Dynamics of Dirac quasi-particles in lattice vibration and anomalous phonon frequency shift of graphene

By exact solution of time-dependent Schrِdinger equation of electron in graphene under interaction with E 2 g phonons, we investigate the dynamical behavior of Dirac quasi-particle in the process of lattice vibration. Due to the global geometric phases acquired by electron during lattice vibration, an anomalous shift of the vibration frequency is obtained. We calculate the Fermi energy dependence of frequency shift which is in consistence with experiment in case of small doping density.