Quadratic magnetooptic Kerr effect spectroscopy of Fe epitaxial films on MgO(001) substrates
Robin Silber, Ondřej Stejskal, Lukáš Beran, Petr Cejpek, Roman Antoš, Tristan Matalla-Wagner, Jannis Thien, Olga Kuschel, Joachim Wollschläger, Martin Veis, Timo Kuschel, Jaroslav Hamrle
QQuadratic magnetooptic Kerr effect spectroscopy of Fe epitaxial films on MgO(001)substrates
Robin Silber , , , Ondˇrej Stejskal , , Luk´aˇs Beran , Petr Cejpek , Roman Antoˇs , Tristan Matalla-Wagner ,Jannis Thien , Olga Kuschel , Joachim Wollschl¨ager , Martin Veis , Timo Kuschel , Jaroslav Hamrle Nanotechnology Centre, VˇSB-Technical University of Ostrava,17. listopadu 15, 70833 Ostrava, Czech Republic IT4Innovations, VˇSB-Technical University of Ostrava,17. listopadu 15, 70833 Ostrava, Czech Republic Center for Spinelectronic Materials and Devices, Department of Physics,Bielefeld University, Universit¨atsstraße 25, 33615 Bielefeld, Germany Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 12116 Prague, Czech Republic Department of Physics and Center of Physics and Chemistry of New Materials,Osnabr¨uck University, 49076 Osnabr¨uck, Germany (Dated: July 8, 2019)The magnetooptic Kerr effect (MOKE) is a well known and handy tool to characterize ferro-,ferri- and antiferromagnetic materials. Many of the MOKE techniques employ effects solely linear inmagnetization M . Nevertheless, a higher-order term being proportional to M and called quadraticMOKE (QMOKE) can additionally contribute to the experimental data. Here, we present detailedQMOKE spectroscopy measurements in the range of 0.8 – 5.5 eV based on a modified 8-directionalmethod applied on ferromagnetic bcc Fe thin films grown on MgO substrates. From the measuredQMOKE spectra, two further complex spectra of the QMOKE parameters G s and 2 G are yielded.The difference between those two parameters, known as ∆ G , denotes the strength of the QMOKEanisotropy. Those QMOKE parameters give rise to the QMOKE tensor G , fully describing theperturbation of the permittivity tensor in the second order in M for cubic crystal structures. Wefurther present experimental measurements of ellipsometry and linear MOKE spectra, wherefrompermittivity in the zeroth and the first order in M are obtained, respectively. Finally, all thosespectra are described by ab-initio calculations. I. INTRODUCTION
Ferromagnetic (FM) materials have been extensivelystudied due to their essential usage in data storage indus-try. Recently, attention has been attracted to antiferro-magnetic (AFM) materials due to the new possibility tocontrol AFM spin orientation with an electrical current(based on spin-orbit torque effects) [1, 2]. Hence, thereis increasing demand for fast and easily accessible meth-ods for AFM magnetization state characterization. How-ever, most of the methods that are used for FM researchare not applicable to AFM materials due to their lackof net magnetization. Nevertheless, the magnetoopticKerr effect (MOKE) [3] and magnetooptic (MO) effectsin general, which are very powerful tools used in the fieldof FM research, can also be employed in AFM research[4]. Although MOKE linear-in-magnetization (Lin-MOKE), being polar MOKE (PMOKE), longitudinalMOKE (LMOKE), and transversal MOKE (TMOKE)are only applicable to canted AFM and AFM dynam-ics [5–7], it is the quadratic-in-magnetization part of theMOKE (QMOKE) that is employable for fully compen-sated AFM [8].There are several magnetooptic effects quadratic inmagnetization. The QMOKE denotes a MO effect orig-inating from non-zero off-diagonal reflection coefficients( r sp or r ps ), which appear due to the off-diagonal permit-tivity tensor elements (such as ε xy ). On the other hand,magnetic linear dichroism (MLD) and birefringence (also called the Voigt or Cotton-Mouton effect) denote MO ef-fects observed in materials where different propagationand absorption of two linearly polarized modes occur, onebeing parallel and the other perpendicular to the magne-tization vector M (or antiferromagnetic (N´eel) vector L in the case of AFM). These effects originate from differentdiagonal elements of the permittivity tensor (for example ε xx − ε yy for light propagating along the z -direction inthe isotropic sample with in-plane M when M x (cid:54) = M y ).A more comprehensive approach to QMOKE is avail-able, which also takes into account the anisotropy ofQMOKE effects. Individual contributions to QMOKEcan be measured and analyzed, stemming from thequadratic MO tensor G [9], which describes a changeto the permittivity tensor of the crystal in the secondorder in M . The separation algorithm (known as the 8-directional method) has been developed for cubic (001)oriented crystals [10]. It is based on MOKE measurementunder 8 different M directions for different sample orien-tations with respect to the plane of incidence. Althoughapplying this method on AFMs would be considerablychallenging (because magnetic moments of AFMs have tobe reoriented to desired directions), it is not in principleimpossible. Switch of AFM through inverse MO effectsis possible [6, 11] and control of AFM domain distribu-tion was demonstrated by polarization-dependent opticalannealing [12]. But the easiest way to apply above men-tioned separation process on AFMs would be to use easy-plane AFMs such as NiO(111) where sufficiently largemagnetic field will align the moments perpendicular to a r X i v : . [ phy s i c s . op ti c s ] J u l the field direction due to Zeeman energy reduction by asmall canting of the moments [13]. Furthermore, if wetake advantage of exchange coupling to an adjacent FMlayer, e.g. Y Fe O (YIG) [14, 15], the requirement onthe field strength will be substantially lower.Nevertheless, to employ QMOKE measurements on aregular basis, the underlying origin of QMOKE must bewell understood. Although QMOKE effects have beenalready studied, especially in the case of Heusler com-pounds [16–23], all the studies employ single wavelengthsonly. The MOKE spectroscopy together with ab-initiocalculations is an appropriate combination to gain a goodunderstanding of the microscopic origin of MO effects.In the field of LinMOKE spectroscopy, much work hasalready been done, [24–35], but in the field of QMOKEspectroscopy, only few systematic studies have been doneso far [36, 37] where non of those studies is based on ourapproach to QMOKE spectroscopy.Our separation process of different QMOKE contribu-tions is stemming from 8-directional method [10], butwe use a combination of just 4 directions and a sam-ple rotation by 45 ◦ as will be described later in thetext. This approach allows us to isolate QMOKE spec-tra that stem mostly from individual MO parametersand thus subsequently determine spectral dependenciesof G s = G − G and 2 G , denoting MO parametersquadratic in M [17, 38–40]. Therefore, we start our studyon FM bcc Fe thin films grown on MgO(001) substratesto get a basic understanding of QMOKE spectroscopy forfurther studies of AFMs. In the case of FM materials,we can simply orient the direction of M by using a suf-ficiently large external magnetic field and then separatedifferent QMOKE contributions. We present a carefuland detailed study of the MO parameters yielding pro-cess, and discuss all the experimental details that haveto be considered in the process. We also present a com-parison to the ab-initio calculations and values that havebeen reported in the literature so far. Possible sources ofdeviations between reported values are discussed.Note that speaking of MOKE in general within this pa-per, we understand effects in the extended visible spectralrange. There is a vast number of other magnetotransportphenomena in different spectral ranges. In the dc spec-tral range we can mention the well known anomalous Halleffect [41], being linear in M , and anisotropic magnetore-sistance (AMR) [42] together with the planar Hall effect,both being quadratic in M . Recently, in the terahertzregion, an MO effect of free carriers, the so-called opti-cal Hall effect [43], has also received much attention [44].From the x-ray family there is the well known x-ray mag-netic circular (linear) dichroism and birefringence, beinglinear (quadratic) in M [45, 46]. All those (and other)effects (together with LinMOKE and QMOKE) can bedescribed by equal symmetry arguments, predicting thepermittivity tensor contributions of the first and secondorder in M [9]. The same argumentation is valid forother transport phenomena induced, e.g., by heat. Here,thermomagnetic effects such as the anomalous Nernst effect (linear in M )[47–49] and the anisotropic magne-tothermopower together with the planar Nernst effect(quadratic in M )[50–52] define the thermopower (or See-beck) tensor.In the upcoming section II, a brief introduction tothe theory of linear and quadratic MOKE is presented.In section III we describe the sample preparation to-gether with structural and magnetic characterization.Section IV provides the optical characterization and sec-tion V the MO characterization of the samples, be-ing LinMOKE and QMOKE spectroscopy together withQMOKE anisotropy measurements. Finally, in sec-tion VI we compare our experimental findings with ab-initio calculations and the literature. II. THEORY OF LINEAR AND QUADRATICMOKE
The complex Kerr angle Φ s/p for s and p polarizedincident light is defined as [53, 54]Φ s = − r ps r ss = tan θ s + i tan (cid:15) s − i tan θ s tan (cid:15) s ≈ θ s + i(cid:15) s , Φ p = r sp r pp = tan θ p + i tan (cid:15) p − i tan θ p tan (cid:15) p ≈ θ p + i(cid:15) p . (1)Here, θ s/p and (cid:15) s/p are Kerr rotation and Kerr ellipticity,respectively. As | Φ s/p | < ◦ for transition metals [30],one can use small angle approximation in Eq. (1). Thereflection coefficients r ss , r ps , r sp , r pp are the elements ofthe reflection matrix R of the sample described by theJones formalism [54] as R = (cid:20) r ss r sp r ps r pp (cid:21) . (2)These reflection coefficients fundamentally depend on thepermittivity tensor ε (second rank 3 × ε ij of the permittivitytensor are complex-valued functions of photon energy,and its real and imaginary part corresponds to dispersionand absorption of the material, respectively. Changes inthe permittivity tensor with M can be described throughthe Taylor series: ε = ε (0) + ε (1) + ε (2) + ... , where thesuperscript denotes the order in M . In our work weignore all the contributions of third and higher orders in M , expressing the elements of the permittivity tensor ε as ε ij = ε (0) ij + K ijk M k (cid:124) (cid:123)(cid:122) (cid:125) ε (1) ij → LinMOKE + G ijkl M k M l (cid:124) (cid:123)(cid:122) (cid:125) ε (2) ij → QMOKE , (3)where M k and M l are the components of the normalized M . K ijk and G ijkl are the components of the so-calledlinear and quadratic MO tensors K and G of the thirdand fourth rank, respectively [9]. In Eq. (3), the Ein-stein summation convention is used. Thus, the permit-tivity ε up to the second order in M is fully described.The general shape of K and G can be substantially sim-plified using the Onsager relation ε ij ( M ) = ε ji ( − M )and symmetry arguments of the material [9]. The formof these tensors for all crystallographic classes was thor-oughly studied by ˇStefan Viˇsˇnovsk´y [53]. A cubic crystalstructure with inversion symmetry (e.g. bcc Fe as inves-tigated in this work) simplifies the permittivity tensorby ε (0) ij = δ ij ε d , (4a) K ijk = (cid:15) ijk K, (4b) G iiii = G , (4c) G iijj = G , i (cid:54) = j, (4d) G = G = G = G , (4e)with δ ij and (cid:15) ijk being the Kronecker delta and the Levi-Civita symbol, respectively. Hence, ε (0) ij is a diagonal ten-sor described by a scalar ε d for each photon energy. Thelinear MO tensor K is described by one free parameter K whereas the quadratic MO tensor G is determined bytwo free parameters G s = ( G − G ) and 2 G . In theliterature ∆ G = G s − G is also used [17, 39], denotingthe anisotropic strength of the G tensor. The shape ofthese tensors for cubic crystals and its dependence on thecrystal orientation are intensively discussed in the liter-ature [9, 40]. The physical meaning of G s and 2 G isthe following: G s , 2 G denote magnetic linear dichroismwhen magnetization is along the (cid:104) (cid:105) and (cid:104) (cid:105) direc-tions, respectively. Namely, G s = ε (cid:107) − ε ⊥ for M (cid:107) (cid:104) (cid:105) and 2 G = ε (cid:107) − ε ⊥ for M (cid:107) (cid:104) (cid:105) , where the parallel ( (cid:107) )and perpendicular ( ⊥ ) symbols denote the directions oflinear light polarization (i.e. applied electric field) withrespect to the magnetization direction, respectively [40].Let us briefly introduce the most important sign con-ventions. All definitions are based on a right-handed ˆ x , ˆ y ,ˆ z coordinate system as sketched in Fig. 1 with the ˆ z -axisbeing normal to the sample surface (i.e. along Fe[001])and pointing into the sample. The ˆ y -axis is parallel withthe plane of light incidence and with the sample surface,while its positive direction is defined by the direction of k y , being the ˆ y -component of the wave vector of incidentlight. The orientation of the sample is then described byan angle α , being the angle between the Fe [100] directionand the ˆ x -axis of the coordinate system. Transverse, lon-gitudinal and polar components of the normalized mag-netization M T , M L and M P are defined along the ˆ x , ˆ y and ˆ z axes, respectively. Further, sign conventions arediscussed in Appendix A.The analytical approximation for FM layers relatingMOKE with the permittivity of the layer is [17] s x yz P pk p ks M T MM L M L M T xz yAoI(a) F e [ ] M x y(b) (c) FIG. 1. (a) The right-handed coordinate system ˆ x , ˆ y , ˆ z isestablished with respect to the plane of incidence and surfaceof the sample. Components of the in-plane normalized mag-netization M T and M L are defined along the axes ˆ x and ˆ y of the coordinate system, respectively. (b) Definition of theright-handed Cartesian system ˆ s , ˆ p , ˆ k of incident and reflectedbeam. (c) Definition of positive in-plane rotation of the sam-ple and magnetization within the ˆ x , ˆ y , ˆ z coordinate system,described by angle α and µ , respectively. Φ s = − r ps r ss = A s (cid:18) ε yx − ε yz ε zx ε d (cid:19) + B s ε zx , Φ p = r sp r pp = − A p (cid:18) ε xy − ε zy ε xz ε d (cid:19) + B p ε xz , (5)with the weighting optical factors A s/p and B s/p beingeven and odd functions of the angle of incidence (AoI),respectively.In the following, we limit ourselves to in-plane normal-ized magnetization M (cid:107) M (cid:107) = M T M L = cos µ sin µ , (6)where µ is the angle between the M direction and ˆ x -axisof the coordinate system (see Fig. 1). From Eqs. (3)–(6),the dependence of Φ s/p on K , G s , 2 G and on the angles α and µ can be derived as [17, 39, 55]Φ s/p = ± A s/p (cid:26) G (cid:2) (1 + cos 4 α ) sin 2 µ − sin 4 α cos 2 µ (cid:3) + G s (cid:2) (1 − cos 4 α ) sin 2 µ + sin 4 α cos 2 µ (cid:3) (cid:27) ∓ A s/p K ε d sin 2 µ ± B s/p K sin µ. (7)A change of the sign ± is related to the incident s/p polarized light beam. From this expression, measurement
62 64 66 682 [deg]0100200 C o un t s [ a r b . un i t s ] (a)
10 nm12.5 nm15 nm20 nm25 nm30 nm (b)0 25 50 75Intensity [arb. unit]
FIG. 2. (a) XRD Θ – 2Θ scans of the samples with a nominalthickness >
10 nm. Thinner samples do not provide suffi-cient peak intensity. (b) An off-specular XRD scan (Euler’scradle texture map) is presented for the Fe { } peaks at2Θ=44.738 ◦ of the sample with a nominal thickness of 20 nm.The measurement was performed for full 360 ◦ sample rota-tion (angular axis of the plot) with the tilt of the sampleΨ= (cid:104) ◦ ,50 ◦ (cid:105) (radial axis of the plot). sequences providing MOKE spectra originating mostlyfrom individual MO parameters are developed [10] andpresented in Section V. III. PREPARATION, STRUCTURAL ANDMAGNETIC ANISOTROPYCHARACTERIZATION OF THE SAMPLES
A series of epitaxial bcc Fe(001) thin films with var-ious thicknesses were prepared in an Ar atmosphere of2.1 · − bar using magnetron sputtering. The Fe layerwas directly grown on the MgO(001) substrate with agrowth rate of 0.25 nm/s. To prevent oxidation, theFe layer was capped with approximately 2.5 nm of sili-con under the same conditions and with a growth rateof 0.18 nm/s. A reference sample of the MgO substratewith only silicon capping was prepared in order to deter-mine the optical parameters of the capping layer indepen-dently. The sample set contains 10 samples with a nom-inal thicknesses of the Fe layer ranging from 0 nm to 30nm as shown in Tab. I. Furthermore, an additional set ofFe samples grown by molecular beam epitaxy (MBE) onMgO(001) substrates and capped with Si were preparedto investigate the influence of the deposition process onthe magnetooptic properties of Fe. Their preparationand comparison with the sputtered samples is discussedin Appendix C.To verify crystallographic ordering and quality,Phillips X’pert Pro MPD PW3040-60 using a Cu-K α source was employed. X-ray diffraction (XRD) Θ – 2Θscans were performed around 2Θ = 65 ◦ , which is theposition of the characteristic Fe(002) Bragg peak. Thin-ner samples provide very weak peaks due to the lack of
20 nm12.5 nm7.5 nmrefer. capping sample
MgOFe (5-30 nm)Si (1.6 nm)SiOx (0.9 nm) I n t e n s i t y [ a r b . un i t s ] XRR dataSimulation
FIG. 3. Selected XRR scans (blue dots) and their simulation(red line) for several samples from the series. The periodic-ity of oscillations is well described, providing us with reliableinformation about the thickness of the layers in the samples.The damping of oscillations is low, suggesting low roughnessof the interfaces within the samples. The curves are shiftedvertically for clarity.Nominal d Fe d cap σ MgO σ Fe σ cap thickness [nm] [nm] [nm] [nm] [nm] [nm]0.0 nm - 3.4 0.2 - 0.32.5 nm 2.5 2.1 0.4 0.0 0.05.0 nm 4.7 2.4 0.0 0.4 0.27.5 nm 6.9 2.5 0.0 0.3 0.410.0 nm 9.4 2.7 0.2 0.0 0.612.5 nm 11.5 2.5 0.1 0.3 0.315.0 nm 14.0 2.5 0.0 0.2 0.120.0 nm 18.4 2.6 0.1 0.0 0.525.0 nm 23.3 2.4 0.1 0.2 0.630.0 nm 28.3 2.5 0.0 0.0 0.6TABLE I. Thicknesses and roughnesses of the samples, as de-termined from XRR. Thicknesses of Fe layer d Fe and cappinglayer d cap are very robust parameters of the fit and the valuefor the error bars is ± σ in tens of percent provide insignificant change to theresult, it is clear that the roughness is very low as suggested inFig. 3, and the estimated value for the error bars is ± the material in the thin layers as presented in Fig. 2(a).Furthermore, for the sample with a nominal thickness of20 nm, an off-specular texture mapping was performedusing a Euler cradle (Fig. 2(b)). During this scan theFe { } peak at 2Θ = 44 . ◦ was used and we scannedΨ in the range of 40 − ◦ with full 360 ◦ rotation of ϕ ,where Ψ and ϕ are the tilt angle of the Euler cradle andthe rotation angle of the sample around its surface nor-mal, respectively. The result implies that the Fe layerwithin the sample is of good crystalline quality, showinga diffraction pattern in four-fold symmetry.The thickness of each layer and the roughness of each K e rr r o t . [ m d e g ] Fe[100]Fe[110] M a g n e t i c r e m a n e n c e [ m d e g ] (b)(a) F e [ ] F e [ ] Fe [100] H [mT] S a m p l e � o r i e n t a t i o n � α � [ d e g ]
20 15 10 5 0 5 H [mT]555045403530 K e rr r o t . [ m d e g ] H [mT]303540455055 K e rr r o t . [ m d e g ] FIG. 4. Magnetic characterization of the sample with a nom-inal thickness of 12.5 nm. (a) The LMOKE hysteresis loopsat Fe[100] and Fe[110] external field directions. About 75 mTis sufficient to saturate the sample in the in-plane hard axis.(b) In-plane magnetic remanence, with the in-plane magneticeasy and hard axes along Fe (cid:104) (cid:105) and Fe (cid:104) (cid:105) directions, re-spectively. interface was characterized by x-ray reflectivity (XRR)using the same diffractometer as for the XRD measure-ments. To analyze the XRR curves, the open-source pro-gram GenX [56] based on the Parratt algorithm [57] wasused. XRR scans are shown for selected samples in Fig. 3.The periodicity of the oscillations is described very wellby the model, providing reliable information about thethickness values of the Fe layers d Fe and the capping lay-ers d cap . The densities of the layers were fixed parame-ters of the fit and all values were taken from the literature[58, 59]. The thickness of the native silicon oxide couldnot be clearly determined by the XRR technique as Siand SiO x have very similar densities. Hence, the thick-ness of the oxide was estimated (0.9 nm) with respectto the growth dynamics of the native silicon oxide [60].Tab. I summarizes all the values of the thickness androughness provided by the XRR data fit.LMOKE hysteresis curves with an external mag- { d } (a)0 1 2 3 4 5 6 7E [eV]010203040506070 { d } (b) FIG. 5. The (a) real and (b) imaginary part of ε d of Fe layers.Black dashed lines are the ε d of Palik [61] and were used asan initial guess for the fit of the ε d of the Fe layers for all thesamples (full, coloured lines). netic field along Fe[100] and Fe[110] directions mea-sured at λ =670 nm (1.85 eV) are shown in Fig. 4(a).The anisotropy of the magnetic remanence (an averagevalue of positive and negative remanence) presented inFig. 4(b) indicates the fourfold cubic magnetocrystallineanisotropy with the magnetic easy and hard axes alongthe Fe (cid:104) (cid:105) and Fe (cid:104) (cid:105) directions, respectively. Fig-ure 4(b) further suggests that magnetic easy and hardaxes are rotated slightly counter-clockwise with respectto Fe (cid:104) (cid:105) and Fe (cid:104) (cid:105) directions, respectively. Thiscould be explained by a slight misalignment of the samplein the setup with respect to α = 0 ◦ , but more probably byadditional QMOKE contributions to the LMOKE loopsas identified in the inset of Fig. 4(a). The magnetic fieldof ≈
75 mT is enough to saturate the sample in a magneticin-plane hard axis, hence the in-plane magnetic field of300 mT used within QMOKE spectroscopy is more thansufficient to keep the sample saturated with any in-plane M direction. IV. OPTICAL CHARACTERIZATION
The Mueller matrix ellipsometer Woolam RC2 was em-ployed to determine spectral dependencies of ε d for allthe layers within the investigated samples in the spectralrange 0.7 – 6.4 eV. Spectra of ε d of Fe were determined bya multilayer optical model [62], processed using Comple-teEASE software [63]. The thicknesses and roughnessesof the constituent layers were determined by XRR mea-surements. The permittivity of MgO and native SiO x was taken from the literature [61]. From the measure-ment of the reference sample (MgO with the Si cappingonly, with nominal Fe thickness 0 nm), the permittivityof the Si layer was obtained. Hence, for all the remainingsamples, ε d of the Fe layer was the only unknown andfree variable of the fit.The spectra of the imaginary part of ε d for Fe and Silayers were described by B-spline [64], while complemen-tary spectra of the real part were determined throughKramers-Kronig relations. The B-spline is a fast andsturdy method for determining spectra of ε d , but doesnot provide direct information about the electronic struc-ture of the material. The resulting spectra of the real andimaginary part of Fe layers are presented in Figs. 5 (a)and (b), respectively. The sample with a nominal thick-ness of 2.5 nm is deviating from the others, probably dueto low crystallographic quality of the film. Although thecharacteristic peak at 2.5 eV in ε d imaginary spectra ofthe Fe layer is not present in the spectra of Fe by Pa-lik [61], the position of this peak is consistent with otherreports as shown in section VI. V. MAGNETOOPTIC CHARACTERIZATION
Three in-house built MOKE setups were employed tomeasure the LinMOKE and QMOKE response on the sample series. One setup (located at Bielefeld Univer-sity) detects the MOKE with variation of the sample ori-entation α for a fixed photon energy 1.85 eV. Two othersetups detect spectra of MOKE for a fixed sample orien-tation, measuring in the spectral range of 1.6 – 4.9 eV(Charles University in Prague) and 1.2 – 5.5 eV (Tech-nical University of Ostrava), respectively, with perfectagreement of spectra obtained from both setups. Thesample with a nominal thickness of 12.5 nm was later re-measured with an enhanced spectral range of 0.8 – 5.5 eV.A detailed description of the spectroscopic setup at theUniversity of Ostrava can be found in the literature [65].We now describe the QMOKE spectra measurementprocess. Using Eq. (7) (describing the Kerr effect depen-dence on the angles α , µ and the MO parameters K , G s and 2 G ) we derive a measurement procedure separatingMOKE contributions originating mostly from individualelements of the linear and quadratic MO tensors, K and G , respectively. With the specified AoI and sample ori-entation α , we measure MOKE with several in-plane M directions [10]. To rotate M in the plane of the sam-ple, a magnetic field of 300 mT is used and secures thatthe sample is always in magnetic saturation as proven inFig. 4. Three MO contributions can be separated:QMOKE ∼ G s = Q s : Φ µ =45 ◦ s/p + Φ µ =225 ◦ s/p − Φ µ =135 ◦ s/p − Φ µ =315 ◦ s/p ≈ ± A s/p (cid:32) G s − K ε d (cid:33) , α = 45 ◦ AoI = 5 ◦ . (8a)QMOKE ∼ G = Q : Φ µ =45 ◦ s/p + Φ µ =225 ◦ s/p − Φ µ =135 ◦ s/p − Φ µ =315 ◦ s/p ≈ ± A s/p (cid:32) G − K ε d (cid:33) , α = 0 ◦ AoI = 5 ◦ . (8b)LMOKE ∼ K : Φ µ =90 ◦ s/p − Φ µ =270 ◦ s/p ≈ ± B s/p K, α = arb . angleAoI = 45 ◦ . (8c)where ± denotes s/p MOKE effects. The AoI in theequations were chosen with respect to the AoI depen-dence of the optical weighting factors A s/p ∼ cos(AoI)and B s/p ∼ sin(AoI). Hence, the AoI in the Eqs. (8a)–(8c) only affects the amplitude of the acquired spectraand is not essential for the spectra separation process,unlike the sample orientation α and the magnetizationdirections µ that are vital to the measurement sequences.QMOKE and LMOKE spectra were measured at AoI=5 ◦ and 45 ◦ , respectively.QMOKE and LMOKE measurement sequences are de-termined by Eqs. (8a) – (8c), left side, as a difference ofMOKE effects for different magnetization orientations µ at specified sample orientation α . We further use thedenominations Q s and Q for those QMOKE measure-ment sequences in Eqs. (8a) and (8b), respectively. Theright side of Eqs. (8a) – (8c) shows the outcome of thosesequences when using the approximative description ofMOKE, Eq. (5), providing selectivity to G s , 2 G and K within validity of Eq. (5), respectively.The next step is to extract the MO parameters G s ,2 G and K from the measured spectra using the phe-nomenological description of the MOKE spectra by Yeh’s4 × R ofthe multilayer system can be obtained, which allows us tonumerically calculate the MOKE angles of the sample ac-cording to Eq. (1). The thickness and the ε d of each layeris known from XRR and ellipsometry measurements, re-spectively. Nevertheless, the permittivity tensor of theFM layer is described by the sum: ε = ε (0) + ε (1) + ε (2) .Hence, G s , 2 G and K are the unknowns in Yeh’s 4 × G s , 2 G and K is not affected by an approx-imation given by Eq. (5). Finally, we would like to pointout that the condition of proper positive direction of M rotation angle µ must be met. Although the opposite di-rection of M rotation will lead only to the opposite signof experimental spectra, it may lead to completely incor-rect spectra of G s and 2 G parameters upon processing.We have checked that all sign conventions as defined inAppendix A agree with experimental procedures, analyt-ical descriptions, and numerical calculations. For furtherdetails about this issue, please see Appendix B. A. QMOKE Anisotropy
The anisotropy of QMOKE is demonstrated by theso-called 8-directional method [10]. The MOKE sig-nal was detected for 8 in-plane magnetization directions,being µ = 0 ◦ + k · ◦ , k = { , , ..., } . From thosemeasurements, constituent MOKE signals were sepa-rated, being namely LMOKE ∼ M L contribution andtwo quadratic contributions QMOKE ∼ M L M T , andQMOKE ∼ ( M L − M T ). Note that the separation processcould be derived using Eq. (7).The dependences of those three MOKE contributionson the sample orientation α are yielded. In Fig. 6 wepresent all three MOKE contributions measured for thesample with a nominal thickness of 12.5 nm at a pho-ton energy of 1.85 eV and with AoI=45 ◦ . The fourfoldanisotropy of the QMOKE contributions and isotropicLMOKE contribution follow the theory well (see α de-pendence in Eq. (7)). Note that the separation processesof the contributions ∼ M L M T for α = 45 ◦ ,0 ◦ and ∼ M L are identical as described in Eqs. (8a) – (8c), respectively. B. Linear MOKE spectroscopy
The LinMOKE spectra provide the spectral depen-dence of K (after processing by Yeh’s 4 × K /ε d contribu-tion as follows from Eqs. (8a) and (8b). Also, it is appro- K e rr r o t . [ m d e g ] M L M L M T ( M T M L ) C DAB FIG. 6. QMOKE anisotropy measurement at a photon en-ergy of 1.85 eV, with AoI=45 ◦ for the sample with a nominalthickness of 12.5 nm. The dependence of the three MOKEcontributions, being LMOKE ∼ M L , QMOKE ∼ M L M T andQMOKE ∼ ( M T − M L ) on the sample orientation α is demon-strated. Further, several MOKE values are designated in thegraph, being A ○ = A s (cid:16) G − K ε d (cid:17) , B ○ = A s (cid:16) G s − K ε d (cid:17) , C ○ = B s K and D ○ = A s ∆ G . priate to provide the complete spectroscopic descriptionof the samples up to the second order in M within thispaper.The PMOKE spectra for all the samples are presentedin Figs. 7 (a, b). In the Figs. 7(c, d), we present the K spectra obtained from the PMOKE spectra and in thecase of the sample with a nominal thickness of 12.5 nmfrom the LMOKE spectra, as well. The LMOKE spectraare presented in Fig. 7(e). It should be noted that thePMOKE spectra were measured with the magnetic fieldof 1.2 T which is not enough to magnetically saturate thesamples out-of-plane. Nevertheless, the PMOKE spectramultiplied by a factor of 2.2 yield spectra in excellentagreement with the K spectra from the LMOKE spec-troscopy, both measured on the sample with the nominalthickness of 12.5 nm. We find this excellent agreement asthe confirmation of the correctness of the determinationof the optical constants of ε d and K from experimentaldata. Note that all the presented spectra in the followingSection VI are recorded only from MOKE measurementswith in-plane magnetization, where the samples were al-ways magnetically saturated.Finally, the dependence of the PMOKE scaled to themagnetization saturation on the Fe layer thickness at aphoton energy of 1.85 eV is shown in Fig. 7(f). The ex-perimental data follows the predicted dependence well.All the values that were needed for the Yeh’s 4 × P M O K E ( p ) [ d e g ] (a) P M O K E ( p ) [ d e g ] (b) E [eV] { K } (c) E [eV] { K } (d) E [eV] L M O K E [ m d e g ] (e) s s Fe thickness [nm] P M O K E [ d e g ] (f) p exp. p exp. p calc. p calc. FIG. 7. Experimental PMOKE spectra of (a) Kerr rotation θ p and (b) Kerr ellipticity (cid:15) p at AoI=5 ◦ , scaled to magnetizationsaturation. Spectra of the (c) real and (d) imaginary part ofthe MO parameter K yielded from the saturated PMOKEspectra do not differ significantly with the thickness (exceptfor the sample with a nominal thickness of 2.5 nm). The K spectra provided by LMOKE spectroscopy of the sample witha nominal thickness of 12.5 nm agree very well with K spectraobtained from the saturated PMOKE spectra. (e) LMOKEspectra of the sample with a nominal thickness of 12.5 nm.(f) Thickness dependence of PMOKE at a photon energy of1.85 eV at AoI=5 ◦ . value of K was provided by LMOKE spectroscopy, hencethe experimental value at a nominal thickness of 12.5 nmdoes not absolutely follow predicted amplitude as onecan notice in Fig. 7(f)). A small disagreement betweenother experimental and calculated values is due to bothslightly different ε d and K for different Fe thicknesses, aswell as a probable small difference in the scaling factorfor different Fe layer thicknesses. Q s ( s ) [ m d e g ] (a) 403020100 Q s ( s ) [ m d e g ] (b)0 1 2 3 4 5 6E [eV]6050403020100102030 Q ( s ) [ m d e g ] (c) 0 1 2 3 4 5 6E [eV]908070605040302010010 Q ( s ) [ m d e g ] (d) FIG. 8. (a) Rotation and (b) ellipticity of Q s spectra. (c)Rotation and (d) ellipticity of Q spectra. All of them mea-sured with s − polarized incident light. The measured spectrawere digitally processed (smoothed) with a Savitzky-Golayfilter over the photon energy to improve signal-to-noise ratio.The sample with a nominal thickness of 12.5 nm was mea-sured with an extended spectral range of 0.8 – 5.5 eV. Again,the thinnest sample with a nominal thickness of 2.5 nm showsthe largest deviation compared to the other samples of thethickness dependent series. C. Quadratic MOKE spectroscopy
The QMOKE spectra for all the samples were mea-sured according to Eqs. (8a) and (8b). The measuredspectra in the range of 1.6 – 4.8 eV are presented inFig. 8. The sample with a nominal thickness of 12.5 nmwas measured at the setup with an extended spectralrange of 0.8 – 5.5 eV. Recall, measured QMOKE also hasa contribution from the linear term K , being proportionalto K /ε d M L M T provided by cross terms ε yz ε zx /ε d and ε zy ε xz /ε d (Eq. (5)). Let us emphasize, this quadratic-in-magnetization contribution to MOKE arises from opticalinterplay of two off-diagonal permittivity elements, bothbeing linear in magnetization.The deduced spectra of the quadratic MO parameters G s and 2 G are shown in Fig. 9. The shape of thespectra do not substantially change with the thickness,showing that there is no substantial contribution fromthe interface.The only exception (apart from the sample with a nom-inal thickness of 2.5 nm, which is also deviating in all pre-vious measurements) is the real part of the 2 G spectrabelow 2 eV for the sample with a nominal thickness of10 nm. The source of this deviation stems from the in- { G s } (a) { G s } (b)0 1 2 3 4 5 6E [eV]0.1000.0750.0500.0250.0000.025 {2 G } (c) 0 1 2 3 4 5 6E [eV]0.050.040.030.020.010.00 {2 G } (d) FIG. 9. Spectra of the (a) real and (b) imaginary part ofthe quadratic MO parameter G s and the (c) real and (d)imaginary part of the quadratic MO parameter 2 G for allthe samples of the series. terplay of two sources: (i) the ellipticity of Q spectra isalmost twice large in the case of this sample, comparedto others (see Fig. 8(d)). (ii) The value of K is above1 in spectral range below 2 eV (for both the real andimaginary part, and in the absolute value). Thus, thecontribution of K /ε d is the dominant contribution to Q spectra below 2 eV, and therefore a small change inthe Q spectra will substantially affect the yielded 2 G spectra.In Appendix C we further present a comparison of K , G s and 2 G spectra of the sample with a nominal thick-ness of 12.5 nm (prepared by magnetron sputtering) andthe sample prepared by MBE. Analogous instability ofthe 2 G parameter can actually be observed here as well.A rather small difference in yielded K spectra and mea-sured Q spectra provides a significant change of resultin yielded 2 G spectra. Otherwise, the spectra of sam-ples grown by two different techniques follow the samequalitative progress, but deviate slightly in the magni-tude, probably due to small differences in the crystallinequality.In Figs. 10 (a) and (b) we present the measured andcalculated Fe layer thickness dependence for Q s and Q , respectively. The dependence is for a photon en-ergy of 1.85 eV and the calculations are provided byYeh’s 4 × ◦ (being the AoIused within the experiment), where ε d , K , G s and 2 G were taken from the sample with a nominal thickness of12.5 nm. The theoretical dependence slightly differs fromexperimental results for thinner Fe layers. This could beexplained by slightly different ε d , K , G s and 2 G for the Q s [ m d e g ] (a) s (exp.) s (exp.) s (calculated) s (calculated) Q [ m d e g ] (b)0 1 2 3 4 5 6E [eV]0.100.050.000.05 G E [ e V ] (c) RealImaginary
FIG. 10. Thickness dependence of (a) Q s and (b) Q for aphoton energy of 1.85 eV. Lines were provided by Yeh’s 4 × ◦ . (c) The spectral dependence of the real and imaginarypart of ∆ G = G s − G represents the anisotropy strengthof the quadratic MO tensor across the whole spectral range.Every point is weighted by its photon energy for clarity. thinner samples as shown in Figs. 5, 7 and 9, respectively,as well as slightly different material properties of cappinglayers in each sample. Strong deviation could be seen inthe case of the experimental value of Q for the samplewith a nominal thickness of 10 nm, as already discussedabove.The parameter ∆ G = G s − G provides informationabout the anisotropy strength of the quadratic MO ten-sor G [40]. Its spectral dependence for the sample witha nominal thickness of 12.5 nm is presented in Fig. 10(c), shown in the form ∆ G · E , i.e. multiplied by photonenergy. VI. COMPARISON OF EXPERIMENTALSPECTRA WITH CALCULATIONS AND THELITERATURE
In this Section, we discuss the comparison of exper-imental spectra with ab-initio calculations and the lit-erature. All the representative experimental data withinthis section are from the sample with a nominal Fe thick-ness of 12.5 nm. Further, all the spectra in this section are0 { d E [ e V ] (a)0 1 2 3 4 5 6E [eV]010203040506070 { d } E [ e V ] (b) Ab-initioExperiment (12.5 nm)Yolken and KrugerJohnson and ChristyWeaver et al . Bolotin et al .Oppeneer et al . ( = 0.68 eV)Oppeneer et al . ( = 0.41 eV)Buchmeier et al .Liang et al . FIG. 11. Experimental (markers) and ab-initio calculated in-terband spectra (lines) of (a) real and (b) imaginary part of( ε d − · E [eV]. Experimental spectra acquired in this workhave marker every 10 experimental points (blue bullets). Theremaining spectra are taken from literature [26, 38, 66–68]. expressed in the form multiplied by photon energy E , be-ing an alternative expression of the conductivity spectra.Note that this is analogous to the well-known relationof conversion between complex permittivity and complexconductivity tensor ε ij = δ ij + iσ ij (cid:126) / ( ε E ), where E isthe photon energy and δ ij the Kronecker delta.The electronic structure calculations of bcc Fe [69] wereperformed using the WIEN2k [70] code. The used latticeconstant for all calculations was the bulk value, being2.8665 ˚A. The electronic structure was calculated for two M directions parallel to Fe[100] and Fe[011], respectively.We used 90 = 729000 k -points in the full Brillouin zone.The product of the smallest atomic sphere and the largestreciprocal space vector was set to R MT K max = 8 with themaximum value of the partial waves inside the spheres, l max = 10. The largest reciprocal vector in the chargeFourier expansion was set to G max = 12 Ry / . The ex-change correlation potential LDA was used within all cal-culations. The convergence criteria were 10 − electrons { K } E [ e V ] (a)0 1 2 3 4 5 6E [eV]1012345 { K } E [ e V ] (b) ExperimentAb-initioKrinchik et al .van Engen et al .Ferguson et al .Oppeneer et al . ( = 0.68 eV)Oppeneer et al . ( = 0.41 eV)Buchmeier et al .Liang et al . FIG. 12. Experimental (markers) and ab-initio calculated in-terband spectra (lines) of the (a) real and (b) imaginary partof K · E [eV]. Experimental spectra acquired in this work havea marker every 5 experimental points (blue bullets). The re-maining spectra are taken from the literature [24–26, 38, 68]. for charge convergence and 10 − Ry=1 . − eV for en-ergy convergence. The spin-orbit coupling is included inthe second variational method.The Fermi level was determined by temperature broad-ened eigenvalues using broadening 0.001 Ry (0.014 eV).The optical properties were determined within electricdipole approximation using the Kubo formula [26, 71].The Drude term (intraband transitions) is omitted in theab-initio calculated optical and MO properties. We dis-cuss possibilities of how to handle the Drude contributionin Appendix D. By broadening the spectra and applyingKramers-Kronig relations, we obtain a full permittivitytensor ε for each direction of M . The spectra for K , G s and 2 G are obtained directly from the permittivitytensors ε [72]. K = 12 (cid:16) ε ([100]) yz − ε ([100]) zy (cid:17) , (9a) G s = ε ([100]) xx − ε ([100]) yy , (9b)2 G = ε ([011]) yz + ε ([011]) zy , (9c)where the superscript denotes the M direction in thecrystallographic structure.1 { G s } E [ e V ] (a) ab-initioExp.(12.5 nm)Sepulveda et al . (multip. by 5)Buchmeier et al .Liang et al . { G s } E [ e V ] (b) E [eV] {2 G } E [ e V ] (c) E [eV] {2 G } E [ e V ] (d) FIG. 13. The experimental G s spectra (a, b) and the experimental 2 G spectra (c,d) compared with the ab-initio calculations. G s spectra are calculated for (cid:126)M (cid:107) [100] and 2 G spectra calculated for (cid:126)M (cid:107) [011] both with smearing of FWHM=1.2 eV andwith 90 = 729000 k -points in the full Brillouin zone. Further, we show a comparison with data taken from the literature[36, 38, 68]. The spectra taken from Sep´ulveda et al. [36] have been multiplied by a factor of 5 to be comparable with ourexperimental spectra. Figures 11(a) and (b) present experimental spectra of ε d − et al. [26] in thesame figure. The imaginary (absorption) part of the di-agonal permittivity, (cid:61) ( ε d ) is dominated by the absorp-tion peak at 2.4 eV. This peak originates from transitionsof mostly-3d down electrons above and below the Fermilevel. The ab-initio calculated peak position is very stableregarding small changes of the lattice constant, magne-tization direction, and small distortion of the Fe lattice.On the other hand, the peak position is determined bythe selected exchange potential, where LDA provides theclosest match to the experimental results, while otherpotentials (GGA, LDA+U, GGA+U) display larger de-viation from the experimental peak position. Thereforewe choose the LDA exchange potential to calculate theelectronic structure of bcc Fe, although LDA still overes-timates the width of the occupied 3d bands. The widthof the occupied 3d bands can be corrected using dynam-ical mean-field theory (DMFT) [73]. Further, note thatthe peak amplitude depends on the smearing parameter[26], and we chose smearing δ = 0 . ε d to adjust the peak height.Figures 12 (a) and (b) show a comparison between ex-perimental and ab-initio calculated spectra of K , demon-strating excellent agreement. Note the absorption partcorresponds to (cid:60) ( K ), with two peaks at 2.0 and 1.1 eV. The amplitude of (cid:60) ( K · E ) is about -2.5 eV, i.e. about4% of the maximal value of (cid:61) ( ε d · E ) being about 60 eV.Although in both figures (Figs. 11 and 12) absolute val-ues differ by dozens of percent for some photon energies,the peaks and courses of spectra, being characteristic forthe given material, are very similar for all the presenteddata, both experimental and theoretical (note that dis-agreement with the reported values at single wavelength[38, 68] is probably due to sign inconsistency). Further,the d.c. limit of the imaginary part of the K spectra cor-responds to the anomalous Hall conductivity. Its valueextracted from the ab-initio calculation is 512 (Ωcm) − (760 (Ωcm) − without broadening) agreeing with thevalue provided in Ref.[74]. Finally, note that sign of ab-initio (Wien2k ver. 17) calculated K -spectra is reversed,to agree with the sign of the experimental K -spectra (thissign error was corrected in Wien2k ver. 19.1).Figure 13 shows experimental spectra of the real (a)and imaginary (b) part of G s spectra, compared with theab-initio calculations. The fundamental (imaginary) partof G s has a pronounced peak at 1.6 eV with the amplitudein the experimental spectra being (cid:61) ( G s · E ) = − .
11 eV.The main features of G s are well-described by ab-initiospectra. However, the ab-initio calculated peak at 1.6 eVhas about half that amplitude. Figures 13 (c) and (d)show the real and imaginary part of the experimentalspectra of 2 G , respectively, compared to the ab-initiocalculations. In the case of the fundamental part of 2 G (cid:60) ( G s ), (cid:60) (2 G )and their ab-initio descriptions (particularly for smallphoton energies) could be due to the missing Drude term,which is omitted in the ab-initio calculations, and whichmainly contributes to the real part of the permittivity atsmall photon energies. Finally, note that in the ab-initiocalculations, convergence (for example on density of the k -mesh) of 2 G is much better compared to G s , as G s iscalculated as a small change of the diagonal permittivities(Eq. (9b)) whereas 2 G is calculated from off-diagonalpermittivity (Eq. (9c)).Further, we show the comparison of the spectral de-pendence of G s and 2 G from Sep´ulveda et.al. [36].The spectra had to be multiplied by a factor of 5 to becomparable to our experimental and the ab-initio spec-tra. Then, the agreement is perfect for the real part ofboth G s and 2 G in the spectral range 1.5–4.0 eV. Thedisagreement of spectral dependence under 1.5 eV can beexplained by different sample quality; as the same be-haviour was already experienced for 2 G in the case ofthe sample with a nominal thickness of 10 nm and alsoin the case of the sample prepared by the MBE, whichis discussed in a previous section and in Appendix C,respectively. The comparison of the imaginary part of G s and 2 G between our data and the scaled data ofSep´ulveda et al. [36] provide very similar behaviour ex-cept for some offset and also different amplitude of peaks,especially in case of the (cid:61) (2 G ) peak at 1.5 eV. We donot know wherefrom the scaling factor 5 between ourdata and data of Sep´ulveda et.al. is stemming. In thecase of Sep´ulveda et.al. the data were obtained from ex-perimental measurement of variation of reflectivity withquadratic dependence on magnetization. The poor qual-ity of the samples can be ruled out, as in the case of poly-crystalline material ∆ G = 0, i.e. G s = 2 G , which isnot the case here. However, note that our optical spectraof ε d , K , 2 G and G s well describe their experimentalreflectivity spectra using our numerical model. VII. CONCLUSION
We provided a detailed description of our approachto the QMOKE spectroscopy, which allows us to obtainquadratic MO parameters in the extended visible spectralrange. The experimental technique stems from the 8-directional method that separates linear and quadraticMOKE contributions.The quadratic magnetooptic parameters G s and 2 G of bcc Fe (expressing magnetic linear dichroism of permit-tivity along the [100] and [110] directions, respectively)were systematically investigated. The spectral depen-dence of G s and 2 G is experimentally determined in thespectral range 0.8 – 5.5 eV, being acquired by QMOKEspectroscopy and numerical simulations using Yeh’s 4 × K in the spectral range 0.8 – 5.5 eVand the diagonal permittivity ε d in the spectral range 0.7– 6.4 eV were also acquired.Further, all measured permittivity spectra are com-pared to ab-initio calculations. The shapes of those spec-tra are well described by electric dipole approximation,with the electronic structure of bcc Fe calculated usingDFT with LDA exchange-correlation potential and withspin-orbit coupling included. However, to describe G s and 2 G , a fine mesh of 90 × ×
90 is used as G s is cal-culated as a small variation of diagonal permittivity ε ii with magnetization direction.With the measurement process well established, thetechnique is ready to be used on other ferromagnetic ma-terials, and also tested on antiferromagnetic materials.A suitable candidate could be the easy-plane AFM NiOgrown on a ferri- or ferromagnetic support in order tocontrol the AFM by the exchange coupling to the addi-tional ferri- or ferromagnetic layer which can be magnet-ically aligned by an external field. In such a bilayer, thecontribution of the ferri- or ferromagnetic layer has to bestudied separately in the same manner as we have donehere for bcc Fe. ACKNOWLEDGMENTS
The authors thank G¨unter Reiss, Jarom´ır Piˇstora, Ger-hard G¨otz and John Cawley for support, assistance anddiscussion. This work was supported by Czech ScienceFoundation (19-13310S) and the Deutsche Forschungs-gemeinschaft (DFG Re 1052/37-1). The work was alsosupported by the European Regional Development Fundthrough the IT4Innovations National SupercomputingCenter - path to exascale project, project number CZ.02 . . / . / . /
16 013 / . . / . / . /
15 003 / Appendix A: Sign conventions
Within the fields of optics and magnetooptics, thereis a vast amount of conventions. As there is no gener-ally accepted system of conventions, we define here allconventions adopted within this work.To describe reflection from a sample, three Cartesiansystems are needed, one for incident light beam, one forreflected light beam and one for the sample. All thoseCartesian systems are right-handed and defined in Fig. 1of the main text.3
Time convention:
The electric field vector of an electromagneticwave is described by negative time convention as E ( r , t ) = E ( r ) e − iωt , providing permittivity in theform ε = (cid:60) ( ε ) + i (cid:61) ( ε ), where the imaginary part ofcomplex permittivity (cid:61) ( ε ) > Cartesian referential of the sample:
The Cartesian system describing the sample is theright-handed ˆ x , ˆ y , ˆ z system, where ˆ z -axis is nor-mal to the surface of the sample, and points intothe sample. The ˆ y -axis is parallel with the plane oflight incidence and with the sample surface, whileits positive direction is defined by the direction of k y , being the ˆ y -component of the wave vector ofincident light as shown in Fig. 1. In this system,rotations of the crystallographic structure and mag-netization take place. Cartesian referential of light:
We use the right-handed Cartesian system ˆ s , ˆ p , ˆ k for description of the incident and reflected lightbeam. The direction of vector ˆ k defines the direc-tion of propagation of light. Vector ˆ p lies in theincident plane, i.e. a plane defined by incident andreflected beam. The vector ˆ s is perpendicular tothis plane and corresponds to ˆ x . This conventionis the same for both incident and reflected beams(Fig. 1). Convention of the Kerr angles:
The Kerr rotation θ is positive if azimuth θ of thepolarization ellipse rotates clockwise, when lookinginto the incoming light beam. The Kerr ellipticity (cid:15) is positive if temporal evolution of the electric fieldvector E rotates clockwise when looking into theincoming light beam. Convention of rotation of the sample, themagnetization and the optical elements :
The rotation is defined as positive if the rotatedvector pointing in the ˆ x (ˆ s ) direction rotates to-wards the ˆ y (ˆ p ) direction. The sample orientation α = 0 corresponds to the Fe[100] direction beingparallel to the ˆ x -axis and, when looking at the topsurface of the sample, the positive rotation of thesample is clockwise. Likewise, the magnetizationdirection µ = 0 corresponds to M being in the pos-itive direction of the ˆ x -axis and, when looking atthe top surface of the sample, the positive rotationof magnetization M is clockwise. Further, whenlooking into the incoming beam, the positive ro-tation of the optical elements is counter-clockwise,in contrast to the positive Kerr angles, defined byhistorical convention. Appendix B: Consequences of the MOKE signdisagreement between the experimental andnumerical model
The correct sign of LMOKE and QMOKE spectra isgiven by the conventions used. Nevertheless, to obtainthe correct spectra of MO parameters K , G s and 2 G ,the same conventions must be adopted within the numer-ical model and the experiment. One would intuitivelyexpect only the reversed sign of yielded MO parameters,when the sign conventions of the experiment and the nu-merical model do not comply. However, completely in-correct values are yielded in this case for the quadraticMO parameters.There are numerous points in the experiment where wecan go wrong and thus measure the MOKE spectra of theincorrect sign according to our conventions, e.g. wrongdirection of in-plane M rotation (i.e. µ → − µ ), wrongdirection of positive external field and thus opposite di-rection of M (i.e. µ → µ + 180 ◦ ), error in the calibrationprocess of the setup itself (note that the positive directionof the optical element rotation and the positive directionof the Kerr rotation have opposite conventions) or somequirk in the processing algorithm of the measured dataitself (usually we measure change of intensity, which hasto be converted to Kerr angles). Further, we can alsomake a sign error in the code of the numerical model.The correct sign of the numerical model output can bechecked for by comparison to the simple analytic modelthat exist for some special cases. E.g. PMOKE effect Φat the normal angle of incidence for a vacuum/FM(bulk)interface within our sign convention:Φ = ε xy (cid:113) ε ( F M ) d (1 − ε ( F M ) d ) (B1)Various sign mistakes in the experiment will not al-ways lead to the same error, e.g. the wrong directionof positive external magnetic field will affect the sign ofLMOKE spectra but not the QMOKE spectra. On theother hand the wrong direction of M rotation will pro-duce a wrong sign of both spectra, LMOKE and QMOKEalike - see the Eqs.(8a)–(8c).In the following, we will discuss a consequence of thelatter case, when the direction of M rotation has theopposite direction, µ (cid:48) → − µ leading to a wrong sign ofexperimental spectra measured according to Eqs.(8a)–(8c). While linear MO parameter K (cid:48) , yielded from theLMOKE spectra with a reversed sign, will only have theopposite sign compared to the true MO parameter K ,the quadratic MO parameters G (cid:48) s and 2 G (cid:48) , yielded fromthe Q s and Q spectra with the opposite sign, will becompletely different from the true MO parameters G s and 2 G , respectively. This is due to the contributionof K /ε d to the Q s and Q spectra, which are invariantto the sign of K itself. Thus, the MO parameters yieldedfrom sign-reversed experimental spectra are bound withthe true MO parameters by following equations.4 K E [ e V ] (a) { K } (correct){ K } (correct) { K } (wrong){ K } (wrong) G s E [ e V ] (b) { G s } (correct){ G s } (correct){ G s } (wrong){ G s } (wrong) G E [ e V ] (c) {2 G } (correct){2 G } (correct){2 G } (wrong){2 G } (wrong) FIG. 14. Comparison of the spectra of MO parameters of (a) K , (b) G s and (c) 2 G to the MO parameters yielded fromthe experimental spectra with the wrong (reversed) sign of µ ( K (cid:48) , G (cid:48) s and 2 G (cid:48) ). K (cid:48) = − K (B2) G (cid:48) s = − G s + 2 K ε d (B3)2 G (cid:48) = − G + 2 K ε d (B4)In Fig. 14 we show the wrong MO parameters K (cid:48) , G (cid:48) s and 2 G (cid:48) compared to the true MO parameters K , G s and 2 G .Note that neither the shape nor the sign of the trueMO parameters is given by the convention used. Anysign conventions can be adopted, but the crucial pointis that the conventions used in real experiments and innumerical calculus are the same. Obviously, this issueapplies to any error in the experimental setup or the nu-merical code that would unintentionally reverse the signof the measured or calculated MOKE spectra, respec-tively. Appendix C: Comparison of the samples grown bymolecular beam epitaxy and by magnetronsputtering
Fe and Si films were prepared on a single crystallineMgO(001) substrate via molecular beam epitaxy (MBE).Prior to deposition, the substrates were annealed at400 ◦ C for 1h in a 1 · − mbar oxygen atmosphere to re-move carbon contamination and obtain defined surfaces.Fe films were deposited by thermal evaporation from apure metal rod at a substrate temperature of 250 ◦ C. Sili-con capping layers were evaporated at room temperatureusing a crucible. The deposition rates of 1.89 and 0.3nm/min for Fe and Si, respectively, were used and con-trolled by a quartz microbalance next to the source. Thebase pressure in the UHV chamber was 10 − mbar.The XRD and XRR were measured as described insection III. A thickness of 12.6 nm was determined byXRR for the MBE prepared Fe layer and 7.0 nm for theSi+SiO x capping layer. The thickness of the referencesample with only Si+SiO x capping was 8.1 nm. The XRDΘ – 2Θ scan was performed around 2Θ = 65 ◦ and showedthat the samples are of good crystallinity. Further, theellipsometry, LMOKE and QMOKE spectroscopy weremeasured on the sample to feed the Yeh’s 4 × K , G s , and 2 G obtained by numerical calculationsare presented and compared to the spectra of the sputter-deposited sample with a nominal thickness of 12.5 nm inFigs. 15(a)–(c), respectively. The behaviour of the spec-tra of both samples is very similar, except for the realpart of 2 G spectra at lower photon energies. Neverthe-less the same discrepancy has already been discussed inthe section V C for the case of the 10 nm sample. Oth-erwise the differences of absolute values across spectraare not surprising, as the reported experimental valuesof MO parameters differ for different samples preparedby different deposition techniques and different groups(as shown in Figs. 11, 12 and 13), probably being con-nected with slightly different crystalline qualities of theFe layer. Appendix D: The Drude contribution
The contribution of intraband transitions in the di-agonal permittivity could be described by the classicalphenomenological Lorentz-Drude model (in the follow-ing, called the Drude term) ε D = 1 − E p E + i Γ E , (D1)where E is the photon energy, E p = (cid:126) ω p is the plasmaenergy describing the strength of the oscillator, with ω p being the plasma frequency and Γ = (cid:126) τ − the dampingconstant, and 1 stands for the relative vacuum permit-tivity.5 K E [ e V ] (a)0.200.150.100.050.000.05 G s E [ e V ] (b) , Sputtering, Sputtering , MBE, MBE G E [ e V ] (c) FIG. 15. Comparison of the spectra of the MO parameters of(a) K , (b) G s and (c) 2 G of two samples, one prepared bymagnetron sputtering and the other by MBE. For each of thesamples, all the data used within numerical calculations wereobtained from the parameters of the particular sample. In order to include the Drude term into G -spectra,first recall that G s and 2 G express magnetic lineardichroism (MLD), G s = ε (cid:107) − ε ⊥ for M (cid:107) (cid:104) (cid:105) and2 G = ε (cid:107) − ε ⊥ for M (cid:107) (cid:104) (cid:105) where parallel ( (cid:107) ) andperpendicular ( ⊥ ) denote the direction of the appliedelectric field (i.e. linear light polarization) with respectto the magnetization direction. Second, we assume both ε (cid:107) and ε ⊥ are described by the Drude model Eq. (D1),however with plasma energy E p and damping constant Γslightly different for both (cid:107) and ⊥ directionsMLD D = ε (cid:107) − ε ⊥ = ∆ E p ∂∂E p ε D + ∆Γ ∂∂ Γ ε D = ε D (cid:34) E p E p − i ∆Γ − E + i Γ (cid:35) , (D2)where MLD D denotes the Drude contribution to mag- netic linear dichroism, with ∆ E p = E p, (cid:107) − E p, ⊥ and∆Γ = Γ (cid:107) − Γ ⊥ being differences of the plasma energyand the damping constant between parallel and perpen-dicular magnetization directions, respectively. Due tothe anisotropy of G -spectra, ∆ E p and ∆Γ have differentvalues for M (cid:107) (cid:104) (cid:105) and M (cid:107) (cid:104) (cid:105) .The number of free parameters in the Eqs. (D1) and(D2) can be reduced from four to two if values of d.c.conductivity and AMR are known. One part of the sam-ple with a nominal thickness of 12.5 nm was patternedinto a Hall bar with a top down process using UV lithog-raphy and Argon milling. Four point conductivity mea-surements were performed for various applied currentsin the [100] direction from 50 to 500 µ A. The charac-teristic dimensions of the Hall bar are length = 635 µ m,width = 80 µ m and height = 11.5 nm. The resistivityand thus the conductivity was determined by performinga linear fit to the data. One obtains conductivity valuesof σ (cid:107) = 0 . · S/m and σ ⊥ = 0 . · S/m, be-ing the conductivity with M parallel and perpendicularto the current, respectively. The AMR value of 0.45%correspond well with the literature [75].The conductivity σ and relative permittivity ε are re-lated by ( ε − ε = iσ (cid:126) /E , where ε is the vacuum per-mittivity and (cid:126) is the reduced Planck constant. Hence,at E = 0, d.c. conductivity is σ = ε (cid:126) E p Γ (D3)and the (d.c.) anisotropy magnetoresistance isAMR = σ ⊥ − σ (cid:107) = ε (cid:126) lim E → (cid:104) iE ( ε (cid:107) − ε ⊥ ) (cid:105) = σ (cid:34) E p E p − ∆ΓΓ (cid:35) . (D4)In order to discuss the Drude contribution to G s and 2 G , we first determine the Drude contribution to( ε d − E p = 4 .
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