States of Local Moment Induced by Nonmagnetic Impurities in Cuprate Superconductors

By using a model Hamiltonian with $d$-wave superconductivity and competing antiferromagnetic (AF) orders, the local staggered magnetization distribution due to nonmagnetic impurities in cuprate superconductors is investigated. We show that the net moment induced by a single impurity corresponds to a local spin with $S_z=0$ or $1/2$ depending on the strength of the AF interaction $U$ and the impurity scattering strength $ϵ$. Phase diagram of $ϵ$ versus $U$ for the moment formation is presented. We discuss the connection of this result with the Kondo problem. When two impurities are placed at the nearest neighboring sites, the net moment is always zero, unusually robust to parameter changes. For two neighboring strong impurities, separated by a Cu-ion site, the induced net moment has $S_z=0$, $1/2$, or 1.

Figure 1

Spatial variations of the staggered magnetization $M_i^s$ [(a), (c), and (e)], and LDOS spectra [(b), (d), and (f)]. The impurity site is at (16,16). The solid lines in panels (b), (d), and (f) are for (16,16), dashed lines for (17,16), dashed-dotted lines for (17,17), and dotted lines for (16,1). The upper panels [(a),(b)], the central panels [(c),(d)], and the lower panels [(e),(f)] are for $(U=2.0,ϵ=3)$, $(U=2.0,ϵ=100)$, and $(U=2.35,ϵ=100)$, respectively.

Figure 2

Phase diagram of $ϵ$ versus interaction strength $U$ for various phases near the impurity.

Figure 3

Impurities configuration [(a),(d)] and staggered magnetization $M_i^s$ [(b),(c) and (e),(f)]. The left panels [(a)–(c)] and the right panels [(d)–(f)] are for two nn impurities and two next nn impurities, respectively. Panels [(b),(c)] and [(e),(f)] correspond to $U=2.0$ and $U=2.4$, respectively.

Figure 4

Impurities configuration (a) and spatial profiles of staggered magnetization $M_i^s$ [(b)–(e)] in a $32\times 32$ lattice. Their corresponding parameters are (b) ($U=1.9$), (c) ($U=2.0$), (d) ($U=2.4$), and (e) ($U=2.2,t^{\prime }=-0.3$).