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Basic Training 2009 L02.Pptx Basic Training 2009– Lecture 02 Competing Ferroic Orders The magnetoelectric effect Craig J. Fennie Cornell University School of Applied and Engineering Physics “I would found an institution where any [email protected] person can find instruction in any study.” − Ezra Cornell, 1868 Basic Training in Condensed Matter Theory 2009 Module Outline 1. Overview and Background • Ferro ordering, the magnetoelectric effect 2. ME revisited, and basic oxide physics • ME effect revisited: Toroidal moments • Complex oxides basics: Types of insulators (i.e., ZSA classifications), Coordination chemistry Basic Training 2009– Lecture 02 2 1 The linear magnetoelectric effect • Symmetry properties: broken time T and space I inversion, but the combination TI can be symmetry operation = α E H FME ij i j 1 1 1 α = α0δ + # τ +[ (α + α ) α δ ] ij 3 ij ijk k 2 ij ji − 3 0 ij 0 α = αii Pseudo scalar Pseudo vector Rank 2 traceless, symmetric Don’t know, θe2 Toroidal may be worth δij moment thinking about ∼ 2πh τ " α k ≡ ijk ij From Joel Moore’s talk yesterday I learned this is related to “axion electrodynamics,” 3-d topological insulators, and Berry phase (or if your from Rutgers: Vanderbilt) theory of polarization Basic Training 2009– Lecture 02 3 Cr2O3 ME effect 2 C/m -5 Polarization 10 1 Telsa = 10 kOe Basic Training 2009– Lecture 02 4 2 From Max Mostovoy Linear magnetoelectric effect !"#$"#%&'()*+,-.+/-- 0$12#&' 345#67898:;# !"#$% %"#<"#=+>?*@;#0$12#&& AB5#6783B: P = χe E + αH M = αE + χm H C"1"#D(E* 2DF#&% GG9#6783H: Basic Training 2009– Lecture 02 5 Ferroelasticity Energy F(η) = αη2 + βη4 - ησ Ordering of long wavelength lattice modes→ spontaneous strain Strain orientation, η Spontaneous symmetry breaking Symmetry properties (-t) = + (t) η η Actually a rank 2 polar* tensor η(-x) = +η(x) *polar tensor transforms like a product of coordinate under rotations or rotations-inversions. Basic Training 2009– Lecture 02 6 3 Ferro recap ? Nature 449, 702-705 (11 October 2007) Observation of ferrotoroidic domains Bas B. Van Aken et al. Basic Training 2009– Lecture 02 7 Missing “Ferro”-icity? What would this missing “ferro”-icity require? OP Coupling Field Interaction energy Ferromagnetism: M H F~ -M⋅H Ferroelectricty: P E F~ -P⋅E Ferroelasticity η σ F~ -η⋅σ Basic Training 2009– Lecture 02 8 4 Can we construct what this missing invariant should look like? (don’t take this seriously, just a lot of hand waving) Sounds like the • Must be odd under space inversion Magnetoelectric tensor • Must be odd under time inversion • Symmetry properties: broken time T and space I inversion, but the combination TI can be symmetry operation = α E H τk "ijkαij FME ij i j ≡ Becomes, slightly more illuminating if we write our free energy in terms the dynamic variables that order: = γ L P M FME ijk k i j The antisymmetric part γ L t˜ " t ≡ Toroidal moment ijk k → k kij Whatever this toroidal moment is it seems to be related to antiferromagnetism Basic Training 2009– Lecture 02 9 Toroidal moment a Torus with an even number of current windings forming a toroidic moment, T. b, The magnetoelectric effect is illustrated by the current loops being shifted by the magnetic field (H), thus inducing an electric polarization (P). By rotating the figure by 90°, the asymmetry (ij = -ji) becomes obvious. Note that ij corresponds to a toroidal moment Tk (i j k). For localized spins Basic Training 2009– Lecture 02 10 5 Nature 449, 702-705 (11 October 2007) Ferrotoroidicity Observation of ferrotoroidic domains Bas B. Van Aken et al. Used SHG to map out ferrotoroidic domains in LiCoPO4 a Torus with an even number of current windings forming a toroidic moment, T. b, The magnetoelectric effect is illustrated by the current loops being shifted by the magnetic field (H), thus inducing an electric polarization (P). By rotating the figure by 90°, the asymmetry (ij = -ji) becomes obvious. Note that ij corresponds to a toroidal moment Tk (i j k). Basic Training 2009– Lecture 02 11 Missing “Ferro”-icity? What would this missing “ferro”-icity require? OP Coupling Field Interaction energy Ferromagnetism: M H F~ -M⋅H Ferroelectricty: P E F~ -P⋅E Ferroelasticity η σ F~ -η⋅σ FerroToroidicty T ∇×H F~ -T⋅(∇×H) T≡ Toroidization Basic Training 2009– Lecture 02 12 6 Honest recap on FerroToroidicty Presented a relatively new (or rediscovered) ferro-toroidal ordering. What is this? Don’t know yet, here is what I know: Electromagnetism allows the existence of some moment referred to as a toroidal moment (toroidal because it has the symmetry of of toroidal solenoid). Expanding the interaction energy between energy the magnetic density and the magnetic field, one arrives at a coupling between the toroidal moment and the field which has the same symmetry as the antisymmetrical component of the magnetoelectric tensor, i.e., a toroidal moment would lead to an antisymmetric ME effect. That’s it, that is all I presently understand. Questions are plentiful. Basic Training 2009– Lecture 02 13 Multifunctional multiferroics Multiferroic: combine more than one “ferro” property: Ferroelectricity, ferroelasticity, and/or magnetism (Hans Schmid, 1973) Polarization, P Magnetization, M - + N S e.g., ferroelectric ferromagnet Basic Training 2009– Lecture 02 14 7 Multifunctional multiferroics Multifunctional: response to more than one external perturbation: Electric and magnetic fields Polarization, P E H Magnetization, M - + N S P M e.g., ferroelectric ferromagnet Basic Training 2009– Lecture 02 15 Multifunctional magnetoelectrics (Generalized) Magnetoelectric: cross coupled response to electric and magnetic fields Polarization, P E H Magnetization, M - + N S P M i.e. control of the magnetic M (electric P) phase with an applied electric E (magnetic H) field Basic Training 2009– Lecture 02 16 8 Multiferrroic and the Linear ME effect Magnetic-ferroelectrics Polarization, P Magnetization, M - + N S Symmetry in some cases then allows a Linear Magnetoelectric effect, Eint = γ E H P = γ H γ ≡ linear magnetoelectric M = γ E coefficient* But… intrinsic γ small *Landau and Lifshitz, “Electrodynamics of continuous media”; Dzyaloshinskii, JETP 1957; Astrov JETP 1960 -> Cr2O3 Basic Training 2009– Lecture 02 17 Why mutiferroics and magnetoelectric effet 1 1 (E, H)= ! E E + µ H H + α E H + F 2 ij i j 2 ij i j ij i j 4παij √#ijµij ≤ Basic Training 2009– Lecture 02 18 9 We are we going? – although symmetry allows the existence of a ME effect, it tells us nothing about its size. This is not a question of optimizing parameters to increase the magnitude of the effect in e.g., Cr2O3, but requires entirely new physical ideas. – An idea that is prevalent in materials physics today (some would argue throughout nature) is the idea where the competition between different ordered ground states leads to new emergent phenomena (sometimes the adjective “colossal” is attached the observed effect). – What if we have a system where ferroelectricity and (anti)ferromagnetism compete with each other, will it lead to “colossal” ME effects. Note this doesn’t mean we simply add ferroelectricity and magnetism, i.e., multiferroics, we need to think of microscopic mechanisms that would COUPLE the two order parameters in a nontrivial way . – Can we provide this coupling through the lattice? So in the remainder of this module we will • Understand the origin of ferroelectricity and how it couples to the lattice • Understand the origin of (anti)ferromagnetism and how it couples to the lattice • Combine these ideas to produce the stated goal. But first, perovskites are ubiquitous materials that display an amazing variety of different phenomena, lets focus on them Basic Training 2009– Lecture 02 19 Emergence of new macroscopic phenomena Where do we look for new phenomena and why do we look there? Basic Training 2009– Lecture 02 20 10 Emergence of new macroscopic phenomena “More is Different,” Phil Anderson, Science 1972 As the complexity of the crystalline motif increase, new properties have the potential to emerge! Elements binary tertiary Quaternary 100 10,000 1,000,000 100,000,000 Life ≈ Elements/ 1 2 3 4 ~10 unit cell Si GeTe ZnCr2O4 YBCO Fe BiFeO3 Hg Based on a talk at Rutgers by P. Coleman Basic Training 2009– Lecture 02 21 Oxides oh plenty! Except for the core, most of the earth is made of oxides http://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements Note, Rare earth’s arent so “rare”, why the name? Also, the chemistry obeys “goldilock” theorm: not too ionic, not too colavent Basic Training 2009– Lecture 02 22 11 Complex oxides: ABO3 Perovskites Nearly any physical property Same prototypical structure Dielectric CaTiO3, SrTiO3, (CaCu3)(Ti4)(O4)3 A Ferroelectric BaTiO3, LiNbO3, PbTiO3 Magnetoelectric TbMnO3, BiFeO3 Antiferroelectric PbZrO3 B Piezoelectric PbZrxTi1-xO3 Antiferromagnetic LaMnO3 Ferromagnetic SrRuO3 Superconducting doped-SrTiO 3 O Colossal Magneto-resistance (La,Ca)MnO3 The most abundant mineral within the earth is a perovskite, which one? Basic Training 2009– Lecture 02 23 Perovskites and the Period Table Substitutions on A, B or both (A1-xA’x)(B1-yB’y)O3 Random distribution or ordered Basic Training 2009– Lecture 02 24 12 ABO3 Perovskites 3-d network of corner shared octahedra Basic Training 2009– Lecture 02 25 Layered Perovskites - Nature’s superlattices Bi O [A’ B O ] A2 [A’n-1BnO3n+1] A [A’n-1BnO3n+1] 2 2 n-1 n 3n+1 Ruddlesden-Popper Dion-Jacobson Aurivillius e.g., n=2 Sr3Ru2O7 CsBiNb2O7 ? SrBi2Ta 2O9 Basic Training 2009– Lecture 02 26 13 Two kinds of insulators (that I’ll be discussing) Structural and optical properties of paraelectric SrTiO3 3329 It is to be noted that for the interpretation of the optical spectra of systems, it does not seem realistic to give a single transition assignment to the peaks present in a crystal reflection spectrum since many transitions (direct to indirect) may be found in the band structure with an Basicenergy Training corresponding 2009– Lecture to the 02 peak and since states away from the lines and point of symmetry 27 could contribute to the reflectivity.
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