de Haas–van Alphen Oscillations in a Superconducting State at High Magnetic Fields

Low-temperature quantum oscillations of the de Haas–van Alphen (dHvA) amplitude are shown to persist far below the upper critical field of a strongly type-II superconductor, due to the gapless nature of the BCS quasiparticle spectrum in high fields. The dHvA amplitude in the superconducting state is smaller than its normal state counterpart by the factor $\sim \left[\max \right(T,\gamma )/\Delta {]}^2$, where $\gamma$ is the damping. This factor reflects the presence of a small gapless portion of the Fermi surface, surrounded by regions where the BCS gap is large. The agreement with recent experimental data on ${\mathrm{V}}_3\mathrm{Si}$ is very good.