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2D Quantum Well and kvaluesby Janeabc
Tags: quantum well 
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#1
May2112, 10:09 PM

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It is a classic 2DE quantum well problem. I do not understand few things. I haven't seen the solution anywhere on the new, so this is why i ask here. It is a quantum well  Lx is it's dimension along xaxis Ly is it's dimension along the yaxis.There is a potential as in picture lets call it Uo. I have found from post here that the kvalue is quantized. If i understood it correctly, the Kz is quantized. I am not sure what does this means. Can actually the particle in the quantum well even move along the zaxis? What are possible values of wavevector Kz ? As i seen in my lecture there are bound condition  so as far as i understand, Kx and Ky are quantized as well. Kx*Lx = 2*n*Pi Ky*Ly = 2*n*Pi Does it change something, that in the lecture is described not single quantum well, but a periodic potential  like in a GaAs based heterostructure. 


#2
May2212, 03:32 PM

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P: 1,659

The picture and your description of it are somewhat misleading. Let us see whether we can clarify.
Looking at your picture, it definitely does not show a quantum well. The picture is also somewhat misleading. On the left and right it says "bulk", but you see what looks like a surface. Either this is really a surface (not really probable) or that is just a means to better illustrate where the bulk lies on the x and yaxes and the bulk goes on in zdirection. In that case the structure shown is rather a quantum wire with confinement in x and y direction (and correspondingly quantized kvalues in x and ydirection), but not in zdirection. If it is supposed to picture confinement in all three directions, you have a quantum, dot or quantum box with confinement and quantized kvalues in all three dimensions. In that case you can get for example tunneling of particles from one quantum well to the next that depends on the distance between quantum wells. 


#3
May2212, 05:08 PM

P: 4

Thanks very much for answer . I uploaded another (i hope) better picture. It is a AlGaAsGaAsAlGaAs heterostructure.
So Kz is quantified in this case, Kx and Ky have continual values. So the electron is free to move along xaxis and yaxis (using conduction level), but when he decided to move along the zaxis he bumps into potential barrier, so it is confined along zaxis. On previous as well on this picture Lx and Lx are both infinite (or equal to the dimension of material).As electron can move freely along xaxis, But another problem i have had is density of states in a quantum well. If Kx and Ky are both continual values the the density of states is infinite. I have read this in the post of yours. 


#4
May2312, 05:37 AM

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P: 1,659

2D Quantum Well and kvalues
Also, I would like to stress that the density of states given in that post was the one in momentum space. Typically more interesting is the density of states in energy. Starting from what is given above, the density of states in kspace can be expressed as [tex] g(k)dk=2\frac{2\pi\leftk\right}{V_{2D}}dk [/tex] where V2d gives the total volume of momentum space (just (2 pi/L)^2). To get to the more common representation of density of states in energy space you need to take the typical relation for energy: [tex] E(k)\frac{\hbar^2 k^2}{2 m} [/tex] and solve for k: [tex] k(E)\frac{\sqrt{2 m E}}{\hbar}. [/tex] You also need to replace dk in the density of states by dE: [tex] \frac{dk(E)}{dE}=\frac{2 m}{2\hbar \sqrt{2 m E}}. [/tex] Plugging all that into your equation for the density of states and inserting the volume in kspace, you get: [tex] g(E) dE=\frac{m}{\pi \hbar^2}dE. [/tex] Which is the famous result that the density of states does not depend on the energy for a single quantum well energy level (=single value of kz). As you have several energy levels in a QW, you will then get the famous picture of a constant DOS that increases steplike at several certain energies when a new energy level comes into play due to the next kz coming into play. 


#5
May2312, 05:57 AM

P: 4

I got everything except V2d=(2*Pi/L)*2. What is L.
It only make sense if L is a x dimension of the material, but if so, the states in x and y direction are quantized as well, Just Lx and Ly are much larger that Lz. So z quantization is the most important. 


#6
May2312, 06:50 AM

Sci Advisor
P: 1,659




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