K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Superexchange-Driven Magnetoelectricity in Hexagonal Antiferromagnets Kris T. Delaney Materials Research Laboratory University of California, Santa Barbara Collaborators: Maxim Mostovoy, University of Groningen Nicola A. Spaldin, UCSB Acknowledgements Funding/Computing: National Science Foundation California Nanosystems Institute San Diego Supercomputer Center Moments and Multiplets in Mott Materials program at the KITP, UCSB

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Magnetoelectricity No permanent M or P required Uses: Low-power, reduced-size technologies; magnetic memory elements, sensors, transducers Often weak: Use DFT to design new, strong magnetoelectrics E M H P

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 /  Free-energy  Induced polarization/magnetization:  Size limit (in bulk):  Our aims:  Strong spin-lattice coupling through superexchange  Increase μ through geometric frustration Linear Magnetoelectric Coupling

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Outline  Route to strong magnetoelectric:  Geometric frustration  Magnetic superexchange  Combination → magnetoelectricity  A periodic model: hexagonal manganite  Avoiding self compensation  Density functional calculations  Further enhancements

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Outline  Route to strong magnetoelectric:  Geometric frustration  Magnetic superexchange  Combination → magnetoelectricity  A periodic model: hexagonal manganite  Avoiding self compensation  Density functional calculations  Further enhancements

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Geometric Frustration ? Drives non-collinear spin order to minimize energy Heisenberg Hamiltonian: Antiferromagnetic Spins (J>0): M=0

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Symmetry & ME Response Symmetry of spin triangle with 120° spin ordering. Reference to radial axis with  :  General Form: Invariants: F=F 0 -   (E x H x +E y H y ) Invariants: F=F 0 +   (E x H y -E y H x ) C 3, m

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Outline  Route to strong magnetoelectric:  Geometric frustration  Magnetic superexchange  Combination → magnetoelectricity  A periodic model: hexagonal manganite  Avoiding self compensation  Density functional calculations  Further enhancements

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Superexchange  Hubbard Model for 2 states:  Virtual hopping (2 nd -order perturbation)  exchange energy, J  Direct d-d exchange often weak in transition metal oxides  Superexchange – hop through ligand J=4t 2 /U

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Superexchange pypy pxpx J<0 J>0 FM Superexchange AFM Superexchange Superexchange:  Ligand mediates exchange (e.g., oxygen)  Effective hopping d 3z 2 -r 2 d 3x 2 -r 2 pypy

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 /  e.g., Mn-O-Mn:  Superexchange magnetoelectricity: Superexchange θ S1S1 S2S2 Anderson-Kanamori-Goodenough rules: J(θ=90º)<0 (FM) J(θ=180º)>0 (AFM) E = 0 E

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Outline  Route to strong magnetoelectric:  Geometric frustration  Magnetic superexchange  Combination → magnetoelectricity  A periodic model: hexagonal manganite  Avoiding self compensation  Density functional calculations  Further enhancements

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Frustration + Superexchange →  d) Triangle acquires a net magnetization. b) Applied electric field displaces oppositely- charged Mn and O in opposite directions. a) M = 0 state of frustrated AFM triangle. Spins coupled by superexchange through O ligands. c) J(  ) changes due to ion displacements. Spins rotate to new ground state. V Sc = 7.3 Å 3

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Periodicity: Triangular Lattice Hexagonal Manganites M. Fiebig et al., J. Appl. Phys. 93, 8194 (2003) R Mn O

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Periodicity: Triangular Lattice Hexagonal Manganites Beware of response cancellation! e.g., Hexagonal RMnO 3 in A 2 or B 1 AFM state

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Periodicity: Kagomé Lattice e.g., Iron jarosite “Antimagnetoelectric”

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Breaking Self Compensation

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Prevent Kagomé Cancellation a) Basic Kagome lattice of Mn ions b) choose O ligand (red) locations so that ME response of orange and green triangles do not cancel. Each Mn ion is now in the center of an O triangle. c) Each MnO triangle can be transformed into a trigonal bipyramid (c.f. YMnO3 – a real material) d) Multiple layers connect through apical oxygen ions. The layers are rotated 180º to account for AFM interlayer coupling. e) Counter ions are introduced to ensure Mn 3+ and O 2- so that trigonal bipyramids are correctly bonded. Final material: CaAlMn 3 O 7

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Layer 1 vs Layer 2 Layer 1 Layer 2

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Symmetry of Structure Inversion center between Mn planes Magnetic state breaks I leaving IT valid

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / KITPite Structure

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Outline  Route to strong magnetoelectric:  Geometric frustration  Magnetic superexchange  Combination → magnetoelectricity  A periodic model: hexagonal manganite  Avoiding self compensation  Density functional calculations  Further enhancements

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / DFT Calculation Details  First principles computation of ME response:  Density functional theory  Finite electric fields through linear response  Density functional theory (DFT)  Vienna Ab initio Simulation Package (VASP) [1]  Plane-wave basis; periodic boundary conditions  Local spin density approximation (LSDA)  Hubbard U for Mn d electrons (U=5.5 eV, J=0.5 eV) [3]  PAW Potentials [2]  Non-collinear Magnetism  No spin-orbit interaction [1] G. Kresse and J. Furthmüller, Phys. Rev. B 54, (1996). [2] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [3] Z. Yang et al, Phys. Rev. B 60, (1999).

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Zero-Field Electronic Structure Ground-state magnetic structure from LSDA+U 120º spin ordering in ground state Net magnetization = 0 μ B Expected crystal-field splitting and occupations for high-spin Mn 3+...No orbital degeneracy Local moment = 4μ B /Mn d xz d yz 3d d x 2 -y 2 d xy dz2dz2 Primitive unit cell for simulations with periodic boundary conditions occupied unoccupied EFEF

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / No Spin-Orbit Coupling  Uniform rotation of ALL spins does not change energy  degenerate in   Calculations do not distinguish between toroidal and non-toroidal arrangements:  Calculate only  0 

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 /  Force on ion in an applied electric field:  where  Compute force-constant matrix (by finite difference):  Equilibrium under applied field (assume linear): Applied Electric Field: Linear Response Z* = Born Effective Charge i,j = degrees of freedom a = ion index P = Berry Phase Polarization: R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Results: Magnetoelectric Coupling Magnetoelectric response of “KITPite”: Cr 2 O 3 : the prototypical magnetoelectric J. Íñiguez, Phys. Rev. Lett. 101, (2008) M E Kris T. Delaney et al., arXiv: accepted in Phys. Rev. Lett.

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / A Makeable Material? YMnO 3 structure 25% alloy Mn with diamagnetic cation Ordered? (c.f. double perovskite) V Sc = 7.3 Å 3 V Mn = 6.7 Å 3

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Outline  Route to strong magnetoelectric:  Geometric frustration  Magnetic superexchange  Combination → magnetoelectricity  A periodic model: hexagonal manganite  Avoiding self compensation  Density functional calculations  Further enhancements

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 /  Three factors:  Magnetic susceptibility  Forces induced by spin change  Rigidity of (polar mode) lattice for response  J’/J small - need to approach small J limit  Lower ordering temperatures Further Enhancements V = volume per Mn ~ 60 Å 3 J’/J ~ 3.3 Å -1 [1] K ~ 6 eV/Å [2] (3x DFT result) [1] Gontchar and Nikiforov, PRB 66, (2002) [2] Iliev et al., PRB 56, 2488 (1997)

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Methods: Applied H Collinear Magnetism Non-collinear Magnetism  F HyHy

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / KITPite Magnetic Susceptibility Comparison of methods for magnetoelectric response: Applied-E (linear response) + Stable, robust - Slow, many calculations Improve with symmetry Applied-H + Fast, few calculations - Hard to stabilize; small energy scale Constant susceptibility (AFM) Huge field  small magnetization Spin system too stiff

K R I S T. D E L A N E Y ( M R L, U C S B ) | S U P E R E X C H A N G E D R I V E N - M A G N E T O E L E C T R I C I T Y | A P S M A R C H M E E T ING 0 3 / 1 9 / Conclusions: Magnetoelectrics  Demonstrate strong magnetoelectricity:  Superexchange  Frustration  non-collinear magnet  Key: avoid cancellation of microscopic response in periodic systems  Future:  Increase  H  Improve J’/J