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integral. - - % TODO just replace these in the formulas themselves As well, to simplify the interim equations we will assign the constants \( A = \frac{1}{2} \sqrt{\frac{L}{\pi \hbar}} \) and \( B = \frac{L}{2 \pi \hbar} \). @@ -590,13 +588,14 @@ axis([0 1e-9 0 0.04]); \int _{-\infty} ^{\infty} \Phi (p')^2 dp = \int _{-\infty} ^{\infty} A^2 \left[ sinc\left( B(p'+p_1) \right)^2 + 2 sinc \left( B(p'+p_1) \right) sinc \left( B(p'-p_1) \right) + sinc \left( B(p'-p_1) \right) \right] dp \end{equation*} - Given property (26) of the \( sinc \) function, we can simplify the left and right elements. % TODO better word than elements - Using a change of variable \( p'' = p_1 - p' \), and properties (27) and (28), we can further evaluate the central element. + Given property (26) of the \( sinc \) function, we can simplify the left and right terms. + Using a change of variable \( p'' = p_1 - p' \), and properties (27) and (28), we can further evaluate the central term. \begin{align} \int _{-\infty} ^{\infty} \Phi (p')^2 dp = \int _{-\infty} ^{\infty} A^2 \left[2 sinc \left( B(2p_1 - p'') \right) sinc \left( B(-p'') \right) \right] dp + A^2 \frac{2}{B} \\ &= \int _{-\infty} ^{\infty} A^2 \left[ 2 sinc \left(B(2p_1 - p'') \right) sinc(B p'') \right] dp + A^2 \frac{2}{B} \\ &= \frac{A^2}{B} \left[ sinc(2p_1) + 2 \right] \\ + &= \frac{1}{2} \left[ sinc(2p_1) + 2 \right] \end{align} \item %3bv @@ -679,36 +678,47 @@ axis([0 1e-9 0 0.04]); \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \item %4a + \begin{minipage}[t]{\linewidth} \begin{figure}[H] \centering - \begin{subfigure}{0.5\textwidth} + \begin{subfigure}[b]{0.5\textwidth} \centering \includegraphics[width=\textwidth]{q4a_e1.png} \caption{} \end{subfigure}% - \begin{subfigure}{0.5\textwidth} + \begin{subfigure}[b]{0.5\textwidth} \centering - \includegraphics[width=\textwidth]{q.png} + \includegraphics[width=\textwidth]{q4a_e2.png} \caption{} \end{subfigure} - \caption{(a),(b) Previous plots but with the classical momentum marked in red.} + \caption{Probability Density Plots for first two energy levels.} \end{figure} + \end{minipage} - % TODO I really don't like this explanation... An \( a \) value of \( 0.53 \angstrom \) was chosen in order to provide an adequately shaped graph without sacrificing too much computation time and ensure that the first two numerical energies correspond to the given experimental results. The experimental results are \(\SI{0.14395}{\electronvolt}\) and \(\SI{0.43185}{\electronvolt} \) for the first and second energy levels - respectively, and the numerical results with our chosen \(a\) are \(\SI{0.0.14386}{\electronvolt}\) and \(\SI{0.43140}{\electronvolt} \) + respectively, and the numerical results with our chosen \(a\) are \(\SI{0.14386}{\electronvolt}\) and \(\SI{0.43140}{\electronvolt} \), + which are in agreement. + - % maybe talk about normalization? \item %4b \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %4bi - \lipsum[1] + + The energies were used were \( 0.14395eV \) and \( 0.43185eV \) for the first and second energy levels, + respectively. + + \begin{minipage}[H]{\linewidth} + \centering + \includegraphics[width=\textwidth]{q4bi_I-V.png} + \captionof{figure}{Current vs. Voltage.} + \end{minipage} + \item %4bii Between \( 0V \) and \( 0.25V \), only the first energy level is carrying any current. @@ -716,14 +726,13 @@ axis([0 1e-9 0 0.04]); energy level drops to 0, meaning no electrons can transfer. Between \(0.4V\) and \(0.65V\), only the second energy level is carrying current. - This energy level stops dropping current because it's shifted energy drops below + This energy level stops conducting current because it's shifted energy drops below the threshold where the contacts have any coupling with it. \item %4biii - + Negative Drain Resistance (NDR) is present in this design from a \(V_D\) of - approximately \(0.27V\) to \(0.45V\), with a mostly linear region from - aroung \(0.28V\) to \(0.3V\). + approximately \(0.27V\) to \(0.45V\), as well as from \(0.65V\) to \(0.8V\) \end{enumerate} diff --git a/PS2/q4b_main.m b/PS2/q4b_main.m index 1557ba3..4d4b04d 100644 --- a/PS2/q4b_main.m +++ b/PS2/q4b_main.m @@ -19,7 +19,7 @@ U0 = 0.025; kBT = 0.025; mu = 0; cal_E1 = 0.2879 / 2; -cal_E2 = 0.2879 * 3 / 2; +cal_E2 = 0.2879 * 3 / 2 - 0.1; % Capacitance parameters diff --git a/PS2/q4bi_I-V.png b/PS2/q4bi_I-V.png index 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