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Phonon study of Jahn-Teller distortion and phase stability in NaMnO$_2$ for sodium-ion batteries

Cathode materials undergo various phase transitions during the charge/discharge process, and the structural transitions significantly affect the battery performance. Although phonon properties can provide a direct clue for structural stability and transitions, it has been less explored in sodium cathode materials. Here, using the first-principles calculations, we investigate phonon and electronic properties of various layered NaMnO$_2$ materials, especially focusing on the dependency of the Jahn-Teller distortion of Mn$^{3+}$. The phonon dispersion curves show that the O$'$3 and P$'$2 structures with the Jahn-Teller distortion are dynamically stable in contrast to undistorted O3 and P2 structures. The structural instability of O3 and P2 structures is directly observed from the imaginary phonon frequencies, as so-called phonon soft modes, whose corresponding displacements are from O atoms distorting along the local Mn-O bond direction in the MnO$_6$ octahedra. This is consistent with the experimental stability and a structural transition with the Jahn-Teller distortion at the high Na concentration. Furthermore, the orbital-decomposed density of states presents the orbital redistribution by the Jahn-Teller distortion such as $e_g$-band splitting, and the stability of O$'$3 and P$'$2 is not sensitive to the electron-electron correlation. Our results demonstrate the importance of phonon analysis to further understand the structural stability and phase transitions in cathode materials.