Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 8

In BCS theory, an electron distorts the positively charged lattice, and the retarded response can attract another electron. Within a phonon-energy shell around the Fermi surface, the effective attraction can overcome the screened Coulomb repulsion and destabilize the normal Fermi sea toward Cooper pairs. In conventional elemental superconductors the pairs predominantly join time-reversed states of opposite momentum and spin, forming a spin singlet.

In the weak-coupling approximation,Here $N(0)$ is the appropriate single-spin density of states and $V_{\mathrm{eff}}>0$ the effective attractive interaction in a consistent normalization. Both the phonon scale and the exponential coupling matter. A large Fermi-level density of states, often associated with partially filled narrow $d$ bands, can raise $T_c$; weak coupling or a small density of states can suppress it even in a good conductor. With force constants and coupling fixed, ${\omega }_D\propto M^{-1/2}$ gives the conventional isotope trend. This is not a universal monotonic ranking of elements by atomic mass: phonon stiffness, electron–phonon matrix elements, Coulomb effects and magnetism vary between elements. Magnetic pair breaking helps explain why some elements do not follow a naive density-of-states expectation.

Let $m,q,n_s$ denote pair mass, charge and number density, respectively. A uniform-phase condensate in the stated gauge has zero canonical momentum $m{\mathbf{v}}_s+q\mathbf{A}=0$. ThereforeFor static fields, $\nabla \times \mathbf{B}={\mu }_0{\mathbf{j}}_s$ and $\nabla \cdot \mathbf{B}=0$. Taking a curl and using the vector identity for a double curl givesFor electronic pairs, $m\simeq 2m_e$ and $q=-2e$, while $n_s$ is the pair density, not twice that density. In a general gauge the canonical momentum is $\hslash \nabla \theta$; taking the curl away from vortices gives the same London equation.

The competition between magnetic screening over the London penetration depth $\lambda$ and condensate recovery over the superconducting coherence length $\xi$ is measured by the Ginzburg-Landau parameter $\kappa =\lambda /\xi$. In the single-component local theory the boundary is $\kappa =1/\sqrt{2}$. For $\kappa <1/\sqrt{2}$, a normal–superconducting interface has positive energy: a type-I superconductor favours macroscopic regions and a thermodynamic critical field $H_c$. Demagnetizing fields may create an intermediate state of normal and superconducting domains.

For $\kappa >1/\sqrt{2}$ the interface energy is negative. A type-II superconductor admits flux vortices between the lower and upper critical fields, with normal-like cores of size $\xi$ and circulating screening currents over $\lambda$. Each singly quantized vortex carries ${\Phi }_0=h/(2e)$; a vortex lattice and pinning-controlled irreversible response can occur. Pairing strength, carrier density, Fermi velocity and impurity scattering affect these lengths; $T_c$ alone does not determine the type. The stated boundary is the standard Ginzburg–Landau result near $T_c$.

For a laterally infinite slab in a parallel field there is no demagnetizing correction. The field has only its applied direction, varies with $x$, and satisfies $B^{\prime \prime }=B/{\lambda }^2$. Reflection symmetry gives $B^{\prime }(0)=0$, and continuity gives $B(\pm a)=B_a$. ThusUniformly for $|x|\le a\ll \lambda$, expanding numerator and denominator to second order yieldsOnly a small fraction of the applied field is screened in this limit; the almost uniform internal field is the origin of the enhanced parallel critical field.

Average the exact slab solution across its thickness:For $u\ll 1$, $M\simeq -B_a{a}^2/(3{\mu }_0{\lambda }^2)$. The zero-field condensation free energy per volume is $-B_C^2/(2{\mu }_0)$. Neglecting normal-state magnetic susceptibility, the Gibbs free-energy difference at fixed applied field follows from $d(G_s-G_n)/d{B}_a=-M$:Equating it to zero gives the constant-$\lambda$ estimate for the parallel critical field of a thin London slabThe enhancement is equivalently $\sqrt{12}\lambda /d$ for total thickness $d=2a$. It is an estimate assuming an intact, spatially uniform condensate density up to the transition. Field-induced pair suppression, the paramagnetic limiting field, and failure of bulk local electrodynamics in an extremely thin film limit an unlimited extrapolation as $a\to 0$.