DOI: 10.1007/s00339-004-3129-5

Appl. Phys. A 80, 1011–1020 (2005)

Materials Science & ProcessingApplied Physics A

h.-c. mertins1,�

s. valencia2

a. gaupp2

w. gudat2

p.m. oppeneer3

c.m. schneider4

Magneto-optical polarization spectroscopywith soft X-rays1 University of Applied Sciences, Münster, Stegerwaldstr. 39, 48565 Steinfurt, Germany2 BESSY GmbH, Albert-Einstein-Str. 15, 12489 Berlin, Germany3 Department of Physics, Uppsala University, Box 530, 75121 Uppsala, Sweden4 IFF Research Center Jülich, 52425 Jülich, Germany

Received: 5 April 2004/Accepted: 22 October 2004Published online: 23 February 2005 • © Springer-Verlag 2005

ABSTRACT X-ray magneto-optical polarization spectroscopy isa relatively new ellipsometric technique with which the com-plete polarization state of X-ray radiation after its interactionwith magnetic matter can be measured. This comprises rotationand ellipticity, which fully quantify the light’s polarization. Em-ploying this technique, the complete magneto-optical constantsin the X-ray regime can directly be obtained, in contrast to themore commonly used intensity-only measurements. The Fara-day and magneto-optical Kerr effects, being odd with respectto magnetization reversal, can be used for the examination offerromagnetic (FM) materials. This we demonstrate here withFaraday and Kerr rotation and ellipticity spectra, measured atthe 2p edges of Fe, Co, and Ni. The Voigt effect and mag-netic linear dichroism (XMLD), which are even with respect tomagnetization reversal, are applied to probe both FM and anti-ferromagnetic (AFM) materials. Using a new XMLD-reflectionspectroscopy we present results obtained on technologically im-portant buried AFM NiO layers.

PACS 78.20.Ls; 78.70.Dm; 75.70.-i

1 Introduction

Magneto-optical spectroscopies in the X-ray range,such as for example X-ray magnetic circular dichroism(XMCD) [1–3] or magnetic linear dichroism (XMLD) [2, 4],have matured into leading-edge research techniques for theinvestigation of both fundamental and applied aspects ofmagnetism. These techniques allow for element-selectiveinvestigations of magnetic properties of constituents of mag-netic materials as well as for the element-selective imagingof magnetic domain structures [5–8], which on such a de-tailed level are not possible with visible light. A relatively newdevelopment in magneto-X-ray spectroscopies is magneto-optical polarization spectroscopy, which can provide the fullpolarization state of the light after its interaction with thesample. To these belong the X-ray Faraday effect [9, 10],X-ray L-MOKE [2, 11, 12], and the X-ray Voigt effect [13].In contrast to plain absorption or intensity measurements,the conducted polarization analysis reveals additional infor-mation, for example the phase of the monochromatic light

� Fax: +49-2551-962-787, E-mail: mertins@fh-muenster.de

in addition to its intensity. In the visible energy range therelated magneto-optical techniques are widely exploited stan-dard tools for the examination of the magnetic electronicstructure of materials [14] as well as for characterization ofmagnetism and microscopy of magnetic domains [15]. Whilethese magneto-optical effects are usually small in the vis-ible energy range, much larger effects are observed in thesoft X-ray range due to the resonant enhancement occurringat the 2p edges of 3d-transition metals [1, 2, 16] and the 3dedges of rare-earth elements [17, 18]. Thus, the developmentof polarization-sensitive techniques in the soft X-ray rangewould open up new avenues for the element-selective inves-tigation and microscopy of magnetic materials. With regardto the classes of magnetic materials of current interest, it isevident that antiferromagnetic materials (AFMs) have gainedappreciable importance lately, because they meet key appli-cations in novel spintronic devices such as magneto-resistive,spin-valve, and exchange-biased materials and nanostruc-tures [19–21]. In these devices, which often comprise buriedAFM layers, coupled to ferromagnetic (FM) layers, the spin-dependent electronic structure of AFMs plays an importantrole. Therefore, spectroscopic techniques are required whichallow one to investigate sensitively the electronic structure ofthe magnetic layers as well as at their interfaces. Here we dis-cuss a promising magneto-X-ray spectroscopy, related to theVoigt effect, which can be applied to study buried AFM layers.

The paper is organized as follows: in Sect. 2 an introduc-tion to X-ray polarization spectroscopy is given and magneto-optical spectroscopies are classified according to their appli-cability to investigate ferro- and antiferromagnetic materials.In Sect. 3 experimental polarization techniques are reported.Section 4 describes the effects which allow for probing ferro-magnetic materials, while Sect. 5 describes techniques for theexamination of AFMs.

2 Polarization spectroscopy

2.1 Polarized light

For the description of the polarized light the elec-trical wave field is conveniently decomposed into two orth-ogonal components Es and Ep that are perpendicular (Es) andparallel (Ep) to the scattering plane. The field propagates inthe z direction with frequency ω, wave vector q, and phaseφs,p. The interaction of the light with a medium can be de-

1012 Applied Physics A – Materials Science & Processing

scribed by the complex coefficients of reflection rs,p , transmis-sion ts,p, and induced phase shift φs, φp. Measurable quantitiesare the reflectance R, the transmittance T , and the phase retar-dation, which are given by

Rs,p = rs,p rs,p∗ , rs,p = |rs,p|eiφs,p ,

Ts,p = ts,p ts,p∗ , ts,p = |ts,p|eiφs,p ,

φs −φp . (1)

The polarization of the light can be described by the Stokesvector S′ = (S′

0, S′1, S′

2, S′3), where the Stokes parameter can

be expressed in terms of the components Es and Ep by [22, 23]

S′0 = [

(Ep)2 + (Es)

2] /2 ,

S′1 = [

(Ep)2 − (Es)

2] /2 ,

S′2 = Ep Es cos(φp −φs) ,

S′3 = −Ep Es sin(φp −φs) . (2)

In the following, we use the normalized Stokes parametersSj = S′

j/S′0 normalized to the intensity S′

0 with the degreeof polarization described by S1 (linear polarization with re-spect to a fixed plane), S2 (linear polarization in a plane ro-tated by 45◦ with respect to the S1 plane), and S3 (circularpolarization). The degree of linear polarization is given byPLin = [

S12 + S2

2]1/2

and the total degree of polarization P

is given by P = [S1

2 + S22 + S3

2]1/2 ≤ 1. From these data the

polarization ellipse can be constructed. Its ellipticity ε and thetilt θ of the main axis with respect to the polarization of theincoming light follow from

sin 2ε = S3 ,

tan 2θ = S2/S1 . (3)

With these quantities the polarization state of the light is com-pletely characterized.

2.2 Interaction of polarized light with magnetic matter

The interaction of light with a periodic material isgoverned by the dielectric tensor ε specific to the material,the particular symmetry of which is determined by the crystalsymmetry, and the individual scattering geometry [24]. Thescattering depends mathematically on the light’s polarizationand its orientation with respect to a preferred axis, e.g. themagnetization direction of the matter. From the dielectric ten-sor the magneto-optical index of refraction n is deduced usingthe Fresnel equation. This index of refraction characterizescompletely the magneto-optical properties. A part of this in-formation can be gained even without polarization analysisfrom simple intensity measurements when one exploits thepolarization dependence of the incident light. However, thephase information (1) is detectable only by a polarization an-alysis. The individual scattering geometries and the relationsbetween Stokes parameters and intensity data are discussed indetail in Sects. 4 and 5.

The scattering of polarized light off isotropic, magneticmatter can be treated within a scattering-type formalism, inwhich, in a Taylor expansion, three terms occur [25]. The scat-tering amplitude f can be expressed as

f ≈ (e′ · e)F0 − i((e′ × e) · M)F1 + (e′ · M)(e · M)F2 , (4)

where e′ and e are the unit polarization vectors of the scat-tered and incident light, respectively. The first and largestterm contains the charge contribution and it is independentof both the magnetization and the polarization of the incidentlight. The second term, which is odd in the magnetization M,scales with (M · q · S3), i.e. the degree of circular polariza-tion and the component of the magnetization parallel to thelight-propagation vector. The third term is – for linearly polar-ized light – proportional to (E · M)2. This term is even in Mand the largest values are obtained for parallel orientation be-tween the electric field E and the magnetic moment while thewave vector is perpendicular to M. For non-isotropic materi-als a term related to the optical anisotropy occurs in addition.Usually the term (M ·q · S3) dominates the magneto-opticaleffects. This term is present for scattering from each individ-ual atom; therefore, in a solid on average 〈M〉 is measured,which is non-zero only for ferro- and ferrimagnetic order-ing. The magneto-optical effects that are sensitive to 〈M〉 are,for example, XMCD (X-ray magnetic circular dichroism) andT-MOKE (transversal magneto-optical Kerr effect), which areintensity measurements, and the Faraday effect as well asthe L- and P-MOKE (longitudinal and polar magneto-opticalKerr effects), which can be obtained only from a polariza-tion analysis. Effects that are dominated by the term (E · M)2

are important for the investigation of antiferromagnetic ma-terials, since this term is on average proportional to 〈M2〉 ina solid. To this class of effects belong the Voigt effect, XMLD,and XMLD-type reflectometry. The latter spectroscopy issuited to investigate technologically important magnetic de-vices such as buried antiferromagnetic layers [26]. A some-what different (but in its essence equivalent) approach to clas-sify magneto-optical effects follows from the parity of theelectric permittivity tensor ε with respect to the magnetiza-tion M. Such classification leads in general to spectroscopiesthat are invariant, odd, or even in M. The above-mentionedexpansion of the scattering in powers of M corresponds thusto the lowest, leading orders of the effects odd or even in M.A summary of these techniques according to their respectivemagnetization dependence is given in Fig. 1.

3 Experiment

3.1 Polarized light sources

Up to date X-ray sources for polarized light are el-liptical undulators employed in an electron-storage ring [27].Examples are the APPLE-II-type undulators UE56 and UE52at BESSY [28–30] producing synchrotron radiation with po-larization tuneable from circular to linear in an energy rangefrom about 100 eV to 1500 eV. These are built of four iden-tical rows of permanent magnets, two on top and two belowthe stored electron beam. Two of these rows can be movedalong the direction of the electron beam quantified by theso-called ‘shift parameter’. Tuning from linear to circular po-larization is done by a parallel shift of the magnetic rows [30].A special feature of the UE52 undulator is the tunability ofthe orientation of the linear polarization from 0◦ to 90◦ (withrespect to the horizontal laboratory plane) accomplished byan antiparallel shift of the magnetic rows. The degree of lin-ear polarization and the intensity are stable within 4% [29].This feature is important for measurements of effects that are

MERTINS et al. Magneto-optical polarization spectroscopy with soft X-rays 1013

FIGURE 1 Survey of X-ray magneto-optical spectroscopy techniques,classified according to three criteria: (i) intensity measurement or polar-ization analysis, (ii) using linearly or circularly polarized light, and (iii)dependence on magnetization, i.e. sensitivity to M or M2

even in M. Here difference measurements for two perpendicu-lar orientations between the electric field E of the light and themagnetization M have to be performed for the same locationof the sample. The light focus is held stable within 2 µm whentuning the polarization plane. This accuracy is not achievedby the mechanical technique in which the sample is rotatedaround the light axis.

3.2 Polarization detectors

The linear polarization of the incident or reflectedbeam is analyzed by rotating a W/Si reflection multilayer(150 periods of 1.12 nm each). It is operated at the Braggangle of incidence, which is close to the Brewster angle. Themultilayer is rotated around the azimuthal angle γ (see Fig. 2),while the reflected intensity is monitored by a GaAs:P diode.This device is about 60-mm long and about 30-mm wideworking with UHV (ultra high vacuum)-compatible steppermotors and a microgear system manufactured by LIGA Pro-cessing. It allows for an angular resolution of 5/100 degree,which is necessary to resolve the Bragg peak. The analyzercan be moved on the reflected beam and aligned in situ, whichallows the polarization analysis at any angle of incidence. Al-

FIGURE 2 Experimental set-up of an X-ray polarization detector based ona multilayer mirror. It is operated at the Bragg angle, which is close to theBrewster angle. The polarization analysis is performed by rotating mirror anddiode about the light axis γ . Stepper motors allow for an in situ alignment ofthe analyzer on the light axis

ternatively, for intensity measurements the reflected intensitycan be monitored by moving a diode on the beam. In situ ex-change and removal of samples enables a quasi-simultaneousintensity or polarization analysis of the incident and the re-flected beams. This device is installed in the BESSY po-larimeter [31] with which all reported experiments have beenperformed.

3.3 Polarization analysis

The experimental procedure of the polarization an-alysis is discussed in some detail for the case of the Faradayeffect observed on a magnetized 50-nm Fe film (Fig. 3). Thephoton energy was set to 705 eV, which is just above the Fe2p3/2 absorption edge. The normalized angular distribution ofthe incident beam as a function of the analyzer angle γ (seeFig. 2) shows a maximal intensity at γ = 0◦ and 180◦, cor-responding to a polarization in the horizontal plane. Since theincident radiation is known to be completely linearly polar-ized [30] the polarizing power of the analyzer is found to bePA = (Rs − Rp)/(Rs + Rp) = 0.98, which was deduced fromreflectance measurements in s- and p-polarization geometry,respectively. After inserting the Fe sample at grazing inci-dence, the polarization plane is rotated by θ (more specifically,here it is θF). The rotation angle and the ellipticity of the lightbehind the sample are obtained by a least-squares fit to theintensity I(γ) = IR[1 + PLin PA cos 2(γ + θ)], with the fit pa-rameters IR, PLin, and θ . This fit formula can be written interms of Stokes parameters as I(γ) = IR[1 + PA(S1 cos 2γ +S2 sin 2γ)] with fit parameters IR, S1, and S2. Assuming fullypolarized light, which is valid for synchrotron radiation [30],the degree of circular polarization is calculated as S3 = 1 −S1

2S22. According to (3) the ellipticity ε and the rotation θ are

directly deduced. The example given in Fig. 3 illustrates the

FIGURE 3 Illustration of the Faraday measurement technique. The filledsymbols show the response obtained for a Faraday measurement on an Fesample at the Fe 2p absorption edge, whereas the open symbols show thepolarization measurement on the incident light. The continuous line is a least-square fit. The rotation angle and ellipticity are indicated

1014 Applied Physics A – Materials Science & Processing

change of the polarization of the light after its interaction withthe magnetized Fe sample. The linear polarization decreasesfrom PLin = 1 to PLin = 0.75 and the circular polarization in-creases to Pcirc = 0.65, resulting in a decreased modulationdepth of the analyzer spectrum. The rotation of the polariza-tion plane is found from the shift of the curves. To determinethe energy dependence of rotation and ellipticity several suchγ scans across the absorption edge of interest have to be per-formed.

4 Effects odd in M

4.1 Faraday effect

Upon transmission through a magnetized samplethe Faraday effect leads to a rotation of the polarization planeand to the change from linear to elliptical polarization of thelight. For a description, the incident linearly polarized lightbeam is decomposed into two circularly polarized waves ofopposite helicity. The interaction of these two waves with themagnetic material is described by its complex optical con-stants n± (+ and − refer to the orientation of photon spinaligned parallel and antiparallel to the magnetic moments ofthe sample). For the polar geometry, where the magnetizationand the propagation direction of the light are parallel, the opti-cal constants can be written as

n± = 1 − (δ0 ±∆δ)+ i(β0 ±∆β) , (5)

with δ0 the real, dispersive part and β0 the absorptive imagi-nary part for linearly polarized light, respectively. We remarkthat the optical constants can be expressed equally well interms of the forward-scattering amplitude

nλ = 1 +(

2πn0

q2

)fλ , (6)

where n0 is the density of scatterers and λ denotes thelight’s polarization state. While it is common to express thepolarization-dependent magneto-optical effects in terms ofthe optical constants, it would also be possible to express themin terms of the forward-scattering amplitude. The circulardichroism, which is related to the difference of the optical con-stants for left and right circularly polarized light, is expressedby

∆n = n+ −n− = −∆δ+ i∆β . (7)

Upon transmission a phase change between the left- and right-handed polarized waves appears, which leads to a rotation ofthe polarization plane. In addition, the dichroism ∆β in theabsorption leads to a transmitted beam with elliptical polar-ization (see Fig. 4). The complex Faraday equation can bewritten as

θF + i tan εF ≈ (ω/2c)d(iεxy/εxx1/2) . (8)

The off-diagonal permittivity εxy is the magneto-optically ac-tive component, which can be shown to be antisymmetric inthe magnetization. For an incident linearly polarized waveboth the ellipticity and the rotation angle of the transmitted

FIGURE 4 The Faraday effect is observed with linearly polarized light,which is decomposed into two circularly polarized waves of opposite helic-ity. Upon transmission through a magnetized sample a phase change betweenthese waves appears, which leads to the Faraday rotation. The dichroism inthe absorption leads to a transmitted beam with elliptical polarization

light are related directly to the optical constants, as describedin first-order approximation by [16, 24]

θF = (ω/c)d∆δ tan φi ,

tan εF = (2ω/c)d∆β tan φi , (9)

with ω the frequency of the light, φi the angle of grazingincidence, and d the sample thickness. While the first-orderapproximation is well valid in the soft-X-ray range, we remarkthat it is possible to derive – and work with – an exact expres-sion which relates both θF and εF to the optical constants [24].

The experimental energy dependence of the Faraday ro-tation and ellipticity of a 50-nm FeNi sample is shown inFig. 5 (left-hand ordinates). Each data point was obtained byan analyzer scan as depicted in Fig. 3. The directly deducedmagneto-optical constants ∆δ and ∆β are plotted at the right-hand ordinates.

A quantity related to the ellipticity can be measured aswell but with circularly polarized light. In XMCD experi-

FIGURE 5 Faraday effect on 50-nm FeNi on 100-nm SiN measured withlinearly polarized light. Experiment (points) and result of ab initio cal-culations (lines). Top: rotation (left-hand ordinate) and directly deduceddichroism ∆δ of the optical constant (right-hand ordinate). Bottom: Fara-day ellipticity and XMCD asymmetry parameter A, the latter deduced fromtransmission experiments with circularly polarized light (left-hand ordinate)and the directly deduced dichroism ∆β of the optical constant (right-handordinate)

MERTINS et al. Magneto-optical polarization spectroscopy with soft X-rays 1015

ments the dichroism is measured, which is given by the asym-metry parameter A, defined as

A = (T+ − T−)/(T+ + T−) ,

A ≈ (2ω/c)d∆β , (10)

where T± = exp{−2(ω/c)d(β ±∆β)} are the transmissioncoefficients of left- and right-handed circularly polarizedX-rays. For small values the asymmetry parameter is di-rectly proportional to the Faraday ellipticity as shown inFig. 5. In XMCD experiments one measures only one of thedichroic quantities (∆β) while the other has to be derivedfrom a Kramers Kronig transformation with the possible un-certainties due to the extrapolation of absorption spectra. InFaraday experiments, on the other hand, one measures bothdichroic quantities (∆δ and ∆β) in a single experiment withthe advantage of using only linearly instead of circularly po-larized light. The experimental data are in good agreementwith theoretical results of ab initio band-structure calcula-tions [16]. These magneto-optical constants of pure Fe, Co,and Ni serve as a data set to simulate thin magnetic films ormultilayers or devices (see Sect. 4.2 and [11, 12, 32, 33]).

For magnetically saturated 3d transition metal films of Fe,Co, and Ni, huge rotation constants of up to 2 ×105 degrees/mm were found [9, 10, 16, 34]. These constants are one orderof magnitude larger than those observed in the visible en-ergy range [35] and in the hard X-ray range [36]. This findinggives rise to technological applications as polarization mod-ulators [34] with possible application in the field of X-raymicroscopy.

4.2 L-MOKE

To treat L-MOKE (longitudinal magneto-opticalKerr effect), which is the reflection analogue to the Faraday ef-fect, the incident linearly polarized light is decomposed intoequal-amplitude left- and right-circularly polarized waves.The propagation and reflection of these waves are describedfor magnetic materials by the complex magneto-optical con-stants ((4) and (5)). The magnetic dichroism may be definedby the Voigt parameter Q, which adopts for L-MOKE theform Q = (n+ −n−)/n sin φt, with n = n+ +n− and the com-plex angle of refraction φt. This leads to the Kerr equationsfor either s- or p-polarized incident light, which can be ex-pressed by the non-magnetic reflection coefficients rss and themagnetization-dependent coefficients rps and rpp [24]:

θs + iεs = −rps/rss ≈ −in0nQ

(n2 −n20)

cos φi tan φt

cos(φi −φt),

θp + iεp = −rps/rpp ≈ −in0nQ

(n2 −n20)

cos φi tan φt

cos(φi +φt). (11)

The reflected light is elliptically polarized, quantified by εs,p,and the polarization plane is tilted over an angle θs,p with re-spect to that of the incident light (Fig. 6). The rotation is dueto unequal damping of the circularly polarized waves, whilethe ellipticity is due to unequal phase advances, both varyingwith the angle of incidence φi (measured to the surface nor-mal). Viewing in terms of linearly polarized waves, additionalfield components Ep (Es) are introduced to the pure Es (Ep)

FIGURE 6 Experimental set-up of a Kerr measurement using s-polarizedincident light. Upon reflection on a magnetized sample the polarization planeof the incident light is rotated and the polarization changes to elliptical. Thepolarization ellipse (top) is determined by a full polarization analysis usingthe detector shown in Fig. 2

components of the incident light. The Kerr data can be approx-imated in terms of the optical constants. For small angles ofgrazing incidence (90◦ ≥ φi ≥ 70◦) and assuming that φt ∼ φi

we find for s-polarized light

θs ≈ ∆δ(β1 −β0)−∆β(δ0 − δ1)

(δ0 − δ1)2 + (β1 −β0)2,

εs ≈ ∆δ(δ0 − δ1)−∆β(β1 −β0)

(δ0 − δ1)2 + (β1 −β0)2. (12)

These equations clearly show the origin of the Kerr effect. Therotation is observed only for absorbing materials, but the el-lipticity is observed even for non-absorbing materials. This isin contrast to the Faraday effect. In particular, the refractiveand absorptive parts of the optical constant remain separate inthe Faraday data (9), while they are not separate in the Kerrdata. The Faraday spectrum directly reveals the Q parameter,namely the individual magnetic contributions ∆δ and ∆β aswell as β0, peaking at the respective absorption edges 2p3/2and 2p1/2.

The Kerr rotation and ellipticity spectra for a 30-nm-thickCo film are plotted in Fig. 7. Each data point was obtainedby an analyzer scan as shown in Fig. 3. The shape of therotation spectrum results from the interplay of three individ-ual contributions: (i) at the 2p3/2 and 2p1/2 absorption edgesthe structures are induced by the magnetic contributions ∆δ

and ∆β of the optical constant, as in Faraday spectra. (ii) Inbetween the 2p edges the Kerr rotation is dominated by singu-larities due to the denominator n2 −n0

2 (see (11)). This effectis not observed in Faraday spectra since these do not con-tain this denominator. (iii) The singularity at 778.5 eV must berelated to interference effects related to small reflection coef-ficients rss. At this low energy no corresponding structure isfound in the magneto-optical data, nor in ∆δ or ∆β, nor in β1

or δ1 [11, 16].For a quantitative modeling of the Kerr spectra a computer

code based on the formalism for magnetic multilayer systems,given by Zak et al. [37], has been developed. It takes intoaccount all mentioned contributions (i)–(iii) and it uses the ex-perimentally determined optical constants. A good agreementwith the experimental data is obtained, even in the pre-edgeregion where the mentioned interference effects are domin-ant. With this computer code a separation of structural effects

1016 Applied Physics A – Materials Science & Processing

FIGURE 7 The reflectance and X-ray L-MOKE spectra measured at the 2pabsorption edge of an amorphous Co sample. Top: reflectance spectrum Rs.Middle: longitudinal X-ray Kerr rotation spectrum. Bottom: X-ray Kerr el-lipticity spectrum. The filled points are experimental data, while the openpoints result from calculations utilizing independently determined opticaland magneto-optical refractive indices. The error bars are as indicated orsmaller than the symbols

of the layered system from the pure magneto-optical effectsis possible. Thus, despite the complex interplay of all men-tioned contributions, the Kerr effect has advantages over theFaraday effect. While the Faraday effect probes the sample’svolume the Kerr spectroscopy can be tuned to become a moresurface sensitive technique suitable for depth profiling of lay-ered structures by varying the incidence angle of light. TheKerr data at the 2p edge exceed those obtained in the visi-ble by two orders of magnitude [38, 39]. Thus, a transfer ofthe MOKE technique to the soft X-ray range would open upnew avenues for the element-selective investigation and mi-croscopy of magnetic domains.

4.3 Magnetic reflection

Resonant magnetic scattering and specular reflec-tometry of circularly and linearly polarized soft X-rays [40–43] have gained increasing importance, since the resonantlyenhanced cross sections at absorption edges lead to large mag-netic responses which can exceed that observed in XMCD ab-sorption experiments. Mostly employed reflection-type spec-troscopies without polarization analysis are XMCD and thetransversal magneto-optical Kerr effect (T-MOKE), both oddin the magnetization. For T-MOKE the reflectance RT± oflinearly p-polarized light is measured for two inverted mag-netization directions each transversal to the scattering plane.

The magnetic signal is defined by the normalized differenceAT = (RT+ − RT−)/(RT+ + RT−). For XMCD in reflectionthe magnetic signal is given by AC = (RC+ − RC−)/(RC+ +RC−), where RC± is the reflectance of left- or right-circularlypolarized light or, equivalently, two inverted magnetizationdirections with components parallel to the light’s wave vectorof one helicity type.

The longitudinal Kerr parameters θs and εs for s-polarizedincident light, that are obtained by polarization analysis,can be related to T-MOKE and reflection XMCD intensity-type measurements. We can use analytical expressions forT-MOKE and XMCD in reflection [40] to derive the followingrelations between the different dichroic quantities:

θs = ATRe{

cos(φi −φt)/4 cos φi}

≈ ATRe{√

n2 −n20/4n +n0/4n cos φi

}, (13)

εs ≈ 1

2AC + cos φiIm

{rsp

√n2 −n2

0/n0r2ss

}. (14)

These are valid at grazing incidence where approximationsfor small angles are applied. The relations reveal that thelongitudinal Kerr rotation is proportional to the T-MOKE sig-nal, which is obtained with an entirely different experimentaltechnique. Analogously, the longitudinal Kerr ellipticity canbe related to XMCD in reflection, which is obviously alsomeasured in a different set-up using circularly polarized light.A detailed experimental discussion is given in [11]. The sub-stitution of the more involved polarization measurement of theKerr data by two independent T-MOKE and XMCD reflec-tion spectra can be of practical importance if a polarizationanalysis is not possible.

5 Effects even in M

Magneto-optical effects that are even in M are theVoigt effect [13] and linear magnetic dichroism (XMLD)[4, 13, 44, 45]. These X-ray magneto-optical spectroscopiesthat are sensitive to M2 hold great potential for the fu-ture investigation of antiferromagnetic materials, somethingwhich is not possible with X-ray spectroscopies that are lin-ear in M. However, the quadratic magneto-optical effectsare more difficult to observe since they are usually muchsmaller than those that are to lowest order linear in M. Sofar only a few optical measurements of XMLD have been re-ported and therefore very little is known about the physicsgoverning the magneto-X-ray effects that are proportionalto M2.

5.1 Voigt effect and XMLD

The Voigt effect is detected in the geometry shownin Fig. 8, in which the incident light is normal to the surface,and is linearly polarized at an angle α with respect to the mag-netization that is perpendicular to the light. For α = π/4 onehas the classical Voigt geometry [46]. In the Voigt effect thepolarization plane of the incident, linearly polarized light be-comes rotated over an angle θV, the so-called Voigt rotationupon transmission. In addition, the light becomes ellipticallypolarized by an amount εV (Fig. 8).

MERTINS et al. Magneto-optical polarization spectroscopy with soft X-rays 1017

FIGURE 8 The X-ray Voigt effect is observed with linearly polarized lightupon transmission through a sample with magnetic moment perpendicular tothe light direction. Largest effects are detected for an orientation α = 45◦ be-tween polarization plane and magnetization. The phase difference betweenthe electric field components leads to the Voigt ellipticity; the dichroism inthe absorption leads to an intensity difference and thus to the Voigt rotationof the polarization plane after the sample

For α = π/4 the incident light can be decomposed into twoequal components, one parallel and one normal to the magne-tization. The two refractive indices of the material are n‖ andn⊥ corresponding to the propagation of linear modes havingthe electrical field vector parallel and perpendicular, respec-tively, to the magnetization. When the two refractive indicesare unequal, which happens in an isotropic material only be-cause of the magnetization, the complex Voigt effect can beexpressed by

θV − iεV ≈ (ωd/2ic)[n‖ −n⊥]≈ (ωd/4icn)

[ε‖ − ε⊥ − ε2

od/ε⊥]

, (15)

where d is the thickness of the film and ε‖ and ε⊥ are thediagonal components of the permittivity tensor for orien-tations of the E vector parallel and normal, respectively,to the magnetization. The average refractive index is givenby n and εod is the off-diagonal permittivity of the ferro-magnetic material [24]. It follows from (15) that the Voigteffect is even in the magnetization: the diagonal compo-nents are even in M, since ε‖,⊥(−M) = ε‖,⊥(+M), whereasthe off-diagonal element is odd, εod(−M) = −εod(M). ForAFM materials εod vanishes, but ε‖ − ε⊥ remains. There-fore, the Voigt effect has the advantage over the Faradayand Kerr effects that it is suitable for the study of AFMs aswell.

Because the Voigt effect is about 1–2 orders of magni-tude smaller than the Faraday effect it is measured in a dif-ferent way, which is more sensitive to small Voigt rotationangles. A polarization analysis of the transmitted light asa function of the sample’s azimuth angle α at a fixed an-alyzer angle γ for each photon energy is performed. Foran analyzer angle γ = π/4 the intensity reflected from theanalyzer is most sensitive to small rotation angles θV (seeFigs. 3 and 4). The intensity monitored by the polariza-tion analyzer is the product of the (constant) incident in-tensity, the transmission T of the sample, and the reflectionR of the analyzer. A typical spectrum is shown in Fig. 9for 200-nm Co on 100-nm SiN. We obtain the ellipsomet-ric quantities by fitting the normalized intensity RT(α) usingthe following functional forms for T and R: the transmis-sion T can be expressed as T = T0 [1 + A cos(2α)], with A theXMLD parameter (see below) and T0 the average transmis-sion, T0 = (T‖ + T⊥)/2 (see Fig. 9). The reflectivity is givenby R = R0(1 + (Rp − Rs)/(Rp + Rs)PLin sin 2θ), where R0 is

FIGURE 9 Voigt measurement on amorphous 200-nm Co on 100-nm SiNat fixed energy at the Co 2p edge. Top: detected signal T ∗R as function ofthe sample azimuth α. Center: transmission from which the XMLD asymme-try A = (T‖ − T⊥)/(T‖ + T⊥) can directly be deduced. Bottom: Voigt rotationand ellipticity deduced from a fit to the measurement (top) as described inSect. 5.1

the average reflectance, Rs, Rp are the known s, p reflectancesof the analyzer, that are independent of α, and PLin is thedegree of linear polarization after the sample. PLin is givenby PLin ≈ 1–2ε2; thus to first order in ε we have PLin ≈ 1.We first perform the fit for PLin = 1, which gives us the α

dependence of θV . This yields directly θV(α) ∼ sin 2α (seeFig. 9), which shows that the measured magneto-optical ro-tation obeys the theoretically expected dependence of theVoigt rotation on the angle α. This excludes that the meas-ured magneto-optical angle is a Faraday rotation induced bymisalignment because, if there were a magnetization compon-ent along the propagation direction, a constant α dependencewould result.

In a second step we perform a fit in which we determine|ε|, since only ε2 enters the fit formula. We note that |ε| obvi-ously cannot be fitted with the same accuracy as θV and alsothat |ε| cannot be determined when θ = 0. We obtain for |ε| thebest fit result with a sin(2α) dependence (see Fig. 9), which isexactly what is expected for the Voigt ellipticity. The meas-ured X-ray Voigt rotation spectrum is shown in Fig. 10 for50-nm Fe. Each data point for θV is the result of an α scan. Themaximum rotation values are one order of magnitude largerthan those observed in the visible energy range [46]. As a fur-ther test we measured under inverted magnetic field the Voigtrotation, which we found indeed to be independent of theparallel or antiparallel magnetization direction. From the ex-perimental spectrum the magneto-crystalline anisotropy candirectly be deduced according to (15). It is plotted at the right-hand ordinate in Fig. 10.

A quantity related to the Voigt effect is the XMLD asym-metry parameter A, which is measured as an intensity differ-ence of light transmitted with polarizations perpendicular T⊥

1018 Applied Physics A – Materials Science & Processing

FIGURE 10 Voigt rotation spectrum across the Fe 2p edge of 50-nm Fe de-duced from polarization analysis as shown in Fig. 9 and XMLD asymmetrydeduced from transmission experiments. Both spectra should agree accordingto theory. The results of ab initio calculations computed with core exchangepolarization are given by the full lines. The right-hand ordinate shows themagneto-crystalline anisotropy in the optical constants, as deduced from (15)

and parallel T‖ to the magnetization (see Fig. 9, center). It isrelated to the Voigt rotation by

A = (T‖ − T⊥)/(T‖ + T⊥) ≈ 2θV . (16)

For comparison to the X-ray Voigt rotation, we determinedthe corresponding XMLD asymmetry. As expected, the twoquantities agree well (Fig. 10).

Further information about the Voigt effect follows fromour ab initio calculations, which are described in detail in [13].The calculations revealed a strong sensitivity of the X-rayVoigt effect to the spin polarization of the core states. InFig. 10 we show the experimental Voigt rotation spectra to-gether with the ab initio calculated spectra of fcc Co andFe, which were computed including the spin polarization ofthe 2p core states. Obviously, the Voigt rotation predictedwith the 2p3/2 exchange splitting included and the measuredspectra are in excellent agreement, astonishing in view ofthe smallness of the magneto-optical quantity. Conversely,the Voigt effect calculated without core exchange splitting,but with exchange-split valence states, is practically zero (notshown) and bears no correspondence to the experiment [13].This demonstrates that the X-ray Voigt effect depends crit-ically on the core spin splitting, which is unexpected, sincethe spin splitting of the 3/2 and −3/2 sublevels of Co is only0.79 eV.

5.2 Density of states

The theoretical understanding of the even magneto-X-ray effects is just beginning to develop [13, 47]. For a qual-itative understanding of the XMLD structure it is instructiveto consider the excitation process of polarized light, the ex-change splitting ∆ex of the 2p core levels and the exchange-split d-density of states, adopting the assumption that thecore hole is well screened and that the d-density of states

FIGURE 11 Top: the XMCD spectrum is directly correlated to the imaginarypart of the dielectric constant Im{ε+ −ε−}, which is directly proportional tothe exchange polarization of the unoccupied d-density of states (D↑ – D↓).Bottom: the XMLD spectrum is proportional to Im{ε‖ −ε⊥}, which in first-order approximation scales with the derivative of (D↑ – D↓). On the left thisis shown schematically, while on the right corresponding measured spectraare shown, obtained at the Fe 2p edge of 100-nm FeCo on 100-nm SiN

is thus not significantly perturbed by the core hole beingpresent. The comparison between measured and spectra com-puted under this assumption prove this assumption to be validfor the L-edges of the late 3d transition metals. The XMCDspectrum, obtained with circularly polarized light, gives theimaginary part of the dielectric constant Im{ε+ − ε−}, whichis directly proportional to the exchange polarization of the un-occupied d-density of states (D↑ − D↓) multiplied with theFermi function (Fig. 11). In XMLD experiments, using lin-early polarized light, the difference of two spectra measuredwith perpendicular and parallel orientations of electric fieldand magnetic moments give the imaginary part Im{ε‖ − ε⊥}.The relation to the d-density of states is obtained from an ex-pansion of expressions for the dielectric tensor componentsin single-particle formulation. Adopting spherical symmetryand accounting for the core splitting ∆ex, one obtains [26]

Im{ε‖ − ε⊥} ∼ ∆exd/dE[D↑ −D↓] . (17)

This provides already the typical double-lobe structure ofXMLD spectra, which are proportional to the derivative ofthe XMCD spectra. This is illustrated with spectra of 100-nmFe20Co80 at the Fe 2p edge (Fig. 11). Thus, the XMLD spec-troscopy gives direct access to the spin-dependent electronicstructure, namely the exchange-split d-density of states.

5.3 XMLD in reflection

The above-mentioned methods are suited to inves-tigate the spin-resolved electronic structure of AFM materialson an element-selective level. However, the Voigt effect is ap-plied in transmission and thus restricted to thin (d < 100 nm)transmission samples. The XMLD based on photoelectronspectroscopy is on the other hand restricted to near-surface

MERTINS et al. Magneto-optical polarization spectroscopy with soft X-rays 1019

FIGURE 12 XMLD on ferro-magnetic Co measured in trans-mission (left) and reflectiontechniques (right). The XMLDis even in the magnetization, asshown by the vanishing differ-ence of two transmission spec-tra with inverted magnetization(left, bottom, open symbols).Both techniques lead to the typ-ical XMLD structure (bottom,left, right). The relative strengthof the XMLD reflection signalexceeds that obtained in trans-mission by one order of magni-tude, resulting in an increasedsignal to noise ratio

layers because the detected secondary electrons have an es-cape depth of only 2–4 nm. Therefore arbitrary objects, or no-tably industrially manufactured devices such as deeply buriedAFM layers, cannot be examined.

In this section we describe a spectroscopy with whichthe XMLD effect can be exploited in reflection. In particu-lar, the large penetration depth of soft X-rays (20–100 nm)allows for the investigation of AFM buried layers. To de-tect the magnetic signal in a XMLD-reflection experimentthe reflection of linearly polarized incident light is meas-ured twice: once with the electric field parallel (R‖) andonce with the electric field perpendicular (R⊥) to the spinaxis of the sample while the incidence angle is held fixed.The difference ∆R = R⊥ − R‖ is proportional to the fun-damental quantity Re{ε‖ − ε⊥} [26, 48, 49]. To demonstrate

FIGURE 13 Top: X-ray reflection spectrum of a buried antiferromagneticNiO layer of a NiO/Co/Ta/Cu system for φi = 60◦ at the Ni 2p edge fortwo perpendicular orientations between electric field and magnetization axis.Bottom: resulting XMLD reflection signal ∆R for two different angles ofincidence

that the novel XMLD-reflection effect can indeed be ob-served we measured the reflection spectrum of ferromag-netic 30-nm Co, capped with 3-nm Al. The XMLD spec-trum obtained in reflection shows a nearly identical double-lobe structure to that of a 50-nm Co film, grown on 100-nmSiN, obtained with the classical XMLD-transmission tech-nique (see Fig. 12). The XMLD in reflection is about oneorder of magnitude larger than that observed in transmis-sion. Moreover, a clear signal could be obtained over a widerange for angles of incidence, so that depth profiling becomesfeasible.

The applicability of the XMLD-reflection effect to AFMsis demonstrated with spectra measured across the Ni 2p edgeof exchange-biased NiO/Co/Ta/Cu (Fig. 13). Large signalswith ∆R/R up to 0.2 can be obtained for incidence angles of72◦ and 60◦. The spectral shape depends sensitively on theangle of incidence. This is typical for reflection spectra re-sulting from interference occurring on layered structures [40].However, the fundamental XMLD spectrum (ε‖ − ε⊥) itselfdoes not depend on the angle of incidence. For a detailedanalysis and a separation between magnetic and structuralfeatures, the above-mentioned computer code (Sect. 4.2) hasto be extended to treat AFMs. With this a quantitative depthprofiling will be able to investigate even deeply buried AFMlayers.

6 Conclusions

Polarization analysis of synchrotron radiation afterits interaction with magnetic material allowed us to com-pletely determine the magneto-optical constants. Utilizingthe Faraday effect, ferromagnetic transmission samples havebeen investigated, probing the sample volume. The Kerr ef-fect, operated in reflection, allowed for a near-surface inves-tigation as well as for magnetic depth profiling. Antiferro-magnetic materials can be examined by the Voigt and XMLDeffects, which are even in the magnetization. The Voigt ef-fect, which needs a polarization analysis, is suited to probethe magneto-crystalline anisotropy in the optical constants.In combination with ab initio calculations it provides a pow-erful tool to investigate the core exchange splitting. The re-cently developed XMLD-reflection spectroscopy allowed usto investigate deeply buried antiferromagnetic layers. Thetechnique can be readily applied to arbitrary samples such

1020 Applied Physics A – Materials Science & Processing

as industrially manufactured devices with oxidized or even‘dirty’ surfaces that cannot be probed by photoelectron-basedmethods.

ACKNOWLEDGEMENTS We acknowledge the growth of sam-ples by O. Zaharko, H. Grimmer, R. Kaltofen, and S. Zotova. This work wassupported financially by the BMBF (Nos. 05KS1 IPB/8 and 05KS1BD1/0)and the Sonderforschungsbereich 463, Dresden, Germany.

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