Superexchange-Driven Magnetoelectricity in Hexagonal Antiferromagnets

Kris T. Delaney Materials Research Laboratory University of California, Santa Barbara Acknowledgements Collaborators: Maxim Mostovoy, University of Groningen Nicola A. Spaldin, UCSB • • • • 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 / 2 0 0 9

Magnetoelectricity

E M H P 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 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 / 2 0 0 9

Linear Magnetoelectric Coupling

  Free-energy

F



F

0  

ij E i H j

Induced polarization/magnetization:

P i M j

  

ij H



ij E i j

  Size limit (in bulk):

α ij

2 ≤

ε ii μ jj

Our aims:  Strong spin-lattice coupling through superexchange  Increase

μ

through geometric frustration 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 / 2 0 0 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 / 2 0 0 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 / 2 0 0 9

Geometric Frustration

Heisenberg Hamiltonian:

H x

 

S i

.

S



j i

, 

j

Antiferromagnetic Spins (J>0):

Drives non-collinear spin order to minimize energy ?

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 / 2 0 0 9

Symmetry & ME Response

Symmetry of spin triangle with 120° spin ordering.

Reference to radial axis with f : f  p/2 C 3 , m C 3 , m Invariants:

F=F 0 -

 0

(E x H x +E y H y )



ij

   0    1 0 0 1    Invariants:

F=F 0 +

 0

(E x H y -E y H x )



ij

   0   0  1 1 0   General Form: 

ij

   0    cos sin     sin cos       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 / 2 0 0 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 / 2 0 0 9

Superexchange

 Hubbard Model for 2 states:

H

 

t i

,  

j



c j

 

c i

 

c i

 

c j

  

U



i n i



n i

  Virtual hopping (2 nd -order perturbation)  exchange energy, J J=4t 2 /U  Direct d-d exchange often weak in transition metal oxides  Superexchange – hop through ligand 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 / 2 0 0 9

Superexchange

Superexchange:  Ligand mediates exchange (e.g., oxygen)  Effective hopping

t dd

~

t

2

pd

/  FM Superexchange p y AFM Superexchange d 3z2-r2 p x d 3x2-r2 p y d 3z2-r2 d 3z2-r2 J<0 J>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 / 2 0 0 9

Superexchange



e.g., Mn-O-Mn:

S 1 θ S 2

E



J

  

S

1 .

S

 2 Anderson-Kanamori-Goodenough rules: J(θ=90º)<0 (FM) J(θ=180º)>0 (AFM) 

Superexchange magnetoelectricity:

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 / 2 0 0 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 / 2 0 0 9

Frustration + Superexchange

→

 a) M = 0 state of frustrated AFM triangle. Spins coupled by superexchange through O ligands.

b) Applied electric field displaces oppositely charged Mn and O in opposite directions.

c) J(  ) changes due to ion displacements.

Spins rotate to new ground state.

V Sc

= 7.3 Å 3 d) Triangle acquires a net magnetization.

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 / 2 0 0 9

Periodicity: Triangular Lattice Hexagonal Manganites

R Mn O M. Fiebig et al., J. Appl. Phys.

93

, 8194 (2003) 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 / 2 0 0 9

Periodicity: Triangular Lattice Hexagonal Manganites

Beware of response cancellation!

e.g., Hexagonal RMnO 3 in A 2 or B 1 AFM state f  0 f  2 p 3 f  4 p 3 

ij

f  0   0    1 0  0 1   

ij

f  2 p / 3   0     1  2 3 2 2 1 2 3     

ij

f  4 p / 3   0     1 2 3 2  2 1 2 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 / 2 0 0 9

Periodicity: Kagomé Lattice

e.g., Iron jarosite “Antimagnetoelectric” f  0 f  p f  p f  0 f  0 

ij

f  f

ij

 0  p    0  0     1 0  1 0 0 1    0 1   f  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 / 2 0 0 9

Breaking Self Compensation

f  0 f  p f  p f  0 f  0 f  p

E



J

  

S

1 .

S

 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 / 2 0 0 9

Prevent Kagomé Cancellation

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.

a) Basic Kagome lattice of Mn ions 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 / 2 0 0 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 / 2 0 0 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 / 2 0 0 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 / 2 0 0 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 / 2 0 0 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, 11169 (1996).

[2] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).

[3] Z. Yang et al, Phys. Rev. B 60, 15674 (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 / 2 0 0 9

Zero-Field Electronic Structure

Expected crystal-field splitting and occupations for high-spin Mn 3+ ...No orbital degeneracy d z2 3d d x2-y2 d xz d yz d xy Local moment = 4μ B /Mn

Ground-state magnetic structure from LSDA+U

occupied

E F

unoccupied 120º spin ordering in ground state Net magnetization = 0 μ B

Primitive unit cell for simulations with periodic boundary conditions 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 / 2 0 0 9

No Spin-Orbit Coupling

  Uniform rotation of ALL spins does not change energy  degenerate in f Calculations do not distinguish between toroidal and non-toroidal arrangements: f  p/2  

ij

   0    cos sin     Calculate only  0 sin cos       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 / 2 0 0 9

 

Applied Electric Field: Linear Response

Force on ion in an applied electric field:

F j

  

i Z



ij

, 

E i

Z* = Born Effective Charge i,j = degrees of freedom a = ion index where 

Z ij

,   

P i



R j

,  P = Berry Phase Polarization: R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).

 Compute force-constant matrix (by finite difference):

C ij

,   

F i

,  

R j

,   Equilibrium under applied field (assume linear): 

R j

,   

i

 , 

C

 1

ij

, 

F i

,  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 / 2 0 0 9

Results: Magnetoelectric Coupling

Magnetoelectric response of “KITPite”: Cr 2 O 3 : the prototypical magnetoelectric

E

M

  3 .

7  10  4 CGS   1 .

0  10  5 CGS Kris T. Delaney et al., arXiv:0810.0552

accepted in Phys. Rev. Lett.

J. Íñiguez, Phys. Rev. Lett.

101

, 117201 (2008) 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 / 2 0 0 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 / 2 0 0 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 / 2 0 0 9

Further Enhancements

 Three factors:    Magnetic susceptibility Forces induced by spin change Rigidity of (polar mode) lattice for response  ~  2

eM J



J



V

~ 10  3 (3x DFT result)

V

= volume per Mn ~ 60 Å 3 J’/J ~ 3.3 Å -1 [1] K ~ 6 eV/Å [2] [1] Gontchar and Nikiforov, PRB 66, 014437 (2002) [2] Iliev et al., PRB 56, 2488 (1997) 

J

’/

J

small - need to approach small

J

limit  Lower ordering temperatures 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 / 2 0 0 9

Methods: Applied H

Collinear Magnetism

M

 

B



N

 

N

  Non-collinear Magnetism

H y

 

E KS

  0 

B



H

 

M

 F 

B

 0

H

  

F

2 

V

   

V

 

V

 

V

 

V

      0 

B

 

H x H z



iH y H x

 

H iH z y

  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 / 2 0 0 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 / 2 0 0 9



Conclusions: Magnetoelectrics

Demonstrate strong magnetoelectricity:   Superexchange Frustration  non-collinear magnet  Key: avoid cancellation of

microscopic response

in periodic systems    Future: Increase c H Improve

J’/J

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 / 2 0 0 9