Electron transport through correlated molecules computed using the time-independent Wigner function: Two critical tests

We have implemented the linear response approximation of a method proposed to compute the electron transport through correlated molecules based on the time-independent Wigner function [P. Delaney and J. C. Greer, Phys. Rev. Lett. 93, 036805 (2004)]. The results thus obtained for the zero-bias conductance through a quantum dot both without and with correlations demonstrate that this method is neither quantitatively nor qualitatively able to provide a correct physical description of the electric transport through nanosystems. We present an analysis indicating that the failure is due to the manner of imposing the boundary conditions and that it cannot be simply remedied.

Figure 1

(Color online) Results for the normalized conductance $G/G_0$ and lowest excitation energy $\delta {\epsilon }_t$ (tunneling splitting) obtained by exact diagonalization in the uncorrelated system $\left(U=0\right)$ for chains with 63 sites, $t=1$, and $t_d=0.4$. Note the logarithmic scale on the ordinate and the square root of the dot energy ($=\text{energy}$ barrier) ${\epsilon }_g$ on the abscissa. The linearity of the two curves at larger ${\epsilon }_g$ confirms the interpretation within the tunneling model.

Figure 2

(Color online) The difference $\delta f\left(q_R,p\right)\equiv f\left(q_R,p\right)-f_0\left(q_R,p\right)$ (in arbitrary units) plotted versus gate potential ${\epsilon }_g$ for 63 sites, $t=1$, $t_d=0.4$. The nonvanishing values at the right interface $q_R$ and positive momenta $p$ indicate that the Wigner distribution function of outgoing electrons is free. The values of $k$ $\left(p=k\pi /n_{q_L}\right)$ are given in the legend.

Figure 3

(Color online) Results on the normalized conductance $G/G_0$ obtained by exact diagonalization and the method based on the Wigner function for 63 sites in the absence of correlations $U=0$ (same parameters as in Fig. 1).

Figure 4

(Color online) Results for the conductance $G/G_0$ and the bias-induced condensation energy $\delta \mathcal{W}$ (the latter in arbitrary units) obtained by means of the SWF method for boundaries chosen at $q_R=-q_L=2$, 3, 4, and 5 (values increasing upwards for the $G$ curves and downwards for the $\delta \mathcal{W}$ curves at, say, ${\epsilon }_g=0.5$). Noteworthy, the conductance does not sensitively depend on the choice of boundaries.

Figure 5

(Color online) SFW results for the conductance $G/G_0$ of the chain with 63 sites at $t=1$ and $t_d=0.4$. The curves computed without imposing the continuity equation exhibit a strong site-dependent current and substantially depart from that labeled by uniform, computed by imposing this equation. Nor the average taken along the chain (label average) represents a satisfactory approximation of the latter. The number of site $q$ in the chain is specified in the legend. The dot is located at $q=0$.

Figure 6

(Color online) Dot occupancy $n_d$ computed by exact diagonalization for 7- and 11-site clusters (solid lines and points, respectively) plotted versus dot energy ${\epsilon }_g$. Notice the small difference between the results for 7 and 11 sites, indicating that finite-size effects play a reduced role.

Figure 7

(Color online) Coulomb blockade peaks of the zero-bias conductance for seven-site clusters as predicted by the SWF method. The solid, dotted, and dashed lines correspond to the values $t_d=0.125$, 0.25, and 0.5 respectively. The values of $U$ are given in the legend and $t=1$. Notice that, unphysically, the maximum conductance is predicted to increase with decreasing $t_d$ even beyond the ideal value $G_0$ $\left(G/G_0>1\right)$.