Merge branch '04-addon' into 'main'

04 pn junction update

See merge request !3

04_pn_junction.tex

@@ -13,7 +13,7 @@ resulting in a new equilibrium (\autoref{label:fig:pn_carrier_profile_equilibriu
     \hfill
     \begin{subfigure}[b]{.45\textwidth}
         \includegraphics[width=\textwidth]{imgs/pn_fermi_level_band_bending.png}
-        \caption{Resulting carrier profile in thermal equilibrium}
+        \caption{Energy band diagrams of pn-junction in thermal equilibrium}
         \label{label:fig:pn_fermi_level_band_bending}
     \end{subfigure}
 \end{figure}
@@ -27,7 +27,7 @@ the energy levels for conduction and valence bands bend, whereas the fermi level
 We assume p and n regions quasi-neutral,
 and the intermediate space charge region to be completely depleted of carriers.
 We further assume all transitions are expressed as step-functions.
-This allows the following simplified equations:
+This allows the following simplified equations: (depicted in \autoref{label:fig:space-charge-region})
 \begin{align}
     \rho(x) & = \begin{dcases}
                     0     & x<-x_p   \\
@@ -62,6 +62,21 @@ Where $\phi_B$ is the built-in potential (see \autoref{label:eq:boltzman:phi_B})
 Also, it is the less heavily doped region that defines the junction width.
 It is also in the less heavily doped region that the depletion zone extends farther.
 
+Therefore we get the total width of space charge region:
+\begin{equation}
+    W_{dep} = x_{d} = x_n + x_p = \sqrt{\frac{2  \varepsilon \phi_B (N_a+N_d)}{q N_a N_d }}
+\end{equation}
+and the maximal electric field: (at metallurgical junction)
+\begin{equation}
+    \left|E_0\right| = \sqrt{\frac{2 q \phi_B N_a N_d}{\varepsilon(N_a+N_d)}}
+\end{equation}
+
+\begin{figure}[H]
+    \centering
+    \includegraphics[width=.4\textwidth]{imgs/space_charge_region.pdf}
+    \caption{Charge density, electric field and the potential in the space charge region.}\label{label:fig:space-charge-region}
+\end{figure}
+
 \subsection{Contact potential}
 Although there is a potential accross the diode, it cannot be measured because there are the metal semi-conductor junctions for both p and n regions.
 \begin{equation}