proof read part 1, added design drawing

coursework.lyx

@@ -140,7 +140,7 @@ f=\frac{c}{\lambda}
 \end_layout
 
 \begin_layout Standard
-In order to find the 
+Therefore in order to find the 
 \begin_inset Formula $E$
 \end_inset
 
@@ -172,14 +172,15 @@ Returning to the specifications, this allows 1.55μm to be expressed as 1.28x10
 \end_layout
 
 \begin_layout Standard
-This energy value will be the same as the total band gap for the well from
- the first hole energy level to the first electron enery level, shown as
+This energy value will be the same as the total interband transition for
+ the well from the first confined hole energy level to the first confined
+ electron enery level,
 \end_layout
 
 \begin_layout Standard
 \begin_inset Formula 
 \begin{equation}
-\varSigma E_{g}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8eV\label{eq:Energy-Gap-Sum}
+E_{g,transition}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8\unit{eV}\label{eq:Energy-Gap-Sum}
 \end{equation}
 
 \end_inset
@@ -221,7 +222,7 @@ status open
 
 \begin_layout Plain Layout
 Band structure of an AlGaAs/GaAs/AlGaAs quantum well including discrete
- energy levels 
+ confined energy levels 
 \begin_inset CommandInset citation
 LatexCommand cite
 key "ieee_s6824198"
@@ -253,18 +254,18 @@ name "fig:Well-Band-structure"
 \begin_inset Formula $E_{g}$
 \end_inset
 
- should be the dominant term in this equation and as such in investigating
- suitable materials, the bulk band gap should be close to but lower than
+ should be the dominant term in this equation and as such when investigating
+ suitable materials the bulk band gap should be close to but lower than
  0.8eV.
 \end_layout
 
 \begin_layout Standard
-None of the binary III-V indium based alloys have bulk band gaps in a suitable
- range, as such ternary alloys were investigated.
+Ternary alloys were investigated in order to allow precise control over
+ the lattice constants and band gap by varying the composition ratio.
 \end_layout
 
 \begin_layout Standard
-indium gallium arsenide (In
+Indium gallium arsenide (In
 \begin_inset script subscript
 
 \begin_layout Plain Layout
@@ -327,7 +328,7 @@ Lattice matching is the process of ensuring that two crystalline structures
  between the two materials.
  This is particularly important for quantum wells formed through epitaxial
  growth as strain introduced between such thin layers can cause defects
- ultimately negatively affecting it's electronic properties.
+ which ultimately negatively affect it's electronic properties.
 \end_layout
 
 \begin_layout Standard
@@ -487,7 +488,7 @@ name "tab:Lattice-constants"
 In order to compute a compound lattice constant for InGaAs, Vegard's law
  can be applied.
  Vegard's law provides an approximation for the lattice constant of a solid
- solution by finding the weighted average the individual lattice constants
+ solution by finding the weighted average of the individual lattice constants
  by composition ratio and is given by:
 \end_layout
 
@@ -742,8 +743,8 @@ Width Calculation
 \begin_layout Standard
 Having found two materials that are lattice matched with a suitable band
  gap value, the final calculation is that of the quantum well width.
- In order to calculate this value, the equation for energy levels within
- an infinite quantum well will be used,
+ In order to calculate this value, the equation for confined energy levels
+ within an infinite quantum well will be used,
 \end_layout
 
 \begin_layout Standard
@@ -787,7 +788,7 @@ noprefix "false"
 \begin_layout Standard
 \begin_inset Formula 
 \[
-\varSigma E_{g}=0.8\unit{eV}=E_{1h}+E_{g}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
+E_{g,transition}=0.8\unit{eV}=E_{1h}+E_{g,InGaAs}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g,InGaAs}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
 \]
 
 \end_inset
@@ -797,16 +798,16 @@ noprefix "false"
 
 \begin_layout Standard
 With the experimentally determined value for 
-\begin_inset Formula $E_{g}$
+\begin_inset Formula $E_{g,,InGaAs}$
 \end_inset
 
- this equation can be condensed to,
+ this equation becomes
 \end_layout
 
 \begin_layout Standard
 \begin_inset Formula 
 \[
-0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
+0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75\unit{eV}+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
 \]
 
 \end_inset
@@ -1064,7 +1065,7 @@ which reduces to a well length of 14.87nm.
 \end_layout
 
 \begin_layout Subsection
-Energy Level Calculations
+Confined Energy Level Calculations
 \end_layout
 
 \begin_layout Standard
@@ -1113,8 +1114,8 @@ E_{1e}=6.65\times10^{-21}\unit{J}=0.041\unit{eV}
 \end_layout
 
 \begin_layout Standard
-This equation shows that energy values are proportional to the square of
- 
+This equation shows that confiend energy level values are proportional to
+ the square of 
 \begin_inset Formula $n$
 \end_inset
 
@@ -1198,6 +1199,63 @@ E_{2h}=5.45\times10^{-21}\unit{J}=0.034\unit{eV}
 \end_inset
 
 
+\end_layout
+
+\begin_layout Standard
+With the dimensions and first confined energy levels calculated, the final
+ design for the quantum well can be seen in figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:quantum-well-design"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename well-design.png
+	lyxscale 30
+	width 85col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+InP/InGaAs/InP quantum well design
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:quantum-well-design"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Section
@@ -1239,7 +1297,7 @@ with
 \end_layout
 
 \begin_layout Standard
-Where 
+Here 
 \begin_inset Formula $A$
 \end_inset
 
@@ -1337,6 +1395,17 @@ noprefix "false"
 \end_inset
 
  has been set to 1 for the ground state.
+ This function for the first excited state can be seen in figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Probability-plot-n-2"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
 \end_layout
 
 \begin_layout Standard
@@ -1350,7 +1419,7 @@ status open
 \begin_inset Graphics
 	filename probability-plot.png
 	lyxscale 30
-	width 100col%
+	width 75col%
 
 \end_inset
 
@@ -1396,7 +1465,7 @@ status open
 \begin_inset Graphics
 	filename probability-plot-with-n-2.png
 	lyxscale 30
-	width 100col%
+	width 75col%
 
 \end_inset
 
@@ -1461,7 +1530,16 @@ noprefix "false"
 
 \end_inset
 
- gives the final probability function for the entire well:
+ gives the final probability function for a distance across the well from
+ 
+\begin_inset Formula $x=0$
+\end_inset
+
+ to 
+\begin_inset Formula $x=x_{0}$
+\end_inset
+
+:
 \end_layout
 
 \begin_layout Standard
@@ -1476,15 +1554,7 @@ P\left(0\leq x\leq x_{0}\right)=\frac{1}{L}\left(x_{0}-\frac{L}{2n\pi}\sin\left(
 \end_layout
 
 \begin_layout Standard
-Where 
-\begin_inset Formula $x_{0}$
-\end_inset
-
- is an arbitrary distance across the well.
-\end_layout
-
-\begin_layout Standard
-For an interval across the well, this becomes:
+For an arbitrary interval across the well, this becomes:
 \end_layout
 
 \begin_layout Standard
@@ -1498,6 +1568,37 @@ P\left(a\leq x\leq b\right)=\frac{1}{L}\left(\left(b-a\right)-\frac{L}{2n\pi}\le
 
 \end_layout
 
+\begin_layout Standard
+This equation can be utilised in order to find the probability of finding
+ the electron between 
+\begin_inset Formula $2\unit{nm}$
+\end_inset
+
+ and 
+\begin_inset Formula $4\unit{nm}$
+\end_inset
+
+ and between 
+\begin_inset Formula $6\unit{nm}$
+\end_inset
+
+ and 
+\begin_inset Formula $8\unit{nm}$
+\end_inset
+
+, the intervals for which can be seen plotted in figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Probability-plot-with-bounds"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+\end_layout
+
 \begin_layout Standard
 \begin_inset Float figure
 wide false
@@ -1509,7 +1610,7 @@ status open
 \begin_inset Graphics
 	filename probability-plot-with-bounds.png
 	lyxscale 30
-	width 100col%
+	width 75col%
 
 \end_inset
 
@@ -1576,7 +1677,7 @@ P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\uni
 \begin_layout Standard
 \begin_inset Formula 
 \[
-P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)\thickapprox0.132
+P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)\thickapprox0.0955
 \]
 
 \end_inset
@@ -1620,7 +1721,7 @@ P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\uni
 \begin_layout Standard
 \begin_inset Formula 
 \[
-P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)\thickapprox0.132
+P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)\thickapprox0.263
 \]
 
 \end_inset