Superconductivity and electron-phonon coupling in lithium at high pressures

Using a first-principles pseudopotential approach we study the origin of superconductivity in lithium under pressure. A recently developed Wannier interpolation based technique that allows for ultradense sampling of electron-phonon parameters throughout the Brillouin zone was employed. The electron-phonon coupling strength as a function of pressure was calculated, precisely resolving many of the fine features of its distribution. The contributions to coupling arising from the Fermi surface topology, phonon dispersions, and electron-phonon matrix elements were separately analyzed. It is found that of the constituent components, the electron-phonon matrix elements are the most sensitive to pressure changes, and a particular phonon is responsible for high values of coupling. Additionally, the distribution of matrix elements over the Fermi surface is seen to be non-uniform and possesses a two-peak structure. Analysis of the Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ shows a considerable increase in spectral weight in the low-frequency region with the application of pressure. We estimate the superconducting transition temperature and find that the obtained values are in good accord with experiment.

Figure 1

(Color online) (a) The Fermi surface nesting function ${\xi }_{\vec{Q}}$, (b) phonon dispersions ${\omega }_{\vec{Q}}$, (c) average matrix element squared $M_{\vec{Q}}^2$, and (d) electron-phonon coupling ${\lambda }_{\vec{Q}}$ for fcc Li along the path inside the Brillouin zone. (b)–(d) are given for lower transverse mode $T1$. In (c) the dimensions of matrix elements are $me{V}^2\times {10}^3$.

Figure 2

(Color online) Magnitude density distribution of the electron-phonon matrix elements $M_{\vec{k},\vec{k}+\vec{Q}}$ for the lower transverse mode $T1$ of fcc Li with $\vec{Q}=\left(0.6,0.6,0.0\right)$. Only matrix elements for the $k$ states close to Fermi level are included. The magnitude density distribution is a histogram which plots the number of occurrences of a particular matrix element magnitude throughout the Brillouin zone as a function of that magnitude.

Figure 3

(Color online) Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ for different values of pressure. Effective sampling of $50\times 50\times 50$ $k$ and $q$ points inside the BZ was utilized. (Inset) Phonon densities of states $F\left(\omega \right)$ for the same pressures.

Figure 4

(Color online) Superconducting temperature vs pressure phase diagram for Li. Experimental data from Refs. 5, 6, 7 are shown as well as calculated results based on Allen-Dynes equation with ${\mu }^{\ast }=0.13$. Proposed phase-stability boundaries from Ref. 9 are indicated by vertical lines.