Hierarchical Energy Relaxation in Mesoscopic Tunnel Junctions: Effect of a Nonequilibrium Environment on Low-Temperature Transport

We develop a theory of far from the equilibrium transport in arrays of tunnel junctions. We find that if the rate of the electron-electron interactions exceeds the rate of the electron-phonon energy exchange, the energy relaxation ensuring the charge transfer may occur sequentially. In particular, cotunneling transport in arrays of junctions is dominated by the relaxation via the intermediate bosonic environment, the electron-hole excitations, rather than by the electron-phonon mechanism. The current-voltage characteristics are highly sensitive to the spectrum of the environmental modes and to the applied bias, which sets the lower bound for the effective temperature. We demonstrate that the energy gap in the electron-hole spectrum which opens below some critical temperature $T^{*}$ due to long-range Coulomb interactions gives rise to the suppression of the tunneling current.

Figure 1

(a) The effective circuit for the tunnel junction subject to bias $V$ and with the environment having the impedance $Z$. (b)–(c) Diagrammatic expansion of $P^{<}$ to the first and the second orders in $\rho$, respectively. The solid lines represent propagation of electrons, the dashed lines denote the environment excitations. The vertex with the two electron lines and one dashed line carries a factor $G_T\rho (\omega )/\omega$, the “two dashed-lines vertex” corresponds to $G_T\rho (\omega )\rho ({\omega }^{\prime })/(\omega {\omega }^{\prime })$.

Figure 2

(a) The single electron two-islands’ circuit. (b)–(e) Diagrams describing the forward inelastic cotunneling rate. The “up” arrows stand for the $e\mathrm{\text{−}}h$ pairs excited during the cotunneling and the “down” arrows correspond to the recombination of the $e\mathrm{\text{−}}h$ pairs. The vertices shown by boxes are proportional to the probability of an elemental $e\mathrm{\text{−}}h$ pair excitation, $\rho (\omega )/\omega$.