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## Homework Statement

a) A quantum well contains electrons at a sheet carrier density of [itex]n_s =2 \times 10^{16}m^{-2}[/itex]. The electron effective mass is [itex]0.1m_e^*[/itex]. Calculate the Fermi energy of the carrier distribution in the well. You may assume the spacings between sub-bands in the quantum well is very much greater than the Fermi energy.

b) If instead the spacing between the lowest two sub-bands is [itex]25meV[/itex], deduce the resultant occupancy in meV of each of the two sub-bands.

## Homework Equations

For a) I used

[tex]\varepsilon_F=\frac{\hbar^2\pi}{m_e^*}n_s[/tex]

which was derived from the 2D density of states.

## The Attempt at a Solution

So I understand part a), but part b) has me confused.

I interpreted the problem as shown in the image below.

I assume the thing that I need to find is the energies I denoted as [itex]\Delta E_{1_o}[/itex] and [itex]\Delta E_{2_o}[/itex], and when I asked my lecturer about the question he said to look at it as a geometry problem but I just cant see, would I literally just need to calculate the energys [itex]E_1[/itex] and [itex]E_2[/itex] and use that and the fermi energy to determine the width?

Using;

[tex]25meV=\frac{\hbar^2\pi^2}{2m_e^*d^2}\left( 2^2-1^2 \right)[/tex]

I found the QW width and then used the following relations to get the occupancy;

[tex]\Delta E_{1_o}=\varepsilon_F -E_1[/tex]

[tex]\Delta E_{2_o}=\varepsilon_F -E_2[/tex]

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