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Jan 10, 2008 - Freescale Semiconductor, Inc., 15 Bozhenko Street, Kiev 03680, Ukraine. Received 2 October 2007; revised manuscript received 22 November ...
PHYSICAL REVIEW B 77, 024410 共2008兲

Theory of interlayer magnetic coupling in nanostructures with disordered interfaces Victor F. Los Institute of Magnetism, National Academy of Sciences of Ukraine, 36-b Vernadsky Boulevard, Kiev 03142, Ukraine

Andrei V. Los Freescale Semiconductor, Inc., 15 Bozhenko Street, Kiev 03680, Ukraine 共Received 2 October 2007; revised manuscript received 22 November 2007; published 10 January 2008兲 Multiple scattering theory of interlayer coupling in nanostructures with disordered alloylike interfaces is developed. Interface-localized potential responsible for the specular electron reflection from a perfect interface 共potential step兲 is found in the framework of multiple scattering theory based on Green’s functions approach. This allowed obtaining of a general expression relating the interlayer coupling energy to electronic density of states change due to interferences of electronic waves resulting from multiple reflections at disordered interfaces. In the case of a small spacer electron confinement 共small scattering兲, the interlayer exchange coupling between two ferromagnets separated by a nonmagnetic spacer is found in the coherent potential approximation for electrons scattering at the disordered interfaces. The influence of disorder on the oscillating with a metallic spacer thickness exchange coupling is analyzed. The main effect is found to be a phase shift and amplitude change of these oscillations with the change of the parameters describing specular and diffuse spin-dependent electron scattering caused by the interfacial disorder. DOI: 10.1103/PhysRevB.77.024410

PACS number共s兲: 73.21.Ac, 72.25.Mk, 73.40.Gk, 85.75.⫺d

I. INTRODUCTION

Artificial structures consisting of alternating magnetic and nonmagnetic layers, each of a few atomic layers thick, have been attracting a good deal of interest especially since the discovery of the giant magnetoresistance 共GMR兲1 and tunneling magnetoresistance 共TMR兲2,3 effects. These structures exhibit GMR and TMR under certain conditions. One of them is a possibility to switch between antiparallel and parallel alignments of magnetization vectors of ferromagnetic layers in a multilayer or a sandwich. The antiferromagnetic coupling of Fe films separated by a Cr spacer was first observed in Ref. 4. Later, a spectacular periodic dependence of the sign of the interlayer exchange interaction between ferromagnetic layers on the metallic nonmagnetic layer thickness was discovered in Refs. 5 and 6. The latter effect as well as the effect of interlayer exchange coupling across nonmetallic layers, first studied in Ref. 7 for Fe films separated by amorphous Si, have been a subject of intensive research in the last years. In contrast to the case of a metal spacer, the coupling through a nonmetal spacer is nonoscillatory, decays with the spacer thickness, and increases with temperature increase. A great number of studies have been performed on the interlayer coupling via a metal spacer with special attention to its oscillatory behavior. We cite here the most comprehensive, from our point of view, theoretical investigations by Bruno8 and Stiles,9 where all other suitable references can be found. In these papers, a general approach to the problem of interlayer coupling in terms of the quantum interference due to the 共spin-dependent兲 specular reflections of electron Bloch waves at the perfect paramagnet-ferromagnet interfaces has been proposed. This interference of the electron waves results in a change of the density of states in the spacer and leads to the oscillatory behavior of interlayer coupling energy with the spacer thickness 共for metal spacer兲. In addition to being very physically transparent, this approach embodies 1098-0121/2008/77共2兲/024410共14兲

all the previously suggested models 关e.g., Rudermar-KittelKasuya-Yosida 共RKKY兲 model10兴 and allows treating on the equal footing of both the metal and insulator spacer cases8 共coupling through a tunneling barrier was first considered by Slonczewski11兲. It should be stressed, however, that the basic approach by Bruno and Stiles treats the interfaces in nanostructures as the perfect ones. In this case the electron reflections at and transmissions through the interfaces are k储 conserved due to the translational invariance of the interface 共k储 is an electron wave vector parallel to the interfaces兲. However, the real interfaces are disordered. They are characterized by roughness 共long-range imperfections, compared to the electron mean free path l, which do not break the in-plane translational invariance兲 and intermixing 共short-range imperfections breaking the in-plane translational invariance兲. The interface roughness can be associated with long-range deviation of either the thickness or the crystallographic orientations of the layers, whereas the intermixing of different layers in the interface region can be modeled by an alloy of short-range scatterers. There have been a number of theoretical and experimental studies of interface roughness and its influence on the GMR and other phenomena. For instance, the influence of interface roughness on GMR was studied in Refs. 12 and 13, where the authors introduced a model treating interfaces as nonflat ones; ab initio calculations for the interlayer exchange coupling in the presence of an interfacial roughness and interdiffusion were performed in Ref. 14. However, to the best of our knowledge, no general theoretical treatment of the problem of interlayer coupling in the case of disordered interfaces has been published to date. The essential problem is then how to generalize the basic approach of Refs. 8 and 9 to the case of disordered interfaces, and whether and how the predictions of these theories for the perfect interfaces are affected by interfacial disorder. Advances in both molecular dynamics simulations and experimental techniques have provided crucial information on

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©2008 The American Physical Society

PHYSICAL REVIEW B 77, 024410 共2008兲

VICTOR F. LOS AND ANDREI V. LOS

the structure of multilayer interface regions.15,16 These studies show that the interfaces in a number of technologically important multilayers can be viewed as locally crystalline with interdiffusion of different atomic species within a few layers of the interface. In a recently published paper,17 the authors stress that for a proper description of the interface structure and interpretation of the experimental data on the GMR effect and interlayer coupling, the atomic scale disorder, which is typically caused by an alloying process 共intermixing of different materials at the interface兲, should be taken into consideration. They suggest a microscopic mechanism of the interface alloying, and provide a description of interface roughness and its correlation with the GMR effect and interlayer coupling in Fe/ Cr multilayers. In the present paper, we develop a general approach to the problem of interlayer coupling in real nanostructures with disordered interfaces. Based on the above-mentioned results, we adopt for the interface structure a model of disordered alloy made of atoms of the adjacent layers. When an interfacial disorder is present, the in-plane component of electron wave vector may no longer be conserved and the influence of disorder on the specular and arising diffuse scattering of electrons at the interface needs to be studied. Transmission and reflection amplitudes for an electron striking a disordered interface at an arbitrary angle have been found in a closed analytical form in Ref. 18 in the self-consistent single-site coherent potential approximation 共CPA兲 for interfacial disorder and the effective-mass approximation 共EMA兲 for the electron spectra in different layers. The EMA is suitable for s and majority d electrons in some important ferromagnetic materials 共Co, Ni兲 and now actively studied diluted magnetic semiconductors 共DMS兲, which are viewed as promising materials for spintronics. In Sec. II of this paper, the general, based on the Green functions formalism, scattering theory approach to the reflection of spacer electrons from two interfaces is outlined. Necessary for this approach, localized 共at an interface兲 potential, responsible for reflection from a perfect interface 共potential step兲, is obtained in Sec. II A. This further allows introducing a combined potential describing the scattering due to both interface potential profile change and disorder caused by the atoms of adjacent layers mixing at the interface. As a result, a general expression for the electron energy change 共the coupling energy兲 caused by the interferences of electronic waves due to multiple reflections of the spacer electrons from the disordered interfaces is obtained in Sec. II B. In Sec. III, this result is used to get the exchange coupling energy 共in the case of small reflection amplitudes兲 for a trilayer made of two ferromagnets divided by a nonmagnetic spacer. The coherent potential approximation 共CPA兲 is used for electron scattering at a disordered interface. Parameters describing the influence of disorder on the exchange coupling are introduced, and the change of the oscillating with the spacer thickness exchange coupling 共in the case of a metallic spacer兲 as a function of these parameters is analyzed and plotted. The results are summarized in Sec. IV.

electrons’ energy 共thermodynamic potential兲 change due to quantum interferences of electronic waves in a spacer layer. Consider a multilayer made of 共generally different兲 magnetic layers separated by nonmagnetic layers. The thicknesses of magnetic layers are supposed 共for simplicity兲 to be larger than the electronic mean free path l, whereas the spacer 共nonmagnetic layer兲 thickness is less than l. In this case we may consider the interlayer coupling between magnetic layers through the spacer taking into account interferences of electronic waves 共as a reason for coupling兲 in the spacer only. Thus it is sufficient to consider a system of two magnetic layers parallel to the 共x , y兲 plane with interfaces at z = 0 and z = d and separated by a nonmagnetic spacer 共with thickness d兲. Electrons traveling in the spacer experience repeated reflections from the nonmagnetic-magnetic interfaces which may lead to quantum interferences of electronic waves and to a corresponding change of the electron density of states and energy. The latter leads to the interlayer coupling. The appropriate 共for considering the electrons’ interferences兲 Hamiltonian of this system may be written as

where H0 is the spacer electrons’ Hamiltonian and V0 and Vd represent the scattering potentials of the interfaces located at z = 0 and z = d, respectively. These potentials stem from the conduction-band steplike profile U共z兲 of the trilayer 共forming a potential well or barrier for spacer electrons兲 and interfacial disorder which will be modeled by the short-range scattering centers located in the interface plane 共x , y兲. For the perfect interfaces, the scattering potential caused by U共z兲 depends only on the coordinate normal to the layers, and the in-plane component k储 of the electron wave vector is a good quantum number. Thus from the symmetry consideration, the electron reflection amplitudes for an electron state of wave vector k = 共k⬜ , k储兲 depends only on the perpendicular to the interfaces component k⬜, which, of course, depends on the electron energy and k储. Therefore for each k储 共angle of electron incidence on the interface兲 we have a one-dimensional problem of the electron multiple scatterings from the interfaces 共the scattering amplitudes are diagonal in the k储 representation兲. For disordered interfaces, when k储 may no longer be conserved, the electron reflection 共and transmission兲 amplitude is not diagonal in the k储 representation, but the needed average 共over all configurations of the mixed at an interface atoms兲 electronic density of states remains diagonal in the k储 representation. It follows 共see below兲 that one may consider a one-dimensional problem of the quantum interferences in the spacer for an electron with given k储 共leading to the electron density of states change兲 and then sum up the result over in-plane electron wave vectors to get a three-dimensional problem solution. The thermodynamic grand potential change ⌬⌽ associated with the electronic density of states change ⌬n共␧兲 at temperature T is

II. SCATTERING THEORY APPROACH TO REFLECTION FROM INTERFACES

⌬⌽ = − kBT

The general approach developed in Refs. 9 and 8 suggests, quite naturally, to relate the interlayer coupling to the

共1兲

H = H0 + V0 + Vd ,





−⬁

where ␧F is the Fermi energy.

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冋 冉 冊册

⌬n共␧兲ln 1 + exp

␧F − ␧ k BT

d␧,

共2兲

PHYSICAL REVIEW B 77, 024410 共2008兲

THEORY OF INTERLAYER MAGNETIC COUPLING IN…

The density of states n共␧兲 is conveniently expressed in terms of the Green function G共␧兲 corresponding to the system Hamiltonian 共1兲 as n共␧兲 = −

1 Im 兺 Tr具G共␧ + i0+兲典, ␲ k储

G共␧兲 = 共␧ − H0 − V0 − Vd兲−1 ,

共3兲

where 具¯典 denotes a configurational average over all the interface scatterers 共impurities兲 configurations and i0+ is infinitesimal imaginary energy. Note that the averaged Green function is diagonal with respect to k储 关we do not show in Eq. 共3兲 explicitly the corresponding matrix element of 具G共␧ + i0+兲典 in k储 representation for brevity兴 because the averaging restores the translational invariance of a disordered interface. Treating V0 + Vd as perturbation, the Green function G共␧兲 may be expressed as G共␧兲 = G0共␧兲 + ⌬G0共␧兲 + ⌬Gd共␧兲 + ⌬G0d共␧兲.

共4兲

Here, G0共␧兲 = 共␧ − H0兲−1 , ⌬G0共d兲共␧兲 = G0共␧兲T0共d兲共␧兲G0共␧兲,

means that the energy ␧ has an infinitesimal imaginary part 共i0+兲. Thus to get the interlayer coupling energy 共8兲, one needs to have the “free” Green function in a spacer G0共␧兲 and to obtain the t-matrix T0共d兲共␧兲 for a single interface located at z = 0 or z = d, which according to Eq. 共5兲 is defined by G0共␧兲 and perturbation V0共d兲 caused by the potential profile U共z兲 and the random scatterers located at the interfaces 共for disordered interfaces兲. In the framework of the adopted scattering formalism it is needed first to get the part of V0共d兲 due to the potential steplike profile U共z兲. A. Perfect interfaces

The goal of this section is to define the scattering matrix T0共d兲 caused by the potential steplike profile U共z兲 for a single perfect interface which then will be generalized to the case of a disordered alloylike interface. We will use the approach of Ref. 19 but extended to the case when the electronic spectra of materials are treated in the EMA with different effective masses in different layers 共in Ref. 19 the free electron spectra are used兲. For the case of material and spin dependent electron effective mass m*共z兲 and arbitrary conduction-band profile U共z兲 共axis z is directed perpendicular to the interface兲 the Schrodinger equation for a single-electron state in the EMA reads



⌬G0d共␧兲 = G0T0共1 − G0TdG0T0兲−1G0Td共1 + G0T0兲G0 + G0Td共1 − G0T0G0Td兲−1G0T0共1 + G0Td兲G0 , T0共d兲共␧兲 = V0共d兲 + V0共d兲G0共␧兲V0共d兲 + V0共d兲G0共␧兲V0共d兲G共␧兲V0共d兲 + ¯ = V0共d兲关1 − G0共␧兲V0共d兲兴−1 ,

共5兲

where G0共␧兲 represents the electron propagation in the spacer material, the t-matrix T0共d兲共␧兲 describes multiple reflections from the interface potential V0共d兲, ⌬G0共d兲共␧兲 expresses the effect of the interface potential V0共d兲 on the “free” Green function G0共␧兲, and ⌬G0d共␧兲 gives the contribution of the multiple reflections from both interfaces. The density of states change ⌬n共␧兲 responsible for the interlayer coupling is given by ⌬G0d共␧兲, Eq. 共5兲, and therefore as it follows from Eq. 共3兲, ⌬n共␧兲 = −

1 Im 兺 Tr具⌬G0d共␧ + i0+兲典. ␲ k储

共6兲

Using the definitions 共5兲 for G0共␧兲 and T0共d兲共␧兲, it may be shown that Tr ⌬G0d共␧兲 =



1 ⌬⌽ = Im 兺 ␲ k储



⌿共z, ␳兲 = 兺 ␺k储共z兲␾k储共␳兲, k储

冋冉 冊 冉 冊

冑A ,

共10兲

2



共11兲

Here, the perpendicular wave vector k⬜共z ; ␧ , k储兲 is defined as 2

k⬜ 共z;␧,k储兲 =

2m*共z兲 关␧ − U共z兲兴 − k2储 . ប2

共12兲

Corresponding to Eq. 共9兲, equation for the retarded Green function is



−⬁

where f共␧兲 is the Fermi-Dirac distribution and superscript ⫹

eik储␳

k⬜ 共z;␧,k储兲 1 ⳵ ⳵ + ␺k储共z兲 = 0. ⳵z m*共z兲 ⳵z m*共z兲

d␧f共␧兲 共8兲

␾k储共␳兲 =

where ␳ = 共x , y兲 is a two-dimensional vector in the plane of the interface and A is the area of the interface, Eq. 共9兲 may be reduced to the following one-dimensional equation for the longitudinal wave function ␺k储共z兲,



⫻Tr具ln关1 − G+0 共␧兲T0+共␧兲G+0 共␧兲Td+共␧兲兴典,

共9兲

where U共z兲 is a homogeneous potential taking different values at both sides of an interface 关as well as m*共z兲兴. By taking advantage of the two-dimensional homogeneity of the system 关the in-plane wave vector k储 = 共kx , ky兲 is a good quantum number兴 and expanding the wave function in the complete set of the transverse wave functions ␾k储共␳兲,

d Tr ln关1 − G0共␧兲T0共␧兲G0共␧兲Td共␧兲兴. 共7兲 d␧

Making use of Eqs. 共6兲 and 共7兲, the potential change 共2兲 caused by the electron interferences in the spacer after integrating by parts may be represented as



1 ប2 ⵜ ⵜ + U共z兲 ⌿共x,y,z兲 = ␧⌿共x,y,z兲, 2 m*共z兲

␧+



1 ប2 ⵜ * ⵜ − U共z兲 Gz0+共r,r⬘ ;␧兲 = ␦共r − r⬘兲, 2 m 共z兲 共13兲

where the energy has an infinitesimal positive imaginary part and the index z0 points out that the case of a single interface

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PHYSICAL REVIEW B 77, 024410 共2008兲

VICTOR F. LOS AND ANDREI V. LOS

located at z = z0 is being considered 共for a trilayer z0 = 0 or d兲. The Green function for the perfect interface and the ␦ function may be presented like Eq. 共10兲 as Gz0+共r,r⬘ ;␧兲 =

1 兺 eik储共␳−␳⬘兲Gz0+共z,z⬘ ;␧,k储兲, A k储

␦共r − r⬘兲 =

␦共z − z⬘兲 A

eik 共␳−␳⬘兲 , 兺 k

共14兲





and Eq. 共13兲 reduces to the one-dimensional equation for the Green function Gz0+共z , z⬘ ; ␧ , k储兲,

冋冉 冊 冉 冊



⬜2

2 1 ⳵ ⳵ k 共z兲 z + + * G 0 共z,z⬘ ;␧,k储兲 = 2 ␦共z − z⬘兲. ⳵z m*共z兲 ⳵z m 共z兲 ប 共15兲

This equation corresponds to the homogeneous Schrodinger equation 共11兲. The solution to Eq. 共15兲 may be found in a standard way 共see, e.g., Ref. 20兲 in terms of solutions to the Schrodinger equation 共11兲 ␺kL储共z兲, ␺kR储共z兲, satisfying boundary conditions on the left and right sides, respectively. For the considered retarded Green function, ␺kL储共z兲 vanishes for z → −⬁ and ␺kR储共z兲 vanishes for z → ⬁. This solution is Gz0+共z,z⬘ ;␧,k储兲 =

Gz0+共z,z⬘ ;␧,k储兲 =

W共z⬘兲 = ␺L共z⬘兲

2 ␺ 共z兲m*共z⬘兲␺R共z⬘兲 L

,

z ⬍ z⬘ ,

2 ␺R共z兲m*共z⬘兲␺L共z⬘兲 , ប2 W共z⬘兲

z ⬎ z⬘ ,

W共z⬘兲

ប2

冋 册 d␺R共z兲 dz

z=z⬘

− ␺R共z⬘兲

冋 册 d␺L共z兲 dz

interface situated at z0 = 0 or z0 = d, v⬎共⬍兲 = បk⬎共⬍兲 / m⬎共⬍兲, m⬎共⬍兲 and U⬎共⬍兲 are the electron effective mass m*共z兲 and potential U共z兲 in the material to the right 共left兲 side of the interface, and rL共R兲, tL共R兲 are the amplitudes of reflection from and transmission through the interface for an electron incoming to the interface from the right 共left兲. In order to obtain the reflection and transmission amplitudes one should match the wave functions 共17兲 and their derivatives at the interface. Integration of Eq. 共11兲 over z in a close vicinity of any interface results in the following continuity relations for derivatives:

冋 册

1 d␺L共R兲 m⬎ dz

rL =

1

−ik⬎共z−z0兲

␺ 共z兲 = L

␺R共z兲 =

␺R共z兲 =

1

冑v ⬍



+

1

冑v ⬍ t L e 1

2m⬎共⬍兲

−ik⬍共z−z0兲

,

ik⬎共z−z0兲

,

rR =

ik⬎共z−z0兲

1

,

冑v ⬍ r R e

2im⬎ t, ប

,

Gz0+共z,z⬘兲 =

1

te iប冑v⬎v⬍

1

2im⬍ t, ប

共19兲

−ik⬍共z−z0兲 ik⬎共z⬘−z0兲

e

,

teik⬎共z−z0兲e−ik⬍共z⬘−z0兲, z⬘ ⬍ z0 ,

1 关eik⬍兩z−z⬘兩 + rRe−ik⬍共z+z⬘−2z0兲兴, iបv⬍

Gz0+共z,z⬘兲 =

z ⬍ z 0, Gz0+共z,z⬘兲 =

Here, k⬎共⬍兲 = 关␧ − U⬎共⬍兲兴 − k2储 is the perpendicular to ប2 the interface wave vector 共12兲 taken to the right 共left兲 of the

2冑v⬎v⬍ , v⬎ + v⬍

z⬘ ⬎ z0 ,

iប冑v⬎v⬍ z ⬎ z 0,

z ⬎ z0 ,

z ⬍ z0 . 共17兲

tL = tR = t =

W⬍ =

z ⬍ z 0,

, z=z⬘

z ⬍ z0 ,

−ik⬍共z−z0兲

共18兲

, z⬍

where W⬎共⬍兲 corresponds to W共z⬘兲 at z⬘ taken to the right 共left兲 side of the interface. Substituting Eqs. 共17兲 and 共19兲 into Eq. 共16兲, we get the following Green functions in different regions of z and z⬘ 共suppressing the arguments ␧ and k储 for brevity兲: Gz0+共z,z⬘兲 =

z ⬎ z0 ,

冋 册

v⬍ − v⬎ , v⬎ + v⬍

z⬘ ⬍ z0 ,

1 关eik⬎兩z−z⬘兩 + rLeik⬎共z+z⬘−2z0兲兴, iបv⬎ z ⬎ z 0,

冑v ⬎ t R e

eik⬍共z−z0兲 +

1

冑v ⬎ r L e

v⬎ − v⬍ , v⬎ + v⬍

W⬎ =

where we omitted the index k储 of the wave functions for brevity. The result given by Eq. 共16兲 generalizes the corresponding formula of Ref. 19 to the case of different effective masses in different layers. The solutions to Eq. 共11兲 may be defined as

冑v ⬎ e

z⬎

1 d␺L共R兲 m⬍ dz

where z⬎共⬍兲 denotes the coordinate of the interface 共z = 0 or z = d兲 with an infinitesimal shift to the right 共left兲. Then, the continuity relations for ␺L, ␺R and their derivatives result in

共16兲

␺L共z兲 =

=

z⬘ ⬎ z0 .

共20兲

Now, we can identify 关in accordance with Eq. 共5兲兴 the “free” Green function 共propagator兲 in the spacer sandwiched between interfaces located at z0 = 0 and z0 = d. The last two lines of Eq. 共20兲 represent the Green function for an electron incoming to a single interface from the left side 共third line兲 and from the right side 共fourth line兲. For the trilayer under consideration, we will adopt the following definitions for the material and spin dependent electron’s effective mass and potential profile:

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THEORY OF INTERLAYER MAGNETIC COUPLING IN…

m*共z兲 = m1, m*共z兲 = m2,

U共z兲 = U1,

T0+共␧,k储兲 =

0 ⬍ z ⬍ d,

U共z兲 = U2,

m*共z兲 = m3,

z ⬍ 0,

U共z兲 = U3,

z ⬎ d,

1 ik 兩z−z⬘兩 = e 2 , iបv2

0 艋 z,

z⬘ 艋 d,

共22兲

2

where vi = mii and ki = ប2 i 共␧ − Ui兲 − k2储 共i = 1 , 2 , 3兲. The contribution to the “free” propagator 共22兲 due to the specular scattering from the perfect interface may be written as ⌬G0共z,z⬘ ;␧,k储兲 =

⌬Gd共z,z⬘ ;␧,k储兲 =

1 d −ik 共z+z⬘−2d兲 re 2 , iបv2 z⬘ ⬍ d,

共23兲

where ⌬G0共d兲 is the contribution due to the scattering from the interface situated at z = 0共d兲 and r0共d兲 is the corresponding reflection amplitude from the left 共right兲 interface 关see Eq. 共19兲兴 r0 =

v2 − v1 , v1 + v2

rd =

v2 − v3 . v2 + v3

共24兲

On the other hand, by expanding G共r , r⬘ ; ␧兲, Eq. 共14兲, into V0共z兲 = W0␦共z兲 or Vd共z兲 = Wd␦共z − d兲, where W0共d兲 is the potential 共that we are looking for兲 responsible for specular reflection of an electron from a perfect interface located at z = 0共d兲, one gets 关in accordance with Eq. 共5兲兴 ⌬G0共d兲共z,z⬘ ;␧,k储兲 =



dz1G0共z,z1 ;␧,k储兲

⫻T0共d兲共z1 ;␧,k储兲G0共z1,z⬘ ;␧,k储兲, 共25兲 where the t-scattering matrix is

W0 , 1 − G0共0,0;␧,k储兲W0

Td共z1 ;␧,k储兲 = Td共␧,k储兲␦共z1 − d兲, Td共␧,k储兲 = Using Eq. 共22兲, we have

Wd . 1 − G0共d,d;␧,k储兲Wd

Td+共␧,k储兲 = iបv2rd .

共28兲

Finally, Eqs. 共27兲, 共28兲, and 共24兲 yield W0 =

iប 共v2 − v1兲, 2

Wd =

iប 共v2 − v3兲, 2

共29兲

where the potentials W0 and Wd define the quantum reflection of a spacer electron from the left and right interface, respectively, and depend 关according to Eq. 共29兲兴 on ␧ and k储 共through the electron velocities兲. Formulas 共29兲 represent the main result of this section: we have managed to find the potential localized at the interface coordinate z0, 共30兲

which in the framework of the general scattering theory allows finding the reflection amplitude by calculating the t-scattering matrix 共26兲 and then using Eq. 共28兲. Now, we can obtain an exact formula for the coupling energy 共8兲 in the considered case of a trilayer with perfect interfaces. Making use of Eqs. 共22兲 and 共28兲 and taking a trace over z coordinates, we have ⌬⌽ =

1 Im 兺 ␲ k储





d␧f共␧兲ln关1 − r0rde2ik2d兴,

共31兲

−⬁

where r0, rd are defined by Eq. 共24兲 and dependent on ␧ and k储 共via the electron perpendicular velocities vi, i = 1, 2, and 3兲 as well as the perpendicular to the interfaces electron vector k2 in a spacer. The symbol 具¯典 is absent for the case of perfect interfaces. The expression 共31兲 was earlier obtained in a different way and analyzed in Ref. 8. Our derivation differs by using the EMA for electronic spectra 共with different effective masses in different layers兲, but more importantly, we have obtained the potentials V0共d兲 关defined by Eqs. 共29兲 and 共30兲兴, which in the adopted scattering theory approach describe the quantum reflections of electrons from the steplike potential U共z兲. In addition to being interesting from the general quantum mechanical point of view, this will allow us to generalize the adopted approach and formula 共31兲 to the case of disordered interfaces. B. Disordered interfaces

T0共z1 ;␧,k储兲 = T0共␧,k储兲␦共z1兲, T0共␧,k储兲 =

iបv2Wd . 共27兲 iបv2 − Wd

Vz0共z;␧,k储兲 = Wz0共␧,k储兲␦共z − z0兲,

1 0 ik 共z+z⬘兲 re 2 , iបv2

0 ⬍ z,

T0+共␧,k储兲 = iបv2r0,

共21兲

冑m

បk

Td+共␧,k储兲 =

Comparing Eq. 共23兲 with Eq. 共25兲 and using Eq. 共22兲, we get

where indexes 1, 2, and 3 correspond to the layer at z ⬍ 0, the spacer, and the layer at z ⬎ d, respectively. Thus for a “free” spacer electron incoming to the interfaces the corresponding propagator is G+0 共z,z⬘ ;␧,k储兲

iបv2W0 , iបv2 − W0

共26兲

In this section we will obtain the t-scattering matrix T0共d兲 for a single disordered interface needed for determining the coupling energy 共8兲. In contrast to a perfect interface, the in-plane electron wave vector k储 is no longer conserved. We will model the interfacial disorder by the short-range scattering centers located in the interfaces planes 共x , y兲 and giving rise to the corresponding scattering potentials Vzi 0共r兲 = Vzi 0共␳兲␦共z0兲, where ␳ = 共x , y兲 and z0共=0 , d兲 is the position of the interface. Thus we will present the interfacial scattering potential V0共d兲 as a sum of the potential caused by the potential profile 共30兲 and Vzi 0共r兲,

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PHYSICAL REVIEW B 77, 024410 共2008兲

VICTOR F. LOS AND ANDREI V. LOS

Tz0共z;k储,k⬘储 兲 = Tz0共k储,k⬘储 兲␦共z − z0兲,

V0共␳,z兲 = 关W0 + V0i 共␳兲兴␦共z兲, Vd共␳,z兲 = 关Wd + Vdi 共␳兲兴␦共z − d兲.

共32兲

It is useful to introduce the translationally invariant generally non-Hermitian single-site coherent potential ␴z0 defining the corresponding translationally invariant interfacial reference medium. Then, the interfacial scattering potential V0共d兲 i 共␳兲 may be presented as V0i 共␳兲 = 兺 共␴0 + ˜␥␣0 兲␦共␳ − ␳␣兲, ␣

Vdi 共␳兲 = 兺 共␴d + ˜␥␣⬘兲␦共␳ − ␳␣⬘兲. d

z

˜ z0共k储兲␦ ˜ 0 Tz0共k储,k⬘储 兲 = 关W k储k⬘储 + Vik k⬘兴 储

˜ z0 ˜ z0共k储兲␦ + 兺 关W k储k1储 + Vik储k 储兴 1

k1储

˜ z0 ˜ z0共k 储兲␦ ⫻ G0共z0,z0 ;k1储兲关W 1 k1储k⬘储 + Vik 储k⬘兴 + ¯ , 1 储 共37兲 where ˜ 0共d兲共k储兲 = W0共d兲共k储兲 + n0共d兲␴0共d兲 , W i

共33兲

␣⬘

Here, a summation is over all randomly distributed ␦ scatterers located at the interfaces with the transverse positions ␳␣ 共␳␣⬘兲 and strengths ␥␣0共d兲 = ˜␥␣0共d兲 + ␴0共d兲, where ˜␥␣0共d兲 = ␥␣0共d兲 − ␴0共d兲. If disorder is present at the interface only 关as it is assumed by Eq. 共33兲兴, then the values ␥␣0 can be estimated via the on-site potentials in the materials separated by the interface, i.e., ␥␣0共d兲 ⬃ 共兩U⬍ − U⬎兩 / 2兲共a / 2兲3, where a is the lattice constant. the In the presence of the interface impurity potential V0共d兲 i reflection from the interface may in general be both specular 共conserving k储兲 and diffuse 共not conserving k储兲. It is convenient to describe these kinds of scatterings in the k储 representation in which the diagonal in this representation t-matrix will define the specular scattering and the nondiagonal part of T0共d兲共z ; ␧ , k储 , k⬘储 兲 will correspond to the diffuse scattering. The k储 representation of the Green function describing the electron transport for the case of a single disordered interface situated at z = z0 may be defined as 关compare with Eq. 共14兲兴 Gz0+共r,r⬘ ;␧兲 =

1 兺 ei共k储␳−k⬘储 ␳⬘兲Gz0+共z,z⬘ ;␧,k储,k⬘储 兲. 共34兲 Ak k 储

⬘储

To get the scattering t-matrix T0共d兲共z ; ␧ , k储 , k⬘储 兲 for a spacer electron reflecting from a single interface, we will transform 关with the help of Eq. 共34兲兴 the expansion of the Green function Gz0+共r , r⬘ ; ␧兲 in the r representation 共␧-dependence is suppressed兲, G0共d兲共r,r⬘兲 = G0共r,r⬘兲 + +

冕 冕 dr1



˜V0共d兲 = 兺 共V ˜ 0共d兲兲 , k储k⬘储 ␣ ik k⬘ 储

Gz0共z,z⬘ ;k储,k⬘储 兲 = G0共z,z⬘ ;k储兲␦k储k⬘储 +

˜ z0 + ˜Vz0兲 Tz0 = 共W i

共1 −

− Gz00˜Vzi 0

共39兲 ˜ z0 and ˜Vz0 are defined by Eqs. where the matrix elements of W i z0 共37兲 and 共38兲 and G0 is the Green operator diagonal in the k储 representation 具k储兩Gz00兩k⬘储 典 = G0共z0 , z0 ; k储兲␦k储k⬘储 = iបv21共k 兲 ␦k储k⬘储 关see Eq. 共22兲 for the “free” electron propagator兴. Using the identity for two arbitrary operators A and B 储

1 1 1 1 , = − B A+B A A A+B

共40兲

the series 共39兲 may be rearranged as 1

˜ z0 T z0 = W

1−

˜ z0 Gz00W 1

+ 1− dz1G0共z,z1 ;k储兲

˜ z0 Gz00W

1

+

˜ z0 Gz00W

⫻ Gz00˜Vzi 0

共36兲

Here, the scattering 共from the interface兲 t-matrix in k储 representation is defined as

1 ˜ z0兲 Gz00W

˜ z0 + ˜Vz0兲Gz0共W ˜ z0 + ˜Vz0兲Gz0共W ˜ z0 + ˜Vz0兲 + ¯ , + 共W 0 0 i i i

共35兲

⫻T 共z1 ;k储,k⬘储 兲G0共z1,z⬘ ;k⬘储 兲. z0

共38兲

˜ z0 + ˜Vz0兲 + 共W ˜ z0 + ˜Vz0兲Gz0共W ˜ z0 + ˜Vz0兲 = 共W 0 i i i

1−





n0共d兲 = N0共d兲 / A is the interface impurity density 关N0共d兲 is the i i i number of interfacial defects兴. Scattering matrix Tz0共k储 , k⬘储 兲, Eq. 共37兲, may be viewed as the matrix element 具k储兩¯兩k⬘储 典 of the following operator:

dr2G0共r,r1兲V0共d兲共r1兲G0共r1,r2兲

into the expansion in k储 representation,



1 0共d兲 −i共k储−k⬘兲␳ ˜ 0共d兲兲 储 ␣, ␥ e 共V k储k⬘储 = ˜ ␣ A ␣

dr1G0共r,r1兲V0共d兲共r1兲G0共r1,r⬘兲

⫻V0共d兲共r2兲G0共r2,r⬘兲 + ¯ ,



1

+

˜Vz0 i ˜Vz0 i

1−

˜ z0 Gz00W 1

1−

˜ z0 Gz00W

˜Vz0 i

1 ˜ z0 1 − Gz00W

Gz00˜Vzi 0

1 ˜ z0 1 − Gz00W

1 ˜ z0 1 − Gz00W

1 ˜ z0 1 − Gz00W

Gz00˜Vzi 0

1 ˜ z0 1 − Gz00W

+ ¯ , 共41兲

˜ z0 共both where we have taken into account that Gz00 and W diagonal in k储 representation兲 commute. It follows from Eqs.

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PHYSICAL REVIEW B 77, 024410 共2008兲

THEORY OF INTERLAYER MAGNETIC COUPLING IN…

共22兲, 共29兲, and 共38兲 that the first term of expansion 共41兲, diagonal in k储 representation, is equal to ˜ z0共k储兲 W ˜ z0共k储兲 1 − G0共z0,z0 ;k储兲W

˜ z0共k储兲, = iបv2共k储兲r

共42兲

where ˜r z0共k储兲 is the specular reflection amplitude from the “perfect” interface 共located at z0 = 0 or z0 = d兲 with the homogeneous effective potential ␴z0, 2i 0 ⌺ v2 − v3 − ប d ˜r0共k储兲 = ˜ , r 共k储兲 = 2i v2 + v1 + ⌺0 v2 + v3 + ប v2 − v1 −

2i d ⌺ ប , 2i d ⌺ ប 共43兲

1 ˜ z0共k储兲 1 − G0共z0,z0 ;k储兲W

Another

˜ z0+共k储兲. = iបv2共k储兲G

共44兲

˜ z0+共k储兲 is the effective-medium retarded Green funcHere, G tion 共propagator兲 ˜ 0+共k储兲 = G

˜ d+共k储兲 = G



2



2

2i iប v1 + v2 + ⌺0 ប

2i iប v2 + v3 + ⌺d ប



,



,

共45兲

共46兲

共47兲

⌬⌽ =

1 Im 兺 ␲ k储





d␧f共␧兲

−⬁ ˆ

共50兲

where the matrix elements of the operators r0共d兲 are i ikˆ2d ik2d defined by Eq. 共47兲, 具k储兩e 兩k⬘储 典 = e ␦k储k⬘储 and k2

冑m 2

= ប22 共␧ − U2兲 − k2储 is the perpendicular to the interfaces component of the spacer electron wave vector. Formula 共50兲 is the main formal result of the paper which generalizes Eq. 共31兲 to the case of disordered interfaces. It differs from Eq. 共31兲 by averaging over all the arrangements of the mixed atoms at the disordered interfaces 共inner brackets 具¯典兲 and by matrix elements of the reflection amplitudes 具k储兩rzi 0共␧兲兩k⬘储 典 which are generally nondiagonal and describe both the specular and diffuse scatterings. The averaging procedure applied in Eq. 共50兲 to the logarithmic function is not generally an easy one to perform. It may be performed, for example, by expanding the logarithmic function and then taking the average of each term of this series expansion. In this paper, we will restrict ourselves to the first term of the logarithm expansion which is easiest for evaluation. At the same time, this is a good approximation 共provides the main contribution兲 if the reflection amplitudes are small 共small confinement in the spacer兲. The latter is the case, e.g., in the transition metal heterostructures, where the difference between on-site potentials in different layers 共electronic band profile兲, which defines the intensity of both the quantum reflection from perfect interfaces and from defects located at the interfaces, is small as compared to the Fermi energy 共for more details see below兲.

where z0 ˜ z0 储 兲, ˜ z0共k储兲␦ ˜ z0 Gz0共k储,k⬘储 兲 = G 储 兲G 共k ⬘ k储k⬘储 + G 共k 储兲Ti 共k 储,k⬘

共49兲

Returning to the general formula 共8兲 for the coupling energy caused by the multiple scattering of the spacer electrons from the disordered interfaces, we see that it may be found by making use of the expression 共22兲 for the “free” electron propagator in a spacer and z-dependent t-scattering matrices T0共z ; k储 , k⬘储 兲 = T0共k储 , k⬘储 兲␦共z兲, Td共z ; k储 , k⬘储 兲 = Td共k储 , k⬘储 兲␦共z − d兲, where Tz0共k储 , k⬘储 兲 is defined by Eq. 共49兲 关the trace in Eq. 共8兲 assumes the integrations over z variables兴. As a result we have 关compare with Eq. 共31兲兴

ˆ

On the other hand, the complete reflection amplitude rz0共k储 , k⬘储 兲 accounting for the electron specular and diffuse scattering of a spacer electron from the disordered interface may be expressed through the k储 representation of the complete Green function 共34兲, describing the electron propagation through a disordered interface, as18 rzi 0共k储,k⬘储 兲 = iប冑v2共k储兲v2共k⬘储 兲Gz0共k储,k⬘储 兲 − ␦k储k⬘储 ,

˜ z0共k储兲 is given by Eq. 共45兲. The specular part of the and G reflection amplitude 共47兲 is given by its diagonal part, Eq. 共46兲, and the diffuse part is defined by the electron scattering on the potential fluctuations 共from the effective-medium potential ␴z0兲 ˜Vzi 0. Thus with accounting for Eqs. 共41兲, 共42兲, 共44兲, and 共46兲– 共48兲, Tz0共k储 , k⬘储 兲, Eq. 共37兲, finally takes the form 关compare with Eq. 共28兲兴

⫻具k储兩具ln关1 − r0i 共␧兲eik2drdi 共␧兲eik2d兴典兩k储典,

defining 共as it is shown in Ref. 18兲 the specular part of the transmission through the disordered interface 共located at z0 = 0 , d兲 which is characterized by the effective potential ⌺z0. In fact, the functions 共45兲 are the k储 representation of the Green function 共34兲 for the case of “perfect” interfaces with a homogeneous effective potential ⌺z0 关compare with the first two lines of Eq. 共20兲兴. The specular reflection 关of a spacer electron with a velocity v2共k储兲兴 amplitude 共43兲 from such a “perfect” interface may also be expressed via functions 共45兲 as ˜ z0共k储兲 − 1. ˜r z0共k储兲 = iបv2共k储兲G

k1储

Tz0共k储,k⬘储 兲 = iប冑v2共k储兲v2共k⬘储 兲rzi 0共k储,k⬘储 兲.

⌺z0 = nzi 0␴z0.

which was first obtained in Ref. 18, and factor entering Eq. 共41兲 is

z z0 ˜ z0 + ¯, 共48兲 ˜ z0共k 储兲V Tzi 0共k储,k⬘储 兲 = ˜Vik0 k⬘ + 兺 ˜Vik G 1 ik1储k⬘储 储k1储 储 储

III. INTERLAYER EXCHANGE COUPLING

We now consider the small confinement case when 兩r0i rdi 兩 is small. Then, the leading term of Eq. 共50兲 is 024410-7

PHYSICAL REVIEW B 77, 024410 共2008兲

VICTOR F. LOS AND ANDREI V. LOS

⌬⌽ = −

1 Im 兺 ␲ k储





ˆ

˜ z0+ 兺 t z0+ T zi 0+ = 兺 t␣z0+ + 兺 t␣z0+G ␤

ˆ

d␧f共␧兲具具k储兩r0i 共␧兲eik2drdi 共␧兲eik2d兩k储典典.



−⬁

˜ z0+ 兺 t z0+G ˜ z0+ 兺 t z0+ + ¯ , + 兺 t␣z0+G ␤ ␥

共51兲 The average over the scatterers configurations of the disordered interfaces integrand may be presented 关with the help of Eqs. 共47兲 and 共48兲兴 as

␤⫽␣



˜ z0+˜Vz0 + ¯ ˜ z0+˜Vz0 + ˜Vz0G ˜ z0+˜Vz0G t␣z0+ = ˜V␣z0 + ˜V␣z0G ␣ ␣ ␣ ␣ 共55兲

˜ z0+ are given by Eqs. Here, the matrix elements of ˜V␣z0 and G 共38兲 and 共45兲, respectively. Thus the matrix element of the scattering matrix 共54兲 in the single-site approximation is

˜ 0共k储兲具T0共k储,k储兲典G ˜ 0共k储兲 ˜0共k储兲r ˜d共k储兲 − iបv2共k储兲G = 兵r i ˜ d共k储兲具Td共k储,k储兲典G ˜ d共k储兲 − iបv2共k储兲G i

T zi 0+共k储,k⬘储 兲 = 兺 t␣z0+共k储,k⬘储 兲,

˜ 0共k储兲具T0共k储,k储兲典G ˜ 0共k储兲G ˜ d共k储兲 + 关iបv2共k储兲兴2G i



˜ d共k储兲具Td共k储,k储兲典G ˜ d共k储兲G ˜ 0共k储兲其e2ik2d + 关iបv2共k储兲兴2G i ˜ 0共k储兲G ˜ 0共k⬘储 兲G ˜ d共k储兲G ˜ d共k⬘储 兲 + 兺 关iប冑v2共k储兲v2共k⬘储 兲兴2G

t␣z0+共k储,k⬘储 兲 =

k⬘储

共52兲

˜ 0共d兲共k储兲 and ˜r0共d兲共k储兲 are given by Eqs. 共45兲 and 共46兲, where G respectively. As it has been shown in Ref. 18 the specular scattering from a disordered interface 共located at z = z0兲 may be completely described by the Green function 共45兲 for the effective 共reference兲 medium ⌺z0 if we choose the coherent potential ⌺z0 in such a way that the averaged diagonal element of the t-scattering matrix satisfies the condition 具Tzi 0+共⌺z0+,k储,k储兲典 = 0,

共54兲

where the single-site scattering operator is

˜ z0+兲−1˜Vz0 . = 共1 − ˜V␣z0G ␣

ˆ ˆ 具具k储兩r0i 共␧兲eik2drdi 共␧兲eik2d兩k储典典

⫻ 具T0i 共k储,k⬘储 兲Tdi 共k⬘储 ,k储兲典ei共k2+k2⬘兲d ,

␥⫽␤

␤⫽␣



␥␣z0 − ␴z0 e−i共k储−k⬘储 兲␳␣ , 1 ˜ z0+共k储 兲 1 − 共␥␣z0 − ␴z0兲 兺 G 1 A k储1 共56兲

where a two-dimensional vector ␳␣ specifies the positions of scatterers at the disordered interface 关see Eq. 共33兲兴. The average over scatterers’ configuration 具t␣z0+共k储 , k⬘储 兲典 is diagonal in k储 space, equal to t␣z0+共k储 , k储兲, and therefore does not depend on the positions of imperfections ␳␣. It may be seen explicitly if we adopt, e.g., the following configuration average procedure:

冋兿 冕 册 Ni

具f共␳1, . . . , ␳N兲典 =

共53兲

where the superscript ⫹ refers to the fact that we selected the retarded Green functions for the present consideration. Equation 共53兲 determines the conventional coherent potential approximation 共CPA兲 in the theory of disordered systems. We see from Eq. 共52兲 that in this approximation the contribution of the specular reflections from the interfaces to the coupling energy 共51兲 is completely defined by the specular reflection amplitudes ˜r z0共k储兲 from an interface with translationally invariant impurity potential ⌺z0+ = nzi 0␴z0+ 关thus we defined the non-Hermitian potential in Eq. 共33兲 as ␴z0+兴. The last term in Eq. 共52兲 describes the double scattering of electrons from the interfaces with nonconserving k储 共diffuse scattering兲. It is necessary to define more explicitly the coherent potential ⌺z0+ fixed by Eq. 共53兲. Since we defined ␴z0+ as a single-site translationally invariant potential 共independent of the site ␣ and k储兲, the self-consistency condition 共53兲 should also be considered in the single-site approximation, i.e., T zi 0+ in this equation should be replaced with the t-scattering matrix in the single-site approximation. The scattering t-matrix T zi 0, Eq. 共48兲, in the operator form may be represented in terms of a single-site scattering t-matrix t␣z0 as 共see, e.g., Ref. 20兲

1 A

␣=1

d␳␣ f共␳1, . . . , ␳Ni兲, A

共57兲

where f共␳1 , . . . , ␳Ni兲 is some function of the scatterers coordinates. Then, Eq. 共53兲 for the coherent potential ⌺z0+ reads

兺␣

␥␣z0 − ␴z0 = 0. z0 z0 1 z0+ ˜ 1 − 共␥␣ − ␴ 兲 兺 G 共k储兲 A k储

共58兲

Now, we can see that the last 共diffuse兲 term in Eq. 共52兲 is equal to zero because the average procedure should be applied to each interface independently 关the positions of scattering centers at z = 0共␳␣兲 and z = d共␳␣⬘兲 cannot coincide兴 and therefore according to Eqs. 共57兲 and 共58兲 具T0i 共k储,k⬘储 兲Tdi 共k⬘储 ,k储兲典 = 具T0i 共k储,k⬘储 兲典具Tdi 共k⬘储 ,k储兲典 = 0 共59兲 in the adopted coherent potential approximation for electron scattering at disordered interfaces. Thus the coupling energy 共51兲 takes the following form: ⌬⌽ = −

1 Im 兺 ␲ k储





˜0共k储兲r ˜d共k储兲e2ik2d d␧f共␧兲r

共60兲

−⬁

and is defined completely by the specular reflection amplitudes from disordered interfaces 共43兲. It should be stressed,

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THEORY OF INTERLAYER MAGNETIC COUPLING IN…

however, that these amplitudes are affected by diffuse scattering 共see below兲. Formula 共60兲, obtained for the case of small reflection amplitudes, provides a relatively simple starting point for further consideration. It accounts for interference between an incident wave and a reflected wave resulting from only one round trip of an electron in the spacer and thus neglects the interference effects of the higher order originated from n ⬎ 1 round trips 共2n reflections from interfaces兲. Use of the single-site CPA 共58兲 implies neglecting of scattering by clusters of defects at each interface, which is reasonable for the considered small scattering case assuming weak scattering by interfacial disorder also. The latter also means that the coherent potential, defined by Eq. 共58兲, should be taken in the the weak scattering limit 关see Eq. 共79兲 and comments below兴. Restriction to only one round trip of an electron in the spacer also results in neglecting the correlation effects between scatterings by defects occurring when an electron strikes the same disordered interface more than once 共these effects arise in the next terms of the logarithm expansion in the small reflection amplitudes兲. Expression 共60兲 is valid for arbitrary materials making up the trilayer. We will now consider the case of a paramagnetic spacer 共metallic or insulator兲 sandwiched between two ferromagnetic films. The quantum interferences of the electronic waves in the spacer induce an interlayer interaction between the ferromagnetic layers. In the ferromagnetic configuration of the magnetic layers 1 ⌬⌽F = − Im 兺 ␲ k储





−⬁

˜0↑共k储兲r ˜d↑共k储兲 d␧f共␧兲关r

˜d↓共k储兲兴e2ik2d , + ˜r0↓共k储兲r

共61兲

while in the antiferromagnetic configuration ⌬⌽AF = −

1 Im 兺 ␲ k储





−⬁

˜0↑共k储兲r ˜d↓共k储兲 d␧f共␧兲关r

˜d↑共k储兲兴e2ik2d , + ˜r0↓共k储兲r

共62兲

z0 ↑共↓兲

where ˜r stands for the reflection amplitude of a spacer electron with the spin parallel 共antiparallel兲 to the magnetization direction of the magnetic layer at the ferromagnetparamagnet interface situated at z = z0. Evidently, ˜r z↑0 ⫽˜r z↓0. Taking the difference of these two expressions and replacing the sum over k储 by the integral, we arrive at the following exchange coupling energy per unit interface area: EF − EAF = −

1 Im ␲3

冕 冕



˜0共k储兲⌬r ˜d共k储兲e2ik2d , d␧f共␧兲⌬r

dk储

For simplicity, we take the ferromagnetic 共semi-infinite兲 layers to be identical, i.e., 0 d ˜r↑共↓兲 = ˜r↑共↓兲 = ˜r↑共↓兲,

˜r↑共↓兲 =

+ v2 + v↑共↓兲 + i⌫↑共↓兲

v2 =

បk2 , m2

ប2k22 ប2k2储 = ␧ + i0+ − − U2 , 2m2 2m2

˜ z0 = ⌬r

˜r

− ˜r 2

共64兲

共67兲

v↑共↓兲 is the electron perpendicular velocity in the ferromagnetic layer with the spin parallel 共antiparallel兲 to the magnetization direction v↑共↓兲 =

បk↑共↓兲 m↑共↓兲

,

ប2k2↑ ប2k2储 = ␧ + i0+ − , 2m↑ 2m↑

ប2k2↓ ប2k2储 = ␧ + i0+ − − ⌬, 2m↓ 2m↓

共68兲

⌬ is the exchange split of spectrum in the ferromagnet, and 2 + 2 + + = ⌺↑共↓兲 = ni␴↑共↓兲 ⌫↑共↓兲 ប ប

共69兲

is defined by a coherent potential of an intermixed ferromagnet-paramagnet interface 共dependent generally of an electron energy ␧兲 which is “felt” by a spacer electron with the spin parallel 共antiparallel兲 orientation relative to the ferromagnet magnetization direction. Note that the effective masses of spin-up and spin-down electrons m↑共↓兲 are generally different and the condition 共65兲 and the definition 共69兲 imply that the intermixing of atoms at both ferromagnet0+ d+ + paramagnet interfaces are identical, i.e., ⌺↑共↓兲 = ⌺↑共↓兲 = ⌺↑共↓兲 . As it follows from Eqs. 共66兲–共68兲, the reflection amplitudes depend on the perpendicular to the interfaces components of electron velocity 共wave vector兲, which, of course, depend on the electron energy ␧ and the parallel to the interfaces component of the wave vector k储 关see Eqs. 共67兲 and 共68兲兴. It is convenient to replace integration in Eq. 共63兲 over ប 2k 2 ␧ and k储 by integration over ␧2 = 2m22 + U2 and k储, which may now be viewed as the independent variables of integration 共see Ref. 8兲. Then, after integration over k储 the exchange coupling energy 共63兲 takes the form EF − EAF = −

2m2 kBT Im ␲ 2ប 2





˜兲2e2ik2d d␧2共⌬r

−⬁

冋 冉 冊册

⌬⌽

z0 ↓

共66兲

.

Here, v2 is the spacer electron velocity perpendicular to an interface,

where EF = A F , EAF = AAF , and the spin-dependent asymmetries of the reflection amplitudes

are introduced.

+ v2 − v↑共↓兲 − i⌫↑共↓兲

−⬁

z0 ↑

共65兲

According to Eq. 共43兲, the reflection amplitudes from ferromagnetic layers may be defined as

共63兲 ⌬⌽

˜0 = ⌬r ˜d = ⌬r ˜. ⌬r

⫻ln 1 + exp

␧F − ␧2 k BT

,

共70兲

˜ 共as well as k2兲 depends only on ␧2 共we assume that where ⌬r the effective mass of a spacer electron is close to the electron masses in the ferromagnet, i.e.,

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兩m2−m↑共↓兲兩

m2+m↑共↓兲

Ⰶ 1兲. The integral

PHYSICAL REVIEW B 77, 024410 共2008兲

VICTOR F. LOS AND ANDREI V. LOS

共70兲 may easily be evaluated at large spacer thickness d when the most important contribution to the coupling energy comes from the neighborhood of the Fermi energy ␧F 共due to the rapidly varying exponential factor e2ik2d兲. By expanding the exponential factor near ␧F and performing the change of variable



x = exp



␧F − ␧2 , k BT

共71兲

the integral 共70兲 may be reduced to the tabulated integral 共see also Ref. 8兲. The final result is EF − EAF = 共EF − EAF兲T=0

2␲kBTd/បvF , sinh共2␲kBTd/បvF兲

共72兲

same as for the case of perfect interfaces: the coupling decreases with temperature for a metallic spacer and increases with T in the case of an insulating spacer 共see Ref. 8兲. Since the self-energy 共coherent potential兲 plays an important role in our calculations, we will now consider it in more detail. In Eq. 共58兲 for a binary-alloy-type interface with atoms of type 1 共ferromagnetic atoms兲 and of type 2 共spacer atoms兲 intermixed, ␥␣ = ␥1,␥2, where ␥1 = ␥↑共↓兲 is the strength of scattering of a spacer electron 关with the spin parallel 共antiparallel兲 to magnetization of the interface ferromagnet兴 by magnetic atoms, and ␥2 is the strength of scattering by the nonmagnetic atoms. Equation 共58兲 may be rewritten, suppressing the interface index z0, as follows: ¯␥↑共↓兲 − ␥↑共↓兲␥2

where 共EF − EAF兲T=0 =

m2vF2 F ˜F兲2e2ik d兴 Im关共⌬r 2 ␲ 2d 2

is the exchange coupling energy at zero temperature, hkF , v = m2 F

F

k =



2m2 共␧F − U2兲 ប2

共74兲

˜F is is the Fermi velocity and wave vector for a spacer, ⌬r defined as ˜F = ⌬r

vF↑ =

ប m↑

+F F − i⌫↑共↓兲 vF − v↑共↓兲 ˜rF↑ − ˜rF↓ F = F , ˜r↑共↓兲 +F , F 2 + i⌫↑共↓兲 v + v↑共↓兲



2m↑ ␧ F, ប2

vF↓ =

ប m↓



2m↓ 共␧F − ⌬兲, ប2

␴↑共↓兲 =

共73兲

共75兲

k BT ⬍

1 , 2d Im បvF

冉 冊

共76兲

which imposes no restriction for a metallic spacer but restricts temperatures to the sufficiently low ones in the case of an insulating spacer. The result given by Eqs. 共72兲 and 共73兲 differs from the one obtained in Refs. 11 and 8 by substitution ⌬rF = 共rF↑ F F F − rF↓ 兲 / 2, where r↑共↓兲 = 关vF − v↑共↓兲 兴 / 关vF + v↑共↓兲 兴 is the spindependent reflection amplitude for a perfect interface, with ˜F, Eq. 共75兲, defined for the disordered interface. This the ⌬r difference is very substantial because the coherent potential F F F F ⌫↑共↓兲 is a complex quantity, i.e., ⌫↑共↓兲 = Re ⌫↑共↓兲 + i Im ⌫↑共↓兲 共see Ref. 18 and below兲. The first consequence of this fact is F that the real part of ⌫↑共↓兲 leads to the emergence 共for a metallic spacer兲 of the cosinelike oscillation of the interlayer coupling 共73兲 in addition to the conventional sinelike behavior observed for perfect interfaces 共when ⌬rF is real兲. The temperature dependence of the interlayer coupling 共72兲 is the

1 ˜ + 共k储兲 1 − 关␥↑共↓兲 + ␥2 − ␴↑共↓兲兴 兺 G A k储 ↑共↓兲

,

共77兲

where ¯␥↑共↓兲 = 1 / Ni⌺␣␥␣ = 关c1␥↑共↓兲 + c2␥2兴 is the average strength of the interface scatterers due to disorder, c1,2 = N1,2 / Ni are concentrations of the first 共ferromagnetic兲 and second 共nonmagnetic兲 materials mixed at the interface 共c1 + c2 = 1兲. The strength of scatterers may be estimated via the difference of the one-site potentials for electrons at both sides of the interface. For a spacer electron ␥↑ ⬃ U2a3, ␥↓ ⬃ 兩U2 − ⌬兩a3, and ␥2 = 0, where we have used the model spec˜ +共k储兲 in Eq. tra 共68兲. The effective-medium Green function G 共77兲 is defined by Eq. 共45兲, ˜ + 共k储兲 = G ↑共↓兲

+F ⌫↑共↓兲

and denotes the coherent potential 共69兲 taken at ␧F. For a metallic spacer ␧F ⬎ U2 and vF共kF兲 is real, while for an insulating spacer ␧F ⬍ U2 and vF共kF兲 is imaginary. It should be added that the integral 共70兲 converges and the result 共72兲 is only valid for

1 兺 G˜+ 共k储兲 A k储 ↑共↓兲

2 + iប关v↑共↓兲 + v2 + i⌫↑共↓兲 兴

,

共78兲

where v↑共↓兲 and v2 are defined by Eqs. 共67兲 and 共68兲 共taken at ␧ = ␧F兲, respectively, and summation 共integration兲 over the parallel to interface wave vector includes both propagating and evanescent intermediate states. The evanescent states F come from integration over k储 ⬎ min共kF , k↑共↓兲 兲, where kF is

冑m 2

冑m 2

defined by Eq. 共74兲, kF↑ = ប2↑ ␧F , kF↓ = ប2↓ 共␧F − ⌬兲 , kF↑ ⬎ kF↓ , kF, and one of the electron velocities is purely imaginary. The summation in Eq. 共77兲 over evanescent states F should be cutoff at a wave vector ␣ min共kF , k↑共↓兲 兲 共␣ ⬎ 1兲 to account for the finite range of the scattering potential and to avoid an ultraviolet divergence which is a consequence of using the ␦-like impurity potentials 共33兲. If ¯␥↑共↓兲 ⫽ 0, the expression for the coherent potential ␴↑共↓兲 may be obtained by iterating Eq. 共77兲 using the small param+ F eter 兩 ប2 ⌺↑共↓兲 兩 / v↑共↓兲 Ⰶ 1 共weak scattering兲. Actually, for consistency of the adopted approximation 共small reflection amplitudes兲 we should take the coherent potential in the lowest approximation in this weak scattering parameter. In the first + + approximation, ␴↑共↓兲 = ¯␥↑共↓兲, and therefore ⌺↑共↓兲 = Re ⌺↑共↓兲 + = ni¯␥↑共↓兲. In this approximation Im ⌺↑共↓兲 = 0, which means that the electron diffuse scattering at a disordered interface, defined by the imaginary part of the coherent potential,18 is absent. In the next approximation

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THEORY OF INTERLAYER MAGNETIC COUPLING IN…

+ 2 2 ⌺↑共↓兲 = ni¯␥↑共↓兲 + ni关␥↑共↓兲 − ¯␥↑共↓兲 兴

1 兺 G˜+ 共k储兲. A k储 ↑共↓兲

共79兲

2 2 ˜ + 共k储兲 is Here, ¯␥↑共↓兲 = c1␥↑共↓兲, ␥↑共↓兲 = c1␥↑共↓兲 共␥2 = 0兲, and G ↑共↓兲 + defined by Eq. 共78兲, where ⌺↑共↓兲 is substituted with ni¯␥↑共↓兲. The self-energy 共coherent potential兲 is now complex as the second term in Eq. 共79兲 has both the real and imaginary parts. In the case under consideration, the fluctuations of the scattering potential of the disordered interfaces, given by the second term in Eq. 共79兲, are small and therefore the real part of the scattering potential, contributing to the electron specular scattering at a disordered interface,18 is much larger than the imaginary part. However, if we wish to take into account the diffuse scattering, the imaginary part of the coherent potential, given by the second term of Eq. 共79兲, should be taken into consideration. If ¯␥↑共↓兲 = 0, then in the first approximation in the small + + + + F parameter 兩 ប2 ⌺↑共↓兲 兩 / v↑共↓兲 , Re ⌺↑共↓兲 ⬃ Im ⌺↑共↓兲 , and Re ⌺↑共↓兲 is due to the contribution of evanescent states into the abovementioned sum over k储 共for more details see Refs. 21 and 18兲. Equations 共72兲–共77兲 solve, under the accepted approximations, the problem of the interlayer exchange coupling in multilayers with disordered interfaces. The key element ˜F, Eq. 共75兲, may be rewritten as stipulating this coupling, ⌬r

˜F = vF ⌬r

+F 共˜vF↓ − ˜vF↑ 兲 + i共Re ⌫+F ↓ − Re ⌫↑ 兲 , 共80兲 F 共vF + ˜vF↑ + i Re ⌫+F vF↓ + i Re ⌫+F ↑ 兲共v + ˜ ↓ 兲

+F F F where ˜v↑共↓兲 = v↑共↓兲 − Im ⌫↑共↓兲 . Using the analytical properties ˜ + 共k储兲, it may be shown that of retarded Green functions G

during fabrication of the multilayers and difficulties in controlling this process 共see, e.g., Ref. 17兲, it seems reasonable to consider the parameter ␣ as an adjustment parameter which may be evaluated experimentally. Anyway, one can see that the contribution of disorder to the interfacial coupling may be comparable with that of an electron confinement in a spacer. It is convenient to introduce the dimensionless complex parameter +F +F +F F F ␣ = ␣⬘ + i␣⬙ = 关Re共⌫+F ↓ − ⌫↑ 兲 + i Im共⌫↓ − ⌫↑ 兲兴/共v↓ − v↑ 兲

共81兲 and to rewrite Eq. 共80兲 in terms of it. The result is ˜F = ⌬r



冊冉 冊 冊 册冋 冉 冊 册

关共1 − ␣⬙兲 + i␣⬘兴 1 − i ⫻

冋 冉

Re ⌫F 1 + F ↑F v + ˜v↑

2

Re ⌫F↑ vF + ˜vF↑

1−i

Re ⌫F↓ vF + ˜vF↓

Re ⌫F 1 + F ↓F v + ˜v↓

2

.

共82兲 The real part of Eq. 共81兲 ␣⬘ describes the contribution of the specular scattering at disordered interfaces while the imaginary part ␣⬙ defines the contribution of the diffuse scattering. It is possible to simplify Eq. 共82兲 in the case of small interF F facial scattering, 兩⌫↑共↓兲 兩 / v↑共↓兲 Ⰶ 1, using the following restrictions on the parameters ␣⬘ and ␣⬙: F F F F 兩/兩vF + v↑共↓兲 兩 Ⰶ 兩␣⬘兩 Ⰶ 兩1 − ␣⬙兩兩vF + v↑共↓兲 兩/兩⌫↑共↓兲 兩. 兩⌫↑共↓兲

↑共↓兲

+ 艋 0. The electron velocity difference vF↑ − vF↓ , definIm ⌫↑共↓兲 ing the specular reflection amplitude at a perfect interface, is ⌬ of the 21 vF↑ ␧F order 共we assumed that ␧⌬F Ⰶ 1, which is the case for the transition metal heterostructures and is in agreement with the adopted approximation of a small electron confinement in a spacer兲. As formula 共80兲 shows, the contribution of the interfacial disorder, caused by the intermixing of magnetic and nonmagnetic atoms at an interface, is defined by the difference of the interface coherent potential for spin-up +F and spin-down electrons ⌫+F ↑ − ⌫↓ , i.e., by asymmetry of spin-dependent scattering due to the interfacial disorder. It +F follows from the above estimates that 兩⌫+F ↑ − ⌫↓ 兩 N N 1 1 1 1 1 ⬃ ប nic1⌬a3 = ប A ⌬a3 = ប c1a⌬, where A is the density of magnetic atoms at the mixed interface. Thus the relative con+F F F tribution of an interface disorder 兩⌫+F ↑ − ⌫↓ 兩 / 兩v↑ − v↓ 兩 F ⬃ c1ak↑ . Evidently, when the concentration of magnetic atoms at a mixed interface goes to zero, the interface becomes perfect, i.e., made up from the nonmagnetic atoms only, and the contribution of interfacial disorder vanishes. On the other hand, when c1 → 1, the interface also tends to the perfect one but made up from the magnetic atoms only and the contribution of disorder should also vanish. Then, one may expect that the relative contribution of atoms’ mixing at an interface to the exchange coupling would depend on the concentration +F F F F of defects as 兩⌫+F ↑ − ⌫↓ 兩 / 兩v↑ − v↓ 兩 = 兩␣兩 ⬃ c1共1 − c1兲ak↑ . Taking into account the complexity of the interface alloying process

vF共vF↓ − vF↑ 兲 共vF + ˜vF↑ 兲共vF + ˜vF↓ 兲

共83兲 Then, the needed square of Eq. 共82兲 takes the form ˜F兲2 = 共⌬rF兲2关共1 − ␣⬙兲2 − ␣⬘2 + 2i␣⬘共1 − ␣⬙兲兴, 共84兲 共⌬r where ⌬rF =

vF共vF↓ − vF↑ 兲

共vF + vF↑ 兲共vF + vF↓ 兲

共85兲

is the reflection amplitude difference ⌬rF = 共rF↑ − rF↓ 兲 / 2 for the perfect interface. Condition 共83兲 implies that 兩1 − ␣⬙兩 should not be too small, meaning that the diffuse scattering is not very prominent, and 兩␣⬘兩, describing specular scattering, is not very small in the considered case of small interfacial F scattering defined by the coherent potential ⌫↑共↓兲 . Particularly, condition 共83兲 may be satisfied when fluctuations of the scattering potential are small 共兩␣⬙兩 Ⰶ 1兲 and the electron scattering by disordered interface is caused mainly by the real F 共average interface potential兲. part of ⌫↑共↓兲 For the case of a metallic spacer, when vF is real, the ˜F兲2 given by Eq. interlayer exchange coupling 共73兲 with 共⌬r 共84兲 may be represented as

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2

共EF − EAF兲T=0 =

2

m 2v F k F 共⌬rF兲2 f共␣⬘, ␣⬙ ;kFd兲, 2␲2

PHYSICAL REVIEW B 77, 024410 共2008兲

VICTOR F. LOS AND ANDREI V. LOS

f共␣⬘, ␣⬙ ;kFd兲 =

1 兵关共1 − ␣⬙兲2 − ␣⬘2兴sin共2kFd兲 共k d兲2 F

+ 2␣⬘共1 − ␣⬙兲cos共2kFd兲其.

共86兲

Equation 共86兲 shows that in the case of the disordered interfaces there are the conventional, existing at ␣ = 0 when interfaces are perfect, sinelike term in the exchange coupling, and an additional cosinelike term, caused mainly by the real part of the parameter 共81兲 共the real part of the interface coherent potential, responsible for the electron specular scattering by the interfacial disorder18兲. The behavior of Eq. 共86兲, as a function of spacer thickness d, depends on the values of the parameters ␣⬘ and ␣⬙ and their signs. Thus it can be expected that the experimental data on the interfacial exchange coupling in the real multilayers with disordered 共alloylike兲 interfaces may by described, at least in the case of weak scattering, by the formula 共86兲 with the properly selected fitting parameters ␣⬘ and ␣⬙. As will be shown below, a similar expression for interfaces with different intermixings includes four fitting parameters. Let us qualitively consider parameter ␣ in more detail. It +F seems natural to relate ⌫+F ↑ − ⌫↓ to the spin-up-spin-down density of states difference at the Fermi energy ␧F. In the adopted EMA for the spectra of electrons, Eqs. 共67兲 and 共68兲, such a difference 共per atom兲 is

␳F↑ − ␳F↓ =

a3 m⌬ , 2␲2ប3 vF↑

共87兲

where we assumed for simplicity that m↑ = m↓ = m and ␧⌬F Ⰶ 1. On the other hand, according to the simple estimate given above 关for the case of weak scattering at interfacial F disorder when the real part of ⌫↑共↓兲 is given by the first term F F of Eq. 共79兲兴, Re共⌫↑ − ⌫↓ 兲 ⬃ c1共1 − c1兲a⌬ / ប, i.e., this difference is defined by the spectrum exchange splitting in a ferromagnet as well as the density of state difference 共87兲. Then, we may give the following estimate: +F Re共⌫+F ↑ − ⌫↓ 兲 ⬃

2␲2ប2vF↑ c1共1 − c1兲共␳F↑ − ␳F↓ 兲. a 2m

共88兲

One can see from this very simplified consideration that the sign of the difference between the strengths of spin-up and spin-down electron scatterings at a disordered interface is naturally defined by the corresponding difference between their densities of states at the Fermi level. In the considered EMA, ␳F↑ − ␳F↓ ⬎ 0, vF↑ − vF↓ ⬇ 21 ␧⌬F ⬎ 0, and thus ␣⬘ ⬎ 0; but, in many rather typical situations the spin-down electrons scatter more heavily, i.e., ␳F↑ − ␳F↓ ⬍ 0 and ␣⬘ may change sign. This follows from the fact that the EMA usually is not good, e.g., for spin-down electrons in such ferromagnets as Co, Ni, and Fe which are typically used for making magnetic multilayers exhibiting GMR 共EMA may be good for the spin-up electrons in, e.g., Co and Ni, and for Cu electrons at the Fermi energy兲. Thus the parameter ␣⬘ may generally have different signs. As to ␣⬙, from the physical point of view, it should be positive since the diffuse scattering at an interface may not increase the contribution to the interface coupling. These parameters may also have different values depending on the

FIG. 1. Interlayer coupling parameter f, Eq. 共86兲, for two ferromagnetic layers separated by a nonmagnetic metallic spacer with disordered interfaces as a function of normalized spacer thickness and 共a兲 specular scattering parameter ␣⬘ for ␣⬙ = 0 共no diffuse scattering兲, and 共b兲 diffuse scattering parameter ␣⬙ for ␣⬘ = 1 / 2. The surface grid is made up of lines of constant kFd, and 共a兲 ␣⬘ or 共b兲 ␣⬙. Note that in the adopted case of small scattering, valid values of ␣⬘ and ␣⬙ are defined by conditions 共83兲.

specific materials making up the multilayer. In the considered simple model we can see from Eqs. 共87兲 and 共88兲, and 1 ⌬ vF↑ − vF↓ ⬇ 2 ␧F that ␣⬘ ⬃ c1共1 − c1兲kF↑ a. Equation 共82兲 shows that at 兩␣兩 Ⰶ 1 the influence of disorder on the interfacial coupling is small and the phase and amplitude of the coupling oscillations are almost the same as for the perfect interfaces. We will consider the situation when 兩␣⬘兩 satisfies the condition 共83兲, i.e., it is not too small 共we have seen that this is possible and stipulated, e.g., by the average scattering potential of a disordered interface兲. When 兩␣兩 grows, the scattering by disordered interfaces leads to a decrease of the conventional 共existing at ␣ = 0 when interfaces are perfect兲 sinelike term in the exchange coupling 关first term in Eq. 共86兲兴 and to an emergence of the cosinelike term 关兩1 − ␣⬙兩 is also not too small according to Eq. 共83兲兴. The real part ␣⬘, if large enough, may also change the sign of the sinelike term. Thus when 兩␣兩 grows, the phase and amplitude of the dependence of exchange coupling oscillation on the spacer thickness change. Figure 1 illustrates dependencies of the dimensionless function f 共86兲 on normalized spacer thickness and real and imaginary parts of the disordered interface scattering parameter ␣. The amplitudes of both terms in Eq. 共86兲 become equal when 兩共1 − ␣⬙兲2 − ␣⬘2兩 = 2兩␣⬘共1 − ␣⬙兲兩. When ␣⬘ ⬍ 0 and the amplitudes of both terms in Eq. 共86兲 become approximately equal, their superposition behaves as 冑2 sin共2kFd − ␲4 兲 or −冑2 cos共2kFd − ␲4 兲 depending on the sign of the first term. For ␣⬙ Ⰶ 1, the former realizes when ␣⬘ ⬇ −0.4, and the latter when ␣⬘ ⬇ −2.4. Thus the cosine term leads to the oscillation phase shift which grows with 兩␣⬘兩 reaching the value of −␲ / 4 at ␣⬘ ⬇ −0.4, and further the value of −3␲ / 4 at ␣⬘ ⬇ −2.4, when the sign of the first term in Eq. 共86兲 becomes negative. It is also seen from Eq. 共86兲 that the amplitude of oscillations grows with 兩␣⬘兩. For ␣⬘ ⬎ 0, the phase shift goes in the opposite direction 共at given above values of 兩␣⬘兩, the phase shifts are ␲ / 4 and 3␲ / 4, respectively兲. For 兩␣⬘兩 = 兩1 − ␣⬙兩, the phase shift is ±␲ / 2 and the conventional sin共2kFd兲 oscillations shift to the ±cos共2kFd兲 ones. When ␣⬙ Ⰶ 1, this realizes for 兩␣⬘兩 ⬇ 1 关see Fig. 1共a兲兴. As ␣⬙, which character-

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THEORY OF INTERLAYER MAGNETIC COUPLING IN…

izes the diffuse scattering, grows, the amplitudes of both terms in Eq. 共86兲 become smaller and the corresponding phase shifts are reached at smaller values of 兩␣⬘兩. Thus larger diffuse scattering leads to larger phase shift 共at the same value of ␣⬘兲 and smaller oscillation amplitudes 关see Fig. 1共b兲兴. The phase shift may reach ␲ when 兩␣⬘兩 Ⰷ 兩1 − ␣⬙兩. Some experiments provide data for the phase shifts of the exchange coupling oscillations. For example, in Fe/ Cr multilayers the phase of interface exchange oscillations was found to be opposite to that predicted theoretically.22 In reality, the intermixing of atoms at different interfaces 0+ d+ ˜0 ⫽ ⌬r ˜ d. ⫽ ⌺↑共↓兲 and, consequently, ⌬r are different, i.e., ⌺↑共↓兲 It is easy to generalize Eq. 共86兲 to this case. The result is 共EF − EAF兲T=0 =

m2vF2 共⌬rF兲2兵关共1 − ␣0⬙兲共1 − ␣d⬙兲 − ␣0⬘␣d⬘兴 2 ␲ 2d 2 ⫻sin共2kFd兲 + 关␣0⬘共1 − ␣d⬙兲 + ␣d⬘共1 − ␣0⬙兲兴cos共2kFd兲其,

共89兲

⬘ and ␣0共d兲 ⬙ are the real and imagiwhere the parameters ␣0共d兲 nary parts of the parameter ␣0共d兲 defined as Eq. 共81兲 for each interface located at z = 0 and z = d, respectively. For validity of Eq. 共89兲 these generally different parameters should satisfy condition 共83兲 at each interface. Four parameters entering Eq. 共89兲 provide more flexibility for adjusting the obtained expressions to the experiment. IV. CONCLUSION

The theory of interlayer coupling in the nanostructures with the disordered alloylike interfaces is developed in the framework of the scattering theory based on the Green functions approach. We found it to be the most convenient formalism for treating the interlayer coupling through the basic mechanism of the electron waves interference resulting from multiple scattering of spacer electrons at both interfaces. This approach required reconsidering the specular electron reflection from a perfect interface in order to find the local potential responsible for such reflection. Conventionally, transmission through and reflection from an interface is considered by matching the electron wave functions and their derivatives at both sides of the interface. This approach becomes problematic for the interlayer coupling in the case of disordered interfaces when in addition to the specular scattering from a steplike potential, the diffuse scattering emerges and k储 is not conserved. The obtained interfacelocalized potential responsible for the specular scattering at

1 M.

N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 共1988兲. 2 R. Julliere, Phys. Lett. 54A, 225 共1975兲. 3 P. LeClair, J. S. Moodera, and R. Meservay, J. Appl. Phys. 76, 6546 共1994兲.

the perfect interface given by Eqs. 共29兲 and 共30兲 allows considering multiple scattering at potential steps 共interfaces兲 in the framework of general scattering theory. This result also seems to be interesting from the general quantum mechanical point of view. Using the local potential 共29兲 and 共30兲 due to the potential profile and adding the short-range potential 共33兲 describing the interfacial scattering by a reference translationally invariant medium and the fluctuations of the scattering disorder potential, we obtained the general expression 共50兲 for the interlayer coupling in the trilayer with disordered interfaces. In contrast to the similar expression for perfect interfaces 共31兲 共see Ref. 8兲, expression 共50兲 contains reflection amplitudes 共47兲 for disordered interfaces 共describing both the specular and diffuse scatterings兲 with averaging over all configurations of the atoms mixed at the interfaces. To get further analytical results, the small reflection amplitude case 共small confinement兲 was considered, which is a good approximation for many practically important systems such as the transition metal based heterostructures 关more complicated situations, when the next terms of the Eq. 共50兲 expansion may be important, are planned to be considered兴. In the single-site coherent potential approximation for the t-scattering matrix, the interlayer coupling 共60兲 is obtained in terms of amplitudes of specular reflection Eq. 共43兲 from the reference medium fixed by the CPA condition 共53兲. This result was used to get the interlayer exchange coupling 共72兲 between two ferromagnets divided by the nonmagnetic metal or insulator spacer. The contribution of disorder to this exchange coupling is defined by the difference of the interface coherent potential for spin-up and spin-down electrons ⌫+F ↑ − ⌫+F ↓ , i.e., by asymmetry of spin-dependent scattering due to the interfacial disorder 关see Eq. 共80兲兴. This contribution is described by the introduced dimensionless complex parameter ␣ 共81兲, which may be viewed as the adjustment parameter for relating the obtained expressions to experimental data. It has been estimated that the contribution of disorder to the interlayer coupling may be comparable with that of an electron confinement in a spacer with perfect interfaces and defined mainly by the real part of the complex coherent po+F F F tential difference ␣⬘ = Re共⌫+F ↓ − ⌫↑ 兲 / 共v↓ − v↑ 兲, which is related to the specular reflection from a disordered interface. This results in an emergence of a cosinelike term in addition to the conventional sinelike term in the exchange coupling for a metallic spacer. The primary effect of the superposition of these two terms is substantial, depending on the value of ␣, phase shift, and amplitude change in the oscillating exchange coupling in the GMR trilayer.

4 P.

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W. Zhou and H. N. G. Wadley, J. Appl. Phys. 84, 2301 共1998兲. V. M. Uzdin and W. Keune, J. Phys.: Condens. Matter 19, 136201 共2007兲. 18 V. F. Los, Phys. Rev. B 72, 115441 共2005兲. 19 D. A. Stewart, W. H. Butler, X.-G. Zhang, and V. F. Los, Phys. Rev. B 68, 014433 共2003兲. 20 E. N. Economou, Green’s Functions in Quantum Physics 共Springer, Berlin, 1979兲. 21 A. Brataas and G. E. W. Bauer, Phys. Rev. B 49, 14684 共1994兲. 22 C. M. Schmidt, D. E. Burgler, D. M. Schaller, F. Meisinger, and H.-J. Guntherodt, Phys. Rev. B 60, 4158 共1999兲. 16 X. 17

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