Magnetic polarization induced by nonmagnetic impurities in high Tc cuprates

The magnetic polarization induced by nonmagnetic impurities such as Zn in high Tc cuprate compounds is studied by the variational Monte Carlo simulation. The variational wave function is constructed from the eigenstates obtained from Bogoliubov de Gennes mean field Hamiltonian for the twodimensional t − J model. A Jastrow factor is introduced to account for the induced magnetic moment and the repulsion between holes and the impurity. A substantial energy gain is obtained by forming an antiferromagnetic polarization covering 4 or 5 lattice sites around the impurity. We also found the doping dependence for the induced magnetic moment consistent with experiments. PACS number: 74.72.-h Typeset using REVTEX 1 Recently a number of experiments, the neutron scattering, nuclear-magnetic-resonance (NMR) and scanning tunneling microscopy (STM)4–6, have been carried out to study the impurity effect on the electronic transport and magnetic properties in high Tc cuprate compounds. These studies provide a detail information about the relationship between magnetism and superconductivity in high Tc cuprates. The nonmagnetic impurity Zn was found to suppress Tc more strongly than magnetic impurity Ni, even though both replace Cu in the CuO2 plane . The amazingly accurate measurement of the local density of states (LDOS) by STM4–6 also provides very different spectra for Zn and Ni. The spin dynamics studied by the neutron scattering experiments reveals that the low-energy spin fluctuations are strongly enhanced near the impurity and the magnetic excitation at the antiferromagnetic wave vector (π, π) disappears with Zn doping in the underdoped region. It is interesting to find from the NMR and SQUID experiments that both the nonmagnetic Zn and the magnetic Ni impurities induce a local magnetic moment on Cu sites surrounding the impurity in the normal state. The broadening of Cu and O NMR lines has been attributed to a distribution of magnetic moments or a spatially inhomogeneous spin polarization extending over several lattice sites around the impurity. On the other hand some experiments found no evidences of the existence of local magnetic moments, at least in the optimum and overdoped samples. More careful theoretical and experimental efforts to exmine the magnetic polarization are needed to clarify this issue. So far most of the theoretical work has been based upon phenomenological BCS type models with emphasis on understanding of the LDOS. The observed nearly-zero-energyresonance peak near Zn impurity was explained very early by Balatsky, Salkola and coworkers11–14 by assuming Zn to be an unitary impurity. Studies15–20 based upon t− J type models have also successfully explained the LDOS. There are only few studies about the structure of magnetic polarization induced by the magnetic moment binded to the nonmagnetic impurity and the screening of this moment by other electrons. However in a recent paper Tsuchiura et al. use Gutzwiller approximation and the Bogoliubov-de Gennes (BdG) approach for the t−J model and they find no evidence of the existence of the local moments 2 around the Zn impurity. They also concluded that the electron avoids the impurity instead of being binded to it. A much more careful examination of the effect of a non-mgnetic impurity in the t− J model is needed to resolve the controversy. Comparing with other phenomenological models, the t − J model has much stronger magnetic correlation and it may lead to a different picture about the magnetic polarization around the impurity. However, previous studies of the t-J model use the BdG approach with or without the Gutzwiller approximation and the no-doubly-occupied constraint imposed by the t − J model is only taken into account on the average or approximately. It very likely underestimates the antiferromagnetic correlation inherent in the t−J model. Another issue has not been addressed adequately before is the doping dependence of the induced magnetic moment. Very different results reported by NMR experiments may be related to the doping dependence. In this paper we will impose the constraint rigorously by using the variational Monte Carlo approach to study the effect of nonmagnetic Zn impurity on the ground state of the t − J model. The ground state trial wave function is first constructed by assuming d-RVB order parameters in the BdG approach. Then the variational wave function is shown to be greatly improved by adding a Jastrow factor to account for the strong magnetic correlation. We found a large energy gain by having an antiferromagnetic polarization around the impurity with size about 4 to 5 lattice sites as observed in Cu NMR data in the underdoped region. The significant suppression of the magnitude of the induced moment and its polarization size as doping increases to optimum doping is also consistent with experimental observations. In addition, our result also provides a reason to explain the similarity between results measured for Li and Zn. Contrary to the work reported in Ref.(21) we show that electrons are always attracted to the impurity. But the effect gets weaker when number of holes increases. The model we consider is the dilute impurity limit of the two-dimensional t− J model. The interaction between impurities is neglected. Zn[3d] has total spin S = 0 and its second ionization energy is about 18eV . Near chemical potential the conduction electron is 3 estimated to encounter a repulsive local potential U0 ≈ 18.9eV 24 when it scatters with the Zn impurity. This is much larger than the bandwidth (2eV ) of the dx2−y2 band of 3d Cu 2+ electrons. Thus, the nonmagnetic impurity Zn can be described roughly by a spin vacancy in the unitary limit. We start from the Hamiltonian, H = −t ∑ ,σ PG(c † iσcjσ + h.c.)PG + J ∑ (Si · Sj − 1 4 ninj) + ∑ i (U0δi,I − μ)niσ, (1) where I labels the site of the impurity. In the standard notation, the < ij > means the summation over nearest neighbors and PG = ∏ i(1 − ni↑ni↓) is the Gutzwiller’s projection operator that prohibits double occupancy. Within the mean field approximation, the BdG equation is derived