# Estimating the required magnitude for the critical dimensions of a quantum dot

I'm having trouble with my approach to the following problem, featured below:

"We want to construct a detector of radiation of wavelength 10μm based on a quantum dot structure and treating the quantum dot as an infinite potential well containing one electron. Estimate the required magnitude of the critical dimension of the dot (i.e., the length of the well)."

Here was my thought process:

• I can relate wavelength to well length with the following equation (where $E$ is the quantized amount of energy the electron in the well can have, $h$ is Planck's constant, $n$ is the quantum number, $m$ is mass of an electron, and $L$ is length of the well, where the electron can be found.

## $$E = \frac{h^2 n^2}{8mL^2}$$

• Since I'm given a wavelength, I can find energy with the relationship $E = \frac{hc}{\lambda}$

• The critical dimension of the dot likely means the smallest length the cavity can have, so that perhaps implies a quantum number of $n = 1$.
• Using these intuitions, I now have all values except for L, which I can then solve for.

However, this proved to be wrong, and the solution from my lecturer used the following steps to solve for L:

## $$E_2 - E_1 = \frac{h^2}{8mL^2}(2^2-1^2) = \frac{hc}{\lambda}$$

From there one needs to simply plug in values and solve for L. However, I'm not sure how this was done. I'm clearly not understanding the whole picture of what's happening here. Why was $n^2$ given to be $2^2-1^2$? Does that imply $n_f^2 - n_i^2 = (2^2 - 1^2)$? I've never seen that used before in this formula.

The notes for the answer key before going into the maths state:

"We need a well with a structure that allows for transitions that emit/absorb a photon with $\lambda = 10\mu m$"

I suppose my question is, what exactly is this question asking for, why are my assumptions wrong, and what were my lecturer's thought processes, given the calculations? Given the statement above, why did my algorithm not satisfy that condition? I know this formula gives us the energy states an electron can have inside a well with infinite energies outside of 0-L, so its $\lambda_{db}$ can only be set values since it must have nodes at either end of the well. However, I don't know how to reconcile this to answer the question I'm given.

Your electron must have an energy level provided by the equation for $E_n$. If you assume it starts in the ground state ($n=1$) then to absorb a photon of a given energy, the electron must be able to transition to a new energy level, where the energy difference is the photon energy. The next available energy level for the electron is $n=2$, taking an energy $E_2 - E_1$ to reach. This is the reasoning behind your lecturers answer.
Remember that $n=0$ cannot be a valid energy level for this system. Using $n=0$ would imply that the wave-function is zero (as it is proportional to $\sin{(\frac{n\pi x}{L})}$) and non-normalisation. Therefore the lowest available level is $n=1$.