Valley polarization induced second harmonic generation in graphene

The valley degeneracy of electron states in graphene stimulates intensive research of valley-related optical and transport phenomena. While many proposals on how to manipulate valley states have been put forward, experimental access to the valley polarization in graphene is still a challenge. Here, we develop a theory of the second optical harmonic generation in graphene and show that this effect can be used to measure the degree and sign of the valley polarization. We show that, at the normal incidence of radiation, the second harmonic generation stems from imbalance of carrier populations in the valleys. The effect has a specific polarization dependence reflecting the trigonal symmetry of electron valley and is resonantly enhanced if the energy of incident photons is close to the Fermi energy.