Quantum transport simulation scheme including strong correlations and its application to organic radicals adsorbed on gold

We present a computational method to quantitatively describe the linear-response conductance of nanoscale devices in the Kondo regime. This method relies on a projection scheme to extract an Anderson impurity model from the results of density functional theory and nonequilibrium Green's functions calculations. The Anderson impurity model is then solved by continuous-time quantum Monte Carlo. The developed formalism allows us to separate the different contributions to the transport, including coherent or noncoherent transport channels, and also the quantum interference between impurity and background transmission. We apply the method to a scanning tunneling microscope setup for the 1,3,5-triphenyl-6-oxoverdazyl (TOV) stable radical molecule adsorbed on gold. The TOV molecule has one unpaired electron, which when brought in contact with metal electrodes behaves like a prototypical single Anderson impurity. We evaluate the Kondo temperature, the finite-temperature spectral function, and transport properties, finding good agreement with published experimental results.

Figure 1

Schematic of the 1,3,5-triphenyl-6-oxoverdazyl (TOV) molecule.

Figure 2

Schematic representation of the method for the calculation of the transport properties in presence of interacting Anderson impurities.

Figure 3

Schematic representation of the two-terminal device comprising the left lead, the extended molecule (EM), and the right lead. We denote the Hamiltonian matrix of the semi-infinite left (right) lead by $H_L$ ($H_R$), the one of the EM as $H$, and the coupling Hamiltonian matrix as $H_{LM}$ ($H_{RM}$). The EM is further subdivided in the interacting region (IR) and the extended interacting region (ER). The IR includes a set of interacting molecular orbitals $\left\{{\psi }_i\right\}$, with $i\in [1,N_{\mathrm{AI}}]$ ($N_{\mathrm{AI}}$ is the number of interacting molecular orbitals; in the schematic figure $N_{\mathrm{AI}}=4$), forming the Anderson impurity (AI). The ER consists of all orbitals in the EM that have finite overlap with the IR orbitals, so that the IR orbitals are always also part of the ER. In total the system is therefore subdivided into four subspaces of decreasing size: EM, ER, IR, and AI.

Figure 4

Left: top view of the TOV molecule, indicating also the electron density isosurface of the TOV SOMO (green bubbles) extending over the nitrogen atoms; the atoms N1, N2, C3, N4, N5, and C6 are explicitly indicated. Right: side view of the optimized geometry for TOV with the two energetically equal conformations, the negatively charged TOV and H-TOV. Color code: C atoms, yellow; H atoms, cyan; N atoms, gray; O atom, red. Note that the N atoms are not visible in the top view as they are surrounded by the green isosurface.

Figure 5

Energy barrier separating the configurations ${\mathrm{CFG}}_1$ (left) and ${\mathrm{CFG}}_2$ (right).

Figure 6

(a) Scanning tunneling microscope (STM) setup used in the electron transport simulations, and (b) projected density of states (PDOS) on the atoms of the molecule on the Au surface (positive values are for spin-up states, negative values for spin-down states).

Figure 7

Transmission for different molecule-substrate separations, given by about 3.1 Å+ $\mathrm{\Delta }h$.

Figure 8

Transmission for no applied scissor operator (SCO), and with a SCO applied non-self-consistently for the converged LDA charge density [SCO (non-scf)], and with a SCO applied self-consistently [SCO (scf)].

Figure 9

Real and imaginary parts of the hybridization function ${\overline{\mathrm{\Delta }}}_{\mathrm{AI}}$ [Eq. (34)].

Figure 10

Top panel: mean-field (MF) and CTQMC onsite occupation $\langle \widehat{n}\rangle =\langle {\widehat{n}}_{\uparrow }\rangle +\langle {\widehat{n}}_{\downarrow }\rangle$ and mean-field local magnetic moment $\langle {\widehat{m}}_z\rangle =\langle n_{\uparrow }\rangle -\langle {\widehat{n}}_{\downarrow }\rangle$ as function of $U$. Bottom panel: MF and CTQMC squared local magnetic moment $\langle {\widehat{m}}_z^2\rangle =\langle {({\widehat{n}}_{\uparrow }-{\widehat{n}}_{\downarrow })}^2\rangle$ and the corresponding square local magnetic moment in the limit of no correlations $\langle {\widehat{m}}_z^2\rangle {|}_0=\langle \widehat{n}\rangle -{\langle \widehat{n}\rangle }^2/2$. Results are for $\theta =20$ K. Error bars in the CTQMC estimates are smaller than the symbols.

Figure 11

Mean-field spectral function at zero temperature and for $U=0$, 0.5, 1, and 2 eV. For $U=0$ eV the spectral function for spin up and down are superimposed as the systems is spin degenerate.

Figure 12

Comparison between the zero-temperature mean-field spectral function (black and red lines for spin up and down, respectively), and the spectral function obtained by performing the analytic continuation of the mean-field Matsubara Green's function at $T=20$ K (green line).

Figure 13

Spectral function for several $U$ and at $\theta =20$ K. The noise is a well-known drawback of the SO method [145]. In principle, it can be greatly reduced by averaging the results of many independent optimizations for the same data set, in practice, however, it cannot be completely removed. The presented results are obtained by averaging over 250 independent runs.

Figure 14

Spectral function for $U=2$ eV and at several different temperatures $\theta$. The inset shows a zoom for the 0.4-eV range around the Fermi energy.

Figure 15

Computed Kondo temperature as function of $U$ for $\theta =20$ K. The CTQMC results obtained with Eqs. (100) (blue circles) and (103) (orange squares) are compared to the experimental value (horizontal green dashed line). The black curve represents the fit of the CTQMC data from Eq. (100) in the range of $U⩽1.5$ by using the model (104) for the asymmetric SIAM (with $\epsilon =-1.42U/2$). This fit allows the extrapolation of the calculated ${\theta }_{\mathrm{K}}$ to larger $U$. For comparison, the pink dashed-dotted line shows the expected ${\theta }_{\mathrm{K}}$ for the symmetric SIAM ($\epsilon =-U/2$).

Figure 16

Transmission including correlations for different values of $U$: (a) total elastic transmission ($T$), (b) background transmission ($T_{\mathrm{B}}$), (c) interference terms of the transmission ($T_{\mathrm{I}}$), (d) elastic transmission of the Anderson impurity ($T_{\mathrm{AI}}$), and (e) total transmission of the Anderson impurity ($T_{t,\mathrm{AI}}$).

Figure 17

Transmission including correlations for $U=2.5$ eV for a tip with a single atom at the top (one-atom tip), and a tip with three atoms at the top (three-atom tip). The different subfigures correspond to the ones described in Fig. 16.