  The interaction of electron-hole pairs with lattice vibrations exhibits a
wealth of intriguing physical phenomena. The Kohn anomaly is a renowned example
where electron-phonon coupling leads to non-analytic phonon dispersion at
specific momentum nesting the Fermi surface. Here we report evidence of another
type of phonon anomaly discovered by low temperature Raman spectroscopy in
bilayer graphene where the charge density is modulated by the electric field
effect. This anomaly, arising from charge-tunable modulations of particle-hole
pairs that are resonantly coupled to lattice vibrations, is predicted to
exhibit a logarithmic divergence in the long-wavelength optical-phonon energy.
In a non-uniform bilayer of graphene, the logarithmic divergence is abated by
charge density inhomogeneity leaving as a vestige an anomalous phonon
softening. The observed softening marks the first confirmation of the phonon
anomaly as a key signature of the resonant deformation-potential
electron-phonon coupling. The high sensitivity of the phonon softening to
charge density non-uniformity creates significant venues to explore the
interplay between fundamental interactions and disorder in the atomic layers.
