Competing Ferroic Orders The magnetoelectric effect

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1 Competing Ferroic Orders The magnetoelectric effect Cornell University I would found an institution where any person can find instruction in any study. Ezra Cornell, 1868 Craig J. Fennie School of Applied and Engineering Physics fennie@cornell.edu Basic Training in Condensed Matter Theory 2009

2 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 3. Structure and Ferroelectricity Basics of space groups Soft mode theory, lattice dynamics, group theoretic methods Competing lattice instabilities microscopic mechanisms, improper FE Modern theory of polarization (Berry Phase) 4. Magnetism Basics, exchange interactions, superexchange, Dzyaloshinskii-Moria How spins couple to the lattice! Phenomenology and microscopics (spin-phonon, spin-lattice, etc) Competing magnetic orders Systems: ZnCr2O4, EuTiO3, SeCuO4, TeCuO4 2

3 Magnetism and how it couples to the lattice Isotropic Exchange Tuning AFM FM The Goodenough-Kanamori rules Spin-phonon coupling ZnCr 2 O 4 spin-induced phonon anisotropy EuTiO3 magneto-capacitance Dzyaloshinskii- Moria Exchange Spin-lattice coupling Weak ferromagnetism Fe2O3 Spin spiral Lifshitz invariant 3

4 Magnetic properties of localized systems Magnetism arises from an incomplete shell Eu 2 O 3 Eu: 4f 7 5d 0 6s 2 Eu 3+ : 4f 6 5d 0 6s 0 0 ˆµ 0 = g J µ B 0 Ĵ 0 =0 Question: is Eu 2 O 3 magnetic? What do I mean by magnetic? For now I mean, Does a magnetic field couple to the fairly large S=3µ B, NO! why? Well magnetic field couples to J= S-L Hund s Rules 1) Max S =3 2) Max L =3 3) J= S-L =0 less than half filled (J= S+L if more ) But is it magnetic? Yes! χ = N V ( 2µ 2 B n 0 L z + gs z 0 2 E n E 0 e2 mu 0 6m e ) Z ri 2 i=1 Van Vleck Paramagnetism diamagnetism Both terms, small and temperature independent (χ VV ~ 1/Δ) 4

5 Magnetic properties of localized systems Currie Weiss Susceptibility 1 χ T θ CW -Θ CW χ -1 0 T N AFM PM Θ CW = T c Temperature FM Magnetic exchange interactions E = ij k B θ CW = 2 3 J ij S i S j + gµ B S i H J n z n S(S + 1) n i Mean-field 5

6 Properties of common Antiferromagnets 6

7 Effect of strong magnetic field (AFM) Susceptibility below T N and Spin flop Hard vs easy axis 7

8 Origin of easy/hard axis Spin-orbit interaction λ L S 1. Single-ion anisotropy E-field 1. Crystal field couple to charge density 2. Charge density couple to spin 2. Anisotropic exchange coupling overlap Different energy 8

9 General form of spin-spin coupling Isotropic symmetric exchange (l=0) Dzyaloshinskii-Moriya Antisymmetric exchange (l=1) D and K both spin-orbit effects D λj ( g/g)j k λ 2 J ( g/g) 2 J H = S 1 J S 2 H = J S 1 S 2 + D ( S 1 S 2 ) +k x S 1x S 2x + k y S 1y S 2y + k z S 1z S 2z Anisotropic symmetric exchange (l=2) 9

10 Dzyaloshinskii-Moriya Relativistic correction to Anderson s superexchange E = J S i S j + D ij S i x S j Collinear AFM D = 0 Spin canting D 0 M Direction of canting determined by the sign of D 10

11 Weak ferromagnetism vs ME effect % #! &! "! ' " Cr 2 O 3!! " L = M 1 -M 2 +M 3 -M 4 # $ " # $ " ' % #! & ' " Fe 2 O 3 L = M 1 +M 2 -M 3 -M 4 Neel temperature ~500K Morin temp ~260K # $ " # $ " '

12 Exchange: Background Hund s rule like FM Anti-bonding Covalent bond like AFM bonding 12

13 Exchange: Background Direct exchange If they are the same orbital AFM If they are the different orbital FM Superexchange 90 SE 90 Hund s rule J_H coupling on the anion TM-d O TM-d 13

14 Exchange: Background NiO Competing interactions 1. Strong AFM Ni- O-Ni Weaker FM 90 SE 14

15 Exchange: Background Arbitrary angle J =J 90 sin 2 θ + J 180 cos 2 θ SeCuO3 FM TeCuO3 AFM Empirically

16 ACr2X4: Cubic Spinel Structure Cr 3+ : 3d 3 4s 0 Superexchange pathways e g t 2g What is Moment? µ ~ S =3/2 Why -> orbital dof quenched S=3/2 16

17 ACr2X4: Cubic Spinel Structure Network of edge sharing octahedra 17

18 Background: Spinels Cr 3+ : 3d 3 4s 0 ACr 2 X 4 A= Zn, Cd, Hg X= O, S, Se e g t 2g S=3/2 a (Å) Theory Exp. CdCr 2 S CdCr 2 Se HgCr 2 Se ZnCr 2 O CdCr 2 O Ferromagnetic Insulators T c ~100K, T Θ ~200K Anti-ferromagnetic Insulators T N ~10K, T Θ ~ 100K 18

19 Exchange interactions Direct Cr-Cr exchange AFM 90 Cr-O-Cr SE FM AFM FM a) as lattice constant increases b) as electronegativity of anion decreases oxygen 19 chromium 19

20 Huge spin-phonon coupling AFM-ZnCr 2 O 4 Symmetry lowering at Tn O h D 4h Cubic to tetragonal Phonon splitting at Tn T1u A2u Eu Singlet A 2u 3-fold T 1u phonon mode Doublet E u T n = 12.5K Temperature (K) Exp: λ =6-10 cm -1 20

21 Spin-phonon coupling Phonon modulated exchange interaction Baltensperger and Helman, Helvetica physica acta J(u) E = E 0 +E phonon + E spin E ph = 1/2 ω 02 u 2 E sp = - J ij S i S j J(u) J(0) + 1/2 2 J/ u 2 S i S j u 2 J(u+δ) ω 2 ω J/ u 2 S i S j renormalized bare magnetic contribution phonon phonon e.g. can understand large spin-phonon coupling in ZnCr 2 O 4 Fennie and Rabe, Phys. Rev Lett. May

22 Spin-phonon coupling Phonon modulated exchange interaction Baltensperger and Helman, Helvetica physica acta J(f u ) J(f u +δ) e.g. can understand large spin-phonon coupling in ZnCr 2 O 4 Fennie and Rabe, Phys. Rev Lett. May 2006 to lowest order in S i S j 22

23 Origin of large anisotropy f 3 has significant anisotropy in force constant matrix f3: One set of T1u Partner functions z y x z-mode Direct-exchange length scale, α -1 Exp: α=8.9 Å -1 Theory: α=9.05 Å -1 x-mode (or equivalently y-mode) Cr ions spin up spin down 23

24 Bulk Eu 2+ Ti 4+ O 3 : Ground state antiferromagnetic paraelectric r(eu 2+ ) ~ r(sr 2+ ); Cubic perovskite Eu 2+ J=S=7/2; T n ~ 5.5K, G-type AFM T N 24

25 Perovskites and the Period Table Substitutions on A, B or both (A 1-x A x )(B 1-y B y )O 3 Random distribution or ordered 25

26 Phonon anomaly at Tc FM- CdCr 2 S 4 Symmetry lowering at Tc O h O h Cubic to cubic (ignoring LS) No phonon splitting at Tc T 1u T 1u 26