[Springer Tracts in Modern Physics] Magnetic Heterostructures Volume 227 || Exchange Coupling in Magnetic Multilayers

4

Exchange Coupling in Magnetic Multilayers

Bretislav Heinrich

Physics Department Surion Fraset University Burnaby, BC, V5AIS6, Canada

Abstract. Spintronics devices employ a wide range of magnetic multilayerstructures. In this chapter the coupling between the magnetic layers throughnon-magnetic spacers will be reviewed. It starts with phenomenology of interlayermagnetic coupling. This is followed by experimental techniques and a detailed in-terpretation of measurements. The main emphases will be given to the static and rfmeasurement techniques. Theory of magnetic coupling covers the basic ideas of inter-layer exchange coupling and dipolar coupling. The additional contributions causedby imperfections in realistic samples are discussed in sessions on orange peel cou-pling, pinhole coupling, and biquadratic exchange coupling. Experimental resultsand their interpretation were carried out to emphasize the richness of magneticinteractions in a wide range of systems. A particular attention was given to theFe/Ag,Au/Fe, Co/Cu/Co and Fe/Cr,Mn,Pd/Fe structures allowing one to demon-strate the importance of interfaces on magnetic coupling in multilayer systems. Fi-nally, it is shown that the time retarded response of interlayer exchange interactionleads to an entirely new coupling which is based on spin pumping and spin sinkconcepts. This contribution arises only during spin dynamics and compared to thestatic interlayer exchange coupling is long ranged allowing one in principle move in-formation by means of a spin current via a mechanism that does not directly involvea net transport of electron charge.

4.1 Introduction

Spintronics and high density magnetic recording employ a number of magneticlayers separated by non magnetic spacers. The magnetic layers are coupledthrough a non-magnetic spacer. The coupling between the magnetic layerscan be caused by intrinsic and extrinsic mechanisms and be of static anddynamic origin. The purpose of this Chapter is to review the basic conceptsof magnetic coupling in 3d transition elements. A number of excellent reviewarticles have been published by Hathaway [1], Slonczewski [2], Stiles [3], andEdwards and Umerski [4]. An extensive review of the behavior of spin densitywaves in Fe/Cr/Fe trilayers and multilayers is provided by Fishman [5].

B. Heinrich: Exchange Coupling in Magnetic Multilayers, STMP 227, 185–250 (2007)

DOI 10.1007/978-3-540-73462-8 4 c© Springer-Verlag Berlin Heidelberg 2007

186 B. Heinrich

The Chapter is organized in the following way: Section 4.2 describesthe phenomenology of magnetic coupling. Section 4.3 deals with the basictechniques employed in the study of magnetic coupling. A brief summaryof theoretical concepts of magnetic coupling is carried out in Sect. 4.4. Asummary of experimental results on Fe/Ag,Au/Fe(001), Co/Cu/Co(001), andFe/Cr,Mn,Pd/Fe(001) systems is presented in Sect. 4.5. The dynamic couplingwill be described in Sect. 4.6.

4.2 Phenomenology of Magnetic Coupling

The description of magnetic coupling will be mostly restricted to ultrathinmagnetic films. In ultrathin films magnetic variations across the thicknessof the film are mainly suppressed. This means that the magnetic momentson lattice sites across the film thickness are nearly parallel to each other.This is not exactly correct, but greatly simplifies the treatment of magneticproperties. The limits of this concept are described in [6] and [7]. The filmcan be fairly well considered as ultrathin when its thickness does not muchexceed the exchange length δ, see [6],

δ =(

A

2πM2s

)0.5

, (4.1)

where A is the exchange stiffness coefficient (for Fe A =2×10−6 ergcm−1) andMs is the saturation magnetization. The exchange length in Fe is 3.2 nm.

The simplest form of magnetic coupling per unit area of an ultrathin filmtrilayer structure of two ferromagnetic films (FM) separated by a normal metal(NM) spacer, FM1/NM/FM2, can be described by a bilinear form

E1 = −J1n1 · n2, (4.2)

where J1 is the exchange coupling coefficient in erg cm−2 and n1 and n2

are unit vectors along the magnetic moments in layers 1 and 2, respectively.Another common coupling equation has a biquadratic form

E2 = +J2 (n1 · n2)2 , (4.3)

where J2 describes the strength of biquadratic coupling. Cases of the bilinearcoupling J1 are found having either a + or a–sign. For a + sign the minimumof the bilinear energy term is reached for a parallel orientation of the magneticmoments; for a–sign the minimum energy corresponds to antiparallel magneticmoments. J2 is almost exclusively found to be positive and the minimumof the biquadratic energy term is reached when the magnetic moments areoriented perpendicularly to each other. A detailed description of bilinear andbiquadratic coupling terms will be carried out in Sect. 4.4.

The equilibrium of the magnetic moments and their dynamic response canbe found by using the Landau-Lifshitz-Gilbert (L.L.G) equation of motion [8]

4 Exchange Coupling in Magnetic Multilayers 187

1γ1,2

∂M1,2

∂t= − [M1,2 × Heff ,1,2] +

G1,2

γ21,2Ms,1,2

[M1,2 × ∂n1,2

∂t

], (4.4)

where γ1,2, M1,2, and G1,2 are the the absolute values of the electron gyro-magnetic ratios, magnetic moments per unit area , and Gilbert damping pa-rameters of layers 1 and 2, respectively. The damping parameter is expressedoften in a dimensionless parameter α = G/γMs. The first term on the right-hand side of (4.4) represents the precessional torque per unit area and thesecond term represents the well known Gilbert damping torque per unit area.The effective fields Heff ,1,2 are given by the derivatives of the magnetic Gibbsenergy density, F, with respect to the components (Mx,1,2,My,1,2,Mz,1,2) ofthe magnetization vector densities M1,2, see [9],[10],[6], and [11],

Heff = − ∂F∂M

. (4.5)

F includes the Zeeman energy of the dc applied magnetic field, demagnetiz-ing fields Hdem, rf magnetic field h, magnetic anisotropies, and the inter-layermagnetic coupling energy. In order to carry out appropriate partial derivativesin (4.5) the magnetic bilinear and biquadratic coupling terms (4.2) and (4.3)are rewritten by replacing n1,2 by M1,2/Ms,1,2. In thick films (∼100 nm) theinternal rf magnetic field has to be evaluated by using Maxwell’s equations inthe presence of the externally applied rf magnetic field. This means that therf field and rf magnetization vary across the film thickness, see e.g. [7, 12]. Inthe ultrathin film approximation the variations across the film thickness areneglected.

For small precession angles (| m |� Ms corresponding to an angle ofprecession less than a few Degrees) the magnetization vector can be linearizedby setting M = m+Ms, where Ms and m are the longitudinal and transversecomponents of M, see Fig. 4.1. This allows one to linearize the equation ofmotion (4.4).

4.3 Experimental Techniques for Magnetic CouplingMeasurements

The magnetic coupling can be determined by measuring the dependence ofthe net magnetic moment on the applied dc field. That can be done using lowand high-frequency techniques. SQUID, sample vibrating magnetometer, andMagneto Optical Kerr Effect (MOKE)[13] are the most common techniquesallowing one to measure the total magnetic moment as a function of the ap-plied field. X-ray Magnetic Circular Diochroism (XMCD)[14] is an elementsensitive technique allowing one to follow the field dependence of a particularlayer in a multilayer structure. The neutron scattering is a very powerful toolfor investigating the structural and magnetic properties of multilayer films[15, 16]. Polarized Neutron Reflection (PNR) measurements allow one to in-vestigate the spatial distribution of the magnetic moment inside the sample

188 B. Heinrich

Fig. 4.1. The coordinate system for the instantaneous magnetization M and exter-nal field H. X,Y, and Z axes are oriented along the principle crystallographic axes. θand ϕ are the polar and azimuthal angles with respect to the crystallographic axes.The x,y,z coordinate system is oriented along the magnetization M with the x axisdirected along the saturation magnetization moment. For a small precessional anglethe components My , Mz � Mx

[17, 18, 19]. Photo Electron Emission Microscopy (PEEM) [20], Low EnergyElectron Microscopy (LEEM) [21, 22] and Secondary Electron Microscopywith Polarization Analysis (SEMPA) [23] are other techniques suitable for thestudy of magnetic coupling. Ferromagnetic Resonance [10] and Brillouin LightScattering (BLS) [24] are rf techniques allowing one to determine quantita-tively all magnetic macroscopic parameters including interlayer and intralayerexchange couplings.

In this Section the discussion will be limited to the most common dc andrf techniques: SQUID, vibrating sample magnetometer, MOKE, FMR, andBLS. The experimental details can be found in the above reference. Here thediscussion will be limited to a quantitative interpretation of the experimentalmeasurements. SQUID and vibrating sample magnetometer measure the totalsample response. At FMR the microwave penetration (skin) depth is ∼ 100nm. In MOKE and BLS the total depth of studies is given by the depthof penetration of the laser beam into metallic samples. For visible light thisis approx. 15 nm. The depth resolution of MOKE signal was investigatedby Hamrle et al. [25] using magnetic multilayers. They determined both therotational and elliptical contributions to MOKE as a function of depth.


4.3.1 Dc Magnetometry

In SQUID and vibrating sample magnetometers the total magnetic momentis measured, usually along the direction of the applied field. In MOKE themagnetic moment is commonly measured in the polar and longitudinal con-figurations. In the longitudinal configuration the dc field is applied parallelto the film surface and it is usually applied in the plane of incidence of thelaser beam. In the longitudinal configuration one is sensitive to three contribu-tions. The first contribution arises from the magnetic moment that is parallelto the dc field (longitudinal Kerr effect), the second contribution comes fromthe magnetic component perpendicular to the film surface (polar Kerr effect),and in a non collinear configuration of magnetic moments the third contri-bution arises from the product of the parallel and perpendicular componentsof the magnetic moments with respect to the applied field. This contributionis called the quadratic magneto-optic effect. It arises from the second ordermagneto-optical effect [26]. This effect is the reflection analogue of the Voigteffect [27]. The magnetization components in the quadratic magneto-opticeffect are confined to the film surface. The quadratic magneto-optic effectcontribution often becomes important in magnetization measurements usingeither antiferromagneticaly coupled films or with the field applied along thehard magnetic axis. The quadratic magneto-optic contribution is a nuisancein MOKE measurements. The quadratic magneto-optic effect is an even func-tion of the applied magnetic field. It does not change its sign with reversalof the magnetic moment and creates a profound asymmetry in magnetizationmeasurements as a function of applied field. It can be removed by adding themeasured MOKE signal to its inverted counterpart. The signal is invertedaround a point that is the intersection of the MOKE signal axis with a lineparallel to the field axis and located midway between the saturated MOKEsignals for positive and negative magnetic fields [6]. The even part, propor-tional to the quadratic effect, can be obtained by subtracting the measuredand inverted signals.

The discussion in this Section will be limited to the micromagnetics oftrilayer structures consisting of a crystalline ultrathin film FM1/NM/FM2structure. For simplicity these calculations will be limited to cubic materialswith the film surface oriented in the (001) plane, see Fig. 4.1. The films of3d transition group elements are often but not exclusively grown on (001)templates. This is a particular case but it involves all the ingredients neededto formulate calculations for any other configurations involving an arbitrarydirection of the applied field and crystalline orientation. The discussion belowshould be viewed as an example taken from a user’s manual.

The total Gibbs energy per unit area for ultrathin films in a distortedcubic material with the saturation magnetic moments, Ms1 and Ms2, andthe applied field H can be written in the following form

190 B. Heinrich

F =∑

i=1,2

[− K

‖1,eff,i

2(α4

X,i + α4Y,i

)− K⊥1,eff,i

2α4

Z,i −

− K‖u,eff,i

(nu,i ·Mi)2

M2s,i

−K⊥u,eff,iα2

Z,i − Mi ·H]di − J1m1 · m2 +

+ J2 (m1 · m2)2 , (4.6)

where αX,i,αY,i and αZ,i are the directional cosines between the magne-tization vector Mi and the crystallographic axes [100], [010], and [001], re-spectively. K‖

1,eff,i,K⊥1,eff,i, K

‖u,eff,i, and K⊥

u,eff,i are the parameters of thein-plane effective four-fold (cubic), perpendicular effective four-fold, in-planeuniaxial, and perpendicular uniaxial anisotropies, respectively. nu,i are the di-rections of the in-plane uniaxial axes and di are the film thicknesses. m1 andm2 are unit vectors directed along the magnetizations of the coupled films:J1 and J2 are the bilinear and biquadratic coupling coefficients. The indicesi =1 and 2 describe the properties of the layers FM1 and FM2, respectively.In ultrathin films the magnetic moments across the film are locked togetherby intralayer exchange coupling and they can be considered to be giant mag-netic molecules [6]. For ultrathin films the role of the interface anisotropiesof a uniformly magnetized sample can be included in the effective anisotropyparameter,

K‖1,eff = K

‖1,bulk +

K‖1,s

d

K‖u,eff = K

‖u,bulk +

K‖u,s

d

K⊥u,eff = −2πM2

s +K⊥

u,s

d, (4.7)

where the subscript s represents the interface contributions. The units of theinterface anisotropies are erg/cm2, see [6]. The energy expression in (4.6) isvalid for a wide range of magnetic ultrathin films such as Fe on Ag(001) [6]and GaAs(001) [28, 29, 30] templates. One can easily generalize it by using theappropriate film symmetry An example for an arbitrary orientation of mag-netic moments can be found in [31]. The expression for K⊥

1,eff is not includedin (4.7). It originates in variations of the perpendicular uniaxial anisotropy,K⊥

u,s, across the film surface. It leads to a higher order dependence on 1/d,see [32] and the end of subsection Biquadratic exchange coupling.

The field dependence of the magnetic moment can be found by minimizingthe total Gibbs energy. The static equilibrium is found by minimizing the totalenergy with respect to the angles ϕ1, ϕ2, θ1, and θ2 for the given angles ϕH

and θH , see Fig. 4.1. In this Chapter the calculations will be restricted to thein-plane geometry, this means θ1 = θ2 = θH = π/2.

There are a number of minimization procedures available and they are usu-ally implemented by individual groups according to their liking. One should


realize that minimum energy solutions can exhibit metastable states. Usuallyone looks for the lowest energy state. This means no hysteresis is present inmagnetization measurements. This is not often the case in experiments es-pecially those using films grown on GaAs(001) substrates [29]. Therefore itis imperative to carry out MOKE measurements in the lowest energy state.The lowest energy state for a given magnetic field can be achieved by cyclingthe magnetic state at the given applied field with a transverse ac magneticfield which increases to some preselected maximum and then the amplitude isgradually decreased to zero. This has to be repeated for each applied dc field.An example of such procedure can be found in [29] using exchange coupledGaAs/Fe/Au/Fe/Au(001) structures.

Several examples of hysteresis loops using exchange coupled magnetic bi-layers FM1/NM/FM2 are shown in Figs. 4.2 and 4.3. In order to avoid com-plexities associated with a general direction of the applied field one shouldemploy two magnetic films having different thicknesses and with the externalfield applied along the magnetic easy axis of the thicker film. In that case thethicker film remains oriented close to the easy axis. It is the thinner film thatundergoes a full angular dependence, see the insets in Fig. 4.2. Since the twomagnetic moments are different it is easy to identify the contributions fromthe individual layers.

The family of curves describing the role of biquadratic magnetic couplingis shown in Fig. 4.2. When the biquadratic magnetic coupling becomes com-parable to the bilinear coupling one is not able to achieve an antiparallelconfiguration of magnetic moments in low magnetic fields. The biquadraticcoupling can lead in zero field to an angle between the two magnetic momentsranging from–180 to 90 degrees. The biquadratic magnetic coupling can alsoaffect the approach to saturation. For even small J2 the saturation point isreached without a jump in the magnetic moment; this is also true even in thepresence of magnetic anisotropies, see Fig. 4.2. The interpretation of the datawhen there is no discontinuity in the magnetization becomes quite simple.The deviation from saturation can be treated using a small angle expansion.It is easy to show that the saturation field along the easy direction can bedescribed by the simple expression

Hsat +2Keff

Ms= −J1 − 2J2

Ms

(1d1

+1d2

), (4.8)

where it has been assumed that the saturation magnetization Ms is the samefor both films. Keff is an effective anisotropy field obtained from minimiza-tion of (4.6), and d1 and d2 are the thicknesses of the layers FM1 and FM2,respectively.

In samples where one is not able to get both easy axes along the samedirection the magnetization curves can be kept relatively simple if one orientsthe field along the easy axis of the film having a large in-plane anisotropy. Anexample is shown in Fig. 4.3. Notice that the thinner film 8Fe with a large

192 B. Heinrich

Fig. 4.2. Simulations of magnetization curves using a set of exchange couplingparameters. The lines 1, 2, 3, 4 and 5 correspond to J2=0.0, 0.01, 0.04, 0.06,and 0.08 erg/cm2, respectively. The bilinear coupling was kept constant, J1=–0.1. The calculations were carried out for the magnetic parameters obtainedin GaAs/8Fe/Au/16Fe/Au(001) structures [30], where the integers represent the

number of atomic layers. 16Fe: K‖1,eff=3.1×105 erg/cm3 ; 8Fe:K

‖1,eff=1.33×105

erg/cm3. 4πMs=21.5 kOe. In-plane uniaxial anisotropies were omitted in order tokeep the easy magnetic axis in both films along the [100] crystallographic direction.The applied field was oriented along the magnetic easy axis [100]. The inset showsthe field dependence of the magnetization angle for J1=–0.1 erg/cm2 and J2=0. Thedashed line corresponds to the 16Fe film and solid line to the 8Fe film. Note thatthe first jump brings the magnetic moment of the thinner 8Fe film from the parallelorientation over the first hard axis, [110], close to the second easy axis, [010]. Thesecond jump corresponds to pulling the magnetic moment over the second hard axis,[110] resulting in an antiparallel configuration of the magnetic moments

in-plane uniaxial anisotropy rotates from the easy axis by a small angle, whilethe thick 16Fe film undergoes a large angular rotation.

In FM1/NM/FM2 structures one can measure by MOKE only negativebilinear J1 (antiferromagnetic coupling). One is able to measure a posi-tive (ferromagnetic) bilinear coupling by using spin “engineered structures”.An additional FM0 layer with a normal metal spacer creating a large an-tiferromagnetic coupling between FM0 and FM1 is needed [33, 34], e.g.FM0/NM0/FM1/NM/FM2. In these structures the magnetic moments in


Fig. 4.3. Simulations of magnetization curves for the magnetic anisotropies cor-responding to GaAs/8Fe/Au/16Fe/Au(001) structures [30], the integers repre-sent the number of atomic layers. The following magnetic parameters were used.16Fe: K

‖1,eff=3.1×105 erg/cm3, K

‖u,eff=3.3×104erg/cm3; 8Fe: K

‖1,eff=1.33×105

erg/cm3, K‖u,eff=–1.14×106 erg/cm3. The hard axis of the in-plane uniaxial

anisotropy in the 8Fe film is oriented along the [110] direction. 4πMs=21.5 kOe.The solid, dashed, and dotted lines in (a) correspond to J1=0.0, –0.1, and –0.3erg/cm2, respectively. The applied field was oriented along the magnetic easy axis,[110], of the 8Fe film. Note that this film has a large in-plane uniaxial anisotropy.This means that for the 16Fe film the applied field was oriented along the hard axis.(b) shows the field dependence of the magnetization angle with respect to the [100]axis for J1=–0.3 erg/cm2. The dotted line corresponds to the 16Fe film and the solidline to the 8Fe film. Note that below 1.5 kOe the magnetic moments are orientedantiparallel to each other with the magnetic moment of the 8Fe film oriented alongits easy magnetic axis [110](45 Degree), and the magnetic moment of the 16Fe filmoriented along its hard magnetic axis [110] (–135 Degree)

FM1 and FM2 in zero applied field are oriented antiparallel to that in FM0.One needs to apply a dc field along the direction of the magnetic moment ofthe FM0 layer to overcome the ferromagnetic coupling and thus to orient themoments in FM1 and FM2 in an antiparallel configuration. Strong antifer-romagnetic coupling can be achieved by using ultrathin Ru spacers [33], seeSect. 4.5.

4.3.2 FMR and BLS Techniques

The magnetic coupling can be measured by rf techniques, see [6, 10, 24, 35, 36,37]. In FMR one usually sets the microwave frequency and sweeps the field.

194 B. Heinrich

However in this age of network analyzers (NA) this is not limitation; one canset the field and sweep the frequency. In BLS one sets the applied field andsweeps the frequency. When the field is held constant the angle between themagnetic moments is fixed. This is a simpler situation compared to a regu-lar FMR measurement (holding constant frequency and sweeping the field)where the angle between the magnetic moments changes in non-collinear con-figurations. For a saturated sample the difference between constant field andconstant frequency sweeps is minimal. The main difference is that in BLS onemeasures spinwaves having an in-plane wave-vector q-component corespond-ing to that of the in-plane wavelength of the incoming laser light whereas inFMR the measured spinwave mode corresponds to q�0 (homogeneous mode).This means that the ferromagnetic films in BLS are always coupled by a dipo-lar interaction, see Subsection 4.2. Interpretation of FMR and BLS results canbe carried out by using rf solutions of (4.4). Let us briefly outline a methodfor setting up the equations of motion for the rf magnetization components.The coordinate system for an arbitrary orientation of the magnetic momentwith respect to the crystallographic axes is shown in Fig. 4.1.

Further discussion will be limited to the in-plane configuration where thesaturation moment and external field are oriented in the X-Y plane, θi = π/2.For a noncollinear configuration with static magnetic moments in the filmplane the directional cosines αX,i,αY,i, and αZ,i with respect to the crystal-lographic axes are given by

αX,i =Mx,i

Mscos(ϕi) − My,i

Mi,ssin(ϕi)

αY,i =Mx,i

Ms,isin(ϕi) +

My,i

Mscos(ϕi)

αZ,i =Mz,i

Ms, (4.9)

where Mx,i,My,i and Mz,i are the instantaneous magnetization componentsin the coordinate systems with the x-axis parallel to the saturation magneti-zation. The effective fields in the frame of the moments FM1 and FM2 can beobtained by inserting (4.9) into (4.6) and using (4.5). However (4.6) is not thedensity of energy. This requires conversion of the interlayer coupling energyto the energy density. This conversion is well described on pp. 569–570 in [6].Assuming an even sharing of the interlayer coupling by all atomic layers insidethe film the density of energy, Ui, for the individual layers FM1 and FM2 aregiven by

Ui = −K‖1,eff,i

2(α4

X,i + α4Y,i

)−K‖u,eff,i

(ni ·Mi)2

M2s,i

+

+ K⊥u,eff,iα

2Z,i − MiH−

(J1m1 ·m2 − J2 (m1 · m2)2

)/di

Heff,i = − ∂Ui

∂Mi. (4.10)


The partial derivatives are taken with respect to Mx,i,My,i and Mz,i. InFMR and BLS the rf transversal components are negligible compared to themagnetization component parallel to the x-axis (static effective fields). Thereplacement of the x-component of the magnetization by the saturation mag-netization Ms can be done only after the partial derivatives have been carriedout. The equations of motion (torque equations) for FM1 and FM2 are ob-tained by using the above effective fields in the LLG (4.4). One can ignore allterms which are quadratic in the small rf components. Notice that the perpen-dicular 4-fold anisotropy,K⊥

1,eff,i, is not included in the parallel configuration;it leads to 3rd order terms in the small rf magnetic componentMz,i. This leadsto a coupled system of equations for the transverse magnetization componentsfor FM1 and FM2. The z-components of the torque include also expressionswhich contain only large static magnetization Ms,i components. These ex-pressions correspond to static equilibrium and are equal zero. They providesolutions for ϕi(H) for the applied field H. The static equilibrium can bealso obtained from minimizing the total energy given by (4.6). The coupledequations for the rf magnetization components leads to two solutions. Theprecessional motions are coupled and result in an acoustic mode in which themagnetic moments in the two layers precess in phase and in an optical mode inwhich the magnetic moments precess in antiphase. The simplest interpretationof the coupling can be obtained in the saturated case when the dc magneticmoments are parallel to the applied field. The isolated films FM1 and FM2must have different resonant frequencies (fields) in order to be able to observeacoustic and optical modes. If the resonance frequencies (fields) of the twofilms are exactly same than the strength of the optical mode is zero. Differ-ent resonant fields are easy to establish by choosing different film thicknessesfor the films FM1 and FM2. The interface anisotropies scale with 1/d andin consequence the isolated films exhibit two separate resonance frequencies(fields) and the optical mode becomes observable. The sign of coupling canbe determined from the relative positions of the acoustic and optical modes.In FMR the microwave frequency is usually fixed. For antiferromagnetic cou-pling (J1 < 0) and ferromagnetic coupling (J1 > 0) the optical modes arelocated at higher and lower fields than the acoustic modes, respectively. Thepositions and intensities of these two modes are nontrivial functions of themagnetic anisotropies and the strength of the interlayer coupling. Howeverthe resonance spectrum can be easily evaluated by using the coupled L.L.G.equations of motion, see Fig. 4.4.

In the saturated state (collinear magnetic moments) the overall strengthof the interlayer coupling, Jeff is given by the superposition of bilinear andbiquadratic interlayer couplings,

Jeff = J1 − 2J2 . (4.11)

The optical mode has its magnetization components out of phase and con-sequently a homogeneous rf driving field inside the ultrathin films makes

196 B. Heinrich

Fig. 4.4. Simulations of acoustic and optical resonance peaks at f=36 GHzas a function of bilayer exchange coupling in an FM1/NM/FM2 structure. Inpanel (a) J1=0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 ergs/cm2. In panel (b) J1=0.0,–0.1, –0.2, –0.3, and –0.4 ergs/cm2. Note that the antiferromagnetic interlayer cou-pling moves the resonant peaks to larger fields. For the antiferromagnetic cou-pling the acoustic and optical peaks move to higher magnetic fields at a fixedFMR frequency. The acoustic peaks keep increasing their intensity with increasingcoupling while the optic peaks get weaker with increasing coupling. The acous-tic peaks gradually approach a fixed point which is located between the reso-nance peaks of the uncoupled films. Calculations were carried out for the mag-netic parameters obtained in GaAs/8Fe/Au/16Fe/Au(001) structures [30], wherethe integers represent the number of atomic layers. The following magnetic pa-rameters were used: 16Fe: K

‖1,eff=3.1×105 erg/cm3, K⊥

u,s=0.88 erg/cm2, and

K‖u,eff=3.3×104erg/cm3; 8Fe:K

‖1,eff=1.33×105 erg/cm3, K⊥,s=0.82 erg/cm2, and

K‖u,eff=–1.14×106erg/cm3. 4πMs=21.5 kG, g=2.09, and α=0.009. The in-plane uni-

axial easy axes for the 16Fe and 8Fe films were along the [110] and [110] directions,respectively. The applied field was oriented along the [110] crystallographic axis.The damping parameter was increased approx. 3 fold, compared to the measuredvalues, to make the FMR lines wide for easy viewing

excitation of optical modes ineffective. The optical mode signal rapidly de-creases with the strength of the interlayer coupling, see Fig. 4.4. It is relativelyeasy to measure the strength of the interlayer coupling up to 0.5 ergs/cm2

[38]. In the saturated state one is not able to measure the interlayer couplingstrength if the two films have the same magnetic properties. The difference inthickness does not help. However in a non-collinear configuration of the mag-netic moments one can measure the exchange coupling even in films havingthe same magnetic properties. In that case in FMR one gets only one resonantmode which depends strongly on the exchange coupling, see Fig. 4.5. This isstrictly only true for the rf field oriented perpendicular to the dc applied field.


Fig. 4.5. The dependence of the FMR absorption peak on the bilinear magneticcoupling. Simulations were carried out at 10 GHz for a FM/NM/FM structure. Themagnetic films were of the same thickness. The magnetic anisotropies were assumedto be zero,4πMs=21.5 kG, g=2.09, and α=0.009. The numbers above the absorptionpeaks represent the strength of bilinear magnetic coupling. One needs to use a lowenough microwave frequency to bring the FMR resonance to low fields where themagnetic moments are not parallel (the unsaturated state). In the saturated statethe FMR signal does not depend on the interlayer coupling

The effectiveness of the coupling between a homogeneous rf field and theoptical mode can be increased if the magnetic moments in the two films arenoncollinear, see Fig. 4.6. It was shown by Z. Zhang et al. [31] that for therf field oriented parallel to the dc field one gets the projected rf field compo-nents in phase with the optical rf magnetization components resulting in anenhancement of the optical resonance. Note in Fig. 4.6 that the acoustic peakis completely absent for the rf field parallel to the dc field while the opticalpeak reaches its maximum. The effective rf field components (perpendicularto the dc magnetic moments) in the magnetic layers are antiparallel. This wayone is not coupled to the acoustic mode but the optical peak is fully excited.

The strength of biquadratic coupling can not be measured independentlyin the saturated state, see (4.11). However in a non-collinear state the con-tributions of bilinear and biquadratic interlayer couplings in FMR and BLSmeasurements can be separated, see Fig. 4.7 and [39].

There is an alternative approach to evaluate the resonance modes using theSmit and Beljers method which is based on the partial derivatives of the Gibbsenergy with respect to the magnetization angles. The details of this approachcan be found in [35]. An excellent theoretical treatment of rf excitations ina wide range of multiayers with complex spin configurations can be found inthe review article by Camley and Stamps [40].

198 B. Heinrich

Fig. 4.6. The dependence of the FMR signal on magnetic coupling in a non-collinearconfiguration. Simulations were carried out at 10 GHz for a FM/NM/FM structure.The magnetic films were of the same thickness. The magnetic anisotropies wereassumed to be zero, 4πMs=21.5 kG, g=2.09, and α=0.009. (a) J1=0.0 (b) J1=–0.4erg/cm2. The rf magnetic field is perpendicular to the applied dc field. Only theacoustic mode is excited. (c) J1=–0.4 erg/cm2. The rf field is oriented 45 Degreeswith respect to the dc applied field. Note that with this rf driving one can seeboth the acoustic and optical modes. (d) J1=–0.4 erg/cm2. The rf field is orientedparallel to the dc applied dc field. Only the optical mode is excited. For (b), (c) and(d) the magnetic moments are non-collinear. Their magnetic moments are cantedsymmetrically away from the dc magnetic field. The FMR signal in (a) is 4.5x largerthan those in (b), (c) and (d)

In order to obtain a reliable interpretation of the magnetic coupling inMOKE and FMR measurements one needs to know reasonably well the mag-netic anisotropies. Independent measurements of the magnetic anisotropies inindividual films are extremely useful. The interlayer coupling parameters arethen the only parameters left to fit the measured data obtained for a pair ofcoupled thin films.

4.4 Theory

4.4.1 Interlayer Exchange Coupling

The first successful model of interlayer exchange coupling was introduced byMathon, Villeret and Edwards [41]. They correctly pointed out that exchangecoupling is primarily a property of the normal metal (NM) spacer and is


Fig. 4.7. BLS spectra for an FM1/NM/FM2 structure. The in-plane magneticanisotropies are assumed to be zero. The upper curves correspond to acousticpeaks, and the lower curves correspond to optical peaks. The calculations werecarried out for J1=–0.2 ergs/cm2 (solid line), and J1=–0.1 ergs/cm2 and J2=0.05ergs/cm2 (dashed line) providing an identical coupling in the saturated state toJ1=–0.2 ergs/cm2, see (4.11). Note that the resonant frequencies are indeed iden-tical for fields greater than that required to align the magnetizations in the twofilms (H=1.38 kOe). However significant differences in resonant frequencies occur inthe non-collinear state allowing one to separate the contributions from the bilinearand biquadratic exchange couplings. Similar behavior would be obtained for FMRmeasurements carried out as a function of microwave frequency at fixed magneticfield

related to the confinement of Fermi surface electrons in the NM. This modelwas quickly extended to include the spin dependent electron reflectivity at theFM/NM interfaces [42, 43, 44]. One has to include the itinerant nature of the3d, 4sp electrons in the FM layers. The interlayer bilinear coupling, J, is givenby the difference in energy between the antiparallel and parallel alignment ofthe magnetic moments in FM/NM/FM structure,

J =1

2A(E↑↓ − E↑↑) , (4.12)

where A is the area of the film. Calculations of energy differences are simpli-fied by using the force theorem. The main problem is how to treat electroncorrelations self consistently. The force theorem says that the energy differ-ence between the two configurations is well accounted for by taking the dif-ference in single particle energies. It is adequate to take an approximate spindependent potential and to calculate the single particle energies in the parallel

200 B. Heinrich

and antiparallel configurations. This difference in energy is very close to thatobtained from self-consistent calculations, see the further discussion in [3]. Infact this section follows closely Stiles’s Sect. 4.4 in [3]. This procedure signifi-cantly simplifies the calculation of exchange coupling and interface magneticanisotropies. In calculations of the interlayer exchange coupling one does notcreate a big error by neglecting spin orbit interactions, while in calculationsof the interface anisotropies spin orbit coupling is the crucial ingredient. Sin-gle particle energy calculations require one to evaluate the electron energyin four quantum well states (QWS), see Fig. 4.8. For thick FM layers onefinds large energy contributions. However these large contributions cancel outin the difference (4.12). In order to avoid mistakes in this procedure it isbetter to calculate the cohesive energy of the QWS by subtracting the bulkcontributions,

ΔEQWS = Etot − EFMVFM − ENMVNM , (4.13)

where VFM,NM and EFM,NM are the total volumes and bulk energies for FMand NM layers, respectively.

Quantum interference

Let us consider a simple one dimensional model in which an electron with awave vector k⊥ travels inside the NM spacer and is partially reflected at theFM/NM (interface A) and NM/FM (interface B) interfaces with reflectioncoefficients RA,B = rA,Bexp(iφA,B). After multiple interference the electrondensity of states (EDS) changes. The phase of the wavefunction after a roundtrip changes by

Δφ = 2k⊥d+ φA + φB . (4.14)

Ferromagnet Spacer

E

EF

k

Unoccupiedstates

k

↑

ParallelAlignment

AntiparallelAlignment

Spin ↑M ↑M ↑ M ↓ M ↑

EF

Spin ↓M ↑ M ↑M ↑ M ↓

EF

↓

M ↑

Fig. 4.8. Quantum wells employed in the calculation of the interlayer ex-change coupling. These spin dependent potentials correspond reasonably well toa Co/Cu/Co(001) system. On the left side the four panels show quantum wellsfor spin up and spin down electrons for parallel and antiparallel alignment of themagnetic moments. The grey regions show the occupied states


The amplitude after multiple reflections is given by a sum of round trips is

∞∑1

[rArBeiΔφ]n =rArBe

iΔφ

1 − rArBeiΔφ. (4.15)

The denominator becomes small when one obtains a constructive interfer-ence Δφ=2nπ. For energies less than the potential barrier at the interfacerA=rB=1 and one gets perfect QWS. For energies above the barrier hight theQWS become broader resonances by partly transmitting its amplitude to thesurrounding FM layers. By changing the NM spacer thickness these states passthrough the Fermi energy, see Fig. 4.9, which leads to an oscillatory behaviorof the cohesive energy and consequently to an oscillatory interlayer exchangecoupling. The first clear experimental observation of QWS was presented byHimpsel and Ortega [45, 46] using photoemission and inverse photoemissionusing a nonmagnetic layer on top of a magnetic layer.

In first approximation the change in the density of states due to interfer-ence, Δ n(ε), should be proportional to rArBcos (Δφ) and to the spacer widthd and the density of states per unit length, (2/π)(dk⊥/dε) [44]. Therefore Δn(ε) per spin can be written as

Δn(ε) � 2dπdk⊥dε

rArBcos(Δφ) =1πIm

(i2d

dk⊥dε

rArBeiΔφ

). (4.16)

For multiple scattering one has to use the expression in (4.15). It is relativelyeasy to show that 4.16 can be generalized to [47]

Δn(ε) = −1πIm

d

dε

[ln(1 − rArBe

iΔφ)], (4.17)

note that (4.17) equals to (4.16) for small reflection coefficients.The cohesive energy is than given by

Ecoh = −∫ EF

−∞dε(ε− EF )Δn(ε) . (4.18)

D + 2π 2kFD + π / 2kFD

E F

/

Fig. 4.9. Evolution of quantum well (QW) states as a function of the film thickness.The solid lines represent bound states (localized in the QW) and resonance statesare shown in “fuzzy ellipses”. EF is the Fermi energy

202 B. Heinrich

Using integration by parts one gets

Ecoh =1πIm

∫ EF

−∞dεln(1 − rArBe

iΔφ) . (4.19)

For fixed thickness d the integral oscillates rapidly as a function of k⊥. Onlycontributions close to the Fermi level will leave non-zero contributions, seeFig. 4.10. It can be shown that in these regions for large d [3]

Ecoh =�vF

2πd

∑n

1nRe((rArB)neinΔφ(kF )

). (4.20)

For small reflection coefficients

Ecoh � �vF

2πdrArBcos(2kF d+ φA + φB) . (4.21)

The interlayer exchange energy, J, is then given by adding all cohesive energiesin Fig. 4.9, assuming the same reflection coefficients at the A and B interfaces

J � �vF

4πdRe(R↑R↓ +R↓R↑ −R2

↑ −R↓2)ei2kF d = −�vF

4πdRe(R↑ −R↓)2ei2kF d .

(4.22)

The exchange coupling in this simple one dimensional limit is inverselyproportional to the film thickness, d, and its oscillatory period is given bythe Fermi spanning vector 2kF . In 3D space one has to take into account thek-vectors parallel to the surface. These k-vectors due to the lattice periodicityare conserved in going from FM to NM regions. In this 3D case the onedimensional QWS have additional k-wave-vectors parallel to the interface.The total cohesive energy per unit area involves integration of the QWS overthe interface Brillouin zone. In the limit of large d [3] (asymptotic limit)

Ecoh � �vF

2πd

∫

IBZ

d2k

(2π)2Re(ei2kF z(k)dRR(k)RL(k)

), (4.23)

where RR(k), RL(k) are the reflectivity coefficients at the right and left in-terfaces, and kF2 kFz is the perpendicular k-vector at the Fermi surface. Theintegrand in (4.23) oscillates rapidly with the argument 2kFz(k)d except onareas of the Fermi surface where opposite sheets of the Fermi surface arenearly parallel, see Fig. 4.10.

The vector connecting these parts of the Fermi surface are called criticalspanning vectors. The spanning k-vectors for (001) interfaces for simple metalssuch as Cu and complex spin density Cr are shown in Figs. 4.14 and 4.21.

The exchange coupling involves the difference in cohesive energies forparallel and antiparallel configuration of magnetic moments. In its asymp-totic form this coupling can be written as


Inte

gran

d

Thickness

2kF

2π/2kF

Fig. 4.10. The right hand side shows a slice through a spherical Fermi surface. Thesolid double headed arrows represent spanning k vectors. The left hand side showstheir oscillatory contributions to the cohesive energy. The sum of these contributionsis shown by a heavy line. The constructive interference (heavy line) comes mostlyfrom the belly area of the Fermi surface. Note that this constructive interferencedecreases with increasing film thickness

J �∑

i

�vi⊥κ

i

16π2d2Re((Ri

↑ −Ri↓)

2eiqi⊥deiχi

), (4.24)

where the vi⊥ are Fermi velocities at the spanning vectors, qi

⊥ is the lengthof a critical spanning wave-vector, κi is the phase associated with the type ofthe critical point, and Ri

↑, Ri↓ are corresponding reflectivities. The periods of

the observed exchange coupling oscillations as the film thickness is varied arein good agreement with those obtained in de Haas-van-Alphen measurements,see Table 4.1 in [3]. A detailed discussion of calculations of exchange couplingfor Co/Cu/Co(001), Fe/Au/Fe(001) and Fe/Ag/Fe(001) systems can be foundin [3]. The quantitative agreement for the exchange coupling between theoryand experiment is far from being good. The main reason is that the interfacesin real samples are far from being ideal and measurements are often not carriedout in the asymptotic limit.

4.4.2 Dipolar Coupling

In measurements involving an inhomogeneous distribution of magnetizationone has to include dipolar coupling. BLS measurements in the backscatteredconfiguration [24, 37, 48] represents perhaps one of the best defined cases ofdipolar couplings. In this case the in-plane k-vector of the rf magnetization isgiven by k = 2qcos(θ), where q is is the length of the laser wave-vector, and θ isthe angle of the incoming laser beam with respect to the film surface. For a filmwith its normal oriented along the z-axis, the in-plane dc magnetic momentoriented along the x-axis, and the rf magnetization components in form ofmyexp(i(k‖x+k⊥y)), mzexp(i(k‖x+k⊥y)), the spatially averaged dipolar fieldcomponents inside the film in the limit as kd→ 0 are given by

204 B. Heinrich

Table 4.1. Comparison of oscillation periods measured in magnetic multilayers withthose expected from the critical spanning extracted from Fermi surfaces measuredin de Haas-van Alphen measurements (dHvA). This Table is a copy of Table 4.1in reference [3], see further references contained therein

interface Period (ML) Period (ML) Technique

Ag/Fe(100) 2.382.37±0.07

5.585.73±0.05

dHvA SEMPA

Au/Fe(100) 2.512.48±0.05

8.68.6±0.3

dHvA SEMPA

Cu/Co(100) 2.562.6±0.052.58 to 2.77

5.888.0±0.56.0 to 6.17

dHvA MOKE SEMPA

Cr/Fe(100) 11.112±112.5

dHvA SEMPA MOKE

Cr/Fe(112) 14.415.4

dHvA MOKE

hx = −2πmy

(k‖k⊥k2

)kdei(k‖x+k⊥y)

hy = −2πmy

(k⊥k

)2

kdei(k‖x+k⊥y)

hz = (−4πmz + 2πmzkd) ei(k‖x+k⊥y) , (4.25)

and k = (k2‖ + k2

⊥)0.5

Outside the film for z ≥ d:

hx = −[2πmy

(k‖k⊥k2

)+ 2πimz

(k‖k

)]kdei(k‖x+k⊥y)e−k(z−d)

hy = −[2πmy

(k⊥k

)2

+ 2πimz

(k⊥k

)]kdei(k‖x+k⊥y)e−k(z−d)

hz = −[2πimy

(k⊥k

)− 2πmz

]kdei(k‖x+k⊥y)e−k(z−d) . (4.26)

Outside the film for z ≤ 0.:


hx = −[2πmy

(k‖k⊥k2

)− 2πimz

(k‖k

)]kdei(k‖x+k⊥y)ekz

hy = −[2πmy

(k⊥k

)2

− 2πimz

(k⊥k

)]kdei(k‖x+k⊥y)ekz

hz =[2πimy

(k⊥k

)+ 2πmz

]kdei(k‖x+k⊥y)ekz . (4.27)

k‖ and k⊥ being in-plane wave-vector components parallel and perpendicularto the saturation magnetization, see Fig. 4.11. Notice that the dipolar fieldoutside the film decays exponentially with the decay length of 1/k. A gen-eral treatment of dipolar field can be found in [49, 50]. Dipolar fields play animportant role in rf measurements using coplanar and slotted transmissionlines. The distribution of k-vectors depends on the geometry of the trans-mission line. These inhomogeneous dipolar fields lead to both a shift of theresonant frequency and a broadening of the FMR line [51, 52].

Orange peel coupling

Rough interfaces lead to a surface magnetic charge density and consequently todipolar coupling. This coupling was introduced by Neel [53]. It is often called“orange peel” coupling [54, 55]. It leads to an additional dipolar magneticcoupling. Figure 4.12 indicates that the interface roughness creates a lowerenergy for the parallel orientation of the film magnetic moments than that forthe antiparallel configuration. Usually the surface roughness is slowly varyingand its amplitude is much smaller than the film thickness. Calculations thenbecome simple. The surface charge can be distributed over a flat surface [3].Assuming that the surfaces vary along the x-direction as zs = Δzcos(2πx/L)

Fig. 4.11. The coordinate system and the film geometry corresponding to dipolarfields generated by a spinwave with the wave vector k. The magnetic layer has itsnormal in the z-direction. d is the layer thickness. The saturation magnetization andspin wave vector k are oriented in the film surface. The k−vector propagates withthe angle ϕ with respect to the saturation magnetization Ms

206 B. Heinrich

----- - - - --

+++++----

++++-

+ +

++++++++++

-----+++++

-----+

- -

(b)

++++++++++

-----+++++

----+

- -

++++++++++

-----+++++

-----+

- -++++++++++

-----+++++

-----+

- -

Fig. 4.12. A schematic picture demonstrating the presence of interface effectivemagnetic charges for an in-phase corrugated interface roughness. The solid shortarrows represent the local induced magnetic dipoles. For the parallel orientation ofthe film magnetic moments the magnetic dipoles form a closed magnetic pattern. Inthe antiparallel configuration this pattern changes to a head to head and tail to tailconfiguration

and zs = d+Δzcos(2πx/L), see Fig. 4.12. For the magnetization perpendicularto corrugation the ferromagnetic coupling strength is given by [3]

J1,dip ∼ 4πM2s

Δz2

Le−2πd/L . (4.28)

When the interface roughness is completely uncorrelated the bilinear dipolarexchange coupling goes to zero.

Pinhole coupling

Magnetic coupling can arise from pinholes. Basically parts of the FM filmsare in a direct contact that results in an overall ferromagnetic coupling [56].However this coupling is not homogeneously distributed over the surface. Fluc-tuations of pinhole coupling over the film surface can result in a contributionto biquadratic exchange coupling.

4.4.3 Biquadratic Exchange Coupling

The presence of biquadratic exchange coupling was observed at the sametime by Heinrich et al. [57] on Co/Cu/Co(001) and by Ruehrig et al. [58] onFe/Cr/Fe(001). The evidence for biquadratic exchange coupling in [57] wasfound in the magnetization loops. In order to properly explain the observedcritical fields one needed to use an angular dependent bilinear exchange cou-pling in the form of

J(θ) = J1 − J2cos(θ) , (4.29)

where θ is the angle between the magnetic moments. Consequently the corre-sponding exchange energy was given by

E(θ) = −J(θ)cos(θ) = −J1cos(θ) + J2cos2(θ) . (4.30)


Ruehrig et al. observed a perpendicular orientation of the magnetic momentsin an Fe/wedge Cr/Fe(001) sample in which the Cr interlayer was grownwith a linearly variable thickness. They explained the observed perpendicularconfiguration by using

E(θ) = −J(θ)cos(θ) = −J1cos(θ) − J2sin2(θ) . (4.31)

Clearly these two concepts are identical. Slonczewski soon after that pro-posed a theoretical interpretation [59]. He realized that fluctuations in theinterlayer thickness could result in an additional coupling term. The ferro-magnetic layers at different parts of the sample have different thicknesses andconsequently different strengths of coupling, see Fig. 4.13. Short-wavelengthoscillations can even result in changing the coupling from ferromagnetic toantiferromagnetic. His model is applicable when lateral variations in the bi-linear coupling strength are on a shorter length scale than the lateral ex-change length. This means that local angular variations from the averagedirection of the magnetic moments are small so that in this case the prob-lem can be treated by perturbation theory. The magnetic moments are frus-trated across the film surface by a variable interlayer coupling. Consequentlythere is an additional energy term which prefers to orient the magneticmoments in the FM layers perpendicularly to each other. This additionalcoupling has then an angular dependence given by cos2(θ), for which the

Fig. 4.13. A schematic picture demonstrating variations of the local magnetic mo-ment across a film surface. The local magnetic moments (solid black arrows) arepartly rotated away from the average direction of the magnetic moment (light greyarrows) in an attemp to decrease the overall interlayer exchange coupling energy. Asa result, moments in FM coupled regions rotate a little towards each other whereasin AFM coupled regions the magnetizations rotate away from each other

208 B. Heinrich

name “biquadratic exchange coupling” was coined. Its strength is given bythe competition between variations in the interlayer exchange coupling field,ΔJ1/Msd , and the in-plane intralayer exchange field, 2Ak2/Ms. The lengthof the k-vector, k, is given by the average lateral variations of the inter-layer exchange coupling, and ΔJ1 is the average variation of the interlayerexchange coupling. Slonczewski has shown that the exchange coupling fluc-tuations are decreased by a factor due to exchange averaging. In the sim-plest form the strength of the biquadratic exchange coupling can be ex-pressed by

J2 =4πΔJ1

ΔJ1Msd

2Ak2

Ms

. (4.32)

Notice that the large fraction describes the exchange averaging effect. A moregeneral description can be found in [59]. The above expression shows thatbiquadratic coupling has only a positive sign that encourages a perpendicularorientation of the magnetic moments. The angle between the magnetic mo-ments is given by a competition between the bilinear, biquadratic magneticcouplings, and the magnetic anisotropies, see Sect. 4.3.1 In zero field this anglecan range continuously from 0 to π.

It is often believed that biquadratic exchange coupling occurs only fromshort wavelength exchange coupling oscillations where the exchange couplingchanges its sign between two subsequent atomic terraces. This is not cor-rect. Any lateral variations in magnetic coupling strength (including ferro-magnetic coupling) will result in biquadratic exchange coupling. Once themagnetic moments are in a non-collinear state the magnetic frustrations dueto an inhomogeneous magnetic coupling strength result in biquadratic mag-netic coupling.

Further details about biquadratic exchange coupling can be found inDemokritov’s review article on “Biquadratic exchange coupling in layeredmagnetic systems” [60].

The Slonczewski idea of additional energy terms due to imperfect inter-faces is more general. It was shown [32] that “in any system that exhibits alateral inhomogeneity, one can expect additional energy terms. It originatesfrom intrinsic magnetic energy terms that fluctuate in strength across thesample interface. These additional terms have a next higher angular powercompared with that of the intrinsic term, and they should have only one sign.The power of a higher order angular term has to satisfy the requirements ofsample symmetry including time inversion symmetry. Variations of the in-terlayer exchange coupling results in a cos2(θ) angular term; variations in auniaxial interface anisotropy results in an angular dependent term having theform cos4(ϑ), where ϑ is the angle between the magnetic moment and the filmnormal.


4.5 Experimental Studies

Early studies

The early stages of interlayer coupling are well described in review articles[3, 6]. The first measurements of interlayer coupling were carried out by Gru-enberg’s group [61]. Using BLS measurements they showed that Cr can coupleFe layers antiferromagnetically. This result was expected considering that Crcontains a spin-density wave having a period of approximately 2 ML. It wasnot clear that one could expect antiferromagnetic coupling through simplemetal spacers such as Cu. The first indication that the exchange couplingthrough Cu could be antiferromagnetic was found by Cellobata et al. [62]in superlattices of fcc [6Co/8Cu](001) and [9Co/5Cu](001) using spin polar-ized neutron scattering and magnetometric techniques. Soon after that sev-eral FMR and BLS experiments established a cross-over from ferromagneticto antiferromagnetic coupling through bcc Cu(001). The first cross-over wasobserved at 8 ML of Cu and the first antiferromagnetic maximum at 11 ML[63, 64]. These measurements were quickly followed by a number of measure-ments that identified exchange coupling terms having long range oscillatoryperiods of 10 ML and 5.5–8 ML for bcc and fcc Cu(001) [38, 65, 66, 67, 68, 69],respectively.

Systematic studies of multilayers grown by means of sputtering revealedoscillatory couplings having oscillation periods in the range of 0.9 nm to 1.2nm for V,Nb,Mo,Rh,Ru,Ta,W,Re and Ir spacer layers [70, 71, 72, 73, 74]. TheCo/Ru/Co and Co/Rh/Co systems became very useful in forming antipar-allel pinned spin valves that are employed in GMR read out heads [75], andMagnetic Random Access Memories(MRAM) using the spin tunelling effect.In Co/Ru/Co and Co/Rh/Co structures the first antiferromagnetic couplingwas found at 0.3 and 0.8 nm with the strength of 5 and 1.6 ergs/cm2 forRu and Rh, respectively [70]. These results were obtained by monitoring theGiant Magneto Resistance (GMR) effect. The resistance of an FM/NM/FMstructure increases for the antiparallel orientation of the magnetic moments(antiferromagnetic coupling) due to the GMR effect. By following the maximaof the resistance one can determine the regions of antiferromagnetic couplingas a function of the spacer layer thickness [76]. The multilayer structures forGMR studies are mostly prepared by means of sputtering. In the work carriedout by the Strasbourg group [72] crystalline Co/Ru/Co(0001) hcp films wereprepared using MBE. The interlayer exchange coupling strength was investi-gated using FMR. The strongest coupling was found to be 6 ergs/cm2 observedat 0.5 nm of Ru at RT. The period of oscillations in the coupling strengthwas found to be 1.2 nm. Preparation of the films using sputtering resulted ina weaker exchange, see above. This indicates that smoother interfaces resultin a stronger coupling.

Ru is used in antiferromagnetic coupled multilayers which are attractive foruse as high density recording media. [Co(0.4)/Pt(0.7)]X−1 multilayer struc-tures, where X represents the number of repetitions, and the numbers are

210 B. Heinrich

in nm, possess a strong perpendicular uniaxial anisotropy that allows theCo magnetic moment to be oriented perpendicular to the film surface. In[[Co(0.4)/Pt(0.7)]X/Co(0.4/Ru(0.9)]]N structures one can create vertical andlaterally coherent antiferromagnetic films by changing X [77].

The strong antiferromagnetic coupling through Ru requires large appliedfields to saturate FM/Ru/FM trilayers. For a Py(5 nm)/Pd(.5 nm)/Py(5 nm)structure the magnetic field required to achieve saturation of the magneticmoments is in excess of 10 kOe at RT, [70, 76]. A FM/Ru/FM film havingzero net magnetic moment can be effectively pinned by an exchange biasfield from an antiferromagnet (AFM) [75, 76]. Such a hard magnetic layercomposed of AFM/Fe/Ru/Fe is extensively used in spin valve structures.

The presence of short wavelength oscillations in the exchange couplingwere observed for the first time using perfect single crystals of Fe whiskers assubstrates. Whiskers were prepared by means of chemical vapor deposition us-ing FeCl2 and H2 as a transport gas. Under correct conditions which requireda proper temperature and a proper flow of hydrogen gas one could sometimesget single crystals of Fe in the form of whiskers having {001} crystalline facets.Whiskers were usually between several mm to 1–2 cm long and 10–100 μmacross. The facets were smooth with atomic terrace sizes in excess of severalμm. Fe whiskers proved to be ideal templates for the observation of shortwavelength oscillations. Approximately at the same time the NIST group ofUnguris et al. [78], and Purcell et al. [79] (Philips group), observed short wave-length oscillations having a period of 2 ML. The exchange coupling throughspin-density wave Cr will be highlighted in detail in Sect. 4.5.3. After realizingthat short wavelength oscillations can be observed in carefully prepared sam-ples a large number of papers were devoted to Co/Cu/Co films oriented alongall the main (001), (110) and (111) crystallographic orientations. A detailedaccount of this work can be found in [6].

In the following Section the emphasis will be put on several prototypes ofexchange coupling covering the simple metal spacers Cu, Ag, Au, Cr and Mn.

4.5.1 Simple Interlayers: Cu, Ag and Au

The observation of short wavelength oscillations required a very smooth inter-face. Convincing evidence of short-wavelength oscillations was presented bythe Philips group [80]. Fcc Co/Cu/Co(001) grown on Cu(001) bulk substratesand bcc Fe/Cu/Fe(001) grown on Fe whiskers were investigated by means ofMOKE. The thickness dependence of the exchange coupling was achieved byusing a Cu wedge grown between the ferromagnetic layers. The spacer thick-ness varied continuously from 4 to 19 atomic layers. In the fcc Co/Cu/Co(001)system the variation of the magnetic coupling with Cu thickness was describedby a superposition of two oscillatory terms having periods of 2 and 8 atomiclayers.

In bcc Fe/Cu/Fe(001) grown on a Ag(001) crystal the observed interlayercoupling oscillated with a period of 2 atomic layers. One does not have to use


Fe whiskers to be able to observe two monolayer exchange coupling oscillationsusing bcc Cu(001) spacers. The interface smoothness of Fe/Cu/Fe(001) sys-tems was significantly improved by growing an Fe film on a Ag(001) singlecrystal substrate at 415 K [81]. The terrace size was increased from 3 to 15nm and resulted in the presence of short wavelength oscillations. The FMRand MOKE data were interpreted by the simultaneous use of bilinear and bi-quadratic exchange coupling terms [6, 81]. The exchange coupling was foundto have maxima of ferromagnetic coupling at 9,11 and 13 atomic layers. Themaxima for antiferromagnetic coupling occurred for 10 and 12 atomic layers ofCu. No ferromagnetic coupling was observed in the Philips data. The maximafor antifferomgnetic coupling were observed at 12,14 and 16 atomic layers inthe Philips work. There was an obvious difference reported in the phase ofthe coupling between the Phillips and B. Heinrich et al. (SFU groups).

Comprehensive studies of exchange coupling and its relationship to quan-tum well states, QW, were carried out by the Qiu group at the University ofCalifornia at Berkeley [82] (see references within) using wedged Cu Co/Cu/Co(001)structures grown on Cu(001) single crystal substrates. This systemwas particularly convenient for such studies because Cu has a simple Fermisurface whose sp bands can be easily separated from the other energy bands,see the Fermi surface of Cu in Fig. 4.14. Cu and Co can be grown in the (001)orientation with atomically flat interfaces. Angular resolved PhotoemissionSpectroscopy (ARPES) of QW states was carried out at the Advanced LightSource (ALS) of the Lawrence Berkeley National Lab: the orientation of the

Fig. 4.14. The (110) cross-section of the fcc Cu Fermi surface. The hexagon ofstraight lines outlines the first Brillouin zone. The solid dots represent reciprocal k-vectors. All three important orientations are present. The critical spanning vectorsin the extended Brillouin zone are outlined by the solid arrows. Along the [001]direction the two critical spanning vectors are located at the belly and neck of theFermi surface

212 B. Heinrich

magnetic moments was determined using Magnetic X-ray Linear Diochroism(MXLD) from the Co 3p level, and the the coupling strength was determinedby means of the MOKE technique. The density of states (DOS) is signifi-cantly increased at energies corresponding to the QW states, see also [46].This allows one to follow the QW states as a function of energy for differentCu thicknesses.

In Fig. 4.15 ARPES measurements show the formation of QW states cor-responding to the belly direction of the fcc Cu Fermi surface, see Fig. 4.14.The study was carried out for 20 ML thick Co grown on a Cu(001) substrateand with a Cu wedge grown on top of the Co layer. The ARPES oscillationshave clearly shown the QW states corresponding to the sp electrons in the Culayer. The periodicity of the oscillations was found to be 5.88 atomic layersat the Fermi level and this is exactly the periodicity of the long period ofthe interlayer exchange coupling in Co/Cu/Co(001) systems. Photoemissionintensity along the belly and the neck directions (with k vectors oriented 30Degrees with respect to the film surface) of the Cu Fermi surface are shownin Fig. 4.16.

The belly, 5.88, and neck, 2.67, atomic layer periodicities can be explainedby employing the extended Brillouin zone picture, see the arrowed solid linesin Fig. 4.14. In this case one subtracts from the regular spanning vector insidethe first Brillouin zone a k vector with the atomic layer periodicity (4π/a).The oscillatory period in Cu is given by

2kedCu − φA − φB = 2πn , (4.33)

Fig. 4.15. Photoemission spectra obtained along the surface normal correspondingto the belly direction of the fcc Cu Fermi surface [82]. Oscillations in intensity as afunction of the Cu layer thickness and electron energy demonstrate the formationof quantum well states (QW)


Fig. 4.16. Photoemission intensity along the belly direction (a) and neck direction(b) of the fcc Cu Fermi surface, see Fig. 4.14, as a function of the film thickness andelectron energy below the Fermi level. Two distinct oscillatory periods are present.The dotted curves are calculated using the phase accumulation method [82]

where ke = kBZ − k, kBZ = 2π/a is a Brillouin-zone vector, n is an integer,and φA,B are the phase shifts of electron wavefunctions upon reflection atthe two boundaries of the potential well formed by the Cu layer surroundedby Co and vacuum, and a is the lattice spacing of Cu. Equation (4.33) ex-plains the long and short wavelength oscillation periods by the belly and neckspanning k vectors, respectively. It is caused by evaluating the strength ofthe exchange coupling at the discrete atomic layer separations. This is of-ten called aliasing effect. Simple calculations using an image potential andthe work function at the Cu/vacuum interface allow one to determine thephase shift at the Cu/vacuum interface. The phase shift at the Co/Cu inter-face is determined by the confinement of Cu electrons by the minority spinenergy band of Co. The Cu sp conduction band can be approximated by anearly-free-electron model. The solutions of (4.33) are shown by dotted linesin Fig. 4.16. Clearly this simple model can account well for the QW states inCu. The QW states are the underlying basis for the presence of the interlayerexchange coupling. To insure the direct comparison of the exchange couplingperiodicity with the QW states as a function of the Cu spacer thickness a halfof the wedge sample was covered by 3 ML thick Co. MXLD measurementsare only surface sensitive [83] and consequently the FM and AFM couplingcan be determined by monitoring the MXLD signal coming from the 3 MLthick Co. Images of the DOS (by photo-emission measurements) at the bellyand neck of the Fermi surface were obtained by scanning the photon beamacross the Cu wedge on the Co/Cu side of the wedge. Figure 4.17c shows theobserved MXLD signal with maxima and minima intensities correspondingto AFM and FM couplings respectively. Clearly long and short wavelengthoscillations are easily visible.

214 B. Heinrich

Fig. 4.17. (a) QW states at the belly of the Cu Fermi surface. (b) QW states atthe neck of the Cu Fermi surface. (c) XMLD from the top 3 atomic layers of Coevaporated over the Cu wedge spacer. See further details in [82]. The dark and lightregions correspond to ferromagnetic and antiferromagnetic coupling. (d) Calculatedinterlayer coupling. Notice remarkable agreement between theoretical predictionsand experiment for the sign of the exchange coupling

The coupling between the layers is determined by the energy differencebetween the parallel (P) and antiparallel (AP) alignment of the magneticmoments

2J ∼ EP − EAP =∫ EF

−∞EΔDdE , (4.34)

where ΔD = DP − DAP is the difference of the DOS between P and APalignment of the magnetic moments. For the P configuration of the magneticmoments the minority spins are confined and form well defined QW states.At the neck of the Fermi surface the minority spins are completely confinedby the spin potential of Co. At the belly of the Fermi surface they are onlypartially confined. Whenever the energy of a QW state crosses the Fermi levelit adds energy to EP making the P configuration of the magnetic momentsunfavorable. Fitting of the MXLD data with

J = −A1

d2sin

(2πΛ1

+ Φ1

)− A2

d2sin

(2πΛ2

+ Φ2

), (4.35)

resulted in Λ1=5.88 ML, Λ2=2.67 ML, A1/A2=1.2, Φ1=–86π, and Φ2=64π.MXLD is not able to determine the strength of the coupling. The couplingstrength was investigated using MOKE [82]. Only saturation fields were givenallowing one to estimate of the strength of the AFM coupling in (4.35). At d=6


ML JAFM �0.1 erg/cm2. Surprisingly this is a weak coupling considering thatthe QW states were so well defined. In addition the MOKE results mostly haveshown oscillations with a periodicity of Λ1=5.88 ML. The short wavelengthoscillations are very weakly present with saturation fields less than 100 Oeimplying that J<0.06 erg/cm2. Clearly the interface roughness is a big factorin the exchange coupling measurements but is much less pronounced in theQW state studies.

The recent studies of interlayer exchange coupling in Cu/Ni/Cu/Ni/Cu(001)and Ni/Cu/Co/Cu(001) structures by FMR were carried out by J. Lindnerand K. Baberschke [36]

Ag and Au:

Ag, Au, and Cu have similar Fermi surfaces. Excellent work using Ag and Auspacers was carried out by the NIST group using Fe whiskers as substrates[84, 85]. They used wedged samples with the Au and Ag spacers ranging inthickness from 0 to 15 nm (equivalent of 75 atomic layers). The NM spacerswere covered by 1–2 nm of Fe. Again these structures were oriented along(001) and displayed an excellent crystalline quality and large atomic terraces.The magnetic state of the top Fe film was monitored using scanning electronmicroscopy with polarization analysis (SEMPA). SEMPA is a surface sensitivetechnique that allows one to measure all three magnetization components [23].

The exchange coupling was found to oscillate between FM and AFM cou-pling over a range of 50–65 atomic layers.

This long range of oscillations allowed one to determine with a high degreeof accuracy the short and long wavelength periods. SEMPA measurements re-vealed the short and long wavelength periodicities 2.38, 5.73 ML and 2.48, 8.6ML for the Fe/Ag/Fe(001) and Fe/Au/Fe(001) films, respectively. See furtherdetails in Table 4.1 in Stiles review chapter [3]. These oscillatory periods are invery good agreement with those measured using de Haas-van Alphen (dHvA)and cyclotron resonance measurements of the Fermi surface extremal areas[3, 86]. The strength of the bilinear coupling in the Fe whisker/Au/Fe(001)system as a function of the Au spacer thickness is shown in Fig. 4.18

Theoretical estimates of the asymptotic value of the coupling at 5 ML ofAu is about a factor 3 higher than that measured. Stiles discusses this dif-ference in detail in his review article [3]. One should note that the measuredexchange coupling is always substantially smaller than theoretical predictions.In my view the interlayer coupling is very sensitive to interface structure andthis is the origin of the discrepancy between theory and experiment. Averagingexchange coupling by taking a statistical distribution of terraces only partlyexplains this difference. Interface mixing can even affect in a profound way thesign of the coupling, see Sect. 4.5.3. The heights of the atomic steps on Au(001)are poorly matched to the Fe(001)interlayer spacing, thus atomic steps lead toa strong vertical mismatch. The aliasing mechanism is no longer effective andcan even wipe out long wavelength oscillations [88]. In Fe/Ag/Fe(001) trilayers

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0 5 10 15 20 25 30Au spacer thickness (layers)

- J

(

mJ/

m )

avg

2

10.0

1.00

0.10

0.01

0.001

best fit

measurement

Fig. 4.18. (a) The solid circules represent the measured values of the antiferromag-netic interlayer exchange coupling in Fe whisker/Au/Fe(001)using MOKE and thegrey areas show the best fit function over a wide range of coupling strength andspacer layer thickness using short, λ1=2.48 ML, and long wavelength, λ1=8.6 ML,oscillatory periods [87]

entirely grown at room temperature (RT) the ferromagnetic exchange couplingdecreased rapidly with increasing thickness. It reached zero at 7 ML [38].Growing the Fe layers at elevated temperatures, Tsub ∼ 180◦, significantlydecreased the density of atomic steps at the Fe/Ag interface and recovered theoscillatory behavior of the exchange coupling [89]. Good quantitative agree-ment with theory is very hard to reach. The NIST group results are unique;they were able to establish a well defined oscillatory behavior of the interlayerexchange coupling over a wide range of thicknesses. This allowed them to de-termine the true exchange coupling periods and showed that well defined QWstates exist in simple metals even for large spacer thicknesses. Quantitativeagreement with theory proves to be more elusive.

The spin-dependent potential in multilayer films creates electron confine-ment and resonant states which are responsible for the oscillatory behaviorof the exchange coupling. According to theoretical calculations [42, 90] suchstates are not restricted to the nonmagnetic spacers, but are also present insidethe ferromagnetic layers, and the coupling cannot be entirely described as aninteraction that is localized at the interfaces. The energy terms coming fromthe electron confinement in the ferromagnetic layers due to multiple reflectionsand the interference of such states with the states inside the spacer layer re-sults in a variation of the interlayer exchange coupling with the ferromagneticlayer thickness. Calculations and experimental studies on the Co/Cu/Co(001)system [74] have shown that the exchange coupling contains a component thatoscillates as a function of the ferromagnetic layer thickness. However, the os-cillatory part is smaller than the total strength of the exchange coupling sothat the sign of the coupling is determined by the thickness of the nonferro-magnetic spacer layer.


One can expect that the quantum well states affect also other magneticproperties. Bayreuther et al. [91] using Ni(0.73 nm) covered by a Au over-layer with a variable thickness (from 0.4 to 8 nm), have shown that the Curiepoint Tc in the Ni layer oscillates with the Au thickness overlayer in agree-ment with QW states in Au. The observed variations in Tc were as largeas ±20 K. Oscillations in Tc of 40 K were also observed by Ney et al. [92]in 2.8Co/2-4Cu/4.8Ni trilayers. The integers represent the number of atomiclayers. These results are supported by theoretical calculations by Pajda et al.[93] of the ordering temperature Tc for a single atomic layer of Fe or Co cov-ered by a variable thickness of Cu(001). The Green’s function method usingthe Random Phase Approximation have shown that the QW states in Cusignificantly affect Tc. The amplitude of oscillations in Tc could be as largeas 30 K which is 10 percents of the average value of Tc. The thickness depen-dence of Tc is particularly spectacular in fcc Fe(001) grown on fcc Cu(001).In these studies by Vollmer et al. [94] the Fe and Cu layer thicknesses werecontinuously changed between 0 to 12 and 0 to 9 atomic layers, respectively,using a double wedge Fe/Cu(001) structure. It is interesting to note that theoscillations in Tc were completely suppressed after the lattice transformationat 10 atomic layers from the fcc structure to the stable bcc Fe structure.

4.5.2 Temperature Dependence of the Exchange Coupling

The temperature dependence of the interlayer exchange coupling provides agreat theoretical challenge. It is always difficult to account correctly for therole of spin fluctuations in any system and nanosystems provide a great palateranging from 1D to 3D behavior including topological complexities relatedto the interface roughness. The temperature dependence of the bilinear andbiquadratic exchange coupling has usually been investigated for magnetic tri-layer FM/NM/FM systems. These studies were usually carried out in systemshaving a significant degree of interface roughness such that the bilinear cou-pling was substantially smaller than that in samples having atomically flatinterfaces, see above. This roughness can be expected to affect the role ofspin fluctuations at the FM/NM and NM/FM interfaces. Small values of theinterlayer exchange coupling implies a strong cancellation of short and longwavelength oscillatory exchange contributions. It can be expected under thesecircumstances that the the bilinear exchange coupling would be particularlyaffected by thermal fluctuations having a wavelength comparable to the aver-age separation between atomic terraces. The temperature dependence of theexchange coupling was measured over the interval from 77 to 400 K by Celin-ski et al. [95] using the system Ag-substrate/9Fe/10,12bccCu/16Fe(001) andby Lindner and Baberschke [35] over the temperature range from 55 to 350 Kusing the system fcc Cu-substrate/7Ni/5,9Cu/2Co(001). The integers repre-sent the number of atomic layers. It was found that the strength of the bilinearexchange coupling was increased by a factor of 3 to 4 times to a maximum of0.18 erg/cm2 in the Fe/Cu/Fe system as the temperature was reduced from

218 B. Heinrich

400 K to 77 K, [95]. In the Ni/Cu/Co system the bilinear exchange couplingstrength increased by a factor of 4 to a maximum of 0.06 erg/cm2 as the tem-perature was reduced from 350 K to 55 K; moreover, this increase was foundto scale with an inverse 3/2 power law in temperature [35]. It is interestingto note that in the Ag-substrate/9Fe/10,12bccCu/16Fe(001)samples no evi-dence of a 3/2 power law was found. In the Cu-substrate/7Ni/5,9Cu/2Co(001)samples the Co layer was only 2 atomic layers thick and therefore a strongdependence on thermal fluctuations would be expected. In fact, in a recentpaper by Schwieger et al. [96] it has been shown that an extended Heisenbergmodel accounts for 75% of the reduction of the bilinear exchange couplingstrength from 0 K to RT. However a similar increase in the bilinear exchangecoupling strength with decreasing temperature using relatively thick Fe lay-ers [95] suggests that there are some other contributions which are significantand not caused by thermal fluctuations. In fact for such thick Fe films onecan’t expect to have a strong temperature dependence since the saturationmagnetization changes by less than 10%. This view is even more supportedby measurements by Milton et al. [97] on the Fe-whisker/Cr/Fe(001) sampleshaving the best available interfaces. Here the temperature dependence of thebilinear coupling strength is a linear function of the temperature and it in-creased by a factor of 2 over the temperature interval from 300 K to 110 K. Inthe Ag-substrate/9Fe/10,12bccCu/16Fe(001) samples the temperature depen-dence was measured independently for the bilinear and biquadratic couplingstrengths. Surprisingly the biquadratic coupling strength was found to be lin-early dependent on the bilinear exchange coupling strength over the abovetemperature interval. This result does not explicitly follow the prediction bySlonczewski’s formula 4.32. In this formula the biquadratic exchange couplingstrength is proportional to the square of the mean deviation of variations of thebilinear coupling strength and therefore one would expect that the biquadraticexchange coupling strength should follow a quadratic power law in the bilinearexchange coupling strength. The biquadratic exchange coupling is definitelycaused by lateral variations in the bilinear coupling strength. In many samplesit has been found that the biquadratic exchange coupling strength decreasessignificantly more rapidly with decreasing terrace size than the bilinear cou-pling strength, see [6, 81], in agreement with formula 4.32. Therefore theobserved linear dependence of the biquadratic exchange coupling strength ontemperature implies that the exchange averaging factor in (4.32) is nearlytemperature independent.

Co(3.2 nm)/Ru(0.9 nm)/Co(3.2 nm)(0001) trilayer structures prepared bymeans of MBE by Zhang et al. [31] have provided good samples for testing thetemperature dependence of the exchange strength in samples having a largeinterlayer exchange coupling strength. In these samples the exchange couplingstrength increased from RT to He temperatures by 30%. This temperaturedependence was well described by the theoretical predictions of Edwards [98]and Bruno [86]. Agreement with theory indicated that the interlayer exchange


coupling through Ru was closely related to the spatial confinement of d holesin the Ru layer due to spin splitting of the d band in the Co layers.

The magnetic order in a [2Fe/13(VHx)]x200 superlattice was investigatedby Leiner et al. [99] as a function of the hydrogen content. It was shown thatthe interlayer exchange coupling strength could be continuously and reversiblychanged via absorption of hydrogen into the nonmagnetic V. The magneticphase diagram of a [2Fe/13(VHx)]x200 superlattice exhibited a critical concen-tration xc �0.022 at which the magnetic order changed from antiferromagneticto ferromagnetic order. In the vicinity of xc �0.022 the transition temperaturestrongly depended on x and decreased continuously as the concentration ap-proached xc �0.022. The hydrogenation of V was used by Paernaste et al. [100]to investigate the magnetic order in a 3Fe/14.4VHx/3Fe trilayer as a func-tion of the oscillatory interlayer exchange coupling strength through the VHx

spacer. A pure V interlayer showed a 2D FM (X-Y) behavior which is consis-tent with a weak interlayer coupling strength. As a result of the introductionof H into the V the critical point Tc exhibited an oscillatory dependence onan increasing H content. The minima and maxima of Tc were expected to bedirectly associated with an oscillatory interlayer exchange coupling strength.An FM exchange coupling could be expected to be associated with an increasein Tc, and AF exchange coupling associated with a decrease of Tc.

4.5.3 Spin Density Wave (SDW) in Cr

Fe-whisker/Cr/Fe(001) wedge structures, see Fig. 4.19, have played a crucialrole in the study of interlayer exchange coupling.

Unguris, Celotta, and Pierce studies [78, 101] using scanning electronmicroscopy with polarization analysis (SEMPA) on Fe-whisker/Cr/Fe(001)samples showed that the exchange coupling oscillates with a short-wavelengthperiod of �2 monolayers ( ML). The SEMPA images revealed in a very explicitway the presence of both short-wavelength and long-wavelength oscillationsin the thickness range from 5 to 80 ML, see Fig. 4.20.

The period of the short-wavelength oscillations, λ=2.11 ML, was found tobe slightly incommensurate with the Cr lattice spacing; the period of the longwavelength oscillations was found to be 12 ML. These two basic periods areexpected for the complex Cr Fermi surface, see Fig. 4.21. The incommensuratenature of the short wavelength oscillations in Cr, λ=2.11 ML, results in thephase slips of exchange coupling at 24 and 44 atomic layers of Cr, see Fig. 4.20and further details in [23].

Heinrich and co-workers (SFU group) carried out quantitative exchangecoupling measurements on Fe-whisker/Cr/Fe(001) samples [6, 102, 103]. Theobjective of the SFU group was to grow samples having the best available in-terfaces, to measure quantitatively the strength of the exchange coupling, andto compare these coupling strengths with ab initio calculations that explic-itly include the presence of spin-density waves in the Cr. The requirementof smooth interfaces limited our study to samples which were grown on

220 B. Heinrich

Fig. 4.19. A schematic view of Fe whisker/Cr/Fe(001) sample containing a wedgedCr spacer employed by the NIST group in SEMPA measurements. The arrows showthe directions of the magnetization. The Fe whisker is demagnetized by a single 180degree domain wall. The direction of the magnetization in the Fe film follows thesign of the local interlayer exchange coupling

Fe-whisker templates with the Cr spacers terminated at an integral numberof Cr atomic layers. It was found that the strength of the exchange couplingthrough the Cr(001) spacer even for these smooth Cr layers was extremelysensitive to small variations in the growth conditions. Cr ultrathin films donot possess a robust spin-density wave; the spatial variation of the spin mo-ments is extremely sensitive to the interfaces. In our studies we concentratedon samples for which the Cr thickness ranged from 4 to 13 atomic layerswhere the role of interfaces was most pronounced. The measured exchangecoupling was found to be reproducible only in those samples that exhibitedtrue layer by layer growth. The existence of unattenuated RHEED inten-sity oscillations of the RHEED specular spot during the growth of the Crspacer did not guarantee reproducible values of the exchange coupling be-tween a thin Fe(001) film and the whisker(001) substrate. It was necessary toestablish conditions such that new layers were initiated at a repeatable pat-tern of nucleation sites and subsequent attachment of adatoms to the atomicsteps of the newly formed atomic islands. This condition was monitored byexamining RHEED intensity oscillation amplitudes together with the widthof the RHEED specular spot profile. The best results were obtained whenthe RHEED intensity oscillations exhibited sharp cusps at the RHEED in-tensity maxima (filled atomic layers)and the RHEED spot profile oscillatedrepeatedly between narrow peaks (filled atomic layers) and wider, split inten-sity peaks (half filled layers), see [103]. These requirements were achieved bymaintaining the substrate temperature in a narrow range of temperatures, 280C<Ts,opt <320 C. The presence of the specular spot splitting in a direction


Fig. 4.20. The direction of the magnetic moment in the Fe film grown over a Crwedge is shown in the second SEMPA image. The top SEMPA image shows thepresence of a magnetic moment in the Cr wedge using an Fe whisker/Cr(001) sam-ple. White and black contrast indicate parallel and antiparallel orientations of themagnetic moment with respect to the Fe whisker magnetization. Note in the bottomSEMPA image that at the boundaries between the parallel and antiparallel orienta-tion of the magnetic moments a perpendicular component of the magnetic momentis present. This is caused by magnetic frustrations due to a partial completion ofthe top Cr/Fe interface which results in a strong biquadratic exchange coupling

parallel with the RHEED streaks for half filled atomic layers indicated thatnew atomic layers were formed from nucleation centers which were separatedby a well defined mean distance of ∼ 80–90 nm. In most depositions of Fe lay-ers the first 5 ML were grown at 100 C. The substrate temperature was thenincreased to 120–240 C. All samples prior to their removal from the vacuumsystem were covered by a 20-ML-thick epitaxial Au (001) layer. The growth ofAu exhibited well defined RHEED intensity oscillations and the surface wasterminated by a 5×1 reconstruction typical for the Au(001) surface.

Magnetic studies

A great review of magnetic studies on Fe/Cr/Fe systems that includes exper-iments and underlying theories can be found in the review paper by Fishman[5]. Here I will summarize mostly studies on the best available samples whichwere prepared on Fe whiskers.

Quantitative BLS studies of the exchange coupling in Fe-whisker/Cr/Fe(001) have been discussed in [6, 103, 104]: the main results are shown inFig. 4.22. These studies showed that the exchange coupling through Cr(001)contains both oscillatory bilinear J1 and positive biquadratic J2 exchange

222 B. Heinrich

k100

k010

k100

k010

Fig. 4.21. The left panel shows a representation of the Cr Fermi surface in theparamagnetic state. The gray shaded areas are ellipsoids centered at the N points inthe bcc reciprocal lattice. The right panel shows a slice through the Fermi surface asindicated in the left panel. The gray shaded arrows are the critical spanning vectorsat the N-centered ellipsoids and the white arrows indicate the nested parts of theFermi surface that give rise to spin density wave antifferomagnetism [3]

Fig. 4.22. The thickness dependence of the bilinear J1 (•) and biquadratic J2 (+)exchange coupling. The biquadratic coupling can be measured only for AFM coupledsamples. The values of J2 for FM coupled samples (10 and 12 ML) were assumed tobe the same as those for the AFM coupled samples having 9,11, and 13 ML of Cr.Note that the coupling becomes AFM for thicknesses greater than 4 ML and thethickness dependence of J1 has a broad maximum around 7 ML of Cr. This behavioris caused by long wavelength oscillations. Large short wavelength oscillations appearfor Cr thicknesses greater than 9 ML


coupling terms. The exchange coupling first becomes antiferromagnetic at 4ML of Cr. For Cr spacer thicknesses dCr <8 ML the strength of the short-wavelength oscillations is quite weak, ∼ 0.1 ergs/cm2. The exchange cou-pling in this range is antiferromagnetic. This is due to the presence of anantiferromagnetic (AFM) long-wavelength bias. This AFM bias is peaked inthe range between 6 to 7 ML. It is interesting to note that the strength ofthe long-wavelength AFM bias is very nearly the same as that observed inFe/Cr/Fe(001) epitaxial multilayers prepared by sputtering where relativelylarge interface roughness annihilates the presence of the shortwavelength oscil-lations [105]. Exchange coupling in the sputtered films showed long-wavelengthoscillations with a rapidly decreasing strength of the coupling for thicknessesgreater than 10 ML of Cr. For Cr spacers thicker than 8 ML, dCr >8 ML, theexchange coupling in films grown on a whisker substrate is dominated by theshort-wavelength oscillations. In this thickness range the samples are antiferro-magnetically coupled for an odd number of Cr atomic layers, Jtot = |J1-2J2| ∼1.5 erg/cm2, and ferromagnetically (FM) coupled for an even number of Cratomic layers. It is interesting to note that the strength of the exchange cou-pling, 1 erg/cm2 in our samples, is very close to that measured by the NISTgroup, see Fig. 4.23, indicating that the Fe-whisker treatment and growthprocedure carried out by the SFU and NIST groups were similar.

MOKE measurements on Fe/wedgeCr/Fe(001) samples were carried outby the NIST group [106]. They covered a wide range of Cr thicknesses, seeFig. 4.23, showing that antiferromagnetic coupling (AFM) was observed atan odd number of atomic layers of Cr. This changes after the first phase slipbetween 24–25 ML of Cr.

Interface atom exchange (interface alloying)

The coupling between the Fe and Cr atoms at the Fe/Cr interface is expectedto be strongly antiferromagnetic [107] and consequently the spin-density wavein Cr is locked to the orientation of the Fe magnetic moments. Since the periodof the short-wavelength oscillations is close to 2 ML one would expect AFcoupling for an even number of Cr atomic layers and FM coupling for an oddnumber of Cr atomic layers. For the period 2.11 ML the first phase slip inthe shortwavelength coupling is predicted to occur at 20 ML. Surprisingly theBLS and SEMPA measurements showed clearly that the phase of the short-wavelength oscillations is exactly opposite to that expected.

It is also important to note that the strength of the exchange coupling wasfound to be much less than that obtained from the first-principles calculations,J1 ∼30 ergs/cm2 [108]. Our studies showed that the strength of the bilinearexchange coupling J1 is very sensitive to the initial growth conditions: a lowerinitial substrate temperature resulted in a larger exchange coupling strength.The bilinear exchange coupling could be changed by as much as a factor of5 by varying the substrate temperature during the growth of the first Cratomic layer [109]. This behavior led us to believe that the atomic formation

224 B. Heinrich

Fig. 4.23. The saturation field as a function of Cr thickness using an Fe-whisker/Crwedge/15Fe(001) sample. The integer represents the number of atomic layers. Thesemeasurements were carried out using MOKE measurements [106]. These measure-ments were possible only for AFM coupling. Note that antiferromagnetic couplingcorresponds to an odd number of Cr atomic layers in agreement with the SFU mea-surements. No quantitative interpretation of the bilinear and biquadratic couplingwas provided

of the Cr layer was more complex than had been previously acknowledged.Angular resolved Auger spectroscopy (ARAES) [110, 111], STM [112], andproton induced Auger electron spectroscopy (ARAES) [113] have shown thatthe formation of the Fe/Cr(001) interface is strongly affected by an interfaceatom exchange mechanism (interface alloying).

The above studies revealed that the Cr adatoms undergo interface mix-ing when the substrate temperature is adjusted for optimum growth. TheARAES data showed that the interface mixing was confined mainly to twoFe atomic layers; see Fig. 4.24, and interface alloying during the growth al-ready starts at low substrate temperatures, Tsub ∼100 C. Using temperaturesless than Tsub ∼100 C one was not able to establish a proper layer by layergrowth even by subsequently raising the substrate temperature. The inter-face alloying increases with increasing substrate temperature; see Fig. 4.24. Itshould be noted that interface alloying due to the atom exchange mechanismis not, in general symmetric; it occurs chiefly at one interface [114]. Interface


Fig. 4.24. The substrate temperature dependence of the fraction of Cr atoms inthe first layer deposited on an Fe whisker substrate (�), the fraction of Cr atomscontained in the whisker surface layer (•), and the fraction of Cr atoms in the firstwhisker subsurface layer (�). The fractional coverages were obtained from fitting theangular dependence of the Auger Cr (529 eV) peak intensity using angular resolvedAuger electron spectroscopy [110]

alloying is driven by the difference in binding energies between the substrateand adatoms. The binding energies are proportional to the melting pointsof the solids. Interface alloying has been observed in systems for which thesubstrates have lower melting points than do the adatom solids [114, 115].

Freyss, Stoeffler, and Dreysse [116] investigated the phase of the exchangecoupling for intermixed Fe/Cr interfaces. The calculations were carried outusing a tight-binding d-band Hamiltonian and a real-space recursive methodfor two mixed layers: Fe(001)/CrxFe1−x /Cr1−xFex /Crn, where n representsthe number of pure Cr atomic layers. This simulates our experimental stud-ies which were carried out on specimens for which the first few atomic Crlayers were grown at lower substrate temperatures where the surface alloy-ing was mainly confined to the two interface atomic layers. The calculationswere able to account for two important experimental observations. First, thecrossover to antiferromagnetic coupling and onset of short-wavelength oscilla-tions was predicted to occur at 4 to 5 ML of Cr, in good agreement with ourobservations, see Fig. 4.22, and in agreement with the NIST studies using theSEMPA imaging technique. Second, the phase of the coupling was reversedfor a concentration x>0.2. AFM and FM coupling was obtained at an oddand an even number of Cr atomic layers, respectively, in perfect agreementwith experiment.

We tested the role of interface mixing by terminating the Cr growth withthe co-deposition of Cr plus Fe to produce a Cr-Fe alloy. Two alloy concen-trations were used: Cr 85%-Fe 15% and Cr 65%-Fe 35%. BLS and MOKEmeasurements revealed that the sign of the exchange coupling between theiron film and the whisker substrate was not changed by this alloyed atomic

226 B. Heinrich

layer. The sign was given by the number of Cr atomic layers deposited with-out any Fe. The system behaved as if the Cr alloy layer formed part of theiron film. This result strongly supports the idea that interface alloying atthe Fe-whisker/Cr interface leads to the observed phase reversal of the short-wavelength oscillations in the strength of the exchange coupling relative to asystem having perfect interfaces.

A large number of measurements showed that a part of J2 was propor-tional to the measured value of J1. We found that J2 ∼0.1+0.16|J1 |. Thisis an interesting result that requires a brief comment. Stoeffler and Gautierpredicted the presence of a biquadratic exchange coupling term for Fe layerscoupled through Cr(001)[108]. This could indicate that the main part of thebiquadratic coupling is of an intrinsic origin. However in subsequent calcu-lations by the Strasburg group [117] for a spin system that was completelyrelaxed the exchange coupling through Cr(001) spacers with ideal interfaceswas formally similar to Slonczewski

′s proximity (torsion) model [118] in which

the exchange energy increases quadratically with the angle between the mag-netic moments of the Fe layers:

E = J(Δϕ− π)2 , (4.36)

where in zero field Δϕ=0 and 180◦ for the odd and even number of Cr atomiclayers, respectively. For AF coupling ϕ varies between 180◦ and 0◦, conse-quently the total magnetic moment approaches saturation gradually; thereis no torque free solution in high fields. The BLS and MOKE measurementsusing Fe-whisker/Cr/Fe(001) samples having a low density of atomic stepsexhibited a much weaker coupling than that predicted by theory and at thesame time it was possible to fully saturate the magnetic moments in thesesamples in sufficiently large external fields. All our MOKE and BLS mea-surements carried out on Fe-whisker/Cr/Fe(001) samples that were preparedwith the best possible interfaces (low density of atomic steps) are consistentwith the use of bilinear and biquadratic exchange coupling terms. In partic-ular, we found that in MOKE and BLS measurements for fields somewhatabove the critical field H2, see Fig. 4.26, the iron film and whisker momentswere parallel, whereas for fields slightly less than the critical field H1 thethin film and whisker moments were antiparallel. The discrepancy betweenexperiment and theory was very likely caused by the presence of interfacealloying at the Fe/Cr interface. Interface alloying not only significantly de-creases the exchange coupling but even changes the angular dependence ofthe coupling.

Neutron-diffraction and MOKE studies by the Bochum group [119] usingFe/Cr/Fe(001) samples having a high density of atomic steps, showed that theground state in Fe/Cr/ Fe(001) multilayers exhibited a noncollinear orienta-tion of the magnetic moments for which the approach to saturation could bedescribed by the Slonczewski magnetic proximity model in which the exchangeenergy depends quadratically on the angle between the magnetic moments ofthe ferromagnetic layers.


However one should realize that dc magnetometry can be affected by sam-ple inhomogeneities. The approach to saturation at the critical field H2 asobserved using MOKE, and the onset of the antiferromagnetic configurationat the critical field H1 , see Fig. 4.25, is more gradual than is expected usingthe sum of bilinear and biquadratic exchange coupling terms; see Fig. 4.26,curve A.

The calculated approach to saturation clearly exhibits a well defined kinkat the field H2, while the experimental measurements usually show a con-cave gradual approach to saturation; see Fig. 4.25. According to the calcu-lations the antiferromagnetic configuration of the Fe magnetic moments isreached via a first-order phase jump; the experimental measurements show amore gradual s-shaped change. However these experimental MOKE featurescan be explained by an inhomogeneous distribution of the exchange couplingstrength. A 10% variation in the exchange coupling across the measured areawould result in hysteresis loops that are very similar to those observed usingMOKE; compare Fig. 4.25 with Fig. 4.26. In fact an inhomogeneous distribu-tion of the exchange coupling also explains the observed differences betweenvalues of the critical fields H2 that have been obtained using the MOKE andthe BLS techniques. The BLS measurements on Fe/Cr/Fe(001) always yieldlower values for the critical field H2, with corresponding lower values of theexchange coupling strength, compared with those obtained using MOKE. TheBLS thin-film resonant modes were also visibly broadened for external fieldsgreater than the saturation field H2 , where the Fe film magnetic momentis parallel to the Fe-whisker moment. The Fe film resonance mode was ob-served to be much narrower at a field midway between H1 and H2 where theFe film and whisker magnetizations are nearly orthogonal [104]. The differ-

Fig. 4.25. The longitudinal MOKE signal for the sample Fe-whisker/11Cr/20Fe(001) as a function of applied field.The integers represent the number of atomiclayers. The saturation field measured using MOKE was 300 Oe higher than thatmeasured using BLS. The difference between the MOKE and BLS measurementswas caused by lateral inhomogeneities in the exchange coupling [103]

228 B. Heinrich

Fig. 4.26. Calculated MOKE signal for a 20 ML Fe(001) film exchange coupledto a bulk Fe(001) substrate and assuming an inhomogeneous distribution of thebilinear coupling strength J1. The biquadratic coupling strength J2 was set to 0.3ergs/cm2 for all curves. The spread of bilinear coupling was assumed to satisfyGauss’s distribution with 〈J1〉=–1 ergs/cm2 , and the distribution width Δ J. CurveA Δ J=0, B Δ J=0.2 ergs/cm2, and C Δ J=0.5 ergs/cm2

ence between critical fields measured using BLS and MOKE, as well as thebroadening of the BLS signal for fields greater than H2 , can be explainedby inhomogeneous distributions of J1 and J2. The BLS technique measuresthe frequencies of the rf resonance modes. The mode corresponding to thethin Fe film covering the Cr spacer exhibits a resonance at a field that isthe algebraic average of local inhomogeneous resonance fields, correspond-ing to the distribution of local exchange coupling strengths. The broadeningof this BLS resonance peak is related to the distribution of the local reso-nance fields. On the other hand, in the MOKE studies a complete saturationis observed only after the external field reaches the value corresponding tothe maximum value of the H2 critical field distribution. The observed differ-ences between the MOKE and BLS measurements clearly indicate that thecoupling through Cr is not even homogeneous across the small area only afew micrometers in diameter corresponding to the laser spot size in the BLSmeasurements.

Role of multiple scattering

The role of electron interface scattering was tested using heterogeneous Crspacers. Two specimens were grown with a Cu interface layer between the Crspacer and the Fe thin film: Fe-whisker/ 11Cr/1Cu/20Fe(001)/20Au and Fe-whisker/11Cr/2Cu/20Fe(001)/20Au, where the integers represent the number


of atomic layers. Two specimens were grown with a silver interface layer: Fe-whisker/11Cr/1Ag/20Fe(001)/20Au and Fe-whisker/11Cr/2Ag/20Fe(001)/20Au. Measurements carried out using BLS and MOKE revealed that theexchange coupling in the Fe-whisker/ 11Cr/1−2Cu/Fe(001) was found to in-crease twofold compared to that observed in samples having a simple 11 ML Crspacer layer. No significant difference was found for the samples having a Aginterlayer where the exchange coupling decreased by 20%. Mirbt and Johans-son [120] presented calculations that are in accord with these results. Their cal-culations showed that the enhanced coupling strength in Fe/Cr/Cu/Fe(001)samples is due to a change in the spin dependent reflectivity of the Cr spacerelectrons at the Cr/Cu/Fe interface. The presence of the Cu atoms changesthe spin dependent interface potential due to hybridization of the Cu electronstates with the Fe electron states. Since the Fe majority spin band lies closestto the Fermi level the effect of the hybridization will be most pronounced forthe majority spin Fe band. The hybridization with Cu results in a downwardenergy shift that moves the Fe majority spin band below the Fermi level. Anenergy gap is created at the Cu/Fe interface, and consequently the majorityspin electrons in Cr undergo a nearly perfect reflection. The states for minorityspin electrons are very little affected by the Cu, and therefore their reflectivityis left unchanged. It follows that the spin reflection asymmetry is increasedleading to an increased coupling. The effect of a Ag spacer on the couplingin Fe/Cr/Ag/Fe films is less dramatic. Calculations show that the spin asym-metry in reflectivity is somewhat decreased leading to an overall decrease inthe exchange coupling. The theoretical calculations of Mirbt and Johanssonsuggested that a proper model for exchange coupling through spin-densitywaves in Cr has to include two contributions: (a) a spin dependent potentialdue to the magnetic moments on the antiferromagnetic Cr atoms; (b) a spindependent potential at the Fe/Cr and Cr/Fe interfaces. The first contribu-tion for Cr layers thinner than 24 ML can be described by a Heisenberg-likeHamiltonian [108] with AFM coupling between the Cr magnetic moments onadjacent (001) planes and a strong AFM coupling between the Cr and Featomic moments at the interfaces. The experimental result implies that thespin-density and multiple scattering contributions act in phase to increase thetotal exchange coupling.

Dependence of exchange coupling on the thickness of the Fe layer

Okuno and Inomata [121, 122] reported a strong oscillatory dependence oniron thickness of the exchange coupling in Fe/Cr/ Fe(001) multilayer speci-mens. The period of the oscillation was 6 ML. However, the specimens usedin their work were multilayers characterized by rough interfaces, and the ex-change coupling exhibited no short period ∼2 ML variations with Cr thick-ness. We have measured the dependence of the exchange coupling strengthon iron film thickness using an Fe(001) whisker substrate, an 11 ML layerof Cr, and a wedge-shaped Fe film deposited on the Cr. This structure,

230 B. Heinrich

Fe-whisker/11Cr/wedge Fe(001), was then capped by 20 ML layers of gold.The Cr was deposited using optimum conditions for a smooth growth. Noevidence was found for either a short or a long wavelength oscillatory de-pendence of the coupling strength on the iron film thickness. We concludedthat Fe/Cr/Fe/Au(001) samples having a low density of interfacial steps,and that exhibit 2 ML oscillations in the exchange coupling as a function ofthe Cr layer thickness, displayed no measurable variations of the exchangecoupling strength as a function of the Fe film thickness. It appears thatthe exchange coupling between Fe layers separated by a Cr spacer can beascribed to interactions that are localized to the interfaces. However, oursubsequent experiments did indicate that the exchange coupling is sensi-tive to the Fe film thickness when the iron film is capped with Cr. For awhisker/11Cr/20Fe/20Au structure J1 =–0.82 erg/cm2, J2 =0.3 erg/cm2;and for a whisker/11Cr/20Fe/11Cr/20Au(001) J1 =–1.6 erg/cm2, J2 =0.33erg/cm2. Clearly the insertion of Cr between the Fe and Au films caused asubstantial increase in the coupling strength. Note that the biquadratic cou-pling term is essentially the same for both specimens. Since the exchangecoupling for Cr/Fe/Cr films depends upon the Fe film thickness, it followsthat the Fe/Cr interfaces support the formation of electron resonance statesin the Fe films. On the other hand, the Fe/Au interfaces tend to suppresselectron resonance states.

Neutron studies

Extensive work on Cr has been carried out by a large number of groups. Iwould like particularly to point out the work done using neutron scattering.Neutron studies have been carried out on thick Cr layers ranging from 2.0nm to 300 nm and over a wide range of temperatures down to cryogenictemperatures where one can study the Cr SDW phase diagram. The conclu-sion of these studies is that the SDW magnetism in Cr films is complex andexquisite. The SDW in Cr films is not a rigid property but depends on thefilm thickness, the temperature, and is crucially affected by interface rough-ness, and interface exchange coupling. Theoretical and experimental investi-gations have been mainly limited to the (001) orientation. The first reliableSDW data in Cr films were reported in papers by Schreyer et al. [123, 124]on [Fe/Cr/Fe](001) superlattices prepared using MBE. Soon after neutronstudies on superlattices grown in the [001], [011], and [111] crystallographicorientations and prepared by sputtering were performed by Fullerton et al.[125] and Adenwalla et al. [126]. The results on the MBE and sputtered sam-ples agree on a few points. The MBE grown superlattices exhibited bothincommensurate collinear (I) and commensurate spiral SDW (C) magneticstates for a Cr spacer thicker than 3.5 nm [127]. The non-collinear commen-surate C state results from magnetic frustration at the Fe/Cr interfaces. Thismode is equivalent to the H SDW in Fishman’s review article, see Fig. 4.15in [5]. Schreyer et al. suggested that the exchange interlayer coupling energy


in C follows the proximity (torque) model suggested by Slonczewski [118],see (4.36) and (4.37). This torque model resulted in a spiral configurationof the Fe magnetic moments. For dCr <4.5 nm only the C state exists. Thetransition temperature from I to C starts from 0 K at 4.5 nm and then grad-ually increases to 300 K with increasing dCr. The Neel ordering temperatureTN �500 K was found to be independent of the Cr thickness, dCr. See thefull phase diagram in [127].

The studies by Fullerton et al. on sputtered specimens [125] yielded differ-ent results. They found a collinear I SDW state with the SDW nodes near theFe/Cr and Cr/Fe interfaces. No C and H SDW states were observed. The pres-ence of SDW nodes at the interfaces can result in the loss of magnetic contactwith the neighboring Fe layers. The Cr layer is either in an incommensuratestate for thick enough layers or becomes paramagnetic with decreasing filmthickness or with increasing temperature. One needs at least one SDW periodto create a collinear I SDW. The SDW period was found to be nearly thesame as the bulk value. The Neel ordering temperature TN vanished for filmshaving a thickness less than a critical thickness of about 30 ML [128].

Clearly the different results obtained using MBE and sputtered samplesis due to the different interface roughness. The sputtered samples have mostlikely a greater interface roughness. The SDW behavior in Cr is very adaptableand easily affected by the interfaces.

The spin flip transition from a transverse to a longitudinal SDW observedin bulk Cr at T=123 K is not observed in superlattices in the thickness range sofar studied. Although in thick Cr layers (above 30 nm) covered by an Fe(001)layer the transverse SDW with the q-vector parallel to the film surface andwith the magnetic moment of Cr perpendicular the film surface was observed[129, 130]. In this case the interface frustrations are minimized by flippingthe magnetic moment of Cr perpendicular to the Fe magnetic moment. Thisis analogous to the biquadratic exchange coupling mechanism in interlayerexchange coupling.

Layer specific measurements of the magnetization reversal in AFM cou-pled Fe/Cr/7Fe/Cr/Fe(001) layers were carried out by L’abbe et al. [131] usingthe isotope (7Fe) and nuclear resonant scattering of circularly polarized syn-chrotron radiation. This work provided interesting results. The total magnetichysteresis loop with the field applied along the magnetic easy axes indicated atypical antiferromagnetic coupling in the presence of 4-fold in-plane magneticanisotropy, compare Fig. 4.14b in [131] with line 3 in Fig. 4.2. The absence ofa jump in order to reach full saturation indicated that biquadratic exchangecoupling was present, but the existence of a well defined jump at low fieldsindicated that the bilinear coupling was strong enough to orient the magneticmoments antiparallel to each other and parallel to the magnetic easy axes.However the resonant scattering results in [131] suggested that the magneticmoment in the central Fe film was oriented away from the magnetic easyaxis and this observation indicated a non-trivial configuration of magneticmoments.

232 B. Heinrich

Magnetic domains in the remanent state in [Fe/Cr] superlattices wereinvestigated by means of Polarized Neutron Reflectivity (PNR) [132] and Syn-chrotron Moessbauer spectroscopy techniques [133].

It is not possible to discuss these studies in greater detail within the spaceavailable for this chapter. There are excellent review articles by Zabel [134],Fullerton et al. [135], Boedeker et al. [129], and Fishman [5] that cover boththeory and experiment including SWD phase diagrams in Cr films; see alsothe references contained therein.

4.5.4 Antiferromagnetic Mn

The growth of Mn is challenging by virtue of its ability to adopt various formsof crystalline structures ranging from the bulk αMn (with many atoms perunit cell) to bcc and fcc structures stabilized by alloying or by epitaxial growthon suitable substrates [136]. Kim et al. [137] found that Mn grows on Fe(001)in a strained body-centered tetragonal (bct) structure with lattice spacing pa-rameters a=0.2866 nm and c=0.3228 nm for thicknesses greater than 4 atomiclayers of Mn. The measured RT values of the Mn magnetic moment for a thinfilm on Fe range from 1.7 to 4.5 μB [138, 139]. The magnetic state of Mn filmsthinner than 5 ML is rather complicated. Magnetic Circular X-ray Dichro-ism (MCXD indicates that the fully filled atomic layer of Mn is magneticallycompensated [139, 140]. For submonolayer coverage the Mn moments wereoriented antiparallel to the Fe magnetic moments [140]. This observation is inagreement with calculations by Wu and Freeman [141]. The ground state of aMn atomic layer is a c(2 × 2) magnetically compensated state. The situationis simpler for Mn films thicker than 4 atomic layers. SEMPA measurementsby the NIST group showed that Mn(001) planes are ferromagnetically orderedwith the adjacent atomic layers oriented antiferromagneticaly, thus the mag-netization exhibits 2 ML oscillations [142]. This conclusion is also supportedby spin polarized STM (spSTM) studies [143, 144, 145]. Schlickum et al. [145]found that in regions where Mn overgrows Fe substrate steps a frustration ofthe antiferromagnetic order results in a 180◦ domain wall in the Mn film. Thewidth of the domain wall increases linearly with the Mn layer thickness andreaches 7 nm in a 20 ML thick film of Mn. The ability of spSTM to observethe ferromagnetic order in the top Mn layer is not obvious: it is a weak effect(a few %). Calculations indicate that in ballistic transport the spin currentasymmetry can be attributed mainly to the symmetry breaking at the surface[146].

The exchange coupling in Fe whisker/Mn/Fe(001) was investigated by theNIST group using SEMPA. The SEMPA images indicated that for the firstthree Mn atomic layers the magnetization of the top Fe layer pointed in thesame direction as the Fe whisker magnetic moment, ie. FM coupling. For Mnthicknesses greater than 4 ML the magnetic moment of the Fe film was notcollinear with the magnetic moment of the Fe whisker. Between 4 to 8 layersof Mn the direction of the Fe film magnetic moment lay at an angle of 60–80◦


relative to the magnetization of the Fe whisker. Beginning at the 9th Mn layer,the direction of the coupling oscillated with a two ML period between 90◦−φand 90◦+φ, where φ was sample dependent. Values of φ were found between10 to 30◦. Similar behavior was found by means of MOKE measurementsusing Fe/Mn/Fe wedged samples deposited on a GaAs/Fe/Ag(001) template[147]. For Mn thicknesses greater than 1.2 nm (7 ML) a non-collinear con-figuration (φ=0) was observed. Hysteresis loops in these measurements werewell described by Slonczewski′s proximity magnetism model which accountsfor interface roughness in strong antiferromagnets. The total coupling energyE is given by

E = C+θ2 + C−(θ − π)2 , (4.37)

where θ is the angle between the magnetic moments of the two Fe layers.The strength of such proximity coupling showed weak oscillations having a2 ML period. It should be pointed out that Mn can not be grown with in-terfaces as smooth as Cr, and therefore it is not surprising that Mn spacersprovide an excellent system where the consequences of magnetic frustrationare fully demonstrated. It is known that even small concentrations of Mn inCr results in a strong and commensurate antiferromagnetism [148]. It wastherefore thought to be of interest to investigate the phase and the strengthof the exchange coupling between Fe layers separated by layers of a Cr-Mnalloy. To this end the SFU group attempted to grow Fe/Cr-Mn/Fe structures.Unfortunately we found that the Mn atoms have a very strong tendency tosegregate on the surface of Cr during the growth. It was necessary to maintaina substrate temperature greater than 200 C in order to obtain a good layer bylayer growth. At this temperature the top surface layer contained a stronglyenhanced concentration of Mn (∼50%). In view of this surface segregation,and since the interfaces play a very crucial role in exchange coupling Heinrichet al. [103] decided to grow pure layers of Mn between the Cr and the Felayers. Eleven and twelve ML of Cr were grown on an Fe(001) whisker usinggrowth conditions optimizing layer-by-layer growth. Mn layers were depositedon the Cr at a substrate temperature of 120◦C. The substrate was allowedto cool to room temperatures and 20 ML of Fe(001) were deposited on theMn layer, and a protective layer of 20 ML of Au(001) were deposited on theiron. At 120 C the deposition of the first two atomic layers of Mn proceededin a good layer by layer growth with large RHEED intensity oscillations atthe second anti-Bragg scattering condition. At substrate temperatures wellbelow 100 C the Mn does not segregate on Fe. In this way one is able togrow well defined Fe/Cr/Mn/Fe(001) structures having smooth and abruptinterfaces. It should be pointed out that the top Cr surface atomic layer issmooth, with large atomic terraces corresponding to those of the Fe whisker,and is unaffected by interface alloying during the deposition of the Mn. There-fore variations in the exchange coupling due to the addition of the Mn layersare primarily due to the presence of the Mn atomic layers and their magneticstate. There is no intermediate Cr-Mn mixed region similar to the Cr-Fe mixed

234 B. Heinrich

region that occurs at the Fe-whisker/Cr interface. The following samples werestudied: Fe(001)/11,12Cr/ 1,2,3Mn/20Fe(001)/20Au. This selection of sam-ples allowed one to determine the effect of Mn on the exchange coupling. Thecoupling was found to be AFM for the 11 ML Cr spacer and FM for the 12ML Cr spacer. Therefore the phase of the coupling was only determined bythe thickness of the Cr layer. Assuming that the Mn is an AFM state with anuncompensated magnetic moment in the (001) plane the coupling should havechanged its sign as each Mn atomic layer was added. Such a phase reversalwas not observed. The exchange coupling oscillated with the same phase andthe same periodicity of 2 ML as was observed using pure Cr spacer layers. Theassumption that the magnetic state of Mn in Fe/Cr/ Mn/Fe(001) structurescan be described by commensurate antiferromagnetism with uncompensated(001) planes is not necessarily correct. In recent calculations by Krueger et al.[149] the magnetic structure of bct Mn in bulk was studied as function ofc/a, a measure of the tetragonal distortion. The calculations showed that bctMn grown on Fe(001) is in a magnetic state that is at the border line be-tween the AFM1 configuration, in which the magnetic moments are parallelin the (001) planes, and the AFM3(110) configuration, having ferromagneticplanes oriented along (110), and fully compensated (001) planes having zeronet magnetic moment. The AFM3(110) state would not lead to an alternatingsign of exchange coupling with increasing Mn thickness in agreement with ourexperimental observations.

4.5.5 Loose Spins

Fe atoms can be dispersed in NM interlayers [150, 151] and result in additionalcoupling between two ferromagnetic films. Slonczewski developed a modelincorporating the exchange interaction between the FM layers and “loosespins” inside the NM spacer [152]. He assumed that the exchange field onthe “loose spins” is given by H1,2 = H1,2(z, d− z)m1,2, where m1,2 are unitvectors along the magnetic moments of layers FM1 and FM2 and z, and d− zdetermine the distance of a loose spin from the interfaces. The strength of theexchange field, H1,2(z) can be estimated from the thickness dependence of theinterlayer exchange coupling. The energy levels of a loose spin are given byεm = −Um/S with m=–S, –S+1,...S, where

U(θ) =(U2

1 + U22 + 2U1U2cos(θ)

)1/2. (4.38)

U1,2 are energies associated with the exchange fields H1,2. U is expected tobe weak and the total energy of a loose spin is given by thermal excitations.The total energy is then multiplied by the concentration of loose spins. Thisresults in bilinear and biquadratic contributions. Heinrich et al. [153] andSchaefer et al. [154] investigated the role of loose spins in Fe/Cu/Fe(001) andFe/Ag/Fe(001) structures, respectively, by inserting less than 1 ML of Fe inwell prescribed positions inside the NM spacer. The strengths of U1 and U2


were estimated from the thickness dependence of the interlayer coupling usingpure Cu spacers. We found that the presence of an alloyed Cu-Fe ML inside theCu spacer significantly decreased the strength of the bilinear coupling whileleaving the biquadratic coupling almost unchanged. This decrease in the bi-linear coupling can then result in the coupled Fe magnetic moments becomingoriented in mutually perpendicular directions. Both groups have shown thatthe consideration of loose spins to describe the data requires the inclusion ofclusters of Fe atoms inside the NM. The application of Slonczewski’s theoryin the work by Heinrich et al. [153] required the inclusion of a molecular ex-change field in the alloyed atomic layer in order to account for the measuredtemperature dependence of the bilinear exchange coupling.

Recently the magnetic state of Cr spacers in [Fe/Cr/Fe](001) superlatticeswas investigated by means of perturbed angular correlation (PAC) spectrom-etry [155]. These measurements revealed that above the blocking temperatureof Cr the Cr magnetic moment exhibits superparamagnetic spin fluctuations.In this temperature regime the interlayer coupling in these samples is mostlygiven by the biquadratic contribution. It was suggested [155] that the su-perparamagnetic clusters acted as “loose spins” and were responsible for thepresence of biquadratic exchange coupling. In my view this is a rather strongclaim. “Loose spins” as treated using the Slonczewski model create a strongbilinear coupling term that surpasses the strength of the biquadratic contri-bution. In order to obtain a dominant biquadratic exchange coupling contri-bution one needs to compensate the bilinear coupling contribution by someother mechanism. The absence of interlayer coupling below the Neel orderingtemperature in [Fe/Cr/Fe] superlattices is caused by the nodes in the Cr mag-netic moment around the Fe-Cr interfaces, see the above section on Neutronstudies. In the Cr paramagnetic regime one can get a direct coupling as iscommonly observed in other simple metal spacers. In that case the interfaceroughness will significantly decrease the bilinear exchange coupling strengthin paramagnetic Cr and will introduce biquadratic coupling due to interfacemagnetic frustrations. Most likely the biquadratic coupling strength observedin [Fe/Cr/Fe] superlattices is a combination of all of these contributions.

We found a typical loose spin-like behavior in Co/Cu/Co(001)/Fe andFe/Pd/Fe(001) systems. The sample 4Co/6Cu/4Co(001) grown on Cu(001)was found to be coupled antiferromagnetically. With the addition of a 3 MLfilm of Fe deposited on the top of the second Co layer the top Fe surface devel-oped a strong reconstruction and the exchange coupling became ferromagnetic[38, 57]. Moreover, the temperature dependence of the exchange coupling fol-lowed a Curie-Weiss type of dependence, proportional to 1/T . In our viewthe strong lattice reconstruction resulted in lattice defects and subsequentpenetration of Co atoms into the Cu spacer. These Co atoms then acted as“loose spins” inside the Cu spacer. These loose spins were subjected to theexchange field of the surrounding FM Co layers and their contribution to theoverall exchange coupling scaled with their magnetic moments. The magneticmoments of true loose spins are expected to follow a Curie-Weiss law.

236 B. Heinrich

4.5.6 Lattice Strained Pd in Fe/Pd/Fe(001) Structures

In the Fe/Pd(001) system, a lattice strain results from the 4.2% mismatchbetween the bcc Fe(001) and the fcc Pd(001) surface nets. Ultrathin Pd(001)films grown on Fe(001) are expanded laterally to match the Fe mesh. Thestructure and magnetism of fcc Pd in Fe/Pd/Fe (001) trilayers grown onAg(001) substrates were studied in [156, 157]. Using RHEED and x-ray diffrac-tion, Fullerton et al. [157], have shown that the Pd ultrathin films grew onFe(001) with a 4.2% latterly expanded lattice accompanied by an out-of-planecontraction of 7.2% (c/a=0.89). The ultrathin films of Pd grown on Fe(001)had a pronounced face centered tetragonal (fct) structure. Theoretical ab ini-tio calculations of ultrathin film Pd layers on an Fe(001) template have shownthat the ground state of the lattice strained Pd(001) has a fct structure suchthat the bulk Pd atomic volume is maintained [157]. This theory is in goodagreement with the results of RHEED and x-ray diffraction measurements. Po-larized neutron reflectivity measurements on an Fe(5.6 ML)/Pd(7 ML)/Au(20ML) sample determined the average moment per Fe atom to be 2.66 μB [157].Ab initio spin density calculations for the same structure showed that thisvalue is consistent with the observed induced Pd polarization. It is interestingto point out that a 4.2% lattice expansion of bulk metallic fcc Pd would resultin long range ferromagnetic order. The main conclusion of the magnetic mea-surements and calculations was that the Pd was ferromagnetic only for twoadjacent Pd atomic layers at the Fe/Pd and Pd/Fe interfaces. By increasingthe thickness of the Pd by one additional atomic layer (a total thickness of5 ML) the long range ferromagnetic order in Pd was lost. Clearly the latticevertical relaxation has to be taken into account in order to explain the realmagnetic properties of strained epitaxial structures. The exchange couplingthrough Pd exhibited an oscillatory behavior as a function of the Pd thickness[156]. The period of oscillations was 4 ML. This period is close to the 3 MLperiod corresponding to the large Pd 4d belly Fermi surface sheets [158]. Thetemperature dependence of the coupling for 5 ML thick Pd showed a perfectCurie Weiss dependence. A strong temperature dependence is also observedfor 6 ML Pd. For thicker Pd layers the temperature dependence is weak. Thisagain indicates the presence of fluctuating magnetic moments in the Pd layersin the range of thicknesses from 5 to 6 ML. In my view the above two systems,Co/Cu/Co/Fe(001) and Fe/Pd/Fe(001), represent the best examples of selfassembled loose spins.

4.6 Time Dependent Exchange Coupling

4.6.1 Multilayers

The interlayer exchange coupling had so far no explicit time dependent be-havior. The spin dynamics studies in magnetic FM/NM/FM trilayers revealedthat the coupling between the magnetic layers can also have a purely dynamic


character. The role of interface damping has been investigated in high qualitycrystalline Au/Fe/Au/Fe(001) structures grown on GaAs(001) substrates, seethe details in [30]. The in-plane FMR experiments were carried out using 10,24, and 36 GHz systems [159]. The in-plane resonance fields and resonancelinewidths were measured as a function of the azimuthal angle ϕ between theexternal dc magnetic field and the Fe in-plane cubic axes.

Single Fe ultrathin films with thicknesses of 8, 11, 16, 21, and 31 mono-layers (ML) were grown directly on GaAs(001). They were covered by a 20ML thick Au(001) cap layer for protection under ambient conditions. FMRmeasurements were used to determine the in-plane four-fold and uniaxial mag-netic anisotropies, K‖

1,eff and K‖u,eff , and the effective demagnetizing field

perpendicular to the film surface, 4π Meff (=4πMs-2K⊥u,s/Ms), as a function

of the film thickness d [30, 160]. The magnetic anisotropies were well describedby a linear dependance on 1/d. The constant and linear terms represent thebulk and interface magnetic properties, respectively. The ultrathin Fe filmsgrown on GaAs(001) have magnetic properties nearly equal to those of bulkFe, modified only by sharply defined interface anisotropies; this indicates thatthe Fe layers are of a high crystalline quality with well defined interfaces.The lineshapes of the FMR peaks are Lorentzian and the FMR linewidthsare small (less than 100 Oe for our microwave frequencies) and only weaklydependent on the film thickness. The reproducible magnetic anisotropies andsmall FMR linewidths provided an excellent opportunity for the investigationof non-local relaxation processes in magnetic multilayer films.

The ultrathin Fe films which were studied in the single layer structureswere regrown as a part of magnetic double layer structures. The thin Fe film(F1) was separated from the second thicker layer (F2), 40 ML thick, by a 40ML thick Au spacer. The magnetic double layers were covered by a 20 MLAu(001) layer for protection under ambient conditions. The thickness of theAu spacer layer was much smaller than the electron mean free path in gold(38 nm) [161], and hence ballistic spin transfer between the magnetic layersis allowed.

The interface magnetic anisotropies separated the FMR fields of F1 andF2 by a large margin ( 1 kOe), see [30]. That separation allowed us to carryout FMR measurements on F1 without exciting a large response in F2: theangle of precession in F2 was negligible compared to that in F1. The FMRlinewidths in single and double layer structures were only weakly dependenton the azimuthal angle ϕ of the saturation magnetization with respect to thein-plane crystallographic axes.

The 16Fe film (F1) in the single and double layer structures had the sameFMR field showing that the interlayer exchange coupling [6] through the 40ML thick Au spacer was negligible, and the magnetic properties of the Fe filmsgrown by MBE on well prepared GaAs(001) substrates were fully reproducible.

The FMR linewidth in the thin films always increased in the presence ofthe thick layer F2. The additional FMR linewidth, ΔHadd, followed an inverse

238 B. Heinrich

Fig. 4.27. The FMR linewidth measurements (half width half maximum (HWHM))in the parallel configuration. (a) The dependence of the additional FMR linewidthΔHadd = ΔHd − ΔHs on 1/d at f = 36 GHz. ΔHd and ΔHs represent the FMRlinewidths for the Fe films in the double and single layer magnetic structures, re-spectively. d is the thickness of the F1 layer in a GaAs/nFe/40 Au/40Fe/20Au(001)structure. The integers in the structure notation represent the number of atomic lay-ers. (b) The frequency dependence of the FMR linewidth in the 16 ML Fe layer: ΔHd

(•) in a GaAs/16Fe/40Au/40Fe/20Au(001) structure; ΔHs (◦) in a single magneticlayer GaAs/16 Fe/20Au(001) structure; and the additional FMR linewidth ΔHadd

is shown in ()

dependence on the thin film thickness d, see Fig. 4.27a. The non-local dampingoriginates at the film interface (F1/Au). The linear dependence of ΔHadd onthe microwave frequency for both the parallel and perpendicular configurationwith negligible zero-frequency offset, see Fig. 4.27b is equally important. Thismeans that the additional contribution to the FMR linewidth can be describedby an interface Gilbert damping. The additional Gilbert damping for the 16Fefilm was found to be weakly dependent on the crystallographic direction,Gadd = 1.2 × 108 s−1 along a cubic axis.

Discussion of the interface torque

Tserkovnyak, Brataas and Bauer [162, 163] showed that interface dampingcan be generated by a spin current flowing from a ferromagnet into adjacentnormal metallic layers. The spin current is generated by a precessing magneticmoment in F1. The spin current was calculated using Brouwer’s scatteringmatrix [164] which evolves under a time dependent parameter (phase angleof precession). Normal metal (NM) layers surrounding a magnetic layer wereviewed as reservoirs in common thermal equilibrium as result of contact withan infinite thermal bath. The calculations were carried out assuming that thereservoirs acted as ideal spin sinks. The resulting spin current per unit areais given by

jspin =�

4π

(ReA↑↓n× ∂n

∂t+ ImA↑↓ ∂n

∂t

), (4.39)


where n is a unit vector along the magnetic moment M, and A↑↓ is the complexspin-pumping conductance per unit area given by the difference between thereflection (gr

↑↓) and transmission (gt↑↓) mixing conductances per unit area,

A↑↓ = gr↑↓ − gt

↑↓ . (4.40)

The spin dependent reflection and transmission matrices for the ferromagneticfilm are given by

gr↑↓ =

∑m,n

(δm,n − r↑m,nr

↓∗m,n

)

gt↑↓ =

∑m,n

t↑m,nt↓∗m,n , (4.41)

where r↑↓m,n are the reflection parameters for spin up and spin down elec-trons in the NM reservoirs, and t↑↓m,n are the transmission parameters into thereservoirs. The indices m and n in (4.40) label the modes (channels) corre-sponding to k‖,⊥ wave-vectors (parallel and perpendicular to the interface)at the Fermi energy. The Gilbert damping is given by the conservation of thetotal spin momentum per unit area

jspin − 1γ∂Mtot

∂t= 0 , (4.42)

where Mtot is the total magnetic moment of F1. After simple algebraic stepsone obtains an expression for the dimensionless damping parameter

α =G

γ1Ms

=(

αbulk +gμB

4πMs

1d1Re(A↑↓)

)

1γeff

=1γ

(1 − gμB

4πMs

1d1Im(A↑↓)

), (4.43)

where d1 is the thickness of the ferromagnetic layer F1. The imaginary partof A↑↓ arises from the

∑m,n r

↑m,nr

↓∗m,n and

∑m,n t

↑m,nt

↓∗m,n sums and is very

close to zero due to cancellation of phases. Therefore spin-pumping mostlyeffects the damping, the renormalization of the gyromagnetic ratio γ due tospin pumping is very small. The inverse dependence of the Gilbert dampingon the film thickness clearly testifies to its interfacial origin. The layer F1 actsas a spin pump. Now another important point has to be answered: how isthe generated spin current dissipated? This answer can be found in the recentarticle by Stiles and Zangwill in Anatomy of spin transfer torque [165], seeSect. 4.4.1. They showed that the transverse component of the spin currentin a normal layer (NM) is entirely absorbed at the NM/FM interface. Forsmall precessional angles the spin current jspin is almost entirely transverse.This means that the N/F2 interface acts as an ideal spin sink, and providesan effective spin brake for F1, see Fig. 4.28(a). For ferromagnetic layers that

240 B. Heinrich

Fig. 4.28. A cartoon representing the dynamic coupling between two magneticlayers which are separated by a non-magnetic spacer. (a) represents two magneticlayers with different FMR fields. F1 is at resonance, and F2 is nearly stationary. Abow like arrow in the normal spacer describes the direction of the spin current. Thedashed line represents the instantaneous direction of the spin momentum. F1 actsas a spin pump, F2 acts as a spin sink, and consequently F1 acquires an additionalGilbert damping. (b) represents a situation when F1 and F2 resonate at the samefield. Both layers act as spin pumps and spin sinks. In this case the net spin mo-mentum transfer across each interface is zero. No additional damping is present asthe precession is in phase

are thicker than the spin lateral coherence length( 2–4 A), and the electronscattering at the interfaces is partly diffuse, the coefficient A↑↓ is nearly equalto the number of transverse channels in the normal metal NM,

∑m,n δm,n,

see [166], [167], [168]. In simple metals this sum per unit area is given by

A↑↓ =k2

F

4π= 0.85n2/3 , (4.44)

where kF is the Fermi wavevector and n is the density of electrons per spinin the normal metal N. The spin current has the form of Gilbert damping. Itcan be shown that the coefficient A↑↓ is proportional to the interface mixingconductance, A↑↓ � h

e2 gcond↑↓ , see [166], [167].

The layers F1 and F2 act as mutual spin pumps and spin sinks. Theequation of motion for F1 can be written as

1γ∂M1

∂t= − [M1 × Heff,1] +

G1

γ2M2s,1

[M1 × ∂M1

∂t

]

+�

4πd1g↑↓,1n1 × ∂n1

∂t− �

4πd1g↑↓,2n2 × ∂n2

∂t, (4.45)


where g↑↓,1,2 are the real parts of A↑↓,1,2 for the layers F1 and F2. M1 is themagnetization vector of F1, n1,2 are the unit vectors along M1,2, and d1,d2

are the thicknesses of layers F1 and F2. The exchange of spin currents is asymmetric concept and the equation of motion for the layer F2 is obtained byinterchanging the indices 1 ⇔ 2.

Equation (4.45) can be tested by investigating the FMR linewidth aroundan accidental crossover of the resonance fields for F1 and F2 [169]. In thiscase the resonant field of F1 approaches the resonant field of F2. When theyreach the same resonant field the rf magnetization components of F1 and F2are parallel with each other. Each precessing magnetization creates its ownspin current which is pumped across the NM spacer. The electron mean freepath in Au thick films is 38 nm [161], and consequently the spin transportis purely ballistic even in an 80 ML thick Au spacer. At the same time bothinterfaces F1/NM and NM/F2 act as spin sinks. It follows that the net flowof the spin current through each interface can be zero, and the contributionof spin pumping to the FMR linewidth can disappear, see Fig. 4.28(b). Thebulk Gilbert damping in F1 and F2 were very nearly equal which resulted inthe marked disappearance of the additional FMR linewidth at the crossoverof the resonance fields, see Fig. 4.29. The good agreement between theoryand experiment clearly shows the validity of the spin pumping theory basedon (4.45). Even in the absence of static interlayer exchange coupling the mag-netic layers are coupled by dynamic interlayer exchange. The increase in theadditional FMR linewidth for 16Fe in Fig. 4.29 which appears for angles nearthe accidental crossover in the resonance fields shows that the spin pumpingeffect can be enhanced when the rf magnetic moments are partly out of phase.This effect is present in ferromagnetic films that are exchange coupled. In theexchange coupled case the optical mode exhibits a larger linewidth than thatexpected from simple spin pumping in which the spin sink has a negligibleprecessional amplitude, see [170, 171].

The quantitative comparison between experiment and spin pumping the-ory is very good [170]. First principles electron band calculations [168] resultedin g↑↓ ≈1.1×1015cm−2 for a clean Cu/Co(111) interface. By scaling this valueto Au using (4.44) one obtains a value for the Gilbert damping parameterGsp,cal =1.4×108s−1 for a 16 ML thick Fe film; this value is very close tothe experimentally observed value Gsp,exp =1.2×108s−1 measured at RT us-ing FMR. This agreement is amazing considering the fact that calculationsof the intrinsic damping in bulk metals have been carried out over the lastthree decades and have not been able to produce a comparable agreementwith experiment.

The spin pumping can be also found in single Fe films surrounded bynormal metal layers provided that the spin current diffuses away from theFM/NM interface. Interface damping was observed in NM/Py/NM sand-wiches by Mizukami et al. [172] where NM=Pt,Pd and Ta non-magnetic layerswere surrounding a permalloy (Py) layer.

242 B. Heinrich

Fig. 4.29. The FMR field and linewidth at 24 GHz as a function ofthe angle ϕ of the applied dc field. The measurements were done onGaAs/16Fe/40Au/40Fe/20Au(001), where the integers represent the number ofatomic layers. The upper Figure shows the FMR fields, the symbols () correspondto 40Fe and (◦) symbols to 16Fe. Notice that the 16Fe film grown on GaAs(001)exhibits a strong in-plane uniaxial anisotropy. The presence of the in-plane uniaxialanisotropy allows one to get an accidental crossover of the resonance fields at theangle ϕ=112◦. The lower Figure shows the FMR linewidth. The solid lines wereobtained from calculations using (4.45). The symbols (•) correspond to F1 (16Fe)and () correspond to F2(40 ML). Note that the FMR linewidth for the thinnersample, 16Fe, first increases before it reaches its minimum value corresponding toits single GaAs/16Fe/20Au(001) layer structure. Note also that the additional linebroadening due spin pumping scales inversely with the film thickness

Spin pumping allows one to take a new look at the new field of spintronics.One can in principle move information by means of a spin current at frequen-cies in the GHz range via a mechanism that does not directly involve a nettransport of electron charge. This potentially represents an approach to elec-tronics that is truly different from that employed using semiconductors.

The spin pump model is a rather exotic theory for those who are usedto magnetic studies. One would expect that there should be a more di-rect connection with common concepts used to understand the behavior


of magnetic multilayers. The obvious choice would be a generalization ofinterlayer exchange coupling. One would expect that a dynamic part of ex-change coupling could create magnetic damping. A ferromagnetic sheet sur-rounded by a normal metal can be investigated using a contact exchangeinteraction between the ferromagnetic spins and the electrons in the normalmetal [173, 174]. A similar model was used by Yafet [175] in order to calculatethe static interlayer coupling. One can extend Kubo’s linear response theory[176] in order to treat a slow precessional motion using a linear approximationfor the retarded magnetic moment,

S(t− τ) ∼= S(t) − τ∂S(t)∂t

, (4.46)

where S(t) is the spin moment of the magnetic sheet at the instantaneoustime t and τ is the time delay of the retarded response. The induced momentin the N metal at the F/N interface results in an effective damping field whichis proportional to the imaginary part of the rf transverse susceptibility of Nand the time derivative of the magnetic moment

Hsddamp ∼

[∂

∂ω

∫ ∞

−∞

dq

2πImχ(q, ω)

]

ω→0

dM(t)dt

. (4.47)

This damping term also satisfies the phenomenology of Gilbert damping. Byusing the same interaction potential it was shown [173, 174] that the Gilbertdamping in dynamic interlayer exchange coupling, Gs−d, is identical to thatcalculated using the spin-pumping theory [162] combined with a perfect spinsink. This leads to an important conclusion: The spin pumping theory is di-rectly related to the dynamic response of the interlayer exchange coupling.

Acknowledgements

The author would like to thank his colleagues Professor J.F. Cochran, Mr.B. Kardasz, and Mr. O. Mosendz for stimulating discussions and invaluablehelp in the preparation of this Chapter. I also owe a debt of gratitude to allthose with whom I have held extensive discussions and who provided valuablehelp during the preparation of this manuscript: Dr. M. Stiles, Dr. J. Unguris,Professor Z. Qiu, Dr. E. Fullerton, Professor H. Zabel, and Referees of thisChapter. Without this help and these discussions the Chapter would have beenless complete. I would like to thank Professor Dr. M. Stiles (Figs. 4.8, 4.9, 4.10and 4.21), Dr. J. Unguris (Figs. 4.18, 4.19, 4.20 and 4.23), Professor Z. Qiu(Figs. 4.15, 4.16 and 4.17), Dr. G. Woltersdorf (Fig. 4.1), and Prof. P. Bruno(Fig. 4.14) for allowing me to include their Figures in this chapter. I thankDr. Stiles for allowing me to include some of his presentation which beautifullydemonstrates theory of interlayer exchange coupling (section Quantum in-terference). The field of magnetic coupling is a very diversified field covering

244 B. Heinrich

at least 2–3 decades. It is not possible for a single Chapter to describe all thecomplexities of this field. I apologize to those who contributed to this fieldand whose results are not explicitly included in this chapter. The omissionis not intentional but merely reflects my inability to include all studies in aChapter that has to be constrained to a reasonable length.

I would like to thank especially the Natural Sciences and Engineering Re-search Council of Canada (NSERC), and the Canadian Institute for AdvancedResearch (CIAR) for continued research funding which makes my work pos-sible. I would also like to express my thanks to the Alexander von HumboldtFoundation and Professor J. Kirschner, Max Planck Institute in Halle, for pro-viding me with generous support during my recent summer research semestersin Germany where this manuscript was partly prepared.

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