= Solution
\b[Equilibrium junction.] Put the p-type side on the left and n-type side on the right. When they are joined, <Electrons> diffuse from n to p and <electron holes> diffuse from p to n. This leaves negative ionized acceptors on the p-side and positive ionized donors on the n-side of the <depletion region>. The resulting built-in <electric field> points from n to p, opposing further diffusion. At equilibrium there is one constant <Fermi level>, and drift and <diffusion currents> cancel for each carrier species.
In the nondegenerate, fully ionized limit, the neutral-region <carrier densities> are
$$
\begin{array}{c|cc}
&\text{p side}&\text{n side}\\ \hline
\text{holes }p&N_A\quad\text{majority}&n_i^2/N_D\quad\text{minority}\\
\text{electrons }n&n_i^2/N_A\quad\text{minority}&N_D\quad\text{majority}.
\end{array}
$$
The <conduction band> and <valence band> lie higher on the p-side and bend down towards the n-side. The gap between them stays fixed for a homojunction. If $\psi$ increases towards the n-side, both edges shift by $-e\psi$, while
$$
n=N_Ce^{-(E_C-E_F)/(k_BT)},\qquad p=N_Ve^{-(E_F-E_V)/(k_BT)},\qquad np=n_i^2.
$$
Thus $n$ rises and $p$ falls continuously across the junction; neither mobile population is an exact step function. Their depletion exposes fixed dopant <charge density>, rather than an absence of every kind of charge. The built-in voltage is
$$
\boxed{V_{\rm bi}=\frac{k_BT}{e}\ln\frac{N_AN_D}{n_i^2}.}
$$
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2018-b2-equilibrium-junction.png]
{title=Equilibrium band bending and carrier concentrations across a p-n homojunction}
{description=Original symmetric illustrative model, with the p-region on the left. A constant Fermi level crosses neither band; log-scale carrier profiles satisfy $np=n_i^2$ and exchange majority and minority roles. The profile is schematic rather than a numerical Poisson solution.}
\b[Light emission.] Forward bias lowers the barrier and injects <minority carriers> across the <p-n junction>. In a <light-emitting diode>, <electron-hole recombination> converts some of their energy into <photons>. A <direct band gap> permits efficient radiative transitions with nearly conserved <crystal momentum>; nonradiative channels compete. Near-edge emission has $\hbar\omega\simeq E_g$ and $\lambda\simeq hc/E_g$. Material composition and confinement select the emission colour. Reverse-bias saturation is not the ordinary LED operating mode.
\b[Reverse current.] Take $V=V_R>0$ and $I_R>0$ to mean reverse voltage and reverse current magnitudes. The <Shockley diode equation> in this convention is
$$
I_R=I_S\left(1-e^{-eV_R/(k_BT)}\right).
$$
The bias-independent $I_S$ term is the supply of thermally excited <minority carriers> from the neutral regions, swept across the junction by its <electric field>. The opposing injection current is reduced by the extra barrier energy $eV_R$ and is $I_S e^{-eV_R/(k_BT)}$. Their equality gives zero net current at zero bias; for $eV_R\gg k_BT$, the reverse current approaches $I_S$ until leakage or breakdown invalidates this ideal law. The collecting field need not create the minority carriers; their thermal supply limits the diffusion-dominated current.
For an ideal long junction of area $\mathcal A$, steady diffusion with recombination gives the <diode minority-carrier diffusion current>
$$
\boxed{I_S=e\mathcal A\left(\frac{D_nn_i^2}{L_nN_A}+\frac{D_pn_i^2}{L_pN_D}\right),\qquad L_n=\sqrt{D_n\tau_n},\quad L_p=\sqrt{D_p\tau_p}.}
$$
It depends on doping, geometry, carrier <diffusion coefficients> and lifetimes, and especially the <intrinsic carrier concentration>. <Depletion-region generation current> and surface leakage can add nonideal contributions to the measured reverse current.
\b[Band-gap estimate from the data.] Read a nearly flat region well above the small-bias knee and below the high-bias upturn, rather than estimating $I_S$ in the breakdown region. For a diffusion-dominated junction with nearly temperature-independent gap and effective masses,
$$
n_i^2=N_CN_Ve^{-E_g/(k_BT)},\qquad N_CN_V\propto T^3,
$$
so $I_S=C(T)T^3e^{-E_g/(k_BT)}$, where $C(T)$ includes transport and doping factors. <Extracting a semiconductor band gap from diode saturation current> amounts to measuring the slope of $\ln[I_S/(C(T)T^3)]$ against $1/T$: it is $-E_g/k_B$. If the prefactor is taken approximately constant over a short interval,
$$
E_g\simeq k_B\frac{\ln[I_S(T_h)/I_S(T_l)]}{1/T_l-1/T_h}.
$$
Approximate readings from the original PDF plateaus are
$$
\begin{array}{c|c|c}
T\ ({}^\circ\mathrm C)&T\ (\mathrm K)&I_S\ (\mathrm A)\\\hline
30&303.15&2.5\times10^{-6}\\
0&273.15&1.7\times10^{-7}\\
-45&228.15&\text{roughly }1.2\times10^{-8}.
\end{array}
$$
Using the two warmer curves and $k_B=8.6173\times10^{-5}\,\mathrm{eV/K}$ gives
$$
E_g^{\rm app}=k_B\frac{\ln(2.5\times10^{-6}/1.7\times10^{-7})}{1/273.15-1/303.15}
\simeq0.64\,\mathrm{eV}.
$$
Keeping the $T^3$ factor but taking the residual transport factor constant changes this estimate to approximately $0.57\,\mathrm{eV}$. The justified graphical estimate is therefore
$$
\boxed{E_g\ \text{of order }0.6\,\mathrm{eV}\quad\text{from the two warmer curves}.}
$$
The colder curve is a genuine limitation of the supplied data: the $0$ to $-45\,{}^\circ\mathrm C$ pair gives only about $0.32\,\mathrm{eV}$ with a constant prefactor, and the endpoint pair gives about $0.42\,\mathrm{eV}$. Thus all three cannot support one precise diffusion-only activation slope. A <depletion-region generation current>, approximately proportional to $n_i$ rather than $n_i^2$, can have a roughly half-gap activation energy; other leakage and temperature-dependent transport can also alter the slope. The PDF alone does not identify their relative contributions. A reliable precision gap measurement would use more temperatures and verify the transport regime instead of forcing one fit through these three readings.
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