3 2 0 2 c e D 7 l l a h s e m t a m d n o c 1 v 8 3 5 4 0 2 1 3 2 v i X r a Direct and indirect spin current generation and spin orbit torques in ferromagnet nonmagnet ferromagnet trilayers V P Amin 1 G G Baez Flores 2 A A Kovalev 2 and K D Belashchenko2 1Department of Physics Indiana University Indianapolis IN 46202 2Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience University of Nebraska Lincoln Lincoln Nebraska 68588 USA Dated December 8 2023 Spin orbit torques in ferromagnet nonmagnet ferromagnet trilayers are studied using a combina tion of symmetry analysis circuit theory semiclassical simulations and first principles calculations using the non equilibrium Green s function method with supercell disorder averaging We focus on unconventional processes involving the interplay between the two ferromagnetic layers which are classified into direct and indirect mechanisms The direct mechanism involves spin current generation by one ferromagnetic layer and its subsequent absorption by the other In the indirect mechanism the in plane spin polarized current from one ferromagnetic layer leaks into the other layer where it is converted into an out of plane spin current and reabsorbed by the original layer The direct mechanism results in a predominantly dampinglike torque which damps the magneti zation towards a certain direction sd The indirect mechanism results in a predominantly fieldlike torque with respect to a generally different direction sf Similar to the current in plane giant mag netoresistance the indirect mechanism is only active if the thickness of the nonmagnetic spacer is smaller than or comparable to the mean free path Numerical calculations for a semiclassical model based on the Boltzmann equation confirm the presence of both direct and indirect mechanisms of spin current generation First principles calculations reveal sizeable unconventional spin orbit torques in Co Cu Co Py Cu Py and Co Pt Co trilayers and provide strong evidence of indirect spin current generation I INTRODUCTION The discoveries of the giant magnetoresistance 1 2 and tunnel magnetoresistance 3 7 effects led to the development of many spintronic devices requir ing the control of magnetization For example spin transfer torque magnetic random access memories STT MRAM employ the tunnel magnetoresistance to read out the state of an individual bit 8 while writing is done using spin transfer torque 9 10 The latter effect relies on the transfer of angular momentum between layers and requires a relatively large charge current flow perpendicular to the layer planes Later it was discovered that the magnetiza tion of a ferromagnetic layer can also be controlled by running an in plane charge current through an adjacent nonmagnetic layer with sufficient spin orbit interaction this process is called spin orbit torque 11 Like spin transfer torques spin orbit torques allow all electrical control of magnetization while offering additional advantages such as higher en durance and the potential for better power efficiency 12 13 Spin orbit torques in magnetic nanostructures are attributed to several mechanisms including the spin Hall effect 14 18 Rashba Edelstein effect 19 22 orbital Hall effect 23 24 planar Hall ef fect 25 26 magnetic spin Hall effect 27 28 and interfacial spin current generation 29 30 Unfor tunately the multitude of mechanisms responsible for spin orbit torques lead to difficulties in inter preting experimental results On the other hand understanding these mechanisms is crucial to opti mize the switching efficiency of spin orbit torque based devices Further efficient field free magneti zation switching in layers with perpendicular mag netization requires additional modifications such as employing systems with lower structural 31 or crys tallographic 32 33 symmetry or ferromagnetic tri layers 34 35 Field free switching has already been demonstrated in ferromagnetic trilayers 35 36 38 The spin diffusion model is often suffi cient for identifying various mechanisms of spin orbit torque 39 41 On the other hand it is well un derstood that the current in plane giant magnetore sistance effect involves interlayer scattering with a length scale on the order of a mean free path a pro cess beyond the spin diffusion model 42 The inter play between such processes and spin orbit coupling and their contributions to spin orbit torques in fer romagnetic trilayers are the subject of the present paper that have no analogue In this work we classify and calculate spin orbit torques in ferromagnet normal metal ferromagnet trilayers in nonmag net ferromagnet bilayers including those not de scribable by spin diffusion models We refer to such torques as nonlocal because they require coupling between the two ferromagnetic layers via spin cur rents We identify two previously unrecognized phe nomena due to nonlocal torques First we find a disorder dependent dampinglike torque that exceeds typical torques in heavy metal ferromagnet bilay ers when in the parallel magnetization configuration but is strongly reduced in the antiparallel configura tion We argue this behavior is the spin orbit torque analogue of the current in plane giant magnetoresis tance and call it giant magneto torquance Second when the magnetizations of the two ferromagnetic layers are orthogonal to each other we identify un conventional fieldlike torques generated by the inter play of spin orbit coupling and interlayer scattering In general the nonlocal torques may be catego In the rized into direct and indirect mechanisms direct mechanism one ferromagnetic layer emits a spin current than is absorbed by the other ferromag netic layer In the indirect mechanism the in plane spin polarized current from one ferromagnetic layer leaks into the other ferromagnetic layer where it is converted into an out of plane spin current and reabsorbed by the original layer The main results of this paper are the following 1 A symmetry based characterization of spin orbit torques in ferromagnetic trilayers 2 A classification of the physical mechanisms of nonlocal spin orbit torques including those beyond the spin diffusion model 3 Semiclassical calculations qualitatively illus trating nonlocal spin orbit torques 4 Ab initio calculations showing that nonlo cal spin orbit torques in ferromagnetic trilay ers have comparable strength to conventional spin orbit torques in bilayers The paper is organized as follows Section II gives a brief background on