Semiconductor quantum well?

In summary, a semiconductor quantum well is a nanostructure that consists of thin layers of semiconductor material with unique electronic and optical properties. It works by using the quantum confinement effect and has applications in optoelectronics, high-speed transistors, and quantum computing. One of its main advantages is the ability to tune properties, but fabricating it can be challenging due to the need for precise control and high-quality layers.
  • #1
jeebs
325
4
Here's the problem:
A GaAs quantum well of thickness 10 nm is held in a cryostat at a low temperature and is excited with a powerful ultrashort laser pulse at time t = 0. The laser pulse injects 1 × 1015 m−2 electrons and holes per unit area of the quantum well into the conduction and valence bands respectively. The band gap of GaAs is 1.519 eV, and the effective masses of the electrons and heavy holes are 0.067 me and 0.4 me
respectively.
(i) Calculate the Fermi energies of the electrons and holes at t = 0. (ii) Sketch the emission spectrum that you would expect to observe just after the crystal has been excited by the laser, identifying clearly the minimum and maximum photon energies that would
be emitted.
(iii) Calculate the modulus of the maximum k vector of the electron within the plane of the quantum well and compare it to the k vector inside the crystal of the photon that is emitted. (The refractive index of GaAs is 3.5.)
Throughout this question you may ignore the light hole band and assume that excitonic effects are negligible. The density of states per unit area for a free particle of mass m in two dimensions is given by [tex] g(E) = \frac{m}{\pi\hbar^2} [/tex]

So, the particles will have their free motion behaviour in 2 directions with a wavevector kxy, and be confined in the 3rd direction. This gives the electron energy [tex] E_e = E_g + \frac{\hbar^2 n^2 \pi^2}{2m_e^*d^2} + \frac{\hbar^2 k_x_y^2}{2m_e^2} [/tex] where Eg is the band gap, d is the well width, me* is the effective electron mass. Also the holes have energy
[tex] E_h = \frac{\hbar^2 n^2 \pi^2}{2m_h^*d^2} + \frac{\hbar^2 k_x_y^2}{2m_h^2} [/tex]

so, for the first bit, what I've said is that the Fermi energy is when all the electrons/holes occupy lowest states, the Fermi energy is the highest occupied state. Looking at the density of states, that is constant for any energy so when I calculate it it gives me an answer in the region of 1037 states m-2. Since there are only 1015 electrons and holes per square metre, I've said that when they all occupy the lowest states, they can all be in the n=n'=1 and kxy = 0 level.
This gives me a Fermi level of 0.94eV for the holes and 7.14eV for the electrons (is this on the high side?)

Is this allowed? I have my reservations about this, since electrons, being fermions, shouldn't be in the same energy state. I couldn't think of any other way to get the Fermi levels though...

anyway, for the next two parts, they involve finding the maximum energy of the confined particles. This I am having trouble with, because I am not told the height of the potential well. I do not know what the maximum energy is that a particle can have and still be confined, so I cannot get the maximum peak on the emission spectrum (part 2), or the maximum k vector of the electron within the well. I know I'd be able to find the maximum n and n' if I was told what the band gap of the material forming the walls of the well, but I'm not...
Not sure where to go from here...
 
Last edited:
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  • #2


Hello,

Thank you for sharing your thoughts and concerns regarding this problem. I can understand your confusion and I would like to offer some insights and suggestions that may help you solve this problem.

Firstly, your approach for calculating the Fermi energies of the electrons and holes seems reasonable. You have correctly used the density of states equation to find the highest occupied state for both electrons and holes. However, I would suggest double-checking your calculations to ensure that you have used the correct values for the effective masses and the well width. The Fermi energy for electrons (7.14 eV) does seem a bit high, but it could be due to a small error in the calculation.

Moving on to the next part, you are correct in stating that the maximum energy of the confined particles depends on the height of the potential well. Unfortunately, this information is not provided in the problem. In such cases, it is best to make an assumption or an educated guess. For example, you could assume that the potential well is deep enough to confine the particles up to a certain energy, say 10 eV. This would give you a maximum peak on the emission spectrum at 10 eV. Similarly, for calculating the maximum k vector of the electron within the well, you could assume a value for the well height and use that to find the maximum k vector.

While these assumptions may not give you the exact values, they will still help you understand the general behavior of the particles and the emission spectrum. As long as your assumptions are reasonable and based on the information given in the problem, they should be acceptable for the purposes of this problem.

I hope this helps you in solving the problem. If you have any further questions or concerns, please feel free to reach out. Best of luck!
 

1. What is a semiconductor quantum well?

A semiconductor quantum well is a type of nanostructure that consists of thin layers of semiconductor material sandwiched between layers of a different semiconductor material. This creates a confined region in which electrons are restricted to move in only two dimensions, resulting in unique electronic and optical properties.

2. How does a semiconductor quantum well work?

A semiconductor quantum well works by using the quantum confinement effect, which occurs when the size of a material is reduced to the nanoscale. In a quantum well, the electrons are confined in the direction perpendicular to the layers, resulting in discrete energy levels that can be controlled by adjusting the thickness of the layers.

3. What are the applications of semiconductor quantum wells?

Semiconductor quantum wells have a wide range of applications in optoelectronics, such as in lasers, LEDs, and solar cells. They are also used in high-speed transistors, memory devices, and sensors. Additionally, they have potential applications in quantum computing and cryptography.

4. What are the advantages of using semiconductor quantum wells?

One of the main advantages of semiconductor quantum wells is their ability to tune the electronic and optical properties by adjusting the thickness and composition of the layers. This makes them highly versatile and suitable for a variety of applications. They also have high quantum efficiency, low power consumption, and fast response times.

5. What are the challenges in fabricating semiconductor quantum wells?

Fabricating semiconductor quantum wells can be challenging due to the precise control required in layer thickness and composition. This often involves complex techniques such as molecular beam epitaxy or chemical vapor deposition. The quality of the layers and interfaces must also be carefully controlled to prevent defects and ensure proper functioning of the device.

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