Spin currents in noncollinear magnetic structures: when linear response goes beyond equilibrium states

1. Spin currents in noncollinear magnetic structures: when linear response goes beyond equilibrium states Peter M Levy New York University

2. Spin currents across noncollinear multilayers Current induced switching of magnetic layers by spin polarized currents can be divided in two parts: Creation of torque on background by the electric current, and reaction of background to torque. Latter epitomized by Landau-Lifschitz equation; micromagnetics

3. The inverse of GMR-Current Induced Magnetic Switching Nanopillar Structure Au I V AP F P H F = Co, Py Cu F V I Cu Dimensions ~ 60 nm x 130 nm

4. SAMPLES MSU Nanopillars ~130 nm I H H ~70nm Au Au tCu 2.5 – 6 nm Py or Co Py or Co Py or Co Py or Co 30 nm Cu Cu I Antiferromagnetically-coupled F layers: tCu = 10 nm Uncoupled F layers: tCu = 10nm Ferromagnetically-coupled F layers: tCu = 2.6 nm

5. Hysteretic AP I = 0 P Uncoupled Co/Cu/Co Nanopillar: l1 ~ 130 nm; l2 ~ 70 nm. 295K AP AP MR  5% P P AP R(I) H = 0 P Is

6. Definition of spin current as the expectation value of a quantum mechanical operator The rub is the states or distribution over which one evaluates this operator. 1st: Diffusive regime In the Boltzmann description the components of the spin current are written as

7. Equations of motionin steady state for elements of the spinor density matrix for each momentum on the Fermi surface are Longitudinal Transverse

8. Diagonal elements represent occupancy of state. Off diagonal elements are coherences between state on Fermi surface and another state of opposite spin. Before we discuss the Boltzmann approach let’s look at the Diffusion equation. It’s simpler as its for densities (x) rather than Distributions (k,x).

9. How the electric field drives a spin system out of equilibrium Normal metal mz z Efield

10. While spin accumulations may be discontinuous, e.g., about interfaces, the spin current iscontinuous.

11. Spin currents From: S. Zhang and P.M. Levy, Phys. Rev. B65, 052409 (2002).

12. Spin Accumulation-left layer-current reversed Spin Accumulation from left layer z z j j How reversal in current directions changes alignment of layers

13. Spin Accumulation from central layer z j

14. One thick-One thin layer

15. First attempt: Spin diffusion equation Spinor current

16. Accumulation of spin near interfaces alters equilibrium densities Asya Shpiro et al. Phys. Rev.B67,104430 (2003). Part due to band structure Contribution from diffusive processes

17. Accumulation of spin near interfaces alters equilibrium densities Asya Shpiro et al. Phys. Rev.B67,104430 (2003). Part due to band structure Contribution from diffusive processes Equation of motion for spin accumulation

18. Equation for spin diffusion Caveat: we have assumed in above Md is constant in magnitude and direction. Changes in Md occur at interfaces.

19. Stationary solutions for spin accumulation Longitudinal Transverse

20. We define the spin torque and effective field as: b=effective field a=spin torque

21. Torques Effective field-”b” Spin torque-”a”

22. Spin transport in magnetic multilayers • Linear response: • only electrons close to Fermi surface contribute to conduction • only equilibrium band structure is necessary to describe effect of electric field in electrons. The case for spin accumulation; does it enter in linear response?

23. Layer by layer approach to transport in metallic structures: • Due to screening in metals transport in each layer can be modeled by equilibrium band structure. • Solve for distribution function ( statistical density matrix) in each layer by using the Boltzmann equation. • The distribution functions describing the out of equilibrium transport across layers are connected by the scattering matrices at the interfaces.

24. Conclusion: Attendant to current driven across inhomogeneous media there is charge and spin redistribution so as to maintain a steady state current. As seen from the Boltzmann equation this out of equilibrium accumulation enters in linear response. In the Kubo approach it alters the local electric field seen by the electrons from that externally applied. For magnetic media the effective field for spin channels are different. This has been sufficient to describe transport in collinear magnetic structures, but it is in sufficient when the magnetic layers are noncollinear?

25. Spin transport in noncollinear magnetic multilayers Does one have to alter the layer by layer approach to transport in metallic structures that has worked so well for collinear magnetic multilayers? Yes How

26. If a spin current transverse to the magnetization of a 3d transition-metal did enter how would it decay? • Here we have different scenarios: • Averaging over wavefunctions • Stiles and Slonczewski has shown that when one averages the phase, • associated with the transverse component of the spin current, over the • Fermi surface it vanishes on the length scale • Averaging over distribution functions • First one finds the Boltzmann distribution f(x,px) for each momentum p • with the boundary condition at the interface that electrons have their • spins polarized at an angle to the magnetization. When this is • averaged over momentum space ( the Fermi surface) • the transverse component of the spin current vanishes on the length • scale d=hvF/J~3nm.

27. Equations of motionin steady state for elements of the spinor density matrix for each momentum on the Fermi surface are Longitudinal Transverse

29. Transverse components of spin current

31. Current induced coherences Spin distributions transverse to the magnetization are described by correlations between up and down spin states. In the one electron spin polarized picture of ferromagnetic band structure there are no correlations between spin split bands in equilibrium. For these correlations to exist they must be induced by the current. They can only be induced by the spin-flip scattering at the interfaces. This only happens if the symmetry is lowered by the presence of spin polarized currents. For the same reason that spin accumulation enters in linear response the effect of this current induced coherence also enters linearly.

32. In a layer by layer approach to calculating the overall conductance or resistance of a multilayer it is necessary to relate the distribution functions for each layer across their interface with an adjacent layer, by using reflection and transmission coefficients. Note that by matching distribution functions, rather than wavefunctions we lose some information, i.e., we lose the coherence of the wavefunction across the multilayer.

33. If the distance between ferromagnetic layers exceeds the interlayer coupling distance due to the equilibrium RKKY-like coupling then there is a unique direction for the magnetization at each N/F interface for the system in equilibrium. This dictates that in equilibrium the scattering amplitudes are diagonal in spin space when referred to this unique axis, and that the transmission and reflection coefficients can only transmit the component of an incoming spin current that is parallel to the magnetization of the ferromagnetic layer, i.e, there is no transmission of transverse spin currents. However, when a current flows across a magnetic multilayer the spin accumulation created at one N/F interface is superimposed on other N/F interfaces that are within a spin diffusion length of it. When the layers are noncollinear the symmetry at the N/F interfaces is lowered so that scattering amplitudes contain off diagonal components in spin space, and transverse components of the spin current are transmitted.

34. Two semi-infinite magnetic layers of same material Current ZL YL Spacer ZR Local coordinate YR XL XR

35. Steady state calculationLocal axis coordinates

36. Methodology: Boltzmann equation using the layer-by-layer approach Boltzmann equation for spin currents in ferromagnetic metals. See Jianwei Zhang et al., PRL 93, 256602 (2004).

38. Equations of motion for distribution functions Longitudinal Transverse Transient response is crucial to understanding states off the Fermi surface contribute to conduction in linear response.

39. Band structure of Co

40. Definition of transverse spin current, in the steady state

41. In a statistical density matrix, e.g., the Boltzmann distribution • function, there are diagonal matrix elements which represent • populations, and the off diagonal which are coherences • between states. • For noncollinear multilayers one must be mindful of • coherences. • In equilibrium magnetic layers are not magnetically coupled; • in the presence of a spin current across a normal spacer the • scattering at the opposite interfaces of the spacer interact • with one another, e.g., see Valet and Fert PRB 48, 7099 (93). • CISP’s is our way of introducing in a steady state calculation • transients that admix excited k states into the ground state • so as to arrive at the correct steady state.

42. Solution for multilayer is to find distribution function in each layer by using Boltzmann equation. To determine unknown constants one has to match functions across layers by using the transmission and reflection coefficients. For example, for transverse distribution function

43. Connection formulae across interfaces see P.M. Levy and Jianwei Zhang,PRB 70, 132406 (2004)

44. This leads to the “mixing conductance in the conventional view.

46. A is the new current induced spin-flip term

48. Interference pattern- y polarization

49. Interference pattern - x polarization

50. No spin-flip at interfaces