Nonequilibrium transport through parallel double quantum dots in the Kondo regime

We extend a perturbative, nonequilibrium renormalization group approach to multiorbital systems and apply it for studying transport through two parallel quantum dots coupled electrostatically. In general, the conductance shows pronounced Kondoesque peaks at three voltages. One of these peaks disappears if, as in some experiments, one of the dots is decoupled from one of the two leads. For an asymmetric coupling to the leads, also negative differential conductances are possible. This is a genuine nonequilibrium effect, accompanying the Kondoesque peaks. Moreover, a criterion to distinguish spin and orbital Kondo effect in such a system is discussed.

Figure 1

Sketch of the parallel double quantum dot configuration. Each dot is coupled to its own leads; the coupling between the dots is only via the Coulomb interaction (not indicated in the figure).

Figure 2

One-loop Feynman diagrams for the renormalization of the vertex functions (a) ${\tilde{g}}_{m_1{m}_2}^{{\lambda }_1{\lambda }_2}\left(\omega \right)$ and (b) $g_{m_1{m}_2}^{{\lambda }_1{\lambda }_2}\left(\omega \right)$. The full line denotes the electron and the dashed the localized state of the Kondo model. Note that the second (hole) diagram of (b) does not contribute to (a) since it cancels with the corresponding first (particle) diagram. For the vertex renormalization, the electron line is restricted to a small interval $\mid dD\mid$ at the band edges, as indicated by the dash.

Figure 3

Vertex corrections and self-energy diagrams beyond the one-loop approximation of Fig. 2.

Figure 4

A similar combination of vertex corrections and self-energy diagrams as in Fig. 3 emerges if we calculate generalized spin and orbital susceptibilities.

Figure 5

Renormalized vertex functions to the left lead at $T∕T_K^0=0.5$, $V∕T_K^0=20$, $\delta ∕T_K^0=65$ ($g_{i\left(12\right)}$ in the legend box stands for $g_{i\left(12\right)}^{LL}$). Part (a) is for symmetric initial couplings $g_i^{\lambda {\lambda }^{\prime }}=0.04$, $g_{12}^{\lambda {\lambda }^{\prime }}=0.06$ for all $\lambda ,{\lambda }^{\prime }$. In part (b) the right lead of dot 2 was removed, i.e., parameters as in (a) except for $g_{2\left(12\right)}^{RR}=g_{2\left(12\right)}^{LR}=0$. In part (c) also the coupling of dot 1 is asymmetric: $g_1^{LL}=0.04$, $g_1^{RR}=0.06$, $g_2^{LL}=0.04$, $g_2^{RR}=0$, $g_{12}^{LL}=0.08$.

Figure 6

Scattering rates as a function of bias voltage $V$ at $T∕T_K^0=0.2$ and $\delta ∕T_K^0=40$. The initial couplings are symmetric: $g_1^{\lambda {\lambda }^{\prime }}=0.04$, $g_2^{\lambda {\lambda }^{\prime }}=0.01$, $g_{12}^{\lambda {\lambda }^{\prime }}=0.06$. For large bias voltage, the scattering rates are larger than $T$ (dashed line) but much smaller than $V$.

Figure 7

Difference of the occupation of the two orbitals, $⟨n_1-n_2⟩$, vs bias voltage $V$ at different orbital splitting $\delta ∕T_K^0=60,40,20,10,4,1$ (from top to bottom), temperature $T∕T_K^0=2$, and symmetric bare couplings $g_i^{\lambda {\lambda }^{\prime }}=0.04$, $g_{12}^{\lambda {\lambda }^{\prime }}=0.06$.

Figure 8

Differential conductance $G_i$ vs bias voltage $V$ for symmetric initial couplings $g_i^{\lambda {\lambda }^{\prime }}=0.04$, $g_{12}^{\lambda {\lambda }^{\prime }}=0.06$, temperature $T∕T_K^0=2$, and different orbital splittings $\delta$. Here and in the following figures, $G_0=2e^2∕h$ denotes the conductance quantum. Inset: temperature dependence of the differential conductance for $\delta ∕T_K^0=20$ ($T∕T_K^0=2$, dashed line; $T∕T_K^0=1$, solid line; $T∕T_K^0=0.1$, dotted line).

Figure 9

Differential conductance $G_i$ vs bias voltage $V$ for symmetric initial couplings $g_1^{\lambda {\lambda }^{\prime }}=0.04$, $g_2^{\lambda {\lambda }^{\prime }}=0.01$, $g_{12}^{\lambda {\lambda }^{\prime }}=0.06$, temperature $T∕T_K^0=2$, and different orbital splittings $\delta$.

Figure 10

(a) and (c) are the same as in Fig. 9, but now for asymmetric initial couplings $g_1^{LL}=0.04$, $g_1^{RR}=0.01$, $g_2^{LL}=0.01$, $g_2^{RR}=0$, and $g_{12}^{LL}=0.12$. The Kondoesque peak at $V=-\delta$ is now missing. Instead of the conductance $G_2$, which is zero, (b) and (d) show the difference in the occupation of the two dots.

Figure 11

Scattering process which connects the Fermi energy of the left lead (at $+V∕2$) and dot 2 (upper level) and that of the right lead (at $-V∕2$) with dot 1 (lower level); there is a separate Fermi level for each dot as indicated. The process displayed results, together with the reverse scattering process, in a Kondoesque resonance at $V\approx \delta$. For $V>\delta$ the scattering process shown is still possible but not the reverse one.

Figure 12

Same as on the right-hand side of Fig. 10 but now for $g_1^{LL}=0.01<g_1^{RR}=0.04$, and $g_{12}^{LL}=0.06$. ($g_2^{LL}=0.01$, $g_1^{RR}=0$ as before); shown are two different values of $\delta$ and temperature. For these parameters, the differential conductance is negative after the Kondoesque peak at $V\approx \delta$ which can be explained by the inversion of the occupation numbers, shown in part (b).

Figure 13

Part (a): Linear conductance $G_1\left(\delta \right)$ as a function of orbital splitting $\delta$ for symmetric coupling (see legend; $T∕T_K^0=2$, note that $T_K^0$ depends on $g$). Part (b) shows $\partial G_1\left(\delta \right)∕\partial \delta$ vs $\delta$ which is symmetric (antisymmetric) for a predominantly spin (orbital) Kondo effect.