Crespo, Adrian

De Sancho, David

Leliaert, Jonathan

Pancaldi, Matteo

Vavassori, Paolo

English

Ghent University Academic Bibliography

www.nanogune.eu   CIC nanoGUNE Tolosa Hiribidea, 76. E-20018 Donostia - San Sebastián    |    +34 943 574 000    |    nano@nanogune.eu   Matteo Pancaldi1, Jonathan Leliaert2, David De Sancho1,3, Adrian Crespo1, Paolo Vavassori1,3 1. CIC nanoGUNE, E-20018 Donostia-San Sebastian, Spain  2. Dept. of Solid State Sciences, Ghent University, Ghent, Belgium 3. IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain Matteo Pancaldi m.pancaldi@nanogune.eu    Introduced in 2006, Artificial Spin Ice (ASI) systems are metamaterials consisting of arrays of dipolar-coupled nanomagnets arranged in frustrated geometries [1].  [1] R. F. Wang et al., Nature 439, 303 (2006). [2] J. M. Porro et al., New J. Phys. 15, 055012 (2013). [3] I. Gilbert et al., Nature Phys. 10, 670 (2014). [4] Y. Perrin et al., Nature 540, 410 (2016). [5] C. Castelnovo et al., Nature 451, 42 (2008). [6] S. Zhang et al., Nature 500, 533 (2013). [7] A. Farhan et al., Phys. Rev. Lett. 111, 057204 (2013). [8] Z. Budrikis et al., New J. Phys. 14, 035014 (2012). [9] A. B. Bortz et al., J. Comp. Phys. 17, 10 (1975). [10] A. Vansteenkiste et al., AIP Advances 4, 107133 (2014).  We acknowledge support from the Basque Government (Project No. PI2015-1-19) and from MINECO, the Spanish Ministry of Economy and Competitiveness (Project No. FIS2015-64519-R and Grant No. BES-2013-063690). A multiscale approach to Artificial Spin Ice thermal simulations μm-scale: thermal protocol and kMC Introduction: Artificial Spin Ice nm-scale: micromagnetism References Acknowledgements ? 6 states (2+1) 2 states (3) E Frustration: the inability of a physical system to satisfy all the interactions simultaneously, leading to a large degeneracy of low-energy states. Square lattice ASI [2] Shakti lattice ASI [3] Interesting features • 2D model systems in which the lowest-energy state can be obtained at room temperature and imaged through lab-scale imaging techniques (e.g., MFM, MOKE, LTEM); • Advanced fabrication techniques (EBL) allow for the systematic variation of the key parameters governing the system (size, distance, arrangement, height offset, …). Key structures • Base element: Ising-like macrospin nanostructure        • Base unit: Vertex (4 interacting nanostructures) 𝑀 𝑀 E 𝐸 =12𝜇0𝑁𝑏𝑀2𝑉 𝑏 ∆𝐸 𝑎 2 stable states: 𝐸 =12𝜇0𝑁𝑎𝑀2𝑉 Assuming a magnetization reversal by coherent rotation, the energy barrier is given by  ∆𝐸 =12𝜇0 𝑁𝑏 − 𝑁𝑎 𝑀2𝑉 Thermal protocol [2,6,7]: from T > TB to T < TB Demagnetizing protocol [2] VS The equlibrium configurations are usually simulated by Monte Carlo techniques [8], considering that • Key structures: Ising-like nanostructures (≤ 500 nm). • Key process: formation of low-energy states in extended systems (≈ 20 – 50 μm, so up to 104 nanostructures). The key process in ASI systems consist in the formation of low-energy states (rarely observed): (1) the system is melted and (2) the frozen state (equilibrium) is recorded. … … … … … … … … … … Kinetic Monte Carlo Micromagnetism Our idea was to move a step further (beyond the macrospin approximation) and consider a multiscale approach (μm- and nm-scale): • Kinetic Monte Carlo (kMC) combined with a more detailed analysis of the switching process; • Micromagnetic analysis of the thermally-induced magnetization reversal (T ≠ 0 K). Kinetic Monte Carlo [9] + precomputed switching frequencies • The considered event is the magnetization reversal of a single nanostructure (single flip); • The base unit is composed by 7 nanostructures (center, first n.n. and second n.n.); • The 7 nanostructures are uniformly magnetized and the magnetization reversal in the central one occurs by coherent rotation, but the actual shape of the nanostructure is taken into account for calculating the energy barriers. The input for the kMC consists in 64 x 4 switching frequencies 𝜈𝑆𝑤𝑖 = 𝜈0exp −Δ𝐸𝐶𝑅𝑖𝑘𝐵𝑇: Right to Left via Top RLT Left to Right via Top LRT Right to Left via Bottom RLB Left to Right via Bottom LRB Switching frequency corresponding to ∆𝐸0 (energy barrier for the coherent reversal of an isolated nanostructure) 8 hours 350 K H = 0 H T2 T1 10 μm This approach allowed us to analyze experimental data available in literature [7]: • Ni83Fe17 • 470 nm x 170 nm x 3 nm • 130 nm edge-to-edge distance • M = 350 kA/m • Sample saturated in a magnetic field applied along [11] and measured by XMCD T1 Vertices T2 Vertices T3 Vertices Experimental data [7] μm-scale multiscale approach Experimental data [7] KMC from [7], where ∆𝐸0 and 𝜏0 have been set to 1.05 eV and 2·10-12 s to get the right time scale μm-scale multiscale approach: ∆𝐸0 turns out to be 2.15 eV The difference in the kMC time scale (given by the energy barrier’s difference) led us to consider a better approach for the magnetization reversal modelling at 𝐓 ≠ 0 K. Micromagnetism • Stadium-like Permalloy nanostructures (150 nm x 100 nm x 3 nm) with 5 x 5 x 3 nm3 cell volume; • Analysis of the central nanostructure time evolution with fixed magnetization in the 6 n.n.; • 5th order Runge-Kutta scheme with 4th order error correction [10] and adaptive timestep; • At each temperature, 10 μs have been simulated and MS has been accordingly adjusted. E − EMin (J) From Right to Left (RLT + RLB) From Left to Right (LRT + LRB) 650 K 700 K MXMS  t μs  650 K no applied magnetic field 650 K 700 K Configuration (𝒊) Transition 𝝉𝟎𝒊  (ns) 𝒄𝒊 LRB 3.5 ± 0.6 0.61 ± 0.03 LRT 2 ± 1 0.63 ± 0.12 RLB 2.6 ± 0.7 0.62 ± 0.07 RLT - - LR 2.6 ± 0.8 0.63 ± 0.04 RL 1.6 ± 0.7 0.74 ± 0.08 B 2.0 ± 0.3 0.71 ± 0.03 T 2.2 ± 0.6 0.51 ± 0.04 - 3.0 ± 0.7 0.59 ± 0.03 𝟐. 𝟑𝟐𝟎 ± 𝟎. 𝟎𝟎𝟑 𝟎. 𝟔𝟑𝟐𝟐 ± 𝟎. 𝟎𝟎𝟎𝟐 𝜏𝑖 = 𝝉𝟎i exp𝒄𝑖∆𝐸𝐶𝑅𝑖𝑘𝐵𝑇, where ∆𝐸𝐶𝑅𝑖  is the coherent reversal energy barrier for configuration 𝑖 (0) (9) (16) (19) Conclusions • We showed that, at high temperature (T close to TB), the energy barrier can be about 63 % of the coherent reversal energy barrier and 𝜏0 lies in a range consistent with literature data. Anyway, a general analysis including room T is still missing (future outlook). • The whole work points out the necessity of considering both the actual nanostructure’s shape and the effect of temperature in order to calculate the proper energy barrier. x T1 T2 T3 T4 MFM Double-height square ASI [4] same degeneracy as Spin-ice compounds [5] Bigger ground-state domains with respect to the demagnetizing protocol Example of kMC input file Sting regime Domain regime Ground state String Domain Ground state 

A multiscale approach to Artificial Spin Ice thermal simulations

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A multiscale approach to Artificial Spin Ice thermal simulations

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