# Few-electron system in double Quantum Dots

Our research focuses in the low-dimensional electronic nanostructures, more precisely in the so-called *designer atoms* or *quantum dots*
(QDs). Modern techniques in constructing ultrasmall semiconductor
devices have made it possible to confine electrons in the three
dimensions, so that the quantization of its energy and charge is easily
observable. The fabrication of these devices can be performed, for
example, by combining pure and doped semiconductor thin layers. In
normal semiconductors, such as GaAs or AlGaAs, the electrons move
freely in all dimensions, and there is no quantization of the energy
levels. In the QDs, electrons are usually trapped into a two
dimensional (2D) gas in the interface between two differently doped
materials. Additionally to the 2D gas there might appear a effective
confinement wich will make the electrons behave like if they were in an
*artificial atom*.

## Few-Body Model

In
order to solve the nonrelativistic SchrÃ¶dinger equation that describes
or quantum system, we make use of the mathematical model named *configuration interaction* (CI) also known as *exact diagonalization method*.
This method is limited by the number of particles present in the
system. Actually we are not applying it to systems bigger than four
electrons with the resulting (sparse) hamiltonian matrix being of the
order of 10^{5}.

The
first step is to write the many-body Hamiltonian taking into account
the interparticle interactions (electrostatic potential), the external
fields (magnetic field is present), Rashba coupling and

where â„‹^{*} is the single-particle Hamiltonian operator explicitly written as:

The confinement potential is modeled by a group of Gaussians[1]

with all the unkown parameters being positive. It should be appreciated the broken symmetry between *x* and *y*
axes, so we are working with elliptically shaped QDs. In the contiguous
figure, we have plotted our actually studied system. It has an interdot
barrier of 3 meV and a separation of 80 nm between dots and spread of
the same magnitude for each of the single dot.
The part in the Hamiltonian wich takes into account the interaction
(Coulomb interaction) is, computationally, the toughest bottleneck[2],
as it includes the evaluation of fourfold summations with up to
thousands of elements.

## References

- Xuedong Hu & Das Sarma: Phys. Rev. A 61, 062301 (2000).
- Jaime Zaratiegui Garcia: J. Math. Phys. 46, 122104 (2005) (5 pages).

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21.6.2006, Jaime Zaratiegui Garcia