Fundamentals of Magnetism – 1 J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. Introduction; basic quantities 2. Magnetic Phenomena. SUMMER SCHOOL 3. Units Comments and corrections please: [email protected] www.tcd.ie/Physics/Magnetism ! Lecture 1 covers basic concepts in magnetism; Firstly magnetic moment, magnetization and the two magnetic fields are presented. Internal and external fields are distinguished. Magnetic energy and forces are discussed. Magnetic phenomena exhibited by functional magnetic materials are briefly presented, and ferromagnetic, ferrimagnetic and antiferromagnetic order introduced. SI units are explained, and dimensions are provided for magnetic, electrical and other physical properties. An elementary knowledge of vector calculus and electromagnetism is assumed. Books Some useful books include: • J. M. D. Coey; Magnetism and Magnetic Magnetic Materials. Cambridge University Press (2010) 614 pp An up to date, comprehensive general text on magnetism. Indispensable! • S. Blundell Magnetism in Condensed Matter, Oxford 2001 A good, readable treatment of the basics. • D. C. Jilles An Introduction to Magnetism and Magnetic Magnetic Materials, Magnetic Sensors and Magnetometers, 3rd edition CRC Press, 2014 480 pp Q & A format. • R. C. O’Handley. Modern Magnetic Magnetic Materials, Wiley, 2000, 740 pp Q & A format. •J. Stohr and H. C. Siegman Magnetism: From fundamentals to nanoscale dynamics Springer 2006, 820 pp. Good for spin transport and magnetization dynamics. Unconventional definition of M • K. M. Krishnan Fundamentals and Applications of Magnetic Material, Oxford, 2017, 816 pp Recent general text.. Good for imaging, nanoparticles and medical applications. IEEE Santander 2017 1 Introduction 2 Magnetostatics 3 Magnetism of the electron 4 The many-electron atom 5 Ferromagnetism 6 Antiferromagnetism and other magnetic order 7 Micromagnetism 8 Nanoscale magnetism 9 Magnetic resonance 10 Experimental methods 11 Magnetic materials 12 Soft magnets 13 Hard magnets 14 Spin electronics and magnetic recording 614 pages. Published March 2010 15 Other topics Available from Amazon.co.uk ~€50 Appendices, conversion tables. www.cambridge.org/9780521816144 IEEE Santander 2017 1. Introduction; Basic quantities SUMMER SCHOOL IEEE Santander 2017 ! Magnets and magnetization 0.03 A m2 1.1 A m2 17.2 A m2 3 mm 10 mm 25 mm m is the magnetic (dipole) moment of the magnet. It is proportional to volume m = MV Suppose they are made of Nd2Fe14B magnetic moment volume -1 magnetization (M ≈ 1.1 MA m ) What are the moments? Magnetization is the intrinsic property of the material; Magnetic moment is a property of a particular magnet. IEEE Santander 2017 Magnitudes of M for some ferromagnets M (MAm-1) Permanent Nd2Fe14B 1.28 magnets BaFe12O19 0.38 Temporary Fe 1.71 magnets Co 1.44 Ni 0.49 NB: These are the values for the pure phase IEEE Santander 2017 Magnetic moment - a vector Each magnet creates a field around it. This acts on any material in the vicinity but strongly with another magnet. Nd2Fe14B The magnets attract or repel depending on their mutual orientation ↑ ↑ Weak repulsion ↑ ↓ Weak attraction ← ← Strong attraction ← → Strong repulsion tetragonal easy axis IEEE Santander 2017 Field due to a magnetic moment m m m Hr Hθ Hϕ Michael Faraday er The Earth’s magnetic field is roughly that of a geocentric dipole e r m ϕ tan I = Br /Bθ = 2cotθ dr/rdθ = 2cotθ m Solutions are r = c sin2θ Equivalent forms m H Note. The dipole field is scale- independent. m ~ d3; H ~ 1/r3 1 Hence H = f(d/r). H Result: 60 years of hard disc recording ! IEEE Santander 2017 Magnetic recording perpendicular A billion-fold TMR increase in recording 22 density 11µ µmm GMR Magnetic recording is the partner of semiconductor technologyAMR in the information revolution. It provides the permanent, nonvolatile storage of information for computersyear andcaapcity the internet. platters size rpm ~ 10 exobit (1021bits) of data is stored 1955 40 Mb 50x2 24 1200 2005 160 Gb 1 2.5 18000 AMR GMR TMR IEEE Santander 2017 14 Porquerolles September 2012 How can you tell which way a magnet is magnetized? Use the green paper! It responds (turns dark) to the perpendicular component of H 1 3-6 2 This tells us the magnetic axis, but not the direction of vector m We can decide the relative orientation of two magnets from the forces, but the direction of the arrow is a matter of convention. IEEE Santander 2017 Equivalence of electricity and magnetism; Units What do the units mean? Michael Faraday m m – A m2 M – A m-1 Ampère,1821. A current loop or André-Marie Ampère 1 A m2 coil is equivalent to a magnet ∈ m m = IA 10,000 turns C 1A area of the loop ∈ Permanent magnets ! win over electro- magnets at small m = nIA 10,000 A sizes number of turns Right-hand corkscrew IEEE Santander 2017 Magnetic field H – Oersted’s discovery H I r δℓ The relation between electric current and magnetic field was discovered by Hans-Christian Øersted, 1820. H ∫ Hdℓ = I Ampères law H = I/2πr I -1 Units of H; A m If I = 1 A, r = 1 mm Right-hand corkscrew H = 159 A m-1 C Earths field ≈ 40 Am-1 IEEE Santander 2017 Field due to electric currents We need a differential form of Ampère’s Law; The Biot-Savart Law 1 H δℓ 1 I δBH H B r j Hj Right-hand corkscrewC (for the vector product) IEEE Santander 2017 42 Magnetostatics Figure 2.12 B A B(H ′)hysteresisloop. H' electrically polarized one. This field is the electrical displacement D,relatedto the electrical polarization P (electrical dipole moment per cubic metre) by D ϵ0 E P. (2.44) = + 2 The constant ϵ0, the permittivity of free space, is 1/µ c 8.854 0 = × 10 12 CV 1 m 1.Engineersfrequentlybrandthequantity J µ M as the − − − = 0 magnetic polarization,whereJ ,likeB,ismeasuredinteslasothattherelation between B, H and M can be writted in a superficially similar form: B µ H J. (2.45) = 0 + This is slightly misleading because the positions of the fundamental and auxil- iary fields are reversed in the two equations. The form (2.27) H B/µ M = 0 − is preferable insofar as it emphasizes the relation of the auxiliary magnetic field to the fundamental magnetic field and the magnetization of the material. Maxwell’sMagnetostatics equations in a material medium are expressed in terms of all four fields: 43 2.4 Magnetic field calculations Maxwell’s Equations D ρ, (2.46) In a medium; ∇ · = B 0, J. C. Maxwell(2.47) InB magnetostatics µ0H there is no time dependence of B, D or ρ. We only ∇ · = Electromagnetism have magnetic material and circulating conduction currents in a steady state. E ∂ B/∂t, with no time- (2.48) Conservation of electric charge∇× is expressed= − by the equation, H j ∂ D/∂t. dependence (2.49) ∇×j = ∂ρ+/∂t, (2.50) Here ρ is the local electric charge∇ · density= − and ∂ D/∂t is the displacement current. andIn in magnetostatics, a steady state ∂ρ we/∂ havet 0only. The magnetic world of material magnetostatics, and circulating which currents we explore Aconsequenceofwritingthebasicequationsofelectromagnetisminthisneat= furtherin conductors, in Chapter all 7, in is a therefore steady state. described The fields by three are produced simple equations: by the magnets and& the easy-to-remember currents form is that the constants µ0, ϵ0 and 4π are invisible. But they inevitably cropj up0 elsewhere,B for0 exampleH in thej. Biot–Savart law (2.5). The fields naturally∇ · form= two∇ pairs:· =B and∇×E are= a pair, which appear in the LorentzIn order expression to solve problems for the in force solid-state on a charged physics particle we need (2.19), to know and theH responseand D are ofthe the other solid pair, to the which fields. are The related response toIEEE the Santander is represented field 2017 sources by – the freeconstitutive current density relationsj and free charge densityMρ,respectively.M(H), P P(E), j j(E), = = = portrayed by the magnetic and electric hysteresis loops and the current– voltage characteristic. The solutions are simplified in LIH media, where M χ H, P ϵ0χ E and j σ E (Ohm’slaw), χ being the electrical sus- = = e = e ceptibility and σ the electrical conductivity. In terms of the fields appearing in Maxwell’s equations, the linear constitutive relations are B µH, D ϵ E, = = j σ E,whereµ µ (1 χ)andϵ ϵ0(1 χ ). Of course, the LIH = = 0 + = + e approximation is more-or-less irrelevant for ferromagnetic media, where M M(H)andB B(H)arethehysteresisloopsintheinternalfield,which = = are related by (2.33). 2.4 Magnetic field calculations In magnetostatics, the only sources of magnetic field are current-carrying con- ductors and magnetized material. The field from the currents at a point r in space is generally calculated from the Biot–Savart law (2.5). In a few situations with high symmetry such as the long straight conductor or the long solenoid, it is convenient to use Ampere’s` law (2.17) directly. To calculate the field arising from a piece of magnetized material we are spoilt for choice. Alternative approaches are: (i) calculate the dipole field directly by integrating over the volume distribu- tion of magnetization M(r); (ii) use the Amperian` approach and replace the magnetization by an equivalent distribution of current density j m; (iii) use the Coulombian approach and replace the magnetization by an equiv- alent distribution of magnetic charge qm. The three approaches are illustrated Fig. 2.13 for a cylinder uniformly mag- netized along its axis. They yield identical results for the field in free space outside the magnetized material but not within it. The Amperian approach gives B and H fields in free space; permeability of free space µ0 When illustrating Ampère’s Law we labelled the magnetic field created by the current, measured in Am-1 as H.