Kohn anomaly of optical zone boundary phonons in uniaxial strained graphene: role of the electronic band structure

One of the unique properties of graphene is its extremely high mechanical strength. Several studies have shown that the mechanical failure of graphene sheet under a tensile strain is due to the enhancement of the Kohn anomaly of the zone boundary transverse optical phonon modes. In this work, we derive an analytical expression of the Kohn anomaly parameter $\alpha_{\vec{K}}$ of these phonons in graphene deformed by a uniaxial strain along the armchair direction. We show that, the tilt of Dirac cones, induced by the strain, contributes to the enhancement of the Kohn anomaly under a tensile deformation and gives rise to a dominant contribution of the so-called {\it outer} intervalley mediated phonon processes. Moreover, the Kohn anomaly is found to be anisotropic with respect to the phonon wave vectors around the K point. This anisotropy may be at the origin of the light polarization dependence of the Raman 2D band of the strained graphene. Our results uncover, not only, the role of the Kohn anomaly in the anisotropic mechanical failure of the graphene sheet, under strains applied along the armchair and zigzag directions, but shed also light on the doping induced strengthening of strained graphene.