Effect of anisotropic spin absorption on the Hanle effect in lateral spin valves

Transcription

1 SUPPLEMETARY IFORMATIO Effect of anisotropic spin absorption on the Hanle effect in lateral spin valves H. Idzuchi,*, Y. Fuuma,3, S. Taahashi 4,5, S. Maeawa 5,6 and Y. Otani,* Institute for Solid State Phsics, Universit of Too, Kashiwa , Japan Center for Emergent Matter Science, RIKE, - Hirosawa, Wao , Japan 3 Frontier Research Academ for Young Researchers, Kushu Institute of Technolog, Kawazu, Iizua , Japan 4 Institute for Materials Research, Tohou Universit, Sendai , Japan 5 CREST, Japan Science and Technolog, Too , Japan 6 Advanced Science Research Center, Japan Atomic Energ Agenc, Toai 39-95, Japan on-local resistance in a lateral spin valve When a magnetic field B = (0, 0, B z ) is applied perpendicular to the plane of a spin injection and detection device consisting of a nonmagnetic metal () connected to the ferromagnets of the injector (F) and the detector (F) with the magnetizations (white arrows) along the direction, the injected spins in the electrode precess around the z axis parallel to B, as shown in Fig. S. i When the spin-current I se i polarized along the e i direction ( i = x, ) is injected from F into i i at x = 0 ( I s > 0) through the st junction and the spin current I se i is absorbed b F at x = L ( I < 0) through the nd junction, the motion of the spin densit S due to the spin accumulation i s is governed b the diffusion-modified Bloch-Torre equation [,43] S S h I h I = γ + D + δ( x) + δ( x) t e A e A x s s es B S ex e τ sf h I h I + ( x L) + eδ( x L), ea x s s exδ ea (S) where τ sf is the spin lifetimes, D is the spin diffusion constant, and A is the cross-sectional area of electrode, e i is the unit vector of the i-th direction, and the spin current is taen to be the same unit as charge current. In perpendicular magnetic fields smaller than the demagnetization field, the out-of-plane component S z of the spin densit is small and is disregarded for simplicit. In the stead state ( S / t = 0), Equation (S) is solved to ield the spin densit S = (S x, S, 0) in

2 Fig. S. Precession of accumulated spins in in a lateral spin valve in the presence of perpendicular magnetic field B where the spin accumulation (spin densit) S rotates during the travel of distance L between the injector F and the detector F. The projection of S (S ) along the magnetization of F is detected b F as output voltage V. the complex representation [] Sx % h () = S() x+ is() x= % Ie % + I% e (S) λ / x x L / x s s 4eDA x with the complex representation of spin currents I % = I + ii and x I % = I + ii through junction and and λ s s s s s s % =, + iτ (S3) L sf λ where L = γ e B z is the Larmor frequenc, γe = μb/ h is the gromagnetic ratio of conduction electrons and λ = Dτsfbeing the spin diffusion length in the absence of the perpendicular magnetic field. Since is the complex quantit, exp( x / ) exhibits a damped oscillation as a function of x. We note that (S) is rewritten as [44] I% I% Sx % h h ( ) = dtp ( xte, ) + dtp ( x Lte, ), (S4) ea s ilt s ilt em em 0 0 ea where P (, ) em x t is the transit-time distribution function:

3 x /(4 Dt) t/ τsf Pem( x, t) = e e. (S5) 4π Dt When describing the spin transport in the presence of spin precession, it is convenient to use the complex spin accumulation δμ( x) = (/ ) S% x % h ( x)/ ( εf) = δμ( x) + iδμ ( x) given b where e δμ% =, (S6) λ / x x L / ( x) I% λ s e λ + I% s e σ A σ is the electrical conductivit of the electrode and (ε F ) is the densit of states (per spin) at the Fermi energ. The absolute value δμ% ( x) corresponds to the splitting in the electrochemical potentials (ECP) of the up and down spin electrons. The charge transport is described b the average of the ECPs: μ = ( ei / σ A) x for x < 0 and μ = 0 (ground level of ECP) for x > 0. In equation (S6), the first term represents the increase of spin accumulation due to spin injection from F and the second term is the decrease due to spin absorption b F. ote that the charge current is absent and the pure spin current flows in the region of x > 0. The spin current densit flowing in the x direction is given b the complex representation % s ( ) σ j ( ), x = δμ% x (S7) e Since the thicnesses of the F and F ( 0nm) are much larger than the spin diffusion length ( λ ~5 nm), we ma tae the spin-dependent ECPs in F and F close to the interfaces in F the forms of vertical transport along the z direction [4] μ μ eλ ( z) = μ ( + ) I P I e, z/ λf ( s ) ( ) c F F F ( ) F σ F AI = + eλ ( ) c F z/ λf F ( z) μf ( ) I e, ( ) s σ F AI (S8) μ = ( ei / σ A ) z + ev represents the EPC of charge in F, A I is the contact area of the c where F F I c -th interface, μ F = ev ev taes a constant potential with no charge current in F, V and V are the voltage drops across junctions and, respectivel, λf is the spin-diffusion length of F and F, and PF = ( σf σf )/ σf where ( σf = σf + σf ) is the spin polarization of the F. The interfacial spin-dependent currents across the -th junction ( =, ) with the 3

4 polarizations parallel ( I % ) and antiparallel ( I% ) to the magnetization direction of F are [,4,45,46] F(0) ( x ) ( x ) I% μ δμ% δμ% = GI Re iai GI Im, e e e μf (0) δμ ( x ) ( ) δμ x I% % % = GI + Re + iai GI Im, e e e (S9) where G σ is the spin-dependent interface conductance of -th junction, G is the transverse I interface conductance per area, so-called the spin-mixing conductance with the dimension Ω m, and x = 0 and x = L. These enable to address the effect of spin absorption not onl for longitudinal spin accumulation [] but also for transverse one. We note that the complex representations (S9) are equivalent to the vector representation in [44]. The total charge and spin currents across the -th interface are I I = % + I %, ( I = I, I = 0) and I% s I = % I%. The above interfacial currents are applicable to junctions from tunneling to transparent regime. Using the boundar conditions that the spin and charge currents are continuous at the interfaces of the junctions and, we can derive the matrix equation for the interface spin currents I I s I I 0 s PI R PF R F x I = + P I R PF R 0 s Xˆ, x I 0 s (S0) with the matrix [ ] r + Re[ λ] Re[ λe ] Im[ λ] Im λ e ˆ Re[ λe ] r + Re[ λ] Im[ λe ] Im[ λ] X =, Im[ λ] Im[ λe ] r + Re[ λ] Re[ λe ] Im[ λe ] Im[ λ] Re[ λe ] r + Re[ λ] (S) where λ = λ and / r = R + R, r =, ( =,) I F. PI R PF R RAI G 4

5 Here, RI / GI R ( G = G + G ) is the interface resistance (conductance) of junction, = I I I = ( ρ λ / A ) and R F = ( ρ F λ F / A I ) are the spin resistances of and F electrodes, ρ and ρ F are the resistivities, and P I = ( G I G I )/( G I + G I ) is the interfacial spin-current polarization. The boundar conditions also lead to the non-local voltage V due to the spin accumulation detected b F, V P R P R = + F F I I PF R PI R R I s, (S) where the minus sign indicates the absorption of spin current b F. Using the solution of the matrix equation (S0), we obtain the non-local resistance V PF RF PI R I PF RF PI R I C = R +, + I P ˆ F R PI R PF R PI R det( X ) (S3) where det( Xˆ ) is the determinant of the matrix ˆX in (S) and C is the (, ) component of the cofactors of ˆX, Re[ λe ] Im[ λe ] Im[ λ ] C = det Im[ λ] r + Re[ λ] Re [ λe ]. Im[ λe ] Re[ λe ] r + Re[ λ] (S4) When junctions and are tunnel junctions (R I >> R, R F ), (S3) reduces to [,44] V I ( λ λ ) = PPR I I Re % / exp( ). (S5) In the absence of perpendicular magnetic field, (S3) reduces to the previous result of [4]. 5

6 Simulated Hanle curves for a graphene based lateral spin valve In order to underscore the validit of our analsis, we fit the reported Hanle signal for a graphene based lateral spin valve with transparent junctions [8]. As shown in Fig. S and table SI, the deduced spin lifetime and diffusion constant are consistent with those from tunnel junctions. w F = 50 nm, w = 00 nm, λ F = 60 nm, ρ F = 6 μωcm, R I = 85 Ω, σ = 0.35 ms are taen from ref. [8]. Table SI: Adjusting parameters for Hanle signals for a graphene based lateral spin valve with Co/Graphene junction. L (μm) P F P I(Co/Graphene) τ sf (ps) D (cm /s) G (m - Ω - ) Fig. S. Simulated Hanle curve of graphene based lateral spin valve with transparent junctions. Dots (experimental data) are adapted from ref. [8] and blue lines are calculated from equation (S3). 6

7 Spin valve measurement In order to mae the analsis simple, we used the same widths of P wires. The switching field of each P wire was controlled b the domain-wall nucleation, i.e., the injector had a large domain wall reservoir at the edge, producing lower switching field than the detector as shown in Fig. S3 [47]. The initial state of the Hanle measurements was set as follows. For the parallel state, firstl P was initialized b the large field (~ 000 Oe), and then the field was set to zero. For the antiparallel state, firstl P was initialized b the large field, and secondl the field was decreased to over the first switching field (~ -00 Oe), finall the field was set to zero. Fig. S3. Spin valve measurement (red lines). Blue line indicates the initialization of P magnetizations for antiparallel configuration for the Hanle measurements. Magnetic field was applied in parallel with the eas axis of P. Bold arrows show the magnetization states of injector and detector ferromagnets. 7

8 The quantitative evaluation of τ sf on the change of G Here we discuss the effect of anisotrop on the evaluation of spin lifetime. In the case of isotropic spin absorption, G = /A J {/(R I )+/(R F )}, as shown in (S). For the Ag based lateral spin valve with P/Ag junctions with L = 3 μm, the derived τ sf with isotropic spin absorption is 0 % smaller than the one with anisotropic spin absorption because of underestimation of spin absorption. In contrast, for the graphene based lateral spin valve with transparent junction (R I = 85 Ω) [8], the derived τ sf with isotropic spin absorption is almost same, which is consistent with derived G is onl 4 % different from /A J {/(R I )+/(R F )}. The small effect of anisotrop is attributed to higher junction resistance compared the one with Ohmic contact in metallic sstem. Additional References [43] M. Johnson, and R. H. Silsbee, Phs. Rev. B 37, 53 (988). [44] F. J. Jedema, M. V. Costache, H. B. Heersche, J. J. Baselmans, and B. J. van Wees, Appl. Phs. Lett. 8, 56 (00). [45] T. Valet, and A. Fert, Phs. Rev. B 48, 7099 (993). [46] X. Wang, G. E. Bauer, B. J. van Wees, A. Brataas, and Y. Tserovna, Phs. Rev. Lett. 97, 660 (006). [47] H. Idzuchi, Y. Fuuma, L. Wang and Y. Otani, Appl. Phs. Exp. 3, (00). 8