Simple concepts of magnetization reversal --------------- II. - PowerPoint PPT Presentation

Simple concepts of 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 Néel, Grenoble, France , France. . Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/

Was your head whirling? Institut Néel, Grenoble, France , France. . Institut Néel, Grenoble Recas, Sep7th 2009 http://perso.neel.cnrs.fr/olivier.fruchart/slides/

Chamois, nearby Grenoble Institut Néel, Grenoble, France , France. . Institut Néel, Grenoble June 1st, 2008 http://perso.neel.cnrs.fr/olivier.fruchart/slides/

NON-SINGLE DOMAIN EFFECTS – General table of content 1. Dipolar energy 2. Coercivity in patterned elements 3. Manipulation of domain walls 4. Interfacial effects Olivier Fruchart – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.4 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/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 – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.5 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/

NON-SINGLE DOMAIN EFFECTS – Origins of magnetic energy Echange energy Magnetocrystalline anisotropy energy   E J S . S 1 2 Ech 1 , 2 M    2 A ( )   sin 2 E K ( ) mc Hext Zeeman energy (enthalpy) Dipolar energy 2 1 1    E M . H S d    d 0 E M . H 2 S Z 0 Olivier Fruchart – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.6 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/

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    Modulus is constant m m m 1 x y z (hypothesis in micromagnetism) Mean-field approach possible: M s = M s ( T ) Olivier Fruchart – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.7 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/

NON-SINGLE DOMAIN EFFECTS – Treatment of dipolar energy (1/3) Density of dipolar energy   1  E ( r ) M ( r ). H ( r ) d 0 d 2  By definition   . As curl ( H ) 0 we have (analogy with electrostatics): div ( H ) div ( M ) d d  div [ m ( r ' )].( r ' r )  3   H ( r ) M d r ' d s 3 space   4 r r '   ( 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 sample     4 r r ' 4 r r '     is called the surface density of magnetic charges, ( r ) M m ( r ) . n ( r ) s where n(r) is the outgoing unit vector at boundaries Do not forget boundaries between samples with different M s Olivier Fruchart – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.8 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/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.    1  M.H . V d E 0 d 2 sample Bottle-neck of micromagnetic calculations Usefull theorem for finite samples:   2   1   1  M.H . d V H . d V E 0 d 0 d 2 2 sample space  E is always positive  Significance of (BHmax) for permanent magnets    1     1  ( M H ) .H . d V B.H . d V Cf: M. Coey 0 d d 0 d 2 2 sample sample  2  1  H . d V 0 d 2 space \ sample Energy available outside the sample, ie usefull for devices Olivier Fruchart – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.9 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/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 – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.10 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/

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   4 r r '  n ( r ' ).( r ' r )  2 i  M m d r ' s i 3 sample   4 r r '  3   1  H ( r ). M . d r E d 0 d 2 sample  n ( r ' ).[ m .( r ' r )] 2  3  2 1 i    M m d r d r ' 0 s i 2 3 sample sample   4 r r '  n ( r ' ).( r ' r ) i j j  3  2   K m m d r d r ' d i j 3 sample sample   4 r r '   K VN m m K V m . . N m E d d ij i j d See more detailed approach: M. Beleggia and M. De Graef, J. Magn. Magn. Mater. 263, L1-9 (2003) Olivier Fruchart – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.11 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/

NON-SINGLE DOMAIN EFFECTS – Demagnetizing coefficients (2/3) t   K N m m K m . . N m N is a positive second-order tensor E d d ij i j d    H ( r ) M N . m …and can be defined and diagonalized d s for any sample shape 2 2 2    K ( N m N m N m ) E   d d x x y y z z N 0 0 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 – Simple concepts of magnetization reversal – European School on Magnetism – Timisoara Sept 2009 – p.12 Institut Néel, Grenoble, France , France Institut Néel, Grenoble http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/