From 0f88146af5448836e8e16e3f7a9840131a66717a Mon Sep 17 00:00:00 2001 From: aj Date: Tue, 12 Nov 2019 11:43:05 +0000 Subject: [PATCH] proof read part 1, added design drawing --- coursework.lyx | 175 +++++++++++++++++++++++++++++++++++++---------- coursework.pdf | Bin 501255 -> 555247 bytes well-design.png | Bin 0 -> 71615 bytes well-diagram.odg | Bin 0 -> 13075 bytes 4 files changed, 138 insertions(+), 37 deletions(-) create mode 100644 well-design.png create mode 100644 well-diagram.odg diff --git a/coursework.lyx b/coursework.lyx index 8db1733..f26de3c 100644 --- a/coursework.lyx +++ b/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 diff --git a/coursework.pdf b/coursework.pdf index d74ead95d9226ccc6ea816ae4f95f08983690972..b8e918902afcd16ea8def0d8540aa14a1aa30e35 100644 GIT binary patch delta 122310 zcmaI7WmFwax2_EVLV_kZ1b26LcXxMpcW6ig1oz$9^W{B z*67h)tE*~O^*yh9u3kZ-xMvpxQN$lu855#M!O4L<4XIdA{nz%1ipeo$lfEyI;@>59 zDrHH_S0xo)TkAL$cco~`Acwau?Au%1XCw!d(!&(hLt!QD;13d3v^g!l0 z)%2>ti{6~wW+=tXtGTK{5^=V?IIMh-k&%&{e->Q};t^gIW@OH8m)YURU6fb~)bih& zFqeRF9-=l2VD0t#faqNHTV6M}gyB|Mj69iVO{k4|naNsFaE`czuHC3;l6*?s+?k;^ zMnOuNMEBr!>*jKqs=(B(y=5{Kp0QDNa1Ol;2$D^-a`tBv`ddp@VS@wibq^FP38{Oj 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