Magnetic Excitation Spectrum of Doped and Undoped Spin Ladders

We present inelastic neutron scattering measurements of three spin-ladder compounds. All are based on two-dimensional copper oxide layers where the copper ions form two-leg, spin-1/2, spin-ladders. Strong antiferromagnetic interactions couple the spin moments along the legs and rungs of the ladder, weaker higher order four-spin ring exchange terms exist, while interladder coupling is weak and frustrated. Two-leg spin-1/2 spin-ladders are characterized by gapped, well-defined one-magnon excitations and multi-magnon continuum excitations. CaCu$_{2}$O$_{3}$ has a weak rung interaction (compared to the leg) and a strong ring interaction which act to drive the system gapless and quantum critical. La$_{4}$Sr$_{10}$Cu$_{24}$O$_{41}$ has a stronger rung and is gapped. Both compounds are undoped and their magnetic excitation spectrum is compared to theoretical models. The third material, Sr$_{2.5}$Ca$_{11.5}$Cu$_{24}$O$_{41}$, is similar to La$_{4}$Sr$_{10}$Cu$_{24}$O$_{41}$ but with holes on the ladder. The holes give rise to significant changes in the excitations which are discussed and compared to theory and the excitation spectrum of high-T$_{c}$ superconductors.