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Coexistence of phononic Weyl, nodal line, and threefold excitations in chalcopyrite CdGeAs$_{2}$ and associated thermoelectric properties

Realization of topologically protected quantum states leads to unprecedented opportunities for fundamental science and device applications. Here, we demonstrate the coexistence of multiple topological phononic states and calculate the associated thermoelectric properties of a chalcopyrite material CdGeAs$_2$ using first-principles theoretical modeling. CdGeAs$_{2}$ is a direct bandgap semiconductor with a bandgap of $0.65$ eV. By analysing the phonon spectrum and associated symmetries, we show the presence of nearly isolated Weyl, nodal line, and threefold band crossings in CdGeAs$_2$. Specifically, the two triply degenerate points (TDPs) identified on the $k_{z}$ axis are formed by the optical phonons bands 7, 8, and 9 with type-II energy dispersion. These TDPs form a time reversal pair and are connected by a straight nodal line with zero Berry phase. The TDPs formed between bands 14, 15, and 16 exhibit type-I crossings and are connected through the open straight nodal line. Our transport calculations show a large thermopower exceeding $\sim$500 and $200$ $\rm μV/K$ for the hole and electron carriers, respectively, above 500 K with a carrier doping of 10$^{18}$ cm$^{-3}$. The large thermopower in $p$-type CdGeAs$_{2}$ is a consequence of the sharp density of states appear from the presence of a heavy hole band at the $Γ$ point. We argue that the presence of topological states in the phonon bands could lead to low lattice thermal conductivity and drive a high figure-of-merit in CdGeAs$_{2}$.