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The coupling of the electrons and phonons is governed wisely by the symmetry properties of the crystal structures. In particular, for two-dimensional (2D) systems, it has been suggested that the electrons do not couple to phonons with pure out-of-plane distortion, as long as there is a $\sigma_h$ symmetry. Over here, we show such a statement is correct when constituents of the unit-cell layer are only located in the $\sigma_h$ symmetric plane; a prominent example of such a system is graphene. For those of 2D crystals where atoms are vertically located away from the horizontal symmetric plane (e.g., 1H TMDs), acoustic flexural modes do not couple to the electrons up to linear order, while optical flexural phonons, which preserve $\sigma_h$ symmetry, couple with the electrons.
Our conclusions are supported by an analytic argument together with numerical calculations using density functional perturbation theory.
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