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.
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