add depletion region width and metal junction potential

04_pn_junction.tex

@@ -30,17 +30,40 @@ We further assume all transitions are expressed as step-functions.
 This allows the following simplified equations:
 \begin{align}
     \rho(x) & = \begin{dcases}
-        0     & x<-x_p   \\
-        -qN_a & -x_p<x<0 \\
-        qN_d  & 0<x<x_n  \\
-        0     & x_n<x
-    \end{dcases} \\
+                    0     & x<-x_p   \\
+                    -qN_a & -x_p<x<0 \\
+                    qN_d  & 0<x<x_n  \\
+                    0     & x_n<x
+                \end{dcases}                            \\
     E(x)    & =\begin{dcases}
-        0                                & x<-x_p    \\
-        -\frac{qN_a}{\varepsilon}(x+x_p) & - x_p<x<0 \\
-        \frac{qN_d}{\varepsilon}(x-x_n)  & 0<x<x_n   \\
-        0                                & x_n<x
-    \end{dcases}
+                   0                                & x<-x_p    \\
+                   -\frac{qN_a}{\varepsilon}(x+x_p) & - x_p<x<0 \\
+                   \frac{qN_d}{\varepsilon}(x-x_n)  & 0<x<x_n   \\
+                   0                                & x_n<x
+               \end{dcases}
 \end{align}
 Where $E$ is found using \eqref{label:eq:def_electric_field_integral}.
 
+\subsection{Electrostatic potential - (Width of the depletion zone)}
+Because of overall charge neutrality
+\begin{equation}
+    qN_ax_p = qN_cx_n
+\end{equation}
+and continuity of the potential at the junction interface
+\begin{equation}
+    \phi_p+\frac{qN_a}{2\varepsilon}x_p^2 = \phi_n-\frac{qN_d}{2\varepsilon}x_n^2
+\end{equation}
+we can find $x_n$ and $x_p$:
+\begin{align}
+    x_n & = \sqrt{\frac{2\varepsilon\phi_BN_a}{qN_d(N_a+N_d)}} \\
+    x_p & = \sqrt{\frac{2\varepsilon\phi_BN_d}{qN_a(N_a+N_d)}}
+\end{align}
+Where $\phi_B$ is the built-in potential, which is the potential over the junction.
+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.
+
+\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}
+    \phi_B = \phi_mn+\phi_mp
+\end{equation}
\ No newline at end of file