REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 71, NUMBER 3 MARCH 2000 REVIEW ARTICLE Surface magneto-optic Kerr effect Z. Q. Qiu Department of Physics, University of California at Berkeley, Berkeley, California 94720 S. D. Badera) Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 Received 20 April 1999; accepted for publication 19 November 1999 The surface magneto-optic Kerr effect SMOKE has significantly impacted research on magnetic thin films. This is due to its sensitivity, local probing nature, and experimental simplicity. The polar and longitudinal Kerr effects are characterized by a complex rotation of the plane of polarization of linearly polarized incident light upon reflection from the surface of a ferromagnetic material. The rotation is directly related to the magnetization of the material within the probing region of the light. Light penetrates into metals 20 nm deep, but the SMOKE technique derives its surface sensitivity from the limited thickness of the deposited magnetic film, which can be as thin as one atomic layer. Basic principles, experimental arrangements, and applications of SMOKE are reviewed in order to acquaint the nonspecialist with the technique and place it into perspective. © 2000 American Institute of Physics. S0034-6748 00 00103-9 I. INTRODUCTION quiring a more powerful electromagnet on Sept. 18, he con- A. Discovery of the magneto-optic effect tinued the experiment with such zeal that he filled twelve pages of his lab notebook in one day and concluded with the Today magneto-optic effects are widely applied in mag- statement: ``An excellent day's work.'' He verified that the netic research. However, in the last century magneto-optics1 effect of the magnet was to rotate the polarization plane of was discovered somewhat unexpectedly while physicists the transmitted light by an angle that depended on the were searching for relationships between light and various strength of the magnet. other forces. Early searches were first conducted to find the The magneto-optic Kerr effect MOKE 5 was discovered interaction of light with electrical fields. It was believed that by the Rev. John Kerr in 1877 while he was examining the the effect of electrical fields should be stronger than that of polarization of light reflected from a polished electromagnet magnetic fields. But in 1825 null results were reported when pole. Kerr ultimately received the Royal Medal in 1898 for Sir John Herschel examined the propagation of a beam of research that ranked among the most important subsequent to polarized light along the axis of a helix carrying an electric Faraday's. When he was presented with the Royal Medal, his current.2 Even Michael Faraday's original search was fo- presenter said it was a wonder that Kerr learned so much cused on the relation between light and electricity. Faraday with the ``comparatively simple and ineffectual apparatus at kept a detailed lab diary, and on August 30, 1845 he re- his disposal.'' Kerr responded, ``Simple it may be, but not corded his failure to find a change in the polarization of light ineffectual; rude, but not crude.'' 6 This statement might rep- passing through a liquid that was undergoing electrolysis.3 It resent the nature of this technique as used in the present day was only when he substituted magnetic for electric forces on adaptations that are the subject of this article, especially Sept. 13 of that year, using an electromagnet with an iron when compared to many of the elegant techniques of modern core, that he discovered the magneto-optic effect. He re- surface science and nonlinear optics. But simplicity is a ma- corded in his lab notebook: ``A piece of heavy glass, which jor reason that this technique has in the last decade been so was 2 in. by 1.8 in. and 0.5 of an inch thick, being a silico- widely embraced to study magnetic thin films. borate of lead, was experimented with... when contrary mag- The application of the magneto-optic Kerr effect to sur- netic poles were on the same side there was an effect pro- face magnetism, better known by its acronym the surface duced on the polarized ray, and thus magnetic force and light magneto-optic Kerr effect SMOKE , began in 1985. The were proved to have relations to each other. This fact will first experimental system studied was ultrathin Fe films most likely prove exceedingly fertile, and of great value in grown epitaxially onto a single crystalline substrate of the investigation of conditions of nature force.'' 4 After ac- Au 100 .7 Hysteresis loops for the Fe film with atomic layer sensitivity were successfully obtained as a function of film a thickness and temperature. Since then SMOKE has been ap- Author to whom correspondence should be addressed; electronic mail: bader@anl.gov plied to address various issues in low-dimensional magne- 0034-6748/2000/71(3)/1243/13/$17.00 1243 © 2000 American Institute of Physics 1244 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Z. Q. Qiu and S. D. Bader tism. Additional interest in SMOKE derives from the recent give different dielectric constants correspondingly. Thus, it is commercialization of high-density magneto-optic informa- the Lorentz force of the external magnetic field that gener- tion storage media,8 and especially by studies of candidate ates the Faraday effect. materials for next-generation media based on Co/Pt Quantum descriptions of the magneto-optic effect have superlattices.9 The present work provides a general back- focused on the explanation of the unusual large magneto- ground to the basic principles and experimental setup of the optic effect in ferromagnetic materials. Early attempts to ex- SMOKE technique, and also highlights contemporary topics plain the much stronger magneto-optic effect in ferromag- in order to provide an appreciation for its value in applica- netic materials assumed that there exists an effective field, tions related to basic research on magnetic thin films. rather than the applied field, that determines the Faraday ro- tation in ferromagnetic materials. In fact, Voigt found that B. Origin of the magneto-optic effect the effective field is of the order of 106­ 107 Oe to produce the observed Faraday rotation. This magnitude is of the order Magneto-optics is presently described in the context of of the Weiss field that was postulated to account for the either macroscopic dielectric theory or microscopic quantum existence of ferromagnetism. The nature of the Weiss field theory.10 Macroscopically, magneto-optic effects arise from remained unexplained until Heisenberg developed the theory the antisymmetric, off-diagonal elements in the dielectric that ascribed the origin of magnetism to the exchange inter- tensor. Microscopically, the coupling between the electrical action among electrons. Although Heisenberg's exchange in- field of the light and the electron spin within a magnetic medium occurs through the spin-orbit interaction. In the fol- teraction correctly reveals the origin of magnetism as an ef- lowing, we give a brief review of the microscopic descrip- fective magnetic field to align the individual spins, this field tion of the magneto-optic effect, and leave the detailed mac- alone cannot be used to explain the Faraday effect. This is roscopic description to the next section. because it is not coupled to the electron motion which deter- As it is well known, the optical properties of a medium mines the dielectric properties of a material. This difficulty are determined by a dielectric tensor that is determined by was solved in 1932 by Hulme11 who pointed out that it is the the motion of the electrons in the medium. Thus, a micro- spin-orbit interaction that couples the electron spin to its mo- scopic description of the magneto-optic effect concerns the tion to give rise to the large Faraday rotation in a ferromag- different response of the electrons to left- and right-circularly netic. Spin-orbit coupling, ( V p)*s, results from the polarized electromagnetic waves. In the proceedings of the interaction of the electron spin with the magnetic field the Royal Society, Sir William Thomson, in 1856, offered a electron ``sees'' as it moves through the electric field V ``microscopic'' explanation of the Faraday effect by arguing with momentum p inside a medium. This interaction couples that the particles in the medium under an external magnetic the magnetic moment of the electron with its motion, thus, field follow different circular paths, depending on their di- connecting the magnetic and optical properties of a ferro- rection relative to the magnetic field. From a modern view- magnet. Indeed, to a certain extent, the spin-orbit interaction point, this explanation is conceptually correct if we identify can be thought of as an effective magnetic field vector po- Thomson's ``particles'' as being electrons although the tential s V acting on the motion of the electron. For electron had not yet been discovered at that time . nonmagnetic materials, this effect is not strong, although the It is worthwhile to first discuss the classical motion of spin-orbit interaction is present, because the equal number of electrons to point out the physical origin of the magneto- spin-up and spin-down electrons cancels the net effect. For optic effect. As a beam of light propagates through a me- ferromagnetic materials, however, the effect manifests itself dium, the electrical field of the light generates the motions of because of the unbalanced population of electron spins. the electrons in the medium. Without an external magnetic Hulme calculated the two refraction indices R and L field, it is obvious that a left-circularly polarized electric field polarized using the Heisenberg model of a ferromagnet, and will drive the electrons into left circular motion, and a right- the Kramers­Heisenberg dispersion formula. This approach circularly polarized electric field will drive the electrons into left circular motion, and a right-circularly polarized electric represents the refraction index in terms of the eigen energy field will drive the electrons into right circular motion. The and matrix elements of the dipole moment operator with re- radius of the electron orbit for left and right circular motion spect to the eigenfunctions of the system. Hulme accounted will be the same. Since the electric dipole moment is propor- for the difference of the two refraction indices by the energy tional to the radius of the circular orbit, there will be no splitting due to the spin-orbit interaction. He neglected, how- difference between the dielectric constants for the left- and ever, the change of the wave function due to the spin-orbit right-circularly polarized electromagnetic waves. Thus, there interaction. This theory is unsatisfying because the quench- will be no Faraday rotation. After an external magnetic field ing of the orbital angular momentum in ferromagnets gives is applied in the propagation direction of the electromagnetic no energy splitting. Kittel showed12 that it is the change of wave, there will be an additional Lorentz force acting on the wave functions due to the spin-orbit interaction that gives each electron. This force points toward or away from the rise to the correct order of magnitude of the difference of the circle's center for left or right circular motion. Thus, the two refraction indices. Argyres13 later gave a full derivation radius for left circular motion will be reduced and the radius of the magneto-optic effect in a ferromagnet using perturba- for right circular motion will expand. The difference in the tion theory. Subsequent works were performed thereafter to radii of the left- and right-circularly polarized modes will calculate the magneto-optic effect in different regimes.14­16 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Surface magneto-optic Kerr effect 1245 II. MACROSCOPIC FORMALISM FOR MAGNETIC breaks time reversal symmetry could, in principle, generate MULTILAYERS antisymmetric elements of the dielectric tensor, and, thus, a Faraday rotation. Macroscopic descriptions of the magneto-optic effect are Since most magnetic materials of interest are metals that based on an analysis of the dielectric properties of a medium. strongly absorb light, it is convenient to experimentally mea- In analogy with mechanical vibrations of a particle, Maxwell sure the reflected light in order to probe the magneto-optic expressed linearly polarized light as being a superposition of effect. Therefore, the general macroscopic formalism is for two circularly polarized components, and realized that the the magneto-optic Kerr effect although the formalism can be Faraday effect is a result of the different propagating veloci- readily extended to include the Faraday effect. Zak et al.19,20 ties of the two circular modes.17 This explanation remains developed a general expression for the Kerr signal based on the phenomenological explanation of the Faraday effect this method. We will outline the theoretical structure and given in introductory physics textbooks. Looked at in greater results here. detail, there are actually two processes taking place for light For a given magnetic multilayer the refraction tensor for propagating in a magnetized medium. First, the two circu- each layer can be expressed by a 3 3 matrix. The goal is to larly polarized modes gain different phase shifts due to their calculate the final reflectivity along different polarization di- different propagating velocities, resulting in a rotation of the rections. The general method is to apply Maxwell's equa- polarization plane. This process is the conventional Faraday tions to the multilayer structure, and to satisfy the boundary rotation. Second, the different absorption rates of the me- conditions at each interface. The essential part of the theory dium for the two circularly polarized modes affects the ellip- is to derive two matrices which relate the electric fields at ticity. In general, both effects exist in a magnetized medium. each interface. The first matrix A is the 4 4 medium The 3 3 dielectric tensor of a medium, ij with i,j boundary matrix. It relates the tangential components of the 1,2,3, can be decomposed into a symmetric part and an electric and magnetic fields with the s and p components of antisymmetric part, ij ( ij ji)/2 ( ij ji)/2. The sym- the electric field. The second matrix D is the 4 4 medium metric part can be diagonalized by an appropriate rotation of propagation matrix. It relates the s and p components of the the coordinate system, thus it does not give rise to the Fara- electric field at the two surfaces of a film of thickness d. day effect. Since the symmetric part of ij is unimportant to With the A and D matrices see Appendix for details , one the Faraday effect, we will always assume that it is isotropic can calculate the magneto-optic effect under any conditions. with dielectric constant 0 . To see the effect of the antisym- Consider a multilayer structure that consists of N indi- metric part of the dielectric tensor, let us consider the fol- vidual layers, and a beam of light impinging on the top of the lowing dielectric tensor: structure from initial medium i. After multiple reflections, 1 iQ there will be a reflected beam backscattered into medium i, z iQy and a transmitted beam that emerges from the bottom layer iQz 1 iQx . 1 into the final medium f Fig. 1 . The electric fields in medium iQy iQx 1 i and f can be expressed The two normal modes are left-circularly polarized light with i i refraction index nL n(1 12 Q*k ), and right-circularly po- ESEi i p ESEp larized light with refraction index n P R n(1 12 Q*k ), where i Er r i r i n is the average refraction index, Q (Q S ssES spE p x ,Qy ,Qz) is Er r i r i called the Voigt18 vector, and k is the unit vector along the p i psES ppE p direction of the light propagation. Thus, the complex Fara- and day rotation of the polarization plane after the light has trav- i i i eled a distance L through the medium is ES tspEp Ei tssESti t i L Ln P p psES ppE p , 3 f 0 0 nL nR Q*k . 2 0 0 f The real part of Eq. 2 gives the rotation, and the imaginary part gives the ellipticity. It is interesting to ask why an ex- where r and t are reflection and transmission coefficients of ternal magnetic field has a stronger effect on the polarization the corresponding components, and superscripts i and r de- plane of light than an external electrical field. Phenomeno- fine the incident and reflected waves at each boundary be- logically, this can be answered by a simple argument based tween two layers. If Pm is the field component at the bottom on time reversal symmetry. Under the time reversal opera- surface in the mth layer, then the relation between Pi and Pf tion, the displacement D and electric field E vectors remain can be expressed as unchanged, but the magnetic field H changes sign. Thus, A 1A Onsager's relation gives iPi A1D1P1 A1D1A1 1P1 i j(E,H) ji(E, H). By expand- ing 1 i j up to terms linear in E and H it becomes obvious that A1D1A1 A2D2P2 the antisymmetric part of ij is generated by the magnetic N field. The magnetic field is only one special case of time- ... A 1 A reversal symmetry breaking. In general, any quantity that mDmAm f P f . 4 m 1 1246 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Z. Q. Qiu and S. D. Bader n r f cos i ni cos f pp n , f cos i ni cos f 4 n r i cos i ps ni cos i nf cos f nf cos i ni cos f cos 2 m m f dmnmQz nfni sin i dmQy , m m 8 4 n r i cos i sp ni cos i nf cos f nf cos i ni cos f cos 2 m m f dmnmQz nfni sin i dmQy . m m Here ni , i , and nf , f are the refractive indices and the incident angles of the initial and final media, z is the surface normal direction, and y is in the plane of incidence and in the film plane. Equation 8 provides a basis for an additivity law for multilayers in the ultrathin limit. This law states that the total Kerr signal is simply a summation of the Kerr signals from each magnetic layer, and is independent of the nonmag- netic spacer layers in the multilayer structure. This additivity law is true only in the limit where the total optical thickness of the layered structure is much less than the wavelength of the incident beam. For thick films, it is obvious that the additivity law must break down because the light attenuates and will not penetrate to the deeper layers of the structure. The additivity law provides a focus for examining data in the FIG. 1. a Schematic of a multilayer structure; b definitions of the s and ultrathin limit. Altered optical constants in the ultrathin limit p directions for the incidence and reflection waves at the boundary between and interfacial roughness, of course, can also give rise to new two media. behavior that cannot be described within the context of the additivity law. If this expression is put in the form of Pi TPf , where N T A 1 1 i AmDmAm Af G H , 5 III. EXPERIMENTAL SETUP m 1 I J An experimental SMOKE setup has the advantage of then the 2 2 matrices G and I can be used to obtain the simplicity, especially for ultrahigh vacuum UHV in situ Fresnel reflection and transmission coefficients measurement. There are several ways to build a SMOKE setup. Here we introduce one of the simplest, which we use G 1 tsstsp t and IG 1 rssrsp . 6 frequently. Before discussing the instrumental setup, it is pstpp rpsrpp necessary to first discuss the working principle of the experi- The Kerr rotation and ellipticity for s-and p-polarized mental method. Consider linear p-polarized light reflected light are then given by from a sample surface. If the sample is nonmagnetic, the reflected light is purely p polarized. If the sample is ferro- r r magnetic then the reflection beam should consist of an s ps sp s s i s r and p p i p . 7 component (E ss rpp s) in addition to the dominant p component (Ep), with Es /Ep being the Kerr rotation. Therefore, mea- In the ultrathin limit the magneto-optic expressions sim- suring this s component will be the goal of the experimental plify further. For ultrathin films the total optical thickness of setup. Experimentally, the measurement of the s component the film is much less than the wavelength of the light, could be realized by placing a linear polarizer in front of the inidi . If the initial and final media are nonmagnetic, photodetector to eliminate the p component. However, this then the 2 2 matrices of G and I in Eq. 5 yield the fol- measurement geometry has the following disadvantage. lowing reflection coefficients: First, since the photodetector measures the light intensity ( E n s 2), the measured quantity is proportional to the r i cos i n f cos f square of the magnetization. Second, it is difficult to quantify ss n , i cos i n f cos f the absolute value of the Kerr rotation. This disadvantage Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Surface magneto-optic Kerr effect 1247 FIG. 2. Schematic drawing of a SMOKE setup. can be circumvented by setting the polarizer at a small angle from the p axis. In this way, the intensity measured by the photodetector after the polarizer is FIG. 3. A SMOKE loop taken from a 6 ML Fe film grown on a Ag 100 I Ep sin Es cos 2 Ep Es 2. 9 substrate. Recall that the expression Es /Ep i gives the Kerr rotation and ellipticity . Then Eq. 9 becomes length, and 4 cm width in the gap. With 300 turns in the coil, the magnetic field in the gap can reach 1.5 kOe at a I Ep 2 i 2 current of 20 A. Wire of 12 or 13 gauge for the coil can carry 10­20 A of current without a heating problem. For in situ 2 E measurements, the magnet is located inside the vacuum p 2 2 2 I0 1 10 chamber. During the measurement, data are taken as a func- with tion of magnetic field to generate an hysteresis loop. Because of possible drift of the laser intensity, it is recommended that I0 Ep 2 2 11 the average of many loops be taken. representing the intensity at zero Kerr rotation. Since both For in situ measurements, the UHV windows used as and are linearly proportional to the magnetization, the viewports usually produce a birefringence, w i w , that measured intensity as a function of H yields the magnetic prevents the realization of the optical extinction condition. In hysteresis loop. The saturation Kerr rotation this situation, a quarter-wave plate is usually placed before m can be de- termined by the relative change of the Kerr intensity I ob- the analyzing polarizer to cancel the window birefringence. tained upon reversing a field value that is equal to or greater Then the measured Kerr intensity becomes than its saturation value 2 I E 1 , 13 I p 2 2 2 I0 m 4 * I . 12 0 i.e., the relative Kerr intensity determines the Kerr ellipticity In the SMOKE experiment, a laser is usually used as the rather than the rotation in this case. The effect of the quarter- light source. Typically a low-power few mW laser suffices. wave plate is to produce a /2 phase difference between the It is highly desirable to use an intensity-stabilized laser, es- s and p components so that the analyzing polarizer will see pecially for monolayer studies. Otherwise, the fluctuations of i( i ) i , i.e., the rotation and ellipticity are the laser intensity may overwhelm the Kerr signal. The effect interchanged. Then to measure the rotation, a half-wave plate of the light intensity drifting during a hysteresis loop mea- could be used to replace the quarter-wave plate. Then the surement say, of 1­10 s duration causes a distortion of the reflected intensity as a function of the external magnetic field hysteresis loop. This effect cannot be eliminated by lock-in can be used to generate a magnetic hysteresis loop. Figure 3 techniques. While it is recommended that the SMOKE ex- shows an example of an hysteresis loop measured by periment be performed in a reduced vibration environment SMOKE for a 6 ML Fe/Ag 100 film, where ML denotes an optical table is not necessary, so it is easy to adapt monolayer. It quite apparent that SMOKE can readily SMOKE to an UHV system. Crystal prism polarizers are achieve monolayer sensitivity. useful both for defining the polarization and as an analyzer in front of the photodiode detector. Sheet polarizers can be IV. THE MOKE FAMILY used, but have a lower extinction ratio when crossed than While the present work is narrowly focused on the prism polarizers, and so are not optimal for monolayer stud- SMOKE approach and applications, this section introduces a ies. Finally, the magnet shown in Fig. 2 consists of two split- broader MOKE ``family'' in order to provide a useful per- coil solenoid pairs. Energizing either pair would generate spective and serve as a pointer to the literature. While a either a field in the film plane or perpendicular to it for lon- number of recent reviews have covered SMOKE,21­24 his- gitudinal or polar measurements, respectively. The dimen- torically the primary MOKE spectroscopy involves the de- sions of such a magnet are about 13 cm in height, 15 cm in termination of wavelength dependent properties.25 Commer- 1248 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Z. Q. Qiu and S. D. Bader cial as well as lab-built spectrometers have been used for from the signals associated with the diffracted beams. This such purposes. The main instrumental addition is a white should continue to provide a wealth of information in the light source and monochrometer. Representative work from characterization of such nanostructures, as well as challenges the laboratory of Schoenes and co-workers includes basic in the modeling of the relevant optical and magnetic pro- descriptions of the spectrometers.26 A recent example in- cesses. volves the giant magneto-optic Kerr rotation observed in A traditional application of MOKE is in magneto-optic single crystals of cerium compounds at low temperatures.27 imaging of magnetic domain structures. An example is in the In this work a Faraday modulator is used in conjunction with characterization of Fe/Cr magnetic multilayers.51 The tech- lock-in detection for noise suppression. Faraday modulators nique of differential polarization microscopy was recently have been utilized in other novel configurations,28 including introduced that utilizes a Wollaston prism to provide im- the quest for anions in high-temperature superconductors.29 proved image contrast.52 Wollaston prisms split the signal Lock-in detection techniques have not been discussed according to polarization content and can be used quite ef- with respect to SMOKE because modern computer- fectively in all types of MOKE measurements. Transparent controlled experiments can provide the signal averaging magneto-optic indicator films are also used quite effectively needed to improve the signal-to-noise level. In an early to image magnetic domain structures.53 In this approach a SMOKE publication30 it was even demonstrated that the hys- transparent Faraday film is placed on top of the sample of teresis loop collected by means of lock-in detection of the interest. The fringe fields emanating from the sample cause photodiode output and use of an incident beam whose polar- the Faraday-rotation contrast of the indicator film in order to ization is modulated by a commercial photoelastic modulator yield the image. This method is sensitive to the perpendicu- was no better than that obtained by the dc method outlined in lar magnetic response of the sample. However, samples with this review. The place where lock-in techniques and polar- in-plane magnetization have been imaged in this manner, ization modulation are especially valuable is in the simulta- with the clever addition of drilling holes in the sample, from neous magneto-optic characterization of the rotation and el- which perpendicular stray fields emerge. In addition to these lipticity as a function of wavelength. A recent example of types of magneto-optic microscopy, there are ideas under this approach is described in the work of Osgood et al.31 but discussion for the extension of such techniques into the near- such work follows the basic outlines provided by pioneers in field region using plasma-resonant Ag particles as probes.54 the field.32­35 Polarization modulation and related optical Synchrotron techniques are also popular and of great techniques are described in a general reference.36 The influ- value in magneto-optic characterizations. A description of ence of imperfections in the polarizer and analyzer were re- synchrotron methods to study magnetic systems has recently cently analyzed.37 Polarization modulation has also been become available; the overview stresses the opportunities used to characterize the Curie temperature of Gd films via in provided by third-generation synchrotron sources those for situ MOKE studies.38 which undulator insertion devices are used to intensify and To fill out the range of parameter space and the con- focus the beam .55 An advantage is that there is elemental comitant phenomena that become physically accessible, a sensitivity since the photon energy can be tuned to exploit number of variants are now mentioned. New magneto-optic the response associated with a specific atomic core level. transitions were identified in monolayer-range Fe films using Synchrotron-based magnetic circular dichroism MCD of- wavelength dependent measurements in the visible region.39 fers selection rules such that the spin and orbital magnetic The measurements were performed ex situ and the films were moments can be separately determined. X-ray MOKE sandwiched between Au layers to protect them from oxida- XMOKE is rapidly becoming another standard synchrotron tion. Oscillatory effects have been observed in SMOKE sig- technique to characterize magnetic films. nals of wedged structures and attributed to spin-polarized The magneto-optic imaging of antiferromagnetic domain quantum size effects.40­42 Dynamic scaling of the magnetic structure has always been a challenging task.56­58 A recent hysteresis was studied in the monolayer range for tour de force experiment that combines the use of a third- Fe/Au 001 ,43 Co/Cu 001 ,44 and Fe/W 110 45 by sweeping generation synchrotron source with x-ray magnetic linear di- the applied magnetic field at rates up to 1 kHz and monitor- chroism spectroscopy and the spatial resolution of a photo- ing the loop area. Magnetic field modulation has been used electron emission microscope permitted the imaging of the to obtain ac susceptibilities using an in situ MOKE antiferromagnetic structure at the surface of NiO 100 .59 apparatus.46 Time-resolved MOKE has been extended to the Magnetic linear dichroism of antiferromagnets depends on a picosecond range. A 30 ps pulsed dye laser was used in a second-order effect in the magnetization as does the trans- pump-probe experiment in order to study magnetization re- verse Kerr effect in ferromagnets . Second-order magneto- versal dynamics in magneto-optic storage materials.47 Dif- optic effects in anisotropic thin ferromagnetic films and the fraction MOKE is another recent addition due to interest in analysis of asymmetric hysteresis loops was the subject of a magnetic nanostructures that consist of arrays of holes48 or recent article that is largely based on the Ph.D. thesis re- lines gratings .49,50 The patterned arrays are of a size and search of Osgood.60 spacing that are comparable to the wavelength of visible Techniques based on nonlinear magneto-optics and sec- light and, thus, they serve as diffraction gratings. The trans- ond harmonic generation SHG and that utilize pump-probe verse MOKE signals from diffracted beams can be compared spectroscopies are rapidly becoming valuable tools in the to that from the specular beam. Enhanced magneto-optic ef- exploration of magnetic surfaces and interfaces. The identi- fects and unique magnetic hysteresis loops are extracted fying acronym is SHMOKE. Fortunately an excellent book Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Surface magneto-optic Kerr effect 1249 FIG. 4. RHEED oscillations taken during the growth of a Co 9.5 ML /Cu 16 ML n superlattice grown on a Cu 100 substrate. was recently published that encompasses both experimental and theoretical aspects of the field.61 Although the signal is weak, so that applications in magnetic recording are not fea- FIG. 5. The Kerr ellipticity measured for different samples. The solid lines sible, the enhanced sensitivity to interfaces, and the impres- are theoretical calculations. sive experimental successes,62 make this an exciting field to monitor in the future. oscillation represents the growth of an atomic layer. The per- Analytical and computational techniques have also un- sistence of the oscillations indicates a stable, well defined dergone a resurgence of activity that has broadened the growth mode. breadth and scope of MOKE studies. First, there are analyses The Kerr ellipticities of the films were measured in situ that are specifically used to motivate new experimental using a He­Ne laser. The angle of incidence was 17°. Re- methods for collecting and analyzing data.63­65 Then there sults are plotted in Fig. 5. The ellipticity for Co on polycrys- are macroscopic expressions that are useful for tal Cu was also measured and shown for comparison. We simulations,66­70 and approximations for multilayers based first concentrate on the magneto-optic behavior of a single on single-layer equivalences71 and effective-medium Co overlayer on a Cu substrate. The ellipticity data for the approaches.72 And finally there are microscopic methods overlayers increase linearly in the ultrathin regime, reach a based on first principles to calculate Kerr spectra including maximum at 120 Å of Co, and approach a constant value surface effects,73 and for transition metals and for 400 Å of Co. The initial rise is expected since the Kerr multilayers,74­76 rare earths77 and actinide systems.78 effect is sensitive to the increasing amount of Co. In the thick regime, 400 Å of Co, the signal saturates since the absorption of light limits the depth sensitivity. In the inter- V. VERIFICATION OF THE MACROSCOPIC FORMULAS mediate regime, the maximum in the ellipticity at 120 Å of Co is attributed to an optical effect: the reflectivity changes A specific example is now considered for the verification from being dominated by Cu to Co. Since Cu has a higher of the macroscopic formulas presented in Sec. II. This work reflectivity than Co, it acts as a mirror to enhance the signal. was accomplished by investigating Co overlayers and Co/Cu Similar behavior was also observed in the Fe/Au system.80 It superlattices.79 The films were grown epitaxially onto is also worth noting that the ellipticity is found to be inde- Cu 100 and Cu 111 single-crystal substrates in UHV base pendent of crystalline orientation in the thickness range stud- pressure of 2 10 10 Torr . The UHV chamber is equipped ied. with reflection high-energy electron diffraction RHEED , To analyze the data quantitatively, the formalism de- low-energy electron diffraction LEED , and Auger electron scribed in Sec. II was applied to simulate the results. The spectroscopy. The Cu substrate single-crystal disks were 1 refractive indices used were obtained from tabulations in the cm in diameter, and were mechanically polished down to a literature:81 nCu 0.249 3.41i and nCo 2.25 4.07i. The 0.25 m paste finish, and then ultrasonically cleaned in values of Q1 and Q2 , where Q Q1 iQ2 , for Co were left methanol before being put into the UHV chamber. Cycles of as free parameters to best fit the experimental curves; the 3 keV Ar sputtering and annealing at 650 °C were used to values Q1 0.043 and Q2 0.007 were obtained. The calcu- clean the Cu substrate surfaces in situ. After this treatment, lated curves, depicted as the solid lines in Fig. 5, are in good well ordered Cu surfaces were formed as indicated by overall agreement with the experimental data. In particular, RHEED and LEED. Auger spectroscopy confirmed the the peaked behavior of the overlayer data are faithfully re- cleanliness of the films. The RHEED intensity also was produced. The ellipticities of three epitaxial Co/Cu superlat- monitored during the growth of the film on the Cu 100 sub- tices were also measured in situ after each Co/Cu bilayer was strate in order to follow the process and to calibrate the grown. The superlattices used were Co(16 Å)/Cu(28 Å) n thickness monitor. Figure 4 shows the RHEED oscillations grown on Cu 100 , and Co(11 Å)/Cu(31 Å) n and during the growth of the Co/Co 100 superlattice. Over 200 Co(18 Å)/Cu(35 Å) n both grown on Cu 111 . The ellip- RHEED oscillations were observed during the growth. Each ticities of the superlattices appear in Fig. 5 plotted as a func- 1250 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Z. Q. Qiu and S. D. Bader the existence of magnetic long-range order LRO in 2D. Mermin and Wagner proved that quantum fluctuations in a 2D isotropic Heisenberg lattice destroys the LRO at finite temperature. Experimentally, however, the Curie tempera- ture (TC) in most 2D magnetic films is finite. This seeming contradiction suggests that, in addition to the Heisenberg ex- change interaction, there must exist other energy terms in magnetic thin films. Such energy terms are referred to as the magnetic anisotropies; these terms favor electron spin orien- tations in particular directions. In a lattice with cubic sym- metry fcc and bcc, for example , it can be shown that the lowest-order term in the anisotropy energy is quartic in the magnetization M. However, when translational symmetry along one direction is broken, a larger, square-term anisot- ropy can be generated along the surface normal direction. This term is often referred to as the magnetic surface anisot- ropy. A discussion of the origin of the magnetic anisotropy is provided in the next section. Nevertheless, a real magnetic FIG. 6. The additivity law shows that the Kerr signal in the ultrathin regime depends only on the thickness of the magnetic layers. thin film should be better described by an isotropic Heisen- berg exchange, a magnetic surface anisotropy (KS), and a tion of the total superlattice thickness. Again the ellipticities shape anisotropy ( 2 M2) which originates from the short- initially increase linearly in the ultrathin region, and then range part of the dipole­dipole interaction. The direction of saturate in thick regime, although there is no maximum at the easy axis of magnetization is determined by the sign of intermediate thickness as for the overlayer cases. The lack of the effective surface anisotropy Keff KS /d 2 M2, where d a maximum in the intermediate thickness regime is because is the film thickness. For systems with KS 0, a magnetiza- the reflectivity is not evolving from that of Cu to that of Co, tion perpendicular to the film plane can be stabilized at low as in the overlayer cases. Instead, the reflectivity maintains temperature and below certain thicknesses. Changing tem- itself at an average value between the two limits, since both perature or thickness can cause Keff to vanish at some point Co and Cu remain within the penetration depth of the light, below TC , and for the film to approach more closely the no matter how thick the superlattice becomes. Using the Q ideal realization of an isotropic 2D Heisenberg system. At value obtained from the Co overlayers, the Kerr ellipticities Keff 0 the spin-reorientation transition should occur wherein for the superlattices were calculated and plotted in Fig. 5. M changes its direction from perpendicular to in plane. The The agreement with the experimental data is obvious. question of interest regards the presence or absence of LRO To test the additivity law, the experimental data from at the transition. Early theoretical studies83 suggested that in Fig. 5 were replotted in Fig. 6 as a function of the thickness the vicinity of the SRT, there is a region in temperature TR of only the magnetic Co layers, as opposed to the total su- wherein the magnetic LRO is lost. perlattice thickness. All the data in the ultrathin regime then Several groups have carried out experiments on this sub- fall onto a single straight line. This result confirms the addi- ject. The first experiments were reported by Pappas Ka¨mper, tivity law that the total Kerr signal in the ultrathin regime is and Hopster84 using spin-polarized electron spectroscopy to a summation of the Kerr signal from each individual mag- characterize the systems Fe/Cu 100 and Fe/Ag 100 . They netic layer and is independent of the thickness of the non- found that at low temperature the easy axis was normal to the magnetic spacer layers. surface plane, at high temperature it was in plane, and in the SRT region there was a temperature gap 20 K wide within which the magnetic remanence vanished. Then the Fe/ VI. APPLICATIONS OF SMOKE IN TWO-DIMENSIONAL Ag 100 system was studied via SMOKE as a function of MAGNETIC THIN FILMS both temperature and film thickness. It was found that the In addition to the simplicity of the SMOKE setup as an magnetization is not identically zero in the transition region, in situ magnetic measurement technique, the two great ad- but is markedly reduced and exhibits structure in a ``pseudo vantages of the SMOKE are its high sensitivity and local gap'' that resembles an asymmetric ramping toward zero probe nature. These two characteristics make SMOKE a with increasing temperature or film thickness.85 Thus, the popular choice to address issues in thin film magnetism. Two gap, if it exists, must be at least an order of magnitude examples are included below to highlight the application of smaller than the 20 K reported by Pappas and co-workers. SMOKE. To illustrate the advantage of the local probe nature of the SMOKE technique, we limit the discussion to address the A. Spin-reorientation transition thickness dependent data only. To explore the detailed fea- The investigation of the two-dimensional 2D spin- tures within the SRT region, many samples with different reorientation transition SRT was originally motivated as a film thicknesses are needed. This is because sample-to- test of the Mermin­Wagner theorem.82 This theorem ad- sample variations are known to occur throughout surface sci- dresses the most fundamental issue in thin film magnetism- ence and thin-film growth that cannot be adequately con- Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Surface magneto-optic Kerr effect 1251 sented by Allenspach and Bischof who applied the SEMPA technique to study the SRT in the Fe/Cu 100 system.89 SEMPA is a highly surface-sensitive magnetic imaging technique that stands for ``scanning electron microscopy with polarization analysis.'' They observed that the single domain structure of the film breaks into stripe domains 1 m size in the gap region. Results of dynamic properties of the SRT are also consistent with a stripe domain structure.90 B. Magnetic anisotropy and lattice symmetry breaking The Heisenberg exchange interaction is invariant under spatial rotation. In a real lattice, however, the electrons that FIG. 7. Spin reorientation transition as a function of film thickness. contribute to the magnetization usually obey the lattice sym- metry in their wave functions due to the crystal fields. Thus, trolled. This difficulty can be overcome by the use of wedge- the spin-orbit interaction can transfer the lattice symmetry shaped samples. Wedged samples provide an essentially from the electron wave functions to the electron spins to continuous change of film thickness within a single specimen break the spin isotropy. Thus, energy terms exist that favor so that, as a local probe scans across the sample, the magne- special directions for the electron spins. This energy is called tization can be measured for different thicknesses in a sys- the magnetocrystalline anisotropy. Given that it originates tematic manner. SMOKE measures the magnetization within from the spin-orbit interaction,91 the magnetic anisotropy the confines of the laser spot. When applied to the SRT, its must obey the symmetry of the lattice. Understanding how thickness resolution can achieve the 0.04 ML level for a lattice symmetry breaking induces magnetic anisotropy is of typical wedge with, say, a 2 ML mm slope. Figure 5 illus- fundamental importance. trates the perpendicular magnetic remanence (M To isolate the lattice symmetry effect from the electronic ) deter- mined from the polar signal, and the parallel remanence effect, a few groups have performed experiments on mag- (M netic thin films grown on stepped 001 substrates.92­94 The ) determined from the longitudinal signal, at room tem- perature for different Fe film thicknesses. The existence of idea is that the atomic steps on the 001 surface break the the SRT is evident as a function of the film thickness. At low fourfold rotational symmetry to induce a uniaxial magnetic thicknesses, M anisotropy in the film plane. Experimentally, stepped sur- maintains its saturation value, and at large thicknesses, M faces consist of low Miller-index terraces uniformly sepa- retains its saturation value. However, in the SRT region d in Fig. 7 , M is greatly suppressed from its rated by atomic steps, and are created by polishing a sample saturation value. This region is similar to that observed by surface that is misaligned by a few degrees from the terrace Pappas, Ka¨mper, and Hopster in Fe/Cu 100 84 and Fe/ normal direction. Such surfaces are also referred to as vicinal Ag 100 via polarized electron scattering.86 But the SMOKE surfaces, because crystallographically they are oriented in the measurements definitively show that M is not zero in this vicinity of fundamental, low Miller-index faces. To experi- region. Thus, this region is not associated with a loss of mentally explore the relationship between induced magnetic LRO, but instead with a pseudo gap that suggests the pres- anisotropy and lattice symmetry breaking, many substrates ence of complex magnetic structure. with different vicinal angles would be needed. In practice, it The formation of magnetic domains within the pseudo- is difficult to prepare multiple surfaces under identical con- gap region provides a possible explanation for the suppres- ditions. To overcome this difficulty, ``curved'' substrates sion of M but without a loss of magnetic LRO. Yafet and have been introduced to provide a continuous gradation in Gyorgy were first to recognize that a stripe domain structure the step density. Substrates of 001 orientation and 1 cm in has a lower energy than a single domain structure in a 2D diameter are used in the examples we cite below. Half of the system with perpendicular, uniaxial anisotropy.87 It was surface is polished to retain its 001 orientation and serve as found that the domain size increases almost exponentially as a reference, while the other half is polished with a curvature the effective surface anisotropy departs from zero. Therefore, such that the vicinal angle varies continuously from 0° to the stripe domains are observable only in the vicinity of the 10°. SMOKE has a distinct advantage for this study because SRT where the effective surface anisotropy is nearly zero the reflection angle of the SMOKE laser beam simulta- and the domain size is less than the sample size. This ex- neously determines the local vicinal angle so that the relation plains why there is a gapped region in the SRT within which between the step-induced magnetic anisotropy and the step the magnetic remanence is greatly suppressed. Stripe do- density can be systematically explored from a single curved mains form a one-dimensional 1D ordered state which it- sample. self is unstable against thermal fluctuations. Indeed, Kashuba Results for three representative systems are discussed: and Pokrovsky88 found that the stripe domain structure is Fe/W 001 , Co/Cu 001 , and Fe/Pd 001 . In Fe/W and Fe/ equivalent to a 2D liquid crystal system in that it possesses Pd, the steps are parallel to the 100 direction of the Fe. For orientational order but no spatial order. An experimental ob- the Co/Cu system, the steps are parallel to the 11¯0 direc- servation of the stripe domains in the SRT region was pre- tion of the Co. The magnetization is in the film plane for all 1252 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Z. Q. Qiu and S. D. Bader FIG. 8. Hysteresis loops for a 2 ML Fe film grown on a stepped W 001 surface miscut by 4.7°. The square loop is for H perpendicular to the step edges, and the split loop is for H parallel to the step edges. three systems, thus only longitudinal hysteresis loops are re- ported. Figure 8 shows hysteresis loops for a 2 ML Fe film grown on stepped W 001 with a 4.7° vicinal angle. The stepped Fe film shows a square loop with 100% remanence for the applied magnetic field oriented perpendicular to the step edges, and two split subloops with zero remanence for the field oriented parallel to the step edges. This behavior indicates that the atomic steps indeed induce an in-plane, uniaxial magnetic anisotropy with the easy magnetization axis perpendicular to the step edges. The easy axis of the step-induced anisotropy depends on the physical system. While the Fe/W 001 95 and Fe/Pd 001 96 systems have their easy axis perpendicular to the step edges, the Co/Cu 001 97 and Fe/Ag 001 98 systems have their easy axis parallel to the step edges. Nevertheless, the splitting field Hs , as defined in Fig. 8 for the hard-axis loop, is proportional to the strength of the step-induced anisotropy. Figure 9 shows the relation between Hs and the vicinal angle which is proportional to the step density for the three systems. Fitting Hs n the solid lines in Fig. 9 yields an exponent n 2 for the Fe/W system, but n 1 for the Co/Cu and Fe/Pd systems. To understand why there is different dependence of Hs on for different systems, one has to examine how the sym- metry of the lattice at the step edges is broken for bcc and fcc FIG. 9. Hs from Fig. 8 vs vicinal angle for a Fe/W 001 , b Co/ Cu 001 , and c Fe/Pd 001 . The solid lines are results of a power-law structures. In the Ne´el pair-bonding model99 the magnetic fitting yielding a quadratic relation between Hs and for the Fe/W system, anisotropy is generated by nearest-neighbor bonds. For bcc but a linear relation for the Co/Cu and Fe/Pd systems. and fcc lattices, there is no uniaxial anisotropy because con- tributions from all nearest-neighbor bonds cancel out the u square-term anisotropy. At the step edges, however, the y sin uz cos , into to the film xyz frame with the x and y axes in the film plane parallel and perpendicular to the step missing atoms break this cancellation so that uniaxial anisot- edges, respectively. The anisotropy for small transforms ropy will be manifest. to E 2 2u2) for a bcc lattice, and E For a bcc lattice with steps parallel to the 100 direc- a K( uyuz 2uy z a K(2 u2 3 u2 2& u tion, the anisotropy due to the missing atoms should have the y z yuz) for a fcc lattice. There- fore the in-plane, step-induced anisotropy (u form E z 0) is Ea a (K/L)u u , where L is the terrace length, u is K 2u2 for the bcc case, and E 2 for fcc. This the unit vector of the magnetization M, and , , and are y a 2K uy provides an explanation for why Fe/W and Co/Cu exhibit the 100 , 010 , and 001 axes, respectively. For fcc lattice quadratic and linear dependences, respectively, for their with steps parallel to the 11¯0 direction, the anisotropy due step-induced anisotropies. The most interesting result is the to the missing atoms has the form E 2 2 a KL(2u 3u linear dependence in the Fe/Pd system. Fe has a bcc structure 2&u u ), where 11¯0 , 110 , and 001 . but Pd has fcc structure. It was shown that the Pd at the Note that the normal direction z axis of the stepped surface interface of Fe/Pd is ferromagnetic due to the Fe spin makes a small vicinal angle to the 001 axis so that polarization.100,101 Since Pd has a much stronger spin-orbit 1/L . The crystal frame of reference has to be trans- interaction than Fe, the Pd is expected to dominate the mag- formed from u ux , u uy cos uz sin , and u netic anisotropy in the Fe/Pd system. We believe that is why Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Surface magneto-optic Kerr effect 1253 the Fe/stepped Pd 001 system exhibits an -linear depen- It is easy to see that D, B or H and k are perpendicular to dence of the step-induced anisotropy as for a fcc lattice. each other. The E vector, however, has a component parallel to the wave vector k. Using the familiar s- and p- polariza- VII. DISCUSSION tion modes, the electric field can be written as The basic principles, experimental setup, and two appli- cations of the SMOKE technique were outlined. Although E Eses Epep i Q*epEs Q*esEp ek . A3 SMOKE is a powerful technique, it has certain drawbacks. For example, it cannot distinguish surface or interface mag- Here es , ep , and ek are unit vectors along the s, p, and k netism from that arising from the interior layers. This is an directions. Es and Ep are the s and p components of the area where nonlinear second harmonic MOKE or electric field, and their equations of motion are SHMOKE has major advantages. SMOKE also cannot, in general, distinguish an antiferromagnetic phase from a non- i 2 Q*ek magnetic phase. These weaknesses leave many experimental 2 c2 k2 Es c2 Ep 0 challenges for the future. Also, concerning theoretical chal- . A4 i 2 Q*e lenges, a microscopic understanding of magneto-optics in the k monolayer regime is needed since macroscopic continuum c2 Es 2 c2 k2 Ep 0 theory must ultimately break down. Experimentally, it is also important to enhance both spatial and time resolution so that To first order in Q, it is easy to show that the two normal small scale processes, such as domain wall dynamics, can be modes are right R - and left L -circularly polarized modes investigated. Possible ways to realize this goal involve com- with bining SMOKE with other techniques, such as near-field op- tical spectroscopy, scanning tunneling microscopy, and/or kR,L k 1 12 Q*ek or nR,L n 1 12 Q*ek . A5 pump-and-probe methods. In the present article a sense of history as well as of future opportunities was invoked to Here k c and n are the wave vector and refraction stimulate interest both in the SMOKE technique and in its index, respectively, without the magnetization. After obtain- impact on modern thin-film and surface magnetism. ing the two normal modes, any mode of the electromagnetic wave can be viewed as their superposition. ACKNOWLEDGMENTS Now, we consider an electromagnetic wave propagating inside a magnetic multilayer structure. At each boundary be- The authors thank the many colleagues that we have tween two layers, the boundary conditions involve Ex , Ey , worked with on the projects cited in the reference list. With- Hx and Hy , where x and y axis are in the film plane and out them this work would not be possible. This work was perpendicular and parallel to the incident plane, respectively. financially supported by the Office of Basic Energy It is more convenient to express these four quantities with Sciences-Materials Sciences of the United States Department the s and p components of the electric field. The x compo- of Energy under Contract Nos. W-31-109-ENG-38 at Ar- nents are easy to write because they are parallel to the s gonne and DE-AC03-76SF00098 at Berkeley and by Na- direction tional Science Foundation Contract No. DMR-9805222 at Berkeley . E i r x Es Es , A6 APPENDIX where the superscripts i and r denote the incident and re- To derive the matrices A and D, it is important to first flected waves, respectively. For the y components, one has to describe the normal modes of the electromagnetic waves in a keep in mind that the electric field has a component i( Q magnetic medium. To obtain them, consider a wave *epEs Q*esEp)ek parallel to the k direction, and that the L eik*x i t propagating in a medium whose dielectric tensor and R modes have different refraction indices and incident is described by Eq. 1 . Since the magnetic response of the angles. Then Ey can be expressed as medium is attributed to the Voigt vector Q in the dielectric tensor, we can assume that the magnetic permeability is 1. E i,L i,R i i i i y E p cos L Ep cos R i Q*epEs Q*esEp Then the relationship between D and E, and B and H is sin Er,L r,R p cos L E p cos R D E i E Q and B H. A1 i Q*er Er Q*erEr sin . A7 Then Maxwell's equations give p s s p k*E ik* E Q 0 Using the relations k E L L c H iEs . A2 EpER iER , A8 k*H 0 p s nR sin R nL sin L n sin k H c E iE Q Ey can be expressed as 1254 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Z. Q. Qiu and S. D. Bader i in E i i i y 2 Qy tan 1 cos2 Qz sin2 Es Hy n cos Es 2 Qy tan Qz Ep i in cos iQ i r r x sin E p 2 Qy tan 1 cos2 n cos Es 2 Qy tan Qz Ep . A10 Q r r Therefore, we obtain the relation between the x and y com- z sin2 Es cos iQx sin Ep . A9 ponents of E and H with s and p components of the electric Hx and Hy can be derived in a similar way from the expres- field. This relation can be expressed as a matrix product sion k E /cH: E i x E EsEi in y A p A11 H i i H Er x 2 Qy sin Qz cos Es nEp x s H r y Ep in r r with the 4 4 matrix A known as the medium boundary 2 Qy sin Qz cos Es nEp , matrix 1 0 1 0 i i 2 Qy tan 1 cos2 Qz sin2 cos iQx sin 2 Qy tan 1 cos2 Qz sin2 cos iQx sin A in in . 2 Qy sin Qz cos n 2 Qy sin Qz cos n in in n cos 2 Qy tan Qz n cos 2 Qy tan Qz A12 The next step is to derive the propagation matrix which relates E and H at the two surfaces of a film of thickness d. kd Since each incident and reflected beam is composed of L- 2 Qy tan Qz and R-circularly polarized modes, we use 1 and 2 to denote U exp ikdcos i . A16 the L and R modes of the incident beam at both surfaces, and kd 3 and 4 to denote the L and R modes of the reflected beam at r 2 Qy tan Qz both surfaces. Then we have the following relations: E1,2 1,2 1 M. Faraday, Trans. R. Soc. London 5, 592 1846 . A EB exp ik1,2d cos 1,2 2 E3,4 3,4 The Effects of a Magnetic Field on Radiation-Memoirs by Faraday, A EB exp ik3,4d cos 3,4 . A13 Kerr and Zeeman, edited by E. P. Lewis American, New York, 1990 , The relation between E preface, p. v. s and E p at boundaries A and B can 3 M. Faraday, Diary, 30 August 1845, Vol. 4, pp. 7434, and 7437­7444. then be expressed by a matrix product 4 M. Faraday, Diary, 13 September 1845, Vol. 4, p. 7504. 5 J. Kerr, Philos. Mag. 3, 339 1877 ; 5, 161 1878 . Ei i 6 s See, Molecular Electro-Optics, Part I, Theory and Methods, edited by C. Ei i T. O'Konski Dekker, New York, 1976 , p. 517. p EsEp 7 E. R. Moog and S. D. Bader, Superlattices Microstruct. 1, 543 1985 ; S. Er D r , A14 s Es D. Bader, E. R. Moog, and P. Gru¨nberg, J. Magn. Magn. Mater. 53, L295 Er r 1986 . p E A p B 8 S. Klahn, P. Hansen, and F. J. A. M. Greidanus, Vacuum 41, 1160 where D is 4 4 matrix known as the medium propagation 1990 . 9 K. Nakamura, S. Tsunashima, S. Iwata, and S. Uchiyama, IEEE Trans. matrix. Magn. 25, 3758 1989 ; S. Hashimoto, H. Matouda, and Y. Ochiai, Appl. Phys. Lett. 56, 1069 1990 ; S. Hashimoto, Y. Ochiai, and K. Aso, J. U cos i U sin i 0 0 Appl. Phys. 67, 2136 1990 . 10 U sin L.D.LandauandE.M.Lifshtz,ElectrodynamicsofContinuousMedia D i U cos i 0 0 Pergamon, London, 1960 . 0 0 U 1 cos 11 H. R. Hulme, Proc. R. Soc. London, Ser. A 135, 237 1932 . r U 1 sin r 12 C. Kittel, Phys. Rev. 83, 208 A 1951 . 0 0 U 1 sin 13 r U 1 cos r P. N. Argyres, Phys. Rev. 97, 334 1955 . A15 14 H. S. Bennett and E. A. Stern, Phys. Rev. 137, A448 1965 . 15 Y. R. Shen, Phys. Rev. 133, A51 1964 . with 16 J. E. Erskine and E. A. Stern, Phys. Rev. B 8, 1239 1973 . Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Surface magneto-optic Kerr effect 1255 17 J. C. Maxwell, Electricity and Magnetism 1873 , Vol. 2, Chap. 21. 57 V. V. Eremenko, N. F. Kharchenko, and L. I. Beliy, J. Appl. Phys. 50, 18 W. Voigt, Magneto- und Elektro-optic Teuner, Leipzig, 1908 ; Hand- 7751 1979 . book der Elektrizita¨t und des Magnetismus Barth, Leipzig, 1915 , Vol. 58 V. V. Eremenko and N. F. Kharchenko, J. Magn. Soc. Jpn. 11, Supple- IV. 2, p. 39. ment, No. S1, 27 1987 . 19 J. Zak, E. R. Moog, C. Liu, and S. D. Bader, J. Magn. Magn. Mater. 89, 59 J. Sto¨hr, A. Scholl, T. Regan, S. Anders, J. Lu¨ning, M. R. Scheinfein, H. 107 1990 . A. Padmore, and R. L. White, Phys. Rev. Lett. 83, 1862 1999 . 20 J. Zak, E. R. Moog, C. Liu, and S. D. Bader, Phys. Rev. B 43, 6423 60 R. M. Osgood III, S. D. Bader, B. M. Clemens, R. L. White, and H. 1991 . Matsuyama, J. Magn. Magn. Mater. 182, 297 1998 . 21 S. D. Bader, J. Magn. Magn. Mater. 100, 440 1991 . 61 Nonlinear Optics in Metals, edited by K. H. Bennemann Clarendon, 22 J. L. Erskine and S. D. Bader, in Ultrathin Magnetic Structures II, edited Oxford, 1998 . by B. Heinrich and J. A. C. Bland Springer, Berlin, 1994 , pp. 297­325. 62 See, for example, T. Rasing, in Ref. 61. 23 Z. Q. Qiu and S. D. Bader, in Nonlinear Optics in Metals, edited by K. H. 63 J. M. Florczak and E. D. Dahlberg, J. Appl. Phys. 67, 7520 1990 ; Phys. Bennemann Clarendon, Oxford, 1998 , pp. 1­41. Rev. B 44, 9338 1991 . 24 Z. Q. Qiu and S. D. Bader, J. Magn. Magn. Mater. 200, 664 1999 . 64 A. Berger and M. R. Pufall, Appl. Phys. Lett. 71, 965 1997 ; J. Appl. 25 G. S. Krinchik and V. A. Artem'ev, Sov. Phys. JETP 26, 1080 1968 . Phys. 85, 4583 1999 . 26 J. Schoenes, Phys. Rep. 66, 187 1980 . 65 M. R. Pufall, C. Platt, and A. Berger, J. Appl. Phys. 85, 4818 1999 . 27 R. Pittini, J. Schoenes, and P. Wachter, Phys. Rev. B 55, 7524 1997 . 66 W. A. McGahan and J. Woollam, Appl. Phys. Commun. 9, 1 1989 . 28 J. F. Dillon, Jr. E. M. Gyorgy, F. Hellman, L. R. Walker, and R. C. 67 M. Mansuripur, J. Appl. Phys. 67, 6466 1990 . Fulton, J. Appl. Phys. 64, 6098 1988 . 68 S. Visnovsky, M. Nyvlt, V. Prosser, R. Lopusnik, R. Urban, J. Ferre´, G. 29 K. B. Lyons, J. Kwo, J. F. Dillon, Jr., G. P. Espinosa, M. McGlashan- Pe´nissard, D. Renard, and R. Krishnan, Phys. Rev. B 52, 1090 1995 . Powell, A. P. Ramirez, and L. F. Schneemeyer, Phys. Rev. Lett. 64, 2949 69 K. R. Heim and M. R. Scheinfein, J. Magn. Magn. Mater. 154, 141 1990 . 1996 . 30 S. D. Bader and E. R. Moog, J. Magn. Soc. Jpn. 11, Supplement No. S17 70 C.-Y. You and S.-C. Shin, J. Appl. Phys. 84, 541 1998 . 1987 . 71 R. Atkinson, J. Magn. Magn. Mater. 95, 61 1991 ; 95, 69 1991 . 31 R. M. Osgood III, K. T. Riggs, A. E. Johnson, J. E. Mattson, C. H. 72 C.-Y. You, S.-C. Shin, and S.-Y. Kim, Phys. Rev. B 55, 5953 1997 . Sowers, and S. D. Bader, Phys. Rev. B 56, 2627 1997 . 73 M. Kim, A. J. Freeman, and R. Wu, Phys. Rev. B 59, 9432 1999 . 32 S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instrum. 40, 761 1969 ; 74 P. M. Oppeneer, T. Maurer, J. Sticht, and J. Ku¨bler, Phys. Rev. B 45, erratum 41, 152 1970 10924 1992 . 33 J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, J. Opt. 8, 373 75 G. Y. Guo and H. Ebert, Phys. Rev. B 50, 10377 1994 . 1977 . 76 T. Gasche, M. S. S. Brooks, and B. Johansson, Phys. Rev. B 53, 296 34 K. Sato, Jpn. J. Appl. Phys. 20, 2403 1981 . 1996 . 35 P. Q. J. Nederpel and J. W. D. Martens, Rev. Sci. Instrum. 56, 687 77 V. N. Antonov, B. N. Harmon, A. Y. Perlov, and A. N. Yaresko, Phys. 1985 . Rev. B 59, 14561 1999 . 36 D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics 78 V. N. Antonov, B. N. Harmon, A. N. Yaresko, and A. Y. Perlov, Phys. and Spectroscopy Academic, Boston, 1990 . Rev. B 59, 14571 1999 . 37 C.-Y. You and S.-C. Shin, Rev. Sci. Instrum. 68, 3519 1997 . 79 Z. Q. Qiu, J. Pearson, and S. D. Bader, Phys. Rev. B 46, 8195 1992 . 38 M. Farle, W. A. Lewis, and K. Baberschke, Appl. Phys. Lett. 62, 2728 80 E. R. Moog, S. D. Bader, and J. Zak, Appl. Phys. Lett. 56, 2687 1990 . 1993 . 81 J. H. Weaver, in CRC Handbook of Chemistry and Physics, 69th ed., 39 Y. Suzuki, T. Katayama, S. Yoshida, K. Tanaka, and K. Sato, Phys. Rev. edited by R. C. Weast, M. J. Astle, and W. H. Beyer CRC, Boca Raton, Lett. 68, 3355 1992 . 1988 , p. E-387ff. 40 A. Carl and D. Weller, Phys. Rev. Lett. 74, 190 1995 . 82 M. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 1966 . 41 R. Me´gy, A. Bounouh, Y. Suzuki, P. Beaunillain, P. Bruno, C. Chappert, 83 D. Pescia and V. L. Pokrovsky, Phys. Rev. Lett. 65, 2599 1990 . B. Lecuyer, and P. Veillet, Phys. Rev. B 51, 5586 1995 . 84 D. P. Pappas, K.-P. Ka¨mper, and H. Hopster, Phys. Rev. Lett. 45, 8169 42 Y. Suzuki, T. Katayama, P. Bruno, S. Yuasa, and E. Tamura, Phys. Rev. 1992 . Lett. 80, 5200 1998 . 85 Z. Q. Qiu, J. Pearson, and S. D. Bader, Phys. Rev. Lett. 70, 1006 1993 . 43 Y.-L. He and G.-C. Wang, Phys. Rev. Lett. 70, 2336 1993 . 86 D. P. Pappas, C. R. Brundle, and H. Hopster, Phys. Rev. B 45, 8169 44 Q. Jiang, H.-N. Yang, and G.-C. Wang, Phys. Rev. B 52, 14911 1995 . 1992 . 45 J.-S. Suen and J. L. Erskine, Phys. Rev. Lett. 78, 3567 1997 . 87 Y. Yafet and E. M. Gyorgy, Phys. Rev. B 38, 9145 1988 . 46 A. Berger, S. Knappmann, and H. P. Oepen, J. Appl. Phys. 75, 5598 88 A. Kashuba and V. L. Pokrovsky, Phys. Rev. Lett. 70, 3155 1993 ; Phys. 1994 . Rev. B 48, 10335 1993 . 47 D. Guarisco, R. Burgermeister, C. Stamm, and F. Meier, Appl. Phys. 89 R. Allenspach and A. Bischof, Phys. Rev. Lett. 69, 3385 1992 . Lett. 68, 1729 1996 . 90 A. Berger and H. Hopster, Phys. Rev. Lett. 76, 519 1996 . 48 P. Vavassori, V. Metlushko, R. M. Osgood III, M. Grimsditch, U. Welp, 91 D. Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett. 70, 869 1993 , and G. Crabtree, W. Fan, S. R. J. Brueck, B. Ilic, and P. J. Hesketh, Phys. references therein. Rev. B 59, 6337 1999 . 92 A. Berger, U. Linke, and H. P. Oepen, Phys. Rev. Lett. 68, 839 1992 . 49 Y. Souche, V. Novosad, B. Pannetier, and O. Geoffroy, J. Magn. Magn. 93 W. Weber, C. H. Back, A. Bischof, Ch. Wu¨rsch, and R. Allenspach, Mater. 177­181, 1277 1998 . Phys. Rev. Lett. 76, 1940 1996 . 50 D. van Labeke, A. Vial, V. Novosad, Y. Souche, M. Schlenker, and A. D. 94 J. Chen and J. Erskine, Phys. Rev. Lett. 68, 1212 1992 . Dos Santos, Opt. Commun. 124, 519 1996 . 95 H. J. Choi, Z. Q. Qiu, J. Pearson, S. J. Jiang, D. Li, and S. D. Bader, 51 M. Ru¨hrig, R. Scha¨fer, A. Hubert, R. Mosler, J. A. Wolf, S. Demokritov, Phys. Rev. B 57, R12713 1998 . and P. Gru¨nberg, Phys. Status Solidi A 125, 635 1991 . 96 H. J. Choi, R. K. Kawakami, E. J. Escorcia-Aparicio, Z. Q. Qiu, J. Pear- 52 W.-H. Yeh, J. Carriere, and M. Mansuripur, Appl. Opt. 38, 3749 1999 . son, J. S. Jiang, Dongqi Li, and S. D. Bader, Phys. Rev. Lett. 82, 1947 53 L. H. Bennett, R. D. McMichael, L. J. Swartzendruber, S. Hua, D. S. 1999 . Lashmore, A. J. Shapiro, V. S. Gornakov, L. M. Dedukh, and V. I. 97 R. K. Kawakami, M. O. Bowen, H. J. Choi, E. J. Escorcia-Aparicio, and Nikitenko, Appl. Phys. Lett. 66, 888 1995 . Z. Q. Qiu, Phys. Rev. B 58, R5924 1998 . 54 M. R. Pufall, A. Berger, and S. Schultz, J. Appl. Phys. 81, 5689 1997 . 98 R. K. Kawakami, E. J. Escorcia-Aparicio, and Z. Q. Qiu, Phys. Rev. Lett. 55 J. B. Kortright, D. D. Awschalom, J. Stohr, S. D. Bader, Y. U. Idzerda, S. 77, 2570 1996 . S. P. Parkin, I. K. Schuller, and H.-C. Siegmann, J. Magn. Magn. Mater. 99 L. Ne´el, J. Phys. Radium 15, 225 1954 . 207, 7 1999 . 100 O. Rader, E. Vescovo, J. Redinger, S. Blu¨gel, C. Carbone, W. Eberhardt, 56 J. F. Dillon, Jr., E. Y. Chen, N. Giordano, and W. P. Wolf, Phys. Rev. and W. Gudat, Phys. Rev. Lett. 72, 2247 1994 . Lett. 33, 98 1974 . 101 C. Liu and S. D. Bader, Phys. Rev. B 44, 2205 1991 .