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}
\subsection{Behaviour in thermal equilibrium}
\subsubsection{Thermal equilibrium}
\begin{align}
\begin{align}
\lambda&\equiv\text{mean free path} [cm] \\
\lambda&\equiv\text{mean free path} [cm] \\
\tau_c &\equiv\text{mean time between collisions} [s^{-1}] \\
\tau_c &\equiv\text{mean time between collisions} [s^{-1}] \\
...
@@ -39,7 +36,7 @@ Looking at the equations in \autoref{label:sec:fermi} we see that the higher the
...
@@ -39,7 +36,7 @@ Looking at the equations in \autoref{label:sec:fermi} we see that the higher the
\end{align}
\end{align}
\subsubsection{Drift velocity}
\subsection{Drift velocity}
Quick electromag recap: (for holes use + and $m_p$)
Quick electromag recap: (for holes use + and $m_p$)
\begin{align}
\begin{align}
F & = -qE \\
F & = -qE \\
...
@@ -51,7 +48,7 @@ Average drift velocity:
...
@@ -51,7 +48,7 @@ Average drift velocity:
v_d = \pm\frac{qE\tau_c}{2m_{n,p}}
v_d = \pm\frac{qE\tau_c}{2m_{n,p}}
\end{equation}
\end{equation}
\subsubsection{Mobility}
\subsection{Mobility}
For the sake of simplicity, let's define mobility for both holes and electrons.
For the sake of simplicity, let's define mobility for both holes and electrons.
(These values are usually found in diagrams.)
(These values are usually found in diagrams.)
\begin{align}
\begin{align}
...
@@ -61,7 +58,7 @@ For the sake of simplicity, let's define mobility for both holes and electrons.
...
@@ -61,7 +58,7 @@ For the sake of simplicity, let's define mobility for both holes and electrons.
\mu_n & >_mu_p
\mu_n & >_mu_p
\end{align}
\end{align}
\subsubsection{Drift current}
\subsection{Drift current}
For the net drift current density slap together velocity, density and charge.
For the net drift current density slap together velocity, density and charge.
\begin{equation}
\begin{equation}
label{eq:drift_current}
label{eq:drift_current}
...
@@ -79,7 +76,7 @@ Which gives us different resistances for n and p type semiconductors.
...
@@ -79,7 +76,7 @@ Which gives us different resistances for n and p type semiconductors.
\end{align}
\end{align}
\subsubsection{Diffusion current}
\subsection{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.
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}
\begin{align}
F_n & = -D_n\frac{\mathrm{d} n}{\mathrm{d} x}\\
F_n & = -D_n\frac{\mathrm{d} n}{\mathrm{d} x}\\
...
@@ -94,12 +91,12 @@ Which gives us the diffusion current density:
...
@@ -94,12 +91,12 @@ Which gives us the diffusion current density:
\end{align}
\end{align}
\subsubsection{Einstein relation between mobility and diffusion coefficient}
\subsection{Einstein relation between mobility and diffusion coefficient}