We study theoretically AB-stacked honeycomb bilayers driven by light in resonance with an infrared phonon within a tight-binding description. We characterize the phonon properties of honeycomb bilayers with group theory and construct an electronic time-dependent tight-binding model for the system following photo-excitation in resonance with an infrared phonon. We adopt an atomically adiabatic approximation, introduced by Mohantya and Heller PNAS 116, 18316 (2019) to describe classically vibrating nuclei, but obtain the Floquet quasienergy spectrum associated with the time-dependent model exactly. We introduce a general scheme to disentangle the complex low-frequency Floquet spectrum to elucidate the relevant Floquet bands. As a prototypical example, we consider bilayer graphene. We find that light in the low-frequency regime can induce a bandgap in the quasienergy spectrum in the vicinity of the K points even if it is linearly polarized, in contrast with the expectations within the Born-Oppenheimer approximation and the high-frequency regime. Finally, we analyze the diabaticity of the driven electron and driven phonon processes and found contrasting effects on the autocorrelation functions at the same driving frequency: driven phonons preserve the character of the initial state while driven electrons exhibit strong deviations within a few drive cycles. The procedure outlined here can be applied to other materials to describe the combined effects of low-frequency light on phonons and electrons.
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