Smoke

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 71, NUMBER 3 MARCH 2000

REVIEW ARTICLE

Surface magneto-optic Kerr effectZ. Q. QiuDepartment 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 magneticthin films. This is due to its sensitivity, local probing nature, and experimental simplicity. The polarand longitudinal Kerr effects are characterized by a complex rotation of the plane of polarization oflinearly polarized incident light upon reflection from the surface of a ferromagnetic material. Therotation 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 sensitivityfrom 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 toacquaint the nonspecialist with the technique and place it into perspective. ©2000 AmericanInstitute of Physics.@S0034-6748~00!00103-9#

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I. INTRODUCTION

A. Discovery of the magneto-optic effect

Today magneto-optic effects are widely applied in manetic research. However, in the last century magneto-op1

was discovered somewhat unexpectedly while physicwere searching for relationships between light and variother forces. Early searches were first conducted to findinteraction of light with electrical fields. It was believed ththe effect of electrical fields should be stronger than thamagnetic fields. But in 1825 null results were reported whSir John Herschel examined the propagation of a beampolarized light along the axis of a helix carrying an electcurrent.2 Even Michael Faraday’s original search was fcused on the relation between light and electricity. Farakept a detailed lab diary, and on August 30, 1845 hecorded his failure to find a change in the polarization of ligpassing through a liquid that was undergoing electrolysis3 Itwas only when he substituted magnetic for electric forcesSept. 13 of that year, using an electromagnet with an icore, that he discovered the magneto-optic effect. Hecorded in his lab notebook: ‘‘A piece of heavy glass, whiwas 2 in. by 1.8 in. and 0.5 of an inch thick, being a silicborate of lead, was experimented with... when contrary mnetic poles were on the same side therewas an effect pro-duced on the polarized ray, and thus magnetic force and lighwere proved to have relations to each other. This fact wmost likely prove exceedingly fertile, and of great valuethe investigation of conditions of nature force.’’4 After ac-

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quiring a more powerful electromagnet on Sept. 18, he ctinued the experiment with such zeal that he filled twepages of his lab notebook in one day and concluded withstatement: ‘‘An excellent day’s work.’’ He verified that theffect of the magnet was to rotate the polarization planethe transmitted light by an angle that depended onstrength of the magnet.

The magneto-optic Kerr effect~MOKE!5 was discoveredby the Rev. John Kerr in 1877 while he was examining tpolarization of lightreflectedfrom a polished electromagnepole. Kerr ultimately received the Royal Medal in 1898 fresearch that ranked among the most important subsequeFaraday’s. When he was presented with the Royal Medal,presenter said it was a wonder that Kerr learned so mwith the ‘‘comparatively simple and ineffectual apparatushis disposal.’’ Kerr responded, ‘‘Simple it may be, but nineffectual; rude, but not crude.’’6 This statement might represent the nature of this technique as used in the presenadaptations that are the subject of this article, especiwhen compared to many of the elegant techniques of modsurface science andnonlinearoptics. But simplicity is a ma-jor reason that this technique has in the last decade beewidely embraced to study magnetic thin films.

The application of the magneto-optic Kerr effect to suface magnetism, better known by its acronym the surfmagneto-optic Kerr effect~SMOKE!, began in 1985. Thefirst experimental system studied was ultrathin Fe filgrown epitaxially onto a single crystalline substrateAu~100!.7 Hysteresis loops for the Fe film with atomic layesensitivity were successfully obtained as a function of fithickness and temperature. Since then SMOKE has beenplied to address various issues in low-dimensional magil:

3 © 2000 American Institute of Physics

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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 rececommercialization of high-density magneto-optic informtion storage media,8 and especially by studies of candidamaterials for next-generation media based on Cosuperlattices.9 The present work provides a general bacground to the basic principles and experimental setup ofSMOKE technique, and also highlights contemporary topin order to provide an appreciation for its value in applictions related to basic research on magnetic thin films.

B. Origin of the magneto-optic effect

Magneto-optics is presently described in the contexteither macroscopic dielectric theory or microscopic quanttheory.10 Macroscopically, magneto-optic effects arise frothe antisymmetric, off-diagonal elements in the dielecttensor. Microscopically, the coupling between the electrifield of the light and the electron spin within a magnemedium occurs through the spin-orbit interaction. In the flowing, we give a brief review of the microscopic descrition of the magneto-optic effect, and leave the detailed mroscopic description to the next section.

As it is well known, the optical properties of a mediuare determined by a dielectric tensor that is determinedthe motion of the electrons in the medium. Thus, a micscopic description of the magneto-optic effect concernsdifferent response of the electrons to left- and right-circulapolarized electromagnetic waves. In the proceedings ofRoyal Society, Sir William Thomson, in 1856, offered‘‘microscopic’’ explanation of the Faraday effect by arguinthat the particles in the medium under an external magnfield follow different circular paths, depending on their drection relative to the magnetic field. From a modern viepoint, this explanation is conceptually correct if we identThomson’s ‘‘particles’’ as being electrons~although theelectron had not yet been discovered at that time!.

It is worthwhile to first discuss the classical motionelectrons to point out the physical origin of the magneoptic effect. As a beam of light propagates through a mdium, the electrical field of the light generates the motionsthe electrons in the medium. Without an external magnfield, it is obvious that a left-circularly polarized electric fiewill drive the electrons into left circular motion, and a righcircularly polarized electric field will drive the electrons inleft circular motion, and a right-circularly polarized electrfield will drive the electrons into right circular motion. Thradius of the electron orbit for left and right circular motiowill be the same. Since the electric dipole moment is proptional to the radius of the circular orbit, there will be ndifference between the dielectric constants for the left- aright-circularly polarized electromagnetic waves. Thus, thwill be no Faraday rotation. After an external magnetic fieis applied in the propagation direction of the electromagnwave, there will be an additional Lorentz force actingeach electron. This force points toward or away fromcircle’s center for left or right circular motion. Thus, thradius for left circular motion will be reduced and the radifor right circular motion will expand. The difference in thradii of the left- and right-circularly polarized modes w


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give different dielectric constants correspondingly. Thus, ithe Lorentz force of the external magnetic field that genates the Faraday effect.

Quantum descriptions of the magneto-optic effect hafocused on the explanation of the unusual large magnoptic effect in ferromagnetic materials. Early attempts to eplain the much stronger magneto-optic effect in ferromanetic materials assumed that there exists an effective firather than the applied field, that determines the Faradaytation in ferromagnetic materials. In fact, Voigt found ththe effective field is of the order of 106– 107 Oe to producethe observed Faraday rotation. This magnitude is of the oof the Weiss field that was postulated to account forexistence of ferromagnetism. The nature of the Weiss firemained unexplained until Heisenberg developed the thethat ascribed the origin of magnetism to the exchange inaction among electrons. Although Heisenberg’s exchangeteraction correctly reveals the origin of magnetism as anfective magnetic field to align the individual spins, this fiealone cannot be used to explain the Faraday effect. Thibecause it is not coupled to the electron motion which demines the dielectric properties of a material. This difficuwas solved in 1932 by Hulme11 who pointed out that it is thespin-orbit interaction that couples the electron spin to its mtion to give rise to the large Faraday rotation in a ferromanetic. Spin-orbit coupling,;(¹V3p)•s, results from theinteraction of the electron spin with the magnetic field telectron ‘‘sees’’ as it moves through the electric field2¹Vwith momentump inside a medium. This interaction couplethe magnetic moment of the electron with its motion, thconnecting the magnetic and optical properties of a fermagnet. Indeed, to a certain extent, the spin-orbit interaccan be thought of as an effective magnetic field vectortential ;s3¹V acting on the motion of the electron. Fononmagnetic materials, this effect is not strong, althoughspin-orbit interaction is present, because the equal numbespin-up and spin-down electrons cancels the net effect.ferromagnetic materials, however, the effect manifests itbecause of the unbalanced population of electron spins.

Hulme calculated the two refraction indices~R and Lpolarized! using the Heisenberg model of a ferromagnet, athe Kramers–Heisenberg dispersion formula. This approrepresents the refraction index in terms of the eigen eneand matrix elements of the dipole moment operator withspect to the eigenfunctions of the system. Hulme accounfor the difference of the two refraction indices by the enersplitting due to the spin-orbit interaction. He neglected, hoever, the change of the wave function due to the spin-ointeraction. This theory is unsatisfying because the quening of the orbital angular momentum in ferromagnets givno energy splitting. Kittel showed12 that it is the change ofthe wave functions due to the spin-orbit interaction that givrise to the correct order of magnitude of the difference oftwo refraction indices. Argyres13 later gave a full derivationof the magneto-optic effect in a ferromagnet using perturtion theory. Subsequent works were performed thereaftecalculate the magneto-optic effect in different regimes.14–16


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1245Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Surface magneto-optic Kerr effect

II. MACROSCOPIC FORMALISM FOR MAGNETICMULTILAYERS

Macroscopic descriptions of the magneto-optic effectbased on an analysis of the dielectric properties of a mediIn analogy with mechanical vibrations of a particle, Maxwexpressed linearly polarized light as being a superpositiotwo circularly polarized components, and realized thatFaraday effect is a result of the different propagating veloties of the two circular modes.17 This explanation remainsthe phenomenological explanation of the Faraday efgiven in introductory physics textbooks. Looked at in greadetail, there are actually two processes taking place for lpropagating in a magnetized medium. First, the two cirlarly polarized modes gain different phase shifts due to thdifferent propagating velocities, resulting in a rotation of tpolarization plane. This process is the conventional Fararotation. Second, the different absorption rates of the mdium for the two circularly polarized modes affects the ellticity. In general, both effects exist in a magnetized mediu

The 333 dielectric tensor of a medium,e i j with i , j51,2,3, can be decomposed into a symmetric part andantisymmetric part,e i j 5(e i j 1e j i )/21(e i j 2e j i )/2. The sym-metric part can be diagonalized by an appropriate rotatiothe coordinate system, thus it does not give rise to the Fday effect. Since the symmetric part ofe i j is unimportant tothe Faraday effect, we will always assume that it is isotrowith dielectric constante0 . To see the effect of the antisymmetric part of the dielectric tensor, let us consider the flowing dielectric tensor:

e5eS 1 iQz 2 iQy

2 iQz 1 iQx

iQy 2 iQx 1D . ~1!

The two normal modes are left-circularly polarized light wirefraction indexnL5n(12 1

2 Q• k), and right-circularly po-larized light with refraction indexnR5n(11 1

2 Q• k), wheren5Ae is the average refraction index,Q5(Qx ,Qy ,Qz) iscalled the Voigt18 vector, andk is the unit vector along thedirection of the light propagation. Thus, the complex Faday rotation of the polarization plane after the light has treled a distanceL through the medium is

u5pL

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pLn

lQ• k. ~2!

The real part of Eq.~2! gives the rotation, and the imaginarpart gives the ellipticity. It is interesting to ask why an eternal magnetic field has a stronger effect on the polarizaplane of light than an external electrical field. Phenomelogically, this can be answered by a simple argument baon time reversal symmetry. Under the time reversal opetion, the displacementD and electric fieldE vectors remainunchanged, but the magnetic fieldH changes sign. ThusOnsager’s relation givese i j (E,H)5e j i (E,2H). By expand-ing e i j up to terms linear inE andH it becomes obvious thathe antisymmetric part ofe i j is generated by the magnetfield. The magnetic field is only one special case of timreversal symmetry breaking. In general, any quantity t


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breaks time reversal symmetry could, in principle, generantisymmetric elements of the dielectric tensor, and, thuFaraday rotation.

Since most magnetic materials of interest are metalsstrongly absorb light, it is convenient to experimentally mesure the reflected light in order to probe the magneto-oeffect. Therefore, the general macroscopic formalism isthe magneto-opticKerr effect although the formalism can breadily extended to include the Faraday effect. Zaket al.19,20

developed a general expression for the Kerr signal basethis method. We will outline the theoretical structure aresults here.

For a given magnetic multilayer the refraction tensoreach layer can be expressed by a 333 matrix. The goal is tocalculate the final reflectivity along different polarization drections. The general method is to apply Maxwell’s equtions to the multilayer structure, and to satisfy the boundconditions at each interface. The essential part of the theis to derive two matrices which relate the electric fieldseach interface. The first matrix~A! is the 434 mediumboundary matrix. It relates the tangential components of telectric and magnetic fields with thes andp components ofthe electric field. The second matrix~D! is the 434 mediumpropagation matrix. It relates thes andp components of theelectric field at the two surfaces of a film of thicknessd.With the A and D matrices~see Appendix for details!, onecan calculate the magneto-optic effect under any conditio

Consider a multilayer structure that consists ofN indi-vidual layers, and a beam of light impinging on the top of tstructure from initial mediumi. After multiple reflections,there will be a reflected beam backscattered into mediumi,and a transmitted beam that emerges from the bottom lainto the final mediumf ~Fig. 1!. The electric fields in mediumi and f can be expressed

Pi5S ESi

Epi

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and

Pf5S ESi

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i

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wherer and t are reflection and transmission coefficientsthe corresponding components, and superscriptsi and r de-fine the incident and reflected waves at each boundarytween two layers. IfPm is the field component at the bottomsurface in themth layer, then the relation betweenPi andPf

can be expressed as

Ai Pi5A1D1P15A1D1A121A1P1

5A1D1A121A2D2P2

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~AmDmAm21!Af Pf . ~4!


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If this expression is put in the form ofPi5TPf , where

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then the 232 matricesG and I can be used to obtain thFresnel reflection and transmission coefficients

G215S tsstsp

tpstppD and IG215S r ssr sp

r psr ppD . ~6!

The Kerr rotationf8 and ellipticity f9 for s-andp-polarizedlight are then given by

fs5fs81 ifs95r ps

r ssand fp5fp81 ifp95

r sp

r pp. ~7!

In the ultrathin limit the magneto-optic expressions siplify further. For ultrathin films the total optical thickness othe film is much less than the wavelength of the lig( inidi!l. If the initial and final media are nonmagnetithen the 232 matrices ofG and I in Eq. ~5! yield the fol-lowing reflection coefficients:

r ss5ni cosu i2nf cosu f

ni cosu i1nf cosu f,

FIG. 1. ~a! Schematic of a multilayer structure;~b! definitions of thes andp directions for the incidence and reflection waves at the boundary betwtwo media.


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r pp5nf cosu i2ni cosu f

nf cosu i1ni cosu f,

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l

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~ni cosu i1nf cosu f !~nf cosu i1ni cosu f !

3S cosu f(m

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~m!2nfni sinu i(m

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~8!

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~ni cosu i1nf cosu f !~nf cosu i1ni cosu f !

3S cosu f(m

dmnm2 Qz

~m!1nfni sinu i(m

dmQy~m!D .

Here ni , u i , and nf , u f are the refractive indices and thincident angles of the initial and final media,z is the surfacenormal direction, andy is in the plane of incidence and in thfilm plane. Equation~8! provides a basis for an additivity lawfor multilayers in the ultrathin limit. This law states that thtotal Kerr signal is simply a summation of the Kerr signafrom each magnetic layer, and is independent of the nonmnetic spacer layers in the multilayer structure. This additivlaw is true only in the limit where the total optical thickneof the layered structure is much less than the wavelengtthe incident beam. For thick films, it is obvious that thadditivity law must break down because the light attenuaand will not penetrate to the deeper layers of the structuThe additivity law provides a focus for examining data in tultrathin limit. Altered optical constants in the ultrathin limand interfacial roughness, of course, can also give rise tobehavior that cannot be described within the context ofadditivity law.

III. EXPERIMENTAL SETUP

An experimental SMOKE setup has the advantagesimplicity, especially for ultrahigh vacuum~UHV! in situmeasurement. There are several ways to build a SMOsetup. Here we introduce one of the simplest, which wefrequently. Before discussing the instrumental setup, itnecessary to first discuss the working principle of the expmental method. Consider linearp-polarized light reflectedfrom a sample surface. If the sample is nonmagnetic,reflected light is purelyp polarized. If the sample is ferromagnetic then the reflection beam should consist of ascomponent (Es) in addition to the dominantp component(Ep), with Es /Ep being the Kerr rotation. Therefore, measuring thiss component will be the goal of the experimentsetup. Experimentally, the measurement of thes componentcould be realized by placing a linear polarizer in front of tphotodetector to eliminate thep component. However, thismeasurement geometry has the following disadvantaFirst, since the photodetector measures the light inten(;uEsu2), the measured quantity is proportional to thsquare of the magnetization. Second, it is difficult to quanthe absolute value of the Kerr rotation. This disadvanta

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can be circumvented by setting the polarizer at a small an~d! from the p axis. In this way, the intensity measured bthe photodetector after the polarizer is

I 5uEp sind1Es cosdu2'uEpd1Esu2. ~9!

Recall that the expressionEs /Ep5f81 if9 gives the Kerrrotationf8 and ellipticity f9. Then Eq.~9! becomes

I 5uEpu2ud1f81 if9u2

'uEpu2~d212df8!5I 0S 112f8

d D ~10!

with

I 05uEpu2d2 ~11!

representing the intensity at zero Kerr rotation. Since bf8 andf9 are linearly proportional to the magnetization, tmeasured intensity as a function ofH yields the magnetichysteresis loop. The saturation Kerr rotationfm8 can be de-termined by the relative change of the Kerr intensityDI ob-tained upon reversing a field value that is equal to or grethan its saturation value

fm8 5d

4•

DI

I 0. ~12!

In the SMOKE experiment, a laser is usually used aslight source. Typically a low-power~few mW! laser suffices.It is highly desirable to use an intensity-stabilized laser,pecially for monolayer studies. Otherwise, the fluctuationsthe laser intensity may overwhelm the Kerr signal. The effof the light intensity drifting during a hysteresis loop mesurement~say, of 1–10 s duration! causes a distortion of thhysteresis loop. This effect cannot be eliminated by locktechniques. While it is recommended that the SMOKEperiment be performed in a reduced vibration environman optical table is not necessary, so it is easy to adSMOKE to an UHV system. Crystal prism polarizers auseful both for defining the polarization and as an analyzefront of the photodiode detector. Sheet polarizers canused, but have a lower extinction ratio when crossed tprism polarizers, and so are not optimal for monolayer sties. Finally, the magnet shown in Fig. 2 consists of two spcoil solenoid pairs. Energizing either pair would genereither a field in the film plane or perpendicular to it for logitudinal or polar measurements, respectively. The dimsions of such a magnet are about 13 cm in height, 15 cm

FIG. 2. Schematic drawing of a SMOKE setup.


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length, and 4 cm width in the gap. With;300 turns in thecoil, the magnetic field in the gap can reach;1.5 kOe at acurrent of 20 A. Wire of 12 or 13 gauge for the coil can car10–20 A of current without a heating problem. Forin situmeasurements, the magnet is located inside the vacchamber. During the measurement, data are taken as ation of magnetic field to generate an hysteresis loop. Becaof possible drift of the laser intensity, it is recommended ththe average of many loops be taken.

For in situ measurements, the UHV windows usedviewports usually produce a birefringence,fw8 1 ifw9 , thatprevents the realization of the optical extinction condition.this situation, a quarter-wave plate is usually placed befthe analyzing polarizer to cancel the window birefringenThen the measured Kerr intensity becomes

I 5uEpu2~d212df9!5I 0S 112f9

d D , ~13!

i.e., the relative Kerr intensity determines the Kerr ellipticirather than the rotation in this case. The effect of the quarwave plate is to produce ap/2 phase difference between ths andp components so that the analyzing polarizer will si (f81 if9)52f91 if8, i.e., the rotation and ellipticity areinterchanged. Then to measure the rotation, a half-wave pcould be used to replace the quarter-wave plate. Thenreflected intensity as a function of the external magnetic fican be used to generate a magnetic hysteresis loop. Figushows an example of an hysteresis loop measuredSMOKE for a 6 ML Fe/Ag~100! film, where ML denotesmonolayer. It quite apparent that SMOKE can readachieve monolayer sensitivity.

IV. THE MOKE FAMILY

While the present work is narrowly focused on thSMOKE approach and applications, this section introducebroader MOKE ‘‘family’’ in order to provide a useful perspective and serve as a pointer to the literature. Whilnumber of recent reviews have covered SMOKE,21–24 his-torically the primary MOKE spectroscopy involves the dtermination of wavelength dependent properties.25 Commer-

FIG. 3. A SMOKE loop taken from a 6 ML Fe film grown on a Ag~100!substrate.


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cial as well as lab-built spectrometers have been usedsuch purposes. The main instrumental addition is a wlight source and monochrometer. Representative work frthe laboratory of Schoenes and co-workers includes bdescriptions of the spectrometers.26 A recent example in-volves the giant magneto-optic Kerr rotation observedsingle crystals of cerium compounds at low temperature27

In this work a Faraday modulator is used in conjunction wlock-in detection for noise suppression. Faraday modulahave been utilized in other novel configurations,28 includingthe quest for anions in high-temperature superconductor29

Lock-in detection techniques have not been discuswith respect to SMOKE because modern computcontrolled experiments can provide the signal averagneeded to improve the signal-to-noise level. In an eaSMOKE publication30 it was even demonstrated that the hyteresis loop collected by means of lock-in detection ofphotodiode output and use of an incident beam whose poization is modulated by a commercial photoelastic modulawas no better than that obtained by the dc method outlinethis review. The place where lock-in techniques and poization modulation are especially valuable is in the simuneous magneto-optic characterization of the rotation andlipticity as a function of wavelength. A recent examplethis approach is described in the work of Osgoodet al.31 butsuch work follows the basic outlines provided by pioneersthe field.32–35 Polarization modulation and related optictechniques are described in a general reference.36 The influ-ence of imperfections in the polarizer and analyzer werecently analyzed.37 Polarization modulation has also beeused to characterize the Curie temperature of Gd films viinsitu MOKE studies.38

To fill out the range of parameter space and the ccomitant phenomena that become physically accessiblnumber of variants are now mentioned. New magneto-otransitions were identified in monolayer-range Fe films uswavelength dependent measurements in the visible regio39

The measurements were performedex situand the films weresandwiched between Au layers to protect them from oxition. Oscillatory effects have been observed in SMOKE snals of wedged structures and attributed to spin-polariquantum size effects.40–42 Dynamic scaling of the magnetihysteresis was studied in the monolayer rangeFe/Au~001!,43 Co/Cu~001!,44 and Fe/W~110!45 by sweepingthe applied magnetic field at rates up to 1 kHz and moniing the loop area. Magnetic field modulation has been uto obtain ac susceptibilities using anin situ MOKEapparatus.46 Time-resolved MOKE has been extended to tpicosecond range. A 30 ps pulsed dye laser was usedpump-probe experiment in order to study magnetizationversal dynamics in magneto-optic storage materials.47 Dif-fraction MOKE is another recent addition due to interestmagnetic nanostructures that consist of arrays of holes48 orlines ~gratings!.49,50 The patterned arrays are of a size aspacing that are comparable to the wavelength of vislight and, thus, they serve as diffraction gratings. The traverse MOKE signals from diffracted beams can be compato that from the specular beam. Enhanced magneto-opticfects and unique magnetic hysteresis loops are extra


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from the signals associated with the diffracted beams. Tshould continue to provide a wealth of information in thcharacterization of such nanostructures, as well as challein the modeling of the relevant optical and magnetic pcesses.

A traditional application of MOKE is in magneto-optiimaging of magnetic domain structures. An example is incharacterization of Fe/Cr magnetic multilayers.51 The tech-nique of differential polarization microscopy was recenintroduced that utilizes a Wollaston prism to provide improved image contrast.52 Wollaston prisms split the signaaccording to polarization content and can be used quitefectively in all types of MOKE measurements. Transparemagneto-optic indicator films are also used quite effectivto image magnetic domain structures.53 In this approach atransparent Faraday film is placed on top of the sampleinterest. The fringe fields emanating from the sample cathe Faraday-rotation contrast of the indicator film in orderyield the image. This method is sensitive to the perpendilar magnetic response of the sample. However, samplesin-plane magnetization have been imaged in this manwith the clever addition of drilling holes in the sample, frowhich perpendicular stray fields emerge. In addition to thtypes of magneto-optic microscopy, there are ideas undiscussion for the extension of such techniques into the nfield region using plasma-resonant Ag particles as probe54

Synchrotron techniques are also popular and of grvalue in magneto-optic characterizations. A descriptionsynchrotron methods to study magnetic systems has recebecome available; the overview stresses the opportunprovided by third-generation synchrotron sources~those forwhich undulator insertion devices are used to intensify afocus the beam!.55 An advantage is that there is elemensensitivity since the photon energy can be tuned to expthe response associated with a specific atomic core leSynchrotron-based magnetic circular dichroism~MCD! of-fers selection rules such that the spin and orbital magnmoments can be separately determined. X-ray MO~XMOKE! is rapidly becoming another standard synchrotrtechnique to characterize magnetic films.

The magneto-optic imaging of antiferromagnetic domastructure has always been a challenging task.56–58 A recenttour de forceexperiment that combines the use of a thirgeneration synchrotron source with x-ray magneticlinear di-chroism spectroscopy and the spatial resolution of a phelectron emission microscope permitted the imaging ofantiferromagnetic structure at thesurface of NiO~100!.59

Magnetic linear dichroism of antiferromagnets depends osecond-order effect in the magnetization~as does the transverse Kerr effect in ferromagnets!. Second-order magnetooptic effects in anisotropic thin ferromagnetic films and tanalysis of asymmetric hysteresis loops was the subjectrecent article that is largely based on the Ph.D. thesissearch of Osgood.60

Techniques based on nonlinear magneto-optics andond harmonic generation~SHG! and that utilize pump-probespectroscopies are rapidly becoming valuable tools inexploration of magnetic surfaces and interfaces. The idefying acronym is SHMOKE. Fortunately an excellent boo


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was recently published that encompasses both experimand theoretical aspects of the field.61 Although the signal isweak, so that applications in magnetic recording are notsible, the enhanced sensitivity to interfaces, and the impsive experimental successes,62 make this an exciting field tomonitor in the future.

Analytical and computational techniques have alsodergone a resurgence of activity that has broadenedbreadth and scope of MOKE studies. First, there are analthat are specifically used to motivate new experimenmethods for collecting and analyzing data.63–65 Then thereare macroscopic expressions that are usefulsimulations,66–70 and approximations for multilayers baseon single-layer equivalences71 and effective-mediumapproaches.72 And finally there are microscopic methodbased on first principles to calculate Kerr spectra includsurface effects,73 and for transition metals anmultilayers,74–76 rare earths77 and actinide systems.78

V. VERIFICATION OF THE MACROSCOPICFORMULAS

A specific example is now considered for the verificatiof the macroscopic formulas presented in Sec. II. This wwas accomplished by investigating Co overlayers and Cosuperlattices.79 The films were grown epitaxially ontoCu~100! and Cu~111! single-crystal substrates in UHV~basepressure of 2310210Torr!. The UHV chamber is equippewith reflection high-energy electron diffraction~RHEED!,low-energy electron diffraction~LEED!, and Auger electronspectroscopy. The Cu substrate single-crystal disks were;1cm in diameter, and were mechanically polished down t;0.25 mm paste finish, and then ultrasonically cleanedmethanol before being put into the UHV chamber. Cycles3 keV Ar1 sputtering and annealing at 650 °C were usedclean the Cu substrate surfacesin situ. After this treatment,well ordered Cu surfaces were formed as indicatedRHEED and LEED. Auger spectroscopy confirmed tcleanliness of the films. The RHEED intensity also wmonitored during the growth of the film on the Cu~100! sub-strate in order to follow the process and to calibratethickness monitor. Figure 4 shows the RHEED oscillatioduring the growth of the Co/Co~100! superlattice. Over 200RHEED oscillations were observed during the growth. Ea

FIG. 4. RHEED oscillations taken during the growth of@Co~9.5 ML!/Cu~16 ML!#n superlattice grown on a Cu~100! substrate.


tal

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oscillation represents the growth of an atomic layer. The psistence of the oscillations indicates a stable, well defingrowth mode.

The Kerr ellipticities of the films were measuredin situusing a He–Ne laser. The angle of incidence was 17°.sults are plotted in Fig. 5. The ellipticity for Co on polycrytal Cu was also measured and shown for comparison.first concentrate on the magneto-optic behavior of a sinCo overlayer on a Cu substrate. The ellipticity data for toverlayers increase linearly in the ultrathin regime, reacmaximum at;120 Å of Co, and approach a constant valfor .400 Å of Co. The initial rise is expected since the Keeffect is sensitive to the increasing amount of Co. In tthick regime,.400 Å of Co, the signal saturates since tabsorption of light limits the depth sensitivity. In the intemediate regime, the maximum in the ellipticity at;120 Å ofCo is attributed to an optical effect: the reflectivity changfrom being dominated by Cu to Co. Since Cu has a higreflectivity than Co, it acts as a mirror to enhance the signSimilar behavior was also observed in the Fe/Au system.80 Itis also worth noting that the ellipticity is found to be indpendent of crystalline orientation in the thickness range stied.

To analyze the data quantitatively, the formalism dscribed in Sec. II was applied to simulate the results. Trefractive indices used were obtained from tabulations inliterature:81 nCu50.24913.41i and nCo52.2514.07i . Thevalues ofQ1 andQ2 , whereQ5Q11 iQ2 , for Co were leftas free parameters to best fit the experimental curves;valuesQ150.043 andQ250.007 were obtained. The calculated curves, depicted as the solid lines in Fig. 5, are in goverall agreement with the experimental data. In particuthe peaked behavior of the overlayer data are faithfullyproduced. The ellipticities of three epitaxial Co/Cu superltices were also measuredin situ after each Co/Cu bilayer wagrown. The superlattices used were@Co(16 Å)/Cu(28 Å)#n

grown on Cu~100!, and @Co(11 Å)/Cu(31 Å)#n and@Co(18 Å)/Cu(35 Å)#n both grown on Cu~111!. The ellip-ticities of the superlattices appear in Fig. 5 plotted as a fu

FIG. 5. The Kerr ellipticity measured for different samples. The solid linare theoretical calculations.


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tion of the total superlattice thickness. Again the ellipticitiinitially increase linearly in the ultrathin region, and thesaturate in thick regime, although there is no maximumintermediate thickness as for the overlayer cases. The laca maximum in the intermediate thickness regime is becathe reflectivity is not evolving from that of Cu to that of Coas in the overlayer cases. Instead, the reflectivity maintitself at an average value between the two limits, since bCo and Cu remain within the penetration depth of the ligno matter how thick the superlattice becomes. Using theQvalue obtained from the Co overlayers, the Kerr ellipticitifor the superlattices were calculated and plotted in FigThe agreement with the experimental data is obvious.

To test the additivity law, the experimental data froFig. 5 were replotted in Fig. 6 as a function of the thickneof only the magnetic Co layers, as opposed to the totalperlattice thickness. All the data in the ultrathin regime thfall onto a single straight line. This result confirms the adtivity law that the total Kerr signal in the ultrathin regimea summation of the Kerr signal from each individualmag-netic layer and is independent of the thickness of the nmagnetic spacer layers.

VI. APPLICATIONS OF SMOKE IN TWO-DIMENSIONALMAGNETIC THIN FILMS

In addition to the simplicity of the SMOKE setup as ain situ magnetic measurement technique, the two greatvantages of the SMOKE are its high sensitivity and loprobe nature. These two characteristics make SMOKEpopular choice to address issues in thin film magnetism. Texamples are included below to highlight the applicationSMOKE.

A. Spin-reorientation transition

The investigation of the two-dimensional~2D! spin-reorientation transition~SRT! was originally motivated as atest of the Mermin–Wagner theorem.82 This theorem ad-dresses the most fundamental issue in thin film magnetis

FIG. 6. The additivity law shows that the Kerr signal in the ultrathin regidepends only on the thickness of the magnetic layers.


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.

su-n-

-

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—

the existence of magnetic long-range order~LRO! in 2D.Mermin and Wagner proved that quantum fluctuations in2D isotropic Heisenberg lattice destroys the LRO at fintemperature. Experimentally, however, the Curie tempeture (TC) in most 2D magnetic films is finite. This seemincontradiction suggests that, in addition to the Heisenbergchange interaction, there must exist other energy termmagnetic thin films. Such energy terms are referred to asmagnetic anisotropies; these terms favor electron spin ortations in particular directions. In a lattice with cubic symmetry ~fcc and bcc, for example!, it can be shown that thelowest-order term in the anisotropy energy is quartic inmagnetizationM. However, when translational symmetralong one direction is broken, a larger, square-term aniropy can be generated along the surface normal directThis term is often referred to as the magnetic surface aniropy. A discussion of the origin of the magnetic anisotropyprovided in the next section. Nevertheless, a real magnthin film should be better described by an isotropic Heisberg exchange, a magnetic surface anisotropy (KS), and ashape anisotropy (22pM2) which originates from the shortrange part of the dipole–dipole interaction. The directionthe easy axis of magnetization is determined by the signthe effectivesurface anisotropyKeff5KS/d22pM2, wheredis the film thickness. For systems withKS.0, a magnetiza-tion perpendicular to the film plane can be stabilized at ltemperature and below certain thicknesses. Changing tperature or thickness can causeKeff to vanish at some poinbelow TC , and for the film to approach more closely thideal realization of an isotropic 2D Heisenberg system.Keff50 the spin-reorientation transition should occur wherM changes its direction from perpendicular to in plane. Tquestion of interest regards the presence or absence ofat the transition. Early theoretical studies83 suggested that inthe vicinity of the SRT, there is a region in temperatureDTR

wherein the magnetic LRO is lost.Several groups have carried out experiments on this s

ject. The first experiments were reported by Pappas Ka¨mper,and Hopster84 using spin-polarized electron spectroscopycharacterize the systems Fe/Cu~100! and Fe/Ag~100!. Theyfound that at low temperature the easy axis was normal tosurface plane, at high temperature it was in plane, and inSRT region there was a temperature gap;20 K wide withinwhich the magnetic remanence vanished. Then theAg~100! system was studied via SMOKE as a functionboth temperature and film thickness. It was found thatmagnetization is not identically zero in the transition regiobut is markedly reduced and exhibits structure in a ‘‘pseugap’’ that resembles an asymmetric ramping toward zwith increasing temperature or film thickness.85 Thus, thegap, if it exists, must be at least an order of magnitusmaller than the;20 K reported by Pappas and co-worke

To illustrate the advantage of the local probe naturethe SMOKE technique, we limit the discussion to addressthickness dependent data only. To explore the detailedtures within the SRT region, many samples with differefilm thicknesses are needed. This is because samplesample variations are known to occur throughout surfaceence and thin-film growth that cannot be adequately c


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trolled. This difficulty can be overcome by the use of wedgshaped samples. Wedged samples provide an essencontinuous change of film thickness within a single specimso that, as a local probe scans across the sample, the mtization can be measured for different thicknesses in atematic manner. SMOKE measures the magnetization withe confines of the laser spot. When applied to the SRTthickness resolution can achieve the 0.04 ML level fotypical wedge with, say, a 2 ML mm slope. Figure 5 illutrates the perpendicular magnetic remanence (M') deter-mined from the polar signal, and the parallel remane(M i) determined from the longitudinal signal, at room temperature for different Fe film thicknesses. The existencethe SRT is evident as a function of the film thickness. At lothicknesses,M' maintains its saturation value, and at larthicknesses,M i retains its saturation value. However, in thSRT region~Dd in Fig. 7!, M is greatly suppressed from itsaturation value. This region is similar to that observedPappas, Ka¨mper, and Hopster in Fe/Cu~100!84 and Fe/Ag~100! via polarized electron scattering.86 But the SMOKEmeasurements definitively show thatM is not zero in thisregion. Thus, this region isnot associated with a loss oLRO, but instead with apseudo gapthat suggests the presence of complex magnetic structure.

The formation of magnetic domains within the pseudgap region provides a possible explanation for the suppsion of M but without a loss of magnetic LRO. Yafet anGyorgy were first to recognize that a stripe domain structhas a lower energy than a single domain structure in asystem with perpendicular, uniaxial anisotropy.87 It wasfound that the domain size increases almost exponentiallthe effective surface anisotropy departs from zero. Therefthe stripe domains are observable only in the vicinity ofSRT where the effective surface anisotropy is nearly zand the domain size is less than the sample size. Thisplains why there is a gapped region in the SRT within whthe magnetic remanence is greatly suppressed. Stripemains form a one-dimensional~1D! ordered state which it-self is unstable against thermal fluctuations. Indeed, Kashand Pokrovsky88 found that the stripe domain structureequivalent to a 2D liquid crystal system in that it possesorientational order but no spatial order. An experimentalservation of the stripe domains in the SRT region was p

FIG. 7. Spin reorientation transition as a function of film thickness.


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sented by Allenspach and Bischof who applied the SEMtechnique to study the SRT in the Fe/Cu~100! system.89

~SEMPA is a highly surface-sensitive magnetic imagitechnique that stands for ‘‘scanning electron microscowith polarization analysis.’’! They observed that the singldomain structure of the film breaks into stripe domains~;1mm size! in the gap region. Results of dynamic propertiesthe SRT are also consistent with a stripe domain structur90

B. Magnetic anisotropy and lattice symmetrybreaking

The Heisenberg exchange interaction is invariant unspatial rotation. In a real lattice, however, the electrons tcontribute to the magnetization usually obey the lattice symetry in their wave functions due to the crystal fields. Ththe spin-orbit interaction can transfer the lattice symmefrom the electron wave functions to the electron spinsbreak the spin isotropy. Thus, energy terms exist that faspecial directions for the electron spins. This energy is cathe magnetocrystalline anisotropy. Given that it originafrom the spin-orbit interaction,91 the magnetic anisotropymust obey the symmetry of the lattice. Understanding hlattice symmetry breaking induces magnetic anisotropy isfundamental importance.

To isolate the lattice symmetry effect from the electroneffect, a few groups have performed experiments on mnetic thin films grown on stepped~001! substrates.92–94 Theidea is that the atomic steps on the~001! surface break thefourfold rotational symmetry to induce a uniaxial magneanisotropy in the film plane. Experimentally, stepped sfaces consist of low Miller-index terraces uniformly seprated by atomic steps, and are created by polishing a samsurface that is misaligned by a few degrees from the terrnormal direction. Such surfaces are also referred to asvicinalsurfaces, because crystallographically they are oriented invicinity of fundamental, low Miller-index faces. To expermentally explore the relationship between induced magnanisotropy and lattice symmetry breaking, many substrawith different vicinal angles would be needed. In practiceis difficult to prepare multiple surfaces under identical coditions. To overcome this difficulty, ‘‘curved’’ substratehave been introduced to provide a continuous gradationthe step density. Substrates of~001! orientation and 1 cm indiameter are used in the examples we cite below. Half ofsurface is polished to retain its@001# orientation and serve aa reference, while the other half is polished with a curvatsuch that the vicinal angle varies continuously from 0°10°. SMOKE has a distinct advantage for this study becathe reflection angle of the SMOKE laser beam simulneously determines the local vicinal angle so that the relabetween the step-induced magnetic anisotropy and thedensity can be systematically explored from a singlecurvedsample.

Results for three representative systems are discusFe/W~001!, Co/Cu~001!, and Fe/Pd~001!. In Fe/W and Fe/Pd, the steps are parallel to the@100# direction of the Fe. Forthe Co/Cu system, the steps are parallel to the@110# direc-tion of the Co. The magnetization is in the film plane for a


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three systems, thus only longitudinal hysteresis loops areported. Figure 8 shows hysteresis loops for a 2 ML Fe filmgrown on stepped W~001! with a 4.7° vicinal angle. Thestepped Fe film shows a square loop with;100% remanencefor the applied magnetic field orientedperpendicularto thestep edges, and two split subloops with zero remanencethe field orientedparallel to the step edges. This behaviindicates that the atomic steps indeed induce an in-plauniaxial magnetic anisotropy with the easy magnetizataxis perpendicular to the step edges. The easy axis ofstep-induced anisotropy depends on the physical sysWhile the Fe/W~001!95 and Fe/Pd~001!96 systems have theieasy axisperpendicularto the step edges, the Co/Cu~001!97

and Fe/Ag~001!98 systems have their easy axisparallel to thestep edges. Nevertheless, the splitting fieldHs , as defined inFig. 8 for the hard-axis loop, is proportional to the strengof the step-induced anisotropy. Figure 9 shows the relabetweenHs and the vicinal anglea ~which is proportional tothe step density! for the three systems. FittingHs;an ~thesolid lines in Fig. 9! yields an exponentn52 for the Fe/Wsystem, butn51 for the Co/Cu and Fe/Pd systems.

To understand why there is different dependence ofHs

on a for different systems, one has to examine how the symetry of the lattice at the step edges is broken for bcc andstructures. In the Ne´el pair-bonding model99 the magneticanisotropy is generated by nearest-neighbor bonds. Forand fcc lattices, there is no uniaxial anisotropy because ctributions from all nearest-neighbor bonds cancel outsquare-term anisotropy. At the step edges, however,missing atoms break this cancellation so that uniaxial aniropy will be manifest.

For a bcc lattice with steps parallel to the@100# direc-tion, the anisotropy due to the missing atoms should haveform Ea52(K/L)uhuz , whereL is the terrace length,u isthe unit vector of the magnetizationM , andj, h, andz arethe @100#, @010#, and@001# axes, respectively. For fcc latticwith steps parallel to the@110# direction, the anisotropy dueto the missing atoms has the formEa52 K

L(2uh213uz

2

12&uhuz), where j5@110#, h5@110#, and z5@001#.Note that the normal direction~z axis! of the stepped surfacmakes a small vicinal anglea to the @001# axis ~so thata'1/L!. The crystaljhz frame of reference has to be tranformed from uz5ux , uh5uy cosa1uzsina, and uz5

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 forH perpendicular to the stepedges, and the split loop is forH parallel to the step edges.


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2uy sina1uzcosa, into to the filmxyzframe with thex andy axes in the film plane parallel and perpendicular to the sedges, respectively. The anisotropy~for small a! transformsto Ea52K(auyuz2a2uy

21a2uz2) for a bcc lattice, andEa

52K(2auy213auz

212&auyuz) for a fcc lattice. There-fore the in-plane, step-induced anisotropy (uz50) is Ea

5Ka2uy2 for the bcc case, andEa522Kauy

2 for fcc. Thisprovides an explanation for why Fe/W and Co/Cu exhiquadratic and lineara dependences, respectively, for thestep-induced anisotropies. The most interesting result islinear dependence in the Fe/Pd system. Fe has a bcc strubut Pd has fcc structure. It was shown that the Pd atinterface of Fe/Pd is ferromagnetic due to the Fe spolarization.100,101 Since Pd has a much stronger spin-orinteraction than Fe, the Pd is expected to dominate the mnetic anisotropy in the Fe/Pd system. We believe that is w

FIG. 9. Hs ~from Fig. 8! vs vicinal anglea for ~a! Fe/W~001!, ~b! Co/Cu~001!, and ~c! Fe/Pd~001!. The solid lines are results of a power-lafitting yielding a quadratic relation betweenHs anda for the Fe/W system,but a linear relation for the Co/Cu and Fe/Pd systems.


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the Fe/stepped Pd~001! system exhibits ana-linear depen-dence of the step-induced anisotropy as for a fcc lattice.

VII. DISCUSSION

The basic principles, experimental setup, and two apcations of the SMOKE technique were outlined. AlthouSMOKE is a powerful technique, it has certain drawbacFor example, it cannot distinguish surface or interface mnetism from that arising from the interior layers. This isarea where nonlinear~second harmonic! MOKE ~orSHMOKE! has major advantages. SMOKE also cannot,general, distinguish an antiferromagnetic phase from a nmagnetic phase. These weaknesses leave many experimchallenges for the future. Also, concerning theoretical chlenges, a microscopic understanding of magneto-optics inmonolayer regime is needed since macroscopic continutheory must ultimately break down. Experimentally, it is alimportant to enhance both spatial and time resolution sosmall scale processes, such as domain wall dynamics, cainvestigated. Possible ways to realize this goal involve cobining SMOKE with other techniques, such as near-fieldtical spectroscopy, scanning tunneling microscopy, andpump-and-probe methods. In the present article a senshistory as well as of future opportunities was invokedstimulate interest both in the SMOKE technique and inimpact on modern thin-film and surface magnetism.

ACKNOWLEDGMENTS

The authors thank the many colleagues that we hworked with on the projects cited in the reference list. Wiout them this work would not be possible. This work wfinancially supported by the Office of Basic EnergSciences-Materials Sciences of the United States Departmof Energy under Contract Nos. W-31-109-ENG-38~at Ar-gonne! and DE-AC03-76SF00098~at Berkeley! and by Na-tional Science Foundation Contract No. DMR-9805222~atBerkeley!.

APPENDIX

To derive the matrices A and D, it is important to firdescribe the normal modes of the electromagnetic wavesmagnetic medium. To obtain them, consider a wa;eik•x2 ivt propagating in a medium whose dielectric tensis described by Eq.~1!. Since the magnetic response of tmedium is attributed to the Voigt vectorQ in the dielectrictensor, we can assume that the magnetic permeabilityThen the relationship betweenD andE, andB andH is

D5«E1 i«E3Q and B5H. ~A1!

Then Maxwell’s equations give

5k•E1 ik•~E3Q!50

k3E5v

cH

k•H50

k3H52v«

c~E1 iE3Q!

. ~A2!


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.-

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aer

1.

It is easy to see thatD, B ~or H! andk are perpendicular toeach other. TheE vector, however, has a component paralto the wave vectork. Using the familiars- andp- polariza-tion modes, the electric field can be written as

E5Eses1Epep1 i ~2Q•epEs1Q•esEp!ek . ~A3!

Here es , ep , and ek are unit vectors along thes, p, and kdirections.Es and Ep are thes and p components of theelectric field, and their equations of motion are

H S v2«

c2 2k2DEs1iv2«Q•ek

c2 Ep50

2iv2«Q•ek

c2 Es1S v2«

c2 2k2DEp50

. ~A4!

To first order inQ, it is easy to show that the two normamodes are right~R!- and left ~L!-circularly polarized modeswith

kR,L5k~16 12 Q•ek! or nR,L5n~16 1

2 Q•ek!. ~A5!

Herek5 vcA« andn5A« are the wave vector and refractio

index, respectively, without the magnetization. After obtaing the two normal modes, any mode of the electromagnwave can be viewed as their superposition.

Now, we consider an electromagnetic wave propagatinside a magnetic multilayer structure. At each boundarytween two layers, the boundary conditions involveEx , Ey ,Hx and Hy , wherex and y axis are in the film plane andperpendicular and parallel to the incident plane, respectivIt is more convenient to express these four quantities wthe s and p components of the electric field. Thex compo-nents are easy to write because they are parallel to thsdirection

Ex5Esi 1Es

r , ~A6!

where the superscriptsi and r denote the incident and reflected waves, respectively. For they components, one has tkeep in mind that the electric field has a componenti (2Q•epEs1Q•esEp)ek parallel to thek direction, and that theLand R modes have different refraction indices and incideangles. ThenEy can be expressed as

Ey5Epi ,L cosuL1Ep

i ,R cosuR1 i ~2Q•epi Es

i 1Q•esi Ep

i !

3sinu2Epr ,L cosuL2Ep

r ,R cosuR

1 i ~2Q•epr Es

r1Q•esrEp

r !sinu. ~A7!

Using the relations

H EpL51 iEs

L

EpR52 iEs

R

nR sinuR5nL sinuL5n sinu, ~A8!

Ey can be expressed as


s-


Ey5i

2@2Qy tanu~11cos2 u!1Qz sin2 u#Es

i

1~cosu1 iQx sinu!Epi 1

i

2@Qy tanu~11cos2 u!

1Qz sin2 u#Esr1~2cosu1 iQx sinu!Ep

r . ~A9!

Hx andHy can be derived in a similar way from the expresion k3E5v/cH:

Hx5in

2~Qy sinu1Qz cosu!Es

i 2nEpi

1in

2~Qy sinu2Qz cosu!Es

r2nEpr ,

ich

fotn

at

n


Hy5n cosuEsi 1

in

2~Qy tanu1Qz!Ep

i

2n cosuEsr2

in

2~Qy tanu2Qz!Ep

r . ~A10!

Therefore, we obtain the relation between thex andy com-ponents ofE andH with s andp components of the electricfield. This relation can be expressed as a matrix product

S Ex

Ey

Hx

Hy

D 5AS Esi

Epi

Esr

EprD ~A11!

with the 434 matrix A known as themedium boundarymatrix

A5S 1 0 1 0

i

2@2Qy tanu~11cos2 u!1Qz sin2 u# cosu1 iQx sinu

i

2@Qy tanu~11cos2 u!1Qz sin2 u# 2cosu1 iQx sinu

in

2~Qy sinu1Qz cosu! 2n

in

2~Qy sinu2Qz cosu! 2n

n cosuin

2~Qy tanu1Qz! 2n cosu 2

in

2~Qy tanu2Qz!

D .

~A12!

ay,

4.

ans.l..

a

The next step is to derive the propagation matrix whrelatesE andH at the two surfaces of a film of thicknessd.Since each incident and reflected beam is composed oL-andR-circularly polarized modes, we use 1 and 2 to dentheL andR modes of the incident beam at both surfaces, a3 and 4 to denote theL andR modes of the reflected beamboth surfaces. Then we have the following relations:

H EA1,25EB

1,2exp~ ik1,2d cosu1,2!

EA3,45EB

3,4exp~2 ik3,4d cosu3,4!. ~A13!

The relation betweenEs andEp at boundaries A and B cathen be expressed by a matrix product

S Esi

Epi

Esr

EprD

A

5DS Esi

Epi

Esr

EprD

B

, ~A14!

whereD is 434 matrix known as themedium propagationmatrix.

D5S U cosd i U sind i 0 0

2U sind i U cosd i 0 0

0 0 U21 cosd r 2U21 sind r

0 0 U21 sind r U21 cosd r

D~A15!

with

ed 5

U5exp~2 ikd cosu!

d i5kd

2~Qy tanu1Qz!

d r5kd

2~Qy tanu2Qz!

. ~A16!

1M. Faraday, Trans. R. Soc.~London! 5, 592 ~1846!.2The Effects of a Magnetic Field on Radiation—Memoirs by FaradKerr and Zeeman, edited by E. P. Lewis~American, New York, 1990!,preface, p. v.

3M. Faraday, Diary, 30 August 1845, Vol. 4, pp. 7434, and 7437–7444M. Faraday, Diary, 13 September 1845, Vol. 4, p. 7504.5J. Kerr, Philos. Mag.3, 339 ~1877!; 5, 161 ~1878!.6See,Molecular Electro-Optics, Part I, Theory and Methods, edited by C.T. O’Konski ~Dekker, New York, 1976!, p. 517.

7E. R. Moog and S. D. Bader, Superlattices Microstruct.1, 543~1985!; S.D. Bader, E. R. Moog, and P. Gru¨nberg, J. Magn. Magn. Mater.53, L295~1986!.

8S. Klahn, P. Hansen, and F. J. A. M. Greidanus, Vacuum41, 1160~1990!.

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