Quantum interference in off-resonant transport through single molecules

We provide a simple set of rules for predicting interference effects in off-resonant transport through single molecule junctions. These effects fall into two classes, showing, respectively, an odd or an even number of nodes in the linear conductance within a given molecular charge state, and we demonstrate how to decide the interference class directly from the contacting geometry. For neutral alternant hydrocarbons, we employ the Coulson-Rushbrooke-McLachlan pairing theorem to show that the interference class is decided simply by tunneling on and off the molecule from same or different sublattices. More generally, we investigate a range of smaller molecules by means of exact diagonalization combined with a perturbative treatment of the molecule-lead tunnel coupling. While these results generally agree well with $GW$ calculations, they are shown to be at odds with simpler mean-field treatments. For molecules with spin-degenerate ground states, we show that for most junctions interference causes no transmission nodes, but we argue that it may lead to a nonstandard gate dependence of the zero-bias Kondo resonance.

Figure 1

The molecular junctions setup with a source-drain voltage $V_{\text{sd}}$ applied across the molecule, which is tunnel coupled to leads through $t_s$ and $t_d$. The electrostatic environment is controlled through a backgate voltage $V_g$.

Figure 2

The even and odd quantum interference classes. The signs of the pole of $h_{00}^{\sigma \sigma }$ at ${\varepsilon }_0^h$ and of $p_{00}^{\sigma \sigma }$ at ${\varepsilon }_0^p$ determine the interference class uniquely. The unshaded region indicates the valid regime of $V_g$. The interference nodes can be interpreted as happening between the particle and hole processes (diamonds) or entirely within either the particle or the hole processes (circles). Note the anomalous degenerate cases with only one node, which shows up in some DFT and HT calculations.

Figure 3

Off-resonant cotunneling conductance through the $\pi$ system of various conjugated alternant hydrocarbon molecular junctions. The conductance is calculated from Eq. (2) by exact diagonalization of the full PPP model including intramolecular interactions and setting ${\Gamma }_s{\Gamma }_d=0.01{\left(\text{eV}\right)}^2$. The dashed arms symbolize binding sites between the electrodes and the molecule. The calculated conductance conforms with the starring rule (summarized in the table). When entering and leaving through two sites belonging to different sublattices (star, no star), the result is an even number of nodes. When the two connected sites belong to the same sublattice (e.g., star, star), there is an odd number of nodes. Inset: Corresponding transmission for a simple Hückel model.

Figure 4

The stilbene molecular junction. (a) The molecular junction setup. (b) The stilbene molecule. The dashed arms symbolize possible binding sites between the electrodes and the molecule. The stars refer to the quantum interference classification explained in Fig. 2. (c) Stilbene stability diagrams for the neutral $\pi$ system for ${\Gamma }_s{\Gamma }_d=0.01{\left(\text{eV}\right)}^2$ in different coupling configurations obtained from the current expression in Eq. (1). In the AB' and BB' configurations all the interference nodes in the conductance are marked with arrows. Note that in the BB' configuration the two nodes at nonzero bias are hidden due to inelastic cotunneling processes. (d) Transmission through the $\pi$ system of substituted stilbene calculated by DFT, HT, HF, $GW$, and ED, respectively. The edge of the shaded region marks the charge degeneracy points, while the unshaded regions denote the gate values of the relevant charge state. The mean-field methods (DFT and HF) and the noninteracting HT all predict interference but fail to capture the double node of the even interference class predicted by the starring rule for the BB' configuration.

Figure 5

The zero-bias conductance of a biphenyl molecular junction with a spin doublet ground state in two different configurations using ${\Gamma }_s{\Gamma }_d=0.01{\left(\text{eV}\right)}^2$. The dashed arms symbolize possible binding sites between the electrodes and the molecule. Note how the interference dips appear at different positions for the $W$ and $J$ channels and therefore do not show up in the total current. The configuration in the right panel shows how the exchange term fills in the dip in the potential scattering amplitude, leaving only a shallow local minimum near $V_g=0$.

Figure 6

Relaxed geometry of substituted stilbene in the AA' configuration used for DFT calculations.

Figure 7

Low-bias differential conductance as a function of gate potential for the benzoquinoid structure. The dashed and solid lines denote the LB and Meir-Wingreen results, respectively.