added structure explanation and probability plot
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coursework.lyx
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coursework.lyx
@ -91,11 +91,11 @@ EEE3037 Nanotechnology Coursework
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6420013
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6420013
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\end_layout
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\end_layout
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\begin_layout Section
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\begin_layout Part
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Quantum Engineering Design
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Quantum Engineering Design
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\end_layout
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\end_layout
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\begin_layout Subsection
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\begin_layout Section
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Structure Design
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Structure Design
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\end_layout
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\end_layout
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@ -178,9 +178,9 @@ This energy value will be the same as the total band gap for the well from
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\begin_layout Standard
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\begin_layout Standard
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\begin_inset Formula
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\begin_inset Formula
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\[
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\begin{equation}
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\varSigma E_{g}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8eV
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\varSigma E_{g}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8eV\label{eq:Energy-Gap-Sum}
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\]
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\end{equation}
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\end_inset
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\end_inset
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@ -317,7 +317,7 @@ Ga
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As and as such this was tested first.
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As and as such this was tested first.
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\end_layout
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\end_layout
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\begin_layout Subsubsection
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\begin_layout Subsection
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Lattice Match
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Lattice Match
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\end_layout
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\end_layout
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@ -494,7 +494,7 @@ In order to compute a compound lattice constant for InGaAs, Vegard's law
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\begin_layout Standard
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\begin_layout Standard
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\begin_inset Formula
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\begin_inset Formula
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\[
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\[
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\alpha_{A_{(1-x)}B_{x}}=(1-x)\alpha_{A}+x\alpha_{B}
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\alpha_{A_{(1-x)}B_{x}}=\left(1-x\right)\alpha_{A}+x\alpha_{B}
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\]
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\]
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\end_inset
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\end_inset
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@ -522,18 +522,916 @@ This shows that to 4 significant figures the composition of InGaAs is lattice
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matched to InP to within 0.001Å which is sufficient for this application.
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matched to InP to within 0.001Å which is sufficient for this application.
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\end_layout
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\end_layout
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\begin_layout Subsubsection
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\begin_layout Subsection
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Band Gap
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Band Gap
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\end_layout
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\end_layout
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\begin_layout Subsection
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\begin_layout Standard
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Probability Plot
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Vegard's law can also be used to approximate the band gap of a ternary alloy,
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such as InGaAs.
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The band gaps at 300K for each alloy can be seen in table
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "tab:Band-gaps"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\begin_inset Float table
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\align center
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\begin_inset Tabular
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<lyxtabular version="3" rows="4" columns="2">
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<features tabularvalignment="middle">
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<column alignment="center" valignment="top">
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<column alignment="center" valignment="top">
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<row>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Material
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Band Gap at 300K, E
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\begin_inset script subscript
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\begin_layout Plain Layout
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g
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\end_layout
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\end_inset
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(eV)
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\end_layout
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\end_inset
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</cell>
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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InAs
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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0.35
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\end_layout
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\end_inset
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</cell>
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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GaAs
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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1.42
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\end_layout
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\end_inset
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</cell>
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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InP
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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1.34
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\end_layout
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\end_inset
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</cell>
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</row>
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</lyxtabular>
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Band gaps for prospective well and barrier materials
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\begin_inset CommandInset citation
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LatexCommand cite
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key "new_semiconductor_materials_archive"
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literal "false"
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\end_inset
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\begin_inset CommandInset label
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LatexCommand label
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name "tab:Band-gaps"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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In this case the band gap approximates to,
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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E_{g,In_{0.53}Ga_{0.47}As}\thickapprox0.53\cdotp0.35+0.47\cdotp1.42\thickapprox0.85\unit{eV}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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However the band gap has been experimentally found to be 0.75eV
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\begin_inset CommandInset citation
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LatexCommand cite
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key "aip_complete10.1063/1.322570"
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literal "false"
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\end_inset
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.
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This implies that the linear relationship provided by Vegard's law is not
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accurate enough and in this case a modified version including a bowing
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parameter
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\begin_inset Formula $b$
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\end_inset
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should be used,
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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E_{g,total}=xE_{g,a}+\left(1-x\right)E_{g,b}-bx\left(1-x\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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For this application, however, the experimentally determined value will
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be used.
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This value is ideal for this application as it is comparable to and slightly
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lower than the required 0.8eV energy value.
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\end_layout
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\end_layout
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\begin_layout Subsection
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\begin_layout Subsection
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Width Calculation
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\end_layout
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\begin_layout Standard
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Having found two materials that are lattice matched with a suitable band
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gap value, the final calculation is that of the quantum well width.
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In order to calculate this value, the equation for energy levels within
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an infinite quantum well will be used,
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\end_layout
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\begin_layout Standard
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\emph on
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\begin_inset Formula
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\begin{equation}
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E_{n}=\frac{n^{2}\pi^{2}\text{ħ}^{2}}{2mL^{2}}\label{eq:Energy-levels}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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Referring back to equation
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Energy-Gap-Sum"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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, the terms for the first electron and hole energy levels can each be replaced
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with equation
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Energy-levels"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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as seen below,
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\varSigma E_{g}=0.8\unit{eV}=E_{1h}+E_{g}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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With the experimentally determined value for
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\begin_inset Formula $E_{g}$
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\end_inset
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this equation can be condensed to,
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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0.05\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2L^{2}}\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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L=\sqrt{\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp(0.05\unit{eV})}\cdotp\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right)}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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As a frequently studied composition due to it's favourable structural parameters
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with InP, The charge carrier effective masses of In
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\begin_inset script subscript
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\begin_layout Plain Layout
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0.53
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\end_layout
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\end_inset
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Ga
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\begin_inset script subscript
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\begin_layout Plain Layout
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0.47
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\end_layout
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\end_inset
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As have been found experimentally to be as shown in table
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\begin_inset CommandInset ref
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reference "tab:Effective-masses"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\begin_inset Float table
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\align center
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\begin_inset Tabular
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<lyxtabular version="3" rows="4" columns="2">
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<features tabularvalignment="middle">
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<column alignment="center" valignment="top">
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<column alignment="center" valignment="top">
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<row>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Charge Carrier
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Effective mass ratio in In
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\begin_inset script subscript
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\begin_layout Plain Layout
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0.53
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\end_layout
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\end_inset
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Ga
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\begin_inset script subscript
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\begin_layout Plain Layout
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0.47
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\end_layout
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\end_inset
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As (
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\begin_inset Formula $\frac{m^{*}}{m^{0}}$
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\end_inset
|
||||||
|
|
||||||
|
)
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
</cell>
|
||||||
|
</row>
|
||||||
|
<row>
|
||||||
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||||||
|
\begin_inset Text
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
Electron
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
</cell>
|
||||||
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||||||
|
\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
|
||||||
|
</cell>
|
||||||
|
</row>
|
||||||
|
<row>
|
||||||
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||||||
|
\begin_inset Text
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
Light Hole
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
</cell>
|
||||||
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||||||
|
\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
|
||||||
|
</cell>
|
||||||
|
</row>
|
||||||
|
<row>
|
||||||
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||||||
|
\begin_inset Text
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
Heavy Hole
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
</cell>
|
||||||
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||||||
|
\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
|
||||||
|
</cell>
|
||||||
|
</row>
|
||||||
|
</lyxtabular>
|
||||||
|
|
||||||
|
\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
|
Probability Intervals
|
||||||
\end_layout
|
\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_layout Standard
|
||||||
\begin_inset Newpage pagebreak
|
\begin_inset Newpage pagebreak
|
||||||
\end_inset
|
\end_inset
|
||||||
@ -541,7 +1439,7 @@ Probability Intervals
|
|||||||
|
|
||||||
\end_layout
|
\end_layout
|
||||||
|
|
||||||
\begin_layout Section
|
\begin_layout Part
|
||||||
Application of Nanomaterials
|
Application of Nanomaterials
|
||||||
\end_layout
|
\end_layout
|
||||||
|
|
||||||
|
|||||||
BIN
coursework.pdf
BIN
coursework.pdf
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BIN
probability-plot.png
Normal file
BIN
probability-plot.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 73 KiB |
@ -16,5 +16,67 @@ year = "2014-11"
|
|||||||
@misc{new_semiconductor_materials_archive,
|
@misc{new_semiconductor_materials_archive,
|
||||||
title={NSM Archive - Physical Properties of Semiconductors},
|
title={NSM Archive - Physical Properties of Semiconductors},
|
||||||
url={http://matprop.ru/},
|
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",
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user