\section{Carrier transport}
\subsection{Fermi distribution}
\label{label:sec:fermi}
Fermions are weird particles, see QM II.
Not sure if needed for this course, but heres the probability distribution:
\begin{equation}
    f(E) = \frac{1}{1+e^{(E-E_F)/kT}}
\end{equation}

Electron concentration in conductance band:
\begin{equation}
    n=N_ce^{-(E_c-E_f)/kT}
\end{equation}
Hole concentration in valence band:
\begin{equation}
    p=N_ve^{-(E_f-E_v)/kT}
\end{equation}

In intrinsic silicon ($n=p=n_i$) we have ($E_i$ somewhere in the middle of the bandgap)
\begin{equation}
    E_i=E_f=\frac{E_c+E_v}{2}-\frac{kT}{q}\ln{\frac{N_c}{N_v}}
\end{equation}

Which gives us the useful relation:
\begin{equation}
    n_i = \sqrt{N_cN_v}e^{-(E_c-E_v)/2kT} = \sqrt{N_cN_v}e^{-E_g/2kT}
\end{equation}

\subsubsection{Temperature dependance}
Looking at the equations in \autoref{label:sec:fermi} we see that the higher the temperature or the lower the band-gap, the more electrons and holes are created.

\subsection{Carrier transport}
\subsubsection{Thermal equilibrium}
\begin{align}
    \lambda & \equiv \text{mean free path} [cm]                   \\
    \tau_c  & \equiv \text{mean time between collisions} [s^{-1}] \\
    v_{th}  & \equiv \text{thermal velocity} [cm/s]               \\[1em]
    \lambda & = v_{th}\cdot\tau_c
\end{align}


\subsubsection{Drift velocity}
Quick electromag recap: (for holes use + and $m_p$)
\begin{align}
    F    & = -qE             \\
    v(t) & =-\frac{qE}{m_n}t
\end{align}

Average drift velocity:
\begin{equation}
    v_d = \pm \frac{qE\tau_c}{2m_{n,p}}
\end{equation}

\subsubsection{Mobility}
\begin{align}
    \mu_{n,p} & = \frac{q\tau_c}{2m_{n,p}} \equiv \text{mobility}\ [cm^2/Vs] \\
    amhere
\end{align}