  The effect of electronic current on the atomic motion still poses many open
questions, and several mechanisms are at play. Recently there has been focus on
the importance of the current-induced non-conservative forces (NC) and
Berry-phase derived forces (BP) regarding the stability of molecular-scale
contacts. Systems based on molecules bridging electrically gated graphene
electrodes may offer an interesting test-bed for these effects.
  We employ a semi-classical Langevin approach in combination with DFT
calculations to study the current-induced vibrational dynamics of an atomic
carbon chain connecting electrically gated graphene electrodes. This
illustrates how the device stability can be predicted solely from the modes
obtained from the Langevin equation including the current induced forces. We
point out that the gate offers control of the current independent of bias
voltage which can be used to explore current-induced vibrational instabilities
due the NC/BP forces. Furthermore, using tight-binding and the Brenner
potential we illustrate how Langevin-type molecular dynamics can be performed
including the Joule heating effect for the carbon chain systems. Molecular
dynamics including current-induced forces enables an energy redistribution
mechanism among the modes, mediated by anharmonic interactions, which is found
to be vital in the description of the electronic heating.
  We have developed a semi-classical Langevin equation approach which can be
used to explore current-induced dynamics and instabilities. We find
instabilities at experimentally relevant bias and gate voltages for the carbon
chain system.
