Electron-phonon interaction and full counting statistics in molecular junctions

The full counting statistics of a molecular level weakly interacting with a local phonon mode is derived. We find an analytic formula that gives the behavior of arbitrary irreducible moments of the distribution upon phonon excitation. The underlying competition between quasielastic and inelastic processes results in the formation of domains in parameter space characterized by a given sign in the jump of the irreducible moments. In the limit of perfect transmission, the corresponding distribution is distorted from Gaussian statistics for electrons to Poissonian transfer of holes above the inelastic threshold.

Figure 1

Upper figure: second-order electron-phonon self energy in Keldysh space $\left(\alpha ,\beta =\pm \right)$. (a) Hartree term. (b) Exchange term. lower figure: corresponding bubble expansion of the CGF $S\left(\chi \right)$. (c) Unperturbed term. (d) Hartree term. (e) Exchange term.

Figure 2

(Color online) Sign of the jump in the derivative with respect to voltage of the order $n$ cumulant at phonon energy $\Delta F_n$. Parameter space is $\left\{\alpha ,T\left(0\right)\right\}$, and phonon energy is ${\omega }_0={10}^{-2}\Gamma$. From up to down: $\Delta F_1$ (jump in the conductance), $\Delta F_2$ (jump in the noise), and $\Delta F_3$ (jump in the third moment). The blue (red) color encodes a negative (positive) jump.