#LyX 2.3 created this file. For more info see http://www.lyx.org/ \lyxformat 544 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass article \use_default_options true \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry true \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine biblatex \cite_engine_type authoryear \biblio_style plain \biblatex_bibstyle ieee \biblatex_citestyle ieee \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date true \justification true \use_refstyle 1 \use_minted 0 \index Index \shortcut idx \color #008000 \end_index \leftmargin 2cm \topmargin 2cm \rightmargin 2cm \bottommargin 2cm \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style english \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Title EEE3037 Nanotechnology Coursework \end_layout \begin_layout Author 6420013 \end_layout \begin_layout Part Quantum Engineering Design \end_layout \begin_layout Section Structure Design \end_layout \begin_layout Standard In order to design a quantum well which emits light of wavelength 1.55μm, a well material must be chosen such that an interband electron transition emits photons of this wavelength. \end_layout \begin_layout Standard This band gap energy can be found from the equation \end_layout \begin_layout Standard \begin_inset Formula \[ E=hf \] \end_inset \end_layout \begin_layout Standard When considering photons, \begin_inset Formula $f$ \end_inset can be substituted with \end_layout \begin_layout Standard \begin_inset Formula \[ f=\frac{c}{\lambda} \] \end_inset \end_layout \begin_layout Standard Therefore in order to find the \begin_inset Formula $E$ \end_inset in terms of wavelength \end_layout \begin_layout Standard \begin_inset Formula \[ E=\frac{hc}{\lambda} \] \end_inset \end_layout \begin_layout Standard Returning to the specifications, this allows 1.55μm to be expressed as 1.28x10 \begin_inset script superscript \begin_layout Plain Layout -19 \end_layout \end_inset J or approximately 0.8 eV. \end_layout \begin_layout Standard 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} E_{g,transition}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8\unit{eV}\label{eq:Energy-Gap-Sum} \end{equation} \end_inset \end_layout \begin_layout Standard see figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Well-Band-structure" 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 WellBandStructure.png lyxscale 40 width 60col% \end_inset \begin_inset Caption Standard \begin_layout Plain Layout Band structure of an AlGaAs/GaAs/AlGaAs quantum well including discrete confined energy levels \begin_inset CommandInset citation LatexCommand cite key "ieee_s6824198" literal "false" \end_inset \begin_inset CommandInset label LatexCommand label name "fig:Well-Band-structure" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Formula $E_{g}$ \end_inset 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 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 \begin_inset script subscript \begin_layout Plain Layout \begin_inset Formula $x$ \end_inset \end_layout \end_inset Ga \begin_inset script subscript \begin_layout Plain Layout \begin_inset Formula $(1-x)$ \end_inset \end_layout \end_inset As) as a well material with indium phosphide (InP) as a barrier material would provide a suitable combination assuming that a composition ratio \begin_inset Formula $x$ \end_inset could be found that satisfied the two conditions of having the required bulk band gap and being lattice matched. A common ratio in industry is 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 and as such this was tested first. \end_layout \begin_layout Subsection Lattice Match \end_layout \begin_layout Standard Lattice matching is the process of ensuring that two crystalline structures are of similar dimensions in order to decrease strain at the interface 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 which ultimately negatively affect it's electronic properties. \end_layout \begin_layout Standard The lattice constants between the barrier and well materials should be as close as is deemed acceptable for the application. The lattice constants for the prospective materials are shown in table \begin_inset CommandInset ref LatexCommand ref reference "tab:Lattice-constants" 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 Lattice Constant, α (Å) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout InAs \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 6.0583 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout GaAs \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 5.6532 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout InP \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 5.8687 \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Lattice constants 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:Lattice-constants" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Standard 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 of the individual lattice constants by composition ratio and is given by: \end_layout \begin_layout Standard \begin_inset Formula \[ \alpha_{A_{(1-x)}B_{x}}=\left(1-x\right)\alpha_{A}+x\alpha_{B} \] \end_inset \end_layout \begin_layout Standard Applying this to the prospective well material gives the following, \end_layout \begin_layout Standard \begin_inset Formula \[ \alpha_{In_{0.53}Ga_{0.47}As}=0.53\cdotp6.0583+0.47\cdotp5.6532=5.8679 \] \end_inset \end_layout \begin_layout Standard 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 Subsection Band Gap \end_layout \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 confined 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}\mathcal{\text{\emph{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 \[ E_{g,transition}=0.8\unit{eV}=E_{1h}+E_{g,InGaAs}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g,InGaAs}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}} \] \end_inset \end_layout \begin_layout Standard With the experimentally determined value for \begin_inset Formula $E_{g,,InGaAs}$ \end_inset this equation becomes \end_layout \begin_layout Standard \begin_inset Formula \[ 0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75\unit{eV}+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}} \] \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 Confined 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 confiend energy level 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 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 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 Here \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. 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 \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 75col% \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 Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename probability-plot-with-n-2.png lyxscale 30 width 75col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Probability plot for electron in 1 \begin_inset script superscript \begin_layout Plain Layout st \end_layout \end_inset excited state \begin_inset CommandInset label LatexCommand label name "fig:Probability-plot-n-2" \end_inset \end_layout \end_inset \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 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 \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 For an arbitrary interval across the well, this becomes: \end_layout \begin_layout Standard \begin_inset Formula \[ P\left(a\leq x\leq b\right)=\frac{1}{L}\left(\left(b-a\right)-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi b}{L}\right)-\sin\left(\frac{2n\pi a}{L}\right)\right)\right) \] \end_inset \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 sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename probability-plot-with-bounds.png lyxscale 30 width 75col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Probability plot for electron in ground state with distance intervals \begin_inset CommandInset label LatexCommand label name "fig:Probability-plot-with-bounds" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection \begin_inset Formula $2\unit{nm}$ \end_inset to \begin_inset Formula $4\unit{nm}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)=\frac{1}{L}\left(2\unit{nm}-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi\cdotp\left(4\unit{nm}\right)}{L}\right)-\sin\left(\frac{2n\pi\cdotp\left(2\unit{nm}\right)}{L}\right)\right)\right) \] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\unit{nm}-\frac{14.87\unit{nm}}{2\pi}\left(\sin\left(\frac{2\pi\cdotp\left(4\unit{nm}\right)}{14.87\unit{nm}}\right)-\sin\left(\frac{2\pi\cdotp\left(2\unit{nm}\right)}{14.87\unit{nm}}\right)\right)\right) \] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)\thickapprox0.0955 \] \end_inset \end_layout \begin_layout Subsection \begin_inset Formula $6\unit{nm}$ \end_inset to \begin_inset Formula $8\unit{nm}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)=\frac{1}{L}\left(2\unit{nm}-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi\cdotp\left(8\unit{nm}\right)}{L}\right)-\sin\left(\frac{2n\pi\cdotp\left(6\unit{nm}\right)}{L}\right)\right)\right) \] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\unit{nm}-\frac{14.87\unit{nm}}{2\pi}\left(\sin\left(\frac{2\pi\cdotp\left(8\unit{nm}\right)}{14.87\unit{nm}}\right)-\sin\left(\frac{2\pi\cdotp\left(6\unit{nm}\right)}{14.87\unit{nm}}\right)\right)\right) \] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)\thickapprox0.263 \] \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Part Application of Nanomaterials \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex btprint "btPrintCited" bibfiles "references" options "bibtotoc" \end_inset \end_layout \end_body \end_document