Spin ladders with spin gaps: A description of a class of cuprates

We investigate the magnetic properties of the Cu-O planes in stoichiometric ${\mathrm{Sr}}_{\mathit{n}\mathrm{-}1}$${\mathrm{Cu}}_{\mathit{n}+1\mathrm{}}$${\mathrm{O}}_{2\mathit{n}}$ (n=3,5,7, . . .) which consist of CuO double chains periodically intergrown within the ${\mathrm{CuO}}_2$ planes. The double chains break up the two-dimensional antiferromagnetic planes into Heisenberg spin ladders with ${\mathit{n}}_{\mathit{r}}$=1/2(n-1) rungs and ${\mathit{n}}_{\mathit{l}}$=1/2(n+1) legs and described by the usual antiferromagnetic coupling J inside each ladder and a weak and frustrated interladder coupling J’. The resulting lattice is a new two-dimensional trellis lattice. We first examine the spin excitation spectra of isolated quasi-one-dimensional Heisenberg ladders which exhibit a gapless spectrum when ${\mathit{n}}_{\mathit{r}}$ is even and ${\mathit{n}}_{\mathit{l}}$ is odd (corresponding to n=5,9, . . .) and a gapped spectrum when ${\mathit{n}}_{\mathit{r}}$ is odd and ${\mathit{n}}_{\mathit{l}}$ is even (corresponding to n=3,7, . . .). We use the bond operator representation of quantum S=1/2 spins in a mean-field treatment with self-energy corrections and obtain a spin gap of ≊1/2J for the simplest single-rung ladder (n=3), in agreement with numerical estimates. We also present results of the dynamical structure factor S(q,ω). The spin gap decreases considerably on increasing the width of the ladders. For a double ladder with four legs and three rungs (n=7) we obtain a spin gap of only 0.1J. However, a frustrated coupling, such as that of a trellis lattice, introduced between the double ladders leads to an enhancement of the gap. Thus stoichiometric ${\mathrm{Sr}}_{\mathit{n}\mathrm{-}1}$${\mathrm{Cu}}_{\mathit{n}+1\mathrm{}}$${\mathrm{O}}_{2\mathit{n}}$ compounds with n=3,7,11, . . ., will be frustrated quantum antiferromagnets with a quantum-disordered or spin-liquid ground state.