Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A10

Assume dilute nondegenerate electrons and electron holes, parabolic three-dimensional band edges with the stated scalar band effective masses, spin multiplicity two, one effective valley for each edge, and thermal equilibrium. Take the unchanged host ground state to have unit activity and neglect bound excitons. The effective density of states values areFor a Fermi level $\mu$ between the band edges,The factors of $\mu$ cancel on multiplication. With $E_g=E_c-E_v$, the semiconductor mass-action law isThis density-form equilibrium constant has units ${\mathrm{m}}^{-6}$. A dimensionless thermodynamic constant uses a specified reference number density $n^{\circ }$, giving $K=K_n/(n^{\circ }{)}^2$. The dot representing the host ground state is not a freely translating gas species requiring a further gas partition function. For an intrinsic semiconductor, charge neutrality gives $n_e=n_h=n_i=\sqrt{K_n}$.

The equal supplied masses give equal effective density of states. At the specified temperature,Using the same energy units in the exponential,For equal electron and hole mobilities, both carriers contribute to the intrinsic electrical conductivity:The factor two is essential. A dimensional check givesusing $\mathrm{A}=\mathrm{C}{\mathrm{s}}^{-1}$.

Substitutional arsenic has one more valence electron than germanium, so it acts as a donor impurity. Thermal ionization of its shallow bound state supplies mobile conduction-band electrons and leaves fixed positive donor charge. The resulting n-type semiconductor can have $n_e\gg n_i$ even when only a small fraction of host atoms are replaced. Its electrical conductivity then rises because the mobile carrier count increases; the opposite donor charge preserves macroscopic charge neutrality.

This argument assumes that the donors are substantially ionized at the measurement temperature and that the added impurity scattering does not overwhelm the carrier increase. The screened bound-donor picture and its departures from a simple hydrogenic model are described in institutional semiconductor spectroscopy notes.^[ref]

Retain the inferred mobilities and assume one ionized donor per added arsenic atom, negligible compensation, the same band gap, and the nondegenerate semiconductor mass-action law. Relative to the intrinsic sample, a conductivity factor $f$ requiresFor complete donor ionization, charge neutrality gives $N_D=n_e-n_h$, soThe host atom density, using $M_{\mathrm{Ge}}\simeq 72.63\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, isThus the required atomic fraction isapproximately one donor per $5.8\times 10^4$ host atoms. The electron density must rise to about $2000n_i$, not $1000n_i$, because the original intrinsic sample conducted through both electrons and holes. This is an estimate under unchanged mobility: impurity scattering, incomplete ionization, compensation and band-edge changes can alter it. Here $n_e/N_c\simeq 0.12$, so the nondegenerate treatment is an approximation rather than an arbitrarily dilute limit.

Two derivatives of the free-electron dispersion giveThe band effective mass becomes the ordinary electron mass when no periodic-potential modification of the quadratic dispersion relation is present.

Differentiating the tight binding dispersion produced by the Hückel method givesHenceUnlike a free particle, the band effective mass depends on wavevector and can change sign. At an inflection point $\cos (ka)=0$, the inverse mass vanishes and the curvature-based mass diverges; a quadratic approximation then ceases to describe the dispersion locally. For the usual $\beta <0$ convention, the bottom at $k=0$ has positive mass and the top at $k=\pi /a$ has negative electron mass. The constant $\alpha$ has no effect on curvature.

In a weakly doped n-type semiconductor, the mobile electrons occupy states near the bottom of the conduction band. Their local band curvature is positive, so their band effective mass is positive and approximately constant sufficiently near that minimum. For the usual $\beta <0$ convention, expand about $k=0$:Stronger overlap gives larger $|\beta |$, greater curvature and a smaller mass within this model. Positive $\beta$ simply places the minimum at the zone boundary and again gives a positive edge mass. A negative electron mass near a band maximum instead motivates the positive-mass electron hole description; it is not the usual conduction-edge situation for an n-type semiconductor. No numerical comparison with $m_e$ follows without values of $a$ and $\beta$.