final state we left off on saturday
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PS1/doc.tex
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PS1/doc.tex
@ -370,53 +370,59 @@ ylabel('Current [A]');
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\begin{subfigure}{0.33\textwidth}
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\begin{subfigure}{0.33\textwidth}
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\centering
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\centering
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\includegraphics[width=\textwidth]{q2_0V.png}
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\includegraphics[width=\textwidth]{q2_0V.png}
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\caption{Fermi Functions at $0V$.}
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\caption{Fermi functions at $0V$.}
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\label{fig:q2_0v}
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\label{fig:q2_0v}
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\end{subfigure}%
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\end{subfigure}%
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\begin{subfigure}{0.33\textwidth}
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\begin{subfigure}{0.33\textwidth}
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\centering
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\centering
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\includegraphics[width=\textwidth]{q2_0V05.png}
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\includegraphics[width=\textwidth]{q2_0V05.png}
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\caption{Fermi Functions at $0.05V$.}
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\caption{Fermi functions at $0.05V$.}
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\label{fig:q2_0v}
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\label{fig:q2_0v}
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\end{subfigure}%
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\end{subfigure}%
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\begin{subfigure}{0.33\textwidth}
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\begin{subfigure}{0.33\textwidth}
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\centering
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\centering
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\includegraphics[width=\textwidth]{q2_0V1.png}
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\includegraphics[width=\textwidth]{q2_0V1.png}
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\caption{Fermi Functions at $0.1V$.}
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\caption{Fermi functions at $0.1V$.}
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\label{fig:q2_0v}
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\label{fig:q2_0v}
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\end{subfigure}
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\end{subfigure}
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\begin{subfigure}{0.33\textwidth}
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\begin{subfigure}{0.33\textwidth}
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\centering
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\centering
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\includegraphics[width=\textwidth]{q2_0V2.png}
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\includegraphics[width=\textwidth]{q2_0V2.png}
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\caption{Fermi Functions at $0.2V$.}
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\caption{Fermi functions at $0.2V$.}
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\label{fig:q2_0v}
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\label{fig:q2_0v}
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\end{subfigure}%
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\end{subfigure}%
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\begin{subfigure}{0.33\textwidth}
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\begin{subfigure}{0.33\textwidth}
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\centering
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\centering
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\includegraphics[width=\textwidth]{q2_0V3.png}
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\includegraphics[width=\textwidth]{q2_0V3.png}
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\caption{Fermi Functions at $0.3V$.}
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\caption{Fermi functions at $0.3V$.}
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\label{fig:q2_0v}
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\label{fig:q2_0v}
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\end{subfigure}
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\end{subfigure}
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\caption{stuff and things}
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\caption{(a) depicts I-V curves at $V_G = 0.5$ V (higher curve) and $V_G = 0.25$ V.
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(b) through (f) show Fermi functions and $D(E)$ at various drain voltages.}
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\end{figure}
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\end{figure}
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\subsection*{(b)}
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\subsection*{(b)}
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We can see from these figures that the area under the $f_1(E + U) - f_2(E + U)$ curve
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As the drain voltage increases, we see that the overlapping area
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gradually moves down in energy, this means that a smaller and smaller portion of it is
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under the "curve" of $f_1(E + U) - f_2(E+U)$ and $D(E)$ increases, up
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above $E = 0eV$, which means there are fewer total energy levels through which current
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until the point where the drain only has electrons below the energy
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can flow. From this we can see that with a high enough $V_D$ we will hit saturation, and
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levels available in the channel. The difference function also reaches its maximum height and it widens downward rather than upward.
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be unable to pass more current.
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At this point, the overlapping area no longer
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increases, and therefore the current does not increase either,
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hitting {\em saturation current}.
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% TODO re-word this
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%% WRONG EXPLANATION - kept for posterity :)
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The self-consistent potential will decrease, which is what causes this shift in levels.
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% We can see from these figures that the area under the $f_1(E + U) - f_2(E + U)$ curve
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% gradually moves down in energy. This means that a smaller and smaller portion of it is
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% above $E = 0eV$, which means there are fewer total energy levels through which current
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% can flow. From this we can see that with a high enough $V_D$ we will hit saturation, and
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% be unable to pass more current. \\
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\subsection*{(c)}
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\subsection*{(c)}
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At $V_D = 0.3V$, the energy levels from approximately $0$ to $0.2$ $eV$ are being
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At $V_D = 0.3V$, the energy levels from approximately $0$ to $0.2$ $eV$ are being
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used for electron transport. This is the range where the difference in the contact Fermi
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used for electron transport. This is the range where the difference in the contact Fermi
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functions is more than 0 and the energy is more than 0.
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functions is more than 0 and the energy is more than 0. \\
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The difference between the two contacts is the greatest at $0eV$, because this is the
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The difference between the two contacts is the greatest at $0eV$, because this is the
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point at which their Fermi functions have the greatest difference, and thus will be making
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point at which their Fermi functions have the greatest difference, and thus will be making
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the most "effort" to equalize the channel potential.
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the most "effort" to equalize the channel potential.
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@ -427,8 +433,7 @@ ylabel('Current [A]');
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drain current. The energy levels vanishing at $-0.4eV$ would mean that there is a
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drain current. The energy levels vanishing at $-0.4eV$ would mean that there is a
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greater number of energy levels that would be able to be used for conduction. There
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greater number of energy levels that would be able to be used for conduction. There
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would be a larger {\em area} where there is a non-zero difference in the two Fermi
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would be a larger {\em area} where there is a non-zero difference in the two Fermi
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functions at the contacts and there are energy levels available in the channel.
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functions at the contacts and there are energy levels available in the channel. \\
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This can be contrasted with material B, which would have {\em no} energy levels in
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This can be contrasted with material B, which would have {\em no} energy levels in
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this "conduction" zone and thus no current would be able to flow at all.
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this "conduction" zone and thus no current would be able to flow at all.
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@ -494,7 +499,7 @@ ylabel('Current [A]');
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\begin{equation*}
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\begin{equation*}
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I = 0, N = 0
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I = 0, N = 0
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\end{equation*}
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\end{equation*}
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%
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This is because the only allowed energy level in the channel is at a higher energy
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This is because the only allowed energy level in the channel is at a higher energy
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level than exists in either of the contacts, thus there are no electrons that would
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level than exists in either of the contacts, thus there are no electrons that would
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flow into the channel from either contact, thus no current {\em and} no electrons in the channel.
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flow into the channel from either contact, thus no current {\em and} no electrons in the channel.
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@ -502,15 +507,15 @@ ylabel('Current [A]');
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\subsubsection*{(ii)}
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\subsubsection*{(ii)}
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\begin{equation*}
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\begin{equation*}
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I = 608nA, N = 0.5
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I = 608\:nA, N = 0.5
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\end{equation*}
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\end{equation*}
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%
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Given that we are operating with a single energy level in the channel, we can use
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Given that we are operating with a single energy level in the channel, we can use
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the equations 9 and 10 (provided in the assignment) directly.
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the equations 9 and 10 (provided in the assignment) directly.
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\begin{equation*}
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\begin{equation*}
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I = \frac{q}{\hbar} \cdot \frac{0.005eV \cdot 0.005eV}{0.005eV + 0.005eV}
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I = \frac{q}{\hbar} \cdot \frac{0.005eV \cdot 0.005eV}{0.005eV + 0.005eV}
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\cdot [1 - 0] = 608nA
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\cdot [1 - 0] = 608\:nA
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\end{equation*}
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\end{equation*}
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\begin{equation*}
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\begin{equation*}
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@ -520,15 +525,15 @@ ylabel('Current [A]');
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\subsubsection*{(iii)}
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\subsubsection*{(iii)}
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\begin{equation*}
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\begin{equation*}
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I = 0A, N = 1
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I = 0\:A, N = 1
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\end{equation*}
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\end{equation*}
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%
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Given that we are operating with a single energy level in the channel, we can use
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Given that we are operating with a single energy level in the channel, we can use
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the equations 9 and 10 (provided in the assignment) directly.
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the equations 9 and 10 (provided in the assignment) directly.
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\begin{equation*}
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\begin{equation*}
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I = \frac{q}{\hbar} \cdot \frac{0.005eV \cdot 0.005eV}{0.005eV + 0.005eV}
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I = \frac{q}{\hbar} \cdot \frac{0.005eV \cdot 0.005eV}{0.005eV + 0.005eV}
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\cdot [1 - 1] = 0A
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\cdot [1 - 1] = 0\:A
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\end{equation*}
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\end{equation*}
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\begin{equation*}
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\begin{equation*}
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@ -548,7 +553,7 @@ ylabel('Current [A]');
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\subsubsection*{(ii)}
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\subsubsection*{(ii)}
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For $f_1(E)$ the step point occurs at $E = \mu_1 = 0.25$ eV. Since $\mu_1 = \mu - qV_S$ and $\mu = 0$ eV, $V_S = -0.25$ V.
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For $f_1(E)$ the step point occurs at $E = \mu_1 = 0.25$ eV. Since $\mu_1 = \mu - qV_S$ and $\mu = 0$ eV, $V_S = -0.25$ V.
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For $f_2(E + U)$ the step point occurs at $E = \mu_2 = -0.25$ eV. Since $\mu_1 = \mu - qV_D$ and $\mu = 0$ eV, $V_D = 0.25$ V.
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For $f_2(E)$ the step point occurs at $E = \mu_2 = -0.25$ eV. Since $\mu_1 = \mu - qV_D$ and $\mu = 0$ eV, $V_D = 0.25$ V.
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This assumes $U = 0$ eV.
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This assumes $U = 0$ eV.
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\subsection*{(c)}
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\subsection*{(c)}
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@ -572,7 +577,7 @@ ylabel('Current [A]');
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I = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot [10^4 \cdot 0.2 - 10^4 \cdot -0.1]
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I = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot [10^4 \cdot 0.2 - 10^4 \cdot -0.1]
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\end{equation*}
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\end{equation*}
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\begin{equation*}
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\begin{equation*}
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I = 1.8mA
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I = 1.8\:mA
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\end{equation*}
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\end{equation*}
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\subsubsection*{(iii)}
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\subsubsection*{(iii)}
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@ -596,59 +601,54 @@ ylabel('Current [A]');
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Using equation (5) in the assignment and plugging in the given values we obtain:
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Using equation (5) in the assignment and plugging in the given values we obtain:
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\begin{equation*}
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\begin{aligned}
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U = -q[\frac{C_SV_S+C_GV_G+C_DV_D}{C_E}] + \frac{q^2(N-N_0)}{C_E} \\
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U = -\alpha_SV_S - \alpha_GV_G - \alpha_DV_D + U_0(N - N_0)\:\:[eV] \\
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U = -0*0 - 0.5*0\:eV - 0.5*0.6\:eV + 0.25\:eV*(0.325 - 0)\:\:[eV] \\
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U = -0.21875\:eV
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\end{aligned}
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\end{equation*}
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\begin{align*}
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U &= -q[\frac{C_SV_S+C_GV_G+C_DV_D}{C_E}] + \frac{q^2(N-N_0)}{C_E} \\
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U &= -\alpha_SV_S - \alpha_GV_G - \alpha_DV_D + U_0(N - N_0)\:\:eV \\
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U &= -0 \cdot 0 - 0.5 \cdot 0\:eV - 0.5 \cdot 0.6\:eV + 0.25\:eV\cdot(0.325 - 0)\:\:eV \\
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U &= -0.21875\:eV
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\end{align*}
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%
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Then, starting with Equation \ref{eq:N_new} and plugging in $D(E-U) = \delta(E - \varepsilon)$, we obtain
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Then, starting with Equation \ref{eq:N_new} and plugging in $D(E-U) = \delta(E - \varepsilon)$, we obtain
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\begin{equation*}
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\begin{equation*}
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N = \frac{\gamma_1f_1(\varepsilon + U) + \gamma_2f_2(\varepsilon + U)}{\gamma_1 + \gamma_2}
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N = \frac{\gamma_1f_1(\varepsilon + U) + \gamma_2f_2(\varepsilon + U)}{\gamma_1 + \gamma_2}
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\end{equation*}
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\end{equation*}
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Recalling that the expressions for the contact Fermi functions are (with $\mu = 0$ eV):
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Recalling that the expressions for the contact Fermi functions are (with $\mu = 0$ eV):
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\begin{equation*}
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\begin{aligned}
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f_1(\varepsilon + U) = \frac{1}{1 + e^{frac{\varepsilon + U + qV_S}{k_BT}}} = \frac{1}{1 + e^{frac{0.2 - 0.21875 + 0}{0.025}}} = 0.679 \\
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f_2(\varepsilon + U) = \frac{1}{1 + e^{frac{\varepsilon + U + qV_D}{k_BT}}} = \frac{1}{1 + e^{frac{0.2 - 0.21875 + 0.6}{0.025}}} \simeq 0
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\end{aligned}
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\end{equation*}
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Then, solving for $\gamma_2$:
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\begin{equation*}
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\begin{aligned}
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\gamma_2 = \gamma_2\frac{N - f_1(\varepsilon + U)}{f_2(\varepsilon + U) - N} = 5.45 \times 10^-3 \: eV
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\end{aligned}
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\end{equation*}
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\begin{align*}
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f_1(\varepsilon + U) = \frac{1}{1 + e^{(\varepsilon + U + qV_S)/k_BT}} = \frac{1}{1 + e^{(0.2 - 0.21875 + 0)/0.025}} = 0.679 \\
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f_2(\varepsilon + U) = \frac{1}{1 + e^{()\varepsilon + U + qV_D)/k_BT}} = \frac{1}{1 + e^{(0.2 - 0.21875 + 0.6)/0.025}} \simeq 0
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\end{align*}
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Then, solving for $\gamma_2$:
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\begin{align*}
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\gamma_2 = \gamma_2\frac{N - f_1(\varepsilon + U)}{f_2(\varepsilon + U) - N} = 5.45 \times 10^{-3} \: eV
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\end{align*}
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\subsection*{(e)}
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\subsection*{(e)}
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Assuming the density of states for each molecule can be modeled by $D(E) = \delta(E-\varepsilon)$,
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Assuming the density of states for each molecule can be modeled by $D(E) = \delta(E-\varepsilon)$,
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Equation (12) in the assignment is valid here. Thus the current will be maximized when $[f_1(E) - f_2(E)]|_{E=\varepsilon}$ is maximized.
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Equation (12) in the assignment is valid here. Thus the current will be maximized when $[f_1(E) - f_2(E)]|_{E=\varepsilon}$ is maximized.
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Solving $[f_1(E) - f_2(E)]|_{E=\varepsilon}$ for each molecule, we obtain:
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Solving $[f_1(E) - f_2(E)]|_{E=\varepsilon}$ for each molecule, we obtain:\\
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Molecule A:
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Molecule A:
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\begin{equation*}
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\begin{equation*}
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[f_{1}(E) - f_{2}(E)]|_{E=\varepsilon_A} =
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[f_{1}(E) - f_{2}(E)]|_{E=\varepsilon_A} =
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[\frac{1}{1 + e^{frac{\varepsilon_A}{k_BT_1}}} - [\frac{1}{1 + e^{frac{\varepsilon_A}{k_BT_2}}}]
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\frac{1}{1 + e^{\frac{\varepsilon_A}{k_BT_1}}} - \frac{1}{1 + e^{\frac{\varepsilon_A}{k_BT_2}}}
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= [\frac{1}{1 + e^{frac{0}{0.024}}} - [\frac{1}{1 + e^{frac{0}{0.027}}}] = 0
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= \frac{1}{1 + e^{\frac{0}{0.024}}} - \frac{1}{1 + e^{\frac{0}{0.027}}} = 0
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\end{equation*}
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\end{equation*}
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Molecule B:
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Molecule B:
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\begin{equation*}
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\begin{equation*}
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[f_{1}(E) - f_{2}(E)]|_{E=\varepsilon_B} =
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[f_{1}(E) - f_{2}(E)]|_{E=\varepsilon_B} =
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[\frac{1}{1 + e^{frac{\varepsilon_B}{k_BT_1}}} - [\frac{1}{1 + e^{frac{\varepsilon_B}{k_BT_2}}}]
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\frac{1}{1 + e^{\frac{\varepsilon_B}{k_BT_1}}} - \frac{1}{1 + e^{\frac{\varepsilon_B}{k_BT_2}}}
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= [\frac{1}{1 + e^{frac{-0.05}{0.024}}} - [\frac{1}{1 + e^{frac{-0.05}{0.027}}}] = 0.02493
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= \frac{1}{1 + e^{\frac{-0.05}{0.024}}} - \frac{1}{1 + e^{\frac{-0.05}{0.027}}} = 0.02493
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\end{equation*}
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\end{equation*}
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Molecule C:
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Molecule C:
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\begin{equation*}
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\begin{equation*}
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[f_{1}(E) - f_{2}(E)]|_{E=\varepsilon_C} =
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[f_{1}(E) - f_{2}(E)]|_{E=\varepsilon_C} =
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[\frac{1}{1 + e^{frac{\varepsilon_C}{k_BT_1}}} - [\frac{1}{1 + e^{frac{\varepsilon_C}{k_BT_2}}}]
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\frac{1}{1 + e^{\frac{\varepsilon_C}{k_BT_1}}} - \frac{1}{1 + e^{\frac{\varepsilon_C}{k_BT_2}}}
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= [\frac{1}{1 + e^{frac{-0.1}{0.024}}} - [\frac{1}{1 + e^{frac{-0.1}{0.027}}}] = 0.00877
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= \frac{1}{1 + e^{\frac{-0.1}{0.024}}} - \frac{1}{1 + e^{\frac{-0.1}{0.027}}} = 0.00877
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\end{equation*}
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\end{equation*}
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Molecule B should be chosen.
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Molecule B should be chosen.
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@ -657,16 +657,16 @@ ylabel('Current [A]');
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Referring again to equation 2 in the assignment, we know that $D(E)$ is valid for all $E$, however
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Referring again to equation 2 in the assignment, we know that $D(E)$ is valid for all $E$, however
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since $\gamma_1$ is dependant on the energy level, we can use it to reduce our limits. For material
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since $\gamma_1$ is dependant on the energy level, we can use it to reduce our limits. For material
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A, this means we only care about $E$ above $0eV$, and for B, $E$ below $0eV$.
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A, this means we only care about $E$ above $0eV$, and for B, $E$ below $0eV$.
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%
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Since we know the voltages on the contacts, we can calculate the effective fermi level at each.
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Since we know the voltages on the contacts, we can calculate the effective fermi level at each.
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\begin{equation*}
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\begin{equation*}
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\mu_1 = \mu_0 - V_S = 0 - (-2V) = 2eV
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\mu_1 = \mu_0 - V_S = 0 - (-2V) = 2\:eV
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\end{equation*}
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\end{equation*}
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\begin{equation*}
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\begin{equation*}
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\mu_2 = \mu_0 - V_D = 0 - 1V = -1eV
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\mu_2 = \mu_0 - V_D = 0 - 1V = -1\:eV
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\end{equation*}
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\end{equation*}
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%
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With the fermi levels of each contact, we can adjust the limits of integration further for each
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With the fermi levels of each contact, we can adjust the limits of integration further for each
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material. Material A can be evaluated on $[0eV, 2eV]$, and material B can be evaluated on $[-1eV, 0eV]$.
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material. Material A can be evaluated on $[0eV, 2eV]$, and material B can be evaluated on $[-1eV, 0eV]$.
|
||||||
|
|
||||||
@ -674,31 +674,31 @@ ylabel('Current [A]');
|
|||||||
I_A = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot \int_{0}^{2} [1 - 0] \cdot 10^4 dE
|
I_A = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot \int_{0}^{2} [1 - 0] \cdot 10^4 dE
|
||||||
\end{equation*}
|
\end{equation*}
|
||||||
\begin{equation*}
|
\begin{equation*}
|
||||||
I_A = 12.2mA
|
I_A = 12.2\:mA
|
||||||
\end{equation*}
|
\end{equation*}
|
||||||
|
|
||||||
\begin{equation*}
|
\begin{equation*}
|
||||||
I_B = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot \int_{0}^{2} [1 - 0] \cdot 10^4 dE
|
I_B = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot \int_{0}^{2} [1 - 0] \cdot 10^4 dE
|
||||||
\end{equation*}
|
\end{equation*}
|
||||||
\begin{equation*}
|
\begin{equation*}
|
||||||
I_A = 6.1mA
|
I_A = 6.1\:mA
|
||||||
\end{equation*}
|
\end{equation*}
|
||||||
|
%
|
||||||
Material A should be chosen.
|
Material A should be chosen.
|
||||||
|
|
||||||
\subsection*{(g)}
|
\subsection*{(g)}
|
||||||
|
|
||||||
Using Ohm's law we find the corresponding conductance:
|
Using Ohm's law we find the corresponding conductance:
|
||||||
\begin{equation*}
|
\begin{equation*}
|
||||||
\sigma{max} = \frac{I_{max}}{V} = \frac{500\:nA}{4.3\:mV} = 116\:\mu S.
|
\sigma_{max} = \frac{I_{max}}{V} = \frac{500\:nA}{4.3\:mV} = 116\:\mu S.
|
||||||
\end{equation*}
|
\end{equation*}
|
||||||
|
%
|
||||||
We expect this result to be an integer multiple of $G_0 = 38.76$ $\mu S$, the quantum of conductance. We find
|
We expect this result to be an integer multiple of $G_0 = 38.76$ $\mu S$, the quantum of conductance. We find
|
||||||
|
|
||||||
\begin{equation*}
|
\begin{equation*}
|
||||||
\frac{\sigma{max}}{G_0} = 3.
|
\frac{\sigma_{max}}{G_0} = 3.
|
||||||
\end{equation*}
|
\end{equation*}
|
||||||
|
%
|
||||||
We conclude there are 3 levels.
|
We conclude there are 3 levels.
|
||||||
|
|
||||||
|
|
||||||
|
Loading…
Reference in New Issue
Block a user