Pair momentum distribution in ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_{8+\mathrm{\delta }}$ measured by positron annihilation: Existence and nature of the Fermi surface

We report the first measurement of the positron-electron momentum density in superconducting single-crystal ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_{8+\mathrm{\delta }}$ (${\mathit{T}}_{\mathit{c}}$≊90 K). The observed anisotropy exhibits a twofold (rather than fourfold) symmetry, which is attributed to the superlattice modulation along the b axis of the ${\mathrm{BiO}}_2$ layers. Subtraction of the superlattice contribution also reveals a pair momentum distribution consistent with the ${\mathrm{CuO}}_2$ and ${\mathrm{BiO}}_2$ Fermi surfaces, and in reasonable agreement with the theoretical pair momentum density derived from band theory.