Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 FLAPW calculations of the magneto-optical Kerr effect of BCC Fe Hiromu Miyazawa, Tamio Oguchi* Department of Quantum Matter, ADSM, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan Received 25 August 1998; received in revised form 15 October 1998 Abstract Magneto-optical Kerr effect (MOKE) of BCC Fe is calculated beyond the visible-light energy up to 27 eV by means of the full-potential linear augmented plane wave (FLAPW) method within the local spin-density approximation. Cal- culated MOKE spectra are in good agreement especially for the high-energy region with recently reported experimental results, which have shown a qualitative discrepancy with previous augmented-spherical-wave results. A new peak in the MOKE spectra is found around 18 eV. Convergence properties of several parameters included in our FLAPW scheme are also studied. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magneto-optical Kerr effect; First-principles calculation; BCC Fe; FLAPW method 1. Introduction MOKE from first principles but also a rigorous evaluation of the one-electron approximation and Local spin-density-approximation (LSDA) cal- numerical techniques involved in direct compari- culations of the magneto-optical Kerr effect son with experiment. (MOKE) have been extensively carried out in last In the MOKE calculations so far reported, linear decade for various kinds of ferromagnetic materials muffin-tin orbital (LMTO) and augmented spheri- such as transition metals [1-3], transition-metal cal wave (ASW) methods were used to obtain compounds [4-10] and multilayers [11-14], and one-electron eigenvalues and eigenfunctions. The rare-earth and actinide compounds [15-17]. All of LMTO and ASW methods are known to be very the calculations are based on the Kubo formula of efficient because of using a minimal basis set, com- the optical conductivity proposed by Wang and pared with plane-wave base methods. The cal- Callaway [18,19]. It is now well recognized that the culated MOKE results indeed indicate that the LSDA calculation provides not only a very power- LMTO and ASW methods are highly efficient and ful tool to investigate the microscopic origins of accurate to calculate MOKE for a wide range of ferromagnetic materials within the visible-light en- ergy range. * Corresponding author. Tel.: #81-824-24-7393; fax: #81- Very recently, MOKE spectra have been mea- 824-24-7395; e-mail: oguchi@ipc.hiroshima-u.ac.jp. sured for BCC Fe in a photon energy up to 10 eV 0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 5 3 5 - 6 326 H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 by using a synchrotron radiation source [20] and reflectance or absorption between left and right- shown a qualitatively different result from the pre- circularly polarized lights which are called mag- vious ASW calculations. It is not clear whether the neto-optical effects. In particular, MOKE is the discrepancy arises from LSDA or other numerical magneto-optical effect appearing in the reflectance methods such as ASW. To answer this is one of spectra and used widely as a reading method in our aims in the present paper. We have magneto-optical devices. In order to evaluate formulated the optical conductivity by means of MOKE from first principles, the transition prob- the full-potential linear augmented plane wave ability of electrons excited by a photon should be (FLAPW) method and calculated MOKE for BCC calculated. The optical conductivity tensor due to Fe in the photon energy up to 27 eV. The calculated the interband transition can be obtained by ap- spectra are in good agreement with the recent plying the Kubo formula of the linear response MOKE data, indicating the importance of repres- theory [18] as entation for nearly free-electron-like unoccupied states in evaluating MOKE in a higher energy than 2ie Re[ ? the visible light. Furthermore, it has also been ob- ?@( )"!m #i/ LLY(k) @LYL(k)] k L LY LYL(k) served that the MOKE spectra show an orientation dependence in the (0 0 1)- and (1 1 0)-plane-grown #i Im[ ? , films [20]. We have studied the dependence by LLY(k) @LYL(kZ)] 1 LYL(k)!( #i/ ) changing the magnetization axis from the [0 0 1] to (3) [1 1 0] direction. where is the relaxation time of the excited elec- tron. Assuming an appropriate value for , the real 2. Methods and imarginary parts of ?@( ) are derived directly from Eq. (3). ?LLY(k) is the matrix element of the Within LSDA, one-electron Kohn-Sham equa- component of a momentum operator p "!i : tions are solved self-consistently by using the FLAPW method [21-23]. We employ particularly ?LLY(k)"dr *L(k,r)p ? LY(k,r) (4) the basis-set construction and iterative algorithm proposed by Soler and Williams [24-26]. and A spin-orbit coupling (SOC), H LLY is the energy difference between occupied , which is ne- (n) and unoccupied (n) states glected in scalar-relativistic LSDA calculations [27], is essential to obtain the orbital magnetic LYL(k)" LY(k)! L(k). (5) moment and consequently the magneto-optical ef- fects. The SOC term may be incorporated as a sec- According to our choice of the FLAPW basis set, ond variation in the LSDA calculation. The the wave functions of Eq. (1) can be expressed as relativistic wave functions L(k, r) can be expanded with unperturbed wave functions N H(k, r)" IH(k, r) G (k, r) as l  # [ JlKH(k, rJ)! IJlKH(k, rJ)], (6) L(k, r)" NG(k, r)CGN L(k). (1) J lK G N where I The expansion coefficients C H(k, r) and IJlKH(k, rJ) are the plane-wave GN L(k) are deter- (Fourier) part defined in the whole space and its mined by solving one-electron equations: spherical-wave (lm) expansion inside a sphere [H around atom , respectively, and #H] L(k, r)" L(k) L(k, r), (2) JlKH(k, rJ) is the spherical-wave part defined inside the sphere with with the corresponding relativistic eigenvalues L(k). the cutoff of l The existence of the magnetic moment with SOC . In Eq. (6), l  lK stands for l  l results in certain difference in the phase shift of  Kl K\l. With use of the wave functions in Eq. (6), the matrix element in Eq. (4) may be given by H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 327 the following three terms: contribution will not be considered in the present study. LLY(k)"  LLY#  LLY#  LLY, (7) In order to compute the optical conductivity accurately, the k integration must be greatly taken  LLY"dr I*Lp ILY, (8) care of. We adopt the improved tetrahedron method [28] and investigate the accuracy by l  l  changing the number of k points Nk up to  LLY"dr [ *JlKLp JlYKYLY 18;18;18"5832 in the full Brillouin zone. One J lK lYKY can get the spin magnetic moment within 0.01 ! I* and the Kerr rotation angle and ellipticity up to JlKLp IJlYKYLY], (9) 1 Ry within a few hundredth degrees with N l k"    14;14;14"2744. LLY"dr ( IJlYKYLp [ JlKLY! IJlKLY] J lK lYKY l Next we have checked the convergence about the  number of states included in the second variation in #[ JlKL! IJlKL]*p IJlYKYLY). (10) Eq. (1). It is found that 10 states per spin are enough Since the dipole transition may be allowed only to describe the spectra up to 1.0 Ry while 15 states between the spherical waves with "l!l""1, the are needed for high-energy spectra up to 2 Ry. This matrix elements for l"l fast convergence may be because of the weakness of  and l"l #1 re- main in Eq. (10). Furthermore, it is expected that the perturbation H in comparison to the energy the  difference between the electron states in the high- LLY term may be much smaller than the other two terms because er-energy region where electrons behave like nearly JlKL! IJlKL has a very small magnitude due to the continuity conditions near free electron. the sphere boundary where IJlKL has a maximum. Therefore, we neglect the  LLY term in the present calculation. To check this, the convergence with 3. A test of the spherical-wave expansion respect to l  will be examined in detail in the following section. The matrix element of a mo- In this section, we examine the convergence in mentum operator becomes Hermitian because the optical conductivity and the Kerr spectra with  respect to the maximum angular momentum in LLY is Hermitian obviously and  LLY is Hermitian by virtue of the continuation condition between the spherical-wave expansion l  in Eq. (6). In the present test for BCC Fe, we commonly use the JlKL and IJlKL at the muffin-tin sphere surface. Finally, the Kerr rotation angle lattice constant of a"2.87 As, the muffin-tin radius )( ) and its ellipticity of 1.1 As, the relaxation-rate parameter , / " )( ) are defined as 0.04 Ry, the cutoff energy of 15 Ry for the basis functions (the corresponding cutoff of 60 Ry for the VW( ) )( )#i )( )"! (11) charge density and potential functions) and VV( )(1#i(4 / ) VV( ) Nk"18;18;18"5832. in the case of the magnetization axis and the inci- Figs. 1 and 2 show the optical conductivity and dent light parallel to the z direction. Kerr spectra calculated for BCC Fe by changing Besides the interband term described above, the l intraband contribution to the conductivity must be  from 2-4. It is seen in Figs. 1 and 2 that the difference between l included in metallic cases and may be additionally "3 and 4 is negligibly small and that l treated within a phenomenological expression by "2 may be almost sufficient for qualitative discussions of the Kerr spectra though Drude. However, since an empirical plasma energy the optical conductivity shows somewhat large (at introduces another ambiguity into the shape of the most 20%) difference between l spectrum especially in low-energy regions and its "2 and 3. The negligible difference between l effects on the spectrum have been already investi- "3 and 4 indi- cates that the l*3 states are well expressed by gated in the case of BCC Fe [1], the intraband the plane waves and the  LLY term in the matrix 328 H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 Fig. 1. Calculated optical conductivity of BCC Fe: (a) the real part of VV, (b) the imaginary part of VV, (c) the real part of VW and (d) the imaginary part of VW with l "2 (dash-dotted line), l "3 (solid line) and l "4 (dotted line). An intraband contribution is not included. elements can be neglected. The difference between representation of the l"2 states are definitely in- l "2 and 3 is mostly due to an error caused by sufficient. the neglect of  LLY, assuming the l"3 states are Concerning the description of the electronic well represented by the plane waves. In the case of states, the conventional FLAPW method requires l "2, important d-f or f-d transitions should be a high l given by the   value to connect the wave functions LLY term, because the plane-wave smoothly at the muffin-tin sphere. For example, H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 329 Fig. 2. Calculated spectra of (a) Kerr rotation angle and (b) Kerr ellipticity of BCC Fe with l "2 (dash-dotted line), l "3 (solid line) and l "4 (dotted line). An intraband contribution is not included. Hathaway et al. [29] adopted l "8 for BCC Fe. above may change by choosing another radius. Fig. This is because that discontinuity in the wave func- 3 shows the Kerr spectra calculated with the sphere tions and the charge density may lead to an inac- radius of 1.24 As, which corresponds to the inscribed curate solution in the self-consistent procedure. In sphere radius of the present BCC lattice. No defi- our formulation, l "2 is enough to get the accu- nite difference can be seen in Fig. 3 except for minor rate electronic states since states with l larger than ones around 10 eV. It is, therefore, concluded that l  are expressed by the plane waves. the present FLAPW scheme for calculating the This feature of our FLAPW basis functions pro- Kerr spectra is robust in the sense that the cal- vides another advantage to represent wave func- culated results are independent of the parameters tions properly in high-energy regions where the involved within their physically reasonable and linear method may break down within a single- computationally efficient range. energy window [21,22]. Electronic states in the high-energy region are basically nearly free-elec- tron like. The states with l'l  inside the muf- 4. Results and discussions fin-tin sphere are not restricted by the linear method in our formulation since they can be ex- Theoretical and experimental Kerr spectra are pressed by the plane-wave functions penetrating shown in Fig. 4 for comparison. It can be found into the sphere. The states with l)l  are ex- that general features of the observed spectra are pected to have very small amplitudes within the well reproduced by the present FLAPW calcu- muffin-tin sphere in the high-energy region. The lation. In the Kerr spectrum, peaks at 4.5 and present linear method is suitable to express 6.0 eV coincide with Katayama's data. A shoulder the electronic states precisely and efficiently in the near 7.4 eV is slightly shifted to a higher energy high-energy region. region by 0.7 eV in this work. As shown in Fig. 4, The spherical-wave expansion must depend on the ASW spectrum reveals a deviation above 6 eV. the sphere radius assumed and the results shown It is seen that qualitative agreement of the 330 H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 Fig. 3. Calculated spectra of (a) Kerr rotation angle and (b) Kerr ellipticity of BCC Fe with the muffin-tin radii of 1.24 As (dotted line) and 1.1 As (solid line). An intraband contribution is not included. Fig. 4. Calculated (thin solid line) spectra of (a) Kerr rotation angle and (b) Kerr ellipticity of BCC Fe with experimental results by Katayama (dotted line) [20] and by Krinchik (dashed line) [38]. Thick solid lines denote calculated Kerr spectra with the correction by the virtual refractive-index method along the experimental film structure (Au capping layer [20 As]/Fe film [1000 As]/Au substrate). A previous ASW result [1] is also plotted by a dash-dotted line. In the theoretical spectra, an intraband contribution in Fe is not included while that in Au is considered. H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 331 present FLAPW results with the experimental complex structure of the MOKE spectrum, or the spectra above 6 eV is much better than in the case off-diagonal part of the optical conductivity in the of the ASW result, in which the position of the wide-energy range provides us opportunity to shoulder is shifted by 1.7 eV. However, compared study the excited states in metallic systems. In addi- with Katayama's data, the absolute values of our tion, we can evaluate the approximation involved calculated spectra are almost rigidly shifted by in the first-principles calculation by comparing about 200 m degrees to a positive side in the with experiment directly. Thus it is strongly desired Kerr ellipticity and to a negative side in the Kerr to measure the MOKE spectrum in a wide-energy rotation angle. We examined effects of the capping range. layer and the substrate of Au used in the experi- Katayama reported the orientation dependence ment by the virtual refractive-index method and of the MOKE spectra between the [0 0 1] and found certain improvement by at most 100 m de- [1 1 0] directions [20]. However, almost no ori- grees towards the experimental spectra, as shown in entation dependence has been found. This should Fig. 4. be due to weak H Calculated MOKE spectra in a wide energy  and the isotropic nature of the crystal field of BCC Fe. This kind of poor orienta- range up to 27 eV are shown in Fig. 5. Since the tion dependence has been already reported for shallowest 3p core states are situated far below it, FCC Co theoretically [3] and experimentally [31]. transitions from the core states do not affect Symmetry breaking by a lattice distortion may be the MOKE spectra in this survey. In Fig. 5, the important for describing the variation observed in Kerr spectrum of Fe can be classified into two film-grown samples. regions based on the character of its final states The main difficulty to calculate the MOKE for by inspecting the matrix elements [30]. The metallic systems comes from the existence of the low-energy region of the spectrum less than about free-electron-like response accompanied by the 7 eV is characterized by the final states with d Fermi surface. Generally this response, the in- symmetry, and by those with s, p and f symmetry traband transition, is approximated by the Drude as tails of the neighboring d states. On the other term with one parameter, the plasma energy or the hand, the high-energy region above 7 eV has optical mass. However, the optical mass should be those characterized only by the free-electron-like k dependent in nature as derived by Wang and states. Callaway [18]. In addition to this, the Drude Fig. 5 indicates three remarkable peaks in the term affects MOKE spectra non-linearly in the high-energy region, which have never been dis- low-energy region with large amount, for cussed so far. The largest peak is situated around example under 3 eV in BCC Fe. Thus unless we 18 eV. This peak can be assigned to the optical obtain accurate values for the diagonal elements of transition around H and N points in the Brillouin the optical conductivity, the Drude term may zone, where flat energy bands exist near 17-18 eV trick us even in the case of qualitative estimation above the Fermi energy [30]. Two other small of the spectra. As well known, the LSDA limits peaks are located around 8 and 25 eV. The peak accurate descriptions of the excited states. One around 8 eV has been observed in Katayama's ex- example is the reflectance spectra of Cu, Ag and Au. periment as shown in Fig. 4. This peak may be Theory underestimates the plasma resonance en- attributed to the transitions around N points be- ergy lower by 0.3-0.8 eV as compared with experi- cause there exist flat bands near 7-8 eV above the ment [32]. This error comes from the fact that Fermi energy at N points. The peak around 25 eV LSDA estimates the d-band position shallower also corresponds to the transitions around than the experimental one [33]. Another example N points because of flat bands near 24-25 eV above can be seen in the overestimation of the d-band the Fermi energy at N points. width of transition metals [34]. To overcome these Like these features, there must exist fruitful struc- problems arising from LSDA, one has to evaluate tures in the ultra-violet region in the other mate- the self-energy, for instance, by using the GW rials and such surveys are now undertaken. This method [35-37]. 332 H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 Fig. 5. Calculated Kerr spectra and optical conductivity spectra of BCC Fe up to 27 eV: (a) Kerr rotation angle (solid line) and Kerr ellipticity (dotted line), (b) the real part of VV (solid line) and the imaginary part of VV (dotted line), and (c) the real part of VW (solid line) and the imaginary part of VW (dotted line). An intraband contribution is not included. 5. Summary convergence properties with respect to both k points and second-variation states are well con- We have developed a precise and efficient scheme firmed. Secondly, as for the convergence for the for calculating MOKE by adopting FLAPW spherical-wave expansion, l method within LSDA. We have checked our "2 is enough to obtain the accurate electronic states in our formu- method by applying it to BCC Fe. Firstly, the lation. We need, however, l "3 when neglecting H. Miyazawa, T. Oguchi / Journal of Magnetism and Magnetic Materials 192 (1999) 325-333 333  LLY in the matrix elements since the optical [4] H. Ebert, H. Akai, J. 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