Quantum Oscillations in the Underdoped Cuprate YBa$_{2}$Cu$_{4}$O$_{8}$*

The quantum oscillations (QOs) seen in the underdoped cuprate superconductor YBa$_{2}$Cu$_{4}$O$_{8}$ (Y124) in magnetic fields up to $B$=85T [1] are strong evidence for a well-defined Fermi surface (FS) in Y124 at low temperature $T$ and high $B$. The QO frequency $F$=660$\pm $15T, implies a FS pocket with 2.4{\%} of the full Brillouin zone (BZ) area. Taken with earlier work [2], our data suggest FS pockets are generic to underdoped CuO$_{2}$ planes and give the first hint of doping dependence of the FS. We discuss the carrier concentration implied by the QOs within various models. Comparison of the T-linear specific heat $\gamma $ (from QO quasiparticle mass m*) to $\gamma $ estimated from zero-field specific heat measurements constrains the number of FS pockets present in the BZ and supports a reduced BZ due e.g. to a charge/spin density wave or ordered orbital currents. [1] E. A. Yelland et al, arXiv:0707.0057. [2] N. Doiron-Leyraud et al, Nature 446, 565 (2007)