Quantitative study of the quadratic magneto-optical Kerr effects in Fe films

We present a rotating field method to separate the linear and quadratic magneto-optical Kerr effects (LMOKE and QMOKE) in Fe/GaAs(001) films. The LMOKE is isotropic in crystal orientation, while the QMOKE has both isotropic and anisotropic contributions. The experimental observation is well explained by Yeh’s 4 4 × matrix formalism. We also report the incident angle and the thickness dependences of the LMOKE and QMOKE, and extract the material’s index of refraction n and the magneto-optical coupling constant K and G . The study gives a full description of the Kerr effect in Fe films, and the proposed method can be applied to other magneto-optical coupling systems. ©2015 Optical Society of America OCIS codes: (210.3820) Magneto-optical materials; (240.0310) Thin films; (260.2130) Ellipsometry and polarimetry; (310.6860) Thin films, optical properties; (999.9999) Magneto-optic ellipsometry. References and links 1. J. 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Introduction
The magneto-optical Kerr effect (MOKE) was discovered by John Kerr [1,2] and first applied to study surface magnetism by Moog and Bader in 1985 [3].The MOKE has been widely used to investigate coercivity, magnetic anisotropy, spin dynamics, interlayer exchange coupling, and magnetic domains.Most applications describe the magnetization with the normalized Kerr angle based on the first-order linear magneto-optical Kerr effect (LMOKE), neglecting the higher-order quadratic magneto-optical Kerr effect (QMOKE).However, the QMOKE analysis is important in the following considerations: (i) for materials with a huge QMOKE [4][5][6], this higher-order contribution must be considered to quantitatively calibrate the magnetization measurements; (ii) because the quadratic magneto-optical (MO) coupling is caused by second-order spin-orbit coupling terms [4], the MO coupling parameters related to the QMOKE can be used as a probe for fundamental electronic interactions in ferromagnetic materials; and (iii) optical effects quadratic in magnetization have recently been determined to be important, including magnetization-dependent second-harmonic generation [7], the quadratic X-ray magneto-optical effect [8], and the closely related X-ray Voigt effect [9].A systematic study of the QMOKE therefore can provide a good reference for the study of other related effects.
The first theoretical prediction of the QMOKE in a ferromagnetic material was proposed by G. Metzger et al. in 1965, but the effect was calculated to be very small [10].Experimental evidence of the QMOKE was reported by Zhong et al. for Ni-Fe bilayers without the knowledge that it was a quadratic effect [11,12].The QMOKE was experimentally proven to exist in Fe and Co thin films by Osgood et al., who considered the QMOKE to be related only to the magnetization term L T m m ( L m and T m are the longitudinal and transverse magnetization components, respectively) [13].Then, by considering the crystal permittivity carefully, Postava et al. added a new term, 2 T m , to the QMOKE signal, giving a complete expression for the QMOKE.Thus, the Kerr signal Φ can be expressed as [14,15]: Here, Φ could be either  [16].Although the total QMOKE contribution was quantized, the contributions of LT  Φ and TT Φ could not be distinguished in this way.Moreover, all previous measurements of the QMOKE were undertaken with a small incident angle without knowledge of the incident angle dependence of the QMOKE.It is therefore crucial to propose a better method to quantitatively separate the QMOKE terms and systematically study it.
In this work, we quantitatively separated the linear and quadratic Kerr terms in Fe thin films using a rotating field method.By measuring the Kerr signal as a function of the field orientation, we separated L Φ , LT Φ and TT Φ contributions from the total Kerr signal.By rotating the sample in-plane, we found that the LMOKE is isotropic in crystal orientation, while the QMOKE has both isotropic and anisotropic contributions.We also found that the LMOKE has a much stronger incident angle dependence than the QMOKE.For normal incidence, the QMOKE has a finite value while the LMOKE approaches zero.The film thickness dependences of the LMOKE and QMOKE's isotropic and anisotropic contributions were studied in a wedge-type sample, and the results indicate that the MOKE is primarily due to bulk-type MO coupling.All the experimental observations can be quantitatively explained by a 4 4  × transfer matrix method, and all the MO coupling parameters can be determined through the theoretical fitting.

Experimental details
Single crystalline body-center-cubic (bcc) Fe(001) films were prepared on GaAs(001) substrates in an ultra-high vacuum chamber at room temperature with a base pressure of 10 2 10 − × torr [17][18][19].The GaAs(001) substrates were prepared by a standard Ar + sputtering and annealing process [20].Fe films were epitaxially grown at room temperature with a growth pressure of less than 10 5 10 − × torr.It is well known that Fe can be epitaxially grown on GaAs(001) surfaces with the lattice relationship Fe(001)<100> // GaAs(001)<100> [17,18], which can be proven by the in situ reflection high-energy electron diffraction pattern.A 4-nm Au protecting layer was grown on top before taking the sample out of the growth chamber.The film thickness was determined by the growth rate which was measured by a calibrated quartz thickness monitor.To systematically study the thickness-dependent Kerr signal, the Fe film was grown into a wedge shape with a slope of ~1.8 nm/mm.The MOKE measurements were performed using a multimode laser diode with a wavelength of 670 nm. Figure 1(a) shows the experimental MOKE setup; and magnetic field with a maximum strength of 1.2 kOe along the arbitrary in-plane direction was applied by a vector magnet.Linear polarized light is reflected from the sample surface, and passes through an analyzing polarizer set at 2° from extinction; then, the light intensity is measured by a photodiode.The Kerr rotation angle can be quantitatively determined in this simple way [21].To measure the Kerr ellipticity, a quarter-wave plate is placed before the light passes through the analyzing polarizer.The incident laser beam can be adjusted into either s-polarized or p-polarized light, and the incident angle ϕ can be adjusted manually with a precision of about ±2°.The diameter of the laser spot on the sample is ~0.2 mm, which only covers ~0.36 nm thickness range on the wedge sample.The sample was mounted on an optical mounting that can be azimuthally rotated precisely to adjust the angle α between the Fe [110] axis and the plane of incidence.Thus, the Kerr signal Φ , including Kerr rotation θ and Kerr ellipticity ε can be systematically measured as a function of ϕ , α , and the field angle H φ , as well as the Fe film thickness; then, L Φ , LT Φ and TT Φ can be quantitatively separated using a rotating field method.All the MOKE measurements were performed at room temperature.

The rotating field method
It is known that L m changes its sign, while L T m m and 2 T m stay constant, when reversing the magnetization (see Eq. ( 1)).Taking advantage of this symmetry relation, the LMOKE and QMOKE can be distinguished by the following equations: Here, m represents the unit magnetization vector.Such a method to separate the LMOKE and QMOKE signals is valid only in a system without broken inversion symmetry, thus, the magnetization can reverse its direction when the strong external field changes its sign.The separation procedure is take in several steps.First, we obtain the total MOKE signal Φ when rotating the magnetic field H in the film plane (shown in Fig. 2(a)).The ( ) Φ and TT Φ can be well quantified.In previous studies [4][5][6]15], the QMOKE contributions were separated by the eight-field method, which assumes the magnetizations lie along the field directions, i.e. m H φ φ = ; this assumption could be questionable if the external field is not strong enough compared with the magnetic anisotropy.In the rotating-field method applied in this work, the magnetization direction was determined experimentally by considering the influence of the magnetic anisotropy on the magnetization orientation, providing a general method to determine the QMOKE signal.
In our study, only the longitudinal Kerr effect was considered in the Fe/GaAs(001) system.The contribution from the polar Kerr effect should be negligibly small, since the Fe film has the strong in-plane magnetic anisotropy, and the demagnetization field is also very strong.In our measurement, the applied field was also carefully adjusted to rotate within the film plane.Moreover, the polar Kerr signal should be antisymmetric to the magnetization direction, thus it will not influent the QMOKE signal determined by Eq. (3).The transverse Kerr effect was not considered in our measurement, since the transverse Kerr effect will not contribute to the polarization rotation of the incident light [21].As a result, there is only longitudinal MOKE contribution in our measurements.Φ , LT Φ and TT Φ can be obtained using the rotating field method described in Fig. 2. We systematically rotated the sample orientation, and quantitatively measured the crystal-orientation-dependent ( )

Anisotropic QMOKE
. Figure 4(a) shows the Kerr rotation θ as a function of α for s-polarized incident light.The LMOKE signal L θ is independent of the crystalline orientation; however, the two quadratic terms, LT θ and TT θ , oscillate with α , and both have a 90° period due to the in-plane fourfold symmetry of the Fe(001) surface.This result is consistent with the literature reports [4][5][6]15].As shown in Fig. 4, we separated the QMOKE and LMOKE signals for both Kerr rotation and Kerr ellipticity with s-and p-polarized light.The LMOKE signals are always independent of the crystalline orientation, and the two QMOKE terms oscillate with similar amplitudes.For both s-and p-polarizations, the TT θ and TT ε always oscillate around zero, but LT θ and LT ε have an obvious offset.To obtain a better understanding of the experimental observation, a theoretical description of the MO effect was undertaken.It was based on an analysis of the dielectric properties of a medium.The tensor element ij ε can be expanded in a power series of the Cartesian direction cosines i m of the magnetization [16]: where ijk K and ijkl G are the linear and quadratic MO coupling constants, respectively.For cubic symmetry, the permittivity tensor (see Eq. ( 4)) is completely defined by five quantities: the nonmagnetic part of the permittivity 0 .By solving Maxwell's equations and the standard boundary conditions, the complex Kerr angle Φ can be qualitatively expressed as [4,15]: where  The incident angle ϕ is one of the factors that can have an impact on the parameters A and B in Eq. ( 5).Previous studies on the QMOKE were only performed with a fixed incident angle, regardless of the effect from the optical surroundings [4-6, 15, 16].We performed systematic measurements as a function of incident angles ϕ ranging from 10° to 55°.For the isotropic contributions, the experimental L iso θ in Fig. 5(a) shows a nearly linear increase with ϕ , while the LT iso θ in Fig. 5(b) has a relatively weak dependence on ϕ ; the L iso θ approaches zero, while the LT iso θ seem to remain at finite values for normal incidence ( 0 ϕ = °).

Incident angle dependence of LMOKE and QMOKE
For the anisotropic contributions, the LT ani θ and TT ani θ always have the same value and remain nonzero at normal incidence, consistent with Eq. ( 5).Therefore, the QMOKE signal could dominant the total Kerr signal with a small incident angle.A non-zero Kerr effect requires the presence of both spin-orbit coupling and exchange interactions [22].For oblique incidence, the first-order contribution to spin-orbit coupling ( SO E L S ξ = ⋅ ) is dominant, giving a large LMOKE contribution.However, for normal incidence, the first order of spin-orbit coupling is zero ( 0 L S ξ ⋅ = ) [4], so the LMOKE quenches and the remaining Kerr effect is the QMOKE, which originates from higher-order spin-orbit coupling.The MO effects linear in magnetization were also calculated to be zero at normal incidence using electromagnetic wave theory [23,24].

Thickness dependence of LMOKE and QMOKE
We quantitatively studied the Fe-thickness dependence of the LMOKE and QMOKE from a wedge sample with a laser incident angle of 45°.The thickness dependence of the experimental LMOKE signal is plotted in Fig. 6(a); triangles and squares indicate Kerr rotation θ and Kerr ellipticity ε , respectively.Both θ and ε increase linearly with Fe thickness only for Fe t < 4 nm, and the additivity law [21,25] of the Kerr signal is no longer valid for Fe thicknesses above 4 nm, although this thickness is well below the penetration length of the incident light.One possible reason is that reflection of the substrate has a more important role in the Kerr signal contribution.A general numerical treatment of the MOKE, undertaken by solving Maxwell's equations and considering standard boundary conditions, was developed by Yeh [26] and Visnovsky et al. [27,28].For an anisotropic medium, the permittivity tensor ε , which contains information about the refractive index n , MO coupling constant K , 11 G , 12 G , and 44 G , describes the MO properties.The evaluation of the MO quantities starts from a calculation of the eigenmodes in this magneto-optical medium.The eigenmode characteristic equation is a fourth-order equation for the normal component of the wave-vector.The relation between the amplitudes of the four eigenmodes in the multilayer is represented by a 4 4 × matrix, which is given by the matrix product of the dynamic and propagation matrices [27,28].Then, the elements of the Jones matrix, which can be expressed with the 4 × 4 matrix elements, determine the magnitude of the complex Kerr angle: s ps ss r r Φ = , p sp pp r r Φ = . Here, the subscripts s and p represent incident waves with s-polarization and p-polarization.Finally, the contributions of L Φ , LT Φ , and TT Φ can be separated in a process similar to that used in our experiment.
We have fitted our experimental data employing the full 4 4  × matrix formalism specified above using fixed refractive indices 3.81 0.17 GaAs n i = + and 0.17 3.61 [29] and treating the refractive index and the MO coupling of Fe as free parameters.The best fit lines are shown in Figs. 5 and 6, and are fully consistent with the experimental data.G G − from both the incident angle-dependent and thickness-dependent measurements are very similar to each other, indicating that the optics are very similar for the two samples.Comparing the parameters of Fe/GaAs films in Table 1 with those of Fe/Ag films in [16], it is clear that the refractive indices of Fe are extremely close, but the MO coupling constants are not quite the same.This difference may be attributed to the structure dependence of the growth condition.Buchmeier et al. attributed the differences in bcc-Fe MO coupling constants between their research [16] and Postava's result [15] to the strong structural dependence.The situation in our experiment and [16] is similar; the interfaces of Fe/GaAs and Fe/Ag result in different strains: −1.6% for Fe/GaAs and 0.7% for Fe/Ag.These opposing strains may lead to a different anisotropic permittivity tensor, causing differences in MO coupling parameters.
Equation (1) indicates that the different behaviors of the LMOKE and QMOKE are related to the different time-reversal symmetries, i.e. the QMOKE has time-reversal symmetry while the LMOKE breaks it.The rotating field method makes it possible to quantitatively separate these two effects, and thus it is believed to be a promising technique in traditional magnetic film investigation, especially for the normal incident case, when the effects linear with m vanish.For example, it may be used in nearly-normal incidence magneto-optical microscopy for in-plane magnetic domain observation.It should be noted that the electric dipole field effect, e.g.ferroelectric and strain effects, also induce the significant Kerr effect without breaking the time reversal symmetry, but such effects can be separated out by the rotating field method.Moreover, the time-reversal symmetry is also preserved in the antiferromagnetic (AFM) system, thus, as long as the AFM spins can be rotated respected to the plane of incidence, the quadratic term of Kerr effect is also expected in AFM system, and may provide similar information as the X-ray linear dichroism effect.In [8], Valencia et al. studied the quadratic X-ray magneto-optical effect in 3d-transition metals, and found it comparable to the X-ray linear dichroism effect.

Conclusion
The LMOKE and QMOKE of thin Fe(001) films have been quantitatively measured with a rotating field method.Two QMOKE terms ( L T m m and 2 T m ) were quantitatively separated.The LMOKE is isotropic, while the QMOKE has both isotropic and anisotropic contributions with crystal rotation.Through incident angle-dependent measurements, we found that the QMOKE stays nonzero while the LMOKE quenches in normal incidence.The thickness dependence of the LMOKE and QMOKE and a general numerical calculation were undertaken.Using the bulk-type optic and magneto-optical constants, the calculation shows good quantitative agreement of the experimental data, suggesting that the thickness dependence of the MOKE is primarily due to bulk-type magneto-optical coupling.Finally, the QMOKE is of interest for fundamental research relating to higher-order spin-orbit interaction in ultra-thin films.

Fig. 1 .
Fig. 1.(a) Schematic of the MOKE setup with the vector magnet.(b) Schematic of the directions of the applied field H and normalized magnetization m ; H φ / m φ refer to the

Fig. 3 .
Fig. 3. (a) LMOKE and (b) QMOKE in different sample orientations; α is the angle between the Fe [110] direction and the incident plane.Lines in (b) are fitting lines.

Figure 3
Figure 3 shows the

Fig. 4 .
Fig. 4. Linear and quadratic Kerr angle as a function of sample orientation α with incident angle 40 ϕ = ° on the sample Fe(12nm)/GaAs(100).(a) and (b) show the Kerr rotation with sand p-polarization of incident light, respectively; (c) and (d) show the Kerr ellipticity with sand p-polarized incident light, respectively.So we systematically studied L Φ , LT Φ and TT Φ as a function of the in-plane crystalline orientation.For a given in-plane crystal orientation α , L the so-called MO anisotropy parameter, α is the angle between the Fe [110] axis and the plane of incidence, and A and B are the complex optical weighting factors, which are determined by the incident angle, film thickness and optical indices of both film and substrate.It can be concluded from Eq. (5) that G Δ causes anisotropy with rotating α .Therefore, .The experimental results in Fig.4show excellent agreement with Eq. (5): L Φ shows no α dependence, TT Φ oscillates around zero and LT Φ oscillates around a nonzero value.It should be noted that LT Φ and TT Φ have the same fourfold anisotropic amplitude, proven by the experimental results in Fig. 4. By fitting the α-dependent results in Fig. 4 with Eq. (5), it is easy to quantify the L iso Φ , TT ani Φ , LT iso Φ and LT ani Φ contributions.

Fig. 5 .
Fig. 5.The isotropic and anisotropic contributions of (a) LMOKE and (b) QMOKE with various incident angles with s-polarized incident light on the sample Fe(12 nm)/GaAs(100).The solid lines are the fitted results.

Figures 6 (
Figures 6(c) and 6(e) show the measured QMOKE signals as a function of Fe thickness for both Kerr rotation and Kerr ellipticity, which show different Fe thickness dependences from that of the LMOKE signal in Fig. 6(a).The LT iso θ of the Kerr rotation signal reaches the peak position at 10 nm and changes sign at an Fe thickness of ~20 nm, while Kerr ellipticity LT iso ε becomes saturated at 15 nm and no sign change happens.For the anisotropic QMOKE contribution, both LT ani Φ and TT ani Φ present the same value for all Fe thicknesses, which is Kerr rotation or Kerr ellipiticity, L Φ is the LMOKE contribution, and LT Φ and TT Φ are the QMOKE contributions.Recently, the eight-field method has been used to separate contributions [4-6, 15] on magnetic films with separated thicknesses.Buchmeier et al. further studied the thickness dependence of the LMOKE and QMOKE on Fe films by fitting the hysteresis loops with the Stoner-Wohlfarth model L Φ , LT Φ and TT Φ

Table 1 .
[16]revious studies on the linear Kerr effect[16], the Voigt parameter Q was