Scattering Theory of Current-Induced Forces in Mesoscopic Systems

We develop a scattering theory of current-induced forces exerted by the conduction electrons of a general mesoscopic conductor on slow “mechanical” degrees of freedom. Our theory describes the current-induced forces both in and out of equilibrium in terms of the scattering matrix of the phase-coherent conductor. Under general nonequilibrium conditions, the resulting mechanical Langevin dynamics is subject to both nonconservative and velocity-dependent Lorentz-like forces, in addition to (possibly negative) friction. We illustrate our results with a two-mode model inspired by hydrogen molecules in a break junction which exhibits limit-cycle dynamics of the mechanical modes.

Figure 1

(a) Limit cycle (blue solid line) and its approach (blue dotted line) at large bias vs stable oscillations at low bias (red asterisk) in the Langevin dynamics without fluctuating force. (b) Several periods of typical trajectories with fluctuating forces for the parameters of the limit cycle in (a). (c) Fourier transform of the correlation function $⟨X_1(t)X_2(t+\tau )⟩$. The limit cycle is signaled by a single peak, as opposed to two peaks in the absence of a limit cycle. (d) The same signature appears in the current-current correlation function, making the onset of limit-cycle dynamics observable in experiment. These zero-temperature results, based on analytical results for the current-induced forces, are obtained for ${\Lambda }_1={\lambda }_1{\sigma }_0$ and ${\Lambda }_2={\lambda }_2{\sigma }_x$ with ${\lambda }_1/{\lambda }_2=3/2$. In units of ${\lambda }^2/(M{\omega }_0^2)$ [where $\lambda =({\lambda }_1+{\lambda }_2)/2$], the elastic modes are degenerate with $\hslash {\omega }_0=0.014$, ${\Gamma }_{L,R}=\frac{1\pm 0.8}{2}({\sigma }_0\pm {\sigma }_z)$, and the hopping between the orbitals is $w=0.9$. The dimensionless coordinates are $x_i=(M{\omega }_0^2/\lambda )X_i$.