Many-body theory of electronic transport in single-molecule heterojunctions

A many-body theory of molecular junction transport based on nonequilibrium Green’s functions is developed, which treats coherent quantum effects and Coulomb interactions on an equal footing. The central quantity of the many-body theory is the Coulomb self-energy matrix ${\Sigma }_C$ of the junction. ${\Sigma }_C$ is evaluated exactly in the sequential-tunneling limit, and the correction due to finite tunneling width is evaluated self-consistently using a conserving approximation based on diagrammatic perturbation theory on the Keldysh contour. Our approach reproduces the key features of both the Coulomb blockade and coherent transport regimes simultaneously in a single unified transport theory. As a first application of our theory, we have calculated the thermoelectric power and differential conductance spectrum of a benzenedithiol-gold junction using a semiempirical $\pi$-electron Hamiltonian that accurately describes the full spectrum of electronic excitations of the molecule up to 8–10 eV.

Figure 1

(Color online) A schematic of detailed balance. The left-hand and right-hand sides of Eq. (22) are represented in (a) and (b), respectively.

Figure 2

The correction to the Coulomb self-energy $\Delta {\Sigma }_C$ is given in the self-consistent Hartree-Fock approximation by the sum of (a) the Hartree (direct) term and (b) the Fock (exchange) term. The wavy line represents the Coulomb interaction, which has the form $U_{nm}\delta \left(\tau -{\tau }^{\prime }\right)$ on the Keldysh time contour.

Figure 3

(Color online) Self-consistent Hartree-Fock correction to the Coulomb self-energy matrix of a hypothetical C-C diatomic molecule versus lead chemical potential, shown in the vicinity of the HOMO resonance. Here ${\mu }_0=\left({\epsilon }_{\text{HOMO}}+{\epsilon }_{\text{LUMO}}\right)/2$ is the chemical potential of the isolated molecule and $T=300\text{K}$. The molecular junction parameters $U_{11}=U_{22}=8.9\text{eV}$, $U_{12}=4.4\text{eV}$, $t_{12}=2.0\text{eV}$, and ${\Gamma }_1={\Gamma }_2=0.2\text{eV}$ were used (see Sec. 3).

Figure 4

(Color online) Equilibrium correlation functions near the HOMO resonance for a diatomic molecule (same parameters as in Fig. 3). Top: total molecular charge $⟨N⟩$ in three different approximations: $\Sigma ={\Sigma }_C^{\left(0\right)}$ (isolated molecule in the grand canonical ensemble); $\Sigma ={\Sigma }_C^{\left(0\right)}+{\Sigma }_T$ (elastic cotunneling approximation); and $\Sigma ={\Sigma }_C^{\left(0\right)}+{\Sigma }_T+\Delta {\Sigma }_C^{\text{HF}}$ (self-consistent HF correction). The charging step is broadened in the elastic cotunneling approximation, compared to that of the isolated molecule, and acquires a “knee” near resonance. In the self-consistent HF approximation, the charge imbalance $\Delta N=⟨N⟩-⟨N⟩{\mid }_{{\Sigma }_T\to 0}$ is strongly screened near resonance, leading to a charging step as steep as that of the isolated molecule, but asymptotically approaches the elastic cotunneling result away from resonance. Bottom: the correlation function $⟨d_{1\sigma }^{†}{d}_{2\sigma }⟩$ in the same three approximations.

Figure 5

(Color online) Transmission probability and phase calculated with and without the self-consistent HF correction. Note that the transmission peaks and nodes are not shifted by $\Delta {\Sigma }_C^{\text{HF}}$ but the width of the transmission resonance is reduced asymmetrically. (Note: linestyles reversed in lower panel.)

Figure 6

(Color online) Thermoelectric power of a BDT-Au junction at $T=300\text{K}$ as a function of the lead-molecule chemical potential mismatch for three different tunneling widths. Comparison to the experimental value (Ref. 30) $7.0\pm 0.2\mu \text{V}/\text{K}$ fixes ${\mu }_{\text{Au}}={\mu }_0-\left(3.22\pm 0.04\right)\text{eV}$ (see the inset showing close-up of experimentally relevant parameter range).

Figure 7

(Color online) Linear-response conductance versus chemical potential (top panel) and differential conductance versus chemical potential and bias voltage (bottom panel) for a benzene(1,4)dithiol-Au junction. The calculation was carried out within the elastic cotunneling approximation using Eq. (28) with ${\Gamma }_1={\Gamma }_2=0.63\text{eV}$ and $T=300\text{K}$. Here $G_0=2e^2/h$ is the conductance quantum. Note the asymmetric Fano-like line shapes and the transmission nodes (arising from destructive interference) in the top panel. The numbers within the central diamonds of the bottom panel indicate the quantized charge on the molecule (relative to neutral benzene). The full spectrum is shown for completeness.

Figure 8

(Color online) Differential conductance spectrum of a benzene(1,4)dithiol-Au junction (same parameters as in Fig. 7) focusing on the vicinity of the HOMO resonance. The arrows indicate features corresponding to resonant tunneling through excitonic states of the junction.