Quantum interference in single molecule electronic systems

We present a general analytical formula and an ab initio study of quantum interference in multibranch molecules. Ab initio calculations are used to investigate quantum interference in a benzene-1,2-dithiolate (BDT) molecule sandwiched between gold electrodes and through oligoynes of various lengths. We show that when a point charge is located in the plane of a BDT molecule and its position varied, the electrical conductance exhibits a clear interference effect, whereas when the charge approaches a BDT molecule along a line normal to the plane of the molecule and passing through the center of the phenyl ring, interference effects are negligible. In the case of oligoynes, quantum interference leads to the appearance of a critical energy $E_c$ at which the electron transmission coefficient $T(E)$ of chains with even or odd numbers of atoms is independent of length. To illustrate the underlying physics, we derive a general analytical formula for electron transport through multibranch structures and demonstrate the versatility of the formula by comparing it with the above ab initio simulations. We also employ the analytical formula to investigate the current inside the molecule and demonstrate that large countercurrents can occur within a ringlike molecule such as BDT, when the point charge is located in the plane of the molecule. The formula can be used to describe quantum interference and Fano resonances in structures with branches containing arbitrary elastic scattering regions connecting nodal sites.

Figure 1

A multibranch structure with nodal sites $L$ and $R$ (on the left and right) connecting external current-carrying leads, by hopping matrix elements $-{\alpha }_L$ (on the left) and $-{\beta }_R$ (on the right), and to internal branches ($l$), by hopping matrix elements $-{\alpha }_l$ and $-{\beta }_l$, respectively. The energies of the nodal sites are ${\varepsilon }_L^0$ and ${\varepsilon }_R^0$. The site energy and hopping matrix element of branch $l$ are ${\varepsilon }_l$ and $-{\gamma }_l$, respectively.

Figure 2

A single impurity in a one-dimensional chain.

Figure 3

A schematic of the nodal atoms connected by an atomic chain.

Figure 4

Transmission functions for increasing lengths of molecular wire using the general formula. The system is modeled using the following parameters: in the leads, ${\varepsilon }_L={\varepsilon }_R=0.0$ and ${\gamma }_L={\gamma }_R=5.0$; for the contacts, ${\varepsilon }_L^0={\varepsilon }_R^0=-0.4$; in the chain, ${\varepsilon }_1=0.5$ and ${\gamma }_1=6.0$; and the coupling between the wire and the electrodes are defined by ${\alpha }_L={\beta }_R=3.0$. All curves intersect at $E_c\approx 0.5\hspace{0.5em}\mathrm{eV}$. Close inspection reveals that at $E\approx -0.2\hspace{0.5em}\mathrm{eV}$, the curves approach each other, but do not intersect at a single energy.

Figure 5

Transmission curves from the smeagol simulations of oligoynes of varying lengths.

Figure 6

Molecular structure used in the transport simulations with the potassium point charge in configuration C1 (a) and the calcium point charge in configuration C2 (b).

Figure 7

Transmission functions obtained from smeagol simulations with the potassium point charge moving closer to the molecule through (a)–(d). The solid curve corresponds to configuration C1 and the dashed curve to configuration C2.

Figure 8

Transmission functions obtained from smeagol simulations with the calcium point charge moving closer to the molecule through (a)–(d). The solid curve corresponds to C1 and the dashed curve to C2.

Figure 9

Transmission curves for the tight-binding model. The solid line corresponds to the case where the charge is located along a line perpendicular to the ring and which passes through its center. The dashed line corresponds to the case where the charge is closer to one arm. (a) shows the transmission curves when the charge is away from the molecule. In (b)–(d), the on-site energy parameters are changed to simulate the charge moving toward the molecule.

Figure 10

(a) Total current through the molecule, (b) the current through branch 1, (c) branch 2 of the molecule. Parameters used are exactly the same as the dashed red line in Fig 9d, namely, ${\varepsilon }_R={\varepsilon }_L=4.0$, ${\varepsilon }_L^0={\varepsilon }_R^0=1.0$, ${\varepsilon }_1=-1.8$, ${\varepsilon }_2=-0.9$, ${\gamma }_L={\gamma }_R=4.0$, ${\gamma }_1={\gamma }_2=1.0$, and ${\alpha }_L={\beta }_R=1.5$.

Figure 11

The case of a molecule with dangling branches. (a) and (b) show two equivalent representations of the same system.