From 9da9a4bae6d549af1a09ba930f3292d344c2ca74 Mon Sep 17 00:00:00 2001 From: aj Date: Mon, 11 Nov 2019 20:00:41 +0000 Subject: [PATCH] added structure explanation and probability plot --- coursework.lyx | 920 ++++++++++++++++++++++++++++++++++++++++++- coursework.pdf | Bin 301349 -> 382359 bytes probability-plot.png | Bin 0 -> 74470 bytes references.bib | 64 ++- 4 files changed, 972 insertions(+), 12 deletions(-) create mode 100644 probability-plot.png diff --git a/coursework.lyx b/coursework.lyx index 4a6fd4a..46dacb9 100644 --- a/coursework.lyx +++ b/coursework.lyx @@ -91,11 +91,11 @@ EEE3037 Nanotechnology Coursework 6420013 \end_layout -\begin_layout Section +\begin_layout Part Quantum Engineering Design \end_layout -\begin_layout Subsection +\begin_layout Section Structure Design \end_layout @@ -178,9 +178,9 @@ This energy value will be the same as the total band gap for the well from \begin_layout Standard \begin_inset Formula -\[ -\varSigma E_{g}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8eV -\] +\begin{equation} +\varSigma E_{g}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8eV\label{eq:Energy-Gap-Sum} +\end{equation} \end_inset @@ -317,7 +317,7 @@ Ga As and as such this was tested first. \end_layout -\begin_layout Subsubsection +\begin_layout Subsection Lattice Match \end_layout @@ -494,7 +494,7 @@ In order to compute a compound lattice constant for InGaAs, Vegard's law \begin_layout Standard \begin_inset Formula \[ -\alpha_{A_{(1-x)}B_{x}}=(1-x)\alpha_{A}+x\alpha_{B} +\alpha_{A_{(1-x)}B_{x}}=\left(1-x\right)\alpha_{A}+x\alpha_{B} \] \end_inset @@ -522,18 +522,916 @@ This shows that to 4 significant figures the composition of InGaAs is lattice matched to InP to within 0.001Å which is sufficient for this application. \end_layout -\begin_layout Subsubsection +\begin_layout Subsection Band Gap \end_layout -\begin_layout Subsection -Probability Plot +\begin_layout Standard +Vegard's law can also be used to approximate the band gap of a ternary alloy, + such as InGaAs. + The band gaps at 300K for each alloy can be seen in table +\begin_inset CommandInset ref +LatexCommand ref +reference "tab:Band-gaps" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +Material +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Band Gap at 300K, E +\begin_inset script subscript + +\begin_layout Plain Layout +g +\end_layout + +\end_inset + + (eV) +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +InAs +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.35 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +GaAs +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +1.42 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +InP +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +1.34 +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Band gaps for prospective well and barrier materials +\begin_inset CommandInset citation +LatexCommand cite +key "new_semiconductor_materials_archive" +literal "false" + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "tab:Band-gaps" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +In this case the band gap approximates to, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +E_{g,In_{0.53}Ga_{0.47}As}\thickapprox0.53\cdotp0.35+0.47\cdotp1.42\thickapprox0.85\unit{eV} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +However the band gap has been experimentally found to be 0.75eV +\begin_inset CommandInset citation +LatexCommand cite +key "aip_complete10.1063/1.322570" +literal "false" + +\end_inset + +. + This implies that the linear relationship provided by Vegard's law is not + accurate enough and in this case a modified version including a bowing + parameter +\begin_inset Formula $b$ +\end_inset + + should be used, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +E_{g,total}=xE_{g,a}+\left(1-x\right)E_{g,b}-bx\left(1-x\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +For this application, however, the experimentally determined value will + be used. + This value is ideal for this application as it is comparable to and slightly + lower than the required 0.8eV energy value. \end_layout \begin_layout Subsection +Width Calculation +\end_layout + +\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, +\end_layout + +\begin_layout Standard + +\emph on +\begin_inset Formula +\begin{equation} +E_{n}=\frac{n^{2}\pi^{2}\text{ħ}^{2}}{2mL^{2}}\label{eq:Energy-levels} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Referring back to equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Energy-Gap-Sum" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, the terms for the first electron and hole energy levels can each be replaced + with equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Energy-levels" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + as seen below, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\varSigma E_{g}=0.8\unit{eV}=E_{1h}+E_{g}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +With the experimentally determined value for +\begin_inset Formula $E_{g}$ +\end_inset + + this equation can be condensed to, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +0.05\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2L^{2}}\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +L=\sqrt{\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp(0.05\unit{eV})}\cdotp\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right)} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +As a frequently studied composition due to it's favourable structural parameters + with InP, The charge carrier effective masses of In +\begin_inset script subscript + +\begin_layout Plain Layout +0.53 +\end_layout + +\end_inset + +Ga +\begin_inset script subscript + +\begin_layout Plain Layout +0.47 +\end_layout + +\end_inset + +As have been found experimentally to be as shown in table +\begin_inset CommandInset ref +LatexCommand ref +reference "tab:Effective-masses" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +Charge Carrier +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Effective mass ratio in In +\begin_inset script subscript + +\begin_layout Plain Layout +0.53 +\end_layout + +\end_inset + +Ga +\begin_inset script subscript + +\begin_layout Plain Layout +0.47 +\end_layout + +\end_inset + +As ( +\begin_inset Formula $\frac{m^{*}}{m^{0}}$ +\end_inset + +) +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Electron +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.041 +\begin_inset CommandInset citation +LatexCommand cite +key "aip_complete10.1063/1.90860" +literal "false" + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Light Hole +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.051 +\begin_inset CommandInset citation +LatexCommand cite +key "aip_complete10.1063/1.92393" +literal "false" + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Heavy Hole +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.2 +\begin_inset CommandInset citation +LatexCommand cite +key "aip_complete10.1063/1.101816" +literal "false" + +\end_inset + + +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Effective masses of charge carriers in +\begin_inset CommandInset label +LatexCommand label +name "tab:Effective-masses" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +As the electrical and optical properties of the valence band are governed + by the heavy hole interactions, this effective mass ration will be used. +\end_layout + +\begin_layout Standard +Substituting these ratios into the above provides, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +L=\sqrt{\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp(0.05\unit{eV})\cdotp m_{e}}\cdotp\left(\frac{1}{0.2}+\frac{1}{0.041}\right)} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +which reduces to a well length of 14.87nm. +\end_layout + +\begin_layout Subsection +Energy Level Calculations +\end_layout + +\begin_layout Standard +With all the parameters of the well ascertained the first and second confined + electron and hole energy levels can be found by utilising equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Energy-levels" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +For confined electron states: +\end_layout + +\begin_layout Standard + +\emph on +\begin_inset Formula +\[ +E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp m_{e}^{*}\cdotp\left(14.87\unit{nm}\right)^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard + +\emph on +\begin_inset Formula +\[ +E_{1e}=6.65\times10^{-21}\unit{J}=0.041\unit{eV} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +This equation shows that energy values are proportional to the square of + +\begin_inset Formula $n$ +\end_inset + +, the principal quantum number or energy level. + As such: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +E_{2e}=2^{2}\cdotp E_{1e} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard + +\emph on +\begin_inset Formula +\[ +E_{2e}=2.66\times10^{-20}\unit{J}=0.17\unit{eV} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +For confined hole states: +\end_layout + +\begin_layout Standard + +\emph on +\begin_inset Formula +\[ +E_{1h}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp m_{h}^{*}\cdotp\left(14.87\unit{nm}\right)^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard + +\emph on +\begin_inset Formula +\[ +E_{1h}=1.36\times10^{-21}\unit{J}=0.0085\unit{eV} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +E_{2h}=2^{2}\cdotp E_{1h} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard + +\emph on +\begin_inset Formula +\[ +E_{2h}=5.45\times10^{-21}\unit{J}=0.034\unit{eV} +\] + +\end_inset + + +\end_layout + +\begin_layout Section +Probability Plot +\end_layout + +\begin_layout Standard +The probability of finding an electron in a quantum well is given by +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +P=\int_{0}^{L}\psi^{*}\psi dx\label{eq:wave-function-probability} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +with +\begin_inset Formula $\psi$ +\end_inset + + in the case of an infinite quantum well being given by, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\psi\left(x\right)=A\sin\left(kx\right)=A\sin\left(\frac{n\pi}{L}x\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Where +\begin_inset Formula $A$ +\end_inset + + acts as a normalisation constant to satisfy the conditions +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int_{{\textstyle all\:space}}\psi^{*}\psi dV=1 +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +in this case providing the wave function +\begin_inset Formula $\psi$ +\end_inset + + as +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\psi\left(x\right)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)\label{eq:wave-function} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Importantly, the above conditions are for an infinite quantum well where + an assumption is made that the well has a barrier region of infinite potential + such that the wavefunction is confined to the well. + A real quantum well is unable to satisfy this leading to the wavefunction + +\begin_inset Quotes eld +\end_inset + +spilling +\begin_inset Quotes erd +\end_inset + + into the barrier region. + For the purposes of plotting the probability density, however, it is a + reasonable assumption to make. +\end_layout + +\begin_layout Standard +Considering equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:wave-function-probability" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, if the probability can be found by integrating +\begin_inset Formula $\psi^{*}\psi$ +\end_inset + +, or in this situation +\begin_inset Formula $\psi^{2}$ +\end_inset + + then the probability can be shown by plotting +\begin_inset Formula $\psi^{2}$ +\end_inset + +, see figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Probability-plot" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + Here the well stretches from 0 to the blue line along the +\begin_inset Formula $x$ +\end_inset + + axis and +\begin_inset Formula $n$ +\end_inset + + has been set to 1 for the ground state. +\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 probability-plot.png + lyxscale 30 + width 100col% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Probability plot for electron in ground state +\begin_inset CommandInset label +LatexCommand label +name "fig:Probability-plot" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section Probability Intervals \end_layout +\begin_layout Standard +Combining equations +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:wave-function-probability" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:wave-function" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + gives the final probability function for the entire well: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(0\leq x\leq x_{0}\right)=\frac{1}{L}\left(x_{0}-\frac{L}{2n\pi}\sin\left(\frac{2n\pi x_{0}}{L}\right)\right) +\] + +\end_inset + + +\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 \begin_inset Newpage pagebreak \end_inset @@ -541,7 +1439,7 @@ Probability Intervals \end_layout -\begin_layout Section +\begin_layout Part Application of Nanomaterials \end_layout diff --git a/coursework.pdf b/coursework.pdf index 7f54d0444ac82ddfcf037be8b77333023abbae90..56ded819a25a0a25324ab25ba294296b42989b4e 100644 GIT binary patch delta 248727 zcmZ^JWmq2B(ru6cf#B|z5Zv9}J-BOdcWr_P2<{pP?(XhRAh^4`JNM0G&YW+~%>98p zO~1W&)n2t~)oOAoQ4)$UBS|4xS?Lq>uffQH9PFX!h2FDP-44xibkvS!dj3NDNK}#c z%Inb^G+TOiKJQH=VkLY!yP6(u_GWyDFsV2O0uoJJ7=t zT-zv7keGt^VYwObXvkOPl)BL3YEJA>PKl3tB3EpXj~i_(R;D?bgcvSshL)FnN`)mpqzWSk?{N z_-Wj6+y3p1Z~$K$L-~(GsD={79e?Og*H91|>yoc+GTHDj)Atm&jvQB;7HUQYfm_)2 z-7#UpBCuKA9_$JFYr&1DBOHnK0jRR01zwn;dj$g6SphPQJYJU~77J`rf@^d^MzR8& zXB5B0pvUn^o%4J%{(uz>aWtl0p)y$7U~Upxe>$CUxa^T zPRvVzpX8=QVWb=u#vo1khfBlu0x7pXOj4`6LU0$fURicYjGQ-cs_MBla-a 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HcmV?d00001 diff --git a/references.bib b/references.bib index a2acea0..c0b7de0 100644 --- a/references.bib +++ b/references.bib @@ -16,5 +16,67 @@ year = "2014-11" @misc{new_semiconductor_materials_archive, title={NSM Archive - Physical Properties of Semiconductors}, url={http://matprop.ru/}, -journal={New Semiconductor Materials Archive}, publisher={Ioffe Institute} +journal={New Semiconductor Materials Archive}, +publisher={Ioffe Institute} } + +@article{aip_complete10.1063/1.322570, +abstract = "Very uniform In 0.53 Ga 0.47 As was grown on InP by liquid phase epitaxy. The electron mobility is 8450 cm 2 /V sec at 300 K and 27700 cm 2 /V sec at 77 K. The mobility increases with decreasing temperature from 300 to 77 K in contrast to the results of In 1− x Ga x As grown directly on GaAs by vapor phase epitaxy. The energy gap of this high‐mobility material is 0.750 eV at room temperature.", +author = "Takeda, Yoshikazu and Sasaki, Akio and Imamura, Yujiro and Takagi, Toshinori", +issn = "0021-8979", +journal = "Journal of Applied Physics", +language = "eng", +number = "12", +pages = "5405,5408", +publisher = "American Institute of Physics", +title = "Electron mobility and energy gap of In 0.53 Ga 0.47 As on InP substrate", +volume = "47", +year = "1976-12", +} + +@article{aip_complete10.1063/1.90860, +abstract = "The band-edge effective mass for conduction electrons in Ga x In 1-x As y P 1-y has been determined for several different alloy compositions covering the complete range of alloys grown lattice-matched on InP. Measurements show that the effective mass varies nearly linearly with alloy composition.", +author = "Nicholas, R. J. and Portal, J. C. and Houlbert, C. and Perrier, P. and Pearsall, T. P.", +issn = "0003-6951", +journal = "Applied Physics Letters", +keywords = "Galliumarsenid ; Drei-Fuenf-Verbindung ; Indiumphosphid ; Effektive Masse;", +language = "eng", +number = "8", +pages = "492,494", +publisher = "American Institute of Physics", +title = "An experimental determination of the effective masses for Ga x In 1-x As y P 1-y alloys grown on InP", +volume = "34", +year = "1979", +} + +@article{aip_complete10.1063/1.92393, +abstract = "We report the use of optical pumping in p -type Ga x In 1-x As y P 1-y nearly lattice-matched to InP. Analysis of the conduction‐electron spin‐polarized photoluminescence has been used to deduce the valence‐band light‐hole effective mass as a function of alloy composition. Our results are in good agreement with masses calculated using the k·p approximation.", +author = "Hermann, Claudine and Pearsall, Thomas P.", +issn = "0003-6951", +journal = "Applied Physics Letters", +keywords = "Halbleiterverbindung ; Galliumarsenid ; Indiumphosphid ; A3-B5-Verbindung ; Optisches Pumpen ; Effektive Masse ; Defektelektron ; Valenzband ; Halbleitersubstrat ; P-Halbleiter ; Leitungselektron ; Photolumineszenz ; Spinorientierung;", +language = "eng", +number = "6", +pages = "450,452", +publisher = "American Institute of Physics", +title = "Optical pumping and the valence-band light-hole effective mass in Ga x In 1-x As y P 1-y (y approx. 2.2x)", +volume = "38", +year = "1981", +} + +@article{aip_complete10.1063/1.101816, +author = "Lin, S. Y. and Liu, C. T. and Tsui, D. C. and Jones, E. D. and Dawson, L. R.", +issn = "0003-6951", +journal = "Applied Physics Letters", +keywords = "Materials Sciencegallium Arsenides ; Holes ; Indium Arsenides ; Cyclotron Resonance ; Effective Mass ; Electronic Structure ; Light Transmission ; Superlattices ; Arsenic Compounds ; Arsenides ; Gallium Compounds ; Indium Compounds ; Mass ; Pnictides ; Resonance;", +language = "eng", +number = "7", +pages = "666,668", +publisher = "American Institute of Physics", +title = "Cyclotron resonance of two-dimensional holes in strained-layer quantum well structure of (100)In 0.20 Ga 0.80 As/GaAs", +volume = "55", +year = "1989-08-14", +} + + +

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