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The plateau in both source plots is approximately $10^{18.5}{\mathrm{m}}^{-3}$, soThese are graphical estimates, not high-precision fitted material parameters. To extract activation energies consistently with part (a), retain the temperature powers in the effective density of states factors.

On the semiconductor carrier freeze-out tail, put $x=1/T$. ThenFor example, the right-hand graph gives roughly ${\log }_{10}[n/({\mathrm{m}}^{-3})]\simeq 17.7$ at $x=0.10{\mathrm{K}}^{-1}$ and $13.0$ at $x=0.30{\mathrm{K}}^{-1}$. Subtract the known prefactor contribution before interpreting the slope:Thus $\boxed{E_i\simeq 9\mathrm{meV}}$, to the accuracy of the drawing. Ignoring the weak prefactor gives about $9.3\mathrm{meV}$, a similar rough estimate.

For the intrinsic branch, useThe left-hand graph gives $n(100\mathrm{K})\simeq 2.5\times 10^{21}{\mathrm{m}}^{-3}$ and $n(500\mathrm{K})\simeq 1.5\times 10^{24}{\mathrm{m}}^{-3}$, both far above $N_D$. ThereforeHere $k_B=8.6173\times 10^{-5}\mathrm{eV}{\mathrm{K}}^{-1}$. Omitting the $T^{3/2}$ prefactor over this wide temperature interval would instead give approximately $0.14\mathrm{eV}$, so it is not a negligible correction for the band-gap extraction from intrinsic carrier activation. The source graph warrants an approximate narrow band gap, rather than several significant figures.

Band edges and optical transitions. A direct band gap has the conduction minimum and valence maximum at the same crystal momentum. A near-edge photon then connects the two almost vertically on an energy-versus-wavevector plot, because its momentum is small on the Brillouin zone scale. An indirect band gap has the extrema at different momenta, so an additional momentum transfer, commonly supplied by phonon absorption or emission, is required for a near-edge radiative transition. This reduces its probability relative to an allowed direct transition.

Near the zone centre, an isotropic direct-gap model has $E_c(k)=E_c+{\hslash }^2{k}^2/(2m_e^{\ast })$ and downward valence dispersions $E_{v,j}(k)=E_v-{\hslash }^2{k}^2/(2m_{h,j}^{\ast })$. The heavy-hole and light-hole bands are the upper branches with large and small positive hole masses: the heavy branch is flatter. In common cubic semiconductors these arise from the $j=3/2$ manifold, while spin-orbit coupling places the $j=1/2$ split-off valence band lower by ${\Delta }_{\mathrm{so}}$. Its thermally generated hole population is suppressed by approximately $e^{-{\Delta }_{\mathrm{so}}/(k_{B}T)}$ relative to upper-valence holes, apart from density-of-states factors. Neglect is justified when ${\Delta }_{\mathrm{so}}\gg k_{B}T$; it is not a universal exclusion of this band at arbitrary temperature.

For an indirect material such as Si, the conduction minimum need not be at $\Gamma$. In applying the carrier-density formulae, $E_c$ must mean the actual minimum of the included conduction valley. The supplied scalar effective masses and densities of states constitute a simplified band-edge model; they do not specify Si's full valley anisotropy or multiplicity.

Carrier populations. Let $f(E)$ be the Fermi-Dirac distribution. In the nondegenerate conduction tail, $f(E)\simeq e^{-(E-\mu )/(k_{B}T)}$. Set $x=E-E_c$ and integrate the given density of states:The integral is $\Gamma (3/2)(k_{B}T{)}^{3/2}=\sqrt{\pi }(k_{B}T{)}^{3/2}/2$. Defining the effective density of states givesThe factor two is already included in the supplied density of states; it must not be added a second time.

A hole is an unoccupied valence state. For $E$ sufficiently below $\mu$, $1-f(E)\simeq e^{(E-\mu )/(k_{B}T)}$. Substitution of $x=E_v-E$ gives in exactly the same wayTheir product eliminates the chemical potential:This is the semiconductor mass-action law at thermal equilibrium. In an intrinsic semiconductor, charge neutrality additionally givesIn a doped semiconductor the product still applies in this nondegenerate equilibrium model, but the two populations need not be equal.

Intrinsic chemical potential. Taking logarithms of $n=p$ yields the intrinsic-semiconductor chemical potentialFor unequal band degeneracies, use the first form with the full $N_c,N_v$. For the supplied mass ratio $m_h^{\ast }/m_e^{\ast }=1/2$, the shift from midgap at $300\mathrm{K}$ is approximately $-13.4\mathrm{meV}$, toward the valence edge.

Three temperature regimes. The original plot runs from high temperature on the left to low temperature on the right. At high temperature, thermal excitation across the gap dominates and $n\simeq p\simeq n_i$ rises rapidly with temperature. In the middle, donors and acceptors are effectively ionized, the material is extrinsic, and $n\simeq N_D-N_A$ makes the plateau. At low temperature, dopant freeze-out binds increasing numbers of electrons to donor levels, reducing the mobile-electron population as the sample cools. Its activation scale is a donor binding energy, not the full band gap. Compensation and incomplete ionization affect that regime.

Gap estimate from the original figure. On the dashed intrinsic branch, $n_i=C{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$. Hence band-gap extraction from intrinsic carrier activation uses the slope of $\ln (n_i/T^{3/2})$ versus $1/T$. With $x=1000\mathrm{K}/T$ and $k_B$ expressed in eV/K, two points giveThe very compressed intrinsic branch is only approximately readable: $n_i\sim 10^{15}\mathrm{c}{\mathrm{m}}^{-3}$ near $x\sim 1.8$ and $n_i\sim 10^{13}\mathrm{c}{\mathrm{m}}^{-3}$ near $x\sim 2.5$. These give $E_g\sim 1.0\mathrm{eV}$, rather than a precision determination. Neglecting the thermal prefactor gives about $1.13\mathrm{eV}$ from those same rough points. Thus a gap of order $1$–$1.1\mathrm{eV}$ is justified by this schematic plot; its resolution does not support distinguishing a few hundredths of an electronvolt. The declining freeze-out branch must not be used to infer the full gap.

Room-temperature densities. At $300\mathrm{K}$, the supplied masses giveIn the fully ionized extrinsic plateau, put $N=N_D-N_A=10^{15}\mathrm{c}{\mathrm{m}}^{-3}$. Charge neutrality and the mass-action law implySince $n_i\ll N$, $n\simeq 10^{15}\mathrm{c}{\mathrm{m}}^{-3}$ is insensitive to the imprecise gap. The minority density is much more sensitive:Using $E_g=1.0\mathrm{eV}$ gives $n_i\simeq 5.8\times 10^{10}\mathrm{c}{\mathrm{m}}^{-3}$ and $p\simeq 3.3\times 10^6\mathrm{c}{\mathrm{m}}^{-3}$. Using $1.1\mathrm{eV}$ instead gives $n_i\simeq 8.3\times 10^9\mathrm{c}{\mathrm{m}}^{-3}$ and $p\simeq 7.0\times 10^4\mathrm{c}{\mathrm{m}}^{-3}$. These are explicitly conditional estimates within the stated mass model. The plot alone cannot fix a precise minority density because the uncertain gap occurs inside an exponential.