--- abstract: 'We have studied the effects of the inter-dot coupling strength on the transport characteristics of a double quantum dot (DQD) system using experimentally-suitable parameters. Our transport calculations were based on a model which is physically motivated by the double dot structure of the measured sample, yet includes a realistic description of the interactions between electrons in the dots. We have also used a phenomenological model to capture some of the effects of the coupling that are not included in our more detailed calculation. We find that the addition of a second energy level, and the consequent inter-dot coupling, plays a crucial role in determining the interdot tunneling current that is possible in our sample.' author: - 'A. D. Armour' - 'W. J. Skocpol' - 'M. T. Peitik' - 'B. D. Thurber' - 'M. L. Leadbetter' - 'B. S. Kumar' title: 'Electron-electron interactions in coupled double quantum dots: from simple models to reality' --- Introduction {#sec:intro} ============ The double quantum dot (DQD) system has in recent years been of great experimental interest. Such systems are made of two quantum dots, connected together and controlled by nearby electrodes. The current through these systems has been successfully explained in terms of the theoretical model originally introduced by Göres [@gores_conductance_1992], and then refined to include more general coupling strengths by Schmid.[@schmid_theory_1998] The Hamiltonian for a DQD system is given by: $$\begin{aligned} \hat H & = & \sum_{i,j} E_i c_{i\sigma}^{\dag} c_{j\sigma} + U_0\sum_{i\sigma} c_{i\sigma}^{\dag} c_{i\sigma} \nonumber \\ &+& \sum_{i,\sigma} t_{i\sigma} c_{i\sigma}^{\dag} c_{i+1\sigma} + \sum_{i,\sigma} t_{i-1\sigma} c_{i-1\sigma}^{\dag} c_{i\sigma} + h.c. \label{eq:Hamiltonian}\end{aligned}$$ The model includes the Hubbard interaction $U_0$ for the electrons, which is a function of the gate voltages applied to the system. The interdot tunneling $t$ is also a function of these gate voltages. Finally, we also include electron-phonon coupling $\Gamma$, but in the presence of the applied magnetic field we choose the coupling to be diagonal in the spin space. This system will be represented schematically in Fig. \[fig:schematic\]. The voltage drop across the system occurs in an asymmetric fashion, as one might expect given the asymmetry of the sample itself. Experiments {#sec:experiment} ----------- Experiments on this system are still few in number. It was realized early on that a DQD system could be used to implement a resonant-tunneling experiment in which the energy level difference between the two dots could be finely adjusted.[@kouwenhoven_electron_1997] The experiment was performed in a two-dimensional electron gas defined in a GaAs/AlGaAs heterostructure, in the presence of a magnetic field. It was possible to control the number of electrons in each dot by applying gate voltages, to move them closer together or further apart. The energy difference between the levels in the dots was adjusted by changing the gate voltages. A typical experimental value for the inter-dot tunneling coupling strength was $\Gamma \approx 2\mbox{ meV}$.[@gores_conductance_1992] The authors of Ref.  report transport measurements at temperatures of $\approx 80$ mK, and use $U_0 = -U_{\infty} = 0.25$ meV, where $U_{\infty}$ is the on-site Coulomb repulsion energy between electrons. The DQD experiment was later repeated at lower temperatures, with the goal of observing the Kondo effect [@goldhaber-gordon_tunneling_1998] and of obtaining more detailed control of the tunneling coupling. [@schmid_kondo_1998] In order to reduce background effects due to the phonon density of states in the leads, this experiment was carried out at a very low temperature $T=100$ mK. In addition to the same set of parameters as Ref. , these authors also considered the effects of an asymmetry between the two dots. The energy difference between the two dots was changed in order to make the coupling between the two dots negligible. The tunneling parameter between the dots was varied in order to change the inter-dot coupling. This experiment was performed with $U_0=U_{\infty}=0.24$ meV. Later, Schmid[@schmid_theory_2000] performed calculations using a model with the same parameters to explain how the conductance of the DQD system changes as the voltage on one of the electrodes was increased. A second experimental realization of this system was reported by Untiedt [*et al.*]{}, who observed phonon-assisted Andreev transport through a DQD. [@goldhaber-gordon_kondo_1998] The sample was formed by defining a single quantum dot in a GaAs/AlGaAs heterostructure, with similar parameters to those described in Refs.  and . The experiment was done at $T \approx 100$ mK, with a coupling between the dot and the leads that could be tuned using a nearby gate. An external magnetic field was used to tune the system into the Kondo regime. The sample was biased to give a current of $0.07$ nA (see Ref. ). Models {#sec:models} ------ Several theoretical models have been used to describe such systems. There is a simple model for the DQD, first presented by Göres and Schmid.[@gores_conductance_1992] For this model, the conductance of the DQD system is governed by the transmission probabilities, $|t_{i\sigma}|^2$, and the energy levels of the dots. The energy levels can be described as two parabolas with a gate voltage dependent separation. These parabolas will each describe an anti-crossing when one or both of the dots are occupied. With a symmetric tunneling interaction, the ground states will be filled one by one, as more electrons are added to the DQD system. A schematic of the resulting energy levels is shown in Fig. \[fig:schematic\]. Recently, two of the present authors have used this model to explain the low-temperature Kondo effect in a DQD, for some sample parameters. [@schmid_theory_2000] It was found that the ground state is antiferromagnetic when the total occupancy is odd, and ferromagnetic when the total occupancy is even. It was also found that the lowest energy excited states in both cases have a significant contribution from the higher-energy level. This occurs because the higher-energy level is closer to the Fermi level and thus has more available electrons. Using this model, it was shown that if one of the dots is occupied by an odd number of electrons, the spin of the system is fixed to be up. If the dots are both occupied by an even number of electrons, then spin flips occur with a rate that is approximately constant in $B$, but which is strongly dependent on the magnetic flux $f$ per magnetic period. The spin flip rate was found to be inversely proportional to the number of electrons in the higher-energy level, and to be proportional to the magnetic field, and to the separation between the two dots. A simple model for the phonons coupling to the electrons in the DQD system was developed by Schmid.[@schmid_theory_2000] This included the effects of the gate voltage. The phonons were modeled using a Debye density of states. The strength of the electron-phonon coupling was set by the coupling to acoustic phonons, as calculated using Fermi’s golden rule. This model reproduced some of the effects of the higher-energy level, when the occupation of the system was fixed so that it had an odd number of electrons. The rate at which the two ground states change due to changes in gate voltage, or magnetic flux, was found to vary as $B^{3/2}$. A phenomenological model was presented by Blick [*et al.*]{}[@blick_double_1999] They added a new parameter to the original model, namely a tunneling interaction between electrons in the dots and a second energy level in the leads. This second energy level was represented by two parabolas. It was suggested that these might be associated with a nearby impurity level. The new parameter was chosen to be small in magnitude, such that the overall energy level separation in the system should remain close to what is observed experimentally. The experimental system of interest was then described by including a coupling to one of the energy levels in the leads, together with an overall electron-electron interaction strength $U_0$. They found that it was possible to get current flow from a DQD system to the leads using a small difference in the energy levels of the electrons in the two dots. Their focus, however, was on phonon-assisted processes which have nothing to do with the ground state, so it would not be appropriate to apply their work directly to the issues we have been considering here. In particular, it would not have been necessary to include the full