First-principles calculation of the electron-phonon coupling in ultrathin Pb superconductors: Suppression of the transition temperature by surface phonons

We present a first-principles calculation of the electron-phonon coupling for thin layered lead systems. Using our Wannier-Fourier approach we show that the superconductivity in these systems is accounted for by traditional isotropic Migdal-Eliasberg theory through phonon-induced electron pairing. It is found that the transition temperature of Pb is suppressed by the presence of surface phonons which have been stiffened by an increased electronic spring constant. Superconductivity persists in ultrathin layered Pb despite the suppression from a decreased electronic density of states and weakened coupling to surface phonons.

Figure 1

(Color online) Phonon density of states for bulk Pb fcc (dashed line) and 2 ML Pb film (solid line). The transverse modes of the 2 ML film are split and the out-of-plane TA peak is stiffened by 2 meV. The highest frequency LA peak is stiffened by 3 meV in the 2 ML calculation as compared to the Pb-fcc data. For comparison, the curves are normalized such that $\int F\left(\omega \right)d\omega =3$.

Figure 2

(Color online) Eliashberg spectral function ${\alpha }^{2}F$ for bulk Pb fcc (dashed line) and 2 ML Pb film (solid line). Here, the total electron-phonon coupling $\lambda =2\int {\alpha }^{2}F\left(\omega \right){\omega }^{-1}d\omega$ is decreased from the bulk value of 1.41 to the 2 ML calculated value of 1.05. As can be seen from the above comparison of the spectral functions, the lower coupling in the 2 ML system results from a depression of spectral weight which has also been shifted to higher frequencies.

Figure 3

(Color online) Superconducting transition temperature as a function of layer thickness for the experimental points of Refs. 8, 33, and the theoretical model [Eq. (2)]. The transition temperature was calculated through the modified McMillan equation [Eq. (34) of Ref. 29]. Frequency moments have been fixed for differing $\lambda$’s and the Coulomb repulsion parameter is taken to be ${\mu }^{\ast }=0.14$, to obtain the experimental bulk $T_c=7.2\text{K}$.