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(1)Faculty of Physics and Applied Computer Science. Doctoral thesis Wojciech Tabiś. Structural changes in magnetite in vicinity of the Verwey transition observed with various x-ray diffraction methods. Supervisors:. dr hab. Andrzej Kozłowski dr Jose Emilio Lorenzo-Diaz. Centre National de la Recherche Scientifique, Grenoble. Krakow 2010.

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(3) This Thesis is dedicated to prof. J. M. Honig: my idea about the Verwey transition could only be gained thanks to his great single crystals of magnetite.

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(5) Acknowledgements In this acknowledgements section of my thesis, I wish to thank all the people who, in one way or another, helped me during my Ph.D. studies. First of all I would like to thank my supervisor Andrzej Kozłowski for his continuous help, support and encouragement. I am sincerely thankful to him for teaching me a lot about science and the passion of being a scientist. In many ways, he was the most influential person both in my life and science. Being a supervisor was his commitment but being my friend and a “scientific father” was his own choice. Unlike most of the Ph.D. students, I’ve been most fortunate to have a second supervisor, Emilio Lorenzo. His deep knowledge of the RXS technique and his ability to answer questions in a very precise way have always been of great help. His continuous good mood and humor has always been welcome. I also want to thank him for making my visit at the CNRS possible and fruitful. I am grateful for very pleasant scientific cooperation with Zbigniew Tarnawski, Kim Tarnawska, Nicolas Jaouen, Zbigniew Kąkol, Joachim Kusz and Grzegorz Król. I have to also admit that every experiment conducted with Tomek Kołodziej was a great pleasure. In my opinion, these were the most enjoyably measurements ever. Apart from great results we got experience in many experimental techniques we used. Asia Stępień was a person I always could talk to. I am grateful for all the coffees and teas we had during our scientific and personal discussions. Thank you for all the lunches we had together. Apart from Asia and Tomek there are other “pizza team” members I want to thank. Marta Borowiec, Krystyna Schneider and Marcin Sikora were the people it was always nice to meet. For a scientific cooperation and for her true friendship I want to thank Magda Makarewicz with whom I did all the preparations before the exams. I am glad our friendship survived the distance. I wish to thank Piotr Żygieło for his continuous help and software support he provided me with. The nice company of my colleagues made our office more than just a work place. Sharing my room with Waldek Tokarz and Marcin Kowalik was an excellent experience, both from the scientific point of view and concerning the amicable atmosphere. Here I want to thank my French friends Agnès Thévenot 5.

(6) and Nicolas Martin for their hospitality and for letting me stay at their place for couple of weeks during my visit to Grenoble. It was great experience to meet French lifestyle and to learn about France at all. I appreciated everyday lift to CNRS Nico was giving me with his “jolie Renault”. I also want to also thank Nico and Kiryll for all the coffees we had at the ILL during busy nights and for the “last beer” we had. My social life in Grenoble would not be so nice if there weren’t also my other friends: Romain, Marine, Justin, Eric and many others. An important part of studies was also free time I loved to spend with my friends Eurydyka Podkówka and Grześ Orłowski. I want to thank them for all the wonderful bicycle trips we had as well as for the events that will come. Finally, but most importantly, I thank my family for supporting me all the years. I am fully aware of this privilege.. 6.

(7) Contents. Contents Introduction ____________________________________________ 9 1. The Verwey transition in magnetite ____________________ 12 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8. Verwey model ___________________________________________________ 14 Electron-electron correlations _______________________________________ 16 Electron-lattice interactions_________________________________________ 16 Magnetic interactions _____________________________________________ 19 Structural and electronic orders______________________________________ 20 Verwey transition “in statu nascendi”_________________________________ 26 The aim ________________________________________________________ 27 Samples ________________________________________________________ 28. 2 Structural changes in magnetite observed with coherent x-ray radiation _______________________________________________ 29 2.1 2.2 2.3 2.4 2.5 2.6 2.7. Introduction _____________________________________________________ 29 Coherence of the synchrotron radiation _______________________________ 31 Influence of the sample on coherence _________________________________ 33 Scattering of coherent x-rays on two-phase systems _____________________ 35 Set-up of the TROIKA beamline ID10A at the ESRF ____________________ 38 Samples ________________________________________________________ 40 Studies of the Verwey transition by observation of the (1 1½ 2) peak growth: “553” sample _________________________________________________________ 42 2.8 X-ray photon correlation spectroscopy studies at the low-T phase of magnetite: “110” sample ____________________________________________________ 46 2.8.1 Theory_____________________________________________________ 46 2.8.2 Experiment and discussion _____________________________________ 48 2.9 Conclusions _____________________________________________________ 54. 3. Resonant x-ray scattering studies _____________________ 56 3.1. Resonant X-ray Scattering technique _________________________________ 56 3.1.1 Theory_____________________________________________________ 57 3.1.2 Resonant x-ray scattering in magnetite____________________________ 61 3.1.2.1 Tensor elements of the atomic scattering factor _____________________ 61 3.1.2.2 High-temperature cubic phase, Fd-3m ____________________________ 62 3.1.2.3 Low-temperature phase: Pmca symmetry _________________________ 64 3.1.3 Experimental ________________________________________________ 69 3.1.4 The ID-20 beamline set-up _____________________________________ 70 3.2 The experimental data treatment _____________________________________ 73 3.2.1 Correction of a Bragg peak due to absorption ______________________ 73 3.2.1.1 Linear absorption coefficient (E) determination from fluorescence or absorption __________________________________________________ 77 3.2.1.2 Linear absorption coefficient (E) determination from calculation ______ 78 7.

(8) Contents. 3.2.1.3 Absorption coefficient Abs(E) calculation__________________________80 3.2.2 Influence of the sample surface on the scattering spectrum ___________ 81 3.2.2.1 Dead layer __________________________________________________81 3.2.3 Influence of the anomalous scattering factor on a peak position________ 85 3.2.4 Calculation of the correlation length from FWHM of a Bragg reflection _ 86 3.2.5 Other corrections ____________________________________________ 87 3.2.6 d spacing – calculations_______________________________________ 89 3.3 Experimental Results presentation___________________________________ 89 3.3.1 Samples presentation _________________________________________ 89 3.3.2 RXS results ________________________________________________ 92 3.3.2.1 Stoichiometric magnetite (“458 Kim_Darm_2008” sample) - unpolished _94 3.3.2.2 Stoichiometric magnetite (“458 Kim_Darm_2008” sample) - polished ___97 3.3.2.3 Fe3-xZnxO4, x = 0.0085 (“459J#3” sample) _______________________100 3.3.2.4 Fe3-xZnxO4, x = 0.012 (“459-1#2B2” sample) ______________________103 3.3.2.5 Fe3-xZnxO4, x = 0.022 (“460-2#1” sample) ________________________106 3.3.2.6 Fe3(1- )O4, = 0.0025 _________________________________________109 3.3.2.7 Fe(1- )3O4, = 0.0035 _________________________________________112 3.3.2.8 Fe3(1- )O4, = 0.0066 _________________________________________116. 4. Data analysis and discussion________________________ 120 4.1 4.2 4.3. Influence of the sample polishing on the RXS results ___________________ Charge and orbital orderings in magnetite____________________________ The origin of the (003) reflection splitting____________________________ (003) PEAK SPLITTING IN STOCHIOMETRIC MAGNETITE. ___________ (003) PEAK SPLITTING IN Fe3-xZnxO4. _______________________________ 4.4 Do CO and OO exist? ___________________________________________ 4.5 Characteristic temperatures in the vicinity of the Verwey transition________ 4.6 Ordered patches above TV ________________________________________ 4.7 The second order magnetite. ______________________________________. 121 123 126 127 129 132 134 135 140. 5. The Verwey transition scenario ______________________ 141. 6. Conclusions ______________________________________ 146. Streszczenie (Summary in Polish)________________________ 148 References___________________________________________ 150. 8.

(9) Introduction. Introduction Magnetite (Fe3O4) is the most magnetic of all the naturally occurring minerals on Earth. It was used by early navigators to locate the magnetic North Pole and is synthesized naturally in living species such as fish and birds to be used as magneto-receptors in their navigation. Due to its noninvasive influence on a living organisms magnetite nanoparticles have found great interest in medicine and pharmacology as a drug delivery system. New area of application of magnetite nanoparticles relies on their usage in biomaterials engineering. Polymer-based biomaterials as implants can only be visible in a living organism by magnetic resonance method (MRI) when are doped with magnetic nanoparticles. Magnetite typically carries the dominant magnetic signature in rocks, and so it has been a critical tool in paleomagnetism, a science important in discovering and understanding plate tectonics and as historic data for magnetohydrodynamics and other scientific fields. Magnetite also can store data in a form of magnetized magnetic domains in artificial magnetic memory storage devices. It is also a promising candidate for spin electronics, due to its spin-polarized transport. All these issues are sufficient for continuing the active research on magnetite among the physics and the material science community. An accurate description of this compound is thus mandatory for the understanding of its properties as well as similar effects in related materials. In addition to the variety of applications, yet another effect was added to this list that has triggered hot discussions lasting until today: the spectacular phase transformation at TV of ca 120 K, where practically all physical properties have anomalies. In particular, the Verwey transition, so known after the author of the first systematic studies, results in the huge peak in heat capacity, proving the transition’s first order character, and a drop of resistance by two orders of magnitude while heating above the Verwey temperature TV. This phase transition, evidently electronic states-related, is simultaneous with the change in the structure symmetry, from high-T cubic Fd3 m to monoclinic Cc that, among other factors, leads to the doubling of the unit cell along one of <100> axes (that becomes the monoclinic c axis). As usual in such a case of simultaneous “electronic” and structural transition, the question remains if the structural changes drive the electronic transition, or vice versa, or finally, if both transformations (or more, since many more aspects are visible) are more subtly interwoven. In fact, this last idea was followed by Verwey (1939) and Anderson (1956), who described the 9.

(10) Introduction. transition as a freezing out of strongly correlated electrons. Those electrons, one from each of two iron octahedral (B) positions, were supposed to travel relatively freely between iron B cations at T > TV (resulting in rather high conductivity) and freeze at particular positions at T < TV, according to the formula: high-T: Fe3+[Fe2.5+]O4 → low-T: Fe3+[Fe3+Fe2+]O4; here brackets mark the octahedral positions. Although this idea was proved to be oversimplified, the identification of the factor triggering the avalanche of processes is open. The observation of structural changes, while the transition develops (i.e. while the temperature stays constant, as in usual first order phase transformation when the latent heat is delivered), is the first aim of the present Thesis. The main outcome is that the structural aspects of transition and “electronic” ones (as marked by the AC magnetic susceptibility changes and the temperature plateau) seem to be partially decoupled. The experimental technique used here was a single crystal x-ray diffraction, utilizing partial coherence of synchrotron radiation, Coherent X-ray Diffraction (CXD), and magnetic AC susceptibility. The CXD experiment has been performed at ID-10A beamline in the ESRF where also other aspect – stability of low-temperature phase of magnetite has been studied. The X-ray Photon Correlation Spectroscopy (XPCS) data analysis show that the structure of magnetite is not stable after the sample quenching to 110K: the structural domains created at the transition have strong reorganization tendency with the correlation time (measured by the observation of the (2 2 ½) low-temperature superlattice peak) of the order of 10 min at this temperature. The notion of exact iron of +2 and +3 valence mentioned above is now disproved and the charge ordering with charge disproportionation of 0.2 electron, instead of 1 as in original Verwey model, is firmly established. However, the dominant role of electron-electron interactions is still considered to be vital and, together with electron-phonon correlations, have been suggested to drive the transition. Moreover, those combined electron-electron and electronphonon interactions resulted also in octahedral iron orbital ordering. Recently, a new fact of the decoupled structural changes was found by the Resonant X-ray Scattering (RXS). Charge and orbitals were found to order separately: at different temperatures and in the different way as the accompanying structural distortion. This crucial finding was immediately questioned based on the same type of experiment and suggested to be associated to the particular sample and not to magnetite in general. Since this fact of nonsimultaneous change of lattice and charge/orbital ordering was in accord with Author’s first observation from CXD, it was decided to check this finding on different stochiometric samples and to proceed the studies on Zn doped and nonstoichiometric magnetite, where the transition may be, first, shifted to lower T and, finally, changed to the second order. The RXS experiments were conducted at ID20 beamline of the 10.

(11) Introduction. ESRF and it was found that the lattice distortion process, the charge and orbital orderings are all decoupled in both stochiometric, nonstoichiometric and Zn doped magnetite. As a result, the scenario of the Verwey transition was suggested. This part constitutes the main body of the Thesis. The Thesis is organized in the following way: First, in Chapter 1, the system under studies is described based on the existing ample literature. In particular, the basics of the AC susceptibility observation of the Verwey transition, the literature data on the intervening interactions, structure of magnetite and charge/orbital ordering will be given. The results of Coherent X-ray Diffraction (CXD) and the X-ray Photon Correlation Spectroscopy (XPCS) measurements are described, in Chapter 2, together with the ID10A setup. As already mentioned, the main outcome is that the structure, when heated, shows the transition already completed, while the temperature still remains constant, suggesting the ongoing transition, the fact also supported by the AC susceptibility. In Chapter 3 the Resonant X-ray Scattering (RXS) is introduced and the method for data elaboration is discussed. Here also the samples for measurements are presented and the bare results of RXS measurements are shown. These data prove sharp lattice distortion changes at TV for the samples with the first order Verwey transition and the stretched, on a temperature scale, development of the (001) and, e.g (007/2) reflections, linked to electronic orderings. Chapter 4 comprises the RXS data analysis and the discussion. The Verwey transition scenario, formulated from these results is presented separately in Chapter 5. Finally, Chapter 6 contains conclusions.. 11.

(12) The Verwey transition in magnetite. 1 The Verwey transition in magnetite Despite the diversity of applications of magnetite this Thesis is particularly focused on the Verwey transition, the first order phase transformation at TV = 124 K signaled by the drop of resistivity by two orders of magnitude while heating above TV (see Fig. 1.1a). Surprisingly, the Verwey transition and its different sorts of "application" mentioned in the Introduction are strictly interwoven. For instance, in spin electronics cooling down magnetite below TV results in a drastic drop of electric conductivity; thermal cycling across the transition greatly diminishes coercive force, i.e. magnetic memory (see e.g. [Muxworthy2000]). In other words, the Verwey transition in magnetite is vital for several applications of this material. But this is also the fascinating problem on its own and its full explanation will certainly help resolve many problems in condensed matter physics. Not only resistivity jumps at TV; spectacular anomalies are observed practically in all physical characteristics. In particular, there is a huge peak in heat capacity, proving first order character of the transition (Fig. 1.1b). The step in AC magnetic susceptibility is also observed (Fig. 1.1c) and, finally, the structure symmetry changes from high-temperature cubic to lowtemperature monoclinic (Fig. 1.1e), which is seen in the splitting of some of cubic the reflections, as shown in Fig. 1.1d. The high-temperature symmetry of magnetite was found to be of inverse spinel-type, with Fd3 m symmetry, with Fe residing in tetrahedral (or A) and octahedral (or B) positions, as shown in Fig. 1.4. Up to the early eighties conflicting experimental data about the Verwey transition were accumulated. Most of these controversies were rationalized when it was found at Purdue University (see Fig. 1.2) [Kakol2000#1], that even very small departures from the ideal 3:4 cation to anion ratio may greatly alter the nature of the transition. Namely, the nonstoichiometry of the level 3 < 0.012 in Fe3(1- )O4 linearly lowers the transition temperature and the same was proved when some other perturbation to the lattice, as doping with different types of cations in Fe3-xMxO4 (where M = Ti, Zn, Ga, Ni, Mg, Co, Al), was introduced. Among these dopands, zinc and titanium play a special role: even though the number of "additional" electrons or holes created by these elements (due to their different valence) and the place they enter are different, a striking universal compositional correspondence 12.

(13) The Verwey transition in magnetite. (b). (a) 7.5. 100000. Fe3O4. 4.5 C(mJ/K*g). log (ρ) (Ωcm). 6.0. 3.0 1.5 0.0. 10000. 1000. -1.5 -3.0. 4. 8. 12. 16. 20. 24. 75 100 125 150 175 200 225 250 275 300 325. -1. T(K). 1000/T (K ). (d). (c). Counts. 0.0012 χAC(a.u.). 0.0010 0.0008. 125. TIP T (K) 130 135. <444>. <440>. 50. Fe3-xZnxO4. 40. x = 0.0072. 30 20 10. 122 118 114 110 T (K) 106 102. 0.0006 0.0004. 350 300 250 200 150 100 50. 122 118 114 110 106 102 T (K) 71000 71500 72000. 80. 100. 120 140 T (K). 160. 180. Fig. 1.1 Physical properties of magnetite influenced by the Verwey phase transition; (a) resistivity, (b) specific heat [Tarnawski2004] (c) magnetic AC susceptibility [Balanda2005]; in the inset the peak marking the isotropy point TIP is shown (d) crystallographic structure observed by the (440) and (444) peaks evolution [Kozlowski1999] and (e) the relationship between high-T cubic phase (bold black) and low-T phase monoclinic (red) unit cells. Counts. χAC(a.u.). 0.0014. t-o-f (µs). 58000. 58500. 59000. t-o-f (µs). (e). x ⇔ 3δ was found [Aragon1988] for these materials with respect to the transition temperature TV (Fig. 1.3). Also, when 3 = xZn = xTi > 0.012 the nature of the Verwey transition is changed from first to second order (what is reflected in the widespread transition region, Fig. 1.2, and in the characteristic change of TV vs. x linear relation, see Fig. 1.3); for concentration higher than 0.036 13.

(14) The Verwey transition in magnetite. the transition disappears altogether. The attribution of the Verwey transition temperature changes to the number of defects is, however, not justified, since other dopands (see Fig. 1.3 again) do not follow such a clear universal dependence. No universal explanation for the disturbing effect of extraneous elements on the Verwey transition was suggested so far. 140. Fe3(1-δ)O4. 130. Fe3-xNixO4. Fe3-xTixO4. Fe3-xCoxO4 Fe3-xMgxO4. 120. Tv ( K ). Fe3-xGaxO4. Fe3-xZnxO4. 110 100 90 80 0,00. 0,02. 0,04. 0,06. 3δ, X. Fig. 1.2 Variation of resistivity with temperature in magnetite Fe3(1-δ)O4 for various nonstoichiometry parameter values [Aragon1986]. (a) = -0.00053; (b) = -0.00017; (c) = 0.00021; (d) = 0.00035; (e) = 0.0017; (f) = 0.0035; (g) = 0.0050; (h) = 0.0068; (i) = 0.0097;. Fig 1.3 Dependence of the Verwey temperature on magnetite nonstoichiometry or sample doping [Kakol2000#1]. Note the characteristic drop of linear TV vs. defect number relation, dividing first-order and continuous Verwey transition regimes. With such an interesting phenomenon as the Verwey transition and the possibility to change its character by a simple dopand manipulation, it was natural that models explaining it were formulated very shortly.. 1.1. Verwey model. Already in the first papers [Verwey1939, Anderson1956] the mechanism of the Verwey transition, that survived for half a century, was formulated. According to this, magnetite was regarded as the ionic [Waltz2002] compound in which, in equivalent octahedral positions of the inverse spinel lattice (see Fig. 1.4a, b), both Fe+2 and Fe+3 were present. Due to postulated interionic strong Coulomb repulsion [Anderson1956], in each tetrahedron of octahedral positions two of them were supposed to be occupied by Fe+2, and the other two by Fe+3 (so called Anderson criterion, see Fig. 1.4c) in a well specified (although still not known for sure) order. On heating above TV, "additional" electrons (taken from Fe2+ ions, leaving Fe3+) were supposed to resonate between adjacent octahedral positions (order-disorder transformation), what resulted in a drastic increase of electric conductivity (see Fig. 1.1a). The high-T electron movement was also regarded as strongly correlated [Anderson1956] (the Anderson criterion was supposed to be still obeyed): the short range order was believed to be present even above TV. 14.

(15) The Verwey transition in magnetite. (a). (b). (c). Tetrahedral (A) sites. Fe3+ T>TV. 2-. O. A sites,Fe3+ B sites, Fe2.5+. Fe2.5+ Octahedral (B) sites. T<TV ⇒ Fe+3-Fe+2. Fig. 1.4 (a) Elementary unit cell of the low-T phase of magnetite (inverse spinel). (b) Tetrahedral and octahedral sites in the structure of magnetite (c) Atomic arrangement, suggested originally by Verwey (and not realized in practice) on octahedral sites in magnetite: all positions are equivalent, Fe+2.5, at T>TV and differ at T<TV. This picture of strongly correlated electrons freezing at TV was partially supported by the existing literature data. First, the entropy released at the transition was Rln2 [Shepherd1991], instead of 2Rln2 in case of no order at all was present above TV. Second, in the older NMR results [Mizoguchi1966] two frequencies observed below TV coalesced into one above this point. Finally, and it was a strong argument in favor for the ionic model, the experimentally determined total magnetic moment of magnetite formula unit agreed well with that calculated from this model [Kakol1989]. On the other hand, in spite of 60 years intensive studies no realistic low-T ionic arrangement was proposed and in early 70-ties it was found [Rakheha1978] that even though the total magnetic moment agrees with ionic value, the individual iron moments are almost identical, and different from the ionic picture. Finally, NMR and the x-ray resonance scattering experiments carried out after a year of 2000 [Garcia2001, Novak2000, Garcia2004, Subias2004] questioned both isolated ion notion as well as fast electronic jump in the disordered phase. However, even though the Verwey simple model was found to be too simplified, strong electron correlations were still regarded as a driving force of the transition.. 15.

(16) The Verwey transition in magnetite. 1.2. Electron-electron correlations. The strong electron-electron correlations were the basis of Honig and Spałek [Honig1992] ionic model, where the different configurations of octahedral Fe2+-Fe3+ pairs simulated the strongly correlated system. Using the different pair excitation levels, both I and II order phase transformation could be obtained with the proper entropy change. This phenomenological model was further developed in [Kloor1994] and [Song1995]. The realistic one-electron picture was first proposed by Yanase [Yanase1984] (APW method) and by Zhang [Zhang1991] (LSDA). Both these treatments have shown that for T > TV minority spin electrons are metallic and that there is a gap at Fermi energy for majority spin electrons. The experimentally observed [Schrupp2005] gap formation below the Verwey temperature could only be obtained [Leonov2004, Jeng2004] if both strong Coulomb correlations and the monoclinic structure were assumed. In such a case, the substantial difference of 0.5 in electron density on t2g, orbitals was found [Leonov2004] between octahedral "Fe+3 and Fe+2 ions". In other words, the original Verwey idea of charge ordering (although to much lower extend than 1, as in the Verwey model) was augmented by the orbital ordering. Due to screening effects, the net charge difference is only 0.2, and the Anderson criterion is violated, both facts well supported by the experiment [Wright2002]. This problem will be further elaborated on in section 1.5. The strong electron repulsion was also the basis of Ihle-Lorentz treatment [Ihle1986] that was successful in explaining high-temperature resistivity; here, however, the second strong interaction, electron-lattice, was needed.. 1.3. Electron-lattice interactions. Apart from Coulomb electron correlations, also the lattice certainly intervenes in the Verwey transition, as is already clear from the change of crystal symmetry at TV. Not only lattice itself, but also lattice dynamics changes at the transition, as the data accumulated within the last 30 years indirectly showed. This fact was first observed by the diffuse scattering of neutrons that showed two critical phenomena above TV [Shapiro1976] (Fig. 1.5). First, the critical quasielastic ("spot-like scattering", vanishing 4 K above TV, Fig. 1.5a) was found for those reciprocal lattice vectors that become Bragg peaks below TV. The second type of critical scattering ("diffuse" scattering, Fig. 1.5a) was also observed for incommensurate Q vectors and at temperatures as large as 80 K above TV. Both types of critical scattering vanish below TV and are strikingly. 16.

(17) The Verwey transition in magnetite. different for the first and the second order samples [Aragon1993] (Fig. 1.5b). This phenomenon is still being investigated [Bosak2010]. 800. 1,25. (8, 0 , 3/4) X3 I (8 0 0.75) (counts/2 min). Fe3O4. 1,00 1/I (arb. units). E=0.0 meV Q=(4,0,0.5). 0,75. X3 ∆5. 0,50. Q=(8,0,0.75). 0,25 0,00 100. 600. δ = 0.000 δ = 0.006. 400. 200 background. TV 120. Fe3(1-δ)O4. 140 160 Temperature (K). 180. 200. 0. 100. 150. 200. Temperature (K). Fig. 1.5a Temperature dependence of inverse intensity at specified vectors of reciprocal lattice; the dotted curve represents “spot like scattering”, while the points diffuse scattering (after [Shapiro1976]). Fig. 1.5b Diffuse scattering at (8, 0, ¾) reciprocal lattice position, suggesting two different lattice dynamics for first and second order magnetite (after [Aragon1993]). Similarly different lattice dynamics below TV for samples undergoing first and second order Verwey transition were observed in heat capacity studies [Kozlowski1996] (Fig.1.6). Although heat capacities for first and second order samples1 are very nearly equal above their transition temperatures, CP for the first order samples is much lower than that of the second order below the transition temperature. In other words, the lattice gets more rigid below TV for the first. 0.8. NIS. Fe3-xZnxO4. 0.4 0.3. 500. 0.2 0.1 40. x=0 x = 0.010 x = 0.028. 60. 80. 450. 80 100 120 Temperature ( K ). 1.0 296 K 140 K 120 K 100 K 25 K. 100. 120. 0.5. 140. 140. Temperature (K). Fig. 1.6 Cp/T vs. T for Fe3-xZnxO4 exhibiting Verwey transition of different order; the inset shows temperature dependence of characteristic Debye temperature θD, suggesting lattice stiffening below TV. 1. 1.5. 2. 550. -3. 0.5. θD ( K ). 2. Cp/T (J/mol K ). 0.6. 4. g(E)/E [10 meV ]. 0.7. 0.0 0. 5. 10 15 Energy [meV]. 20. 25. Fig. 1.7 Low energy dependence of phononic density of states, g(E)/E2, for magnetite thin film (partially after [Handke2005]); points mark phonon modes proved [Piekarz2007] to be vital for the transition. Samples undergoing a first order phase transition and a second order phase transition are called first and. second order samples respectively. 17.

(18) The Verwey transition in magnetite. order magnetite, while remains intact for higher order Verwey transition. In Fig.1.6, this fact is additionally accentuated by the presentation of the respective temperature dependence of the effective Debye temperature θD(T) and its jump of 50 K at the transition for first order magnetite. Finally, by the observation of nuclear inelastic scattering the lattice stiffening below TV was proved directly (Fig. 1.7) [Handke2005]: for high temperatures, the phononic density of states is high in the vicinity of 7 meV and jumps to lower values (maximum is transferred to the higher energies) while the systems orders at the Verwey transition. Not only diffuse neutron scattering marks the fluctuations at much higher temperatures than TV but also c44 elastic constant that, even for first order magnetite, softens on cooling above TV (Fig. 1.8), showing typical Landau relation for second order phase transitions [Schwenk2000]. These facts suggest that there exist thermal fluctuations that drive the system to the second order phase transition and they are rapidly broken by the nucleation of a new phase that has no relation to the critical high-temperature behavior. The crucial role of electron-phonon and electron-electron correlations in the mechanism of the transition was confirmed in the recent theoretical analysis [Piekarz2006, Piekarz2007]. Three vital phonon modes were identified that, together with strongly coupled electrons, drive the transition: (i) A X3 phonon optic mode, mainly composed of iron vibrations, has been shown to considerably lower the electronic energy, leading ultimately to the change of structure i.e. the transition. (ii) A zone center optic mode of T2g symmetry is related to c44 elastic constant, i.e. behave critically already 80 K above TV . (iii) The critical behavior of the third important mode ∆5 is observed just a few K above TV.. 1,1. c44(T)/c44(270 K). x=0 x=0.02. 1,0. Fe3-xZnxO4. 0,9 0,8 0,7 0,6. x=0.032 0. 50. 100. 150. 200. 250. 300. 350. Temperature (K). Fig. 1.8 The temperature dependence of c44 elastic constant for Zn doped magnetite showing similar approach to the transition, irrespective to its order. (after [Schwenk2000]). 18. Fig. 1.9 Magnetite total energy diminution with X3 and ∆5 phonon mode amplitude (after [Piekarz2007]).

(19) The Verwey transition in magnetite. The analysis showed very persuasively that the low-temperature insulating phase of magnetite is due to instability driven by the electron-phonon coupling in presence of strong electron interactions.. 1.4. Magnetic interactions. Magnetic interactions were also postulated to stabilize low-temperature electronic long range order. Indeed, in the ionic model, the same 3d electrons from octahedral iron cations that freeze at the Verwey transition also convey magnetic interactions by superexchange mechanism. However, no significant, higher than 0.1 %, change in magnetization was observed at the transition temperature and a common opinion is that magnetic degrees of freedom have no effect on the transition, although magnetic phenomena may interact with the crystal lattice, i.e. may effect the transition in an indirect way. T>130K (a). βM. 130K>T>TV (b). T<TV. (c) cM. bM aM Fig. 1.10 Relationship between easy (green), intermediate (blue) and hard (red) magnetization axis in magnetite at three characteristic temperatures. The mechanism justifying this last conjecture is following: Fe3O4 is a ferrimagnetic material, with spin-only magnetic moment of a total value 4.06 µB /f.u. At high temperatures, magnetic easy, intermediate and hard axes are along cubic <111>, <110> and <001> directions, respectively (Fig. 1.10a). This situation changes at the isotropy point of TIP = 130K (anisotropy energy diminishes to zero) and below this temperature magnetic moment points toward <100> direction [Novak2000, Aragon1992], as shown in Fig. 1.10b. This easy direction is preserved when cooling magnetite below TV, Fig. 1.10c, where cubic symmetry turns to monoclinic. However, since each of cubic <100> can become an easy axis, the material breaks into several structural domains (twins) unless the external magnetic field H > 2 kOe along particular [001] is applied [Kakol1989, Calhoun1954]. On the other hand, when particular easy axis has been established and the magnetite sample is magnetized along other <100> direction, then at 19.

(20) The Verwey transition in magnetite. temperatures TAS, slightly lower than TV, a reorientation of magnetic moments, i.e. axis switching, may take place and this <100> direction can become a new easy axis. Thus, the domain structure linked to different magnetic easy axis direction and, in particular, the axis switching mechanism, is some indication of the magnetism affecting the Verwey transition. Apart from that, the creation of structural domains constitutes a real problem for single crystal x-ray diffraction studies and in the experiments presented in this Thesis considerable effort was undertaken to prevent this to happen (samples were cooled in external magnetic field from the permanent magnet). On the other hand, since magnetic domain structure and structural domains are intimately linked [Balanda2005] any technique that utilizes magnetic domain walls movement (e.g. AC magnetic susceptibility) can be efficiently used to monitor the appearance of the Verwey transition. In particular, smaller lattice domains at T < TV curtail, to some extend, domain wall movement, largely diminishing the AC susceptibility signal. For these reasons, AC magnetic susceptibility will be used for the present data description as a major indication of the sample quality and to locate the Verwey transition temperature.. 1.5. Structural and electronic orders. Magnetite crystal structure above the Verwey transition (see Fig. 1.4a) is well known to be the cubic inverse spinel (space group Fd3 m ), represented by (Fe3+)[Fe2+Fe3+]O4, where () denote 8(a) tetrahedral (point group .-43m) while [] 16(d) (point group .-3m) octahedral iron positions. Oxygen ions 32(e) positions deviate slightly from their ideal arrangement. Octahedral cation sites form tetrahedra that share corners with each other (Fig. 1.4b); all the tetrahedra on, say, the xy plane, are isolated.(Fig. 1.4c) The crystal structure of magnetite below the Verwey transition was found to be monoclinic [Iizumi1982] (space group Cc), with the exception of the magnetoelectric effect measurements [Miyamoto1993] that suggested a breaking of the ac mirror plane symmetry, implying triclinic symmetry, and x-ray topography [Medrano1999] studies with the same conclusion. However, in spectroscopic experiments [Gasparov2000] triclinic symmetry was not confirmed. Also all direct crystal structure measurements point to monoclinic structure as indicated by the characteristic features of the observed diffraction pattern [Iizumi1982]: i. a rhombohedral distortion of the cubic unit cell, ii. a doubling of the cell along c axis, iii. the existence of a c-glide.. 20.

(21) The Verwey transition in magnetite. (a). (b). Fig. 1.11 (a) Unit cells used for structural properties description of magnetite (after [Wright2002]) (b) Simplified low-T elementary cell P2/c shown on top of the real Cc cell and high-temperature cubic cell. The basis vectors aM, bM, and cM of the monoclinic unit cell roughly coincide with the [1 1 0], [110], and [001] directions of original cubic lattice, as shown in Figs. 1.1e and 1.11b. The monoclinic cM-axis is actually tilted ~0.20° away from the vertical towards the -aM direction (see Fig. 1.1e), due to rhombohedral elongation along the [ 1 11] or [ 1 1 1 ] axes. The displacements of atoms, as compared to cubic symmetry, are reported to be of the order of 0.1 Å. Lattice parameters for magnetite at room temperature and at low-T structure, from available data, are presented in Tab. 1.1. after [Wright2002] lattice parameter. HRPD. after [Iizumi1982] HRPD. BM16. BM16. neutron diffraction,. T =90 K. T=130 K. T=90 K. T=130 K. 10K. a/Å. 11.8838 (2). 8.3939(2). 11.8888 (4). 8.390405(3). 11.868 (2). b/Å. 11.8502 (2). 11.84944. 11.851 (2). 16.77508. 16.752 (4). 90.2365(2). 90.20 (3) °. (4) c/Å. 16.7716 (1) (3). 0. 90.255(5). Tab. 1.1. Lattice parameters of magnetite revealed during different experiments; after [Wright2002]. 21.

(22) The Verwey transition in magnetite. All the information, possibly relevant to the studies presented below, concerning lattice parameters change with temperature and composition (for Zn doped magnetite; to the best of Author’s knowledge, no data for nonstoichiometric magnetite is available) are shown in Fig.1.12. 16.790. 11.89 aM. 11.88. Fe3O4. T=70K. Fe3-xZnxO4 16.785. 11.87. cM (Å). aM, bM. (Å). T=70K. 11.86. 16.780. 11.85 bM. 11.84. 0.00. 0.01. 0.02 composition x. Fe3O4. Fe3O4 Fe3-xZnxO4 0.03. 0.04. Fig. 1.12a Zn contents dependence of monoclinic aM and bM lattice parameters of zinc doped magnetite (after [Owoc2007]). Fe3-xZnxO4 16.775. 0.00. 0.01. 0.02 composition x. 0.03. 0.04. Fig. 1.12b Zn contents dependence of monoclinic cM lattice parameter of zinc doped magnetite (after [Owoc2007]). Here and also throughout the Thesis, all the reflections will be labeled according to cubic lattice parameters. Although magnetite low-temperature structure is probably Cc, the unit cell is much too large and the departures from simpler monoclinic symmetry P2/c too small to be able to fit the powder diffraction data to this symmetry. For these reasons, some other unit cells, usually P2/c with Pmca (see Fig. 1.11a) orthorhombic symmetry constraints on the atomic positions are used. In orthorhombic space group Pmca, there are 56 atoms per unit cell and the lattice parameters are ≈ ac/√2 x ac/√2 x 2ac, where ac is the cubic lattice constant. In this unit cell, there are 2 nonequivalent tetrahedral iron positions and 4 different octahedral positions, B1, B2, B3, and B4. Whereas iron ions in positions B1 and B2 are at the centers of their respective octahedra, the iron atoms in B3 and B4, are slightly off center. These position shifts are responsible for the doubling of the cell along the c axis, whereas the B1 and B2 ordering has no effect on the cell doubling (this will be further treated in Chapter 3.1.2). In the monoclinic P2/c group (the unit cell has approximately the same dimensions) identical orthorhombic B1 and B2 Fe sites split and there are altogether six nonequivalent sites, namely, B1a, B1b, B2a, B2b, B3, and B4. In the real, Cc, structure of magnetite, four times larger than P2/c unit cell, tetrahedral iron cations reside on 8, while octahedral cations on 16 nonequivalent positions. Valence charges calculations on these B Fe ions within the lowtemperature crystal structure P2/c refined by Wright [Wright2002] (based on his combined 22.

(23) The Verwey transition in magnetite. neutron and x-ray studies), show that the B Fe ions can be divided into two groups according to their valence charges (Figs. 1.11b and 1.13). The electron-rich B1 and B4 Fe ions have a lower ionicity of +2.4 and the electron-poor B2 and B3 Fe ions have a higher ionicity of 2.6. Such a charge order is clearly different from the ideal Verwey arrangement where Fe+2 and Fe+3 ions were supposed to form some ordered pattern. The structural evidence for charge ordering below the transition, found there, was based on estimation of mean B-site-to-oxygen distance (bond valence sum analysis). This small charge disproportionation occurs with modulation vectors Q = (001) and Q = (001/2) in units of 2 /a.. (a). (b). Fig. 1.13 (a) The arrangement of the atoms in low-T phase simplified unit cell in P2/c symmetry (b) the arrangement of orbitals. (after [Jeng2004]). Fig. 1.14 Spin-resolved density of state of Fe3O4 in (a) the high-T cubic structure and (b) the low-T monoclinic structure (upper panel) and a schematic energy level diagram for the spin-down B d orbitals in the monoclinic phase (lower panel). (after [Jeng2006]). The suggestion of a small charge disproportionation is supported by numerous band structure calculations. In one of them, Jeng [Jeng2004], using the structure refined by Wright, has performed calculations within local density approximation (LDA) with generalized gradient correction (GGA) plus on-site Coulomb interaction U (LDA+U). For high-temperature cubic magnetite, the metallic state was found with the Fermi level sitting in the middle of the spin23.

(24) The Verwey transition in magnetite. down B Fe t2g band (Fig. 1.14) The spin-up B Fe 3d band and the spin-down A Fe 3d band are completely filled, in accordance with Hund’s rule and in agreement with the Verwey model: spin-up states are composed solely from B iron 3d electrons and spin down states from 3d electrons on tetrahedral sites and "additional" electrons on octahedral sites. For the monoclinic structure, an energy gap of the order of 0.2 eV is opened at the Fermi level, consistent with experimental values found in photoemission and optical measurements [Park1998] (note that a gap of 50 meV was found by Schrupp [Schrupp2005]). Densities of states are grouped into those of “Fe+2” character (5.6 3d electrons, B1a, B1b, B4 positions) and of “Fe+3” character (5.4 3d electrons, B2a, B2b, B3 positions). The B Fe+2 ions, have a narrow spin-down 3d-t2g band right below the Fermi level with a bandwidth of 0.5 eV, whereas such a sharp band is absent in the B Fe3 ions. These bands of the B1a, B1b, and B4 Fe2+ ions below the Fermi level are, respectively, of predominant dyz, dxz, and dxy characters, what is schematically shown in the lower panel of Fig. 1.14. This indicates the formation of the spin-down t2g orbital-ordered state: each Fe2+ ion has one spin-down t2g orbital occupied, whereas all the spin-down t2g orbitals of the B Fe3+ ions are empty (Fig. 1.13(b)). In the century not only the charge order in magnetite B-sublattice was partially defined (as that proposed by Wright, or by Jeng, as described above), but also a strong arguments against any charge ordering were raised. First, Novak et al. [Novak2000] in their NMR studies could not see different spin-lattice relaxation time for Fe+3 and Fe+2 (the relaxation time for those should be much shorter). Then, a very strong criticism of possibility of charge order experimental observation were raised by the Garcia group from Saragossa. In [Garcia2000, Garcia2001, Subias2004] RXS studies on stoichiometric magnetite were reported. The reflections that were observed are: those that come from the doubling of cubic cell, as the superlattice (441/2) peak, those that result from the lost fcc translation, as the (003) and (210) peaks, those, as (002) and (006), forbidden in high-T phase due to the diamond glide plane in the spinel structure and those weak reflections, as (442) that are forbidden for iron atoms but allowed for the oxygen. It was shown, by the examination of the (002) and (006) forbidden reflections, above and below TV, that all the resonant spectra are identical and the conclusion drawn was that no charge ordering occurs in the low-temperature phase [Garcia2000]. Also, [Subias2004], the observation of energy dependence of (00l) reflections (see Fig. 1.15) prove that (003/2) peak does not exist, and (003) peak is larger off- than at-resonance; according to the authors, these facts also lead to the conclusion that magnetite is not charge ordered on octahedral Fe sites, contrary to the common belief lasting more than 60 years. 24.

(25) The Verwey transition in magnetite. Fig. 1.15 Logarithmic intensity vs. the momentum transfer around the (00l) reflection of magnetite measured at energies of EP = 7.112 keV (upper line) and EW = 7.125 keV (lower line) at T= 20 K. Left inset: x-ray absorption spectra at the Fe K edge. Arrows indicate the two selected energies for the l scans. Right inset: detail of the l scan around the (003) reflection at EW = 7.125 keV. (after [Subias2004]). Garcia’s point of view was contested in [Lorenzo2008, Joly2008, Nazarenko2006]. It was first proved that not all Bragg reflections are sensitive to charge ordering (see Fig.1.16), and the failure in detecting CO signatures in some of those peaks is not a proof that the charge order does not exist altogether. By observation of (hkl) peaks with h + k = odd reflections (that do not exist in the Pmca unit cell and are thus pure Cc reflections) a signature of the charge ordering was found at the Fe B sites, with the free electron unequally shared between two adjacent Fe (and the corresponding O), Fe+2.5± , with. . 0.12 e the charge separation. It has also been. experimentally observed that the (001/2) peak, studied at the Fe L3-edge, arises from an orbital order at the octahedral B sites, i.e. it was confirmed that the low-temperature structure is also a t2g orbitally ordered. Fig. 1.16 Some of the experimental and calculated resonant x-ray diffraction peak spectra in Fe3O4 at T = 50 K. Experimental data (black dots) were fitted to the model with (red lines), or without (blue dots) charge order. Note that for some peaks, as (-1 1 0), the fit is conclusive, while other peaks (e.g. (-4 4 2)) do not differentiate between the models. (after [Joly2008]). 25.

(26) The Verwey transition in magnetite. Fig. 1.17 The energy variation of three Bragg peaks intensities in stoichiometric magnetite (after [Lorenzo2008]). Fig. 1.18 Temperature evolution of three reflections from Fig. 1.17, sensitive to lattice distortion, charge and orbital orderings. (after [Lorenzo2008]). In a recent RXS work, of outmost importance to the results presented below, Lorenzo et al. have shown [Lorenzo2008], by the observation of the (001), (003) and (001/2) peaks, that both orbital and charge orders do not disappear at TV, but rather at a higher temperature, TIP. . 130 K. (see Fig. 1.18) This completely new result has opened a new direction for the understanding the occurrence of the Verwey transition in magnetite.. 1.6. Verwey transition “in statu nascendi”. All those contradicting results presented above, some confirming, some fiercely opposing charge ordering in magnetite, strongly call for more studies of this material, especially by crystallographic methods, to clarify the existence of charge order in general and, the last findings of “three Verwey transitions”, in particular. On top of that, the Author was impressed by the variety of effects that show up in just one phenomenon like a Verwey transition and the abundance of interwoven interactions that should be taken into account in explaining it. For these reasons he decided, first, to engage in a more detailed studies of the process of how the transition develops, already started in Krakow laboratory [Tabis2007]. The experiment was aimed to observe how AC magnetic susceptibility and electrical resistivity changed at the transition, i.e. when the sample temperature stayed constant (heat of transition was delivered to the sample but sample temperature was unchanged as for the first order transition; the physical response of the sample was observed as a function of time). As a result of these experiments it was first shown that although susceptibility varied, as described above, it was a side effect, connected to the changing magnetic domain wall movement, rather than the proof of magnetic interaction participating in the transition, as explained above. Indeed, the step in to zero while the external magnetic field was applied (see Fig. 1.19). 26. AC. was forced.

(27) The Verwey transition in magnetite. 126,0. 10 8. 124,5. χ'(t), Hext=0 6 χ'(t), Hext=2 kOe. 124,0. Hext=2 kOe. 123,5 123,0 200. 125,0. 4. T(t), Hext=0 T(t), Hext=2 kOe 2. χAC. 10. 124,9. 8. 124,8 124,7 124,6 124,5. 6. Temperature 4. resistance. 2. 124,4 124,3. 400. 600. 800. 1000. 1200. 1400. 2000. resistance (a.u.). 125,0. 12. Temperature (K). Hext=0 kOe. 125,1. χAC (a.u.). Temperature (K). 125,5. 12 14. 4000. 0 6000. time (s). time (s). Fig. 1.19 The comparison of magnetic AC susceptibility with and without external magnetic field. The temperature plateau indicates that the transition is in progress. Fig. 1.20 Temporal evolution of temperature, resistance and magnetic AC susceptibility at the Verwey transition. Similar experiment showed (Fig. 1.20) that while the sample temperature stayed constant, the resistivity gradually raised with time, as for susceptibility, although the exact time dependence of both characteristics are not similar and the repeated measurements showed slightly different behavior. Since the melting of the electronic pattern that was observed by susceptibility and resistivity temporal changes are linked to the lattice, adding the lattice observation to these experiments might be very interesting and seems natural.. 1.7. The aim. Thus, one of the aims of this Thesis is the direct detailed observation of the proliferation of the new high symmetry phase (on heating), or a new low-T phase (on cooling) at the transition i.e. while the temperature stays constant. To do so, the experiment on ID10A of the ESRF (experiment HS3274) was conducted, where partial coherence of synchrotron x-ray radiation was exploited. Despite numerous studies of the Verwey transition, no experiments aimed to see the details of the structural transformation process were conducted before the completion of this Thesis. The second set of aims concerns the problem of “three Verwey transitions” found by Lorenzo. The results of two experiments, HE3065 and HE3335, on ID20 in ESRF are described (actually, part of the data was also collected during HS3981, also on ID20). In these set of experiments anomalous x-ray scattering on stoichiometric magnetite, but also on Zn-doped and nonstoichiometric magnetite single crystals were measured. The reason that Zn doped magnetite was measured is that substitution of Zn2+ for Fe3+ in the tetrahedrally coordinated cation sites. 27.

(28) The Verwey transition in magnetite. needs to be compensated by one hole per Zinc ion on the octahedrally coordinated Fe-ions. Therefore large variations on the charge and orbital orderings features might be expected.. 1.8. Samples. Since very small doping or nonstoichiometry levels greatly affect the Verwey transition it is obvious that proper sample preparation procedures are crucial to the reliability of the experimental results. All the measurements were performed on single crystalline samples grown at Purdue University by the skull melter, crucibleless technique [Harrison1978]. This method is based on radiofrequency induction heating of a material in the cold crucible and its main advantage is that the unmelted material, “skull”, isolates the melt from the crucible, rendering the crystal free from crucible contaminations. The technique also allows for the control of the oxygen partial pressure during growth, thereby ensuring that the melt remains within the stability range of the material. The crystals were then subjected to subsolidus annealing under CO/CO2 gas mixtures to establish the appropriate metal/oxygen ratio [Aragon1983]. If the ratio of CO and CO2 partial pressures is fixed (e.g. by fixing the flow ratios) the oxygen partial pressure remains constant for stable temperature. The oxygen partial pressure pO2 was monitored by the use of ZrO2/Y2O3 transfer cell, which, in analogy to galvanic cells with liquid electrolytes, consist of two electrodes separated by solid ionic conductor (ZrO2) acting as electrolyte. If the atmospheres of two different oxygen partial pressures (pure oxygen served as a reference) were separated by the cell, the ensuing EMF allowed for the control of pO2 to within 1% of log pO2. This was enough to assure that stoichiometry parameter δ in Fe3(1-δ)O4 was controlled to 10-4. After annealing the crystals were quenched to room temperature and the surface regions were trimmed to remove partially oxidized outer layers. The preparation technique for single Fe3-xZnxO4 crystals was basically similar, but it required different annealing conditions established based on [Wang1990]. Zinc concentration was established from the universal TV vs. xZn curve from Fig. 1.3. The above steps rank among the most stringent precautions that have been reported to date. The samples for individual measurements and the check of their quality will be presented in appropriate sections.. 28.

(29) Structural changes in magnetite observed with coherent x-ray radiation. 2 Structural changes in magnetite observed with coherent x-ray radiation 2.1. Introduction. When a crystalline sample of magnetite is cooled down across the Verwey transition, in spite of continuous heat transfer from the sample, its temperature starts to be stable when the phase transformation begins (first order phase transition): due to huge latent heat of the material (965 J/mol), the energy released during the transition is just the energy taken from the sample. This fact was used by the Author for a precise observation of the transition in statu nascendi. In the previous Author’s research, simultaneous magnetic AC susceptibility and resistance variation at the transition has been studied [Tabis2007]. By a precise heat transfer control it was possible to stay at the transition for couple of minutes and observe these parameters changing at a constant temperature. Thermal evolution of the magnetic susceptibility and resistance are shown in Fig. 2.1a. Although, all the details concerning the transition are hidden in a step at TV, the evolution of these parameters, shown as a function of time, gives more specific information (Fig. 2.1b). 126,0. 100. 20. χmax. 14. 125,5. 1. 8. 0,1 120. 125. 130. 125,0. 124,0. 4 135. 123,5. Fig. 2.1a Magnetic AC susceptibility and resistance vs. temperature. 5. T(t). 6. Temperature (K). 10 R(t). χAC(t). 124,5. χmin. 5000. R,χAC (a.u.). 10. T (K). R(T). 115. 15. 12. χAC(T). χAC(a.u.). R (a.u.). 10. 5200. 5400 5600 time (s). 5800. 0. 6000. Fig. 2.1b The same parameters as in Fig 2.1a but observed as a function of time. The temperature has been measured with small Pt thermometer glued on the sample surface. Apart from those shown above, there is yet another, obvious, physical property of magnetite that is characteristic to the transition – the crystal structure. As the crystal symmetry changes at the transition from cubic Fd 3 m into monoclinic Cc some additional Bragg peaks 29.

(30) Structural changes in magnetite observed with coherent x-ray radiation. show up. It is, thus, natural to add to the above picture the temporal behavior of the structure and the (1 3/2 2), and (2 2 ½) Bragg peaks, allowed only in the low-T phase of magnetite (Cc symmetry), were chosen to study at the ID10 beamline of the ESRF. The way the superstructure peaks increase at the transition, while the sample is cooled down, gives information about the nature of the transition. Observation of the structural domains, while the transition develops, provides some information if the structure and electronic system (observed by susceptibility changes) undergo the transition simultaneously, or one precedes the other. These studies are also important from a general point of view: there is a continuous interest in observation of how the fascinating phenomenon of first order phase transition proceeds (for water/ice phase transition, see e.g. [Grimsditch1996], while Brihuega et al. [Brihuega2005] show how Pb/Si(111) monolayer changes its phase). In these studies the aim was to figure out if the structure is stable once created, and what the dynamics of the created domains is. To make the results comparable with those shown above in Fig. 2.1, magnetic AC susceptibility and the temperature were simultaneously observed with the superstructure peak intensity. As described in Chapter 1, low Cc symmetry of magnetite below TV guarantees the presence of structural domains. Their rearrangement and their interplay at the transition has been studied using coherent x-ray diffraction technique (CXD), in Author’s opinion the most appropriate method to study spatial and temporal reorganization of the structure. Stability of structural domains arrangement at low-T phase of magnetite has been also studied using the coherent x-ray beam. The x-ray photon correlation spectroscopy (XPCS), used here, is sensitive to the structural domains walls movement in the material. In this experiment, the sample was quenched across the transition to 110 K and using a CCD camera the temporal fluctuations of the (2 2 ½) were observed in equilibrium state. Coherent x-ray diffraction and speckle analysis is a subtle tool to study the static and dynamics of phase transformation as is clear from recent increased interest in this technique (see e.g. [Holt2007], [Ravy2007]). For more comprehensive information see [Stadler2005], [Dufresne1995] and [Fluerasu2003]. As a coherence of the synchrotron radiation is a key here, this issue is discussed first. The main features and the formation of the coherent beam are described in the next section. In the following, influence of the sample on coherence in the scattering process is also introduced. The description of the experimental setup at the ID10A beamline is given. The main part of this Chapter comprises the results and the discussion that are summarized at the end.. 30.

(31) Structural changes in magnetite observed with coherent x-ray radiation. 2.2. Coherence of the synchrotron radiation. There are two main types of photon sources2 (i) chaotic (or Gaussian) light sources such as thermal sources, synchrotron radiation storage rings, x-ray tubes and lasers below threshold. They are characterized by spontaneous emission in many different modes. (ii) coherent sources such as one-mode lasers with stabilized amplitude operated above threshold. The emission process is dominated in this case by stimulated emission in one mode. The synchrotron x-ray beam from an undulator is a chaotic light, because the emission processes are spontaneous and thus, each photon is independent. However, the most advanced currently available light sources in the x-rays range, as the third generation synchrotron radiation sources, have brilliance more than 10 orders of magnitude higher than that of x-ray tubes. The photons emitted by highly relativistic electrons are extremely well collimated. The beam from a high- undulator is only 1 mm in size at 40 m from the source. So although synchrotron source produces chaotic x-rays, owing to high brilliance of the radiation it is possible to consider synchrotron radiation as a partially coherent and, under specific conditions, conduct experiments in a coherence regime. Thus, high brilliance may cause majority of photons, coming to a small part of the sample, to have the same energy, momentum and phase; in this sense synchrotron radiation may be coherent. On the other hand, when the coherence conditions in the diffraction experiment are fulfilled, the individuality of the defects arrangement in a sample shows up as speckle pattern in the scattered intensity. Otherwise, a configurational average washes out the speckle and only diffuse scattering and possibly Bragg reflections will survive. The loss of interference may also be caused by the finite detection time or due to the finite detector pixel size. This subchapter is aimed to describe the main features of coherence in the x-ray physics. For classical x-ray scattering experiments, the diffraction of a large beam with many coherent volumes is observed. In each coherence volume Vi, a wave Ai(q) is diffracted. Only an incoherent sum of the intensities over a large number of domains N is measured: N. r r 2 I (q ) = ∑ Ai (q ). (2.1). i =1. However, when the coherence volume Vi of a single ray is similar in size to an elemental diffracting volume (e.g., single structural domain) in the sample, the incident beam, considered as a sum of elemental rays, may be treated as highly coherent. The coherence volume of a ray. 2. The general description of coherence follows the papers by [Madsen2006] and [Lengeler2001]. 31.

(32) Structural changes in magnetite observed with coherent x-ray radiation. has a longitudinal dimension, parallel to the propagation direction and two, orthogonal, transverse dimensions. The longitudinal coherence length, is defined by a longitudinal coherence time ll = c. 0, where 0. 0. as. is related to the bandwidth ∆ of the light by ∆=1/ 0. Finally, the longitudinal. coherence length, defining a distance over which the phase of the field amplitude undergoes no fluctuations, is expressed by the expression: ll =. λ2 π ⋅ ∆λ. (2.2). .. The longitudinal coherence length is a property of the radiation source (the monochromator) and is a measure of its monochromaticity. For well monochromatized x-ray beam, a bandwidth is typically ∆/ = 10-4, therefore a longitudinal coherence length of about 0.5 m. Every real light source has a finite size and when the source is chaotic this will lead to reduction in interference contrast. The corresponding degree of spatial coherence can be studied by means of a Young double-slit experiment. Photons from each source point generate an interference pattern (Fig. 2.2a). If the extended source is a chaotic source then the interference patterns from the different source points will tend to smear out (Fig. 2.2b).. Fig. 2.2a Interference pattern in a Young double-slit experiment. Fig. 2.2b Loss of the interference contrast due to finite size of the light source. The transverse coherence length depends on the source via its spatial extension and on the geometrical arrangement of the whole measuring set-up via the source-aperture distance L. . For a source size d, a source-aperture distance L and a wavelength , the transverse coherence length is defined as: lt =. λL , π ⋅d. (2.3). An ESRF high -undulator with an electron beam size of 35 m FWHM in the vertical and 700 m in the horizontal has, for 12.4 keV photons at 40 m from the source, lateral coherence lengths of 72 m in the vertical and 3.6 m in the horizontal directions. For ID-10 beamline, where the experiments with the coherent x-rays were conducted, the lateral coherence lengths were: 10 µm in the vertical and 1 µm in the horizontal directions, for the wave length used in the experiment. 32. . = 1.5 Å.

(33) Structural changes in magnetite observed with coherent x-ray radiation. The key parameter which characterizes the sources is the degeneracy parameter ∆c. It is defined as the average number of photons per mode. Since a mode is an eigenstate of these photons are indistinguishable. Equivalently, ∆c is also the average number of photons in the coherence volume and may be expressed by: ∆ c = Br. λ3 , π ⋅c. (2.4). where Br is brilliance of the beam in the coherence volume. Chaotic sources can be made as coherent as unimodal laser radiation but only at the expense of the photon flux expressed by the degeneracy parameter ∆c. The degeneracy parameter for unimodal laser is ∆c = 107, whereas ∆c = 10-3 for an ESRF undulator. Experiments with coherent radiation are possible at the ESRF due to high brilliance of the beam. Finally, the detector collection time T should be properly chosen in terms of the dynamical processes in the sample. When the collecting time is shorter than the detector response time (about 10–6 s for the fastest detectors, such as avalanche diodes and streak cameras) than the detected interference pattern will be averaged over many different configurations of the structure in the sample. This will result with loosing speckle pattern in the scattered radiation. 2.3. Influence of the sample on coherence. Coherence of the incident beam does not guarantee the speckle pattern detection in the detector. The coherence may be lost during the scattering process due to disorder in the sample. Influence of disorder in the sample on the speckle pattern washout may be readily explained using results of a computer simulation presented below. A diffraction pattern from a disordered system is proportional to S(Q), describing a probability of a photon momentum change Q and is expressed as a squared modulus of a sum of amplitudes:. S (Q ) =. 2. ∑f. j. exp(iQ ⋅ r j ) .. (2.5). j. For a quadratic crystal with a total of N = M2 atoms, the lattice constant a and the atomic form factor f, where fraction of atoms is replaced by substitutional impurities with form factor f + f, the average scattering factor is:. f =.  1 ∑ f j  ,  N  j . (2.6). where the atomic form factor in the lattice position j is f j = f + δf j . 33.

(34) Structural changes in magnetite observed with coherent x-ray radiation. With an assumption that the structural factor is a result of coherent scattering, Eq. 2.5 may be rewritten as a sum of three components: S (Q ) = f. 2. ∑ exp(iQ(r. j. − ri )) +. ij.   +2 Re f *∑ δf j exp(iQ(r j − ri )) + ∑ δf i * δf j exp(iQ(r j − ri )) ij  ij . .. (2.7). The first term describes Bragg reflection, NSB, from the ideal average lattice. The second and third terms describe diffuse scattering and the speckle superposed to it. For an ideal lattice only the first term is nonzero. A). B). Fig. 2.3 Two sample areas with equal average periodic lattice, equal defect densities, equal short and longrange order but different individual arrangement of the defects. Fig. 2.3 shows two sample areas, A and B, with equal defects concentration and equal short and long range order. Nevertheless, the areas A and B have a different arrangement of defects. The structure factor S(Q) computed, according to Eq. 2.7, for such two quadratic lattices (with two different defect configurations), with N = 100 x 100, where 1 % of the original atoms (with atomic factor f) have been replaced statistically by impurities with the form factor 1.5 f is shown in Fig. 2.4. The results show that: -The Bragg peak along Q = (1,1) for the ideal average lattice is the same for both configurations. Note that side lobes result from the finite size of the crystal. -Defects generate diffuse scattering and speckle. Two different configurations of the defects show different speckle but the same diffuse scattering. -Averaging over many particular defect distributions washes out the speckle pattern. This is equivalent to illuminating a sample volume which is much larger than the coherence volume.. 34.

(35) NS(∆Q). Structural changes in magnetite observed with coherent x-ray radiation. 10. 8. 10. 7. 10. 6. 10. 5. 10. 4. 10. 3. 10. 2. 10. 1. 10. 0. 10. -1. 10. -2. 10. -3. 0.0. Fig. 2.4 Structure factor S(∆Q) (around Q = (1,1)) for two defect configurations in a square lattice with 100×100 atoms, 99% of which have form factor f and 1% have form factor 1.5 f. Both configurations have the same Bragg peak (1,1) and the same Laue diffuse scattering (solid horizontal line) but different speckle (black and red lines). 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ∆Q (2π/a). When the illuminated sample volume Vill is smaller than the coherence volume, Vi, then the individuality of an area gives rise to speckle in the Bragg and diffuse scattering. Its granular structure reflects the particular distribution in the defect distribution. If the defect distribution is changed without changing the average lattice, then the speckle pattern is changed whereas the Bragg reflections are unaltered. The short-range disorder in real space is reflected in a higher Qvector in the reciprocal space. This is how the structure can be studied by means of speckle pattern. In principle, the static configuration of defects in a sample can be studied by speckle spectroscopy but what is interesting from this Thesis point of view is to study the dynamical processes in the sample related to structural phase transition and the structural domain walls movement. 2.4. Scattering of coherent x-rays on two-phase systems. As shown in the previous section the crystallographic structure defects influence the structural factor S(Q) and this is reflected in the diffraction pattern. Thus, the speckle pattern contains information about short-range disorder. However, the disorder related to the coexistence of two lattices can also be studied with the coherent radiation. These two lattices result from the doubling of the unit cell in c direction caused by the Verwey transition; this is reflected by the appearance of superstructure peak, a useful tool to study the transition: once it appears, it means that the transition took place. Coexistence of two phases simultaneously, as in case of the structural transition, results in a perturbation of that superstructure peak. In this section the results of the diffraction pattern simulation for two-phase systems are presented.. 35.

(36) Structural changes in magnetite observed with coherent x-ray radiation. The structural factor S(Q), Eq. 2.7, was computed for sets of lattices simulating the low-T and the high-T structures of magnetite. For simplicity the simulation was performed on two dimensional systems of atoms. Magnetite has a cubic symmetry above TV, thus the quadratic lattice of two kinds of atoms, with two different values of fj, has been taken as the high-T structure. This lattice with the elementary unit cell marked with black square is shown in Fig 2.5a. The low-T phase has been, exaggeratingly, simulated by shifting every forth row by 1/2a, where a is the interatomic distance. This lattice, with the elementary unit cell marked by black rectangle, is presented in Fig 2.5b.. Fig. 2.5a 2D Simulation of high-T magnetite phase. Fig. 2.5b 2D Simulation of low-T, single domain magnetite phase; the lattice period is doubled in c direction. The diffracted spectra for the structures are presented in Fig. 2.6a and 2.6b, respectively. The simulations has been performed on quadratic lattices with N = 50 x 50 atoms with fb = 1 (black in Fig. 2.5) and fr = 2 (red in Fig. 2.5). No atomic defects has been implemented in the structure.. Fig. 2.6a Simulation of a diffraction pattern of the structure from Fig. 2.5a. Fig. 2.6b Simulation of a diffraction pattern of the structure from Fig. 2.5b. In both patterns (Fig. 2.6a and 2.6b) the main Bragg peak is observed for small values of (Qx,Qy) vector. Additionally, due to presence of two kinds of atoms in the structure, the period of the basic lattice increases in both dimensions in respect to monatomic structure, resulting in appearance of additional satellites in Q = ( ,0), (0, ) and ( , ) positions. The low-T phase, with 36.

(37) Structural changes in magnetite observed with coherent x-ray radiation. lattice doubling in the c-direction caused by atomic displacement, shown in Fig. 2.5b, causes the appearance of the superstructure peak at Q = (0, /2) position in the reciprocal space, as expected. The simplified picture of the transition, while it occurs on cooling, has been modeled and the result of the (0, /2) superstructure peak growth is shown in Fig. 2.7 (along Q=(l,l) direction). The starting point for the calculation was the high-T phase represented by the lattice in Fig. 2.5a. The transition has been modeled by the shift of atoms in the structure, as shown in Fig. 2.5b. The shape of the superstructure peak has been computed at a few stages of the transition, simulating the time flow under cooling. The intensity of the peak increases with increasing the fraction of the low-T phase. As before, the side lobes in the peak result from the finite size of the crystal.. Fig. 2.7 Development of the (0, /2) reflection (measured along Q = (l,l)) as the transition occurs (i.e. vs. cooling time). The simulations has been performed for 17%, 33%, 50%, 67%, 83%, and 100% fraction of the low-T phase in the high-T phase. The appearance of the structural domains is related to lowering of the structure symmetry that additionally modify the observed diffraction pattern. Thus, apart from the coexistence of the high-T and low-T phases at the transition, the superstructure peak is also influenced by the structural domain of the low-T phase. Such a combination of three phases (high-T and two domains with different doubling direction) is shown in Fig 2.8. The red line separates two phases whereas the black line imparts two structural domains. The shape of the diffracted intensity of the superstructure peak calculated for such atom arrangement is plotted in Fig 2.9 (along Q = (l,l) direction). The splitting of this reflection is caused by the difference in spatial arrangement of the structural domains. All above mentioned structural events contribute to the scattered intensity. First, each defects configuration (already described in Section 2.3) gives rise to a characteristic pattern. Then the coexistence of two the phases and structural domains that develop at the transition leads to static superstructure peaks. Finally, the superstructure peak pattern may contain information. 37.

(38) Structural changes in magnetite observed with coherent x-ray radiation. Fig. 2.8 The simulation of a transition development with high-T phase still present and two different low-T domains (with marked unit cells). Fig. 2.9 (0, /2) superstructure reflection (measured along Q=(l,l)) for a structure from Fig. 2.8 with two low-T domains and high-T phase still present. about dynamics of ordering processes. Thus, the diffraction pattern resulting from the experiment with coherent x-ray beam performed on real material is rich in structural information, but is very complex and the separation of each component might be very difficult. 2.5. Set-up of the TROIKA beamline ID10A at the ESRF. The experiments with coherent radiation have been carried out at the ID10A beamline of the European Synchrotron Radiation Facility in Grenoble, shortly described in this section. The ID10A beamline is a part of the TROIKA beamline located at the insertion device port ID10. It comprises two experimental stations: TROIKA I and TROIKA III. The undulator sources, the Front-End and the Optics-Hutch are common for both beamline branches (ID10A and ID10B). For the experiments the beam was yielded from two undulators3 installed in series in the ID10 high- straight section: one with 27 mm magnetic field periodicity undulator (U27) and the second one with 35 mm periodicity (U35), each of them 1.6 m long with the magnetic field 2 T. The schematic layout of the TROIKA beamline is shown in Fig. 2.10.. Fig. 2.10 Schematic layout of the ESRF TROIKA beamline (after [ID10_2007]). 3. 38. Currently there are three undulators.