The Pennsylvania State University The Graduate School Department of Materials Science and Engineering

Transcription

1 The Pennsylvania State University The Graduate School Department of Materials Science and Engineering PHASE TRANSITIONS AND DOMAIN STRUCTURES IN MULTIFERROICS A Dissertation in Materials Science and Engineering by Eftihia Vlahos c 011 Eftihia Vlahos Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 011

2 The dissertation of Eftihia Vlahos was reviewed and approved by the following: Venkatraman Gopalan Dissertation Advisor Chair of Committee Professor of Materials Science and Engineering Clive A. Randall Professor of Materials Science and Engineering Long-Qing Chen Professor of Materials Science and Engineering Zhiwen Liu Associate Professor of Electrical Engineering Joan M. Redwing Professor of Materials Science and Engineering Chair of the Graduate Program in Materials Science and Engineering Signatures are on file in the Graduate School.

3 Abstract Thin film ferroelectrics and multiferroics are two important classes of materials interesting both from a scientific and a technological prospective. The volatility of lead and bismuth as well as environmental issues regarding the toxicity of lead are two disadvantages of the most commonly used ferroelectric random access memory (FeRAM) materials such as Pb(Zr,Ti)O 3 and SrBi Ta O 9. Therefore lead - free thin film ferroelectrics are promising substitutes as long as (a) they can be grown on technologically important substrates such as silicon, and (b) their T c and P r become comparable to that of well established ferroelectrics. On the other hand, the development of functional room temperature ferroelectric ferromagnetic multiferroics could lead to very interesting phenomena such as control of magnetism with electric fields and control of electrical polarization with magnetic fields. This thesis focuses on the understanding of material structure-property relations using nonlinear optical spectroscopy. Nonlinear spectroscopy is an excellent tool for probing the onset of ferroelectricity, and domain dynamics in strained ferroelectrics and multiferroics. Second harmonic generation was used to detect ferroelectricity and the antiferrodistortive phase transition in thin film SrTiO 3. Incipient ferroelectric CaTiO 3 has been shown to become ferroelectric when strained with a combination of SHG and dielectric measurements. The tensorial nature of the induced nonlinear polarization allows for probing of the BaTiO 3 and SrTiO 3 polarization contributions in nanoscale BaTiO 3 /SrTiO 3 superlattices. In addition, nonlinear optics was used to demonstrate ferroelectricity in multiferroic EuTiO 3. Finally, confocal SHG and Raman microscopy were utilized to visualize polar domains in incipient ferroelectric and ferroelastic CaTiO 3. iii

4 Table of Contents List of Figures List of Tables Acknowledgments ix xxiii xxv Chapter 1 Introduction and Background Ferroelectrics Definition Phase Transitions Landau-Devonshire theory Second Order Transitions First Order Transitions Polarization Hysteresis Symmetry Arguments Ferroelectric Crystal Structures Ferroelectric Domains Strain-enabled Ferroelectricity Overview of Experimental Techniques Optical Second Harmonic Generation Experimental SHG set-up Thesis Organization Bibliography Chapter Nonlinear Optical Probing of Anisotropically Strained Ferroelectric SrTiO Introduction Theoretical Predictions Thin Film Growth and Characterization iv

5 . Dielectric Measurements Second Harmonic Generation (SHG) Measurements SHG Analysis Polarization Along Single Direction Multi-domain Models Conclusions Bibliography Chapter 3 Ferroelectricity in CaTiO 3 Thin Films and Single Crystal Polar Surfaces Probed by Nonlinear Optics and Raman Spectroscopy Introduction Twinning in the Orthorhombic Structure Incipient Ferroelectric Ferrielectric Twin Walls in CaTiO Defect Chemistry of CaTiO Theoretical Predictions of Strain-coupling to Ferroelectricity Thin Film Growth SHG Measurements Determination of Polar Symmetry Light Propagation in Anisotropic Media Experimental Determination of Γ ω Dielectric Measurements SHG Imaging of CaTiO 3 Surfaces SHG Imaging Micro-Raman Imaging Suggestions on the Origin of the Polar Signal Conclusions Bibliography Chapter 4 Phase transitions and domain structures in SrTiO 3 /BaTiO 3 superlattices Introduction Theoretical Predictions Thin Film Growth Determination of Phase Transitions by SHG Nonlinear SHG Tensors Determination of T C with SHG Determination of T C with UV Raman Spectroscopy Comparison of the Two Techniques v

6 4.4.5 Polarization Distribution by SHG Conclusions Bibliography Chapter 5 Strain-enabled Ferroelectric Ferromagnetic EuTiO Introduction Multiferroic Materials Theoretical Predictions Thin Film Growth Determination of Ferroelectric State Comparative SHG Studies of EuTiO 3 thin films under different strain states Electric Field Poling of SHG Signal Determination of Ferromagnetic State SQUID and MOKE Measurements Conclusions Bibliography Chapter 6 Conclusions and Future Work Conclusions Outstanding Issues and Future Work Bibliography Appendix A Glazer Notation of Tilted Octahedra 144 A.1 Introduction A. Effect of octahedral tilts Appendix B Beam diameter measurement 147 B.1 Introduction B. Experiment B.3 Data Analysis Appendix C mm multi-domain SHG intensity derivation 149 C.1 Case I: Point group mm with Z 3 x,y C. Domain variant contributions vi

7 C..1 Domain Variant 1, X C.. Domain Variant, X C..3 Domain Variant 3, X C..4 Domain Variant 4, X C..5 Domain Variant 5, Y C..6 Domain Variant 6, Y C..7 Domain Variant 7, Y C..8 Domain Variant 8, Y C.3 SHG intensity calculations C.4 Case II: Point group mm with Z 3 x,y Appendix D SHG derivations for CaTiO 3 thin films and surfaces 157 D.1 Introduction D. Space Group 33 (Ref 56), Pna D..1 SHG derivations without birefringence D.. SHG derivations with birefringence effects D.3 Space Group 6 (Ref 57), Pmc D.3.1 SHG derivations without birefringence D.3. SHG derivations withbirefringence effects D.4 Space Group 6 (Ref 57), Pmc 1, Case (b) D.4.1 SHG derivations without birefringence D.4. SHG derivations with birefringence effects D.5 Space Group 31 (Ref 58), Pmn D.5.1 SHG derivations without birefringence D.5. SHG derivations with birefringence effects D.6 Space Group 31 (Ref 58), Pmn 1, Case (b) D.6.1 SHG derivations without birefringence D.6. SHG derivations with birefringence effects D.7 Space Group 7 (Ref 59), Pc D.7.1 SHG derivations without birefringence D.7. SHG derivations with birefringence effects D.8 Space Group 7 (Ref 59), Pc, Case (b) D.8.1 SHG derivations without birefringence D.8. SHG derivations with birefringence effects D.9 Space Group 4 (Ref 60), P D.9.1 SHG derivations without birefringence D.9. SHG derivations with birefringence effects D.10 Space Group 4 (Ref 60), P 1, Case (b) D.10.1 SHG derivations without birefringence vii

8 D.10. SHG derivations with birefringence effects D.11 Space Group 6 (Ref 61), Pm D.11.1 SHG derivations without birefringence D.11. SHG derivations with birefringence effects D.1 Triclinic Group, D.1.1 SHG derivations without birefringence D.1. SHG derivations with birefringence effects Appendix E Dielectric measurements of CaTiO 3 thin films 34 E.1 CMB E. MDB E.3 CMB viii

9 List of Figures 1.1 Second-order phase transition (a). The free energy, F is plotted as function of polarization for T>T c, T=T c, and T<T c. The right top panel is a schematic of the spontaneous polarization P s (T), and the bottom right panel illustrates the temperature dependence of the susceptibility, χ(t) and its inverse χ 1 (T). (b) First-order phase transition. The free energy, F is plotted on the left panel as function of polarization for T>T c, T=T c, and T=T 0 <T c. The right top panel is a schematic of the spontaneous polarization P s (T), and the bottom right panel shows the temperature dependence of the susceptibility, χ(t) and its inverse χ 1 (T). Modified from [4] Schematic of a polarization hysteresis loop for a typical ferroelectric indicating the coercive field E c, the spontaneous polarization, P s, and the remanent polarization, P r. The material can be switched between two stable states, as indicated by the saturated polarization at the highest positive and negative fields, by the application of an external electric field. The direction of the electric field is indicated by the small arrows in the P-E loop Classification of 3 point group crystal classes into piezoelectric, pyroelectric (polar), and ferroelectric point groups. Ferroelectrics are a subset of piezoelectric, and pyroelectric materials which exhbit a P-E loop. Conversely, a ferroelectric must be both pyroelectric and piezoelectric ix

10 1.4 Schematic of the ABO 3 perovskite structure. The larger cations A, shown in gray, occupy the corners of the unit cell, and the smaller cations B are surrounded by 6 oxygen atoms (not shown here) which form networks of connected octahedra. Here the octahedra (shown in purple) are connected at their vertices (edge sharing). Breaking of the inversion symmetry typically occurs when the smaller B cations move away from the geometrical center of the oxygen octahedra leading to the creation of spontaneous polarization Orthorhombic DyScO 3 crystal structure with (a) pseudocubic unit cells (dark lines) within the orthorhombic unit cell (dashed lines), and (b) tilting of the ScO 6 octahedra. Modified from [13] Experimental second harmonic generation (SHG) set-up in typical transmission geometry. GT: Glan-Taylor prism, BS: beamsplitter, PD: photodiode, λ/: half waveplate, ND: neutral density filters, F1: long-pass filter, L: lens, F: blue filters, A: analyzer, PMT: photomultiplier tube. The sample is mounted on a heating, cooling, or rotation stage, depending on the experiment (a) Room temperature crystal structure of cubic SrTiO 3, (b) alternative representation of the cubic cell showing the TiO 6 octahedra encompassing the Ti ions. Note: the ionic radii are not drawn to scale Theoretical predictions for the range of the ferroelectric Curie temperature T c versus in-plane strain ε 0 for SrTiO 3 after taking into account the AFD transition. The thin film is predicted to be ferroelectric for strains achievable with commercially available substrates. In addition, T c is predicted to be in the room temperature range under moderate tensile strain. Theoretical predictions by Dr. Y. Li [7] Temperature dependent dielectric permittivity at 5 KHz of 50 Å thick SrTiO 3 thin film measured for several angles showing two peaks (T 1 and T ) as function of angle. 0 is aligned with the [010] p axis and 90 is aligned with the [100] p axis. A smaller anomaly T 3 occurs at approximately 165 K. Measurements by Dr. M. D. Biegalski [15] x

11 .4 (a) Temperature dependence of the in-plane switchable polarization along the long [010] p and short [100] p axis, (b) switchable polarization as function of angle from the long axis at 70K for the 50 Å thick SrTiO 3 thin film. Measurements by Dr. M. D. Diegalski [15] The top schematic depicts the second-harmonic generation (SHG) setup where light of frequency ω and linear polarization at an angle θ with respect to the x axis, is converted to light of frequency ω, after interacting with the sample. By varying the angle θ, and fixing the analyzer along the x, y, x, and y, directions, SHG intensity polar plots, i.e., I ω x, I ω y, I ω x, and I ω y are obtained. Panels (a) through (f) show SHG polar plots at 175 K. These compare the theoretical models (thin solid red lines) to the experimental data (open circles). In particular, the comparison is the following: (a) Model I fit, I ω x, (b) model I fit, I ω y, (c) model II, case I fit, I ω x, (d) model II, case I fit, I ω y, (e) model II, case II fit, I ω x, and (f) model II, case II fit, I ω y. The thicker solid red lines in panels (c) through (f) represent intensities which must be reciprocal in order for the fitting models to be valid. Clearly, this is true only for (c) and (d), which are based on model II, case I [15] Temperature dependence of the nonlinear coefficient ratio d 31 /d 33. Shown here are the quantities that correspond to left- and right-hand sides of Equation (.7) (black solid lines) and Equation (.8) (red dashed lines). For all measured temperatures, Equation (.7) is valid since the leftand the right-hand side quantities are equal (black squares and stars), whereas Equation (.8) is not valid. Since, Equation (.7) is satisfied, we can conclude that the in-plane polarization is along 100 p and not along 110 p directions [15] Shown here is the temperature dependence of (a) second-harmonic intensity with a fixed analyzer along the y direction and θ =0, i.e., I ω y (0) and (b) the absolute ratio of the net polarization along the y axis to the net polarization along the x axis, P y net /Px net. Both (a) and (b) are in good agreement with the electrical measurements of the polarization. (c) shows the temperature dependence of the nonlinear coefficient ratio d 15 /d 33 and (d) shows the offset angle φ, from the polar plot fitting, as a function of temperature. Both show an anomaly at 175 K that can be most likely attributed to the AFD transition [15] xi

12 3.1 Top illustration: structure of the orthorhombic CaTiO 3 with exaggerated oxygen octahedra tilts. The titanium and oxygen ions are located at the center and corners of the TiO 6 octahedra, whereas the calcium ions are located outside the oxygen octahedra [8]. (a) Depiction of the orthorhombic and pseudocubic CaTiO 3 unit cell. (b) Most common form of twinning involving rotation about normal to the (110) with composition plane (110), (c) twinning by rotation of 90 about normal to (110) with composition plane (110), and (d) twinning by rotation of 180 about normal to (11) with composition plane (11) [9] Temperature dependence of the relative dielectric constant ε(t)/ε(rt) in SrTiO 3, KTaO 3, CaTiO 3, and TiO where ε(rt) is the dielectric constant at room temperature [1] First principles calculations of epitaxially strained CaTiO 3.(a) Total energy per five atom formula unit for various epitaxially constrained structures as function of misfit strain. At each strain, the energy of the c epbnm structure is taken as the zero of the energy. The connecting lines are guides for the eyes. (b)two possible epitaxial orientations and the primitive perovskite substrate plane. On the left the ab epbnm phase and on the right the c epbnm phase [0] Schematic of experimental SHG geometry. The polarization of the fundamental light is rotated in the xy plane by an angle θ, and SHG polar plots are collected along the x, y, x, and y directions. φ denotes the sample tilt angle, and light enters the sample film side first Possible epitaxial orientations of thin film CaTiO 3 based on [4]. The orthorhombic unit cell of CaTiO 3 is shown in solid lines, outlined by its pseudo cubic mesh, whereas the allowed polarization directions are indicated in red. The following symmetries are illustrated:(a)space group Pna 1,(b)space group Pmn 1,(c) alternative orientation of space group Pmn 1,(d) space group Pmc 1,(e) alternative orientation of Pmc 1, (f) space group P 1, (g) space group Pm, (h) space group Pc, and (i) triclinic P xii

13 3.6 SHG measurements of strained CaTiO 3 films. (a) Temperature dependence of the SHG signal for 0 nm CaTiO 3 /NdGaO 3 (circles), 0 nm CaTiO 3 /LSAO (squares), and bare CaTiO 3 substrate (rhombs). Only CaTiO 3 on NdGaO 3 shows a low temperature transition at 150 K. The top inset shows the temperature dependence of d 33 / d 31. Also shown is a schematic of the SHG set-up. The tilt dependence of Iy ω (θ=90,φ) is shown in (b) for T = 100 K, and (c) for T = 5 K for sample CMB611. Experimental data are shown as open circles, and fits as solid lines. SHG polar plots and their fits collected at φ=0 and φ=40 are also shown for both temperatures. Panels (d) and (e) shows the same SHG quantities obtained from sample CMB Normal incidence SHG polar plots at T = 5K for CaTiO 3 /NdGaO 3 with (a) horizontal analyzer (parallel to x), (b) vertical analyzer (parallel to y), (c) analyzer at 45, and (d) analyzer at -45. Data: open circles, fits: solid lines Measurement of sample induced phase delay Γ ω. Transmitted 400 nm intensity as function of analyzer rotation angle θ a for (a) without sample, and (b) with the sample in the beampath. The fits (solid) lines are based on Equation (3.10) Dielectric data for 0 nm CaTiO 3 grown on NdGaO 3 measured at 0 K with interdigitated electrodes. Inset shows P-E hysteresis loop measured at 150K, and 10 K at 10 KHz Schematic of the scanning SHG microscope. The input fundamental light is directed towards the microscope with a two mirror periscope. The dichroic mirror efficiently reflects the 800 nm light towards the microscope objective which focuses the light onto the sample. Any generated SHG signal is detected in the reflection geometry using a photomultiplier tube (PMT). Polarization dependent studies can be carried out by changing the input polarization of the fundamental beam with a half waveplate, and selecting SHG polarization components with an analyzer before the (PMT) xiii

14 3.11 SHG area scans of CaTiO 3 crystal surface showing large polar domains obtained under various polarization conditions. (a) Input 800 nm polarization: parallel, output 400nm polarization: parallel, (b) input polarization: parallel, output polarization: perpendicular, (c) input polarization: perpendicular, output polarization: parallel, and (d) input polarization: perpendicular, output polarization: perpendicular SHG area scans of CaTiO 3 crystal surface under various polarization conditions. The scanned area is the same as the one shown in Figure (a) Input 800 nm polarization: -45, output 400 nm polarization: -45, (b) input polarization: -45, output polarization: +45, (c)input polarization: +45, output polarization: -45, and (d)input polarization: +45, output polarization: Representative SHG intensity polar plots obtained with the confocal SHG microscope from the area scan shown in the center. The experimental data are shown as open points whereas the fits (shown as lines) are based on Pm symmetry. The schematic illustrations of the sample axes were provided by Dr. T. T. A. Lummen Schematic of the confocal scanning Raman microscope. A 514 nm laser light is coupled into the microscope with an optical fiber, and is then focused on the sample via a microscope objective. The confocal microscope geometry allows for detection of backscattered light with a spectrometer. Polarization dependent studies are carried out with a polarizer which controls the polarization of the incident light, and an analyzer which selects the polarization components of the backscattered light (a) Polarized Raman spectra of CaTiO 3 crystal, where X denotes horizontal polarization direction, and Y vertical polarization. The dark lines point to the A 1g and B 1g Raman peaks at 155 cm 1 and 6 cm 1 respectively. (b) Raman intensity area map centered around the A 1g peak, and (b) Raman intensity area map centered around the B 1g peak. Complimentary domains light up for each configuration indicating the lattice structure rotates by 90 between the domains. Data collected and analyzed by Dr. T. T. A. Lummen xiv

15 3.16 Raman center of mass maps for (a) input excitation polarization at - 45 and Raman polarization -45, (b) input excitation polarization at -45 and Raman polarization at 45, (c) input excitation polarized along 45 and Raman polarization along 45 and(d) input excitation polarized along 45 and Raman polization along -45. The images reveal additional subdomains within the (11) twins. Data collected and analyzed by Dr. T. T. A. Lummen domain walls. (a) Ising type, (b) Bloch type, (c) Néel type, and (d) Mixed Ising-Néel type walls. A mixed Bloch-Néel-Ising wall is a combination of (b) and (d). This figure is taken from [16] Phase diagram of BaTiO 3 films as a function of temperature and substrate in-plane strain, calculated by Dr. Y. Li. The dashed lines show the BaTiO 3 strain states for the tested superlattices [0] Temperature dependence of SHG intensity for BaTiO 3 /SrTiO 3 SLs. (a) Normal incidence geometry with AOI = 0. Sample A53 (SL on SrTiO 3 ) is not shown here because it did not give any SHG signal at any temperature. (b) AOI = 45 for all samples except A69 (SL on NdScO 3 )and A99 (SL on PrScO 3 ) where both BaTiO 3 and SrTiO 3 are under tensile strain (a) Room temperature Raman spectra of the BaTiO 3 /SrTiO 3 superlattices grown on SrTiO 3, DyScO 3, SmScO 3, and PrScO 3 substrates compared to the spectra of bulk SrTiO 3 (The symbols R label the structure modes due to the rotation of the Ti-O octahedra) and three ferroelectric phases of bulk BaTiO 3. Arrows mark the phonon peaks of the SLs, asterisks (*) indicate the peaks of the substrates. (b) and (c) show the temperature evolution of the Raman spectra of the SLs grown on DyScO 3 and SmScO 3 respectively. The dash-dotted lines are guides to the eye. Figure taken from [3] xv

16 4.5 Experimental epitaxial strain phase diagram of BaTiO 3 /SrTiO 3 superlattices. The solid squares (blue and red) are the T c s determined by SHG measurements, while the open circles (blue and red) are the transitions temperatures determined with UV Raman spectroscopy. P x, P y, are the in-plane polarization components, and P z is the out-of-plane polarization component of the superlattices. UV Raman measurements were carried out by the Tenne group. The connecting lines are guides to the eye Schematic representation of SHG experimental geometries for probing the in-plane (AOI = 0 ) and out-of-plane (AOI = 45 ) polarization components of the BaTiO 3 /SrTiO 3 superlattices. 5 K polar plots for p (b) and s (c) output polarizations at AOI = 45 for sample A53 (superlattice on SrTiO 3 ). The solid lines are the fits based on 4mm symmetry Experimental SHG intensity polar plots (points) for AOI = 0, (a) I x (θ) and (b) I y (θ) output polarization at 5 K for sample A136 (superlattice on DyScO 3 ). The solid lines are fits based on mm symmetry with polarization along 100 p. Also shown are experimental SHG polar plots (points) for AOI = 45 at 5 K for (c) p, and (d) s output polarization. The fits (solid lines) are based on 4mm symmetry Experimental SHG intensity polar plots (points) for AOI = 0 for (a) I x (θ) and (b) I y (θ) output polarization at 5 K for sample A95 (superlattice on TbScO 3 ). The solid lines are the fits based on mm symmetry with polarization along 100 p. Also shown are experimental SHG polar plots (points) for AOI = 45 at 5 K for (c) p, and (d) s output polarization. The fits (solid lines) are based on 4mm symmetry Experimental SHG intensity polar plots (points) for AOI = 0 for (a) I x (θ) and (b) I y (θ) output polarization at 5 K for sample A138 (superlattice on GdScO 3 ). The solid lines are fits based on mm symmetry with polarization along 100 p ; the fitting is invalid because it does not satisfy the intensity reciprocity conditions given in 4.4. Also shown are experimental SHG polar plots (points) for AOI = 45 at 5 K for (c) p, and (d) s output polarization. The fits (solid lines) are based on the 4mm symmetry xvi

17 4.10 Experimental SHG intensity polar plots (points) for AOI = 0 for (a) I x (θ) and (b) I y (θ) output polarization at 5 K for sample A98 (superlattice on EuScO 3 ). The solid lines are fits based on mm symmetry with polarization along 110 p. Also shown are experimental SHG polar plots (points) for AOI = 45 at room temperature for (c) p, and (d) s output polarization. The solid lines are based on 4mm symmetry Experimental SHG intensity polar plots (points) for AOI = 0 for (a) I x (θ) and (b) I y (θ) output polarization at 5 K for sample A48 (superlattice on SmScO 3 ). The solid lines are fits based on mm symmetry with polarization along 100 p. Also shown are experimental SHG polar plots (points) for AOI = 45 at 5 K for (c) p, and (d) s output polarization. The fits (solid lines) are based on 4mm symmetry Experimental SHG intensity polar plots (points) for AOI = 0 at 5 K for (a) I x (θ) and (b) I y (θ) output polarization for sample A48 (superlattice on NdScO 3 ). The solid lines are fits based on mm symmetry with polarization along 100 p. Also shown are experimental SHG polar plots (points) for AOI = 0 at 5 K for (c) I x (θ), and (d) I y (θ) output polarization. The fits (solid lines) are based on mm symmetry with polarization along 100 p (a) Bulk unstrained EuTiO 3 unit cell, where the tilted arrows indicate the magnetic spins of the Eu + ions. Note: the ionic radii are not drawn to scale (b) First principles phase diagram for strained EuTiO 3 indicating the predicted paraelectric (PE), antiferromagnetic (AFM), ferroelectric, and ferromagnetic (FM) phases. The substrates listed on the top are the ones tested in this work, corresponding to film strain values of -0.9%, 0%, and +1.1%. First Principles Calculations by Dr. C. J. Fennie [10], [15] SHG measurements of EuTiO 3 thin films grown on DyScO 3, SrTiO 3 and LSAT substrates. Temperature dependence of the SHG intensity Ix ω (0 ) indicating a polar phase transition at approximately 60 K (heating cycle) only for the sample grown on DyScO 3, The schematic of the experimental SHG geometry is shown in the inset xvii

18 5.3 5 K SHG polar plots obtained from nm thick EuTiO 3 grown on DyScO 3 substrate. Experimental SHG polar plots with the analyzer (a) parallel to x and y and (b) parallel to x and y with theoretical fits based on single domain mm with polarization along y. Panels (c) and (d) show the same experimental data with theoretical fits based on single domain mm with polarization along x. For all figures, experimental data are shown as open points, and theoretical fits as solid lines K SHG polar plots obtained from nm thick EuTiO 3 grown on DyScO 3 substrate. Experimental SHG polar plots with the analyzer (a) parallel to x and y and (b) parallel to x and y with theoretical fits based on multidomain mm model with polarization parallel to 110 p. Experimental data are shown as open points, and theoretical fits as solid lines K SHG polar plots obtained from nm thick EuTiO 3 grown on DyScO 3 substrate. Experimental SHG polar plots with the analyzer (a) parallel to x and y and (b) parallel to x and y with theoretical fits based on single domain m. Experimental data are shown as open points, and theoretical fits as solid lines K SHG polar plots obtained from nm thick EuTiO 3 grown on DyScO 3 substrate. Experimental SHG polar plots with the analyzer (a) parallel to x and y and (b) parallel to x and y with theoretical fits based on single domain m. Experimental data are shown as open points, and theoretical fits as solid lines SHG hysteresis loops for JHL143C ( nm EuTiO 3 /DyScO 3 ) at 5 K. The electric field is applied along the y direction, and Iy ω (90 ) was monitored as function of the applied field. The SHG intensity data are shown in red, whereas the converted P-E loop is shown in blue. Subtraction of a small but finite SHG background at the minima of the SHG-E loop was performed in order to make the P-E loop symmetric about the y axis. This small background can be attributed to incomplete cancellation of antiparallel domains in the probed region xviii

19 5.8 Magnetic measurements for EuTiO 3 thin films. (a) MOKE measurements at K of EuTiO 3 /DyScO 3 (red), EuTiO 3 /SrTiO 3 (blue), EuTiO 3 /LSAT (green), and bare DyScO 3 substrate (gold). (b) Temperature dependence of magnetization determined both with MOKE and SQUID measurements. The inset shows the isothermal SQUID magnetization curves at T = 1.8 and 3.8 K. The red data points with the error bars show the temperature dependence of the remanent θ Kerr for EuTiO 3 /DyScO 3 (JHL143C). Measurements carried out by L. Fang, and Dr. X. Ke. Figure taken from [15] Suggested flow chart for polar plot fitting for orthorhombic symmetry mm with polarization along < 100 > p. A similar approach can be followed for mm systems with polarization along < 110> p or m A.1 Illustration of the Glazer notation of perovskite oxygen octahedra rotations. (a) No octahedra tilting about a, b or c therefore the Glazer notation is a 0 a 0 a 0. (b) in-phase tilting about the c-axis. The black octahedra are stacked on top of the blue octahedra and the two layers rotate in the same direction. Since there is no tilting about the a or b axis the Glazer notation is a 0 a 0 c +.(c) anti-phase tilting about the c-axis where the two octahedra layers rotate in opposite directions. The Glazer notation of (c)is a 0 a 0 c B.1 Measured beam divergence. Squares: experimental data, Solid lines: fits. 148 C.1 Schematic of 8 possible domain variants for point group mm D.1 Two relative orientations of the film lattice vectors with respect to the primitive substrate planes: (a) c axis parallel to substrate plane, and (b) c-axis perpendicular to substrate plane. Figures modified from Reference [8] of Chapter xix

20 D. Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 33 (Ref 56). (a) Side view schematic of the actual experimental set-up, and (b) pseudo-top view useful for the derivation of the SHG tilt expressions D.3 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 6 (Ref 57), Pmc 1 with (a) top view schematic of the SHG experimental set-up, and (b) rotationtop view useful for the derivation of the SHG tilt expressions as function of ϕ D.4 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 6 (Ref 57) Pmc 1, Case (b) D.5 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 31 (Ref 58), Pmn 1 with (a) top view schematic of the SHG experimental set-up, and (b) rotationtop view useful for the derivation of the SHG tilt expressions as function of ϕ D.6 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 31 (Ref 58) Pmn 1, Case (b) D.7 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 7 (Ref 59). (a) side view schematic of the actual experimental set-up, and (b) pseudo-top view useful for the derivation of the SHG tilt expressions D.8 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 7 (Ref 59), Case(b). (a) side view schematic of the actual experimental set-up, and (b) pseudotop view useful for the derivation of the SHG tilt expressions xx

21 D.9 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 4 (Ref 60), Pmn 1 with (a) top view schematic of the SHG experimental set-up, and (b) rotationtop view useful for the derivation of the SHG tilt expressions as function of ϕ D.10 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 4 (Ref 60), P 1, Case (b) with (a) top view schematic of the SHG experimental set-up, and (b) rotation-top view useful for the derivation of the SHG tilt expressions as function of ϕ D.11 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on Space Group 6 (Ref 61), Pm, Case (b) with (a) top view schematic of the SHG experimental set-up, and (b) rotation-top view useful for the derivation of the SHG tilt expressions as function of ϕ D.1 Experimental SHG set-up with input electric field orientation, and sample crystal physics axes based on triclinic symmetry 1, Case (b) with (a) top view schematic of the SHG experimental set-up, and (b) rotationtop view useful for the derivation of the SHG tilt expressions as function of ϕ E.1 Spatial distribution of P-E loops in sample CMB611. The loops were collected at 10 K. Measurements carried out by Dr. M. D. Biegalski E. Fatigue of CMB611.(a) Origin loop, (b) Loop after 10 6 cycles. Measurements carried out by Dr. M. D. Biegalski E.3 Frequency dependence of capacitance and loss for sample CMB611. Measurements carried out by Dr. M. D. Biegalski E.4 Spatial distribution of P-E loops in sample MDB46 at T = 10 K. Measurements carried out by Dr. M. D. Biegalski xxi

22 E.5 Temperature dependence of P-E loops in CMB419. Hysteresis loop at (a) T= 50 K, (b) T = 150 K, (c) T= 100 K, (d) T = 75 K, (e) T = 50 K, and (f) T = 5 K. The loops evolve from slim relaxor type behavior to double-hysteresis type loop at the lowest temperature. Measurements carried out by Dr. M. D. Biegalski xxii

23 List of Tables 3.1 d 33 / d 31, d 3 / d 31 and d 4 / d 31 fit parameters for CaTiO 3 on NdGaO 3 thin films based on symmetries Pmn 1 and Pmc I ω x (θ) and I ω y (θ) fit parameters for CaTiO 3 crystal domains of Group A, based on Pm symmetry using Equation (3.1) I ω x (θ) and I ω y (θ) fit parameters for CaTiO 3 crystal domains of Group B, based on Pm symmetry using Equation (3.1) Epitaxial strain values for MBE grown [(BaTiO 3 ) n /(SrTiO 3 ) m ] p superlattices. For all of the superlattices, m = 4, n = 8, and p = 40, except for A53 for which p = Summary of experimentally measured T c for BaTiO 3 /SrTiO 3 superlattices as determined by SHG and UV Raman Spectroscopy (Tenne, et. al.) where T 1 c corresponds to the T c of the in-plane signal, and T c corresponds to the T c of the out-of-plane signal Polarization dependent SHG data for [(BaTiO 3 ) 8 /(SrTiO 3 ) 4 ] 40 superlattices grown on various substrates. Listed here is the epitaxial strain of each layer (SrTiO 3 and BaTiO 3 ), the in-plane symmetry, in-plane polarization direction, and out-of plane symmetry as determined by SHG measurements xxiii

24 5.1 Extracted nonlinear ratios from independent polar plot fits based on the m symmetry for EuTiO 3 /DyScO 3 (JHL143C) C.1 Summary of domain variant contributions D.1 Summary of space groups considered for the SHG data fitting obtained from thin film CaTiO 3 and single crystal CaTiO 3 surfaces. a, b, and c refer to the CaTiO 3 orthothombic axes with a = Å, b = Å, and c = Å. Reference numbers based on [10] xxiv

25 Acknowledgments First, I would like to thank my adviser, Dr. Venkatraman Gopalan, for his guidance and mentorship throughout my graduate studies. I also greatly appreciate his help and insight when research woes strike, as well as continuous motivation for good work when things go smoothly. Next, I would also like to thank the committe members: Dr. Randall, Dr. Chen, and Dr. Liu; each being an expert in their own field allows for an invaluable external perspective of my work. This work would have been impossible without collaborators providing high quality samples; therefore I would like to thank Dr. Schlom and his students June, Charles, and Kevin for providing MBE-grown strained thin films ranging from the exotic EuTiO 3 to the frustrating CaTiO 3. I would also like to thank Dr. Soukiassian for providing the BaTiO 3 /SrTiO 3 superlattices, and Dr. Biegalski from Oak Ridge National Lab for performing the electrical measurements on many of the samples included in this work. I would also, like to thank Dr. Tenne and his group for their extensive UV Raman measurements of the same superlattices. Many thanks to the MRI and MRL support staff especially Maria from the polishing lab, Jeff helping with all things electrical, Chris from the machine shop, and Trudy from GTS Welco; whenever I talk to her I know my liquid helium will be delivered on time! I would also like to thank all our group members, past and present, and especially Dr. xxv

26 Sava Denev and Dr. Tom Lummen for their continuous patience, help, and advice. Last but not least, I would like to thank my friends, and family; without their support this would not have been possible. xxvi

27 Dedication To my favorite human, A. T. B. xxvii

28 1 Chapter 1 Introduction and Background 1.1 Ferroelectrics Definition Ferroelectrics are materials with nonzero spontaneous electric polarization that can be switched between two stable states by the application of an external electric field (coercive field) E C, smaller than the breakdown field, E B [1]. Reversal of electrical polarization by an external electric field (P-E switching) was first observed by Valasek in the earlier part of the twentieth century in a material called Rochelle salt (NaKC 4 H 4 4H 0) []. Monopotassium phosphate, KH PO 4 or KDP and KDP-related ferroelectrics were discovered in the 1930 s by Busch and Scherrer [3]. In addition, newly synthesized materials such as (NH 4 )H PO 4 or ADP became important in World War II as the material of choice for underwater transducers and submarine detectors [3]. Approximately ten years later, ferroelectricity was discovered in the perovskite oxides: BaTiO 3 (1945,1946), KNbO 3 and KTaO 3 (1949), LiNbO 3 and LiTaO 3 (1949), and PbTiO 3 (1950) [1]. In general, ferroelectrics undergo phase transitions from a high temperature paraelectric (higher symmetry) prototype phase to a low temperature (lower symmetry) ferroelectric phase. The high temperature phase is also known as the non-polar phase, whereas the low temperature phase is the polar phase of the ferroelectric. The ap-

29 pearance of spontaneous polarization and the change in crystal symmetry occurs at a temperature called the Curie temperature, T c Phase Transitions Landau-Devonshire theory According to the Landau-Devonshire theory, the free energy F of bulk ferroelectric systems with spatially uniform polarization, is a function of temperature (T ), polarization (P), electric field (E) and stress (σ). While T is a scalar quantity, both P and E are vector quantities with three components, and σ is a six component tensor. Thermodynamics dictate that the free energy at equilibrium is a function of ten variables: three from polarization, six from stress, and one from the temperature dependence. For an unstrained crystal(σ = 0), and assuming polarization only along one direction, the free energy can be expressed as: F = 1 a 0(T T 0 )P bp cp6 EP (1.1) where a 0 and c are assumed to be positive. Depending on the sign of the b coefficient one can distinguish two types of transitions: f irst order and second order transitions. The second order transition is discussed first Second Order Transitions Second order (or continuous) phase transitions occur when b > 0. Neglecting the higher power terms (c = 0), and taking the partial derivative of Equation (1.1) with respect to P, we get the following: E = a 0 (T T 0 )P+bP 3 (1.)

30 3 For zero external electric field (E = 0), Equation (1.) has three roots: P = 0, and P=±P s. In particular, P s = ( a ) 0 1/ b (T T 0) (1.3) As shown in Figure 1.1(a), P s decreases continuously as the temperature approaches T 0 ; therefore T 0 = T c. The dielectric constant ε=ε 0 (1+χ), where χ is the susceptibility, can be easily found to be: ε=ε 0 (1+ ) 1 a 0 (T T c ) (1.4) The temperature dependence of F(P), for a second order transition, is shown on the left side of Figure 1.1(a). For T > T c, F(P) has parabolic shape with a single minimum at P = 0. This corresponds to a high temperature paralectric phase. As the temperature decreases, and T < T c, F exhibits two minima for finite P; these correspond to the two ground states of a ferroelectric which have spontaneous polarizations in opposite direction. It turns out that for ferroelectrics a moderate electric field can overcome the energy barrier between the two states, and the material can be switched between the two states. Ferroelectric switching is discussed in more detail in a subsequent section of this chapter. The temperature dependence of spontaneous polarization P s, susceptibility, χ and its inverse χ 1 are also shown on the right side of Figure 1.1(a). The temperature dependence of P s follows Equation (1.3). Note the linear temperature dependence of χ 1 in accordance with the Curie-Weiss law. Also for T < T c the absolute value of the slope for χ 1 (T) is twice than that for χ 1 (T) when T > T c. At T = T c, χ 1 becomes zero, so in theory the susceptibility diverges. For real materials, the ferroelectric transition is signified by a large but finite peak in the dielectric constant.

31 4 (a) P 0 T>T 0 T=T 0 F χ -1 T 0 χ T P T<T 0 T 0 (b) T>T c P 0 F T=T c T 0 T c T P χ -1 χ T=T 0 <T c T 0 T c Figure 1.1. Second-order phase transition (a). The free energy, F is plotted as function of polarization for T>T c, T=T c, and T<T c. The right top panel is a schematic of the spontaneous polarization P s (T), and the bottom right panel illustrates the temperature dependence of the susceptibility, χ(t) and its inverse χ 1 (T). (b) First-order phase transition. The free energy, F is plotted on the left panel as function of polarization for T>T c, T=T c, and T=T 0 <T c. The right top panel is a schematic of the spontaneous polarization P s (T), and the bottom right panel shows the temperature dependence of the susceptibility, χ(t) and its inverse χ 1 (T). Modified from [4].

32 First Order Transitions First order (discontinuous) phase transitions occur when b < 0, and c > 0. For zero external field(e = 0), and taking the partial derivative of Equation (1.1) with respect to P the following expression is obtained: a 0 (T T 0 )P b P 3 + cp 5 = 0 (1.5) Equation (1.5) has three roots: P = 0 and two non-zero solutions from cp 4 b P + a 0 (T T 0 )=0. As illustrated on the left side of Figure 1.1(b), the polarization is discontinuous across the phase transition, but now T 0 T c. The temperature dependence of the free energy, F(P), susceptibility, χ(t) and its inverse are also shown in Figure 1.1(b) Polarization Hysteresis Ferroelectrics consist of electrical dipoles that can be reoriented under an external electric field. A typical polarization versus electric field (P-E) hysteresis loop is shown in Figure 1.. Here we assume that the applied electric field has a triangular waveform with no DC offset. The shape of the P-E loop can be explained as follows: initially, the material has no net polarization because it consists of randomly orientated dipoles. As the electric field begins to increase the random dipoles gradually align towards the direction of the applied field. Since reversible domain wall displacements are easily achieved in this low field region, the material behaves like a linear dielectric. As the electric field is further increased, the polarization reaches saturation due to nucleation of domains parallel to the field and irreversible domain wall motion. At this point all the electrical dipoles are fully aligned in the direction of the applied field. The spontaneous polarization (P s ) of the material corresponds to the value of the saturation polarization.

33 6 P P r P s E c E Figure 1.. Schematic of a polarization hysteresis loop for a typical ferroelectric indicating the coercive field E c, the spontaneous polarization, P s, and the remanent polarization, P r. The material can be switched between two stable states, as indicated by the saturated polarization at the highest positive and negative fields, by the application of an external electric field. The direction of the electric field is indicated by the small arrows in the P-E loop. When the electric field reaches the peak of the triangular waveform and its amplitude begins to drop, the polarization does not immediately reverse. Instead it stays aligned with the increasing cycle of the applied field. Because of this delay in the response of the material, when the electric field is further reduced and becomes zero, the polarization has a non-zero value; this is known as the remanent polarization P r. Next, as the applied field becomes negative (it is applied in the opposite direction compared to the positive cycle), the electrical dipoles begin to rotate towards the opposite direction as well. This leads to a reduction of the net polarization, reaching a zero value at E c. E c is known as the coercive field, and it is a measure of how easy or hard it is to switch a ferroelectric. As the field becomes more negative, the polarization also becomes negative and reaches negative saturation. Once again, as the electric field reaches the maximum negative

34 7 value and starts to increase, the polarization does immediately respond; it eventually reaches positive saturation when the electric field reaches its maximum positive value. The same P-E loop is traced for all subsequent cycles of the electric field Symmetry Arguments Ferroelectricity is allowed only in certain materials which are non-centrosymmetric. Out of a total of 3 point groups 1 are non-centrosymmetric. These are: 1,, m,, mm, 4, 4, 4, 4mm, 4m, 3, 3, 3m, 6, 6, 6, 6mm, 6m, 3, 43, and 43m. In addition, all but one of the non-centrosymmetric point groups are piezoelectric (group 43 is the exception). Piezoelectricity refers to the linear coupling between stress and electric polarization (direct piezoelectric effect), or equivalently the linear coupling between strain and applied electric field (converse piezoelectric effect). A special subset of piezoelectrics are polar or pyroelectric; these are materials where the electrical polarization changes as function of temperature. There are 10 polar groups and these are: 1,, m, mm, 3, 3m, 4, 4mm, 6, 6mm. For a material to be ferroelectric, it must be non-centrosymmetric, piezoelectric, pyroelectric and a P-E loop must be observed. In addition, the spontaneous polarization is allowed only along certain directions known as the polar axes or within polar planes. Figure 1.3 summarizes the symmetry requirements for ferroelectrics Ferroelectric Crystal Structures The main groups of ferroelectric materials important for optical applications are the following: oxygen-octahedra ferroelectrics and phosphate, arsenate, sulphate, selenate structure ferroelectrics [5]. Oxygen-octahedra ferroelectrics (thin films) are extensively studied in this work, whereas phosphate, etc., type ferroelectrics are not the focus of

35 8 3 point groups centrosymmetric (11) non-centrosymmetric (1) non-piezoelectric (1) piezoelectric (0) non-polar (10) polar (10) pyroelectric (10) no P-E loop P-E loop non-ferroelectric ferroelectric Figure 1.3. Classification of 3 point group crystal classes into piezoelectric, pyroelectric (polar), and ferroelectric point groups. Ferroelectrics are a subset of piezoelectric, and pyroelectric materials which exhbit a P-E loop. Conversely, a ferroelectric must be both pyroelectric and piezoelectric. this work. Oxygen-octahedra ferroelectrics can be further classified into perovskite, Aurivillius-phase, Tungsten-bronze (TB) phase, and LiNbO 3 structure ferroelectrics. Typically, ferroelectricity is triggered by the Jahn-Teller effect which causes the displacement of the small metal cation (with valence of 4+ or 5+) from the center of an oxygen octahedron leading to its distortion as well as the creation of finite dipole moment. Perovskite oxide ferroelectrics have a prototype ABO 3 structure as shown in Figure 1.4. ABO 3 ferroelectrics are widely studied, experimentally and theoretically, in bulk and thin film form. Some of the most important ABO 3 ferroelectrics are the well known BaTiO 3, KNbO 3, PbTiO 3, Pb 1 x La x (Zr y Ti 1 y ) 1 0.5x V B 0.5x O 3 (PLZT), and Ba 1 x Sr x TiO 3 [5]. In this structure, the larger A cations occupy the corners of the unit

36 9 cell, whereas the smaller B cation is surrounded by 6 oxygen anions occupying the face centers of the unit cell. In the paraelectric phase, the B cation sits in the center of the oxygen octahedron. Spontaneous polarization appears either when A or B, or both, displace relative to the oxygen octahedra. In addition, since the oxygen octadehra are interconnected (via face, edge, or corner sharing) their rotations or tilts have significant effects on the properties of the structure. For example, they can suppress or facilitate ferroelectricity depending on whether neighboring octahedra oppose or favor rotations/tilts induced by their nearest neighbors. Aurivillius phase ferroelectrics consist of perovskite-like (Me n 1 R n O 3n+1 ) layers alternating with (Bi O ) + layers, where Me can be Bi +, Ba +, Sr +, Ca +, and so on, and R can be Ti 4+, Nb 5+, Ta 5+, Mo 6+, and so on [6]. Their general formula is Bi Me n 1 R n O 3n+3. Examples of such ferroelectric structures are: Bi 4 Ti 3 O 1, SrBi Nb O 9, SrBi Ta O 9, BaBi Nb O 9, etc. Another group of stacked ferroelectric structures are the Ruddlesden-Popper (RP) phases [7]. Their general formula is given by A n+1 B n O 3n+1 where n is the number of perovskite unit cells sandwiched by AO layers. Extensive experimental studies were recently undertaken on a series of strained Sr n+1 Ti n O 3n+1 samples for n = 1-6. The structures appear to be ferroelectric for n = 3 through 6. Finally tungsten bronze (TB) structures include ferroelectrics such as Sr 5 x Ba x Nb 10 O 30 (SBN), etc [5]. Lithium niobate structure ferroelectrics such as LiNbO 3 and LiTaO 3 are exceedingly useful optical ferroelectrics very commonly used in applications such as high speed electro-optics, acousto-optics, beam steering, focusing, frequency conversion, etc. The oxygen octahedra BO 6 are face-sharing along the c-axis, and their geometric centers are sequentially occupied by Li, Nb or Ta, vacancy, Li, Nb or Ta, vacancy, and so on. When the structure transforms from the high temperature paraelectric phase 3m to the low temperature ferroelectric phase 3m, with T c 1480 K, the Nb or Ta ions, are displaced from

37 10 c x z y a b Figure 1.4. Schematic of the ABO 3 perovskite structure. The larger cations A, shown in gray, occupy the corners of the unit cell, and the smaller cations B are surrounded by 6 oxygen atoms (not shown here) which form networks of connected octahedra. Here the octahedra (shown in purple) are connected at their vertices (edge sharing). Breaking of the inversion symmetry typically occurs when the smaller B cations move away from the geometrical center of the oxygen octahedra leading to the creation of spontaneous polarization. their equilibrium positions, along the c axis. Due to electrostatics, their motion causes a similar displacement (along the c axis) for the Li ions. The latter are displaced outside their original octahedra and fill the vacancy sites of the neighboring octahedrons. These ionic displacements create significant spontaneous polarization along the c axis; at room temperature LiNbO 3 has spontaneous polarization of 71 µc/cm, whereas LiTaO 3 has a somewhat smaller polarization of 50 µc/cm [3] Ferroelectric Domains Ferroelectrics consist of regions of uniform polarization known as domains; the boundaries between domains are called domain walls. Domains are formed as a way of minimizing the total energy of the material but they can also be created due to defects, non-uniform strain, etc. A simple model of a material with net electrical dipoles is that of a rectangle with opposite charges on two ends. Charge neutrality is satisfied as a

38 11 whole but the interfaces can be seen as locally charged. Electrical neutrality of the interfaces is achieved with the creation of surface charges; these create an electric field which is opposite to that created by the electric dipoles of the material; this is known as the depolarizing f ield. The material will try minimize this depolarizing field by splitting up into domains. However since the formation of domains requires certain energy expenditure, due to the creation of new domain walls, the total energy of the material is a fine balance between the decrease in the depolarizing field, and the increase in energy required to maintain more domains. For thin films this energy competition leads to the formation of multidomain ferroelectric surfaces [4]. Domain walls have descriptive names which relay information about the directions of polarization on either side of the wall. For example, an 180 wall refers to a wall which separates polarizations that are parallel to the wall and pointing in opposite directions. A 90 wall separates polarizations which form a 90 angle between them; the wall is perpendicular to both of them. Therefore, a 90 wall is equivalent to crystallographic twinning [5]. Somewhat more complicated are the 10 and 60 domain walls; these occur when the polarization is along the face diagonals of a cube. Again the wall is perpendicular to the polarizations on either side, but now these form 10 and 60 angles respectively. In addition, the number (and types) of domain walls are determined by the symmetry of the ferroelectric material: the lower the symmetry, the more degrees of freedom for the allowed polarization directions; thus a larger number of domain variants is allowed to form. For instance, monoclinic symmetry allows for 4 types of walls, whereas orthorhombic symmetry allows only 4: 60, 10, 90, and 180 walls. Similarly, tetragonal symmetry allows for 90 and 180 walls, whereas trigonal, and hexagonal structures can only have 180 domain walls.

39 1 1. Strain-enabled Ferroelectricity The effect of strain on ferroelectric properties has been extensively investigated in recent years. A strong coupling between strain and ferroelectricity exists and as a result significant enhancement of ferroelectric properties such as T c and P s. has been predicted and observed when straining ferroelectrics (for example BaTiO 3 ) [8]- [9]. In addition, theory predicts that certain non-ferroelectric materials turn ferroelectric when strained. This has been experimentally verified for a number of different systems including SrTiO 3 and EuTiO 3 [10]-[1]. Thin films are ideal candidates for straining because they can withstand large amount of strain (tensile or compressive) compared to bulk materials. They are grown on top of substrates, which are either single crystals or wafers that are chemically and structurally compatible with the film, with techniques such as Molecular Beam Epitaxy (MBE) or Pulsed Laser Deposition (PLD). The film becomes strained either through the mismatch of its lattice parameters with respect to the substrate lattice parameters or from defects during the growth process. It is important to work only with fully coherent films (not relaxed) in order to directly correlate the effect of strain on their properties. Tensile strain refers to positive (+) strain which causes the unit cell of the film to stretch compared to its bulk form; whereas, as the name implies, compressive (-) strain causes the unit cell of the film to compress (compared to its bulk form). For the special case where both the film and the substrate have ABO 3 perovskite structures the chemical bonding occurs at the oxygen terminated layer of the substrate; hence the film and the substrate become connected at the oxygen vertices of the corner sharing octahedra. As mentioned previously, the larger A ions are centered between the oxygen octahedra, and the smaller B ions are centered within the octahedra. Some common substrates used for thin film growth are YAlO 3, LaSrAlO 4, LaAlO 3,

40 13 LaSrGaO 4, NdGaO 3, (LaAlO 3 ) 0.9 -(Sr 0.5 Al 0.5 TaO 3 ) 0.71 (LSAT), LaGaO 3, SrTiO 3, and KTaO 3. High quality single crystals of DyScO 3, GdScO 3, SmScO 3 and NdScO 3 were recently grown using the Czochralski technique [15]. These rare earth scandates (ReScO 3 ) are ideal substrate materials since they have very good mechanical and chemical stability over a wide range of temperature, similar thermal expansion coefficients as those of the films, and have perovskite-like structures. In particular, room temperature ReScO 3, where Re = Ho to La, have orthorhombic GeFeO 3 structure with space group Pnma [15]. The unit cell of orthorhombic DyScO 3 is shown in Figure 1.5(a). The dark lines connecting the scandium ions indicate the pseudocubic unit cell which has particular importance for thin film growth, because it forms a close-to-cubic (pseudocubic) mesh on the (101) face. The pseudocubic lattice constants can be easily calculated. The room temperature lattice constants for orthorhombic DyScO 3 are a = 5.717() Å, b = 7.901() Å, and c = 5.443() Å ; hence the longer pseudocubic axis is b/ or 3.950(5) Å, and the shorter axis is (a/) +(c/) or 3.946(8) Å. In general, rare earth scandates have pseudocubic lattice constants ranging from 3.93 to 4.05 Å ; as a result they can impart moderate strains to important ferroelectrics such as (Ba,Sr)TiO 3, BiMnO 3, BiFeO 3, PbTiO 3, BaTiO 3 and so on [16]. Figure 1.5(b) shows the ScO 6 octahedra of DyScO 3. In terms of the pseudocubic unit cell, the Glazer tilts are a b + a [14], which is equivalent to the orthorhombic a b + c 0. These corresponds to an in-phase tilt along the b axis, and an antiphase tilt along the a axis. For more information on the Glazer notation please refer to Appendix A. The effect of strain on ferroelectrics can be quantified when strain dependent terms are added to Equation (1.1). In particular, the free energy F will consist of two terms: the polarization dependent term F p from (1.1), and a strain dependent term F η. For a

41 14 (a) c a b Sc 3+ Dy 3+ (b) O - c a b Figure 1.5. Orthorhombic DyScO 3 crystal structure with (a) pseudocubic unit cells (dark lines) within the orthorhombic unit cell (dashed lines), and (b) tilting of the ScO 6 octahedra. Modified from [13]. uniaxial ferroelectric, F η is given by [4]: F η = 1 Kη + QηP +... ησ (1.6) where η is a component of the strain. K is an elastic constant (Hooke s law), P is the

42 15 polarization, and σ is the external stress. Equilibrium properties are found by minimizing F both with respect to P and η. In particular the partial derivative of F with respect to the strain η can be shown to be: F(P, η) η = Kη+QP σ=0 (1.7) For no external stresses, σ = 0, we have η= QP /K. Substituting this in Equation (1.1) the following expression is obtained: F = 1 a 0(T T 0 )P (b Q /K)P cp6 EP (1.8) When comparing Equation (1.8) to (1.1), the only term that is diffent is the quadratic coefficient. For the case of first-order transitions (b < 0) T 0 is increased, and when Q /K > b>0 a second order transition occurs. Even though the Landau-Devonshire theory is valid for bulk ferroelectrics with uniform polarization, qualitatively similar results can be derived for thin film, multidomain systems. In general, enhancement of T 0 is expected, and experimentally observed, as the strain increases. 1.3 Overview of Experimental Techniques This thesis focuses heavily on optical Second Harmonic Generation (SHG) measurements Optical Second Harmonic Generation Nonlinear optical effects can be accounted in terms of the Lorentz oscillator model which describes the displacement x of an electron under an electric field E(t). Ideally,

43 16 the potential energy of an electron can be modeled as that of a perfect spring or a harmonic oscillator i.e. V(x) = 1/ mω 0 x. For more realistic effects (non-ideal spring behavior), extra power terms are included in V(x), such as V(x)=1/ mω 0 x + Ax 3 + Bx The nonlinear oscillator model can be written as: x t + ω 0x+ax = e E(t) (1.9) m where ω 0 is the electron oscillation frequency, a takes into account nonlinear effects and is equal to 3A/m, e is the electron charge, and m is the electron mass. Solving Equation (1.9) for E(t)=R(E ω cos(ωt)), a solution of the following form is obtained: x(t)= e/m ω 0 ω E ω cos(ωt) a a ( 1 e/m ω 0 4ω ω 0 ω ω 0 ( e/m ω 0 ω ) E ω cos(ωt) ) E ω (1.10) This tells you that the displacement of the electron x oscillates both at the fundamental excitation frequency ω, as well as at ω. The second term is frequency independent (DC) and corresponds to a process known as optical rectification [17]. Here we are interested in the third term which oscillates at the second harmonic frequency ω. Since, polarization P is defined as the dipole moment per unit volume and is given by P = Nex where N is the number of atoms per unit volume, by substituting the nonlinear displacement x from Equation (1.10), we can obtain the nonlinear polarization, P (). This is given by: P () = Nae 3 m (ω 0 4ω )(ω 0 ω ) E ω (1.11)

44 17 From the Maxwell equation: E E ε 0 µ 0 t = µ P 0 t (1.1) we can see that any polarization P gives rise to electric field E. Hence, P () must generate electric field at ω. This is what we call second-harmonic generation (SHG). An intuitive way of looking at SHG is the following: when very intense light interacts with matter the electric polarization P is no longer linearly proportional to the electric field, E. If one includes higher (nonlinear) effects, P can be expressed as P=χ (1) E+ χ () EE+ χ (3) EEE+... (1.13) where χ is the electric susceptibility. The second order susceptibility or nonlinear susceptibility χ () is a polar third rank tensor, and is nonzero for crystals lacking inversion symmetry. If we consider two input electric fields, of the same frequency, of the form ( E(r,t)=E 0 e ikr iωt + cc ), then substituting in Equation (1.13), we get: P () = χ () E 0+ χ E 0 ( ) e ikr iωt + cc (1.14) The second term of this equation is referred to as frequency doubling or second harmonic generation and it is consistent with the result obtained from the anharmonic oscillator model. The second term of Equation (1.13), the nonlinear electrical polarization, can be written as P (), and it is given by: P () i = χ i jk E je k (1.15) j,k

45 18 where χ () i jk are tensor components. The explicit form of (1.15) is the following: P x () P y () P () z = χ () xxx χ () xyy χ () xzz χ () xyz χ () xzy χ () xzx χ () xxz χ () xxy χ () xyx χ () yxx χ () yyy χ () yzz χ () yyz χ () yzy χ () yzx χ () yxz χ () yxy χ () yyx χ () zxx χ () zyy χ () zzz χ () zyz χ () zzy χ () zzx χ () zxz χ () zxy χ () zyx E x E y E z E y E z E z E y E z E x E x E z E x E y E y E x (1.16) i, j, k are indices related to the orthogonal axes x, y, z. For the special case of second harmonic generation (SHG) the two excitation waves are the same, E j = E k ; therefore χ () i jk = χ() ik j. Using the convention i=1,,3 to denote the x, y, and z axes, the polarization components can be written as: P x () P y () P () z E y d 11 d 1 d 13 d 14 d 15 d 16 = E d 1 d d 3 d 4 d 5 d z 6 E y E z d 31 d 3 d 33 d 34 d 35 d 36 E x E z E x E y E x (1.17) where d i j is the short-hand notation of d i jk and is known as the nonlinear optical coefficient. In particular, xx = 11 = 1, yy = =, zz = 33 = 3, yz = 3 = 3 = 4, xz=13=31=5, and xy=1=1=6.

46 19 Since ferroelectrics lack inversion symmetry (they are non-centrosymmetric) they should be SHG active. Therefore, observation of SHG signal is a necessary but not sufficient condition for a system to be ferroelectric. In other words, ferroelectrics are a subgroup of SHG active systems. In order to establish ferroelectricity, an SHG intensity versus electric field loop must be also collected. For ferroelectrics, this should yield an I ω E butterfly loop which can be converted to the standard P E hysteresis loop Experimental SHG set-up SHG signal is produced when matter interacts with very intense light. Therefore it was not a coincidence that the first successful nonlinear optics experiment was performed after the discovery of the laser. Ultrafast lasers are excellent candidates for creating nonlinear optical phenomena because of their high peak power output. In the case of pulsed lasers, one must make a distinction between the average power and the peak power. Average power[w] is defined as the ratio of the pulse energy[j] over the inverse repetition rate[s] of the laser, whereas peak power is defined as the ratio of pulse energy [W] per pulse duration. As the pulse duration is decreased, the peak power becomes significant. For example, for pulse energy of J and 80 fs pulse duration, the peak power is approximately 76 kw. On the other hand, for the same pulse energy and a repetition rate of 8 MHz, the average power is approximately 1 W. All the SHG experiments described in this work were carried out using two laser sources: the Spectra Physics Tsunami, and the Spectra Physics Spitfire. The Tsunami is a mode-locked Ti:sapphire laser with a tunable output from nm, pulse duration of approximately 80 fs, and average output power of 1.1 W. A titanium-doped sapphire (Ti:sapphire) rod is pumped by the Spectra Physics Millenia 53 nm Continuous Wave (CW) laser at 8. W (optical power). Ti:sapphire is a solid state medium which has a broad absorption band in the visible regime from approximately nm, and a very

47 0 broad emission band from approximately nm (near infrared). For proper laser operation the Ti:sapphire rod must be water cooled, so the user must start the Lytron chiller first (temperature setpoint = 18 C) first. The beam divergence is less than < 0.6 mrad and 1/e beam diameter approximately mm. In order to obtain even higher peak pulse power, part of the Tsunami output is used as the input to the Spitfire regenerative amplifier. The Spitfire operates on the principle of Chirped Pulse Amplification (CPA) where an ultrafast pulse is first stretched into longer time duration, amplified, and then compressed back to shorter duration. In particular, the femtosecond pulses from the Tsunami are stretched into ns duration using a pair of diffraction gratings. When the beam is incident on the diffraction grating, its frequency components are dispersed. The grating can be configured such that the blue frequencies travel a longer path than the red frequencies, causing the pulse to stretch. Next, the ns pulses are allowed to enter the laser amplification cavity (via Pockel cell 1), which consists of a Ti:sapphire rod that is pumped by a 53 nm CW laser (Spectra- Physics Evolution 15). Once again, the Ti:sapphire rod must be water cooled, so a second Lytron chiller must be turned on before operating the Spitfire (temperature setpoint = C). The Ti:sapphire rod emits light in the near infrared band which adds constructively with the weak ns pulses. After a certain number of roundtrips in the laser cavity, optimum amplification is achieved, and the pulses are allowed to exit (via Pockel cell ). Adjusting the timing of the second Pockel cell can optimize the output power. Finally, the amplified ns pulses go through the compressor grattings, which reverse the stretching, and compress the pulses to approximately 130 fs. The settings of the SDG II fast electronic module are the following: Synch Enable: ON, Continuous: ON, OUT DELAY 1 ns = 106, OUT DELAY ns = 73, SYNCH OUT DELAY = 99 ns. In addition, photodiode 1 and (PD 1, ) which monitor the stability of the Tsunami must be both on. The Spitfire output is linearly polarized (horizontal) with a beam diameter 6-8

48 1 mm, pulse energy 1mJ, and centered at approximately 800 nm. A typical SHG set-up is shown in Figure nm (fundamental) light is incident on the sample under investigation. A Glan- Taylor calcite polarizer (Thorlabs GTO10B) with an extinction ratio of 100,000:1 provides extremely pure linear polarization polarized along the x axis. Next, the beam passes through an achromatic λ/ waveplate (Thorlabs AHWP05M-950) which is mounted on a motorized rotation stage (Newport UE16cc). This allows precise rotation of the input polarization. Any SHG signal that might have been generated by previous optics is filtered by a highpass filter (Thorlabs FEL0750). An f =100 mm lens, mounted on a linear translation stage, focuses the light onto the sample. The translation stage allows for fine control of the location of the focus. Another f =100 mm lens collimates the second harmonic light whereas the fundamental is rejected by stacked bandpass filters (Newport BG40). Precise SHG polarization selection is achieved with a polarizer (Newport 10LPVIS) mounted on a manual rotation stage. The weak SHG intensity is detected with a photomultiplier tube (Hamamatsu R7899) equipped with a narrow bandpass filter (Thorlabs FEL400-10), and the signal is measured with a lock-in amplifier (SRS 830) which is connected to a computer controlled data acquisition card. For all experiments, the PMT is operating in its linear regime. Reflection or transmission SHG geometries are used depending on whether the sample is transparent or opaque, the direction of film polarization, as well as the optical transparency of the substrate. Furthermore, temperature dependent studies are carried out with a stainless steel heater, a nitrogen cooled stage (Instec HCS6V-F8) and two helium cryostats equipped with windows: Oxford 5110 and Janis ST-300S. The combined temperature range achievable is K. Finally, electrical feedthroughs on both of the cryostats allow low temperature electric field dependent studies.

49 Laser ω GT BS λ/ ND F1 ω L f=10cm sample L f=10cm ω ω F A ω PMT ω PD heating stage, cooling stage, rotation stage, etc Figure 1.6. Experimental second harmonic generation (SHG) set-up in typical transmission geometry. GT: Glan-Taylor prism, BS: beamsplitter, PD: photodiode, λ/: half waveplate, ND: neutral density filters, F1: long-pass filter, L: lens, F: blue filters, A: analyzer, PMT: photomultiplier tube. The sample is mounted on a heating, cooling, or rotation stage, depending on the experiment. 1.4 Thesis Organization This thesis consists of 6 chapters. Chapter 1 serves as an introduction with a brief overview of ferroelectricity, nonlinear optics, SPM, as well as a detailed discussion of the experimental apparatus. Chapters and 3 focus on SHG studies of phase transitions and domain structures of ferroelectric thin films, such as SrTiO 3 and CaTiO 3, as well as confocal SHG studies of polar CaTiO 3 surfaces. Finally, multiferroic EuTiO 3 is discussed in Chapters 5. Chapter 6 is a brief summary of the results, conclusions, as well as suggestions for future work. Numerous appendices are added in order to keep this work informative, and useful to the reader.

50 3 Bibliography [1] J. F. Scott, Ferroelectric memories, Springer-Verlag, Berlin Heidelberg, (000). [] N. A. Hill, Why are there so few magnetic ferroelectrics, J. Phys. Chem. B 104, (000). [3] M. E. Lines, and A. M. Glass, Principles and applications of ferroelectrics and related materials, Clarendon, Oxford (1977). [4] K. M. Rabe, C. H.Ahn, and J. -M. Triscone (Eds), Physics of Ferroelectrics. A modern perspective., Springer-Verlag, Berlin Heidelberg (007). [5] M. C. Gupta, and J. Ballato, The handbook of photonics, CRC Press (007). [6] J. M. Perez-Mato, M. Aroyo, A. Garcia, P. Blaha, K. Schwarz, J. Schweifer, and K. Parlinski, Competing structural instabilities in the ferroelectric Aurivillius compound SrBi Ta 0 9, Phys. Rev. B 70, (004). [7] C. J. Fennie, and K. M. Rabe, Structural and dielectric properties of Sr TiO 4 from first principles, Phys. Rev. B 68, (003). [8] K. J. Choi, M. D. Biegalski, Y. L. Li, A. Sharan, J. Schubert, et. al., Enhancement of ferroelectricity in strained BaTiO 3 thin films, Science, 306, (004). [9] D. A. Tenne, P. Turner, J. D. Schmidt, M. D. Biegalski, Y. L. Li, L. Q. Chen, A. Soukiassian, S. Trolier-Mckinstry, D. G. Schlom, X. X. Xi, D. D. Fong, P. H. Fuoss, J. A. Eastman, G. B. Stephenson, C. Thompson, and S. K. Streiffer, Ferroelectricity in ultrathin BaTiO3 films: Probing the size effect by ultraviolet raman spectroscopy, Phys. Rev. Lett (009). [10] M. D. Biegalski, Y. Jia, D. G. Schlom, S. Trolier-McKinstry, S. K. Streiffer, V. Sherman, R. Uecker, and P. Reiche, Relaxor ferroelectricity in strained epitaxial SrTiO 3 thin films on DyScO 3 substrates, Appl. Phys. Letters, 88, (006). [11] Y. L. Li, S. Choudhury, J. H. Haeni, M. D. Biegalski, A. Vasudevarao, A. Sharan, H. Z. Ma, J. Levy, V. Gopalan, S. Trolier-McKinstry, D. G. Schlom, Q. X. Jia, and L. Q. Chen, Phase transitions and domain structures in strained pseudocubic (100) SrTiO 3 thin films, Phys. Rev. B 73, (006).

51 4 [1] J. H. Lee, L. Fang, E. Vlahos, X. Ke, Y. W. Jung, L. Fitting-Kourkoutis, et. al., Creating a Strong Ferroelectric Ferromagnet via Spin-Phonon Coupling, accepted to Nature (010). [13] M. D. Biegalski, Epitaxially Strained Strontium Titanate, Ph.D. dissertation, (006). [14] A. M. Glazer, The classification of tilted octahedra in perovskites, Acta Crystallographica, B 8, 3384 (197). [15] B. Velickov, V. Kathlenberg, R. Bertram, and M. Bernhagen, Crystal chemistry of GdScO 3, DyScO 3, SmScO 3, and NdScO 3, Z. Kristallogr, (007). [16] D. G. Schlom, L. Q. Chen, C. B. Eom, K. M. Rabe, S. K. Streiffer, and J.-M. Triscone, Strain Tuning of Ferroelectric Thin Films, Annu. Rev. Mater. Res., 73, (007). [17] P. W. Milonni, and J. H. Eberly, Lasers, Wiley & Sons, Inc., 1988.

52 5 Chapter Nonlinear Optical Probing of Anisotropically Strained Ferroelectric SrTiO 3.1 Introduction Room temperature strontium titanate (SrTiO 3 ) is an ABO 3 cubic perovskite with point group symmetry m 3m (space group Pm 3m) and lattice constant a = Å. At room temperature, the oxygen anions sit at the 3c Wyckoff positions (1/, 1/, 0), the titanium cation is at the 1b site (1/,1/,1/) and the strontium cation is at the 1a site (0,0,0) [1]. At approximately 105 K cubic SrTiO 3 undergoes a nonpolar antiferrodistortive phase (AFD) transition, attributed to its TiO 6 octahedra rotations, and it transforms to tetragonal phase with point group 4/mmm (space group I4/mcm). The Glazer notation which describes octahedra distortions is given by a α b β c γ where a, b, and c correspond to the crystallographic axes and the superscripts describe the tilts along a, b, and c respectively. In particular, a superscript of 0 means no tilt, + describes inphase tilt, and describes out-of-phase (anti-phase) tilt []. The Glazer notation for cubic SrTiO 3 is given by a 0 a 0 a 0, whereas the Glazer notation for the tetragonal phase is a 0 a 0 c. The unit cell of cubic SrTiO 3 and its oxygen octahedra are shown in Figure.1. In addition, SrTiO 3 exhibits a polar phonon instability at low temperature. This soft

53 6 (a) x z y Sr + Ti 4+ O - (b) z x y Figure.1. (a) Room temperature crystal structure of cubic SrTiO 3, (b) alternative representation of the cubic cell showing the TiO 6 octahedra encompassing the Ti ions. Note: the ionic radii are not drawn to scale. mode is representative of a transverse optical mode corresponding to the vibration of an atom relative to the oxygen octahedron center. Freezing or hardening of the mode means that the atom moves away from the geometrical center of the octahedron and a spontaneous dipole is created. Typically, soft mode freezing is an indication of paralectric to ferroelectric phase transitions. For SrTiO 3, the tendency for soft mode freezing is reflected by the temperature dependence of its dielectric properties. Its dielectric con-

54 7 stant increases from 300 at room temperature to more than 0000 at 5 K, and it follows the Curie-Weiss law with T c 40 K [3]. However, for temperatures less than 10 K the dielectric constant levels, and in fact the ferroelectric transition does not occur. The suppression of the transition is attributed to quantum fluctuations; hence SrTiO 3 is refer.1.1 Theoretical Predictions The effect of epitaxial strain on SrTiO 3 has been investigated extensively both theoretically and experimentally [1], [6]-[11]. Here is a brief summary of the established results. First principles calculations predict that even a small amount of epitaxial strain can turn SrTiO 3 ferroelectric [1]. In particular, for tensile strain greater than 0.54% SrTiO 3 becomes ferroelectric with orthorhombic point group symmetry mm (space group Amm ) and spontaneous polarization P s along 110 p. Similarly, for compressive strain greater than 0.75% the structure becomes ferroelectric with tetragonal point group symmetry 4mm (space group P4mm) and P s along [001]. First principles calculations are complemented by phase field simulations which can predict ferroelectric transition temperatures, domain structures, etc. The ferroelectric transition temperature range versus in-plane strain is shown in Figure. [7]. The right side of this diagram is of particular interest because it predicts T c to be in the room temperature range for strains achievable with commercially available substrates. Extensive experimental work has verified that strained SrTiO 3 thin films are ferroelectric close to room temperature [8], and are in fact relaxor ferroelectrics [6]. Very briefly, the distinguishing features of relaxors, when compared to typical ferroelectrics, are: (1) strong frequency dependence and broad peak of the dielectric constant below the temperature of the dielectric constant maximum (T max ), () switchable polarization which persists well above T max, (3) slim P-E hysteresis loops near T max, and (4) no macroscopic volume or structure change

55 8 at T max [1]. This work discusses the effects of non-uniform biaxial strain on the the properties of the strained thin film. Figure.. Theoretical predictions for the range of the ferroelectric Curie temperature T c versus in-plane strain ε 0 for SrTiO 3 after taking into account the AFD transition. The thin film is predicted to be ferroelectric for strains achievable with commercially available substrates. In addition, T c is predicted to be in the room temperature range under moderate tensile strain. Theoretical predictions by Dr. Y. Li [7]..1. Thin Film Growth and Characterization A 50 Å thick SrTiO 3 thin film was grown on a(101) DyScO 3 substrate with molecular beam epitaxy (MBE) using a shuttered growth technique by Mr. Charles Brooks [13], [14]. The film was fully coherent with the substrate and biaxially strained with 1.06 ± 0.03% strain along the longer DyScO 3 in-plane direction[010] and 1.03±0.03% along the shorter in-plane direction [10 1]. The out-of-plane direction corresponded to [001] p SrTiO 3. The epitaxial relationship is the following: (001) p SrTiO 3 (101) DyScO 3

56 9 and [010] p SrTiO 3 [010] DyScO 3. The high quality of the film is verified by the experimental rocking curve which has a Full Width Half Max (FWHM) of less than 7 arc sec. Here, the crystallographic description of DyScO 3 is based on the conventional space group #6, Pnma instead of the non-standard Pbnm [15]. The conversion from the Pnma a, b, c axes, and directions[a b c] to a, b, c axes, and directions[a b c ] in Pbnm is given by a = c, b = a, and c = b.. Dielectric Measurements Measurements of the in-plane dielectric properties of the film were carried out by Dr. M. Biegalski using interdigitated (IDT) electrodes deposited directly on the film. The IDTs were aligned at 0, 15, 30, 45, 60, and 90 from the long axis. The temperature and angle dependence of the dielectric constant and loss tangent tan(δ) at 5 KHz are shown in Figure.3. Two distinct peaks appear as function of IDT angle. When the IDT is aligned with the [010] axis or 0, a strong peak is observed at T 1 60 K. When the IDT is aligned with the [10 1] axis or 90 another peak is observed at T 0 K. A mix of the two peaks is observed at all other angles. The shape of the peaks is characteristic of relaxor ferroelectrics. Finally, a very small permittivity anomaly is observed at T K; this could correspond to the antiferrodistortive transition which has been predicted by thermodynamic calculations to be between 10 and 175 K for films under the same average strain. In addition, the switchable polarization was measured as function of temperature along the long y and short x axes. This is shown in Figure.4(a). The more strained y axis develops polarization at 40 K higher than the short x axis. These measurements suggest that T 1 corresponds to the development of a ferroelectric phase with polarization

57 Figure.3. Temperature dependent dielectric permittivity at 5 KHz of 50 Å thick SrTiO 3 thin film measured for several angles showing two peaks (T 1 and T ) as function of angle. 0 is aligned with the [010] p axis and 90 is aligned with the [100] p axis. A smaller anomaly T 3 occurs at approximately 165 K. Measurements by Dr. M. Biegalski [15]. 30

58 31 Switchable Polarization (µc/cm ) longer axis shorter axis (a) Temperature (K) Switchable Polarization (µc/cm ) (b) Experiment <100> No a 1 -a switching <110> Complete a 1 -a switching longer axis Angle ( ) shorter axis Figure.4. (a) Temperature dependence of the in-plane switchable polarization along the long [010] p and short [100] p axis, (b) switchable polarization as function of angle from the long axis at 70K for the 50 Å thick SrTiO 3 thin film. Measurements by Dr. M. Diegalski [15].

59 3 along the long axis, and T corresponds to a second transition where the polarization either rotates away from the longer axis or polarization develops along the shorter axis [15]. At lower temperatures, where both axes are ferroelectric, the angular dependence of switchable polarization shows a peak at approximately 45, as shown in Figure.4(b). In order to explain this behavior two models are proposed. One model assumes that polarization is along 100 p but it has uneven components along [100] p and [110] p, and the second model assumes polarization is along 110 p. It turns out both of them fit the angular dependence of polarization equally well (dashed lines in Figure.4(b)). Polarization dependent Second Harmonic Generation (SHG) measurements, and appropriate symmetry analysis, can clear up this ambiguity..3 Second Harmonic Generation (SHG) Measurements SHG measurements were carried out in order to determine(i) the point group symmetry of the strained film, and (ii) the direction of the in-plane polarization. As described in more detail in Chapter 1, SHG is a nonlinear process which involves the conversion of a fundamental light of frequency ω to light of frequency ω, as determined by the nonlinear polarization P ω i d i jk E j E k where d i jk is a third rank nonlinear tensor with form similar to that of the piezoelectric tensor [16]. The subscripts i, j, k refer to the orthogonal crystal physics axes of the sample. The sample was placed in a liquid Helium cryostat with angle of incidence, AOI = 0, and the 800 nm fundamental light was incident on the film side first. No SHG signal was observed from either the cryostat windows or the substrate. An achromatic polarizer (analyzer) was used to sample the generated SHG signal along certain polarization directions, and the weak SHG intensity was detected with a photomultiplier tube. SHG polar plots were recorded as function of temperature along the x[100] p, y[010] p,

60 33 x [110] p, and y [ 110 ] p, directions..3.1 SHG Analysis The orthorhombic point group mm is considered as a possible symmetry group for the strained thin film based on the fact that both x-ray diffraction and phase-field simulations indicate three different lattice parameters, two in the substrate plane and the third normal to the film surface. According to Neumann s principle, the only contributing (non-zero) nonlinear d i jk coefficients are d 15, d 4, d 31, d 3, and d 33. Assuming the two-fold rotation (polar) axis is in the plane of the film, two general models describing the direction of ferroelectric polarization are considered. Model I. The film has in-plane polarization along one direction only: either along [100] p (short axis), or along [010] p (long axis), or at an arbitrary angle ϕ away from the short axis. Model II. The film is composed of multiple ferroelectric domains with polarization orthogonal to each other, and the probing beam samples a mixture of different types of domains. Two cases are considered: Case I: in-plane ferroelectric domains with polarization along [100] p (short axis), and [010] p (long axis). Case II: ferroelectric domains with polarization along [110] p and[010] p. The validity of each of these models is examined next Polarization Along Single Direction This model assumes the ferroelectric polarization is along only one axis (short or long axis). For polarization along the long axis y, the polarization dependent SHG intensities are the following: I ω x = d 15 sin ((θ ϕ)) (.1)

61 34 and I ω y =((d 31 cos (θ ϕ)+d 33 sin (θ ϕ) ) (.) where θ is the angle of polarization of the fundamental beam, and ϕ describes polarization rotations in the plane of the film that can be sensitive to antiferrodistortive octahedra rotations [10]. In this case, ϕ is assumed to be small; therefore (.1) and (.) are simplified versions of the equations derived in Appendix C. Similar expressions can be derived when the polarization is only along x. Figure.5(a) and (b) show polar plots at T = 175 K with the analyzer parallel to the x and y directions respectively. Clearly, this model does not fit the experimental data Multi-domain Models As discussed in Chapter 1, formation of domains is one of the mechanisms that reduce the total energy of a ferroelectric. Here, we assume that the thin film consists of ferroelectric domains with polarizations orthogonal to one another. The SHG polar plot intensity can be written as I ω j = K 1, j sin ((θ ϕ))+k, j (sin (θ ϕ)+k 3, j cos (θ ϕ) ) + K 4, j (sin (θ ϕ)+k 3, j cos (θ ϕ) ) sin((θ ϕ)) (.3) where j = x, y, x, y and K i, j are constants. In addition, the ratio of the net polarization along y and x, Py net /Px net are given by ( P net) 4 y Px net ( K1x K y K x K 1y ) K 3y (.4)

62 35 and the ratio of the nonlinear coefficients, d 15 /d 33 d 15 4 d 33 ( K1x K 1y = K x K y ) K 3y (.5) where d 15 = d 15+ d 4. Case I. Each domain has polarization either along x[100] p or y[010] p. It can be easily shown that the following condition should be satisfied: d 33 d 31 = K 3x = K3y 1 (.6) where d 31 =(d 31+ d 3 )/ and. Since K 3 j = Iω j must also be satisfied: (0 )/I ω j (90 ) the following equality I ω x (0 )/I ω x (90 )=Iy ω (90 )/Iy ω (0 ) (.7) Figure.5(b) and (c) show polar plots at T = 175 K with the analyzer parallel to x and y respectively. The experimental data (circles) fit very well with the theoretical model (solid red line). In addition, the ratios I ω x I ω y (90 )/Iy ω (0 ) (1.00) are in excellent agreement. (0 )/Ix ω (90 ) (1.0) and Case II. The domain polarization is either along x [110] p or y [ 110 ]. The following p condition is obtained: I ω x (45 )/I ω x (135 )=I ω y (135 )/I ω y (45 ) (.8) Figure.5 (e) and (f) show polar plots at T = 175 K with the analyzer parallel to the x and y directions respectively. Even though the shape of the fit is good overall, the equality condition defined in (.8) is not satisfied, since I ω x (45 )/I ω x (135 ) (1.3)

63 36 and I ω y (135 )/I ω y (45 ) (0.74). Figure.6 plots the left and right side of Equation (.8) as a function of temperature; the equality is not satisfied. Therefore we can conclude that P s is along 100 p. A quick note on the polar plots fitting. Initially, each plot was fitted independently with equation (.3), and the extracted K 3, j coefficients were compared between related sets. However, due to the large number of fitting coefficients, the possibility of multiple solutions cannot be excluded; hence a more reliable fitting scheme was devised by Dr. T.A. Lummen where related sets of polar plots are fitted simultaneously while obeying the reciprocity condition. The results are consistent with either scheme. The temperature dependence of Iy ω (0 ) is shown in Figure.7(a), where y corresponds to the more strained long axis of the substrate or [010] p. Its temperature dependence is very similar to that of the measured spontaneous polarization as discussed at an earlier section. The difference in the temperatures at which the dielectric constant along the y axis has a peak (T 1 = T m ax) and where the system becomes nonpolar (T b 310 K) can be attributed to the relaxor nature of strained SrTiO 3 ; hence T b could correspond to the relaxor Burns temperature, and the small SHG signal can be attributed to glassy nanopolar regions, which exist in relaxor systems between T m ax and T b. The temperature dependence of Py net /Px net is also shown in Figure.7(b). For 5 < T < 175 K, the area probed by the laser beam has approximately equal fractions of domains with polarization along x[100] p and y[010] p. For T > 175 K, P net y greater than P net x T > 60K, P net y becomes, and the ratio reaches a peak of 7:1 at approximately 60 K. For starts to drop, and at room temperature it becomes 5:1. This /P net x trend can be explained as follows: at the lowest temperature, both axes, x and y, are ferroelectric; therefore Py net 1. As the temperature starts to increase, the least /P net x strained short axis, x becomes less ferroelectric faster than the most strained long axis y. This explains the increase in Py net for 175 K<T<60 K. At approximately 60 /P net x

64 37 ω E θ ω E x x' y y' (a) x[ 100] p (b) x[ 100] p y[ 010] p y[ 010] p (c) x[ 100] p (d) x[ 100] p y[ 010] p y[ 010] p (e) x' [ 110] p (f) x' [ 110] p [ 1 ] p y' 10 [ 1 ] p y' 10 Figure.5. The top schematic depicts the second-harmonic generation (SHG) setup where light of frequency ω and linear polarization at an angle θ with respect to the x axis, is converted to light of frequency ω, after interacting with the sample. By varying the angle θ, and fixing the analyzer along the x, y, x, and y, directions, SHG intensity polar plots, i.e., I ω x, I ω y, I ω x, and I ω y are obtained. Panels (a) through (f) show SHG polar plots at 175 K. These compare the theoretical models (thin solid red lines) to the experimental data (open circles). In particular, the comparison is the following: (a) Model I fit, I ω x, (b) model I fit, I ω y, (c) model II, case I fit, I ω x, (d) model II, case I fit, I ω y, (e) model II, case II fit, I ω x, and (f) model II, case II fit, I ω y. The thicker solid red lines in panels (c) through (f) represent intensities which must be reciprocal in order for the fitting models to be valid. Clearly, this is true only for (c) and (d), which are based on model II, case I [15].

65 38 d 31 '/d [100] p [010] p [110] p [110] p Temperature (K) Figure.6. Temperature dependence of the nonlinear coefficient ratio d 31 /d 33. Shown here are the quantities that correspond to left- and right-hand sides of Equation (.7) (black solid lines) and Equation (.8) (red dashed lines). For all measured temperatures, Equation (.7) is valid since the left- and the right-hand side quantities are equal (black squares and stars), whereas Equation (.8) is not valid. Since, Equation (.7) is satisfied, we can conclude that the in-plane polarization is along 100 p and not along 110 p directions [15]. K, the x axis is weakly or no longer ferroelectric, and P net y /Px net reaches a maximum. For 60 K<T<35 K, the ratio decreases with increasing temperature because the thin film approaches the ferroelectric to paralectric transition for the y axis, and the structure becomes non-polar. The ratio of the nonlinear coefficients, d 15 /d 33, which is an intrinsic property of the film, is plotted as function of temperature in Figure.7(c). There is a clear anomaly at 175 K, which could be associated with the antiferrodistortive transition (AFD) of

66 39 SrTiO 3. Furthermore, Figure.7(d) shows the temperature dependence of the offset an- I y ω (0 o ), (arb.u.) net P y /P x net d 15 '/d φ, ( o ) Temperature (K) Figure.7. Shown here is the temperature dependence of (a) second-harmonic intensity with a fixed analyzer along the y direction and θ =0, i.e., I ω y (0) and (b) the absolute ratio of the net polarization along the y axis to the net polarization along the x axis, P y net. Both (a) and /P net x (b) are in good agreement with the electrical measurements of the polarization. (c) shows the temperature dependence of the nonlinear coefficient ratio d 15 /d 33 and (d) shows the offset angle φ, from the polar plot fitting, as a function of temperature. Both show an anomaly at 175 K that can be most likely attributed to the AFD transition [15].

67 40 gle ϕ used in the polar plot fitting from (.3). For T > 00 K the offset is negligible. However, for T < 175 K, a small offset of 1.6 appears, and it persists down to the lowest temperature. This small rotation could be also attributed to the SrTiO 3 AFD transition, predicted between 10 K and 175 K [17],[18]. This offset arises from the coupling of the antiferrodistortion to the SHG signal [10]..4 Conclusions In summary, SHG studies were carried out in order to complement electrical measurements on an anisotropically strained SrTiO 3 thin film. Temperature dependent dielectric measurements reveal two distinct peaks at 60 K and 10 K. The first one was observed along the longer in-plane y [010] p axis and the second one along the shorter in-plane x [100] p axis. Switchable polarization, and SHG measurements confirm that these peaks are real. In particular, the higher temperature peak corresponds to the development of ferroelectric polarization along [010] p, while the lower transition corresponds to the development of ferroelectric polarization along [100] p. At temperatures less than 10 K, both the x and y axes are ferroelectric. Finally, the antiferrodistortive transition of SrTiO 3, near 170 K, was observed with both (dielectric and SHG) techniques.

68 41 Bibliography [1] A. Antons, J. B. Neaton, K. M. Rabe, and D. Vanderbilt, Tunability of the dielectric response of epitaxially strained SrTiO 3 from firs principles, Phys. Rev. B 71, 0410 (005). [] A. M. Glazer, The classification of tilted octahedra in perovskites, Acta Crystallographica, B8, 3384 (197). [3] K. A. Muller, and R. Burkard, SrTiO 3 : An intrinsic quantum paraelectric below 4 K, Phys. Rev. B 19, (1979). [4] O. E. Kvyatkovskii, Theory of isotope effect in displacive ferroelectrics, Solid State Commun. 117, 455 (001). [5] W. Zhong, and D. Vanderbilt, Effect of quantum fluctuations on structural phase transitions in SrTiO 3 and BaTiO 3, Phys. Rev. B 53, 5047 (1996). [6] M. D. Biegalski, Y. Jia, D. G. Schlom, S. Trolier-McKinstry, S. K. Streiffer, V. Sherman, R. Uecker, and R. Reiche, Relaxor ferroelectricity in strained epitaxial SrTiO 3 thin films on DyScO 3 substrates, Appl. Phys. Lett. 88, (006). [7] Y. L. Li, S. Choudhury, J. H. Haeni, M. D. Biegalski, A. Vasudevarao, A. Sharan, H. Z. Ma, J. Levy, V. Gopalan, S. Trolier-McKinstry, D. G. Schlom, Q. X. Jia, and L. Q. Chen, Phase transitions and domain structures in strained pseudocubic (100) SrTiO 3 thin films, Phys. Rev. B 73, (006). [8] A. Vasudevarao, A. Kumar, L. Tian, J. H. Haeni, Y. L. Li, C. -J. Eklund, Q. X. Jia, R. Uecker, P. Reiche, K. M. Rabe, L. Q. Chen, D. G. Schlom, and V. Gopalan, Multiferroic Domain Dynamics in Strained Strontium Titanate, Phys. Rev. Lett., (006). [9] A. Vasudevarao, S. Denev, M. D. Biegalski, Y. L. Li, L. Q. Chen, S. Trolier- McKinstry, D. G. Schlom, and V. Gopalan, Polarization rotation transitions in anisotropically strained SrTiO 3 thin films, Appl. Phys. Lett. 9, 1990 (008). [10] S. Denev, A. Kumar, M. D. Biegalski, H. W. Jang, C. M. Folkman, A. Vasudevarao, Y. Han, I. M. Reaney, S. Trolier-McKinstry, C. -B. Eom, D. G. Schlom,

69 4 and V. Gopalan, Magnetic Color Symmetry of Lattice Rotations in a Diamagnetic Material, Phys. Rev. Lett., 100, (008). [11] H. W. Jang, A. Kumar, S. Denev, M. D. Biegalski, P. Maksymovych, C. W. Bark, C. T. Nelson, C. M. Folkman, S. H. Baek, N. Balke, C. M. Brooks, D. A. Tenne, D. G. Schlom, L. Q. Chen, X. Q. Pan, S. V. Kalinin, V. Gopalan, and C. B. Eom, Ferroelectricity in Strain-Free SrTiO 3 Thin Films, Phys. Rev. Lett (010). [1] C. J. Stringer, N. J. Donnelly, T. R. Shrout, and C. A. Randall, Dielectric Characteristics of Perovskite-Structured High-Temperature Relaxor Ferroelectrics: The BiScO 3 -Pb(Mg 1/3 Nb /3 )O 3 -PbTiO 3 Ternary System, J. Am. Ceram. Soc., (008). [13] D. G. Schlom, L. Q. Chen, X. Q. Pan, A. Schmehl, and M. A. Zurbuchen, A thin film approach to engineering functionality into oxides, J. Am. Ceram. Soc (008). [14] J. H. Haeni, C. D. Theis, and D. G. Schlom, RHEED intensity oscillations for the stoichiometric growth of SrTiO 3 thin films by reactive molecular beam epitaxy, J. Electroceram (000). [15] M. D. Biegalski, E. Vlahos, G. Sheng, M. Bernhagen, P. Reiche, R. Uecker, S. K. Streiffer, L. Q. Chen, V. Gopalan, D. G. Schlom, and S. Trolier- McKinstry, Influence of anistotropic strain on the dielectric and ferroelectric properties of SrTiO 3 thin films on DyScO 3 substrates, Phys. Rev. B, (009). [16] Y. R. Shen, The Principles of Nonlinear Optics, Wiley, New York, 003. [17] N. A. Pertsev, A. K. Tagantsev, and N. Setter, Phase transitions and strain-induced ferroelectricity in SrTiO 3 epitaxial thin films, Phys. Rev. B 61 R85 (000). [18] J. H. Haeni, P. Irvin, W. Chang, R. Uecker, P. Reiche, Y. L. Li, S. Choudhury, W. Tian, M. E. Hawley, B. Craigo, A. K. Tagantsev, X. Q. Pan, S. K. Streiffer, L. Q. Chen, S. W. Kirchoefer, J. Levy and D. G. Schlom, Room-temperature ferroelectricity in strained SrTiO 3, Nature (004).

70 43 Chapter 3 Ferroelectricity in CaTiO 3 Thin Films and Single Crystal Polar Surfaces Probed by Nonlinear Optics and Raman Spectroscopy 3.1 Introduction The discovery of new materials is at the heart of advances in fundamental condensed matter physics and technological applications. One route to the discovery of novel phases with distinct structure and properties is epitaxial strain engineering, in which the high strain achievable in materials grown as thin films on lattice-mismatched substrates or in atomic-scale superlattices drives the system through a phase boundary; useful functional behavior can be exhibited at the phase boundary as well as by the new phase itself [1]. One specific realization of great recent interest is the phenomenon of epitaxial-strain-induced ferroelectricity in materials with non-polar bulk phases, as reported theoretically and experimentally in SrTiO 3 [] in which the epitaxial strain acts directly to favor the polar distortion through the large polarization-strain coupling characteristic of many perovskite oxides. CaTiO 3 is one such candidate material. Similar to other titanates, such as SrTiO 3

71 44 and BaTiO 3, it is commonly used in a variety of applications including capacitors, microwave components, varistors, electro-chemical oxygen separators, as well as it is an important component of materials used for nuclear waste containment [3]. At room temperature CaTiO 3 has orthorhombic Pbnm symmetry, and at higher temperatures it first becomes tetragonal (space group I4/mcm), and then cubic (space group Pm 3m) [4]. Based on recent neutron diffraction data, the transition temperatures have been determined to be 151 ± 13 K, and 1636 ± 1 K respectively [5]. However, earlier neutron diffraction studies suggest the existence of an intermediate high temperature orthorhombic phase Cmcm [6] with a possible sequence of phase transitions: Pbnm 1380K? Cmcm 1500K I4/mcm 1580K Pm 3m (3.1) Up to this date there is no consensus whether the orthorhombic Pmcm phase exists or not. The room temperature lattice parameters of CaTiO 3 are a=5.378 Å, b=5.436 Å, and c=7.638 Å, and the angle between the a and b axes is [9]. As with many perovskites, the orthorhombic unit cell can be described in terms of the pseudocubic unit cell with a pc = 3.83 Å and c pc = Å, where the subscript pc refers to pseudocubic. In addition, the orthorhombic structure exhibits significant TiO 6 octahedral tiltings which are out-of-phase along the a and the b axes, and in-phase along the c axis; these are described with the standard Glazer notation as a a c + [4], [5]. Similarly, the octahedra tiltings for the suggested intermediate orthorhombic phase Cmc are a 0 b c + [7], and that of the tetragonal phase I4/mcm are a 0 a 0 c [4]. Since the symmetry of bulk orthorhombic CaTiO 3 is Pbnm, it is neither polar nor ferroelectric. It is however ferroelastic meaning that it has two or more orientational states differing in spontaneous strain, that can be transformed reversibly from one state

72 45 to another by the application of an external mechanical stress [10]. Finally, it is worth pointing out that CaTiO 3 exhibits strong optical birefringence; for example at 590 nm the refractive indices along the a, b, and c axes are.30,.34 and.38 respectively [11] Twinning in the Orthorhombic Structure Since CaTiO 3 has close to ideal perovskite structure some of its crystallographic directions (for example 110 and 001 or 100, 010, and 11 ) are quasi-equivalent and hardly distinguished by X-ray diffraction. This leads to a tendency for twinning during crystal growth. According to [9], three types of twins have been observed in orthorhombic CaTiO 3 : twins mirrored about a (110) plane, where the orientation of the c-axis is the same for both but the a, and b axes are mirrored about the (110) boundary. This type of twin is the most common in CaTiO 3. Another type of twinning occurs by rotation of 90 about the normal to (110), and the third twin type involves mirroring about the (11) boundary planes. Figure 3.1 shows illustrations of the the room temperature CaTiO 3 structure as well as schematics of the orthorhombic twins [8], [9] Incipient Ferroelectric Very much like SrTiO 3, CaTiO 3 is also an incipient ferroelectric [1]. Its dielectric constant increases from 168 at room temperature to 331 at 4. K where it reaches saturation. Figure 3. shows the temperature dependence of the relative dielectric constant ε(t)/ε(rt) for SrTiO 3, KTaO 3, CaTiO 3, and TiO. Even though the low temperature values of ε are much smaller than those of the other well known incipient ferroelectrics, the general behavior of ε(t) for CaTiO 3 follows a similar trend. In addition, frequency dependent measurements carried out from 10 Hz to 1 MHz [1] show that there is no dispersion of the dielectric constant. The lack of frequency dependence of the dielectric

73 46 Figure 3.1. Top illustration: structure of the orthorhombic CaTiO 3 with exaggerated oxygen octahedra tilts. The titanium and oxygen ions are located at the center and corners of the TiO 6 octahedra, whereas the calcium ions are located outside the oxygen octahedra [8]. (a) Depiction of the orthorhombic and pseudocubic CaTiO 3 unit cell. (b) Most common form of twinning involving rotation about normal to the (110) with composition plane (110), (c) twinning by rotation of 90 about normal to (110) with composition plane (110), and (d) twinning by rotation of 180 about normal to (11) with composition plane (11) [9]. constant of CaTiO 3 for a broader frequency range is strengthened by the fact that the reported value of ε at 10 GHz is 169, which is virtually the same value as that at the low frequency regime [13].

74 Ferrielectric Twin Walls in CaTiO 3 Recent theoretical studies investigate possible polar ordering in the ferroelastic walls of CaTiO 3 [14]. According to this study considerable polarization can be found in the orthorhombic (110) twin walls of orthorhombic CaTiO 3. In particular, simulations show nonzero polarization distributions at and close to the walls, including a strong antiferroelectric contribution, as well as possible trapping of oxygen vacancies that could lead to pinning of these walls [15]. Figure 3.. Temperature dependence of the relative dielectric constant ε(t)/ε(rt) in SrTiO 3, KTaO 3, CaTiO 3, and TiO where ε(rt) is the dielectric constant at room temperature [1].

75 Defect Chemistry of CaTiO 3 It is well known, that defects significantly affect material properties. For example, the dielectric and piezoelectric properties of polycrystalline samples depend both on intrinsic and extrinsic mechanisms: the intrinsic contributions originate from the relative ion motion of the ferroelectric structure, whereas the extrinsic contributions are due to domain walls and defect dipoles [16]. In particular, studies on soft PZT have shown that less than 50 % of the total magnitude of the dielectric constant at room temperature comes from intrisic contributions; the rest is due to extrinsic mechanisms [17]. According to [18], the behavior of the electrical conductivity of undopped CaTiO 3 at 13 K, as function of the oxygen partial pressure, points to three distinct defectchemistry models: for (i) extremely reducing conditions, (ii) reducing conditions, and (iii) oxidizing conditions. The classification of each regime is based on the slope of the ratio of logσ over log p(o ) where σ is the electrical conductivity and p(o ) is the oxygen partial pressure. This value is described by the symbol m σ, and for extremely reducing conditions it is equal to -6, for reducing conditions it is -4, and for oxidizing conditions m σ = 4. Here defect models from regimes (i) and (ii) are discussed in more detail. Extremely reducing conditions lead to the reduction of CaTiO 3. Based on the Kroeger Vink [19] notation the reduction can be written as: O x O 1 O +VÖ+ e (3.) where O O is the oxygen in its lattice site, VÖ is a doubly ionized oxygen vacancy and e is an electron. At higher oxygen partial pressures, corresponding to the reducing conditions regime, additional defects (besides the doubly ionized oxygen vacancies) are formed: singly ionized oxygen vacancies, tri-valent Ti ions on Ti sites, and excess Ca

76 49 vacancies leading to the formation of CaO layers within the CaTiO 3 blocks; these are the so called Ruddlesden-Popper type phases. The formation of the singly ionized oxygen vacancies can be described by the following equilibrium: O x O 1 O +VȮ+ e (3.3) whereas the tri-valent Ti ions are formed according to the following reaction: Ti Ti + O O Ti Ti+VÖ+ 1 O (3.4) Finally, the formation of Ruddlesden-Popper type phases involve excess Ca ions accomodated in-between the CaTiO 3 blocks, can lead to the creation of Ca and oxygen vacancies according to: Ca Ca + O O V Ca +V Ö +(CaO) layer (3.5) Another possibility is anti-site defects consisting of Ca + ions in the Ti sites. Their formation can be described with the following reaction: CaO Ca Ca +Ca Ti+ O O +VÖ (3.6) By applying charge neutrality conditions on all the defect models, the electron concentration n can be related to the oxygen partial pressure p(o ) as n p(o ) 1/4. Hence, the dependence of the electrical conductivity σ (which depends on n) on p(o ) cannot distinguish the dominating defects, because all of them, as function of the oxygen partial pressure, affect σ in a similar manner. Ref. [18] suggests positron annihilation as a promising tool in the studies of defect chemistry of CaTiO 3.

77 Theoretical Predictions of Strain-coupling to Ferroelectricity A recent first-principles study [0] suggests that CaTiO 3 can be driven into a ferroelectric phase by experimentally accessible tensile epitaxial strains with in-plane polarization along 110 with respect to the primitive perovskite lattice vectors, and point group mm. Figure 3.3(a) shows the epitaxial strain dependence of the total energy per formula unit. Two general epitaxial CaTiO 3 structures, shown in Figure 3.3(b), labeled as ab epbnm and c epbnm were considered: for the ab epbnm structure the c-axis lies in the plane of the substrate, and for the c epbnm the c-axis is normal to the substrate plane. According to theoretical calculations, the ab epbnm structure is favored for epitaxial strains up to 1.5% but at higher strains the c epbnm phase becomes more favorable. Unlike SrTiO 3, for which compressive and tensile strains are equally effective [1], the persistent rotation patterns in CaTiO 3 appear to play an important role, leading to suppression of the polarization under compressive strain. This intriguing observation raises important questions that can only be addressed by experimental investigation. As shown in Figure 3.3(a) additional lower energy symmetries are revealed when symmetry breaking distortions are taken into account. These are: Pmc 1 with polarization along a, Pmn 1 with polarization along b, P 1 with polarization along c, and Pm with polarization along the ab diagonal. Only the first two structures were found to be ferroelectric with predicted polarization of 18 µc/cm at +1.5% strain [0] Thin Film Growth Strained CaTiO 3 thin films were grown using reactive molecular-beam epitaxy (MBE) by Mr. Charles Brooks. In particular, 0 nm thick CaTiO 3 films were grown on (001) LaSrAlO 4, (001) SrTiO 3, (110) NdGaO 3 and (110) DyScO 3 substrates with the shuttered growth technique at 90 K and partial oxygen pressure of Torr with 10%

78 51 (a) (b) ab-epbnm c-epbnm Figure 3.3. First principles calculations of epitaxially strained CaTiO 3.(a) Total energy per five atom formula unit for various epitaxially constrained structures as function of misfit strain. At each strain, the energy of the c epbnm structure is taken as the zero of the energy. The connecting lines are guides for the eyes. (b)two possible epitaxial orientations and the primitive perovskite substrate plane. On the left the ab epbnm phase and on the right the c epbnm phase [0]. O 3 in order to reduce oxygen vacancies. Film growth was monitored in-situ by reflection high-energy electron diffraction (RHEED).The room temperature in-plane lattice parameter of tetragonal LaSrAlO 3 is a= 3.756Å and that of SrTiO 3 is a=3.905 Å. These correspond to in-plane strains of approximately -1.7% and +.%. On the

79 5 other hand, when the CaTiO 3 films are grown on orthorhombic substrates they are subjected to anisotropic biaxial strain. Based on the room temperature lattice parameters of NdGaO 3 [] and DyScO 3 [], the average in-plane biaxial strains are calculated to be 1.1% and 3.4% respectively. For the 0 nm CaTiO 3 /NdGaO 3 samples, the outof-plane lattice constant was determined to be Å by means of x-ray diffraction (XRD).These measurements were performed using a high-resolution Phillips XPert Pro MRD diffractometer with a PreFix Hybrid Monochrometer on the incident side and Triple Axis/Rocking Curve attachment on the diffracted side. Rocking curve full-width at half max values for the films are on the order of SHG Measurements This section focuses on the SHG studies of 0 nm thick CaTiO 3 thin films grown on (110) NdGaO 3 and (001) LaSrAlO 4. Optical Second Harmonic Generation (SHG) is used to study phase transitions and domain structures of polar materials. It is a nonlinear process which involves the conversion of two photons of frequency ω into a single photon of frequency ω via the induced nonlinear polarization P ω i d i jk E ω j Eω k,where E ω j and E ω k are the fundamental electric fields of frequency ω, and the nonlinear d i jk tensor is a material property [10]. In particular, d i jk is nonzero for materials which lack inversion symmetry; a condition necessary but not sufficient for ferroelectricity. Experiments were carried out with a Ti:sapphire laser amplifier which produces 140 fs pulses centered at 800 nm with a repetition rate of 1 khz. The polarization of the fundamental light was continuously rotated by an angle θ from the x-axis, as shown in Figure 3.4. The average power density of the fundamental light was maintained at approximately 16 W/cm for all measurements and at these intensity levels no signal was observed from the substrates. While normal incidence geometry (φ = 0 ) can probe only in-plane

80 53 polarization components, non-normal (tilted) incidence geometry (φ 0 ) probes both in-plane E θ ω ϕ x ω E x y y Figure 3.4. Schematic of experimental SHG geometry. The polarization of the fundamental light is rotated in the xyplane by an angle θ, and SHG polar plots are collected along the x, y, x, and y directions. φ denotes the sample tilt angle, and light enters the sample film side first. and out-of-plane polarization components. The normal incidence SHG temperature scan (open circles) for CaTiO 3 /NdGaO 3 is shown in Figure 3.6(a). The SHG intensity becomes nonzero at approximately 150 K, indicating a paraelectric to polar phase transition consistent with the theoretical prediction for a polar ground state under biaxial tensile strain between +1% and +%. No transition to a polar state is observed either for CaTiO 3 /LaSrAlO 4 (squares) or bulk CaTiO 3 crystal (rhombs); these results are consistent with the prediction of non-polar ground states for biaxial strain smaller than +1% Determination of Polar Symmetry Group theoretical analysis [4] can predict which ferroelectric phases can coexist with a a c + octahedral rotations of the CaTiO 3 structure, as shown in Figure 3.5 (a) through (i). The orthorhombic unit cell of CaTiO 3 is shown in solid lines, outlined by its pseudo cubic mesh, whereas the allowed polarization directions are indicated in red. Here the

81 54 orthorhombic lattice parameters are based on the Pbnm symmetry. In particular, Figure 3.5 (a) corresponds to space group Pna 1 with polarization along the c-axis. Figure 3.5 (b) shows space group Pmn 1 with polarization along the a-axis, whereas Figure 3.5 (c) is an alternative orientation of space group Pmn 1 with polarization along the a-axis as well. Similarly, Figures 3.5 (d) and (e) show alternative orientations of space group Pmc 1 with polarization along the b-axis. Finally, Figure 3.5 (f) is an illustration of space group P 1 with polarization along the c-axis. For the remaining three symmetries shown in Figures 3.5 (g), (h), and (i) the polarization direction is not restricted to a single direction but it lies in the ab plane for space group Pm, or in the ac or bc planes for space group Pc. Figure 3.5 (i) corresponds to triclinic group P1 where there are no symmetry restrictions on the polarization direction. Finally, in terms of epitaxial growth, the substrate growth surface corresponds to the horizontal plane of the pseudo cubic cells. Next, we determine the group symmetry and the orientation of spontaneous polarization in the CaTiO 3 /NdGaO 3 film system. SHG polar plots (SHG intensity, I ω (θ) for a fixed φ =0 ) with the SHG signal polarized along one of the four orthorhombic directions of the substrate, namely x[001] o, y [ 110 ] o, x [ 111 ] o and y [ 1 11 ] were collected o at various temperatures. Here we use the Pbnm setting to describe the crystallography of the NdGaO 3 substrate. For space groups Pmn 1 and Pmc 1 the SHG intensities along the x and y directions were found to be: lx ω (θ,φ)=k 1,x (sinφ+k,x cosφ) sin θ l ω y (θ,φ)=k 1,y [ K,y cos θ+ ( K 3,y sin φ+k 4,y cos φ+k 5,y sinφ ) sin θ ] (3.7) where K 1,x = I 0 tx f x fz d 15, K,x = f y / f z, K 1,y = I 0 ty d 1, K,y = fx ( ) f z + f y, K3,y = ( ) f z d / d 1 + f y d 3 / d 1, K4,y = fy ( ) f z d / d 1 + f y d 3 / d 1, and f z

82 55 Pna 1 Pmn 1 Pmn 1 (a) (b) (c) c a b c a b c a b Pmc 1 Pmc 1 P 1 (d) (e) (f) c a b c a b c a b Pm Pc P1 (g) (h) (i) c a b c a b c a b Figure 3.5. Possible epitaxial orientations of thin film CaTiO 3 based on [4]. The orthorhombic unit cell of CaTiO 3 is shown in solid lines, outlined by its pseudo cubic mesh, whereas the allowed polarization directions are indicated in red. The following symmetries are illustrated:(a)space group Pna 1,(b)space group Pmn 1,(c) alternative orientation of space group Pmn 1,(d) space group Pmc 1,(e) alternative orientation of Pmc 1, (f) space group P 1, (g) space group Pm, (h) space group Pc, and (i) triclinic P1. K 5,y = f y f z ( f z + f y ) d 4 / d 1. Here I o is the fundamental intensity, t x and t y are the transmission Fresnel coefficients for x and y polarized SHG light at the substrate-air interface respectively, whereas f x,y,z and f x,y,z are the linear and nonlinear transmission Fresnel coefficients which depend on the refractive indices of the film, the substrate, and the angle of incidence [5]. The d i j coefficients are transformed tensor elements that can be

83 56 related to the standard mm tensor elements as follows: d 15 = d 15 /, d 1 = d 31 /, d =(d 4 + d 3 + d 33 )/, d 3 =( d 4 + d 3 + d 33 )/, and d 4 =( d 3 + d 33 )/. More details on this derivation are given in Appendix D. The inset on the left of Figure 3.6(b) shows the theoretical fits for SHG intensity polarized along the x and y directions versus θ, for φ = 0, at 100 K based on Equation (3.7) for CaTiO 3 /NdGaO 3 (sample CMB611). Similarly, the polar plot set on right was collected for φ = 40 at 5 K, and also fitted with the same equations. The insets of Figure 3.6(b) show the corresponding polar plots for T = 5 K. In addition, the main panels of Figures 3.6 (b) and (c) shows the SHG intensity Iy ω (θ=90 ) as function of the tilt angle φ at 100 K and 5 K respectively. The experimental data are shown as open circles and the fits as solid (blue) lines. For both temperatures, the fits are in very good agreement with the experimental SHG intensity tilt dependence. Figures 3.6(d) and (e) show similar measurements carried out for another 0 nm CaTiO 3 sample grown on NdGaO 3 (sample CMB543). For both samples, the SHG data can be fitted with space groups Pmn 1 and Pmc 1. Similar, analysis was carried out for space group Pm. The SHG intensities along the x and y directions as function of the polarization rotation angle θ and the tilt angle φ were found to be: lx ω (θ,φ)=k 1,x (sinφ+k,x cosφ) sin θ l ω y (θ,φ)=k 1,y [ K,y cos θ+ ( K 3,y sin φ+k 4,y cos φ+k 5,y sinφ ) sin θ ] (3.8) where K 1,x = I0 t x f x fx fy d 15, and K,x = f z d 16 / ( ) f y d 15, K1,y = I0 t y d 31, K,y = fx ( ) f z d 1 / d 31 + f y, K3,y = fz ( ) f z d / d 31 + f y d 3 / d 31, K 4,y = fy ( ) ( ) f z d 3 / d 31 + f y d 3 3/ d 31, and K5,y = f y f z f z d 4 / d 31 + f y d 34 / d 31. and d 15 = ( d 15 + d 16 )/, d 16 =(d 15 + d 16 )/, d 1 =(d 1 + d 31 )/,

84 ( d =(d + d 3 + d 4 + d 3 + d 33 + d 34 )/ ), ( d 3 =(d + d 3 + d 4 d 3 d 33 d 34 )/ ), ( d 3 =(d + d 3 d 4 + d 3 + d 33 d 34 )/ ), ( d 33 =(d + d 3 d 4 d 3 d 33 + d 34 )/ ), ( d 4 = (d d 3 + d 3 d 33 )/ ) (, and d 34 = (d d3 d 3 +d 33 )/ ). Derivation of these expressions can be found in Appendix D. The tilt fits are also shown as solid (red) lines in Figures 3.6 (b) and (c), and they are also in excellent agreement with the experimental data. SHG intensity expressions as function of the polarization and the tilt angle were also derived for the each of the other space groups shown in Figure 3.5. Space groups Pmn 1 and Pmc 1 with the c axis out-of-the plane, as well as Pc were rejected because they do not fit the normal incidence polar plots. The tilted SHG data were then used to narrow down the remaining symmetries. For symmetries Pna 1 and P 1 the polarization is along the c axis and lies in the substrate plane. For both symmetries, the SHG intensity expressions have the same general form. For Pna 1 the following expression is obtained for I ω y when θ=90 : 57 l ω y (90,φ)=K 1,y ( sin φ+k,y cos φ+k 3,y sinφ ) (3.9) where K 1,y = I 0 t y f y f 4 z d 31, K,y = ( f y d 33 ) / ( f z d 31 ), and K3,y = ( f z f y d 15 ) / ( f y f z d 31 ). Tilt fits based on this symmetry are shown as solid (green) lines in Figures 3.6 (b)-(e). This symmetry fits the CMB611 data quite well but not the CMB543 data. Therefore, the highest symmetries which fit all the experimental data are Pmn 1 or Pmc 1. Extraction of unique nonlinear d i j ratios is difficult when dealing with complicated fitting models such as the Frensel model. Nonetheless, the d 33 / d 31 ratio, which is a material property, is shown in the inset of Figure 3.6(a) as function of temperature. Ba-

85 58 SHG Intensity (arb.u.) (a) CaTiO 3 /NdGaO 3 ϕ = 0 d 33 /d ω E θ ϕ ϕ = 0 x E y ω 5 CaTiO 3 /LSAO CaTiO 3 ϕ = 45 0 ϕ = Temperature (K) SHG Intensity (arb. u.) 8 (b) T = 100 K Pmn 1, Pmc 1 6 Pna 1, P 1 Pm ϕ = 0 (c) ϕ = 40 I ω x I ω y T = 5 K 4 ϕ = 0 ϕ = Tilt angle φ, degrees SHG Intensity (arb. u.) (d) T = 100 K ϕ = 0 (e) T = 5 K ϕ = 40 Pmn 1, Pmc 1 Pna 1, P 1 Pm ω x ω y ϕ = Tilt angle φ, degrees I I Figure 3.6. SHG measurements of strained CaTiO 3 films. (a) Temperature dependence of the SHG signal for 0 nm CaTiO 3 /NdGaO 3 (circles), 0 nm CaTiO 3 /LSAO (squares), and bare CaTiO 3 substrate (rhombs). Only CaTiO 3 on NdGaO 3 shows a low temperature transition at 150 K. The top inset shows the temperature dependence of d 33 / d 31. Also shown is a schematic of the SHG set-up. The tilt dependence of Iy ω (θ=90,φ) is shown in (b) for T = 100 K, and (c) for T = 5 K for sample CMB611. Experimental data are shown as open circles, and fits as solid lines. SHG polar plots and their fits collected at φ = 0 and φ = 40 are also shown for both temperatures. Panels (d) and (e) shows the same SHG quantities obtained from sample CMB543.

86 59 Table 3.1. d 33 / d 31, d 3 / d 31 and d 4 / d 31 fit parameters for CaTiO 3 on NdGaO 3 thin films based on symmetries Pmn 1 and Pmc 1. Sample Temperature (K) d 33 / d 31 d 3 / d 31 d 4 / d 31 CMB ± ± ±0.18 CMB ± ± ±0.094 CMB ± ± ±0.305 CMB ± ± ±0.309 CMB ± ± ± Extracted from normal incidence polar plots. sed on the highest fitting symmetry (mm) the extracted d 3 / d 31 and d 4 / d 31 ratios are given in Table 3.1. Coefficients from a second 0 nm CaTiO 3 /NdGaO 3 sample (CMB543) are also included for comparison. 3.. Light Propagation in Anisotropic Media Of particular interest is the interaction of the second harmonic light with optically anisotropic materials. Optical propagation through anisotropic media is an entire separate field of study, so here we focus on two issues: (i) the effects of the thin film anisotropy, and (ii) the effects of substrate anisotropy. Case (i) has been already covered extensively in past; the multidomain SHG models discussed earlier in this thesis take into account thin film birefringence and introduce an appropriate mixing term between orthogonal nonlinear polarization contributions. If the films are very thin, and the refractive index difference between the two orthogonal directions is small then the mixing term can be effectively ignored. On the other hand, Case (ii) deals with the effects of substrate anisotropy on the propagating SHG signal (here we assume that the fundamental light is focused on the film side first). For isotropic substrates, such as cubic SrTiO 3, the randomly polarized SHG signal encounters the same refractive index in every direction; hence, the optical indicatrix which relates the refractive index to the crystal axes is

87 60 a sphere, where all the propagation directions are equivalent. Orthorhombic substrates are optically biaxial with three distinct refractive indices along the a, b and c axes. Hence, their optical indicatrix can be described as a three dimensional ellipse where the principal axes are equal to n a, n b, and n c corresponding to the refractive indices parallel to the a, b, and c axes. For a typical (110) substrate orientation, the orthogonal edges of the substrate correspond to x[001] o, and y [ 110 ], with dis- o tinct refractive indices n c and n = { / [ (1/n a ) +(1/n b ) ]} 1/. The phase difference between two orthogonal polarizations polarized along these directions after propagation of thickness L is given by Γ where Γ ω = nωl/c where n= n c n at frequency ω, and c is the speed of light in vacuum. The induced nonlinear polarizations P ω i generated by the film can be treated as source terms, and here for the sake of simplicity we assume single domain symmetry and ignore birefringence effects due to the small film thickness. For the space group symmetries described in Figure 3.5(a), (b), (d),(f), and (g), at normal incidence, P ω i can be derived as P ω x = Asinθ and P ω y beginning of the substrate (L=0), we have Px ω (L=0) = Px ω Here the x and y directions correspond to the [001] o and y [ 110 ] o = B ( sin θ+c cos θ ). At the and Py ω (L=0) = Py ω. directions of the substrate. After propagation through substrate thickness L, the same quantities can be written as P ω x (L) = P ω x e ( inω c SHG intensities are given by I ω x L/c) and Py ω (L) = Px ω e ( i(nω ) L/c). The detected = Px ω ( ) P ω x = A sin θ and Iy ω = Py ω ( ) P ω y = B ( sin θ+c cos θ ). For SHG polar plots collected at ±45 from the x-axis we obtain I±45 ω = ( Px ω ) + ( ) P ω ± y cosγω Px ω Py ω. Representative SHG polar plots from CaTiO 3 / NdGaO 3 (CMB611) are shown in Figure 3.7 (a) through (d). The polar plots were collected at T = 5 K, and the solid lines are fits based on the equations described above with Γ ω = 1.40 rad or 80.3.

88 61 (a) (b) (c) (d) Figure 3.7. Normal incidence SHG polar plots at T = 5K for CaTiO 3 /NdGaO 3 with (a) horizontal analyzer (parallel to x), (b) vertical analyzer (parallel to y), (c) analyzer at 45, and (d) analyzer at -45. Data: open circles, fits: solid lines Experimental Determination of Γ ω In order to compare the value of Γ ω obtained from the fits with an experimentally measured value the following experiment was carried out. Its schematic is shown on the inset of Figure nm fundamental light was focused on a piece of z-cut LiNbO 3 which generated 400 nm signal. Next, a polarizer was used to select linearly polarized light at 45, and a computer controlled analyzer was placed after the sample in order to record the transmitted intensity as function of the analyzer rotation angle. The sample was placed in a cryostat and cooled to 5 K and special care was taken in order to make sure that its edges were at 45 from the 45 linearly polarized 400 nm light. In order to take into account the finite extinction of the rotating analyzer, a reference scan without the sample was collected first, and it is shown in Figure 3.8(a).The minimum intensity was determined to be 0.15 micro volts, and this value was subtracted as background from all other scans. The dependence of the transmitted 400 nm intensity on the analyzer

89 6 rotation angle, with the sample in the beampath, is shown in Figure 3.8(b). The theoretical expression for the transmitted intensity after the analyzer can be easily found by multiplying the Stokes vector of the linearly polarized light at 45 by the Mueller matrices of the sample (retarder) and a rotating analyzer. This was found to be: I(θ a )=I 0 (1+sinθ a cosγ ω ) (3.10) where θ a is the analyzer rotation angle. The fit is shown as a solid line in Figure, with Γ ω = 1.31 rad or 75. which is in very good agreement with Γ ω = 1.40 rad which was used to fit the SHG polar plots. input x y θ a sample detector (a) no sample fit (b) with sample fit Figure 3.8. Measurement of sample induced phase delay Γ ω. Transmitted 400 nm intensity as function of analyzer rotation angle θ a for (a) without sample, and (b) with the sample in the beampath. The fits (solid) lines are based on Equation (3.10).

90 Dielectric Measurements Dielectric measurements on CaTiO 3 thin films were carried out by Dr. Mike Biegalski at Oak Ridge National Lab. The in-plane dielectric properties of the CaTiO 3 thin films were measured using interdigitated electrodes. These electrodes were aligned along the 110 o substrate direction to measure the ferroelectric transition in the film. The dielectric data plotted in Figure 3.9 shows a peak in the capacitance at 175 K indicating the ferroelectric transition temperature. This is corroborated by the existence of a P-E hysteresis loop by 150 K as seen in the inset of Figure 3.9. At lower temperatures the P-E loop turns into a double antiferroelectric-like hysteresis loop. Similar behavior has been observed in other MBE grown CaTiO 3 samples on the same substrate, and this behavior is currently not well understood, but they are likely associate with extrinsic effects. A sampling of the dielectric measurements is included in Appendix E. Capacitance (pf) Polarization ( µc/cm ) K 10 K Field (V/cm) Capacitance tan(δ) Temperature (K) Tan(δ) Figure 3.9. Dielectric data for 0 nm CaTiO 3 grown on NdGaO 3 measured at 0 K with interdigitated electrodes. Inset shows P-E hysteresis loop measured at 150K, and 10 K at 10 KHz.

91 SHG Imaging of CaTiO 3 Surfaces As mentioned earlier in this chapter, bulk CaTiO 3 is centrosymmetric with orthorhombic point group Pbnm and it is not polar. However, here we report preliminary experimental results that reveal large, regular, polar (SHG active) crystallographic domains on CaTiO 3 single crystals. Optical SHG mapping of the CaTiO 3 surface was performed using an 80-fs, 800 nm fundamental beam generated from a Ti-sapphire laser coupled to a commercially available confocal scanning microscope (Witec Alpha 300 S). A detailed configuration of the experimental set-up is shown in Figure The fundamental light was coupled into the microscope using a two mirror periscope, and directed towards the sample with a dichroic mirror. The dichroic mirror has a dual purpose of reflecting the infrared light towards the sample, and transmitting any generated 400 nm SHG signal towards the detector. The fundamental light was focused onto the sample with either an 100X (NA = 0.90) or 40X (NA = 0.6) microscope objective depending on the required resolution (the exact type of objective used will be explicitly mentioned when discussing the experimental data). Depending on the sample transparency the microscope can be operated in transmission or reflection mode. For all of the experiments described here, we monitored reflected 400 nm light which was coupled back into the objective and detected with a photomultiplier tube equipped with blue bandpass filters. Finally, polarization dependent SHG studies were undertaken with a computer controlled motorized half-waveplate, which continuously rotates the polarization of the fundamental light, and an analyzer placed after the dichroic mirror. This configuration is equivalent to the tabletop SHG set-ups used in other experiments described in this thesis. Scanning SHG microscopy allows for hight quality image scans with resolution of approximately 300 nm when an 100 X objective is used.

92 65 Figure Schematic of the scanning SHG microscope. The input fundamental light is directed towards the microscope with a two mirror periscope. The dichroic mirror efficiently reflects the 800 nm light towards the microscope objective which focuses the light onto the sample. Any generated SHG signal is detected in the reflection geometry using a photomultiplier tube (PMT). Polarization dependent studies can be carried out by changing the input polarization of the fundamental beam with a half waveplate, and selecting SHG polarization components with an analyzer before the (PMT) SHG Imaging Commercially grown single crystal CaTiO 3 substrates were tested with the SHG microscope. X-ray measurements show that the substrates are orientated with the orthorhombic c-axis out of plane; however the substrates have highly twinned areas consisting of (110) and (11) twins. Polarization dependent SHG area scans were collected at these

93 66 areas and they are shown in Figures 3.11 and 3.1. Light colors correspond to high SHG intensity and dark colors to low SHG signal. The arrows around the images show the polarization state of the fundamental light as well as the analyzer direction for the SHG signal. Hence, for all four images shown in Figure 3.11, the electric field polarization directions (fundamental and induced) are parallel or perpendicular to the horizontal direction of the page. The images with the highest contrast are (a) and (d), and they highlight opposite domains: for (a) the left (vertical) domains are bright, and the right (horizontal) domains are dark, whereas for (b) the left (vertical) domains is dark, and the right (horizontal) are bright. Given, that the geometry of the experiment allows the detection of in-plane polarization only, these results are consistent with the existence of horizontal polarization (or projection of) in the left domains, and vertical polarization (or projection of) in the right domains. In addition, the zigzag boundaries between the two types of domains form nearly 90 angles and their overall appearance is consistence with that of (11) twins encountered in perovskites [6]. Similarly, Figure 3.1 shows SHG scans collected with the input polarization and analyzer parallel or perpendicular at 45 with respect to the horizontal of the paper. The contrast of the domains is not as sharp as that of the previous scans; this was to be expected since the 45 signal is a mix of both horizontal and vertical contributions that washes out most of the domain contrast. This result is consistent with the previous estimate of the horizontal and vertical polarization directions. On the other hand, these area scans highlight the boundaries between the domains (shown as dark lines) with minimum SHG intensity. In summary, visual inspection of the SHG area scans reveal two major types of domains with orthogonal polarization directions parallel and perpendicular to the horizontal direction. Literature search suggests that these domains are (11) type twins with (110) subdomains [9], [6]. Since the boundary plane between consecutive (110) twins is parallel to the orthorhombic c axis, we can conclude that for

94 nm 400 nm 400 nm 800 nm (a) (b) (c) (d) 800 nm 400 nm 800 nm 400 nm Figure SHG area scans of CaTiO 3 crystal surface showing large polar domains obtained under various polarization conditions. (a) Input 800 nm polarization: parallel, output 400nm polarization: parallel, (b) input polarization: parallel, output polarization: perpendicular, (c) input polarization: perpendicular, output polarization: parallel, and (d) input polarization: perpendicular, output polarization: perpendicular.

95 nm 400 nm 400 nm 800 nm (a) (b) (c) (d) 800 nm 400 nm 400 nm 800 nm Figure 3.1. SHG area scans of CaTiO 3 crystal surface under various polarization conditions. The scanned area is the same as the one shown in Figure (a) Input 800 nm polarization: -45, output 400 nm polarization: -45, (b) input polarization: -45, output polarization: +45, (c)input polarization: +45, output polarization: -45, and (d)input polarization: +45, output polarization: +45. the vertical domains on the left the c axis is in the plane and orientated along the vertical

96 69 direction and for the horizontal domains on the right the c axis is also in the plane and orientated along the horizontal direction. Also, the a and b axes must be orientated at 45 from the sample surface. Nonetheless, micro X-ray mapping would be an ideal tool for direct probing of the local structure. In addition to SHG mapping, detailed polarization dependent studies were undertaken in the same area. SHG polar plots were collected within each of the domains. Representative SHG polar plots, along with the corresponding SHG area scan, are shown in Figure The experimental data are shown in open circles, and the polar plots in black were collected with the analyzer parallel to the horizontal of the paper or x direction, while the polar plots in red were collected with the analyzer parallel to the vertical of the paper or y direction. In order to analyze the SHG data, polar symmetries based on the CaTiO 3 oxygen octahedra pattern were considered. These were the same as the ones considered when trying to fit the SHG data from the CaTiO 3 thin films. Initial attempts to fit the experimental data, considering single domain geometries, were not successful for any point group symmetry including triclinic group 1 (no symmetry). Since the SHG microscope is well calibrated with respect to a reference sample, we know that the shape of polar plots is not an experimental artifact; therefore additional effects must be taken into account. As mentioned before, one of these effects is sample birefringence which leads to mixing of polarization components and distorting SHG polar plots. Whereas previously the birefringence of the thin films was ignored, here we are dealing with a bulk crystal where the fundamental light propagates a considerable depth within the material before generating the SHG signal. In turn, the backwards propagating SHG signal also experiences birefringence before it reaches the detector. The maximum depth at which the SHG signal is created dependents on the absorption of the fundamental light in the material. Here for simplicity, we assume that the SHG signal is generated from a single depth which corresponds to the most intense part

97 70 I ω x (θ) I ω y (θ) I ω x (θ) I ω y (θ) θ θ I ω x (θ) I ω y (θ) θ y E ω x y E ω x 45 ω I (θ) x I ω y (θ) θ Figure Representative SHG intensity polar plots obtained with the confocal SHG microscope from the area scan shown in the center. The experimental data are shown as open points whereas the fits (shown as lines) are based on Pm symmetry. The schematic illustrations of the sample axes were provided by Dr. T. T. A. Lummen. of the focused beam, and we ignore birefringence effects on the backwards propagating SH signal. An upper depth limit can be estimated from the SHG coherence length of CaTiO 3. In general, the coherence length describes the optimum length for constructive interference of the generated second harmonic signal, and it is given by: l c = π κ = λ ω (n ω n ω ) (3.11)

98 71 For an average value of CaTiO 3 refractive index at 800 nm of.5, and.4 at 400 nm, the coherence length (and upper depth limit) can be calculated to be approximately.5 µm. After taking into account birefringence of the fundamental, two symmetries can successfully fit the data; these are either monoclinic symmetry Pc with the mirror plane perpendicular to a or b axis or triclinic symmetry 1. The SHG fitting expressions for m perpendicular to a are the following: l ω x (θ)=a x ( (sin θ+b x cosγ ω cos θ+c x cosγ ω sinθ ) + ( Bx sinγ ω cos θ+c x sinγ ω sinθ ) ) l ω y (θ)=a y ( (sin θ+b y cosγ ω cos θ+c y cosγ ω sinθ ) + ( By sinγ ω cos θ+c y sinγ ω sinθ ) ) (3.1) where for A x = d 1+d + d 16, B x = d 33 d 31 +d 3, C y = d34 d 31 +d 3, and Γ ω = nωl c d 3 d 1 +d +d 16, C x = (d4 +d 15 ) d 1 +d +d 16, A y = d 31+d 3, B y =, where n is the refractive index difference between the 001 o and 110 directions. Here we assume that the monoclinic c axis lies o completely in the plane, and the a, and b axes are at 45 from the surface. Therefore the polarization can lie anywhere either in the ac or bc plane which is at 45 to the surface. As mentioned before two major domain groups can be identified: one group (A) includes domains 1,, 4, 6, 9, 13, 14, and the other group (B) includes domains 3, 5, 7, 8, 10, 11, 1, 15, 16, and 17. Group A can be divided into two subgroups one including domains 1, 4, 9, 13 and the other including domains, 6, and 14. Similarly Group B can be divided into two subgroups containing domains 3, 7, 8, 11, 17 and 5, 10, 1, 15 and 16 respectively. The summary of the polar plot fitting, with independent fit parameters, is shown in Table 3. for Group A and Table 3.3 for Group B. The value of Γ ω changes slightly between the two polarizations of the same subgroup, as well as it is somewhat

99 7 Table 3.. I ω x (θ) and I ω y (θ) fit parameters for CaTiO 3 crystal domains of Group A, based on Pm symmetry using Equation (3.1). Domain A x B x C x Γ ω x 1.089± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.05 Domain A y B y C y Γ ω y 1.763± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.077 different between Groups A and B. This can be attributed to non-optimal sample alignment or slight distortion of the unit cells within the subdomains. Based on an average value of Γ ω of approximately 90.5, the depth of generation of the SHG signal can be estimated to be approximately 3.1 µm, which is somewhat higher than the coherence length but is still a reasonable depth estimate. In conclusion, the group symmetry of the CaTiO 3 surface, based on SHG measurements, has been determined to be either Pm or 1. For Pm symmetry the polarization can be lie in the ac or bc planes which are at 45 from the sample surface; however based on experimental observations it seems that for Group A the in-plane polarization projection direction is close to the horizontal, and for Group B it is close to the vertical direction.

100 73 Table 3.3. I ω x (θ) and I ω y (θ) fit parameters for CaTiO 3 crystal domains of Group B, based on Pm symmetry using Equation (3.1). Domain A x B x C x Γ ω x ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.061 Domain A y B y C y Γ ω y ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Micro-Raman Imaging Raman spectroscopy is an important non-destructive technique used for the study of materials. It involves the inelastic scattering of light resulting from the excitation of vibrations within materials. In Raman spectroscopy, the sample is irradiated by an intense laser light, and the scattered light is detected with a spectrometer. The scattered light consists of two components: elastically (Rayleigh) scattered light which is very strong and has the same frequency as the irradiated light, and inelastically scattered (Raman) light which is much weaker, typically in the order of 10 5 when compared

101 74 to the incident beam, and has a slightly different frequency. Based on the conservation of energy, the Raman process can be described as ω i = ω s ± Ω where ω i is the incident photon energy, and ω s is the scattered phonon energy. The term Ω describes the energy of phonons created (Stokes scattering) are destroyed (anti-stokes scattering) during the scattering process, and the Raman shift Ω is typically measured in units of inverse centimeters (cm 1 ). Raman spectroscopy can be used to probe the local symmetry of material as well as tract phase transitions in ferroelectrics, because the breaking of the paraelectric inversion symmetry leads to Raman active first-order peaks [7]. Confocal micro-raman experiments were carried out as follows: a continuous 514 nm laser was focused with an 100X objective on the CaTiO 3 sample in the area where the SHG scans were collected. The schematic of the confocal Raman system is shown in Figure Polarized Raman spectra were collected with a polarizer placed in front of the Raman laser, and an analyzer before the spectrometer which was used as a detector. A laser line filter centered at 514 nm cuts out most of incident intensity and allows for the detection of the weak Raman peaks. Similar to the SHG area scans, polarized Raman area scans were collected; however each pixel corresponds to a full Raman spectrum. According to group theory analysis, the Raman active vibrational modes of orthorhombic CaTiO 3 are 7A g +7B 1g +5B g +5B 3g [8], [9], [30]. The polarized Raman spectra for various polarization configurations are shown on Figure 3.15 (a) and they are in good agreement with previous experimental work [9]. These were collected and analyzed by Dr. T. T. A. Lummen. Identification of the Raman peaks is a time consuming process requiring the collection of various polarized Raman spectra from which the polarization dependence of each peak can be extracted. In particular, a Raman peak intensity I k is proportional to e i R k i j e j where e i and e s are the unit polarization vectors of the incident and scattered light and R k i j is the Raman scattering tensor which contains information about the local symmetry [31]. For orthorhombic symmetry the Raman scattering ten-

102 75 Figure Schematic of the confocal scanning Raman microscope. A 514 nm laser light is coupled into the microscope with an optical fiber, and is then focused on the sample via a microscope objective. The confocal microscope geometry allows for detection of backscattered light with a spectrometer. Polarization dependent studies are carried out with a polarizer which controls the polarization of the incident light, and an analyzer which selects the polarization components of the backscattered light. sors are given by [3]: a d e A g : 0 b 0 ;B 1g : d b 0 ;B g : ;B 3g : 0 0 f (3.13) 0 0 c 0 0 c e f 0 Here we focus on the A 1g and B 1g peaks which are the Raman peaks corresponding to

103 cm 1, and 6 cm 1. Figure 3.15(b) shows an intensity area map for integrated Raman signal between cm 1 which includes the A 1g peak. Similarly, Figure 3.15(c) shows mapping for integrated wavenumbers between 0-36 cm 1 which includes the B 1g peak. The dark colors correspond to low signal intensity and light colors to high intensity. Two major types of domains are identified; these are the (11) domains previously detected with SHG. Additional, Raman analysis based on orthorhombic symmetry Pbnm identifies the directions of the three lattice parameters within these domains: two of them are at 45 to the surface and one lies completely in the plane. In particular, for the highlighted domains in the A 1g scan the in-plane lattice parameter is parallel to the vertical direction, and for the highlighted domains in the B 1g scan the inplane lattice parameter is parallel to the horizontal direction. This behavior is consistent with (11) twinning. Any additional information contained in the Raman data is buried within the broad second order Raman background, or in minute changes in the sharper first order Raman peaks. By plotting the center of mass of the entire Raman spectrum finer domain are revealed. Examples of such Raman maps are shown in Figure In particular, Figure 3.16(a) is obtained when the excitation laser is polarized along -45 and the Raman analyzer is set at -45 ; Figure 3.16(b) when the excitation laser is polarized along -45 and the Raman analyzer is set at 45 ; Figure 3.16(c) when the excitation laser is polarized along 45 and the Raman analyzer is set at 45, and finally Figure 3.16(d) when the excitation laser is polarized along 45 and the Raman analyzer is set at -45. Careful analysis of the polarized Raman data can, in theory, relate the source of the domain contrast to certain unit cell distortions; however assignment and interpretation of subtle Raman features with respect to perovskite lattice distortions is a difficult task, due to the complexity of the Raman spectra in such crystal structures.

104 77 (a) 8000 Raman Intensity (arb. u) 6000 A g Z ( Y Y ) Z 4000 Z (45 45) Z 000 B 1g Z ( X X ) Z Z (-45-45) Z Wavenumber (cm -1 ) (b) (c) Figure (a) Polarized Raman spectra of CaTiO 3 crystal, where X denotes horizontal polarization direction, and Y vertical polarization. The dark lines point to the A 1g and B 1g Raman peaks at 155 cm 1 and 6 cm 1 respectively. (b) Raman intensity area map centered around the A 1g peak, and (b) Raman intensity area map centered around the B 1g peak. Complimentary domains light up for each configuration indicating the lattice structure rotates by 90 between the domains. Data collected and analyzed by Dr. T. T. A. Lummen.

105 78 (a) (b) (c) (d) Figure Raman center of mass maps for (a) input excitation polarization at -45 and Raman polarization -45, (b) input excitation polarization at -45 and Raman polarization at 45, (c) input excitation polarized along 45 and Raman polarization along 45 and(d) input excitation polarized along 45 and Raman polization along -45. The images reveal additional subdomains within the (11) twins. Data collected and analyzed by Dr. T. T. A. Lummen. Nevertheless, a robust conclusion from the Raman maps of Figure 3.16 is that there exist small crystallographic distortions within the subdomains that give rise to the Raman contrast. If all subdomains were identical no Raman contrast would have been observed.

106 Suggestions on the Origin of the Polar Signal The observation of polar domains in bulk CaTiO 3 is surprising, and additional experimental work must be carried out in order to determine its origin. One possible explanation is that inhomogeneous strains around domain walls or defects distort the crystal structure leading to lowering of the local symmetry. For example, in BaTiO 3 inhomogeneous strains around domain walls have been shown to lower the local symmetry [33]. Similarly, recent work on tetragonal BaTiO 3 correlates the existence of metastable monoclinic domains in areas of high local strains [34]. 3.4 Conclusions SHG measurements were carried out on thin film and single crystal CaTiO 3. CaTiO 3 films grown on NdGaO 3 become polar for temperatures less than 150 K, whereas no such transition was observed for CaTiO 3 films grown on LaSrAlO 4. This observation is in excellent agreement with recent theoretical predictions, and highlights the importance of the persistent oxygen octahedral rotations of the CaTiO 3 structure which are likely responsible for the suppression of ferroelectric ground states in samples under compressive strain. In addition, polarization dependent SHG studies determine the highest symmetry of CaTiO 3 /NdGaO 3 as mm with polarization along the a or b orthorhombic axes. Ferroelectricity is established by dielectric measurements and the observation of switchable P-E loops in the same samples. Considerable variation in the shape of the P-E loops, both spatially and as function of temperature was observed, and further work is needed in order to explain this behavior. On the other hand, SHG imaging studies were carried out on single crystal CaTiO 3 using a confocal microscopy. They reveal large, highly-regular, crystallographic polar

107 80 domains at room temperature, with in-plane polarization components delineated by twin walls. SHG analysis indicates that the highest symmetry of the polar surface is m or space group Pc with polarization in the m plane. Polarized Raman studies, carried out in the same regions, are in excellent agreement with the SHG observations and indicate that the local symmetry is lower than the expected orthorhombic Pbnm. The underlying mechanisms for the origin of the polar signal has not been determined yet, and the future direction of this project is discussed in more detail in Chapter 6.

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111 84 Chapter 4 Phase transitions and domain structures in SrTiO 3 /BaTiO 3 superlattices 4.1 Introduction A superlattice is a heterostructure consisting of alternating layers of two (or more) materials. Superlattices of ferroelectric thin films have been predicted and reported to have enhanced properties when compared to homogeneous films of the same composition [1]- [7]. BaTiO 3 /SrTiO 3 superlattices are of particular interest; researchers have successfully grown such heterostructures, both with reactive molecular beam epitaxy (MBE) and pulsed-laser deposition (PLD), and their Curie transition temperatures (T c ) have been measured with UV Raman spectroscopy [10]. Here we focus on the SHG studies of strained BaTiO 3 /SrTiO 3 superlattices, which were also probed by UV Raman spectroscopy in order to determine their T C as function of epitaxial strain, the polar symmetry of the layers, and the direction of ferroelectric polarization.

112 85 4. Theoretical Predictions As discussed in Chapter 1 epitaxial strain can turn non-ferroelectric materials ferroelectric or enhance properties of materials which are already ferroelectric. One such example, of a non-ferroelectric turning ferroelectric when strained, is thin film SrTiO 3 [11]. On the other hand, strained BaTiO 3 exhibits both enhanced T c and increased remanent polarization when compared to bulk BaTiO 3 [1]. Since epitaxial strain is proven to be beneficial for both SrTiO 3 and BaTiO 3, one of the questions raised is how does it affect the properties of a structure consisting of both SrTiO 3 and BaTiO 3 when one or both layers are under strain. According to first principles calculations by Neaton and Rabe,[9] for a short period BaTiO 3 /SrTiO 3 superlattice grown on (001) p (p for pseudocubic) SrTiO 3, when the BaTiO 3 composition is larger than 40 %, the spontaneous polarization of the heterostructure becomes greater than that of bulk tetragonal BaTiO 3. In addition, the unstrained SrTiO 3 layers are predicted to become tetragonal and polar due to electrostatic effects from the nearby polarized BaTiO 3 layers [9]. Experimental work on [(BaTiO 3 ) n /(SrTiO 3 ) m ] p (for n =, 5, 8, m = 4, and p = 10, 5, 40) superlattices grown on (001) p SrTiO 3 has verified these predictions, i.e. that the superlattices are ferroelectric, and that the SrTiO 3 layers are polarized [10]- [14]. Finally, based on the experimental work of,[10] it was found that the T C of the superlattices can be tuned from 151 K to 638 K by simply varying n and m. The experimentally observed variation of T c as function of n, and m is in excellent agreement with phase field simulations of multidomain BaTiO 3 /SrTiO 3 superlattices [15].According to Li, et. al., the BaTiO 3 layers consist of tetragonal c domains separated by 180 domain walls, whereas the unstrained SrTiO 3 layers are also polar with weaker polarization. If P x, P y, and P z denote the in-plane and out-of-plane polarization components of the superlattice, P z was found to have a maximum in the middle

113 86 of the BaTiO 3 layer which decreases to a minimum in the middle of the SrTiO 3 layer. On the other hand, P x and P y have small, but finite values around the 180 domain walls which become maximum at the BaTiO 3 /SrTiO 3 interfaces. This means that the polarization rotates continuously across the interfaces. Finally, according to the same study, the polarization of the unstrained SrTiO 3 layers is the result of an induced dipolar field produced by the nearby BaTiO 3 layers; also known as a ferroelectric proximity effect [15] which is in agreement with previous theoretical predictions [9]. Recent theoretical work on multilayer heterostructures consisting of BaTiO 3 and SrTiO 3 layers under compressive (-0.79 %) and tensile (+1.63 %) strains respectively, explored the spatial ferroelectric polarization distribution within the BaTiO 3 /SrTiO 3 layers as well as the structure of the domain walls and interfaces of the superlattice [16]. In agreement with previous theoretical studies, [15]- [17] the ferroelectric polarization was found to alternate between in-plane (SrTiO 3 layer) and out-of-plane (BaTiO 3 layer) as well as forming polarization vortices leading to polarization rotation and tumbling at 180 domain walls. This is an unusual prediction because the typical 180 ferroelectric wall is an Ising type wall where the polarization switches direction while remaining parallel to the wall. In fact, the predicted heterostructure polarization distribution reminds of magnetization vortices which occur at 180 Bloch or Néel type walls in magnetic materials. In a Bloch wall the magnetization vector rotates in a plane parallel to the domain wall, whereas in a Néel wall the mangetization rotates (tumbles) in a plane perpendicular to the domain wall. Figure 4.1 is an illustration of such 180 walls from [16]. Finally, as one might expect, the composition and layering of the superlattice affects the (ferroelectric) properties of the structure. Ongoing theoretical studies predict that for an increasing number of SrTiO 3 layers, while keeping the number of BaTiO 3 layers fixed, ferroelectric properties such as the coercive field is substantially reduced, while the structure retains significant tunability [18].

114 87 Figure domain walls. (a) Ising type, (b) Bloch type, (c) Néel type, and (d) Mixed Ising-Néel type walls. A mixed Bloch-Néel-Ising wall is a combination of (b) and (d). This figure is taken from [16]. 4.3 Thin Film Growth Commensurate BaTiO 3 /SrTiO 3 superlattices were grown using reactive MBE by Dr. Soukiassian. From here on the abbreviation SL will be used instead of the term superlattice. A BaO monolayer was grown first on an atomically flat substrate followed by BaTiO 3 ; the top layer of the SL was SrTiO 3 terminated with a TiO monolayer. Molecular beams of strontium and barium were produced by sublimating elemental strontium and barium contained in titanium crucibles using low-temperature effusion cells, whereas the titanium flux was supplied by a Ti-Ball titanium sublimation pump [19]. The SLs were grown by sequential shuttered deposition of the constituent monolayers in a background pressure of Torr of molecular oxygen and a substrate temperature of 650 [19]. SLs were grown on 8 different substrates imparting various strain states on the films; these are listed in Table 4.1 with the corresponding strain values for the BaTiO 3 and SrTiO 3 layers. The strains are calculated as ( ) a a 0 /a0

115 88 Table 4.1. Epitaxial strain values for MBE grown [(BaTiO 3 ) n /(SrTiO 3 ) m ] p superlattices. For all of the superlattices, m = 4, n = 8, and p = 40, except for A53 for which p = 10. Sample Substrate Strain SrTiO 3 % Strain BaTiO 3 % A53 (001) SrTiO A136 (101) DyScO A95 (101) TbScO A138 (101) GdScO A98 (101) EuScO A48 (101) SmScO A69 (101) NdScO A99 (101) PrScO where a is the measured in-plane lattice parameter of the SL, and a 0 is the cubic lattice parameter of BaTiO 3 (3.99 Å ) and SrTiO 3 (3.905 Å ). The SLs can be described as [(BaTiO 3 ) n /(SrTiO 3 ) m ] p where [(BaTiO 3 ) 8 /(SrTiO 3 ) 4 ] 40 except for the SL grown on SrTiO 3 which has p = 10 in order to avoid film relaxation. 4.4 Determination of Phase Transitions by SHG SHG is a convenient tool for tracking phase transitions in BaTiO 3 /SrTiO 3 SLs because the contribution of the in-plane and out-of-plane polarization components can be separated. In general, under compressive strain BaTiO 3 becomes tetragonal with 4mm point group symmetry and out-of-plane polarization whereas under tensile strain it becomes orthorhombic with point group symmetry mm and in-plane polarization [0]. A more detailed BaTiO 3 temperature-strain phase diagram, based on phase field simulations, is shown in Figure 4.. The letters T, O, and M are abbreviations for tetragonal, orthorhombic, and monoclinic whereas the superscripts f and p stand for ferroelectric and paraelectric. Clearly, for strains ε 0 less than -1 % and greater than 0.75 % BaTiO 3 becomes tetragonal and orthorhombic as mentioned before. However, for -1 %<ε 0

116 89 SrTiO 3 EuScO 3 NdScO 3 GdScO 3 PrScO SmScO 3 DyScO 3 3 TbScO 3 Figure 4.. Phase diagram of BaTiO 3 films as a function of temperature and substrate in-plane strain, calculated by Dr. Y. Li. The dashed lines show the BaTiO 3 strain states for the tested superlattices [0]. < 0.75 % the situation is somewhat more complicated since multiple phases both with in-plane and out-of-plane polarization components appear. As indicated by the dashed lines, for many of the tested superlattices, the BaTiO 3 layers are under strain values which fall within mixed phases that are predicted to have both in-plane, and out-ofplane polarization components. On the other hand, the situation for SrTiO 3 is somewhat more straightforward since under tensilve strain it is predicted to be orthorhombic with point group mm and in-plane polarization [11]. In summary, even though it might be tempting to assign the out-of-plane polarization components to the BaTiO 3 layers, and the in-plane polarization components to the SrTiO 3 layers, which could be the case under certain strain states, caution is needed when analyzing experimental SHG data because the technique gives information only about the existence of in-plane and outof-plane polarization components, and not about their origin.