PX262 - L3 - p-n junction

it occurs at the boundary of one region doped with p-type impurities and another with n-type impurities
the current flow is given by:

I = i_{s} (\exp (\frac{e V}{k_{B} T}) - 1)

where, $i_{s}$ is the saturation current

in equilibrium, there is no net current through the junction, but carriers cross back and forth
when first connected, the many holes in the p-region diffuse across the junction, into the n-region, where they can recombine with some of the many free electrons
similarly, electrons diffuse from the n to the p-region
this sets up an electric field, ${\vec{ε}}_{e l e c}$
the potential energy associated with this field raises the energy levels in the p-region relative to those in the n-region so that the chemical potentials align
the upward shift (AB) $≃ Δ$

in equilibrium, there are four currents across the junction
the diffusion processes lead to a recombination of holes and electrons: $i_{p r}$ and $i_{n r}$
at the same time, electron-hole pairs are generated by thermal excitation: $i_{p g}$ and $i_{n g}$
the electric field, ${\vec{ε}}_{e l e c}$ sweeps them out of the junction
at equilibrium, these currents balance:

\begin{matrix} i_{p r} + i_{p g} = 0 \\ i_{n r} + i_{p r} = 0 \end{matrix}

note:
- $i_{n g} \propto \exp (- Δ / k_{B} T)$ , ie. the probability of exciting a minority carrier in the p-region ( $e^{-}$ )
- $i_{n r} \propto \exp (- Δ_{A B} / k_{B} T)$ , ie. the probability of a majority carrier flowing from the n-region (hole), or, the probability that a majority carrier has enough energy to surmount the barrier, $Δ_{A B}$ , where $Δ ≃ Δ_{A B}$
applying a forward bias (positive potential difference) across the junction
the electric field, ${\vec{ε}}_{e l e c}$ , and the difference between the energy levels in the p-region and the n-region are reduced by: $Δ E = - e V_{a p p l i e d}$

⟹ Δ_{A B} = Δ - e V_{a p p l i e d}

it becomes easier for the electrons in the n-type region to climb the potential barrier and diffuse into the p-region, and for the holes in the p-region to diffuse into the n-region
both recombination currents $i_{n r}$ and $i_{p r}$ are increased by: $\exp (e V_{a p p l i e d} / k_{B} T)$
the generation currents, $i_{n g}$ and $i_{p g}$ , do not change
the net hole current:

i_{p, t o t} = i_{p r} + i_{p g} = | i_{p g} | (\exp (\frac{e V_{a p p l i e d}}{k_{B} T}) - 1)

i_{n, t o t} = i_{n r} + i_{n g} = | i_{n g} | (\exp (\frac{e V_{a p p l i e d}}{k_{B} T}) - 1)

I = i_{s} (\exp (\frac{e V_{a p p l i e d}}{k_{B} T}) - 1)

where, $i_{s} = | i_{p g} | + | i_{n g} |$

therefore, this diode effectively allows current in the forward direction only
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