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

We investigate the magnetic properties of the Cu-O planes in stoichiometric Sr₍-₁Cu₍+₁O₂₍ (n=3, 5, 7,. . . ) which consist of CuO double chains periodically intergrown within the CuO₂ planes. The double chains break up the two-dimensional antiferromagnetic planes into Heisenberg spin ladders with nₑ=1/2 (n-1) rungs and n₋=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 nₑ is even and n₋ is odd (corresponding to n=5, 9,. . . ) and a gapped spectrum when nₑ is odd and n₋ 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, ). 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 Sr₍-₁Cu₍+₁O₂₍ compounds with n=3, 7, 11,. . . , will be frustrated quantum antiferromagnets with a quantum-disordered or spin-liquid ground state.