
\sect{Shot noise}

%based on Baruch's  D26.

The discreteness of the electron charge ${e}$ implies that the
current is not uniform in time and is a source of noise. Consider
a vacuum tube in which electrons are emitted from the negative
electrode and flow to the positive electrode; the probability of
emitting any one electron is independent of when other electrons
are emitted. Suppose that the current meter has a response time
$\tau$. If $T_e$ is the average time between the emission of two electrons, then the average current
is  $\langle I\rangle=e/T_e=\frac{e}{\tau}\eta$, where $\eta=\tau/T_e$ is the transmission probability,
$0\leq \eta\leq 1$.
\Dn


\begin{itemize}
\item [(a)]
Show that the fluctuations in $I$ are $\langle\left(\Delta I\right)^{2}\rangle =\frac{e^2}{\tau^2}\eta(1-\eta)$. Why would you expect the fluctuations
to vanish at both $\eta=0$ and $\eta=1$?
[Hint: For each $\tau$ interval ${n_{i}}$ is the number of electrons hitting the positive electrode. Therefore, it can be equal to ${n_{i}=0}$ or ${n_{i}=1}$ which results in an average
$\langle n_i\rangle=\tau/T_e$; discretize time in units of $\tau$.]

\item [(b)]
Consider the meter response to be in the range 
${0<|\omega |<2\pi/\tau }$. Show that for $\eta\ll 1$ the fluctuations in the frequency domain are $\langle\left(\Delta I\right)^{2}\rangle = e\langle I\rangle$ . What is the condition for this noise to dominate over the
Johnson-Nyquist noise in the circuit?

\item[(c)] Show that the 3rd order commulant is
$\langle (I-\langle I\rangle)^3\rangle=\frac{e^3}{\tau^3}\eta(1-\eta)(1-2\eta)$.
\end{itemize}

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