPT at the $\overline{\Gamma }$ point induced by frozen phonon displacements. The contour lines of the resulting CDOs with respect to the static charge density show the corresponding increase (red lines) or decrease (blue lines), respectively.