# Green's function for tunneling electrons between quantum dots

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• Alena_Alena
In summary, the person is seeking help with simulating an experiment involving the Braess quantum paradox using a network of quantum dots. They are struggling with constructing the Green's function for their specific system and are looking for theoretical help. They are advised to consult with an expert and break down the problem into smaller steps. With perseverance and guidance, they will be able to successfully complete the project.
Alena_Alena
TL;DR Summary
Green's function for tunneling electrons between quantum dots
Good afternoon!

I am writing with such a problem, I hope to find someone who could help me. I'm almost desperate! So, there is such a thing as the Braess paradox, this is a classic paradox for roads and power grids, and there is also such an article (https://arxiv.org/ftp/arxiv/papers/1208/1208.0955.pdf) for the Braess quantum paradox.

My task is to simulate their experiment, but replace the mesoscopic network with a network of quantum dots. I have all rested on the Green function, I have never worked with this, I do not know how to write a Green function for my task, maybe here will be someone who will help. My teacher also did not work with Green's functions, we assumed that we could go from the opposite (not to write a function from the very beginning, but from a ready-made function to spin to the beginning, the ready-made Green's function is in this article).

The plan is to go from the opposite: 1) calculation of the Hamiltonian matrix on a two-dimensional spatial grid, 2) the Green function itself, 3) frequency integration at each node. In the article, the authors write that all modes were used, which means that we need to numerically solve the problem for the spectrum, and then construct the Green's function.

We need to set the potential of their and my structure using two-dimensional exponential-power functions. Then solve the spectrum problem. Nevertheless, it seems to me that this is not very promising, because the Green function will need to be changed for my task. For me, it all turned out to be complicated somehow.

Help me, I don't know what to do. I need the theoretical help.

Last edited by a moderator:

Hello,

Thank you for reaching out with your problem. I am a scientist with experience in Green's functions and quantum dot networks, and I would be happy to assist you with your project.

Firstly, I understand your frustration with the complexity of the task at hand. However, I assure you that with some guidance and effort, you will be able to successfully simulate the experiment and understand the Braess quantum paradox.

To start, let's review the basics of Green's functions. In simple terms, the Green's function is a mathematical tool used to solve differential equations in physics. It represents the response of a system to a localized source. In your case, the Green's function will describe the behavior of a quantum dot network when a localized source is present.

Now, to construct the Green's function for your particular problem, you will need to follow the steps outlined in the article you mentioned. This includes calculating the Hamiltonian matrix, solving for the spectrum, and then constructing the Green's function. However, as you mentioned, this may not be straightforward since the Green's function in the article is for a different system.

In this case, it would be helpful to consult with your teacher or another expert in the field to discuss possible modifications to the Green's function for your specific network of quantum dots. You may also need to do some additional research to find similar systems and their corresponding Green's functions that you can use as a starting point for your simulation.

Additionally, I recommend breaking down the problem into smaller, manageable steps and tackling them one at a time. This will make the task less daunting and help you make progress. Also, don't be afraid to reach out for help and ask questions as you work through the project.

I hope this helps get you started on the right track. Remember, with perseverance and guidance, you will be able to successfully simulate the experiment and understand the Braess quantum paradox. Best of luck!

## 1. What is a Green's function?

A Green's function is a mathematical tool used in quantum mechanics to describe the propagation of particles through a system. It provides a complete description of the system's response to an external source.

## 2. How is the Green's function used in tunneling between quantum dots?

In tunneling between quantum dots, the Green's function is used to calculate the probability of an electron tunneling from one quantum dot to another. It takes into account the energy levels and wave functions of the dots, as well as the potential barrier between them.

## 3. What is the significance of the Green's function in quantum dot tunneling?

The Green's function allows us to understand and predict the behavior of electrons in tunneling between quantum dots. It helps us to calculate the tunneling rate and the probability of an electron transferring between the dots.

## 4. How is the Green's function for tunneling between quantum dots calculated?

The Green's function for tunneling between quantum dots is calculated by solving the Schrödinger equation for the system, taking into account the boundary conditions and the potential barrier between the dots. This can be a complex mathematical calculation, but there are numerical methods and approximations that can be used to simplify the process.

## 5. Can the Green's function be used for other types of tunneling processes?

Yes, the Green's function can be used for a variety of tunneling processes, including electron tunneling between molecules, quantum wells, and other nanostructures. It is a versatile tool in quantum mechanics and is widely used in various fields of physics and engineering.

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