Conserving $GW$ scheme for nonequilibrium quantum transport in molecular contacts

We give a detailed presentation of our recent scheme to include correlation effects in molecular transport calculations using the nonequilibrium Keldysh formalism. The scheme is general and can be used with any quasiparticle self-energy, but for practical reasons, we mainly specialize to the so-called $GW$ self-energy, widely used to describe the quasiparticle band structures and spectroscopic properties of extended and low-dimensional systems. We restrict the $GW$ self-energy to a finite, central region containing the molecule, and we describe the leads by density functional theory (DFT). A minimal basis of maximally localized Wannier functions is applied both in the central $GW$ region and the leads. The importance of using a conserving, i.e., fully self-consistent, $GW$ self-energy is demonstrated both analytically and numerically. We introduce an effective spin-dependent interaction which automatically reduces self-interaction errors to all orders in the interaction. The scheme is applied to the Anderson model in and out of equilibrium. In equilibrium at zero temperature, we find that $GW$ describes the Kondo resonance fairly well for intermediate interaction strengths. Out of equilibrium, we demonstrate that the one-shot $G_0{W}_0$ approximation can produce severe errors, in particular, at high bias. Finally, we consider a benzene molecule between featureless leads. It is found that the molecule’s highest occupied molecular orbital–lowest unoccupied molecular orbital gap as calculated in $GW$ is significantly reduced as the coupling to the leads is increased, reflecting the more efficient screening in the strongly coupled junction. For the $I\text{−}V$ characteristics of the junction, we find that Hartree–Fock (HF) and $G_0{W}_0\left[G_{\mathrm{HF}}\right]$ yield results closer to $GW$ than does DFT and $G_0{W}_0\left[G_{\mathrm{DFT}}\right]$. This is explained in terms of self-interaction effects and lifetime reduction due to electron-electron interactions.

Figure 1

Before the coupling between the three regions is established, the three subsystems are in equilibrium with chemical potentials ${\mu }_L$, ${\mu }_C$, and ${\mu }_R$, respectively.

Figure 2

The $GW$ and second Born self-energies, ${\Sigma }_{GW}$ and ${\Sigma }_{2B}$, can be obtained as functional derivatives of their respective $\Phi$ functionals, ${\Phi }_{GW}\left[G\right]$ and ${\Phi }_{2B}\left[G\right]$. Straight lines represent the full Green’s function $G$, i.e., the Green’s function in the presence of coupling to the leads and interactions. Wiggly lines represent the interactions.

Figure 3

The extended central region $\left(C2\right)$ is chosen so large that it comprises all perturbations in the effective DFT potential arising from the molecular contact. The central region $\left(C\right)$ can be a proper subregion of $C2$, but it must be so large that there is no direct coupling across it. We solve for the self-consistent Kohn–Sham potential within $C2$, but we replace the static xc potential by the $GW$ self-energy inside $C$.

Figure 4

(Color online) Spectral function of the central site for $\Gamma =0.65$, $U=4.0$, and different values of ${\epsilon }_c$. The inset in the lower panel is a zoom of the $GW$ spectral peak around $\omega =0$.

Figure 5

(Color online) FWHM of the Kondo resonance as calculated in the $GW$ approximation and from the analytical result Eq. (69). The interaction strength is $U=4$ and ${\epsilon }_c$ is varied in the Kondo regime.

Figure 6

(Color online) Spectral function for $U=4.0$, ${\epsilon }_c=-U∕2$, and three different values of $\Gamma =2.0$, 1.0, and 0.5 corresponding to strong, intermediate, and weak coupling to the leads.

Figure 7

(Color online) Differential conductance, $dI∕dV$, as a function of the central site energy, ${\epsilon }_c$, for different applied biases, $U=4$ and $\Gamma =0.65$.

Figure 8

(Color online) Differential conductance as a function of applied bias for $U=4$, $\Gamma =0.65$, and ${\epsilon }_c=-4$. For these parameters, the nonmagnetic HF solution is stable for bias voltages smaller than $\sim 1.6$. The $G_0{W}_0$ approximation yields different currents at the left and right interfaces $(\Delta I\ne 0)$ and yields negative differential conductance at finite bias.

Figure 9

(Color online) Upper panel: Occupation of the central site as function of ${\epsilon }_c$ for $U=4$, $\Gamma =0.65$, and bias $V=0.8$. Notice that the HF solution breaks the spin symmetry for some ${\epsilon }_c$ values. Middle panel: Current calculated in self-consistent $GW$ and $G_0{W}_0[G_{\mathrm{HF},\uparrow },G_{\mathrm{HF},\downarrow }]$. Lower panel: Violation of the continuity equation measured as the difference between the currents in the left and right leads.

Figure 10

(Color online) Same as Fig. 9, but for bias voltage $V=4.0$.

Figure 11

(Color online) (a) Illustration of a benzene molecule coupled to featureless electrodes with different chemical potentials. (b) Isosurfaces for the 18 partially occupied Wannier functions used as basis functions in the calculations. The WFs are linear combinations of Kohn–Sham eigenstates obtained from a DFT-PBE plane-wave calculation.

Figure 12

Density of states for a benzene molecule weakly coupled to featureless leads $(\Gamma =0.05)$. The common Fermi levels of the leads is indicated. Notice the characteristic opening of the band gap when going from DFT-PBE to HF and the subsequent (slight) reduction when correlations are included at the $GW$ level.

Figure 13

(Color online) The HF and $GW$ HOMO-LUMO gap of the benzene molecule as a function of the coupling strength $\Gamma$. The difference between the curves represents the reduction in the gap due to the correlation part of the $GW$ self-energy. This value increases with the coupling strength as screening by electrons in the leads becomes more effective.

Figure 14

(Color online) Differential conductance of the benzene junction for ${\Gamma }_L={\Gamma }_R=0.25\mathrm{eV}$. Notice the strong $G_0$ dependence of the $G_0{W}_0$ result.

Figure 15

(Color online) Equilibrium DOS for the benzene molecule coupled to wide-band leads with a coupling strength of ${\Gamma }_L={\Gamma }_R=0.25\mathrm{eV}$. Upper panel shows DFT-PBE and HF single-particle approximations, while the lower panel shows the self-consistent $GW$ result as well as the one-shot $G_0{W}_0$ results based on the DFT and HF Green’s functions, respectively.

Figure 16

The transition region, $T$, is defined as the part of the lead beyond which the lead Hamiltonian becomes periodic.