First-principles study of the double-dome superconductivity in the kagome material ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ under pressure

Recent high-pressure experiments discovered abnormal double-dome superconductivities in the newly synthesized kagome materials ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ $(A=\mathrm{K}, \mathrm{Rb}, \mathrm{Cs})$, which also host abundant emergent quantum phenomena such as charge density wave (CDW), anomalous Hall effect, nontrivial topological property, and so on. In this work, by using first-principles electronic structure calculations, we studied the CDW state, superconductivity, and topological property in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ under pressures $(< 50 \mathrm{GPa})$. Based on the electron-phonon coupling theory, our calculated superconducting $T_{\text{c}}\mathrm{s}$ are consistent with the observed ones in the second superconducting dome at high pressure, but are much higher than the measured values at low pressure. The further calculations including the Hubbard $U$ indicate that with modest electron-electron correlation the magnetism on the V atoms exists at low pressure and diminishes gradually at high pressure. We thus propose that the experimentally observed superconductivity in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ at ambient/low pressures may still belong to the conventional Bardeen-Cooper-Schrieffer (BCS) type, but is partially suppressed by the V magnetism, while the superconductivity under high pressure is fully conventional without invoking the magnetism. We also predict that there are a second weak CDW state and topological phase transitions in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ under pressures. Our theoretical assertion calls for future experimental examination.

Figure 1

Phonon spectra of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ at (a) 0, (b) 5, (c) 12, (d) 25, and (e) 35 GPa, respectively. The line colors in the panels (b) to (d) correspond to the moment- and mode-resolved electron-phonon coupling strength ${\lambda }_{\mathbf{q}\nu }$. The $B_{1u}$ and $A_g$ modes at the $M$ point and the $B_{1u}$ mode at the $L$ point are labeled. (f) Pressure evolution of the phonon frequencies of the $B_{1u}$ (black squares) and $A_g$ (red dots) modes at the $M$ point and the $B_{1u}$ mode (blue triangles) at the $L$ point. (g,h) Charge density wave (CDW) distortions for the imaginary $A_g$ mode at the $M$ point under ambient pressure, so called the (g) SD and (h) ISD patterns. The blue, red, and yellow balls represent the Cs, V, and Sb atoms, respectively. The distortion of the $B_{1u}$ mode at the $L$ point is equivalent to a layer by layer stacking of the panels (g) and (h). (i) Distortion for the imaginary $B_{1u}$ mode at the $M$ point under 35 GPa, which is mainly contributed by the movements of Cs atoms along the $c$ axis.

Figure 2

(a) Unfolded band structure of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ in the $2\times 2\times 2$ CDW state with the alternately stacked ISD and SD distortions at 0 GPa. Band structures of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ at (b) 26 GPa and (c) 40 GPa. (d) Calculated topological invariants $Z_2$ for the occupied states below the gap 1 and the gap 2 under different pressures, respectively. (e,f) Enlarged views of the parity inversion in panels (b) and (c), respectively, where “${\mathrm{\Gamma }}_7^{-}$”, “${\mathrm{\Gamma }}_8^{+}$”, “${\mathrm{\Gamma }}_5^{-}$”, and “${\mathrm{\Gamma }}_5^{+}$” are named by the bands' irreducible representations of the corresponding point groups.

Figure 3

(a) Superconducting $T_{\text{c}}$ from our EPC calculations (black dots and circles) and previous experiments [35, 36, 37] (red, purple, and blue dots) under pressure. The black dots and circles represent the calculation results when the ${\mu }^{*}$ is set to 0.08 and 0.10, respectively. (b) Calculated total EPC constant $\lambda$ and logarithmic frequency ${\omega }_{\text{log}}$. (c) Calculated electronic density of states at the Fermi level $N\left(E_F\right)$ before (solid line) and after (dash line) the CDW distortion.

Figure 4

(a)–(d) Calculated Eliashberg spectral function $\left[{\alpha }^{2}F\left(\omega \right)\right]$ and the phonon density of states (PHDOS) for ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ at (a) 5 GPa, (b) 12 GPa, (c) 25 GPa, and (d) 43 GPa, respectively.

Figure 5

(a) The pressure-dependent energy differences between the ferrimagnetic state and the nonmagnetic state $[\mathrm{\Delta }E$ = $E\left(\text{FIM}\right)-E\left(\text{NM}\right)]$ calculated with different Hubbard $U$ values. (b) The corresponding total magnetic moment in a ferrimagnetic unit cell.

Figure 6

A schematic superconducting phase diagram of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. The green region represents the original CDW state. The red and blue regions label the experimentally observed double super- conducting domes, respectively. The orange line represents the conventional EPC-derived superconducting $T_{\text{c}}$ without the magnetism. The black dashed line outlines the pressure-suppressed magnetic fluctuation. The yellow region represents the magnetism-suppressed superconductivity. The pink, shaded region covers our predicted second CDW dome.