Magnetization reversal --------------- II. Non-single-domain effects: Interactions, nanostructures and domain walls Olivier Fruchart Institut Néel (CNRS-UJF-INPG) Grenoble - France http://neel.cnrs.fr Institut N Institut Né éel, Grenoble el, Grenoble, France , France. . Institut Néel, Grenoble, France. http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/ http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/
NON-SINGLE DOMAIN EFFECTS – General table of content II. Non-single-domain effects 1. Dipolar energy 2. Coercivity in patterned elements 3. Manipulation of domain walls 4. Interfacial effects Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.2 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http:// http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/
NON-SINGLE DOMAIN EFFECTS – Dipolar energy I.1. Dipolar energy 1. Treatment of dipolar energy 2. Some consequences of dipolar energy on hysteresis loops 3. Dipolar energy and collective effcts in assemblies Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.3 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http:// http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/
NON-SINGLE DOMAIN EFFECTS – Origins of magnetic energy Echange energy Magnetocrystalline anisotropy energy = − E J S . S 1 2 M = ∇ θ Ech 1 , 2 = θ 2 A ( ) sin 2 E K ( ) mc Hext Zeeman energy (enthalpy) Dipolar energy 2 = − µ 1 1 E M . H = − µ S d d 0 E M . H 2 S Z 0 Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.4 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/ http://
NON-SINGLE DOMAIN EFFECTS – Notations Magnetization M m x x = = M M M m Magnetization vector M y s y M m Can vary in time and space. z z 2 + 2 + 2 = m m m 1 Modulus is constant x y z (hypothesis in micromagnetism) Mean-field approach possible: M s = M s ( T ) Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.5 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http:// http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/
NON-SINGLE DOMAIN EFFECTS – Treatment of dipolar energy (1/3) = − µ 1 Density of dipolar energy r M r H r E ( ) ( ). ( ) d 0 d 2 = = − curl H 0 By definition H M ( ) div ( ) div ( ) . As we have (analogy with electrostatics): d d − div [ m ( r ' )].( r ' r ) = − 3 H ( r ) M d r ' d s ∫∫∫ 3 space π − 4 r r ' ρ = ( r ) - M div[ m ( r ) ] is called the volume density of magnetic charges s To lift the divergence that may arise at sample boundaries a volume integration around the boundaries yields: − − div [ m ( r ' )].( r ' r ) [ m ( r ' ). n ( r ' )].( r ' r ) = − 3 + 2 H ( r ) M d r ' d r ' d s ∫∫∫ ∫∫ 3 3 space π − sample π − 4 r r ' 4 r r ' σ = ( r ) M m ( r ) . n ( r ) is called the surface density of magnetic charges, s where n(r) is the outgoing unit vector at boundaries Do not forget boundaries between samples with different M s Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.6 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http:// http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/
NON-SINGLE DOMAIN EFFECTS – Treatment of dipolar energy (2/3) Some ways to handle dipolar energy Integrated dipolar energy: Notice: six-fold integral over space: non-linear, long-range, time-consuming. = − µ 1 M.H . V d 0 ∫∫∫ d 2 E sample Bottle-neck of micromagnetic calculations Usefull theorem for finite samples: = − µ = µ 2 1 1 M.H H . d V . d V 0 ∫∫∫ d 0 ∫∫∫ d 2 2 E sample space � E is always positive � Significance of (BHmax) for permanent magnets − µ + = − µ 1 1 ( M H ) .H . d V B.H . d V 0 ∫∫∫ d d 0 ∫∫∫ d 2 2 sample sample = µ 2 1 H . d V 0 ∫∫∫ d 2 space \ sample Energy available outside the sample, ie usefull for devices Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.7 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http:// http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/
NON-SINGLE DOMAIN EFFECTS – Treatment of dipolar energy (3/3) Examples of magnetic charges Notice: no charges and E =0 for infinite - + - - + + cylinder - + + + + + + + + + + + + + + + + + + + + + + + Charges on surfaces - - - - - - - - - - - - - - - - - - - - + + + + Surface and volume charges + + + + + + + + - - - - x Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.8 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/ http://
NON-SINGLE DOMAIN EFFECTS – Demagnetizing coefficients (1/3) ( ) ≡ = + + = M ( r ) M M m x m y m z M m u Assume uniform magnetization s x y z s i i − [ m . n ( r ' )].( r ' r ) = 2 H ( r ) M d r ' d s ∫∫ 3 sample π − 4 r r ' − n ( r ' ).( r ' r ) = 2 i M m d r ' s i ∫∫ 3 sample π − 4 r r ' = − µ 3 1 H r M r ( ). . d d 0 ∫∫∫ d 2 E sample − n ( r ' ).[ m .( r ' r )] = − µ 2 3 2 1 i M m d r d r ' 0 s i ∫∫∫ ∫∫ 2 3 sample sample π − 4 r r ' − r n ( ' ).( r ' r ) i j j = − 3 2 K m m d r d r ' d i j ∫∫∫ ∫∫ 3 sample sample π − r r 4 ' = = K VN m m K V m . . N m d d ij i j d E See more detailed approach: M. Beleggia and M. De Graef, J. Magn. Magn. Mater. 263, L1-9 (2003) Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.9 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http:// http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/
NON-SINGLE DOMAIN EFFECTS – Demagnetizing coefficients (2/3) = = t K N m m K m . . N m d d ij i j d N is a positive second-order tensor E < >= − H ( r ) M N . m d s …and can be defined and diagonalized for any sample shape = 2 + 2 + 2 K ( N m N m N m ) d d x x y y z z N 0 0 E x = < >= − N 0 N 0 H ( r ) M N Valid along main axes only! y d, i s i 0 0 N + + = N N N 1 z x y z What with ellipsoids??? Self-consistency: the magnetization must be at equilibrium and therefore fulfill m // H eff Assuming H applied and H a are uniform, this requires H d ( r ) is uniform. This is satisfied only in volumes limited by polynomial surfaces of order 2 or less: slabs, cylinders, ellisoids (+paraboloïds and hyperboloïds). J. C. Maxwell, Clarendon 2, 66-73 (1872) Olivier Fruchart – Non-single-domain effects – European School on Magnetism – Cluj Sept 2007 – p.10 Institut N Né éel el, , Grenoble Grenoble, France , France Institut Institut Néel, Grenoble, France http://lab lab- -neel.grenoble.cnrs.fr neel.grenoble.cnrs.fr/ /themes themes/couches/ /couches/ext ext/ /slides slides/ / http://lab-neel.grenoble.cnrs.fr/themes/couches/ext/slides/ http://
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