Pat t erned magnet ic st ruct ures f rom f undament al micromagnet ism t o micron-scale applicat ions Olivier Fruchart - Laboratoire Louis Néel, Grenoble, France. Olivier Fruchart - Laboratoire Louis Néel, Grenoble, France. Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > Table of cont ent Micromagnetism (fundamental) The background Magnetostatics The fundamental issues of micromagnetism Coherent reversal Domains and walls Characteristic length scales Multidomains : theory ( ) and real life( ) Applications for ‘large’ microstructures Magnetic recording heads (general) Magnetic recording heads (alditech : tapes) Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.2 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.2 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > ref erences Micromagnetism = Continous media theory describing the magnetization distribution inside samples ! Classical theory ! Atomic structure of matter is ignored ! Analytical as well as numerical approach Magnet ic domains, A. Hubert and R. Schäf er, Springer Verlag, 1998. Practical although rigourous approach to micromagnetism. More imaging. An int roduct ion t o t he t heory of f erromagnet ism, A. Aharoni, Clarendon Press, 2001. A more mathematical approach. More historical (math.) concepts. Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.3 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.3 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > exchange Exchange energy Ferromagnetic order comes from quantum mechanics Pauli exclusion principle + Spins do not ignore each other Electrostatic forces Exchange energy = − e J S . S For ferromagnetic substances : parallel alignement is favored 1 2 ex 1 , 2 ! Magnetic moment, M(T), etc. ( ) ( ) ≈ ∇ = ∂ ∂ e A θ 2 A θ / x 2 for 1D situation ex Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.4 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.4 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > magnet ocryst alline anisot ropy Magnetocrystalline anisotropy energy Electronic cloud Atom nucleus (crystal structure) Spin-orbit coupling ! the energy of both spin and orbital moment depends on orientation Series development on an angular basis: Anisotropy energy Normalized magnetization components = + + e K m 2 K m 4 ... Uniaxial z z Alignement of magnetization mc 2 4 = + + + is favored along e K m 2 m 2 m 2 m 2 m 2 m 2 ( ) ... x y y z z x mc 4 given axes of the crystal Cubic … (Derived f rom slide of A. Thiaville – CNRS/ Or say) Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.5 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.5 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > Zeeman energy Zeeman energy External magnetic field (applied by magnets, earth, etc.) Analogy : a compass needle in the earth’s magnetic field Zeeman energy = − Alignement of magnetization e µ M . H S is favored parallel to the external field Z 0 Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.6 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.6 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > dipolar energy Dipolar energy Magnetic moments (spin or orbital) are assimilated to microscopic currents " they create long-range dipolar fields H " What is the effect of these fields ? The dipolar energy is the Zeeman energy of the sample in the dipolar field H d created by all its spins Mutual energy should be counted only once ! = − = − E µ µ µ . H µ . H 2 1 1,2 0 0 1 2 2 ( ) 1 = − + µ µ H µ H . . 2 1 0 1 2 2 1 Local dipolar energy 1 = − e µ M . H S d d 0 2 (per unit volume) Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.7 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.7 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > dipolar energy Cone of alignment Mutual energy of two magnetic dipoles : µ 3 = − E 0 µ . µ ( µ . r ).( µ . r ) 1,2 1 2 1 2 π r 3 r 2 4 Let us assume two magnetic dipoles 2 with vertical direction, either ‘up’ or ‘down’ : θ [ ] µ θ = µ µ − θ θ = 2 0 cos 2 E ( ) 1 3 cos ( ) 1 / 3 π 1 1,2 1 2 C 3 4 r < ≈ ° θ θ Parallel alignment is favored for 54 . 74 C > ≈ ° θ θ Antiparallel alignment is favored for 54 . 74 C 1/ r 3 decay: the dipolar interaction is long ranged Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.8 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.8 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > dipolar energy Cone of alignment How to use the ‘cone of alignment’ to predict the effect of dipolar fields ? Situation 1 : M perpendicular Most of the spins are in the antiparallel cone " not favorable Situation 2 : M parallel Most of the spins are in the parallel cone " favorable The favored magnetization direction is along the long axis of the sample Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.9 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.9 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > magnet ost at ics laws Electrostatics / Magnetostatics parallel Electrostatics ‘Electric charge’ Maxwell’s equations : ρ P d P ρ 3 ( ) u ∫∫∫ PM = = M div E E ( ) ε πε PM 2 4 Magnetostatics 0 0 = div B 0 P d P 3 div [ M ( )] u ∫∫∫ PM = − M = − H ( ) H M div div π PM 2 4 B = + H M µ ‘Magnetic charge’ 0 For a finite size sample : after integration over the entire space, a new term arises due to the magnetization discontinuity at the sample’s surface: ‘Volume charges’ ‘Surface charges’ ∂ ∂ ∂ M M P d P P M 3 M u M n div [ ( )] ( ). ∫∫∫ ∫∫ PM = + y + z = − + x with : M dS div M H ( ) ∂ ∂ ∂ d x y z π PM π PM 2 2 4 4 sample' s sample Local dipolar energy surface 1 The dipolar field coming from a sample = − e µ M . H S d can be calculated from these ‘magnetic charges’ d 0 2 Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.10 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.10 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > st ray- and demagnet izing f ields Example - + - + Let us assume a uniformly magnetized prism body : - M + - + - + Note: a free dipole aligns itself parallel to the Magnetic charges stray field H of the magnet Stray field = - + Field created - + - S N + outside the sample - + - + Long-range : dipole-like Demagnetizing field = - + - + N S - Field created + H - d + - inside the sample + (acting from the sample on itself) (I mages f rom A. Thiaville – CNRS/ Or say) Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.11 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.11 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > demagnet izing f ields Demagnetizing fields How to use the ‘surface charges’ model to predict the effect of dipolar fields ? Situation 1 : M perpendicular + + + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - - Many surface charges : high dipolar fields " not favorable Situation 2 : M parallel - + - - + + - + Few surface charges : low dipolar fields " favorable The favored magnetization direction is along the long axis of the sample « Shape anisotropy » Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.12 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.12 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
Micromagnet ism > shape ef f ect Hypothesis : Uniformly magnetized body with arbitrary shape See validity for real samples, later in the course It can be shown that : ( ) 1 = + + e µ N M 2 N M 2 N M 2 x x y y z z d 0 2 + + = ≥ M N N N N 1 , 0 With : x y z i (see analogy with This is the ‘Shape anisotropy energy’ magnetocrystalline…) Notes and consequences : 1 1 ∫∫∫ = = e max e d τ µ M 2 max . d d 0 S V 2 sample Even if M is assumed to be The ‘shape’ energy is uniaxial uniform in the system, H d is in general not uniform, except for special shapes. N i is higher along short sample directions ! see examples = + K K K Effective anisotropy energy: eff mc d Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.13 ] Olivier Fruchart - LLN-CNRS. [ 09/10/2001 / p.13 ] Slides on- line: http:/ / ln-w 3 .polycnrs-gre.fr/ them es/ couches/ ext/
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