02 - Magnetic Anisotropy of Heterostructures (Jürgen Lindner)

8/20/2019 02 - Magnetic Anisotropy of Heterostructures (Jürgen Lindner)

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Magnetic Anisotropy of Heterostructures

Jürgen Lindner and Michael Farle

Fachbereich Physik and Center for Nanointegration, Universität Duisburg-Essen,Lotharstr. 1, 47048 Duisburg, [email protected]

Abstract. The chapter provides a detailed introduction to magnetic anisotropy of ferromagnetic ultrathin films and its analysis by ferromagnetic resonance on a tuto-rial level. While the microscopic origins of the magnetic anisotropy as well as recentdevelopments in its theoretical description are shortly discussed, emphasis is put ona phenomenological description using the free energy of the system together withits symmetries. The formalism is used to describe ferromagnetic resonance exper-iments which present an extremely sensitive method to experimentally investigatemagnetic anisotropy in thin film heterostructures. Expressions for the free energyand the resonance equations are derived for the most widely used crystal symmetries

such as cubic, tetragonal and hexagonal. The general equations are illustrated bygiving selected examples of current research on thin metallic films on diff erent kindsof substrates (MgO, GaAs and Cu).

2.1 Introduction

Magnetic thin films have provided a highly successful test ground for un-derstanding the microscopic mechanisms which determine macroscopicallyobservable quantitities like the magnetization vector, diff erent types of mag-netic order (ferro-, ferri- and antiferromagnetism), magnetic anisotropy andordering temperatures (Curie, Néel temperature). The success has beenbased on the simultaneous development of the following techniques: a) thepreparation of single-crystalline mono- and multilayers on diff erent types of substrates in ultrahigh vacuum systems, b) the development of vacuum com-patible, monolayer-sensitive magnetic analysis techniques, c) the advance incomputing power to provide first-principles calculations of magnetic groundstate properties [1]. Aside from these basic research orientated investigationsthe technological exploitation of thin film magnetism has lead to huge in-

creases in the hard disk’s magnetic data storage capacities [2] and new typesof magneto-resistive angle and position sensitive sensors in the automotiveindustry, for example.

J. Lindner and M. Farle: Magnetic Anisotropy of Heterostructures, STMP 227, 45–96 (2007)

DOI 10.1007/978-3-540-73462-8 2 c Springer-Verlag Berlin Heidelberg 2007


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46 J. Lindner and M. Farle

The purpose of this article is twofold: a) an introductory level overview onmagnetic anisotropy, b) characteristic examples of current research using mag-netic resonance techniques to quantitatively determine magnetic anisotropyand explore its microscopic sources.

Various aspects of ultrathin film magnetism [3] have been discussed in ex-tensive reviews and book chapters over the last few years (see for example[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). There is no way to summarize allthese issues in such limited space and the reader is referred to the reviewsmentioned before. For an overview on the technological relevance or the manyexperimental techniques and methods that have been developed to investi-gate magnetic heterostructures the reader is referred to the series of booksby Heinrich and Bland [6, 7, 8]. This review also excludes laterally structuredsamples (for excellent reviews see e.g. [9, 10, 11]) and epitaxially grown films

comprised of two or more elements (double- tri- and multilayers, see the articleof B. Heinrich in this book or [6, 7, 8, 14]) in which coupling eff ects lead tophenomena like the tunneling magneto-resistance (TMR), the giant magneto-resistance (GMR) or spin current related eff ects (see e.g. [16, 17, 18] for adetailed discussion) such as current induced switching [19], current induceddomain wall movement or spin torque induced magnetic damping [20, 21, 22].The examples which will be discussed here are strictly restricted to the thick-ness and temperature dependent magnetic anisotropy of single element ferro-magnetic metallic monolayers on diff erent kind of single crystalline substrates(metals, semiconductors and insulators). It will be shown that epitaxial filmsconsisting of few atomic layers provide an interesting playground for artifi-cially controlling magnetic properties and hence improving the understandingof the underlying physical mechanisms.This chapter is divided into two basic sections. While the first will shortly dis-cuss the sources of magnetic anisotropy energy (MAE), explain FerromagneticResonance (FMR) and introduce a phenomenological description of the MAEand its influence on the FMR resonance equations in terms of the magneticpart of the free energy of the system, the second section will give examples of FMR investigations of heterostructures. In the framework of a tutorial descrip-

tion, the prototype systems Fe/MgO(001), Fe/GaAs(001) and Ni/Cu(001)were chosen.

2.2 Origin of Magnetic Anisotropy

Magnetic anisotropy describes the fact that the energy of the ground state of a magnetic system depends on the direction of the magnetization. The eff ectoccurs either by rotations of the magnetization vector with respect to the

external shape of the specimen (shape anisotropy) or by rotations relative tothe crystallographic axes (intrinsic or magneto-crystalline anisotropy). Thedirection(s) with minimum energy, i.e. into which the magnetization pointsin the absence of external fields are called easy directions. The direction(s)


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2 Magnetic Anisotropy of Heterostructures 47

with maximum energy are called hard direction. The MAE between twocrystallographic directions is given by the work W MAE needed to rotatethe magnetization from an easy direction into the other direction. TheMAE is a small contribution on the order of a few µeV/atom to the to-

tal energy (several eV/atom) of a bulk crystal. To estimate the magnitudeof the MAE one can use as a rule of thumb, that the lower the sym-metry of the crystal or of the local electrostatic potential (crystal or lig-and field) around a magnetic moment, the larger the MAE is. This be-comes evident, if one remembers that in a crystal field of cubic symmetrythe orbital magnetic moment is completely quenched in first approximation[23]. Only by calculating in higher order (2nd) or by allowing a slight dis-tortion of the cubic crystal a small orbital magnetic moment, i.e. a non-vanishing expectation value of the orbital momentum’s z component is re-

covered. Without the presence of the orbital momentum which couples thespin degrees of freedom to the spatial degrees of freedom the MAE wouldbe zero, since the exchange interaction is isotropic. One should also note,that the easy axis can deviate from crystallographic directions as for ex-ample in the case of Gd whose easy axis is temperature dependent andlies between the c-axis and the basal plane at T = 0 K [24]. Table 2.1gives an overview about easy and hard axes and on the magnitude of theMAE for some elementary ferromagnetic materials with diff erent crystalsymmetry.There are fundamentally two sources of magnetic anisotropy: (i) spin-orbit(LS)interaction and (ii) the magnetic dipole-dipole interaction. From the pointof view of quantummechanics both interactions are relativistic correctionsto the Hamilton-operator of the system that lift the rotational invarianceof the quantization axis. Despite the fact that the dipole-dipole interactionas well as the LS coupling are much weaker than the exchange interaction(≈ 1 − 100 µeV/atom compared to ≈ 0.1 eV/atom), they link the magnetic

Table 2.1. Anisotropic orbital moments, direction of easy axis of the magnetization,

and magnetic anisotropy energy at T = 0 K for the four elemental ferromagnets astaken from standard references like Landolt-Börnstein [32]. ∆µL is the diff erence of the orbital magnetic moment measured along the easy and hardest magnetizationdirection. µtot is the sum of orbital and eff ective spin moment. The latter includesthe so-called T z contribution entering the sum rule analysis of x-ray magneticcircular dichroism measurements

Material Easy axis |MAE| (µeV/atom) ∆µL/µtot

bcc Fe 100 1.4 1.7 · 10−4hcp Co

0001

65 4.5 · 10−4

fcc Co 111 1.8fcc Ni 111 2.7 1.8 · 10−4hcp Gd tilted hcp 50 ≈ 10−3


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moments to position space. In the following we will use the term magneticanisotropy energy or MAE for the magnetic anisotropy energy density, re-sulting from spin-orbit interaction which includes the so-called magneto-crystalline and also the magneto-elastic contributions [15]. Contributions orig-

inating from the magnetic dipole-dipole interaction will be called shape ormagneto-static anisotropy.We note that the important aspect of magnetic domain formation is not dis-cussed here, since we restrict our discussion to the intrinsic contributions tothe magnetic anisotropy and the main technique which will be discussed indetail in this chapter is ferromagnetic resonance (FMR). This technique is –in most cases – performed in large enough fields that drive the magnetic filminto a single domain state. The reader should keep in mind, however, that theinterplay of magnetostatic energy, exchange energy and magnetic anisotropy

in general leads to an energetically favored multi-domain state. Especially,the analysis of hysteresis loops is complicated due to domain formation atsmall magnetic fields. Special imaging techniques have been developed [25] toobtain quantitative understanding of the many domain configurations. Due tothe single-domain description used in this chapter, aspects of configurationalmagnetic anisotropy [26] which appear in submicron sized nanomagnets dueto small deviations from the uniform state will not be discussed. Similarly,so-called exchange anisotropies which may arise from diff erent exchange cou-pling constants along diff erent crystallographic directions in a crystal will notbe specifically addressed, since the experimental observations can be well de-scribed by the phenomenological approach presented in Sect. 2.3.2. Also adiscussion of unidirectional anisotropy or exchange bias is beyond the scopeof this chapter. Excellent overviews can be found in [27, 28, 29, 30]. Finally, wenote that the aspect of a non- homogeneous magnetization across the thick-ness of a several nanometer thick film does not enter the following discussion.An excellent overview on the magnetization profile across a thin film and itsdependence on the film’s morphology has been recently given by Jensen andBennemann [31].

2.2.1 Spin-orbit Interaction

In a classical picture the orbital motion of the electrons in a perfect crystal isdefined by the potential that is predetermined by the crystal lattice. In casethat there is an interaction between the orbital motion and the spin of theelectrons (i.e. when spin-orbit interaction is present), the spins and thus themagnetization become coupled to the lattice. By using perturbation theory inwhich the LS coupling is described as perturbation of the exchange splittingBruno [33] showed that the energy correction and the orbital moment of the

minority spins are related as ∆E LS ∝ − 14ξ Ŝ ·L↓. Here ξ is the radial part of the spin-orbit interaction, Ŝ is the unit vector along the spin direction, deter-mining the magnetization direction and L↓ the orbital angular momentum of


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the minority spin band, so that Ŝ ·L↓ is the projection of L↓ on the magneti-zation direction. The magnetic anisotropy energy for uniaxial symmetry, forwhich the energy diff ers between the directions parallel () and perpendicular(⊥) to one anisotropy axis, is given by the anisotropy of the orbital angularmomentum ∆L↓ = L↓⊥ −L↓ (or, respectively, the anisotropy of the orbitalmoment ∆µ↓L = µ

↓⊥L − µ↓ L ):

MAE = ∆E ⊥LS −∆E LS ∝ −ξ

4∆L

↓ = − ξ 4µB

∆µ↓L . (2.1)

According to Bruno the easy axis is the one, where the orbital moment islargest. This fact which is experimentally often overlooked yields informationon the intrinsic origin of the macroscopically measured MAE by straight-forward SQUID magnetometry measurements along diff erent crystallographicaxes. The saturation magnetizations along the easy and hard axes are dif-ferent! The eff ect is very small, in bulk crystals–on the order of 10−4, butmeasurable, and well documented (see Table 2.1 and [34]). Here, one shouldnote that shape anisotropy is not involved. That is to say, that when takingthe shape anisotropy (see Sect. 2.3.2) into account, the equilibrium (easy)direction of magnetization in zero field may be a hard magnetocrystallineanisotropy direction with the smaller orbital moment. While Bruno assumeda fully occupied majority spin band in his model (exchange splitting muchlarger than the bandwidth), this restriction was dropped in the later work of van der Laan [35], who extended Bruno’s relation by including the majority

spin band orbital moment µ↑L:

MAE ∝ − ξ 4µB

∆µ

↓L −∆µ↑L

. (2.2)

Although approaches that employ perturbation theory have the advantageof being less complex, they often yield wrong results on a quantitative ba-sis (in most cases too large values). Ab initio theories that consider the LScoupling within a fully relativistic ansatz lead to a clear improvement. Theyare, however, much more elaborate as the precision of the calculation of the

total energy of the system has to be very high. The reason is that the over-all total energy is of the order of 1 eV/atom, while the MAE is very muchsmaller and of the order of several µeV/atom. Nevertheless, a considerableprogress on ab-initio calculations of the MAE was achieved within the last10−15 years (see e.g. [31, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]).The correlation between the anisotropy of the orbital angular momentum andthe MAE becomes strikingly evident in experimental results on ultrathin films[50] and few atom nanostructures of Co [51], for which orbital anisotropies upto ∆µL/µL ≈ 20% have been experimentally confirmed. In general, however,one should note that a direct proportionality between ∆µL and the MAE isonly correct for ∆µL → 0 [41].


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2.2.2 Dipole-Dipole Interaction

The magnetic field produced by a dipole µi at the position ri is given by:

H i (ri) = 3 (ri · µi) · ri

r5i − µi

r3i . (2.3)

Due to this field a second dipole µj in the distance rij with respect to thefirst has the energy E Dip = −µj ·H i:

E Dip = µi · µj

r3ij− 3 (rij · µi) · (rij · µj)

r5ij. (2.4)

Since the dipoles are placed on periodic positions within a crystal lattice, the

axis rij connecting the two dipoles is linked to the crystallographic directionsand, in fact, the interaction energy is connected to the relative orientation of the crystallographic axes and the direction of the magnetic moments. This inturn leads to magnetic anisotropy.

2.3 Models of Magnetic Anisotropy

Phenomenologically, the crystallographic easy axis of the magnetization is

determined by the minimum of the free energy F 1

. Before the explicit ex-pressions of F for various crystal symmetries are discussed, the microscopicorigins that explain magnetic anisotropy will be shortly described.

2.3.1 Single Ion Anisotropy

The single ion anisotropy is determined by the interaction between the orbitalstate of a magnetic ion and the surrounding crystalline field, when the crystalfield is very strong. The anisotropy is the result of the quenching of the orbital

moment by the crystalline field. As this field has the symmetry of the crystallattice, the orbital moments can be strongly coupled to the lattice. This inter-action is transferred to the spin moments via the spin-orbit coupling, giving aweaker electron coupling of the spins to the crystal lattice. When an externalfield is applied the orbital moments may remain coupled to the lattice whilstthe spins are more free to turn. The magnetic energy depends on the orienta-tion of the magnetization relative to the crystal axes.

1 In the chapter by B. Heinrich in this book, the Gibbs free energy U is used insteadof F . As long as no external energy contributions are incorporated into F , thefree energy is the appropriate thermodynamic potential. Upon inclusion of theZeeman energy of the external field, however, U is the relevant potential. However,to avoid confusion, we will use the term F throughout the whole chapter.


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In a magnetic layer, the single-ion anisotropy is present throughout its volume,and contributes in general to the volume part of the MAE. In transition met-als this contribution is usually much smaller than the shape anisotropy butcan be comparable in magnitude in rare earth metals. However, in some cases

also for 3d metals or alloys with small magnetization (and consequently shapeanisotropy) the single ion contribution might overcome the shape anisotropy,leading to perpendicular magnetic anisotropy, for which the easy axis of mag-netization is aligned perpendicular to the film surface.The single-ion anisotropy can also contribute to the surface anisotropy viaNéel interface anisotropy [52], where the reduced symmetry at the interfacestrongly modifies the anisotropy at the interface compared to the rest of thelayer [53]. One may question the relevance of this purely phenomenological ap-proach, which has also been extended in terms of two-ion anisotropy contribu-

tions. The importance lies in the fact that a simple model for the temperaturedependence of the MAE and its correlation to the temperature dependenceof the magnetization can be established. More recently, Mryasov et al. [54]have put this model on solid ground by showing that a model of magneticinteractions on the basis of first-principles calculations of non-collinear mag-netic configurations in FePt eff ectively contains the observed single-ion andtwo-ion contributions and explains the observed unusual scaling exponent Γbetween the magnetization and the MAE (K (T ) ∝ M (T )Γ, see also Sect. 2.4).Later on, in Sect. 2.5, we discuss that for ultrathin Fe films a single-ion model(Γ = 3) yields an excellent explanation for the measured correlation of M (T )and K (T ).

2.3.2 Free Energy Density

As stated above the magnetic anisotropy energy is the work W MAE neededto rotate the magnetization between two diff erent directions. If this rotationis performed at constant temperature T , the MAE is given by the diff erenceof the free energy F of the system with the magnetization pointing alongthe two directions. This is easy to see when one considers that for a closed

system (no exchange of particles) dF = −dW − S dT , S being the entropy, atconstant T reduces to dF = −dW . Setting dW ≡ dW MAE this in turn yieldsF 2 − F 1 =

21

dW MAE = MAE, where 1 and 2 denote the initial (e.g. easy)and the final direction of the magnetization. Provided that an expression forthe magnetic part of F is given for the system under consideration, it can beused to interpret the FMR data on thin magnetic films. The phenomenologicalexpression for F is usually found by symmetry considerations. In the followingwe will summarize the expressions for F of the most often used symmetries.We discuss cubic, tetragonal as well as hexagonal symmetry as these are widely

found in thin film systems. As special case of hexagonal symmetry the uniaxialone is introduced, which in form of shape anisotropy always occurs in thin filmsand which due to stray field minimization favors an easy in-plane alignment of the magnetization. We will use anisotropy constants having suffixes according


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to the symmetry they describe, e.g. K 2 for uniaxial (second order) and K 4for cubic symmetry. This seems for us to be more transparent than to justnumber the constants in a row. Note that the latter numbering is used by otherauthors, so that care has to be taken when comparing results of anisotropy

constants. A first order cubic anisotropy constant (denoted as K 4 within ournomenclature) is often denoted as K 1 by other authors.An expression for the anisotropy field follows from considering the torqueexerted on the magnetization by the eff ective magnetic field within the sample.We assume for simplicity that the free energy of the system only dependson the angle θ of the magnetization with respect to an anisotropy axis. If θB is the angle of the external field relative to this axis, the free energycan be written as F = F a − M · B cos(θ − θB), where the first term is theanisotropy energy and the second the Zeeman contribution of the external

field. The equilibrium angle of M can be found from

∂ F

∂ θ = 0 =

∂ F a

∂ θ + M ·B sin(θ − θB) = ∂ F a

∂ θ + |M ×B|. This equation means that in equilibrium

the torque M ×B due to the external field is balanced by the torque due tothe magnetic anisotropy field given by −∂ F a

∂ θ (the opposite sign indicates that

the torques are antiparallel). When B causes a turn of M of δθ, the torquedue to the anisotropy field is proportional to δθ and given by −∂ F a

∂ θ = c · δθ.

Thus, for δθ → 0 c = −∂ 2F a/∂ θ2δθ=0 and the equilibrium condition becomes

c ·δθ+ M ·B sin δθ ≈ − ∂ 2F a/∂ θ2

δθ=0·δθ+ M ·Bδθ = 0. From this equation

the anisotropy field is found to be

Ba = − 1M

· ∂ 2F a

∂ θ2

δθ=0

. (2.5)

Note that the derivative has to be taken at the equilibrium angle, for whichδθ = 0.It is a quite common though not the only possibility to expand the free energyof a magnetically saturated single crystal (i.e. no domain walls in the crystal)as a series of the direction cosines αi of the magnetization vector relative to

a rectangular Cartesian system of coordinate axes. The direction cosines arethe projections of M onto the three unit vectors defining the crystal latticeand given by αi = M /M · ei (i = 1, 2, 3) where the ei are the unit vectors.According to Birss one can write [55]:

F = biαi + bijαiαj + bijkαiαjαk + bijklαiαjαkαl + . . . . (2.6)

This series is a direct consequence of Neumann’s principle stating that anytype of symmetry which is exhibited by the point group of the crystal, i.e.by the group of symmetry operations that describe the symmetry of the

unit cell of the crystal, is possesed by every physical property tensor. Thus,the limitations of crystal symmetry must be reflected by the tensors bijk....Note, that the higher order terms make F to oscillate rapidly with the an-gular orientation of the direction of magnetization. Since this is contray to


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experimental observations the higher order terms must be very small. Thecomponents of the tensors bijk... transform under a rotation of coordinateaxes according to the relations bijk...n = lipljq lkr . . . lnub pqr...u (in many casesthis transformation of a tensor is used as definition of the tensor as a phys-

ical object). Note that in the equation the Einstein notation is used, i.e.when a letter occurs as a suffix twice in the same term on one side of theequation, summation with respect to that suffix is to be understood, i.e.bijk...n =

3 p=1

3q=1

3r=1 . . .

3u=1 lipljq lkr . . . lnub pqr...u. The matrix [lip]

describes the symmetry operation. As special case the equation contains thetransformation of a vector given by bi =

3 p=1 lipb p = lipb p. As example a

right-handed rotation of 180◦ about the z-axis is described by the matrix

[lip] = cos 180◦ 0 0

0 cos 180◦ 00 0 cos 0◦

= −1 0 0

0

−1 0

0 0 1 . (2.7)

With this it follows that the requirement that bijk... is a property tensorand invariant under all permissible symmetry operation appropriate to theparticular crystal class is equivalent to the requirement that the componentsbijk...n satisfy the set of equations:

bijk...n = σipσjqσkr . . .σnub pqr...u . (2.8)

All the matrices [σ] correspond to permissible symmetry operations. It can be

shown that there are only 9 so-called generating matrices that are needed todescribe all crystal classes, i.e. all symmetry operations of the point groups canbe described by these matrices and multiplications of them. The matrices are:

σ(unit)

=

1 0 00 1 0

0 0 1

σ(inv) =

−1 0 00 −1 0

0 0 −1

σ(2⊥z)

=

−1 0 00 1 0

0 0 −1

σ

(2z)

=

−1 0 0−1 0 0

0 0 1

σ(2̄⊥z)

=

1 0 00 −1 0

0 0 1

σ(2̄z) =

1 0 00 1 0

0 0 −1

(2.9)

σ(3z)

=

− 12 12

√ 3 0

− 12

√ 3 − 12 00 0 1

σ(4z) =

0 1 0−1 0 0

0 0 1

σ(4̄z) = 0 −1 01 0 00 0 −1

σ(3dia) = 0 1 00 0 11 0 0

.The first matrix is the unity matrix, the second describes a point inversionthrough the unit cell of the crystal. The next two matrices describe a twofold


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rotation parallel to the z-axis and to an axis perpendicular to the z-axis,respectively. The other matrices describe in analogy fourfold and threefoldrotational axes (a bar on top denoting a rotation followed by a point inversion).Finally, the last matrix describes a threefold rotation parallel to a cube-body

diagonal. Other symmetry operations can be written as multiplications of theabove matrices. E.g. a six-fold rotation parallel to the z-axis can be describedby

σ(inv)

σ(2z)

σ(3z)

.

In the following expressions for the free energy in uniaxial, hexagonal,cubic and tetragonal crystals are derived. The cartesian coordinate systemchosen to describe the crystal is shown in Fig. 2.1. The system is chosen sothat the z-axis coincides with the film normal. Consequently, the x- and y-axesare located within the film plane. To obtain expressions that are a functionof the external magnetic field B0 and the magnetization M , the polar angles

θB and θ as well as the azimuthal angles ϕB and ϕ are introduced. In caseof an additional distinguished direction (like the direction of step edges), theangle δ is defined with respect to the x-axis.

Cubic Symmetry

For crystals of cubic symmetry the generating matrices are:σ(inv)

,σ(4[001])

and

σ(4[111])

. While the first matrix describes the fact that the cubic unit

cell is centro-symmetric, the second and third matrix describe fourfold ro-

tational axes parallel to a cube-body edge and diagonal, respectively. Us-ingσ(inv)

within (2.8) yields bijk...n = −bijk...n for all tensors of odd

rank (n is an odd number), thus making them vanish. We note that this

z

y

x

B0qB

jB

j

qM

d

Fig. 2.1. Cartesian coordinate system used to derive the expressions for the freeenergy for the diff erent crystal symmetries


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also follows directly from time inversion symmetry, which is valid as magne-tocrystalline anisotropy is a static property. Using the matrices

σ(4[001])

and

σ(4[111])

leads to relations showing that for the tensor bij all compo-

nents, for which i = j vanish, while b11 = b22 = b33 which expresses therequirement that for cubic symmetry the energy must not change upon ex-changing two αi (change of equivalent cubic axes). The first non-vanishingterm is thus given by bijαiαj = b11α

2x + b11α

2y + b11α

2z . Similarly one ob-

tains the allowed terms for bijkl. Using again the symmetry matrices one getsb1111 = b2222 = b3333 and b1122(6) = b2233(6) = b1133(6) ((6) means the 6 per-mutation that can be made from the term). Then, one has bijklαiαjαkαl =b1111

α4x + α

4y + α

4z

+ 6b1122

α2xα

2y + α

2yα

2z + α

2zα

2x

. The factor 6 arises from

the multiplicity implicit in the second relations of the bijkl. The first non-vanishing contributions for cubic crystals therefore are:

F cub = b11α2x + α

2y + α

2z

+ b1111

α4x + α

4y + α

4z

+ 6b1122

α2xα

2y + α

2yα

2z + α

2zα

2x

+ . . . . (2.10)

Using the relations α2x + α2y + α

2z = 1 and 1 − 2

α2xα

2y + α

2xα

2z + α

2yα

2z

=

α4x + α4y + α

4z yields:

F cub = K 0 + K 4α2xα

2y + α

2xα

2z + α

2yα

2z

+ K 6α

2xα

2yα

2z . . . . (2.11)

Here the anisotropy constants K 0 = b11 + b1111 and K 4 = 6b1122−

2b1111 weredefined. Note that also the next higher order term was introduced, for whichK 6 = 3b111111 − 45b111122 + 90b112233 (see [55] for details).

Cubic Crystals with (001)-orientation

Considering the coordinate system of Fig. 2.1 the direction cosines can bewritten as αx = sin θ cosϕ, αy = sin θ sinϕ, αz = cosθ. In the following weidentify the z-axis with the [001]-direction, the x(y)-axis with the [100]([010])-direction of the cubic crystal. Inserting the expressions for the αi into (2.11)

one obtains:

F 001 = K 4

sin2 θ cos2 ϕ sin2 θ sin2 ϕ + sin2 θ sin2 ϕ cos2 θ + sin2 θ cos2 ϕ cos2 θ

= K 4

sin2 θ cos2 θ + sin4 θ cos2 ϕ sin2 ϕ

. (2.12)

This expression can be transformed into another equivalent form:

F 001 = K 4

sin2 θ cos2 θ + sin4 θ cos2 ϕ sin2 ϕ

(2.13)

= K 4 sin2 θ 1 − sin2 θ + sin4 θ cos2 ϕ 1 − cos2 ϕ= K 4

sin2 θ − sin4 θ cos4 ϕ− cos2 ϕ + 1


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[001]

[100]

[110]

[001]

[100]

[110]

[001]

[110]

Fig. 2.2. Free energy for a cubic crystal with (001)-orientation with the 100-(left panel ), the 111- (middle panel ) and the 110-axes (right panel ) being theeasy ones

= K 4

sin2 θ − sin4 θ

1

8 cos 4ϕ +

1

2 cos 2ϕ +

3

8 − 1

2 − 1

2 cos 2ϕ

= K 4 sin2 θ − 1

8K 4 (cos 4ϕ + 7) sin

4 θ.

The equation shows that for K 4 > 0 the 100-directions are the easy ones,whereas for K 4


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(001) (011) (001) (111)

z

y

x, x’

z’

y’

45°

45°

[ 0 1 1

]

[ 0 1 1 ]

[001]

[100]

[010]

z

y

x

z’

[ 1 1 1 ]

x’

y’

45°

45°

[ 1 1 0 ]

[ 1 1

2 ]

[001]

[100]

[010]

Fig. 2.3. Rotation of the (001)-coordinate system

with its z

-axis parallel to any of the {110}-planes. As example in Fig. 2.3this rotation is shown for the case that the z

-axis becomes parallel to the[011]-direction. Still the direction cosines in the new systems are given byαx = sin θ cosϕ, αy = sin θ sinϕ, αz = cosθ. θ is now defined withrespect to the z-axis ([011]-direction), ϕ is measured against the x-axis([100]-direction). For (2.11), however, the direction cosines with respect tothe (x, y, z)-coordinate system are needed, i.e., the direction cosines withinthe (x, y, z)-system have to be expressed by means of the direction cosineswithin the (x, y, z)-system. From Fig. 2.3 the direction cosines with respectto the axes of the two systems can be deduced. The relation are summa-rized in Table 2.3. This yields αx = αx = sin θ cosϕ, αy = 1/

√ 2 αy +

1/√

2 αz = 1/√

2 (sinθ sinϕ + cos θ) , αz = −1/√

2 αy + 1/√

2 αz =1/

√ 2 (− sin θ sin ϕ + cosθ). Using these direction cosines within (2.11) one

derives for the free energy of an (011) oriented cubic crystal:

F 011 =

K 4

4

cos

4

θ + sin

4

θ

sin

4

ϕ + sin

2

(2ϕ)

+ sin

2

(2θ)

cos

2

ϕ − sin2 ϕ

2

.(2.14)

We note that this equation describes the same polar plot that (2.12) does withthe diff erence that it has been rotated such that the [011]-direction forms the

Table 2.3. Direction cosines between (001) and (011)-coordinate system

x y z

x

cos0◦ = 1 cos 90◦ = 0 cos 90◦ = 0y

cos 90◦ = 0 cos 45◦ = 1/√

2 cos 135◦ = −1/√ 2z

cos 90◦ = 0 cos 45◦ = 1/√

2 cos 45◦ = 1/√

2


14/52


15/52


Table 2.4. Direction cosines between the (001) and (111)-coordinate system

x y z

x

cos 114.1◦ =−

1/√

6 cos 114.1◦ =−

1/√

6 cos 35.3◦ =√

2/√

3

y cos 45◦ = 1/√ 2 cos 135◦ = −1/√ 2 cos 90◦ = 0z

cos 54.7◦ = 1/√

3 cos 54.7◦ = 1/√

3 cos 54.7◦ = 1/√

3

respect to the z-axis ([111]-direction), ϕ the one with respect to the x-axis([112̄]-direction). The direction cosines between the two systems are given inTable 2.4. In analogy to the (011)-plane one gets expressions for the directioncosines within the (x, y, z)-system as function of the α’s within the rotatedsystem:

αx = − 1√ 6αx +

1√ 2αy +

1√ 3αz = −

sin θ cosϕ√ 6

+ sin θ sinϕ√

2+

cos θ√ 3

= 1√

3

cos θ −

√ 2sin θ sin

ϕ +

π6

αy = − 1√ 6αx −

1√ 2αy +

1√ 3αz = −

sin θ cosϕ√ 6

− sin θ sinϕ√ 2

+ cos θ√

3

= 1

√ 3 cos θ

−

√ 2sin θ cosϕ +

π

3

αz =

√ 2√ 3αx +

1√ 3αz =

√ 2√ 3

sin θ cosϕ + 1√

3cos θ

(2.15)

= 1√

3

cos θ +

√ 2sin θ cosϕ

.

Here the trigonometric expressions for sin (x ± y) = sin x cos y ± cos x sin y,

cos(x ± y) = cos x cos y ∓ sin x sin y and sin(π/6) = cos(π/3 ) = 1/2 andcos(π/6) = sin(π/3) =

√ 3/2 have been used. Equation (2.11) yields for the

free energy of a (111)-oriented cubic crystal:

F 111 = K 4

1

3 cos4 θ +

1

4 sin4 θ −

√ 2

3 sin3 θ cos θ cos3ϕ

. (2.16)

F 111 is visualized in Fig. 2.4 (middle panel). Now the [111]-direction coin-cides with the z-axis from Fig. 2.1. In the azimuthal plane the free energy is

isotropic (see right plot of the middle panel of Fig. 2.4). An angular depen-dence within the azimuthal plane does only occur, when the next higher orderterm (K 6-term in (2.11)) is considered (see also Sect. 2.3.4 for the azimuthaldependence of F ).


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Cubic Films with In-plane Magnetization

For thin layers of a cubic material and for the case that the magnetizationis confined to the film plane, the direction cosines for the (001)-, (011)- and

(111)-plane take the forms listed in Table 2.5. For the free energy also listed inthe table in addition to the K 4-term the next higher order term (K 6) accordingto (2.11) was considered. Table 2.5 shows that for the (001)-orientation afourfold symmetry in the plane exists, while the (011)-orientation shows alsotwofold terms. For (111)-oriented films the in-plane anisotropy in first ordervanishes and only to the next higher order shows a sixfold symmetry, which,however, in most cases is very small.

Tetragonal Symmetry

In case of tetragonal symmetry only the (001)-orientation will be discussedas this is the one most widely dicussed one in literature. The transforma-tion to other orientations can be performed in the same way as discussedfor cubic symmetry. The symmetry matrices for tetragonal systems are:σ(inv)

,σ(2⊥z)

and

σ(4z)

. The rotational axis perpendicular to the [001]-

direction now presents only a twofold symmetry and thus the terms in theexpansion of the free energy reflect this lowering of symmetry. The first al-lowed term is b11

α2x + α2y

+ b33α2z and thus, the twofold symmetry does notvanish as for cubic symmetry. The first terms are given by:

F tet = b11α2x + α

2y

+ b33α

2z + b1111

α4x + α

4y

+ b3333α

4z +

+ 6b1122α2xα

2y + 6b1133

α2xα

2z + α

2yα

2z

+ . . . . (2.17)

Table 2.5. Direction cosines and free energy for the case that the magnetization isconfined in the film plane

Plane Direction cosines Free energy

(001)αx = cos ϕαy = sin ϕαz = 0

F 001 = K 4 cos2 ϕ sin2 ϕ

= K44 sin2 2ϕ

= K48 (1 − cos4ϕ)

(011)

αx = cos ϕαy =

1√ 2

sin ϕ

αz = −1√

2 sin ϕ

F 011 = K44

sin4 ϕ + 4 sin2 ϕ cos2 ϕ

+K64 sin

4 ϕ cos2 ϕ= K4

4

sin4 ϕ + sin2 2ϕ

+K616 sin

2 ϕ · sin2 2ϕ= K432 (7 − 4cos2ϕ − 3cos4ϕ)

+ 1128

K 6 (2 − cos2ϕ − 2cos4ϕ + cos 6ϕ)

(111)αx = −

cosϕ

√ 6 + sinϕ

√ 2αy =

− cosϕ√ 6

− sinϕ√ 2

αz =√ 3√ 2

cos ϕ

F 111 = K44

+ K654

9cos2 ϕ − 24 cos4 ϕ + 16 cos6 ϕ

= K44 + K6108 (1 + cos 6ϕ)


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One can see that for b11 = b33, b1111 = b3333 and b1122 = b1133, i.e. for thecase where x-,y- and z-axes are equivalent, the expression for cubic symmetryis retained. Using the relations α2z = 1 −

α2x + α

2y

and

α2xα

2z + α

2yα

2z

=

12 − 1

2 α4x + α4y + α4z− α2xα2y the equation transforms to:F tet = K

0 − K 2⊥α2z + K 4

α4x + α

4y

+ K 4α

2xα

2y + K

4⊥α

4z + . . . , (2.18)

with K 0 = b11 + 3b1133, K 2⊥ = b11 − b33, K 4 = b1111 − 3b1133, K 4 =

6 (b1122 − b1133) and K 4⊥ = b3333 − 3b1133. A further simplification is madethrough the relation α4x+α

4y =

α2x + α

2y

2 −2α2xα2y or α2xα2y = 12 α2x + α2y2 −12

α4x + α

4y

, leading to:

F tet = K 0 − K 2⊥α2z − 1

2K 4 α

4x + α

4y−

1

2K 4⊥α4z + . . . , (2.19)

with K 0 = K 0 + 1

2K 4 = b11 + 3b1122, K 2⊥ = K

2⊥ − K 4 = b11 − b33 +

6 (b1122 − b1133), K 4 = −2K 4 + K 4 = −2b1111 + 6b1133 + 6 (b1122 − b1133)and K 4⊥ = −2K 4⊥ − K 4 = −2b3333 + 6b1133 − 6 (b1122 − b1133). Using thepolar coordinates according to Fig. 2.1 finally yields:

F tet = −K 2⊥ cos2 θ − 12

K 4⊥ cos4 θ − 18

K 4 (3 + cos 4ϕ)sin4 θ . (2.20)

Uniaxial and Hexagonal Symmetry

The symmetry matrices for hexagonal systems areσ(inv)

,σ(2⊥z)

,σ(2z)

and

σ(3z)

. The matrix σ(3z) describes threefold rotational symmetry about

the z-axis (note that the sixfold symmetry of the hexagonal unit cell can bedescribed by combinations of the matrices). As for the other cases, we have acentrosymmetrical unit cell (due to

σ(inv)

) and thus all odd rank tensors van-

ish. The use of σ(2⊥z)

within (2.8) further shows that the σ‘s are separately

non-zero only if i = p or j = q or . . . In addition, the product σipσjqσkr . . .σipis –1 when the subscript 2 appears an odd number of times (note that the

number of σ’s within the product must be even). Since this means thatdijkl... = −dijkl... all coefficients, in which the subcript 2 appears an oddnumber of times, must vanish. Similarly, if

σ(2z)

is used, it can be shown

that all coefficients, in which the subscript 3 appears must vanish. These tworestrictions then imply that the coefficients, in which any subscript appears anodd number off times vanish. Thus, the first non-vanishing term is the sameas for tetragonal symmetry, i.e. bijαiαj = b11

α2x + α

2y

+ b33α

2z . The last ma-

trix yields several relations between the remaining dijkl...’s, leading finally to

bijklαiαjαkαl = b1122 α2x + α2y2

+ 6b1133 α2x + α2yα2z + b3333α4z (see [55] fordetails of the calculation). Taking even the next higher order term one obtains:

F hex = K 0 + K 2⊥α2x + α

2y

+ K 4⊥

α2x + α

2y

2+ K 6⊥

α2x + α

2y

3+

+ K 6α2x − α2y

α4x − 14α2xα2y + α4y

. (2.21)


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One sees that for K 4


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films. This situation was theoretically studied by Bruno [60], who modeled theroughness by two parameters σ and ξ , the former being the mean vertical de-viation from a reference plane, the latter describing the average lateral size of flat areas. The calculation shows that the stray field energy for perpendicular

magnetization of a rough film is given by:

F shaperough = 0.5µ0M 2S d

1 − σ

2df

σ

ξ

. (2.24)

Here d is the average thickness, i.e. the thickness of the reference plane. M S is the saturation magnetization. The function f has the value 1 for a flat film,σ = 0, and it approaches 0 for increasing roughness, σ/ξ → 1. This modelshows that film roughness gives rise to a small dipolar surface anisotropy con-tribution of magnitude ∝ σ/ξ that favors an out-of-plane easy axis of magne-

tization. Such a contribution was indeed found in [61] for rough Ni films onCu(001).Another point raised by Heinrich and Cochran [4] and already earlier byBenson and Mills [62] is that for very thin films of only a few atomic layers thecontinuum approximation fails to describe the dipolar shape anisotropy. Thediscreteness of the atomic moments results in a variation of the dipolar fieldacross the sample which depends on the number of atomic layers involved. Thedipolar field of a given layer decreases exponentially away from its surface witha decay length corresponding to the in-plane lattice spacing. Thus, the dipo-lar field inside the film decreases when approaching the sample surface frominside the film and the value of the average dipolar field decreases stronglywhen the thickness of the film is reduced towards the monolayer regime. Thelarger the lattice spacing, the stronger this eff ect will be. The reduced dipolarfield will appear as a reduced shape anisotropy and the reduction can be writ-ten as a reduced demagnetizing factor. For bcc(001) films the demagnetizingfactor is given as N ⊥ = 1 − 0.425N , while for the more densely packed fcc(001)surface N ⊥ = 1 − 0.234N , N being the number of atomic planes of the film. Thecase of hcp structure is discussed in [24]. Except for very thin films with athickness of few monolayers this correction is less than 1% and will thus not

be considered in the following.

Uniaxial Symmetry – Surface Anisotropy

In thin films the presence of the symmetry breaking surface and interface tothe substrate also introduces an uniaxial anisotropy term that can be writtenin the form:

F ⊥uni = K 2⊥ sin2 θ = K 0 − K 2⊥ cos2 θ = K 0 − K 2⊥α2z , (2.25)

with K 0 = 1 2

. When an uniaxial distortion of the crystal lattice is presentin the volume of the material, such a term may also contribute to volume

2 Note that as a potential energy contribution F is defined as energy diff erence, sothat adding constant (angular independent) terms has no influence on the value


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anisotropy. In general, magnetic anisotropies of thin films can be decomposedin volume contributions, that are independent of thickness and surface stateand can be explained as a superposition of shape, magnetocrystalline andresidual strain anisotropies, and surface contributions, which scale with 1/d

and depend sensitively on the state of the surface. In some cases Néel’s phe-nomenological anisotropy model has provided a useful connection betweendiff erent components of surface anisotropies. For example, the role of epi-taxial strain for the anisotropies of ultrathin films was demonstrated for thecase of Co(0001) films on W(110) by measurements of anisotropies using tor-sion oscillation magnetometry combined with measurement of the strain byhigh-angular-resolution low-energy electron diff raction. Up to a thickness of d = 2 nm, the films were found to grow in a state of constant strain, governedby pseudomorphism with a growth relation [11̄00]Co [11̄0] W, which resultsin a true volume-type strain anisotropy. Above 2 nm, a relaxation of strainis observed which scales roughly with 1/d and, therefore, results in an appar-ent surface-type contribution to strain anisotropy, superimposed on a reducedvolume contribution [63].Besides uniaxial anisotropy parallel to the film normal uniaxial contributionsfrequently appear along a direction in the film plane. This can be caused e.g.by preferential interactions due to oriented hybridization at the film-substrateinterface or an uniaxial in-plane distortion in the volume. Such a term can bedescribed by:

F uni

= K 2

sin2 θ cos2 (ϕ−

δ) , (2.26)

where ϕ is measured with respect to the [100]-direction (x-axis). To includeany possible in-plane easy axis the angle δ was defined as shown in Fig. 2.1.One should note the following: The anisotropy due to F ⊥uni, F

shapeuni (and in

many cases also F uni) are direct results of the fact that the specimen has the

shape of a thin film with interfaces that break the translational symmetry of the system. Thus, all these contributions are inherently connected with thefilm surface itself and do not depend on the crystallographic direction of thefilm normal. Consequently, for other crystallographic orientations of the film,

no transformation of F ⊥uni, F shapeuni and F uni has to be made, in contrast tothe case of crystalline anisotropy resulting from the volume symmetry of thefilm material as discussed for cubic symmetry in Sect. 2.3.2.

In order to separate the diff erent anisotropy contributions into volume andsurface contributions one needs also to perform thickness dependent measure-ments. The thickness dependence of each anisotropy constant can be fittedby a constant term representing a volume contribution (K vi ) and an eff ective

surface/interface contribution (K s,eff i ) being proportional to 1/d where d isthe thickness of the film.

of F . Therefore, the constant term K 0 does not contribute to magnetic anisotropyand can be made to vanish upon normalizing F (i.e. subtracting the constant K 0).


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K i = K vi +

K s,eff id

i = 2, 4. (2.27)

Magnetoelastic Contributions

pt Once one realizes that the MAE is a quantity describing the interactionbetween the electron spin and the lattice, it is intuitively clear that changes of the lattice constant will aff ect the magnetic properties. In a magnetized bodyone has energy terms that depend both on the strain and the magnetizationdirection: the magneto-elastic energy. Although resulting from the same ori-gin, namely the spin-orbit interaction, magneto-elastic anisotropies only existwhen stress is exerted on the magnetic system. For instance in iron, the eff ectof tension on a single crystal is to create a preferred direction of magnetization

parallel to the direction of stress. The experimentally obtained magneto-elasticconstants are significantly larger than the crystalline anisotropy constants [64].As a consequence, even small strains may give rise to an important anisotropycontribution. Moreover, this phenomenon may be of importance in epitaxialstructures, where considerable strains may result from the epitaxial growth of the film on a substrate or adjacent layers having a diff erent lattice parame-ter. With respect to the film material, the strain in epitaxial films is given byη = (asub − afilm) /afilm, i.e. the misfit is determined by the lattice constantsai, which describe the atomic distances on the relevant surface orientation. If the lattice mismatch is not too large, below a critical thickness dc (coherentregime), the misfit is accommodated by introducing a tensile strain η in onelayer and a compressive strain in the other such that both adopt the samein-plane lattice magnetic anisotropy parameter. A reasonable epitaxial matchof the film lattice with respect to the substrate one can also be achieved bythe rotation of the two lattices against each other. This happens e.g. for bcc-Fe(001) growing on fcc-Ag(001), for which the lattices are rotated by 45◦ withrespect to each other. For relatively thin films the strain and the magneto-elastic coupling are independent of thickness. Above the critical thickness dc,it becomes energetically more favorable to introduce misfit dislocations, which

partially accommodate the lattice misfit, allowing the uniform strain to be re-duced (incoherent regime). In the incoherent regime, the contribution to themagneto-elastic energy contains a reciprocal thickness dependence [65]. Thefilm strain can be isotropic in the plane of the film (i.e. 11 = 22 = η, wherethe ii are the in-plane component of the strain tensor along two axes that areperpendicular to each other) or also anisotropic (11 = 22) The latter mightoccur when preferentially oriented misfit dislocations have formed or a stronginterface hybridization between the film and substrate along specific directionsis present. According to continuum elasticity the in-plane strain leads to an

out-of-plane variation of the lattice. From the requirement of a minimum of theelastic energy one can calculate the strain component 33 perpendicular to thefilm plane. Table 2.6 lists the results for several symmetries of the film lattice.The corresponding change in volume is given by ∆V /V = (11 + 22 + 33).


22/52


Table 2.6. Calculated out-of-plane strain 33 as function of in-plane strain compo-nents 11 , 22 (from [15]). The elastic stiff ness constants cij are tabulated in [66]

Cubic (001) Cubic (011) Cubic (111) Hex. (0001)1:[100], 2:[010] 1:011, 2:[100] 1,2:⊥ arbitrary axes 1,2:⊥ basal plane−c12(11+22)

c11

−(c11+c12−2c44)(11+2c1222)(c11+c12+2c44)

−(c11+2c12−2c44)(11+22)(c11+2c12+4c44)

−c13(11+22)c33

The magneto-elastic part to the magnetic anisotropy for a cubic system canbe written as [15]:

F cubMEL = B111α

2x + 22α

2y + 33α

2z

(2.28)

+ 2B2 (αxαy12 + αyαz23 + αxαz31) + . . . .

For hexagonal systems the magneto-elastic contribution is [15]:

F hexMEL = B111α

2x + 22α

2y + 12αxαy

+ B2

1 − α2z

33 + B3

1 − α2z

(11 + 22) (2.29)

+ B4 (αyαz23 + αxαz13) + . . . ,

where the ij (i, j = 1, 2, 3) are the strain components, the Bi the magneto-elastic coupling constants and the αi the direction cosines (see coordinatesystem in Fig. 2.1) given by α1 = sin θ cosϕ, α2 = sin θ sin ϕ and α3 = cosθ.One should note that while in general the B

i of ultrathin films are diff erent

than in the respective bulk material one finds in the case of Ni films that thestrain induced anisotropy contributions can be explained – even as a functionof temperature – by the respective bulk values [5]. In Sect. 2.5 we will, however,show that the Bi-values for Fe films on GaAs diff er from the ones of Fe bulk.In order to calculate the magneto-elastic part one needs to measure the Bi(volume values for 3d ferromagnets are found in e.g. in [15]). In (2.29) if i = j ,the strain is along the cubic 100 axes, for i = j the strain is along the 110axes. While the former type of strain leads to a change of the volume of theunit lattice cell, the latter is equivalent to a shearing of the lattice keepingthe volume constant.

2.3.3 Landau-Lifshitz Equation of Motion and General ResonanceEquation

Magnetic excitations from the ground state that occur in the microwaveregime and are detected within an FMR experiment (see Sect. 2.5 for de-tails on the FMR technique) are usually described within the Landau-Lifshitz

formalism. Due to the high number of spins that take part in the absorptionprocess and the large quantum numbers associated with it classical and quan-tummechanical description lead to identical results [67] and thus a classicalformulation of the process is usually considered.


23/52


Upon considering the total angular momentum J n as a classical vector withcontinuous possibilities of adjustments, the time dependence of J n is – accord-ing to Newton’s law – given by the torque Dn =

dJ ndt . In our case the torque

is given by a magnetic field acting on the magnetic moment µn that results

from J n. Considering a magnetic part of the rf-field due to the microwaveexcitation brf µn , an external field B0 and also an internal field B

intµn

within the

sample (e.g. magnetic anisotropy fields) as contributions to the overall mag-

netic field, the torque is given by µn ×B

intµn

+B0 + brf µn

. Taking the two

expression for the torque together and using µn = −γ J n with γ = gJ µB beingthe gyromagnetic ratio (gJ : Landé g-factor) one obtains:

1

γ

dµndt

= −µn ×b

intµn

+ b0 + brf µn

. (2.30)

Summing up all magnetic moments yields the macroscopic internal fieldBint =

nB

intµn

, that acts on the total magnetic moment µt =

µn. Takinginto account that the magnetization M is defined as magnetic moment perunit volume (µt/V ) and assuming a homogeneous microwave field brf over thesample, one has:

dM

dt = −γ M ×Beff . (2.31)

Here the abbreviation Beff = Bint +B0 + brf was used. The latter equationis known as the Landau-Lifshitz(LL)-equation. We note that it can be also

derived in the framework of quantummechanics [68]. The LL equation canbe extended to include magnetic damping, leading to a finite linewidth of the FMR signal. However, throughout this paper, which focusses on magneticanisotropy and thus on the field needed for resonance (resonance field) only,damping will be neglected.

A straightforward but complex way to describe FMR is to solve the LLequation for given anisotropy fields. There is, however, an alternative route,which uses the LL equation to formulate a general equation on the basis of the magnetic part of the free energy of the system. As this approach is rather

general, it is described in some detail in the following. The method to calculatethe resonance frequency or, equivalently, the resonance field for the uniformFMR mode (collective precession of all magnetic moments) was introducedby Smit and Beljers and independently by Suhl ([69, 70]). In this formalismthe equation of motion is described by the free energy F . The same resultwas obtained by Gilbert by solving a Lagrange-Equation for the motion of M [71]. The magnetization is considered as classical gyroscope with moment of inertia I . Figure 2.6 shows the transformation from the laboratory (x, y, z)-coordinate system to another cartesian one (x

, y

, z

), in which the z

-axisrotates with the magnetization. The transformation is uniquely given by thethree Euler Angles ϕ, θ,ψ.

The kinetic energy of the system is given by E kin = I

2

ψ̇ + ϕ̇ cos θ

2.

From this the Lagrangian function of the system follows to beL = E kin − E pot (ϕ, θ), E pot being the potential energy. The Langrangian


24/52


z

y

x

z’

M

y’

x’

Fig. 2.6. Euler Angles to describe the rotation of the coordinate system

equation of motion then is ddt

∂ L∂ q̇i

− ∂ L∂ qi

= 0, where the q i are the generalizedcoordinates, which in our case are given by ϕ, θ and ψ. This approach yieldsthe following three equations:

d

dt

I

ψ̇ + ϕ̇ cos θ

cos θ

+ ∂ E pot∂ϕ

= 0

I

ψ̇ + ϕ̇ cosθ

ϕ̇ sin θ + ∂ E pot∂ θ

= 0 (2.32)

ddt

I

ψ̇ + ϕ̇ cos θ

= 0.

As e.g. shown in [72] the term I ( ψ̇+ ϕ̇ cos θ) describes the angular momentumof the magnetization within the (x

, y

, z

)-system and thus, the last equationshows the time invariance of the angular momentum. Considering the LLGequation of motion the angular momentum of the magnetization is given byM S /γ . Therefore, the two remaining Lagrangian equations yield:

M S

γ

θ̇ sin θ = ∂ F

∂ϕ

(2.33)

−M S γ

ϕ̇ sin θ = ∂ F

∂ θ .

Here it was taken into account that E pot is given by the free energy F of the system. We now assume that the precession angle of the magnetization issmall, so that only small variations δθ and δϕ with respect to the equilibriumorientation

θ0,ϕ0

occur. This approach must be modified for high power

microwave excitations which cause large precession angles and non-linear re-sponses. In the small angle regime we have θ = θ0 + δθ, ϕ = ϕ0 + δϕ and onecan expand the first derivatives of F around the equilibrium position into aseries of δϕ and δθ, in which only the linear terms have to be considered:

F θ = F θθδθ + F θϕδϕ , F ϕ = F ϕθδθ + F ϕϕδϕ . (2.34)


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The derivatives have to be taken at the equilibrium positions, i.e. F θθθ0 and

so on. Periodic solutions δθ, δϕ∝ exp(iωt) have to be found, as the deviationsaround the equilibrium position are driven by the periodic excitation due tothe microwave field with a frequency ω. This yields θ̇ = iωδθ and ϕ̇ = iωδϕ.

Taking into account that for small deviations sinθ

0

+ δθ

= sin θ0

cos δθ +cos θ0 sin δθ ≈ sin θ0 + cosθ0δθ ≈ sin θ0, one has from (2.33):

iωM S γ

sin θ0 − F ϕθ

δθ − F ϕϕδϕ = 0 (2.35)

− iωM S γ

sin θ0 − F ϕθ

δϕ − F θθδθ = 0 . (2.36)

In matrix formulation this can witten as:

F ϕθ − iωγ · M S sin θ0 F ϕϕ

F θθ F ϕθ + iωγ · M S sin θ

0 .

·δθδϕ

= 0 . (2.37)

The condition for a solution is F 2θϕ

− F θθF ϕϕ +ω2γ −2M 2S sin2 θ0 = 0, yieldingthe following equation for the resonance frequency:

ω

γ

2−

F θθF ϕϕ − F 2θϕ

M 2S sin2 θ0

= 0 ⇒ ωγ

= 1

M S sin θ0

F θθF ϕϕ − F 2θϕ

.

(2.38)If one has an expression for the free energy, from which also the equilibrium

angles ϕ0 and θ0 can be determined, (2.38) yields the resonance conditionω(B) or an expression for the resonance field Bres as function of the anglesof the external field for a fixed frequency. Equation (2.38) shows that FMR issensitive to the curvature of the free energy surface. As this surface stronglydepends on the anisotropy fields, FMR is a very useful tool to quantitativelydetermine magnetic anisotropy. We finally note that (2.38) presents a sin-gularity at the angle θ0 = 0. This problem is removed from the resonance

equation by adding the first derivatives as discussed in [73], the improvedform being:

ω

γ

2=

1

M 2S

F θθ

F ϕϕ

sin2 θ0 +

cos θ0

sin θ0F θ

−

F θϕ

sin θ0 − cos θ

0

sin θ0F ϕ

sin θ0

2 .

(2.39)

For θ0 = π/2, i.e. for the in-plane configuration (see Fig. 2.1) the latter res-onance equation has the same form as the original one, since the prefactor

cos θ0/ sin θ0 vanishes. Also for other angles, except θ0 = 0, the original formis still numerically correct, since at equilibrium the first derivatives F θ andF ϕ are zero. The original equation is, however, not convenient anymore as the


26/52


diff erent terms of the free energy are mixed, covering the symmetries. Theform by Baselgia et al. [73] is therefore favorable compared to the older ver-sion of the resonance equation, in particular when out-of-plane dependenciesare considered.

2.3.4 Resonance Equations for Angular and FrequencyDependent FMR

In the following the resonance equations for tetragonal, cubic and hexagonalfilms will be explicitly given. As for thin films shape anisotropy and uniaxial in-plane terms play an important role, they are included (see (2.26) and (2.23)).Moreover, the Zeeman energy F Zee = −M · B due to the presence of the ex-ternal field is included. Only the equations for the so-called ‘saturated’ modes

are given, being solutions, for which the precessional motion of the magneti-zation is mainly determined by the external field. So called ‘unsaturated’ (ornot aligned) modes are solutions, for which the motion of the magnetizationvector is strongly influenced by internal fields (e.g. anisotropy fields). Suchmodes usually occur for small external magnetic fields and only a numericaldescription is possible. One should keep in mind that in the following res-onance equations not aligned modes are excluded. This implies that we setϕB = ϕ

0 for the out-of-plane geometries (meaning that the magnetization isconfined to the same plane, in which the external magnetic field is varied) and

θB = θ

0

for the in-plane geometry.

Cubic and Tetragonal Symmetry: Out-of-plane Geometry

For the out-of-plane geometry the external magnetic field is varied in a planethat comprises the film normal and one principal in-plane crystallographicaxis (see also Fig. 2.1 for the coordinate system in use).

(001)-Orientation

The free energy used to derive the resonance equations for tetragonal films(of which cubic ones are a special case) is (see Sect. 2.3.2):

F = − M B0 (sin θ sin θB cos (ϕ − ϕB) + cos θ cos θB)+ K 2 sin

2 θ cos2 (ϕ − δ) −µ0

2

N ⊥ − N

M 2 − K 2⊥

sin2 θ

− 12

K 4⊥ cos4 θ − 18

K 4 (3 + cos 4ϕ)sin4 θ. (2.40)

The equation includes cubic systems, as can be seen by setting K 4⊥ = K 4 =K 4. Except for a constant term, which does not lead to anisotropy this yieldsthe expression for cubic symmetry as given in Sect. 2.3.2. Then, (2.39) for


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the out-of-plane geometry, for which the external field is varied from the filmnormal [001] to the [100]-direction (ϕ0 = ϕB = 0) yields:

ωγ

2

= Bres⊥ cos∆θ + M eff + K 4⊥

M − K 4

M cos2 θ0

+

K 4⊥

M +

K 4M

cos4 θ0 + u1

(2.41)

·

Bres⊥ cos∆θ +

M eff −

4K 4M

cos2 θ0

+

2K 4⊥

M +

2K 4M

cos4 θ0 +

2K 4M

+ u2

− u3,

with ∆θ = θ0

−θB and M eff =

2K 2⊥

M −µ

0N ⊥ − N M denoting the eff ectiveout-of-plane anisotropy field. For M eff 0) the easy axis of the system

lies in (normal to) the film plane. Note that according to Fig. 2.1 the anglesθ of the magnetization and θB of the external field are measured with respectto the film normal, while the in-plane angles ϕ and ϕB were defined withrespect to the [100]-direction. The terms ui resulting from an uniaxial in-planeanisotropy are listed in Table 2.7. The set of the ui being appropriate to agiven out-of-plane geometry is determined by the in-plane angle of the externalmagnetic field ϕB (being equal to the equilibrium angle of the magnetizationϕ0 for the reasons mentioned at the beginning of Sect. 2.3.4). For cubic systems

one has to set K 4⊥ = K 4 = K 4 within the equation.For the out-of-plane geometry, for which the external field is varied from thefilm normal [001] to the [11̄0]-direction (ϕ0 = ϕB = −π/4) the followingequation results:

ω

γ

2=

Bres⊥ cos∆θ +

M eff +

K 4⊥M

− K 42M

cos2 θ0

+ K 4⊥

M +

K 42M cos4 θ

0 + u1 (2.42)·

Bres⊥ cos∆θ +

M eff +

K 4M

cos2 θ0

+

2K 4⊥

M +

K 4M

cos4 θ0 − 2K 4

M + u2

− u3.

Again the replacement K 4⊥ = K 4 = K 4 leads to the special case of cubicsymmetry.

(011) and (111)-Orientation

In this case the same free energy expression is used as for the (001)-orientation(2.40) with the only diff erence that the cubic anisotropy contribution beingproportional to K 4 is now given by (2.14) for (011)-oriented films and by (2.16)


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in case of (111)-orientation. This yields for the case that the external field isvaried from the film normal ([011]- or [111]-direction) to the in-plane direction([011̄] or ϕB = 90

◦ in case of (011)-orientation and [11̄0] or ϕB = 90◦ for the(111)-orientation):

ω

γ

2=

Bres⊥ cos∆θ + M eff cos 2θ0 + a + u1

(2.43)

·

Bres⊥ cos∆θ + M eff cos2 θ + b + u2

− u3.While the ui are the same as given by Table 2.7, the terms a and b are listedin Table 2.8. For the case that the external field is varied from the film normal([011]-direction) to the [100]-direction (ϕB = 90

◦) diff erent values for a and bresult which are also given in Table 2.8.

Cubic and Tetragonal Symmetry: In-plane Geometry

Using the free energy expressions according to Table 2.5 within the generalresonance equation ((2.39)), one obtains for the case that the magnetizationis restricted to the film plane:

ω

γ

2=

Bres cos∆ϕ − M eff + a − u1

Bres cos∆ϕ + b − u2

− c2 ,

(2.44)with ∆ϕ = ϕ0−ϕB. The relations for a, b and c are summarized in Table 2.9. If an uniaxial in-plane anisotropy is present, the terms u1 =

2K 2M

cos2ϕ0 − δ

and u2 = 2K 2M

cos2ϕ0 − δ have to be added.

One should note that the angles ϕ and θ are measured with respect todiff erent crystallographic axes for the diff erent orientations, i.e. θ is measuredeither against the [111]-, the [011]- or the [001]-direction, ϕ against the [100]-direction in case of the (011)- and (001)-orientation and with respect to the[112̄]-direction in case of the (111)-orientation.

Table 2.7. Uniaxial in-plane terms contributing to the resonance equations for atetragonal (cubic) thin film with (001)-orientation. The equilibrium angle θ0 fromminimizing the free energy given by (2.40)

ϕB =ϕ0 u1 u2 u3

0 2K2

M cos2 δ cos2θ0

2K2M

cos2δ cos2θ0 − cos2δ

K2

2

M 2 cos2θ0sin22δ

π4 2K2

M

cos2 π4

+δ cos2θ0 2K2M

cos2 π4

+δ cos2θ0+sin2δ K22M

2 cos2θ0cos22δ π

2

2K2M

cos2 δ cos2θ0 2K2

M

sin2δ cos2θ0 + cos 2δ

K22

M 2 cos2θ0sin22δ


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Table 2.8. Cubic terms within the resonance equations for (011)- and (111)-orientedcubic systems. The equilibrium angle θ0 from minimizing the free energy given by(2.40)

Plane a b

[111]to[11̄0]

− K43M cos θ0

16√

2sin θ0 cos2 θ0

−10√ 2sin θ0 + 28 cos3 θ0−27cos θ0 − K4

M

−K43M

cos θ0

4√

2sin θ0 cos2 θ0

−10√ 2sin θ0 + 7 cos3 θ0 − 3cos θ0[011]

to[100]K4M

12 cos4 θ0 − 13cos2 θ0 + 2 K4

M

3cos4 θ0 − 7cos2 θ0 + 2

[011]to[011̄]

− 2K4M

8cos4 θ − 8cos2 θ + 1 −K4

M

3cos4 θ − 3cos2 θ − 1

Hexagonal Symmetry

For hexagonal symmetry and (0001)-oriented films the resonance equation forthe in-plane variation of the external field (in the plane perpendicular to thec-axis) is given by the same equation as for tetragonal symmetry ((2.44)) witha and b listed in Table 2.9. For the out-of-plane geometry one can in most casesneglect the very small sixfold anisotropy in the azimuthal plane given by K 6and only consider the out-of-plane constant of highest order (K 2). Then, theresonance equation has the form of (2.42) and (2.43) when one sets K 4i = 0.

2.4 Temperature Dependence of Magnetic Anisotropy

The macroscopic anisotropy energy density is temperature dependent. Thisstatement holds for the anisotropy contributions due to dipole-dipole and spin-orbit interaction. The shape anisotropy which is proportional to the squareof the magnetization (see Sect. 2.3.2) vanishes at the Curie temperature T C .

Table 2.9. Resonance equations for a cubic thin film with diff erent crystallographicorientations as well as for an (001)-oriented tetragonal and a (0001)-oriented hexag-onal system. The equilibrium angle ϕ0 from minimizing the free energy given by(2.40)

Plane a b c

(001) K42M

cos4ϕ0 + 3

2K4M

cos4ϕ0 0(001)tetra.

K4

2M

cos4ϕ0 + 3

2K4M

cos4ϕ0 0

(011) K4M 3cos

4 ϕ0 + cos2 ϕ0 − 2 K4M 12 cos

4 ϕ0− 11cos2 ϕ0 + 1 0(111) −

K4M 0 −

√ 2K4M sin3ϕ

0

(0001)hex.

− 4K4⊥+6K6⊥+6K6 sin 6ϕ0

M − 36K6

M sin6ϕ0 0


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Also, the intrinsic magneto-crystalline (spin-orbit) MAE is temperature de-pendent (see e.g. [5, 12, 34, 55]) and vanishes at T C . This has been often over-looked in the comparison of theoretical studies (usual performed at T = 0 K)and experimental investigations (usually conducted at room temperature). In

regard to the microscopic origin of MAE, i.e. the anisotropy of the orbital mo-ment, it is surprising to experimentally measure a temperature dependence of the MAE. When one considers that the spin-orbit interaction (approximately70 meV in 3d ferromagnets) is temperature independent, and smearing outthe exchange split states at the Fermi level (order of eV) [38] does not aff ectthe easy direction of the magnetization, one has to conclude that the diff er-ence of the orbital magnetic moment along the easy and the hard magneticaxis persists above T C . Unfortunately, there is no direct evidence for this,since the magnetic moment fluctuates too vividly in space and time above

T C . Most techniques will measure an averaged magnetic moment only. How-ever, susceptibility measurements and paramagnetic resonance measurements(which actually measure the susceptibility at microwave frequencies) proof the existence of atomic magnetic moments above T C even in an intinerantferromagnet like Ni. As the magnetic moment above T C is the same (exceptfor polarization of the conduction electrons) as the one measured below T C (for T = 0 K), it is reasonable to conclude that the orbital magnetic momentis unchanged in the paramagnetic state. A direct proof of the existence of the orbital magnetic moment and its anisotropy in the paramagnetic stateis obtained by angular dependent measurements in the paramagnetic phaseof a ferromagnet in magnetic fields of several kOe. To our knowledge suchmeasurements can be performed by electron spin resonance (ESR, EPR) only[74]. Here, the deviations of the spectroscopic splitting factor which is pro-portional to the ratio of orbital to spin magnetic moment was found to bediff erent for diff erent crystallographic directions and could be well describedin the framework of crystal field theory.How can one resolve the conceptual problem that the macroscopically mea-sured MAE is temperature dependent while its microscopic origin is not? Theclassical theory of the temperature dependence of the intrinsic anisotropy

(see for example [75] and references therein) was worked out based on theassumption that around each lattice site there exists a region of short-rangemagnetic order in which the local anisotropy constants are temperature in-dependent. Due to thermal motion, the local instantaneous magnetizationsof these regions will be distributed randomly, and they produce the averagemagnetization of the crystal as a whole which vanishes at T C . This does notmean that the magnetic moment vector vanishes, but it fluctuates so quicklyand uncorrelated to other moments that the spatially and timely averagedmoment vanishes. Hence, also the macroscopically measurable MAE vanishes,

it averages out above T C . This hand-waving argument has been quantifiedby expanding the MAE in a series of spherical harmonics Y lm(θ,ϕ), whichreflects the role of crystal field and spin-orbit interaction with temperaturedependent coefficients k2l,m(T ):


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The relationship between the temperature variations of the anisotropy coeffi-cients ki and the magnetization M was theoretically [76] and experimentally[77] found to have the form:

k2l,m

(T )

k2l,m(0) ∝

M (T )Γ

M (0) , (2.49)

where Γ = l(l + 1)/2, l being the order of the sperical harmonics. This givesfor example k2,m ∝ M (T )3, k4,m ∝ M (T )10. The Callen-Callen model doesnot identify the microscopic origin of the anisotropy coefficients, but includesthe contributions from magneto-elastic as well as magnetostrictive proper-ties entering into the spin hamiltonian through the combination of spin-orbitcoupling and crystal field splitting.

As the relation above holds for the anisotropy coefficients k2l,m, one has to

be careful when comparing to temperature dependencies of the experimentallymeasured anisotropy constants K i. The relations between the k2l,m and theK i can be found from (2.46), (2.47) and 2.48). Assuming a typical tempera-ture dependence of the magnetization one can plot the anisotropy coefficientsas shown in Fig. 2.7. One sees that the k2,0 and k4,0 decrease monotonicallywith increasing temperature and vanish at T C . If one confuses these tem-perature dependent ki,0 with the usual magnetic anisotropy parameters K i,one would draw the conclusion that a temperature change of the easy axis of magnetization is not possible [5]. However, one finds that if one rearranges

the cos and sin terms in the Spherical harmonics in terms of increasing pow-ers that the new parameters K i (the ones used in the experimental analysis)can vary in sign so that their temperature dependent change of sign in Co

0 100 200 300 400-2

0

2

TC

K2

k4

k2

K4

Temperature (K)

K i /arb.units

K 2(T) = 1.47 k 2 - 3.3 k 4(T)

K 4(T) = 3.85 k 4(T)

K 2

K ( a

r b . u

n i t s

)

i

Fig. 2.7. Temperature dependence of the coefficients k2l,m used when expanding thefree energy into spherical harmonics and the expected experimental K i anisotropyparameters that are coefficients of an expansion into direction cosines (reproducedfrom [5])


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or Gd can be quantitatively understood. An illustrative example is plotted inFig. 2.7 showing that K 2⊥ changes its sign. This behavior becomes also clearfrom (2.47), where one can see that k2,0 is a linear combination of K 2⊥ andK 4⊥ (for K 6⊥ = 0).

There are only few experimental results on the power law dependenceof the second order normalized MAE on the magnetization. Γ = 2.6(5)for 5.6 monolayer (ML) Fe on Cu(100) [78] and Γ = 6.5 for W(110)/Fe6 nm/W(110) [79] was reported. While the value for Fe on Cu(100) showsreasonable agreement with the theory, the reason for the large value in thecase of Fe on W(110) is unclear. One may speculate that higher order con-tributions of the anisotropy were not properly accounted for. An unusualexponent Γ = 2.1 was also reported for bulk like FePt films [80, 81] andfound to be the result of delocalized induced Pt moments leading to a two-ion

anisotropy. This result has been explained by ab initio electronic structure the-ory for L10 ordered FePt [47, 54]. The importance of separating higher orderterms from second order terms for this type of analysis was recently shown byZakeri et al. [83]. Here, the thickness of Fe layers on GaAs(001) was tuned toa critical thickness so that higher order anisotropies present in thicker layersvanished. In this case perfect agreement was found within the error bar of theexperiment with the theoretical prediction Γ = 3 (Fig. 2.8). A linear power lawcorrelation with Γ = 2.9 is observed in the experiment. The inset shows thedeviations from the linear behaviour, i.e. deviations from Γ = 2.9, for otherfilm thicknesses in which K 4 contributions become important. Similarly, the

a r b

. u

n i t s

Fig. 2.8. Temperature dependence of the uniaxial out-of-plane anisotropy K 2⊥( filled circles ) and the magnetization M (open squares ) for 5 ML Fe/GaAs(001).The inset shows the dependence for Fe films from 5 ML to 20 ML (‘Reprintedfigure with permission from Kh. Zakeri et al., Phys. Rev. B, Vol. 73, 052405 (2006).Copyright (2006) by the American Physical Society’)


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temperature dependence of the surface anisotropy and its relation to the sur-face magnetization following a diff erent temperature dependence than the bulkone have been analyzed [5].

2.5 Selected Experimental Results

Before discussing the experimental results, a typical FMR-signal ist shortlyexplained. Within an FMR experiment, the specimen is placed in a cavity, intowhich microwaves are coupled that excite the magnetic system. To generateresonant absorption from the microwave field inside the cavity, the experi-ment is performed in an external magnetic dc-field that is varied while themicrowave frequency is kept constant. Detailed descriptions of various setupsmay be found elsewhere [4, 5, 7, 14]. FMR absorption spectroscopy measures

the imaginary part of the high frequency susceptibility χ = mrf /hrf . mrf is the dynamic contribution of the magnetization that is created due to thehigh frequency magnetic field hrf of the microwaves and, thus, χ determinesthe response of the magnetic system to the excitation (see [4, 5, 7, 14] fordetails).A typical FMR signal from a thin film measured at a microwave frequency

of 9 GHz is shown in Fig. 2.9. While the main plot shows the derivative of the signal obtained from the lock-in detection procedure, the inset shows theintegral, i.e. the absorption signal itself. Three pieces of information can bedirectly extracted: (i) The resonance field Bres that includes information onthe internal fields, such as anisotropy fields. (ii) The linewidth ∆B that yieldsinformation on magnetic damping and the distribution of internal magnetic

250 300 350 400 450 500 550 600-0.6

-0.4

-0.2

0

0.2

0.4

0.6

B (mT)0||

d

’ ’ / d B ( a r b

. u n i t s )

c

DBpp

Bres A300 400 500 6000

B (mT)0||

c ’

’

( a r

b .

u n

i t s

) Bres

DBpp

I s o t r o p i c

r e s o n a n c e f i e l d

Fig. 2.9. Typical FMR spectrum of a thin film. The spectrum is measured asderivative of the high frequency susceptibility with respect to the external magneticfield The inset shows the integral of the spectrum (reproduced from [82])


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fields and (iii) the intensity of the signal that is proportional to the number of magnetic moments taking part in the resonance absorption. In the followingwe focus only on the analysis

02 - Magnetic Anisotropy of Heterostructures (Jürgen Lindner)

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