Note: part of the lecture was made on the blackboard, and does not appear here Olivier Fruchart SPI SPINTEC, Uni niv. Gr Grenob oble le Al Alpes / CN / CNRS S / / CE CEA-INAC, Fr Fran ance www.spintec.fr email: olivier.fruchart@cea.fr Slides: http://fruchart.eu/slides Possible reading: Lecture notes in nanomagnetism: http://fruchart.eu/olivier/publications Lecture in Sevilla, Spain
M OTIVATION – M ATERIALS Magnetic domains Numerous and complex shape of domains History: Weiss domains Practical: improve material properties 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
M OTIVATION – S PINTRONIC DEVICES Magnetic bits on hard disk drives Underlying microstructure Co-based hard disk media : bits 50nm and below B. C. Stipe, Nature Photon. 4, 484 (2010) 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
T ABLE OF CONTENTS Motivation Dynamics Basics Statics 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
B ASICS – C ONTROL MAGNETIZATION REVERSAL The hysteresis loop S pontaneous ≠ S aturation Magnetization reversal under magnetic Spontaneous magnetization field The most widespread characterization Remanent magnetization Coercive field Losses 𝑋 = 𝜈 0 ර 𝐈. d𝐍 𝐂 = 𝜈 0 𝐈 + 𝐍 Magnetic induction 𝐊 = 𝜈 0 𝐍 Another notation 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
B ASICS – S OFT AND HARD MAGNETIC MATERIALS Soft magnetic material Hard magnetic material Transformers Magnetic recording Magnetic shielding, flux guides Permanent magnets Magnetic sensors What determines hysteresis loops? Material composition and crystal structure Microstructure 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
B ASICS – D OMAINS , FROM BULK TO NANO Mesoscopic scale Bulk material Nanoscopic scale Numerous and Small number of domains, Magnetic single domain complex shape of domains simple shape FeSi soft magnetic sheet Microfabricated elements Nanofabricated dots Kerr microscopy MFM Sample courtesy: I. Chioar, A. Hubert, Magnetic domains A. Hubert, Magnetic domains N.Rougemaille Nanomagnetism ≈ Mesomagnetism 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
T ABLE OF CONTENTS Motivation Dynamics Basics Statics 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – M ICROMAGNETISM FORMALISM Magnetization 𝑁 𝑦 𝑛 𝑦 Magnetization vector M 𝑛 𝑧 Continuous function 𝑁 𝑧 𝐍(𝐬) = = 𝑁 s 𝑛 𝑨 𝑁 𝑨 May vary over time and space 2 + 𝑛 𝑧 2 + 𝑛 𝑨 2 = 1 𝑛 𝑦 Modulus is constant and uniform (hypothesis in micromagnetism) 𝑁 s = 𝑁 s 𝑈 Mean field approach is possible: Exchange interaction (total energy, J) ℰ = − 𝐾 𝑗,𝑘 𝐓 𝑗 . 𝐓 𝑘 Atomistic view 𝑗≠𝑘 2 𝜄 𝑗,𝑘 𝐓 𝑗 . 𝐓 𝑘 = 𝑇 2 cos(𝜄 𝑗,𝑘 ) ≈ 𝑇 2 1 − Micromagnetic view 2 2 𝜖𝑛 𝑗 𝐹ex = 𝐵 𝛂. m 2 = 𝐵 𝜖𝑦 𝑘 𝑗,𝑘 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – T HE VARIOUS TYPES OF MAGNETIC ENERGY Exchange energy Magnetocrystalline anisotropy energy 2 𝜖𝑛 𝑗 𝐹ex = 𝐵 𝛂. m 2 = 𝐵 𝜖𝑦 𝑘 𝐹 mc = 𝐿 𝑔(𝜄, 𝜒) 𝑗,𝑘 Zeeman energy (→ enthalpy) Magnetostatic energy 𝐹 d = − 1 𝐹 Z = −𝜈 0 𝐍. 𝐈 2 𝜈 0 𝐍. 𝐈 d 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – D IPOLAR ENERGY Usefull expressions Analogy with electrostatics ℰ d = − 1 2 𝜈 0 ම 𝐍. 𝐈 d d𝑊 div 𝐈 d = −div 𝐍 Maxwell equation → Sample div 𝐧 𝐬 ′ . (𝐬 − 𝐬 ′ ) ℰ d = 1 𝐈 d 𝐬 = −𝑁 s ම d𝑊′ 𝟑 d𝑊 2 𝜈 0 ම 𝐈 d 4𝜌 𝐬 − 𝐬 ′ 3 space Space Always positive To lift the singularity that may arise at boundaries, a Zero means minimum volume integration around the boundaries yields: 𝐈 d 𝐬 = ම 𝜍 𝐬 ′ . 𝐬 − 𝐬 ′ 4𝜌 𝐬 − 𝐬 ′ 3 d𝑊 ′ + 𝜏 𝐬 ′ . 𝐬 − 𝐬 ′ 4𝜌 𝐬 − 𝐬 ′ 3 d𝑇 ′ Hd depends on shape, not Magnetic charges size 𝜍(r)= − 𝑁 S div 𝐧(𝐬) Synonym: 𝜏(r)=𝑁 S 𝐧 𝐬 . 𝐨(𝐬) dipolar, magnetostatic 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – M AGNETIC CHARGES Examples of magnetic charges Note for infinite cylinder: no charge ℰ = 0 Charges on side surfaces Surface and volume charges Take-away message Dipolar energy favors alignement of magnetization with longest direction of sample 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – D IPOLAR FIELDS Vocabulary Generic names Magnetostatic field Dipolar field Inside material Demagnetizing field Oustide material Stray field 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – T ENDENCY FOR FLUX - CLOSURE DOMAINS Films with easy axis out-of-the-plane: Kittel domains Principle: compromise between gain in dipolar energy, and cost in wall energy C. Kittel, Physical theory of ferromagnetic domains, Rev. Mod. Phys. 21, 541 (1949) Nanostructures with in-plane magnetization Van den Berg theorem Principle: Reduce dipolar energy to zero H. A. M. van den Berg, J. Magn. Magn. Mater. 44, 207 (1984) 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – M AGNETIC LENGTH SCALES The dipolar exchange length The anisotropy exchange length When: anisotropy and exchange compete When: anisotropy and exchange compete 2 2 𝐹 = 𝐵 𝜖𝑛 𝑗 𝐹 = 𝐵 𝜖𝑛 𝑗 + 𝐿 d sin 2 𝜄 + 𝐿 sin 2 𝜄 𝜖𝑦 𝑘 𝜖𝑦 𝑘 Dipolar Anisotropy Exchange Exchange 𝐿 d = 1 2 2 𝜈 0 𝑁 s J/m 3 J/m 3 J/m J/m Δ u ≃ 1 nm → 100 nm Δu = 𝐵/𝐿 2 Δd = 𝐵/𝐿 d = 2𝐵/𝜈 0 𝑁 s Hard Soft Δ d ≃ 3 − 10 nm Critical single-domain size, relevant for small particles made of soft magnetic materials Sometimes called: Bloch Often called: exchange length parameter, or wall width Note: Other length scales can be defined, e.g. with magnetic field 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – D OMAIN WALLS AND DIMENSIONALITY Bloch wall in the bulk (2D) Domain walls in thin films (towards 1D) Bloch wall 𝑢 ≳ 𝑥 No magnetostatic energy Néel wall 𝑢 ≲ 𝑥 Δu = 𝐵/𝐿 Width Implies magnetostatic energy 𝛿w = 4 𝐵𝐿 Energy No exact analytic solution Other angles & L. Néel, C. R. Acad. Sciences 241, 533 (1956) anisotropy Vortex (1D → 0D) Bloch point (0D) F. Bloch, Z. Phys. 74, 295 (1932) Point with vanishing Constrained walls (eg in strips) magnetization Permalloy (15nm) Strip width 500nm W. Döring, T. Shinjo et al., JAP 39, 1006 (1968) Science 289, 930 (2000) 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATICS – W ALLS AND TOPOLOGY (B LOCH POINT ) What is a Bloch point? A magnetization texture with local cancellation of the magnetization vector R. Feldkeller, Z. Angew. Physik 19, 530 (1965) W. Döring, J. Appl. Phys. 39, 1006 (1968) 2 Bloch-point wall, theory 𝐸 ≳ 7𝛦 d Bloch-point wall, experiments Experiment Simulation WIRE SHADOW Shadow XMCD-PEEM H. Forster et al., J. Appl. Phys. 91, 6914 (2002) A. Thiaville, Y Nakatani, Spin dynamics in confined S. Da-Col et al., PRB (R) 89, 180405, (2014) magnetic structures III, 101, 161-206 (2006) 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
S TATIC – W ALLS AND TOPOLOGY ( SKYRMIONS ) The Dzyaloshinskii-Moriya interaction Magnetic skyrmions Usual magnetic exchange ℰ 𝑗,𝑘 = −𝐾 𝑗,𝑘 𝐓 𝑗 . 𝐓 𝑘 Promotes ferromagnetism (or antiferromagnetism) Bulk FeCoSi 90 nm Lorentz microscopy The DM interaction Claims and facts ℰ DMI = −𝐞 𝑗,𝑘 . 𝐓 𝑗 × 𝐓 𝑘 Requires: loss of inversion symmetry Promotes spirals and cycloids I. Dzyaloshiinsky, J. of Phys. Chem. Solids 4, 241 (1958) O. Boulle et al., T. Moriya, Phys. Rev. 120, 91 (1960) Nat. Nanotech., A.Fert and P.M.Levy, PRL 44, 1538 (1980) 11, 449 (2016) 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
T ABLE OF CONTENTS Motivation Dynamics Basics Statics 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
D YNAMICS – T HE L ANDAU -L IFSHITZ EQUATION LLG equation Pioneering experiment of precessional magnetization reversal Describes: precessional dynamics of magnetic moments Applies to magnetization, with phenomenological damping d𝐧 d𝑢 = 𝛿 0 𝐧 × 𝐈 + 𝛽𝐧 × d𝐧 d𝑢 𝛿 0 = 𝜈 0 𝛿 < 𝟏 Gyromagnetic ratio 𝛿 𝑡 = 28 GHz/T 𝛽 > 0 Damping coefficient 𝛽 = 0.1 − 0.001 C. Back et al., Science 285, 864 (1999) 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
D YNAMICS – M OTION OF DOMAIN WALLS Precessional dynamics under field d𝐧 d𝑢 = 𝛿 0 𝐧 × 𝐈 + 𝛽𝐧 × d𝐧 d𝑢 12 th June 2017 Olivier FRUCHART Magnetization textures Lecture Sevilla, Spain
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