fixes ex4

src/14_doping_and_devices.md

@@ -249,17 +249,19 @@ For that we consider a doped semiconductor in the extrinsic regime.
 ### Exercise 3: Performance of a diode
 
 ### Exercise 4: Quantum well heterojunction in detail
-A quantum well is formed from a layer of $GaAs$ of thickness $L$, surrounded by layers of $Al_{x}Ga_{1−x}As$
-_add fig Fig. 18.2 from the book_. You may assume that the band gap of the $Al_{x}Ga_{1−x}As$ is substantially larger than that of $GaAs$.The electron effective mass in GaAs is 0.068 $m_{e}$ whereas the hole effective mass is 0.45 $m_{e}$ with $m_{e}$ the mass of the electron.
+A quantum well is formed from a layer of $GaAs$ of thickness $L$, surrounded by layers of $Al_{x}Ga_{1−x}As$.
+
+![Quantum Well](https://commons.wikimedia.org/wiki/File:Quantum_well.svg#/media/File:Quantum_well.svg)
+
+You may assume that the band gap of the $Al_{x}Ga_{1−x}As$ is substantially larger than that of $GaAs$.The electron effective mass in GaAs is 0.068 $m_{e}$ whereas the hole effective mass is 0.45 $m_{e}$ with $m_{e}$ the mass of the electron.
 
 
 1. Sketch the shape of the potential for electrons and holes
 2. If we want to design a bandgap 0.1$eV$ larger than that of bulk $GaAs$, what size of $L$ do we need?
-3. Write down the Schrödinger's equation for electrons and holes (separating $\bf{k}$ in its three components $k_x$
-, $k_y$ and $k_z$)
+3. Write down the Schrödinger's equation for electrons and holes
 ??? hint
     It is a 2D electron gas with confined levels in the third direction
-4. Find the eigenvalues
+4. Find the eigenvalues  separating $\bf{k}$ in its three components $k_x$ , $k_y$ and $k_z$)
 5. Calculate the density of state of electron and holes in the quantum well
 6. Why could this structure be more useful as a laser than a normal pn-junction?
 7. What would be the advantage of doping the $Al_{x}Ga_{1−x}As$ compared to the $GaAs$?