These notes come primarily from reading an article by R.C. Ashoori of MIT which was published in the journal Nature, Volume 379, February 1996. The article is entitled “Electrons in Artificial Atoms“.

In an artificial atom, the effects of electron-electron interaction are more important than in a normal atom! This is because orbital energies are far lower in artificial atoms than in real ones. The opposing effect of the spreading out of the electrons in space is not as large. Thus the relative importance of electron-electron interactions increases.

Energy resolution of new spectroscopic techniques is only limited by the sample’s temperature.

Ashoori describes a setup where a QD is close enough to one contact (capacitor plate) that an electron would be able to quantum tunnel between them. The other capacitor plate is too far away to tunnel. When an electron is successfully added to the QD, you can detect a tiny change in charge (typically about half an electron charge) on the surface of the farther capacitor plate.

A neat extension of this idea is to attach an AC current to the DC gate voltage that is driving electrons onto the QD. This makes it possible for an oscillation to occur at specific DC gate voltages. This is when the electron tunnels to and from the QD with each oscillation of the AC part of the gate voltage. This allows for synchronous detection by the farther capacitor plate as its charge changes slightly in time with these oscillations. This is known as single-electron capacitance spectroscopy (SECS).

Gated transport spectroscopy (GTS) seems to be what we do with our double quantum dots. We maintain a voltage difference between the source and drain contacts, and thus we can measure the current flow changing with changing conditions such as a changing gate voltage. When this article was written (1996), no one had yet successfully conducted GTS with fewer than about 10 electrons on the dot.

There are two main effects that make it more difficult to add extra electrons to a QD. The first is electron-electron interactions. They obviously push each other away. This is called the charging energy. Then there is the quantum energy levels. In order for an electron to be present in the dot, it needs to be occupying a quantum level. Due to the Pauli exclusion principle, it is not possible for more than two electrons to occupy the exact same quantum level. The factor of two is due to different possible spins of the electrons. This review claims that the charging energy is about five times the quantum level spacing for the samples described in the paper.

There is a geometric factor that connects the value of the gate voltage with the actual amount of energy needed to add an electron to the dot. This paper seems to be claiming that they simply use the geometry of the sample to estimate this. In the case of their perfectly symmetric doubly-contacted QD, they claim that this geometric factor is 0.5.

## Modeling

The author sketches out the basics of the parabolic potential well assumption. They assume that the z-direction is completely constrained, and that the x and y directions are governed by the parabolic potential well. The potential is circularly symmetric.

I have also read elsewhere that this model matches up rather well with observations. Even in 1996 this was apparently already known.

Introduction of a constant magnetic field to the analysis breaks the degeneracy in the quantum number *l*. Now positive and negative *l’*s have have slightly different energies due to the contributions of the magnetic moment interactions with the magnetic field. Note that we are talking about the magnetic moment created by the evolution of the electron’s wavefunction such that the electron can be considered to be moving in a circle around the center of the potential well. Magnetic field applied along the z axis enhances confinement in the dot. The magnetic field also introduces Zeeman (spin) splitting due to the magnetic moments of the electrons.

With the introduction of the magnetic field to the discussion, the author began to refer to the quantum levels as Landau levels. Strong magnetic fields can cause only one side of the level, let’s say the side with positive *l*, to be populated.

There is an interesting plot demonstrating the zig-zag effect as electrons end up populating different levels as the magnetic field strength increases. At some point, the zig zag stops because all electrons are in the lowest Landau level (and on one side of the *l* range I believe). This is very striking when seen in a plot.

Another important effect of increasing magnetic field is the fact that all of the radii of the different *l* levels shrink. This makes sense in light of the observation we made above that the magnetic field increases the confinement strength.

Example with 2 electrons in ground singlet state (*l = 1, s = +-1)*. With increasing B-field, it is possible to increase the Zeeman energy enough that one of the electrons gets promoted to *l* = 1.

Interactions of magnetic moments alone is not sufficient to produce the observed spin flips and *l* transitions. We must take into account coulomb interaction between the electrons in order to get the right answer. As the magnetic field increases, the *l* radius decreases and eventually it becomes possible for an electron to jump into the *l *= -1 state from the *l* = 0 state.

Once in the lowest Landau level, more changes can still occur. The lowest energy of all the electrons if they are in the lowest Landau level would nominally be when they are paired off with each pair having opposing spins to each other. As the magnetic field increases however, eventually higher-*l *states become lower-energy than either the spin up or spin down states, depending on the direction of the magentic field. This means that we will continue to see bumps in the spectrum as electrons will flip spins and move to more distance *l *values. Self-consistent calculations can reproduce some of these effects, but they seem to overstate the number of flips that happen at low field, and underestimate the number of flips at high field. Something is obviously still missing.

The Hartree-Fock technique takes into account the repulsion between many different electron wavefunctions. It reproduces much of the correct behaviour. An interesting result is that there is actually a short-range attraction force acting on electrons that are in different *l *bands. Adjacent bands tend to be preferentially populated. This can be compared to the fact that electrons in the same *l* band tend to be on opposite sides of the dot from one another. By moving to the higher *l* bands, it seems that the electrons can both be in a lower energy state and be closer to one another, an apparent paradox.

When the magnetic field becomes even higher, eventually it becomes energetically favourable for gaps to form in the *l *spectrum. The lowerest energy states involving gaps typically involve the gaps being adjacent to one another. Thus we don’t end up with a smattering of *l *gaps throughout our states. We end up with one big block of empty *l’*s.

It is known however that the Hartree-Fock calculations leave something out. They do not take into account the known electron correlation. In this area, some other techniques such as “exact diagonalization” seem to be better. However, the authors do not mention a successful combination of these factors into one theoretical model.

Their final section mentions some exciting further work that is being pursued, or will likely be pursued, in the field of quantum dots. One of the more interesting points for me is that a single-electron transistor obviously has a ‘fan-out’ problem. That is, normally the result of a piece of digital logic can be used to set off a cascade of other logic operations. This is obviously difficult if the result of your digital logic operation is the movement of a single electron. It seems however that people are finding ways around this. Perhaps I will find out more about this in the future.