The formation of bound states at surfaces of materials with an
energy gap in the bulk electron spectrum is a well known physical phenomenon.
At superconductor surfaces, quasiparticles with energies inside the
superconducting gap Δ may be trapped in bound states in quantum wells,
formed by total reflection against the vacuum and total Andreev reflection
against the superconductor. Since an electron reflects as a hole and sends a
Cooper pair into the superconductor, the surface states give rise to resonant
transport of quasiparticle and Cooper pair currents, and may be observed in
tunnelling spectra. In superconducting junctions these surface states may
hybridize and form bound Andreev states, trapped between the superconducting
electrodes. In d-wave superconductors, the order parameter changes sign under
90° rotation and, as a consequence, Andreev reflection may lead to the
formation of zero energy quasiparticle bound states, midgap states (MGS). The
formation of MGS is a robust feature of d-wave superconductivity and provides
a unified framework for many important effects which will be reviewed: large
Josephson current, low-temperature anomaly of the critical Josephson current,
π-junction behaviour, 0→π junction crossover with temperature,
zero-bias conductance peaks, paramagnetic currents, time reversal symmetry
breaking, spontaneous interface currents, and resonance features in subgap
currents. Taken together these effects, when observed in experiments, provide
proof for d-wave superconductivity in the cuprates.