From 7646bc9d710936530c088ea585cbcc40f72d7d7a Mon Sep 17 00:00:00 2001 From: David Lenfesty Date: Sat, 6 Feb 2021 21:50:07 -0700 Subject: [PATCH] fix q4c image --- PS1/doc.tex | 58 ++++++++++++++++++++++++++++++++++++++++++---------- PS1/q4c.jpg | Bin 524357 -> 353539 bytes 2 files changed, 47 insertions(+), 11 deletions(-) diff --git a/PS1/doc.tex b/PS1/doc.tex index 604b74b..52e4a50 100644 --- a/PS1/doc.tex +++ b/PS1/doc.tex @@ -541,7 +541,7 @@ ylabel('Current [A]'); $\mu_1 = 0$ eV. Since $\mu_1 = \mu - qV_S$ and $\mu = 0$ eV, $V_S = 0$ V. For $f_2(E + U)$ the step point occurs at $E = \mu_2 - U = -0.25$ eV. Since $U = -0.25$ eV, it follows that $\mu_2 = -0.5$ eV. Since $\mu_1 = \mu - qV_D$ and $\mu = 0$ eV, $V_D = 0.5$ V. - + \subsubsection*{(ii)} 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. @@ -549,34 +549,70 @@ ylabel('Current [A]'); This assumes $U = 0$ eV. \subsection*{(c)} - + \subsubsection*{(i)} - \begin{figure} + \begin{figure}[H] \centering - \includegraphics[width=\textwidth]{q4c.jpg} - \caption{} + \includegraphics[width=0.7\textwidth, angle=90, origin=c]{q4c.jpg} + \caption{Visualisation of energy levels.} \label{fig:q4c} \end{figure} \subsubsection*{(ii)} + Starting with equation 2 in the assignment we get: + + \begin{equation*} + I = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot \int_{-0.1}^{0.2} [1 - 0] \cdot 10^4 dE \\ + \end{equation*} + \begin{equation*} + I = \frac{q}{\hbar} \cdot \frac{0.005^2}{0.01} \cdot [10^4 \cdot 0.2 - 10^4 \cdot -0.1] + \end{equation*} + \begin{equation*} + I = 1.8mA + \end{equation*} + + (Note that 1) + + \subsubsection*{(iii)} - + \subsection*{(d)} Using equation (5) in the assignment and plugging in the given values we obtain: \begin{equation*} - U = -q[\frac{C_SV_S+C_GV_G+C_DV_D}{C_E}] + \frac{q^2(N-N_0)}{C_E} - U = \alpha_SV_S + \alpha_GV_G + \alpha_DV_D + U_0(N - N_0)\:\:[eV] - U = 0*0 + 0.5*0\:eV + 0.5*0.6\:eV + 0.25\:eV*(0.325 - 0)\:\:[eV] - U = 0.38125 eV$ $ + \begin{aligned} + U = -q[\frac{C_SV_S+C_GV_G+C_DV_D}{C_E}] + \frac{q^2(N-N_0)}{C_E} \\ + U = -\alpha_SV_S - \alpha_GV_G - \alpha_DV_D + U_0(N - N_0)\:\:[eV] \\ + U = -0*0 - 0.5*0\:eV - 0.5*0.6\:eV + 0.25\:eV*(0.325 - 0)\:\:[eV] \\ + U = -0.21875\:eV + \end{aligned} + \end{equation*} + + Then, starting with Equation \ref{eq:N_new} and plugging in $D(E-U) = \delta(E - \varepsilon)$, we obtain + + \begin{equation*} + N = \frac{\gamma_1f_1(\varepsilon + U) + \gamma_2f_2(\varepsilon + U)}{\gamma_1 + \gamma_2} + \end{equation*} + Recalling that the expressions for the contact Fermi functions are (with $\mu = 0$ eV): + \begin{equation*} + \begin{aligned} + 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 \\ + 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 + \end{aligned} + \end{equation*} + Then, solving for $\gamma_2$: + \begin{equation*} + \begin{aligned} + \gamma_2 = \gamma_2\frac{N - f_1(\varepsilon + U)}{f_2(\varepsilon + U) - N} = 5.45 \times 10^-3 \: eV + \end{aligned} \end{equation*} - Then, starting with Equation \ref{eq:N_old} and plugging in $D(E-U) = \delta(E - U - \varepsilon)$, we obtain + \subsection*{(e)} \subsection*{(f)} \subsection*{(g)} diff --git a/PS1/q4c.jpg b/PS1/q4c.jpg index 1c8718bf0bb3d5d72068db50bb3878e644daa068..cb6290357ce0305e579bd0be3db21f8aaa1baf16 100644 GIT binary patch literal 353539 zcmb@t2UJtb7e0CdgkFMl2oM1Ur6@?18tH;kl-`@d4ZU+gN{~*df)oL1qI57o=sgta zD7|-)j#3o##oOTb*8i=y-dgWuWisoO%*>uW-`=zLcYaO$S^yyTl~k1g5C{N3xDW7a z36DorUfxVoTSG}zUGc9Ac#HtogzJVd^_ql%$3oAG0f9wB;U)=v5 zjs5<202t-}KV1KRJ4Iq`<7S1sX%F{xcf~c1>#Q}dPW$xly7%w8<=^$_-*wl=+VZ%2 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