Theory of Multiphoton Magnetoabsorption in Semiconductors

We calculate direct interband transitions induced by a strong oscillating electric field, in the presence of a longitudinal or transverse magnetic field, including the largest, intraband effects of the electric field in the original wave functions. This is an extension of the Keldysh treatment to include the magnetic field, and also to include absorption of a second optical electric field, where we work in the effective-mass approximation, which is more suitable for the magnetic field case. The intraband effects, combined with an interband electric perturbation obtained from the effective-mass theory, induce Zener or photon-assisted (Franz-Keldysh) tunneling in parallel fields as the frequency of the strong electric field becomes small, and multiphoton transitions for relatively high frequencies and moderate electric fields. When a one-photon $\Delta n=0\mathrm{}$ transition is allowed, the intraband effects induce multiphoton transitions with the same selection rule, except that in a transverse magnetic field H, $\Delta n=\pm m$ transitions are allowed, but reduced by a factor proportional to $H^m$. The transition energies are those predicted earlier by Lax, where for a longitudinal magnetic field the electric-field energy shift is that obtained by Keldysh for zero magnetic field, and is modified for a transverse field by the cyclotron resonance frequencies.