= Solution
An ideal diffusion-dominated <p-n junction> obeys the <Shockley diode equation>
$$
\boxed{I(V,T)=I_s(T)\left[e^{eV/(\eta k_BT)}-1\right],}
$$
with <diode ideality factor> $\eta=1$ in the simplest model. Forward bias gives exponential current growth; reverse bias gives a nearly constant $-I_s$ until <diode reverse breakdown>, outside this ideal expression.
For a long abrupt <p-n junction> in low <carrier injection>, the <diode saturation current> is
$$
\boxed{I_s=e\mathcal A n_i^2\left(\frac{D_p}{L_pN_D}+\frac{D_n}{L_nN_A}\right)=C(T)n_i^2.}
$$
Here $\mathcal A$ is area, $N_D,N_A$ are ionized dopant densities, and $D,L$ are minority-carrier diffusion coefficients and diffusion lengths in the appropriate regions. At each temperature, extract $I_s$ either from the reverse-saturation plateau below breakdown or from a forward exponential fit using its measured <diode ideality factor>.
<Extracting a semiconductor band gap from diode saturation current> requires correcting for the transport prefactor:
$$
\boxed{\log\frac{I_s(T)}{C(T)T^3}=\text{constant}-\frac{E_g}{k_BT}.}
$$
A plot against $1/T$ has slope $-E_g/k_B$ when the gap and effective masses are nearly temperature-independent. Alternatively divide the measured $I_s$ by a known $C(T)$ to recover the full $n_i^2(T)$. The prefactor cannot generally be assumed constant over an arbitrarily broad range. Junction-generation or recombination-dominated current can scale as $n_i$ instead, so the diffusion regime should be identified from the current-voltage behaviour.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2022-q6-junction-and-activation.png]
{title=Ideal diode characteristic and extraction of a semiconductor activation energy}
{description=Original model sketches. The normalized current-voltage curve shows forward exponential growth and reverse saturation. The activation panel plots a model proportional to $n_i^2/T^3$ for a fixed illustrative gap; it shows the slope to infer after correcting transport prefactors, not measured device data.}
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