Figure 1

Barrier DOS as a function of the Falicov-Kimball interaction $U$. The different line widths and styles denote different $U$ values, as detailed in the legend. Note how the DOS initially evolves as in the bulk, with the DOS being reduced near $\omega =0$, and the bandwidth increasing. But as we pass through the Mott transition, we see that the double-peak Mott-Hubbard bands appear, but so does a low-energy (interface-localized) band near $\omega =0$, which looks like a low-weight metallic band for large $U$.Reuse & Permissions

Figure 2

Lead DOS for an $N=5$ barrier device with $U=2$. The different panels show the DOS in the first metal plane to the left of the barrier, in the second, the third, the tenth, and the thirtieth. Note how the system approaches the bulk cubic DOS as it moves further from the interface, as expected. A careful examination of the panels shows that the “flat” region with $\mid \omega \mid <2$ shows a half-period oscillation for each unit of distance from the current plane to the interface, but the amplitude shrinks dramatically as we move further from the interface.Reuse & Permissions

Figure 3

Lead DOS for an $N=5$ barrier device with $U=4$. The different panels show the DOS in the first metal plane to the left of the barrier, in the second, the third, the tenth, and the thirtieth. Note how the amplitude of the oscillations increases as $U$ increases.Reuse & Permissions

Figure 4

Lead DOS for an $N=5$ barrier device with $U=6$. The different panels show the DOS in the first metal plane to the left of the barrier, in the second, the third, the tenth, and the thirtieth. Note how the amplitude of the oscillations is even larger here. A careful examination shows there are also oscillations (with the same kind of increase in the number of half periods with the distance from the interface) in the region $\mid \omega \mid >2$.Reuse & Permissions

Figure 5

DOS at specific values of $\omega$ as a function of the plane position in the device. We plot only the left-hand piece of the plots, since the right-hand piece is a mirror image of the left-hand piece. Note that the $U=6$ panel is a semilogarithmic plot. The four values of $\omega$ for $U=2$ are 0.0, 3.0, 4.0, and 5.0. The barrier thicknesses are $N=1$, 5, 10, 20, 40, and 80. The four values of $\omega$ for $U=4$ are 0.0, 2.5, 3.5, and 5.0. The barrier thicknesses are $N=1$, 5, 10, 20, 40, and 80. The four values of $\omega$ for $U=6$ are 0.0, 0.2, 0.4, and 1.0. The thicknesses are $N=1$, 4, 7, 10, 15, and 20. Note how all curves lie on top of each other in the metallic lead, indicating the structure in the metallic lead is frozen in for an $N=1$ barrier, and does not significantly change with increasing $N$. In the barrier, we only have oscillations at the interface, and then the curves either are flat with thickness ($U=2$ and 4), or exponentially decreasing or flat $(U=6)$. The little tails that stick out for the lowest two frequencies with $U=6$ show that the middle plane of the barrier does not follow the same exponential decay as the other planes do. But the exponent of the exponential decay is frozen in starting at $N\approx 1$.Reuse & Permissions

Figure 6

(Color online) False-color plot of the DOS for a $N=1$ barrier plane device with $U=6$. The barrier plane is just the lowest plane at the bottom of the figure, while the 30 metallic planes lie on top. Note how the ripples of the Friedel oscillations are most visible in the central region, where the DOS has a plateau.Reuse & Permissions

Figure 7

(Color online) False-color plot of the DOS for a $N=20$ barrier plane device with $U=6$. The barrier planes are the lower 10 planes, while the 30 metallic planes lie on top. Note how the ripples of the Friedel oscillations agree with those in Fig. 6. In the barrier, the DOS decreases rapidly on this linear scale, and shows few oscillations, but one can see some small oscillations near the band edges in both regions.Reuse & Permissions

Figure 8

Semilogarithmic plot of the imaginary part of the self-energy on the central plane of the barrier at small frequency for five different thickness barriers ($N=1$, 4, 7, 10, and 15). Note how the imaginary part of the self-energy becomes very small for frequencies close to $\omega =0$, but as we approach $\omega =0$, a sharp delta-function-like peak develops that narrows as the barrier is made thicker. It is precisely this structure that is hard to reproduce with numerical calculations. Note that this kind of a self-energy is very similar to what is seen in the hypercubic lattice in infinite dimensions.Reuse & Permissions

Figure 9

Thouless energy for a $U=4$ (diffusive, but very strongly scattering metal) barrier as a function of the barrier thickness $L=Na$. The different curves correspond to different temperatures. The top panel multiplies the Thouless energy by $L^2$ to try to isolate the prefactor for the diffusive transport, while the bottom panel plots the Thouless energy on a semilogarithmic plot. Note that the temperature dependence of the constant, seen for thick barriers in panel (a), arises from the fact that the $U=4$ DOS has significant low-energy structure, because there is a dip that develops near the chemical potential, so the temperature dependence is both stronger than expected for normal metals, and anomalous because many more states are involved as $T$ is increased, i.e., it behaves more like an insulator.Reuse & Permissions

Figure 10

Thouless energy for a $U=6$ (Mott-insulating) barrier as a function of temperature on a log-log plot. The different curves correspond to different thicknesses of the barrier, ranging from $N=1$ for the top curve to $N=2$, 3, 4, 5, 7, 10, 15, and 20 as we move down the plot. Note how the Thouless energy picks up dramatic temperature dependence here. The dashed line is the curve where $E_{\mathit{Th}}=T$. We find that when the Thouless energy equals the temperature, interesting effects occur (see below).Reuse & Permissions

Figure 11

Resistance-area product for nanostructures with $U=2$, 4, 5, and 6, and various thicknesses. Panel (a) is a semilogarithmic plot, while panel (b) is a linear plot. The temperature is $T=0.01$ in both panels. Note how the correlated insulator $(U=6)$ has an exponential growth with thickness as expected for a tunneling process, but it turns over at the thickest junction, indicating a crossover to the incoherent transport regime. The $U=5$ data, which is close to the critical point for a MIT, has neither linear, nor exponential growth of its resistance-area product. The metallic cases ($U=2$ and 4) have perfect linear scaling of the resistance with current, with a nonzero intercept corresponding to the contact resistance. This may be surprising for $U=4$, because it is so strongly scattering (with a mean free path much less than a lattice spacing), that one would not think a semiclassical approach should apply there. The constant satisfies ${\sigma }_0=2e^2∕h{a}^2$.Reuse & Permissions

Figure 12

Resistance-area product for nanostructures with (a) $U=4$ and (b) $U=6$ as a function of temperature [panel (a) is on a linear scale, and panel (b) is a log-log plot]. In panel (a) we include results for $N=1$, 2, (lowest two curves), 5, 10, 15, 20, 40, 60, and 80. Note how at each temperature there is a linear dependence of the resistance-area product with the thickness of the junction. Note further, that these junctions have anomalous temperature dependence for a metal (they actually look insulating in their dependence). In panel (b), we show the results for $U=6$ with $N=1–10$, 15, and 20. Note at low temperature we have tunneling, as the resistance-area product is weakly dependent on temperature, and the steps are equally spaced as a function of thickness, indicating exponential dependence on the thickness. At higher temperatures, there is a crossover to the incoherent transport regime, with the resistance-area product picking up a strong $T$ dependence, and scaling linearly with the thickness. The dotted line that connects the solid dots is a plot of the resistance-area value at the temperature where $E_{\mathit{Th}}=T$ which determines the crossover.Reuse & Permissions