(La,Th)H$_{10}$: the potential high-$T_{c}$ superconductors stabilized thermodynamically below 200 GPa

The recent high-pressure experimental discovery of superconductivity in (La,Y)H$_{10}$, (La,Ce)H$_{9}$, (La,Ce)H$_{10}$, (Y,Ce)H$_{9}$, and (La,Nd)H$_{10}$ shows that the ternary rare-earth clathrate hydride can be promising candidate for high-temperature superconductor. In this work, we theoretically demonstrate that the combination of actinide-metal thorium (Th) and rare-earth-metal lanthanum (La) with hydrogen can also form some ternary hydrides with cage-like structures to be stable at 200 GPa. Using the evolutionary algorithms combined with the first-principles calculations, we have predicted the pressure-dependent ternary phase diagram of La$_{x}$Th$_{y}$H$_{z}$, particularly including the case of (La$_{1-x}$Th$_{x}$)H$_{n}$ [or designated as (La,Th)H$_{n}$ for simplicity]. Our calculations show that the hydrogen-rich phases such as (La,Th)H$_{9}$ (only including $P\bar{6}m2$-LaThH$_{18}$) and (La,Th)H$_{10}$ (including $I4/mmm$-La$_{3}$ThH$_{40}$, $R\bar{3}m$-LaThH$_{20}$, and $I4/mmm$-LaTh$_{3}$H$_{40}$) with H$_{29}$ and H$_{32}$ cages can be thermodynamically stable below 200 GPa. However, the phase decomposition can happen to only (La,Th)H$_{9}$ when the pressure is above 150 GPa. More importantly, the electron-phonon coupling (EPC) calculations show that the (La,Th)H$_{10}$ series could the potential superconductors, of which $I4/mmm$-La$_{3}$ThH$_{40}$ at 200 GPa exhibits the large EPC constant $\lambda$ = 2.46 with a highest transition temperature ($T_\mathrm{c}$) of 210 K. Since there are few previous studies on ternary hydrides composed of actinide metals, the present work would greatly stimulate the further discovery of this type of ternary hydrides and provide useful guidance for the high-pressure experimental studies on them.