Quantum Dots vs Quantum Wells

[vtahmoorian]

Hi everyone
I am doing some researches on Quantum Dots and Quantum Wells, especially on their quantum optical properties such as nonlinear effects( refractive index, absorption) and also their interactions with deriving classical/ non classical fields and cavities ..
but whenever it comes to write the Hamiltonian or density matrix equations of motion I can't see any difference between Quantum Dots and Quantum Wells?
Is there anyone who helps me get more clear on this?

 

[vanhees71]

As far as I know a quantum dot is some structure, usually manufactured in semiconductors, to confine quasiparticles in a 3D small volume, where small means that it's smaller than a typical length scale of the system (e.g., the Bohr radius of excitons), and a quantum well is a similar structure restricting the quasiparticles effectively in a 2D space, but I'm not an expert in nano physics.

 

[Cthugha]

Well, the Hamiltonian indeed is not too different.

QWs consist of a thin layer of one material, surrounded by layers of a different material with a wider band gap. This essentially traps the carriers inside the central region with the lower band gap. The thickness of the layer is usually comparable to the de-Broglie wavelength of the carriers, so you get discrete energy levels in the direction of confinement just like in the standard qm particle in a box problem, but the carriers are free to move inside the plane of the layer.

Going to QDs, just means that you now surround a small dot of one material with a material of wider band gap in all dimensions. So there is no free motion at all and there will only be discrete energy levels. This change leads to a very different density of states.

What does the Hamiltonian in your book look like? Quite often, the authors include some external potential term V(r) to include the external potential created by the two materials used and it is this term that changes when going from QWs to QDs.

 

[vtahmoorian]

Thank you Vanhees71 and thank you Cthugha too, regarding to your question Cthugha !actually in an article, I faced a Hamiltonian for a triple coupled QDs as below:
H = H0 + H1 + H2 +H3
H0 = ℏωl0><0l+ ℏω1l1><1l+ ℏω2l2><2l+ ℏω3l3><3l;
H1 = −ℏΩe−iωtl1><0l+H.c;
H2 = P1εl0><1l+H.c;
H3 = TA(l2><1l+l1><2l)+TB(l3><2l+l2><3l);
which is based on Bardeen's approach and TA/TB are tunneling factors and P1 is dipole moment of atomic transition corresponding to pumping from l0>to l1> and the electric field ε implies the the electrical amplitude of the field...and related to this Hamiltonian the writers had obtained density matrix equations of motion which were achievable, my question is if we change these coupled QDs by QWs and Hamiltonian doesn't change and followed by that density matrix equations of motion also don't change,then how will we show that we are studying QWs not QDs? I mean where does V(r) shows up in our calculations?
Thank you in advance

 

[Cthugha]

Please make it a habit of yours to cite your sources adequately. Nobody can know what the notation will mean exactly without actually reading the paper.

My guess is that |0>, |1>, |2> and |3> are discrete QD levels which immediately makes it clear that we are dealing with QDs here as you would not get discrete levels for QWs.

Please note that no research paper actually shows a Hamiltonian starting from scratch. For coupled QDs one would start with the discrete levels. You simply do not have enough space in the manuscript to explain how you arrive at the discrete levels and their spectral position. It is also not very interesting.

 

[vtahmoorian]

Sorry the source is "Optically controllable switch for light propagation based on triple coupled quantum dots" 10April 2014/Vol.53,NO.11/APPLIED OPTICS
But still my question is there? that is:"Where does V(r) show up in our calculations? does it appear just in TA/TB calculations?"
On the other hand you said :"My guess is that |0>, |1>, |2> and |3> are discrete QD levels which immediately makes it clear that we are dealing with QDs here as you would not get discrete levels for QWs".!(AS YOU WOULD NOT GET DISCRETE LEVELS IN QWs)!
As far as I have studied and also you mentioned in your previous post in the confinement direction we have discrete energy levels so we can have discrete levels in quantum wells as well, therefore how could you guess that it was QD?
And that's why I would like to know how to enter confinement direction in Hamiltonian when there is no room for it, at least in the article that I read and the Hamiltonian I wrote!
Again thank you for your time and answers I am now getting more clear...

 

[Cthugha]

Sorry the source is "Optically controllable switch for light propagation based on triple coupled quantum dots" 10April 2014/Vol.53,NO.11/APPLIED OPTICS

Ok, thanks for reference. Indeed they already start from discrete levels.

But still my question is there? that is:"Where does V(r) show up in our calculations? does it appear just in TA/TB calculations?"

It does not show up explicitly. Really nobody goes back to these very basics. One starts assuming that the problem of finding the energy levels of the bare quantum dots is already solved and the result are the levels they give. For QWs you would not get these discrete levels as the density of allowed states will turn out to be different. See the figure at the bottom of the following page for a sketch of how the DOS looks like in different dimensions:
http://britneyspears.ac/physics/dos/dos.htm

As far as I have studied and also you mentioned in your previous post in the confinement direction we have discrete energy levels so we can have discrete levels in quantum wells as well, therefore how could you guess that it was QD?

In a QW you have confinement in one direction only, so the energy according to motion in this direction is quantized. However, there is no confinement in the other two directions and the kinetic energy corresponding to motion in these two directions is continuous and can take any value. As the total energy is the sum of all these parts, the range of possible energy states will be continuous as long as there is at least one direction without confinement.

And that's why I would like to know how to enter confinement direction in Hamiltonian when there is no room for it, at least in the article that I read and the Hamiltonian I wrote!
Again thank you for your time and answers I am now getting more clear...

As I said, these guys do not start from scratch. They start from the levels seen for isolated quantum dots and are interested in what changes when tunneling comes into play. They do not explicitly solve for the quantum dot levels, but assume that problem is already solved. IN QWs things might get messy because the range of possible transitions might become huge.

 

[vtahmoorian]

Thanks a lot for your time, I have now new insights on this context...and also thanks for the webpage address, Although I had seen this figure somewhereelse before, but this time with your explanation ,the figure made everything more tangible to me.
Have a nice time

 

[Dotsaway]

greeting I am a founder of biotech and we are looking for a matter expert in QD theoretical physics to join our scientific team. Frederick.abraham@xtest.us