*Experiments in Double Quantum Dots

Figure: Left: Schematic diagram of a `double gate single electron transistor' (double quantum dot) by Fujisawa and Tarucha (1997). The 2DEG is located 100 nm below the surface of an AlGaAs/GaAs modulation-doped heterostructure with mobility $8\cdot 10^5$ cm$^2$ (Vs)$^{-1}$ and carrier concentration $3\cdot 10^{11}$ cm$^{-2}$ at 1.6 K in the dark and ungated. Ga focused ion beam implanted in-plane gates and Schottky gates define the dot system. A double dot is formed by applying negative gate voltages to the gates GL, GC, and GR. The structure can also be used for single-dot experiments, where negative voltages are applied to GL and GC only.

We now shortly describe of a recent experimental realization of a two-level system in a semiconductor structure. This experiment has been performed in coupled artificial atoms, that is coupled quantum dots, by Fujisawa and co-workers at the Technical University of Delft (Netherlands) in 1998.

The double quantum dot is realized in a 2DEG AlGaAs-GaAs semiconductor heterostructure, see Fig.(3.4). Focused ion beam implanted in-plane gates define a narrow channel of tunable width which connects source and drain (left and right electron reservoir). On top of it, three Schottky gates define tunable tunnel barriers for electrons moving through the channel. By applying negative voltages to the left, central, and right Schottky gate, two quantum dots (left $L$ and right $R$) are defined which are coupled to each other and to the source and to the drain. The tunneling of electrons through the structure is large enough to detect current but small enough to have a well-defined number of electrons ($\sim 15$ and $\sim 25$) on the left and the right dot, respectively. The Coulomb charging energy ($\sim 4$ meV and $\sim 1$ meV) for putting an additional electron onto the dots is the largest energy scale, see Fig.(3.4).

Figure: Left: Current at temperature $T=23$mK as a function of the energy difference $\varepsilon=\varepsilon_L-\varepsilon_R$ in the experiment by Fujisawa and co-workers. The total measured current is decomposed into an elastic and an inelastic component. If the difference $\varepsilon$ between left and right dot energies $\varepsilon_L$ and $\varepsilon_R$ is larger than the source-drain-voltage, tunneling is no longer possible and the current drops to zero. The red circle marks the region of spontaneous emission of phonons. Phonons are the quanta of the lattice vibrations of the substrate (the whole semiconductor structure), in the same way as photons are the quanta of the electromagnetic (light) field. The spontaneous emission is characterized by the large `shoulder' for $\varepsilon>0$ with an oscillation-like structure on top of it.

By tuning simultaneously the gate voltages of the left and the right gate while keeping the central gate voltage constant, the double dot switches between the three states $\vert\rangle =\vert N_L,N_R\rangle$, $\vert L\rangle=\vert N_L+1,N_R \rangle$, and $\vert R\rangle =\vert N_L,N_R+1\rangle$ with only one additional electron either in the left or in the right dot (see the following section, where the model is explained in detail).

The main experimental trick is to keep the system within these states and to change only the energy difference $\varepsilon=\varepsilon_L-\varepsilon_R$ of the dots without changing too much, e.g., the barrier transmissions. The measured average spacing between single-particle states ($\sim 0.5$ and $\sim0.25$ meV) is still a large energy scale compared to the scale on which $\varepsilon$ is varied. The largest value of $\varepsilon$ is determined by the source-drain voltage which is around $0.14$ meV. The main findings are the following:

1. At a low temperature of $23$ mK, the stationary tunnel current $I$ as a function of $\varepsilon$ shows a peak at $\varepsilon=0$ with a broad shoulder for $\varepsilon>0$ that oscillates on a scale of $\approx 20-30\mu$eV, see Fig.(3.5).

2. For larger temperatures $T$, the current increases stronger on the absorption side $\varepsilon<0$ than on the emission side. The data for larger $T$ can be reconstructed from the $23$ mK data by multiplication with Einstein-Bose factors (the Planck radiation law) for emission and absorption.

3. The energy dependence of the current on the emission side is between $1/\varepsilon$ and $1/\varepsilon^2$. For larger distance of the left and right barrier ($600$ nm on a surface gate sample instead of $380$ nm for a focused ion beam sample), the period of the oscillations on the emission side appears to become shorter.

Those who are interested in more details on this fascinating experiment can read the article: T. Fujisawa, T. H. Oosterkamp, W. G. van der Wiel, B. W. Broer, R. Aguado, S. Tarucha, and L. P. Kouwenhoven, Science 282, 932 (1998).