Density of states in a quantum well

In summary: E1. The electron will start at E2 and jump to E1 when it exceeds the energy of E1. However, because the electron is no longer confined to the QW, it will stay at E1 for a finite amount of time (aka 'step'), and then it will start decaying back to E2. So you would expect the DOS to be zero between E1 and E2, but because it's not, there is a finite 'step' between them.In summary, the density of states between two quantised energy levels in a quantum well is not zero, due to the finite 'step' that electrons take between the levels.
  • #1
McKendrigo
26
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Hi there, I'm a bit confused as to the meaning of the following figure, for the quantum well case:

Image441.gif


I understand that with the quantum dot, since the states are completely quantised you get the delta functions - states only exist at certain energies so the DOS at these energies is non zero (and zero at all other energies).

The quantum well case confuses me though. I understand that the energy levels within a quantum well to be quantised in one dimension. However, if the energy levels are quantised, why is the DOS not zero between these energy levels? In other words, why are there flat 'steps' rather than a delta function?

I can follow the maths behind the derivation of the DOS for a quantum well or dot, but without really grasping the physical meaning of what it tells us for the QW. Any help would be appreciated!
 
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  • #2
I've realized that the difference between the two is something to do with the fact that in the QW the electrons are only confined in 2 directions, which explains why there are allowed energy levels in between the quantised levels.

In other words, in the Z direction the energy levels are quantised, but not in the X and Y directions.

I'm still not fully grasping what's going on though, so if anyone has a nice simple explanation I'd be glad to hear it!
 
  • #3
Funny, I would expect the same thing you did for the well. If I understand correctly the concept of density of states (which now I'm pretty sure I don't) each point in this graphic means "this energy value can hold this many states", and if the system can only assume certain energy levels, the gaps in the energy spectrum should have zero density of states.
 
  • #4
I Wrote the above reply before reading your second post. I think you've nailed the problem. This is a 3d quantum well
 
  • #5
First, it is important to understand that you are discussing the density of states per energy. The density of states per wave vector k in momentum space is a bit easier to grasp.

In 3D k-space, the whole k-space volume at some arbitrary k is given by a sphere of volume 4/3 Pi k^3. All states with the same energy will also have the same magnitude of the wave vector. Therefore all states of the same energy will form a surface of that sphere. and the density of states will be proportional to 4 Pi k^2.

In 2D one of the k-values is quantized. The k-space volume will be a circle around that fixed k-value. All states with the same energy will lie on the circumference of that circle. The density of states will be proportional to 2 Pi k.

In 1D two of the k-values are quantized. The k-space volume will just be a line of length 2 k. The states with the same energy will be points on a line in k-space. The corresponding density in k-space is just a constant.

In all cases the density of states is the derivative of the volume in terms of k.

To get to the density of states in energy space you just need to know that the energy E is proportional to (k^2)/c where c is just a constant. So you just need to take the k-space volumes from before and replace k by sqrt(E) c. Now you have expressed the volume in terms of the energy. Now you just need to calculate the derivative of that term in terms of the energy and you will get the dependencies which are shown in your picture.
 
  • #6
Hi Cthugha, thanks for your reply.

I'm able to follow the procedure you mention in terms of deriving the expressions that give the graphs shown in my first post. What I'm not clear on is a mental picture of what physically is happening.

I can understand the 1D quantum dots. Only quantised energy levels are allowed, so you get this series of delta functions occurring at the allowed energy levels. This I can understand easily.

The quantum well has quantised energy levels too, but in only one direction. Obviously, the reason we see the 'steps' and not the delta function spikes is in some way due to the fact that the electrons are not confined in the other two directions...but I'm not really sure why. I can follow the derivation of why, without really understanding properly what it actually means :/

I can see that when you increase the energy of an electron to the point it exactly reaches one of the quantised levels in the QW, more states are now available to it so the DOS 'step' increases accordingly.
It's the flat part of the steps I'm unsure of, if you follow me. Suppose there are quantised energy levels in the QW called E1 and E2, and E2>E1. Now suppose we inject an electron into the QW with energy E, such that E2>E>E1. Naively I'd assumed that electrons could only take on an energy of E2 or E1, and not something in between, but the figure above does not suggest that.

Why can the electrons take on an energy <E2 and >E1? Or, put another way, where are these allowed states which seem to exist in between the QW quantised energy levels? Do these states exist in the 2 unconfined directions?
 
  • #7
McKendrigo said:
I can understand the 1D quantum dots. Only quantised energy levels are allowed, so you get this series of delta functions occurring at the allowed energy levels. This I can understand easily.

QDs are 0D. I assume this is a typo.

McKendrigo said:
I can see that when you increase the energy of an electron to the point it exactly reaches one of the quantised levels in the QW, more states are now available to it so the DOS 'step' increases accordingly.
It's the flat part of the steps I'm unsure of, if you follow me. Suppose there are quantised energy levels in the QW called E1 and E2, and E2>E1. Now suppose we inject an electron into the QW with energy E, such that E2>E>E1. Naively I'd assumed that electrons could only take on an energy of E2 or E1, and not something in between, but the figure above does not suggest that.

The energy levels in a QW are not quantized. The wave vector in one direction is. Let me assume it is [tex]k_z[/tex].
In effective mass approximation with effective mass [tex]m^\ast[/tex], the energy will be something like
[tex]E=\frac{\hbar^2 k^2}{2 m^\ast}[/tex]

with [tex]k=\sqrt{k_x^2 + k_y^2 + k_z^2}[/tex]

As only [tex]k_z[/tex] is quantized, but the other two components of the vector are free, the total energy is not quantized.
 
  • #8
Cthugha said:
In effective mass approximation with effective mass [tex]m^\ast[/tex], the energy will be something like
[tex]E=\frac{\hbar^2 k^2}{2 m^\ast}[/tex]

with [tex]k=\sqrt{k_x^2 + k_y^2 + k_z^2}[/tex]

As only [tex]k_z[/tex] is quantized, but the other two components of the vector are free, the total energy is not quantized.

I think it's clear now. Thanks for your help!

And yes, "1D quantum dots" was a typo...
 
  • #9
density of state in 3D between k and k+dk is,


=k^2 dk/(2* pi^2)

what happen if we double the k value??

K=2k
dK=2dk

then new density of state in 3D between k and k+dk is.
=8*K^2 dk/(2* pi^2)

is it correct?
 

1. What is a quantum well?

A quantum well is a structure in quantum mechanics that consists of a narrow region of a material sandwiched between two wider regions. This confinement of electrons in the narrow region leads to unique properties and behaviors, such as discrete energy levels and changes in the density of states.

2. How does the density of states in a quantum well differ from that of a bulk material?

The density of states in a quantum well is different from that of a bulk material because the confinement of electrons in the narrow region leads to a quantization of energy levels. This means that the density of states in a quantum well is not continuous, but instead consists of discrete energy levels.

3. How does the width of a quantum well affect the density of states?

The width of a quantum well has a direct effect on the density of states. As the width decreases, the energy levels become more spaced out, leading to a higher density of states. Conversely, as the width increases, the energy levels become closer together, resulting in a lower density of states.

4. What is the significance of the density of states in a quantum well?

The density of states in a quantum well is significant because it determines the number of allowed energy levels for electrons within the well. This, in turn, affects the electronic and optical properties of the material, making quantum wells useful for various applications in electronics and photonics.

5. How is the density of states in a quantum well experimentally measured?

The density of states in a quantum well can be experimentally measured using various techniques, such as capacitance-voltage measurements, photoluminescence spectroscopy, and tunneling spectroscopy. These methods involve measuring the energy levels and their corresponding populations within the quantum well under different conditions.

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