Spin and Orbital Magnetic Moments of Fe
3O
4D. J. Huang,1,2C. F. Chang,1H.-T. Jeng,3G. Y. Guo,4,1H.-J. Lin,1W. B. Wu,2,1H. C. Ku,5A. Fujimori,6 Y. Takahashi,7and C. T. Chen1
1National Synchrotron Radiation Research Center, Hsinchu 30077, Taiwan 2Department of Electrophysics, National Chiao-Tung University, Hsinchu 300, Taiwan
3Physics Division, National Center for Theoretical Sciences, Hinchu 300, Taiwan 4Department of Physics, National Taiwan University, Taipei 106, Taiwan 5Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan
6Department of Complexity Science and Engineering and Department of Physics, University of Tokyo, Tokyo 113-0033, Japan 7Graduate School and Faculty of Science, Himeji Institute of Technology, Hyogo 678-1297, Japan
(Received 9 January 2004; published 11 August 2004)
We present measurements of the spin and orbital magnetic moments of Fe3O4 by using SQUID and magnetic circular dichroism in soft x-ray absorption. The measurements show that Fe3O4 has a noninteger spin moment, in contrast to its predicted half-metallic feature. Fe3O4also exhibits a large unquenched orbital moment. Calculations using the local density approximation including the Hubbard Umethod and the configuration interaction cluster-model suggest that strong correlations and spin-orbit interaction of the 3d electrons result in the noninteger spin and large orbital moments of Fe3O4.
DOI: 10.1103/PhysRevLett.93.077204 PACS numbers: 75.50.Ss, 71.28.+d, 75.25.+z, 78.70.Dm
Magnetite (Fe3O4) exhibits many interesting properties such as charge ordering, mixed valence, and metal-insulator transition known as the Verwey transition [1], in which the conductivity decreases by 2 orders of mag-nitude upon cooling through the transition temper-ature TV 120 K. In spite of intensive studies on its electronic structure, surprisingly, no consensus has been reached concerning the electronic nature of Fe3O4. Experimental studies, including neutron diffuse scatter-ing [2], NMR [3], and x-ray scatterscatter-ing [4,5], indicate that Fe3O4should be considered as an itinerant magnet rather than a fluctuating mixed-valence material. According to band theory, Fe3O4is a half-metal above TV; its minority-spin electrons are conducting, whereas the majority-minority-spin ones are insulating [6]. In addition, Fe3O4would have an integral spin moment per formula unit (f.u.), i.e., 4:0B;
the orbital moment of metallic Fe3O4would be quenched. On the other hand, charge ordering of the octahedral (B-site) Fe in Fe3O4 has been suggested by the refine-ments of x-ray and neutron diffraction data [7], implying that the 3d electrons of Fe3O4 have a strong localized character. Fe2 in Fe
3O4 is thus expected to exhibit a large unquenched orbital moment, like Fe2in FeO [8]. Theoretical and experimental works show that localiza-tion of the 3d electrons of transilocaliza-tion-metal compounds leads to giant orbital moments. For example, FeO [8], CoO [9], Fe impurities in alkali metals [10], and Fe nitridometalates [11] are shown to have giant or un-quenched orbital moments. In addition, calculations based on atomic multiplet theory show that the localized nature of the open 3d shell of Fe3O4 sets a limit of 66:7% on the spin polarization of conduction electrons [12], rather than 100% predicted by band theory. Results of spin-resolved photoemission from epitaxial
thin films and single crystals of Fe3O4 support the con-clusion of multiplet calculations [13–16], in contrast to the conclusion from spin-resolved photoemission of Fe3O4111 thin films grown on W(110) [17].
Measurements of orbital and spin moments therefore provide an opportunity to explore the character of 3d electrons in Fe3O4 [18,19]. Examining whether Fe3O4 has a quenched orbital moment and an integral spin mo-ment is important in revealing its electronic nature.
In this Letter, we present studies of the spin and orbital moments of Fe3O4single crystals by combining magnetic circular dichroism (MCD) in soft x-ray absorption spec-troscopy (XAS) and measurements using a superconduct-ing quantum interference device (SQUID) magnetometer. In addition, we performed cluster-model calculations in the configuration interaction (CI) approach and band-structure calculations in the local spin density approxi-mation including the on-site Coulomb interaction U (LDA U) [20,21] to unravel the underlying physics of the magnetic moments of Fe3O4.
Single crystals of Fe3O4 were grown by the floating zone method and fully characterized by x-ray diffraction. Temperature-dependent measurements of the resistance of the crystal show an abrupt change at 120 K, as plotted in Fig. 3(a), revealing the Verwey transition of Fe3O4. We measured the total magnetic moment of a 21.59-mg Fe3O4 single crystal with an applied field of 1 T along the [111] direction using a SQUID magnetometer.
We measured MCD in XAS on Fe3O4 at various tem-peratures under an applied magnetic field of 1 T along the [111] direction using the Dragon beam line at the National Synchrotron Radiation Research Center in Taiwan. XAS spectra of Fe3O4 were taken in the total electron yield (TEY) mode with a photon-energy resolution of 0.2 eV VOLUME93, NUMBER7 P H Y S I C A L R E V I E W L E T T E R S 13 AUGUST 2004week ending
and an incident angle of 60. The degree of circular polarization of the incident light was 80%. The crystal was freshly cleaved in an ultrahigh vacuum at 90 K; the fracture plane of the sample is normal to the [110] direc-tion. We take our MCD measurements to be representative of bulk Fe3O4, because the probing depth of the TEY method is around 50 A or deeper.
The sum rules of MCD in x-ray absorption permit an element-selective separation of the spin and orbital con-tributions to the total magnetic moment of materials [22 – 26]. The total orbital moment morbper formula of Fe3O4 can be expressed as [27] morb 4 3 R L2;3 d! R L2;3 d! Nh; (1)
in which and are the absorption cross sections taken with the projection of spin of incident photons parallel and antiparallel to those of the majority of 3d electrons, respectively. In addition, ! is the photon en-ergy; Nh is the total number of Fe 3d holes per formula
unit.
XAS recorded with the TEY method suffers typically from the saturation effects, leading to an inaccurate mea-sure of orbital moments [28]. The meamea-sured absorption
ITEY in a TEY measurement is reduced by a factor of
f 1=1 e=xcos, where xand e are the photon
penetration depth and the electron sampling depth, re-spectively, and is the incidence angle of x ray with respect to the surface normal [28]. ewas estimated to be
50 A for Fe3O4 [29]. By using quasitransmission mea-surements of XAS, we have determined the photon-energy-dependent x of Fe3O4 to correct our XAS and MCD measurements for the saturation effects [30]. x at the L3 and L2 edges are, respectively, 254 and 653 A.
Figure 1 displays Fe L2;3-edge XAS and MCD spectra of Fe3O4single crystals measured at 88 K using the TEY method. Our XAS and MCD spectra are similar to those of epitaxial Fe3O4 thin films [31,32]. The XAS back-ground shown in Fig. 1 is composed of an arctangentlike edge-jump function and a linear function. With Nh
13:5 [33] and taking the geometric effect in absorption and the degree of circular polarization of incident pho-tons into account, we obtained morbof Fe3O4 at various temperatures, as summarized in Table I. The uncertainty in determining morb originates mainly from the back-ground function of XAS. Our measurements unravel that Fe3O4 exhibits an unquenched orbital moment. For example, the measured orbital moment morb is 0:65 0:07 at T 145 K. In other words, the average orbital moment per B-site Fe is 0:33 0:04B, because the orbi-tal moment of A-site Fe3 is insignificant according to Hund’s rule and the local density approximation (LDA) and the local density approximation with Hubbard U (LDA U) calculations described later; such an un-quenched orbital moment is much larger than that of Fe metal, 0:09B [25].
To comprehend the underlying physics of an un-quenched orbital moment of Fe3O4, we performed band-structure calculations on its cubic phase using the all-electron full potential linear muffin-tin orbital method including the spin-orbit interaction [35] within the LDA and LDA U schemes. Both LDA [6] and LDA U calculations [36] conclude that cubic Fe3O4 is half-metallic and has a spin moment of 4:0B per f.u. as summarized in Table II. Also the orbital moment of A-site Fe ions is insignificant ( 0:02B), as expected from Hund’s coupling of a half-filled Fe3. LDA calculations give rise to a nearly quenched orbital moment of Fe3O4. On the other hand, an unquenched orbital moment of 0:21B per B-site Fe atom was obtained by the LDA U calculations [37], indicating that the Coulomb interac-tions of 3d electrons lead to the unquenched orbital mo-ment. To demonstrate such an effect, we calculated the occupation numbers and charge densities of the B-site 3d
Intensity (arb. units)
740 730
720 710
700
Photon energy (eV)
T = 88 K
( - ) 2
MCD integration × 1/5
FIG. 1. Fe L2;3-edge XAS and MCD spectra of Fe3O4 with correction for the saturation effect. Top: XAS spectra with spin of photons parallel and antiparallel to that of Fe 3d majority electrons. The XAS background is depicted in a thin dotted line. Middle: MCD spectrum, i.e., 2. Bottom: integration of MCD. Spectra of MCD and MCD integration are plotted with different vertical offsets for clarity.
TABLE I. Measured morb of Fe3O4 from MCD at various temperatures.
T (K) 88 100 145 200
morb 0:76 0:09 0:66 0:07 0:65 0:07 0:67 0:08
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down-spin states projected to orbitals with different mag-netic quantum numbers m. Without inclusion of the Coulomb interaction U, the projected occupation number of B-site 3d electrons distributes almost evenly onto orbitals of different magnetic quantum numbers; the or-bital moment is thus quenched. With inclusion of U, on the other hand, the occupation number of the m 1 state is drastically enhanced, and that of the m 1 state is suppressed, resulting in a strong orbital polarization and a large orbital moment of B-site Fe, as presented in Fig. 2.
We also calculated the magnetic moments of octahe-dral Fe using a configuration interaction (CI) cluster model including spin-orbit interaction [32,39]. The re-sults suggest that the octahedral Fe2in a FeO10
6 cluster exhibits a spin moment of 3:74Band an orbital moment of 0:97B, implying an average orbital moment of
0:48B per B-site Fe atom in Fe3O4 [32]. The measured average orbital moment of 0:33 0:04B per B-site Fe
thus indicates that the 3d electrons of Fe3O4 have a strongly correlated electronic nature even at temperatures above TV.
To quantitatively determine the total spin moment mspin per formula unit of Fe3O4, we measured also the total magnetic moment of Fe3O4 at the temperatures between 80 and 200 K by using a SQUID magnetometer, as plotted in Fig. 3(b). Combining the MCD and SQUID
measure-ments, we then obtained mspin at various temperatures, because both LDA and LDA U calculations conclude that the orbital moment of oxygen is negligible. Our measurements indicate that the spin and orbital moments of Fe3O4do not change significantly around the tempera-ture TV. In addition, Fe3O4 exhibits a noninteger spin moment. For example, the total magnetic moment of Fe3O4 at T 145 K is 4:33B; with the measured morb of 0:65 0:07, remarkably mspin per f.u. of Fe3O4 is 3:68 0:09B, as displayed in Table II, in contrast to
the integral spin moment of 4:0B as a result of half-metallic behavior predicted by band theory. With CI calculations, we found that the spin moment of octahedral Fe2 in a FeO10
6 cluster is suppressed by 5% if the strength of the spin-orbit interaction of 3d electrons is doubled, whereas the integral spin moment of Fe3O4 obtained from LDA U calculations is rather insensitive to the strength of spin-orbit coupling. This observation suggests that the observed noninteger spin moment is beyond the Bloch electron picture and might result from a combined effect of the spin-orbit interaction and strong correlations of the 3d electrons in Fe3O4.
With measurements of SQUID and MCD in soft x-ray absorption, we can also study the spin moment of oxygens in Fe3O4. The spin sum rule of MCD [23] correlates the total spin moment mFe
spinof Fe in Fe3O4to the MCD data as mFespin 7hTzi 6p4qI N
h, in which p and q are defined
as MCD integrationsRL
3 d! and
R
L3L2
d!, respectively, and I as the XAS integration. In addition, hTzi is the expectation value of magnetic dipole TABLE II. Calculated and measured (at T 145 K)
mag-netic moments of Fe3O4. Total spin (mspin) and orbital (morb) moments per f.u. of Fe3O4, and average orbital moment (mBorb) per B-site Fe atom are displayed in units of B.
mspin morb mBorb morb=mspin
LDA 4.0 0.06 0.04 0.015
LDA U 4.0 0.43 0.21 0.108
Expt. 3:68 0:09 0:65 0:07 0:33 0:04 0.18
FIG. 2. Charge densities of cubic Fe3O4versus radial distance in units of atomic radius a0. The charge densities projected to different orbitals with magnetic quantum number m were obtained from (a) LDA and (b) LDA U calculations.
0.1 1 10 100 Resistance ( 5.0 4.0 3.0 2.0 1.0 0.0 Magnetic moment ( B /f.u.) 200 180 160 140 120 100 80 60 Temperature (K) orbital moment spin moment total moment (a) (b)
FIG. 3. (a) Resistance of a Fe3O4single crystal in the vicinity of the Verwey transition. (b) Total magnetic moment, spin, and orbital moments of Fe3O4at various temperatures.
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operator. By using the spin sum rule with correction for its deviation resulting from the intermixing between the
L2 and L3 edges [40], we obtained a value of 3:55B for
mFe
spin 7hTzi at T 145 K. Our LDA U calculations
disclose that 7hTzi per B-site Fe atom is 0:155B, whereas
7hTzi of the A-site Fe atoms is 0:0001B. The
signifi-cant 7hTzi value of the B-site Fe atoms is caused by their
strong orbital polarization as shown in Fig. 2. The total spin moment mFe
spin of Fe in Fe3O4 is therefore 3:24B,
leading to a spin moment of 0:44B originating from oxygen atoms per Fe3O4, i.e., a spin moment of 0:11B per O atom in Fe3O4. This deduced spin moment of oxygen is close to the calculated spin moment of 0:07B per O atom from our LDA U calculations and consistent with LDA calculations [6,42].
In conclusion, we have measured the orbital and spin magnetic moments of Fe3O4 by combining SQUID and MCD. We found that Fe3O4has a noninteger spin moment, in contrast to its predicted half-metallic feature, and that the average orbital moment of B-site Fe in Fe3O4 is significantly larger than that of Fe metal. As evidenced by LDA U calculations, the on-site Coulomb interac-tions of 3d electrons result in the unquenched orbital moment and magnetic dipole moment of Fe3O4. Our results suggest that spin-orbit interaction and electron correlations of 3d electrons play an important role in the spin and orbital moments of Fe3O4. We call for further theoretical work on the magnetic moments of Fe3O4.
We acknowledge S. Kimura for providing Fe3O4 single crystals, A. Tanaka for sharing the computation code of the CI calculation, and C.-M. Huang and C.-H. Hsu for characterizing the orientation of the fracture plane of the Fe3O4sample. We thank L. H. Tjeng, T. Jo, C. H. Chen, and C. S. Hsue for valuable discussions. This work was sup-ported in part by the National Science Council of Taiwan and by a Grant-in-Aid for Scientific Research in Priority Area ‘‘Novel Quantum Phenomena in Transition-Metal Oxides’’ from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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