By using a model Hamiltonian with competing antiferromagnetic (AFM) and $d$-wave superconductivity orders, the impurity-induced structures of orders in $d$-wave superconductors is investigated. We find that the transition between one-dimensional stripe and two-dimensional checkerboardlike modulation around a single nonmagnetic impurity can take place as the strength of the AFM interaction $U$ or the impurity scattering strength $V_0$ is varied. It is also found that the impurity-induced stripe can first transit to checkerboardlike modulation and then disappears with increasing the next-nearest-neighbor hopping strength $\left|t^{\prime }\right|$. Phase diagrams of $V_0$ versus $U$ and $\left|t^{\prime }\right|$ for various modulations of the spin order are presented. In addition, the quantum interference effect on the modulations of orders due to two strong nonmagnetic impurities is briefly examined, and the checkerboardlike and quasistripe patterns can occur depending on the sites where two impurities are placed.

• Received 20 March 2009

DOI:https://doi.org/10.1103/PhysRevB.80.064518