Current-induced nonequilibrium vibrations in single-molecule devices

Finite-bias electron transport through single molecules generally induces nonequilibrium molecular vibrations (phonons). By a mapping to a Fokker-Planck equation, we obtain analytical scaling forms for the nonequilibrium phonon distribution in the limit of weak electron-phonon coupling $\lambda$ within a minimal model. Remarkably, the width of the phonon distribution diverges as $\sim {\lambda }^{-\alpha }$ when the coupling decreases, with voltage-dependent, noninteger exponents $\alpha$. This implies a breakdown of perturbation theory in the electron-phonon coupling for fully developed nonequilibrium. We also discuss possible experimental implications of this result, such as current-induced dissociation of molecules.

Figure 1

(a) Scaling behavior of the phonon distribution width as a function of electron-phonon coupling for representative values of $(eV∕\hslash \omega ,ϵ∕\hslash \omega )$ equal to $a:(3,0)$; $b:(5,1)$; $c:(5,0)$; $d:(7,1)$. (b) Width of the phonon distribution as a function of bias and gate voltage at $\lambda =0.15$ and $k_{B}T=0.005\hslash \omega$. The scaling of the width differs for the plateaus according to A: $q_0\sim {\lambda }^{-1}$, B: $q_0\sim {\lambda }^{-4∕3}$, and C: $q_0\sim {\lambda }^{-3∕2}$.

Figure 2

(Color online) Phonon distributions $P_q$ at bias voltages (a) $eV=3\hslash \omega$ and (b) $eV=5\hslash \omega$, plotted for several coupling strengths $\lambda$ at $ϵ=0$, $k_{B}T=0.05\hslash \omega$. The insets show that these distributions (approximately) collapse to universal curves given by the red (dark gray) smooth curves, which are the solutions of the the Fokker-Planck equation (11) specific to each voltage range $n\hslash \omega <eV∕2<(n+1)\hslash \omega$.

Figure 3

(a) Width of the phonon distribution as a function of electron-phonon coupling strength for several vibrational relaxation times $\tau$ and $eV=3\hslash \omega$, $k_{B}T=0.05\hslash \omega$, $ϵ=0$. For $\lambda >0.25$ all curves show the approximate $q_0\sim {\lambda }^{-1}$ scaling. Below a relaxation-rate-dependent crossover point ${\lambda }_0$, the WHM is strongly suppressed due to direct relaxation. (b) Crossover point ${\lambda }_0$ vs relaxation time $\tau$.

Figure 4

Mean dissociation rate as a function of electron-phonon coupling strength at $k_{B}T=0.005\hslash \omega$ for several bias voltages as obtained by Monte Carlo simulation with a Morse potential containing ten bound states.