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\exnumber{7005}
\heading{Shot noise} 
\auname{Tuval Bendosa and Sarit Nagar} 


{\bf The problem:}

\Dn

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}
\Dn

{\bf The solution:}
\Dn

\begin{itemize}
\item [(a)]
During the response time $ \tau $ there is a probability of ${\frac{\tau}{T}}$ that the electron will hit the cathode. The average of $n_{i}$ is then 
 \[\tag{1} \left\langle n_{i}\right\rangle=1\cdot\frac{\tau}{T_e}+0\cdot\left(1-\frac{\tau}{T_e}\right) =\frac{\tau}{T_e}\equiv \eta \]


The current is simply $I=e\frac{n_{i}}{\tau}$ .The average current is $\left\langle I\right\rangle =\frac{e}{\tau}\langle n_{i}\rangle =\frac{e}{\tau}\eta$.

The fluctuations in I
  are:
  \[\tag{2} \langle\left(\Delta I\right)^{2}\rangle =\langle I^{2}\rangle -\left\langle I\right\rangle ^{2}=\frac{e^{2}}{\tau^{2}}\langle n_{i}^{2}\rangle -\frac{e^{2}}{\tau^{2}}\eta^{2}=\frac{e^{2}}{\tau^{2}}\eta\left(1-\eta\right)   \]

 where $\langle n_{i}^{k}\rangle =\left\langle n_{i}\right\rangle =\eta$ for $ k\in\mathbb{N}$.\\
 \\ 
 In the case where $\eta=0$ or $\eta=1$, there are no fluctuations, since the average current equals the minimum and maximum values that can be attained by I.

\item [(b)]
The power spectrum  $\tilde{C}\left(\omega\right)$
 is the Fourier transform of $C\left(t\right)$, where $ t=t_{2}-t_{1}$
  is the correlation time. Since the particles arrive at random times, the pulses in the current are uncorrelated. Therefore, $C\left(t\right)$ is of the form
 \[\tag{3}C\left(t\right)=\left\langle \Delta I\left(t\right)\Delta I\left(0\right)\right\rangle =\langle(\Delta I\right)^{2})\rangle \delta\left(t\right)\] 
 Calculating $\tilde{C}\left(\omega\right)$  we get, 
\[\tag{4}\tilde{C}\left(\omega\right)=\intop_{-\infty}^{\infty}dt\mbox{e}^{i\omega t}\langle (\Delta I\right)^{2})\rangle \delta\left(t\right)=\langle (\Delta I\right)^{2})\rangle =\frac{e^{2}}{\tau^{2}}\eta\left(1-\eta\right)\]
 We get the fluctuations in the frequency domain from "Wiener-Khinchin theorem".
\[\tag{5} \langle | \Delta I_{\omega} |^{2}\rangle =\tilde{C}\left(\omega\right)\times\tau=\frac{e^{2}}{\tau}\eta\left(1-\eta\right)=e\left\langle I\right\rangle \left(1-\eta\right)\]
 
 where $\tau$
  is the averaging time. Also we used $ \left\langle I\right\rangle =e\eta/\tau$
  from previous result.\\ For $\eta\ll1$
  we finally receive  
  \[\tag{6}\langle | \Delta I_{\omega} |^{2}\rangle  =e\left\langle I\right\rangle \]
  
 This is the shot noise. This noise is white since it does not depend on the frequency.
 \\

The Johnson-Nyquist noise is a thermal noise in a circuit, with power spectrum equals to $2T/R$ in equilibrium, where T is the temperature and R is the resistance.  
 In  non-equilibrium conditions we might have the addition of a shot noise to the thermal noise. Decreasing the temperature, the shot noise becomes more important. At sufficiently low temperature the shot noise dominates the Johnson-Nyquist noise, $e\left\langle I\right\rangle >2T/R$, which implies the condition
  \[\tag{7} e\left\langle I\right\rangle R>2T\]

 


\item[(c)] 

The 3rd order commulant is:
 \[\tag{8} \langle (I-\left\langle I\right\rangle )^{3}\rangle =\frac{e^{3}}{\tau^{3}}(\langle n_{i}^{3}\rangle -3 \langle n_{i}^{2}\rangle \left\langle n_{i}\right\rangle +3\left\langle n_{i}\right\rangle \left\langle n_{i}\right\rangle ^{2}-\left\langle n_{i}\right\rangle ^{3}\right)=\frac{e^{3}}{\tau^{3}}\left(\eta-3\eta^{2}+2\eta^{3}\right)   \]
  \[\tag{9} \langle (I-\left\langle I\right\rangle )^{3}\rangle =\frac{e^{3}}{\tau^{3}}\eta\left(1-\eta\right)\left(1-2\eta\right)   \]

\end{itemize}




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