Magnetite (Fe$_{3}$O$_{4}$) is a mixed valent system where electronic
conductivity occurs on the B-site (octahedral) iron sublattice of the spinel
structure. Below $T_{V}=122$ K, a metal-insulator transition occurs which is
argued to arise from the charge ordering of 2+ and 3+ iron valences on the
B-sites (Verwey transition). Inelastic neutron scattering measurements show
that optical spin waves propagating on the B-site sublattice ($\sim$80 meV) are
shifted upwards in energy above $T_{V}$ due to the occurrence of B-B
ferromagnetic double exchange in the mixed valent metallic phase. The double
exchange interaction affects only spin waves of $\Delta_{5}$ symmetry, not all
modes, indicating that valence fluctuations are slow and the double exchange is
constrained by electron correlations above $T_{V}$.Comment: 4 pages, 5 figure

E. J. W. Verwey

J. M. Honig

M. Yethiraj

P. Metcalf

R. J. McQueeney

S. Chang

T. G. Perring

W. Montfrooij

English

arXiv

Magnetite (Fe3O4) is a mixed valent system where electronic conductivity occurs on the B site (octahedral) iron sublattice of the spinel structure. Below Tv=123 K, a metal-insulator transition occurs which is argued to arise from the charge ordering of 2+ and 3+ iron valences on the B sites (Verwey transition). Inelastic neutron scattering measurements show that optical spin waves propagating on the B site sublattice (similar to 80 meV) are shifted upwards in energy above Tv due to the occurrence of B-B ferromagnetic double exchange in the mixed valent phase. The double exchange interaction affects only spin waves of Δ5  symmetry, not all modes, indicating that valence fluctuations are slow and the double exchange is constrained by short-range electron correlations above Tv. © 2007, American Physical Societ

McQueeney, RJ

Yethiraj, M

Chang, S

Montfrooij, W

Perring, TG

Honig, JM

Metcalf, P

ANSTO Publications Online

Physical Review Letters

Zener double exchange from local valence fluctuations in magnetite.

Received 19 July 2007; published 10 December 2007; publisher error corrected 5 February 2008Magnetite (Fe3O4) is a mixed valent system where electronic conductivity occurs on the B site (octahedral) iron sublattice of the spinel structure. Below TV=123   K, a metal-insulator transition occurs which is argued to arise from the charge ordering of 2+ and 3+ iron valences on the B sites (Verwey transition). Inelastic neutron scattering measurements show that optical spin waves propagating on the B site sublattice (∼80  meV) are shifted upwards in energy above TV due to the occurrence of B-B ferromagnetic double exchange in the mixed valent phase. The double exchange interaction affects only spin waves of Δ5  symmetry, not all modes, indicating that valence fluctuations are slow and the double exchange is constrained by short-range electron correlations above TV.Work is supported by the U. S. Department of
Energy Office of Science under the following contracts: Ames Laboratory under Contract No. DE-AC02-07CH11358, Oak Ridge National Laboratory, which is managed by UT-Batelle LLC, under Contract No. DEAC00OR22725

McQueeney, R. J.

Yethiraj, M.

Chang, S.

Montfrooij, Wouter

Perring, T. G.

Honig, Jurgen M.

Metcalf, P.

University of Missouri: MOspace

Zener Double Exchange from Local Valence Fluctuations in MagnetiteR. J. McQueeney,1,2 M. Yethiraj,3,* S. Chang,2 W. Montfrooij,4 T. G. Perring,5,7 J. M. Honig,6 and P. Metcalf61Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011 USA2Ames Laboratory, Ames, Iowa 50011 USA3Center for Neutron Scattering, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 USA4Department of Physics and Missouri Research Reactor, University of Missouri, Columbia, Missouri 65211 USA5ISIS Facility, Rutherford Appleton Laboratory, Didcot, Oxfordshire, OX11 0QX, United Kingdom6Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 USA7Department of Physics and Astronomy, University College London, Gower Street, London WC1E6BT, United Kingdom(Received 19 July 2007; published 10 December 2007; publisher error corrected 5 February 2008)Magnetite (Fe3O4) is a mixed valent system where electronic conductivity occurs on the B site(octahedral) iron sublattice of the spinel structure. Below TV  123 K, a metal-insulator transition occurswhich is argued to arise from the charge ordering of 2 and 3 iron valences on the B sites (Verweytransition). Inelastic neutron scattering measurements show that optical spin waves propagating on the Bsite sublattice (80 meV) are shifted upwards in energy above TV due to the occurrence of B-Bferromagnetic double exchange in the mixed valent phase. The double exchange interaction affectsonly spin waves of 5 symmetry, not all modes, indicating that valence fluctuations are slow and thedouble exchange is constrained by short-range electron correlations above TV .DOI: 10.1103/PhysRevLett.99.246401 PACS numbers: 71.30.+h, 75.30.Ds, 75.30.Et, 78.70.NxMagnetite (Fe3O4) is the prototypical example of ametal-insulator transition with a charge-ordered (CO) in-sulating ground state (Verwey transition). Since its discov-ery nearly 70 years ago [1], the driving forces behind theVerwey transition are still not completely understood [2,3].Magnetite has a cubic inverse spinel crystal structure con-taining two different symmetry iron sites: the A site residesin tetrahedrally coordinated oxygen interstices and hasstable valence (3d5, Fe3); the two B sites have octahedralcoordination and a fractional average valence of 2:5.The ferrimagnetic structure consists of ferromagneticA- and B-sublattices aligned antiparallel to each other(TC  858 K). Below TV  123 K, magnetite undergoesa metal-insulator transition resulting in a decrease of theconductivity by two orders-of-magnitude. The model thathas persisted over time is that extra electrons forming Fe2ions (3d6) hop to neighboring Fe3 sites on the corner-shared B-sublattice tetrahedral network and give rise toelectrical conductivity. Anderson argued that short-rangeordering of Fe2 and Fe3 exists above TV due to signifi-cant intersite Coulomb repulsion and frustration on the Bsite sublattice [4]. The short-ranged electron correlationsmaintain local charge ‘‘neutrality’’ (2:5 average valenceon each tetrahedron), thereby restricting charge hoppingand conductivity [5]. In the classic picture of the Verweytranstion, Coulomb repulsions win out at low temperatures,resulting in long-range CO of Fe2 and Fe3 [6] in a pro-cess reminiscent of Wigner crystallization [7]. However,elastic and orbital interactions [8] induce monoclinic lat-tice distortions whose complexity has made characteriza-tion of the Verwey state difficult [9]. Even the validity ofthe CO model has been questioned recently [10], butneutron [11] and resonant x-ray scattering measurements[12] now appear to converge on fractional CO.In this Letter, we provide strong evidence forAnderson’s original picture of short-ranged electron corre-lations in the mixed valent (MV) phase. Valence fluctua-tions occurring on the B-sublattice modify the magneticexchange and affect spin waves propagating on Fe B sites.Inelastic neutron scattering measurements reveal that Bsite optical spin waves are shifted up in energy and broad-ened above TV due to ferromagnetic double exchange(DE). Ferromagnetic DE arises from real charge transferprocesses in MV materials, a good example being theferromagnetic metallic state in the manganites [13]. Forfast electron hopping in the band limit, cubic symmetrydemands that the average DE should uniformly affect allB-B pairs. However, our results show that only spin wavesof a particular symmetry are affected by DE, implying thepresence of electron hopping restricted by short-rangedcharge correlations. The observed correlations give indi-rect evidence that intersite Coulomb interactions play arole in the Verwey transition.Inelastic neutron scattering measurements were per-formed on the MAPS instrument at the ISIS facility atRutherford Appleton Laboratory on a single-crystal ofFe3O4 weighing 10 grams (TV  123 K). Details ofsample preparation and characterization are given else-where [16]. The sample was mounted with cubic [HK0]as the primary scattering plane, and data were collectedwith an incident neutron energy of 160 meV and theincident beam at an angle of 25 from the [110] axis.Time-of-flight neutron spectra were collected at 110 K(T < TV) and 130 K (T > TV) and scattered intensitieswere histogrammed into energy transfer (@!) and momen-tum transfer (@Q) bins. Data were subsequently ana-lyzed using MSLICE [17] and TOBYFIT [18] computerprograms.PRL 99, 246401 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 DECEMBER 20070031-9007=07=99(24)=246401(4) 246401-1 © 2007 The American Physical SocietyThe measured neutron scattering spectra from MAPS(and also the HB–3 spectrometer at the High FluxIsotope Reactor at Oak Ridge National Laboratory) wereused to determine the spin wave dispersion in magnetite. Inthe cubic spinel phase above TV , the primitive rhombohe-dral unit cell contains six iron atoms (2A and 4B), leadingto six spin wave branches. Figure 1 shows the spin wavedispersion along [100] as determined from Gaussian fits tospin wave modes [19]. The dispersion is well representedby a Heisenberg model where antiferromagnetic ABsuperexchange (SE), via oxygen, is dominant (JAB 4:8 meV) and is responsible for ferrimagnetism. Ferro-magnetic BB exchange arises from a combination of SE,DE, and direct exchange and is much weaker (JBB 0:69 meV). Weak antiferromagnetic nearest- and next-nearest-neighbor AA exchange is also present (J1AA 0:35 meV, J2AA  0:2 meV). The dispersion calculatedfrom a Heisenberg model with exchange parameters aboveand spins SA  2:5 and SB  2:25 is also shown in Fig. 1.Branch symmetries were identified from the model spinwave eigenvectors and are given the following labels anddescriptions: 1 (acoustic and steeply dispersing optic),02A (optic spin wave on A-sublattice), 02B (optic spinwave on B-sublattice), and 5 (doubly-degenerate opticspin waves on B-sublattice). Modes of 5 and 02Bsymmetry at 80 meV (in the hatched region of Fig. 1)propagate solely on the B-sublattice and were observed tobe very broad and weak, as discussed below.Figures 2(a) and 2(b) show images of scattered inten-sity in the [HK0] plane at 130 and 110 K, respectively.The arcs and rings in the images correspond to the inter-section of the neutron measurement surface and the dis-persion surfaces. Figure 2(c) shows a calculation of theMAPS data using the Heisenberg model described above.Calculations simulate the sampling of spin waves in (Q, !)space performed by MAPS. Features of the calculatedand measured spectra are in good agreement (as antici-pated from Fig. 1) with one notable exception. As indicatedin Fig. 2(a), the measured B site optical spin wave branchat 80 meV is poorly defined in the region between [220]and [420] at 130 K. Comparison with the 110 K spec-trum illustrates two features: (1) the B site modes sharpenup below TV in better agreement with the cubic model(despite the lowering of crystallographic symmetry), and(2) other spin wave branches are essentially unaffected,indicating that JAB and JAA are insensitive to the Verweytransition.To study the B site modes more closely, energy cuts weremade through the MAPS data at different values of K andFIG. 1. Spin wave dispersion of magnetite above TV: Black(gray) symbols are inelastic neutron scattering data from MAPS(HB-3). Lines are results from a Heisenberg model with parame-ters discussed in the main text. The hatched area contains verybroad B site spin waves.FIG. 2 (color). (a) Neutron scattering intensity in the [HK0] plane at 130 K. The color scale indicates neutron intensity (red thehighest). Thin lines show the Brillouin zone boundaries of the cubic phase. Curved, dashed lines show 0, 80, and 120 meV energytransfers (from left to right). The purple circled region indicates extremely broad and weak 5 symmetry optical spin waves at 80 meV.(b) The 110 K data set below TV . In the purple circled region, the 5 mode is more pronounced. (c) Calculation of the MAPS spin wavespectra using a Heisenberg model with JBB  0:44 meV (T  110 K). Data and calculations have been smoothed by a boxcaraveraging procedure. (d) Symmetry of B site optical spin waves in different Brillouin zones at 80 meV; 5 (blue), 02B (green), V(purple). Energy cuts shown in Fig. 3 are indicated by arrows.PRL 99, 246401 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 DECEMBER 2007246401-2different temperatures, as shown in Fig. 3. To improvestatistics, cuts are obtained by averaging neutron intensityover a range of reciprocal space. Figure 3 shows thefollowing cuts made in the range 0:25  L  0:25:[H20] (1:75  K  2:25), [H30] (2:75  K  3:25), and[H40] (3:75  K  4:25). These reciprocal space cutsallow for specific symmetry selection of the B site opti-cal spin waves, as shown in Fig. 2(d). In the constrained(Q, @!) measurement surface of MAPS, H is a function of@!, as can be ascertained from the comparison of Figs. 2and 3. The energy resolution full-width-at-half-maximum(FWHM) is calculated to be 3–4 meV for dispersionlessmodes in the energy range shown (using the TOBYFIT pro-gram). In all cuts, spin wave modes at 60 and 117 meV areindependent of temperature and resolution limited, while80 meV modes are broader than the resolution. Theenergy cut along the [H20] direction in Fig. 3(a) showsthat the 5 symmetry branch at 80 meV shifts up inenergy and becomes severely broadened above TV , asalso apparent in Fig. 2. Gaussian fits to the data indicatethat the mode energy shifts from 79.6(5) to 84(1) meV andthe FWHM broadens by 60% from 17 to 28 meVabove TV .The B site spin waves along [H30] in Fig. 3(b) (V sym-metry [20]) and [H40] in Fig. 3(c) (02B symmetry) alsohave broad line shapes with FWHM of 14 and 10 meV,respectively, but relatively little temperature dependence.We also note the weak peak near 70 meV in Fig. 3(c) islikely to be a phonon, based on examination of higher-Qdata.The anomalous B site spin waves can be explained byconsideration of ferromagnetic DE. The conduction elec-tron associated with the Fe2 ion is forced to remainoppositely aligned to the Fe3 3d5 core spin by the Pauliexclusion principle (effectively JHund ! 1. Since theB-sublattice is already ferromagnetically aligned due tostrong JAB, spin polarized conduction results in DE [21]and leads to an increase in JBB. In turn, the energy of B siteoptical spin waves increases. The enhancement of DEabove TV can be estimated from the energy of the disper-sionless 5 mode, 6JABSB  8JBBSB. Since JAB does notchange [16], the shift of 4.4 meV implies that JBB increasesfrom 0.44 to 0.69 meV. The energy cuts are compared tothe Heisenberg model structure factor calculation inFigs. 3(d)–3(f) with JBB  0:44 meV (T < TV) and0.69 meV (T > TV). Comparison to calculation showsthat (1) all B site spin wave peaks are much weaker andbroader than Heisenberg model predictions, and (2) chang-ing JBB in the model shifts the energy of all mode symme-tries, in disagreement with the observation that only the 5mode has a large temperature dependence. Structure fac-tors in the energy range 75  @!  90 meV and 115 @!  120 meV and L-range 0:25  L  0:25 areshown in Fig. 4 as a function of K. Figure 4(a) furtherillustrates the anomalous 5 mode, which shows a mini-mum in the measured structure factor whereas the calcu-lation predicts a maximum. Figure 4(b) indicates that A siteoptical spin waves have little temperature dependence andagree quite well with Heisenberg calculations. The dis-agreement between data and model structure factor calcu-lations for the B site modes reflects a failure of theHeisenberg model for only the broadest (most stronglycoupled) modes.It is clear that stiffening and extreme broadening of onlythe 5 modes above TV cannot be explained by uniformlyenhanced BB coupling as might be expected from a bandFIG. 4 (color online). (a) Neutron structure factor for B siteoptical spin waves averaged from 75–90 meV. Data are shown at130 K (dark circles, red online) and 110 K (light circles, blueonline). Heisenberg model calculations are shown with JBB 0:69 meV (dark line, red online) and 0:44 meV (light line, blueonline). Horizontal bars indicate dominant symmetry. (b) Struc-ture factor for A site optical spin waves from 115–120 meV. Lineis calculated from a Heisenberg model.FIG. 3 (color online). Inelastic neutron scattering intensity as afunction of energy transfer for cuts along (a) [H20], (b) [H30],and (c) [H40]. Upper (lower) symbols indicate measurementsdone at T  110 K (130 K). Lines are Gaussian fits to therespective data. The dashed line is the background estimatedfrom cuts along [H60]. The 110 K plots are offset vertically forclarity. Panels (d)–(f) show the identical cuts as (a)–(c) ascalculated from a Heisenberg model for magnetite with JBB 0:44 meV (light, blue online) and JBB  0:69 meV (dark, redonline). Spin wave modes are labeled by symmetry.PRL 99, 246401 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 DECEMBER 2007246401-3approach. The failure of a band model is reiterated byestimating the DE arising from metallic conductivity,JDE  nt=22pS2B [22,23] where n 	 1=2 is the concentra-tion of charge carriers in the conduction band and t theelectron hopping integral. The effective one-band hoppingintegral from local spin density approximation (LSDA)calculation is 130 meV [24], leading to an estimate ofJDE  5 meV that is significantly larger than the observedvalue (JDE 	 JBBT > TV  JBBT < TV  0:25 meV).This disagreement emphasizes the importance of chargecorrelation and/or polaronic effects above TV that lead toactivated conduction and a reduction of the effective hop-ping integral. This picture is supported by optical conduc-tivity measurements showing that the 100 meV elec-tronic energy gap below TV (which suppresses DE) isreplaced by a pseudogap above TV [25,26]. In this limit,the Zener DE picture of slow and local charge hoppingbetween distinct valence states above TV is appropriate,and the closing of the electronic gap allows the coupling ofspin waves to valence fluctuations leading to broadenedspin wave line shapes. We also conclude that no features inthe spin wave spectrum below TV can be definitivelyassociated with long-range CO, in agreement with theresults of our previous study [16].The predominant 5 spin wave symmetry implies thatvalence fluctuations have the same symmetry and occuralong the quasi-one-dimensional [110] B site chains, asillustrated in Fig. 5. The spin wave changes the alignmentof neighboring spins, thereby impeding valence fluctua-tions which favor parallel spins. The presence of slow 5valence fluctuations coupling to the lattice has also beenobserved with neutron diffuse scattering and occur wellabove TV [27]. In the MV phase, these local symmetry-breaking charge correlations are short-ranged. While weinterpret our data based on the symmetry assignments formodes along the [100] direction, broadening everywhere inthe Brillouin zone indicates that the coupled B site chargeand spin fluctuations are local. The resulting picture ofvalence fluctuations constrained by short-range ordering ofFe valences is similar to that proposed originally byAnderson [4]. In addition, the implication of slow electronhopping (&80 meV) supports more detailed theories ofmagnetite that invoke polaronic behavior for the Fe2charge carriers [5]. Spin-charge coupling is large andmay contribute to the polaronic binding energy. Finally,although the spin waves are insensitive to the details oflong-range CO below TV , the 5 symmetry of chargecorrelations above TV is similar to the order parameter ofthe Verwey state, which has been argued to be a combina-tion of 5  X3 cubic representations [15]. This indirectlysupports the idea that intersite Coulomb interactions play arole in the Verwey transition.R. J. M. would like to thank S. Satpathy for useful dis-cussions. Work is supported by the U. S. Department ofEnergy Office of Science under the following contracts:Ames Laboratory under Contract No. DE-AC02-07CH11358, Oak Ridge National Laboratory, which ismanaged by UT-Batelle LLC, under Contract No. DE-AC00OR22725.*Current address: Bragg Institute, ANSTO, Lucas Heights,NSW 2234 Australia[1] E. J. W. Verwey, Nature (London) 144, 327 (1939).[2] F. Walz, J. Phys. Condens. Matter 14, R285 (2002).[3] J. Garcı´a and G. Subı´as, J. Phys. Condens. Matter 16,R145 (2004).[4] P. W. Anderson, Phys. Rev. 102, 1008 (1956).[5] D. Ihle and B. Lorenz, J. Phys. C 19, 5239 (1986).[6] E. J. Verwey, P. W. Haayman, and F. C. Romeijn, J. Chem.Phys. 15, 181 (1947).[7] N. F. Mott, Adv. Phys. 16, 49 (1967).[8] I. Leonov et al., Phys. Rev. Lett. 93, 146404 (2004).[9] J. M. Zuo, J. C. H. Spence, and W. Petuskey, Phys. Rev. B42, 8451 (1990).[10] J. Garcı´a et al., Phys. Rev. Lett. 85, 578 (2000).[11] J. P. Wright, J. P. Attfield, and P. G. Radaelli, Phys. Rev. B66, 214422 (2002).[12] E. Nazarenko et al., Phys. Rev. Lett. 97, 056403 (2006).[13] Y. Tokura, Rep. Prog. Phys. 69, 797 (2006).[14] A. Rosencwaig, Phys. Rev. 181, 946 (1969).[15] P. Piekarz, K. Parlinski, and A. M. Oles, Phys. Rev. Lett.97, 156402 (2006).[16] R. J. McQueeney et al., Phys. Rev. B 73, 174409 (2006).[17] R. Coldea, computer code MSLICE: A data analysis pro-gramme for time-of-flight neutron spectrometers, 2004.[18] T. G. Perring, computer code TOBYFIT-Least-Squares Fit-ting to Single Crystal Data on HET, MARI, and MAPS, 2004.[19] Use of Gaussian fits, rather than resolution convolved fits,may lead to small errors in the fitted energy for steeplydispersing modes.[20] V symmetry direction is [10], on the Brillouin zone face.[21] J. Loos and P. Nova´k, Phys. Rev. B 66, 132403 (2002).[22] A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys.Rev. Lett. 74, 5144 (1995).[23] N. Furukawa, J. Phys. Soc. Jpn. 65, 1174 (1996).[24] Z. Zhang and S. Satpathy, Phys. Rev. B 44, 13319 (1991).[25] L. Degiorgi, P. Wachter, and D. Ihle, Phys. Rev. B 35,9259 (1987).[26] S. K. Park, T. Ishikawa, and Y. Tokura, Phys. Rev. B 58,3717 (1998).[27] S. M. Shapiro, M. Iizumi, and G. Shirane, Phys. Rev. B 14,200 (1976).+2+3+2.5+2.5(a) (b)FIG. 5 (color online). Schematic drawing of (a) spin and(b) charge modes on the B-sublattice with 5 symmetry. In(b), the heavy arrow indicates direction of charge fluctuations.PRL 99, 246401 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 DECEMBER 2007246401-4

Zener Double Exchange from Local Valence Fluctuations in Magnetite

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Zener double exchange from local valence fluctuations in magnetite

Zener double exchange from local valence fluctuations in magnetite

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