Transport through a double quantum dot in the sequential tunneling and cotunneling regimes

We study transport through a double quantum dot, both in the sequential tunneling and cotunneling regimes. Using a master equation approach, we find that, in the sequential tunneling regime, the differential conductance $G$ as a function of the bias voltage $\Delta \mu$ has a number of satellite peaks with respect to the main peak of the Coulomb blockade diamond. The position of these peaks is related to the interdot tunnel splitting and the singlet-triplet splitting. We find satellite peaks with both positive and negative values of differential conductance for realistic parameter regimes. Relating our theory to a microscopic (Hund-Mulliken) model for the double dot, we find a temperature regime for which the Hubbard ratio (=tunnel coupling over on-site Coulomb repulsion) can be extracted from $G\left(\Delta \mu \right)$ in the cotunneling regime. In addition, we consider a combined effect of cotunneling and sequential tunneling, which leads to new peaks (dips) in $G\left(\Delta \mu \right)$ inside the Coulomb blockade diamond below some temperature scales, which we specify.

Figure 1

(a) Two coupled quantum dots, with the inter-dot tunneling amplitude $t_0$, attached to Fermi liquid leads at different chemical potentials ${\mu }_L$ and ${\mu }_R={\mu }_L-\Delta \mu$. The tunnel coupling between the dots and leads is characterized at $t_0=0$ by the tunneling amplitudes $t_L$ and $t_R$. (b) A schematic plot of the linear $\left(\Delta \mu \to 0\right)$ conductance $G$ through the double dot (DD) as a function of the gate voltage ${\tilde{V}}_g$, showing the Coulomb blockade (CB) effect, with sequential-tunneling peaks and CB valleys. The number of electrons contained in the DD, $N=N_1+N_2$, is fixed in the valleys between the peaks. The position away from the $N=1,2$ peak is defined by $\delta E$, which takes on negative (positive) values in the $N=1$ $\left(N=2\right)$ CB valley.

Figure 2

(a) The ratio $\tau =\left({\rho }_S+{\rho }_T\right)∕\left({\rho }_{+}+{\rho }_{-}\right)$ plotted versus $\Delta \mu ∕2=\Delta {\mu }_{L,R}$ for $\delta E<0$. (b) The inverse ratio $1∕\tau$ versus $\Delta \mu ∕2$ for $\delta E>0$. For both figures, we choose the units of energy such that $\mid \delta E\mid =1$. We use the following parameters: $2t_0=0.8$, $J=0.2$, $\mathcal{S}=0.5$, and $\varphi =0.4$. The grids above (a) and (b) show the division into the intervals (39, 46), respectively. The insets show the transitions which get switched on at $\Delta \mu ∕2=\mid \delta E\mid$.

Figure 3

With the same parameters as in Fig. 2, the ratio $\beta ={\rho }_T∕{\rho }_S$ versus $\Delta \mu ∕2$ for (a) $\delta E<0$ and (b) $\delta E>0$. The shaded area in (a) shows the inactive region with no population of the $N=2$ sector, see Fig. 2a. We note that this region can be active if a combined effect of sequential tunneling and cotunneling is considered, see Sec. 6.

Figure 4

With the same parameters as in Fig. 2, the ratio $\gamma ={\rho }_{-}∕{\rho }_{+}$ versus $\Delta \mu ∕2$ for (a) $\delta E<0$ and (b) $\delta E>0$. The inset in (b) shows the sequence of transition which takes place at $\Delta \mu ∕2⩾\delta E+2t_0-J$ [compare with the inset in Fig. 2b].

Figure 5

The population probabilities ${\rho }_p$ versus $\Delta \mu ∕2$ calculated from Eqs. (34) with the values of $\tau$, $\beta$, and $\gamma$ given by, respectively, Figs. 2, 3, 4 for (a) $\delta E=-1$ and (b) $\delta E=1$.

Figure 6

The bias dependence of the differential conductance $G$ for (a) $\delta E=-1$ and (b) $\delta E=1$. The ordinate axis is scaled by $G_l=e^2\pi \nu {\mid t_l\mid }^2∕\hslash T$. We use the same parameters as in Fig. 2 and with $t_L=t_R$ $\left(G_L=G_R\right)$.

Figure 7

A gray scale plot of $G$ versus $\delta E$ and $\Delta \mu ∕2=\Delta {\mu }_L=\Delta {\mu }_R$. The white (black) color corresponds to positive (negative) values of $G$; gray stands for $G=0$. Here, we use $2t_0=1$, $J=0.25$, $\mathcal{S}=0.6$, $\varphi =0.3$, and $\eta =1$.

Figure 8

The same as in Fig. 7, but for asymmetric biasing $\Delta \mu =\Delta {\mu }_L$, $\Delta {\mu }_R=0$.

Figure 9

(a) Differential conductance $G$ vs bias in the cotunneling regime for the $N=2$ CB valley. The dashed curve corresponds to the asymptotic value of $G$ at low temperatures. The solid curves are for a finite $T=\left(0.2,1,2\right)T_h$, where $T_h$ is the characteristic temperature of the strong heating regime, see text. For the calculation we used $\mathcal{S}=0.6$, $\varphi =0.316$ $\left(T_h\simeq 0.23J\right)$, $\eta =1$, $E_{-}\left(2\right)=E_{+}\left(2\right)=E_C$, and ${\Gamma }_{L,R}∕E_C=0.1$. (b) The quantity $A∕\left[1-A\left(\Delta \mu -J\right)∕2J\right]$, with $A\equiv -\left(J∕\Delta G\right)\left(dG∕d\Delta \mu \right)$, where $\Delta G=G\left(\Delta \mu \right)-G\left(\infty \right)$, is plotted for the three solid curves of $G\left(\Delta \mu \right)$ shown in Fig. 9a. The solid curve, corresponding to $T=0.2T_h$, saturates with good precision at the value of $8∕\left(2+\kappa \right)=2{\left(1-\varphi \right)}^2∕\left(1+{\varphi }^2\right)$, whereas the other solid curves (not in the strong heating regime) have a smaller saturation value. The dotted curves A, B, and C, are plotted for the $T=0.2T_h$ curve and correspond to choosing the value of $G\left(\infty \right)$ as shown in Fig. 9a by dotted horizontal segments. Only for a correct choice of $G\left(\infty \right)$ the curve has a plateau and saturates at a finite value.

Figure 10

Differential conductance $G$ versus bias for (a) $\delta E=-1$ and (b) $\delta E=1$. Here, we use the following parameters: $t_0=0.1$, $J=0.025$, $\mathcal{S}=0.2$, $\varphi =0.6$, ${\Gamma }_L={\Gamma }_R=0.001$, $T=0.001$, $\Delta {\mu }_{L,R}=\Delta \mu ∕2$. The conductance is scaled by $G_R=e^2{\Gamma }_R∕\hslash T$ on both plots.