We present a comparative ab initio study of Li, Na, and Mg storage in tin, including phononic effects and phase competition between α and β Sn. Mg doping at low concentration is found to stabilize the β phase. On the contrary, Li and Na doping is shown to reverse the stability of the phases at room temperature: Li/Na-doped α-Sn is more stable than Li/Na-doped β-Sn up to a temperature of around 380/400 K. This may rationalize the formation of α-Sn upon lithiation and delithiation of β-Sn anodes reported in experimental studies. The changes in phase stability with Li/Na/Mg doping are directly related to the intercalation energies of Li/Na/Mg in one phase versus the other: at 300 K, Li/Na is easier intercalated in α-Sn (-0.37/-0.08 eV) than in β-Sn (0.06/0.49 eV), while Mg intercalation energy is, although positive (i.e., unfavored intercalation), lower in β-Sn (0.53 eV) than in α-Sn (0.66 eV). The temperature effect is found to affect significantly the intercalation energy, by up to 0.13 eV at 300 K. Analysis of diffusion barriers shows that Li, Na, and Mg diffusion in β-Sn is anisotropic with migration barriers along the (001) direction (respectively, 0.01, 0.22, and 0.07 eV) significantly lower than those in α-Sn (respectively, 0.20, 0.52, and 0.40 eV).
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