T Kuschel

H Bardenhagen

H Wilkens

R Schubert

J Hamrle

J Pištora

J Wollschläger

Vectorial magnetometry using magnetooptic Kerr effect including first and second order contributions for thin ferromagnetic films

published or not. The documents may come

Vectorial magnetometry using magnetooptic Kerr effect including first-and second-order contributions for thin ferromagnetic films

I. INTRODUCTION

The magnetooptic Kerr effect (MOKE) discovered by John Kerr 1 and reported in 1877 is a powerful tool for analysis of magnetic properties of ferromagnetic samples. Incident linear polarized light is reflected from a surface of a magnetized sample. The reflected light is elliptically polarized and its main axis is rotated with respect to the incident polarization.

The angle of rotation and the amount of ellipticity depend on the magnetization of the sample. Thus, after analyzing the reflected light, sample properties like magnetic anisotropy, magnetic domains and the magnetization reversal process can be determined. The first MOKE investigations on thin magnetic films were reported in 1985 by Moog and Bader 2 .

Henceforward, MOKE has become an important analysis tool for investigations on thin magnetic films.

One interesting application for MOKE is vectorial magnetometry, abbreviated vector MOKE. The different components of the magnetization vector can be separately determined by this MOKE technique. There have been several possibilities used to realize this. Either the alignment of the external magnetic field was kept constant and the adjustment of the optical components had to be changed 3,4 or the optical components were not modified and the direction of the external magnetic field had to be varied 5,6 . In this work both methods are combined to have some advantages described below. For a review of the methods used so far some vectorial magnetometry experiments are shortly described.

Florczak and Dahlberg reported on measurements of the in-plane magnetization components with longitudinal MOKE (LMOKE) and transversal MOKE (TMOKE) 3 . LMOKE is generated by the in-plane component of the magnetization parallel to the plane of incidence of light and TMOKE is caused by the in-plane component of the magnetization perpendicular to the plane of incidence of light. The last magnetization component (out-of-plane) generates polar MOKE (PMOKE). Both LMOKE and PMOKE cause a change of polarization of the incident light, while TMOKE changes the intensity of incident parallel polarized light, but no change of polarization.

Usually, the magnetization in thin films is energetically preferred to be in-plane due to the magnetic shape anisotropy. Therefore, an additional signal of the PMOKE can often be neglected. Although the determined Kerr signal for each magnetization component is proportional to its generating magnetization component, the proportionality factors between Kerr signals and magnetization components are different. Therefore, the two Kerr signals of the in-plane components (change of polarization for LMOKE, change of intensity for TMOKE) can only be compared qualitatively, but not quantitatively. For example, an exact in-plane magnetization vector cannot directly be reconstructed from the measured data.

A possible solution to determine the relation between the separate components of the magnetization vector is to normalize the Kerr signals to their saturation values as reported by Vavassori 7 . Therefore, either an external magnetic field in the direction of each measured component is needed to saturate the magnetization signal or the saturation value has to be estimated by other measurements and calculations. Vavassori combined both methods:

The external magnetic field is applied for both in-plane components and the calculation is done for the out-of-plane component to reconstruct the complete magnetization vector in dependence of the external magnetic field 7 . Although this technique is very successful to determine the magnetization for one specific sample, it has the drawback that different samples can only be compared qualitatively due to the normalization to the saturation magnetization.

Another method to obtain the in-plane magnetization components is described by Daboo et al. 5 . Both components are determined with LMOKE. For one of them the external magnetic field is parallel to the plane of incidence of light. For the other one the sample and the external magnetic field is rotated by 90 • . Thus in the latter case, the magnetization component perpendicular to the external magnetic field generates the LMOKE (change of polarization). So, the signals of both in-plane components are obtained with LMOKE and therefore, they are directly comparable without any normalization. Thus, the complete reversal process of the magnetization can be reconstructed and different samples can be compared quantitatively in units of Kerr rotation, only if the out-of-plane magnetization component can be neglected.

If the latter assumption cannot be applied, the out-of-plane component also contributes to the polarization of the detected light and the experimental data may be misinterpreted.

The in-plane and out-of-plane components of magnetization, however, can be separated combining signals from parallel (p) and perpendicular (s) polarized incident light. The sum of the signals delivers the out-of plane component and the difference provides the in-planecomponent parallel to the incidence plane of light 7 .

In this work, we built up a vector MOKE setup to determine all three components of the magnetization vector without normalization to the saturation values and with quantitative comparability in units of Kerr rotation. Therefore, we use a combination of both vector MOKE techniques described above. The components of the magnetization vector, which cause both the LMOKE and the PMOKE are measured with s-and p-polarized light 7 . The third component is measured by a 90 • rotation of the sample and of the external magnetic field 5 . If the out-of-plane component cannot be neglected, the PMOKE is also detected in this case. In order to separate these two components, again measurements with s-and p-polarized light are performed.

In addition, MOKE of the second order in magnetization is detected and included into the analysis. The first theory due to this quadratic MOKE (QMOKE) was proposed in 1965 by Metzger et al. 8 and the first measured magnetization curve including QMOKE was reported in 1990 by Zhong et al. 9,10 for Ni-Fe bilayers without the knowledge that it was a quadratic effect in magnetization. Further investigations on QMOKE are reported for Fe films on MgO 11 , Mo/Al 2 O 3 4 , Au/Fe/GaAs 12 and Ag/Au/GaAs/Ag 13 . Also the magnetization curves of Co films 4,14 and Co-based Heusler alloys 15,16 are influenced by QMOKE.

Since vector MOKE is based on linear MOKE, quadratic contributions to the measured MOKE data have to be separated to determine the magnetization vector from the linear part of the MOKE curves. Exemplary vector MOKE studies on a 7 nm Fe film deposited on MgO(001) are reported in this work. The separating of the linear and quadratic part of the magnetization curves is shown and the analysis of the curves including domain splitting and reconstruction of the magnetization vector is demonstrated. A theoretic description of the data processing for cubic and tetragonal crystal structures is added to show that this vectorial magnetometry method is valid even for films with tetragonal distorted unit cells.

II. EXPERIMENTAL MOKE SETUP AND DATA PROCESSING

The used MOKE setup pictured in Fig. 1 contains a He-Ne Laser with a wavelength of λ = 632.8 nm, a polarizer at 0/90 • (p/s-polarized) and a photoelastic modulator (PEM) with a modulation frequency of f = 50 kHz. After converting the polarization signal into an intensity signal by an analyzer (polarizer at 45 • ), the signal was detected by a photo diode.

The measured intensity of the 2f -signal (sensitive to the Kerr rotation) was obtained using the lock-in technique with the modulation frequency as reference signal. The magnetization curves were measured by varying the external magnetic field µ 0 H between -130 mT and +130 mT with bigger intervals for higher magnetic field values close to the saturation magnetization and smaller intervals for a lower magnetic field to detect the reversal process of the magnetization in more detail. Thus, the characteristic steps during the reversal process of the hysteresis can be better identified. The applied magnetic field was determined by a

Hall probe.

The measured intensity signal was calibrated to the Kerr angle by small simultaneous rotation of modulator and analyzer. The rotation angle of the optic components is similar to the rotation angle of the polarization of the reflected light induced by the Kerr effect.

Thus, this angle can be related to the change of intensity. The external magnetic field is kept constant during calibration to saturate the magnetization.

Two coordinate systems are defined in Fig. 1. The laboratory frame of reference is denoted x 1 , x 2 , x 3 with x 1 and x 2 in-plane aligned perpendicular and parallel to the plane of incidence of light, respectively. The x 3 -axis is out-of-plane and parallel to the plane of incidence of light pointing into the sample. The components of the magnetization vector can here be described by M 1 , M 2 , and M 3 . The crystal frame of reference is defined as

x ′ 1 , x ′ 2 , x ′ 3 with x ′ 1 =Fe[100],
x ′ 2 =Fe[010] and x 3 =x ′ 3 =Fe[001] of the Fe crystal lattice. Here, the magnetization vector is described by M ′ 1 , M ′ 2 , and M ′ 3 . This coordinate system is used in the Appendix A to present the validity of the processing. The azimuthal sample angle α is the angle between x 2 and x ′ 2 (rotation angle of the sample with respect to the plane of incidence of light). The positive s-direction of the linear polarized light is aligned in the x 1 -direction. The positive p-direction is defined so that s, p, and direction of propagation of light are right-handed.

In Fig. 2 the measurement procedure and the processing of the data are demonstrated.

A theoretical description of this processing is presented in the appendices. The complex amplitude of the Kerr rotation including the second order in magnetization is calculated in Appendix A. In Appendix B it is shown that the processing of the measured data is valid also for non-vanishing out-of-plane magnetization component and second order contributions.

Magnetization hysteresis curves for one complete sample rotation denoted by the azimuthal sample angle α are measured with different orientation of the external magnetic field (parallel and perpendicular to the plane of incidence of light) and with different po-larization of the incident light (s and p). The achieved four data sets are indicated with rectangles in Fig. 2. The further data processing (rhombi) include separation of odd and even parts of the curves to extract the linear and quadratic parts of the magnetic process.

The different intermediary results are indicated with ellipses. The different steps are labeled with the numbers of the figures in this article and with the equations in Appendix B.

In order to obtain a sufficient large LMOKE signal, the angle of incidence of 45 • was chosen. For smaller angles of incidence the LMOKE signal decreases and the PMOKE signal increases. Because of the different reflection coefficients for s-and p-polarized light at an angle of incidence of 45 • the measured Kerr effect signals are of slightly different magnitude.

Therefore, the linear parts of the hysteresis curves are scaled to the mean saturation value between s-and p-polarized light using the scaling factors k p and k s (cf. Fig. 2). A detailed definition of k p and k s can be found in Appendix B. These factors are determined from the mean saturation value of the curves obtained with external magnetic field parallel to the plane of incidence of light. The difference and the sum of the scaled linear curves result in the components of the magnetization vector M 1 , M 2 and M 3 . Thus, the comparability of different samples is verified because the hystereses are not normalized to the saturation values.

In case of M 3 = 0 (no out-of-plane component), the magnitude of the magnetization vector and its azimuthal rotation angle with respect to the direction of the positive external magnetic field can easily be determined. Thus, the complete reversal process can be pictured directly in a polar plot of the magnetization vector. The complete calculation of the magnetization vector from the experimental data including the separation of the linear MOKE and the quadratic part of the magnetization curves is performed by an in-house developed analysis software named MOKE Analyzer 17 .

III. SAMPLE PREPARATION AND STRUCTURAL PROPERTIES

The Fe film used for exemplary studies was assembled by molecular beam epitaxy (MBE) in a high vacuum chamber at room temperature. Using electron beam bombardment Fe was sublimated from a rod by heating (applied voltage 1 kV). The Fe deposition rate amounted to 1 nm/min at normal incidence. During evaporation the pressure in the preparation chamber was p = 10 -7 mbar (base pressure 10 -8 mbar).

Before film preparation, the MgO substrate was annealed up to 1000 K for 12 h to obtain a clean and well ordered surface. X-ray photoelectron spectroscopy (XPS) was used to check for contamination. After cooling down the substrate with a cooling rate of 5 K/min the quality of the MgO(001) surface structure was tested with low energy electron diffraction (LEED). The quality of the deposited Fe film was also approved by LEED and XPS.

Thereafter, the Fe film was capped by amorphous Si to avoid oxidation after transfer to ambient conditions to perform the MOKE studies.

The film thickness and structure were also determined ex-situ by X-ray reflectivity (XRR), X-ray diffraction (XRD) and grazing incidence X-ray diffraction (GIXRD). The measurements were performed at HASYLAB (DESY, Hamburg, Germany) by synchrotron radiation in ϑ -2ϑ scattering geometry at beamline W1 with a photon energy of 10.5 keV (λ = 1.18 nm).

These structural techniques confirm the well-known epitaxial properties of Fe on MgO(001) 18 .

Fe grows in the bcc structure and the cubic Fe unit cell is 45 The thickness of the Fe film studied here is 7 nm and its vertical lattice constant a ⊥ amounts to 2.83 Å, which is 1.4% less than the bulk value of 2.87 Å. This is caused by a tetragonal distortion of the Fe film due to its lateral expansion of 1.0% (a = 2.90 Å). Although this tetragonal distortion is relatively small, we probed the validity of the presented data processing for cubic and tetragonal crystal structure in the appendices. At the end of Appendix B it is proved that a tetragonal crystal structure can be analyzed the same way as a cubic one. So, the introduced data processing is valid in both cases. Additional structural details of the exemplary Fe/MgO sample will be published elsewhere.

IV. VECTOR MOKE RESULTS

An exemplary measured magnetization curve of the Fe film is shown in Fig. 3. The measurement was performed with s-polarized incident light and external magnetic field parallel to the plane of incidence of light. The curve is separated into an even and an odd part.

The even part includes an offset, originating from two sources, the offset coming from the experimental setup and the offset from the contributing QMOKE itself. The offset coming from the experimental setup is generated e.g. by small movement of the laser spot on the photo diode after rotating the sample. In principle, it can be separated from the QMOKE offset by sets of measurements, applying an external magnetic saturation field in eight different directions while the sample position is fixed 15,19 . Since only the linear part of the magnetization curve is important for vectorial magnetometry, we concentrate on the odd part of the magnetization curve which equates this linear part in case of no exchange bias effects. The curve represents the typical hysteretic behavior as shown in the inset of Fig. 3 which focuses on small magnetic field strength.

Using Eq. (B3) for the linear part of the magnetization curves for s-and p-polarized light the in-plane components of the magnetization vector can be determined. In case of the external magnetic field parallel to the plane of incidence of light and to a magnetic hard in-plane axis of the sample magnetization curves for s-and p-polarized light are shown in Fig. 4(a). The difference and the sum of both signals with different polarizations result in M 2 and M 3 as pictured in Fig. 4(b). Since the out-of-plane component M 3 does not exist during the entire reversal process, we can describe the complete reversal process using only the in-plane components M 1 and M 2 .

To check the validity of this calculation with a present out-of-plane component, the external magnetic field is rotated by 28 • with respect to the sample surface to induce a nonvanishing component M 3 . The incident angle is reduced to 0.2 • off the normal of the sample surface to get a larger PMOKE signal and to decrease simultaneously the LMOKE signal. In Fig. 5(a) the magnetization curves for s-and p-polarized light are pictured. Small hysteresis shapes are still present combined with a linear slope due to a part of the external magnetic field in out-of-plane direction. The curve of the resulting M 2 component (Fig. 5(b)) shows a typical hysteresis, but the amount of Kerr rotation is reduced due to the small incident angle. The M 3 component does not vanish as it is the case for the set-up with only in-plane external magnetic field (compare Fig. 4(b)). This behavior points to the existence of a forced out-of-plane component M 3 . Going back to the first set-up and the measurements of Fig. 4, thus, we proved that the M 3 component is negligible.

Magnetization curves of different azimuthal directions α are measured, too. After separating linear and quadratic part M 1 and M 2 are obtained from s-and p-polarized signal. In Fig. 6 curves are shown for (a) magnetic easy axis, (b) 5 • off the magnetic hard axis and (c) magnetic hard axis of the sample. Please, note there is still a small difference (< 1 • ) between the measured direction indicated 'magnetic hard axis' and the real magnetic hard axis due to small misalignment of the sample.

All curves for M 2 have the same Kerr rotation for the magnetic saturation of about 72 mdeg, but they differ in the shape of the hysteresis loop. They also vary in the number of magnetization steps for each branch during the reversal process and in the magnitude of the magnetic remanence (magnetization for zero external magnetic field). On the one hand, for the magnetic easy and hard axis only one step in the magnetization curve is observed.

On the other hand, for the direction 5 • off the magnetic hard axis two steps are obtained as shown in the inset of Fig. 6(b). The magnetic remanence of the magnetization curves is large in a magnetic easy axis and small in a magnetic hard axis. Also, the shape of the curves for M 1 in Fig. 6 varies for the different directions. In magnetic easy direction nearly no M 1 component is detectable during the entire reversal process as shown in Fig. 6(a). In magnetic hard direction and 5 • off the magnetic hard axis a M 1 component is obtained, but the shape of theses magnetization curves differs for both crystallographic directions. For example, the coercive field is much smaller for the magnetic hard direction compared to the direction 5 • off the magnetic hard axis. The Kerr rotation for the magnetic saturation vanishes for the magnetic easy as well as hard axis, but for the direction 5 • off the magnetic hard axis a Kerr rotation is still present. Effects like unsaturated magnetization vector or linear background from optical components due to stray field is ruled out as the loops in Fig. 6(a) and (c) do not have this offset.

Magnetization hystereses for many different crystallographic directions (azimuthal sample angle α) have been recorded to obtain more detailed information. In Fig. 7 the magnetic remanence of several in-plane directions for the M 1 component (circles) and the M 2 component (squares) is plotted against the azimuthal sample angle α. The observed fourfold magnetic anisotropy originates in the cubic crystal structure of the Fe bcc lattice 20 . Combining both curves in Fig. 7 one can detect four nearly perfect circles. This nice indication of the fourfold magnetic structure is due to sine and cosine characteristic for the projection of M in magnetic remanence to the direction perpendicular and parallel to the plane of incidence of light, respectively. The directions with low magnetic remanence for M 1 and large magnetic remanence for M 2 are due to the magnetic easy axes, which equate to the For the reconstruction of the reversal process the magnitude | M | = M 2 1 + M 2 2 and the azimuthal rotation angle γ = arctan (M 1 /M 2 ) of the in-plane magnetization vector M are calculated with respect to the positive external magnetic field, since the out-of-plane M 3 component is absent due to the magnetic shape anisotropy as proved by our analysis.

In Fig. 8 the dependence of the magnitude | M | on the external magnetic field is shown for the three crystallographic directions introduced above. The numbers in the curves denote the sequence of the reversal process. The saturation values of all curves amounts to 72 mdeg (sequences no. 1, 3, 4 and 6 in Fig. 8). This is in agreement to the saturation value of the M 2 component, because in magnetic saturation the magnetization vector should be parallel to the external magnetic field. Notice, that the magnitude for the direction 5 • off the magnetic hard axis exhibits the same saturation value as in case of the other loops, although in this case the M 1 component is not saturated.

During the reversal process | M | is nearly constant for stronger magnetic fields. For lower magnetic fields, however, sharp decreases are observed (sequences no. 2 and 5 in Fig. 8(a)-(c)). For the magnetic easy axis one sharp drop for each branch at ±1.0 mT with nearly vanishing magnetization is obtained (inset in Fig. 8(a)). The curve for the direction 5 • off the magnetic hard axis contains two drops for each branch at ±0.8 mT and ±3.5 mT.

Here, the minimum magnetization is approximately 1/ √ 2 times the saturation value (inset in Fig. 8(b)). In magnetic hard direction one drop for each branch occurs. The critical magnetic fields of ±0.8 mT for these sharp decreases of the magnetization are comparable to the one in magnetic easy direction. The minimal value of the magnetization, however, is higher than for the easy direction (inset in Fig. 8(c)). Additional to the drops slowly decreasing magnetization is observed between -50 mT and +50 mT for the magnetic hard axis (sequences no. 1b, 3a, 4b and 6a in Fig. 8(c)). This behavior is related to the transition from magnetic mono domain state into magnetic multi domain state with different magnetization orientations. Therefore, the resulting averaged magnetization vector is smaller in this case.

The azimuthal rotation angle γ of the magnetization vector M , which is defined as the angle between sample magnetization and positive external magnetic field direction, is presented in Fig. 9(a)-(c) during the reversal process for all three crystallographic directions described above. The labeling of the curves is equivalent to the labeling in Fig. 8(a)-(c).

By definition, the reversal process starts in saturation at γ = 0 • (sequence no. 1 in Fig. 9)

with magnetization parallel to the external magnetic field except for the direction 5 • off the magnetic hard axis (Fig. 9(b)). Here, the magnetization is initially tilted by γ = 9 • with respect to the applied external magnetic field due to the offset in saturation for the M 1 component in Fig. 6(b).

During the switching of polarity of the external magnetic field the magnetization rotates by 180 • with a steep step for the curves of the magnetic easy and hard axis (sequence no. 2 in Fig. 9(a) and Fig. 9(c)). Decreasing the external magnetic field for the 5 • off direction, first the magnetization slips to the direction of one magnetic easy axis (dashed line for 45 • ).

This rotation is followed by two steep 90 • steps to other magnetic easy directions (sequences no. 2a and 2b in Fig. 9(b)), before the magnetization rotates towards the direction of the external magnetic field for large strength of the reversed magnetic field (sequence no. 3 in Fig. 9(b)). Similar shallow steps occur in the curve of the magnetic hard axis (sequence no.

1b, 3a, 4b and 6a in Fig. 9(c)). Switching back the polarity of the external magnetic field, the behavior of the magnetization repeats for all magnetic directions until the magnetization reaches the initial orientation. The direction of the rotation is clockwise both for the magnetic easy axis and the hard axis following the sequence numbers of the reversal process. For the direction 5 • off the magnetic hard axis the sense of the rotation is counterclockwise except for the parts where the magnetization rotates backwards into the direction of the external magnetic field. These backwards rotating parts are indicated by sequences no. 3 and 6 in Fig. 10(b). If the external magnetic field is not applied exactly in easy or hard axis the magnetization rotates towards the closest magnetic easy axis with respect to the direction of the external magnetic field. Thus, the magnetization in Fig. 10(a) rotates clockwise because the magnetic easy axis is initially slightly below the direction of the external field. In Fig. 10(b) the magnetic easy axis at about 42.5 • is obviously the next one compared with the magnetic easy axis at -47.5 • . Thus, the magnetization rotates counterclockwise. In Fig. 10(c) the external magnetic field is in the direction of the magnetic hard axis and so approximately between two magnetic easy axes. The one at about -45 • seams to be slightly closer to the direction of the external magnetic field than the one at 45 • , so the magnetization turns clockwise.

V. DISCUSSION

The reversal process of the magnetization vector can be interpreted using mono domain states during maximal magnitude of the magnetization vector and domain splitting if the magnitude of the magnetization vector decreases. During domain splitting phase in most cases the magnetic moments of the individual domains are parallel to magnetic easy axes.

In Fig. 11, 12 and 13 the interpretation for the three discussed magnetic directions of the Fe film is presented. The labeling of the pictures equates the sequence numbers in Fig. 10.

Only the first half of the reversal process is shown, the second half is equivalent due to two-fold symmetry of the process.

In magnetic easy direction a mono domain state exists until the external magnetic field vanishes (sequence no. 1 in Fig. 11). After switching polarity of the external magnetic field and reaching coercive field, two 180 • domain states with antiparallel magnetization exist resulting in a vanishing magnetization vector (sequence no. 2 in Fig. 11). During increase of the reversed external magnetic field, again a mono domain state is generated (sequence no. 3 in Fig. 11).

If the external magnetic saturation field is 5 • off the magnetic hard axis a mono domain state exists initially again. Decreasing the strength of the external magnetic field the magnetization rotates towards the closest easy axis (sequence no. 1 in Fig. 12), which is followed by two 90 • domain states with magnetization perpendicular to each other caused by the negative external magnetic field (sequence no. 2a in Fig. 12). The magnitude of the magnetization vector is in minimum 1/ √ 2 times the saturation value, since half of the magnetic moments are aligned in one magnetic easy direction and the other half is oriented parallel to the other closest magnetic easy direction. This magnetization jump is characterized by a straight line in the polar plot due to simultaneous decrease of magnetic moments in one magnetic easy direction and increase of magnetic moments in the other magnetic easy direction. Here, the magnetic domains flip sequentially from the first magnetic easy direction into the second. Thus, the resulting magnetization vector follows a straight line and cannot be smaller than 1/ √ 2 of the saturation value.

Small increase of the external magnetic field generates a second domain splitting into two 90 • domain states. But in this case the magnetic moments of the domains point to different easy axes as it is the case for the first domain splitting. The result is the second jump of the magnetization vector (sequence no. 2b in Fig. 12). Here again, the characteristic of the magnetization jump is a straight line in the polar plot due to sequential domain flip. After further increase of the external magnetic field, mono domain state is again formed (sequence no. 3 in Fig. 12).

This reversal process is comparable to that one described qualitatively in the literature, e.g. Daboo et al. 21,22 studied one-and two-jump reversal processes and their occurrence in dependence of azimuthal sample angle and strength of uniaxial anisotropy contributions for Fe on GaAs(001). In Ref. 23 a similar reversal process is shown for FeNi films on glass.

However, in contrast to our measurements there is no straight line characteristic in the polar plot observed during the switch of the magnetization. This is due to coherent rotation of the magnetization vector instead of domain splitting. The investigated FeNi films, however, have a two-fold uniaxial magnetic anisotropy compared to the four-fold magnetocrystalline anisotropy for Fe on MgO studied here.

In magnetic hard direction the mono domain state is only existing at the start of the reversal process applying the external magnetic saturation field (sequence no. 1a in Fig. 13).

Domain splitting already occurs, if the external magnetic field is slightly decreased, so that in magnetic remanence the magnetization vector is composed by magnetic moments parallel to two different magnetic easy axes (sequence no. 1b in Fig. 13).

Applying negative external magnetic field the magnetic moments of most domains jump clockwise 90 • further into the next magnetic easy direction. This jump is again characterized by a straight line in the polar plot comparable to the case for the direction 5 • off the magnetic hard axis. But now there are two simultaneous switching processes. One of them is the flip of a bigger amount of magnetic domains clockwise from the 315 • magnetic easy axis to the 225 • axis, which is schematically shown in the lower part of the polar plot. The second flip is performed by a smaller amount of magnetic domains counterclockwise from the third magnetic easy axis (45 • ) to the fourth (135 • ). This is shown in the upper part of the polar plot. Summing up the magnetic moments of all domains during this double switch the resulting minimum of | M | is smaller than 1/ √ 2 of the saturation value. Here, domains exist in all magnetic easy directions with four 90 • domain states (sequence no. 2 in Fig. 13).

Increasing the external magnetic field, only two 90 • domain states remain (sequence no. 3a in Fig. 13) and with maximal external magnetic field the mono domain state is reached again (sequence no. 3b in Fig. 13).

Compared to the vectorial magnetometry techniques described in the introduction, the presented technique has some advantages. Since Florczak and Dahlberg 3 measured LMOKE and TMOKE signal, they were not able to set the obtained magnetization components in relation due to different characteristics of the measured signals (change of polarization for LMOKE, change of intensity for TMOKE). So the reversal process could only be described qualitatively. Furthermore, the measurement is restricted to in-plane magnetization. Daboo et al. 5 used LMOKE for measuring both in-plane magnetization components by rotating both sample and external magnetic field by 90 • . Thus, an in-plane reversal process can completely be described, but again the method is restricted to in-plane magnetization, while the technique presented here is also sensitive to the out-of-plane magnetization.

Using different polarizations (s and p) for the incident light the in-plane and out-of-plane magnetization component can be separated like Vavassori scaling makes it possible to use larger angles of incidence and the processing is still valid as shown in the appendices. Even if there are quadratic contributions of the Kerr effect or tetragonal distortion in the investigated film, the presented method is able to separate the components of the magnetization vector.

VI. CONCLUSION

In this work a combined vectorial magnetometry technique is presented, which reveals the complete reversal process of the magnetization vector in units of Kerr rotation. The method includes separation of linear and quadratic contributions, as well as determination of all three magnetization components. For this purpose s-and p-polarized incident light and an external magnetic field parallel and perpendicular to the plane of incidence of light are used. The curves for s-and p-polarized light are scaled to the mean saturation value in order to separate the in-plane and out-of-plane components of the magnetization vector, even for larger angles of incidence. In comparison to previous vectorial magnetometry techniques the magnetization curves are not normalized to the saturation value. Thus, different samples can be compared quantitatively in units of Kerr rotation. The reversal processes are advantageously visualized as polar plots of M . For an exemplary thin Fe film of 7 nm thickness and three typical crystallographic directions the reversal process is reconstructed.

Multi domain states can easily be identified as parts of the reversal process with reduced magnitude of the magnetization vector. While in magnetic easy direction only 180 • domain states occur, in magnetic hard direction and 5 • off the magnetic hard axis different 90 • domain states exist. Furthermore, in magnetic hard direction domain splitting proceeds before the magnetic remanence is reached. The flip process of the magnetization vector has a straight line characteristic in the polar plot due to sequential flipping of domains from one magnetic easy axis to the next one. In case of the magnetic hard direction two simultaneous switching processes occur for different easy axes. This results again in a straight line characteristic for the magnetization vector, but the minimum magnitude is reduced compared to the 5 • off direction.

Therefore, this careful analysis can be used to obtain the reversal process even for more complex film structures including a present out-of-plane component of the magnetization vector. Since the proportionality factor in-plane (between complex Kerr amplitude and in-plane component of magnetization vector) and the proportionality factor out-of-plane (between complex Kerr amplitude and out-of-plane component of magnetization vector) are not equal, the reversal process can only be reconstructed in units of Kerr rotation. Up to now, the method is restricted to cubic and tetragonal crystal structures and exchange bias effects have to be excluded. Based on the present study, however, experimental methods for analysis of different crystal structures can be investigated.

VII. ACKNOWLEDGMENTS

Portions of this research were carried out at the light source DORIS III at DESY. DESY is a member of the Helmholtz Association (HGF). We would like to thank Wolfgang Caliebe for assistance in using beamline W1.

Ostrava group acknowledges financial support from CZ.1.05/2.1.00/01.0040 and PIEF-GA-2009-254511.

ǫ =         G 11 M ′ 2 1 + G 12 M ′ 2 2 + G 13 M ′ 2 3 + ǫ d 2G 66 M ′ 1 M ′ 2 + K 12 M ′ 3 2G 44 M ′ 3 M ′ 1 -K 31 M ′ 2 2G 66 M ′ 1 M ′ 2 -K 12 M ′ 3 G 12 M ′ 2 1 + G 11 M ′ 2 2 + G 13 M ′ 2 3 + ǫ d 2G 44 M ′ 2 M ′ 3 + K 31 M ′ 1 2G 44 M ′ 3 M ′ 1 + K 31 M ′ 2 2G 44 M ′ 2 M ′ 3 -K 31 M ′ 1 G 31 M ′ 2 1 + M ′ 2 2 + G 33 M ′ 2 3 + ǫ d33         (A5)
where M ′ i denote the components of the magnetization in the crystal frame of reference. For the azimuthal sample angle α = 0 this matrix is also valid in the laboratory frame of reference. In order to determine the dependence of the elements of the permittivity tensor regarding α, the permittivity tensor of the crystal structure has to be transformed using the

rotation matrix R =       cos(α) -sin(α) 0 sin(α) cos(α) 0 0 0 1       (A6)
keeping the magnetization vector M fixed with respect to the laboratory frame of reference. This is done by rotating the permittivity tensor of zeroth order ǫ(0) and both magnetooptic tensors K and Ĝ independent of M using again the Einstein convention and the equations

ǫ (0) ij =R im R jn ǫ (0) ′ mn , K ijk =R im R jn R ko K ′ mno , G ijkl =R im R jn R ko R lp G ′ mnop . (A7)
After this rotation ǫ(0) and K do not change for tetragonal symmetry. However, the quadratic magnetooptic tensor Ĝ changes. Only off-diagonal elements of the permittivity tensor are taken into account for determination of the Kerr effect (cf. Eq. (A1)). Thus, we focus on the off-diagonal elements of the rotated permittivity tensor up to second order. The result The Kerr amplitude for increasing and decreasing external magnetic field is denoted by Φ inc and Φ dec , respectively. Symmetrization and antisymmetrization of Eq. (A9) neglecting the terms of higher order mentioned above yields to

Φ quad s/p = ± A s/p 2G 66 + ∆G 2 (1 -cos (4α)) - K 2 31 ǫ d M 1 M 2 ∓ A s/p ∆G 4 sin (4α) M 2 2 -M 2 1 + B s/p G 44 M 3 M 1 Φ lin s/p = ± B s/p K 31 M 2 -A s/p K 12 M 3 . (B2)
For small angles of incidence no scaling of the curves is necessary, since the Kerr amplitude for s-and p-polarized light are of the same values. In that case, A s = A p ≡ A and B s = B p ≡ B can be assumed. Thus, Φ lin s/p can easily be used to obtain the magnetization curves Φ + (M 3 ) and Φ -(M 2 ) since they are proportional to the components M 3 and M 2 of the magnetization vector, respectively 7 . The result is

Φ + = - Φ lin s + Φ lin p 2 = AK 12 M 3 Φ -= Φ lin s -Φ lin p 2 = BK 31 M 2 . (B3)
In order to determine such curves for larger angles of incidence (as it is the case in this study) the latter assumption is not valid any more. Nevertheless, scaling the magnetization curves Φ lin s/p to the mean saturation value Φ + (M 3 ) and Φ -(M 2 ) are still proportional to M 3 and M 2 , respectively. Applying the external magnetic saturation field Φ lin,sat s/p = ±B s/p K 31 M 2 is valid since it is M 3 = 0 in magnetic saturation. Thus, the scaling factor k s/p can be described by

k s/p = ± Φ lin,sat s -Φ lin,sat p 2Φ lin,sat s/p = B s + B p 2B s/p (B4)
where Φ lin,sat s/p are linear Kerr effects in saturation for s-and p-polarized incident light.

Using again Eq. (B3) for the scaled curves k s/p Φ lin s/p we obtain

Φ + = - k s Φ lin s + k p Φ lin p 2 = B s + B p 4
A s B s + A p B p K 12 M 3 = B s + B p 2 CK 12 M 3 Φ -= k s Φ lin s -k p Φ lin p 2 = B s + B p 4 2K 31 M 2 + A p B p - A s B s K 12 M 3 = B s + B p 2 K 31 M 2 .
(B5) Thus, Φ + is proportional to M 3 independent of A s/p and B s/p , while Φ -is proportional to

M 2 because A s /B s = A p /B p = C.
This comes directly from the definition of the Kerr effect as the polar component is Φ pol s/p = A s/p K 12 M 3 = ∓r pol ps/sp /r ss/pp , whereas the longitudinal one writes Φ lon s/p = ±B s/p K 31 M 2 = ∓r lon ps/sp /r ss/pp . As follows from symmetry arguments and when s-and p-directions are defined as in Fig. 1, r pol sp = r pol ps and r lon sp = -r lon ps . Hence, A s /B s = r pol ps /r lon ps = C, A p /B p = -r pol sp /r lon sp = r pol ps /r lon ps = C. All those assumption are valid independent of the angle of incidence 24 . Thus, there is no dependence on M 3 for Φ - in Eq. (B5) any more. Hence, it allows to separate polar and longitudinal direction, using only one scaling, in direction, where it is easier to saturate the sample (in our case here for in-plane magnetization, but it can be easily generalized for scaling in out-of-plane direction).

However, this is valid in case if complex description of A s/p , B s/p etc. is kept for the analysis. Hence, to determine the pure M 2 signal, one would need to measure the full complex Kerr effect (both Kerr rotation and Kerr ellipticity). However, in our case here, we have measured only Kerr rotation, providing Θ pol,lon

s/p = ℜ(Φ pol,lon s/p ) = ±ℜ(B s/p K 31 )M 2 - ℜ(A s/p K 12 )M 3 with the real part ℜ(z) of z. As in general ℜ(A s/p /B s/p ) = ℜ(A s/p )/ℜ(B s/p ),
the relation used for Φ -in Eq. (B5), A s /B s = A p /B p , is valid only approximately. For example, if only Kerr rotation is measured, and if M 2 is expressed, then for incidence angle of 30 • and 45 • , the 'cross talk' of M 3 to the separated M 2 is 18% and 30%, respectively. This example is numerically calculated for the case of our investigated system Fe(7nm)/MgO, using the light description inside a multilayer structure 26 . Although this cross talk may look large on first sight, one has to keep in mind that the value of the longitudinal Kerr rotation is about 20% to 30% of the polar one, respectively, as calculated for this particular case.

For determination of the signal proportional to the third component of the magnetization vector M 1 the external magnetic field and the sample have to be in-plane rotated 90 • clockwise. Thus, the coordinate system pictured in Fig. 1 also rotates and the x 2 -axis becomes the x 1 -axis and the x 1 -axis becomes the negative x 2 -axis. The whole theoretic description in this appendix is now valid using M 1 instead of M 2 and -M 2 instead of M 1 . For the scaling of the measured curves in this case again the factors k s/p determined for the setup with external magnetic field parallel to the plane of incidence of light are used. The scaling factors k s/p are the same for the two different alignments of the external field because in both cases the same Kerr effects occur (LMOKE for the in-plane components M 1 and M 2 , PMOKE for the out-of-plane component M 3 ). In conclusion, all components of the magnetization vector can be determined in units of Kerr rotation. This separation method is also valid for cubic crystal structure. Here, the number of independent coefficients is reduced using K 12 = K 31 in Eq. (B5). So, independent of the amount of tetragonal distortion, the separation of the signals proportional to M 2 and M 3 is possible (and separation of M 1 and M 3 for 90 • rotated magnet and sample). Only the proportionality factor between the complex Kerr amplitude and the components of the magnetization vector changes compared to the case of tetragonal crystal structure. So, all conclusions are valid for tetragonal and for cubic crystal structure. Furthermore, the definition of the coordinate systems is shown. The laboratory frame of reference is defined as x 1 , x 2 , x 3 with x 1 and x 2 in-plane aligned (perpendicular and parallel to plane of incidence of light, respectively). The crystal frame of reference is defined as

x ′ 1 , x ′ 2 , x ′ 3 with x ′ 1 = [100], x ′ 2 = [010] and x 3 = x ′ 3 = [001].

  • rotated in-plane to the cubic rock salt unit cell of the MgO(001) substrate, so Fe[100] MgO[110] and Fe[001] MgO[001].

  [100], [010], [ 100], and [0 10] directions of the Fe bcc lattice. The magnetic hard axes (low magnetic remanence for M 2 and large magnetic remanence for M 1 ) equate to the [110], [ 110], [ 11 0], and [1 10] directions, respectively.

Fig. 10

 10 Fig. 10 presents a polar plot of the magnetization | M (γ)|, in order to illustrate the complete behavior of the magnetization vector. For comparison to the previous diagrams the numbers of the sequence of the reversal process are included into the polar graphs, too.

  7 reported. Combing the different directions of the external field and the different polarizations of the incident light one can separate all three components of the magnetization vector as shown in this article. In most cases in the literature (e.g. report ofVavassori 7 ) the magnetization curves are normalized to the saturation magnetization. Here, we present a technique to keep the saturation value in units of Kerr rotation by scaling the curves to the mean saturation magnetization of s-and p-polarized signal. Thus, the comparability persists for different samples. Furthermore, this

FIG. 1 .

 1 FIG.1. MOKE setup for the different measurements with s-and p-polarized incident light and external magnetic field parallel and perpendicular to the plane of incidence of light, respectively.

FIG. 2 .FIG. 3 .

 23 FIG. 2. Schematic drawing of the processing (rhombi) of the experimental data (rectangles). The results (ellipses) are labeled with the numbers of the figures in this article and the equations of the Appendix B. , ⊥ denotes parallel, perpendicular to plane of incidence of light, respectively. The two lower dashed-lined boxes are only valid in case of M 3 = 0.

FIG. 4 .

 4 FIG. 4. (a) The linear part of the magnetization curves measured with incident s-and p-polarized light for a magnetic hard axis. (b) Determination of M 2 and M 3 from the magnetization curves in (a). The out-of-plane component M 3 of the magnetization vector does not exist during the reversal process. The insets are close-ups of the regions between -5 mT and +5 mT.

FIG. 6 .

 6 FIG. 6. Magnetization curves for M 1 and M 2 for (a) magnetic easy axis, (b) 5 • off the magnetic hard axis, (c) magnetic hard axis. The insets are close-ups of the regions between -5 mT and +5 mT.

FIG. 7 .

 7 FIG. 7. Magnetic remanence of the M 1 component (circles) and of the M 2 component (squares) of the magnetization vector plotted against the azimuthal sample angle α. The yellow dashed lines denote the magnetic easy axes. Please, note that the third crystallographic direction Fe(001) points away from the observer.

FIG. 8 .

 8 FIG. 8. Magnitude of the magnetization vector | M | during the reversal process calculated from the magnetization curves for M 1 and M 2 shown in Fig. 6 for (a) magnetic easy axis, (b) 5 • off the magnetic hard axis, (c) magnetic hard axis. The numbers in the curves denote the sequence of the reversal process. The insets are close-ups of the regions between -5 mT and +5 mT.

FIG. 9 .

 9 FIG.9. Rotation angle γ of the magnetization vector M during the reversal process calculated from the magnetization curves for M 1 and M 2 shown in Fig.6for (a) magnetic easy axis, (b) 5 • off the magnetic hard axis, (c) magnetic hard axis. The external magnetic field is aligned in γ = 0 • and γ = 180 • direction, respectively (blue dots). The yellow dashed lines denote the magnetic easy axes of the Fe film. The numbers in the curves denote the sequence of the reversal process.

FIG. 10 .

 10 FIG.10. Reversal process of the magnetization vector for (a) magnetic easy axis, (b) 5 • off the magnetic hard axis, (c) magnetic hard axis. The magnitude of the magnetization vector | M | in Fig.8is plotted against the rotation angle γ of the magnetization vector M in Fig.9. The external magnetic field is aligned in γ = 0 • and γ = 180 • direction, respectively (blue dots). The yellow dashed lines denote the magnetic easy axes of the Fe film. The numbers in the curves denote the sequence of the reversal process.

FIG. 11 .

 11 FIG. 11. Analysis of the reversal process of the magnetization vector with an external magnetic field parallel to the magnetic easy axis. Lower left corner: Strength of the external magnetic field. Lower right corner: Possible domain structure. The indication of sequence numbers and lines equate the labeling in Fig. 10(a).

FIG. 12 .

 12 FIG. 12. Analysis of the reversal process of the magnetization vector with an external magnetic field 5 • off the magnetic hard axis. Lower left corner: Strength of the external magnetic field. Lower right corner: Possible domain structure. The indication of sequence numbers and lines equate the labeling in Fig. 10(b).

FIG. 13 .

 13 FIG. 13. Analysis of the reversal process of the magnetization vector with an external magnetic field parallel to the magnetic hard axis. Lower left corner: Strength of the external magnetic field. Lower right corner: Possible domain structure. The indication of sequence numbers and lines equate the labeling in Fig. 10(c).

  The azimuthal sample angle α is the angle between x 2

		experi-	external magnetic field external magnetic field
		mental data	p-polar.	s-polar.	p-polar.	s-polar.
		loop symme-	odd even odd even odd even	odd even
		trisation
		Fig.3 Eq.(B2)	lin.	quad. lin. quad. lin. quad. lin. quad.
		determ.		*	k p	*	k s	*	k p	*	k s
		scaling to the mean sat. value Eq.(B4)	scaled field: lin. pol.:	scaled field: lin. pol.:	scaled field: lin. pol.:	lin. scaled field: pol.:
		vectorial
		component
		determination
		Fig.4/6 Eq.(B5)		M 2	M 3	M 1	M 3
		magnitude and rever-		√ + M 1 2	M 2 2 2 M 2 2		atan(	M 1	/	) M 2
		sal angle γ
		Fig.8/9			M				γ
		reconstr.					polar plot
		of reversal
		process
		Fig.10				reversal process
	and x ′ 2 .	legend	exp. data		processing		results

Appendix A: Complex Kerr amplitude including second order effects for cubic and tetragonal crystal structure

In the following the magnetooptic Kerr effect is calculated using the permittivity tensor up to second order in magnetization. Furthermore, the complete magnetization vector M is considered including its out-of-plane component.

The magnetooptic Kerr effect for s-and p-polarized incident light can be described by a complex Kerr amplitude Φ s/p which consists of the Kerr rotation Θ s/p and the Kerr ellipticity e s/p . This complex Kerr amplitude Φ s/p = Θ s/p -ie s/p depends on the complex reflection coefficients (r ss , r sp , r pp , r ps ) and therefore on the components of the permittivity tensor ǫ ij .

For magnetic thin films the complex Kerr amplitude for s-and p-polarized incident light is 15

Although this relation was originally derived for film thickness t ≪ λ/(4πN ) with the wavelength λ of the incident light and the refractive index N of the magnetic film, it is also valid for a general film thickness 24 . The permittivity of zeroth order of the magnetic film is ǫ d and the complex weighting coefficients A s/p and B s/p depend on the angle of incidence of light and on the optical surroundings (overlayers, underlayers) of the magnetic film.

The components of the permittivity tensor ǫ ij depend on the magnetization vector M = (M 1 , M 2 , M 3 ). The exact dependence is related to the crystal structure of the magnetic material in the film. The Kerr amplitude Φ s/p was calculated in Ref. 15 for cubic crystal structure, in-plane rotation of the sample and the assumption M 3 = 0. Here, we calculate Φ s/p without the assumption of M 3 = 0 to show that our data processing can be performed for each possible orientation of M including second order effects. Furthermore, we study the Kerr effect of tetragonal crystal structure to include films with tetragonal distorted unit cells. Afterwards, the results are compared with the calculations for cubic crystal structure.

We use the cartesian coordinate system x 1 , x 2 , x 3 pictured in Fig. 1 which is the laboratory frame of reference. The x 1 -and x 2 -axes are in-plane aligned with respect to the sample surface and the x 3 -axis corresponds to the surface normal (out-of-plane) pointing into the film. The x 1 -direction is perpendicular and the x 2 -direction parallel to the plane of incidence of light. The components M 1 , M 2 and M 3 are denoted with respect to the x 1 -, x 2 -and x 3 -directions, respectively. Furthermore, a second coordinate system is defined as the crystal frame of reference which is independent of the position of the sample in the setup. Here, the Fe[100], Fe[010] and Fe[001] direction of the crystal are parallel to the x ′ 1 -, x ′ 2 -and x ′ 3 -axis of the cartesian coordinate system of the crystal frame of reference, respectively. The x ′ 3 -axis is parallel to the x 3 -axis.

In the laboratory frame of reference each element ǫ ij of the permittivity tensor ǫ can be expressed as

using the constant permittivity term of zeroth order ǫ (0) ij , the elements K ijk of the linear magnetooptic tensor K, the elements G ijkl of the quadratic magnetooptic tensor Ĝ and the Einstein summation convention. Furthermore, some general symmetry arguments are valid which are 25

The expressions concerning the crystal frame of reference are now defined as ǫ

and G ′ ijkl . Eq. (A2) and (A3) are also fulfilled here. Furthermore, for a tetragonal crystal structure the number of independent tensor elements up to second order in magnetization is reduced to

Consider that the permutation rules of Eq. (A3) are still valid. This yields to a permittivity tensor for tetragonal crystal structure of using Eq. (A2) for the rotated tensors yields to

Here, the magnetooptic anisotropy parameter ∆G = G 11 -G 12 -2G 66 is introduced. These off-diagonal elements are inserted into Eq. (A1) for the complex Kerr amplitude to obtain the complex Kerr amplitude up to second order in magnetization which is

So the Kerr amplitude depends on linear terms (which are proportional to M 2 and M 3 , corresponding to longitudinal and polar MOKE), and second order terms proportional to

1 , and M 3 M 1 . For cubic crystal structure the number of independent coefficients is reduced using K 12 = K 31 , G 12 = G 13 = G 31 and G 44 = G 66 in Eq. (A9).

Appendix B: Theoretic description of the experimental processing

Here, from theoretical point of view, we present the validity of the processing, namely the separation of linear and quadratic part of the curves as well as the determination of the separate components of the magnetization vector.

Following the processing of the experimental data in Fig. 2 the odd and even part of the magnetization curve is separated. For systems without exchange bias it is assumed that the magnetic reversal can be described by M H = -M dec -H with the magnetization vector M inc for increasing external magnetic field and M dec for decreasing external magnetic field. Thus, symmetrization and antisymmetrization of the Kerr amplitude (odd and even part) result in the linear and quadratic MOKE using 15 Φ lin/quad = Φ inc H ∓ Φ dec -H /2 .

(B1)


F Stromberg

C Antoniak

U Von Hörsten

W Keune

B Sanyal

O Eriksson

H Wende

Textured growth of the high moment material Gd(0001)/Cr(001)/Fe(001)

Keywords: numbers: 81.15.Hi, 61.05.jh, 61.05.C, 76.80.+y, 76.50.+g, 75.70.Cn

By magnetic coupling of Fe and Gd via Cr interlayers, the large local moment of Gd can be combined with the high Curie temperature of Fe. The epitaxial growth of a Gd film is studied here by preparing trilayer systems of Fe/Cr/Gd on MgO(100) substrates by molecular-beam epitaxy (MBE). The thickness of the Cr interlayer was varied between 3-5 monolayers. The structural quality of the samples was confirmed by in-situ RHEED and ex-situ XRD measurements. Epitaxial Cr(001)/Fe(001) growth was observed, as expected. By use of 57 Fe-CEMS (Conversion Electron Mössbauer Spectroscopy) in combination with the 57 Fe tracer layer method the Fe/Cr interface could be examined on an atomic scale and well separated Fe/Gd layers for all Cr thicknesses were confirmed. The unusual Gd/Cr crystallographic relationship of Gd(0001) Cr(001), with domains of the hexagonal Gd basal planes randomly oriented in the sample plane and not in registry with the underlying Cr(001) lattice, was found from combined RHEED and X-ray measurements. Annealing of the samples resulted in a remarkable improvement of the crystalline structure of the Gd layers. On the other hand, the appearance of a single line in the CEM spectrum leads to the conclusion that during annealing a small amount of Fe diffuses into the Cr layer. The electronic structure and magnetism of this system is investigated by first principles theory.

Introduction

Materials with high saturation magnetization are important in microelectronic devices such as read heads for high density magnetic recording. The highest magnetic moment per atom among the 3d metals is known for Fe x Co 1-x alloys, where x ∼ 0.65, as predicted by the Slater-Pauling curve [1]. An important combination of high frequency response with high magnetization is realized, for example, in nanogranular systems of the type [2] (Fe x Co 1-x )-Al-O. Other materials, which follow the Slater-Pauling behavior, are full Heusler alloys containing Co, Fe, Rh and Ru [3]. A combination of 4f and 3d metals, which are coupled via a Cr interlayer, may lead to still observable huge magnetic moments on the rare earth atoms even at RT. Through coupling of Gd to a 3d ferromagnet like Fe it was possible to observe nearly temperature independent Gd moments of 6.2-7.2 µ B at the Gd/Fe interfaces in Fe/Gd multilayers and a modified Curie temperature with nearly the same value as for Fe (T C = 1043 K) [4]. It was also shown that the Fe and Gd moments at the interfaces are antiparallel at RT. One idea to align the Fe and Gd moments is to insert the correct number of monolayers (ML) of Cr between Fe and Gd. Cr is an antiferromagnet with a Néel temperature T N of 313 K [5], below which it attains a state of static spin density waves (SDW) along the [001]-direction, which can have complex variations [6]. In bulk Cr the AF order leads to an incommensurate spin density wave (ISDW) with a periodic modulation of the Cr magnetic moment. Since the Fe, Gd and Cr magnetic moments couple antiferromagnetically with each other [7,8], one can achieve a parallel alignment of the Fe and Gd moments by choosing the correct number of Cr ML, i.e. (2n+1), where n is zero or an integer. We have shown recently the parallel coupling of Fe and Gd by Cr interlayers. The results of element specific measurements on this system via XMCD (Xray Magnetic Circular Dichroism) and the theoretical framework concerning the coupling of Fe/Cr/RE (RE: Rare Earth) have been published recently [9]. This work focuses on the structural characterization of Fe/Cr/Gd trilayers. In the past, the common epitaxial relationships for hexagonal (hcp) and body-centered cubic (bcc) systems were reported. Depending on the extent of the lattice mismatch two different epitaxial relationships were found. If the hcp lattice is larger by 13% compared to the bcc lattice constant, the hcp[1100] direction is parallel to bcc[110] and hcp[1120] bcc[001]. This is designated as Nishiyama-Wassermann (NW) orientation [10][11][12][13]. This type of epitaxy is also found for the growth of rare earths on W(110) [14]. On the other hand, if the lattice mismatch is larger than 35%, the [1120] direction is either parallel [111] or [111]. This is the so called Kurdjumov-Sachs (KS) orientation [15,16]. Studies on the growth of Gd on Cr(001) are very rare. The only study which is known to the authors examined Gd/Cr multilayers on MgO(001) and it was found that Gd grows polycrystalline [17]. In the present work we investigate growth and structure of Gd thin films (a hexagonal system) deposited onto the surface of Cr(001) which has 4-fold symmetry. An unusual crystallographic relationship is observed. The electronic and magnetic structure of this system is also reported here, using first-principles theory.

Experimental Procedure

Trilayer systems of the type Fe/Cr/Gd with 3.5 monolayer (ML) thick 57 Fe(001) tracer layers have been grown on commercial MgO(001) substrates by molecular-beam epitaxy (MBE). The sample structure was MgO(001)/15 ML nat Fe/3.5 ML 57 Fe/x ML Cr/15 ML Gd, with x=3, 4, 5, respectively ( nat Fe=iron of natural (∼2%) 57 Fe isotopic abundance). The MgO substrates were rinsed in propanol and dry-cleaned with N 2 before loading into the UHV chamber. Subsequently they were cleaned by low-energy Ar ions (ion energy 0.5 keV, current density 1 µA/cm 2 , 5.5•10 -5 mbar) for times ≥ 1h at 700 • C. After this treatment almost all of the carbon impurities were removed, as confirmed by Auger electron spectroscopy (AES). Reflection high-energy electron diffraction (RHEED) images of the clean substrates proved that the surface was flat and well ordered with no superstructures due to surface impurities. The 57 Fe (95% enriched in 57 Fe), nat Fe (purity of 99.9985 at.%) and Cr (purity of 99.995 at.%) films were deposited by thermal evaporation from Al 2 O 3 crucibles. The Gd (purity of 99.9 at.%) films were evaporated from an Al 2 O 3 crucible, which was lined on the inside with a Mo-sheet. The pressure during 57 Fe and nat Fe deposition was 8•10 -10 mbar (base pressure of 2•10 -10 mbar), with a deposition rate of 0.03 Å/s at a nominal substrate temperature of T s =350 • C, which leads to a minimum surface roughness and large island sizes [18]. The pressure during Cr-evaporation was 4•10 -10 mbar with a deposition rate of 0.02 Å/s and T s ≤ 60 • C. Finally, the Gd-films were evaporated at 4.5•10 -10 mbar, with a deposition rate of 0.03 Å/s at T s ≤ 53 • C. If not otherwise stated, all samples had a Gd thickness of 15 ML, corresponding to 15•c Gd /2 = 43.4 Å and c Gd = 5.781 Å. The deposition rates were monitored with calibrated quartz-crystal oscillators. The 3.5 ML tracer layers of 57 Fe were deposited onto the surface of the 11.5 ML thick nat Fe layers, in order to examine the Fe/Cr interfaces with 57 Fe CEMS (conversion electron Mössbauer spectroscopy) [19]. For protection the samples were covered with 40 Å thick Au or 50 Å Cr layers. The epitaxial growth of the Fe, Cr and Gd layers during growth was investigated by RHEED with the electron beam (15 keV, 30 µA) incident at about 2-3 • relative to the sample plane and along the [100] azimuthal direction of the MgO(100) substrate. The RHEED patterns during the growth process were recorded in real time by a CCD camera for data evaluation. Ex-situ x-ray diffraction (XRD) in θ -2θ geometry was used to study the epitaxial nature of the samples. 57 Fe CEMS spectra were measured at RT using a homemade He/CH 4 -gas-flow proportional counter. A Mössbauer drive system operated in constant acceleration mode combined with conventional electronics and a 57 Co (Rh matrix) source of ≈ 50 mCi activity were used. The least-squares fitting procedure of the CEM spectra was performed with the NORMOS program package by R. A. Brand [20]. Due to the high enrichment of the 57 Fe probe layer (95%) and the low 57 Fe content of the nat Fe layer (∼2%) the CEMS signal originates predominantly from the interfacial 57 Fe probe layer.

Results and Discussion

The RHEED patterns of the Fe, Cr and Gd layers taken along the [100] azimuth of MgO(001) are shown in figure 1. Fe and Cr show well defined streaks, which confirm the epitaxy and the flatness of the surface. As additionally confirmed by LEED, Fe(001) on MgO(100) grows with the relationship Fe[110] MgO[100] (rotation by 45 • ). The Cr film shows the same orientation as the Fe(001) film and even sharper RHEED streaks (figure 1 (b)). The most interesting RHEED pattern is exhibited by Gd (figure 1 (c) and (e)). We observe broadened and faint (but clearly visible) streaks indicating preferential crystallogrpahic growth of Gd. No LEED pattern could be detected for these Gd films. Strikingly, the RHEED pattern from the Gd films was found to be nearly unaffected by azimuthal rotation of the sample (figure 2). This leads us to the assumption that the Gd film consists of tiny (0001)-oriented crystallites, with their c-axes preferentially oriented perpendicular to the film plane (as proven by XRD, see below) and with their hexagonal basal planes randomly oriented in the film plane and not in registry with the underlying Cr(001) lattice. In order to test this assumption we have simulated a RHEED pattern within the kinematic scattering approximation. Figure 1 (d) represents the simulated RHEED pattern from a flat 2-ML thick hexagonal Gd(0001) crystallite of 19 Gd atoms (bottom layer: 12 Gd atoms; top layer: hexagon of 7 Gd atoms; assuming lattice parameters of bulk Gd) that was rotated about the c-axis (perpendicular to the sample plane) in steps of 1 • from an azimuthal angle of 0 • to 59 • , and the obtained 60 individual diffraction patterns were incoherently superimposed. This simplified model simulates the case of tiny Gd(0001) crystallites with their c-axes perpendicular to the film plane and with their hcp basal planes randomly oriented in the film plane. It can be seen that the resulting simulated RHEED pattern (figure 1 (d)) is in fair agreement with the experimental RHEED patterns (figure 1 (c),(e)). This provides evidence that our assumption on the Gd-film structure is justified. The RHEED intensity of the (0,0) streak showed no oscillations during Fe or Gd growth, which is indicative of a Volmer-Weber (island) growth mode. During evaporation of Cr, weak RHEED oscillations could be detected (not shown), which were maintained until 4-5 ML. For higher thicknesses the oscillations faded away. This indicates a Stranski-Krastanov growth mode for Cr with a layer-by-layer growth for the first few ML. The in-plane interplanar distances of all films were evaluated by the simple relationship derived from the Bragg formula d f = (k s /k f )d s , where d f and d s are the in-plane interplanar distances of film and substrate perpendicular to the RHEED beam, respectively, and k f and k s are the position in k-space of the first-order diffraction spots relative to the zero-order reflection of film and substrate, respectively. We used a value of d s = 2.105 Å for bulk MgO. k f and k s were measured in pixels from a recorded video during film growth. The evolution of the obtained in-plane interplanar distances is shown in figure 3. It is observed that the in-plane atomic distance,d F e , of Fe decreases to a minimum of 1.95 Å after evaporation of 3.5 ML Fe. This is followed by a slow increase up to 1.99 Å after a total of 12 ML Fe. Considering a value of a F e / √ 2 = 2.027 Å in Fe bulk, this correponds to a relative difference of -1.33%. Cr shows negligible changes (not shown) and attains an in-plane value of d Cr = 2.00 Å after 4 ML. This results in a difference of -1.8% compared to a Cr / √ 2 = 2.037 Å in bulk Cr. Starting with the evaporation of Gd on Cr, the RHEED pattern showed a transition with additional broad reflections, which were superimposed on the Cr pattern. Therefore, a quantitative evaluation of the Gd reflections was only possible after 8 ML Gd (figure 3). X-ray diffraction indicates that the c-axis of Gd is preferentially oriented perpendicular to the sample surface (figure 4). The in-plane atomic distance of Gd after evaporation of 24.5 Å of Gd is 3.32 Å (figure 3 (b)). Although this is close to 3.14 Å, which would be expected for Gd[1120] Cr[110], azimuthal rotation of the sample relative to the RHEED beam produced only slight changes in the RHEED pattern (figure 2), and the observed interplanar distance for an azimuthal direction of 45 • relative to the RHEED beam (i.e. along Cr[100]) was 3.31 Å (figure 3 (c)) which is nearly identical to the previous value of 3.32 Å. This demonstrates that the average in-plane interplanar distance of the Gd film is isotropic. This implies that the Gd film consists of randomly oriented tiny grains with a c-axis texture perpendicular to the film surface. The interplanar distance from the simulated RHEED pattern of figure 1 (d) was 3.2 Å. Compared to this value we observe an in-plane lattice expansion in the Gd film of about 4%.

The θ -2θ x-ray diffraction (XRD) diagrams of the as grown trilayers indicate a high crystallographic order for the Fe and Cr films. elastically distort their unit cell in order to adapt to the in-plane epitaxial condition with Cr(001). The broad rocking curve can be assigned to a mosaic spread, probably induced by misfit dislocations within a relaxed structure. It can also originate from the reduced lateral coherence length related to the small lateral size of Gd grains. A broad rocking curve was also measured for the Fe layer (not shown). CEM spectra at RT for samples with different Cr thicknesses are shown in figure 6. While the Mössbauer spectrum of pure bcc Fe shows the well-known Zeeman sextet with sharp Lorentzian lines [START_REF] Wertheim | Mössbauer Effect: Principles and Applications[END_REF], the present spectra exhibit asymmetric line broadening and distinct shoulders. Therefore, the CEM spectra were least-squares fitted with two blocks of distributions of hyperfine magnetic fields P(B hf ) [plus a central single line for the annealed sample a 5 ML Cr interlayer 6 (b))] in the range of T (lowfield distribution 1) and 18-36 T (high-field region, distribution 2), which are characteristic for 57 Fe atoms at the Fe/Cr interface [START_REF] Uzdin | [END_REF]. A linear correlation between the hyperfine field B hf and the isomer shift δ had to be assumed in order to achieve satisfying fits. This well known correlation [24] is based on the fact that the electronegativity of Fe is larger than that of Cr. Consequently, the s-electron density of Fe is expected to increase with a higher number of Cr neighbours, leading to a more negative isomer shift. At the same time the higher number of Cr neighboring atoms reduces the magnitude of the hyperfine magnetic field B hf . The line intensity ratios of the sextets, which contribute to the distribution, were fixed at values of 3:4:1:1:4:3, which implies an inplane orientation of the Fe-spins. Distribution 2 exhibits five maxima, which are located Figure 6. Typical RT-CEM spectra of as-prepared samples with 4 ML Cr (a), and with 5 ML Cr after annealing at 210 • C in UHV (b). (a) was least-squares fitted with two blocks of magnetic hyperfine-field distributions P(B hf ) (0-18 T (blue), 18-36 T (green)). (b) was least-squares fitted with two similar blocks of hyperfine-field distributions and a central single line (dark blue). The corresponding hyperfine field distributions, P(B hf ), are inserted below the respective CEM spectrum. at 19.2 T, 22.4 T, 25.6 T, 28.4 T and 32.8 T (figure 6, inserts). They correspond to different atomic surroundings of the 57 Fe atoms at the Fe/Cr interface. Lower hyperfine fields indicate an increased number of Cr neighbours. Very smooth interfaces should also exhibit an enhanced hyperfine field (≈ 34.5 T) originating from the second Fe subsurface [START_REF] Uzdin | [END_REF], but this is not the case for our samples since a mixing of Fe and Cr on the atomic scale was unavoidable. The diffusion of Fe and Cr in the first three samples (table 1) cannot be severe since no paramagnetic component (central single line) with B hf = 0 T is observable. One can conclude that the quality of the Fe/Cr interface is sufficient to produce closed Cr layers, which safely separate the Fe and Gd layers from each other, starting from 3 ML Cr (the lowest thickness which we examined). The lowfield distributions contribute only ∼10.5% to the total spectral area (total integrated 1). The spectrum of the annealed sample with 5 ML Cr had to be fitted with an additional narrow single line with δ=-0.10(2) mm/s and a relative spectral area of 4%. From this finding one can conclude that the annealing procedure created a small amount of paramagnetic 57 Fe impurities, which diffused into the Cr layer during annealing and which only sense Cr nearest neighbours. Since the structure of Cr is cubic, it produces a single line with negative isomer shift in the CEM spectrum. The negative isomer shift of the single line is typical for 57 Fe impurities in a Cr matrix [25,26]. Interestingly, the spectral intensity of 4% stems only from the high-field region (distribution 2), which is reduced from 90% to 86% (table 1). This means that those 57 Fe atoms which have a relatively large number of Fe neighboring atoms, diffuse easier into the Cr layer than Fe atoms with more Cr neighbors.

Theoretical results

We have performed first principles calculations based on density functional theory to study the Gd/Cr/Fe(001) system. The computational unit cell consisted of 1 ML Gd, 1 ML Cr and 5 MLs of Fe. Plane wave projector augmented wave basis was used in the generalized gradient approximation for the exchange correlation potential as implemented in the VASP code [27,28]. Gd 4f states were kept in the core. The plane wave cut off energy used was 400 eV. A 6 × 6 × 2 Monkhorst Pack k-points set was used for the integration in Brillouin zone. Atoms were relaxed until the Hellman and the interface Fe (Fe 1 ) atoms. As 4f states of Gd are kept in the core, they are not seen here. The spin-polarized Gd 5d-orbital is seen with a small exchange splitting. Exchange splittings of Cr and Fe d states are much higher, as seen in the figure. As

Cr is antiparallel to Gd and Fe, the majority and minority spin states are reversed. A considerable hybridization between the Cr and Gd states is observed close to the Fermi level. The reminiscence of bcc Fe electronic structure is observed for the interface Fe atom. The spin-up d-states (upper half, figure 8) are nearly full whereas the Fermi level cuts through a pseudogap in the spin-down DOS.

Conclusions

Trilayer systems of the type Fe/Cr/Gd have been grown on clean MgO(001) substrates by MBE. Tracer layers of 3.5 ML thick 57 Fe(001) on top of 11.5ML nat Fe were inserted, in order to perform 57 Fe CEM measurements and to probe the structural quality of the Fe(001)/Cr(001) interface. The Fe layers were grown on MgO(001) at 350 • C, the other two layers near RT. The growth of Fe and Gd followed the Volmer-Weber growth mode, as indicated by RHEED. The Fe and Cr films exhibit well-ordered surfaces, whereas the Gd layer had a rough surface. The in-plane lattice parameters of Fe and Cr were -1.58% and -1.8% lower than their bulk values. The Gd/Cr epitaxial relationship is unusual and was found to be Gd(0001) Cr(001) with otherwise random orientation in the sample plane. An average in-plane expansion of the Gd lattice of +4% was found from comparison with a simulated RHEED pattern for random in-plane orientation of tiny Gd(0001) crystallites with their c-axes perpendicular to the sample plane. X-Ray rocking curve scans around the Gd(0002) reflection indicate a transition from an elastically distorted Gd film at lower thickness, which adapts to the Cr(001) surface, to a relaxed structure for thicker layers. Annealing at 210 • C led to an improvement of the crystalline quality of the Gd layers and a smoother Gd surface structure. However, the appearance of a weak central single line in the CEM spectrum points to a small fraction of paramagnetic 57 Fe impurities, which diffused into the Cr layer during annealing. First principles calculations for the density of states, the interplanar separations and the layer projected magnetic moments were presented. The Fe/Cr/Gd system studied here has a great potential for sensor applications, since the parallel coupling of Fe and Gd moments combined with the huge Gd moment (observable at RT due to the strongly enhanced Gd Curie temperature [9]) may result in a huge magnetization of this trilayer.

Figure 1 .

 1 Figure 1. Typical RHEED patterns taken with the beam along the [100] azimuthal direction of MgO(001) of 15 ML (21.5 Å) Fe (a), followed by 4 ML (5.8 Å) Cr (b), followed by 15 ML (43.4 Å) Gd (c), and 15ML Gd after annealing at 210 • C in UHV (e). (d) shows a simulated RHEED pattern from a 2-ML thick Gd(0001) crystallite consisting of 19 atoms (12 atoms in the bottom layer, 7 atoms in the top layer). In order to simulate the random in-plane orientation of the hexagonal basal plane, the Gd(0001) crystallite was rotated in the film plane relative to the beam direction from 0 • to 59 • in steps of 1 • , and the obtained 60 diffraction patterns were incoherently superimposed.

Figure 2 .

 2 Figure 2. RHEED patterns of a 45 Å thick Gd film taken with the beam along different azimuthal directions: (a) 0 • , (b) 15 • , (c) 30 • , (d) 45 • . The angles refer to the [100] azimuthal direction of MgO(001).

Figure 4 (Figure 3 .

 43 Figure 3. In-plane interplanar distances as a function of film thickness of Fe (a) and Gd (b),(c): The RHEED beam was parallel to the [100] azimuthal direction of MgO(001) in (a) and (b), and parallel to [110] of MgO(001) in (c). The value of 4.211 Å of bulk MgO[21] was considered as reference for the following evaluation of the in-plane atomic distance in the films.

Figure 4 .Figure 5 .

 45 Figure 4. Typical XRD patterns for 15ML Gd samples as prepared (a) and annealed (b), and for 200 Å Gd film (c). For comparison, magnified sections of (b) and (c) around the Gd(0002) reflection are displayed in (d). The samples are of similar geometrical structure: MgO(001)/15 ML nat Fe/3.5 ML 57 Fe/5 ML Cr/Gd(t), with t=43.4 Å (15ML) or 200 Å. Annealing was performed in UHV at different temperatures and times, the maximum temperature was 210 • C. The position of the reflections for bulk Gd are shown as vertical bars in (d). (Cu-K α radiation).

Figure 7 .

 7 Figure 7. (left) Interplanar separations from surface to bulk. i=1, 2, ... denote Gd, Cr .... layer respectively as indicated in the right panel. (right) Layer projected magnetic moments from surface to bulk.

Feynman

  forces were minimized up to 0.01 eV/ Å. The optimized interplanar separations are shown in the left panel of figure7. As Gd is a bigger atom, the separation between Gd and Cr (next layer) is rather large. The distance between Fe and Cr layers is reduced compared to Fe-Fe separations in the next layers. It is observed that the bulk Fe-Fe interlayer distance is achieved after 4th Fe layer. The right panel of figure7shows the layer projected spin moments. As expected, Gd has 7 µ B moment coming from halffilled 4f shell. The small fractional moment (∼ 0.1µ B ) arises from the 5d shell. The Cr moment is antiparallel to Gd and Fe moments. The interface Fe atom (Fe 1 ) has a reduced moment whereas the moments on the next Fe layers increase until they reach the bulk value. Figure8shows the local DOSs projected on the d-orbitals of Gd, Cr

Figure 8 .

 8 Figure 8. Density of states (DOS) projected on the d-orbitals of interface Fe, Cr and Gd atoms.

Table 1 .

 1 Spectral Mössbauer parameters for samples with different Cr thickness. δ : average isomer shift(relative to α-Fe at RT; only δ for single line), B hf : average hyperfine field, Γ: linewidth (FWHM). Hyperfine-field distribution 1 corresponds to the low-field region (0-18 T), distribution 2 corresponds to the high-field region (18-36 T).

		δ	B hf	Γ	rel. spectral
		(mm/s)	(T) (mm/s)	area (%)
	3 ML Cr (as prepared)
	Distribution 1	+0.02(5)	7.0	0.35	10
	Distribtuion 2	-0.01(2)	29.0	0.32	90
	4 ML Cr (as prepared)
	Distribution 1	+0.12(5)	6.7	0.35	10
	Distribtuion 2	-0.01(1)	28.9	0.32	90
	5 ML Cr (as prepared)
	Distribution 1	+0.22(5)	7.1	0.35	11
	Distribtuion 2	-0.01(1)	29.9	0.32	89
	5 ML Cr (annealed)
	Distribution 1	+0.12(5)	6.7	0.35	10
	Distribution 2	-0.01(1)	28.9	0.32	86
	Single line	-0.10(2)	-	0.35	4
	intensity) in all samples (table

Work supported by Deutsche Forschungsgemeinschaft (Sonderforschungsbereich SFB491). BS is grateful to STINT Institutional grant for young researchers programme for financial support. Also, BS and OE acknowledge financial support from Swedish Research Council and computational support from Swedish National Infrastructure for Computing (SNIC). O.E. acknowledges the ERC for financial support.