Collective Magnetic behavior of Geometrically Frustrated Arrays

1. Collective Magnetic behavior of Geometrically Frustrated Arrays Y. Pan, K. K. Kohli, R. Fraleigh, L. Balk, D. Finkel, S. Zhang, J. Li, I. Gilbert, C.J. Grigas, P. Lammert, R. Misra, V. H. Crespi, P. Schiffer, N. Samarth Department of Physics, Pennsylvania State University

2. Nanomagnets • Permalloy (80% Ni, 20% Fe) • Island size – 220(l) x 80(w) x 25(h) nm • Island moment constrained along long axes • Square lattice spacing from 320 nm to 880 nm • Geometry of the lattice creates frustration amongst individual islands • What is the collective behavior? K. K. Kohli et al. PRB Rapid Comm. 84, 180412(R) (2011)

3. Magneto-Optical Kerr Effect H Laser λ = 632.8 nm Beam spot size ≈ 50 μm Longitudinal MOKE Kerr rotation α sample magnetization Sample Magnet Lens Lens Analyzer Chopper Lens Lock-In Amp. HeNe Mirror Laser Polarizer Detector

4. Simulations SEM Simulations performed using NIST OOMMF v.1.2a4 OOMMF code takes in images Colors define regions Use SEM image to get realistic edge roughness OOMMF vector field display replicates shape and edge variation from images Simulation inputs: Saturation magnetization (Ms = 860 x 103 Am-1) Exchange constant (A = 13 x 10-12 Jm-1) Cell size (5 nm x 5 nm x 5 nm) Image pixel size < cell size (~2 nm x 2 nm)

5. Lattice Dependence • As the strength of interaction decreases, Hc increases • Interaction lowers the Hc, induces earlier collective switching • Cascades?

6. Angle Dependence • Angular dependence peaks at 5°-10° for small lattice spacing  interaction related effect • Large lattice spacing islands don’t see dipolar field from neighboring islands • What is the origin of this peak at 5°? H θ

7. Vertical Islands • When Hext + Hnn > H//island, reversal occurs • Tip bending • Reversed island enhances field on NN • Un-reversed island has reduced field on NN • Lower intrinsic coercivity islands initiate cascades, decreasing Hc of the whole Vertical +y 0 degrees H θ +x -y

8. Horizontal Islands • At θ=0, horizontal islands point , , or vortex • Dictated by edge • Vertical islands experience different fields • At θ=5°, horizontal islands rotate in unison • Vertical islands experience same field • Effective field from horizontal islands causes Hcmax to occur at non-zero angle Horizontal +y 5 degrees H θ +x -y

9. Hexagonal in-plane arrays

10. Angle-Dependence - Hexagonal • Angle dependence as expected for hexagonal arrays • Experiment matches well with simulation H α

11. Hex – 30 degrees • Vertical, horizontal and perpendicular islands • Perpendicular islands rotate in unison • Form chains along horizontal axis first • Initiate cascades of both horizontal and vertical islands • Rotate back to initial state Vertical Perpendicular Horizontal +y H α +x -y

12. Hex – 30 degrees • Immediately prior to reversal • Chains along the field axis • Chains along the horizontal axis • Immediately after reversal • Reversal of chains (cascade behavior) along the field axis • Smooth return to saturation Immediately prior to reversal Immediately after reversal Saturation Saturation +y +x -y

13. Hex – 0 degrees Vertical • Parallel Islands only • No perpendicular islands • Chains in vertical axis along both directions • Forcing horizontal islands to point into each other • Horizontal islands reinforce field on vertical islands while poles are strong • When Hext is large enough, poles of horizontal islands weaken Horizontal Horizontal +y H α +x -y

14. Hex – 0 degrees • Pole of a horizontal island weakens • Breaks the chain, induces cascade along one vertical direction • Cascade breaks chains in the other direction Immediately prior to reversal During reversal Saturation Saturation +y +x -y

15. Conclusions • For square arrays, lower intrinsic coercivity islands initiate cascades, thus decreasing the coercivity of the array • Pole strength of vertical islands is important • An island that has reversed enhances the field • An island that has not reversed reduces the field acting on the neighbors • Maximum Hc occurs for a non-zero angle at which the net effective horizontal field on the vertical islands is zero. • In hexagonal arrays, chains form along vertical and/or horizontal axes • At 30°, cascades along field axis • At 0°, pole strength of horizontal islands is important • Once a horizontal island breaks a chain, cascades occur along vertical axis

16. Thank you for your attention Questions?