Defect-trapped electrons and ferromagnetic exchange in ZnO

Abstract

A model for ferromagnetism observed at ambient temperature in films of oxides such as ZnO is proposed and evaluated. The ferromagnetic moment in the model arises from electrons trapped at negatively charged vacancies in an n-type oxide. These vacancies are capable of trapping either one or two electrons. Trapped electrons are described by a one-band Hubbard Hamiltonian where the Hubbard U is the effective electron-electron repulsion for a pair of electrons in a vacancy. Ferromagnetism is known to exist in the Hubbard model applied to periodic three-dimensional (3D) lattices, provided the Hubbard U parameter exceeds the defect bandwidth W and the filling is away from half or complete filling. Hybrid and local-density approximation density-functional theory calculations are used to evaluate Hubbard model parameters for electrons trapped in defects in ZnO. They are also used to calculate magnetic exchange couplings of well-separated, singly negatively charged defects, which are induced by a conduction band electron. Strong ferromagnetic coupling between defects is found in these total-energy calculations over a range exceeding 10 ˚ A when the defects have a large, positive Hubbard U value. Hubbard U values for oxygen (VO), zinc (VZn), and zinc-oxygen complex (VZnO) vacancies in various charge states are estimated from defect transition levels. V −, the negatively charged ZnO pair vacancy, and V − are proposed as possible sources of magnetic moment in ferromagnetic ZnO films. These vacancies can trap one or two electrons and their charge transition levels lie in the band gap. Some literature values of U and those obtained here for unrelaxed vacancies are large enough to support a Hubbard model for ferromagnetism; however, U values obtained depend strongly on lattice relaxation. The relaxed vacancies considered here have U/W values which are not large enough for ferromagnetism using the simple criterion U/W >1.