Electronic correlations in coherent transport through a two quantum dot system

Coherent electronic transport through a double quantum dot (2qd) system connected in a series with electrodes is studied by means of nonequilibrium Green functions and using the equation of motion method, in which all electron correlations in the 2qd are treated exactly. For moderate Coulomb interactions we predict features in a conductance characteristics resulting from transmission through triplet states, which can be strongly activated for larger source-drain voltages. The analysis of the spin-spin correlation functions shows strong antiferromagnetic correlations arising from transport through the singlet state and a reduction of the total magnetic moment. However, when the transmission channel corresponding to the triplet state becomes activated the antiferromagnetic correlations are much weaker, the spins behave as free electrons spins and strongly fluctuate. We speculate that this effect can be seen in wide range of a gate voltage for a double electron occupancy of the 2qd.

Figure 1

Zero-voltage conductance ${\mathcal{G}}_0$ (top figure) and correlators (bottom figure): $⟨n_{1,+}⟩$—solid curve, $⟨{\mathbf{S}}_1^2⟩$—long-dashed curve, $-⟨{\mathbf{S}}_1\bullet {\mathbf{S}}_2⟩$—short-dashed curve, as a function of the relative position of the energy level ${ϵ}_1-E_F$. The parameters are $t_{12}=1$, $U_0=6$, $U_1=1.6$, ${\Gamma }_L={\Gamma }_R=0.1$, ${ϵ}_2={ϵ}_1$, and $T=0$. The labels above the conductance peaks denote the position when an extra electron is added to the 2qd system. The vertical solid lines show positions of poles of the Green functions corresponding to transmission through the two-electron states: singlet $\left(s\right)$ and the triplet $\left(t\right)$, whereas the dashed lines—transmission through the one-electron states.

Figure 2

Height of the conductance peaks through: (1) the one-electron level ${ϵ}_1-E_F=\mid t_{12}\mid$ and (2) the singlet level ${ϵ}_1-E_F=-\mid t_{12}\mid -(U_0+U_1-\Delta )∕2$, plotted as the function of the intradot coulomb repulsion parameter $U_0$. The interdot coulomb interaction is taken $U_1=U_0∕4$, ${\Gamma }_L={\Gamma }_R=0.1$, ${ϵ}_2={ϵ}_1$.

Figure 3

Zero-voltage conductance ${\mathcal{G}}_0$ vs ${ϵ}_1-E_F$ for two different couplings of the 2qd system with the electrodes ${\Gamma }_L={\Gamma }_R=0.2$ (thick curve) and ${\Gamma }_L={\Gamma }_R=0.05$ (thin curve). The other parameters are the same as in Fig. 1. The dashed curve is the conductance determined form Eq. (5) taking only the interference term (the second one) for ${\Gamma }_L={\Gamma }_R=0.2$. Vertical lines denote position of resonant levels for the two-electron states (solid lines) and for the one-electron states (dashed lines).

Figure 4

Zero-voltage conductance ${\mathcal{G}}_0$ as a function of ${ϵ}_1-E_F$ for different temperatures: $T=0$—solid curve, $T=0.05$—dashed-dot curve, $T=0.1$—dashed curve, and $T=0.2$—dot curve. The coupling to the electrodes is ${\Gamma }_L={\Gamma }_R=0.1$, $t_{12}=1$, $U_0=6$, $U_1=1.6$, and ${ϵ}_2={ϵ}_1$.

Figure 5

Zero-voltage conductance ${\mathcal{G}}_0$ vs the center level position ${ϵ}_0=({ϵ}_1+{ϵ}_2)∕2$ (with respect to $E_F$) and the detuning parameters $\delta =({ϵ}_1-{ϵ}_2)∕2$ plotted as a gray-scale map. The parameters are $t_{12}=1$, $U_0=6$, $U_1=1.6$, ${\Gamma }_L={\Gamma }_R=0.1$, and $T=0$.

Figure 6

Differential conductance $\mathcal{G}$ (top figure) and correlators (bottom figure): $⟨n_{i,+}⟩$—solid curve, $⟨{\mathbf{S}}_i^2⟩$—long-dashed curve, $-⟨{\mathbf{S}}_1\bullet {\mathbf{S}}_2⟩$—short-dashed curve, as a function of the applied source-drain voltage $V$ at ${ϵ}_1-E_F={ϵ}_2-E_F=2$. Note that in the low voltage region the values of $⟨n_{i,+}⟩$ and $⟨S_i^2⟩$ for $i=1$ are larger than those ones for $i=2$, whereas in the high voltage regime the situation is opposite and these quantities are higher for $i=2$. The parameters are $t_{12}=1$, $U_0=6$, $U_1=1.6$, ${\Gamma }_L={\Gamma }_R=0.1$, and $T=0$. For comparison the plot for the equilibrium situation is presented by the thin curves for the zero-voltage conductance ${\mathcal{G}}_0$ and the correlators as a function of ${ϵ}_1-E_F$ (the top axis).

Figure 7

Differential conductance $\mathcal{G}$ (top figure) and correlators (bottom figure): $⟨n_{i,+}⟩$—solid curve, $⟨{\mathbf{S}}_i^2⟩$—long-dashed curve, $-⟨{\mathbf{S}}_1\bullet {\mathbf{S}}_2⟩$—short-dashed curve, as a function of the applied source-drain voltage $V$ at ${ϵ}_1-E_F={ϵ}_2-E_F=-3$. The parameters are $t_{12}=1$, $U_0=6$, $U_1=1.6$, ${\Gamma }_L={\Gamma }_R=0.1$. For comparison the plot for the equilibrium situation is presented by the thin curves for the zero-voltage conductance ${\mathcal{G}}_0$ and correlators as a function of ${ϵ}_1-E_F$ (the top axis).