= Solution
\b[Nondegenerate carrier product.] Assume three-dimensional parabolic <energy bands> near their extrema, <spin> degeneracy two, and no additional uncounted valley multiplicity. Near the conduction edge, a Boltzmann approximation to the <Fermi-Dirac distribution> gives
$$
n=2\int\frac{d^3k}{(2\pi)^3}\exp\left[-\frac{E_c+\hbar^2k^2/(2m_e)-\mu}{k_BT}\right].
$$
Angular integration and the Gaussian radial integral yield
$$
\boxed{n=N_c e^{-(E_c-\mu)/(k_BT)},\qquad N_c=2\left(\frac{m_ek_BT}{2\pi\hbar^2}\right)^{3/2}.}
$$
For valence <electron holes>, the empty-state probability is $1-f(E)\simeq e^{(E-\mu)/(k_BT)}$. Measuring <electron hole> <kinetic energy> downwards from $E_v$ similarly gives
$$
\boxed{p=N_v e^{-(\mu-E_v)/(k_BT)},\qquad N_v=2\left(\frac{m_hk_BT}{2\pi\hbar^2}\right)^{3/2}.}
$$
The factors two count <spin> once in each band. Multiplying cancels the <chemical potential> and proves the <semiconductor mass-action law>:
$$
\boxed{np=N_cN_v e^{-E_g/(k_BT)}=\frac12(m_em_h)^{3/2}\left(\frac{k_BT}{\pi\hbar^2}\right)^3e^{-E_g/(k_BT)}=n_i^2.}
$$
In an intrinsic neutral <semiconductor>, $n=p=n_i$. In a doped equilibrium <semiconductor> the same product holds under the stated nondegenerate assumptions, even when $n\ne p$. Extra valleys or several valence branches must be included in the effective density-of-states factors rather than counted inconsistently.
\b[Equilibrium junction.] Put the p region on the left and the n region on the right. Before contact their majority carriers are <electron holes> and <Electrons>, respectively. <Diffusion> initially sends <Electrons> from n to p and <electron holes> from p to n. This leaves negative ionized acceptors on the p side and positive ionized donors on the n side, creating a <depletion region> and an <electric field> directed from n to p. At equilibrium this field balances <diffusion>, so the net <electric current> vanishes.
For a homojunction the gap remains constant while the <conduction band> and <valence band> edges bend downwards from p to n by $eV_{\rm bi}$. The equilibrium <chemical potential> is flat. Since $E_{c,v}(x)=E_{c,v}^{\rm ref}-e\phi(x)$, the <electrostatic potential> rises from p to n. The carrier profiles satisfy
$$
n(x)=N_c e^{-[E_c(x)-\mu]/(k_BT)},\qquad p(x)=N_v e^{-[\mu-E_v(x)]/(k_BT)},\qquad n(x)p(x)=n_i^2.
$$
With fully ionized nondegenerate dopants, the neutral p side has $p_p\simeq N_A$, $n_p\simeq n_i^2/N_A$, while the neutral n side has $n_n\simeq N_D$, $p_n\simeq n_i^2/N_D$. Thus the <Electron> concentration rises across the junction and the <electron hole> concentration falls, with both small relative to the far-side majority densities in much of the depleted region. The ideal <depletion approximation> neglects these mobile populations when calculating the fixed space <electric charge>, but the actual equilibrium profiles are smooth.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2014-junction-equilibrium.png]
{title=Equilibrium band bending, a flat chemical potential and complementary electron/hole profiles across a homojunction}
The <built-in potential> follows from these Boltzmann factors:
$$
\boxed{V_{\rm bi}=\frac{k_BT}{e}\ln\frac{N_AN_D}{n_i^2}.}
$$
There is no applied terminal bias; the internal <electrostatic potential> variation does not contradict a constant electrochemical potential.
\b[Opposing carrier fluxes under <reverse bias>.] Use $V>0$ for the magnitude of <reverse bias> and $I>0$ for reverse <electric current>, as in the PDF. The junction barrier increases by $eV$. The <Shockley diode equation> in that orientation can be written
$$
I=\underbrace{I_{\rm sat}}_{\text{reverse-directed carrier supply}}-\underbrace{I_{\rm sat}e^{-eV/(k_BT)}}_{\text{opposing thermally activated flux}}.
$$
The reverse-directed contribution is supplied by minority <Electrons> on the p side and minority <electron holes> on the n side. They reach the depletion edge by <diffusion> and are swept across by the built-in field. The opposing contribution is associated with majority carriers thermally crossing the barrier in the forward direction; <reverse bias> suppresses it by the <Boltzmann factor>. At zero bias detailed balance makes the two equal, while at large enough <reverse bias> the opposing flux becomes negligible.
In the ideal neutral-region <diffusion> model,
$$
I_{\rm sat}=e\mathcal A n_i^2\left(\frac{D_n}{L_nN_A}+\frac{D_p}{L_pN_D}\right).
$$
Here $D_n,D_p$ are minority-carrier <diffusion> coefficients, $L_n,L_p$ their <diffusion> lengths, and $\mathcal A$ is junction area. The two terms inside this positive prefactor are the <Electron> and <electron hole> contributions; both flow in the same reverse-current direction. This separates the two carrier-species contributions from the two opposing fluxes in the <Shockley diode equation>. Calling the saturation term a <drift current> refers to its passage across the depleted region, and does not eliminate the minority-carrier <diffusion> needed to supply it.
\b[Reading the measured curves.] At $eV\ll k_BT$, the ideal equation gives $I\simeq I_{\rm sat}eV/(k_BT)$, initially proportional to reverse <electric potential difference>. At $eV\gg k_BT$, it approaches a plateau $I\simeq I_{\rm sat}$. The thermal <electric potential differences> at the three temperatures are about $21$–$26\,\mathrm{mV}$, consistent with a rise at small reverse biases and near saturation at larger ones. Higher <temperature> produces a much larger plateau because thermal electron-hole excitation raises $n_i^2$ exponentially.
The lower-temperature curve has a noticeable residual <electric potential difference> dependence rather than a perfectly flat plateau. Depletion-region generation, surface leakage or other departures from the ideal <diffusion> model can contribute. The sharp upturn at the largest reverse <electric potential differences> is reverse breakdown, outside the unextended <Shockley diode equation>; the plot alone does not identify a unique microscopic breakdown mechanism. The original horizontal axis duplicates the final decade label, so a precise breakdown <electric potential difference> cannot be read reliably. Neither issue prevents estimating the gap from the intermediate-bias saturation region.
\b[Band-gap estimate.] Use plateau <electric currents> at a common intermediate <reverse bias>, away from both the small-bias rise and the breakdown upturn. Approximate readings are $I_{\rm sat}(298\,\mathrm K)\sim2.5\times10^{-6}\,\mathrm A$, $I_{\rm sat}(273\,\mathrm K)\sim1.5$–$2\times10^{-7}\,\mathrm A$, and $I_{\rm sat}(248\,\mathrm K)\sim10^{-8}\,\mathrm A$. Kelvin temperatures are required in the activation law.
For fixed effective masses, $n_i^2\propto T^3e^{-E_g/(k_BT)}$. If the remaining transport prefactor is approximately constant, <extracting a semiconductor band gap from diode saturation current> therefore uses
$$
\ln\frac{I_{\rm sat}}{T^3}=\text{constant}-\frac{E_g}{k_BT},\qquad E_g=\frac{k_B\left[\ln(I_h/I_l)-3\ln(T_h/T_l)\right]}{1/T_l-1/T_h}.
$$
Using the widest <temperature> separation and the rounded readings gives
$$
E_g\simeq\frac{(8.6173\times10^{-5}\,\mathrm{eV\,K^{-1}})\left[\ln(250)-3\ln(298/248)\right]}{1/248-1/298}\simeq0.63\,\mathrm{eV}.
$$
Neglecting all algebraic prefactors and fitting $\ln I_{\rm sat}$ instead gives about $0.70\,\mathrm{eV}$. These differ by roughly $0.07\,\mathrm{eV}$, a material correction on the graph's precision scale. Thus an honest graph-based conclusion is
$$
\boxed{E_g\sim0.6\text{–}0.7\,\mathrm{eV}\quad\text{for germanium, approximately }0.65\,\mathrm{eV}.}
$$
The nonflat low-temperature plateau, rough <electric current> readings, unprovided <temperature> dependence of $D/L$, and possible gap variation prevent a more precise extraction. If depletion-region generation dominated rather than the specified $I_{\rm sat}\propto n_i^2$ mechanism, its activation slope could involve a different <energy> scale; the displayed estimate explicitly uses the diffusion-dominated saturation model requested here.
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