Superconductivity from orbital-selective electron-phonon coupling in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$

Recent experiments have shown that the phase diagrams of the kagome superconductors $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ are strongly impacted by changes in the $c$-axis lattice parameter. Here, we show that $c$-axis deformations impact primarily the Sb apical bonds and thus the overlap between their $p_z$ orbitals. Changes in the latter, in turn, substantially affect low-energy electronic states with significant Sb character, most notably the central electron pocket and the van Hove singularities located above the Fermi level. Based on the orbital-selective character of $c$-axis strain, we argue that these electronic states experience a non-negligible attractive electron-phonon pairing interaction mediated by fluctuations in the apical Sb bonds. We thus propose a multiband model for superconductivity in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ that includes both the Sb pocket and the V-derived van Hove singularities. Upon comparing the theoretical phase diagram with the experimentally observed vanishing of the $T_c$ dome across a Lifshitz transition of the Sb pocket, we propose that either an $s^{+-}$ or an $s^{++}$ state is realized in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$.

Figure 1

[(a), (b)] Schematic illustrations of two neighboring $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ unit cells and of the displacement pattern of the apical Sb promoted by the ${\mathrm{A}}_{1g}$ phonon mode, respectively. We choose a sign convention such that the displacement of apical Sb towards the kagome plane corresponds to positive ${\mathrm{A}}_{1g}$. (c) Changes in the distances between the Sb apical atom and its nearest-neighbor (green triangles) and the V atoms (red circles) as a function of $c$-axis strain for ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. The purple squares give the atomic displacements from the equilibrium structure corresponding to a frozen excitation of the optical ${\mathrm{A}}_{1g}$ phonon mode. (d) Low-energy band structure of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. The thickness of the bands is proportional to the total projection onto the $p_z$ orbitals of the Sb atoms, whereas their color is proportional to the projection onto the planar Sb (blue) and apical Sb (red). The $\mathrm{\Gamma }$ band has contributions from both Sb sites. (e) Modifications in the low-energy band structure as a function of $c$-axis strain. The momentum L is located above M in the hexagonal Brillouin zone; the other momenta are defined in Fig. 2.

Figure 2

(a) Pairing interactions of the four-patch model involving fermions at the ${\mathrm{M}}_i$ and $\mathrm{\Gamma }$ points. The Fermi surface is shown in gray. (b) Tight-binding dispersions, highlighting the Lifshitz transition of the $\mathrm{\Gamma }$-point electron pocket as a function of the parameter $\delta \mu \equiv ({\mu }_{\mathrm{\Gamma }}-{\mu }_c)/{\mu }_c$; $\mathrm{\Lambda }$ is the pairing interaction cutoff. (c) SC phase diagram of the four-patch model (away from the Lifshitz transition) as a function of the interactions shown in panel (a); $s$-wave corresponds to either $s^{++}$ or $s^{+-}$ states depending on the sign of $g_{\mathrm{\Gamma }\mathrm{M}}$. (d) $T_c$ as a function of the parameter $\delta \mu$ that tunes the $\mathrm{\Gamma }$ pocket across the Lifshitz transition at $\delta \mu =0$, as shown in the insets. The interaction parameters, marked by the orange and purple symbols shown in panel (c), are ($g_{\mathrm{M}\overline{\mathrm{M}}},g_{\mathrm{\Gamma }\mathrm{M}},g_{\mathrm{\Gamma }\mathrm{\Gamma }},g_{\mathrm{MM}})=(0.1,0.015,-0.15,0$) for the orange lines and ($0.07,0.1,-0.03,0$) for the purple lines. $T_{c0}$ is the SC transition temperature for the first set of parameters (orange) at ${\mu }_{\mathrm{\Gamma }0}=-3.65\tilde{t}$.