Theory of the NMR relaxation rates in cuprate superconductors with field induced antiferromagnetic order

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Based on a model Hamiltonian with a d-wave pairing interaction and a competing antiferromagnetic interaction, we numerically study the site dependence of the nuclear spin resonance (NMR) relaxation rate T₁⁻¹ as a function of temperature for a d-wave superconductor(DSC) with magnetic field induced spin density wave (SDW) order. In the presence of the induced SDW, we find that there exists no simple direct relationship between NMR signal rate T₁⁻¹ and low energy local density of states while these two quantities are linearly proportional to each other in a pure DSC. In the vortex core region, T₁⁻¹ on ¹⁷O site may exhibit a double-peak behavior, one sharp and one broad, as the temperature is increased to the superconductivity transition temperature Tc, in contrast to a single broad peak for a pure DSC. The existence of the sharp peak corresponds to the disappearance of the induced SDW above a certain temperature TAF which is assumed to be considerably lower than Tc. We also show the differences between T₁⁻¹ on ¹⁷O and that on ⁶³Cu as a function of lattice site at different temperatures and magnetic fields. Our results obtained from the scenario of the vortex with induced SDW is consistent with recent NMR and scanning tunneling microscopy experiments.