sum more fixes

03_pn_junction_basics.tex

@@ -67,8 +67,12 @@ And by extension
 
 Rearranging the above, we find an expression for the potential:
 \begin{align}
-    \phi & =\frac{kT}{q}\ln\frac{n}{n_i}   \label{label:eq:boltzman:phi_n} \\
-    \phi & = -\frac{kT}{q}\ln\frac{p}{n_i} \label{label:eq:boltzman:phi_p}
+    \phi_n & =\frac{kT}{q}\ln\frac{n}{n_i}   \label{label:eq:boltzman:phi_n} \\
+    \phi_p & = -\frac{kT}{q}\ln\frac{p}{n_i} \label{label:eq:boltzman:phi_p} \\
+    \begin{split}
+        \phi_B & = \phi_n-\phi_p                                               \\
+        & = \frac{kT}{q}  \ln\frac{N_A N_D}{n_i^2}
+    \end{split}   \label{label:eq:boltzman:phi_B}
 \end{align}
 For Si at room temperature this is an increase of 60 mV per decade in doping.
 \begin{equation}

05_pn_junction_bias.tex

@@ -32,17 +32,17 @@ Importantly, the SCR resistance ist the most important one and others can be neg
 \subsection{Space charge region (SCR)}
 In essence, applying a forward/reverse bias effects the depletion region:
 \begin{align}
-    \phi_B            & \rightarrow \phi_B-V_{pn}                                 \\
-    x_n(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_a}{q(N_a+N_d)N_d}}   \\
-    x_p(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_d}{q(N_a+N_d)N_a}}   \\
-    x_d(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_dN_a}{q(N_a+N_d)}}   \\
-    \left|E(V)\right| & =\sqrt{\frac{2q(\phi_B-V)(N_aN_d)}{\varepsilon(N_a+N_d)}}
+    \phi_B            & \rightarrow \phi_B-V_{pn}                                    \\
+    x_n(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_a}{q(N_a+N_d)N_d}}      \\
+    x_p(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_d}{q(N_a+N_d)N_a}}      \\
+    x_d(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)(N_a+N_d)}{q N_d N_a}}    \\
+    \left|E(V)\right| & =\sqrt{\frac{2q(\phi_B-V) N_a N_d}{\varepsilon (N_a + N_d)}}
 \end{align}
 
 In the case of a strongly doped $p^+n$ junction,
 we can approximate the SCR since it exists only in the lesser doped region.
 \begin{equation}
-    x_n(V)=x_{n0}\sqrt{a-\frac{V}{\phi_B}}
+    x_n(V)=x_{n0}\sqrt{1-\frac{V}{\phi_B}}
 \end{equation}
 
 
@@ -55,7 +55,7 @@ In reverse bias, the PN junction acts as a capacitor.
 So as a function of the bias voltage, we get
 \begin{equation}
     \begin{split}
-        C_j(V) &= \frac{\varepsilon}{x_c(V)}\\
+        C_j(V) &= \frac{\varepsilon}{x_d(V)}\\
         &=\sqrt{\frac{q\varepsilon N_aN_d}{2q(\phi_B-V)(N_a+N_d)}}\\
         &=\frac{C_{j0}}{\sqrt{1-\frac{V}{\phi_B}}}
     \end{split}