Nonequilibrium distribution functions for quantum transport: Universality and approximation for the steady state regime

We derive a general expression for the electron nonequilibrium (NE) distribution function in the context of steady state quantum transport through a two-terminal nanodevice with interactions. The central idea for the use of NE distributions for open quantum systems is that both the NE and many-body (MB) effects are taken into account in the statistics of the finite size system connected to reservoirs. We develop an alternative scheme to calculate the NE steady state properties of such systems. The method, using NE distribution and spectral functions, presents several advantages, and is equivalent to conventional steady state NE Green's function (NEGF) calculations when the same level of approximation for the MB interaction is used. The advantages of our method resides in the fact that the NE distribution and spectral functions have better analytic behavior for numerical calculations. Furthermore, our approach offers the possibility of introducing further approximations, not only at the level of the MB interaction as in NEGF, but also at the level of the functional form used for the NE distributions. For the single-level model with electron-phonon coupling we have considered, such approximations provide a good representation of the exact results, for either the NE distributions themselves or the transport properties. We also derive the formal extensions of our method for systems consisting of several electronic levels and several vibration modes.

Figure 1

NE distribution functions for the off-resonant regime (${\varepsilon }_0=0.50$) and for different biases $V$. (a) $V=0.2<{\omega }_0$, (b) $V=0.4\sim {\omega }_0$, (c) $V=0.75>{\omega }_0$, (d) $V=1.0\gg {\omega }_0$. The NE distribution $f^{\mathrm{NE}}$ is completely different from the noninteraction NE distribution $f_0^{\mathrm{NE}}$ when $V\ge {\omega }_0$; in this case inelastic processes occur and induce a redistribution of the electron population in the region $C$. The other parameters are ${\gamma }_0=0.09$, ${\omega }_0=0.3$, $t_{0\alpha }=0.15$, $T_{\alpha }=0.017$, ${\eta }_L=1$, ${\varepsilon }_{\alpha }=0$, ${\beta }_{\alpha }=2$.

Figure 2

NE distribution functions for the off-resonant regime (${\varepsilon }_0=0.70$) for different approximations and for different coupling strengths ${\gamma }_0$. (a) ${\gamma }_0=0.03$ (${\gamma }_0/{\omega }_0=0.15$), (b) ${\gamma }_0=0.06$ (${\gamma }_0/{\omega }_0=0.3$), (c) ${\gamma }_0=0.08$ (${\gamma }_0/{\omega }_0=0.4$). Only for weak coupling, all approximated NE distributions provide a good representation of the exact distribution $f^{\mathrm{NE}}$. We recall that for fully self-consistent calculations, the results obtained for $f^{\mathrm{NE}}$ with our method are strictly equivalent to those obtained from NEGF calculations. See text for more detailed comments. The other parameters are $V=1.0$, ${\omega }_0=0.2$, $t_{0\alpha }=0.22$, $T_{\alpha }=0.017$, ${\eta }_L=1$, ${\varepsilon }_{\alpha }=0$, ${\beta }_{\alpha }=2$.

Figure 3

Dynamical conductance $G\left(V\right)=dI/dV$ (in units of quantum of conductance $G_0$) for the off-resonant regime (${\varepsilon }_0=0.70$). $G\left(V\right)$ calculated with the approximated distribution $f_{\left(1\right)}^{\mathrm{NE}}$ (solid line) gives a good representation of the conductance calculated with the exact distribution $f^{\mathrm{NE}}$ (dashed line). The latter is strictly equivalent to NEGF calculations. The other parameters are ${\omega }_0=0.3$, ${\gamma }_0=0.10$, $t_{0\alpha }=0.19$, $T_{\alpha }=0.017$, ${\eta }_L=1$, ${\varepsilon }_{\alpha }=0$, ${\beta }_{\alpha }=2$.

Figure 4

IETS signal $d^{2}I/d{V}^2$, normalized by the conductance $G\left(V\right)$, for the far-off-resonant regime (${\varepsilon }_0=3.70$). The IETS calculated with the approximated distributions $f_{\left(1\right)}^{\mathrm{NE}}$ (solid line) or even $f_{\mathrm{LOE}}^{\mathrm{NE}}$ (squares) gives a good representation of the IETS calculated with the exact distribution $f^{\mathrm{NE}}$ (dashed line). The results obtained with $f^{\mathrm{NE}}$ are strictly equivalent to NEGF calculations. The other parameters are ${\omega }_0=0.3$, ${\gamma }_0=0.12$, $t_{0L}=0.45$, $t_{0R}=0.10$, $T_{\alpha }=0.017$, ${\eta }_L=1$, ${\varepsilon }_{\alpha }=0$, ${\beta }_{\alpha }=2$.