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We perform a systematic ab initio study on phonon-mediated superconductivity in the transition-metal-based superconductors Sc$_6M$Te$_2$ ($M$ = Fe, Co, Ni). Firstly, our charge analysis reveals significant electron transfer from Sc to $M$ due to the substantial difference in the electronegativity, filling the 3$d$ orbitals of $M$ and suppressing magnetic instability. Secondly, we show that Sc$_6$FeTe$_2$ exhibits strong lattice anharmonicity. Moreover, for $M =$ Fe and Co, we find low-frequency soft phonon bands of $M$ which can be interpreted as "rattling phonons" in the framework formed by Sc. While not observed in the case of $M=$ Ni, the rattling phonons give rise to a prominent peak or plateau in the Eliashberg spectral function and enhance the pairing instability. By reproducing the experimental trend of superconducting transition temperatures, our study underscores the potential of designing phonon-mediated superconductors by strategically combining non-superconducting and magnetic transition-metal elements., Comment: 13 pages, 10 figures

Inspired by the recent discovery of a new polar metal LiReO$_3$ by K. Murayama, \textit{et al}, we calculate the temperature($T$)-dependent crystal structures of Li$B$O3 with $B$ = Ta, W, Re, Os, using the self-consistent phonon (SCPH) theory. We have reproduced the experimentally observed polar-nonpolar structural phase transitions and the transition temperatures ($T_c$) of LiTaO$_3$, LiReO$_3$, and LiOsO$_3$. From the calculation, we predict that LiWO$_3$ is a polar metal, which is yet to be tested experimentally. Upon doping electrons to the insulating LiTaO$_3$, the predicted $T_c$ is quickly suppressed and approaches those of the polar metals. Thus, there is a continuous crossover between ferroelectric insulators and polar metals if we dope electrons to the polar metals. Investigating the detailed material dependence of the interatomic force constants (IFCs), we explicitly show that the suppression of $T_c$ in polar metals can be ascribed to the screening of the long-range Li-O interaction, which is caused by the presence of the itinerant electrons.

We present a theory and a calculation scheme of structural optimization at finite temperatures within the quasiharmonic approximation (QHA). The theory is based on an efficient scheme of updating the interatomic force constants with the change of crystal structures, which we call the IFC renormalization. The cell shape and the atomic coordinates are treated equally and simultaneously optimized. We apply the theory to the thermal expansion and the pyroelectricity of wurtzite GaN and ZnO, which accurately reproduces the experimentally observed behaviors. Furthermore, we point out a general scheme to obtain correct $T$ dependence at the lowest order in constrained optimizations that reduce the number of effective degrees of freedom, which is helpful to perform efficient QHA calculations with little sacrificing accuracy. We show that the scheme works properly for GaN and ZnO by comparing with the optimization of all the degrees of freedom.

We formulate a first-principle scheme for structural optimization at finite temperature ($T$) based on the self-consistent phonon (SCP) theory, which accurately takes into account the effect of strong phonon anharmonicity. The $T$-dependence of the shape of the unit cell and internal atomic configuration is determined by minimizing the variational free energy in the SCP theory. At each optimization step, the interatomic force constants in the new structure are calculated without running additional electronic structure calculations, which makes the method dramatically efficient. We demonstrate that the thermal expansion of silicon and the three-step structural phase transitions in BaTiO$_3$ and its pressure-temperature ($p$-$T$) phase diagram are successfully reproduced. The present formalism will open the way to the non-empirical prediction of physical properties at finite $T$ of materials having a complex structural phase diagram.

We formulate a theory of thermal expansion based on the self-consistent phonon (SCP) theory, which nonperturbatively considers the anharmonic effect. We show that the Gr\"unseisen formula holds within the SCP theory by replacing the phonon frequency by the SCP frequency. By comparing it with the quasiharmonic approximation (QHA), we derive explicit formulas of the correction to the QHA result. We show that the phonon anharmonicity gives a small correction of $O(\langle{\hat{U}_4\rangle}/\langle{\hat{U}_2\rangle})$ to the thermal expansion coefficient $\alpha$, where $\hat{U}_2$ and $\hat{U}_4$ are the harmonic and the quartic terms of the potential energy surface. On the other hand, we show that the phonon anharmonicity gives two correction terms to the temperature ($T$)-dependent phonon frequency shift which are comparable to the original QHA term. In strongly anharmonic materials such as NaCl and MgO, these two correction terms tend to cancel out each other, which explains why the QHA sometimes gives reasonable values for the $T$-dependent phonon frequency shift while it fails for thermal expansion.

Based on the Boltzmann transport theory, we study the origin of the anomalous temperature dependence of the Nernst coefficient ($\nu$) due to the phonon-drag mechanism. For narrow-gap semiconductors, we find that there are two characteristic temperatures at which a noticeable peak structure appears in $\nu$. Contrarily, the Seebeck coefficient ($S$) always has only one peak. While the breakdown of the Sondheimer cancellation due to the momentum-dependence of the electron relaxation time is essential for the peak in $\nu$ at low $T$, the contribution of the valence band to the phonon-drag current is essential for the peak at higher $T$. By considering this mechanism, we successfully reproduce $\nu$ and $S$ of FeSb$_2$ for which a gigantic phonon-drag effect is observed experimentally.