Thermoelectric transport parallel to the planes in a multilayered Mott-Hubbard heterostructure

We present a theory for charge and heat transport parallel to the interfaces of a multilayer (ML) of the ABA type, where A and B are materials with strongly correlated electrons. When separated, both materials are half-filled Mott-Hubbard insulators with large gaps in their excitation spectrum. In a ML, the renormalization of the energy bands gives rise to a charge reconstruction which breaks the charge neutrality of the planes next to the interface. The ensuing electrical field couples self-consistently to the itinerant electrons, so that the properties of the ML crucially depend on an interplay between the on-site Coulomb forces and the long range electrostatic forces. Using the Falicov-Kimball model, we compute the Green's function and the local charge on each plane of the ML by inhomogeneous DMFT and find the corresponding electrical potential from Poisson's equation. The self-consistent solution is obtained by an iterative procedure, which yields the reconstructed charge profile, the electrical potential, the planar density of states, the transport function, and the transport coefficients of the device. For the right choice of parameters, we find that a heterostructure built of two Mott-Hubbard insulators exhibits, in a large temperature interval, a linear conductivity and a large temperature-independent thermopower. The charge and energy currents are confined to the central part of the ML. Our results indicate that correlated multilayers have the potential for applications; by tuning the band shift and the Coulomb correlation on the central planes, we can bring the chemical potential in the immediate proximity of the Mott-Hubbard gap edge and optimize the transport properties of the device. In such a heterostructure, a small gate voltage can easily induce a MI transition. Furthermore, the right combination of strongly correlated materials with small ZT can produce, theoretically at least, a heterostructure with a large ZT.

Figure 1

The reconstructed charge, $\delta n_{\alpha }=n_{\alpha }-n_{BG}$ and the electrical potential $V_{\alpha }$ are plotted at various temperatures as functions of the planar index for the band shift $\mathrm{\Delta }{E}_c=-0.9$ in the central planes. The full black line, dashed magenta, full blue, and red lines correspond to $T=0.002,0.001,0.05$, and 0.1, respectively. Left panel: The reconstructed charge $n_{\alpha }$. Right panel: The potential $V_{\alpha }$ given by Poisson's equation. One can clearly see that for temperatures below $T=0.05$, the charging error is small enough that the iterative solution is physically valid.

Figure 2

The reconstructed charge $\delta n_{\alpha }=n_{\alpha }-n_{BG}$ is plotted versus planar index $\alpha$ for various values of the screening parameter ${\epsilon }_{\mathrm{Schott}}$ on the central planes. The other parameters are the same as in Fig. 1. Left panel: $\delta n_{\alpha }$ for $T=0.01$. Right panel: $\delta n_{\alpha }$ for $T=0.1$; the lines a, b, c, d are computed with ${\epsilon }_{\mathrm{Schott}}=1,10,20,50$, respectively. The corresponding charging errors $\delta n$ are $-0.00018 -0.00009, -0.00008$, and $-0.00007$ for $T=0.01$ and $-0.00614, -0.00366, -0.00261$, and $-0.00141$ for $T=0.1$. We do not report transport results for charging errors larger than 0.001.

Figure 3

Left panel: The local DOS, calculated at $k_{B}T=0.003$ (full lines) and $k_{B}T=0.05$ (dashed-dotted lines) is plotted, for $\mathrm{\Delta }{E}_c=-0.9$ and constant ${\epsilon }_{\mathrm{Schott}}$ in the leads and the center, as a function of frequency for several planes of the ML. The black lines show the plane in the lead which is far away from the interface ($\alpha =10$), red lines show the last plane in the left lead ($\alpha =40$), the magenta lines show the leftmost plane in the central part ($\alpha =41$), and the blue lines show the mirror plane of the ML ($\alpha =46$). Central panel: The low-frequency part of the local DOS is shown for the same planes as in the left panel. The inset shows the very-low frequency part of the planar DOS. Right panel: ${\rho }_{\alpha }$ plotted versus $\omega$ for $k_{B}T=0.1,\mathrm{\Delta }{E}_c=-0.9,{\epsilon }_{\mathrm{Schott}}=0.4$ in the leads and various values of ${\epsilon }_{\mathrm{Schott}}$ in the center. The blue, magenta, red, and black curves show the local DOS on the mirror plane ($\alpha =46$), the plane next to the interface ($\alpha =41$ and $\alpha =40$), and the plane deep in the leads ($\alpha =10$), respectively. The values of ${\epsilon }_{\mathrm{Schott}}$ in the center are 1.0 (full lines), 10 (dashed-dotted lines), and 50 (dashed lines), respectively. This panel is shown only to illustrate what happens when there is a charging error.

Figure 4

The transport function calculated at $k_{B}T=0.003$ (full black line) and $k_{B}T=0.06$ (dashed red line) is plotted, for $\mathrm{\Delta }{E}_c=-0.9$, as a function of frequency. Inset: The low-frequency part of ${\mathrm{\Lambda }}_{tr}\left(\omega \right)$ showing the shift of the transport function to higher energies and the reduction of the transport gap at low temperatures.

Figure 5

Left panel: The conductivity of the ML is plotted as a function of temperature for the same parameters as in Fig. 4. Inset: the conductivity plotted on a logarithmic scale as a function of inverse temperature. Right panel: The thermopower of the ML in units of $[k_B/e]$ is plotted as a function of temperature for the same parameters as the conductivity.

Figure 6

Left panel: Lorenz number in units of ${[k_B/e]}^2$ plotted as a function of temperature. Right panel: ZT plotted as a function of temperature. Parameters are the same as in Fig. 5.