J. Mathon

Oscillations in exchange coupling across a nonmagnetic metallic layer

Abstract The exchange coupling between two strong itinerant ferromagnets separated by N atomic planes of a nonmagnetic metal is calculated using a Hubbard-type model. It is shown that for certain positions of the Fermi level the variation of the exchange coupling with N exhibits oscillations of long period. The amplitude of the oscillations falls of as 1/N2 and agrees in order of magnitude with the exchange coupling observed by Parkin et al. in Co/Ru and Fe/Cr multilayers. Further agreement is the finding that antiparallel alignment of the ferromagnetic layers is favoured for small N. The relationship between the coupling found here and one of RKKY type is discussed.

Intrinsic and secondary mechanisms for biquadratic exchange coupling in magnetic trilayers

Abstract It is shown for a simple model that intrinsic biquadratic exchange is associated with higher harmonics in the oscillatory exchange coupling. Consequently it is much smaller than the bilinear exchange and falls off much more rapidly with increasing temperature. Neither the intrinsic mechanism nor Slonczewskis secondary mechanism seems to explain the strong observed biquadratic exchange in Fe/Al/Fe and a suggestion is made for resolving the difficulty.

Self-consistent theory of current-induced switching of magnetization

A self-consistent theory of the current-induced switching of magnetization using nonequilibrium Keldysh formalism is developed for a junction of two ferromagnets separated by a nonmagnetic spacer in the ballistic limit. It is shown that the spin-transfer torques responsible for current-induced switching of magnetization can be calculated from first principles in a steady state when the magnetization of the switching magnet is stationary. A steady state is achieved when the spin-transfer torque, proportional to bias voltage in the linear response regime, is balanced by the torque due to anisotropy fields. The spin-transfer torque is expressed in terms of one-electron surface Green functions for the junction cut into two independent parts by a cleavage plane immediately to the left and right of the switching magnet. The surface Green functions are calculated using a tight-binding Hamiltonian with parameters determined from a fit to anab initio band structure. This treatment yields the spin transfer torques taking into account rigorously contributions from all the parts of the junction. The spin-transfer torque has two components, one with the torque vector Ti in the plane containing the magnetizations of the two magnetic layers and another with the torque vector T’ perpendicular to this plane. It is shown that, in general, Ti and T’ may be comparable in magnitude and they both tend to finite values independent of the spacer thickness in the limit of a thick spacer. T’ is shown to be small when the exchange splitting of the majority- and minority-spin bands in both ferromagnets tends to infinity or in the case when the junction has a plane of reflection symmetry at the center of the spacer. The torques T’ and Ti are comparable for a Co/ Cu/ Cos111d junction when the switching Co layer is one or two atomic planes thick. T’ is <27% of Ti even for a switching Co magnet of ten atomic planes. Depending on material parameters of the junction, the relative sign of T’ and Ti can be negative as well as positive. In particular, T’ / Ti , 0 for Co/ Cu/ Cos111d with switching Co magnet of one atomic plane and T’ / Ti . 0 for two atomic planes of Co. A negative sign of the ratio T’ / Ti has a profound effect on the nature of switching, particularly in the realistic case of easy-plane sshaped anisotropy much larger than in-plane uniaxial anisotropy. To calculate the hysteresis loops of resistance versus current, and hence to determine the critical current for switching, the microscopically calculated spintransfer torques are used as an input into the phenomenological Landau-Lifshitz equation with Gilbert damping. In the absence of an applied magnetic field, an ordinary hysteresis loop is the only possible switching scenario when T’ / Ti . 0. However, for T’ / Ti , 0, a normal hysteretic switching occurs only at relatively low current densities. When the current exceeds a critical value, there are no stable steady states and the system thus remains permanently in a time dependent state. This is analogous to the observed precession of the switching magnet magnetization caused by a dc current in the presence of an applied magnetic field. The present calculations for Co/ Cu/ Cos111d show that the critical current for switching in the hysteretic regime is <10 7 A/c m 2 , which is in good agreement with experiment.

Theory of magnetic multilayers. Exchange interactions and transport properties

Abstract The temperature dependence of the local magnetization in the spin-wave regime is discussed and it is shown that it can be used as a sensitive probe of local exchange interactions in magnetic surfaces and overlayers. The present status of the theory of oscillatory exchange interactions through transition and noble metal spacer layers is reviewed. The theory of the giant magnetoresistance observed recently in some antiferromagnetically coupled multilayers is also discussed. It is shown that both the oscillatory exchange and giant magnetoresistance effects can be understood in terms of different distributions of up and down spin d holes in different parts of the layer structure. The total energy and hence the exchange coupling is determined directly by the d band owing to its high density of states and the transport properties more indirectly through the Mott mechanism in which conduction electrons scatter into d band.

Quantum oscillation of magnetoresistance in tunneling junctions with a nonmagnetic spacer

We make a theoretical study of the quantum oscillations of the tunneling magnetoresistance (TMR) as a function of the spacer layer thickness. Such oscillations were recently observed in tunneling junctions with a nonmagnetic metallic spacer at the barrier-electrode interface. We calculate the TMR ratio for disordered tunneling junctions containing a spacer at which quantum well states are formed. A single-orbital tight-binding model, the linear response theory, and the coherent potential approximation are used for the calculation. As a function of the spacer thickness, calculated TMR ratio shows damped oscillation around zero with a single period given by the Fermi wave vector of the spacer, which is consistent with observed results. It is shown that momentum selection due to the insulating barrier and conduction via quantum well states in the spacer, mediated by diffusive scattering caused by disorder, are essential features required to explain the observed oscillation in the TMR ratio. We further show that calculated results can be reproduced by the stationary phase approximation, which implies that obtained results hold qualitatively in more realistic band models.

Exchange coupling in magnetic multilayers: effect of partial confinement of carriers

The exchange coupling J(l) between magnetic layers across a non-magnetic spacer is observed to oscillate as a function of the spacer thickness l. In an earlier work a theory of the oscillatory exchange was proposed which shows that the oscillation periods are characteristic of the spacer. The theory relied on the assumption of an infinitely large exchange splitting in the magnetic layers, which leads to complete confinement of magnetic carriers of one spin in the ferromagnetic configuration of the sandwich. While this may be valid for strong ferromagnets such as Co or Ni, the complete confinement model is not a realistic approximation for iron which has holes in its majority-spin band. The theory is now generalized to the case of partial confinement of carriers in the spacer appropriate to a sandwich with weakly ferromagnetic layers. An exactly solvable hole-gas model of the coupling as well as numerical tight-binding results are presented which demonstrate that the oscillation period is unaffected but the amplitude and phase depend critically on the degree of confinement. Asymptotic expansions for J(l) valid at finite temperature and for an arbitrary single tight-binding band are also obtained. They show that the period and temperature dependence of the oscillations are directly related to the properties of the spacer Fermi surface but the amplitude and phase depend also on the exchange splitting in the ferromagnetic layers.

Theory of spin waves in magnetic interfaces, superlattices and disordered layer structures

A general recursion method for calculating the exact local spin-wave Green function in an arbitrary ferromagnetic interface, superlattice and disordered layer structure is developed. The method is applied to magnetic insulator structures described by a nearest-neighbour exchange Hamiltonian. It is shown that the complete response function of an arbitrary layer structure can be generated from a single matrix element of the Green function in the surface plane of a magnetic overlayer. The proposed algorithm for overlayers is very simple, computationally stable and extremely accurate. The method is used to determine the exact exchange stiffness D of a ferromagnetic superlattice and of a disordered layer structure. The range of validity of the exact result for D is discussed in the light of recent experiments on the temperature dependence of the surface magnetisation. The application of the method to ferromagnets with long-range exchange interactions and to metallic layer structures is also discussed.

Theory of oscillatory exchange in magnetic multilayers: effect of partial confinement and band mismatch

Abstract In an earlier work a theory of the oscillatory exchange coupling between two ferromagnetic layers across a nonmagnetic spacer was proposed. It relied on size-quantization of the energy of electrons of one spin confined in the spacer (complete confinement model). Detailed tight-binding calculations of the coupling between Fe layers across Cr(001) using five d orbitals and the complete confinement model are reported. The coupling exhibits oscillations with several different periods but the amplitude and phase do not agree with experiment. An exactly solvable hole gas model and new tight-binding results are presented which demonstrate that the oscillation period is unaffected but the amplitude and phase depend critically on the degree of confinement. In particular, the amplitude is reduced dramatically for a weak confinement appropriate to iron. The dependence of the coupling on the occupancy of the spacer layer d band is also investigated. It is shown that a mismatch between the ferromagnet and spacer layer d bands, which increases with increasing number of holes in the spacer, leads to a systematic variation of the coupling strength across the transition metal series in agreement with experiment.

Theory of spin current in magnetic nanopillars for zero-field microwave generation

In a magnetic nanopillar, microwave oscillations of the magnetization of one magnetic layer can be driven by spin-polarized current emitted from another magnetic layer. The conditions for this to occur in zero applied field are formulated in terms of the two components of the spin-transfer torque. One simple route to achieve microwave generation is to ensure that these components have opposite sign. Quantum-mechanical calculations are presented that show how this may be achieved by a suitable choice of the oscillating magnet thickness.

Theory of oscillatory exchange coupling in magnetic multilayers

Abstract It is found experimentally that the exchange coupling J between ferromagnetic layers separated by nonmagnetic transition or noble metal spacers oscillates as a function of the spacer thickness D . For noble metal spacers RKKY theory is applicable but it must be modified to take account of discreteness of the spacer layer. It is shown that Bragg scattering in the spacer is crucial for explaining the observed long oscillation periods. The coupling through a transition metal is attributed to the spatial confinement of holes in the d band leading to size quantization of their energy. An analogy with the dHvA effect is exploited to derive an RKKY-like formula for J valid for D ≥ 5–6 ML. Numerical calculations for a simple tight-binding model of the spacer lead to J of the observed magnitude and to a long oscillation period. Calculations for Cr(100) with all five d bands included confirm the long period but an oscillation with a period of 2 ML is also found. This is in good agreement with the most recent experiments.