Dénes Lajos Nagy, Márton Major, Dávid Visontai KFKI Research Institute for Particle and Nuclear Physics, Budapest Nano-Scale Materials: Growth – Dynamics – Magnetism Grenoble, 6-8 February Nano-Scale Materials: Growth – Dynamics – Magnetism Grenoble, 6-8 February Dynamics of magnetic domains (the pixel model)

Outline  Experimental facts  Native patch-domain formation in antiferromagnetically coupled multilayers  Spontaneous and irreversible growth of the domain size during demagnetisation: the domain ripening  Spin-flop-induced domain coarsening  Supersaturation domain memory effect  Temperature-induced ripening  Pixel model of domains and domain walls; Monte Carlo simulation of domain dynamics

M. Rührig et al., Phys. Stat. Sol. (a) 125, 635 (1991). M. Rührig, Theses, Ripple domains Patch domains Domains in an Fe/Cr/Fe trilayer

Patch domains in AF-coupled multilayers Layer magnetisations: The ‘magnetic field lines’ are shortcut by the AF structure  the stray field is reduced  no ‘ripple’ but ‘patch’ domains are formed.

 /2  -scan: Q z -scan d = 2  /Q z  -scan: Q x -scan  = 1/  Q x Arrangement of an SMR experiment 22  or  H ext x y z k APD from the high-resolution monochromator E

Field dependence of the magnetisation M and of the intensity I AF of the SMR AF reflection (easy direction) H S = 0.85 T

From saturation to remanence: the domain ripening  In decreasing field the domain-wall angle and, therefore, the domain-wall energy as well as its surface density is increasing.  In order to decrease the surface density of the domain-wall energy, the multilayer spontaneously increases the average size of the patch domains (‘ripening’).  The spontaneous domain growth is limited by domain-wall pinning (coercivity).

450 mT300 mT 150 mT9 mT300 mT 600 mT9 mT 600 mT MgO(001)[ 57 Fe(26Å)/Cr(13Å)] 20 +  - scattering JINR Dubna, REMUR QzQz QxQx Domain ripening: off-specular PNR, easy axis

ESRF ID18 Correlation length:  = 1/  Q x   370 nm   800 nm Domain ripening: off-specular SMR MgO(001)[ 57 Fe(26Å)/Cr(13Å)] 20 2 AF reflection, hard axis

From saturation to remanence: the native state  The native domains do not change their shape and size (370 nm) from saturation down to 200 mT.  Between 200 and 100 mT the domain size increases to 800 nm.  The growth stops below 100 mT.  Domain growth is an irreversible process; the domain size does not change up to saturation.

Spin-flop induced domain coarsening (PNR) 7 mT 14.2 mT 35 mT MgO(001)[ 57 Fe(26Å)/Cr(13Å)] 20, easy axis JINR Dubna SPN-1 non-spin-flip scattering S n || M spin-flip scattering S n  M Q x (10 -4 Å -1 ) Q z (Å -1 )

Spin-flop-induced domain coarsening (SMR) MgO(001)[ 57 Fe(26Å)/Cr(13Å)] 20 2 AF reflection, easy axis delayed in remanence after 13 mT prompt delayed in remanence after 4.07 T reflected intensity (% of max.) delayed in remanence after 35 mT QxQx (10 -4 Å )  90  rot. ESRF ID18 Correlation length:  = 1/  Q x Delayed photons before the spin flop  = 800 nm Delayed photons after the spin flop   > 5  m   = 800 nm

Domain coarsening on spin flop  Coarsening on spin flop is an explosion-like 90-deg flop of the magnetization annihilating primary 180-deg walls. It is limited neither by an energy barrier nor by coercivity. Consequently, the correlation length of the coarsened patch domains  may become comparable with the sample size.

Domain ripening: off-specular SMR, hard direction: the ‘supersaturation memory effect’ ESRF ID18

Field dependence of the magnetisation M and of the intensity I AF of the SMR AF reflection HSHS H SS normalised value

JINR Dubna, REMUR Polarised neutron reflectometry on Sample 2

 The lateral distribution of the saturation field in Sample 2 is much narrower than that in Sample 1.  The supersaturation effect is probably due to the still not saturated fraction of Fe. When the field is released from a value H S < H < H SS, the old domain pattern re- nucleates on the residual seeds of very strongly coupled regions. Supersaturation: the explanation J. Meersschaut et al., Phys. Rev. B 73, (2006).M. Major PhD thesis (2006).

Supersaturation memory effect and the lack of ripening at T = 15 K Supersaturation memory effect and the lack of ripening at T = 15 K Initial state: coarsened domains H S = 1.55 T H SS = 3.60 T ESRF ID18

The lack of low-temperature ripening is due to the temperature dependence of the Fe coercive field H c V(110)/[Fe(1.2 nm)/Cr(26 nm) 15 /Fe(1.2 nm)/Cr(10 nm) J. Hauschild et al., Appl. Phys. A 74, S1541 (2002)

Domain ripening with increasing temperature ESRF ID18

Rationale of the pixel model  A full micromagnetic simulation would include about spins  a simplified model is needed.  The bilinear layer-layer coupling  the saturation field H s has a lateral distribution obeying, e.g., a Gaussian statistics.

Rationale of the pixel model  Pixels (small homogeneous regions) are defined on a (rectangular) lattice.  Domain-wall width << pixel << domain size.  Two-sublattice model of the multilayer characterised by one opening angle 2 is used.

The domain-wall angle Domain-wall angle = 2 ( H ) = 2 arccos H/H s (r)

Rationale of the pixel model  The domain-wall energy is proportional to the square of the domain-wall angle: E wall = 4 D 2 In remanence In external field H

Rationale of the pixel model  The total domain-wall energy is the sum of the wall energies with the next 8 neighbours.  The hysteresis loss of a pixel associated with changing the sense of rotation (‘red’ or ‘green’) stems from the field-perpendicular component of the layer magnetisation.

The Monte Carlo ‘movies’  Random values of H s (r) are generated on a rectangular lattice.  H and/or H c (T) are varied step-by-step.  Subsequent pictures of the calculation always differ from each other only by the sense of rotation of a single pixel, the saturation state ( = 0) being considered to have a third, ’neutral’ (yellow) sense of rotation.

The Monte Carlo ‘movies’ ?4 765  On gradually changing H or H c, a pixel will change its sense of rotation if the new state, taken into account the domain-wall energy and the hysteresis loss, will be energetically more favourable.  A Gaussian distribution of the saturation field of expectation value = 0.8 and standard deviation  = 0.13 were used.  The simulation depends from D and H c M only through their ratio D/H c M.

Formation, ripening, supersaturation

Formation, ripening, saturation

Temperature-induced ripening

Conclusions With suitable magnetic field program, it is possible to shape the domain structure of AF-coupled multilayers.  On leaving the saturation region sub-  m native patch- domains are formed in decreasing field.  On further decreasing the field, the domain size spontaneously and irreversibly increases and the domain shape changes (ripening).  The bulk spin flop leads to an explosion-like increase of the domain size (coarsening).  In some samples, the domain structure is erased only in a field significantly higher than saturation, a probable consequence of the existence of very strongly coupled regions.

 Due to the increased coercive field, at low temperature no ripening takes place.  The native domains retained at low temperature in remanence ripen when increasing the temperature (and so decreasing the coercive field). Conclusions

 The Monte Carlo simulation based on the rough and phenomenological pixel model describes with surprisingly high accuracy the o formation of patch domains (without introducing an artificial smoothing to H s (r)), o domain ripening during demagnetisation from saturation, o apparent supersaturation domain memory effect, o domain ripening at remanence with increasing temperature. Conclusions

Acknowledgements to: ESRF Grenoble ILL Grenoble JINR Dubna KFKI RMKI Budapest KU Leuven University Mainz D. Aernout Yu. Nikitenko L. Bottyán O. Nikonov A. Chumakov A. Petrenko B. Croonenborghs V. Proglyado L. Deák R. Rüffer B. Degroote H. Spiering J. Dekoster C. Strohm T.H. Deschaux-Beaume J. Swerts H.J. Lauter E. Szilágyi V. Lauter-Pasyuk F. Tanczikó O. Leupold K. Temst J. Meersschaut V. Vanhoof D.G. Merkel A. Vantomme