  Anharmonic effects in an atomic monolayer thin crystal with honeycomb lattice
structure are investigated by analytical and numerical lattice dynamical
methods. Starting from a semi-empirical model for anharmonic couplings of third
and fourth order, we study the in-plane and out-of-plane (flexural) mode
components of the generalized wave vector dependent Gr\"uneisen parameters, the
thermal tension and the thermal expansion coefficients as function of
temperature and crystal size. From the resonances of the
displacement-displacement correlation functions we obtain the renormalization
and decay rate of in-plane and flexural phonons as function of temperature,
wave vector and crystal size in the classical and in the quantum regime.
Quantitative results are presented for graphene. There we find that the
transition temperature from negative to positive thermal expansion is lowered
with smaller system size and by renormalization of the flexural mode frequency.
Extensive numerical analysis throughout the Brillouin zone explores various
decay and scattering channels. The relative importance of Normal and Umklapp
processes is investigated. The work is complementary to crystalline membrane
theory and computational studies of anharmonic effects in two-dimensional
crystals.
