Interference effects in the conductance of multilevel quantum dots

Using exact-diagonalization techniques supplemented by a Dyson equation embedding procedure, the transport properties of multilevel quantum dots are investigated in the Kondo regime. The conductance can be decomposed into the contributions of each level. It is shown that these channels can carry a different phase, and destructive interference processes are observed when the phase difference between them is $\pm \pi$. This effect is very different from those observed in bulk metals with magnetic impurities, where the phase differences play no significant role. The effect is also different from other recent studies of interference processes in dots, as discussed in the text. In particular, no external magnetic field is introduced here, and the dot-leads hopping amplitude for all levels are the same. However, conductance cancellations induced by interactions are still observed. Another interesting effect reported here is the formation of localized states that do not participate in the transport. When one of these states crosses the Fermi level, the electronic occupation of the quantum dot changes, modifying the many-body physics of the system and indirectly affecting the transport properties. Unusual discontinuities between two finite conductance values can occur as the gate voltage is varied, as discussed here.

Figure 1

Schematic representation of the main system studied in this paper, consisting of one quantum-dot with two levels $\alpha$ and $\beta$. The two possible paths carrying different phases ${\Phi }_{\alpha }$ and ${\Phi }_{\beta }$ are shown.

Figure 2

Conductance for five different values of $\Delta V$. The conductance (in units of $e^2∕h$) is shown in solid lines and the phase difference (divided by $\pi$) in dotted lines. When $\Delta V$ is the largest energy [$\Delta V=2t$ (e)], two structures similar to those found in a single-level QD, separated by $\Delta V$, appear in the conductance. The case (d) is similar, but since $\Delta V$ is not large enough, an extra thin peak appears around $V_g=-1$ (this peak is shown with more detail in Fig. 8). When $\Delta V$ decreases, these structures began to interact giving rise to dips in the conductance, as shown in (b) and (c) for the cases $\Delta V=0.2t$ and $\Delta V=0.5t$ ($\Delta V=0.2t$ is studied in more detail in Fig. 4). Finally, in the limit $\Delta V=0$, the conductance dips are transformed into discontinuities.

Figure 3

Conductance (in units of $e^2∕h$) in a narrow range of $V_g$ and $\Delta V$. The discontinuity at $\Delta V=0$ is transformed into a dip for finite values of $\Delta V$.

Figure 4

(a) Conductance (in units of $e^2∕h$), (b) phase difference (divided by $\pi$), and (c) partial conductances, for $\Delta V=0.2t$. The dips, indicated by “a” and “b” in (a), are in regions where the phase difference jumps to $\pi$ producing an exact cancellation when ${\sigma }_{\alpha }={\sigma }_{\beta }$.

Figure 5

Conductance (in units of $e^2∕h$), phase difference, and partial conductances for $\Delta V=0.2t$ for the “b” dip of Fig. 4a. Thicks arrows shows the gate potential where the conductance cancels, $\Delta \Phi =\pi$ and ${\sigma }_{\alpha }={\sigma }_{\beta }$.

Figure 6

Conductance (in units of $e^2∕h$), total charge of the QD, and charge at each level for the case $\Delta V=0.2t$. The charge of the level $\alpha$ is indicated with a continuos lines, while the charge of $\beta$ is shown with a dashed line. The central peak emerges in the regime where there are two electrons in the dot, in different levels.

Figure 7

Conductance (in units of $e^2∕h$) and charges for the case $\Delta V=t$ as a function of the gate potential. These results should be compared against those of Fig. 6.

Figure 8

Conductance (in units of $e^2∕h$), phase difference, and partial conductances for $\Delta V=t$ for the central peak shown in Fig. 2d. The partial contributions to the conductance (${\sigma }_{\alpha }$ and ${\sigma }_{\beta }$) have the same shape as in Fig. 4, but now the phase difference between them is zero. This phase difference creates a peak where in the other situation appears a dip.

Figure 9

Convergence of the conductance with the size of the exactly solved cluster $L$ for the case $\Delta V=0.2t$. Dotted lines are for $L=4$, continuous thin lines for $L=8$, and dashed lines for $L=12$. Extra points for $L=16$ and $L=20$ are shown with crosses and diamonds, respectively.

Figure 10

Study of the conductance and total charge when $U^{\prime }$ and $J$ are not active. Shown are results for the case $U^{\prime }=J=0$ (solid line) compared with the previously shown results at $U^{\prime }=2∕3U$ and $J=U-U^{\prime }$, both at $\Delta V=0.35$. The effects are qualitative the same in both cases.

Figure 11

Conductance (in units of $e^2∕h$) with large $U^{\prime }=2t$. In (a), shown is the case with a ferromagnetic Hund coupling $J=\mid U-U^{\prime }\mid$ and in (b) with antiferromagnetic $J=U-U^{\prime }$. Both figures are for $\Delta V=0$.

Figure 12

Schematic representation of the different situations that could occur in a two-level QD as the gate voltage changes. For more details see the text.

Figure 13

Conductance (in units of $e^2∕h$) for differents $J$ couplings. (a) corresponds to the case $J=U-U$, (b) is at $J=0$, and (c) is for a large negative case $J=-0.5t$. The central peak disappears when the ferromagnetic spin correlations are eliminated at even larger negative $J$. More detail can be found in the text.

Figure 14

Density-of-states of the bare $L=4$ cluster, for the cases $\Delta V=0.05t$ and $V_g==-1.082t$. The Fermi level lies at $\omega =0.0$. Note the relative change of the weight signs for the poles near $\omega =0$, when comparing the Green’s functions $g_{\alpha N}$ and $g_{\beta N}$.

Figure 15

Comparison between the imaginary parts of $g_{\alpha \alpha }$ and $g_{\alpha N}$, for the cases $\Delta V=0$ and $\Delta V=0.05t$, working at $V_g=-1.082t$ for the $L=4$ cluster. The pole in $g_{\alpha N}$ near the Fermi level disappears (i.e. carries no weight) at $\Delta V=0$.

Figure 16

Nonsymmetric $\alpha \text{−}\beta$ case, where the Coulomb repulsions in both levels are different, i.e., $U_{\beta }=U_{\alpha }+0.1t$. Dips in the conductance are observed even with $\Delta V=0$, due to the explicitly broken symmetry. More details can be found in the text.

Figure 17

(a) Schematic representation of a system of two coupled QDs, one of them side connected. In this case, each dot has only one level. (b) Equivalent system after the transformation Eqs. (19, 20). The two possible paths in the transformed basis are shown.

Figure 18

Conductance (in units of $e^2∕h$), phase difference and partial conductances as a function of the gate potential $V_g$, for the geometry shown in Fig. 17. The parameters used, in units of $t$ (the bandwidth is $W=4t$), are $t_a=0.12$, $t_2=0.25$, and $U_i=0.5$. In (a), two zeros in the conductance as a function of $V_g$ are shown. (b) The phase difference $\Delta \Phi$ near these zeros jumps to $-\pi$. In (c), the partial contributions to the conductance ${\sigma }_{-}$ and ${\sigma }_{+}$ are given. For $V_g=-U_i∕2$, we can observe a constructive interference process where ${\sigma }_{-}={\sigma }_{+}$ and $\Delta \Phi =0$.

Figure 19

Conductance (in units of $e^2∕h$) and charge as a function of $V_g$, for the same parameters used in Fig. 18. Shown are (a) the conductance vs $V_g$, (b) the charge at the bonding and antibonding dimer states, and (c) the charge at the real dots A and S.

Figure 20

Conductance (in units of $e^2∕h$) for three different values of $t_a$. (a) $t_a=0.12$ (reproduced here for comparison), (b) $t_a=0.24$, and (c) $t_a=0.48$. For $t_a=0$ the side coupled dot is not connected and the conductance presents the characteristics of one QD (not shown). Increasing $t_a$, the side coupled dot starts to participate in the transport, and interference effects appear. For large $t_a$, the system—represented by the dimer states—is equivalent to one dot with two levels, separated by an energy $2t_a+U$.

Figure 21

Convergence of the conductance with the cluster size, for the case of the side-connected dot. The parameters are the same as in Fig. 18.

Figure 22

Convergence study of the charge at dot S increasing the size of the cluster exactly solved. For the smaller cluster, we found important changes around $V_g=-U∕2$, but eventually the system converges to a plateau for a sufficiently large cluster. The parameters are the same as in Fig. 18.

Figure 23

Schematic representation of the system of two coupled QDs with two levels. The two possible paths carrying different phases ${\mathbf{\Phi }}_{\alpha }$ and ${\mathbf{\Phi }}_{\beta }$ are shown

Figure 24

Conductance of two coupled quantum dots, each with two levels, for three different cases of $\Delta V$ (large, intermediate, and zero). (a) When $\Delta V$ is the largest energy, two structures separated by $\Delta V$ appear in the conductance, similarly as for two coupled one-level QDs. (b) When $\Delta V$ decreases these two structures start to overlap producing dips in the conductance. (c) A more complex conductance structure with several discontinuities is obtained in the limit $\Delta V=0$

Figure 25

Conductance (in units of $e^2∕h$), phase, and partial conductances for a small $\Delta V=0.01t$, in the case of two coupled two-level QDs. Typical dips in the conductance will be analyzed in Fig. 26. Near the gate potential $V_g=-0.5$, a constructive interference occurs. The phase is zero and the partial conductance becomes $\sim 0.25$, inducing a constructive phase process for electron transport.

Figure 26

First two dips in the conductance for the small $\Delta V$ case shown in Fig. 25. Both partial conductances (${\sigma }_{\alpha }$ and ${\sigma }_{\beta }$) have a zero near the dips, but not at the same time. Then, the dip in the overall conductance is produced by a phase interference process induced by $\Delta \Phi =\pm \pi$.