7.8: So how can we approach the subthreshold limit?
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It is thought that if all the charge in the channel behaves collectively, (i.e. all or none of the charge contributes to current) then it might be possible to switch closer to the limit. Perhaps the best examples of this principle are the voltage-dependent ion channels of biology, in which conformation changes may enable subthreshold slopes as sharp ≈ 10 mV/decade.\(^{†}\)
Below, we show the structure and mechanism of the mechanical change in a voltage dependent \(\ce{K^{+}}\) ion channel, as determined by MacKinnon, et al.\(^{\S}\) The channels sit in a membrane; when open they allow the diffusion of ions from one side of the membrane to the other.
Consider a membrane where there are N closed channels and N* open channels. The ratio of open to closed channels is determined by the Boltzmann relation:
\[ \frac{N^{*}}{N} = \exp \left[ -\frac{U_{open}-U_{closed}}{kT} \right] \label{7.8.1} \]
where \(U_{open}\) and \(U_{closed}\) are the energies of the open and closed conformations respectively. Under an electric field, we assume that Z charges move through a potential of \(\Delta V\), i.e.:
\[ U_{open} = U_{closed} - Zq \Delta V \label{7.8.2}. \]
The current through the ion channel is proportional to the number of open channels, N*.
\[ I \propto N^{*} \label{7.8.3} \]
Since N + N* is a constant
\[ I \propto \frac{N^{*}}{N+N^{*}} \approx \frac{N^{*}}{N} = \exp \left[ \frac{Zq \Delta V}{kT} \right] \label{7.8.4} \]
That is, the subthreshold slope is sharpened by a factor, Z, the effective\(^{†}\) number of charges on the movable paddles.
\[ \frac{\Delta V}{\log_{10} I} = \frac{kT}{Ze} \frac{1}{\log_{10}e} \approx \frac{60}{Z}\text{ mV/decade} \label{7.8.5}. \]
The conclusion is that transistors are possible with subthreshold characteristics superior to those of conventional FETs. The ion channel shows that mechanically-coupling the charges together is one path to achieving the collective behavior that we desire. But the reliability of mechanical devices is questionable. Instead, it is possible that another collective phenomenon, like the switching of a magnetic domain in a ferromagnet, may be exploited to improve switching.
\(^{†}\)Hodgkin and Huxley, J. Physiol. 116, 449 (1952a)
\(\S\)Y. Jiang, A. Lee, J. Chen, V. Ruta, M. Cadene, B.T. Chait and R. MacKinnon. Nature. 423. 33-41 (2003)
\(^{†}\)Note that the effective number of charges is usually less than the actual number of charges on the movable structures in the ion channel because the charges are not usually free to move through the full potential DV across the membrane (the motion of the paddles is somewhat restricted).