5.3: FET Calculations
- Page ID
- 51320
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Unlike the two terminal case, where we arbitrarily set \(E_{F} = 0\) and shifted the Source and Drain potentials under bias, the FET convention fixes the Source electrode at ground. There are two voltage sources: \(V_{GS}\), the gate potential, and \(V_{DS}\), the drain potential. We analyze the influence of \(V_{GS}\) and \(V_{DS}\) on the molecular potential using capacitive dividers and superposition:
\[ U_{E S}=-q V_{G S} \frac{1 /\left(C_{D}+C_{S}\right)}{1 /\left(C_{D}+C_{S}\right)+1 / C_{G}}-q V_{D S} \frac{1 /\left(C_{G}+C_{S}\right)}{1 /\left(C_{G}+C_{S}\right)+1 / C_{D}} \nonumber \]
Simplifying, and noting that the total capacitance at the molecule is \(C_{ES} = C_{S} + C_{D} + C_{G}\):
\[ U_{ES} = -qV_{GS}\frac{C_{G}}{C_{ES}}-qV_{DS} \frac{C_{D}}{C_{ES}} \nonumber \]
We also must consider charging. As before,
\[ U_{C} = -\frac{q^{2}}{C_{ES}}(N-N_{0}) \nonumber \]
Recall that charging opposes shifts in the potential due to \(V_{GS}\) or \(V_{DS}\). Thus, if charging is significant, the switching voltage increases; see Figure 5.3.2.
Adding the static potential due to \(V_{DS}\) and \(V_{GS}\) gives, the potential U in terms of the charge, N, and bias.
\[ U=-qV_{GS} \frac{C_{G}}{C_{ES}}-qV_{DS}\frac{C_{D}}{C_{ES}}+\frac{q^{2}}{C_{ES}}(N-N_{0}) \nonumber \]
We also have an expression for potential for N in terms of U (see Equation (3.7.8))
\[ N = \int^{\infty}_{-\infty} g(E-U) \frac{\tau_{D}f(E,\mu_{S})+\tau_{S}f(E, \mu_{D})}{\tau_{S}+\tau_{D}} dE \nonumber \]
As before, Equations (5.3.4) and (5.3.5) must typically be solved iteratively to obtain U. Then we can solve for the current using:
\[ I = q\int^{\infty}_{-\infty} g(E-U)\frac{1}{\tau_{S}+\tau_{D}} \left(f(E,\mu_{S})-f(E,\mu_{D})\right)dE \nonumber \]