Nonequilibrium Charge Susceptibility and Dynamical Conductance: Identification of Scattering Processes in Quantum Transport

We calculate the nonequilibrium charge transport properties of nanoscale junctions in the steady state and extend the concept of charge susceptibility to the nonequilibrium conditions. We show that the nonequilibrium charge susceptibility is related to the nonlinear dynamical conductance. In spectroscopic terms, both contain the same features versus applied bias when charge fluctuation occurs in the corresponding electronic resonances. However, we show that, while the conductance exhibits features at biases corresponding to inelastic scattering with no charge fluctuations, the nonequilibrium charge susceptibility does not. We suggest that measuring both the nonequilibrium conductance and charge susceptibility in the same experiment will permit us to differentiate between different scattering processes in quantum transport.

Figure 1

Nonequilibrium charge susceptibility ${\chi }_c^{\mathrm{NE}}$ (solid lines) and dynamical conductance $G$ (dashed lines) versus applied bias. (a) Noninteraction case, with ${\epsilon }_0=0.5$ and ${\beta }_{\alpha }=0.7$. (b),(c) With interaction and for different transport regimes (${\beta }_{\alpha }=2$). From off-resonant to resonant: (b) ${\epsilon }_0=0.7$, (c) ${\epsilon }_0=0.5$, and (d) ${\epsilon }_0=0.15$. ${\chi }_c^{\mathrm{NE}}$ is rescaled by $\Gamma ={\Gamma }_{\alpha }({\mu }^{\mathrm{eq}})=t_{0\alpha }^2/{\beta }_{\alpha }$. On this scale, both ${\chi }_c^{\mathrm{NE}}$ and $G$ present the same spectral features: peaks associated with charge fluctuations in the electronic resonances. Calculations are done for symmetric coupling $t_{0\alpha }=0.15$ and asymmetric potential drops ${\mu }_L={\mu }^{\mathrm{eq}}+eV$ and ${\mu }_R={\mu }^{\mathrm{eq}}$. The other parameters are ${\omega }_0=0.3$, ${\gamma }_0=0.21$, and ${\epsilon }_{\alpha }=0$. The energy parameters are in eV, and $G$ is in units of $G_0=e^2/h$.

Figure 2

Nonequilibrium charge susceptibility ${\chi }_c^{\mathrm{NE}}$ (solid lines) and dynamical conductance $G$ (dashed lines) versus $V$. Same parameters as in Fig. 1c, except as otherwise stated. (a) Far beyond the wideband approximation: ${\beta }_{\alpha }=0.7$. (b) Interaction at the Fock level only. (c) Strong coupling to the leads $t_{0\alpha }=0.30={\omega }_0$. (d) Asymmetric coupling to the leads and potential drop, $t_{0L}=0.07$, $t_{0R}=0.15$, ${\eta }_L=t_{0R}/(t_{0L}+t_{0R})=0.68182$, and ${\epsilon }_0=0.70$.

Figure 3

Derivative ${\partial }_V{\chi }_c^{\mathrm{NE}}$ (dashed lines) and IETS signal ${\partial }_{V}G$ (solid lines). ${\partial }_V{\chi }_c^{\mathrm{NE}}$ does not have the peak or dip feature of the IETS at $V\sim {\omega }_0$. (Top) Resonant transport regime. Calculations were done with $t_{0\alpha }=1.50\sim {\beta }_{\alpha }$, ${\epsilon }_0=0.0$, and ${\gamma }_0=0.195$. (Bottom) Off-resonant regime for different ${\epsilon }_0$. ${\partial }_V{\chi }_c^{\mathrm{NE}}$ and ${\partial }_{V}G$ are normalized by ${\chi }_c^{\mathrm{NE}}$ and $G$, respectively. (a) ${\epsilon }_0=0.70$, (b) ${\epsilon }_0=0.99$, and (c) ${\epsilon }_0=1.20$. The arrows point out the position of the electron resonance at $V={\tilde{\epsilon }}_0-{\omega }_0$. Calculations were done with the same parameters as in Fig. 1b.