Mass enhancement parameter in free-standing ultrathin Pb(111) films: The effect of spin-orbit coupling

The influence of spin-orbit interaction on the electron-phonon coupling strength at the Fermi level of thin lead films is investigated using first-principles calculations in the density functional perturbation formalism. The calculations both scalar relativistic and including spin-orbit coupling (SOC) have been carried out for free-standing Pb(111) films consisting of four to ten atomic layers. It is shown that the spin-orbit interaction produces a large enhancement of the electron-phonon coupling strength regardless of the film thickness. This partly reflects a strong SOC-induced softening of the film phonon spectra, and partly a SOC-mediated increase in electron-phonon coupling matrix elements. For thin films, quantum size effects result in pronounced oscillations of the average coupling constant with the number of layers, which become damped for thicker films.

Figure 1

Interlayer relaxations for the distances between the first (top) and second layers ($\Delta d_{12}$) and the second and third layers ($\Delta d_{23}$) as a function of film thickness. The changes are given in percents of the bulk value: $\Delta d_{ij}=(d_{ij}-d_{\mathrm{bulk}})/d_{\mathrm{bulk}}$. The values obtained both without (open circles) and with spin-orbit coupling (full, red circles) are presented. Also shown are the results of scalar relativistic pseudopotential calculations using gradient-corrected density functional theory: Ref. 38 (grey circles) and Ref. 17 (open squares). Black diamonds show $\Delta d_{12}$ and $\Delta d_{23}$ for ultrathin Pb films on Si(111) determined with low-energy electron diffraction.[8]

Figure 2

Mass enhancement parameter $\lambda \left(E_{\mathrm{F}}\right)$ (top) and the density of electronic states at the Fermi level $N\left(E_{\mathrm{F}}\right)$ (middle) as a function of film thickness. The full and open circles correspond to the calculations with and without SOC, respectively. The values of ${\lambda }^{\mathrm{bulk}}\left(E_{\mathrm{F}}\right)$ and $N^{\mathrm{bulk}}\left(E_{\mathrm{F}}\right)$ shown by full (open) triangles were reported in Ref. 27. (Bottom) Electronic band structure of a five-layer free-standing Pb(111) film along high-symmetry directions of the film IBZ. The dashed lines denote the dispersion curves obtained in the semirelativistic calculation while the solid lines represent the calculation including spin-orbit interaction.

Figure 3

(Top) phonon dispersion curves for a four-layer free-standing Pb(111) film calculated both with (solid lines) and without (dashed lines) spin-orbit coupling. The highest optical phonon branches are indicated as ${\varepsilon }_1$ and ${\varepsilon }_2$. The mode labels are taken from Ref. 18. (Bottom) Phonon frequencies of optical modes ${\varepsilon }_1$ (circles) and ${\varepsilon }_2$ (triangles) at the BZ origin ($\overline{\Gamma }$) as a function of film thickness calculated with (full symbols) and without (open symbol) spin-orbit coupling. The experimental data (grey symbols) are taken from Ref. 18 for Pb(111) films on Cu(111). The maximum bulk phonon frequency[49] of 9.04 meV is shown by arrows.

Figure 4

(Top) Phonon density of states $F\left(\omega \right)$ of a four- and ten-layer free-standing Pb(111) films. (Bottom) Eliashberg spectral functions ${\alpha }^{2}F\left(\omega \right)$, calculated at the Fermi energy of a four- and ten-layer free-standing Pb(111) films. The data obtained both with (solid lines) and without (dashed lines) spin-orbit coupling are presented.

Figure 5

Superconducting transition temperature as a function of film thickness. The full and open symbols correspond to the calculations with and without SOC, respectively. The data obtained both with ${\mu }^{*}=0.1$ (squares) and ${\mu }^{*}=0.11$ (circles) are presented. The experimental[46] bulk $T_{\mathrm{c}}=7.2$ K is shown by an arrow.