Recent experiments have established previously predicted ${\mathrm{LaH}}_{10}$ as the highest-temperature superconductor, with $T_C$ up to 250–260 K [Drozdov et al., Nature (London) 569, 528 (2019); Somayazulu et al., Phys. Rev. Lett. 122, 027001 (2019)]. In this work we explore the high-pressure phase stability and superconductivity of lanthanum hydrides ${\mathrm{LaH}}_m$. We predict the stability of the hitherto unreported polyhydride $P6/mmm-{\mathrm{LaH}}_{16}$ at pressures above 150 GPa; at 200 GPa, its predicted superconducting $T_C$ is 156 K, the critical field ${\mu }_0{H}_C\left(0\right)$ is approximately 35 T, and the superconducting gap is up to 35 meV. We revisit the superconductivity of ${\mathrm{LaH}}_{10}$ and find its $T_C$ to be up to 259 K at 170 GPa from solving the Eliashberg equation and 271 K from solving the gap equation within the superconducting density functional theory, which also allows us to compute the Coulomb pseudopotential ${\mu }^{*}$ for ${\mathrm{LaH}}_{10}$ and ${\mathrm{LaH}}_{16}$.

Thermodynamic parameters of the LaH₁₀ superconductor were an object of our interest. LaH₁₀ is characterised by the highest experimentally observed value of the critical temperature: $${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{a}}}={\bf{215}}$$TCa=215 K (p_a = 150 GPa) and $${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{b}={\bf{260}}$$TCb=260 K (p_b = 190 GPa). It belongs to the group of superconductors with a strong electron-phonon coupling (λ_a ~ 2.2 and λ_b ~ 2.8). We calculated the thermodynamic parameters of this superconductor and found that the values of the order parameter, the thermodynamic critical field, and the specific heat differ significantly from the values predicted by the conventional BCS theory. Due to the specific structure of the Eliashberg function for the hydrogenated compounds, the qualitative analysis suggests that the superconductors of the La_δX_1−δH₁₀-type (LaXH-type) structure, where X ∈ {Sc, Y}, would exhibit significantly higher critical temperature than T_C obtained for LaH₁₀. In the case of LaScH we came to the following assessments: $${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{a}}}\in \left\langle {\bf{220}},{\bf{267}}\right\rangle $$TCa∈220,267 K and $${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{b}}}\in \left\langle {\bf{263}},{\bf{294}}\right\rangle $$TCb∈263,294 K, while the results for LaYH were: $${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{a}}}\in \left\langle {\bf{218}},{\bf{247}}\right\rangle $$TCa∈218,247 K and $${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{b}}}\in \left\langle {\bf{261}},{\bf{274}}\right\rangle $$TCb∈261,274 K.

We investigate the possibility of achieving the room-temperature superconductivity in hydrogen sulfide (H₃S) through increasing external pressure, a path previously widely used to reach metallization and superconducting state in novel hydrogen-rich materials. The electronic properties and superconductivity of H₃S in the pressure range of 250–500 GPa are determined by the first-principles calculations. The metallic character of a body-centered cubic Im $$\overline{{\bf{3}}}$$3 ¯ m structure is found over the whole studied pressure. Moreover, the absence of imaginary frequency in phonon spectrum implies that this structure is dynamically stable. Furthermore, our calculations conducted within the framework of the Eliashberg formalism indicate that H₃S in the range of the extremely high pressures is a conventional strong-coupling superconductor with a high superconducting critical temperature, however, the maximum critical temperature does not exceed the value of 203 K.

The pairing mechanism for the high- superconductors based on the electron-phonon (EPH) and electron-electron-phonon (EEPH) interactions has been presented. On the fold mean-field level, it has been proven, that the obtained s-wave model supplements the predictions based on the BCS van Hove scenario. In particular: (i) For strong EEPH coupling and the energy gap ( ) is very weak temperature dependent; up to the critical temperature extends into the anomalous normal state to the Nernst temperature. (ii) The model explains well the experimental dependence of the ratio on doping for the reported superconductors in the terms of the few fundamental parameters. In the presented paper, the properties of the d-wave superconducting state in the two-dimensional system have been also studied. The obtained results, like for s-wave, have shown the energy gap amplitude crossover from the BCS to non-BCS behavior, as the value of the EEPH potential increases. However, for the energy gap amplitude extends into the anomalous normal state to the pseudogap temperature. Finally, it has been presented that the anisotropic model explains the dependence of the ratio on doping for the considered superconductors.