### Introduction

This post is a continuation of the one on the Kondo effect^{1} and is leading towards one presenting the numerical renormalization group and a simple program that implements it. I won’t insist much on details about the quantum dots, but I thought I should mention them because the numerical method is widely used to study them.

As a simple definition, a quantum dot is a small device that confines electrons/holes in a space small enough to exhibit quantum behavior. Typically semiconductors are used because compared with metals, the electrons inside have a quite low Fermi velocity, that is, the De Broglie wavelength for the Fermi surface electrons is bigger than in metals, which makes the quantum dots exhibit quantum behavior at sizes bigger than for the metal case.

They can be used for a lot of applications, ranging from measuring devices, tunable light emitters, spintronic devices and so on up to hopefully quantum computing.

### Construction

There are many ways of obtaining quantum dots, I’ll mention here only three of them. The main idea is to isolate out a small region of space using potential barriers. The ‘region of space’ does not need to be a 3D region.

- For example, one can use a carbon nanotube or a graphene nanoribbon to make a quantum dot. Just hang the nanotube resting with its ends on a source and a drain, respectively, and have some gates underneath. By applying a potential to the gates, a region of the nanotube can be isolated out forming a quantum dot.
One can use a 2D electron gas to make a lateral quantum dot, by using gates to confine the electron gas into a quantum dot.

They can use deposition of doped semiconducting layers, lithography and etching to isolate out a 3D quantum dot having the source at the bottom and the drain on top, being surrounded by gates that allow tuning it.

The actual construction methods do not matter much for our purpose, but the fact that they can be interconnected and various parameters tuned.

### Properties

To illustrate the quantum dot properties I drawn a picture:

Right in the middle there is the potential well that confines the electrons. I represented the energy levels not as straight lines but as having a width because unlike an electron isolated out in an ideal box as in the particle in a box case, the potential well is finite and more, there are potential barriers with a finite width between the quantum dot and leads (marked L and R here, from ‘left’ and ‘right’) allowing electrons to tunnel in and out. This interaction with the leads outside splits out the uni-electron levels so they become continuous bands. As an alternate picture, because of the tunneling the electron life time on a level is limited, which means an indeterminacy in energy. The stronger the coupling with the leads are, the smaller the lifetime and bigger the broadening of the energy level.

The particle in a box model is already quite good to illustrate how spacing of energy levels – the in the figure – can be tuned by adjusting the size and shape of the potential well. In many setups they have several gates that allow them to do that. For the particle in the box case, the spacing is .

But that’s not the only thing that is related with the size. The Coulomb interaction, U, also depends on the size. It takes more energy to fit a second electron into a smaller box than into a big one. This big energy makes the quantum dot exhibit the Coulomb blockade.

In this picture, the potential bias between the left and right leads is in the zero bias limit. By applying a potential difference between them, one can open a window large enough to fit several energy levels in it. The conductance is proportional with the number available channels, for the ideal case being a quantum conductance for a channel. The gate potential can be used to move up and down the energy levels inside the dot (basically by pushing up and down the bottom of the potential well). As they enter or leave the potential window formed by the potential difference between the leads, the conductance changes. As the bias is lowered towards the zero bias, only one level can fit into the window and by varying they obtain peaks in the differential conductance that allows measuring the distance between the levels.

Last but not least, the thermal energy scale must be mentioned. It must be lower than the other important energy scales in the problem. If energy is large enough, an electron from a lead would have enough energy from the thermal excitations to ‘jump’ into a level that is high (even higher than the charging energy) and then go into the other lead, thus having an electric current even with no energy levels in the potential difference window. Also an electron from the quantum dot can gain enough energy to jump out. The quantum effects are washed out.

### There is more

From this description it might not appear that they have so interesting properties. Indeed they expose quantum mechanical effects similarly with an atom, which is why they are also called artificial atoms. They have orbitals inside and electrons that fit inside respect Hund rules and so on. Well, they indeed can be tuned, which maybe is harder to do with an atom, but there is more. In the description above I did not say much about the leads. They can be of various sorts, from semiconductors to ferromagnetic metals to superconductors (left and right of different types, too) and that adds to the interesting effects that can be obtained.

But even without those complexities, the interaction with the leads has something which is hidden in the above description: the Kondo effect^{1}. If levels inside the quantum dot – that are lower than the lead electrochemical potential marked with and – are all filled but one which is occupied by a single electron only, the quantum dot can scatter electrons at the Fermi level in a similar manner as the magnetic impurity in a metal described in the post about the Kondo effect. Unlike the metallic host case, where the impurity can scatter in any direction, the quantum dot can scatter either back into the lead the electron was coming from, or forward into the other lead. This forward scattering leads to an enhanced conductance. And there is even more, they can get the Kondo effect with an even number of electrons in the dot, by applying a magnetic field to have electrons in triplet state. More on quantum dots and the Kondo effect, here^{2}.

### Even more

The above description was for a single quantum dot, but they can be connected – with connections that can be adjusted, too – obtaining artificial molecules. Now there are even more possibilities. Just to point to one of them: Loss–DiVincenzo quantum computer^{3}. Even the Kondo effect gets more complexities, they study for example^{4} two-stage Kondo effects. I bet you guessed by now, there is also a three-stage Kondo effect and in general, a multi-stage one^{5}.

I’ll post more links to info on both Kondo effect and quantum dots when I’ll post about the numerical renormalization group, until then I’ll point again to Michael Sindel Disertation^{6}.

### Conclusion

I briefly presented quantum dots because they can be studied with the numerical renormalization group. I’ll get to it soon and maybe even present a configuration of two connected quantum dots.

The next post will probably be about renormalization groups, though.

- Kondo effect The post about the Kondo Effect. ↩ ↩
- Revival of the Kondo effect Leo Kouwenhoven and Leonid Glazman. ↩
- Quantum Computation with Quantum Dots Daniel Loss and David DiVincenzo. ↩
- Two-stage Kondo effect in T-shaped double quantum dots with ferromagnetic leads Krzysztof P. Wójcik and Ireneusz Weymann. ↩
- Correlation Effects in Side-Coupled Quantum Dots R Zitko and J Bonca. ↩
- Michael Sindel Disertation ↩