PHYSICAL REVIEW B 87, 195310 (2013) Modeling spin injection across diffuse interfaces P. Chureemart, * R. Cuadrado, I. D’Amico, and R. W. Chantrell Department of Physics, University of York, York YO10 5DD, England, United Kingdom (Received 4 March 2013; revised manuscript received 22 April 2013; published 21 May 2013) We propose a model of the injection of spin-polarized current in systems with diffuse interfaces. The effect of the interfaces between a ferromagnet and a nonmagnet and between two ferromagnets on spin injection is investigated. We first generalize the formalism for calculating the spin accumulation by taking the spin accumulation as the difference of spin-up and spin-down density of states, which is necessary for treating the interface between different ferromagnets. Then, we include the effect of atomic species interdiffusion at the interface by using Fick’s law. It is shown that the discontinuity of the spin accumulation at the interface depends strongly on the degree of interface mixing. DOI: 10.1103/PhysRevB.87.195310 PACS number(s): 72.25.Ba, 72.15.−v, 73.40.−c, 75.76.+j I. INTRODUCTION Spin electronics, using the spin of the electron in addi- tion to its charge, is an emergent technology with exciting potential. 1 The development of spin electronics follows the discovery of giant magnetoresistance (GMR). 2,3 The GMR effect is associated with the spin-dependent scattering both at the interfaces and within the magnetic layers. 4–7 GMR, and the subsequent discovery of tunneling magnetoresistance (TMR), 8,9 led rapidly to applications as spin-valve read heads for magnetic recording, giving rise to remarkable increases in storage density and revolutionizing computer applications and efficiency. The high resistance of TMR allowed the development of magnetic random access memory (MRAM), which combines key advantages such as nonvolatility, in- finite endurance, and fast random access, making MRAM an important future technology. However, switching of the magnetic elements is difficult with conventional technology, and present devices predominantly use the spin-transfer torque phenomenon to achieve switching. The spin torque effect proposed by Slonczewski 10 and Berger 11 introduces an entirely new route for the control of the magnetization of magnetic structures 12–14 and for spintronic device concepts. A polarized current is produced by a magnetic film and injected into a second layer, where its polarization is rotated into the new magnetization direction, exerting a reaction torque on the magnetization. Injection of spin-polarized current is in general of great interest for potential new spintronic devices. The concept of spin injection across the interface between a ferromagnet (FM) and a nonmagnet (NM) was first suggested by Aronov 15 and experimentally observed by Johnson and Silsbee. 16 Injecting an electric current into a ferromagnet results in a spin-polarized current, which subsequently flows across the interface into a nonmagnet, giving rise to a spin current in the nonmagnet and spin accumulation close to the interfacial region. The spin accumulation diffuses into the nonmagnet from the interface with a length scale associated with the spin relaxation time. 17–19 Clearly, the full understanding of the spin torque phe- nomenon is important for the development of MRAM and other spin electronic technologies, which are currently a topic of extensive interest at the fundamental and technological levels. The nature of the interface is an important factor in spin injection and consequently in the phenomenon of spin torque, 20,21 which relies on the spin injection. However, the effect of diffuse interfaces has received relatively little attention. Since practical devices are generally produced by sputtering, it must be expected that the interfaces are not atomically smooth, and consequently it is important to develop models of diffuse interfaces. Zhang, Levy, and Fert (ZLF) 22 studied the spin accumulation arising from the injection of a polarized current produced by a pinned FM layer into a second FM layer, which results in a discontinuity of the spin accumulation across the interface. Shpiro, Levy, and Zhang 23 used a similar formalism to develop a semianalytical approach to diffuse interfaces in which the degree of continuity of the spin accumulation was determined by an effective interface resistance. In this letter we generalize these approaches in two important ways. First, we use a definition of the spin accumulation based on the spin-up and spin-down density of states (DOS) rather than deviations from the equilibrium value. While this does not affect the spin torque, it does provide a physically sound basis for the treatment of interfaces between FM layers of different materials. Second, we propose a simple model of the behavior of diffuse interfaces. The model is applied to the study of the transmission of spin current and the development of spin accumulation between two FM layers of different properties and also between FM-NM layers. II. METHODOLOGY A. Spin accumulation In principle the spin accumulation represents the differ- ence between spin-up and spin-down conduction electron populations. However, the common usage defines the spin accumulation as the deviation from the equilibrium value; δm = (n ↑ − n ↑ eq ) − (n ↓ − n ↓ eq ), where n eq refers to the equi- librium (bulk) populations and n ↑(↓) are the local spin-up (spin-down) carrier densities. This definition was used by ZLF 22 in their study of the spin accumulation. In the work of ZLF 22 the spin accumulation is assumed to precess about the local magnetization in the presence of damping with nonconservation of the magnitude of δm. Specifically, the damping term is of the Bloch form −δm/τ SF , with τ SF being the spin relaxation time of the conduction electrons. Here we propose to use the definition m = n ↑ − n ↓ . Clearly m and δm 195310-1 1098-0121/2013/87(19)/195310(6) ©2013 American Physical Society