Ž .Thin Solid Films 313]314 1998 222]227

Theoretical and experimental determination of optical andmagneto-optical properties of LuFe single crystal2

S.J. Lee, R.J. LangeU, S. Hong, S. Zollner1, P.C. Canfield, A.F. Panchula,B.N. Harmon, D.W. Lynch

Ames Laboratory and Department of Physics and Astronomy, Iowa State Uni¨ersity, Ames, IA 50011, USA

Abstract

We have studied the diagonal and off-diagonal optical conductivity of a LuFe single crystal grown by the flux method.2Using spectroscopic ellipsometry we have measured the dielectric function from 1.5 to 5.5 eV. The magneto-optical

Ž .parameters Kerr rotation and ellipticity from 1.4 to 4.0 eV were obtained using a magneto-optical polar Kerr spectrometer attemperatures between 7 and 295 K and applied magnetic fields up to 1.2 T which fully saturates the magnetic moment ofLuFe . We describe the apparatus and evaluate the off-diagonal conductivity components from the data. Theoretical2calculations of optical conductivities and magneto-optical parameters were performed using the tight binding-linear muffin tinorbitals method within the local spin density approximation. When lifetime broadening is included, the calculations agree wellwith the experimental data. Q 1998 Elsevier Science S.A.

Keywords: Kerr effect; LuFe ; Rare earth compounds; Optical conductivity; TB-LMTO2

1. Introduction

Magneto-optical recording technology based onmagneto-optical effects is pursued for its high storage

w xdensity and the rewritability of magnetic media 1 .Magneto-optical effects occur when linearly polarizedlight interacts with magnetic materials. There are twokinds of magneto-optical effects, depending on theexperimental geometry. When the light passes througha magnetized material, it is called the Faraday effectw x2 , and when the light is reflected by a magnetized

w xsurface, it is called the magneto-optic Kerr effect 3 .The reflection geometry is more appropriate in realmagneto-optical recording applications, so much re-search has focused on the Kerr effect. In the Kerr

U Corresponding author. Fax: q1 515 2940689; e-mail:lange@ameslab.gov

1Present address: Motorola SPS, Arizona Technology Laborato-ries, MD M360, 2200 West Broadway Road, Mesa, AZ 85202, USA.Tel.: q1 602 6554655; Fax: q1 602 6555013; E-mail:R37595@email.sps.mot.com

effect the reflected light becomes elliptically polarizedeven at normal incidence. The major axis of thereflected light is rotated from the polarization axis ofthe incident light. The angle of rotation is called theKerr rotation and the ratio of the minor to the majoraxis of the ellipse of polarization is called the elliptic-ity. These two magneto-optical parameters are mea-surable quantities with a magneto-optical polar Kerr

Ž .spectrometer MOPKS .There are three types of Kerr effect configurations:

in the polar Kerr effect, the magnetization is perpen-dicular to the reflecting surface. In the longitudinaland transverse Kerr effect, the magnetization is paral-lel to the interface. In the longitudinal configurationthe magnetization is parallel to the plane of inci-dence, while in the transverse geometry the magneti-zation is perpendicular to it. Among these three con-figurations the polar geometry gives the largest Kerreffect.

Ž .Rare-earth transition metal RE-TM compoundshave very good properties for magneto-optical record-

w xing 4 . The sample we measured is a single crystal of

0040-6090r98r$19.00 Q 1998 Elsevier Science S.A. All rights reservedŽ .P I I S 0 0 4 0 - 6 0 9 0 9 7 0 0 8 2 2 - 5

( )S.J. Lee et al. r Thin Solid Films 313]314 1998 222]227 223

w xLuFe grown by a flux method 5 . Single crystals have2many advantages in magneto-optic measurements overpolycrystalline samples or thin films in that they havea higher purity which is manifested in reproducibilityof data with samples from different growths. Thedisadvantage of single crystals is the usually smallsize, so the reflected intensity is decreased, limitingthe spectral range.

2. Theory

Theoretical studies on the magneto-optical effectsin solids have been performed to understand mag-

w xneto-optic phenomena 6]8 . Some of the magneto-optical properties of 3d-based transition metals orintermetallic compounds have been successfully ex-plained by ab initio calculations based on the local

Ž . w xspin density approximation LSDA 9 . However, theab initio calculation of the magneto-optical character-istics of f-electron compounds has not been so suc-cessful with LSDA due to the strongly correlated

w xf-electrons 10 . It is easier to study the magneto-opti-cal properties for LuFe which has the Cu Mg crystal2 2structure because the 4f shell of Lu is completelyfilled.

We consider the case of the magnetization direc-w xtion along 001 although experiments were per-

Ž .formed on LuFe 111 surfaces. However, the2anisotropy in the Kerr spectra for crystals with high

w xsymmetry is usually negligible. Weller et al. 11 inves-tigated the dependence of the Kerr effect on theorientation of the crystallographic axis of epitaxial fcc

Ž . Ž .Co and the spectra for Co 110 and Co 100 wereexperimentally indistinguishable.

For the calculation we have used a scalar relativis-w xtic tight binding-linear muffin tin orbitals method 12

Ž .TB-LMTO based on the atomic-sphere-approxima-Ž .tion ASA with the inclusion of spin-orbit coupling.

For the exchange-correlation functional, we used thew xvon Barth-Hedin form 13 . The absorptive part of the

off-diagonal conductivity component is proportionalto the difference in the absorption of the left and

Ž .right circularly polarized LCP, RCP light. We usedw xKubo’s linear response theory 14 , which leads to

interband contributions to the conductivity tensor ofthe following forms:

2 2Ž . Ž . Ž . w Ž .xs v s ie rm hV f v 1y f vÝ Ýx x l nkª n ,l

q yª ª2 2< < < <p q p 1 1nl nl= qž /2v vyv q ie vqv q ienl nl nl

2 2Ž . Ž . Ž . w Ž .xs v s e rm hV f v 1y f vÝ Ýx y l nª n ,lk

q yª ª2 2< < < <p y p 1 1nl nl= yž /2v vyv q ie vqv q ienl nl nl

Ž .1

Ž .where f v is the Fermi distribution function and l,nstand for the initial and final energy band states at

ªwave vector k , respectively. The momentum opera-tor is expressed by

ªª ª "p s p q s= = V ,˜24mc

"ª ª ªi Ž .p s p " i p . 2yž /x'2

ªª "Ž .p k represent matrix elements of the LCP andnlRCP components of the momentum operator, respec-tively. In this calculation we have used the lifetimebroadening es0.3 eV. The complex Kerr rotationcan be related to the optical conductivity tensor bythe following equation:

y1r2s̃ 4ps̃x y x x Ž .Csu q i« sy 1q i . 3K K ž /vs̃x x

Furthermore, using the measured data of u , « , andK Koptical constants n and k we can calculate the opticalconductivities using the following formulae:

s ss q is ,x̃ x 1 x x 2 x x

vs s 2nk ,1 x x 4p

v 2 2Ž . Ž .s s 1yn qk , 42 x x 4p

for the diagonal components and

s ss q is ,x̃ y 1 x y 2 x y

Ž .Ž .s sy vr4p Au yB« ,1 x y K K

Ž .Ž . Ž .s sy vr4p Bu qAB« 52 x y K K

for the off-diagonal components, where A and B aregiven by

3 2 3 2 Ž .Asyk q3n kyk , Bsyn q3nk qn 6

3. Experiment

The MOPKS shown in Fig. 1 employs a photoelas-Ž . w xtic modulator PEM 15 . The light source is a high-

pressure 75 W Xe short arc lamp. After passing acalcite polarizer whose transmission axis makes an

( )S.J. Lee et al. r Thin Solid Films 313]314 1998 222]227224

angle of 458 with respect to the vertical axis, i.e. theoptical axis of the PEM, the light becomes linearlypolarized. This linearly polarized light passes throughthe PEM and experiences a periodically varying rela-tive phase shift d between orthogonal amplitudecomponents. The relative phase shift has the formdsd sin v t, where d is the peak relative phase0 0difference and v is the modulation angular frequency

Ž .of the PEM 50 kHz in our experiment . d is propor-0tional to Vrl, where V is the voltage applied to thePEM and l is the wavelength of the incident light.Throughout the entire scan d is kept constant by0varying the voltage. After passing the PEM, the lightis reflected from the magnetized sample or a refer-ence aluminium mirror in the cryostat. The reflectedlight passes through an analyzer whose transmissionaxis is parallel to the plane of incidence. Finally thelight beam goes into a 1r4-m monochromator and isdetected by an S-20 photomultiplier. The amplitudeof the electric field vector transmitted through theanalyzer can be written in a simple form using the

w xJones matrix of each optical element 16 .The angle of incidence is kept below 48. Under the

near normal-incidence condition, the reflection oflight by a magnetized surface can be described by theFresnel reflection coefficients r s r eiu ", where ""̃ "

represents right and left circularly polarized light,w xrespectively 16 . To subtract out Faraday rotation of

the optical windows of the cryostat, we used an Alreference mirror which shows negligible Kerr rotationand ellipticity between 1 and 5 eV, even in highmagnetic fields. Furthermore, to subtract out the me-chanically- and thermally-induced strain birefringence

Ž .we need to measure for both positive and negativefield directions because the strain effect of the win-dows is independent of magnetic field. Therefore theformula used for the Kerr rotation u is given byK

wŽ q q . Ž y y .x Ž .u s u yu y u yu r2, 7K S M S M

where " are positive and negative magnetic fields,and the subscripts S, M designate the sample andreference mirror, respectively. The ellipticity « isK

Fig. 1. Magneto-optical polar Kerr spectrometer.

determined in the same way. Each spectrum we showis the result of four scans taken over a period oftypically 4 h. The polarity of the magnetic field isdetermined using a well known sample such as nickelwhich shows a negative Kerr rotation between 1 and 4

w xeV 17 .The transmitted intensity is given by

D R Ž . Ž .IsI Rq sindqRsin Duq2f cosd 80 ž /2

where for small Kerr angles

< < 2 < < 2 < < 2 < < 2 Ž .Rs r q r r2, D Rs r y r 9Ž .q y q y

and

< < < < Ž .r r fR , Dusu yu . 10q y q y

We perform a Fourier analysis of I using expansionformulae for the sind and cosd terms on the righthand side. We can rewrite I, using Bessel functions,as

Ž .IsI qI sin v tqI cos2v tq ??? , 11DC v 2 v

where

� Ž . Ž .4I sI R 1qJ d sin Duq2fDC 0 0 0

Ž .I sI D RJ dv 0 1 0

Ž . Ž . Ž .I s2 I RJ d sin Duq2f . 122 v 0 2 0

The definition of Kerr rotation and ellipticity in thelimit of small angles is

Ž .u syDur2, « fD Rr4R . 13K K

Ž .Using Eq. 12 , we can determine the Kerr rotationand ellipticity from the experimentally measured val-ues of I , I and I .DC v 2 v

4. Results and discussions

The flux method yielded single crystals of LuFe as2large as 4=2=0.5 mm3. The surface of the plate-likecrystals was mirror-like and did not require furthertreatment. X-ray powder diffraction analysis showedthat the content of secondary phases was less than

˚4%. The measured lattice parameter of 7.221 A is inaccordance with published data and is used for theo-retical calculations. LuFe orders ferrimagnetically2w x18,19 in an fcc structure and has a Curie tempera-

w xture of 570 K 20 . SQUID measurements confirmedthat an applied magnetic field of 1.0 T is sufficient to

Ž .saturate the magnetic moments of LuFe see Fig. 2 .2

( )S.J. Lee et al. r Thin Solid Films 313]314 1998 222]227 225

Fig. 2. Magnetization of LuFe at 5 K. The magnetic field H was2w xapplied parallel to 111 .

The saturation magnetization is 2.46 m rf.u. andBtherefore smaller than the magnetic moment de-

w xtermined by neutron scattering by Givord et al. 21who measured 2.85 m rf.u.B

Using a rotating analyzer ellipsometer we measuredthe dielectric function of LuFe between 1.5 and 5.52eV and calculated the diagonal part of the opticalconductivity from the optical constants. In Fig. 3 weshow the results together with the absorptive part ofs obtained from TB-LMTO where we included a˜lifetime broadening of 0.3 eV. The shape of thecalculated and measured spectrum agrees well. Theamplitude of the experimental conductivity is lowerthan expected due to a native oxide overlayer. Fig. 4displays the experimental Kerr rotation and ellipticitymeasured at different temperatures and magneticfields between 1.4 and 4.0 eV. The minimum Kerrrotation appears near 3 eV. As the field increases, themagnitude of the Kerr rotation increases, but there isno essential change of shape of the Kerr rotationspectrum. The solid lines in Fig. 4 show calculated

Fig. 3. Diagonal component of the optical conductivity of LuFe2measured at room temperature with a rotating analyzer ellipsome-ter. The solid line shows the absoptive part s , the dashed line1 x xthe dispersive component s . The dotted line is s obtained2 x x 1 x xfrom TB-LMTO using a lifetime broadening of 0.3 eV.

Ž . Ž .Fig. 4. Polar Kerr rotation a and ellipticity b of LuFe . The2Kerr effect was measured at 295 K with an applied magnetic field

Ž . Ž . Ž .of 0.4 T ^ and 0.55 T \ , and at 7 K with 1.2 T I ,respectively. The solid line shows the result obtained from TB-LMTO with a lifetime broadening of es0.3 eV.

values of the Kerr rotation and ellipticity. In thiscalculation, we treated the 4f electrons of Lu asvalence electrons. Two narrow spin-orbit split 4f bandsŽ .4f and 4f lie about 4.8 and 3.3 eV below the5r2 7r2

Ž .Fermi level, respectively Fig. 5 . These results aresimilar to those obtained for elemental Lu by Min et

w xal. 22 .When we compare experimental and theoretical

Kerr rotation spectra, we notice in the measuredrange, that the theoretical and experimental valuesagree well in magnitude and shape. The agreement

Žbetween theory and experiment for RFe RsGd,2.Ho, Tb was not as good. This is attributed to the

problem of treating the 4f bands within the LSDAformalism whereby partially occupied 4f bands arepositioned at the Fermi level. Only negligible Lu fconduction electron states are occupied near theFermi level, which indicates that the electronic con-figuration is transition metal-like. Around 3.5 eV,there is a small shoulder in the experimental dataŽ .Fig. 4 . In the theoretical results we see the same flatshoulder appear between 3.3 and 3.7 eV. This occursonly when we treat the 4f electrons of Lu as valenceelectrons. When we treat them as core electrons, theflat shoulder does not appear in the calculated Kerrrotation.

Fig. 6 shows the spectra of vs derived from2 x y

( )S.J. Lee et al. r Thin Solid Films 313]314 1998 222]227226

Ž .Fig. 5. Calculated spin polarized partial density of states DOS forŽ . Ž .Fe a and Lu b using TB-LMTO including spin-orbit interaction.

Ž . Ž .Arrows indicate the DOS for majority ­ and minority x spins,respectively. The DOS for s states is not shown. The total DOS is

Žthe sum of all contributions. Two peaks in the DOS of Lu f ,5r2.f are split by spin-orbit interaction.7r2

Ž . Ž .experimental data of u , « and n, k and theoreti-K KŽ .cal values from TB-LMTO calculations. From Eq. 1 ,

we know that vs is proportional to the difference2 x yof the absorption rates for left- and LCP and RCPlight. vs has a large value at 1.5 eV as shown in2 x y

ŽFig. 6. Off-diagonal conductivity determined from TB-LMTO solid.line and from experimental results obtained from ellipsometry and

Ž .Kerr spectroscopy I . A reduced lifetime broadening of 0.2 eVwas used to enhance the structure of the calculated conductivity.No Drude contributions were included in the theoretical results.

Fig. 6, therefore transitions relating to LCP light arestronger in this region resulting in a large positivevs . From density of states analysis this peak in2 x y

Ž .s near 1.5 eV likely originates from Fe- 4pª3d2 x yŽ .and Lu- 6pª5d interband transitions near E .F

5. Summary

The dielectric function of LuFe single crystal has2been measured between 1.5 and 5.5 eV at 295 K. Themagneto-optic response has been determined between1.4 and 4.0 eV, at 7 and 295 K, in applied magneticfields up to 1.2 T, and compared with theoreticalspectra obtained using the TB-LMTO scheme. Thereis good agreement with the LuFe spectra between2theory and experiment, and one may conclude thatthe TB-LMTO method is suited to describe the nar-row 4f bands involved. For compounds containingrare earths other than Lu, i.e. those with a partiallyfilled 4f shell the localized 4f states cannot be ade-quately treated within the LDA, and the so-calledLDAqU method, which explicitly includes the on-siteCoulomb interaction among highly correlated local-ized electrons, seems to offer an improved description

w xof the MO spectra 23 .

Acknowledgements

Ames Laboratory is operated for the US Depart-ment of Energy by Iowa State University under Con-tract No. W-7405-Eng-82. This work was supported bythe Director for Energy Research, Office of BasicEnergy Science.

References

w x1 A.B. Marchant, Optical Recording, Addison-Wesley, NewYork, 1990.

w x Ž .2 M. Faraday, Phil. Trans. R. Soc. London 136 1846 1.w x Ž .3 J. Kerr, Phil. Mag. 3 1877 321.w x4 P. Chaudhari, J.J. Cuomo, R.J. Gambino, Appl. Phys. Lett. 22

Ž .1973 337.w x Ž .5 P.C. Canfield, Z. Fisk, Phil. Mag. B 65 1992 1117.w x Ž .6 H.S. Bennett, E.A. Stern, Phys. Rev. 137 1965 A448.w x Ž .7 B.R. Cooper, Phys. Rev. 139 1965 A1504.w x Ž .8 P.S. Pershan, J. Appl. Phys. 38 1967 1482.w x9 P.M. Oppeneer, T. Maurer, J. Sticht, J. Kubler, Phys. Rev. B¨

Ž .45 1992 10924.w x Ž .10 D.M. Bylander, L. Kleinman, Phys. Rev. B 49 1994 1608.w x11 D. Weller, G.R. Harp, R.F.C. Farrow, A. Cebollada, J. Sticht,

Ž .Phys. Rev. Lett. 72 1994 2097.w x12 O.K. Andersen, O. Jepsen, D. Gloetzel, in: F. Bassani, F.

Ž .Fumi, M.P. Tosi Eds. , Highlights of Condensed MatterTheory, North-Holland, New York, 1985.

w x Ž .13 U. von Barth, L. Hedin, J. Phys. C 5 1972 1629.w x Ž .14 R. Kubo, J. Phys. Soc. Jpn. 12 1957 570.w x15 PEM-80 Photoelastic Modulator Systems, HINDS Internatio-

nal.

( )S.J. Lee et al. r Thin Solid Films 313]314 1998 222]227 227

w x16 R.M.A. Azzam, N.M. Bashara, Ellipsometry and PolarizedLight, North-Holland, Amsterdam, 1977.

w x Ž .17 G.S. Krinchik, V.A. Artemev, Sov. Phys. JETP 26 1968 1080.w x18 Y. Kasamatsu, J.G.M. Armitage, J.S. Lord, P.C. Riedi, D.

Ž .Fort, J. Magn. Mag. Mater. 140r144 1995 819.w x19 F. Baudelet, C. Brouder, E. Dartyge, A. Fontaine, J.P. Kap-

Ž .pler, G. Krill, Europhys. Lett. 13 1990 751.w x20 E. Belorizky, M.A. Fremy, J.P. Gavigan, D. Givord, H.S. Li, J.

Ž .Appl. Phys. 61 1987 3971.

w x21 D. Givord, A.R. Gregory, J. Schweizer, J. Magn. Mag. Mater.Ž .15r18 1980 293.

w x22 B.I. Min, T. Oguchi, H.J.F. Jansen, A.J. Freeman, Phys. Rev.Ž .B 34 1986 654.

w x23 B.N. Harmon, V.P. Antropov, A.I. Liechtenstein, I.V.Ž .Solovyev, V.I. Anisimov, J. Phys. Chem. Solids 56 1995 1521.