v.2019.AUG D. Magnetic Properties of Materials...

Magnetic Properties 1

D. Magnetic Properties of Materials

• Introduction (8.1)

– magnetic moments (m), magnetization (M), magnetic field (B)

• classification (8.2)

– dia, para, ferro, anti-ferro, ferri

• ferro magnetism

– origin: exchange interaction (8.3)

– magnetic domains (8.5.1), domain walls (8.5.3), wall motion (8.5.5), anisotropy (8.5.2), M-H curves (8.5.6), demag (8.5.7)

• soft & hard magnets (8.6)

• superconductivity (8.9)

References:

S.O.Kasap (3rd Ed) Chapter 8; RJD Tilley, Understanding Solids (2nd Ed.) (2013), Chapter 12

D.R. Askeland and P.P. Phulé, The Science and Engineering of Materials (4th ed.) (2004), Chapter 19

W.F. Smith and J. Hashemi, Foundations of materials science and engineering (5th ed.) (2010), Chapter 16

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v.2019.AUG


• early studies of weak magnetic materials use a Gouy balance (Fig. 12.1) which tilts up

or down depending on how material responds to magnetic field (Fig. 12.2) (I1)

• magnetic field origins: by solids, especially strong magnetic materials (ferromagnet)

(Fig. A), wires carrying current (Fig. B), solenoid carrying current (Fig. C) (I2)

• our main interest: magnetic solids

(I0)8.1 Introduction

+

nucleus

electrons

atom

molecule / lattice

(polycrystalline)

solid

magnetic dipole moment

magnetic dipole moment (mm)

Bohr magneton (mB)magnetic field (B)

magnetization (M)

spin

(ms)

(n,l,ml,ms)

orbital

(l,ml)dipole interaction,

dia/para/ferri

ferro/antiferro

magnetism

(I3)(I4)⏩ (8.2)(I5-I6)

grain/domain


(I1)Weak magnets

• most materials are “non-magnetic” (extremely weak magnets). Gouy balance (Fig. 12.1) shows that in

a magnetic field, most materials weight slightly less (diamagnets), some slightly more (paramagnets)

• diamagnets () repel, paramagnets () attracted to magnetic field

• magnetic field strength H in free space Fig. 12.2(a), inside diamagnets (b), inside paramagnets (c)


B. current through wire

C. current through solenoid B = μ0NI = μ0(n / L)I

Magnetic field B : physical origins

Biot–Savart law (1820)

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magnetic permeability of free space

m/H104 7

0

m

(สภาพใหซึ้มได)้

(conduction electrons)

(conduction electrons)

B

(I2)

A. ferromagnet

(electron spin & orbital motions

around nucleus)

BM

northern lights (aurora borealis)

southern lights (aurora australis)

http://en.wikipedia.org/wiki/File:Manoderecha.svg

http://upload.wikimedia.org/wikipedia/commons/0/0d/VFPt_Solenoid_correct2.svg

http://upload.wikimedia.org/wikipedia/commons/5/57/Magnet0873.png


8.1.1 magnetic (dipole) moment

A

I

mm

un

Definition of a magnetic dipole moment.

B

A

I

B

mm

• dielectric properties (e) of materials are based on (electric) dipole moments (p)

• magnetic properties (m) of materials are based on magnetic (dipole) moments (mm)

current

loop

areanIAu

torque tries to rotate

mm to align with B

magnetic moments originate from the flow of electrons

Cl H+

po

-Q

F = Q E

F

po = aQ

E

+Q

(a) (c)

pav 0 E

(d)(b)

pav = 0

(a) A HCl molecule possesses a permanent dipole moment, po (b) In

the absence of a field, thermal agitation of the molecules results in zero

net average dipole moment per molecule. (c) A dipole such as HCl

placed in a field experiences a torque which tries to rotate it to align po

with the field E. (d) In the presence of an applied field the dipoles try to

rotate to align with the field against thermal agitation. There is now a

net average dipole moment per molecule along the field.

e

mm

magnetic moments (A-m2)mm mo magnetic permeability (H/m)

mm

Definition of (electric) dipole moment

p-E interaction

mm-B interaction

torque tries to rotate

p to align with E

(I3)


I

A

-e

L

r

morb

mz µspin

Bz

Sz

8.1.2 Atomic Magnetic Moments

Origins of mm in atoms

Bohr

magneton

1. Orbiting electrons 2. Spinning electrons

• overall magnetic moments of the electron:

• overall magnetic moments of the atom:

orbspinelectron μμμ

electronelectronsall

atom μμ

Quantum numbers:

n = 1,2,3... l = 0,1,...(n-1)

ml = -l,...-1,0,1...l ms = ½

electrons in closed subshells matom = 0

2

22

orb22

2

rmmvrL

Lm

ererI

e

T

e

dt

dqI

e

m

224

spin

mA10274.9

2

B

e

s

e

z

e

z

e

m

em

m

eS

m

e

Sm

e

m

m

m

L : Orbital angular momentum S : Spin (intrinsic) angular momentum m : magnetic quantum number

matom(I4)

222

Earth mA108 m


8.1.3 magnetization (M), magnetic flux density (B)

I

I

A

(b)

B

I

I

Bo

(a)

M

I

Im

B

M

I

Surface currents

Surface currents

no net

bulk

current

magnetization current Im

conduction current I

surface is effectively a solenoid !

- when placed in a magnetic field (Bo) generated by a solenoid (Fig. A), permanent magnets develop

magnetization (M) (Fig. B)

- magnetization (M) ≡ magnetic dipole moment / volume [unit: (A·m2)/m3 = A/m]

- each atom in the magnet (its internal current loops, Fig. C) experiences a torque (I3) that causes the

dipole moment to align with Bo. Inside, I = 0. Surface, I ≠ 0

- material is said to be magnetized; magnetic flux density in material is due to magnetization (surface)

current Im and conduction current I (Fig. D), thus B > Bo

Surface current

Surface current

(I5)

Fig. A

Fig. B

Fig. D

Fig. C


- field amplification

- susceptibility

- permeability

(a) vacuum:

Magnetizing field (magnetic field strength)

Magnetic field (magnetic flux density)

(b) medium:

l

nIH

HBo om

HB m

mm

m

m

m

mm

H

H

HH

MH

MHB

ro

mo

mo

o

oo

1

field

amplification

permeability

mo mm 1

H

Mm

unit:

[A/m] – common

[A-turns/m] – complete

[Tesla, Wb/m2]

The field B in the material inside the solenoid is due to the conductioncurrent I through the wires and the magnetization current Im on the

surface of the magnetized medium, or B = Bo + moM.

Current I (A)

magnetizing field H (A/m)

mag

net

ic f

ield

mag

net

ic f

lux

den

sity

B(W

b/m

2)

(a)

(b)

polarizability:

relative permittivity :χm Susceptibility สภาพรับไวไ้ด ้(รับสนามแม่เหลก็)m Permeability สภาพใหซึ้มได ้(ใหส้นามแม่เหลก็ซึมผา่นได)้

oer N

P

ee

/1

/

(I6)8.1.3 magnetization (M), magnetic flux density (B)

susceptibility

e Permittivity สภาพยอม (ยอมสนามไฟฟ้า)


8.2 Magnetic Material Classifications

H

Mm para – 1. alongside

2. beyond(parallel)

- Classification based on susceptibility

- prefix in name/type (Greek) indicates effects/materials

Greek – English

dia – 1. across

2. opposite

(diameter)


NS FM

Diamagnetism

• diamagnetic atoms have no elementary magnetic

dipoles ← closed electronics shells and subshells

• materials: group IB (Cu, Ag, Au), Si, NaCl

Origin:

• orbiting electrons try to resist B: dipole moments try to expel B from the materials

• diamagnets experience force toward smaller fields

• effect extremely weak (χm in − ppm) (Table 16.2)

• diamagnets have no practical importance, except

“perfect diamagnet”—superconductors (8.9)

99995.0

00

r

mM

m

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µav = 0 and M = 0

moH

M

µav 0 and M = mH

(b)(a)

(a) In a paramagnetic material each individual atom possesses apermanent magnetic moment but due to thermal agitation there is noaverage moment per atom and M = 0. (b) In the presence of anapplied field, individual magnetic moments take alignments along theapplied field and M is finite and along B.

Paramagnetism

(microscopic) each atom/molecule has net magnetic

dipole moment

(macroscopic) no net magnetic moments due to

thermal agitation

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• paramagnetic atoms have elementary

magnetic dipoles ← unpaired electrons:

incomplete cancellation of mspin and/or morb

• materials: alkali earth (Na,Mg,Ca), transition

metals (Ti,Zr,Mo), others (Al,Cu alloys)

Origin:

• dipoles non-interacting (with/without B)

• individual mm of atoms /molecules

experiences torque → align with B, but

alignment is incomplete due to interaction

with nearby atoms and thermal agitation

• effect very weak (χm in + ppm) (Table 16.2),

and depends on temperature T:

• ferro-magnets and ferri-magnets above TCurie

exhibit paramagnetism

T

C C: Curie constant


Pauli paramagnetism

• most metals are weakly paramagnetic

• not due to unpaired electrons (d and f shells) which

are localized (bound to atoms)

• but due to conduction electrons in CB, delocalized

and mobile throughout solids (non-CB electrons are

in filled shells and do not contribute mm)

Origin:

• conduction electrons are spin-paired, number of spin up

electrons (↑) = spin down (↓) (Fig. 12.11a)

• with B (bias), the spin down increases in number (some

spin up electrons flip), resulting in net mm (Fig. 12.11b)


M

In a magnetized region of a ferromagnetic material such as iron allthe magnetic moments are spontaneously aligned in the samedirection There is a strong magnetization vector M even in theabsence of an applied field.

Ferromagnetism

Fe

• well above TC, ferromagnet obeys Curie-

Weiss law

Ferro | Para

TCurie

0B

0

H = 0 but M 0 (m )

HM m

LatinFerr(um) – iron

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• ferromagnetic atoms have many unpaired

electrons: matom results partly from morb ,

mainly from mspin (spin parallel)

• materials: transition metals (incomplete d

shells—Fe, Co, Ni), lanthanoids, actinoids

(incomplete f shells—Gd, Dy) (Table 12.3)

Origin:

• magnetic ordering (constructive)

• dipoles interact—exchange interaction (8.3),

dipoles align in parallel over a considerable

distance in solids → magnetic domain (8.5.1)

• effect very strong, results in intense external

magnetic field

• dipoles alignment destroyed at high

temperature due to thermal agitation

• above Curie temperature TC, ferromagnet

becomes paramagnet (dipoles disorder)

T

C

: Curie-Weiss constant


In this antiferromagnetic BCC crystal (Cr) the magnetic moment ofthe center atom is cancelled by the magnetic moments of the corneratoms (an eighth of the corner atom belongs to the unit cell).

Anti-FerromagnetismAnti-Ferro | Para

TNeel

• similar in principle to ferro, but dipoles align in opposite direction such that they completely

cancel, effectively no mm

• materials: mostly oxides of transition metals (Table 12.3)

Origin:

• magnetic ordering (destructive)

• (in oxides) superexchange (F5)equal amplitudes

opposite directions

M=0

0B

Cr

opposing

spins

none

MnO

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Illustration of magnetic ordering in a ferrimagnetic crystal. AllAatoms have their spins aligned in one direction and all Batomshave their spins aligned in the opposite direction. As the magneticmoment of an Aatom is greater than that of a Batom, there is netmagnetization, M, in the crystal.

Ferrimagnetism

Origin:

• magnetic ordering (partially destructive)

• (in oxides) double exchange (F5)

A B

M

0B

0

Materials:

- collectively called Ferrites

- typically poorly conducting (ceramic oxides)

- do not much suffer from eddy losses

- used in HF electronics

different amplitudes

opposite directions

Ferri | Para

TCurie

cubic ferrite: FeOFe2O3 (Fe3O4)

hexagonal ferrites: MO6(Fe2O3) (AB12O19)

* [A = Ba, Pb, St] * [B = Al, Ga, Cr, Fe]

garnets: M3Fe5O12

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FeOFe2O3 (or Fe3O4)

magnetite

Magnetite is the first magnet known (origin

of “magnet”), also known as lodestone

dipoles sum to 0✔

✘ ✘


atom has

unpaired electron(s)?

mm interact?

magnetic ordering?

Dia-

Para-

Ferro-Ferri-Anti-Ferro-

@ T > TCurie

Magnet-classification flowchart

N

N

Y

Y

destructive constructive

completely partially

@ T > TNeel

for ferro, ferri

for anti-ferro

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…atom / molecule / unit cell(gas) (liquid) (solid)

lattice

bound (mspin, morb)

unbound (mspin)


H = 0 H 0

Pair up:

ferromagnetism

diamagnetism

paramagnetism

moro mmmm 1

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brief exercise


8.4 Saturation Magnetization and Curie Temperature

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Iron

Msat(T)

Msat(0)

T / TC

Msat = condition at which all atomic moments have been aligned

lattice vibration

TCurie is the temperature at which

thermal energy = potential energy

(from vibration) (from exchange interaction)

Eex = kTC

relative scale

absolute scale


Eex (meV)

50

90

120


8.3 The Origin of Magnetism

• (metals) the origin of ferromagnetism and antiferromagnetism comes from exchange

interaction—the interaction that leads to chemical bonding where unpaired electrons parallel

(ferro) or anti-parallel (anti-ferro) their spins (F1)

• combined Pauli (exclusion principle) & Hund (first rule) → shell filling as shown in (F2)

• 3d elements: electron configuration, exchange energies, and 26Fe atom vs crystal (F3,F4)

• calculation exercise (F4), check answer with properties of ferromagnets (Table 8.3)

• (oxides) the origin of antiferromagnetism comes from superexchange, ferrimagnetism comes

from double exchange interactions (F5)

(F0)

d10

shell

s (l = 0)

p (l = 1)

d (l = 2)

f (l = 3)


(F1)Exchange interaction (exchange energy vs bonding energy)

• systems = nucleus + core electrons + the d shell (upto 10 electrons) (F0,)

• total energy E = (E1 + E2) −(E3 + E4) (highly simplified!)

• minimum energy configuration ↔ most thermodynamically stable

• Coulomb basics (general, regardless of spin):

• electron-electron repulsion decrease system stability (increase system energy) [E1]

• electron-proton attraction increase system stability (decrease system energy) [E2]

• Coulomb basics (specific to two d electrons in same shell, same atom): (F0,)

• opposite spin: tend to occupy overlapping region [E3 is negative]

• same spin: tend to avoid the same region [E3 is positive]

• Bonding basics (specific to outer electrons of adjacent atoms)

• opposite spin: bonding [E4 is positive]

• same spin: antibonding [E4 is negative]

• exchange energy (E3): the total energy E decrease due to spins trying to align in parallel

• bonding energy (E4): the total energy E decrease due to spins trying to align in antiparallel

• whether spins align or oppose depends on which factor dominates (E3 or E4)

• for isolated atoms: (no bonding required) Pauli & Hund (F2)

• for solid: (bonding required) depends on relative size of atoms vs d electrons, Bethe-

Slater curve (F4)

Magnetic Properties

Li

Be B

K

L p

(n=1)

(n=2)s

-1 0 1 = m

H

s

He

K

L p

(n=1)

(n=2)

Electronic configurations for the first five elements. Each box represents

an orbital (n, , m ).

F Ne

s

p

sK

L

s

p

sK

L

C N O

Electronic configurations for C, N, O, F and Ne atoms. Notice that Hund'srule forces electrons to align their spins in C, N and O. The Ne atom hasall the K and L orbitals full.

Fig 3.38

1925: Pauli exclusion principle

No two electrons may occupy the same quantum state

simultaneously. For example, if n, l, and ml are the same,

ms must be different (electrons have opposite spins).

1927: Hund’s First rule

Electrons in the same n, l orbitals prefer their spins

to be parallel (same ms)

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(F2)Shell filling of the first 10 elements

• illustrates the rules of nature (according to Pauli and Hund)


3d6 4s2

n = 3, l = 2

ml = -2,-1,0,1,2

4 unpaired electrons

intrinsic moments = 4mB

(elementary magnetic

dipole moment)

Fe = [Ar]3d64s2

Fe crystal (solid)

• levels (3d, 4s, 4p) broaden into bands (Fig. 12.14)

• 4s-3d overlap reduces the number of unpaired

electrons in each Fe atom from 4 to 2.2

• the d-shell of iron crystal is occupied by 7.78 electrons

(an increase of 7.78-6 = 1.78)

• the increase is from 4s which must decrease by same

amount, leaving 2-1.78 = 0.22 e/atom (conduction

electrons)

Fe atom (gas)

(F3)The 3d elements (Fe, …)

29Cu (dia) vs 26Fe (ferro)

crystal structure: Cu FCC a = 3.61 Å, Fe BCC a = 2.87 Å

atomic density: Cu 8.4912 atoms/cm3, Fe 8.4911 atoms/cm3

conduction electron/atom: Cu ~1, Fe ~0.22

resistivity: Cu 1.68×10−8 W⋅m, Fe 9.71×10−8 W⋅m

Cu ≈ Fe ≈ 8.51022 atoms/cm3


Ex. 19.1 Calculate saturation magnetization of Fe, given that Fe has a BCC lattice structure

with a = 2.866 Å. Compare with measured value of 2.1 T.

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(F4)

dia

para

anti-ferro

ferro

Z

21

22

23

24

25

26

27

2829

Bethe–Slater curve

Ex

cha

ng

e E

ner

gy

outer shell

(form bonds, no mm)

inner shell

(mm)

Cr

Gd


(F5)Superexchange and double exchange interactions

• systems = transition metal oxides (ionic bonding)

• (Fig. 12.19) superexchange leads to antiferromagnetism in NiO

• (Fig. 12.21) double exchange leads to ferromagnetic alignment in part of magnetite (Fe3O4, ferrite)

and some conductivity


The Magnetization State

• Fe is a permanent magnet, but most (macroscopic) Fe samples are non-magnetic or demagnetized

(do not attract/repel each other). Why?

• microstructure affects the macroscopic properties

• structure in the submillimeter length scale, approx 10−4-10−6 m or 1-100 mm

• example: Fe microstructure consists of polycrystal grains and domains (M1)

• each (polycrystal) grain has several (magnetic) domains (8.5.1) (M2)

• domains have magnetic dipoles pointing in different directions, but all are equivalent since

they belong to the easy axis; material exhibits magnetostatic anisotropy (8.5.2) (M3-4)

• the domains are separated by domain walls (8.5.3) (M5)

• to magnetize iron it is necessary to apply external magnetic field (H)

• macroscopically, this causes magnetostriction (8.5.4) (M6)

• microscopically, this causes domain wall motions (8.5.5) (M7-8)

• the magnetization-magnetizing field relationship, the M-H curve (8.5.6) (M9-11)

• to demagnetize, deperm (8.5.7) (M12)

• classification of magnets (soft/hard) based on M-H characteristics (8.6) (N0)

(M0)


source: http://www.ebsd.com/13-solving-problems-with-ebsd

Fe: microstructure

Technique: scanning electron microscopy (SEM)Mode: electron backscatter diffraction (EBSD)

actualschematic

(M1)


Closure domainsClosure domain

SN

N

N S

S

NS

90¡ domain wall

(c) (d)

N

NS

S

Domain wall (180¡)

N

S

M

(b)(a)

8.5.1 Magnetic Domains- steels (Fe + impurities [C]) are polycrystal; microstructure made up of small grains

- each grain consists of domain(s), except small grain

- grains are separated by grain boundaries (polycrystallinity)

- domains are separated by domain walls (magnetism)

- domain walls (DW) are created to reduce magnetostatic energy

Magnetostatic energy:

potential energy stored

in magnetic fields (B)

domain walls formed.

energetically more favorable

(magnetostatic energy ),

some external field lines

closure domains

(magnetostatic energy ),

no external field lines

demagnetized (M = 0)

*** magnetic domains creation continues until

reduction in external (B) potential energy = increase in internal (domain wall) potential energy

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)(J/m2/ 32 mB

magnetized (M ≠ 0)

1 grain

1 domain

1 grain

10 domains

(M2)


Hard

[111]

Medium

[110]

Easy

[100] OA

C

B

D

0

0.5

1

1.5

20 1 2 3 4

0 0.01 0.02 0.03 0.04 0.05 0.06

[100]

[111]

[110]

Applied magnetic field moH (T)

Magnetizing field H (104 A m-1)

P

Msat

Magnetocrystalline anisotrpy in a single iron crystal. M vs. Hdepends on the crystal direction and is easiest along [100] andhardest along [111]

8.5.2 Magnetocrystalline Anisotropy

magnetizations along

OA, OB, OC (easy)

rotate to OD

(difficult)

magnetization along some

planes are easier than others

(easy directions)

Fe (BCC)

MHB

HM

oo

m

mm

notes:

Analogy: optical anisotropy

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(M3)


Table 8.4 Exchange interaction, magnetocrystalline anisotropy energy K, and saturation

magnetostriction coefficient λsat

Material Crystal Eex ≈ kTC

(meV)

Easy Hard K

(mJ cm−3)(hard vs easy)

λsat

(× 10−6 )

Fe BCC 90 <100> <111> 48 20 [100]

−20 [111]

Co HCP 120 // to c axis ^ to c axis 450

Ni FCC 50 <111> <100> 5 −46 [100]

−24 [111]

K: energy required to magnetize a unit volume in a

particular direction w.r.t. the easy directions

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(8.5.4)(8.5.2)(8.3)

Fig.16.16 (Smith)

(M4)

Magnetic Properties

8.5.3 (Bloch) Domain Walls (DW)

31

In a Bloch wall the neighboring spin magnetic moments rotategradually and it takes several hunbdred atomic spacings to rotate themagnetic moment by 180°.

Potential energy

Domain wall thickness,

Exchange energy, Uexchange

Anisotropy energy, Uanisotropy

Domain wall

energy, Uwall

exchange interation (Hund’s rule) prefers

parallel spins ( or ). Hence, thick

walls (ideal = ) adjacent magnetic

dipoles tend to minimise , maximise

anisotropy energy (K) prefers thin walls

(ideal = 0) i.e. z –z in one atomic

spacing in the easy directions (see 8.5.2)

compromise: minimize total potential

energy ~ 0.1mm for Fe

N = /a

(+z)

(-z)

1 /

c.f. easy

direction

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Felix Bloch, Nobel 1952, "for the development of new methods for nuclear magnetic precision measurements"

(M5)


Original Fe crystal

x [100]

y [010]

+

H

Magnetostriction means that the iron crystal in a magnetic fieldalong x, an easy direction, elongates along x and but contracts inthe transverse dirtections

8.5.4 Magnetostriction

= dimensional change (+,-) as a result of magnetization

• caused by change in bond length ( torque)

• magnetic energy mechanical energy

stress ferro crystal a exchange interactions between atomic spins M

H M strain (a ) l l

l

ll

Strain:

magnetostrictive constant (sign

depends on direction, magnitude

of H)—see Fig. 16.16 (M4)

Magnetostriction is responsible

for hum noise near transformers

(l/l vibrate the surroundings)

(the opposite is true for Ni)

(analogy: piezoelectric)

Equilibrium domain configuration (few large domains vs many small domains) is determined by magnetostrictive energy and domain wall energy. (Smith p.895)

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(M6)


A B

[100]

HA B(a) (b)

(a) An unmagnetized crystal of iron in the absence of an applied magneticfield. Domains A and B are the same size and have oppositemagnetizations. (b) When an external magnetic field is applied he domainwall migrates into domain B which enlarges A and B. The result is that thespecimen now aqcuires net magnetization.

magnetized

A AB B

magnetization

results from

movements of

Bloch walls

enlarges A and shrinks B

spins (in walls and B)

gradually rotated by H

(they experience a torque)

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8.5.5 Domain wall motions

Magnetic materials placed in a magnetizing field (H) torque domain walls move

(M7)


Domain

Dislocation

Tension

Tension

Compression

BLOCH WALL

Domain

if wall gets close to dislocation

cancels

lowers strain energy

** Walls usually formed at dislocations

DWs move in response to changing magnetizing field H.

Motions are jerky due to DW pinning by

1. dislocations

Interaction of a Bloch wall with a non-magnetic (no permanentmagnetization) inclousion. (a) The inclusion becomes magnetizedand thereis magnetostatic energy. (b) This arrangement has lowerpotneital energy and is thus favorable.

Bloch wall

DomainDomain

S

N

S

N

ImpurityBloch wall

(a) (b)

2. inclusions / impurities

(M8)


M vs. H behavior of a previously unmagnetized polycrystalline iron specimen. An example grain in the unmagnetized specimen is that at O.

(a) Under very small fields the domain boundary motion is reversible. (b) The boundary motions are irreversible and occur in sudden jerks.

(c) Nearly all the grains are single domains with saturation magnetizations in the easy directions. (d) Magnetizations in individual grains

have to be rotated to align with the field, H. (e) When the field is removed the specimen returns along d to e. (f) To demagnetize the

specimen we have to apply a magnetizing field of Hc in the reverse direction.

Msat

M

O

ab

c

d

e

f

Mr

H

e

-Hc-x

-H

+x

H

H

a Reversible

boundary

motion

H

b Irreversible

boundary

motion

Hc Rotation

of MM

H

M

d Saturation

of M

Oabcd: Initial magnetization curve

discrete jumps due to sudden

jerks in wall motions

M in each grain rotates to

align parallel to the nearest

easy direction

remnant

magnetization

new domains

generated (nucleated

at impurity sites)

coercive fieldrepresents resistance to

demagnetization

Mr: remanant/residual magnetization

Msat: saturation magnetization

Hc: coercivity, coercive field

8.5.6 Polycrystalline materials and M-H curves

Barkhausen

effect

2102308

(a)-(c) do

main

gro

wth

(easy)

(d) d

om

ain ro

tatio

n

(difficu

lt)(M9)


Bsat

B

d

Br

g

-H

-B

-Bsat

-Br

(b)

H

Msat

M

O

d

Mr e

f

g

h

H-H

-M

-Msat

-Mr

-Hc

Hc Hsat-Hsat

i

(a)

Magnetization curves

MMHB oHM

oo mmm

The B vs. H hyterisis loop depends on the magntitude of the appliedfield in addition to the material and sample shape and size.

Bsat

B

H

-H

-B

Magnetized

to saturation

Small cyclic

applied field

Hsat

Bm

Hm

M-H B-H

Ferro- and ferri-

Hysteresis loss (energy dissipated

per unit volume per cycle of field

variation) due to:

- Joules loss (Eddy)

- heat loss (Barkhausen), required to

push DWs back & forth during

mag/demag

Saturation (major) hysteresis loop (datasheet)

2102308

M-H (T)

see (8.4)

(M10)


Definitions of (a) maximum permeability and (b) initial permeability

B

HO

Slope = mrimo

B = moH

(b)

B

HO

Slope = mrmaxmo

B = moH

P

(a)

Permeability

H

B

o

rm

m

maxrm rim

not a constant!

(usually quoted values in material datasheet)

2102308

(M11)

rim


Br

B

H-H

-B

-Hc

e

fe'

O

f'

Br'

8.5.7 Demagnetization

How to demagnetize magnets?

maybe

no B

HH

B

A magnetized specimen can be demagnetized by cycling the fieldintensity with a decreasing magnitude, i.e. tracing out smaller and smallerB-H loops until the origin is reached, H = 0.

Deperming

f e'

small domain walls

motions are reversible

(bounce back)

f ' O

possible only if f ' is

known precisely

2102308

(M12)


8.6 Soft and Hard Magnetic Materials

magnetizing

demagnetizing

B

H

−Hc Hc

- classification based on () readiness to magnetize/demagnetize (,), and power loss ():

- soft = easy to magnetize/demagnetize. Loss (hysteresis, BHmax) is low, suitable for

applications requiring repeated mag/demag cycles (motors, transformers, inductors...)

- hard = difficult to magnetize/demagnetize. Loss is high, suitable for permanent magnets,

data storage (conventional HDD)

- In the past (only), “soft/hard” magnets are physically “soft/hard” (ductile)

- wide range of magnetic materials, v. soft to v. hard (N1): soft (N2-N4), hard (N5-N7)

Energy Product:

energy stored in external

magnetic field (available

to do work)

B

H

−Hc

(B·H) product:

322 m

J

m

C/s

m

sV

m

A

m

Wb

m

AT

(N0)


ferro, ferrite ferro, ferriteReminder:

“ferro” = Fe + …

“ferrite” = Fe2O3 + …oxides

metallic

ceramic

(N1)


8.7 Soft Magnets

eddy*

low-field (H)

high-freq (f)MOFe2O3

(J/m3 per cycle)

(1 MHz)

Ni-Zn ferrite (200 MHz)

* Power loss (eddy) ∝ f 2/r

Mn,Zn,Ni,Fe

(N2)


(N3)


Fe-Si Fe-Ni

q) why silicon?

a) resistivity

Pros: 1. r ↑ (Nordheim) → eddy loss ↓

2. (K↓, mr↑) → (BH)↓

3. l↓→ hum noise ↓

Cons: Bsat ↓, TCurie ↓, ductility ↓→ limit to 4% Si

q) why nickel?

a) magnetostriction

Pros: (mr↑↑) ← (K↓,l↓)

note: l + for Fe, − for Ni

optimized at 78.5% Ni, 21.5% Fe

Ferrite

why ceramics?

Pros: r ↑↑→ eddy loss ↓↓

Cons: brittle

solid core laminated sheets

Ferro Ferrite

solid core

metallic glass

(Fe,Co,Ni) + (B,Si)- glassy (amorphous) state

≠ normal (crystalline) state

- melt → rapid solidification (106 °C/s)

- used in low-loss transformer core, recording head

- very soft (mri↑↑) ← DW moves very easily ←

microstructure contains no GB, no crystal

anisotropy (K)

(N4)


8.8 Hard Magnets

- Hysteresis loops wide and high

- Magnetizing: applied energy (H) converted and

stored as potential energy (Br)

- Demagnetizing: difficult (Hc ↑), of particular

importance is the second quadrant of the B-H curve

(N5)

f + d-electrons

d-electrons


Sm, Nd

4f metals(Lanthanides)

(N6)


Alnico Rare Earth

hard ferrite

Fe + (Al, Ni, Co)

Hc ↑ ← needle shape microstructure (K ↑)

- 4f elements, origin of dipole moments

same as 3d elements

- highest (BH)max – see pic

- Characteristics: tiny motors, large force

- Applications:

- medical devices (size critical):

implant pumps and valves

- automotive (weight critical): starting

motors

(Fe, Co) + (Nd, Sm)

MO6(Fe2O3)

Ba,Sr

Hc ↑ ← needle shape microstructure (K ↑)

(N7)


8.9 Superconductivity

- Superconductors conduct electricity without resistance below a critical temperature TC

- scientists are searching for materials that can superconduct at room temperature

- superconductivity = zero resistance (S1) + Meissner effect (S2)

- Meissner effect = expulsion of magnetic fields (B), causing levitation, applied in Maglev (S3)

- B must be below a critical value BC, otherwise material returns to normal state (S4)

- Type I, II superconductors lose superconducting properties suddenly, slowly (S5)

- Type II develop vortex state (transition state between normal & superconducting) (S6)

- superconducting materials must not carry current beyond a critical current density JC (S7)

- examples of Type I, II superconductors (S8); the most studied superconductor YBCO (S9)

- form of materials in applications: if brittle (ceramic, ionic bond) → only flat tape (S10), if ductile

(metallic bond) → coils (S11-S12), MRI (S13)

- origins of superconductivity in

- (conventional, low TC) Type I superconductors, Coopers pair (S14)

- (unconventional, high TC) Type II superconductors, ??

- record TC, note the (impractical) high pressures in many high-TC superconductors (S15)

(S0)


Temperature, T

Superconductor (e.g. Pb)

Tc

Normal metal

(e.g. Ag)rresidual

00

A superconductor such as lead evinces a transition to zero resisitivityat a critical temperature Tc (7.2 K for Pb) whereas a normal

conductor such as silver does not, and exhibits residual resisitivity atthe lowest temperatures.

1911

8.9.1 Zero resistance and Meissner effect

Tmp of Hg is 234 K (-38.8 degC)

(S1)

- Onnes first discovered zero resistance in solid Hg at liquid helium temperature (~4.2 K)

- normal metals (good conductors, Ag ) do not show superconductivity; at low temperatures they

show residual resistivities limited by scattering from impurities and lattice defects

- poor metals (such as Pb ) have zero resistance below critical temperature TC


Surface current I developed. Magnetization M and external field lines cancel everywhere inside the material.

Switching off field induces EMF (gives surface current) that opposes the change (Lenz’s law)

r = 0 and Meissner

r = 0 only

49

(S2)8.9.1 Zero resistance and Meissner effect


Left: A magnet over a superconductor becomes levitated. The superconductor is a perfect

diamagnet which means that there can be no magnetic field inside the superconductor.

Right: Photograph of a magnet levitating above a superconductor immersed in liquid nitrogen

(77 K). This is the Meissner effect. (SOURCE: Photo courtesy of Professor Paul C.W. Chu.)

N

SMagnet

Surface currents

Superconductor below Tc

Magnet

Superconductor above Tc

N

S

50

1m

Main Applications: Maglev, MRI (see later)

LN2 (77 K)

condensation

superconductormagnet bar

(S3)

http://images.google.com/imgres?imgurl=http://www.rise.org.au/info/Tech/scon/image001.jpg&imgrefurl=http://www.rise.org.au/info/Tech/scon/index.html&usg=__O-qgctMchJ-ynhBP4iDsEwu9S70=&h=496&w=650&sz=40&hl=en&start=15&um=1&tbnid=g2cRrjP-4QpBPM:&tbnh=105&tbnw=137&prev=/images?q=superconductor&hl=en&safe=off&rls=com.microsoft:en-us&sa=N&um=1


0 2 4 6 8 10

0

0.1

Lead

Superconducting

state

Normal state

Tc

Temperature (K)

Bc (Tesla)

The critical field vs temperature in Type I superconductors.

0 2 4 6 8

Temperature (K)

Lead

Tin

0

0.02

0.04

0.06

0.08

Mercury

Bc

(T)

External electric field does enter cladding even when i > c but with exponentially decreasing amplitude

External magnetic field does enter superconductor even below Tc but with exponentially decreasing amplitude:

Near 0 K: l ~ 10 – 100 nm

If B field too high, l > sample size.

At critical field Bc, l . Superconductivity is lost.

l/exp0 xBxB

51

cT

TBBC

2

01

0B

(S4)


Characteristics of Type I and Type II superconductors. B = µoH is the

applied field and M is the overall magnetization of the sample. Field inside the sample,

Binside = µoH + µoM, which is zero only for B < Bc (Type I) and B < Bc1 (Type II).

8.9.2 Type I and Type II superconductors.

Loss is sudden Loss is gradual

52

(S5)


The mixed or vortex state in a Type II superconductor.

Tc0

Normal stateBc2

Bc1

Vortex state

Meisner state

Critical magnetic field

Kasap, Fig 8.51

Applied fields able to pierce through local tubular

(filamentary) regions of normal state embedded in the

superconducting state.

r = 0, B = 0

r = 0, B 0

r 0, B 0

53

Normal state

Superconducting state

Magnetic field lines

Vortex of flux lines

Supercurrent areas:

Meissner state Vortex state

(Type I, II) (Type II only)

(S6)


T

B

J

24.5 T

18 K

~107 A cm-2

Jc

Bc2

Tc

Nb3Sn

The critical surface for a niobium-tin alloy which is a Type IIsuperconductor.

8.9.3 Critical current density (Jc, Ic)

Current through material generates magnetic fields. If current too high, surface magnetic field will exceed Bc

and superconductivity is lost—true for Type I; Type II more complicated..

54

J-B-T surface dictates

the operating limits

Practical superconducting wires are limited to Type II

- Type I: superconductivity destroyed too easily, operation limited to low Ic, Bc

- Type II: superconductivity persist to much greater J-B-T values, hence high fields operation possible

compare values in

table next slide

(S7)

Magnetic Properties

1970s 1987 2000s

552102308

1911

(S8)


YBCO (123)

- most studied superconductor

- can be considered a defective perovskite structure with 3

perovskite unit cells stacked as shown below

- “defective” = oxygen vacancies provide coupling of

electrons in the CuO2 planes

Despite its high Tc, YBCO (a ceramic) is brittle, has low current density applications limited to thin-film electronics

(S9)

Magnetic Properties

These high temperature superconductor (HTS) flat

tapes are based on (Bi2-xPbx)Sr2Ca2Cu3O10-d(Bi-2223).

The tape has an outer surrounding protective metallic

sheath. Right: HTS tapes having ac power loss below

10 mW/m have a major advantage over equivalent-

sized metal conductors, in being able to transmit

considerably higher power loads. Coils made from

HTS tape can be used to create more compact and

efficient motors, generators, magnets, transformers

and energy storage devices.

| SOURCE: Australian Superconductors.

572102308

source: W

iki (A

pr 2

016)

Compound (ceramic) superconductors

Bi-2223

(S10)


A solenoid carrying a current experiences radial forces pushing the coilapart and axis froces compressing the coil.

Mechanical

support structure

Coil windings

Radial forces

Air

Superconductor

Copper matrix

Solenoid

58

Nb3Sn, Tc = 18 K

for mechanical strength +in case superconductor fails (B > Bc, J > Jc, T > Tc)

Typical cross-section of a high-current high-field superconducting coil

used in MRI, Maglev, Particle accelerator (CERN)...

(S11)

Compound (intermetallic) superconductors


Superconducting electromagnets used in MRI.

Operates with liquid He, providing a magnetic field

0.5–1.5 T.

Earth’s magnetic field: 50 mT

Record B in 2012: 15 Tesla

Extra reading:

Spectrum.IEEE.Org Nov 2013

The world’s most powerful MRI…

Current:

1.5-3 T, 1 mm (10000 neurons), s

Future:12 T, 0.1 mm (1000 neurons), 0.1 s

Magnetic Resonant Imaging (MRI)(S10)


http

://ww

w.sp

rawls.o

rg/m

ripm

t/MR

I01

/index

.htm

l

Nuclear Magnetic Resonance (NMR)(S13)

Magnetic Properties

Lattice vibration

1

2

A pictorial and intuitive view of an indirect attraction between twooppositely travelling electrons via a lattice distorsion and vibration.

8.10 Superconductivity origin- 1911: Onnes, first discovery

- 1957: BCS theory (Bardeen, Cooper, Schrieffer)

- Cooper pairs: a pair of oppositely spinning and travelling electrons

- #1 distorts positive nuclei, #2 feels net attractive force

- #1 & #2 do not directly interact, but effectively attracted to each other via lattice distortion

(normally Coulombic repulsion)

- Temperature must be sufficiently low such that random thermal vibrations are weak

- Cooper pair has no net spin, do not obey Fermi-Dirac statistics (Pauli exclusion), and can condense

to lowest energy state, having one wavefunction extending the whole sample

- a crystal imperfection cannot scatter a single Cooper pair since all pairs act as one

- Superconductivity is said to be a macroscopic manifestation of quantum mechanics

612102308

(S14)


Nobel prizes related to Superconductivity (source: http://ieeecsc.org/pages/nobel-laureates-superconductivity)

1913: Onnes 1911, “for his investigations on the properties of matter at low temperatures…”

1972: Bardeen, Cooper, Schrieffer, “theory (BCS theory 1957) of superconductivity” (type I)

1973: Josephson, “for his theoretical predictions of the properties of a supercurrent through a tunnel barrier”

1973: Esaki, Giaever, “experimental discoveries of tunneling phenomena in semiconductors and superconductors”.

1987: Bednorz and Müller, “discovery of superconductivity in ceramic materials”

2003: Abrikosov, Ginsburg, Leggett , “theory of superconductors…” (type II)

Nobel prizes related to nuclear magnetic resonance (NMR) (source: http://www.nobelprize.org/)

1952: Bloch, Purcell, "methods for nuclear magnetic precision measurements“ (1D)2003: Lauterbur, Mansfield, "for discovering magnetic resonance imaging (MRI)” (3D, magnetic field gradient)

source: Wiki (Apr 2016) (S15)


Important Units

T = Wb/m2

Wb = HA = Vs

2102308

A

B

C

D

E

F

G

m

item

H

B is known as:

- magnetic field

- (magnetic) induction

- (magnetic) flux density

Appendix (Z1)


A

B

C

E

F

m

item

H

Appendix (Z2)


source: qdusa.com

Appendix (X3)


http://www.lightningmaps.org

This map shows lightning densities in Europe


https://spacetourismguide.com/northern-lights-sweden/

This map shows the best places to watch northern lights

---- business vs people

What local people? Examining the Gállok mining conflict and the rights of the Sámi population in terms

of justice and power

https://doi.org/10.1016/j.geoforum.2017.08.009

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