2019-11-04 15:34:25 +00:00
|
|
|
|
#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
|
2019-11-14 10:53:43 +00:00
|
|
|
|
\begin_preamble
|
|
|
|
|
|
\rfoot{6420013}
|
|
|
|
|
|
\end_preamble
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\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
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\cite_engine biblatex
|
|
|
|
|
|
\cite_engine_type authoryear
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\biblio_style plain
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\biblatex_bibstyle ieee
|
|
|
|
|
|
\biblatex_citestyle ieee
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\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
|
2019-11-14 10:53:43 +00:00
|
|
|
|
\paperpagestyle fancy
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\begin_layout Part
|
2019-11-04 15:34:25 +00:00
|
|
|
|
Quantum Engineering Design
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\begin_layout Section
|
2019-11-04 15:34:25 +00:00
|
|
|
|
Structure Design
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
Therefore in order to find the
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
2019-11-12 16:23:10 +00:00
|
|
|
|
J or approximately 0.800 eV.
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2019-11-12 11:43:05 +00:00
|
|
|
|
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,
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
\begin_inset Formula
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\begin{equation}
|
2019-11-12 16:23:10 +00:00
|
|
|
|
E_{g,transition}=E_{1h}+E_{g}+E_{1e}\thickapprox0.800\unit{eV}\label{eq:Energy-Gap-Sum}
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\end{equation}
|
2019-11-11 13:39:57 +00:00
|
|
|
|
|
|
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
confined energy levels
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
2019-11-12 11:43:05 +00:00
|
|
|
|
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
|
2019-11-11 13:39:57 +00:00
|
|
|
|
0.8eV.
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2019-11-12 11:43:05 +00:00
|
|
|
|
Ternary alloys were investigated in order to allow precise control over
|
|
|
|
|
|
the lattice constants and band gap by varying the composition ratio.
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2019-11-12 11:43:05 +00:00
|
|
|
|
Indium gallium arsenide (In
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
2019-11-11 23:13:08 +00:00
|
|
|
|
As) as a well material with indium phosphide (InP) as a barrier material
|
2019-11-11 13:39:57 +00:00
|
|
|
|
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
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\begin_layout Subsection
|
2019-11-11 13:39:57 +00:00
|
|
|
|
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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
which ultimately negatively affect it's electronic properties.
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
|
|
|
|
|
<lyxtabular version="3" rows="4" columns="2">
|
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
|
<row>
|
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
Material
|
|
|
|
|
|
\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
|
2019-11-11 15:33:04 +00:00
|
|
|
|
Lattice Constant, α (Å)
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
|
|
|
|
|
InAs
|
|
|
|
|
|
\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
|
|
|
|
|
|
6.0583
|
|
|
|
|
|
\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
|
|
|
|
|
|
GaAs
|
|
|
|
|
|
\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
|
2019-11-11 15:33:04 +00:00
|
|
|
|
5.6532
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
|
|
|
|
|
InP
|
|
|
|
|
|
\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
|
2019-11-11 15:33:04 +00:00
|
|
|
|
5.8687
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
</cell>
|
|
|
|
|
|
</row>
|
|
|
|
|
|
</lyxtabular>
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\begin_inset Caption Standard
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
Lattice constants for prospective well and barrier materials
|
2019-11-11 15:33:04 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
|
|
|
|
|
LatexCommand cite
|
|
|
|
|
|
key "new_semiconductor_materials_archive"
|
|
|
|
|
|
literal "false"
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
solution by finding the weighted average of the individual lattice constants
|
2019-11-11 15:33:04 +00:00
|
|
|
|
by composition ratio and is given by:
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
\begin_inset Formula
|
|
|
|
|
|
\[
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\alpha_{A_{(1-x)}B_{x}}=\left(1-x\right)\alpha_{A}+x\alpha_{B}
|
2019-11-11 15:33:04 +00:00
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
|
|
\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.
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\begin_layout Subsection
|
2019-11-11 13:39:57 +00:00
|
|
|
|
Band Gap
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
|
|
|
|
|
<lyxtabular version="3" rows="4" columns="2">
|
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
|
<row>
|
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
Material
|
|
|
|
|
|
\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
|
|
|
|
|
|
Band Gap at 300K, E
|
|
|
|
|
|
\begin_inset script subscript
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
g
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
(eV)
|
|
|
|
|
|
\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
|
|
|
|
|
|
InAs
|
|
|
|
|
|
\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
|
|
|
|
|
|
0.35
|
|
|
|
|
|
\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
|
|
|
|
|
|
GaAs
|
|
|
|
|
|
\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
|
|
|
|
|
|
1.42
|
|
|
|
|
|
\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
|
|
|
|
|
|
InP
|
|
|
|
|
|
\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
|
|
|
|
|
|
1.34
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
</cell>
|
|
|
|
|
|
</row>
|
|
|
|
|
|
</lyxtabular>
|
|
|
|
|
|
|
|
|
|
|
|
\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.
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Subsection
|
2019-11-11 20:00:41 +00:00
|
|
|
|
Width Calculation
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2019-11-11 20:00:41 +00:00
|
|
|
|
Having found two materials that are lattice matched with a suitable band
|
|
|
|
|
|
gap value, the final calculation is that of the quantum well width.
|
2019-11-12 11:43:05 +00:00
|
|
|
|
In order to calculate this value, the equation for confined energy levels
|
|
|
|
|
|
within an infinite quantum well will be used,
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
|
|
\emph on
|
|
|
|
|
|
\begin_inset Formula
|
|
|
|
|
|
\begin{equation}
|
2019-11-11 23:13:08 +00:00
|
|
|
|
E_{n}=\frac{n^{2}\pi^{2}\mathcal{\text{\emph{ħ}}}^{2}}{2mL^{2}}\label{eq:Energy-levels}
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\end{equation}
|
|
|
|
|
|
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
|
|
|
|
|
\[
|
2019-11-12 11:43:05 +00:00
|
|
|
|
E_{g,transition}=0.8\unit{eV}=E_{1h}+E_{g,InGaAs}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g,InGaAs}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
With the experimentally determined value for
|
2019-11-12 11:43:05 +00:00
|
|
|
|
\begin_inset Formula $E_{g,,InGaAs}$
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\end_inset
|
|
|
|
|
|
|
2019-11-12 11:43:05 +00:00
|
|
|
|
this equation becomes
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
\begin_inset Formula
|
|
|
|
|
|
\[
|
2019-11-12 11:43:05 +00:00
|
|
|
|
0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75\unit{eV}+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
|
|
\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
|
|
|
|
|
|
<lyxtabular version="3" rows="4" columns="2">
|
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
|
<row>
|
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
Charge Carrier
|
|
|
|
|
|
\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
|
|
|
|
|
|
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
|
|
|
|
|
|
</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
|
2019-11-14 10:53:43 +00:00
|
|
|
|
by the heavy hole interactions, this effective mass ratio will be used.
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
Confined Energy Level Calculations
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
This equation shows that confiend energy level values are proportional to
|
|
|
|
|
|
the square of
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
|
|
|
|
|
|
2019-11-12 11:43:05 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
Here
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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.
|
2019-11-12 11:43:05 +00:00
|
|
|
|
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
|
|
|
|
|
|
|
|
|
|
|
|
.
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
width 75col%
|
2019-11-11 20:00:41 +00:00
|
|
|
|
|
|
|
|
|
|
\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
|
|
|
|
|
|
|
|
|
|
|
|
|
2019-11-11 23:13:08 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
width 75col%
|
2019-11-11 23:13:08 +00:00
|
|
|
|
|
|
|
|
|
|
\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
|
|
|
|
|
|
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
2019-11-12 11:43:05 +00:00
|
|
|
|
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
|
|
|
|
|
|
|
|
|
|
|
|
:
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
For an arbitrary interval across the well, this becomes:
|
2019-11-11 23:13:08 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
2019-11-12 11:43:05 +00:00
|
|
|
|
\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
|
|
|
|
|
|
|
2019-11-11 23:13:08 +00:00
|
|
|
|
\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
|
2019-11-12 11:43:05 +00:00
|
|
|
|
width 75col%
|
2019-11-11 23:13:08 +00:00
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-12 16:23:10 +00:00
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\align center
|
|
|
|
|
|
Green: 2nm - 4nm
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\align center
|
|
|
|
|
|
Purple: 6nm - 8nm
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-11 23:13:08 +00:00
|
|
|
|
\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
|
|
|
|
|
|
\[
|
2019-11-12 11:43:05 +00:00
|
|
|
|
P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)\thickapprox0.0955
|
2019-11-11 23:13:08 +00:00
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
|
|
\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
|
|
|
|
|
|
\[
|
2019-11-12 11:43:05 +00:00
|
|
|
|
P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)\thickapprox0.263
|
2019-11-11 23:13:08 +00:00
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-12 16:23:10 +00:00
|
|
|
|
\begin_layout Subsection
|
|
|
|
|
|
Conclusions
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
Considering these two probabilities it is clear that it is more likely for
|
|
|
|
|
|
the electron to be found between 6nm and 8nm than between 2nm and 4nm across
|
|
|
|
|
|
the well.
|
2019-11-14 10:53:43 +00:00
|
|
|
|
This is as expected considering 6nm to 8nm places the interval towards
|
|
|
|
|
|
the center of the well.
|
2019-11-12 16:23:10 +00:00
|
|
|
|
As the probability density function is a
|
|
|
|
|
|
\begin_inset Formula $\sin^{2}$
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
function, the maxium area will be towards the center.
|
|
|
|
|
|
Referring to figure
|
|
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
|
LatexCommand ref
|
|
|
|
|
|
reference "fig:Probability-plot-with-bounds"
|
|
|
|
|
|
plural "false"
|
|
|
|
|
|
caps "false"
|
|
|
|
|
|
noprefix "false"
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
this can be seen graphically as the region created by the purple lines
|
|
|
|
|
|
has a far greater area under the probability density function than the
|
2019-11-14 10:53:43 +00:00
|
|
|
|
region formed by the green lines.
|
2019-11-12 16:23:10 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-11 20:00:41 +00:00
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
\begin_inset Newpage pagebreak
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Part
|
2019-11-04 15:34:25 +00:00
|
|
|
|
Application of Nanomaterials
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-12 16:23:10 +00:00
|
|
|
|
\begin_layout Standard
|
2019-11-14 10:53:43 +00:00
|
|
|
|
The use of albumin protein nanoparticlces has provided a new delivery method
|
|
|
|
|
|
for the highly effective chemotherapy drug, paclitaxel, in turn reducing
|
|
|
|
|
|
side effects caused by previous delivery schemes and increasing circulation
|
|
|
|
|
|
half life around the body.
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Section
|
|
|
|
|
|
Paclitaxel
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Section
|
|
|
|
|
|
Previous Delivery
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Section
|
|
|
|
|
|
Albumin
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Section
|
|
|
|
|
|
NAB-Paclitaxel
|
2019-11-12 16:23:10 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
\begin_inset Newpage pagebreak
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
2019-11-11 13:39:57 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
\begin_inset CommandInset bibtex
|
|
|
|
|
|
LatexCommand bibtex
|
|
|
|
|
|
btprint "btPrintCited"
|
|
|
|
|
|
bibfiles "references"
|
|
|
|
|
|
options "bibtotoc"
|
|
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
2019-11-04 15:34:25 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
|
|
\end_body
|
|
|
|
|
|
\end_document
|
