Singular behavior of the Debye-Waller factor of graphene

It is shown that the mean-square displacement or the exponent of the Debye-Waller factor of graphene has a singularity except at zero temperature. The zero-temperature values of the mean-square displacement are calculated separately for planar and out-of-plane phonon modes for graphene. These values give the Debye-Waller factor that can be used to model various scattering processes at temperatures much lower than the Debye temperature of graphene. Since the Debye temperature of graphene is about 2300 K for planar modes, the calculated values should provide a useful estimate of the Debye-Waller factor at temperatures of practical interest. Finally, it is shown qualitatively that the singularity can be removed by accounting for the finite size of real graphene crystals.