#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
\begin_preamble
\rfoot{6420013}
\end_preamble
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry true
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine biblatex
\cite_engine_type authoryear
\biblio_style plain
\biblatex_bibstyle ieee
\biblatex_citestyle ieee
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date true
\justification true
\use_refstyle 1
\use_minted 0
\index Index
\shortcut idx
\color #008000
\end_index
\leftmargin 2cm
\topmargin 2cm
\rightmargin 2cm
\bottommargin 2cm
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle fancy
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
\begin_layout Title
EEE3037 Nanotechnology Coursework
\end_layout
\begin_layout Author
6420013
\end_layout
\begin_layout Part
Quantum Engineering Design
\end_layout
\begin_layout Section
Structure Design
\end_layout
\begin_layout Standard
In order to design a quantum well which emits light of wavelength 1.55μm,
a well material must be chosen such that an interband electron transition
emits photons of this wavelength.
\end_layout
\begin_layout Standard
This band gap energy can be found from the equation
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E=hf
\]
\end_inset
\end_layout
\begin_layout Standard
When considering photons,
\begin_inset Formula $f$
\end_inset
can be substituted with
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
f=\frac{c}{\lambda}
\]
\end_inset
\end_layout
\begin_layout Standard
Therefore in order to find the energy,
\begin_inset Formula $E$
\end_inset
, in terms of wavelength
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E=\frac{hc}{\lambda}
\]
\end_inset
\end_layout
\begin_layout Standard
Returning to the specifications, this allows 1.55μm to be expressed as 1.28x10
\begin_inset script superscript
\begin_layout Plain Layout
-19
\end_layout
\end_inset
J or approximately 0.800 eV.
\end_layout
\begin_layout Standard
This energy value will be the same as the total interband transition for
the well from the first confined hole energy level to the first confined
electron enery level,
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
E_{g,transition}=E_{1h}+E_{g,bulk}+E_{1e}\thickapprox0.800\unit{eV}\label{eq:Energy-Gap-Sum}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
see figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Well-Band-structure"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename WellBandStructure.png
lyxscale 40
width 50col%
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
Band structure of an AlGaAs/GaAs/AlGaAs quantum well including discrete
confined energy levels
\begin_inset CommandInset citation
LatexCommand cite
key "ieee_s6824198"
literal "false"
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "fig:Well-Band-structure"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $E_{g}$
\end_inset
should be the dominant term in this equation and as such when investigating
suitable materials the bulk band gap should be close to but lower than
0.8eV.
\end_layout
\begin_layout Standard
Ternary alloys were investigated in order to allow precise control over
the lattice constants and band gap by varying the composition ratio.
\end_layout
\begin_layout Standard
Indium gallium arsenide (In
\begin_inset script subscript
\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset
\end_layout
\end_inset
Ga
\begin_inset script subscript
\begin_layout Plain Layout
\begin_inset Formula $(1-x)$
\end_inset
\end_layout
\end_inset
As) as a well material with indium phosphide (InP) as a barrier material
would provide a suitable combination assuming that a composition ratio
\begin_inset Formula $x$
\end_inset
could be found that satisfied the two conditions of having the required
bulk band gap and being lattice matched.
A common ratio in industry is In
\begin_inset script subscript
\begin_layout Plain Layout
0.53
\end_layout
\end_inset
Ga
\begin_inset script subscript
\begin_layout Plain Layout
0.47
\end_layout
\end_inset
As and as such this was tested first.
\end_layout
\begin_layout Subsection
Lattice Match
\end_layout
\begin_layout Standard
Lattice matching is the process of ensuring that two crystalline structures
are of similar dimensions in order to decrease strain at the interface
between the two materials.
This is particularly important for quantum wells formed through epitaxial
growth as strain introduced between such thin layers can cause defects
which ultimately negatively affect it's electronic properties.
\end_layout
\begin_layout Standard
The lattice constants between the barrier and well materials should be as
close as is deemed acceptable for the application.
The lattice constants for the prospective materials are shown in table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Lattice-constants"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
Material
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Lattice Constant, α (Å)
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
InAs
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
6.0583
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
GaAs
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
5.6532
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
InP
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
5.8687
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Lattice constants for prospective well and barrier materials
\begin_inset CommandInset citation
LatexCommand cite
key "new_semiconductor_materials_archive"
literal "false"
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "tab:Lattice-constants"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
In order to compute a compound lattice constant for InGaAs, Vegard's law
can be applied.
Vegard's law provides an approximation for the lattice constant of a solid
solution by finding the weighted average of the individual lattice constants
by composition ratio and is given by:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\alpha_{A_{(1-x)}B_{x}}=\left(1-x\right)\alpha_{A}+x\alpha_{B}
\]
\end_inset
\end_layout
\begin_layout Standard
Applying this to the prospective well material gives the following,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\alpha_{In_{0.53}Ga_{0.47}As}=0.53\cdotp6.0583+0.47\cdotp5.6532=5.8679
\]
\end_inset
\end_layout
\begin_layout Standard
This shows that this combination of InGaAs is lattice matched to InP to
within 0.001Å, a sufficient offset for this application.
\end_layout
\begin_layout Subsection
Band Gap
\end_layout
\begin_layout Standard
Vegard's law can also be used to approximate the band gap of a ternary alloy,
such as InGaAs.
The band gaps at 300K for each alloy can be seen in table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Band-gaps"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
Material
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Band Gap at 300K, E
\begin_inset script subscript
\begin_layout Plain Layout
g
\end_layout
\end_inset
(eV)
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
InAs
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.35
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
GaAs
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.42
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
InP
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.34
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Band gaps for prospective well and barrier materials
\begin_inset CommandInset citation
LatexCommand cite
key "new_semiconductor_materials_archive"
literal "false"
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "tab:Band-gaps"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
In this case the band gap approximates to,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E_{g,In_{0.53}Ga_{0.47}As}\thickapprox0.53\cdotp0.35+0.47\cdotp1.42\thickapprox0.85\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
However the band gap has been experimentally found to be 0.75eV
\begin_inset CommandInset citation
LatexCommand cite
key "aip_complete10.1063/1.322570"
literal "false"
\end_inset
.
This implies that the linear relationship provided by Vegard's law is not
accurate enough and in this case a modified version including a bowing
parameter
\begin_inset Formula $b$
\end_inset
should be used,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E_{g,total}=xE_{g,a}+\left(1-x\right)E_{g,b}-bx\left(1-x\right)
\]
\end_inset
\end_layout
\begin_layout Standard
For this application, however, the experimentally determined value will
be used.
This value is ideal for this application as it is comparable to and slightly
lower than the required 0.8eV energy value.
\end_layout
\begin_layout Subsection
Width Calculation
\end_layout
\begin_layout Standard
Having found two materials that are lattice matched with a suitable band
gap value, the final calculation is that of the quantum well width.
In order to calculate this value, the equation for confined energy levels
within an infinite quantum well will be used,
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\begin{equation}
E_{n}=\frac{n^{2}\pi^{2}\mathcal{\text{ħ}}^{2}}{2mL^{2}}\label{eq:Energy-levels}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
Referring back to equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Energy-Gap-Sum"
plural "false"
caps "false"
noprefix "false"
\end_inset
, the terms for the first electron and hole energy levels can each be replaced
with equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Energy-levels"
plural "false"
caps "false"
noprefix "false"
\end_inset
as seen below,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E_{g,transition}=E_{1h}+E_{g,InGaAs}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g,InGaAs}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}=0.8\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
With the experimentally determined value for
\begin_inset Formula $E_{g,,InGaAs}$
\end_inset
this equation becomes
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75\unit{eV}+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
0.05\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2L^{2}}\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
L=\sqrt{\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp(0.05\unit{eV})}\cdotp\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right)}
\]
\end_inset
\end_layout
\begin_layout Standard
As a frequently studied composition due to it's favourable structural parameters
with InP, The charge carrier effective masses of In
\begin_inset script subscript
\begin_layout Plain Layout
0.53
\end_layout
\end_inset
Ga
\begin_inset script subscript
\begin_layout Plain Layout
0.47
\end_layout
\end_inset
As have been found experimentally to be as shown in table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Effective-masses"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
Charge Carrier
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Effective mass ratio in In
\begin_inset script subscript
\begin_layout Plain Layout
0.53
\end_layout
\end_inset
Ga
\begin_inset script subscript
\begin_layout Plain Layout
0.47
\end_layout
\end_inset
As (
\begin_inset Formula $\frac{m^{*}}{m^{0}}$
\end_inset
)
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
Electron
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.041
\begin_inset CommandInset citation
LatexCommand cite
key "aip_complete10.1063/1.90860"
literal "false"
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
Light Hole
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.051
\begin_inset CommandInset citation
LatexCommand cite
key "aip_complete10.1063/1.92393"
literal "false"
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
Heavy Hole
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.2
\begin_inset CommandInset citation
LatexCommand cite
key "aip_complete10.1063/1.101816"
literal "false"
\end_inset
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Effective masses of charge carriers in
\begin_inset CommandInset label
LatexCommand label
name "tab:Effective-masses"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
As the electrical and optical properties of the valence band are governed
by the heavy hole interactions, this effective mass ratio will be used.
\end_layout
\begin_layout Standard
Substituting these ratios into the above provides,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
L=\sqrt{\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp(0.05\unit{eV})\cdotp m_{e}}\cdotp\left(\frac{1}{0.2}+\frac{1}{0.041}\right)}
\]
\end_inset
\end_layout
\begin_layout Standard
which reduces to a well length of 14.87nm.
\end_layout
\begin_layout Subsection
Confined Energy Level Calculations
\end_layout
\begin_layout Standard
With all the parameters of the well ascertained the first and second confined
electron and hole energy levels can be found by utilising equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Energy-levels"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
For confined electron states:
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1e}=\frac{1^{2}\pi^{2}\text{ħ}^{2}}{2\cdotp m_{e}^{*}\cdotp\left(14.87\unit{nm}\right)^{2}}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1e}=6.65\times10^{-21}\unit{J}=0.041\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
This equation shows that confiend energy level values are proportional to
the square of
\begin_inset Formula $n$
\end_inset
, the principal quantum number or energy level.
As such:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E_{2e}=2^{2}\cdotp E_{1e}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{2e}=2.66\times10^{-20}\unit{J}=0.17\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
For confined hole states:
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1h}=\frac{1^{2}\pi^{2}\text{ħ}^{2}}{2\cdotp m_{h}^{*}\cdotp\left(14.87\unit{nm}\right)^{2}}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1h}=1.36\times10^{-21}\unit{J}=0.0085\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E_{2h}=2^{2}\cdotp E_{1h}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{2h}=5.45\times10^{-21}\unit{J}=0.034\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
With the dimensions and first confined energy levels calculated, the final
design for the quantum well can be seen in figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:quantum-well-design"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename well-design.png
lyxscale 30
width 85col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
InP/InGaAs/InP quantum well design, relative confined energy level heights
are not to scale
\begin_inset CommandInset label
LatexCommand label
name "fig:quantum-well-design"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Probability Plot
\end_layout
\begin_layout Standard
The probability of finding an electron in a quantum well is given by
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
P=\int_{0}^{L}\psi^{*}\psi dx\label{eq:wave-function-probability}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
with
\begin_inset Formula $\psi$
\end_inset
in the case of an infinite quantum well being given by,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\psi\left(x\right)=A\sin\left(kx\right)=A\sin\left(\frac{n\pi}{L}x\right)
\]
\end_inset
\end_layout
\begin_layout Standard
Here
\begin_inset Formula $A$
\end_inset
acts as a normalisation constant to satisfy the conditions
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\int_{{\textstyle all\;space}}\psi^{*}\psi dV=1
\]
\end_inset
\end_layout
\begin_layout Standard
in this case providing the wave function
\begin_inset Formula $\psi$
\end_inset
as
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\psi\left(x\right)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)\label{eq:wave-function}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
Importantly, the above conditions are for an infinite quantum well where
an assumption is made that the well has a barrier region of infinite potential
such that the wavefunction is confined within the well.
A real quantum well is unable to satisfy this leading to the wavefunction
\begin_inset Quotes eld
\end_inset
spilling
\begin_inset Quotes erd
\end_inset
into the barrier region.
For the purposes of plotting the probability density, however, it is a
reasonable assumption to make.
\end_layout
\begin_layout Standard
Considering equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:wave-function-probability"
plural "false"
caps "false"
noprefix "false"
\end_inset
, if the probability can be found by integrating
\begin_inset Formula $\psi^{*}\psi$
\end_inset
, or in this situation
\begin_inset Formula $\psi^{2}$
\end_inset
then the probability can be shown by plotting
\begin_inset Formula $\psi^{2}$
\end_inset
, see figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Probability-plot"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
Here the well stretches from 0 to the blue line along the
\begin_inset Formula $x$
\end_inset
axis and
\begin_inset Formula $n$
\end_inset
has been set to 1 for the ground state.
This function for the first excited state can be seen in figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Probability-plot-n-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename probability-plot.png
lyxscale 30
width 60col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Probability plot for electron in ground state
\begin_inset CommandInset label
LatexCommand label
name "fig:Probability-plot"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename probability-plot-with-n-2.png
lyxscale 30
width 60col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Probability plot for electron in 1
\begin_inset script superscript
\begin_layout Plain Layout
st
\end_layout
\end_inset
excited state
\begin_inset CommandInset label
LatexCommand label
name "fig:Probability-plot-n-2"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Probability Intervals
\end_layout
\begin_layout Standard
Combining equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:wave-function-probability"
plural "false"
caps "false"
noprefix "false"
\end_inset
and
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:wave-function"
plural "false"
caps "false"
noprefix "false"
\end_inset
gives the final probability function for a distance across the well from
\begin_inset Formula $x=0$
\end_inset
to
\begin_inset Formula $x=x_{0}$
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(0\leq x\leq x_{0}\right)=\frac{1}{L}\left(x_{0}-\frac{L}{2n\pi}\sin\left(\frac{2n\pi x_{0}}{L}\right)\right)
\]
\end_inset
\end_layout
\begin_layout Standard
For an arbitrary interval across the well, this becomes:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(a\leq x\leq b\right)=\frac{1}{L}\left(\left(b-a\right)-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi b}{L}\right)-\sin\left(\frac{2n\pi a}{L}\right)\right)\right)
\]
\end_inset
\end_layout
\begin_layout Standard
This equation can be utilised in order to find the probability of finding
the electron between
\begin_inset Formula $2\unit{nm}$
\end_inset
and
\begin_inset Formula $4\unit{nm}$
\end_inset
and between
\begin_inset Formula $6\unit{nm}$
\end_inset
and
\begin_inset Formula $8\unit{nm}$
\end_inset
, the intervals for which can be seen plotted in figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Probability-plot-with-bounds"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename probability-plot-with-bounds.png
lyxscale 30
width 60col%
\end_inset
\end_layout
\begin_layout Plain Layout
\align center
Green: 2nm - 4nm
\end_layout
\begin_layout Plain Layout
\align center
Purple: 6nm - 8nm
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Probability plot for electron in ground state with distance intervals
\begin_inset CommandInset label
LatexCommand label
name "fig:Probability-plot-with-bounds"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
\begin_inset Formula $2\unit{nm}$
\end_inset
to
\begin_inset Formula $4\unit{nm}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)=\frac{1}{L}\left(2\unit{nm}-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi\cdotp\left(4\unit{nm}\right)}{L}\right)-\sin\left(\frac{2n\pi\cdotp\left(2\unit{nm}\right)}{L}\right)\right)\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\unit{nm}-\frac{14.87\unit{nm}}{2\pi}\left(\sin\left(\frac{2\pi\cdotp\left(4\unit{nm}\right)}{14.87\unit{nm}}\right)-\sin\left(\frac{2\pi\cdotp\left(2\unit{nm}\right)}{14.87\unit{nm}}\right)\right)\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)\thickapprox0.0955
\]
\end_inset
\end_layout
\begin_layout Subsection
\begin_inset Formula $6\unit{nm}$
\end_inset
to
\begin_inset Formula $8\unit{nm}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)=\frac{1}{L}\left(2\unit{nm}-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi\cdotp\left(8\unit{nm}\right)}{L}\right)-\sin\left(\frac{2n\pi\cdotp\left(6\unit{nm}\right)}{L}\right)\right)\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\unit{nm}-\frac{14.87\unit{nm}}{2\pi}\left(\sin\left(\frac{2\pi\cdotp\left(8\unit{nm}\right)}{14.87\unit{nm}}\right)-\sin\left(\frac{2\pi\cdotp\left(6\unit{nm}\right)}{14.87\unit{nm}}\right)\right)\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)\thickapprox0.263
\]
\end_inset
\end_layout
\begin_layout 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.
This would be expected considering 6nm to 8nm places the interval towards
the center of the 14.87nm long well.
As the probability density function is a
\begin_inset Formula $\sin^{2}$
\end_inset
function, the majority of the 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
region formed by the green lines.
\end_layout
\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset
\end_layout
\begin_layout Part
Application of Nanomaterials - Abraxane
\end_layout
\begin_layout Standard
The use of albumin protein nanoparticles has provided a new delivery aid
for the highly effective chemotherapy drug, paclitaxel, in turn reducing
side effects and toxicity caused by previous delivery schemes and increasing
circulation half life around the body.
\end_layout
\begin_layout Section
Paclitaxel
\end_layout
\begin_layout Standard
Paclitaxel is a chemotherapy drug in the taxane family which together function
as mitotic inhibitors.
This involves the suppression of mitosis or cell division by preventing
the breakdown of the microtubules helping provide structure to cells.
\end_layout
\begin_layout Standard
This is effective in treating cancer as constant, unmitigated cell mitosis
is how cancer spreads throughout the body, blocking this process causes
the cells to die without reproducing.
\end_layout
\begin_layout Standard
While taxanes are an effective cancer treatment, their use is made less
efficacious due to their practical insolubility in water.
In order to allow intravenous treatment, additional chemicals must be used
as delivery 'vehicles' to improve solubility.
\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 Taxol.svg
lyxscale 30
width 30col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Chemical structure for paclitaxel
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Previous Delivery Methods
\end_layout
\begin_layout Standard
As a result of the poor water solubility of taxanes and paclitaxel, a method
for delivering a solution was required.
Polyethoxylated castor oil (commercially known as Kolliphor EL, formerly
Cremophor EL [CrEL]) combined with dehydrated ethanol provides a suitable
formulation vehicle for many poorly water soluble and lipophilic (tending
to dissolve in lipids or fats) drugs and has been the standard for many
forms of commercially available paclitaxel such as Taxol.
\end_layout
\begin_layout Standard
While this solution has proved to be an effective delivery mechanism there
are significant side effects.
CrEL has been shown to cause severe hypersensitivity reactions and peripheral
neuropathy which are exacerbated by the high volumes of delivery agent
which must be coadministered with the active ingredient
\begin_inset CommandInset citation
LatexCommand cite
key "elsevier_sdoi_10_1016_S0959_8049_01_00171_X"
literal "false"
\end_inset
.
The use of CrEL also affects the behaviour of paclitaxel when administered,
manifesting as undesirable non-linear absorption, distribution, metabolism
and excretion behaviour
\begin_inset CommandInset citation
LatexCommand cite
key "proquest78006535"
literal "false"
\end_inset
, typically referred to as a drug's pharmacokinetic characteristics.
\end_layout
\begin_layout Section
Human Serum Albumin
\end_layout
\begin_layout Standard
Human serum albumin (HSA), sometimes referred to as blood albumin is the
most frequently found protein in the human body
\begin_inset CommandInset citation
LatexCommand cite
key "proquest1881262578"
literal "false"
\end_inset
and is part of the albumin protein family.
HSA is produced by the liver and performs important functions such as maintaini
ng oncotic pressure in the blood vessels (ensuring the right levels of fluids
are found between blood vessels and body tissues) and transporting hormones
and fatty acids around the body.
\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 hsa.jpg
lyxscale 30
width 40col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Crystal structure of human serum albumin with binding sites annotated
\begin_inset CommandInset citation
LatexCommand cite
key "BARBOSA2014345"
literal "false"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Importantly for the application of drug delivery HSA along with the rest
of the albumin proteins are water soluble and HSA effectively binds with
both hydrophobic and hydrophilic chemicals
\begin_inset CommandInset citation
LatexCommand cite
key "proquest1881262578"
literal "false"
\end_inset
.
Critically HSA has been shown to be nontoxic, non-immunogenic (provoking
little response from the immune system), biocompatible and biodegradable
\begin_inset CommandInset citation
LatexCommand cite
key "wos000301045400002"
literal "false"
\end_inset
providing many theoretical advantages over Cremophor EL delivery as a result
of using a native biological subtance.
\end_layout
\begin_layout Standard
While HSA is frequently used due to it's native presence in the body reducing
the chances of an immunologic response, suitable albumin can also be found
in egg whites (ovalbumin [OVA]) and bovine serum (bovine serum albumin
[BSA]) where abundance and low cost are advantages.
\end_layout
\begin_layout Section
NAB-Paclitaxel
\end_layout
\begin_layout Standard
While there are many ways to produce albumin nanoparticles including desolvation
, emulsification and thermal gelation, an albumin specific technology was
developed in order to capture lipophilic drugs in albumin nanoparticles
known as NAB-technology where NAB refers to nanoparticle albumin-bound.
\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 nab-pac.png
lyxscale 30
width 60col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Diagram showing albumin nanoparticles in combination with paclitaxel
\begin_inset CommandInset citation
LatexCommand cite
key "veeda_edge"
literal "false"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The formulation process involves the drug in question being mixed in an
aqueous solution with HSA before being passed through a high pressure jet.
This forms nanoparticles of sizes between 100nm and 200nm
\begin_inset CommandInset citation
LatexCommand cite
key "wos000301045400002"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
The solubility in water of the final product is increased as operating at
the nano-scale increases the surface area of the particles and increases
the dissolution of the formulation.
This protein based delivery solution also has the benefit of allowing higher
doses of paclitaxel than is deemed safe when delivered in combination with
Cremophor.
\end_layout
\begin_layout Section
Abraxane
\end_layout
\begin_layout Standard
Abraxane is a NAB-paclitaxel drug sold by Celgene, a biotechnology company
developing drugs for cancer and inflammoatory diseases.
Abraxane is made up of nanoparticles roughly 130nm in size and represents
the first FDA approved use of a nanotechnology chemotherapy for metastatic
breast cancer
\begin_inset CommandInset citation
LatexCommand cite
key "wos000301045400002"
literal "false"
\end_inset
.
The European Medicines Agency lists three applications for Abraxane
\begin_inset CommandInset citation
LatexCommand cite
key "epar_summary_for_the_public-abraxane_2015"
literal "false"
\end_inset
:
\end_layout
\begin_layout Itemize
Metastatic breast cancer
\end_layout
\begin_deeper
\begin_layout Itemize
Following failure of an initial treatment
\end_layout
\begin_layout Itemize
When a standard treatment including an 'anthracycline' drug is not suitable
\end_layout
\end_deeper
\begin_layout Itemize
Metastatic adenocarcinoma of the pancreas
\end_layout
\begin_deeper
\begin_layout Itemize
In combination with the drug gemcitabine
\end_layout
\end_deeper
\begin_layout Itemize
Non-small cell lung cancer
\end_layout
\begin_deeper
\begin_layout Itemize
In combination with the drug carboplatin
\end_layout
\begin_layout Itemize
When surgery or radiotherapy is not suitable
\end_layout
\end_deeper
\begin_layout Section
Efficacy
\end_layout
\begin_layout Standard
The efficacy of abraxane can be measured by comparing the treatment results
of this nanoparticle based approach with the alternative solvent based
method.
The European Medicines Agency list the results from clinical studies for
each of the cancers listed above
\begin_inset CommandInset citation
LatexCommand cite
key "epar_summary_for_the_public-abraxane_2015"
literal "false"
\end_inset
, with the effectiveness measure defined as whether tumours disappeared
or were reduced by at least 30%.
\end_layout
\begin_layout Standard
Abraxane was found to be 31% effective compared to 16% for the alternative
paclitaxel based treatment for metastatic breast cancer.
\end_layout
\begin_layout Standard
However, when considering only patients who had not previously received
treatment following a metastatic diagnosis, the effectiveness was the same
for both.
\end_layout
\begin_layout Standard
For non-small cell lung cancer it was found to be 33% effective as opposed
to 25% for the alternative.
\end_layout
\begin_layout Standard
With regards to the pancreatic study a combination of Abraxane and gemcitabine
increased overall survival to 8.7 months from 6.7 months with a treatment
of just gemcitabine.
\end_layout
\begin_layout Standard
This indicates that the drug performance is as good or better than alternatives
for all three, an encouraging result for a delivery method that also reduces
side effects and increases efficiency of delivery.
\end_layout
\begin_layout Section
Discussion
\end_layout
\begin_layout Standard
Considering these results the use of protein nanoparticles looks to represent
an effective alternative to solvent based methods in delivering lipophobic
drugs.
In doing so the side effects of the solvent based methods can be avoided.
\end_layout
\begin_layout Standard
The landscape is further broadening with research being completed into applying
NAB-technology to other taxanes such as docetaxel and macrolides such as
rapamycin.
\end_layout
\begin_layout Standard
Drug delivery is one of the largest areas within the field of nanomedicine
with other sectors including direct cancer treatment, medical imaging and
blood purification.
\end_layout
\begin_layout Paragraph*
Part II Word Count: 989
\end_layout
\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "references"
options "bibtotoc"
\end_inset
\end_layout
\end_body
\end_document