Effects of electronic correlations and magnetic field on a molecular ring out of equilibrium

We study the effects of electron-electron interactions on the steady-state characteristics of a hexagonal molecular ring in a magnetic field as a model for a benzene molecular junction. The system is driven out of equilibrium by applying a bias voltage across two metallic leads. We employ a model Hamiltonian approach to evaluate the effects of on-site as well as nearest-neighbor density-density–type interactions in a physically relevant parameter regime. Results for the steady-state current, charge density, and magnetization in three different junction setups (para, meta, and ortho) are presented. Our findings indicate that interactions beyond the mean-field level renormalize voltage thresholds as well as current plateaus. Electron-electron interactions lead to substantial charge redistribution as compared to the mean-field results. We identify a strong response of the circular current on the electronic structure of the metallic leads. Our results are obtained by steady-state cluster perturbation theory, a systematically improvable approximation to study interacting molecular junctions out of equilibrium, even in magnetic fields. Within this framework, general expressions for the current, charge density, and magnetization in the steady state are derived. The method is flexible and fast and can straightforwardly be applied to effective models as obtained from ab initio calculations.

Figure 1

(Top) Illustration of the device setup: A planar molecular ring is connected to two metallic reservoirs. The setup is placed in a perpendicular magnetic field $B$. A bias voltage $V_B$ is applied between the left and right lead. (Bottom) Bare single-particle energy levels of the noninteracting, disconnected molecule. Bias voltages required for the specific levels to contribute to transport are indicated on the right. The HOMO as well as the LUMO are doubly degenerate ($d=2$) for $B=0$.

Figure 2

(Top row) In the left panel, the total transmission current $j_t$ for molecule-lead couplings of $t^{\prime }=-0.05$ eV (blue), $t^{\prime }=-0.6$ eV (green), $t^{\prime }=-0.9$ eV (black), and $t^{\prime }=-1.2$ eV (magenta) is shown for para-connected benzene. In the right panel the circular current $j_c$ is shown for the same parameters. In each panel we visualize results for the noninteracting device and for an on-site interaction strength of $U=9$ eV. (Bottom row) The total transmission current $j_t$ for molecule-lead couplings of $t^{\prime }=-0.05$ eV (blue), $t^{\prime }=-0.2$ eV (red), and $t^{\prime }=-0.8$ eV (green) is shown for para-(left), meta-(center), and ortho-(right) connected benzene in the vicinity of the first threshold voltage. Data shown are for an on-site interaction strength of $U=9$ eV in the zero-field setup. The zoom in the central figure shows the ledge for $t^{\prime }=0.05$ eV in detail. Note that the vertical axis of the transmission current data is scaled by a factor of $\frac{1}{|t^{\prime }{|}^2}$.

Figure 3

Current-voltage characteristics of para-(top), meta-(center), and ortho-(bottom) connected benzene. We plot the total transmission current $j_t/nA$ in the left and the ring current $j_c/\mu A$ in the right column. For every device setup, data for the noninteracting case (blue) is shown in comparison to results for an on-site interaction strength of $U=9$ eV. We compare data for the zero-field case (solid lines) to curves obtained for a magnetic field of $B=2$ T (dash-dotted lines).

Figure 4

(Left) Total transmission current $j_t$ as a function of bias voltage $V_B$ for various interaction strengths $U$. We illustrate data for meta-connected benzene and a magnetic field of $B=1$T. (Right) Threshold voltage $V_T$ at which the signal in the circular current $j_c$ and inflection point in the transmission current $j_t$ occurs in regions I, II, and III (symbols).

Figure 5

Maximal total transmission current $j_t^{\max }$ in the vicinity of the first threshold voltage (I). (Left) As a function of on-site interaction strength $U$ for the meta (solid, circles) and ortho (dashed, triangles) device. For the para setup (thick black line, stars), it is independent of $B$ as a function of $U$. (Right) The same quantity showing saturation as a function of $B$ for various values of on-site interaction strength $U$ in the meta device.

Figure 6

Total transmission current signal $j_t$ (left column) and ring current signal $j_s$ (right column) as a function of bias voltage $V_B$. From top to bottom we compare the current signal in the para-, meta-, and ortho-connected device for the noninteracting case $U=0$ eV (blue) to an on-site interaction strength of $U=9$ eV (red). We show data for zero field $B=0$T (solid), a field of $B=2$T (dashed), as well as $B=15$T(dotted). The signals have been shifted from their respective first threshold voltage (I) on top of each other to $V_B=0$.

Figure 7

Ss local charge density of the para-connected device (left) as well as the meta-, and ortho-connected devices (right). We show data for a device with on-site interaction of $U=9$ eV in a magnetic field of $B=2$T. The drawings of the device at the top serve as a legend for the line colors to identify the respective orbitals. $\Delta n$ denotes the shift of the local density away from $\langle n_i\rangle =1$ in the low-bias region.

Figure 8

Signals in the sts magnetization of the device due to the Zeeman effect. We compare the effects of magnetic fields of $B=1$T (blue), $B=2$T (red), and $B=5$T (green) for a device with an on-site interaction of 9 eV.