  We unravel the critical role of vibrational mode softening in single-molecule
electronic devices at high bias. Our theoretical analysis is carried out with a
minimal model for molecular junctions, with mode softening arising due to
quadratic electron-vibration couplings, and by developing a mean-field
approach. We discover that the negative sign of the quadratic
electron-vibration coupling coefficient can realize at high voltage a sharp
negative differential resistance (NDR) effect with a large peak-to-valley
ratio. Calculated current-voltage characteristics, obtained based on ab initio
parameters for a nitro-substituted oligo(phenylene ethynylene) junction, agree
very well with measurements. Our results establish that vibrational mode
softening is a crucial effect at high voltage, underlying NDR, a substantial
diode effect, and the breakdown of current-carrying molecular junctions.
