Element selective X-ray magnetic circular and linear dichroisms in ferrimagnetic yttrium iron garnet films

ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 3945–3962

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Element selective X-ray magnetic circular and linear dichroisms in ferrimagnetic yttrium iron garnet films A. Rogalev a, J. Goulon a,, F. Wilhelm a, Ch. Brouder b, A. Yaresko c, J. Ben Youssef d, M.V. Indenbom d a

European Synchrotron Radiation Facility (ESRF), B.P. 220, F-38043 Grenoble Cedex, France Institut de Mine ralogie et de Physique des Milieux Condense s, UMR-CNRS 7590, Universite Paris VI-VII, 4 place Jussieu, F-75252 Paris Cedex 05, France c Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany d Laboratoire de Magne tisme de Bretagne, CNRS FRE 2697, UFR Sciences et Techniques, F-29328 Brest Cedex, France b

a r t i c l e in fo

abstract

Article history: Received 24 June 2009 Received in revised form 21 July 2009 Available online 8 August 2009

X-ray magnetic circular dichroism (XMCD) was used to probe the existence of induced magnetic moments in yttrium iron garnet (YIG) films in which yttrium is partly substituted with lanthanum, lutetium or bismuth. Spin polarization of the 4d states of yttrium and of the 5d states of lanthanum or lutetium was clearly demonstrated. Angular momentum resolved d-DOS of yttrium and lanthanun was shown to be split by the crystal field, the two resolved substructures having opposite magnetic polarization. The existence of a weak orbital moment involving the 6p states of bismuth was definitely established with the detection of a small XMCD signal at the Bi M1 -edge. Difference spectra also enhanced the visibility of subtle changes in the Fe K-edge XMCD spectra of YIG and {Y, Bi}IG films. Weak natural X-ray linear dichroism signatures were systematically observed with all iron garnet films and with a bulk YIG single crystal cut parallel to the (111) plane: this proved that, at room temperature, the crystal cannot satisfy all requirements of perfect cubic symmetry (space group: Ia3d), crystal distortions preserving at best trigonal symmetry (R3 or R3m). For the first time, a very weak X-ray magnetic linear dichroism (XMLD) was also measured in the iron K-edge pre-peak of YIG and revealed the presence of a tiny electric quadrupole moment in the ground-state charge distribution of iron atoms. Band-structure calculations carried out with fully relativistic LMTO-LSDA methods support our interpretation that ferrimagnetically coupled spins at the iron sites induce a spin polarization of the yttrium d-DOS and reproduce the observed crystal field splitting of the XMCD signal. & 2009 Elsevier B.V. All rights reserved.

PACS: 75.25.+z 75.70.kw 76.50+g 78.20.Ls Keywords: XMCD XANES

1. Induced magnetism in iron garnets Yttrium iron garnet (YIG) is a generic term for a rich family of ferrimagnetic iron garnets in which yttrium can be substituted with rare earths (RE) or bismuth in variable proportions [1,2]. Over the past decades, much expertise has accumulated on how to grow single crystal garnet films by liquid phase epitaxy (LPE) on oriented gadolinium gallium garnet (GGG) substrates [3–6], while pulsed laser deposition (PLD) methods are now developing very rapidly [7–9]. The experiments described below concern YIG thin films grown by LPE on GGG substrates. More specifically, we shall focus on iron garnet films in which diamagnetic cations such as La3þ , Lu3þ and Bi3þ partly substitute for Y3þ. Indeed, fBi2Lu2Yg IG films represent an important class of materials that are of outstanding technological importance, not anymore for the production of bubble memories, but for the design of nonreciprocal isolators in optical fibers, for ultra-high frequency

 Corresponding author.

E-mail address: [email protected] (J. Goulon). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.07.078

optical modulators [10–12], femtosecond photomagnetic switching [13] as well as for the development of a new generation of monolithic microwave integrated circuits (MMIC) devices [14,15]. Many of the current applications, e.g. as optical isolators, stem from the remarkable property of Bi-substituted iron garnets to be highly transparent in the near IR spectral range while they exhibit a huge specific Faraday rotation [16–19]. Moreover, for given orientations of the substrates, these films may also show giant linear magnetoelectric effects [20] and a fairly large efficiency in second harmonic generation (SHG) [21,22]. It is commonly admitted that the crystal structure of YIG is 3 cubic with space group Ia3d (O10 h ; group N 230) [2]. There are still debates, however, about the relevant crystal structure in epitaxially grown films as well as in single crystals. Typically, the observation below 125 K of the first order (linear) magnetoelectric effect in a high purity YIG single crystal should definitely rule out inversion symmetry [23–25]. In epitaxially grown thin films, symmetry lowering is also to be expected due to growth anisotropy while structural distortions could be caused by a residual mismatch between the lattice parameters of the film and GGG substrate [21]. Moreover, in ferrimagnetically ordered

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phases, the Curie principle as well as Landau’s theory of phase transitions requires that the crystal symmetry can only be a subgroup of the paramagnetic (cubic) phase. In this paper, we shall produce X-ray linear dichroism data which confirm the existence of small structural distortions making rhombohedral subgroups more realistic such as R3 (S6 ; group N 3 148) [26] in the ferrimagnetic phase, or R3m (C3v ; group N 3 160) in the magnetoelectric phase [21]. If one neglects such small distortions for simplicity, then the cubic unit cell of YIG consists of eight formula units: fYg3 ½Fe2 ðFeÞ3 O12 . This formulation emphasizes the role of two non-equivalent sites for iron: the first one (16a sites) has octahedral coordination with oxygen anions, whereas the second one (24d sites) has only tetrahedral coordination with O2 . Below the magnetic ordering temperature ðTc C550 KÞ, the two Fe sublattices get magnetized antiparallel one to each other according to the ferrimagnetic model of Ne el, with an unbalanced magnetization (ca. 5mB ) in favor of the tetrahedral sites. This is classically explained by a strong superexchange interaction between the two iron sublattices: the rather large Fe(a)–O–Fe(d) angle ð126:6 3 Þ is a clear indication that the wavefunctions of oxygen and iron have a substantial overlap so that superexchange may well be mediated by the oxygen anions. This picture describes correctly the spin magnetization at the iron sites, but it does not preclude the existence of weaker, partially unquenched orbital moments that can hardly be seen in neutron diffraction. On the other hand, the detection of satellites in the 57Fe NMR spectra (see Refs. [27–29]) has fed the presumption that induced magnetic moments may well be carried by the non-magnetic yttrium or RE atoms: this is not totally unexpected since those atoms (in 24c sites) have dodecahedral coordination to the same oxygen anions mediating superexchange between the iron spins. In the case of LuIG, this presumption was supported by the observation of a broad NMR signal attributed to the resonance of 175Lu and 176Lu nuclei ðI ¼ 7=2Þ [30]: the rather large hyperfine field (100 kOe) broadened by quadrupole interaction was interpreted as a clear indication of a charge transfer from iron to lutetium. It was conjectured that a similar effect could exist for yttrium but any attempt to detect the 89Y ðI ¼ 1=2Þ resonance failed so far. In YIG thin films, magnetic circular dichroism signatures were nevertheless detected in the vacuum ultraviolet (VUV-MCD), near the N2;3 absorption edge of yttrium [31]: it was concluded that very weak induced moments ð0:01mB =atomÞ could really exist at the ‘‘diamagnetic’’ yttrium sites. This result prompted us to revisit this problem in the X-ray range, taking full advantage of the higher sensitivity of X-ray magnetic circular dichroism (XMCD). In Section 3, we try to resolve the respective contributions of the induced spin and orbital magnetization components in the framework of differential magneto-optical sum-rules. The validity of our interpretation is supported in Section 5 with band structure calculations conducted in accordance with a fully relativistic formalism. The naive view according to which Bi3þ should be diamagnetic because of its closed shell ð1 SÞ electronic structure fails when chemical bonding involves linear combinations of overlapping atomic orbitals that may become magnetically polarized. In the late 1980s, a broad NMR signal assigned to the 209Bi nucleus ðI ¼ 9=2Þ was detected in fBix Y3x gIG films: since the calculated internal field was too high (60 kOe) to be explained only by the dipole field of the ferric ions [32], it was conjectured once again that induced magnetic moments could be located at the Bi sites. Already in the early 1970s, Wittekoek and coworkers [33,34] suggested that the large Faraday effect measured in bismuth substituted garnets could be due to the hybridization of the bismuth 6p orbitals with the oxygen 2p and iron 3d orbitals.

Recent calculations using either a cluster molecular orbital approach [35], or fully relativistic LCAO/LSDA methods [36] confirmed this interpretation. A more favorable energetic overlap would apparently make the hybridization of the Bi 6p orbitals even stronger in the Fe 3d conduction bands, than in the O 2p or Fe 3d valence bands [36]. Thus, spin–orbit coupling in the Bi 6p orbitals would let us expect some orbital magnetization of the p-projected densities of states (DOS) of bismuth: in other terms, XMCD should be detectable for transitions toward localized states involving Bi 6p orbitals and allowed by the electric dipole selection rules. This may be the case at the Bi L1 or M1-edges where {2s,3s} -np transitions are allowed whereas 3d -np transitions are again allowed at the Bi M4;5 -edges. Unfortunately, a variety of technical difficulties listed in Section 2 concur to transform the detection of XMCD signals at any Bi M-edge into a formidable tour-de-force. On the other hand, one may question how far the substitution of Bi3þ cations for Y3þ could induce a detectable perturbation of the XMCD spectra recorded at the Fe K-edge. It will be shown in Section 4 that such subtle effects can be measured at the ESRF beamline ID12 where the high sensitivity of the instrumentation also allowed us to detect—for the first time—a tiny X-ray magnetic linear dichroism (XMLD) signal at the iron K-edge, as predicted by theory.

2. Experimental aspects 2.1. Sample selection and characterization Numerous iron garnet films with variable composition were grown at the Laboratoire de Magne tisme de Bretagne in Brest (France). Oriented GGG substrates (+ 1 in; thickness: 457750 mm) were directly purchased from the Crystal Division of SAINT-GOBAIN Inc. (France). Both faces of each single crystal substrate were optically polished. The films were grown by LPE following well-established procedures and recipes detailed elsewhere [4]. A first screening of the magnetic properties of each film took place in Brest. For X-ray absorption spectroscopy, it was desirable to maximize the film thickness ðhZ8 mmÞ in order to minimize the contamination of the XMCD spectra by unwanted sharp Bragg reflections (glitches) from the underlying crystalline GGG substrate. For obvious sensitivity reasons, we tried to have the highest lanthanum or bismuth concentrations in yttrium substituted films. In this paper, we shall report results collected on only three films: (i) 1 ¼ Y3 Fe5 O12 (YIG). (ii) 2 ¼ fY1:3 La0:47 Lu1:3 gFe4:84 O12 . (iii) 3 ¼ fY1:69 Bi1:31 gFe4:9 Ga0:1 O12 . In this paper, reference will also be made to X-ray linear and circular dichroism measurements performed on a high quality YIG single crystal cut parallel to the pseudo-cubic (111) plane (4). This single crystal was borrowed from Kolomiets in the Department of Condensed Matter Physics at the Charles University, Prague (Czech Republic). The elemental composition of each film was ascertained by X-ray fluorescence analyses performed at the ESRF in a scanning electron microscope. The thickness (h) of the last film clearly did not meet our target value. Typical magnetization curves measured for films 2 are reproduced in Fig. 1 with the field either parallel or perpendicular to the film. Note that a field in excess of 2.5 kOe is required to achieve saturation of the film in perpendicular

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Table 1 Characterization of the films retained for XMCD studies.

0.20 0.15

Film # 2 LaLu 3.2 / GGG (100)

M (a.u.)

0.10 0.05 0.00 -0.05 -0.10 -0.20

M (a.u.)

0.10

-20

-10

0 H (Oe)

10

20

30

0.05 0.00

H

-0.05 -0.10 -3

-2

-1

0 1 H (kOe)

Reference #

Orientation (GGG)

h ðmmÞ

4pMs (G)

BA (G)

1 2 3

YIG 520 LaLu 3.2 Bi 2054

(111) (1 0 0) (1 0 0)

9.8 8.5 4.6

1750 1800 1800

1650 2220 1000

may have been exchanged with Lu3þ cations in the 16a iron antisites as described by Gilleo [2]: this would increase the magnetization and rather favor perpendicular anisotropy as observed experimentally. 2.2. Circularly polarized, monochromatic X-rays

Film # 2 LaLu 3.2 / GGG (100)

-4

Film no.

1 GðemuÞ ¼ 103 Am1 (SI).

H

-0.15

-30

3947

2

3

4

Fig. 1. Characteristic magnetization plots for film 2 ¼ LaLu 3.2: (a) in plane magnetization and (b) perpendicular magnetization.

magnetization geometry whereas the coercivity fields in the hysteresis loop are rather small. The essential characteristics of those films are summarized in Table 1 in which 4pMs is the saturation magnetization in emu units, whereas BA refers to the effective anisotropy field evaluated from FMR spectra. A major distinction has to be made between the uniaxial component ðBu Þ and the cubic component ðBA1 Þ which refer to magnetic anisotropy free energies described with ‘ ¼ 2 or 4 spherical harmonics respectively. In current practice, one may write: Bu ¼ 2Ku1 =Ms  m0 DMs in which Ku1 is the first order uniaxial anisotropy constant and D is the effective demagnetizing coefficient that is shape-dependent and is close to unity only for a film with vanishing aspect ratio. Similarly, one may define: BA1 ½1 1 1 ¼ ð2=3Þð2KA1 =Ms Þ or BA1 ½1 0 0 ¼ 2KA1 =Ms depending on the substrate orientation. For YIG films, the cubic anisotropy is often given as negligibly small at room temperature ðBA1 r50 GÞ. A key issue in growing iron garnet films by LPE is to control the respective contributions of (i) the growth induced anisotropy g e ); (ii) the magnetoelastic contribution (Ku1 ). The first one is (Ku1 most sensitive to the nature of the substituting cations (e.g. La3þ , Lu3þ , Bi3þ ) whereas the second originates from the actual mismatch between the lattice parameter of the GGG substrate ðaS ¼ 12:383 AÞ and the lattice parameter of the deposited film, ˚ e.g. aF ¼ 12:376 A for YIG. ˚ As confirmed by Table 1, all selected films had perpendicular anisotropy. Thus, the growth anisotropy induced by the high-Z cations (La3þ , Lu3þ in 2; Bi3þ in 3) finally dominated the magnetoelastic anisotropy. This was not unexpected because the lattice expansion due to La3þ in 2 was partly compensated by a lattice compression due to Lu3þ [2]. It is well documented that a partial substitution of the 24d iron sites with Ga or Al decreases the magnetization and is often exploited to favor in plane anisotropy [2,37]. In this respect, the small deficit in iron revealed by the elemental analysis of film 2 let us presume that a few Fe3+ cations

All X-ray dichroism spectra discussed in this paper were recorded at the ESRF beamline ID12. This beamline, which was described in detail elsewhere [38,39], is equipped with three powerful helical undulator segments optimized to deliver high fluxes of circularly polarized X-ray photons over the whole energy range 1.8–18 keV. We made use of an Apple-II helical undulator (HU-38) for linear and circular dichroism experiments carried out at the iron K-edge. The Helios-II undulator (HU-52) was selected for the challenging XMCD measurements at the bismuth M1 -edge. For all X-ray absorption spectroscopy (XAS) studies below 3 keV, including XMCD measurements at the yttrium L-edges, the preference was systematically given to another type of hybrid undulator (EMPHU) combining electromagnet/permanent magnet technologies [40]. In all experiments, the two-crystal monochromator was equipped with a pair of Si(111) crystals cooled down to 140 K. What makes the originality of beamline ID12 is that the users have a full control of the polarization of the X-ray photons emitted by the undulator sources: for instance, they have the capability to flip the circular polarization from left (L) to right (R). Even though the Stokes–Poincare circular polarization rate P3 is excellent for the undulator radiation (P3 C0:97), we need to stress that the effective circular polarization rate at the sample is usually far from unity and will drop down to zero if the X-ray monochromator is operated at Bragg angles yB near 453 since the two crystals will act then as a very efficient linear polarizer. This is precisely what did happen at the Bi M4;5 -edges where the Bragg angles were ca. 47:33 and 50:033 respectively. At the corresponding energies, the circular polarization rate P3 could not exceed 1% and 5% respectively and made XMCD measurements fairly difficult if not hopeless. This point is clearly illustrated with Fig. 2 in which we plotted the circular polarization rate P3 of the X-ray beam at the exit port of the monochromator in the energy range of interest for XMCD experiments. We tentatively indicated on the same plot the relative edge jumps measured at the yttrium L-edges and bismuth M-edges. Fig. 2 shows that corrections taking into account the true value of P3 are mandatory at the Bi M1 -edge as well as at the Y L2;3-edges. 2.3. Instrumental energy resolution versus natural width of core levels Due to the excellent optical properties of the undulator sources, energy resolution was very close to the expected theoretical limit of what can be expected for a double crystal monochromator equipped with a pair of Si(111) crystals: DEC1:15 eV at the Fe K-edge; DEC0:6 eV at the Bi M1 -edge; DEC0:32 eV at the Y L-edges. Let us recall, however, that the

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10

1.0

Table 2 FWHM natural width of selected L core levels (after [43]).

8

Y L3

Bi M2

Bi M3

0.8

Y L2

6

4

2

0.6

Bi M4

0.4

Bi M5

Circular Polarization Rate

0

Polarization Rate P3

Fluorescence Intensities If/I0

Bi M1

0.2

Element

Z

GðL1 Þ (eV)

GðL2 Þ (eV)

GðL3 Þ (eV)

Yttrium Lanthanum Lutetium Bismuth

39 57 71 83

4.71 4.06 5.54 12.4

1.65 3.68 4.92 6.67

1.50 3.41 4.68 5.98

Table 3 FWHM natural width of M levels of bismuth (after [44]).

GðM1 Þ

GðM2 Þ

GðM3 Þ

GðM4 Þ

GðM5 Þ

21.7 eV

14.6 eV

10.7 eV

2.88 eV

2.74 eV

0.0 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

Photon Energy (eV) Fig. 2. Stokes–Poincare circular polarization rate P3 at the L-edges of yttrium and M-edges of bismuth when the double crystal monochromator is equipped with a pair of Si(111) crystals.

repeatability of the energy scans of the monochromator had to be better than 10 MeV in order to warrant derivative free XMCD spectra. Note that the instrumental energy resolution at the iron K-edge is better than the natural width (FWHM: full width at half maximum) of the 1s core level: GðKÞ ¼ 1:25 eV [43]. As illustrated with Table 2, the instrumental energy resolution is indeed far better than the natural width of the L levels of yttrium, lanthanum, lutetium and bismuth which are of direct interest for this paper. We also regrouped in Table 3 the natural width G of the M levels in bismuth. From Table 2, we may anticipate that the shorter core hole life time will broaden the white line resonances in the XANES and XMCD spectra recorded at the rare earth L2;3 -edges whereas those resonance should look much sharper at L2;3 -edges of yttrium. Table 3 let us anticipate much bigger difficulties to detect weak XMCD signatures at the bismuth M-edges: whereas the natural widths of the M4;5 -levels are of the same order of magnitude as the instrumental resolution, this is not the case at the M1 -edge where G is ca. 40 times larger than DE and will blur out the weak XMCD signatures we want to detect. We do not expect XMCD measurements to be a lot easier at the Bi L1 -edge (DEC3:88 eV at 16.388 keV): for deep core levels excited at high photon energy, the transition probabilities suffer from the poor radial overlap between the core state and the final states while the relative contribution of electric quadrupole transitions would increase. 2.4. X-ray fluorescence excitation spectra Due to the strong absorption of the samples at low photon energy, X-ray absorption near edge structures (XANES) or XMCD spectra were systematically recorded in the fluorescence excitation detection mode. The X-ray fluorescence photons were collected over a wide solid angle using Si photodiodes designed to maximize the active detection area while preserving a very low capacitance. We had to worry about further complications: (i) The detection sensitivity tends to deteriorate at low energy, i.e. at the Bi M-edges as well as at the Y L-edges, due to lower fluorescence yields. (ii) Raw experimental data had to be carefully corrected for fluorescence reabsorption [41,42], as well as for amplitude distortions caused by multiple fluorescence lines and scattering background in the film itself or in the substrate.

Such corrections turned out to be very important at the L-edges of yttrium and rare earths where the XANES spectra exhibit very sharp ‘‘white lines’’ resonances (see Fig. 2). At low excitation energy, further difficulties may arise from the contamination of the incident X-ray beam with the unwanted higher order harmonics of the monochromator: this well identified problem was solved either on using efficient low pass mirror filters [38], or in slightly mistuning the parallelism of the two crystals of the monochromator. Let us insist that careful normalization procedures of the XANES spectra recorded in the fluorescence excitation mode proved to be also very efficient in minimizing the effects of unwanted harmonics in the analysis of the data. Whereas (Z-1) filters can be used at the iron K-edge to minimize the contribution of elastic scattering (and Bragg glitches), this was not possible at low excitation energies with the practical consequence that the statistical noise was inherently higher. 2.5. Circular versus linear magnetic dichroisms In X-ray magnetic circular dichroism (XMCD) experiments, the magnetic bias field B0 is parallel to the wavevector kcp of the circularly polarized X-ray photons. Whenever the magnetization vector is also parallel to kcp , then XMCD is obtained from the difference: fIFðLÞ =I0ðLÞ  IFðRÞ =I0ðRÞ g. Here, I0 and IF stand for the intensities of the incident beam and of the collected fluorescence photons for left (L) or right (R) circularly polarized incoming X-rays, the ratio IF =I0 defining the relevant reduced fluorescence intensity. Since XMCD is a time-reversal odd property of matter, its sign is expected to change on inverting the direction of the magnetic bias field. In X-ray magnetic linear dichroism (XMLD) experiments, the magnetic bias field B0 is set perpendicular to the wavevector klp of the linearly polarized X-ray photons and is therefore in the polarization plane. Recall that XMLD is a time-reversal even magnetic property of matter: its sign and amplitude should thus remain invariant on inverting the direction of the magnetic bias field. To perform XMLD experiments at the ESRF beamline ID12, we found most efficient to convert the monochromatic, elliptically polarized X-ray photons transmitted by the monochromator into linearly polarized X-rays. This can be easily done with an X-ray quarter wave plate (QWP) with the advantage that one can rotate very rapidly the polarization vector by 903 : a fast digital piezoactuator makes it easy to invert the sign of the angular offset DyQWP that is of the order of only 200 arcsec [39]. Throughout the present paper, linear dichroism is defined with respect to the horizontal (H) and vertical (V) polarizations of the incident beam as the difference: fIFðVÞ =I0ðVÞ  IFðHÞ =I0ðHÞ g. What was measured can

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then be identified with the Stokes–Poincare polarization rate P1 at the absorbing site.

3. XMCD at spin–orbit split edges In this section, we are concerned with the detection of surprisingly large XMCD signatures measured at the L2;3 -edges of yttrium, lanthanum and lutetium which are often given as nonmagnetic elements on making implicitly reference to the magnetic properties of the cations Y3þ , La3þ and Lu3þ . At the corresponding edges, the degeneracy of the photoionized core level is removed by a strong spin–orbit coupling given that the relevant orbital quantum number ‘c a0. As an example, we have reproduced in Figs. 3a and b the XANES and XMCD spectra which were recorded at the L2;3 -edges of yttrium with the YIG film 1. The most prominent extrema of the XMCD spectra were labelled Ei0 with i ¼ 1; 4 at the Y L3-edge, and Ei00 at the Y L2-edge. 3.1. Magneto-optical DOS polarization rules Let us summarize briefly in this section what could be learned from the theory of XMCD at spin–orbit split edges. Indeed, a perennial difficulty is to express the X-ray absorption crosssection sðEÞ on starting from the well-known Fermi’s golden rule written in either the electric dipole or the electric quadrupole approximations. Since the early work of Dill and Dehmer [45], it was found rather convenient to exploit a real space, scattered wave (SW) approach in which the final state is represented as a time-reversed wavefunction of the photoelectron scattered by a cluster of atoms. For the sake of simplicity, Dill and Dehmer adopted a now classical space partition in which the atoms were associated with non-overlapping spherical domains where the local atomic potentials could be spherically averaged, the corresponding ‘‘muffin-tin’’ potentials being still embedded in some constant interstitial potential. In this context, the X-ray absorption cross-section can most easily be calculated in terms of the socalled scattering paths operators si;j which are symbolic supermatrices describing the entire set of scattering paths that begin at atom i and terminate at atom j. The scattering path operators should not be confused with the atomic t-matrices involved in the calculation of s. In an effort to unify the various theories of X-ray absorption spectroscopies, Natoli et al. [46] proved that, within the limits of validity of the optical theorem [47], the multiple scattering approach was strictly equivalent to alternative methods

YIG Thin Film Y L2- Edge

YIG Thin Film Y L3- Edge

2.0 1.5

XANES

1.0

XANES E’2

0.5

E”1

E’3

E”4 XMCD

XMCD

0.0

x 20

10 2. 11 2. 12 2. 14 2. 15 2. 16 2. 17 2. 18 2. 19 2. 20

09

E”2

2.

08

x 20 E”3

E’4

2.

07

E’1

2.

2.

06

-0.5

2.

Reduced Fluorescence Intensity

2.5

Photon Energy (keV) Fig. 3. XANES and XMCD spectra of the YIG thin film 1 (raw data) at the yttrium Ledges: (a) Y L3-edge and (b) Y L2-edge.

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such as Green’s operator approach or the band structure approach in periodic lattices. For circularly polarized X-ray photons with left helicity (‘ ) or right helicity (þ‘ ), the following expression of the X-ray absorption cross-sections can be derived:

s7 ¼ 4pa0 ‘ o

X

k2 ½M7 L ½M7 L0 Im½t00 LL0 

ð1Þ

LL0

in which ½M 7 L is the transition matrix element, k the photoelectron wavenumber and t00 LL0 a scattering path operator which refers to a single or multiple scattering process starting and ending at the absorbing atom labelled 0. In short, the transition matrix elements describe the interaction of the polarized radiation field with matter, whereas the scattering path operator describes how the excited photoelectron could remain confined in a localized final state due to resonant multiple scattering processes by the surrounding potentials. Typically, the whole information concerning chemical bonding and crystal field symmetry is stored in the scattering path operator which is responsible for shape resonances or white lines at low k values and extended X-ray absorption fine structures (EXAFS) at large k values. Magnetic circular dichroism, i.e. Ds ¼ ½s  sþ a0, will arise as the conjunction of three mechanisms: (i) At spin–orbit split L2;3 -edges, the excited photoelectrons are spin-polarized (Fano effect) because the angular momentum of the absorbed photon is converted into spin and orbital moments due to the strong spin–orbit interaction in the core state. Whereas the total angular momentum of the photoelectron is always directed along kcp , its sign and amplitude depend on both the helicity of the photon and on ‘c 7s. (ii) Spin polarized photoelectrons excited by photons of opposite helicity do not reach identical final states as a consequence of electric dipole selection rules: final states will have either s- or d- symmetry because ‘ ¼ ‘c 71 ¼ 0; 2, but the absorption of left- (right-) circularly polarized X-ray photons requires DmJ ¼ 1 ðDmJ ¼ þ1Þ. (iii) As soon as magnetic exchange splitting is turned on, the spindependent final states with d-like symmetry split under the additional constraints of crystal field and/or spin–orbit interactions. Any imbalance in spin or orbital momentum in the final states will be sensed by the spin-polarized photoelectron. Over the past decade, interest in XMCD at L-edges was boosted by the hope that magneto-optical sum-rules could be exploited to determine the spin and orbital magnetic moments in the ground state [48–50]. More details on this subject could be found in recent reviews (e.g. [51,52]). Unfortunately, this approach proved itself to be of limited interest in our problem because the ground-state spin and orbital moments are actually very small at the yttrium sites. In order to promote a more detailed interpretation of the large XMCD signatures seen in Fig. 3, we prefer to exploit a differential formulation of the X-ray magneto-optical sum-rules as discussed by Strange and others [53–55]. In this approach, we may express the X-ray absorption cross-sections Ds in terms of density of states (DOS) operators with a clear physical meaning [54,55]:   CD d 2 7 ð2Þ /Lz Sd þ /Sz Sd þ /Tz Sd DsL3 ¼ d 3Nb dDE 3 3

DsL2 ¼

  CdD d 4 14 /Lz Sd  /Sz Sd  /Tz Sd 6 dDE 3 3

ð3Þ

In these equations, Nb stands for the statistical branching ratio which may slightly deviate from its statistical limit ðNb ¼ 2Þ whenever

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/‘z Sd ¼

d /Lz Sd dDE

/sz Sd ¼

d /Sz Sd dDE

not be always justified [59] and may be even more questionable for heavier elements such as yttrium, lanthanum or lutetium because relativistic effects will increase pZ 4 . Nevertheless, there is no example where the spin–orbit splitting in 4d and 5d final states would exceed the crystal field splitting as this is commonly the case with 4f states. For rare earths cations, the ligand field is usually not cubic and should rather mimic the symmetry of icosahedral groups (K) that have fivefold symmetry but share many proper subgroups with the cubic groups [61,60]. In short, the quadrupole moment of the charge distribution can hardly be very large whereas the magnitude of /Tz S should depend in the first place on the crystal space group that is approximativelycubic (Ia3d). Finally, let us recall that the magnetocrystalline anisotropy free energy of YIG is known to be fairly small along the [111] direction: a large contribution of /Tz S would then be most surprising.

/tz Sd ¼

d /Tz Sd dDE

3.2. Yttrium, lanthanum and lutetium L-edges

there is a strong spin–orbit coupling in the final states. Even though CdD could be expressed in terms of the absorption cross-section per hole in the d-band within the electric dipole approximation (D), we shall simply take it here as a ‘‘constant’’ scaling factor for a given element in a given crystal field. Finally, /Lz Sd and /Sz Sd are the expectation values of the orbital or spin momentum integrated over all excited states of d-symmetry. Defining DE ¼ ERX þ E0  EF as the energy of the photoelectron referred to the Fermi level, we are interested in this paper in the expectation values of a whole family of magnetically polarized, density of states (DOS) operators for d-like symmetry:

so that Eqs. (2) and (3) can be rewritten

DsL3 ¼

DsL2 ¼

CdD



3Nb CdD 6



/‘z Sd þ

/‘z Sd 

2 7 /sz Sd þ /tz Sd 3 3

4 14 /sz Sd  /tz Sd 3 3

 ð4Þ  ð5Þ

One may then envisage to display directly angular momentum resolved density of states (DOS), i.e. /‘z Sd and /sz Sd using a linear combination of the X-ray absorption cross-sections measured at the L2;3 -edges: CdD /‘z Sd ¼ f2Nb DsL3 þ 2DsL2 g

ð6Þ

CdD f2/sz Sd þ 7/tz Sd g ¼ f3Nb DsL3  6DsL2 g

ð7Þ

Indeed, Eqs. (6) and (7) are most easily exploited when /tz Sd can be neglected and when Nb C2. Neither should one forget that Eqs. (4) and (5) are valid only within rather restrictive approximations:

 it is assumed that the probability of transitions toward final   

states with s-like symmetry ð‘ ¼ 0Þ is much lower than for transitions toward final states with d-like symmetry ð‘ ¼ 2Þ; energy dependence of the matrix elements is assumed to be negligible over a limited range of DE; differences between d3=2 and d5=2 wavefunctions are ignored; the total angular momentum j of the core hole is assumed to be a good quantum number (no jj mixing).

Angular-dependent XMCD spectra are required whenever the contribution of /tz S or /Tz S cannot be neglected. Typically, /Tz S reflects (to the lowest order) the asphericity of the spin magnetization resulting from anisotropic charge distributions and spin–orbit interactions [56,57]. Though Tz does not contribute to the Zeeman free energy, it is known to contribute to the magnetocrystalline anisotropy free energy as pointed out by Bruno [58] and Van der Laan [57]. As long as the spin–orbit splitting remains small with respect to the crystal field splitting, TC  ð27Þ Q  uS in which uS denotes the unit vector along the direction of the spin moment, whereas Q ¼ L2  ð13ÞL2 is the quadrupole moment of the charge distribution which is a traceless spherical tensor of rank 2. In cubic crystals, it is tempting to consider that Q should vanish, whereas /Tz S is most often neglected, at least at the L-edges of 3d transition metals. This may

Fig. 4 illustrate how Eqs. (6) and (7) can be used to display the /‘z S4d and /sz S4d DOS using the XMCD spectra recorded at the Y L-edges in the case of the iron garnet films 1 (Fig. 4a) and 2 (Fig. 4b). All spectra reproduced in Fig. 4 are plotted as a function of DE and were artificially normalized for Cd ¼ 1 assuming Nb ¼ 2. Any asphericity of the spin magnetization was neglected ð/tz Sd ¼ 0Þ. To avoid unwanted spectral distortions, the raw data shown in Fig. 3 were carefully corrected for the polarization rate and for fluorescence reabsorption while background effects (e.g. due to multiple fluorescence matrix excitation, elastic scattering, etc.) were minimized. For the sake of comparison, we systematically added into Fig. 4 polarization-averaged X-ray absorption spectra (XAS): these spectra calculated under strictly the same approximations should yield the energy profile of the charge DOS over the whole 4d band but convoluted by the average natural widths of the L core levels. It clearly appears that there is a very sharp resonance (‘‘whiteline’’) associated with the excitation of the photoelectron toward the (nearly) empty 4d states of yttrium (Z þ 1 spectral term: 2 D). Given that the natural width of the yttrium L2;3 core levels is rather narrow (Go1:7 eV), it became possible to resolve a shoulder in the XAS spectra which is assigned to the crystal-field splitting of the 4d band. Dodecahedral coordination to oxygen results into a ligand field that had only D2 point group symmetry: under such conditions, the degeneracy of the d orbitals should be lifted out and the 4d final states could be identified with irreducible representations of types a, b1 , b2 , b3 respectively. In reality, the corresponding final states tend to split into two well resolved bands: both are characterized with derivative-like dichroism but of opposite sign. What may explain a different sign for the circular dichroism measured in these two bands is either spin–orbit or superexchange interactions as discussed in Section 5. Since the two dichroism signatures have nearly the same amplitude but opposite signs, one may easily anticipate that the integral spin moments should more or less cancel out. This explains why the induced spin moment is so small in the d-band of yttrium when it is estimated from standard magneto-optical sum-rules ð/Sz Sd r0:01mB Þ. It also immediately appears from Figs. 4a and b that the measured XMCD signatures should arise essentially from the spin polarization of the d-DOS, because 2/sz S largely exceeds the contribution /‘z S. Note that the contribution of /‘z S is fairly small at energies DEi , except (perhaps) for i ¼ 3. This has important implications regarding the interpretation of X-ray detected magnetic resonance experiments at the yttrium L-edges performed on the same samples and discussed elsewhere [62].

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12 10 Normalized d-DOS Spectra

YIG Film Y L-Edges

XAS

8

3951

{YLaLu} IG Film Y L-Edges

XAS

XAS XAS z

6

z

2s sz

2s sz

4 3 2

2

2

3

0 1

5

1

5

-2

4 -4

-2

4

0 2 4 6 8 10 12 14 -4 Relative Energy: ∆E (eV)

-2

0 2 4 6 8 10 12 14 Relative Energy: ∆E (eV)

Fig. 4. Magnetically polarized /‘z S and 2/sz S 4d-DOS of yttrium. X-ray absorption spectra (XAS) averaged over the spin–orbit split core levels are also plotted as a function of the photoelectron energy DE: (a) YIG film 1(a); (b) (Y,La,Lu) IG film 2.

{YLaLu}IG Film 3.5 La L-edges

XAS

3.0

Normalized d-DOS Spectra

2.5

Lu L-edges

XAS

XAS

2.0

Ga K-edge

z

2ssz

1.5

2,3

2 1.0 3 0.5

x10

x5

0.0 4

-0.5

1

1

4

-1.0 -10

0

10 20 ∆E (eV)

30

40

-10

0

10 20 ∆E (eV)

30

40

Fig. 5. Magnetically polarized /‘z S and 2/sz S 5d-DOS of lanthanum (a) or lutetium (b) as deduced from the relevant L-edge XMCD spectra of film 2.

The XMCD spectra recorded at the L2;3 -edges of lanthanum and lutetium (film 2) were analyzed in the same way. The results are shown in Fig. 5a regarding the 5d-DOS of lanthanum, and in Fig. 5b for the 5d-DOS of lutetium. In the case of lanthanum which has a filled 4d inner shell but a completely empty 4f shell, the spectra exhibit a marked analogy with those of yttrium although the core hole lifetime broadening is much more severe than for yttrium: no shoulder can be resolved anymore in the white line but the two derivative-like signatures (with opposite signs) are still resolved in the 2/sz Sd spectrum. The results are significantly different for lutetium in which the 4f shell is expected to be now

completely filled. Further complications should be taken into account:

(i) The natural width of the L2;3 levels in lutetium is three times broader than in yttrium and starts to smear out the crystal field splitting in the XAS spectrum. (ii) Since Lu3þ cations can substitute either for Y3þ on 24c sites, or for Fe3+ on the 16a antisites, marked differences in the apparent crystal field splitting or in superexchange interactions would not be surprising.

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(iii) The penetration depth of X-ray photons of energy 9.24 keV (Lu L3 -edge) or 10.35 keV (Lu L2 -edge) is much larger than for X-ray photons of 5.48 keV (La L3 -edge) with the practical consequence that the experimental XMCD spectra were strongly contaminated with Bragg glitches from the GGG substrate while the Ga K-edge (10.37 keV) is distinctly excited and is distorting the XAS spectra above the lutetium L2 white line.

3.3. Bismuth M-edges The bismuth M4;5 -edges are indeed split by spin–orbit in the core levels but X-ray should excite transitions toward final states with either p or f symmetry. The corresponding XANES spectra are displayed in Fig. 6. If we assume that the 4f and 5d shells are entirely filled in the Bi3þ cations, then the well resolved pre-edge resonance involving localized final states that should have a strong contribution of the bismuth 6p orbitals whereas the 5f orbitals should contribute to the XANES spectrum beyond the edge. Actually, one may guess that, in the pre-edge resonance, the final states should merely involve linear combinations of bismuth AOs such as f5dxy ; 6pz g which would belong to an irreducible representation b1 in a ligand field of point group symmetry D2, f5dxz , 6py g for representation b2 or f5dyz ; 6px g for representation b3 . Unfortunately, we failed to detect any signal to be unambiguously assigned to XMCD: this is not really surprising because Fig. 2 clearly shows that the circular polarization rate is fairly low at the bismuth M4;5 -edges, while the intensity of the pre-edge resonance is poor compared to the edge jump or to the white lines measured at the L-edges of yttrium. Beyond the edge, there might be a very weak dichroism signature (marked with arrows in Fig. 6) and which seems to change its sign on inverting the magnetic field: the corresponding excitation energy would be more consistent with a multielectron excitation involving bismuth 5d electrons (O4;5 shell photoionization).

Whereas the integral spin moment of d-DOS was found to nearly cancel out at the yttrium sites, this is not anymore the case at the lutetium sites: as a consequence, the magneto-optical sum-rules would let us predict that there should be a spin moment contribution at the lutetium sites in the ground-state d-band. This conclusion may be consistent with the observation of a broad 175 Lu and 176Lu NMR signal in LuIG whereas no zero-field NMR resonance of the 89Y nucleus could ever be detected in YIG. At this stage, let us emphasize that it would be misleading to try to relate the differences that are observable between Figs. 5a and b to strong quadrupolar transitions toward final states involving the empty 4f states of lanthanum. Such an interpretation seems to be clearly ruled out by the marked similarities that are clearly apparent in Figs. 4b and 5a, i.e. if one compares the spectral shapes deduced from the measurements carried out at the L2;3 -edges of yttrium and lanthanum respectively. Indeed, quadrupolar transitions toward 4f states are not expected to contribute to the strong white lines of the yttrium L-edge spectra. It does not mean, however, that there is no contribution of electric quadrupole transitions in the XAS and XMCD spectra recorded at the L-edges of lanthanum: it is our interpretation that the empty 4f-band of lanthanum in sample 2 should be fairly broad and strongly overlapping with the empty 5d band so that the corresponding contribution of the quadrupolar transitions should be very weak and would cause only minor distortions to the white line, the intensity of which being still largely dominated by electric dipole transitions.

4. X-ray dichroisms from ‘c ¼ 0 core levels 4.1. Iron K-edge dichroism spectra of a YIG (111) single crystal We have reproduced in Fig. 7a the iron K-edge XMCD spectrum of the YIG (111) single crystal 4, together with the XANES spectrum obtained on averaging consecutive scans recorded with

Film 3: {Y1.7Bi1.3} Fe4.9Ga0.1O12 1.2

1.0 XANES

Normalized XANES / XMCD

XANES 0.8

0.6

Bi M5 - edge

Bi M4 - edge

0.4 ? ? 0.2

XMCD*50

XMCD*50

0.0

-0.2 2.58

2.60

2.62 2.64 Energy (keV)

2.66

2.68

2.70

2.72 2.74 Energy (keV)

2.76

Fig. 6. XANES and tentative XMCD spectra of film 3 recorded at the bismuth M4;5 -edge: (a) Bi M5 -edge; (b) Bi M4 -edge. Hypothetical multielectron excitations are marked with an arrow.

ARTICLE IN PRESS

XANES

1.0 0.8 0.6

0.4 0.3 0.2

0.2

XLD vs B0

0.0305

0.1 0.0

YIG Single Crystal Fe K-Edge

Asymmetry factor 0.0310

XANES

(V) & (H) Polarizations

0.0300

2.0x10-3

0.4

XLD B0 = 0.0T

1.0x10-3

0.0295

B0 = 0.5T

0.0290

0.0 XMCD x 20

-1.0x10-3

E1

XMLD_2 XMLD_1H XMLD_1V

0.0285 -0.2

Horizontal Magnetic Field

-0.1

71 10 71 20 71 30 71 40 71 50 71 60 71 70

-2.0x10-3

0.0 B0 (T)

0.1

0.2

08 71 10 71 12 71 14 71 16 71 18 71 20 71 22

E2

0.0 -0.2

YIG Single Crystal Fe K-edge

3953

71

Normalized Fluorescence Intensity

1.2

Normalized XANES

A. Rogalev et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3945–3962

Photon Energy (eV)

Photon Energy (eV)

Fig. 7. (a) XANES and XMCD spectra of the YIG (111) single crystal 4 at the Fe K-edge; (b) pre-edge resonance scanned successively with either horizontal (H) or vertical (V) polarization components; (c) zoomed X-ray linear dichroism spectra DsXLD =sejp recorded with or without a transverse magnetic field B0 : evidence for XMLD; (d) magnetic field dependence of the asymmetry factor defined as DsXLD =s and measured at a fixed X-ray photon energy as indicated with the arrow in (c).

left and right circularly polarized incident X-rays. With an absolute intensity of DsXMCD =sejp C4:4  103 normalized to the edge jump ðsejp Þ, the XMCD signal measured at the excitation energy E1 is typically one order of magnitude weaker than the yttrium XMCD signals measured at energies fEi0 ; Ei00 g in Fig. 3. This may look somewhat paradoxical since we found that the groundstate magnetic moments located at the yttrium sites were very small, whereas the whole ferrimagnetic order in YIG stems from exchange splitting in the valence band of iron. As discussed in Section 4.3, there is no contradiction between these different experimental observations which yield a more complete description of induced magnetism in these samples. On the other hand, XANES spectra were also recorded with linearly polarized X-ray photons. A diamond quarter waveplate (QWP) allowed us to set the polarization vector either vertical or horizontal. While the two XANES spectra displayed in Fig. 7b look nearly superimposed, a difference analysis revealed the existence of a small, but fairly reproducible X-ray linear dichroism (XLD) displayed in Fig. 7c. The maximum amplitude of this XLD signal normalized to the edge jump was found to be of the order of 3 DsVH . We shall discuss in the next section what XLD =sejp C1:5  10 may be the origin of such a weak natural linear dichroism measured in the absence of any magnetic bias field. When we turned on a horizontal magnetic bias field ðB0 ¼ 0:5 TÞ oriented perpendicular to the X-ray wavevector, then the use of the QWP allowed us to detect a much weaker X-ray magnetic linear dichroism ðXMLD2 Þ signal, i.e. a difference between the X-ray absorption cross-sections measured with the electric polarization vector set successively along the vertical or along the horizontal directions. At this stage, one should introduce an important distinction between two different types of XMLD signatures [63]: (i) for a given linear polarization, one may first look at the change of the X-ray absorption cross-section measured with or without the external magnetic field: XMLD1H;1V ¼ sðHÞ;ðV Þ ðB0 Þ  sðHÞ;ðV Þ ðB0 ¼ 0Þ This is often referred to as a magnetic linear dichroism of first type. One may alternatively look (ii) for a difference in the linear

dichroism measured with or without external magnetic field when the direction of polarization is rotated by 90 3 : XMLD2 ¼ DsðVHÞ ðB0 Þ  DsðVHÞ ðB0 ¼ 0Þ This is obviously a second order effect. The respective contributions of XMLD1H , XMLD1V and XMLD2 are plotted in Fig. 7c. To the best of our knowledge, this is the first example where XMLD signatures of first and second types could be analyzed at the K-edge of a transition metal. Indeed, the normalized amplitude of the XMLD2 signal is rather weak: Ds=sejp C2:5  104 of the edge jump. As illustrated with Fig. 7d, the asymmetry factor DsXLD =s does not change its sign when the magnetic field is inverted: this is the expected behavior for a time-reversal even effect. 4.2. Natural X-ray linear dichroism at the Fe K-edge The observation of a natural X-ray linear dichroism in the absence of any magnetic bias field is surely a puzzling result if we stick out for a cubic crystal structure with space group Ia3d. In this context, it is noteworthy that such a natural linear dichroism was systematically detected in all iron garnet films, even though the signal was definitely weaker with film 1. It is well known that, within the electric dipole approximation, the X-ray absorption cross-sections are isotropic for all cubic space groups. One may question whether electric quadrupole transitions could cause such a weak linear dichroism. Analytical expressions of the angular dependence of X-ray absorption cross-sections sðEÞ were established by Brouder [64] who included explicitly the contributions of electric quadrupole transitions. For cubic space groups, he pointed out that a small corrective term DsQ due to electric quadrupole transitions should be added: 1

DsQ ðe; kÞ ¼  pffiffiffiffiffiffif4  35 sin2 y cos2 y cos2 c 14

5 sin2 y½cos2 y cos2 ccos4f þ sin2 cð1  cos4fÞ 2 cosysinccoscsin4fgsQ ð4; 0Þ

ð8Þ

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In his derivation, the polarization vector e and the wavevector k were defined with respect to the crystallographic axes as 0 1 sinycosf C e¼B @ sinysinf A cosy 0

cosycosfcosc  sinfsinc

1

B C k ¼ @ cosysinfcosc þ cosfsinc A sinycosc

4.3. Magnetic dichroisms at the Fe K-edge

In cubic crystals, there is the major simplification that DsQ ðe; kÞ depends on only one single spherical tensor component: sQ ð4; 0Þa0 (‘ ¼ 4, m ¼ 0). It is, however, a trivial exercise to check that no linear dichroism is predicted by Eq. (8) when kJ½0 0 1 and y ¼ p=2. Similarly, there is again no linear dichroism if kJ½1 1 1 because DsQ remains invariant for two orthogonal directions of the polarization vector, e.g. for eJ½1 1 2 or eJ½1 1 0. As ¨ pointed out by Drager et al. [65], electric quadrupole transitions will cause a substantial linear dichroism when kJ½1 1 0. In the present paper, we are not concerned with this peculiar geometry which was exploited to detect natural X-ray linear dichroism in a cubic crystal of pyrite [66]. Thus, if we stick out for a cubic structure of YIG, the natural linear dichroism signal of Fig. 7c cannot be caused by electric quadrupole transitions. It is our interpretation that such a dichroism can be observed because structural distortions tend to lower the symmetry of the crystal. For subgroups R3 or R3m [21,26] a linear dichroism should already be expected within the electric dipole transition approximation as soon as the X-ray wavevector is not parallel to the trigonal axis since

DsD ðe; kÞ ¼ p1ffiffi2ð1  3cos2 yÞsD ð2; 0Þ

is expected to be weak if kJ½1 1 1cub because it can be due only to electric quadrupole transitions that have a low probability but cannot be ignored at the iron K-edge [66]. Further work presently under development at the IMPMC in Paris also tends to confirm that the natural linear dichroism signal which we measured on the YIG crystal could not be assigned to any temperature effect on the pre-peak of the XANES spectrum.

ð9Þ

Indeed, Eq. (9) refers to new orthonormal axes, the z axis being taken along the trigonal symmetry axis (C3 ). Typically, Eq. (9) would let us expect a linear dichroism if the wavevector k is directed along the former cubic axis ½0 0 1cub as this was precisely the case of our experiments carried out on the garnet films 2 and 3. In contrast, there cannot be any linear dichroism if kJ½1 1 1cub because the wavevector should be directed along the new z axis: y ¼ 903 in Eq. (7) and DsD ðe; k) is invariant in any rotation of the polarization vector in the perpendicular plane (0 0 1). Thus, we face again a problem with the linear dichroism of Fig. 7c because we checked very carefully that the wavevector could not depart by more than 70:583 from the direction of the trigonal axis. We want to draw attention below onto the fact that a weak linear dichroism can still be observed if we include the anisotropy of the electric quadrupole transitions. For space groups R3 and or R3m, a much more complicate angular dependence of DsQ was predicted by Brouder [64]. In practice, one may considerably simplify the discussion in observing that DsQ pcos2a, in which a would refer to the azimuthal angle of the electric polarization vector in a plane perpendicular to the [111] direction of the crystal, i.e. the direction of the X-ray wavevector. The origin of a is fixed by the fact that a ¼ p=2 when the electric polarization vector is perpendicular to the c axis of the crystal. Unfortunately, we could not determine quantitatively a from our dichroism experiments because the true orientation of the crystal with respect to its [111] direction was not known given that no vacuum-compatible goniometer was available on beamline ID12 for that preliminary experiment. In the geometry of the experiments carried out on film 1 or on the YIG crystal 4, natural linear dichroism is forbidden by symmetry for space group Ia3d, but it may become allowed due to small crystal distortions compatible with rhombohedral space groups R3 or R3m. Nevertheless, the measured natural dichroism

The XMCD spectra recorded at the Fe K-edge on the iron garnet films 1 and 2 do not exhibit any major difference with the XMCD spectrum of the YIG crystal 4. In all cases, the most intense XMCD signal is associated with localized final states contributing to a well resolved pre-edge resonance. It has long been recognized [68,69] that the energy of the latter pre-edge resonance (ca. 7115 eV for YIG) was typical of Fe3þ cations with ðZ þ 1Þ spectral term 5 D. Recall that EPR studies revealed strong structural distortions of both tetrahedral/octahedral coordination sites [67]. As illustrated with Fig. 8, the point group symmetry of the ligand field would be S4 rather than Td in the tetrahedral coordination sites (24d) and S6 rather than Oh in the iron antisites (16a). This should affect the type of atomic orbitals involved in the orbitals of symmetry e and eg respectively. Typically, in tetrahedral coordination sites with S4 point group, the orbitals of symmetry e may be built from either (4px ,4py ) or (3dxz ,3dyz ) atomic orbitals (AOs) of iron, but no combination of 4p and 3d orbitals of iron would be allowed in point group S6 . Since one may expect the spin–orbit interaction to be more efficient for Fe 3d AOs than for the Fe 4p AOs, it is tempting to conclude that, within the electric dipole approximation, the (24d) iron sites featuring tetrahedral coordination should have a dominant contribution to XMCD. However, XLD measurements reminded us that electric quadrupole transitions cannot be ignored at the iron K-edge. Note that a crude estimation of the relative intensities of electric dipole/electric quadrupole transitions was obtained from ligand field multiplet calculations performed by Arrio et al. [68] assuming undistorted tetrahedral and octahedral sites. These calculations confirmed that the pre-edge resonance should largely be dominated by electric dipole/electric quadrupole transitions toward 5 E states in the tetrahedral sites in which 3d–4p admixture is possible, whereas the electric quadrupole transitions toward 5 T2g or 5 Eg states is much weaker in the octahedral sites. When the excited core level has no spin–orbit coupling, i.e. when ‘c ¼ 0, the angular momentum of the absorbed photon is indeed entirely transferred as an orbital momentum to the photoelectron which is not spin polarized anymore. This implies that this photoelectron will only probe the orbital polarization of the final states as first pointed out by Igarashi and Hirai [70]. Accordingly, only orbital DOS operators should be involved in the XMCD differential cross-section [53–55]: 1 Q Cd f/‘z Sd ½DsXMCD ‘c ¼0 ¼ 3CpD /‘z Sp cosyz þ 10

þ13ð5 cos2 yz  3Þ/Ozzz Sd gcosyz

ð10Þ

Here again, Cp should be seen as a constant scaling factor for electric dipole transitions toward final states with p-like symmetry, whereas /‘z Sp refers to the orbital component of the magnetized DOS of p-like symmetry at the absorbing site. The first term of Eq. (10) was originally derived by Igarashi and Hirai [70]. As pointed out by Guo [54], the magnetization of the p-projected orbital DOS will vanish either if there is no spin–orbit

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Octahedral Feantisites (16 a)

Tetrahedral Fe sites (24 d) [100]

3955

[111] [001]

[111]

[100]

[010]

[010]

Y Fe O Fe O

Fig. 8. Ligand field at the Fe (24d) Wyckoff sites and (16a) antisites of the space group Ia3d. Structural distortions would include a significant elongation of the tetrahedrons along the [0 0 1] direction as well as some elongation of the octahedrons along the [111] direction [67].

½DsXMLD ‘c ¼0 ¼

1 D 1 d /Qzz Sp C /qzz Sp ¼ CpD 2 p 2 dDE

ð11Þ

where Qzz is one diagonal element of the rank-2 spherical tensor: Q ¼ L2  1=3L2 which we already introduced as the quadrupole moment of the charge distribution. Keeping in mind that the quadrupole moment of the charge distribution is vanishingly small in a cubic crystal in the absence of any external bias field, it is not really surprising that the XMLD signal measured with a bias field (B0 ? k) should be much weaker than the natural linear dichroism caused by structural distortions. 4.4. XMCD of bismuth substituted films 4.4.1. XMCD at the bismuth M1 -edge As an important step toward the clarification of the magnetooptical properties of bismuth substituted iron garnet films, we succeeded in measuring a weak but quite significant XMCD signal at the bismuth M1 -edge in the iron garnet film 3. As illustrated with Fig. 9, this broad XMCD signal (multiplied by a factor 50) changes its sign when the direction of the magnetic bias field B0 is inverted. To the best of our knowledge, this is the first magnetic circular dichroism ever measured at a bismuth M1 -edge and we like to insist that this experiment, which required long data acquisition times, represented for us a real tour-de-force from the

1.75 Film 3: {Y1.7Bi1.3} Fe4.9Ga0.1O12 1.50

1.25 Normalized Fluorescence Intensities

coupling in the final state, or if there is no spin polarization in the empty states. The second term in Eq. (10) refers to the weaker contribution of electric quadrupole transitions as formulated by Carra et al. [50]. In this formulation, cosyz denotes the direction cosinus of the X-ray wavevector with respect to a unit vector z directed along the magnetization vector. It is not really surprising that the main correction due to quadrupolar transition is simply /‘z Sd , i.e. the orbital component of the magnetized DOS with d-like symmetry at the absorbing site. The additional term /Ozzz Sd is a higher order (octupolar) orbital component which is expected to be negligible for cubic ligand fields. Similarly, the DOS operator associated with X-ray magnetic linear dichroism can only be of a pure orbital nature. Defining DsXMLD ¼ ½s?  sJ , one would obtain [50]

XANES 1.00

0.75 B0 = +0.5 T 0.50

B0 = −0.5 T

0.25

XMCD x 50

0.00

XMCD x 50

-0.25 3.98

3.99

4.00 4.01 4.02 X-ray Photon Energy (keV)

4.03

Fig. 9. Normalized XANES and XMCD spectra of the fY; Big IG thin film 3 recorded at the Bi M1 -edge.

technical point of view. Recall that this challenging experiment cumulated a number of serious handicaps which we already evoked in Section 2: a poor circular polarization rate, a very low

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fluorescence efficiency, a strong background resulting in a very high statistical noise level and, last but not least, a very broad linewidth mainly due to the very short life time of the core hole. With a FWHM natural width GM1 C22 eV for the 3s core level, the XANES spectrum can only exhibit a fairly broad white line. If we admit that the dodecahedral ligand field has still D2 point group symmetry, we may assign this broad white line to unresolved final states belonging to the irreducible representations b1 2b3 involving linear combinations of the bismuth 6p AOs with oxygen 2p orbitals. In reality, additional complication certainly arises from the fact that the ground-state occupation number of the 6p AOs could already be as high a 1:8e due to their hybridization with the atomic orbitals of the ligands. On the other hand, we expect the valence band orbitals belonging to the irreducible representations b1 2b3 to involve hybridized bismuth AOs such as f6dxy ; 6pz g; f6dxz ; 6py g and f6dyz ; 6px g respectively. One may even raise the question to know whether the 5d AOs—which are most often regarded as a closed inner shell—could possibly contribute to a weak hybridization with the valence orbitals because the ionization energy of the (5d)O4;5 shell is rather low (ca. 24–28 eV). Since the M1 -edge refers to the photoionization of the 3s core level with ‘c ¼ 0, the measured XMCD signal could be described again with Eq. (10), with the simplification that the contribution of electric quadrupole transitions should be fairly small at such low photon energies. Our experiment thus produces direct evidence of non-zero orbital DOS /‘z Sp at the bismuth sites. Interestingly, the XMCD signal displayed in Fig. 9 does not exhibit a derivative-like lineshape as observed at the yttrium L-edges or even at the iron K-edge: this implies that the expectation value of the ground-state orbital moment /Lz Sp has no chance to cancel out in the (integral) magneto-optical sum-rule. Unfortunately, there is no simple option to evaluate the scaling factor CpD, the determination of which is a pre-requisite to access to an absolute value of /Lz Sp . What makes such a determination particularly delicate here is the lack of any independent information regarding the true number of holes in the fd; pg valence band. Nevertheless, our experiment leaves absolutely no doubt about our assertion that the bismuth atoms should carry a significant orbital moment.

4.4.2. Changes induced in the Fe K-edge XANES and XMCD spectra on substituting yttrium with bismuth Substituting bismuth for yttrium in the 24c sites does not leave the XANES spectrum recorded at the iron K-edge unchanged. One may question how far the scattering path operator t00 LL0 in Eq. (1) may be sensitive to the substitution of yttrium by a heavier scatterer such as bismuth in the cluster of atomic potentials centered around the absorbing iron atom. In Fig. 10a, we compare the Fe K-edge XANES spectra of the garnet films 3 and 1. The difference XANES spectrum is also reproduced on the same plot in order to make small changes more visible. We have identified with letters ðAÞ; ðBÞ; . . . ; ðEÞ energies at which marked differences were detected. Clearly the largest changes occur at the edge (B) and for shape resonances (C)–(E) beyond the edge. However, on zooming the pre-edge range, it clearly appears in Fig. 10d that quite sensible changes should also affect the localized final states since the onset of the pre-edge resonance is shifted down to lower energy by ca. 0:16 eV, whereas the shift of the maximum of the pre-peak is only ca. 0:11 eV. Even more important are the spectral distortions that show up slightly above the pre-edge resonance at energies marked with label (A) (see Fig. 10d). The unprecedented quality of the data collected at beamline ID12 allowed us to look as well for small changes in the Fe K-edge XMCD spectra. Actually, some changes are perfectly visible in Fig. 10b in which we compare the XMCD spectra of the garnet films 3 and 1. Whereas the largest differences in Fig. 10a were observed beyond the edge, e.g. at energies ðBÞ; . . . ; ðEÞ, it immediately appears from Fig. 10c that the most important differences in the XMCD spectra tend to concentrate in the pre-edge range with structures labelled (1,2,3): in particular, the largest peak (1) is to be correlated with a shift of the XMCD signal of film 3 by ca. 0:088 eV down to lower energy, while weak additional signatures (2) and (3) appear in the energy range marked (A). Indeed, subtle changes can also be detected beyond the edge: for instance, peak (4) should combine the joint effects of a shift and a distortion of the XMCD signal that shows up near the edge at energy (B). It is worth adding here that the difference XMCD signatures shown in Fig. 10c were perfectly reproduced with other fBi; YgIG=YIG samples originating from diverse sources (including a BiIG film prepared by PLD). Indeed, in this exercise, one has to XMCD

XANES

(E)

1.0

(B)

0.8

Fe K-Edge

Pre-EDGE (A) (B) 4 1

2

(C)

3

(D)

(E)

{Bi,Y} IG YIG

Difference {Bi,Y} IG YIG Difference

0.4 0.2

(A)

0.0 7.11 7.12 7.13 7.14 7.15 7.16 7.17 Energy (keV)

0.0010

4

0.0005

(D) 123

0.0000 -0.0005

(C)

(E)

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.004 0.003 0.002 0.001 0.000 -0.001 -0.002 -0.003 -0.004 -0.005

(A)

XAS

XMCD 1

2

{Bi, Y} IG YIG Difference

71 10 71 12 71 14 71 16 71 18 71 20 71 22

0.6

Difference XMCD

Normalized XANES

1.2

0.003 0.002 0.001 0.000 -0.001 -0.002 -0.003 -0.004

Normalized XANES

1.4

Normalized XMCD

(D) (C)

Normalized XMCD

1.6

7.11 7.12 7.13 7.14 7.15 7.16 7.17 Energy (keV)

Energy (eV)

Fig. 10. (a) Difference XANES spectrum ({Y,Bi}IG–YIG) revealing small changes in the Fe K-edge XANES on partly substituting bismuth for yttrium; (b) comparison of the Fe K-edge XMCD spectra of the bismuth substituted garnet film 3 and of the YIG film 1; (c) difference XMCD spectrum ({Y,Bi}IG–YIG); (d) zoomed spectra in the pre-edge energy range to make more visible the subtle changes in the XANES/XMCD spectra of film 3.

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make sure that the XMCD spectra were not contaminated by unwanted residual diffraction glitches. This was checked carefully on collecting successively series of XMCD spectra with small polar tilt angles of the wavevector k with respect to a unit vector perpendicular to the film ðDyk r71:53 Þ. No direct interpretation of difference XANES spectra can yet be given without performing ab-inito multiple scattering calculations in the continuum, e.g. using the FDMNES code of Joly [71]. A perennial difficulty is that the difference XANES spectra are very sensitive to small atomic displacements perturbing the long multiple scattering paths: this point was already raised many years ago by Brouder and Goulon [72]. We would nevertheless suspect that the sequences of nearly aligned atoms Fei ; . . . ; YðBiÞ; . . . ; Fej illustrated with Fig. 8a for tetrahedral iron sites, should contribute to the largest differences in the multiple scattering process due to shadowing effects that are well known in EXAFS. The same remarks probably hold true if we want to analyze the different XMCD signatures in the energy range (B),y, (E). Let us emphasize, however, that the bismuth atoms are involved here as scattering potentials and not anymore as the central absorbing atom as this was the case for the experiment carried out at the M1 -edge. Moreover, the scattered photoelectron is not spin-polarized since it was created by the photoionization of the Fe (1s) core level: this implies that such a photoelectron will essentially probe the orbital polarization of scattering potentials. In other terms, the observation of additional features in the difference-XMCD spectrum 10c would support the claim made in the previous subsection that the bismuth atoms should carry orbital moments /Lz S. Conversely, we would expect the yttrium atoms to have only very little contribution to the iron K-edge XMCD spectrum since we pointed out in Section 3 that /Lz S was negligibly small at the yttrium sites. From a general point of view, the information content of difference-XMCD spectra appears much more detailed than that of XMCD spectra recorded at the bismuth M1 - or L1-edges because the natural width of the iron 1s level is considerably lower, while the quality of the experimental data is much higher. In that respect, a result which deserves more attention is the experimental evidence of additional XMCD signatures (2,3) at energies (A) slightly above the pre-edge resonance. It is our interpretation that hybridized f5d; 6pg orbitals of symmetry b1, b2 or b3 at the bismuth sites may have a significant direct overlap with the iron orbitals of symmetry e in tetrahedral point group symmetry S4 (Fig. 8a). In other terms, the bismuth 6p orbitals seem to directly affect the upper part of the valence band of iron. Of course, band structure calculations are most appropriate to check the consistency of this interpretation.

5. First principle calculations 5.1. Relativistic LMTO-LSDA calculations Fully relativistic spin-polarized band structure calculations were carried out with the linear muffin tin orbital method in the atomic-sphere approximation, including the so-called combined corrections ðASA þ CCÞ. We used the PY-LMTO-LSDA code (relativistic version 5.00.4) developed by Perlov and Yaresko according to principles detailed elsewhere [73–76]. Following MacDonald ^ D can be decomposed as and Vosko [77], the Dirac Hamitonian H ^0 þH ^ m with ^D ¼H H 2

^ 0 ¼ c ½a  = þ c ½b þ VH ðrÞ þ Vxc ðrÞ H 2 i ^ m ¼ m ½b½s  BðrÞ H B

and

Here ½a ¼



0 ½s ½s 0



and ½b ¼



½I 0 0 ½I

3957



are Dirac matrices in which ½s

is the vector built from the three Pauli matrices and ½I is the 2  2 unitary matrix. The exchange-correlation potential Vxc ðrÞ and the effective magnetic field are defined as Vxc ðrÞ ¼

BðrÞ ¼

@Exc @r

@Exc @m

in which rðrÞ and mðrÞ are the electron charge density and the spin-magnetization density. On the other hand Z Exc ðr; mÞ ¼ ½exc ðr; mÞrðrÞd3 r where exc is the exchange-correlation energy per electron in a uniform electron gas with local density rðrÞ. In a one-electron description, one may write for spin-up sðþÞ and spin-down sðÞ : @exc vxc ðsðþÞ Þ ¼ exc ðrsðþÞ ; rsðÞ Þ þ r @rsðþÞ The PY-LMTO code offers a broad choice of options for the parametrization of exc as suggested by von Barth and Hedin [78]; Gunnarson and Lundqvist [79]; Morruzzi–Williams–Janak [80]; Vilk–Nusair–Vosko [81]; Perdew and Zunger [82]; Perdew and Wang [83]. For our calculations, we selected the parametrization of Perdew and Wang. There have been several attempts to extend the standard LMTO method [84] within a rigorous, fully relativistic frame [85,86]. In the PY-LMTO code, one has to build up first the LMTO basis for the ^ 0 and one includes the spin spin-independent Hamiltonian H ^ m at a variational step: this polarization matrix elements of H approach is much simpler to implement and time-effective while the results are as accurate as with fully relativistic codes. In short, ^0 Bloch wavefunctions are built from the solutions Fk;m ðe; rÞ of H and from the energy derivative of Fk;m [76]. Indeed, the radial part of Fk;m involves both the (so-called) large and small components, while the angular part is determined by bi-spinors. Recall that the   relativistic quantum numbers k; m can take the following values: ( ð‘ þ 1Þ for j ¼ ‘ þ 1=2 k¼ and  jrmrj ‘ for j ¼ ‘  1=2

MacDonald and Vosko explicitly prefaced their theory as being ^ m does not include any orbital contribution to the spin-only since H Zeeman energy [77,76]. As a remedy, it was proposed empirically by Brooks and Johansson to add a so-called orbital polarization ^ 0 which was borrowed from atomic physics and term Vop ðrÞ to H was given the same formulation as a spin–orbit term [87]. On the other hand, it has long been recognized that something was also missing in a tight binding model of strongly correlated d and f electrons: Coulomb repulsion is not properly accounted for. Typically, since the energy required to vary the occupation number of the metal sites is underestimated, it is not surprising that the band gap in insulators is most often largely underestimated by standard LSDA calculations. One possible way to cure this deficiency in theory is the socalled LSDA þ U method in which a Coulomb repulsion term EU is explicitly added to the LSDA total energy functional ELSDA [88,89]: ELSDAþU ¼ ELSDA þ EU  Edc

ð12Þ

whereas the so-called double counting term Edc should approximately cancel out the Coulomb interaction already taken into account within the LSDA method. In a rotationally invariant

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LSDA þ U calculation, the Coulomb term is written as EU ¼

lenge [91,92]. The difficulty of the calculations would rapidly increase with the rhombohedral space group R3 in which one had to deal with six non-equivalent iron sites, two non-equivalent yttrium sites and eight non-equivalent oxygens [26]. This convinced us to stick to the standard bcc cubic structure for YIG. Moreover, the LDA+U method was restricted to only the 3d states of the iron atoms using: U ¼ 3:5 eV and J ¼ 0:8 eV, exactly as in Ref. [92]. Nevertheless, quite significant progresses could be made in experimenting a refined space partition in which empty spheres were artificially added in order to give a more realistic description of the interstitial region. In the present calculation, the YIG unit cell was approximated with up to 168 muffin-tin spheres, out of which only 80 did correspond to real atoms: 8 for iron atoms in (16a) Wyckoff sites; 12 for iron atoms in (24d) sites; 12 for yttrium atoms in (24c) sites; 48 for oxygen atoms in (96h) sites. In addition, we defined 24 empty spheres of type ES1 in Wyckoff sites (48g), 48 of type ES2 in sites (96h) and 16 of type ES3 in sites (32e). Such a refined space partition was indeed more demanding in terms of computing resources. A primary outcome of our relativistic LMTO-LSDA calculation is to be illustrated with Fig. 11 which reproduces the spin polarized d-DOS at the iron and yttrium sites as well as the p-projected DOS of oxygen. Although a crude LDA þ U correction was included in our calculation, we have to admit that the direct gap which we found between the occupied/unoccupied states, i.e. DEC1:12 eV, still remains underestimated when compared either to the experimental value (DEC2:85 eV) or to the value calculated by Ching et al. (DEC2:66 eV [92]). This is simply a consequence of the fact that U was kept frozen in order to avoid a highly time-consuming self-consistent optimization. This should not affect the reliability of our conclusions as discussed below. Regarding more specifically XMCD experiments carried out at the Fe K-edge, it immediately appears from Figs. 11a and b that the predicted spin polarization of the empty states is very high in the

1X s ½n ð/m1 ; m3 jV ee jm2 ; m4 S 2 s;fmg m1 ;m2

/m1 ; m3 jV ee jm4 ; m2 SÞnsm3 ;m4 s þnsm1 ;m2 /m1 ; m3 jV ee jm2 ; m4 Sn m3 ;m4 

ð13Þ

s where V ee is the screened Coulomb interaction and n mi ;mj are the elements of the occupation matrix of localized states with spin s at sites i and j. Let us emphasize that the orbital polarization correction is automatically included in Eq. (13) [89]. In principle, the screened Hubbard’s repulsion parameter U, which enters the expressions for the matrix elements of V ee , could be estimated from constrained LSDA calculations. In practice, however, U is most often treated as an adjustable parameter, as well as the Stoner exchange parameter J [75].

5.2. LMTO-LSDA calculations in YIG 5.2.1. Spin-polarized DOS in YIG It was attractive to check how far first-principle calculations would support our interpretation of the XANES and XMCD spectra reported in Sections 3 and 4. In other terms, we were primarily interested in a detailed description of the lowest, localized states lying slightly above the Fermi level. In this respect, relativistic LMTO-LSDA calculations look like a reasonable choice to start with even though one may always worry about various limitations. One of them may concern the difficulty to take into account the perturbation induced by the core hole even though this may not be necessarily a too serious problem as discussed by Brouder et al. [90]. The biggest difficulty is elsewhere: even if one sticks to ˚ the complexthe simplest cubic space group ðIa3d; a ¼ 12:376 AÞ, ity of the YIG crystal structure with 80 atoms in the primitive cell already makes accurate ab-initio calculations a formidable chal-

6 d-DOS (1/eV/atom)

4 Fe (24d)

2 0 -2 d_down d_up

-6

Y (24c)

2 0 -2 d_down d_up

-4 -6

-8

1.0 10

p-DOS (1/eV/atom)

d-DOS (1/eV/atom)

-4

4

d_down d_up

5 0

Fe (16a) -5

O (96h)

0.5 0.0 -0.5 -1.0 p_down p_up

-1.5 -2.0

-6

-4

-2

0 2 4 Energy (eV)

6

8

10

-6

-4

-2

0 2 4 Energy (eV)

6

8

10

Fig. 11. Spin-polarized DOS in the conduction band of YIG. Plots (a) and (b) reproduce the calculated d-projected DOS at the tetrahedral (24d) or octahedral (16a) iron sites: the strong ferrimagnetic spin polarization of the unoccupied d-band is correctly reproduced. As illustrated with plot (c), there is a weaker spin polarization of the yttrium dDOS in sites 24c. As pointed out by two arrows, the oxygen p-states are quite significantly polarized by the two Fe cations.

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conduction band for both the tetrahedral (24d) and octahedral (16a) iron sites. Indeed, the sign of the polarization is quite consistent with the expected ferrimagnetic order. Let us also notice that the spin polarization of the empty states (DE40) is systematically inverted with respect to the spin polarization of the filled states (DEo0): this is, indeed, fully consistent with the magneto-optical sum-rule. As illustrated with Fig. 11c, there is a weaker spin polarization of the yttrium d-DOS which is fully consistent with the existence of characteristic XMCD signatures measured in the white lines of the L2;3 -edge of yttrium. On the other hand, Fig. 11d also clearly confirms that the ferrimagnetic order of the empty bands of iron in the tetrahedral (24d) and octahedral (16a) sites is mediated by a quite significant polarization of the oxygen p-states in the same energy range as a typical illustration of superexchange. A weaker polarization of the oxygen p-states is also associated with the spin-polarization of the yttrium d-DOS. Interestingly, our calculated d-DOS look some more structured in the empty conduction band when compared to the earlier calculations reported by Ching et al. [92]. This prompted us to look at the projections of the same DOS onto the irreducible representations of the relevant local point groups, i.e. S4 for the tetrahedral iron sites (24d) or S6 for the octahedral iron sites (16a). This is precisely illustrated with Figs. 12a and b. Quite in the same way, Fig. 12c reproduces the projection of the yttrium d-DOS on the irreducible representations of group D2 . Typically, Fig. 12b makes it clear that, in the empty conduction band, the expected splitting between the t2g and eg representations of group Oh is well reproduced by our calculation; moreover, the lower symmetry of point group S6 results in a further splitting of the t2g band into well identified representations eg0 and ag . As illustrated with Fig. 12a, a slightly different scenario takes place at the S4 iron sites: this is now the doubly degenerated e-band of group Td that is clearly split into non-degenerated subbands associated with the representations b and a0 , whereas the triply degenerated band t2 of group Td is again split into lowenergy states associated with the fully symmetrical representa-

3959

tion a, plus high energy states associated with the doubly degenerated representation e of group S4 . Note that our results are fully consistent with the distortions of the iron ligand fields anticipated long time ago by Geschwind [67] on extrapolating for YIG the analysis of EPR spectra collected from a single crystal of yttrium–gallium–garnet doped with trace amounts of iron. On the other hand, it clearly appears from Fig. 12c that the calculated d-DOS of yttrium are dominated by the well resolved contributions associated with the irreducible representations b1 –b3 of point group D2 . There are nevertheless two weaker sub-bands, labelled for clarity a11 and a12 , and which both belong to the fully symmetrical representation a of group D2 . For what concerns XAS or XMCD spectra recorded at the iron K-edge, electric dipole selection rules would allow transitions toward final states with symmetry b or e in tetrahedral sites (S4 ); by contrast, in octahedral sites (S6 ), transitions toward final states belonging to the representations ag or eg are allowed only by electric quadrupole selection rules. It is, however, well documented that electric quadrupole transitions are usually much weaker than electric dipole transition: thus, XAS and XMCD spectra recorded at the K-edge are—to some extent—siteselective since the tetrahedral sites are favored by electric dipole transitions. Even though the sub-bands associated with the irreducible representations b or e of group S4 are well resolved in our calculation, there is, unfortunately, no hope left to resolve experimentally the transitions toward the corresponding final states due to the core hole broadening effect which is much sensible at the Fe K-edge. At the yttrium L2;3 -edges, the initial core state should belong to an irreducible representation bj (j ¼ 2; 3) with the predictable consequence that electric dipole transitions should be allowed toward final states belonging to representations a or biaj. Regarding XAS and XMCD spectra recorded at the L2;3 -edges of yttrium, Fig. 12c would let us expect some splitting of the experimental spectra given that the sub-bands associated with the irreducible representation b1 –b3 of group D2 are quite narrow and are separated by several eV. In this respect, one should keep in

4

2 0 Fe S4

Y D2

2

-4 -6

e a

}t

b a'

}

2

0

e DOS (1/eV)

DOS (1/eV/atom)

-2

8 Fe S6 6 4 2

eg

}e

ag e'g

}

a11 a12

-2

b1 b2

g

-4

b3

t2g

0 -6 -2 -6

-4

-2 0 Energy (eV)

2

4

0

2

4 6 Energy (eV)

8

10

Fig. 12. Spin-polarized d-DOS projected on the irreducible representations of the local point group symmetry. Plots (a) and (b) reproduce the calculated DOS projected on the irreducible representations of point groups S4 and S6 that are relevant for the two iron cations subject to a distorted tetrahedral and octahedral ligand field respectively. Similarly, plot (c) reproduces the calculated DOS projected on the irreducible representations of point group D2 that is relevant for the yttrium cations.

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YIG : Simulated Spectra at Y L-edges 3 XAS

Absorption Cross-Sections (a.u.)

7

0.3

2

0.2



6 5

0.1

L3-Edge

4

0.0 1

3 -0.1 2

Induced Magnetic Moments

8

-0.2

1 L2-Edge

4

-0.3

0 0

2 4 6 8 10 Relative Energy ∆E (eV)

12

0

2 4 6 8 10 Relative Energy ∆E (eV)

12

Fig. 13. Spin-averaged and spin-polarized X-ray absorption spectra at the yttrium L-edges. (a) Transition probabilities from the yttrium 2p core levels toward unoccupied dstates as a function of the photoelectron transferred energy; as expected the intensities of the L2;3 -edge white lines are fully consistent with the statistical limit of the branching ratio. (b) Simulated spectra of the effective operators /‘z Sd and /sz Sd ; clearly, the contribution of /sz Sd largely exceeds that of /‘z Sd .

mind that we produced clear experimental evidence in Section 3 that the XAS white line at the yttrium L-edges had a characteristic shoulder whereas the corresponding XMCD spectra exhibited two well resolved signatures (see Fig. 4a). 5.2.2. XAS and spectral magnetization densities At this stage, it was attractive to check how far the calculated d-projected DOS at the yttrium sites could let us reproduce the splitting of the white lines which we observed experimentally at the Y L-edges. This led us to simulate XANES and XMCD spectra using our LMTO-LSDA calculations. In this respect, these simulations proved to be very successful since a splitting is clearly apparent in the spectra reproduced in Fig. 13a in which we plotted the transition probabilities from the 2p core levels toward unoccupied yttrium states of allowed symmetry as a function of the photoelectron transferred energy. Note that the simulated Ledge XAS spectra are fully consistent with the statistical limit of the so-called branching ratio: ½sþ þ s LII 1 C ½sþ þ s LIII 2

ð14Þ

Even more important was the capability of such relativistic LMTO-LSDA calculation to produce simulated spectra for the effective operators of XMCD, i.e. /‘z Sd and /sz Sd . This is precisely illustrated with Fig. 13b. Recall that all spectra displayed in Fig. 13 were artificially convoluted with a freely adjustable broadening function in order to take into account the finite core hole lifetime that tends to smear out the calculated spectral structures. This cannot alter the main result which is that, clearly, the contribution of /sz Sd should largely exceed that of /‘z Sd at the Y L-edges: this conclusion is fully supported by the experimental spectra previously shown in Fig. 4a. However, within the present stateof-the-art of LMTO-LSDA calculations, we could not yet reproduce perfectly the relative amplitudes of the signatures labelled 1–4 in Fig. 4a.

Table 4 Magnetic moments localized at the yttrium and oxygen sites. Yttrium (orbital)

Y (s)

Y (p)

Y (d)

Y (f)

2/Sz S /Lz S 2/Sz S þ /Lz S /Tz S

þ0:00419 þ0:00000 þ0:00419 þ0:00000

0:00072 þ0:00002 0:00070 0.00000

þ0:01228 þ0:00006 þ0:01234 þ0:00000

þ0:01115 0.00011 þ0:01104 þ0:00000

Oxygen (orbital)

O (s)

O (p)

2/Sz S /Lz S 2/Sz S þ /Lz S /Tz S

þ0:00058 þ0:00000 þ0:00058 þ0:00000

0.05980 0.00038 0.06018 0.00002

Magnetic moments in Bohr magnetron mB .

Table 5 Magnetic moments localized at the iron sites with either S6 or S4 coordination symmetry. Fe ðS6 Þ

Fe (s)

Fe (p)

Fe (d)

2/Sz S /Lz S 2/Sz S þ /Lz S /Tz S

0:00753 þ0:00000 0:00753 þ0:00000

0:00410 þ0:00005 þ0:00405 þ0:00000

4:02529 0:01756 4.04285 0.00010

Fe (S4 )

Fe (s)

Fe (p)

Fe (d)

2/Sz S /Lz S 2/Sz S þ /Lz S /Tz S

þ0:01465 þ0:00000 þ0:01465 þ0:00000

þ0:05661 0:00030 þ0:05631 þ0:00000

þ3:90478 þ0:01302 þ3:91780 þ0:00008

Magnetic moments in Bohr magnetron mB .

5.2.3. Integral magnetic moments The PY-LMTO-LSDA code proved to be also extremely useful in calculating the element-selective projections of the magnetic moments /Lz S, 2/Sz S and /Tz S integrated over all unoccupied

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states of s, p or d types at a given site. The corresponding moments are regrouped in Table 4 for the yttrium and oxygen atoms that are in rather low symmetry environments, and in Table 5 for the two non-equivalent iron sites, i.e. the octahedral sites (16a) or the tetrahedral sites (24d). It immediately appears from Table 4 that the induced magnetic moment localized on the yttrium atoms, i.e. 2/Sz S þ /Lz S, is very weak for any DOS projection whereas the total magnetic moment still remains very small ðC0:03mB Þ. There is indeed a net spin polarization of the oxygen ligands but it is very weak ðC0:06mB Þ and it cannot be measured anyhow at the oxygen K-edge. It also appears from Table 5 that the two Fe sublattices get magnetized with their spin ordered antiparallel one to each other and with an unbalanced magnetization in favor of the 12 tetrahedral sites coupled to only eight octahedral sites. The corresponding spin moment is indeed maximum for the d-projected DOS and is of the order of 4mB : this is slightly less than the ca. 5mB which one would commonly expect. Table 5 also provides clear evidence that the integral orbital moment /Lz S is largely quenched since it is only of the order of 0.5% of the spin magnetization, with again a very small unbalanced orbital magnetization in favor of the tetrahedral sites. As often anticipated for a cubic crystal, we did not find any significant contribution of /Tz S for the d-projected DOS of iron nor yttrium: this is extremely encouraging regarding the practical use we made of the magneto-optical sum-rules.

6. Conclusion High quality X-ray dichroism spectra have been recorded on a series of yttrium iron garnet thin films grown by LPE on GGG substrates and in which the yttrium was replaced in variable proportions by lanthanum, lutetium or bismuth. Strong XMCD signals were observed at the L2;3 -edges of yttrium, lanthanum and lutetium. Spin and orbital angular momentum resolved DOS could be derived from the corresponding XMCD spectra and proved to be always largely dominated by the spin DOS component 2/sz Sd . As confirmed by LMTO-LSDA calculations detailed in Section 5, the corresponding 4d band of yttrium or the 5d band of lanthanum is split by the crystal field. For yttrium and lanthanum, integration over the whole d-band of either 2/sz Sd or /‘z Sd yields vanishingly small estimations of the ground-state expectation values of the spin and orbital moments, i.e. 2/Sz Sd and /Lz Sd . At least for the YIG film, the LMTO-LSDA calculations fully supported the assumption we made in our analysis that /tz S and /Tz S could be neglected. Note that the absence of any significant magnetic moments at the yttrium or lanthanum sites may well explain why no zero field NMR signal had ever been measured with either 89Y or 139La nuclei. Natural X-ray linear dichroism (XLD) as well as X-ray magnetic linear dichroism (XMLD) has been detected in the YIG film at the iron K-edge. While the observation of natural X-ray linear dichroism is incompatible with the cubic space group Ia3d, we pointed out that a very weak natural linear dichroism may become allowed if small structural distortions lower the crystal symmetry which may be compatible either with subgroup R3 as suggested by Rodic et al. [26] or R3m. On the other hand, the very weak intensity of the measured XMLD signal is consistent with prediction that the quadrupole moment of the charge distribution should be quite small in a crystal that had nearly cubic symmetry. Further experiments are in progress to investigate the X-ray nonreciprocal dichroism properties of these films to be expected for magneto-electric systems. It has long been conjectured, e.g. on the basis of NMR, that magnetic moments might exist at the bismuth sites in BiIG or

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{Bi,Y,Lu}IG films. The XMCD signal which we measured at the bismuth M1 -edge not only confirmed that the bismuth atoms carried a magnetic moment but it specified that this moment should be of orbital nature. Additional information which may support as well this conclusion was collected on exploiting for the first time the difference XMCD spectrum obtained on combining high quality XMCD spectra recorded at the iron K-edge with the YIG film 1 and the {Bi,Y,Lu}IG film 3. The latter difference spectrum seems to confirm the hybridization of the bismuth 6p orbitals in the Fe 3d conduction band as well as in the Fe 3d valence band. Element selective XMCD experiments thus appear as a powerful new tool to elucidate the magneto-optical properties of these materials of outstanding technological importance.

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