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\begin{document}

\begin{center}
\textbf{Physics 120  \smallskip - \smallskip David Kleinfeld \smallskip - \smallskip Spring 2017 }
\end{center}

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\section{Notes on diode analysis}
\medskip
\subsection{Background}
Mass action says that it is more probable for charge to move from a region with high charge carrier density to
one with lower density. This gives rise to the notion of the diode, where current can flow more strongly in
one direction than the other.  The semiconductor diode is built by taking two blocks of pure semiconductor, like Si,
and doping one block so that it has a net excess of negative carriers (n-side) and doping the other block
so that it has a net excess of positive carriers (p-side) and then bringing the blocks in contact (Fig.)

Let $N_n$ be the number of negative change carriers (electrons) and $N_p$ be the number of
positive charge carriers (holes). Right at the junction, the extra holes in the p-side will be
swept toward the n-side and the extra electrons on the n-side will be swept onto the p-side.
This results in a steady-state situation with a junction that has no free charge and a smooth transition in potential
from one side to the other.  The total potential drop, $\phi$, can be expressed in terms of carrier density
\begin{equation}
\frac{N_p (n-side)}{N_p (p-side)}\;  =  \; e^{-e\phi/k_BT}.
\end{equation}
Now the current of holes that flows is
\begin{eqnarray}
\rm{Current} &\propto& {\rm Hole \; from \, (p-side) \, moving \, to \, (n-side)} \; \; - \; \; {\rm Hole \, from \; (n-side) \,moving \, to \, (p-side)}  \nonumber \\
                    &\propto& N_p {\rm(p-side)}\; e^{-e\phi/k_BT} \; \; - \; \; N_p {\rm (n-side)} \nonumber \\
                    &=&  0 .
\end{eqnarray}

\begin{figure}
  \includegraphics[width=10cm]{JPG/Diode_notes.jpg}
 \centering
\end{figure}

Suppose that we now apply a potential, denoted $V_D$, to the n-side that lowers the potential; this is called the forward-biased condition as the applied potential
is higher on the p-side than the n-side. The current is now
\begin{eqnarray}
\rm{Current}  &\propto& N_p {\rm(p-side)}\; e^{-e(\phi-V_D)/k_BT} \; - \; N_p {\rm(n-side)} \nonumber \\
                    &\propto& N_p {\rm(p-side)}\; e^{-e(\phi-V_D)/k_BT} \; - \; N_p {\rm(p-side)}\; e^{-e\phi/k_BT} \nonumber \\
                    &\propto&  \left[N_p {\rm(p-side)}\; e^{-e\phi/k_BT} \right] \left[e^{eV_D/k_BT} \; - \; 1\right] .
\end{eqnarray}

We can repeat this argument in terms of electrons
\begin{equation}
\frac{N_n (p-side)}{N_n (n-side)} \; =  \; e^{-e\phi/k_BT}.
\end{equation}
The current of electrons that flows with $V_D$ > 0 is again
\begin{eqnarray}
\rm{Current} &=& {\rm Electron \; from \, (n-side) \, moving \, to \, (p-side)} \; \; - \; \; {\rm Electron \, from \; (p-side) \,moving \, to \, (n-side)}  \nonumber \\
                    &\propto&  N_n {\rm(n-side)}\; e^{-(\phi-V_D)/k_BT} \; \; - \; \;  N_n {\rm (p-side)} \nonumber \\
                    &\propto& N_n {\rm(n-side)}\; e^{-(\phi-V_D)/k_BT} \; \; - \; \;  N_n {\rm (n-side)} \; e^{-e\phi/k_BT}\nonumber \\
                    &\propto&  \left[N_n {\rm(n-side)}\; e^{-e\phi/k_BT} \right] \left[e^{eV_D/k_BT} \; - \; 1\right] .
\end{eqnarray}

We can add the two currents up to get
\begin{eqnarray}
I  &\propto& \left[N_p {\rm(p-side)}\; e^{-e\phi/k_BT} + N_n {\rm(n-side)}\; e^{-e\phi/k_BT} \right]  \left[e^{eV_D/k_BT} \; - \; 1\right] \nonumber \\
&\propto& \left[\left(N_p {\rm(p-side)}\; + N_n {\rm(n-side)} \right) e^{-e\phi/k_BT}\right]  \left[e^{eV_D/k_BT} \; - \; 1\right] \nonumber \\
   &=& I_0  \left[e^{eV_D/k_BT} \; - \; 1\right].
\end{eqnarray}

\bigskip

The product $N_n N_p$ can be estimated as follows. In
equilibrium, the numbers of holes and electrons are constant. There is a chance for an electron to leave the valence band and
join the conduction band, thus forming an electron-hole pair.  Thus the production of electron-hole pairs is given by the Boltzmann
factor $exp( -E_{gap}/k_BT )$ where $E_{gap}$ is the gap energy, about 1.1 eV for silicon. These charge pairs will not accumulate, but reach a steady
state through the recombination of electron and hole pairs.  The probability of decay depends on the product $N_n N_p$,
\begin{equation}
N_p \; N_n \approx e^{-E_{gap}/k_BT}.
\end{equation}

\medskip
The next order to business to to use the diode in a circuit. Let's see how we handle the nonlinearity!



\end{document}
