dc conductance of molecular wires

Inspired by the work of Kamenev and Kohn [Phys. Rev. B 63, 155304 (2001)], we present a general discussion of the two-terminal dc conductance of molecular devices within the framework of time dependent current-density functional theory in the linear response regime. We derive a formally exact expression for the adiabatic conductance. For junctions made of long molecular chains that can be either metallic or insulating, we derive the exact asymptotic behavior of the adiabatic conductance as a function of the chain’s length. Our results follow from the analytic structure of the bands of a periodic molecular chain and a compact expression for the Green’s functions. In the case of an insulating chain, not only do we obtain the exponentially decaying factors, but we also obtain the corresponding amplitudes, which depend very sensitively on the electronic properties of the contacts. We illustrate the theory by a numerical study of a simple insulating structure connected to two metallic jellium leads.

Figure 1

(Color online) The geometry of the $\text{chain}+\text{leads}$ structure.

Figure 2

(Color online) [(a) and $\left({\mathrm{a}}^{\prime }\right)$] The position of ${ϵ}_F^{\pm }$ in the complex energy plane and [(b) and $\left({\mathrm{b}}^{\prime }\right)$] the corresponding $k_F^{\pm }$ in the complex $k$ plane for metals (left) and insulators (right). The figure also shows the two branch points [red (full) dots], one at $k=0$ and one at the zone boundary, and their corresponding energies.

Figure 3

(Color online) Schematic of the effective potential of the one-dimensional chain.

Figure 4

(Color online) Schematic of the three-dimensional $\text{chain}+\text{leads}$. The shaded region in the middle indicates the region that is periodically repeated in the construction of the periodic potential $V_0$.

Figure 5

(Color online) (a) Schematic of the QD chain attached to jellium leads. (b) The self-consistent electron density. (c) The self-consistent potential. (d) The local density of states (integrated over the $xy$ coordinates) as function of position along the chain and energy. The black line shows the position of the Fermi level. The numbers represent atomic units.

Figure 6

(Color online) (a) The effective Kohn-Sham potential of the $\text{chain}+\text{leads}$ and the region that is repeated periodically. (b) The periodic potential obtained by repeating the central piece of the upper panel. (c) The difference between the effective and periodic potentials. The numbers represent atomic units.

Figure 7

(Color online) Left panel: The lowest bands of the periodic potential. Right panel: The Riemann surface corresponding to these bands.

Figure 8

(Color online) Complex band calculations. Each panel shows the real (left) and imaginary (right) parts of ${ϵ}_{n,k}$, $n=1,\dots ,4$, for different values of $\mathrm{Im}k$. In the panels from (a) to (f), $\mathrm{Im}\left(k\right)$ was uniformly increased from $0\text{to}0.06\mathrm{a.u.}$ In the panels from (g) to (j), $\mathrm{Im}\left(k\right)=0.1$, 0.11, 0.115, and $0.12\mathrm{a.u.}$, respectively.

Figure 9

(Color online) The differential conductance for $\text{chain}+\text{leads}$ structures containing 2, 3, 4, 5, 6, and 7 QDs, as function of the applied bias potential. The thick vertical dashed line represents the converged position of the Fermi level. The thin dotted lines indicate the edges of the valence and conduction bands for the structure containing seven QDs.