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Calculations of Spin-Spin Correlation Functions Out of Equilibrium for Classical Heisenberg Ferromagnets and Ferrimagnets Spinwaves Symposium, June 2013 T. Ostler, S. Wallace, J. Barker and R. W. Chantrell Dept. of Physics, The University of York, York, United Kingdom.

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Motivation: Ultrafast Demagnetization Currently a lot of interest in the physics behind femtosecond demagnetisation and magnetization process on the fs time- scale. Figure from Radu et al., Nature, 472, 205-208 (2011). Collapse of order seen in the magnetization depends on a number of features (fluence, material etc).

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Spin-Spin Correlation Graves et al., Nature Materials, 12, 293-298 (2013).

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Correlation Function We can study the correlations at different length scales by calculating the correlation function. By this definition the ordered state (T=0K) has the correlation function equal to 1 for all length scales. +ve Correlation z i -z j -ve Correlation z i -z j For the TM and RE sublattices we can calculate how correlations vary within each sublattice.

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Our Approach: Atomistic LLG We use a model based on the Landau-Lifshitz-Gilbert (LLG) equation for atomistic spins. Demagnetisation interpreted as thermal disorder due to thermal excitation. Temporal variations in temperature mean the strength of our stochastic term changes. For the ferrimagnetic calculations we create a super cell to give TM 3 RE 1 (allows use of FFT).

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Two-Temperature Model of Laser Heating [1] Chen et al. International Journal of Heat and Mass Transfer. 49, 307-316 (2006) Electrons e-e- e-e- e-e- Lattice e-e- G el Laser input P(t) We use theTwo-temperature[1] model which defines an electron and phonon temperature (T e and T l ) as a function of time. We couple the electron temperature to the spin system. The change in temperature gives changes in size of the random thermal field.

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Demagnetization Correlation function for ferromagnet reaches equilibrium very quickly, same rate as the magnetization. Correlation function decreases quite uniformly over the system. Similar in ferrimagnets except the rate of each sublattice is different due to different magnetic moments. TM RE

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Transient Ferromagnetic-like State At the start of the transient ferromagnetic- like state long range correlation dissapears. Localized regions of switching of TM against exchange field of RE. Build up of order in TM sublattice drives switching of RE. Collapse and re-emergence of order in TM much faster than RE. Atomistic level Correlated regions with different orientations More information found on arXiv:1207.4092

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Transient Ferromagnetic-like State For higher fluence case we do not see the large precession induced over the macrospin as the increased temperature means correlations are not built up as readily. But the correlation function suggests that it occurs on a small length-scale. Low Fluence High Fluence

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Remagnetization in a ferromagnet It has been demonstrated that when ferromagnets are completely demagnetized, recovery of magnetization is very long. Multi-domain states form on cooling. These domains must also re-order. [1] – Kazantseva et al. EPL 81, 27004 (2008).

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Remagnetization continued Initial results show that ferrimagnetic materials do not get stuck in this state. High frequency excitations associated with AFM interactions drives any competing domains out? Competition between domains means magnetization can take a long time to recover.

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Summary & Conclusions We have compared how correlations change in ferromagnetic and ferrimagnetic materials. Demagnetisation shows similar behaviour and the correlations decay in a time-scale that scales with time-scale of the magnetization. We have observed how the different sublattices in a ferrimagnet change during heat induced switching. These results could give us insight into the size limitations of a system undergoing thermally induced switching. Initial calculations show that remagnetisation in ferrimagnets is faster than ferromagnets due AFM exchange interaction. Requires further investigation into Outlook Further study into the limitations of system size and the key parameters. Analysis of remagnetisation rates in ferro- and ferri-magnets.

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Acknowledgements The Nuffield Foundation for funding studentships. European Community’s Seventh Framework Programme (FP7/2007-2013) Grant No. NNP3-SL-20120281043 (FEMTOSPIN). Thank you for listening.

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