## Musings inspired by Ashoori’s article “Electrons in Artificial Atoms”

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.

## What I learned from Kastner’s “Artificial Atoms” paper from 1993

These are my notes from reading a paper that was published in Physics Today in 1993 entitled “Artificial Atoms” by Marc Kastner of MIT.

### Quantum dots in general

Quantum dots can constrain the motion of electrons to a space that is on the order of 100 nm. Within this space the energy levels of the electrons become quantized similarly to an atom. This is partly why Kastner regards quantum dots as ‘artificial atoms’.

The basic concept of a quantum dot is essentially a quantum well that is localized in all three dimensions. A bit of semiconductor is surrounded by some geometry of insulator.

He presents a different perspective on coulomb blockade than what I had learned before. In retrospect, I had focused on different aspects in my previous learnings about the coulomb blockade effect. This analysis focuses on how the electron experiences capacitance with the entire geometry of the setup. The change in capacitance is an energy that needs to be overcome when adding an electron to a quantum dot. This energy change is $$e^2/C$$, so an energy difference between the Fermi level of the source and the Fermi level of the dot that is smaller than this minimum implies that an electron cannot tunnel.

This is of course assuming that the thermal energy kT is smaller than $$e^{2}/C$$.

### Low temperature current flow

Fairly interesting discussion of the reasons why only specific conditions allow for current flow through a quantum dot near zero temperature. He shows that the energy of the state of a charge is given by:

$$E = Q V_G + Q^2 / 2C$$

Where E is the energy, Q is the charge, $$V_G$$ is the gate voltage, and C is the capacitance with regards to the rest of the system. For this analysis he only considers the capacitance with the quantum dot itself. In a real-life situation there would likely be notable contributions from the gate and contacts as well.

$$Q_0$$ is defined as the charge at which energy is minimized. Since the above equation is parabolic, you can imagine that $$Q_0$$ is the charge at which the minimum of the parabola occurs. $$Q_0$$, just like most charge quantities we talk about, is quantized into units of fundamental charge.

Imagine quantized spots on the parabola of energy separated by one fundamental charge from each other. When there are two degenerate energies corresponding to two spots on a horizontal plane from one another – which might be $$Q_1 = -N e$$  and $$Q_2 = -(N-1) e$$ for example, then current can flow at zero temperature. This is because no energy is needed to switch between the states with different numbers of electrons.

### Analogy to chemistry

Increasing gate voltage in his example leads to large numbers of electrons being constrained in the quantum dot. As gate voltage increases we also observe changes in the behaviour of these electron system. A direct analogy can be drawn to the chemistry of the periodic table. Using gate voltage, we can transform our quantum dot from one element to another. Just as in chemistry, the electronic behaviour can vary substantially depending on the number of electrons present.

### Energy quantization

Energy quantization of the electrons in our artificial atoms. Here Kastner briefly discusses the fact that only a small fraction of electrons in the quantum dot are free. The rest are bound tightly to atoms in the lattice. These free electrons are the ones we are generally talking about when we discuss quantum dots. He briefly describes how different construction techniques tend to allow for different numbers of free electrons to be constrained on the quantum dot. For the purposes of my research, I am already aware that we have a system in which we can easily choose conditions under which the quantum dot(s) will contain zero, one, two, etc free electrons.

It is possible to map out the energy spectrum of a quantum dot by keeping the gate voltage steady and conducting a source-drain bias sweep. If an energy level falls between the Fermi levels of the source and drain, current will flow. If two energy levels fall between, then more current will flow. Some corrections need to be made for the changes in the Fermi energy of the device itself (since it will be somewhere between the source and drain levels), but this is rather straightforward. The energy spectrum can thus be mapped out. Note that this energy spectrum includes multiple electron states as well as excited states of each number of electrons.

Increasing the gate voltage a lot would lead to more electrons being present on the quantum dot. This means that there are more valid energy states to be filled at or below the thermal energy. Thus, it makes sense that Kastner says that increasing the gate voltage leads to a decrease in the energy of confined states.

### Screening length and surface charge

It was here that I ran across the term ‘screening length’. Since I wasn’t 100% sure what it was, I started searching. I quickly found the Wikipedia articles on Debye length and electric field screening. It seems that screening length is referring to the concept also known as the Debye length. Over these distances, plasmas can screen out electric fields. That is, at distances longer than the Debye length, the effect of electric fields is substantially hidden by the movement response of the plasma to compensate.

In the article, Kastner uses the concept of screening length when discussing the all-metal artificial atom. In this case, the metal has a short screening length, so charge added to the quantum dot will reside very close to the surface. This in turn means that the electron-electron interaction is always $$e^2/C$$ regardless of the number of electrons that have already been added to the quantum dot. This does not apply to all types of quantum dots. The discussion seems to be limited in this case to the all-metal quantum dots.

### Experiments vs predictions

The energy levels of a two-probe quantum dot depend strongly on the applied magnetic field. This is not the case for all types of quantum dots. Level spacings in a two-probe quantum dot are irregular due to the effect of charged impurities in the materials used.

In 1993 it seems that the calculation of a full spectrum was not possible yet. I imagine that soon I will be looking at more recent literature in which this is accomplished. The simplest calculation method is using the simple harmonic oscillator potential. They also assume a non-interacting system where the added electrons don’t change the potential shape or strength.

They show at the end of the paper experimental results comparing to their theoretical expectations. Due to some notable discrepancies, they conclude that the constant-interaction model is not quantitatively correct. They claim that this is because it is not self-consistent. I am not totally sure why they claim this. Perhaps it will come to be clear to me in time.

The line shape for electrons on quantum dots is Lorentzian. The following analysis places some constraints on the physical design of the quantum dot such as a minimum width criterion for the barriers.

The last section includes a few of the basic applications that were forseeable at that point in history. It is interesting to me that this article predates the quantum computation fad that has swept much of condensed matter physics and certainly the sub-field of quantum dot physics.