Models in spintronics (Part I)

Transcription

1 Models in spintronics (Part I) OUTLINE : Spin-dependent transport in metallic magnetic multilayers -Introduction to spin-electronics -Spin-dependent scattering in magnetic metal -Current-in-plane Giant Magnetoresistance -Modelling transport in CIP spin-valves -Current-perpendicular-to-plane Giant Magnetoresistance -Spin accumulation, spin current, 3D generalization.

2 Birth of spin electronics : Giant magnetoresistance (1988) Fe/Cr multilayers Fert et al, PRL (1988), Nobel Prize 27 Two limit geometries of measurement: I ~ 8% V I Current-in-plane I V Magnetic field (kg) I I Current-perpendicular-to-plane GMR Antiferromagnetically coupled multilayers RAP RP RP

3 Antiferromagnetic pinning layer NiFe 4Å Ferromagnetic pinned layer Cu 22Å Non magnetic spacer NiFe 7Å Ta 5Å Soft ferromagnetic layer Buffer layer R/R (%) 4 Ultrasensitive magnetic field sensors (MR heads) Spin engineering et al, Phys.Rev.B.(1991)+patent US52659 (1991). (b) -2 substrate IBM Almaden (a) 4 R/R (%) FeMn 9Å M (1-3 emu) Low field GMR: Spin-valves 4 2 H(Oe) 6 8 (c) H(Oe) 4

4 Benefit of GMR in magnetic recording GMR spin-valve heads from 1998 to 24

5 Two current model (Mott 193) for transport in magnetic metals As long as spin-flip is negligible, current can be considered as carried in parallel by two categories of electrons: spin and spin (parallel and antiparallel to quantization axis) Sources of spin flip: magnons and spin-orbit scattering Negligible spin-flip often crude approximation (spin diffusion length in NiFe~4.5nm, 3% spin memory loss at Co/Cu interfaces)

6 Spin dependent transport in magnetic metals (1) Band structure of 3d transition metals In transition metals, partially filled bands which participate to conduction are s and d bands Non-magnetic Cu : Magnetic Ni : E E s (E) s (E) s (E) s (E) D (E) D (E) D (E) D (E) E E D (EF) D (EF) D (EF) = D (EF) Most of transport properties are determined by DOS at Fermi energy Spin-dependent density of state at Fermi energy

7 Spin dependent transport in magnetic metals (2) m*(d) >> m*(s) J mostly carried by s electrons in transition metals Scattering of electrons determined by DOS at EF : Fermi Golden rule : s s d s s d E s (E) 2 P i W f D f (EF ) D (E) s (E) D (E) Most efficient scattering channel Spin-dependent scattering rates in magnetic transition metals Example: Co 1nm; Co 1nm

8 Potential experienced by conduction electrons in magnetic metallic multilayers Parallel magnetic configuration Antiparallel magnetic configuration Co/Cu majority electrons : antiparallel magnetic configuration Co/Cu majority electrons : parallel magnetic configuration Co Cu Co (a) Co/Cu minority electrons : parallel magnetic configuration Cu (c) Co/Cu minority electrons : antiparallel magnetic configuration U(x,z) (b) (d) x z Lattice potential modulation due to difference between Fermi energy and bottom of conduction band (reflection, refraction) Spin-dependent scattering on impurities, interfaces or grain boundaries (Dominant effect in GMR)

9 Simple model of Giant Magnetoresistance Parallel config Antiparallel config Fe Cr Fe Fe Cr Fe ap 1 Equivalent resistances : P 2 AP Key role of scattering contrast 2

10 Modeling current-in-plane transport (semi-classical Boltzman theory of GMR) Approach initiated by Camley and Barnas, PRL, 63, 664 (1989) Gas of independent particles described by distribution f(r, v, t), submitted to force field F (=-ee for electrons in electrical field E). Time evolution of the distribution described by Boltzman equation: Equilibrium function conserved in a volume element drdv along a flow line. t+dt E t df df df In presence of scattering, dt dt F dt scattering Balance between acceleration due to force and relaxation due to scattering f f f f f f f df. v. v. v. a. a.a z x y z x y y z v x v y v z dt F t x f v. r f a. v f ma F with t

11 df df dt F dt scatt In stationary regime, f F v. r f. v f t m f t In single relaxation time approximation ( ) f f df dt scatt Where f is the equilibrium distribution (Fermi Dirac for electrons). Boltzmann equation for electron gas in electrical field E: ee f f v. r f. v f m

12 Modeling current transport in bulk metals I f r, v f r, v g r, v V Fermi-Dirac f r, v I Perturbation due to electric field 1 F exp k BT 1 ee f f v. r f. v f m E x Spatially homogeneous transport r f kz ee x f g (v x ) m v x In k-space, shift in Fermi surface by k ee x k kx ky

13 Modeling current transport in bulk metals (cont d) Current density: 3 j e v x g (v ) d v ne 2 1 j E E E m ne 2 m Well-known expression of conductivity in Drude model n=density of conduction electrons n~1/atom in noble metals such as Cu, Ag, Au n~.6/atom in metals such as Ni, Co, Fe

14 Typical resistivity of sputtered metals Material Measured resistivity 4K/3K Material (ferro) Measured resistivity 4K/3K Cua.5-.7.cm 3-5 Ni8Fe2a 1-15.cm Agf 1.cm 7 Ni66Fe13Co cm 2-23 Aug 2.cm 8 Pt5Mn5e Ni8Cr2 Ruc e 16.cm cm cm 14-2 b 1 Coa,d cm Co9Fe1h 6-9.cm Co5Fe5h 7-1.cm 15-2 Thermal variation of resistivity due to phonon scattering and magnon scattering (in magnetic metals)

15 Modeling current transport in metallic thin films I f r, v f r, v g r, v z V ee f f v. r f. v f m I E x Due to scattering at outer surfaces, the perturbation g is no longer homogeneous: g(z) g z, v g z, v ee f v z v z mvz v x vf General solution : = elastic mean free path f g z, v ee v x Integration constants determined from boundary conditions z 1 A exp v z +(-) refer to electrons traveling towards z> (z<)

16 Boundary conditions for a thin film : g zsup g zinf g zsup zsup g zsup Rg zsup g zinf Rg zinf g zinf Proba R Proba 1-R zinf Proba of specular reflection R Proba of diffuse reflection 1-R Two boundary conditions, two unknowns A+, A-, solvable problem j(z) e v x g v z, z d 3v Current density: t t A exp d j ( z ) 1 2 A exp 1 2 t=layer thickness, = cosine of electron incidence

17 Modeling current transport in metallic thin films (cont d) G G G1 ne 2 G vf m thickness t t Same conductance as in bulk 1 t t 3 2 A 1 exp G1 d 1 A 1 exp 4 G1 contains all finite size effects. Characteristic length in current-in-plane transport=elastic mfp Fuchs-Sondheimer approximate expressions: j(z) Assuming diffuse surface scatt mvf ne for t 8t 4mvF 3ne 2 1 for t t ln / t.423

18 Models in spintronics (Part I) OUTLINE : Spin-dependent transport in metallic magnetic multilayers -Introduction to spin-electronics -Spin-dependent scattering in magnetic metal -Current-in-plane Giant Magnetoresistance -Modelling CIP Giant Magnetoresistance -Current-perpendicular-to-plane Giant Magnetoresistance -Spin accumulation, spin current, 3D generalization.

19 Modeling current in plane GMR in metallic multilayers I Same as for thin films but separately considering spin and spin electrons and solving the Boltzmann equation within each layer with appropriate boundary conditions. V I Bulk spin-dependent scattering described by spin dependent mfp Interfacial scattering described by spin dependent R and T CIP Proba Ti of transmission Layer i+1 Interface i Layer i Proba Ri of reflexion Proba 1-Ri -Ti of diffusion g i 1, Ti g i, Ri g i 1, g i, Ti g i 1, Ri g i,

20 Modeling current in plane GMR in metallic multilayers G G G1 N n t G e 2 i i i ti / i i 1, v F mi i 1, N 3 i G1 4 i Same conductance as if all layers were connected in parallel t i t i d 1 Ai, 1 exp Ai, 1 exp,i i i 1 2 G1 contains all finite size effects and is responsible for GIP GMR. To obtain the GMR, G1 is calculated in parallel and antiparallel configurations. In AP configuration,, R, T and, R, T are inverted in every other layer. CIP GMR comes from second order effect in conductivity (in contrast to CPP GMR) Characteristic lengths in CIP GMR are the elastic mean-free paths

21 Influence of feromagnetic layer thickness on GMR in spin-valves Example of calculated CIP curves For a NiFe tnife/cu 2nm/NiFe tnife Sandwich. R=coeff of specular reflection at lateral edges.3 G ( -1) p=1 R=1.1 NiFe 7 nm, NiFe.8nm, p = R=.5 Cu 12nm, R=1-1 G ( ) p=1 T NiFe / Cu T NiFe / Cu.9. 3 Absolute change of sheet conductance most intrinsic measure of CIP GMR. 2 p= R=. 1 F/Cu25Å/NiFe5Å/FeMn1Å R/R p (%) p=1 R=1 1 5 p= R= t (nm) (Å) 2 25 JMMM (1994) 3 F

22 R/R p (%) R/R (%) Influence of nature of ferro materials on GMR in spin-valves NiFe 2 Fe 1 1 Co tf (nm) tf (nm) F tf/cu 2.5nm/NiFe 5nm/FeMn 1nm, with F=Ni8Fe2, Co and Fe (Dieny, 1991). F tf/cu 2.5nm/NiFe5nm/FeMn 1nm NiFe 7 nm, NiFe.7nm, TNiFe. 85, T / Cu NiFe / Cu.3, Co 9nm, Co.9nm, TCo / Cu.95, TCo / Cu.3, Fe 4.5nm, Fe 4.5nm, TFe / Cu.7, TFe / Cu.3.

23 Influence of non-magnetic spacer layer thickness on GMR in spin-valves NiFe5Å/NM tnm /NiFe5Å/FeMn1Å NiFe 3nm/Cu tcu/nife 3nm 2 p R/R (%) p= p= 5 1 tcutf (nm) 15 Experiments Semi-classical theory JMMM (1994) Two effects as tnm increases: Reduced number of electrons travelling from one ferro layer to the other Increasing shunting of the current in the spacer layer

24 Local current density and magnetic field due to current in a CIP spin-valve t t j ( z ) Ai, exp i Ai, exp i d,i i i 1 Local current density (u.a) Magnetic field due to sense current (Oe) Oersted field due to sense current in a CIP spin-valves J z B ( z ) j ( z ' )dz ' cst <j>=2.17a/cm² Oe field plays an important role in the bias of spin-valve heads

25 Vertical head with CIP MR reader reader writer 1 m Fly height 1nm, MR shield 2 shield 1 substrate Planar coil Pinned Free J 1nm Field to be measured disk rotating at 5 to 1 rpm.

26 Linear response of spin-valves MR heads Energy terms influencing the orientation of free layer magnetization: Zeeman coupling to H, to coupling field through Cu spacer and to Oersted field: E M 1 ( H H SV H J ). sin( ) Uniaxial and shape anisotropy E ( K NM s2 ). cos ²( ) Dipolar coupling with pinned layer E M 1 H dip. sin( ) Minimizing energy yields sin( ) linear in H. Since R varies as M 1.M 2 sin( ) linear R(H) M 1 H H SV H I H dip R H RP RAP RP K NM 1

27 Experimental optimization of spin-valves Spin-valves greatly optimized in for HDD MR heads but replaced in 24 by TMR heads NiFeCr/PtMn 12Å/ CoFe 15Å /Ru 7Å/CoFe 5Å/NOL/CoFe 15Å /Cu 2Å/CoFe1Å/NiFe 2Å /NOL pinned soft 25 2 GMR (%) H (Oe) GMR amplitude significantly improved by increasing specular reflection at boundaries of active part of the spin-valve //(pinned/spacer/free)//

28 Conclusion on CIP GMR : CIP GMR well modeled by semi-classical Boltzmann theory. Local conductivity can only vary on length scale of the order of the elastic mean free path. Quantum mechanical theory of CIP transport were proposed to properly take into account the quantum confinement/reflection/refraction effects induced by lattice potential modulation. Predictions of oscillations in conductivity and GMR versus thickness but hardly seen in experiments due to interfacial roughness CIP GMR amplitude up to 2% in specular spin-valves. Spin-valves were used in MR heads of hard disk drives from 1998 to 24. Later on, they were replaced by Tunnel MR heads.

29 Models in spintronics (Part I) OUTLINE : Spin-dependent transport in metallic magnetic multilayers -Introduction to spin-electronics -Spin-dependent scattering in magnetic metal -Current-in-plane Giant Magnetoresistance -Modelling CIP Giant Magnetoresistance -Current-perpendicular-to-plane Giant Magnetoresistance -Spin accumulation, spin current, 3D generalization.

30 Current Perpendicular to Plane GMR I V I I Much more difficult to measure, Either on macroscopic samples (.1mm diameter) with superconducting leads (R~. thickness / area ~ 1 ) or on patterned microscopic pillars of area < m² (R~ a few Ohms)

31 Serial resistance model for CPP-GMR Without spin-filp, serial resistance network can be used for CPP transport CPP transport through F/NM/F sandwich described by: (a) Parallel magnetic configuration : F t F ARF / NM NM t NM ARF / NM F t F F t F ARF / NM NM t NM ARF / NM F t F (b) Antiparallel magnetic configuration : F t F ARF / NM NM t NM ARF / NM F t F F t F ARF / NM NM t NM ARF / NM F t F

32 Serial resistance model for CPP-GMR: Parallel resistance model for CIP-GMR: CPP GMR of (Co tco/cu tcu) 2 multilayers CIP GMR of (Co tco/cu tcu) 2 multilayers Parallel configuration : Parallel magnetic configuration : Antiparallel configuration : Cot Co Co t Co /Cot Co CuCut Cu/ tcu Cut Cu Cot Co Cut Cu Spin up channel Spin down channel Co tco Cot/Co CuCut Cu / tcu Cot Co Cut Cu Cot Co Cut Cu Antiparallel magnetic configuration : /t t Co Co Co Co CuCut Cu / tcu Cot Co Co tco Cot/Co CuCut Cu/ tcu Cut Cu Cot Co Cut Cu Spin up channel Spin down channel CIP GMR cannot be described by a parallel resistance network only (no change of R between parallel and antiparallel magnetic configuration) Cot Co Cut Cu Cot Co Cut Cu CPP GMR can be «qualitatively» described by a serial resistance network «Models in spintronics» no29 European School Magnetism, Timisoara (without spin-flip, i.e. mixing between uponand down channels)

33 Spin accumulation spin relaxation in CPP geometry F2 F1 e flow 1 J 2 1 J 2 J e flow J lsf lsf z Valet and Fert theory of CPP-GMR (Phys.Rev.B48, 799(1993)) Different scattering rates for spin and spin electrons different spin and spin currents. Larger scattering rates for spin : J >>J far from the interface. In F2: Larger scattering rates for spin : J >>J far from the interface. In F1: Majority of incoming spin electrons, majority of outgoing spin electrons Building up of a spin accumulation around the interface balanced in steady state by spin-relaxation

34 Starting point : Valet and Fert theory of CPP-GMR (Phys.Rev.B48, 799(1993)) Spin-relaxation at F /F interface: : spin-dependent chemical potential In homogeneous material, = F-e Spin-dependent current driven by 1 J e z 1 J 2 1 J 2 J J lsf lsf z Generalization of Ohm law F e F* l SF J J e 2 z 2lSF Spin relaxation : lsf=spin-diffusion length (~5nm in NiFe, ~2nm in Co)) lsf lsf z

35 Interfacial boundary conditions i (1 ) zi 1 i ( ) zi 1 ri 1( ) J i ( ) z i 1 J i (1 ) zi 1 J i ( ) zi 1 (Ohm law at interfaces) (if no interfacial spin-flip is considered) Note: Interfacial spin memory loss can be introduced by : J i (1 ) zi 1 J i ( ) zi 1 3% memory loss as at Co/Cu interface yields =.7

36 Input microscopic transport parameters to describe macroscopic CPP properties : Within each layer : 2 * 1 ( ) -The measured resistivity. -The scattering asymmetry. measured -The spin diffusion length lsf. * 1 2 At each interface : -The measured interfacial area*resistance product rmeasured -The interfacial scattering asymmetry. r 2r * 1 ( ) rmeasured r r r * 1 2 r r

37 Examples of bulk parameters Material Measured resistivity 4K/ 3K Bulk scattering asymmetry lsf Cu.5-.7.cm 3-5 5nm 5-2nm Au 2.cm 8 35nm 25nm Ni8Fe Ni66Fe13Co Co Co9Fe Co5Fe Pt5Mn Ru

38 Examples of interfacial parameters Measured R.A interfacial resistance Interfacial scattering assymetry Co/Cu.21m. m Co9Fe1/Cu Co5Fe5/Cu NiFe/Cu NiFe/Co Co/Ru Co/Ag Material

39 Diffusion equation of spin-accumulation Spin-dependent currents : Spin relaxation : J 1 e z ( generalized Ohm law in the bulk of the layer) J e 2 z 2lSF Diffusion equation diffusion for spin accumulation Solution within each layer: z z A exp B exp l l sf sf z lsf A, B integration constants determined from interfacial boundary conditions 1 J 1 grad e z Drift due to electrical field Diffusion due to local spin accumulation

40 Final expression of CPP resistance for any multilayered structures (N layers) : RCPP 1 d N 2 i * i i ri* 1 i 1 i i 1 di r 1 i li 1 Ai 1 i 1 e i i e J i li di ri 1 i li 1 i Bi 1 i 1 e e J i li Where : iaa iab iba ibb Ai 1 Ai ˆ i JJˆi Bi 1 di Bi * l r i 1 1* i 1 *i i li i li with : l e i iaa i BA i iab ibb d * i l r 1 i 1* i 1 *i e li i li i li and di * i 1l i 1 ri li 1 * e * i li i l i d * i l r 1 i 1* i 1 *i e li i li i l i iaj Jˆ i BJ i with : 1 * ri i i *i 1 l i 1 i 1 i 2 1 * AJ *i School ri» i 29 ieuropean l i 1 Timisoara i «Modelsi in spintronics 1 i 1on Magnetism, 2 ibj

41 Comparison of NiFe and FeCo free layer : influence of lsf.cm m. m² Free layer : NiFe/FeCo or FeCo only 4 Ni Fe 8 2 nm F t l SF NiFe Smooth maximum in CPP GMR versus tf =4.5nm 3 R/R (%) nm 2 Fe Co 5 l 5 SF FeCo =15nm Phenomenologically, R 1-exp(-tF/lSF F) and R/R [1-exp(-tF/lSF F]/(tF+cst) 1 (c) Free layer thickness (nm) Data from Headway Technologies 3

42 Comparison of Cu and Au spacer layers : influence of lsf Cu l =5nm R/R (%) 2 SF Cu Au l =1nm SF Au Spacer layer thickness (nm) 5 Data from Headway Technologies Phenomenologically, decrease of R as exp(-tspacer/lsf spacer) and R/R as exp(-tspacer/lsf spacer)/(tspacer+cst)

43 Generalization of spin-dependent transport to any geometry (colinear magnetization) Inhomogeneous current distributions in numerous experimental situations Current crowding effects in low RA MTJ K.Nagasaka et al, JAP89 (21), 6943 H.Fukazawa et al, IEEE Trans.Mag.4 (24), 2236 See for instance: J.Chen et al, JAP91(22), Metallic CPP heads Shield 1 Spin transfer oscillators in point-contact geometry S.Kaka et al, Nat.Lett.437, 389 (25) F.B.Mancoff et al, Nat.Lett.437, 393 (25) J Shield 2 Current confined paths (CCP) GMR 2nm almost all models of spin-dependent transport assume uniform current

44 Extending Valet-fert theory at 3D in colinear geometry 1 J grad e z 1 J grad e z Electrical current: J el J J With 1 Out-of-equilibrium magnetization energy grad DOS m e B z 1 J el 2 grad 2 m e B z 3D expression of electron current: m B 2 J e 2 m e B

45 Spin current: m 1 1 J s J J grad e e e z B 2 2 Js grad 2 e B e 2 2 B Jm grad e 2 e m z : Spin current m z : Moment current 3D expression of moment current: 2 2 B Jm 2 m e e In colinear magnetization Jm is a vector

46 2 J e 2 m e B 2 2 B Jm 2 m e e mz 2 Unknowns: Transport equations: Diffusion of charge (with conservation of charge) divj e 2 divj m m lsf J derived from Valet/Fert: e 2 z 2lSF Diffusion of spin (without spin conservation due to spin-torque and spinrelaxation lsf=spin-diffusion length 2 Equations: 1 diffusion of e + 1 diffusion of m Can be solved in complex geometry with a finite element solver

47 in 2D CPP pillar with extended electrodes out Cu Cu Mapping of charge current 2nm Charge current amplitude Je 1 times higher current density at corners than in center of CPPCo3/Cu2/Co3nm GMR stack 2nm

48 2D CPP pillar with extended electrodes y z x Cu Large excess of e Mapping of y-spin current Cu Co/Cu/Co ( Jm,yx, Jm,yy) Cu Co/Cu/Co Cu Very large excess of e Large excess of e Large inplane spincurrent Parallel Weak excess of e my Antiparallel

49 Conclusion on CPP transport -Serial resistance model can be used at lowest order of approximation to describe CPP transport. However, does not take into account spin-flip. -With spin-flip, spin accumulation and spin-relaxation play an important role in CPP transport. -Semi-classical theory of transport describes CPP transport fairly well. CPP macroscopic transport properties (R, R/R) can be calculated from microscopic transport parameters (, lsf, r ) -In complex geometry, charge and spin current can have very different behavior. Two contributions to j : drift along electrical field and diffusion along gradient of spin accumulation.

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51 Signal/noise ration in CIP MR heads Signal : V R.I Noise : V Johnson 4k B TR f Signal/Noise : SNR V V Johnson R power R 4 k B T f To maximize the SNR in MR heads, the power dissipated in the head must be as large as possible compatible with reasonable heating and electromigration <j>~4.17a/cm² used in heads with CIP-GMR~15%