Fundamentals of – 1

J. M. D. Coey School of 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 , 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 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 3 Magnetism of the electron 4 The many-electron atom 5 6 Antiferromagnetism and other magnetic order 7 Micromagnetism 8 Nanoscale magnetism 9 Magnetic resonance 10 Experimental methods 11 Magnetic materials 12 Soft 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 . 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

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 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 = I A

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 H – ’s discovery

H I

r δℓ

The relation between 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 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 ∇× 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 . 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 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– 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 , it is convenient to use ’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. This is the ‘magnetic field strength’ Maxwell’s equations have another field, the ‘magnetic flux density’, labelled B, in the equation ∇. B = 0. It is a different quantity with different dimensions and units. Whenever H interacts with matter, to generate a force, an energy or an emf, the constant µ0, the ‘permeability of free space’ is involved. In free space, the relation between B and H is simple. They are numerically proportional to each other

B = µ0H In practice you can never mix them up. Permeability The differ by B-field H-field of free space almost a million! Units Tesla, T Units A m-1 Units TmA-1 (795775 800000)

µ depends on the definition of the Amp. Nicola Tesla 0 -7 -1 It is precisely 4π 10 T m A

IEEE Santander 2017 Magnetic Moment and Magnetization

The atomic magnetic moment ma is the elementary quantity in atomic-scale magnetism. (L. 2) Define a local moment density - magnetization – M(r, t) which fluctuates wildly on an atomic, sub-nanometer scale and on a sub-nanosecond time scale. Define a mesoscopic average magnetization

M = m/ V δm = MδV δ δ

The continuous medium approximation

M could be the spontaneous magnetization Ms within a ferromagnetic domain or fine particle. Most materials are not spontaneously ordered and M = 0. The atomic moments ma then fluctuate rapidly and the time average of any one of them, or the spatial average of an ensemble of atoms is zero. M is then the response of the molecular material to an external magnetic field H. At very low temperatures the spontaneous fluctuations freeze out. Initially the response is linear; M = χH. The dimensionless constant χ is the susceptibility.

IEEE Santander 2017

Magnetization curves - Hysteresis loop

M spontaneous magnetization

remanence

coercivity virgin curve initial susceptibility H

major loop

The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to a magnetic field . It reflects the arrangement of the magnetization in ferromagnetic domains. A broad loop like this is typical of a hard or permanent magnet. The remanent state is normally metastable. IEEE Santander 2017 Magnetization and current density

The magnetization of a solid is somehow related to a ‘magnetization current density’ Jm that produces it.

Since the magnetization is created by bound currents, ∫s Jm.dA = 0 over any surface.

Using Stokes theorem ∫ M.dℓ = ∫s(∇ × M).dA and choosing a path of integration outside the magnetized body, we obtain ∫s M.dA = 0, so we can identify Jm

Jm = ∇ × M

We don’t know the details of the magnetization currents, but we can measure the mesoscopic average magnetization and the spontaneous magnetization of a sample.

IEEE Santander 2017 density - B

Now we discuss the fundamental field in magnetism. Magnetic poles, analogous to electric charges, do not exist. This truth is expressed in Maxwells equation ∇.B = 0. This means that the lines of the B-field always form complete loops; they never start or finish on magnetic charges, the way the electric E-field lines can start and finish on +ve and -ve electric charges.

The same can be written in integral form over any closed surface S

∫SB.dA = 0 (Gausss law).

The flux of B across a surface is Φ = ∫B.dA. Units are Webers (Wb). The net flux across any closed surface is zero. B is known as the flux density; units are Teslas. (T = Wb m-2) 15 Flux quantum Φ0 = 2.07 10 Wb (Tiny)

IEEE Santander 2017 The B-field

Sources of B • electric currents in conductors • moving charges • magnetic moments • time-varying electric fields. (Not in magnetostatics) B = µ0I/2πr In a steady state: Maxwell’s equation

ex ey ez (Stokes theorem; I ∂/∂x ∂/∂y ∂/∂z )

Bx BY BZ Field at center of current loop

IEEE Santander 2017 Forces between conductors; Definition of the Amp

F = q(E + v x B) Lorentz expression. Note the does no work on the charge Gives dimensions of B and E. [T is equivalent to Vsm-2] If E = 0 the force on a straight wire arrying a current I in a uniform field B is F = BIℓ The force between two parallel wires each carrying one ampere is precisely 2 10-7 N m-1. (Definition of the Amp) The field at a distance 1 m from a wire carrying a current of 1 A is 0.2 µΤ

B = µ0I1/2πr

Force per meter = µ0I1I2/2πr

IEEE Santander 2017 The range of magnitude of B in Tesla (for H in Am-1 multiply by 800.000)

• The tesla is a rather largeLargest unit continuous laboratory field 45 T • Largest continuous laboratory(Tallahassee) field ever achieved is 45 T

Surface Magnet Space the Compression at Magnet Space Flux Brain Heart Field Star MagnetMagnet

Human Human InterstellarInterplanetary Earth's Solenoid PermanentSuperconductingHybrid Pulse Explosive Neutron Magnetar

1E-15 1E-12 1E-9 1E-6 1E-3 1 1000 1E6 1E9 1E12 1E15 pT µT T MT TT

IEEE Santander 2017 Typical values of B

Electromagnet 1 T

Permanent magnets 0.5 T Human brain 1 fT Earth 50 µT

Electromagnet 1 T Superconducting 12 Helmholtz coils 0.01 T Magnetar 10 T magnet 10 T

IEEE Santander 2017 Sources of uniform magnetic fields in free space

Helmholtz coils Long solenoid

B =µ0nI

3/2 B = (4/5) µ0NI/a Halbach cylinder

r2

r1 B = µ0Mremln(r2/r1)

IEEE Santander 2017 Why the H-field ?

Ampère’s law for the field is free space is ∇ x B = µ0( jc + jm) but jm cannot be measured !

Only the conduction current jc is accessible.

We showed that Jm = ∇ × M

Hence ∇ × (B/µ0 – M) = µ0 jc

We can retain Ampère’s law is a usable form provided we define H = B/µ0 – M

Then ∇ x H = µ0 jc

And B = µ0(H + M)

IEEE Santander 2017 The H-field.

The H-field plays a critical role in condensed matter. The state of a solid reflects the local value of H. Hysteresis loops are plotted as M(H)

Unlike B, H is not solenoidal. It has sources and sinks in a magnetic material wherever the magnetization is nonuniform. ∇.H = - ∇.M The sources of H (magnetic charge, qmag) are distributed

— in the bulk with -∇.M — at the surface with surface charge density M.en

Coulomb approach to calculate H (in the absence of currents)

Imagine H due to a distribution of magnetic charges qm (units Am) so that 3 H = qmr/4πr [just like ]

IEEE Santander 2017 Potentials for B and H

It is convenient to derive a field from a potential, by taking a spatial derivative. For example E

= -∇ϕe(r) where ϕe(r) is the . Any constant ϕ0 can be added.

For B, we know from Maxwells equations that ∇.B = 0. There is a vector identity ∇. ∇× X ≡ 0. Hence, we can derive B(r) from a vector potential A(r) (units Tm),

B(r) = ∇ × A(r)

The of any scalar f can be added to A (a gauge transformation) This is because of another vector identity ∇×∇.f ≡ 0.

Generally, H(r) cannot be derived from a potential. It satisfies Maxwells equation ∇ × H = jc + ∂D/∂t. In a static situation, when there are no conduction currents present, ∇ × H = 0, and

H(r) = - ∇ϕm(r)

In these special conditions, it is possible to derive H(r) from a magnetic scalar potential ϕm (units A). We can imagine that H is derived from a distribution of magnetic charges± qm. IEEE Santander 2017 Relation between B and H in a material

The general relation between B, H and M is

B = µ0(H + M) i.e. H = B/µ0 - M

H M B

We call the H-field due to a magnet; — stray field outside the magnet

— , Hd, inside the magnet IEEE Santander 2017

Boundary conditions

Conditions on the fields It follows from Gausss law

∫SB.dA = 0 that the perpendicular component of B is

continuous. ( B1- B2).en= 0 It follows from from Ampères law

∫ H.dl = I = 0 that the parallel component of H is continuous. since there are no conduction currents on the

surface. (H1- H2) x en= 0

Conditions on the potentials Since ∫SB.dA = ∫ A.dl (Stokes theorem)

(A1- A2) x en= 0

The scalar potential is continuous ϕm1 = ϕm2

IEEE Santander 2017 Boundary conditions in linear, isotropic homogeneous (LIH) media

In LIH media, B = µ0µrH (µr = 1 + χ ) since B = µ0(H + M) Permeability Hence B1en = B2en susceptiility

H1en = µr2/µr1H2en So field lies ~ perpendicular to the surface of soft iron but parallel to the surface of a superconductor. Diamagnets produce weakly repulsive images. Paramagnets produce weakly attractive images.

Soft ferromagnetic mirror (attraction) Superconducting mirror (repulsion)

IEEE Santander 2017 Paramagnets and diamagnets; susceptibility

Only a few elements and alloys are ferromagnetic. (See the magnetic periodic table). The atomic moments in a ferromagnet order spontaneosly parallel to eachother. Most have no spontaneous magnetization, and they show only a very weak response to a magnetic field. They are paramgnetic, with a small positive susceptibility when there are disordered atomic moments; otherwise they are diamagnetic.

M paramagnet H Here |χ| << 1 diamagnet χ is 10-4 - 10-6

Magnetic moment/volume/field Magnetic moment/mass/field Disordered T > Tc Magnetic moment/mole/field

IEEE Santander 2017 1 H Magnetic Periodic Table 2 He 1.00 4.00

66 Atomic Number Dy Atomic symbol 3 Li 4 Be 162.5 Atomic weight 5 B 6 C 7 N 8 O 9 F 10Ne 9 6.94 9.01 Typical ionic change 3 + 4f 10.81 12.01 14.01 16.00 19.00 20.18 0 2 + 2s0 179 85 1 + 2s Antiferromagnetic TN(K) Ferromagnetic TC(K) 35

11 12 13 14 15 16 17 18 Na Mg Al Si P S Cl Ar 22.99 24.21 26.98 28.09 30.97 32.07 35.45 39.95 1 + 3s0 2 + 3s0 3 + 2p6

19K 20Ca 21Sc 22Ti 23V 24Cr 25Mn 26Fe 27Co 28Ni 29Cu 30Zn 31 Ga 32Ge 33As 34Se 35Br 36Kr 38.21 40.08 44.96 47.88 50.94 52.00 55.85 55.85 58.93 58.69 63.55 65.39 69.72 72.61 74.92 78.96 79.90 83.80 1 + 4s0 2 + 4s0 3 + 3d0 4 + 3d0 3 + 3d2 3 + 3d3 2 + 3d5 3 + 3d5 2 + 3d7 2 + 3d8 2 + 3d9 2 + 3d10 3 + 3d10 312 96 1043 1390 629

37Rb 38Sr 39Y 40Zr 41Nb 42 Mo 43Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50Sn 51Sb 52Te 53I 54Xe 85.47 87.62 88.91 91.22 92.91 95.94 97.9 101.1 102.4 106.4 107.9 112.4 114.8 118.7 121.8 127.6 126.9 83.80 1 + 5s0 2 + 5s0 2 + 4d0 4 + 4d0 5 + 4d0 5 + 4d1 3 + 4d5 3 + 4d6 2 + 4d8 1 + 4d10 2 + 4d10 3 + 4d10 4 + 4d10

55Cs 56Ba 57La 72Hf 73Ta 74W 75Re 76Os 77Ir 78Pt 79Au 80Hg 81Tl 82Pb 83Bi 84Po 85At 86Rn 132.9 137.3 138.9 178.5 180.9 183.8 186.2 190.2 192.2 195.1 197.0 200.6 204.4 207.2 209.0 209 210 222 1 + 6s0 2 + 6s0 3 + 4f0 4 + 5d0 5 + 5d0 6 + 5d0 4 + 5d3 3 + 5d5 4 + 5d5 2 + 5d8 1 + 5d10 2 + 5d10 3 + 5d10 4 + 5d10

87Fr 88Ra 89Ac 223 226.0 227.0 2 + 7s0 3 + 5f0 58Ce 59Pr 60Nd 61Pm 62Sm 63Eu 64Gd 65Tb 66Dy 67Ho 68Er 69Tm 70Yb 71Lu 140.1 140.9 144.2 145 150.4 152.0 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0 4 + 4f0 3 + 4f2 3 + 4f3 3 + 4f5 2 + 4f7 3 + 4f7 3 + 4f8 3 + 4f9 3 + 4f10 3 + 4f11 3 + 4f12 3 + 4f13 3 + 4f14 13 19 105 90 292 229 221 179 85 132 20 85 20 56

90Th 91Pa 92U 93Np 94Pu 95Am 96Cm 97Bk 98Cf 99Es 100Fm 101Md 102No 103Lr 232.0 231.0 238.0 238.0 244 243 247 247 251 252 257 258 259 260 4 + 5f0 5 + 5f0 4 + 5f2 5 + 5f2

Nonmetal Diamagnet Ferromagnet TC > 290K

Metal Paramagnet Antiferromagnet with TN > 290K Antiferromagnet/Ferromagnet with T /T < 290 K Radioactive BOLD Magnetic atom N C

IEEE Santander 2017 Susceptibility of the elements

IEEE Santander 2017 Antiferromagnets and ferrimagnets

A few elements and many compounds are antiferromagnetic. The atomic moments order spontaneously antiparallel to eachother in two equivalent sublattices. Ferrimagnets have two numerically inequivalent sublattices. They respond to a magnetic field like a ferromagnet. M ferromagnet antiferromagnet H For antiferromagnets χ << 1 χ is 10-4 - 10-6

For ferrimagnets and ferromagnets χ 1 Ordered T < Tc

IEEE Santander 2017 Magnetic fields - Internal and applied fields

In ALL these materials, the H-field acting inside the material is not the one you apply. These are not the same. If they were, any applied field would instantly saturate the magnetization of a ferromagnet when χ > 1. M

Consider a thin film of iron. Ms Field applied || to film iron

Field applied ⊥ to film Hʹ

Ms substrate -1 H = H + Hd Ms = 1.72 MAm Internal field Demagnetizing field External field

IEEE Santander 2017 Demagnetizing field in a material - Hd

The demagnetizing field depends on the shape of the sample and the direction of magnetization. For simple uniformly-magnetized shapes (ellipsoids of revolution) the demagnetizing field is related to the magnetization by a proportionality factor N known as the demagnetizing factor. The value of N can never exceed 1, nor can it be less than 0.

Hd = - N M More generally, this is a tensor relation. N is then a 3 x 3 matrix, with trace 1. That is

N x + N y + N z = 1

Note that the internal field H is always less than the applied field H since

H = H - N M

IEEE Santander 2017 Demagnetizing factor N for special shapes.

N = 0 N = 1/3 N = 1/2 N = 1

Long needle, M parallel H’ to the long axis H’ Thin film, H’ Sphere M perpendicular to plane

Long needle, Thin film, M parallel M perpendicular to plane to the long axis

To ro i d , M perpendicular H’ to r

IEEE Santander 2017 The shape barrier

N < 0.1

Daniel Bernouilli Shen Kwa 1060 1743 S N

Gowind Knight 1760 T. Mishima 1931

N = 0.5 New icon for permanent magnets! ⇒ Philips 1952

IEEE Santander 2017 SUMMER SCHOOL

! IEEE Santander 2017 The shape barrier ovecome !

H = - N M d working M (Am-1) point Ms

Mr H H d H d c H (Am-1)

IEEE Santander 2017 Single-domain particles

When ferromagnetic particles are no bigger than a few tens of nanometers, it does not pay to form domains. In very small particles, reversal takes place by coherent rotation of the magnetic moment m. There must be an easy direction of magnetization. If an external field H is applied at an angle ϕ to the easy direction, and the magnetization is at an angle θ to the easy direction, the energy is Etot = Ea + EZ the sum of anisotropy Zeeman 2 terms. The first Ea = KuV sin θ, hence there in an energy barrier KuV to reversal.

2 Etot = KuVsin θ µ0mH cos(ϕ - θ) m The energy can be minimized, and the hysteresis loop calculated θ H numerically for a general angle ϕ. This is the Stoner-Wohlfarth model. ϕ

M M

H H A particle where the magnetization rotates coherently. IEEE Santander 2017 Paramagnetic and ferromagnetic responses

Susceptibility of linear, isotropic and homogeneous (LIH) materials M = χH χis external susceptibility (no units)

It follows that from H = H + Hd that 1/χ = 1/χ - N Typical paramagnets and diamagnets: χ χ (10-5 to 10-3 ) Demag field is negligible. Paramagnets close to the Curie point and ferromagnets:

χ >> χχ diverges as T → T C but χ never exceeds 1/N.

MM M M

M H’ s M Ms/3/3 H H'H

IEEE Santander 2017 Energy of ferromagnetic bodies

Torque and energy of a magnetic moment m in a uniform field,.

Γ = m x B ε = -m.B B θ m Γ = mB sinθ. ε = -mB cosθ. Force

In a non-uniform field, f = -∇ε f = m.∇B

• Magnetostatic (dipole-dipole) forces are long-ranged, but weak. They determine the magnetic microstructure (domains).

2 • ½µ0H is the energy density associated with a magnetic field H -1 6 -3 M 1 MA m , µ0Hd 1 T, hence µ0HdM 10 J m 2 2 • Products B.H, B.M, µ0H , µ0M are all energies per unit volume. • Magnetic forces do no work on moving charges f = q(v x B) [Lorentz force] • No potential energy associated with the magnetic force.

IEEE Santander 2017 Magnetostatic forces

Force density on a magnetized body at constant temperature

Fm= - ∇ G

Kelvin force

General expression, when M is dependent on H is

V =1/d d is the density

IEEE Santander 2017 Some expressions involving B

F = q(E + v × B) Force on a charged particle q

F = B lℓ Force on current-carrying wire

E = - dΦ/dt Faradays law of electromagnetic induction

E = -m.B Energy of a magnetic moment

F = ∇ m.B Force on a magnetic moment

Γ = m x B Torque on a magnetic moment

IEEE Santander 2017 2. Magnetic Phenomena

SUMMER SCHOOL

IEEE Santander 2017

! Functional Magnetic Materials

Magnetism is an experimental science that has spawned generations of useful technology, thanks to the range of useful phenomena associated with magnetically-ordered materials.

Property Effect Magnitude Applications

Hard (Hc < M) Creates stray H ≤ 2Ms Motors, actuators, field flux sources 3 Soft (Hc ≈ 0) Amplifies flux 1 < χ < 10 Electromagnetic drives Magnetostrictive Changes 0 < δℓ/ℓ < 10-3 Actuators length Sensors ℓ = ℓ(H) Magnetoresistive Changes δρ/ρ ~ 1 % or Spin electronics resistivity 100% in Sensors ρ = ρ(H) heterostructures Magnetocaloric Changes δT ~ 2 K Refrigeration T = T (H) temperature

IEEE Santander 2017 Fundamentals of Magnetism Magnets and Energy

Fundamentals of Magnetism What do magnetic field look like?

Fe-Pt medium Nd-Fe-B spindle motor

Nd-Fe-B voice coil motor Spin valve read Fundamentals of Magnetism head

Modern Magnetic Technology – Everywhere!

• Magnetic Material Systems – Spintronics (electron + spin) – “Soft” magnets http:// – “Hard” or permanent magnets www.medscape.com/ Flux distributionviewarticle/712338_7 in a motor • Advanced Applications: 2015 IEEE Summer School – June 14-19, University of Minnesota 26 – Computer technology

– Electrical distribution 2015 IEEE Summer School – June 14-19, University of Minnesota – Sensors, Motors • Other Applications: – Drug delivery – Cancer therapies IEEE Santander 2017 – Biosensors http://www.scs.illinois.edu/suslick/sonochemistry.html

2015 IEEE Summer School – June 14-19, University of Minnesota 8 M Magnetic materials; A 25 B€ market

H

➁ Hard ferrite Nd-Fe-B ➀ Sm-Co Alnico Tapes M Co-Cr-Pt NiFe/CoFe

FeSi H H FeSi(GO) NiFe, CoFe Amorphous M Soft Ferrite

H

Schultz Syposium 22-ix-2014 IEEE Santander 2017 3. Units

SUMMER SCHOOL

IEEE Santander 2017

! A note on units:

Magnetism is an experimental science, closely linked to electricity, and it is important to be able to calculate numerical values of the physical quantities involved. There is a strong case to use SI consistently Ø SI units relate to the practical units of electricity measured on the multimeter and the oscilloscope Ø It is possible to check the dimensions of any expression by inspection. Ø They are almost universally used in teaching of science and engineering Ø Units of H, B, Φ or dΦ/dt have been introduced.

BUT Most literature still uses cgs units. You need to understand them too.

IEEE Santander 2017 SI / cgs conversions:

SI units cgs units

B = μ0(H + M) B = H + 4πM m A m2 emu M A m-1 (10-3 emu cc-1) emu cc-1 (1 k A m-1) σ A m2 kg-1 (1 emu g-1) emu g-1 (1 A m2 kg-1) H A m-1 (4π/1000 0.0125 Oe) Oersted (1000/4π 80 A m-1) B Tesla (10000 G) Gauss (10-4 T) Φ Weber (Tm2) (108 Mw) Maxwell (G cm2) (10-8 Wb) dΦ/dt V (108 Mw s-1) Mw s-1 (10 nV) χ - (4π cgs) - (1/4π SI)

IEEE Santander 2017 592 Appendices

B.3 Dimensions

Any quantity in the SI system has dimensions which are a combination of the dimensions of the five basic quantities, m, l, t, i and θ.Inanyequationrelating combinations of physical properties, each of the dimensions must balance, and the dimensions of all the terms in a sum have to be identical.

B3.1 Dimensions

Mechanical

Quantity Symbol Unit ml t i θ

Area m2 02000 A Vo l u m e V m3 03000 1 Velocity v ms− 01100 2 − Acceleration a ms− 01200 3 − Density d kg m− 1 3000 − Energy ε J12200 1 − Momentum p kg m s− 11100 2 1 − Angular momentum L kg m s− 12100 − Moment of inertia I kg m2 12000 Force f N11200 3 − Force density F Nm− 1 2 200 − − Power P W12300 − Pressure P Pa 1 1 200 2 − − Stress σ Nm− 1 1 200 2 − − Elastic modulus K Nm− 1 1 200 1 − − Frequency f s− 00100 2 1 − Diffusion coefficient D m s− 02100 2 − Viscosity (dynamic) η Nsm− 1 1 100 2 1 − − Viscosity ν m s− 02100 − Planck’s constant ! Js 1 2 100 −

IEEE Santander 2017 Thermal

Quantity Symbol Unit ml t i θ

Enthalpy H J1220 0 1 − Entropy S JK− 12 20 1 1 1 − − Specific heat C JK− kg− 02 20 1 1 − − Heat capacity c JK− 12 20 1 1 1 − − Thermal conductivity κ Wm− K− 11 30 1 1 1 − − Sommerfeld coefficient γ Jmol− K− 12 20 1 1 − − Boltzmann’s constant k JK− 12 20 1 B − − 593 Appendix B Units and dimensions

Electrical

Quantity Symbol Unit ml t i θ

Current I A00010 2 Current density j Am− 0 2010 − Charge q C00110 Potential V V123 10 − − V123 10 E − − C F 1 2420 − − Resistance R " 123 20 − − Resistivity ϱ"m133 20 1 − − Conductivity σ Sm− 1 3320 − − Dipole moment p Cm 0 1 1 1 0 2 Electric polarization P Cm− 0 2110 1 − Electric field E Vm− 113 10 2 − − Electric displacement D Cm− 0 2110 − Electric flux % C00110 1 Permittivity ε Fm− 1 3420 1 − − Thermopower S VK− 123 1 1 2 1 1 − − − Mobility µ m V− s− 10210 −

IEEE Santander 2017 Magnetic

Quantity Symbol Unit ml t i θ

Magnetic moment m Am2 02010 1 Magnetization M Am− 0 1010 2 1 − Specific moment σ Am kg− 12010 1 − Magnetic field strength H Am− 0 1010 − Magnetic flux ' Wb 1 2 2 10 − − Magnetic flux density B T102 10 − − L H122 20 − − Susceptibility (M/H) χ 00000 1 Permeability (B/H) µ Hm− 112 20 − − Magnetic polarization J T102 10 − − A00010 F Magnetic ‘charge’ qm Am 0 1 0 1 0 3 Energy product (BH)Jm− 1 1 200 3 − − Anisotropy energy K Jm− 1 1 200 1 − − Exchange stiffness A Jm− 11200 3 1 − Hall coefficient R m C− 031 10 H − − Scalar potential ϕ A00010 Vector potential A Tm 1 1 2 10 2 1 − − Pm Tm A− 122 20 1 2 − − Reluctance R AT− m− 1 2220 m − − 593 Appendix B Units and dimensions

Electrical

Quantity Symbol Unit ml t i θ

Current I A00010 2 Current density j Am− 0 2010 − Charge q C00110 Potential V V123 10 − − Electromotive force V123 10 E − − Capacitance C F 1 2420 − − Resistance R " 123 20 − − Resistivity ϱ"m133 20 1 − − Conductivity σ Sm− 1 3320 − − Dipole moment p Cm 0 1 1 1 0 2 Electric polarization P Cm− 0 2110 1 − Electric field E Vm− 113 10 2 − − Electric displacement D Cm− 0 2110 − Electric flux % C00110 1 Permittivity ε Fm− 1 3420 1 − − Thermopower S VK− 123 1 1 2 1 1 − − − Mobility µ m V− s− 10210 −

Magnetic

Quantity Symbol Unit ml t i θ

Magnetic moment m Am2 02010 1 Magnetization M Am− 0 1010 2 1 − Specific moment σ Am kg− 12010 1 − Magnetic field strength H Am− 0 1010 − Magnetic flux ' Wb 1 2 2 10 − − Magnetic flux density B T102 10 − − Inductance L H122 20 − − Susceptibility (M/H) χ 00000 1 Permeability (B/H) µ Hm− 112 20 − − Magnetic polarization J T102 10 − − Magnetomotive force A00010 F Magnetic ‘charge’ qm Am 0 1 0 1 0 3 Energy product (BH)Jm− 1 1 200 3 − − Anisotropy energy K Jm− 1 1 200 1 − − Exchange stiffness A Jm− 11200 3 1 − Hall coefficient R m C− 031 10 H − − Scalar potential ϕ A00010 Vector potential A Tm 1 1 2 10 2 1 − − Permeance Pm Tm A− 122 20 1 2 − − Reluctance R AT− m− 1 2220 m − −

IEEE Santander 2017 592 Appendices

B.3 Dimensions

Any quantity in the SI system has dimensions which are a combination of the dimensions of the five basic quantities, m, l, t, i and θ.Inanyequationrelating combinations of physical properties, each of the dimensions must balance, and the dimensions of all the terms in a sum have to be identical.

B3.1 Dimensions

Mechanical

Quantity Symbol Unit ml t i θ

Area m2 02000 A Vo l u m e V m3 03000 1 Velocity v ms− 01100 2 − Acceleration a ms− 01200 3 − Density d kg m− 1 3000 − Energy ε J12200 1 − Momentum p kg m s− 11100 2 1 − Angular momentum L kg m s− 12100 − Moment of inertia I kg m2 12000 Force f N11200 3 − Force density F Nm− 1 2 200 − − Power P W12300 − Pressure P Pa 1 1 200 2 − − Stress σ Nm− 1 1 200 2 − − Elastic modulus K Nm− 1 1 200 1 − − Frequency f s− 00100 2 1 − Diffusion coefficient D m s− 02100 2 − Viscosity (dynamic) η Nsm− 1 1 100 2 1 − − Viscosity ν m s− 02100 − Planck’s constant ! Js 1 2 100 −

Thermal

Quantity Symbol Unit ml t i θ

Enthalpy H J1220 0 1 − Entropy S JK− 12 20 1 1 1 − − Specific heat C JK− kg− 02 20 1 1 − − Heat capacity c JK− 12 20 1 1 1 − − 594 Thermal conductivity κ AppendicesWm− K− 11 30 1 1 1 − − Sommerfeld coefficient γ Jmol− K− 12 20 1 1 − − Boltzmann’s constant k B3.2 ExamplesJK− 12 20 1 B − − (1) Kinetic energy of a body: ε 1 mv2 = 2 [ε] [1, 2, 2, 0, 0] [m] [1, 0, 0, 0, 0] = − = 2[0, 1, 1, 0, 0] [v2] − − = [1, 2, 2, 0, 0] (2) Lorentz force on a moving charge; f qv B − − = × [f ] [1, 1, 2, 0, 0] [q] [0, 0, 1, 1, 0] = − = [v] [0, 1, 1, 0, 0] = − [1, 0, 2, 1, 0] [B] − − = [1, 1, 2, 0, 0] − (3) energy γ w √AK (γ w is an energy per unit area) 1 = [γ w] [εA− ][√AK] 1/2[AK] = = 1 [1, 2, 2, 0, 0] [√A] 2 [1, 1, 2, 0, 0] = − = [1, 1−, 2, 0, 0] - [0,[1,1, 2, 0, 0,2, 0 0,] 0] [√K] 1 − − − − = 2 [1, 0, 2, 0, 0] [1, 0, 2, 0, 0] − = − (4) Magnetohydrodynamic force on a moving conductor F σ v B B = × × (F is a force per unit volume)IEEE Santander 2017 [F ] [FV 1][σ] [ 1, 3, 3, 2, 0] = − = − − [1, 1, 2, 0, 0] [v] [0, 1, 1, 0, 0] = − = − [0, 3, 0, 0, 0] 2[1, 0, 2, 1, 0] [B2] − − −[1, 2, 2, 0, 0] = [1, 2, 2, 0, 0] − − − − (5) Flux density in a solid B µ (H M) (note that quantities added or = 0 + subtracted in a bracket must have the same dimensions) [B] [1, 0, 2, 1, 0] [µ ] [1, 1, 2, 2, 0] = − − 0 = − − [0, 1, 0, 1, 0] [M], [H ] − = [1, 0, 2, 1, 0] (6) Maxwell’s equation H j dD/dt. − − ∇× = + [ H] [Hr 1][j] [0, 2, 0, 1, 0] [dD/dt] [Dt 1] ∇× = − = − = − [0, 1, 0, 1, 0] [0, 2, 1, 1, 0] = − = − [0,1,0,0,0] [0, 0, 1, 0, 0] − − [0, 2, 0, 1, 0] [0, 2, 0, 1, 0] = − = − (7) Ohm’s Law V IR = [1, 2, 3, 1, 0] [0, 0, 0, 1, 0] = − − [1, 2, 3, 2, 0] + − − [1, 2, 3, 1, 0] = − − (8) Faraday’s Law ∂%/∂t E = − [1, 2, 3, 1, 0] [1, 2, 2, 1, 0] = − − − − [0, 0, 1, 0, 0] − [1, 2, 3, 1, 0] = − − 594 Appendices

B3.2 Examples (1) Kinetic energy of a body: ε 1 mv2 = 2 [ε] [1, 2, 2, 0, 0] [m] [1, 0, 0, 0, 0] = − = 2[0, 1, 1, 0, 0] [v2] − − = [1, 2, 2, 0, 0] (2) Lorentz force on a moving charge; f qv B − − = × [f ] [1, 1, 2, 0, 0] [q] [0, 0, 1, 1, 0] = − = [v] [0, 1, 1, 0, 0] = − [1, 0, 2, 1, 0] [B] − − = [1, 1, 2, 0, 0] − (3) Domain wall energy γ w √AK (γ w is an energy per unit area) 1 = [γ w] [εA− ][√AK] 1/2[AK] = = 1 [1, 2, 2, 0, 0] [√A] 2 [1, 1, 2, 0, 0] = − = [1, 1−, 2, 0, 0] [1,1, 2, 0, 0] [√K] 1 − − − − = 2 [1, 0, 2, 0, 0] [1, 0, 2, 0, 0] − = − (4) Magnetohydrodynamic force on a moving conductor F σ v B B = × × (F is a force per unit volume) [F ] [FV 1][σ] [ 1, 3, 3, 2, 0] = − = − − [1, 1, 2, 0, 0] [v] [0, 1, 1, 0, 0] = − = − [0, 3, 0, 0, 0] 2[1, 0, 2, 1, 0] [B2] − − −[1, 2, 2, 0, 0] = [1, 2, 2, 0, 0] − − − − (5) Flux density in a solid B µ (H M) (note that quantities added or = 0 + subtracted in a bracket must have the same dimensions) [B] [1, 0, 2, 1, 0] [µ ] [1, 1, 2, 2, 0] = − − 0 = − − [0, 1, 0, 1, 0] [M], [H ] − = [1, 0, 2, 1, 0] (6) Maxwell’s equation H j dD/dt. − − ∇× = + [ H] [Hr 1][j] [0, 2, 0, 1, 0] [dD/dt] [Dt 1] ∇× = − = − = − [0, 1, 0, 1, 0] [0, 2, 1, 1, 0] = − = − [0,1,0,0,0] [0, 0, 1, 0, 0] − − [0, 2, 0, 1, 0] [0, 2, 0, 1, 0] = − = − (7) Ohm’s Law V IR = [1, 2, 3, 1, 0] [0, 0, 0, 1, 0] = − − [1, 2, 3, 2, 0] + − − [1, 2, 3, 1, 0] = − − (8) Faraday’s Law ∂%/∂t E = − [1, 2, 3, 1, 0] [1, 2, 2, 1, 0] = − − − − [0, 0, 1, 0, 0] − [1, 2, 3, 1, 0] = − − IEEE Santander 2017 Table B SI–cgs conversion Locate the quantity you wish to convert in column A (its units are in the same row); to convert to a quantity in row B (its units are in the same column),multiply it by the factor in the table. Examples are given in appendix F.

Field conversions Susceptibility conversions

SI SI cgs cgs SI SI SI cgs cgs

A BH BH B AB SI χm χmol χ0 cgs χm χ mol 1 3 1 3 1 2 1 1 1 units Am− TOeG units χ m kg− m mol− JT− kg− κ emu g− emu mol− ↓ → ↓ → 1 3 3 3 3 3 SI H Am 1 µ 4π 10 4π 10 SI χ 11/d10− /d 1/µ d1/4π 10 /4πd10/4πd − 0 − − M 0 M 4× 4× 3 1 3 3 3 SI B T1/µ 110 10 SI χ m kg− d1 10− 1/µ d/4π 10 /4π 10 /4π 0 m M 0 M 3 4 3 1 3 3 3 3 6 6 cgs H Oe 10 /4π 10 11 SI χ m mol− 10 d/ 10 / 110/µ 10 d/4π 10 /4π 10 /4π − mol M M 0M M M 3 4 2 1 3 7 4 4 cgs B G10/4π 10− 11 SI χ0 JT− kg− µ0d µ0 10− µ0 110− d 10− 10− 3 6M 4 M cgs κ 4π 4π10− /d4π10− /d 10 /d1 1/d /d M M B–H conversions are valid in free space only 1 3 6 4 cgs χm emu g− 4πd4π10− 4π10− 10 d1 1 3 6M 4 M cgs χ emu mol− 4πd/ 4π10− / 4π10− 10 / d/ 1/ 1 mol M M M M M 1 is molecular weight (in g mol− ), d is density (use SI units in rows 1–4, cgs units in rows 5–7) M Magnetic moment and magnetization conversions

SI SI SI SI cgs cgs cgs cgs

AB m M σσmol m M σσmol 2 1 2 1 2 1 3 1 1 units Am Am− Am kg− Am mol− emu emu cm− emu g− emu mol− ↓ → 24 21 m µB /formula 9.274 10− 5585d/ 5585/ 5.585 9.274 10− 5.585d/ 5585/ 5585 2 × M M 3 3 × 3 M M SI m Am 11/V 1/dV 10− /dV 10 10− /V 1/dV /dV 1 3M 3 3 M SI M Am− V 11/d10− /d10V 10− 1/d /d 2 1 3M 3 3 M SI σ Am kg− dV d1 10− 10 dV 10− d1 2 1 2 3 3 M 6 3 M3 SI σ mol Am mol− 10 dV/ 10 d/ 10 / 110dV/ d/ 10 / 10 3 M 3 M M 3 M M M cgs m emu 10− 10 /V 1/dV 10− /dV 11/V 1/dV /dV 3 3 3 3M M cgs M emu cm− 10− V 10 1/d 10− /d V 11/d/d 1 3 3 3M M cgs σ emu g− 10− dV 10 d1 10− dV d1 1 3 3 3M M cgs σ mol emu mol− 10− dV/ 10 d/ 1/ 10− dV/ d/ 1/ 1 M M M M M M

1 is molecular weight (g mol− ), d is density, V is sample volume (use SI in rows 1–5, cgc in rows 6–9 for density and volume). Note that the quantity 4πM is frequently quoted in the cgs system, M in units of gauss (G) 6 2 2 To deduce the effective Bohr magneton numberp m /µB from the molar CurieIEEE constant, Santander the relation 2017 isC 1.571 10− p in SI, and C 0.125 p in cgs. eff = eff mol = eff mol = eff Review — Lecture 1

You should now know:

Ø Some good books Ø How magnets behave; Magnetic moment m and magnetization M Ø How magnetism is related to electric current Ø The two fields; B and H Ø Demagnetizing effects in solids Ø Energy and force of magnetic moments Ø What magnetic materials are used for Ø The units used in magnetism, and understand what they mean Ø How to check the dimensions of any quantity or equation.

IEEE Santander 2017