Past exam of the condensed matter physics major option course of the University of Oxford / 2018 / 5

In weak-coupling BCS theory, a useful scale isHere the phonon spectrum sets the attractive-interaction cutoff, $N(0)$ is the consistently defined Fermi-level density of states and $V_{\mathrm{eff}}$ the effective pairing attraction after repulsion is accounted for. Light hydrogen supports high vibrational frequencies; strong electron–phonon coupling and favourable metallic density of states can raise $T_c$. Pressure changes bonding, phase stability, electronic structure and phonon frequencies. It is not simply an independent multiplier in this expression. In a real strongly coupled hydride, Eliashberg theory uses the full spectral function and a screened Coulomb pseudopotential; the elementary exponential is a mechanism guide rather than a controlled quantitative fit to $203\mathrm{K}$.

Measure a four-terminal resistance that falls below the instrumental bound, and test that a magnetic field suppresses the transition. Independently detect diamagnetic screening and, where feasible, field-cooled flux expulsion or an AC susceptibility signal with controlled cell background. A resistance drop alone can reflect contact changes or a filamentary path; a consistent magnetic response is important evidence for superconductivity. A reproducible isotope shift upon replacing hydrogen by deuterium tests the involvement of lattice motion, although it is not by itself proof of superconductivity.

Extreme-pressure samples are microscopic and enclosed in a diamond-anvil cell. Reliable separate electrical contacts, homogeneous pressure, pressure calibration during cooling, chemical stability and the tiny magnetic signal relative to the anvils and gasket are practical difficulties. Structural identification matters because a sulfur-rich hydrogen compound formed under pressure need not retain the starting stoichiometry.

Historical correction: the original primary report associates the $203\mathrm{K}$ transition with pressures near $155\mathrm{GPa}$, while roughly $90\mathrm{GPa}$ is the metallization scale, not that same transition point. The printed pairing of $203\mathrm{K}$ with $90\mathrm{GPa}$ should not be treated as an established measured pair. The statement about the highest conventional transition temperature is also a historical statement in this 2018 examination.^[ref]

For an ideal long specimen with negligible demagnetizing field, the Meissner state has $B={\mu }_0(H+M)=0$, so $M=-H$. A type-I superconductor retains this response up to its thermodynamic critical field $H_c(T)$ and then enters the normal state, where the small normal magnetic response can be neglected. Nonzero demagnetizing fields can create an intermediate state and broaden the measured transition.

A type-II superconductor is fully screening only below $H_{c1}(T)$. Between $H_{c1}$ and $H_{c2}$, magnetic flux enters as quantized vortices and the reversible magnetization remains diamagnetic but decreases in magnitude towards zero at $H_{c2}$. Above $H_{c2}$ the bulk is normal. Ginzburg–Landau theory distinguishes the types at $\kappa =1/\sqrt{2}$, not at $\kappa =1$; the later condition in the source merely restricts its estimate.

The $H$–$T$ sketch has two increasing-width regions below $T_c$: Meissner below the lower boundary, mixed vortex state between the two boundaries, and normal state above the upper boundary. Both critical fields vanish at $T_c$. The phase-diagram panel of the original figure in solution shows this topology; its normalized field curves are illustrative, not a fit to the source magnetization loop.

The displayed horizontal variable is ${\mu }_{0}H$ in tesla. The source formulas written with a flux quantum actually have the dimensions of magnetic flux density, so define $B_{c1}={\mu }_0{H}_{c1}$ and $B_{c2}={\mu }_0{H}_{c2}$ when applying them:From the onset of curvature away from the low-field line in panel A, $B_{c1}$ is roughly $0.06\mathrm{T}$ with uncertainty of order $0.01\mathrm{T}$. Panel B's branches approach the normal background over approximately $2$–$3\mathrm{T}$; use $B_{c2}\simeq 2.5\mathrm{T}$ as a coarse central estimate. Resolving irreversible loop closure from the true thermodynamic upper field, and correcting demagnetization, would require better data and geometry. These readings support scales, not precise material constants.

The upper field givesThe leading-log critical-field inversion starts by dividing the two field formulas to eliminate $\xi$:The large-$\kappa$ solution is $\kappa \simeq 6.15$, henceThe penetration-depth range is an indicative graph/model scale, not a propagated confidence interval. There is a mathematical ambiguity worth making explicit: $f(\kappa )=\ln \kappa /(2{\kappa }^2)$ rises from zero at one, peaks at $\sqrt{e}$, then falls. The same central ratio also gives $\kappa \simeq 1.055$ and $\lambda \simeq 12.1\mathrm{nm}$, so the printed restriction $\kappa >1$ alone does not select a unique solution. The leading-log lower-field formula is intended for appreciably large $\kappa$, which motivates the $\kappa \simeq 6$ branch; near one it needs the omitted core-energy correction. Even the larger-branch estimate has systematic uncertainty from that approximation.

$\lambda$ is the London penetration depth, the magnetic screening scale, while $\xi$ is the superconducting coherence length, the recovery scale of the superconducting order parameter after spatial suppression. In the mixed state the complex order parameter vanishes in vortex cores of radius of order $\xi$, recovers between separated cores and winds in phase by $2\pi$ around each singly quantized vortex. Currents and magnetic field spread over scale $\lambda$; vortex density grows with the mean internal field according to $n_v{\Phi }_0=\overline{B}$. Near $H_{c2}$ cores overlap and the order-parameter amplitude becomes small throughout the sample. A schematic core profile appears in solution.

The hysteresis is not the reversible equilibrium magnetization: defects pin vortices, so entry and exit leave different spatial flux distributions. Macroscopic currents oppose changes in flux, trapped vortices give remanence, and depinning and creep depend on history and temperature. Surface barriers can also contribute. The loop width probes an irreversible current scale, but its conversion to current density additionally needs sample geometry and an appropriate critical-state model.

For a long cylindrical wire carrying axial current $I$, Ampère's law gives $B_s={\mu }_{0}I/(2\pi a)$ at its surface. With $a\gg \lambda$, the current resides mainly in a shell of thickness $\lambda$. If “critical” denotes the first equilibrium entry of vortices in an otherwise Meissner wire, set $B_s=B_{c1}$:The dependence on $a$ is necessary because no numerical radius is supplied. The local surface current density at this entry scale is different:Its conversion to the cross-section average is $\overline{J}\simeq (2\lambda /a)j_s$. Treating the shell current density as uniform over the entire thick wire would overestimate its total current by $a/(2\lambda )$.

Meaning of a maximum: in a type-II material, first vortex entry need not be the measured dissipative transport limit. Pinned vortices can sustain greater current, whereas freely moving vortices generate voltage. The hysteresis loop indicates pinning, but no geometry-dependent pinning force or transport criterion is given, so a unique practical maximum cannot be deduced solely from $\lambda ,\xi$ and $a$.

For comparison, the intrinsic homogeneous Ginzburg–Landau depairing scale follows from $j(q)={\Phi }_{0}q(1-{\xi }^2{q}^2)/(2\pi {\mu }_0{\lambda }^2)$. Maximizing at $q=1/(\sqrt{3}\xi )$ givesThis is a local, approximately uniform-flow theoretical scale, not the average current density throughout a thick self-field wire. Surface barriers, nonlinear screening, vortex motion and temperature determine how that scale relates to its actual critical current. Thus the entry estimate, local shell density and intrinsic depairing limit answer distinct physical criteria; the source's word “maximum” does not supply the missing pinning information.