Electric field-induced valley degeneracy lifting in uniaxial strained graphene: evidence from magnetophonon resonance

A double peak structure in the magneto-phonon resonance (MPR) spectrum of uniaxial strained graphene, under crossed electric and magnetic fields, is predicted. We focus on the $\Gamma$ point optical phonon modes coupled to the inter-Landau level transitions $0 \leftrightarrows \pm 1$ where MPR is expected to be more pronounced at high magnetic field. We derive the frequency shifts and the broadenings of the longitudinal (LO) and transverse (TO) optical phonon modes taking into account the effect of the strain modified electronic spectrum on the electron-phonon coupling. We show that the MPR lines acquire a double resonance structure originating from the two-fold valley degeneracy lifting. The latter is due to the different Landau level spacings in the two Dirac valleys resulting form the simultaneous action of the inplane electric field and the strain induced Dirac cone tilt. We discuss the role of some key parameters such as disorder, strain, doping and electric field amplitude on the emergence of the double resonance structure.