add chap2

02_carrier_transport.tex

@@ -52,7 +52,56 @@ Average drift velocity:
 \end{equation}
 
 \subsubsection{Mobility}
+For the sake of simplicity, let's define mobility for both holes and electrons.
+(These values are usually found in diagrams.)
 \begin{align}
     \mu_{n,p} & = \frac{q\tau_c}{2m_{n,p}} \equiv \text{mobility}\ [cm^2/Vs] \\
-    amhere
-\end{align}
\ No newline at end of file
+    v_{dn}    & =-\mu_nE                                                     \\
+    v_{dp}    & = \mu_pE                                                     \\
+    \mu_n     & >_mu_p
+\end{align}
+
+\subsubsection{Drift current}
+For the net drift current density slap together velocity, density and charge.
+\begin{equation}
+    label{eq:drift_current}
+    J^{drift} = J_n^{drift}+J_p^{drift} = q(n\mu_n+p\mu_p)E
+\end{equation}
+From which we can  find Ohm's law:
+\begin{alignat}{2}
+    J    & =\sigma E         &  & = \frac{E}{\rho}                         \\
+    \rho & =\frac{1}{\sigma} &  & = \frac{1}{q \left(n\mu_n+p\mu_p\right)}
+\end{alignat}
+Which gives us different resistances for n and p type semiconductors.
+\begin{align}
+    \rho_n & \approx \frac{1}{qN_d\mu_n} \\
+    \rho_p & \approx \frac{1}{qN_a\mu_p}
+\end{align}
+
+
+\subsubsection{Diffusion current}
+If there is a concentration gradient, the carriers will diffuse to equalize the concentration. Here flux $F \ [cm^{-2}s^{-1}]$ is the number of electrons/holes per unit area per unit time.
+\begin{align}
+    F_n & = -D_n\frac{\mathrm{d} n}{\mathrm{d} x} \\
+    F_p & = -D_p\frac{\mathrm{d} p}{\mathrm{d} x}
+\end{align}
+
+Which gives us the diffusion current density:
+(Defined as density times charge, ergo the double negative for electron diffusion.)
+\begin{align}
+    J_n^{diff} & = qD_n\frac{\mathrm{d} n}{\mathrm{d} x}  \\
+    J_p^{diff} & =- qD_p\frac{\mathrm{d} p}{\mathrm{d} x}
+\end{align}
+
+
+\subsubsection{Einstein relation between mobility and diffusion coefficient}
+\begin{equation}
+    \frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q^2}
+\end{equation}
+
+\subsubsection{Total current}
+\begin{alignat}{2}
+    J_{total} & =J_n+J_p                &  &                                                 \\
+    J_n       & =J_n^{drift}+J_n^{diff} &  & =qn\mu_nE+qD_n\frac{\mathrm{d} n}{\mathrm{d} x} \\
+    J_p       & =J_p^{drift}+J_p^{diff} &  & =qp\mu_pE-qD_p\frac{\mathrm{d} p}{\mathrm{d} x}
+\end{alignat}
\ No newline at end of file

semiconductor_summary.tex

@@ -14,7 +14,7 @@
 
     This summary for
     \href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{Micro and nanoelectronic devices}
-    © 2021 by
+    © 2023 by
     \href{https://gitlab.epfl.ch/sthuer}{Simon Thür}
     is licensed under
     \href{http://creativecommons.org/licenses/by/4.0/}{CC BY 4.0}.