Past exam of the physics A course of the University of Cambridge / 2014 / Paper 2 / Section D

For a plane sound wave, acoustic impedance is $Z=p/u=\rho c$, with pressure amplitude $p$ and particle-velocity amplitude $u$. Continuity of pressure and normal velocity at normal incidence on a lossless interface givesMatching $Z_1=Z_2$ eliminates reflection. A quarter-wave intermediate layer with $Z_m=\sqrt{Z_1{Z}_2}$ transforms the load to the source impedance at its design frequency; its bandwidth is limited. For strings, the analogous mechanical impedance is transverse force divided by velocity, $Z=T{v}^{-1}=\sqrt{T\mu }$; continuity of displacement and transverse force gives the corresponding matching condition. These amplitudes use different force/velocity conventions from acoustic pressure, so reflection signs must be stated consistently.

Reflections from the two film surfaces interfere with propagation phase $4\pi nd\cos {\theta }_t/{\lambda }_0$ plus the relative reflection phase. Reflection from a lower to a higher refractive index contributes a $\pi$ phase reversal. For a higher-index film surrounded by air there is one relative reversal, henceThickness or angle variation therefore produces fringes or colours under broadband illumination. An antireflection coating between indices $n_0,n_s$ instead uses, at normal incidence, $n_f=\sqrt{n_0{n}_s}$ and $n_{f}d={\lambda }_0/4$: the two reflected amplitudes then cancel at the design wavelength. Absorption, polarization, unequal amplitudes and multiple reflections determine the full contrast.

A Michelson interferometer divides light at a beam splitter, sends it to two mirrors, and recombines the returned beams. A compensating plate can equalize glass traversal. For approximately axial rays the optical path difference is $\Delta =2(n_1{L}_1-n_2{L}_2)$, plus fixed instrument phase, andfor coherent, mode-matched fields. Moving one mirror by $\delta L$ in air changes the round-trip path by $2\delta L$, so one fringe corresponds to ${\lambda }_0/2$ mirror motion. It measures displacement, wavelength and refractive-index changes; scanning the delay enables Fourier-transform spectroscopy. Mirror alignment, polarization overlap, vibration and source coherence control visibility. A finite coherence length limits useful arm mismatch.

Donor doping supplies conduction electrons to n-type semiconductor; acceptor doping supplies holes to p-type semiconductor. In the nondegenerate, fully ionized limit, majority densities are approximately $n_n=N_D$ and $p_p=N_A$, while minority densities follow $np=n_i^2$. On contact, majority carriers diffuse across the junction, exposing positive donor ions on the n side and negative acceptor ions on the p side. The depletion-region electric field points from n to p and opposes further diffusion. At thermal equilibrium the Fermi level is uniform and drift balances diffusion, with no net current.

For an abrupt junction with uniform doping and permittivity ${\epsilon }_s$, charge neutrality gives $N_D{x}_n=N_A{x}_p$. Poisson's equation then givesPositive applied $V$ means forward bias: it lowers the barrier and narrows the depletion region. Most depletion width lies in the more lightly doped side. The formula assumes the depletion approximation and fails once its operating assumptions break down.

forward bias increases minority densities at the depletion edges by the factor $e^{qV/k_{B}T}$. In the neutral material, the steady diffusion–recombination equation gives exponentially decaying excess minorities, with lengths $L_n=\sqrt{D_n{\tau }_n}$ and $L_p=\sqrt{D_p{\tau }_p}$. Taking the carrier gradients at the junction yieldsThis ideal diode law assumes low injection, nondegenerate statistics, negligible depletion-region recombination and series resistance, and sufficiently long neutral regions. Its exponential comes from the bias-dependent injection barrier; there is no universal abrupt “turn-on” voltage. reverse bias suppresses injection, giving an approximate $-I_S$ current until additional leakage or breakdown matters.^[ref]

Rectification uses the asymmetric current-voltage characteristic; controlled reverse breakdown enables voltage references, with current limiting. Bias-dependent depletion capacitance enables varactors. photodiodes and solar cells collect optically generated carriers through the junction field and diffusion toward it; efficient light-emitting diodes use radiative electron–hole recombination, preferably in a direct-gap material. Series resistance, high injection, trap-assisted recombination, heating and finite carrier storage explain departures from the ideal steady-state law.