induced phase transition studies in magnetoelectric bifeo3 ... · benjamin thibault ruette...

Induced Phase Transition Studies inMagnetoelectric BiFeO3 Crystals, Thin-

layers and Ceramics

Benjamin Thibault Ruette

Thesis submitted to the Faculty of the Virginia Polytechnic Institute and

State University in partial fulfillment of the requirements

for the degree of :

Master of Science

in Materials Science and Engineering

Dwight Viehland, Chair

Jie Fang Li

Richard O. Claus

may 2003

Blacksburg, Virginia

Keywords : Perovskite, Magnetoelectric, Phase transition, Bismuth Ferrite

Benjamin Thibault Ruette“Induced Phase Transition Studies in Magnetoelectric BiFeO3 Crystals, Thin-layers andCeramics”

Abstract

Bismuth ferrite (BiFeO3) is a magneto-electric material which exhibitssimultaneously ferroelectric and antiferromagnetic properties. We have used high-fieldelectron spin resonance (ESR) as a local probe of the magnetic order in the magnetic rangeof 0-25 Tesla. With increasing magnetic field, an induced transition has been found betweenincommensurately modulated cycloidal antiferromagnetic and homogeneous magnetized spinstate. The data reveal a number of interesting changes with increasing field, including: (i)significant changes in the ESR spectra; (ii) hysteresis in the spectra near the critical field. Wehave analyzed the changes in the ESR spectra by taking into account the magnetic anisotropyof the crystal and the homogeneous anti-symmetric Dzyaloshinsky-Moria exchange.

We have also investigated phase induced transition by epitaxial constraint, andsubstituent and cystalline solution effects.

Variously oriented BiFeO3 epitaxial thin films have been deposited by pulsed laserdeposition. Dramatically enhanced polarization has been found for (001)c, (110)c, and (111)c

films, relative to that of BiFeO3 crystals. The easy axis of spontaneous polarization lies closeto (111)c for the variously oriented films. BiFeO3 films grown on (111)c have a rhombohedralstructure, identical to that of single crystals. Whereas, films grown on (110)c or (001)c areexplained in terms of an epitaxially-induced transition between cycloidal and homogeneousspin states, via magneto-electric interactions.

Finally, lanthanum modified BiFeO3-xPbTiO3 crystalline solutions have been foundto have a large linear magneto-electric coefficient, "P. The value of "P (2.5x10-9 s/m or C/m2-Oe) is -10x greater than that of any other material (cg., Cr2O3 -2.5x10-10 s/m), and manyorder(s) of magnitude higher than unmodified BiFeO3 crystals. The data also reveal: (i) that"P is due to a linear coupling between polarization and magnetization; and (ii) that "P isindependent of dc magnetic bias and ac magnetic field. We show that the ME effect issignificantly enhanced due to the breaking of the transitional invariance of a long-periodspiral spin structure, via randomly distributed charged imperfections.

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Acknowledgements

The author would like to thank the following people :

Drs Dwight Viehland, Jie Fang Li and Shuxiang Dong for their help, valuable advice andsupport during my master thesis;

Dr Richard O. Claus for taking the time to help read my thesis, and serve in my committee.

Dr Anatolii Zvezdin, Alexander Pyatakov and Dr Sergei Zvyagin for their help, advice andcontribution in the experimental and theoritical work; Drs R. Ramesh and M. Wuttig forcollaborations on PLD films; Drs L. Cross and J. Cheng for collaborations on ceramics;

David Berry for their help in the experimental work; Cesar Foschini and Christelle Jullianfor their support and friendship;

et finalement mes parents, Dominique et André Ruette, pour leur inconditionel support,appui et encouragements tout au long de ma vie d’étudiant.

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List of figures

Figure 1.1 : A barium titanate (BaTiO3) unit cell in an isometric projection and viewed looking alongone face, which shows the displacement of Ti4+ and O2- ions from the center of the face.. . . . . . . 3

Figure 1.2 : Typical hysteresis loop for a ferroelectric materials showing domains evolution as theelectric field E is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 1.3 : Local magnetic moment (spin) induced by unpaired electrons of unfilled electron shells.The direction of the electron rotation give rise to two energy levels : spin “up” and spin “down”. 7

Figure 1.4 : Paramagetic (PM), ferromagnetic (FM) and antiferromagnetic (AFM) magneticorderings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 1.5 : Temperature dependence of magnetic susceptibility X for antiferromagnet, ferromagnetand paramagnet ². . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 1.6a : The magnetic moment on the left is converted to the one on the right by the operationof the symmetry element R... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 1.6b : Axial vector representations of the parallel m’ magnetic symmetry operation... . 15

Figure 1.7a : Example of an antiferromagnetic ordering where no spontaneous magnetization andpolarization occur [4]... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 1.7b : An electrical field is applied, unbalancing the magnetic and electric dipole inducingspontaneous polarization and magnetization [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2.1a : Representation of the perovskite unit cell of BiFe03 in the pseudo-cubic system. 25

Figure 2.1b : 3D representation of the BFO perovskite structure in an hexagonal system, thehexagonal direction [001]h being the cubic direction [111]c. For simplication oxygen anions are notrepresented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Figure 2.1c : Side view of BFO perovskite along [100]h.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Figure 2.2 : Variation of the real (g‘) and imaginary (g‘’) part of the dielectric constant of BFO singlecrystal grown along [100] with frequency 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 2.3 : Stereogrpah of the BFO magnetic point group 3m. . . . . . . . . . . . . . . . . . . . . . . . . . 30

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Figure 2.4 : BFO local antoferromagnetic structure expressed in an hexagonal system. Spin momentsare parallel to [110]h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Figure 2.5 : Influence of the orientation of Happlied on the induced polarization. P lies in (001)whatever is the direction of H.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Figure 2.6 : Components of the electric polarization of BFO along [100]h and [010]h (Pa and Pb) at20 °K as a function of the magnetic field applied 45° to the [100].[7].. . . . . . . . . . . . . . . . . . . .38

Figure 4.1 : Hexagonal representation of spin structure of BiFeO3. Both the antiferromagnetic orderand spin rotation planes are shown in this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 4.2 : Energy levels of an electron under an applied magnetic field. . . . . . . . . . . . . . . . . 44

Figure 4.3 : Schematical representation of an Electron Spin Resonance Spectrometer. . . . . . . . 47

Figure 4.4 : ESR spectra for BFO as a function of magnetic field between 0 and 25 Tesla for varioussub-millimeter frequencies between 1.15x1011 and 3.65x1011 Hz. . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 4.5 : Electron spin resonance frequency as a function of magnetic field H. . . . . . . . . . . 50

Figure 4.6 : Representative ESR signal illustrating hysteretic effects present in the intermediate fieldrange of 10< H <18 Tesla. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 4.7 : Polarization electric field behavior of a (001)-oriented BiFeO3 thin layer prepared bypulsed laser deposition, taken at room temperature. Inset (a) shows an enhancement of the inducedpolarization in the remanent state; and Inset (b) shows the magnetoelectric coefficient. These datawere taken from reference [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 5.1 : Reciprocal space scans for various oriented BiFeO3 single crystals. (a) (001)c orientation;(b) (110)c; and (c) (111)c orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 5.2 : Reciprocal space scans for various oriented BiFeO3 films. (a) (001) scan for (111)c film;(b) (110) scan for (111)c film; (c) (111) scan for (111)c film; (d) (001) scan for (110)c film; (e) (110)scan for (110)c film; (f) (111) scan for (110)c film; (g) (001) scan for (001)c film; (h) (110) scan for(001)c film; and (i) (001) scan for (111)c film. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 5.3 : Ferroelectric properties for (001)c, (110)c, and (111)c BiFeO3 films. (a) P-E curves; (b)P-E curves projected on (111)c; and (c) pulsed remanent polarization )P.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Figure 5.4 : Ferromagnetic properties for BiFeO3 films. (a) M-H curves for (001)c, (110)c, and (111)c

films, the red lines are fittings to the superparamagnetic equation of state; and (b) Ms as a function

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of film thickness for (001)c films. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Figure 6.1 : Large amplitude induced polarization and strain as a function of electric field forLa-modified BiFeO3-30%PbTiO3 ceramics. (a) Bipolar P-E response; and (b) bipolar g-E response.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 6.2 : Dielectric properties of La-modified BF-x%PT: (a) as a function of PT at roomtemperature for 0 at% La , taken at various frequencies; (a) as a function of PT at room temperaturefor 20 at% La, taken at various frequencies; and (c) as a function of temperature for BF-30%PT for10 and 20 at% La substitution, data taken a 106 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 6.3 :Magnetoelectric coefficient as a function of ac magnetic field for poled(Bi0.9La0.1)FeO3-30%PbTiO3. The left hand side shows the magnetoelectric polarization coefficient"P, and the right hand side shows the magnetoelectric field coefficient "E. The value of "P and "E

were independent of Hdc for 0<Hdc<3000 Oe. The measurement frequency was 103 Hz. . . . . . . 87

Figure 6.4 : Magnetic field induced polaration and voltage as a function of ac magnetic field forpoled (Bi0.9La0.1)FeO3-30%PbTiO3. The left hand side shows the induced polarization )P, and theright hand side shows the induced voltage )VME. The measurement frequency was 103 Hz. . . 88

Figure 6.5 : Magnetoelectric coefficient as a function of temperature for poled(Bi0.9La0.1)FeO3-30%PbTiO3. The value of "P and "E were independent of Hdc for 0<Hdc<3000 Oe.The measurement frequency was 103 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 6.6 : Magnetoelectric coefficient "P as a function of ac magnetic field for unpoled(Bi0.9La0.1)FeO3-30%PbTiO3. The value of "P was independent of Hdc for 0<Hdc<3000 Oe. Themeasurement frequency was 103 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Figure 6.7 : Magnetoelectric coefficient "P as a function of ac magnetic field for poled variousLa-modifed BF-x%PT compositions. (a) x=30 at% PT, and 20 at% La modification; (b) x=45 at%PT, and 10 at% La modification; and (c) x=20 at% PT, and 10 at% La modification [specimen couldnot be poled due to high Ec]. The value of aP was independent of Hdc for 0<Hdc<3000 Oe. Themeasurement frequency was 103 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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List of Tables

Table 1.1 : Example of perovskite-like structures that exhibit ME effect. Notation : FE - ferroelectric,AFE - antiferroelectric, FM - ferromagnetic, AFM - antiferromagnetic, WFM - antiferromagneticwith weak ferromagnetism, FIM - ferrimagnetic, TC - temperature of electric transition, TN -temperature of magnetic transition. 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Table 1.2 : Terms of the density of stored “free enthalpy” g for magnetic and electric phenomena.Terms of strain-electric and strain-magnetic are omitted [6].. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Table 1.3 : Symmetry of "ij determined by Neuman’s principle [5]. . . . . . . . . . . . . . . . . . . . . . . 22

Table 2.1 : Atomic positional parameters for BiFeO3 expressed in a rhombohedral system determinedby neutron diffraction 7.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Table 5.1 : Summary of interplanar spacings and lateral correlation lengths obtained from small areareciporcal lattice scans for the variously oriented BiFeO3 films and crystal. Peak splitting wasobserved along the (110) and (111), the relative intensities are designated by I, and the weaker peakis designated by brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1 : Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction Ferroelectric and Antiferromagnetic ordering . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Introduction to Ferroelectricity ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1.1 Source of polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1.2 Curie point and Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1.3 Hysteresis and Ferroelectric domains . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Introduction to Antiferromagnetic ordering . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2.1 Source of magnetic moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2.2 Magnetic ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.2.3 Antiferromagnetic ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Magnetoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.1 A review about magnetic symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1.1 Classical symmetry operations . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.1.2 Magnetic symmetry operation . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2 The magnetoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2.1 Magnetoelectricity : a brief historical review . . . . . . . . . . . . . . 141.2.2.2 Thermodynamic considerations . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2.3 Linear magnetoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.2.4 Symmetry considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 2 : The ferroelectromagnet BiFeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Ferroelectric ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Magnetic structure and antiferromagnetic ordering of BFO . . . . . . . . . . . . . . . . . . . 28

2.3.1 Local magnetic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Microscopic spin structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Magnetoelectric coupling in BiFeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 Linear and quadratic magnetoelectric susceptibility tensors for 3m

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2.4.1.1 Transformation law of a quasi-tensor . . . . . . . . . . . . . . . . . . . . 312.4.1.2 Application of Neumann’s principle . . . . . . . . . . . . . . . . . . . . . 322.4.1.3 Reduction of the 2nd rank tensor "ij to a 1st rank quasi-tensor

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 Thermodynamic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.3 Previous experimental studies of [ME]H in BiFeO3 . . . . . . . . . . . . . . . . . 36

Chapter 3 : Purpose of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 4 : Observation of magnetic-field induced phase transition in BiFeO3 by high-field ElectronSpin Resonance: cycloidal to homogeneous spin order . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Basics of Electron Spin Resonance (ESR) method . . . . . . . . . . . . . . . . . . 434.1.2 ESR experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 ESR Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.1 Resonance frequency vs. H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1.1 Low (0< H <10 Tesla) field range . . . . . . . . . . . . . . . . . . . . . . 514.2.1.2 Intermediate (10<H<18 Tesla) field range . . . . . . . . . . . . . . . . 524.2.1.3 High field range of H>18 Tesla . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Analysis according to Landau-Ginzburg theory and supporting evidence . . . . . . . . 554.3.1 Theory of the Field-induced Transition of BiFeO3 . . . . . . . . . . . . . . . . 554.3.2 Theory of the ESR Signal at Fields of H > Hc . . . . . . . . . . . . . . . . . . . . . 594.3.3 Predictions and Supporting Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4.1 Importance of Hc in design of advanced magnetoelectric materials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 5 : Epitiaxial-Induced Transitions in (001)c, (110)c, and (111)c BiFeO3 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 6 : Large Linear Magnetoelectric Effect in Modified BiFeO3 . . . . . . . . . . . . . . . . . . . . 816.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2.1 Dielectric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2.2 Magnetoelectric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

“Induced Phase Transition in Magnetoelectric BiFeO3 Crystals, Ceramics and Thin-Layers” Benjamin Ruette 1

Chapter 1 : Background

1.1 Introduction Ferroelectric and Antiferromagnetic ordering

In this chapter, the basic concepts of ferroelectric and antiferromagnetic orderings

will be introduced to give the necessary background to understand the following chapters.

Electronic, crystal, and microscopic properties, will be reviewed.

1.1.1 Introduction to Ferroelectricity ordering

Ferroelectricity (FE) was discovered in Rochelle Salt in 1921. At that time, it was

called Seignette-electricity, honoring its discoverer. The main properties of a FE material

are: (i) a spontaneous polarization on cooling below the Curie point; (ii) the presence of a

ferroelectric hysteresis loop in the P-E response; and (iii) the presence of ferroelectric

domains. In the 1950s, ferroelectric research was of a huge interest, leading to the modern

day electro-ceramics industry of barium titanate (BaTiO3) for capacitors of Pb(Zr,Ti)O3 for

piezoelectrics transducers. Since then, many other ferroelectric ceramics have been

discovered, amongst them are La-modified Pb(Zr,Ti)O3 or PLZT, and the relaxor

ferroelectric Pb(Mg1/3Nb2/3)O3 or PMN. These materials have found a wide range of

applications. In the last decade, ceramic processing and thin film technology have greatly

improved, and many more new applications have developed.


1.1.1.1 Source of polarization

Ferroelectric materials exhibit a spontaneous polarization Ps, in the absence of an

applied electrical field P. Physically, Ps is the value of the charge on the surface

perpendicular to the axis of the spontaneous polarization per unit area. This polarization is

likely to occur if (i) the crystal is noncentrometric; (ii) there are alternate equivalent atom

positions within the unit cell. These conditions permit the polarization under field, and its

retention after removal of field.

The origin of ferroelectricity can be explained using the well known example of

barium titanate (BaTiO3). The spontaneous polarization is due to the positioning of the Ba2+,

Ti4+ and 02- ions within the unit cell, as shown in Figure 1.1. The Ba2+ cations are located at

the corners of the unit cell, which in this case has symmetry. A dipole moment occurs due

to the relative displacements of the O2- and Ti4+ ions from their symmetrical positions as

shown in the front view of Figure 1.1. The O2- ions are positioned slightly below the center

of the six faces of the cube, whereas the Ti4+ ion is displaced upward from the unit cell.

Therefore, the unit cell has a permanent ionic dipole moment. However, at temperature

above a critical phase transition or Curie Point (120°C for BaTiO3), the ferroelectric order

disappears.

The FE ordering can be defined by its permanent polarization, which is induced by

the ion diplacements. In the example of barium titanate at 250 °C, Ps is parallel to [111]c. The

mathematical expression of Ps is :

[1]P q S d ds = +−12 02

1 2. . .( )

where “12“ is the number of charges lying on the (001) plane; q0 is the elementary charge;


Figure1.1 : A barium titanate (BaTiO3) unit cell in an isometric projection and viewed looking along one face,which shows the displacement of Ti4+ and O2- ions from the center of the face.


S is the surface of the plane (111) (unit area), within the unit cell; and d1 and d2 are the

relative displacements of the Ti4+ and Ba2+ from their symmetrical positions projected on the

plane (111).

1.1.1.2 Curie point and Phase transitions

All FE materials have a phase transition temperature called the Curie Temperature

or Tc. At a temperature T > Tc the crystal does not exhibit ferroelectricity, while for T < Tc

it is ferroelectric. When the temperature decreases through Tc, the crystal undergoes a phase

transition from a non-FE phase to a FE one. This is due to a crystal distortion that develops

at Tc, where the atoms are displaced in such a way that an internal polarization forms. This

results in a structural phase transition and a lowering of symmetry. The low-T phase, is

noncentrometric, and thus allows for spontaneous polarization. As a consequence the

dielectric, elastic, optical, and thermal constants exhibit anomalous behaviour near Tc.

1.1.1.3 Hysteresis and Ferroelectric domains

A P-E hysteresis curve of a ferroelectric is illustrated in Figure 1.2. The insets of this

figure illustrate change in FE domains that occur on cycling the field. A domain is a region

of the crystal where the polarization is aligned in a common orientational variation. Prior to

application of an electric field, the domains are randomly oriented. Thus, no net polarization

(Pnet) exists across the sample. As an electric field is applied and increased, the domains

aligned with E tend to grow in size; whereas the one perpendicular to E decrease in size. This

results in the development of a Pnet. At higher fields, the polarization response saturates.

Then, if E is reduced to zero, the domains that were aligned with E will not disappear.


Figure1.2 : Typical hysteresis loop for a ferroelectric materials showing domains evolutionas the electric field E is applied.


Rather a remanent polarization (Pr) is sustained, and the P-E response is hysteretic. It is

noteworthy that the theoritical Ps [1] is an upper limit since the experimental value is

approached asymptotically. However, when the field is reversed the domains will switch

between equivalent orientational variants. At a critical field, the domain variants have equal

population, and the net polarization is zero. At higher reverse fields, a net polarization will

develop, but in the opposite direction.

1.1.2 Introduction to Antiferromagnetic ordering

Naturally occuring magnetism has been known for centuries, even though the origins

of the phenomena was deemed mysterious. One of the first used natural magnets was the

mineral magnetite (Fe3O4), also called by its mineralogical name Lodestone. It was used in

compasses during the 13th century. In the 1950s, the modern age of synthetic magnets began.

However, in a number of applications, synthetic magnets have had a serious advantage as

most are on metals, and hence have low resistivities, high losses and eddy currents. Magnetic

oxides are known, in particular the ferrite family.

1.1.2.1 Source of magnetic moments

An atom is surrounded by negative charged electrons, rotating around a

positive charged nucleus. This creates an elementary magnetic momen; a spin. In non-

magnetic elements, the electron orbitals are filled, and thus the electrons rotate in pairs

canceling their magnetic moments (one spin is said to be up and the other to be down).

However, in magnetic elements containing unpaired electrons, a net magnetic moment

results, as shown in Figure 1.3. Commonly, in ceramics, the electronic structure of these


Figure 1.3 : Local magnetic moment (spin) induced byunpaired electrons of unfilled electron shells. The directionof the electron rotation give rise to two energy levels : spin“up” and spin “down”.


elements have a partially filled inner electron shell. These correspond to the transition

elements such as Fe, Co, Ni, rare earth elements, and actinide elements. As an example, the

cation Fe3+ (3d5) has five unpaired electrons, and thus a substantial magnetic moment.

However, the presence of magnetic moment does not guarantee that the crystal will exhibit

a net magnetic moment. This will depend on the atomic arrangement within the crystal

structure and the exchange energy.

1.1.2.2 Magnetic ordering

The arrangement of spins within a crystal structure will determine if the material has

a net magnetization or not. Spin arrangement is generally classified into four types of

magnetism: (i) paramagnetism (PM); (ii) ferromagnetism (FM); (iii) antiferromagnetism

(AFM); and (iv) ferrimagnetism (FIM). This classification describes how adjacent spins

interact with each other. In a paramagnetic material, alignment of adjacent spin is not

obversed. Ferromagnetism consist of parallel alignment of adjacent spins. Antiferromagnetic

order consists of antiparallel adjacent spins. And, ferrimagnetic order consists of antiparallel

adjacent spins of different spin moments, resulting in a non-zero net magnetization. Figure

1.4 illustrates PM, FM and AFM type spin orderings.

Magnetic susceptibility (Pij) is often set to characterize magnetic ordering. The

tensor Pij is the quantitative measure of the response of a material to an applied magnetic

field. The mathematical definition is given by the formula

; [2]M Hi ij j= χ .


Figure 1.4 : Paramagetic (PM), ferromagnetic (FM) and antiferromagnetic(AFM) magnetic orderings.


where Mi is the magnetization vector, and Hj is the applied magnetic field. Similar to

ferroelectric materials, magnetic ordering is dependent on temperature. Phase transitions

occurs at critical temperatures. With respect to the understanding of the present thesis, we

will consider antiferromagnetic ordering in more detail below.

1.1.2.3 Antiferromagnetic ordering

In an antiferromagnet, spins are aligned antiparallel. This results in spin cancelation,

and a zero net magnetization. Antiferromagnetic ordering is mathematically defined by the

antiferromagnetic vector, which is a double unit cell vector that join two spins of equal

energies (up and down).

Temperature dependence of the magnetic susceptibility

Figure 1.5 illustrates the temperature dependance of an antiferromagnet, paramagnet

and ferromagnet. These figures contain a significant amount of information concerning

magnetic materials. Contrary to paramagnetism and ferromagnetism, where Pij increases with

decreasing temperature. Antiferromagnetic materials exhibit a decrease of Pij with

decreasing temperature below the transition temperature TN (Néel temperature). This

temperature behaviour is due to the tendency to maintain AFM spin order that opposes

magnetization under the application of an external field. This results in a unique

characteristic temperature dependence of Pij, shown by the curve denoted P* where H is

applied parallel to the spin axis. When H is applied perpendicular to the spin axis,

magnetization rotation occurs as the spins move away from their easy axis. This

perpendicular susceptibility Pzis shown in Figure 1.5. In a polycrystaline AFM material, the


Figure 1.5 : Temperature dependence of magnetic susceptibility X forantiferromagnet, ferromagnet and paramagnet ².


P* + Pzresponses are combined to yield an average AFM response, as shown in Figure 1.5.

This average response has a “peak” at TN. Above TN, the AFM material is in a paramagnetic

state.

Type of antiferromagnet

Another concept has to be introduced to better understand antiferromagnetic order.

Antiferromagnets are subclassified according to their electronic structure. Two kind of AFM

materials have been identified. These are : (i) the Ising; and (ii) Heisenberg-like

antiferromagnets. In a Heisenberg type AFM material, the magnetic ion is a transition metal

one (Fe, Cr...). In a Ising type AFM material, the magnetic ion is a rare-earth (Dy,Ga...).

These two type of AFM states have unique distinguishable magnetization characteristics.

1.2 Magnetoelectric effect

The purpose of this section is to discuss the origin of the magnetoelectric (ME) effect

in crystals. We qualify here the term “ME” effect by the linear magnetoelectric effect.

Linearity, in this case means an induced polarization that is directly proportional to an

applied Hi, or conversly an induced magnetization that is directly proportional to an applied

E. In the following subsections, a brief introduction to magnetic symmetry, and Landau

thermodynamics of the ME effect will be discussed.


1.2.1 A review about magnetic symmetry

1.2.1.1 Classical symmetry operations

It is important to understand magnetic symmetry when dealing with magnetic

properties. Magnetic group symmetry can easily be derived from the “classical” point group

symmetry, simply by the inclusion of time inversion. In a three-dimensional crystal, it is

known that the only possible point group symmetry operations are:

. [3a][3b]N

N m

1 2 3 4 6

1 3 4 6

, , , ,

, , , ,

Amongst these, the first five given in [3a] are called the proper symmetry operations (or

symmetry operations of the first kind). When they operate, an object does not change

handedness or chiralty. The other operations given in [3b] are known as the improper

symmetry operations (or symmetry operations of the second kind). They change the

handedness of an object on which they are operated. These are the 1, 2, 3, 4 and 6-fold roto-

inversions.

1.2.1.2 Magnetic symmetry operation

When a crystal contains ions with magnetic moments, an additional type of symmetry

operation appears in order to define the symmetry of the magnetic structure. This operation

is called time reversal, designated as R. R is defined as a sign changing of the moment, when

it operates on an object that has a magnetic spin. In Figure 1.6a, the operation R transforms


the moment on the left to the one on the right. By applying the R operation on each of the

classical symmetry operators mentioned in equation [3], a new set of magnetic symmetry

elements is formed. These are represented as :

; [4]R n n

R n n

⊗ =

⊗ =

'

'

where the symbol q represents the operation of R on another given symmetry operation. As

an example, in Figure 1.6b, the operation R is applied to a mirror plane, creating the

magnetic symmetry operation m’.

The symmetry operations presented in [3] can be combined in numerous ways about

a point in space. These combinations are the 32 crystallographic point groups. These point

groups do not have R as a symmetry operation, and therefore can not be used to fully

describe the symmetry of crystals that have spin. The magnetic point groups are obtained by

grouping the time reversal operation and the classical symmetry; together, they can be used

to describe the symmetry of “magnetic crystals”.

1.2.2 The magnetoelectric effect

1.2.2.1 Magnetoelectricity : a brief historical review

The first report of magnetoelectricity was by Landau and Lifshitz [1], in the early 60s.

By the use of the Neumann’s principle, they predicted the existence of ME effects, by

determining the symmetry of the magnetic property tensors. The first explicit prediction of

a ME effect in a material was by Dzyaloshinskii [7] in the 60s, he showed that Cr2O3 had a


Figure 1.6a : The magnetic moment on theleft is converted to the one on the right bythe operation of the symmetry element R.

Figure 1.6b : Axial vector representations of the parallel m’magnetic symmetry operation.


Figure 1.7a : Example of an antiferromagneticordering where no spontaneous magnetization andpolarization occur [4].

Figure 1.7b : An electrical field is applied,unbalancing the magnetic and electric dipole inducingspontaneous polarization and magnetization [4].


ME effect, which is allowed by the magnetic symmetry. The ME effect was experimentally

observed for the first time in an unoriented crystal Cr2O3 crystal by Astrov [8-9]. Rado and

Folen [10-13] then the revealed anisotropic nature of the ME effect in oriented Cr2O3 crystals.

These investigations defined what we now call the electric field induced ME effect [(ME)E

effect]. Later, Rado and Folen observed the converse effect, the magnetic field induced ME

effect [(ME)H effect]. Figures 1.7a and 1.7b present a simplified conceptual model of a the

magnetoelectric effect. Table 2.1 presents a list of known magnetoelectric perovskites.

1.2.2.2 Thermodynamic considerations

The equation of state for magnetoelectricity has been derived by thermodynamic

considerations. The magnetoelectric effect is a secondary ferroic effect with a Gibbs energy

of the functional form :

; [5]dG SdT P dE M dH di i i i ij ij= − + + + ε σ

where S is the entropy, Pi the polarization vector, Mi the magnetization vector, ,ij the elastic

strain tensor, Ei the applied electric field, Hi the applied magnetic field, and Fij the stress

tensor.

For an isothermal condition, the term SdT is zero. And, assuming zero applied stress,

the last terms of [5] should also become zero. Using a Maclaurin two-variable expansion

(equivalent to a Newtonian two-variable series at 0), equation [5] becomes


Perovskite

Compound

ABO3

Type of

electric order

Type of

magnetic order

TC (°K) TN (°K)

Pb(Fe2/3W1/3)O3 FE AFM 178 263

Pb(Fe1/2Nb1/2)O3 FE AFM 387 143

BiFeO3 FE AFM 1123 650

Eu1/2Ba1/2TiO3 FE FM 165 4

Pb(Mn2/3W1/3)O3 AFE? AFM 473 203

Pb(Mn1/2Re1/2)O3 AFE? FIM 393 103

Pb(Mn1/2W1/2)O3 AFE? AFM 423 100

Pb(Fe1/2Ta1/2)O3 FE AFM 233 233

Pb(Fe1/2Re1/2)O3 AFE? FIM 433 > 293

Pb(Co1/2Re1/2)O3 AFE? AFM 403 < 77

Pb(Ni1/2Re1/2)O3 AFE? FIM 343 < 77

Pb(CO1/2W1/2)O3 FE WFM 68 9

BiMnO3 AFE FM 773 103

Cd(Fe1/2Nb1/2)O3 AFE? AFM 753 48

Bi2Bi4Fe2Ti3O18 FE FM? 1171 / 1025 723?

Bi9Ti3F5O27 FE WFM 1103 / 1073 363 / 403Table 1.1 : Example of perovskite-like structures that exhibit ME effect. Notation : FE - ferroelectric,AFE - antiferroelectric, FM - ferromagnetic, AFM - antiferromagnetic, WFM - antiferromagneticwith weak ferromagnetism, FIM - ferrimagnetic, TC - temperature of electric transition, TN -temperature of magnetic transition. 3


[6]G P E M H E E H H E H

E H H H E Ei i i i ik i k ik i k ik i k

ijk i j k ijk i j k

= + + + ++ + +

( / ) ( / )( / ) ( / ) ... ;

1 2 1 21 2 1 2

0 0ε ε µ µ αβ γ

where " is the linear magnetoelectric coefficient (2nd rank property tensor which contains 9

independant coefficients), and $ and ( are the quadratic magnetoelectric coefficients (3rd

rank property tensor, 27 coefficients). The ME coefficients "ij is a second rank tensor with

a maximum of nine independent coefficients. In matrix form, "ij is given as

. [7][ ]αα α αα α αα α α

=

11 12 13

21 22 23

31 32 33

Table 2.2 presents the possible couplings between magnetic and polarization ordering. Toderive the resultant polarization Pi change, equation [4] can be differentiate with respect toEk, given as

. [8]∂ ∂ ε ε α

β γG E P E H

H H H Hk k

Sik i ki i

kij i j ijk i j

/ ( / )( / ) ( / )

= + ++ +

1 21 2 1 2

0

The induced polarization is then the sum of the linear and quadratic [ME]h effects for E = 0,given as

. [9]P H H Hks

ki i kij i j= +α β( / )1 2


Terms of the density ofstored “free enthalpy”

- g =

Corresponding phenomena Name of coefficient

+κ io

iE. Ferroelectricity Spontaneous polarization

+X Hio

i. Ferromagnetism Spontaneousmagnetization

+ ( / ). .1 2 X H Hik i kInduced magnetisation Magnetic susceptibility

+αik i kE H. . Magnetoelectric effect Magnetoelectricsusceptibility

+ ( / ) . . .1 6 κ ijk i j kE E E Electro-optic effect Non-linear electricsusceptibility

+ ( / ). . . .1 6 X H H Hijk i j kMagneto-optic effect Non-linear magnetic

susceptibilty

+ ( / ). . . .1 2 α ijk i j kH E E Second ordermagnetoelectric effect (I)

First non-linearmagnetoelectricsusceptibility

+ ( / ). . . .1 2 βijk i j kE H H Second ordermagnetoelectric effect (II)

Second non-linearmagnetoelectricsusceptibility

Table 1.2 : Terms of the density of stored “free enthalpy” g for magnetic and electric phenomena.Terms of strain-electric and strain-magnetic are omitted [6].


1.2.2.3 Linear magnetoelectric effect

Equations for the linear magnetoelectric effect can be derived from equation [4],

given in Einstein summation notation, as

; [10]( ) :( ) : ( / )ME P HME M E

H i ij j

E i ji j

==αα µ 0

where (ME)H is the magnetically induced ME effect, (ME)E the electrically induced ME

effect, and µ0 the permeability of free space. In the SI system, the ME coefficient has the unit

of (s/m), whereas in Gaussian units, " = 4BP/H = 4BM/E, which is a dimensionless quantity.

1.2.2.4 Symmetry considerations

According to Neuman’s principle the symmetry of a crystal is related to the

symmetry of its properties. It is stated as : “The symmetry elements of any physical property

of a crystal must include the symmetry elements of the point group of the crystal.” An ME

effect is likely to be observed if at least one among of the nine coefficients in the

magnetoelectric susceptibility tensor is non-zero. Therefore, by applying the magnetic

symmetry of the crystal to the property tensor, the form of the ME coefficient matrix can be

derived for a particular material. Table 2.3 summarizes the results of applying Neuman’s

principle to the linear ME susceptibility to all different magnetic point groups.


α α αα α αα α α

11 12 13

21 22 23

31 32 33

α αα α

11 12

12 11

00

0 0 0−

α αα α

α

11 12

21 22

33

00

0 0

αα

α

11

11

33

0 00 00 0

0 00 0

0

13

23

31 32

αα

α α

αα

11

11

0 00 00 0 0

−

αα

α

11

22

33

0 00 00 0

0 00 0

0 0 0

12

12

αα−

0 00 0

0 0 0

12

21

αα

αα

α

11

11

11

0 00 00 0

α αα α

α

11 12

12 11

33

00

0 0−

Magnetic Point

Group

Matrix representation

of "ij

Magnetic Point Group Matrix representation of

"ij

1 1, ' 4', ,4'/m’4

2, m’, 2/m 422, 4m’m’, ,4 2' 'm

4/m’m’m’, 32, 3m’, 622,

6m’m’, , 3' 'm 6 2' 'm

6/m’m’m’

2', m, 2'/m 4'22, , 42m 42' 'm

4'/m’mm

222, m’m’2, m’m’m’ 42'2', 4mm, 4'/m’mm, 32',

62'2', 6mm,

6/m’mm, 3m,3'm

,4 2' 'm 6 2' 'm22'2', mm2, m’m2',

m’mm

23, m’3, 432, m’3m’,

4 3' 'm

4, , 4/m’, 3, ,4' 3'

6, , 6/m’6'

Table 1.3 : Symmetry of "ij determined by Neumann’s principle [5].


Chapter 2 : The ferroelectromagnet BiFeO3

In this chapter, background information will be presented concerning the structural,

electrical and magnetic properties of the perovskite-like BiFeO3 system.

2.1 Crystal structure

BiFeO3 or BFO was one of the first ferromagnetoelectric materials reported. It has

a rhombohedral distorted perovskite structure, whose crystal symmetry is R3c (or C63v) for

Tc. The rhombohedral unit cell parameters are ar = 5.61 D and "r = 59°40' (table 2.1 gives

the results of Neutron diffraction studies in rhombohedral phase). The pseudo-cubic

representation of these rhombohedral cell parameters are ac = 3.96 D and "c = 89°28'. The

Figure 2.1a illustrates the pseudo-cubic representation of the BFO crystal structure.

Any rhombohedral unit cell representation can be transformed to an equivalent

hexagonal one. The magnetic and electrical properties of the BFO crystal are often expressed

in the hexagonal cell, the pseudo-cubic direction [111]c corresponding to the hexagonal

[001]h. Figures 2.1b and 2.1c illustrate a 3-D view of the hexagonal cell and its side view

along [100]h, respectively.

In hexagonal representation, at room temperature, the cell parameters are ah = 5.58

D and ch = 13.9 D. The [001]h direction coincides with a 3-fold rotation axis and the [110]h

direction has a mirror plane. The relative positions of Bi and Fe ions are :


At 4.2 °K

Atom Position x y z B(D²)

Fe 0,083333 0 0 0 0.10

Bi 0,083333 0.2802 0.2802 0.2802 0.13

O 6b -0.3248 0.8044 0.2130 0.33

a = 5.617 " = 59.40

At Room Temperature

Atom Position x y z

Fe 0,083333 0 0 0

Bi 0,083333 2797 2797 2797

O 6b -3243 8026 2146

a = 5.616 " = 59.35Table 2.1 : Atomic positional parameters for BiFeO3 expressed in a rhombohedral system determinedby neutron diffraction [20].


Figure 2.1b : 3D representation of the BFO perovskitestructure in an hexagonal system, the hexagonal direction[001]h being the cubic direction [111]c. For simplicationoxygen anions are not represented.

Figure 2.1a : Representation of the perovskite unit cell ofBiFe03 in the pseudo-cubic system.


Figure 2.1c : Side view of BFO perovskite along [100]h.


; [11]Bi s Fe th h( , , / ) ( , , )0 0 14 0 0+

where s and t are the atomic displacements from their centro-symmetric positions. For BFO,

s = 0.05 and t = 0.02. Due to polarization and spin orderings, as to be discussed in the next

sub-chapters, the crystal structure undergoes a sequence of structural phase transitions at TN

= 400 °C and Tc = 830 °C, corresponding to the Néel and Curie points respectively.

2.2 Ferroelectric ordering

Bismuth ferrite undergoes a ferroelectric ordering along [001]h at a Tc of 830 °C. The

polarization is due to the cooperative distortion of the Bi3+ and Fe3+ cations from their centro-

symmetric positions. The spontaneous polarization Ps is oriented along the pseudo-cubic

[111]c, as shown in Figure 2.1b. The characterisation of the dielectric and polarization

properties of BiFeO3 have proven difficult due to a low electrical insulation. During the

crystallisation process, the stoichiometry of iron is variable (Fe2+ vs. Fe3+), and thus sensitive

to oxygen content. The mixed valence condition results in high conductivity, due to valence

band doping. Previous investigations of the P-E behavior of oriented single crystals have

shown a Ps of 0.035 C/m2 along the pseudo-cubic [001]c at 77 °K, this corresponds to a Ps

of 0.061 C/m2 along the pseudo-cubic [111]c.

However, recent investigation of single crystalline thin layers fabricated by pulsed

laser deposition (PLD) have shown a much higher Ps along [001]c and [111]c, approaching

values of 0.6 C/m2 and 1 C/m2 at room temperature, respectively. The resistivity of these

films were quite high, in the range of 109-1010 S/cm. The dielectric constant of [001]c

BiFeO3 crystals is approximately 70-80 at room temperature, as shown in Figure 2.2.


2.3 Magnetic structure and antiferromagnetic ordering of BFO

2.3.1 Local magnetic structure

The magnetic point group of BiFeO3 is 3m, and TN = 400 °C. The magnetic structure

has a 3-fold rotation axis along [001]h and a mirror plane m along [110]h, as illustrated in the

stereographic projection in Figure 2.3. The magnetic order is consequently G-type

antiferromagnetic, it is a Heisenberg type antiferromagnet.

In this structure each Fe 3+ cation is surrounded by six nearest Fe 3+ neighbours, which

have an opposite spin direction. Figure 2.4 illustrates the hexagonal representation of the

BiFeO3 spin structure where (110)h is the spin rotation plane. However, since the bond angle

Fe - O - Fe is not exactly 180°, the spins do not completely cancel. Hence BFO has weak

ferromagnetism at room temperature, due to spin canting. The nearest neighbours of the iron

cations are oxygen anions. Thus, the antiparallel arrangement of the spins is not a direct spin-

spin interaction. Rather, interactions are mediated by a so-called “superexchange”.

Essentially, the spin moments of the Fe3+ species on the opposite side of O2- interact with

each other via the p-orbit of oxygens.

2.3.2 Microscopic spin structure

The spin rotation plane gives rise to a canting of the magnetic moment.

Consequently, microscopically, the antiferromagnetic spin order is not homogeneous for

BFO single crystals. Rather, the spins form a cycloidal structure with a length modulation

of 620 D, which is oriented along the [110]h direction.


Figure 2.2 : Variation of the real (g‘) and imaginary (g‘’) partof the dielectric constant of BFO single crystal grown along[100] with frequency [21]


Figure 2.3 : Stereogrpah ofthe BFO magnetic pointgroup 3m.

Figure 2.4 : BFO local antoferromagnetic structure expressed in an hexagonalsystem. Spin moments are parallel to [110]h.


This result in a net zero value for the antiferromagnetic vector, as the mean value of the

projections cancel. As we will show in the results of this thesis in chapter 4, the cycloidal

structure of BFO is destroyed by the application of high magnetic field, and a homogeneous

antiferromagnetic spin order is induced [20-23, 25].

2.4 Magnetoelectric coupling in BiFeO3

To understand the magnetoelectric effect of BiFeO3, the linear and quadratic

magnetoelectric susceptibility tensors can be simplified by use of Neuman’s principle. It is

assumed that (i) the magnetic point group symmetry is 3m; and (ii) the spin order is of a

homogeneous antiferromagnetic type. Subsequently, in the following sections, the tensor

representation will be used within the Landau thermodynamic framework. Equations for the

magnetic-induced polarization will be given and compared to previous experimental studies.

2.4.1 Linear and quadratic magnetoelectric susceptibility tensors for 3m

2.4.1.1 Transformation law of a quasi-tensor

The linear ME susceptibility tensor is a quasi-tensor. Unlike conventional 2nd rank

property tensors, it has a handness. Therefore, tensor transformation laws are different,

involving a time reversal constant R (either -1 or +1),

; [12]T R a a a Tij ik jl kl' =

where *a* is the determinant of the transformation matrix a.


2.4.1.2 Application of Neumann’s principle

The symmetry of the second rank axial tensor "ij, was determined by a direct

inspection method. The 3m point group consists of one 3-fold rotation axis directed along

[001]c and a mirror plane parallel to [110]c. The 3-fold rotation matrix transformation is

given as :

. [13][ ]aCos SinSin Cos=

° − °° °

( ) ( )( ) ( )120 120 0120 120 00 0 1

Applying this transformation law to the 2nd-rank axial tensor of [12] yields :

; [14]α α αij ik jl kl ija a a' = + =

where . [15]α α αα α α α α

13 23 31

32 11 22 12 21

0 0 00

= = == = = −

, , ,, ,

The symmetry of the tensor can be further simplified by applying the mirror operation along

[110]c. This results in "12 = -"21 and "11 = "22 = 0, summarized in matrix form as,

. [16][ ]αα

α= −

0 00 0

0 0 0

12

12

3m

2.4.1.3 Reduction of the 2nd rank tensor "ij to a 1st rank quasi-tensor

It is important to note that " is an antisymmetric tensor. An antisymmetric tensor


does not have real eigen values. To readily illustrate the concept of an antisymmetric tensor,

it is necessary to perform a mathematical transformation from a 2nd rank tensor

representation to a 1st rank quasi-tensor one. The “Levi-Civita symbol” is defined as,

. [17]ε ijk

if ijk is aneven permutation ofif ijk is anodd permutation of

if any ijk areequal=

+−

1 1 2 31 1 2 3

0

, ,, ,

A vector A = g .["] is assumed. The components of this vector are given as,

. [18]rA

AAA

=

=

1

2

3 12

00

2α

The general transformation vector v = [x, y, z] is then

. [19]r r rv v Aijk. ( / ). . .α ε= 1 2

The right hand side of equation [19] is by definition a cross product. Equation [19] then

simplifies to

. [20]r r rv A v. ( / ).α = ×1 2

Assuming that the vector v is the magnetic field H,

. [21]r r rH A H. ( / ).α = ×1 2

The left hand side of this equation is the magnetic field induced polarization ,P Hi ij j=α


which can be expressed as,

. [22]r r rP A H= ×( / ).1 2

Equation [22] states that the magneto-electric polarization is the cross product of the ME

susceptibility tensor and the magnetic field axial tensor. Since A1 = A2 = 0 and by definition

of a cross product, the polarization will lie in the (001)c plane, independent of the direction

along which a magnetic field is applied. In other words, Pi is independent of the components

of Hi along [001]c as shown in Figure 2.5. In the cartesian system, in which the calculations

were performed, the linear ME tensor has no term in its diagonal. Thus it is antisymmetric.

In an alternative general system, the tensor "ij’would have both antisymmetric and

symmetric nature given as

. [23]α α α' ' '= +S A

This antisymmetric component of "’A represents a toroidal magnetic moment Ti, and the

symmetric part "’S represents the conventional axial magnetic moment.

2.4.2 Thermodynamic considerations

Using the "ij and $ijk tensors for the 3m magnetic point group in the last section,

thermodynamic relationships can be established for the magnetic induced polarization,

. [24]P P H H Hi si

ik k ijk j k= + +α β. ( / ). . .1 2

This equation contains a induced polarization dependent of the both linear and non-linear


Figure 2.5 : Influence of the orientation ofHapplied on the induced polarization. P lies in(001) whatever is the direction of H.


powers of H. The spontaneous polarization is non-zero only along the hexagonal [001]h

direction, given as

. [25]P Psi

s h= ( , , )1 0 0

Under an externally applied magnetic field H = [H1, H2, H3], the complete induced

polarization vector, taking into account the form of the " and $ tensors for the 3m point

group is

[26]

P P H H H H H H HH H

P H H H H H H HH H

P H H

s11

12 2 222 22

1 2 12

223 1 3

232 1 2

2 12 1 222 22

1 2 12

223 2 3

232 3 2

3 322 12

22

1 2 2 1 21 2

1 2 2 1 21 21 2

= + + − − ++= − + + − ++= +

α β ββ

α β βββ

. ( / ). .( . . ) ( / ). . .( / ). . .

. ( / ). .( . . ) ( / ). . .( / ). . .( / ). .( ).

2.4.3 Previous experimental studies of [ME]H in BiFeO3

Although allowed by symmetry, the linear ME effect has yet to be observed at low

magnetic fields in BiFeO3 [20,21].

Due to the cyloidal spin structure, the average value of the projection of the

antiferromagnetic vector is zero over the modulation of the cycloid. Thus, the linear

magnetoelectric effect is averaged out. A small quadratic magnetoelectric effect has been

measured at higher magnetic fields [20,21]. Figure 2.6a shows the components of the induced

polarization of BiFeO3 along the a and b axes as function of H (0 < H < 3.105 Oe) applied

along [110]h at T = 20 °K. Figure 2.6b shows the induced polarization along a and b when

the H field is applied along the [001]c at T = 18 °K. The values of the linear and quadratic

magnetoelectric coefficients have been reported to be


[27]

ααβ

126 2

216 2

22211 2 2

0 029 0 003 100 032 0 003 10

510

= − ±= + ±=

−

−

−

( . . ). / ( . )( . . ). / ( . ). / ( . ).

C m kOeC m kOe

C m kOe

An asymmetry in the linear magnetoelectric tensor can be seen in [27] confirming

then presence of a magnetic toroidal moments ("12 = -"21.). Also, the data illustrate the

presence of a critical field above which there is significant change in polarization. This

demonstrates a magnetic field induced phase transition at high fields. The purpose of this

thesis is to investigate this induced phase transition and its ramifications.


Figure 2.6b : Components of the electricpolarization of BFO along [100]h and [010]h at18°K as a function of the magnetic field appliedalong [001]h. [20]

Figure 2.6a : Components of the electricpolarization of BFO along [100]h and [010]h (Paand Pb) at 20 °K as a function of the magneticfield applied 45° to the [100].[20]


Chapter 3 : Purpose of research

The purpose of this thesis was to investigate induced phase transitions in BiFeO3

single crystals, thin films and ceramics. Previous investigations have shown that the spin

cycloid results in the cancellation of the antiferromagnetic vectors. Therefore, the linear

magnetoelectric effect is averaged out to zero.

Induced phase transitions from the spin cycloid to a homogeneous spin ordered state

may result in significant enhancements in the magnetoelectric behavior, in addition to the

polarization and magnetization responses. In this thesis, three ways have been found to

stabilize a homogeneous spin ordered state. These are: (i) application of high magnetic field

to oriented single crystals; (ii) by epitaxial constraint which fragments the cycloid and (iii)

by aliovalent and crystalline solution effects which frustrate the magnetic exchange.

In chapter 4, a magnetic-field induced transition in BiFeO3 single crystals will be

shown by high-field electron spin resonance. The data demonstrate an induced transition

from an incommensurate cycloidal modulated state to one with homogeneous spin order. In

chapter 5, epitaxial constrained thin films of BiFeO3 will be shown to have dramatically

higher ME coefficients, magnetization, and polarization. In chapter 6, random-fields

associated with aliovalent substituent modification of BiFeO3 ceramics will be shown to

disrupt the spin cycloid, resulting in dramatically higher magneto-electric (ME) coefficients.

The room temperature value of "p (2.5x10-9 C/m²-Oe) is -10x greater than that of any other

material, and many order(s) of magnitude higher than unmodified BiFeO3 crystals.


transition in BiFeO3 by high-field Electron Spin Resonance:

cycloidal to homogeneous spin order

Ferromagnetoelectric materials have two order parameters [26]. These are a

spontaneously polarization (ferroelectric or antiferroelectric) and a spontaneous

magnetization (ferromagnetic or antiferromagnetic). A Landau phenomenological theory of

multiferroic materials has previously been developed by Salje [27]. The magneto-electric

effect is the coupling between the spontaneous polarization Ps and the spontaneous

magnetization Ms. Phenomenologically, this coupling can be either mediated by linear,

biquadratic, or even higher order exchanges [27].

The average crystal lattice of BiFeO3 is a rhombohedrally distorted perovskite

structure [30,32-37], which belongs to the space group R3c (or ), as shown in Figure 4.1.C63ν

The hexagonal unit cell parameters are a = 5.58 Å and c = 13.9 Å. In this structure, along the

3-fold pseudocubic [111]c rotation axis, the Bi3+ and Fe3+ cations are displaced from their

centro-symmetric positions. This centrosymmetric distortion is polar, and results in a

spontaneous polarization (Ps), as illustrated in Figure 4.1. The antiferromagnetic (TN ~ 643

°K) order of BiFeO3 is of the G-type [26,32]. In this arrangement, the Fe3+ cations are

surrounded by six nearest Fe3+ neighbors, with opposite spin directions. Along the

[001]h/[111]c, BiFeO3 has antiferromagnetic order; however a spin rotation plane exists

parallel to the [110]h. The antiferromagnetic order and spin rotation plane are illustrated in

Figure 4.1.


Figure 4.1 : Hexagonal representation of spin structure of BiFeO3. Both theantiferromagnetic order and spin rotation planes are shown in this figure.


Microscopically, the antiferromagnetic spin order is not homogenous for BiFeO3

single crystals. Precise neutron diffraction studies [40,41] have revealed an incommensurately

modulated spin structure which manifests itself as a cycloid with a long wavelength 8 of ~

600 Å. In the incommensurate phase the periodicity of the spin-polarization is

incommensurate with crystallographic lattice parameters. The cycloidal spin structure has

been shown to be directed along the [110]h. The existence of the incommensurate cycloid has

been confirmed by line shape analysis of nuclear magnetic resonance (NMR) spectra [42,43].

In BiFeO3, the spin profile is nearly a linear function of the space coordinate in the direction

of the modulation: it is only slightly anharmonic [44]. However, in general, incommensurate

phases can have spatial profiles that are strongly anharmonic [45], consisting of nearly

commensurate regions separated by domain walls (i.e., solitons) where the phase of the order

parameter changes abruptly.

The intrinsic spin-phonon coupling is generally small [26], as it involves a relativistic

(Dzyaloshinksy-Moria) interaction between the two order parameters Ps and Ms.

Phenomenologically, the linear exchange between the two in the Landau-Ginzburg equation

of state, i.e. Pi = QijMj, inherently involves a coupling between the linear power of a polar

tensor and a linear power of an axial tensor [46-52,6]. This intrinsic linear coupling coefficient

Qij does not have a normal symmetric tensor matrix as a solution. Rather, the linear

magneto-electric exchange must have an anti-symmetric component in its property tensor

[21-28], in addition to the usual symmetric one. In BiFeO3, the conventional

Dzyaloshinksy-Moria interaction is zero. It is forbidden by the space group symmetry.

However, a linear magneto-electric-like Dzyaloshinksy-Moria interaction is allowed by the

3m magnetic point group [49]. Due to the cycloidal spin structure, the average value of the


projection of the antiferromagnetic vector is zero. Thus, the linear magnetoelectric effect is

averaged to zero [53-55].

A quadratic magnetoelectric effect has been measured [56]. Experimental

investigations have focused on the low-field regime. The cycloidal spin order can be changed

by application of magnetic field H. Induced phase transitions have been reported under high

pulse magnetic fields for H between 20 to 25 Tesla [53-55]. Significant changes in spontaneous

polarization have been shown to occur at this H-induced transition, indicating changes in the

magnetoelectric coupling [53-55]. If a homogeneously magnetized state is induced under high

magnetic field, a transition between cycloidal and uniform spin order may occur. Thus, to

understand the intrinsic magnetoelectric coupling of BiFeO3, it is important to understand

the nature of this high H induced phase transition.

In this thesis, we have studied the local spin structure of BiFeO3 as a function of

magnetic field for 0 < H < 25 Tesla using high-frequency/field electron spin resonance

(ESR). We have identified a H-induced transition from the incommensurate cycloidal spin

structure to a homogeneous magnetized state. This transition to a homogeneous magnetized

state has important ramifications on the magneto-electric effect.

4.1 Experimental details

4.1.1 Basics of Electron Spin Resonance (ESR) method

When an electron is placed in a magnetic field, the energy of an electron spin pair

split as shown in Figure 4.2. This splitting is called the Zeeman effect, and is given by the

spin hamiltonian


Figure 4.2 : Energy levels of an electron under anapplied magnetic field.


; [28]r rH g B Ss B z= . . .µ

where B is the magnetic field, Sz is the component of the spin moment in the direction of H,

µB is the Bohr magneton, and g is known as the "g-factor". The parameter g depends only

on the electronic structure of the species. The g-factor gives important structural information,

as it relates the frequency at which a species resonates to an applied H.

The electron spin energy levels (corresponding to the lines in Figure 4.2) can be

found by applying Hs to the electron spin eigenfunctions (ms = ±1/2), given as

. [29]rH g B Es B( / ) . . . ( / ) ( / )± = ± ± = ±±1 2 1

21 2 1 2µ

The energy split between the two induced levels is

. [30a]∆E E E g BB= − =+ − . .µ

This corresponds to a micro-wave energy of

; [30b]h g BB. . .ν µ=

where L is the micro-wave frequency. For example, in the simplest case of a single electron,

if a magnetic field of 15000 G is applied, the corresponding radiation necessary to split the

electron spin energy level is L = 42 GHz.

Under an applied H, a transition between spin states can be induced by absorption

of electromagnetic microwaves or photons. Absorption is proportional to the number of spin

in the lower level, and the emission of photons is proportional to the number of spins in the

higher level. The Electron Spin Resonance (ESR) method consist of measuring the net


absoption (difference between number of photons absorded and emitted) as a function of an

applied magnetic field in the microwave frequency range.

4.1.2 ESR experimental setup

Bismuth ferrite single crystals were grown by a flux method from a

Bi2O3-Fe2O3-NaCl melt. Single crystals were oriented along the [001]c. Specimens were cut

into dimensions of 1x1x0.5 mm3. The structure was rhombohedral (or hexagonal) and the

crystal lattice parameters were a = b = c = 5.61 Å and a = 89.28° (or, a = 5.58 Å and c = 13.9

Å).

ESR is an extremely powerful tool to test magnetic excitation spectra in solids,

providing important information on a magnetic structure and main parameters of the

effective spin Hamiltonian. In order to test the field-induced phase transition in BiFeO3,

high-field ESR measurements were performed using the sub-millimeter facility at the

National High Magnetic Field Laboratory (NHMFL, Tallahassee, FL) [Zvyagin], in fields

up to 25 T as schematically illustrated in Figure 4.3. Investigations have been done in the

frequency range of 115-360 GHz. Quasi-continuously re-tuned sources of millimeter and

sub-millimeter wave radiation, Backward Wave Oscillators, were used. Transmission type

ESR spectrometer with oversized cylindrical waveguides and a sample-holder in the Faraday

configuration (with a wave propagation vector parallel to the external field) was employed.

A high-homogeneity (12 ppm/cm DSV) magnetic field was produced by the 25 T

hysteresis-free resistive Bitter-type W.M. Keck magnet. In our experiment, the magnetic

field was oriented along the pseudo-cubic [001]c direction. Since absorption lines were


Figure 4.3 : Schematical representation of an Electron SpinResonance Spectrometer.


relatively broad, an optical modulation of the microwave power was used. The spectra were

recorded during the magnetic field sweeping. A low noise, wide frequency range, InSb hot

electron bolometer, operated at liquid-He temperatures, served as a detector. Experiments

were performed at a temperature of 4.2 °K.

4.2 ESR Spectra

Figure 4.4 shows the ESR signal as a function of magnetic field for 0 < H < 25

Tesla, taken at various frequencies between 115 < L < 360 GHz. This figure shows dramatic

changes in the ESR spectra for the various frequencies. At lower frequencies L = 115 Ghz,

a single absorption peak was found at lower magnetic fields, as illustrated by a black arrow.

With increasing L, this absorption peak was continuously shifted to higher fields (L ~ H) and

the degree of absorption was increased, as illustrated by black arrows. Additional peaks were

found in higher fields, denoted by blue and red arrows.

Of particular interests was the peak indicated by red arrows in Figure 4.4, which

became apparent for L > 200 GHz. This resonance absorption was quite pronounced, and in

fact the attenuation of the sub-millimeter radiation in the crystal at this resonance was

significant. With increasing n this peak continuously shifted to higher H (L ~ H), the

adsorption became increasingly pronounced, and the peak became increasingly broad.

Clearly, under high magnetic field, a secondary resonance state is induced. The shape of this

secondary resonance peak is noticeably dependent upon H and L.


Figure 4.4 : ESR spectra for BFO as afunction of magnetic field between 0 and 25Tesla for various sub-millimeter frequenciesbetween 1.15x1011 and 3.65x1011 Hz


0 5 10 15 20 250

1x1011

2x1011

3x1011

4x1011

Fre

quen

cy, H

z

Magnetic Field, Tesla

Figure 4.5 : Electron spin resonance frequency as a function of magneticfield H.


4.2.1 Resonance frequency vs. H

The experimental observations of the dependence of the resonance frequencies L on

H are summarized in Figure 4.5. The maximum of the absorption has been chose to indicate

the resonance field. Clearly, there is more than one resonance mode present over the

magnetic field range investigated.

4.2.1.1 Low (0< H <10 Tesla) field range

A low field spin resonance mode was observed that had a linear relationship between

L and H, which had a slope of ~ 27 GHz/Tesla (that correspond to g ~ 2.0) and a slope

intercept of zero. These data are illustrated as black points in Figure 4.5. This is the spin

resonance mode of the incommensurate cycloidal spin structure whose modulation

wavelength is 8 ~ 600 Å. The profile of this cycloidal modulation has previously been

measured by temperature and field dependent NMR studies [42,43]. The spin profile is slightly

anharmonic at 4.2 °K, becoming increasingly sinusoidal with increasing temperature, and

increasingly anharmonic with increasing H.

The quadratic magnetoelectric effect has been shown to be a result of changes in the

spin profile and modulation wavelength of the cycloidal spin structure [54,55]. According to

Landau theory, incommensurate phases are sandwiched in-between the high temperature

prototypic higher symmetry and the lower temperature transformed phases [46]. Changes in

the spatial profile of incommensurate structures [45,57,58], in soliton density [45,59,60], and soliton

dimensionality (1-q vs. 3-q) [61,62] have been reported with increasing magnetic and/or

electric field that result in changes in physical properties.


4.2.1.2 Intermediate (10<H<18 Tesla) field range

The low field spin mode was found to continue until higher fields, maintaining the

same slope value of geµB. However, an "intermediate field range" of 10 < H < 18 Tesla can

be designated, based upon the presence of an additional resonance peak whose resonance

frequency L decreased with increasing H, as shown in Figure 4.5 by blue points. This is an

anomalous result as both the g-factor and magnetic permeability are positive. It is an unusual

spin mode that can be attributed to dynamic effects. Locked incommensurately modulated

structures are known to become de-pinned under high drive [45].

In addition, hysteretic effects became very pronounced between ESR spectra

obtained on increasing vs. decreasing H sweeps. This is further illustrated in Figure 4.6,

which shows a representative ESR signal taken at a measurement frequency of 236 GHz. In

this figure, significant hysteretic effects were found in the intermediate field range of 10 <

H < 18 Tesla. Thermal and electrical/magnetic hysteresis effects are well known in locked

incommensurate structures [45], seen for instance in the field-induced incommensurate phase

of CuGeO3 [63,64]. Unlocking of incommensurate structures under excitation is known to

produce cascade type events [45]. Induced transformations from incommensurate to

commensurate states are known to proceed through cascade events in materials systems

which have significant concentrations of frozen-in defects of the random-field type [65].

The data support a model of an incommensurately modulated cycloidal structure that

is pinned. The cycloidal spin structure is changed by application of H. With increasing H,

the structure becomes un-pinned, resulting in dynamic effects. Unlocking of the

incommensurate structure, and excitation of its amplitude and phase, should result in the


Figure 4.6 : Representative ESR signal illustrating hystereticeffects present in the intermediate field range of 10< H <18Tesla.


eventual destruction of the incommensurate modulation and stabilization of a homogeneous

spin order.

4.2.1.3 High field range of H>18 Tesla

Figure 4.5 clearly demonstrates an induced phase transition near H = 18 Tesla. In the

high field range, a secondary spin resonance mode was observed that had a linear

relationship between v and H. These data points are shown in red in Figure 4.5. The slope

of this secondary spin mode was ~½ of that of the low field one, having a value of 13

GHz/Tesla. In addition, the slope intercept was non-zero. No hysteretic effects were

observed in the ESR spectra, within this secondary mode.

It is important to notice that the primary (low-field) spin mode can be linearly extrapolated

into the high field regime, revealing the presence of a field induced gap state. Due to strong

microwave absorption at higher frequencies, it was found difficult to experimental confirm

the linear extrapolation. However, the ESR spectra at 365 GHz clearly show this primary

resonance persisting, which was of noticeable absorption. The data indicate an induced gap

for H > 18 Tesla. But, it is unclear how far into the high field regime that this gap continues.

It is not clear whether this gap is an equilibrium characteristic of the state, or whether it is

due to the non-equilibrium co-existence of various types of spin order over a field range.

To understand the nature of the secondary resonance mode, it is thus necessary to

develop a theoretical approach. We conjecture that the induced phase transformation is from

the cycloidal to a homogeneously magnetized state. In the following section, we shall present

this theoretical formulation, and show supporting evidence.


4.3 Analysis according to Landau-Ginzburg theory and supporting evidence

4.3.1 Theory of the Field-induced Transition of BiFeO3

The BiFeO3 spin structure is characterized by the unit antiferromagnetic vector rl

that in spherical coordinates can be represent as = (sin2cosn, sin2sinn, cos2), where 2 isrl

the polar angle of the vector, n is the azimuthal angle of , and is the polar axis. Therl

rl $c

Landau-Ginzburg energy [67] of the spin structure is the sum

[32]F F F F FL exch an m= + + +

The first term FL in [32] is the magneto-electric coupling that is linear in gradient (i.e.,

Lifshitz invariant), given as

[33a]F P l l l l PL z x x z y y z z x y= − ⋅ ∇ + ∇ = − ⋅ ∇ + ∇α α θ θ ϕ θ ϕ( ) sin ( cos sin )2

where Pz is spontaneous polarization, and " is the inhomogeneous relativistic exchange

constant (inhomogeneous magneto-electric constant). The Lifshitz invariant is responsible

for the creation of the spatially modulated spin structure in BiFeO3, as will be shown below.

The second term Fexch in [32] is the inhomogeneous exchange energy, given as

[33b]( ) ( )( )F A l Aexch ii x y z

= ∇ = ∇θ + ∇ϕ=∑ ( ) sin

, ,

2 2 2 2θ

where A is a stiffness constant. The third term Fan in [32] is the anisotropy energy, given as


[33c]F K l K l K K Kan u z z u= − − = + −22

42

22

42( ) sin sinθ θ

where Ku is the uniaxial magnetic anisotropy constant, and K2 is the second order anisotropy.

The fourth term Fm in [32] is the magneto-static energy, given as

[33d]FH

mtotal= − ⊥

⊥χ( ),

2

2

where Pz is the magnetic susceptibility of the media in the direction perpendicular to the

antiferromagnetic vector , and Htotal,z is the component of the total magnetic fieldrl

perpendicular to the antiferromagnetic vector . rl

The total field consists of the sum of the externally applied field Happl, and an effective

internal field that originates from the magneto-electric-like Dzyaloshinsky-Moria[ ]r rl D×

DM interaction, given as

[34][ ]r r r rH H l Dtotal = + ×

where , where $ is the homogeneous magneto-electric constant. The( )rD Pz= ⋅0 0, ,β

Dzyaloshinsky-Moria interaction WDM is the antisymmetric exchange, given as

[35]W D m l m lDM y x x y= • −r

( . )

where m is the unit vector of magnetization.

Taking into account that , we can find the magneto-static( )r r r r rH H l H ltotal total total,⊥ = − •

energy to be


[36]( ) ( ) ( ) ( )( )F H P P H H H lm z z x y= − + + − − ⋅⊥χ β θ β θ ϕ ϕ2

22 2 2sin sin sin cos

r r

The Dzyaloshinsky-Moria term in this equation can be attributed to the effective anisotropy

constant K'u

[37]( )

KP

K Kuz

u' = −

⋅+ +⊥χ

β 2

222

The Lagrange equations ; for the case of zeroddr

F

r

Fr

r

∂

∂ ∂θ∂

∂∂θ

− =0 d

drF

r

Fr

r

∂

∂ ∂ϕ∂

∂∂ϕ

− =0

applied field can be written as

[38]( ) ( )( )

( )( )( ) ( )

2 2 2 0

2 2 0

2 22

2

2 2

A Py x

A K K

A P

z u

z y x

⋅ − ⋅ −

− ⋅ ∇ϕ + − =

⋅ + ∇ϕ ⋅ ∇θ + ⋅ − =

∆θ

∆θ

α θ ϕ ∂ϕ∂

ϕ ∂ϕ∂

θ θ

θ θ α θ ϕ∇ θ ϕ∇ θ

sin cos sin sin sin

sin sin sin cos sin

'

The solution of these equations are

[39a] [39b]ϕ

θ θ ξ

=

=

arctgqq

y

x

( )

where > = qxx + qyy . A simple relationship for 2 that describes the space-modulated spin

structure of BiFeO3 with a wave vector q = (qx,qy, 0) was given in [19], which is

[39c]θ= +q x q yx y

Equation [39c] requires that the spins lie in the plane of the wave-vector, as follows from


[39a]. This requirement is in agreement with the experimentally observed cycloidal spin

structure.

Substituting the solution [39] into equation [32] for the volume-averaged free energy under

a magnetic field of H = (Hx, 0, Hz), the following expression can be derived for the free

energy of the cycloidal spin structure

[40]FH P

q AqK

KH H

cycloidz u x z= − −

⋅+ + − + +⊥ ⊥ ⊥χ

αχ ϕ χ

22

2

22

2

2 2 238 4 4

'

cos

Minimization of [39] occurs for q = "qPz/4A and n = B/2 . Substituting these values in to

equation [40] and taking into account equation [37] gives

[41]( ) ( )F

HA

PK

KP H

cycloid zu z z= − − ⋅ + + −

⋅+⊥ ⊥ ⊥χ α χ

βχ

22

2

2 2

21

16 258 4 4

A phase transition to the homogeneous state will occur at critical field Hc, when the energy

of the homogeneous state is equal to that one of the cycloidal one. The energy of the

homogeneous state, n = 2 = B/2 = const, under an external field of H = (Hx, 0, Hz) is

[42]( ) ( )F

HK K

PP Hog u

zz xhom = −

⋅+ + −

⋅− ⋅⊥

⊥ ⊥

χχ

βχ β

2

2

2

2 2

Taking into account that the field was directed along the rhombohedral axis,

, the following quadratic equation for Hc can be obtainedH H H= ⋅( / , , / )2 3 0 3


[43]( ) ( ) ( )H P HK K

PA

Pc z cu

z z2 2 2 2

4 6 692

33

40+ ⋅ − − + ⋅ −

⋅⋅ =β

χ χβ

χβ

4.3.2 Theory of the ESR Signal at Fields of H > Hc

To consider the dynamic properties of the cycloidal modulation of BiFeO3 that are

conditioned by oscillation of the vector, we use a Lagrangian, given asrl

[44]L l H l l Ftotal= − ⋅ ×

−

⊥ ⊥χγ

χγ2 22

2r r r r& &

where g = 1.73x107 (cm/g)1/2 is the gyromagnetic ratio, F is the free energy given in [32],

and Htotal is total field given in [34].

In the high field regime (H > Hc), the cycloidal spin structure is destroyed and a

homogenously magnetized state formed. Accordingly, we can omit all terms in the

Lagrangian that have spatial gradients. As in the experiment Hy = 0, the Lagrangian is then

simplified to

[45]( ) ( )( )

L ddt

ddt

H l l K K

H P H H H

total u

z x x z

=

+

− ⋅ ×

− + +

+ + ⋅ ⋅ − ⋅ +

⊥ ⊥

⊥

12

22

22

2

2 22

24

2 2

χγ

θ ϕ θχγ

θ θ

χβ θ ϕ θ ϕ θ

sin & sin sin

sin sin sin sin cos

'r r r

Lets consider a small deviation of the ferromagnetic vector from the equilibrium state:

and ; where *2( ) ( )θ θ δθx y t x y x y to, , ( , ) , ,= + φ φ δφ( , , ) ( , ) ( , , )x y t x y x y to= +

and *N are small; and the angles corresponding to the equilibrium state are 2o = No = 90o.


The Lagrange equations in the linearddt

L L ddt

L L∂∂θ

∂∂θ

∂∂ϕ

∂∂ϕ&

;&

− =

− =0 0

approximation of *2 and *N can be written as

[46a][46b]( ) ( )

( )

− − ⋅ + − ⋅ − =

− − − ⋅ − =

⊥⊥ ⊥ ⊥

⊥⊥ ⊥ ⊥

χγ

δθ χ β δθ δθ χ β δθ χ δϕ

χγ

δϕ χ δϕ χ β δϕ χ δθ

2

2

2

2 0

20

ddt

P K P H H H

ddt

H P H H H

z u z x x z

x z x x z

( )

( )

Considering only small deviations from the equilibrium sate, in the form of harmonic

oscillations and , the following matrixδθ θ ω( , , )x y t ei t= ⋅ δϕ ϕ ω( , , )x y t ei t= ⋅

representation can be obtained for equations [19]

[47]( ) ( )

( )

ωγ χ

β β

ωγ

β

θϕ

2

2

2

2

22

2

0+ − ⋅ − ⋅ −

− − − ⋅

⋅ =⊥

KP P H H H

H H H P H

uz z x x z

x z x z x

'

By setting the determinate of the matrix equal to zero, the ESR resonance frequency can be

obtained as

[48]ν ωπ

γπ

= = − ± −

2 2

42

2b b c

( ) ( )bK

P P H Huz z x x= − ⋅ − ⋅ −

⊥

222 2

χβ β

( ) ( ) ( )( ) ( )cK

P P H H P H H Huz z x x z x x y= − ⋅ − ⋅

− − ⋅ −

⊥

2 2 2 2

χβ β β


4.3.3 Predictions and Supporting Evidence

Equations [48] describe the magnetic field dependence of the electron spin resonance

frequency for a homogeneous spin state. It is not applicable to the cycloidal spin state, as the

gradient terms were excluded in equation [45]. Equations [48] demonstrate that the resonant

frequency of the homogeneous spin phase is dependent only upon two independent

parameters:

(i) an anisotropic magnetic factor, given as

[49a]Ku

χ⊥

(ii) a polarization factor that is coupled to the magnetization through the linear

magneto-electric coefficient $ of the homogenous spin ordered state, given as

. [49b]H P mDM z= ⋅ =

⊥

βχ

In the homogeneous spin order state, there will be a net magneto-electric effect that is equal

to $.

These parameters can be obtained by fitting to the experimental data. Figure 4.5

shows the result of this fitting, illustrated as dashed lines. It yields values for the anisotropic

magnetic factor of Ku/Pz = 1.41x1010 erg/cm3, and for the Dzyaloshinsky-Moria field of D

= $qPz = 1.19x105 Oe (or 11.9 Tesla). The blue dashed line, corresponding to the solution

involving in equation [48], is in good agreement with the experimental data for− −b c2 4

fields above that of the induced phase transition (~18 Tesla). However, the solution


involving in [48] gives frequencies much greater than that of the upper limit of+ −b c2 4

our experimental range, and thus has not been experimentally verified.

The uniaxial anisotropy constant Ku can also be calculated from the ESR data. Using a

reported value [66] for the perpendicular susceptibility of Pz = 0.6@10-5 the uniaxial anisotropy

constant can be approximated as Ku = 0.85@105 erg/cm3 . Furthermore, the value of the

second order anisotropy constant K2 can be obtained by [43]. Using values of Hc = 1.8@105

Oe, A = 10-7 erg/cm, "qPz = 4Aq, q = 2B/8, and 8 = 620 D [44], the value of the second order

anisotropy constant can be approximated as . K erg cm23 315 10≈ ⋅. /

The agreement of the theory and the high-field electron spin resonance mode

demonstrates that the induced phase is a homogeneous magnetized state. The cycloidal spin

structure is destroyed by increasing magnetic field, resulting in homogeneous spin order. The

term homogeneous does not designate a single crystal, single domain state which has

homogenous order through out the specimen; such a condition only requires Landau terms

to be zero in the free energy expansion. Rather, we use the term homogeneous to designate

that the local pattern of spin arrangement is uniform within the domains. Accordingly, the

spatial profile is uniform (cycloidal structure destroyed) and the magneto-electric

coefficients are not averaged to zero.

4.4 Discussion and Summary

It is generally believed that the intrinsic spin-phonon coupling is small [26,27]. This is

due to the fact that the electronic structure that favors ferromagnetism is antagonistic to that


favoring ferroelectricity [67]. Linear magnetoelectric coefficients are known, mediated by

relativistic DM interactions, but the net effect is small [23]. However, many perovskites are

known to have a local structure which is significantly different than the average lattice

symmetry [68,69]. Complex oxides have spatial variability in the distribution of electronic

environments [70]. Spatially-variable electronic environments could result in compromised

average structures with effective magneto-electric coefficients that are significantly larger

than that of a compositionally/electronically homogeneous system. Clearly, spatially-variable

structures (whether periodically modulated or not) are important to the magnetoelectric

exchange, and other electrical/magnetic/optical properties of complex oxides of perovskite

structuture [70].

Bismuth ferrite at low fields has a cycloidal spin structure [42-44]. The cycloidal

structure is periodically modulated with 8 ~ 600 Å. The spin profile is slightly anharmonic.

To condense the homogeneously magnetized state requires the field-forced destruction of

the long wavelength cycloidal modulation. We have performed investigations of the electron

spin resonance modes of BiFeO3 as a function of magnetic field. Studies were done up to

very high magnetic fields (25 Tesla), sufficient to force the spin alignment, and to condense

a homogeneous spin order.

We summarize the observed changes in spin structure as follows. At low to moderate

fields, a cycloidal spin structure exists. The profile and modulation wavelength of the

cycloidal structure are changed with H. With increasing H, the incommensurate structure

becomes depinned, resulting in hysteretic effects between increasing and decreasing H

sweeps. Above some critical field Hc, a transition to a homogeneously magnetized state is

induced. This transition results in the destruction of the cycloidal spin structure.


Phenomenologically, the order parameter of the transition between the cycloidal and

homogeneously magnetized states is dependent on the inhomogeneous and homogeneous

magneto-electric coefficients (", $, respectively).

The ESR peaks in the homogeneous magnetized state were found to be quite broad,

and in fact became increasingly broad with increasing for H > Hc, as can be seen in Figure

4.4. Peak broadening of ESR and NMR spectra has previously been described using a

multi-soliton theory [45,58-62]. Our results indicate that there may be many numerous regions

of homogeneously magnetized spin order. A high density of soliton walls that separate

homogeneously magnetized (that are also polarized) regions may be very important to

enhanced magnetoelectric coupling, and the coexistence of high values of Ps and Ms.

4.4.1 Importance of Hc in design of advanced magnetoelectric materials

The existence of a critical magnetic field at which a homogeneous spin state is

induced offers a potential thermodynamic means by which to design new and/or modified

materials with enhanced linear magnetoelectric effects. The key lies in the control of the

critical field required to induce the transformation. Normally, in materials design, this is

achieved by variations in composition and temperature space, which to date has revealed

limited success [26]. However, epitaxial constraint is an additional important materials

parameter, which is often used to thermodynamically tune properties and phase stability in

ferroic thin layers. For example, in magnetostrictive thin layers, the phase transition

temperature can be doubled or eliminated by epitaxial strain [71].


Recent investigations of (001)-oriented BiFeO3 epitaxial thin layers grown on

strontium titanate have shown dramatically higher values of the spontaneous (Ps) and

remanent (Pr) polarizations, as shown in Figure 4.6 [38], than for bulk single crystals.

The saturation polarization of the thin layer can be seen to be ~0.52 C/m2, which is

much higher than the value of 0.035 C/m2 previously reported for (001)-oriented single

crystals. In addition, the (001)-oriented BiFeO3 thin layers were reported to have

dramatically higher magnetoelectric (ME) coefficients, than the bulk crystals, as shown in

the Inset of Figure 4.6. The value of the ME coefficient in the remanently polarized and

magnetized states is ~3.5 V/Oe-cm, which is orders of magnitude greater than that previously

reported values for bulk crystals [53]. The significant changes in the magnetoelectric,

ferroelectric, and ferromagnetic behavior of the thin layers can be understood in terms of an

epitaxially-induced homogeneously magnetized state.

From the Dzyaloshinsky-Moria field of equation [49b], the ME coefficient of the

homogeneous spin state can then be estimated from the fitting of the ESR data in Figure 4.5.

Rearrangement of equation [49b] gives the homogeneous ME coefficient 1/$ in units of

V/Oe-m

[24a]1

3β εV

Oe mP

KHz

o DM−

=

where go is the permittivity of free space, and K is the relative dielectric constant which is

equal to ~100 [38]. The value of 1/$3 can be estimated from the measured value of HDM =

1.2x105 Oe in Figure 4.5, and the inherent value of Pz = 0.035 C/m2 of the single crystal [35].

The value of 1/$3 for the homogeneously spin ordered state can then be estimated as


Figure 4.7 : Polarization electric field behavior of a (001)-oriented BiFeO3 thin layerprepared by pulsed laser deposition, taken at room temperature. Inset (a) shows anenhancement of the induced polarization in the remanent state; and Inset (b) showsthe magnetoelectric coefficient. These data were taken from reference [23].


[50b]1 0 035

885 10 100 12 1033

312 5β

=⋅ ⋅ ⋅ ⋅ −

=

−

−

.. .

.V

Oe mV

Oe m

Comparisons of the predicted value of 1/$3 in equation [50b] with the experimentally

observed value in Inset of Figure 4.7 will demonstrate remarkable agreement. In the spin

modulated state, the magneto-electric effect is averaged to zero over 8. But, in the

homogeneous spin ordered state, there is a significant net magneto-electric coefficient of

1/$3.

4.4.2 Summary

In this investigation, we have reported the experimental direct evidence of a high

magnetic field induced phase transformation between a cycloidal and a homogeneous

magnetized state in BiFeO3. The field dependence of electron spin resonance modes have

been theoretically predicted using Landau-Ginzburg theory that includes a relativistic

anti-symmetric exchange, and experimentally confirmed by high field ESR measurements.

The transformation between cycloidal and homogeneous spin orders is related to the linear

magnetoelectric effect. In both states, the magnetoelectric coefficient $ is the same, but the

ME coefficient is only manifested in the homogeneous one.


Chapter 5 : Epitiaxial-Induced Transitions in (001)c, (110)c,

and (111)c BiFeO3 Thin Films

Recently, dramatically increased Ps, Ms, and "E have been reported in epitaxial

thin-films of BiFeO3 grown on (001)c SrTiO3 [6]. For example, the Ps of (001)c BiFeO3 thin

films is ~0.6 C/m2 - which is ~20x larger than that of a bulk crystal projected onto the same

orientation. Hetero-epitaxy induces significant and important structural changes. The lattice

parameters, (c,a)=(4.005, 3.935) Å, of thin films are not the rhombohedral ones of crystals

and ceramics, (ar=3.96) Å. Recent electron spin resonance (ESR) investigations of BiFeO3

crystals under high-magnetic field H have shown an induced phase transition from cycloidal

to homogeneous spin orders along (111)c/(001)H [49]. These results combined with those for

(001)c oriented thin films indicate that transitions to a homogeneously spin ordered state

might be induced along various crystallographic directions by either applied H or epitaxial

constraint.

In this thesis, variously oriented BiFeO3 epitaxial thin films have been deposited by

pulsed laser deposition. Dramatically enhanced polarization has been found for (001)c,

(110)c, and (111)c films, relative to that of BiFeO3 crystals. The easy axis of spontaneous

polarization lies close to (111)c for the variously oriented films. BiFeO3 films grown on

(111)c have a rhombohedral structure, identical to that of single crystals. Whereas, films

grown on (110)c or (001)c are monoclinically distorted from the rhombohedral by epitaxial


constraint. The results are explained in terms of an epitaxially-induced transition between

cycloidal and homogeneous spin states, via magneto-electric interactions.

5.1 Experimental

We have grown phase-pure BiFeO3 (BFO) thin films of 2000 Å thickness by pulsed

laser deposition (PLD) onto (001)c, (110)c, and (111)c single crystal SrTiO3 substrates. The

conducting perovskite oxide electrode, SrRuO3 (SRO) [50], was chosen as the bottom

electrode. Films of SRO of 500 Å were deposited at 600 ºC in an oxygen ambient of 100

mTorr; and followed by the BFO film, deposited at 670 ºC in an oxygen ambient of 20

mTorr at a growth rate of 0.7 Å/sec. The BiFeO3 film thicknesses were all close to 2000 Å,

which is necessary to reduce the influence of film thickness [6], if the out-of-plane lattice

parameters are to be compared. Chemical analysis was carried out by SEM x-ray

microanalysis, indicating a cation stoichiometry in the BFO films of ~1:1. Reciprocal lattice

mapping was performed using a Phillips MPD system. Ferroelectric measurements were

performed using a RT6000 test system (Radiant Technologies). Field-dependent magnetic

measurements were obtained using a vibrating sample magnetometer (Lakeshore

Cryotronics, USA).

5.2 Results

Small area reciprocal lattice contour scans taken about the (001)c, (110)c, and (111)c

orientations for a BiFeO3 crystal are shown in Figures 5.1(a)-(c), respectively. A single peak


Figure 5.1 : Reciprocal space scans for variousoriented BiFeO3 single crystals. (a) (001)c orientation;(b) (110)c; and (c) (111)c orientation.


was found along the (001)c, with d<001>=3.958 Å, which is in agreement with previous reports

of a rhombohedral phase [30,32,9,10,72,73]. A peak splitting was found along the (110)c with

d<110>=2.783 and 2.809 Å, and along the (111)c with d<111>=2.277 and 2.306 Å. The values

of the interplanar spacing and lateral correlation length x- are summarized in Table 5.1 for

the various orientations.

Small area (001), (110) and (111) reciprocal lattice contour scans are shown in

Figures 5.2(a)-(c) for a (111)c oriented BiFeO3 film. Both a sharp peak from the SrTiO3

substrate and a broad peak from the film can be seen in each figure. The values of (HKL)

were determined by referencing to the values of the bulk single crystal. BiFeO3 films were

found to grow epitaxially on (111) SrTiO3 substrates with values of the interplanar spacing

of d<001>=3.959 Å, d<110>=2.810 Å, and d<111>= 2.306 Å. These interplanar spacing are near

exactly equal to those of the bulk single crystals, as can be seen by comparisons in Table 5.1.

Clearly, the (111) BiFeO3 films are in a single domain state with a rhombohedral structure.

The crystal structure of the (110)c and (001)c BiFeO3 films was found to be

monoclinically distorted from the rhombohedral one. Small area (001), (110) and (111)

reciprocal lattice contour scans are shown in Figures 5.2(d)-(f) for (110)c films, and in

Figures 5.2(g)-(i) for (001)c films, respectively. The values of the interplanar spacings and

>z are summarized in Table 5.1. For both the (110)c and (001)c films, the values of d<111> were

equal to those of the rhombohedral phase, whereas d<110> and d<001> were significantly

different. The (110)c film is nearly single domain with d<111>=2.307 Å and d<110>=2.828 Å;

whereas the (001)c film consists of two variants, with the dominate variant having values of

d<111>=2.277 Å and d<110>=2.792 Å, which are both noticeably different than those of the

rhombohedral lattice of the single crystal. For both the (110)c and (001)c oriented films, even


Figure 5.2 : Reciprocal space scans for various oriented BiFeO3 films. (a) (001) scan for(111)c film; (b) (110) scan for (111)c film; (c) (111) scan for (111)c film; (d) (001) scan for(110)c film; (e) (110) scan for (110)c film; (f) (111) scan for (110)c film; (g) (001) scan for(001)c film; (h) (110) scan for (001)c film; and (i) (001) scan for (111)c film.


Crystal

(111)c Film

(110)c Film

(001)c Film

attice spacing

d<001>

d<110>

d<111>

ξ? <001> ξ? <110> ξ? <111>

3.958 Å

2.783 Å (I=0.62) [2.809 Å (I=0.38)]

2.277 Å (I=0.87) [2.306 Å (I=0.13)]

200 Å 750 Å 550 Å

3.959 Å

2.810 Å

2.306 Å (I=1.00)

770 Å 750 Å 600 Å

3.984 Å

2.828 Å

2.307 Å (I=0.98) [2.278 Å (I=0.02)]

750 Å 340 Å 280 Å

4.001 Å

2.792 Å (I=0.72) [2.816 Å (I=0.28)]

2.278 Å (I=0.90) [2.304 Å (I=0.10)]

200 Å 300 Å 420 Å

Table 5.1 : Summary of interplanar spacings and lateral correlation lengths obtained from small areareciporcal lattice scans for the variously oriented BiFeO3 films and crystal. Peak splitting wasobserved along the (110) and (111), the relative intensities are designated by I, and the weaker peakis designated by brackets.


more pronounced deviations from the rhombohedral lattice were found along the (001). The

value of d<001> increased from 3.959 Å for the (111)c film, to 3.984 Å for the (110)c film, to

4.001 Å for the (001)c film.

On (111)c, BiFeO3 films grow in an unconstrained single domain condition. The

crystal structure is rhombohedral and identical to that of single crystals. Films grown on

either (110)c or (001)c are under significant elastic constraint. The crystal structure is

monoclinically distorted from the rhombohedral along the (110) and (001), as evidenced by

the values of d<110> and d<001> in Table 5.1. With respect to the rhombohedral structure, the

lattice of (110)c BiFeO3 films expand along both (110) and (001); whereas that of (001)c

films contract along (110) and expand along (001).

We also investigated the effect of orientation of this constrained crystallographic film

state on the physical properties of BiFeO3. The ferroelectric properties were characterized

by a polarization hysteresis method. Figure 5.3a shows the P-E response for the variously

oriented films. For each orientation, we observed hysteresis loops typical of a ferroelectric.

We found a remanent polarization Pr of ~55 FC/cm2 for (001)c films, ~80 FC/cm2 for (110)c

films, and ~100 FC/cm2 for (111)c films. Figure 5.3b shows %3·P(001)c, %2·P(110)c, and

P(111)c as a function of E for the variously oriented films. In this figure, the values of the

projected polarizations can be seen to be nearly equivalent. This confirms that the direction

of spontaneous polarization lies close to (111)c, and that the values measured along (110)c

and (001)c are simply projections onto these orientations. Clearly, similar to bulk crystals and

ceramics, the spontaneous polarization is oriented close to (111)c. However, Ps is

dramatically increased! The pulsed remanent polarization )P is shown in Figure 5.3c. The


-75 -50 -25 0 25 50 75-150

-100

-50

0

50

100

150

Electric Field, E (MV/m)

(a)(111)(110)

(001)

Pola

rizat

ion

( µC

/cm

2 )

-75 -50 -25 0 25 50 75-150

-100

-50

0

50

100

150

Electric Field, E (MV/m)

(b)

Pola

rizat

ion

( µC

/cm

2 )

P(111) 21/2 P(110) 31/2 P(110)

0 25 50 75

0

50

100

150

200

250(c)

∆P ( µ

C/c

m2 )

Electric Field (MV/m)

[111]c [110]c [001]c

Figure 5.3 : Ferroelectric properties for (001)c,(110)c, and (111)c BiFeO3 films. (a) P-E curves; (b)P-E curves projected on (111)c; and (c) pulsedremanent polarization )P.


switched polarization for (111)c films reached values >200 µC/cm2 for fields >25 MV/m.

This is the highest known value of switched polarization!

The ferromagnetic M-H properties are shown in Figure 5.4a for (001)c oriented films.

Slim quadratic loops were observed with a Mr of ~25 emu/cc, and a remanent magnetization

Mr of essentially zero. The M-H curves were well-fitted to M=Motanh("H), as illustrated by

red lines. Hyperbolic equations have often been used to parameterize M-H curves of spin

cluster states and spin glasses [52,75]. The thickness dependence of Ms for the (001)c film is

shown in Figure 5.4b. The value of Ms increased dramatically with decreasing film thickness

(t). For t<700 Å, Ms was ~175 emu/cc, or ~1 µB/unit-cell. This is a large Ms - almost half of

that of Fe. For t>4000 Å, Ms was ~0, consistent with an antiferromagnetic state, as

previously reported for BiFeO3 crystals [30,31,9,10]. Recent high-field ESR investigations have

shown an induced phase transition from cycloidal to homogeneous spin orders [49]. The ESR

signals were found to be broad in the high field regime, revealing many regions of

homogeneous spin order.

5.3 Discussion and Summary

We can understand the influence of epitaxy by the Landau-Ginzburg (LG) formalism

of [49], given as

[51]F F F F FL exch an m= + + + ;

where FL is the magnetoelectric coupling that is linear in gradient (i.e., Lifshitz invariant),

Fexch is the inhomogeneous exchange energy, Fan is the magnetic anisotropy energy, and Fm

is magneto-static energy. The individual terms in the sum of [51] have previously been given


-8x103 -4x103 0 4x103 8x103-300

-200

-100

0

100

200

300

(a)

M=Mo*tanh(αH)Mo=171±2α=7.7x10-4

R2=0.99

Mag

netiz

atio

n, M

(em

u/cc

)

Magnetic Field, H (Oe)

D

0 100 200 300 400 5000

50

100

150

200(b)

Mag

netiz

atio

n, M

(em

u/cc

)

Film Thickness (nm)

Figure 5.4 : Ferromagnetic properties for BiFeO3 films. (a) M-Hcurves for (001)c, (110)c, and (111)c films, the red lines arefittings to the superparamagnetic equation of state; and (b) Ms asa function of film thickness for (001)c films.


in [49].

To understand the influence of substituent and crystalline solution effects, we add a

perturbation to Fan, given as

[52]( ) ( ) ( )( )F r K r K r ran u i pert i ir

N

= ⋅ − ⋅=∑ ' sin ² sin ² cos² ;θ θ ϕr r r

1

where K’u is the uniaxial magnetic anisotropy constant, and Kpert is the monoclinic

perturbation of the magnetic anisotropy. A summation is taken over all pinning centers, and

the anisotropy axis is allowed to randomly change from one to another. The ratio r in [52]

is given by r=(b3/V)=(1/N), where b is the characteristic volume of the pinning center, V is

the unit volume, and N is the number of pinning centers per unit volume.

The volume-averaged LG free energy density of the cycloid (Fcycloid) can be derived

following [30], as

[53a]( )F P q Aq KP

Kcycloidz

us

pert= − ⋅ + + − −⊥α θ χ

βθ ϕ

2 4

2

² sin ².

sin ² cos²'

where, [53b]( )K

PKu

zu

' = −⋅

+⊥χβ 2

2

where q is the wave vector of the modulation, A is the stiffness constant of the cycloid, a is

the inhomogeneous magnetoelectric constant, $ is the magnetoelectric constant of the

homogeneous spin state, and Pz is the perpendicular magnetic susceptibility. By replacing

the sum in [52] with an integral over the volume in [53], and by minimizing [53] for


q=(".Pz)/4A and substituting the values back into [53], we obtain the volume-averaged

free-energy as

[54]( ) ( )FA

P K P Kcycloid z

u z pert= − ⋅ + −⋅

−⊥116

23 3 6

22

αχ α

π.

We treat the uniaxial anisotropy constant as a sum of that for unmodified BiFeO3

crystals Ku°=8.5x103 J/m3 and a perturbation )Kepi caused by epitaxial constraint. This is

expressed as Ku°+)Kepi = Ku°+Kpert, where we assume that the perturbation )Kepi is of the

same order as the monoclinic perturbation Kpert.

[55]( )F K

PHomogeneous u

s= −⋅

⊥χβ 2

2.

The homogeneous spin state will be favored, when the anisotropy constant fulfills the critical

pertubation condition of

[56]( ) ( )K

P PA

K x Jmpert

c s s13

16 6 16 3

7 102 2

10

43−

> −

⋅+

⋅− ≈⊥π

χβ α

;

where Pz=4.7x10-5, A=8x10-7 erg/cm, "·Pz=4Aq, q=(2B)/8, 8=620 D and $·Ps=1.2x105 Oe,

[17,19]. A transition to a uniform spin state will occur if the critical anisotropy perturbation is

Kcpert$2x105 J/m3. This transition can be induced by epitaxy, only if 1/2 Y·g2

epi>>Kcpert

Kcpert$2x105 J/m3; where Y is Young's modulus, and gepi is the epitaxial strain. We estimate

Y=1011 N/m2, by comparisons to other mixed ferroelectric perovskite crystals under


compressive stress [22], and gepi~10-2. The value of 1/2 C·gepi2 is ~5x106 J/m3, which is >10

Kcpert..

In summary, BiFeO3 films grown on (111)c have a rhombohedral structure, identical

to that of single crystals. Whereas, films grown on (110)c or (001)c are monoclinically

distorted from the rhombohedral by epitaxial constraint. The easy axis of spontaneous

polarization lies close to (111)c for the variously oriented films. The results are explained in

terms of an epitaxially-induced transition from cycloidal to homogeneous spin states, via

magnetoelectric interactions.


Chapter 6 : Large Linear Magnetoelectric Effect in Modified

BiFeO3

The magneto-electric effects have been reported in single phase materials [27-31,75];

however to date, a ME material with significant coupling has yet to be found. The largest

ME coefficient has been reported for single crystals of Cr2O3 [34-40]. The value of the ME

polarization coefficient is "P=2.67x10-10 s/m C/m2-Oe (or the equivalent ME voltage

coefficient is "E~0.02 V/cm-Oe) at room temperature, becoming zero above a Néel

temperature of TN=34°C.

Single crystals of Cr2O3 are antiferromagnetic, but not ferroelectric [34-40]. Significant

ME exchange has always been sought in ferromagnetoelectric materials, but never found [26].

Ferromagnetoelectric materials have two order parameters [26]. These are a spontaneously

polarization (ferroelectric or antiferroelectric) and a spontaneous magnetization

(ferromagnetic or antiferromagnetic). Phenomenologically, this coupling can be either

mediated by linear, biquadratic, or even higher order exchanges [27]. The linear exchange

involves a coupling between the linear power of a polar tensor and a linear power of an axial

tensor [13-20], via a relativistic (Dzyaloshinksy-Moria) interaction between Ps and Ms.

The linear magnetoelectric coupling, although allowed by symmetry, has yet to be

observed in BiFeO3 [29,32]. However, a very small quadratic magnetoelectric effect has been

measured [29,32]. It has previously been established that the 3m magnetic point group allows

for a linear magnetoelectric coupling. However, the antiferromagnetic vector and

magnetoelectric effect are both averaged to zero over 8 of the spiral. In this paper, we will


show that the magnetoelectric exchange of BiFeO3-based materials can be dramatically

enhanced by crystalline solution and substituent effects.

6.1 Experimental

We have developed crystalline solutions of La-modified BF-x%PT, with La

concentrations of 0, 0.1 and 0.2 at% and for 0.2<x<0.3. Starting materials were commercial

reagent-grades of Bi2O3, Fe2O3, La2O3, PbCO3, and TiO2 with 99%+ purity. The oxides were

mixed by ball milling for 24 hours with stabilized ZrO2 media, calcined at 750°C for 4 hours,

and sintered at 1000-1120°C for 0.8 hours in a sealed crucible. BiFeO3-xPbTiO3 (BF-x%PT)

has been reported to form a continuous solution across its entire range, with a

rhombohedral-tetragonal morphotropic phase boundary (MPB) at x~0.4 [33]. X-ray

diffraction revealed that our specimens were rhombohedral, for 0.20#x#0.35. Near the

MPB, the c/a ratio of the tetragonal phase reached a maximum of ~1.17, in agreement with

previous reports [34].

The specimen dimensions were 10.4 mm in diameter and 0.5 mm in thickness. The

disks were poled in a 120°C oil bath at 40 kV/cm for 10 minutes. Specimens were then

electroded using a post-fired silver paste (Dupont 6160). Dielectric measurements were

carried out for 30<T<600°C using a computer controlled HP4284. Ferroelectric hysteresis

loops and strain measurements were made using a modified Sawyer-Tower circuit with a

linear variable differential transducer (LVDT). The piezoelectric properties were examined

using a Berlincourt d33 meter.


The induced magneto-electric voltage )VME was characterized. An electromagnet

was used to apply a dc magnetic bias Hdc of 0 to 3000 Oe. Small Helmoltz coils were used

to excite an ac magnetic field Hac of 0.1 to 0.8 Oe, which was superimposed on . The voltage

across the thickness of the specimen was then measured as a function of Hac using a lock-in

amplifier method. The system had a low noise floor, and is capable of discriminating induced

voltages on the order of 10-9 volts. A measurement frequency of 103 Hz was used. The ME

voltage coefficient "E was then calculated in units of [V/Oe-cm] by dividing )VME by the

specimen thickness and Hac. However, to properly represent the coupling between Ps and Ms,

the ME voltage coefficient must be converted to the ME polarization coefficientα δδE

VH

=

, where K is the relative dielectric constant and go is the relativeα δδ

ε δδP o

PH

K EH

= =

permittivity of free space. The H-induced polarization was calculated from )VME by the

relation , where )t is the specimen thickness.∆∆∆

P KV

toME= ε

6.2 Results

6.2.1 Dielectric properties

Two of the main problems of BiFeO3-based materials that have previously limited

development are (i) a very high coercive field Ec which makes them difficult to pole; and (ii)

a low resistivity D which has made it difficult to obtain good dielectric or magneto-electric

properties. Our La-modified BF-x%PT ceramics had values of D>1012 W-cm. In addition,


-100 -50 0 50 100-40

-20

0

20

40

(a)

(Bi,La)FeO3-0.3PbTiO3

Pol

ariz

atio

n (µ

C/c

m2 )

Electric Field (kV/cm)

0% La 10% La

-100 -50 0 50 100-0,05

0,00

0,05

0,10

0,15

0,20

(b)

(Bi0.9La0.1)FeO3-0.3PbTiO3

Str

ain,

ε%

Electric Field (kV/cm)

Figure 6.1 : Large amplitude induced polarization and strain as afunction of electric field for La-modified BiFeO3-30%PbTiO3ceramics. (a) Bipolar P-E response; and (b) bipolar g-E response.


Ec was pronouncedly decreased from >100 kV/cm to <30 kV/cm with increasing La content

between 0 and 20 at%, as shown in Figure 1a for BF-30%PT. Figure 6.1b shows the

corresponding g-E response, clearly demonstrating significant strains of g~2x10-3. The

piezoelectric constant for (Bi0.9La0.1)FeO3-30%PbTiO3 was determined to be 130 pC/N.

In addition to lowering Ec, La3+ substitution also enhanced the dielectric properties.

Figures 6.2(a) and (b) show K and tan* as a function of x%PT for La concentrations of 0

and 20 at%, respectively. For example, for x=30 at%, K was increased by ~4x on increasing

La content between 0 at.% (K=180, tan*~0.03) and 20 at% (K=750, tan*~0.07). Figure 6.2c

shows K as a function of temperature in the range of 25<T<600°C for BF-30%PT modified

with 10 and 20 at% La. The Curie temperature Tc was ~400-500°C, and was decreased by

La substitution. In addition, the phase transformation characteristics became increasingly

diffuse with increasing La content.

6.2.2 Magnetoelectric properties

The magneto-electric coefficient "P for poled (Bi0.9La0.1)FeO3-30%PbTiO3 is shown

in Figure 6.3. The value of "P was 2.5x10-9 C/m2-Oe. It was found to be independent of

applied dc magnetic bias (Hdc) and ac magnetic field (Hac). Previous investigations have

shown that "P is essentially zero for BiFeO3 single crystals [29,32]. The results presented in

Figure 6.3 unambiguously demonstrate dramatic changes in "P induced by crystalline

solution and substituent effects - the ME effect is enhanced by orders of magnitude relative

to that of unmodified BF.


0,30 0,35 0,40 0,45 0,50 0,550

500

1000

1500

(b) (Bi0.8La0.2)FeO3-xPbTiO3

tanδ 1 KHz

10 KHz 100 KHz

Die

lect

ric C

onst

ant,

KPbTiO3 Content, x mol

0,0

0,1

0,2

0,20 0,25 0,30 0,35 0,40 0,450

100

200

300

400

500

tan δ

1 KHz 10 KHz 100 KHz

Die

lect

ric C

onst

ant,

K

PbTiO3 Content, x at%

0,00

0,05

0,10

(a) BiFeO3-xPbTiO3

0 200 400 6000

2000

4000

6000(c) (Bi,La)FeO 3-30%PbTiO 3

20 at% La

10 at% La

Die

lect

ric C

onst

ant,

K

Temperature (oC)

Figure 6.2 : Dielectric properties of La-modified BF-x%PT: (a) as a function of PT at roomtemperature for 0 at% La , taken at various frequencies; (a) as a function of PT at room temperaturefor 20 at% La, taken at various frequencies; and (c) as a function of temperature for BF-30%PT for10 and 20 at% La substitution, data taken a 106 Hz.


0,0 0,2 0,4 0,6 0,8 1,00

1x10-9

2x10-9

3x10-9

Hac (Oe)

αP (

C/O

e-m

2 )

0,000

0,005

0,010

Independent of Hdc

(Bi0.9La0.1)FeO3-0.3PbTiO3, Poled

αE (V

/Oe-cm

)

Figure 6.3 :Magnetoelectric coefficient as a function of ac magneticfield for poled (Bi0.9La0.1)FeO3-30%PbTiO3. The left hand side showsthe magnetoelectric polarization coefficient "P, and the right handside shows the magnetoelectric field coefficient "E. The value of "Pand "E were independent of Hdc for 0<Hdc<3000 Oe. Themeasurement frequency was 103 Hz.


0,0 0,2 0,4 0,6 0,8 1,00

1x10-9

2x10-9

3x10-9

Hac (Oe)

Pol

ariz

atio

n (C

/m2 )

0

1x10-4

2x10-4

3x10-4

(Bi0.9La0.1)FeO3-0.3PbTiO3, Poled

∆VM

E (V)

Figure 6.4 : Magnetic field induced polaration and voltage as a functionof ac magnetic field for poled (Bi0.9La0.1)FeO3-30%PbTiO3. The left handside shows the induced polarization )P, and the right hand side showsthe induced voltage )VME. The measurement frequency was 103 Hz.


The data in Figure 6.3 shows that "P is independent of Hac for

(Bi0.9La0.1)FeO3-30%PbTiO3. To illustrate linearity, the induced polarization )P was

measured as a function of Hac, as shown in Figure 6.4. In this figure, it can be seen that )P

has an exceptional good linear response to changes in magnetic field for 0.1<Hac<1 Oe. The

magnetoelectric coupling coefficient is then related to the ratio of the electric to magnetic

energy densities, given as

[57]k EH

cKme

p p22

2

2 2 2

= = ≈εµ

αεµ

α;

where kme is the magnetoelectric coupling factor, and c is the speed of light. From the square

root of the ratio, the ME coupling was determined to be ~0.04.

Previously, the largest ME effect in a single phase material was that for Cr2O3

crystals [34-36,32,40], which has a value of "p=2.5x10-10 s/m at TN=34°C. In fact, Cr2O3 has

been considered to have a "giant" magneto-electric effect; having values of "p and TN

significantly higher than those of any other material. However, our value of "p=2.5x10-9

C/m2-Oe for modified BF-PT is ~10x larger than this; furthermore, TN of BF-based materials

is dramatically higher than that of any other known ferromagnetoelectric material. Figure

6.5 shows "p as a function of temperature for (Bi0.9La0.1)FeO3-30%PbTiO3. The value of "p

can be seen to remain very high until T>200°C.

The magnetoelectric properties were also measured for (Bi0.9La0.1)FeO3-30%PbTiO3

in the unpoled condition, as shown in Figure 6.6. The value of "p in the unpoled condition

was 1.4x10-9 C/m2-Oe. This is about ½ the value of the poled condition shown in Figure 6.3.


0 50 100 150 200 2500

2x10-9

4x10-9

6x10-9(Bi0.9La0.1)FeO3-0.3PbTiO3, Poled

α

P (

C/O

e-m

2 )

Temperature (oC)

Figure 6.5 : Magnetoelectric coefficient as a function oftemperature for poled (Bi0.9La0.1)FeO3-30%PbTiO3. The valueof "P and "E were independent of Hdc for 0<Hdc<3000 Oe. Themeasurement frequency was 103 Hz.


0,0 0,2 0,4 0,6 0,8 1,00

1x10-9

2x10-9

3x10-9

4x10-9

(Bi0.9La0.1)FeO3-0.3PbTiO3; unpoled

Independent of Hdc

αP (

C/O

e-m

2 )

Hac (Oe)

Figure 6.6 : Magnetoelectric coefficient "P as a function of acmagnetic field for unpoled (Bi0.9La0.1)FeO3-30%PbTiO3. Thevalue of "P was independent of Hdc for 0<Hdc<3000 Oe. Themeasurement frequency was 103 Hz.


Unpoled BF-x%PT is neither pyroelectric nor piezoelectric. However, a linear coupling

between polarization and magnetization exists, even though neither vectors have a net

moment. Clearly, there is a non-local exchange between Ps and Ms.

Figure 6.7(a) shows the effect of increasing La-content to 20 at% on the ME

coefficient for BF-30%PT. Data are shown as a function of Hac for various magnetic biases.

The value of the ME coefficient was only slightly increased to "p =2.7x10-9 s/m, relative to

that for (Bi0.9La0.1)FeO3-30%PbTiO3 in Figure 6.3. This slight increase was accompanied by

a slight dependence on Hdc. With increasing bias for 0<Hdc<3000 Oe, the value of "p

gradually decreased, approaching the value for (Bi0.9La0.1)FeO3-30%PbTiO3 which was

independent of Hdc. The effect of PT-content on "p was also determined (La content of 10

at%). Figures 6.7(b) and (c) show the effect of both higher (x=45 at%) and lower (x=20 at%)

PT atomic ratio, which can be compared with Figure 6.3 (x=30 at%). The magneto-electric

coefficient was found to increase slightly with increasing PT content. A maximum value of

"p=2.8x10-9 C/m2-Oe was found in the vicinity of the MPB, near x=45 at% PT. However,

"p was lower by a factor of ~½ for x=20 at% PT, due to the fact that this specimen could not

be poled as Ec was too high; in fact its value (1.3x10-9 C/m2-Oe) was close to that of unpoled

(Bi0.9La0.1)FeO3-30%PbTiO3 shown in Figure 6.6.

The magnetoelectric results demonstrate (i) that (Bi,La)FeO3-x%PT has a

dramatically enhanced "P, relative to BF crystals, with a kme~0.04; (ii) that the ME coupling

is linear; (iii) that the ME coupling is present until high temperatures; and (iv) that the ME

coupling is inhomogeneous (non-local) in nature.


0,0 0,2 0,4 0,6 0,8 1,00

1x10-9

2x10-9

3x10-9

4x10-9

(a) (Bi0.8La0.2)FeO3-0.3PbTiO3

αP (

C/O

e-m

2 )

Hac (Oe)

HBias=0 Oe 500 Oe 1000 Oe 1500 Oe 2000 Oe 2500 Oe 3000Oe

0,0 0,2 0,4 0,6 0,8 1,00

1x10-9

2x10-9

3x10-9

4x10-9

(b) x=45 at% PT, 10 at% La

αP (

C/m

2 -Oe)

Hac (Oe)0,0 0,2 0,4 0,6 0,8 1,00

1x10-9

2x10-9

3x10-9

4x10-9

(c) x=20 at% PT, 10 at% La

αP (

C/m

2 -Oe)

Hac (Oe)

Figure 6.7 : Magnetoelectric coefficient "P as a function of ac magnetic field for poled variousLa-modifed BF-x%PT compositions. (a) x=30 at% PT, and 20 at% La modification; (b) x=45 at%PT, and 10 at% La modification; and (c) x=20 at% PT, and 10 at% La modification [specimen couldnot be poled due to high Ec]. The value of aP was independent of Hdc for 0<Hdc<3000 Oe. Themeasurement frequency was 103 Hz.


6.3 Summary and discussion

Microscopically, the antiferromagnetic spin order of BiFeO3 single crystals is not

homogenous. Rather, an incommensurately modulated spin structure is present [30,26,27],

which manifests itself as an incommensurate spiral with a long wavelength 8 of ~600 Å

[28-31]. The spin spiral is directed along the [110]h, and is dependent on applied magnetic field

and temperature [35]. Breaking of the translation symmetry of the spiral spin modulation

might be achieved by aliovalent substitution or crystalline solution effects in perovskites

which have ferroelectric and antiferromagnetic orders [4,36-38]. If the linear magnetoelectric

exchange is averaged to zero by the spiral modulation, then breaking of the translation

symmetry of the modulation would result in significantly enhanced ME effects. We can

understand these changes using the Landau-Ginzburg (LG) the free energy for the spin,

previously given in equation [51].

We then treat the uniaxial anisotropy constant as a sum of that for unmodified BiFeO3

crystals Ku°=8.5x103 J/m3 and a perturbation )KRF caused by substituents and crystalline

solution effects. This is expressed as Ku°+)KRF = Ku°+Kpert, where we assume that the

perturbation )KRF is of the same order as the monoclinic perturbation Kpert. We approximate

)KRF, by considering only imperfections which are charged, given as )KRF = QRFPz [41].

Charged imperfections conjugately coupled to the polarization will subsequently couple to

the spin cycloid, via magnetoelectric interactions, given as

[58]( ) ( )FA

P K P Q Pcycloid z

u z RF z= − ⋅ + −⋅

−⊥116

23 3 6

22

αχ α

π.

Minimization of the energy with respect to the polarization yields


[59]δδ

α χ βπ

FP

EA

P P Qspiral

zz z

RF= = = − − +⊥08

23 6

2 2

;

where the applied electric field E=0. Rearrangement of [59] yields

[60]α α χ βπeff

RF

z

A Q AP

2 2 2163

43

= −

=⊥ ;

where "eff² is the effective inhomogeneous magnetoelectric constant. In a pure material, with

no charged imperfections, "eff² is zero, as the contributions from the inhomogeneous and

homogeneous spin orders cancel over the modulation wavelength 8. However, randomly

distributed charged imperfections may fragment the cycloid, resulting in frustrated spin

order, releasing a fraction of the latent magnetoelectricity of the homogenous spin state that

is trapped within the cycloid.

In summary, La-modified BF-x%PT crystalline solutions have been found to have

a large linear magneto-electric effect. The ME coefficients are ~10x greater than that

previously reported for any other bulk material. The ME coefficient is independent of dc

magnetic bias, and independent of ac magnetic field. The ME effect is enhanced by the

breaking of the translational invariance of the spiral spin structure by substituent and

crystalline solution effects. Enhancement is due to a non-local relativistic exchange between

Ps and Ms.


Conclusions

The focus of this thesis has been on induced phase transitions in BiFeO3 single

crystals, thin-films and ceramics. Previous investigations have shown that the spin cycloid

results in the cancellation of the antiferromagnetic vectors; therefore, averaging out the linear

magnetoelectric effect. Induced phase transitions have the potential to release the latent

magnetoelectric interactions, locked within the spin cycloid.

In this thesis, experimental evidence of a high magnetic field induced phase

transformation between a cycloidal and a homogeneous magnetized state in BiFeO3 has been

found by electron spin resonance. The field dependence of electron spin resonance modes

have been theoritically predicted using Landau-Ginzburg theory that includes a relativistic

anti-symmetric exchange, and experimentally confirmed by high field ESR measurements.

The transformation between cycloidal and homogeneous spin orders is related to the linear

magnetoelectric effect. In both states, the magnetoelectric coefficient $ is the same, but the

ME coefficient is only manifested in the homogeneous one.

Transitions to the homogeneous spin state have been found to also be induced by

epitaxial constraint in oriented thin-layers. Dramatically enhanced polarization,

magnetization, and magnetoelectric coefficients occur. In this thesis, it has been found that

BiFeO3 films grown on (111)c have a rhombohedral structure, identical to that of single

crystals. Whereas, films grown on (110)c or (001)c are monoclinically distorted from the

rhombohedral by epitaxial constraint. The easy axis of spontaneous polarization lies close

to (111)c for the variously oriented films. The results are explained in terms of an epitaxially-


induced transition between cycloidal and homogeneous spin states, via magnetoelectric

interactions.

In this thesis, it has also been found that latent magnetoelectric interactions can be

released by disruption of the translational invariance of the spin cycloid by substituent and

crystalline solution effects. La-modified BF-x%PT crystalline solutions have been found to

have a large linear magnetoelectric effect. The ME coefficients are 10x greater than that

previously reported for any other bulk material. The ME coefficient is independent of dc

magnetic bias, and independent of ac magnetic field.


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Vita

Benjamin Thibault Ruette was born in Marseilles, France on December 6, 1978. He

graduated in 1999 from Marseille Technology University with a B. S. Degree in Mechanics

and Production Engineering. He then enrolled in the Technology University of Compiégne

and obtained his Master in Mechanical Engineering. In 2001, he entered in the graduate

program of Virginia Tech in the Materials Science and Engineering department. He is a

member of the American Ceramic Society. He has co-authored three papers in the litterature.

induced phase transition studies in magnetoelectric bifeo3 ... · benjamin thibault ruette...

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