Spin Transfer Torques induced by Spin Hall Effect
A. Vedyayev, N. Strelkov, M. Chshiev, N. Ryzhanova, B. Dieny
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Spin Transfer Torques induced by Spin Hall Effect
A. Vedyayev,
1, 2
N. Strelkov,
1, 2
M. Chshiev, N. Ryzhanova,
1, 2 and B. Dieny SPINTEC, UMR 8191 CEA-INAC/CNRS/UJF-Grenoble 1/Grenoble-INP, 38054 Grenoble, France Department of Physics, Moscow Lomonosov State University, Moscow 119991, Russia
Spin accumulation and spin transfer torques induced by Spin Hall Effect in bi-layer structurescomprising ferromagnetic and paramagnetic materials are theoretically investigated. The charge andspin diffusion equations taking into account spin-flip and spin Hall effect are formulated and solvedanalytically and numerically for in structures with simplified and complex geometry, respectively. Itis demonstrated that spin torques could be efficiently produced by means of Spin Hall effect whichmay be further enhanced by modifying structure geometry.
INTRODUCTION
In [1, 2] Dyakonov and Perel predicted the existence of Spin Hall Effect (SHE) in paramagnetic metal withoutapplying the external magnetic field under influence of spin-orbit interaction. It was found later a lot of similaritiesbetween the “anomalous” Hall Effect in ferromagnetic metals (FM) and SHE [3, 4], assuming that in both casesthe mechanisms of creation of the transversal electric current is the skew scattering or side jump mechanism due tothe spin-orbit interaction. The more detailed analyses of SHE was presented in [5–7] employing the semi-classicalBoltzmann equation in [5] and Keldish formalism in [6, 7] for the calculation of the transport properties of theparamagnetic metal taking into account the spin-orbit interaction. It is considered that SHE may be very effectivetool for the manipulation with spin current and spin accumulation. The most interesting could be the hybrid structureconsisting of ferromagnetic (FM) metal/paramagnetic (PM) metal with large SHE, if both mechanisms of creation ofthe spin current are involved: SHE and spin polarization by ferromagnetic metal. In this work we investigate spintransfer torques produced by the interplay between these mechanisms in FM/NM bi-layer structures by solving chargeand spin diffusion equations.
MODEL
For the calculation both spin accumulation and spin polarized current in such a structure which may have a complexgeometry, it is necessary to derive spin diffusion equation taking into account both SHE for the paramagnetic metaland processes governing the spin transport in ferromagnetic metal. It is necessary also to develop the convenient codefor the numerical simulations for the system of the complicated geometry in 2D and 3D cases. Following [8–11] wherethe diffusion equation describing spin transport in ferromagnetic multilayered structures where developed and withdiffusion equation obtained in [7] and describing SHE we can write down: ~j e = − σ ~ ∇ ϕ − β σ eν ~ ∇ ( ~U M , ~m ) + a σ SH h ~m × ~ ∇ ϕ i (1) ~j ( i ) m = − βσ ~ ∇ ϕU ( i ) M − σ eν ~ ∇ m ( i ) − σ SH U ( i ) m h ~U m × ~ ∇ ϕ i (2) div ~j e = 0div ~j m ( i ) = − σ e νl J h ~m × ~U M i ( i ) − σ e νl m ( i ) , (3)where σ is the conductivity, β is the spin-asymmetry parameter of conductivity, σ SH is the spin Hall conductivity, ν – density of states, ~U M = ~M /M s , where ~M is magnetization vector in ferromagnetic, ~U M = ~m/ | ~m | , where ~m isspin accumulation vector, index i is a component of vectors ~m , ~j m and ~U M in spin space, l sf – spin diffusion lengthand l J – exchange spin diffusion length. Inside the ferromagnetic metal one have to omit the terms of SH, and inparamagnetic metal β = 0. In equation (1) we added the last term, which is quadratic on the ~ ∇ ϕ as the value of m isproportional to ~ ∇ ϕ , and in (2) we omitted terms corresponding to the contribution of the anomalous velocity (see (2)in [7]). Here we have to mention the article [12], where it was proven that the large σ SH /σ , experimentally observedfor Au doped by F e and
P t impurities [13, 14] and for Cu doped by Ir [15] may be attributed to the resonant electronscattering on impurities if take into account spin-orbit interaction.Let us consider the system, consisting of two flat layers, one of the paramagnetic metal with SHE and secondferromagnetic layer with current in x direction. If to consider the case L x > L y ≫ L Fz + L Pz , where L x and L Fz + L Pz are the lengths of the system in x and z directions, the solution of (3) in the region of x far from ( − L x , L x ) may beeasily found, and expression for ϕ , m (2) ( m (0) = m (3) = 0) are the following: m (1) = − V eν D σ SH σ l sf , L x (cid:20) sinh L l sf , sinh L + 2 z l sf , + σ σ l sf , l sf , (cid:0) − β (cid:1) tanh L l sf , sinh zl sf , (cid:21) (4) m (2) = − V eν D σ SH σ l sf , L x sinh L l sf , cosh L − zl sf , / cosh L l sf , (5) ϕ = V (cid:18) xL x (cid:19) − V ( eν ) (cid:18) σ SH σ l sf , L x (cid:19) sinh L l sf , × " L l sf , cosh L + 2 z l sf , + σ σ l sf , l sf , cosh zl sf , − cosh L x l sf , − cosh L x l sf , tanh L l sf , (6) ϕ = V (cid:18) xL x (cid:19) − V ( eν ) (cid:18) σ SH σ l sf , L x (cid:19) sinh L l sf , × (cid:20) sinh L l sf , + σ σ l sf , l sf , tanh L l sf , (cid:21) − V h sign U (2) M i β D σ SH σ l sf , L x cosh L l sf , − cosh L − zl sf , cosh L l sf , (7) D = sinh L l sf , + σ σ l sf , l sf , (cid:0) − β (cid:1) tanh L l sf , cosh L l sf , , where L and L – are thicknesses of SH-layer and FM-layer respectively. FIG. 1: Schematic of
P t / P y bylayer. Sizes are in “nm”. 1 –
P t layer, 2 –
P y layer, 3 – Cu electrodes. Current is along x axe.Magnetisation of P y is in xy plane at π/ x axe. The spin accumulation m (2) produces the effective field H eff acting on the magnetisation of the ferromagnetic layer,which value is equal H eff = m (2) J sd /µ B , where J sd is s - d exchange integral. It is important to notice that this field isproportional to drop of voltage and not to current density, as in the case of Oersted field created by the current. Soif one choose as a source of SHE dirty paramagnetic metal only due to its higher resistance the value of the inducedby SHE effective field for the constant current density will increase, and besides that the following conclusion of [12]the value of σ SH /σ may increase as well. Another interesting conclusion, following from expression for the potential ϕ , is that SHE produce drop of voltage in z direction perpendicular to the current. If there is no ferromagnetic layerabove SHE structure the drop of voltage is symmetric ( ∝ cosh[ z/l sf ]) and quadratic on the applied voltage V , but inpresence of FM layer this drop has linear on V part and is finite across the thickness of the system. To investigatethe influence of the edge of layers on spin accumulation in the system of finite size we have solved equation of (3)numerically using Comsol Multiphysics for several artificial system, consisting of SHE substrate and ferromagneticlayers or dots situated on the top of this substrate. -500 -480 -460 440 460 480 500-200-1000100200 H , O e x, nm H x H y H z FIG. 2: Effective fields induced by
P t
SHE in
P y near the interface. Adopted values of the parameters: σ Pt = 0 . · nm ) − , l Pt sf = 10 nm , σ Pt SH = 0 . σ Pt , σ Py = 0 . · nm ) − , l Py sf = 6 nm , β = 0 . l J = 1 nm , current density j = 10 A/cm .FIG. 3: Schematic of P t / P y bylayer. Sizes are in “nm”. 1 –
P t layer, 2 –
P y layer, 3 – Cu electrodes. Current is along x axe.Magnetisation of P y is in xy plane at π/ x axe or along z axe. RESULTS
In Fig.1 we give the schematic of
P t / P y bilayer, which was experimentally investigated in [16]. In fig.2 thedependence on x coordinate of the induced by SHE fields H SHE acting on the magnetization inside of
P y layer andnear the
P t / P y interface is shown. It is clear that SHE spin current produce all tree components of the field, meanwhileOersted field H j produced by the current in given geometry has only y -component, and H j = 4 Oe. The z -componentof the SHE field produces torque in the plane of the P y layer, and the ratio H SHE /H j lies within the interval 1 . ÷ . . P y layer while 3 . S/A = 0 .
63 in [16] which gives H SHE /H j = p πM eff /H eff S/A = 1 . π/ x -axes, or it is parallel or antiparallel to y axes. In the case ~M k y only y -component ofinduced fields not zero and its average value does not change under inversion of magnetization direction along y axes.This field is much stronger than Oersted field which is about 5 Oe. Besides that field does not produce any torque butdefines the most energetically favorable direction of magnetization. When this direction is not collinear with y -axes,the SHE produces not only y -component of spin accumulation but all three xyz components of spin accumulation dueto the precession of spin accumulation vector in exchange field of ferromagnet.In fig.5 the dependence on x coordinate of three components of spin transfer torque, produced by spin accumulation -10 -5 0 5 10-1400-1200-1000-800-600-400-2000200400 H , O e x,nm M y =1 M y =-1 FIG. 4: Effective fields in Co / P t structure for the direction of magnetisation parallel and antiparallel to y axe. Only H y component survive. σ Co = 0 . · nm ) − , l Co sf = 10 nm . -10 -5 0 5 10-20020406080100 T , O e x, nm Tx Ty Tz FIG. 5: Effective torques for the case of ~U M = (cos π/
4; sin π/
4; 0) near the interface. Averaged values over volume of Co are < T x > = 15 Oe , < T x > = − Oe , < T z > = 3 Oe . due to the SHE are shown. The torque acting on magnetisation is defined as h ~M × ~H SHE i , and in LLG equationit has to be multiplied by the gyromagnetic ratio γ . For the case U m = (cos π/
4; sin π/
4; 0) the xy part of torquevector ~T = ( T x ; T y ; 0) is lying in xy plane perpendicular to ~U m and may by considered like an additional damping orantidamping term in LLG equation. The torque due to Oersted field has only z -component.For the case of ~M k Oz , T x and T y torque components value close to the interface are shown in fig.6. Thesetorques try to reorient magnetization of FM layer onto xy plane and T y component may be considered like dampingor antidamping term. We checked the results taking the spin Hall conductivity equal to zero and obtained that alltorques besides one produced by Oersted field of the current vanish. -10 -5 0 5 10-140-120-100-80-60-40-20 T , O e x, nm T x T y FIG. 6: Effective torques near the interface in case of M k Oz . Averaged values over volume of Co are < T x > = − Oe , < T y > = − Oe . CONCLUSIONS
We have shown that SHE may represent a powerful tool for the manipulation with magnetization of small ferromag-netic metal dots situated on the surface of the thin paramagnetic metal layer with large value of SHE conductivity.Especially important feature of spin transfer torques created by spin accumulation due to influence of SHE is thatthese torques have component similar to damping or antidamping spin torque produced by current in non-collinearmagnetic multilayers.
ACKNOWLEDGMENTS
This works has been supported by Russian Fund of Fundamental Research, ERC Advanced Grant ”HYMAGINE”and by French National Research Agency Project ANR-10-BLANC ”SPINHALL”. [1] M. I. D’yakonov and V. I. Perel’, JETP , 467 (1971).[2] M. I. Dyakonov and V. I. Perel, Phys. Lett. A , 459 (1971).[3] A. Fert, A. Friederich, and A. Hamzic, Journal of Magnetism and Magnetic Materials , 231 (1981).[4] J. E. Hirsch, Phys. Rev. Lett. , 1834 (1999).[5] S. Zhang, Phys. Rev. Lett. , 393 (2000).[6] R. V. Shchelushkin and A. Brataas, Phys. Rev. B , 045123 (2005).[7] R. V. Shchelushkin and A. Brataas, Phys. Rev. B , 073110 (2005).[8] T. Valet and A. Fert, Phys. Rev. B , 7099 (1993).[9] S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. , 236601 (2002).[10] N. Strelkov, A. Vedyayev, D. Gusakova, L. D. Buda-Prejbeanu, M. Chshiev, S. Amara, A. Vaysset, and B. Dieny, MagneticsLetters, IEEE , 3000304 (2010).[11] N. Strelkov, A. Vedyayev, N. Ryzhanova, D. Gusakova, L. D. Buda-Prejbeanu, M. Chshiev, S. Amara, N. de Mestier,C. Baraduc, and B. Dieny, Phys. Rev. B , 024416 (2011).[12] A. Fert and P. M. Levy, Phys. Rev. Lett. , 157208 (2011).[13] S. Takeshi, H. Yu, M. Seiji, T. Saburo, I. Hiroshi, M. Sadamichi, N. Junsaku, and K. Takanashi, Nature Materials , 125(2008).[14] S. Takeshi, private communication.[15] Y. Niimi, M. Morota, D. H. Wei, C. Deranlot, M. Basletic, A. Hamzic, A. Fert, and Y. Otani, Phys. Rev. Lett. , 126601(2011).[16] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett.106