Cyclotron resonance of the magnetic ratchet effect and second harmonic generation in bilayer graphene

We model the magnetic ratchet effect in bilayer graphene in which a dc electric current is produced by an ac electric field of frequency $\omega$ in the presence of a steady in-plane magnetic field and inversion-symmetry breaking. In bilayer graphene, the ratchet effect is tunable by an external metallic gate which breaks inversion symmetry. For zero in-plane magnetic field, we show that trigonal warping and inversion-symmetry breaking are able to produce a large dc valley current, but not a non-zero total dc charge current. For the magnetic ratchet in a tilted magnetic field, the perpendicular field component induces cyclotron motion with frequency ${\omega }_c$ and we find that the dc current displays cyclotron resonance at ${\omega }_c=\omega$, although this peak in the current is actually smaller than its value at ${\omega }_c=0$. Second harmonic generation, however, is greatly enhanced by resonances at ${\omega }_c=\omega$ and ${\omega }_c=2\omega$ for which the current is generally much larger than at ${\omega }_c=0$.