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added structure explanation and probability plot

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aj 2019-11-11 20:00:41 +00:00
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@ -91,11 +91,11 @@ EEE3037 Nanotechnology Coursework
6420013
\end_layout
\begin_layout Section
\begin_layout Part
Quantum Engineering Design
\end_layout
\begin_layout Subsection
\begin_layout Section
Structure Design
\end_layout
@ -178,9 +178,9 @@ This energy value will be the same as the total band gap for the well from
\begin_layout Standard
\begin_inset Formula
\[
\varSigma E_{g}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8eV
\]
\begin{equation}
\varSigma E_{g}=E_{1h}+E_{g}+E_{1e}\thickapprox0.8eV\label{eq:Energy-Gap-Sum}
\end{equation}
\end_inset
@ -317,7 +317,7 @@ Ga
As and as such this was tested first.
\end_layout
\begin_layout Subsubsection
\begin_layout Subsection
Lattice Match
\end_layout
@ -494,7 +494,7 @@ In order to compute a compound lattice constant for InGaAs, Vegard's law
\begin_layout Standard
\begin_inset Formula
\[
\alpha_{A_{(1-x)}B_{x}}=(1-x)\alpha_{A}+x\alpha_{B}
\alpha_{A_{(1-x)}B_{x}}=\left(1-x\right)\alpha_{A}+x\alpha_{B}
\]
\end_inset
@ -522,18 +522,916 @@ This shows that to 4 significant figures the composition of InGaAs is lattice
matched to InP to within 0.001Å which is sufficient for this application.
\end_layout
\begin_layout Subsubsection
\begin_layout Subsection
Band Gap
\end_layout
\begin_layout Subsection
Probability Plot
\begin_layout Standard
Vegard's law can also be used to approximate the band gap of a ternary alloy,
such as InGaAs.
The band gaps at 300K for each alloy can be seen in table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Band-gaps"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<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.
\end_layout
\begin_layout Subsection
Width Calculation
\end_layout
\begin_layout Standard
Having found two materials that are lattice matched with a suitable band
gap value, the final calculation is that of the quantum well width.
In order to calculate this value, the equation for energy levels within
an infinite quantum well will be used,
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\begin{equation}
E_{n}=\frac{n^{2}\pi^{2}\text{ħ}^{2}}{2mL^{2}}\label{eq:Energy-levels}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
Referring back to equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Energy-Gap-Sum"
plural "false"
caps "false"
noprefix "false"
\end_inset
, the terms for the first electron and hole energy levels can each be replaced
with equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Energy-levels"
plural "false"
caps "false"
noprefix "false"
\end_inset
as seen below,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\varSigma E_{g}=0.8\unit{eV}=E_{1h}+E_{g}+E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+E_{g}+\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
\]
\end_inset
\end_layout
\begin_layout Standard
With the experimentally determined value for
\begin_inset Formula $E_{g}$
\end_inset
this equation can be condensed to,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
0.8\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{h}^{*}L^{2}}+0.75+\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2m_{e}^{*}L^{2}}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
0.05\unit{eV}=\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2L^{2}}\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
L=\sqrt{\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp(0.05\unit{eV})}\cdotp\left(\frac{1}{m_{h}^{*}}+\frac{1}{m_{e}^{*}}\right)}
\]
\end_inset
\end_layout
\begin_layout Standard
As a frequently studied composition due to it's favourable structural parameters
with InP, The charge carrier effective masses of In
\begin_inset script subscript
\begin_layout Plain Layout
0.53
\end_layout
\end_inset
Ga
\begin_inset script subscript
\begin_layout Plain Layout
0.47
\end_layout
\end_inset
As have been found experimentally to be as shown in table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Effective-masses"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<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
by the heavy hole interactions, this effective mass ration will be used.
\end_layout
\begin_layout Standard
Substituting these ratios into the above provides,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
L=\sqrt{\frac{\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp(0.05\unit{eV})\cdotp m_{e}}\cdotp\left(\frac{1}{0.2}+\frac{1}{0.041}\right)}
\]
\end_inset
\end_layout
\begin_layout Standard
which reduces to a well length of 14.87nm.
\end_layout
\begin_layout Subsection
Energy Level Calculations
\end_layout
\begin_layout Standard
With all the parameters of the well ascertained the first and second confined
electron and hole energy levels can be found by utilising equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Energy-levels"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
For confined electron states:
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1e}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp m_{e}^{*}\cdotp\left(14.87\unit{nm}\right)^{2}}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1e}=6.65\times10^{-21}\unit{J}=0.041\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
This equation shows that energy values are proportional to the square of
\begin_inset Formula $n$
\end_inset
, the principal quantum number or energy level.
As such:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E_{2e}=2^{2}\cdotp E_{1e}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{2e}=2.66\times10^{-20}\unit{J}=0.17\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
For confined hole states:
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1h}=\frac{1^{2}\pi^{2}\text{\emph{ħ}}^{2}}{2\cdotp m_{h}^{*}\cdotp\left(14.87\unit{nm}\right)^{2}}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{1h}=1.36\times10^{-21}\unit{J}=0.0085\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
E_{2h}=2^{2}\cdotp E_{1h}
\]
\end_inset
\end_layout
\begin_layout Standard
\emph on
\begin_inset Formula
\[
E_{2h}=5.45\times10^{-21}\unit{J}=0.034\unit{eV}
\]
\end_inset
\end_layout
\begin_layout Section
Probability Plot
\end_layout
\begin_layout Standard
The probability of finding an electron in a quantum well is given by
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
P=\int_{0}^{L}\psi^{*}\psi dx\label{eq:wave-function-probability}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
with
\begin_inset Formula $\psi$
\end_inset
in the case of an infinite quantum well being given by,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\psi\left(x\right)=A\sin\left(kx\right)=A\sin\left(\frac{n\pi}{L}x\right)
\]
\end_inset
\end_layout
\begin_layout Standard
Where
\begin_inset Formula $A$
\end_inset
acts as a normalisation constant to satisfy the conditions
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\int_{{\textstyle all\:space}}\psi^{*}\psi dV=1
\]
\end_inset
\end_layout
\begin_layout Standard
in this case providing the wave function
\begin_inset Formula $\psi$
\end_inset
as
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\psi\left(x\right)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)\label{eq:wave-function}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
Importantly, the above conditions are for an infinite quantum well where
an assumption is made that the well has a barrier region of infinite potential
such that the wavefunction is confined to the well.
A real quantum well is unable to satisfy this leading to the wavefunction
\begin_inset Quotes eld
\end_inset
spilling
\begin_inset Quotes erd
\end_inset
into the barrier region.
For the purposes of plotting the probability density, however, it is a
reasonable assumption to make.
\end_layout
\begin_layout Standard
Considering equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:wave-function-probability"
plural "false"
caps "false"
noprefix "false"
\end_inset
, if the probability can be found by integrating
\begin_inset Formula $\psi^{*}\psi$
\end_inset
, or in this situation
\begin_inset Formula $\psi^{2}$
\end_inset
then the probability can be shown by plotting
\begin_inset Formula $\psi^{2}$
\end_inset
, see figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Probability-plot"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
Here the well stretches from 0 to the blue line along the
\begin_inset Formula $x$
\end_inset
axis and
\begin_inset Formula $n$
\end_inset
has been set to 1 for the ground state.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename probability-plot.png
lyxscale 30
width 100col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Probability plot for electron in ground state
\begin_inset CommandInset label
LatexCommand label
name "fig:Probability-plot"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Section
Probability Intervals
\end_layout
\begin_layout Standard
Combining equations
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:wave-function-probability"
plural "false"
caps "false"
noprefix "false"
\end_inset
and
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:wave-function"
plural "false"
caps "false"
noprefix "false"
\end_inset
gives the final probability function for the entire well:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P\left(0\leq x\leq x_{0}\right)=\frac{1}{L}\left(x_{0}-\frac{L}{2n\pi}\sin\left(\frac{2n\pi x_{0}}{L}\right)\right)
\]
\end_inset
\end_layout
\begin_layout Standard
Where
\begin_inset Formula $x_{0}$
\end_inset
is an arbitrary distance across the well.
\end_layout
\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset
@ -541,7 +1439,7 @@ Probability Intervals
\end_layout
\begin_layout Section
\begin_layout Part
Application of Nanomaterials
\end_layout

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references.bib
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@ -16,5 +16,67 @@ year = "2014-11"
@misc{new_semiconductor_materials_archive,
title={NSM Archive - Physical Properties of Semiconductors},
url={http://matprop.ru/},
journal={New Semiconductor Materials Archive}, publisher={Ioffe Institute}
journal={New Semiconductor Materials Archive},
publisher={Ioffe Institute}
}
@article{aip_complete10.1063/1.322570,
abstract = "Very uniform In 0.53 Ga 0.47 As was grown on InP by liquid phase epitaxy. The electron mobility is 8450 cm 2 /V sec at 300 K and 27700 cm 2 /V sec at 77 K. The mobility increases with decreasing temperature from 300 to 77 K in contrast to the results of In 1− x Ga x As grown directly on GaAs by vapor phase epitaxy. The energy gap of this high‐mobility material is 0.750 eV at room temperature.",
author = "Takeda, Yoshikazu and Sasaki, Akio and Imamura, Yujiro and Takagi, Toshinori",
issn = "0021-8979",
journal = "Journal of Applied Physics",
language = "eng",
number = "12",
pages = "5405,5408",
publisher = "American Institute of Physics",
title = "Electron mobility and energy gap of In 0.53 Ga 0.47 As on InP substrate",
volume = "47",
year = "1976-12",
}
@article{aip_complete10.1063/1.90860,
abstract = "The band-edge effective mass for conduction electrons in Ga x In 1-x As y P 1-y has been determined for several different alloy compositions covering the complete range of alloys grown lattice-matched on InP. Measurements show that the effective mass varies nearly linearly with alloy composition.",
author = "Nicholas, R. J. and Portal, J. C. and Houlbert, C. and Perrier, P. and Pearsall, T. P.",
issn = "0003-6951",
journal = "Applied Physics Letters",
keywords = "Galliumarsenid ; Drei-Fuenf-Verbindung ; Indiumphosphid ; Effektive Masse;",
language = "eng",
number = "8",
pages = "492,494",
publisher = "American Institute of Physics",
title = "An experimental determination of the effective masses for Ga x In 1-x As y P 1-y alloys grown on InP",
volume = "34",
year = "1979",
}
@article{aip_complete10.1063/1.92393,
abstract = "We report the use of optical pumping in p -type Ga x In 1-x As y P 1-y nearly lattice-matched to InP. Analysis of the conduction‐electron spin‐polarized photoluminescence has been used to deduce the valence‐band light‐hole effective mass as a function of alloy composition. Our results are in good agreement with masses calculated using the k·p approximation.",
author = "Hermann, Claudine and Pearsall, Thomas P.",
issn = "0003-6951",
journal = "Applied Physics Letters",
keywords = "Halbleiterverbindung ; Galliumarsenid ; Indiumphosphid ; A3-B5-Verbindung ; Optisches Pumpen ; Effektive Masse ; Defektelektron ; Valenzband ; Halbleitersubstrat ; P-Halbleiter ; Leitungselektron ; Photolumineszenz ; Spinorientierung;",
language = "eng",
number = "6",
pages = "450,452",
publisher = "American Institute of Physics",
title = "Optical pumping and the valence-band light-hole effective mass in Ga x In 1-x As y P 1-y (y approx. 2.2x)",
volume = "38",
year = "1981",
}
@article{aip_complete10.1063/1.101816,
author = "Lin, S. Y. and Liu, C. T. and Tsui, D. C. and Jones, E. D. and Dawson, L. R.",
issn = "0003-6951",
journal = "Applied Physics Letters",
keywords = "Materials Sciencegallium Arsenides ; Holes ; Indium Arsenides ; Cyclotron Resonance ; Effective Mass ; Electronic Structure ; Light Transmission ; Superlattices ; Arsenic Compounds ; Arsenides ; Gallium Compounds ; Indium Compounds ; Mass ; Pnictides ; Resonance;",
language = "eng",
number = "7",
pages = "666,668",
publisher = "American Institute of Physics",
title = "Cyclotron resonance of two-dimensional holes in strained-layer quantum well structure of (100)In 0.20 Ga 0.80 As/GaAs",
volume = "55",
year = "1989-08-14",
}