  Trilayer graphene in both ABA (Bernal) and ABC (rhombohedral) stacking
sequences is shown to exhibit intense infrared absorption from in-plane optical
phonons. The feature, lying at ~1580 cm-1, changes strongly with electrostatic
gating. For ABC-stacked graphene trilayers, we observed a large enhancement in
phonon absorption amplitude, as well as softening of the phonon mode, as the
Fermi level is tuned away from charge neutrality. A similar, but substantially
weaker effect is seen in samples with the more common ABA stacking order. The
strong infrared response of the optical phonons and the pronounced variation
with electrostatic gating and stacking-order reflect the interactions of the
phonons and electronic excitations in the two systems. The key experimental
findings can be reproduced within a simplified charged-phonon model that
considers the influence of charging through Pauli blocking of the electronic
transitions.
