Molecular junctions in the Coulomb blockade regime: Rectification and nesting

Quantum transport through single molecules is very sensitive to the strength of the molecule-electrode contact. Here, we investigate the behavior of a model molecular junction weakly coupled to external electrodes in the case where charging effects do play an important role (Coulomb blockade regime). As a minimal model, we consider a molecular junction with two spatially separated donor and acceptor sites. Depending on their mutual coupling to the electrodes, the resulting transport observables show well defined features such as rectification effects in the $I\text{−}V$ characteristics and nesting of the stability diagrams. To be able to accomplish these results, we have developed a theory which allows us to explore the charging regime via the nonequilibrium Green function formalism, and found full agreement with the master-equation results. Our theory, beyond its experimental relevance, offers a transparent framework for the systematic and modular inclusion of a richer physical phenomenology.

Figure 1

(Color online) The general configuration of a double-site junction. The levels ${ϵ}_{1,2}$ with charging energies $U_{1,2}$ are connected via $t$ and coupled to the electrodes via the linewidth injection rates ${\gamma }_{\alpha }^i$.

Figure 2

(Color online) The stability diagram of a SSJ with ${ϵ}_{\sigma }=2.0\mathrm{eV}$, $U=4.0\mathrm{eV}$, ${\Gamma }_L={\Gamma }_R=0.05\mathrm{eV}$. (a) The incorrect result obtained by means of the widely used formula in Eq. (20) for the lesser GF is not symmetric for levels ${ϵ}_{\sigma }$ and ${ϵ}_{\sigma }+U$. (b) Results obtained by means of our ansatz in Eq. (14) shows correctly symmetric for levels ${ϵ}_{\sigma }$ and ${ϵ}_{\sigma }+U$.

Figure 3

(Color online) The comparison of the master-equation method and our ansatz for the differential conductance of the two level model with ${ϵ}_{\uparrow }=-0.35\mathrm{eV}$, ${ϵ}_{\downarrow }=-0.65\mathrm{eV}$, $U=1.0\mathrm{eV}$, $V_g=1.0\mathrm{V}$, and ${\Gamma }_L={\Gamma }_R=0.05\mathrm{eV}$.

Figure 4

(Color online) The stability diagram (the contour plot of the differential conductance) calculated by our ansatz for the two level model with parameters as in Fig. 3. The latter is indicated with a dash line at $V_g=1.0\mathrm{V}$.

Figure 5

(Color online) The stability diagram of a serial DSJ with ${ϵ}_{1,\sigma }={ϵ}_{2,\sigma }=-0.15\mathrm{eV}$, $U_1=U_2=0.3\mathrm{eV}$, $t=0.05\mathrm{eV}$, ${\gamma }_L^1={\gamma }_R^2=0.02\mathrm{eV}$, ${\gamma }_L^2={\gamma }_R^1=0$, and $V_{\text{bias}}=0.005\mathrm{V}$. The maxima of conductance are observed when the levels of the first site (${ϵ}_{1,\sigma }$ or ${ϵ}_{1,\sigma }+U$) are overlapped with the levels of the second site (${ϵ}_{2,\sigma }$ or ${ϵ}_{2,\sigma }+U$), and with the Fermi energy in the leads. The splitting of the four maxima is due to the hopping between the dots.

Figure 6

(Color online) Current and conductance vs bias voltage of a DSJ far from equilibrium with parameters ${ϵ}_{1,\sigma }=0.5\mathrm{eV}$, ${ϵ}_{2,\sigma }=-0.5\mathrm{eV}$, $U_1=U_2=U=0.2\mathrm{eV}$, $t=0.07\mathrm{eV}$, ${\gamma }_L^1={\gamma }_R^2=0.03\mathrm{eV}$, $V_{{\mathrm{g}}_2}=-V_{{\mathrm{g}}_1}=V_{\text{bias}}∕4$, and $V_R=-V_L=V_{\text{bias}}∕2$. The red curve represents the current, while the blue the conductance. The inset is the blowup for the conductance peak split. The dash and dot-dash curves are for current and conductance with $U=0$, respectively.

Figure 7

(Color online) The processes involved in the transport characteristics in Fig. 6. ${ϵ}_1\equiv {ϵ}_{1,\sigma }$, ${ϵ}_2\equiv {ϵ}_{2,\sigma }$. The red line indicates electron resonant tunneling. (a) The first conductance peak. (b) The second conductance peak. (c) The pseudopeak of conductance. (d) The first current maximum, and the red line indicates resonant tunneling of electrons. (e) The second current maximum for electron resonant tunneling. (f) The dip of conductance.

Figure 8

(Color online) Nested stability diagram of a parallel DSJ with parameters ${ϵ}_{1,\sigma }=-1.8\mathrm{eV}$, ${ϵ}_{2,\sigma }=-0.3\mathrm{eV}$, $U_1=3.6\mathrm{eV}$, $U_2=0.6\mathrm{eV}$, $t=0.001\mathrm{eV}$, ${\gamma }_L^1={\gamma }_R^1=0.04\mathrm{eV}$, ${\gamma }_L^2={\gamma }_R^2=0.05\mathrm{eV}$, $V_{{\mathrm{g}}_2}=V_{{\mathrm{g}}_1}∕2=V_{\mathrm{g}}∕2$, and $V_R=-V_L=V_{\text{bias}}∕2$. See discussion in the text.