Spin orbit torque in a three fold symmetric bilayer and its effect on magnetization dynamics Wuzhang Fang Edward Schwartz Alexey A Kovalev and K D Belashchenko Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience University of Nebraska Lincoln Lincoln Nebraska 68588 USA Field free switching of perpendicular magnetization has been observed in an epitaxial L11 ordered CoPt CuPt bilayer and attributed to spin orbit torque SOT arising from the crystallographic 3m point group of the interface Using a first principles nonequilibrium Green s function formalism combined with the Anderson disorder model we calculate the angular dependence of the SOT in a CoPt CuPt bilayer and find that the magnitude of the 3m SOT is about 20 of the conventional dampinglike SOT We further study the magnetization dynamics in perpendicularly magnetized films in the presence of 3m SOT and Dzyaloshinskii Moriya interaction using the equations of motion for domain wall dynamics and micromagnetic simulations We find that for systems with strong interfacial DMI characterized by the N el character of domain walls a very large current density is required to achieve deterministic switching because reorientation of the magnetization inside the domain wall is necessary to induce the switching asymmetry For thicker films with relatively weak interfacial DMI and the Bloch character of domain walls the deterministic switching with much smaller currents is possible which agrees with recent experimental findings PACS numbers I INTRODUCTION Spin orbit torque SOT 1 provides an e icient way of controlling the magnetization in spintronic devices such as the magnetic random access memories In ferromagnet FM heavy metal HM bilayers the spin Hall effect in a non magnetic layer with strong spin orbit interaction SOI can be used to produce a flow of angular momen tum into neighbouring ferromagnet and thus induce SOT Various mechanisms can contribute to SOT and many experiments are consistent with the spin Hall ef fect 2 6 mechanism of the dampinglike SOT and the inverse spin galvanic effect mechanism 7 10 of the field like SOT Other mechanisms of SOT have also been iden tified such as the orbital Hall effect 11 12 the planar Hall effect 13 14 the magnetic spin Hall effect 15 16 and various interfacial mechanisms e g associated with interfacial spin current generation 17 19 Current induced magnetization switching by SOT has been demonstrated in bilayers with in plane magneti zation 4 Current induced switching of perpendicular magnetization in systems with axial symmetry requires additional symmetry breaking mechanism such as appli cation of external magnetic field 4 5 9 Alternatively to enable the field free switching of perpendicular mag netization one can resort to systems of trilayers 20 21 or systems with lower symmetry 22 25 A possibility of field free switching of perpendicular magnetization in a system with lower crystal symmetry has been demon strated in a recent experiment 24 These two authors contributed equally alexey kovalev unl edu belashchenko unl edu In this work we consider a bilayer system with lower crystal symmetry which as has been demonstrated in Ref 26 can display spin Hall effect with unconventional polarization and unconventional SOT Developing these ideas further one can expand SOT in orthogonal vector spherical harmonics identifying only those contributions that are allowed by symmetry 19 27 The CoPt CuPt bilayer considered here has C3v symmetry which allows the existence of so called 3m dampinglike and fieldlike torques 24 28 We begin by performing first principles calculations of the spin orbit torquances in an L11 ordered CoPt CuPt bilayer and find appreciable 3m field like and damping like torques in addition to standard damping like and field like torques We find that the 3m damping like torque decreases significantly at higher disorder strengths while the 3m field like torque is largely un affected by disorder We then examine the magnetization reversal process in a disordered magnetic system as a result of these torques We use a collective coordinate model to demon strate that the 3m damping like torque results in domain wall expansion in a way analogous to an out of plane ap plied field while the effect of the 3m field like torque is more complicated For N el type domain walls the 3m field like torque produces no net expansion of an isolated domain In contrast Bloch walls experience a finite av erage pressure which is approximately quadratic in the applied current density for arbitrarily small 3m torque Our collective coordinate model is supported by micro magnetic simulations 2 Disorder was simulated within the Anderson model as a uniformly distributed random potential Vi Vm Vi Vm applied on each atomic site i To account for the potential drop in the contacts the electric field in the embedded region is found as E V Le where V is the voltage drop between the left and right leads 1 and R 1 GLB is the Landauer Le R dR dL B ttiker resistance of the active region of thickness L 39 For the Fermi surface contribution we considered three disorder strengths Vm 0 65 0 82 and 0 98 eV which yield resistivities of 13 19 and 26 cm re spectively Using supercells with Ly 1 2 and 4 we found that Ly 2 which corresponds to 12 monolay ers as seen in Fig 1 is su icient to converge the 3m torques which are of central interest here Unless oth erwise noted torquances were calculated using 150 2 supercells Averaging was performed over 150 disorder configurations at Vm 0 65 eV and over 36 configura tions at 0 82 and 0 98 eV To estimate the Fermi sea contribution to the DL and 3m torques we used a 60 1 supercell with 100 disor der configurations at Vm 0 65 eV A finite bias of or der 0 1 mV was applied symmetrically and the torques at a positive and negative bias were subtracted to re move the equilibrium torque associated with disorder dependent magnetic anisotropy The Fermi sea contribu tion involves an integral over the filled states which was evaluated on an elliptical contour shifted vertically by 1 mRy along the imaginary axis A 101 point Legendre quadrature was used on the semi ellipse and a uniform 10 point mesh on each vertical segment The total torquances acting on the magnetization of the whole bilayer will be expressed per unit area of the 1 For the DL interface in units of 105 2e and FL torques this definition is identical to the torque FL that are often used in experimental e