## 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.

## Going home for the holidays

Excited to spend several weeks back in Saskatchewan. Going to see a lot of family and friends.

Also hoping to grab some of my undergrad physics notes that I think might be of value for the masters studying. I find that while my notes might be of relatively low quality compared to a textbook…they are the way that I learned the subject in the first place, and are thus well suited to help refresh the neural pathways that were once there 🙂

Hope everyone is excited for the holidays.