From b1b55ef50a03108180fb98e8cf6ad2ef9af26afb Mon Sep 17 00:00:00 2001
From: Anton Akhmerov <anton.akhmerov@gmail.com>
Date: Thu, 4 Apr 2019 07:06:25 +0000
Subject: [PATCH] minor formatting changes

---
 src/14_doping_and_devices.md | 22 +++++++++-------------
 1 file changed, 9 insertions(+), 13 deletions(-)

diff --git a/src/14_doping_and_devices.md b/src/14_doping_and_devices.md
index afb3c57b..0363c262 100644
--- a/src/14_doping_and_devices.md
+++ b/src/14_doping_and_devices.md
@@ -254,7 +254,7 @@ Consider a pn-junction diode as follows
 
 ??? info "source"
 
-    By Raffamaiden [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]), [Link](https://commons.wikimedia.org/wiki/File:PN_diode_with_electrical_symbol.svg)
+    By Raffamaiden [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0)), [Link](https://commons.wikimedia.org/wiki/File:PN_diode_with_electrical_symbol.svg)
 
 The current flowing through a diode as a function of applied bias voltage is given by the Shockley diode equation:
 
@@ -268,7 +268,7 @@ where $I_s(T)$ is the saturation current.
 3. Based on this, estimate how the saturation current $I_s$ depends on temperature.
 
 ### Exercise 4: Quantum well heterojunction in detail
-Consider a a quantum well formed from a layer of $GaAs$ of thickness $L$, surrounded by layers of $Al_{x}Ga_{1−x}As$.
+Consider a a quantum well formed from a layer of GaAs of thickness $L$, surrounded by layers of Al$_{x}$Ga$_{1−x}$As.
 
 ![Quantum Well](https://upload.wikimedia.org/wikipedia/commons/4/45/Quantum_well.svg)
 
@@ -276,17 +276,13 @@ Consider a a quantum well formed from a layer of $GaAs$ of thickness $L$, surrou
 
     Vectorised by User:Sushant savla from the work by Gianderiu - [Quantum well.svg](https://commons.wikimedia.org/w/index.php?curid=73413676), [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0 "Creative Commons Attribution-Share Alike 3.0").
 
-Assume that the band gap of the $Al_{x}Ga_{1−x}As$ is substantially larger than that of $GaAs$.
+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 $m_{e}$, the hole effective mass is 0.45 $m_{e}$ with $m_{e}$ the mass of the electron.
 
 1. Sketch the band diagram of this quantum well.
-3. Write down the Schrödinger's equation for electrons and holes
-4. Find the energies of electron and holes in the quantum well
-
-    ??? hint
-        Separating $\bf{k}$ in its components $k_z$ and $k_{\perp}$ , with $k_{\perp}^2=k_x^2+k_y^2$
-
-2. If we want to design a quantum well with a bandgap 0.1 eV larger than that of bulk $GaAs$, what thickness $L$ do we need?
-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 $GaAs$ in this quantum well?
+2. Write down the Schrödinger's equation for electrons and holes
+3. Find the energies of electron and holes in the quantum well as a function of $k_x, k_y$.
+4. Calculate the density of states of electron and holes in this quantum well.
+5. If we want to design a quantum well with a bandgap 0.1 eV larger than that of bulk $GaAs$, what thickness $L$ do we need?
+6. Why is this structure more useful for making 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 $GaAs$ in this quantum well?
-- 
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