stoner-wohlfarth theory

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Stoner-Wohlfarth Theory “A Mechanism of Magnetic Hysteresis in Heterogenous Alloys” Stoner E C and Wohlfarth E P (1948), Phil. Trans. Roy. Soc. A240:599–642 Prof. Bill Evenson, Utah Valley University

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Stoner-Wohlfarth Theory. “A Mechanism of Magnetic Hysteresis in Heterogenous Alloys” Stoner E C and Wohlfarth E P (1948), Phil. Trans. Roy. Soc. A240 :599–642 Prof. Bill Evenson , Utah Valley University. E.C. Stoner, c. 1934 . E. C. Stoner, F.R.S. and E. P. Wohlfarth (no photo) - PowerPoint PPT Presentation

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Page 1: Stoner-Wohlfarth Theory

Stoner-Wohlfarth Theory

“A Mechanism of Magnetic Hysteresis in Heterogenous Alloys”Stoner E C and Wohlfarth E P (1948), Phil. Trans. Roy. Soc. A240:599–642

Prof. Bill Evenson, Utah Valley University

Page 2: Stoner-Wohlfarth Theory

TU-Chemnitz 2June 2010

E.C. Stoner, c. 1934

E. C. Stoner, F.R.S.and E. P. Wohlfarth (nophoto)

(Note: F.R.S. = “Fellow of the

Royal Society”)Courtesy of AIP Emilio Segre Visual Archives

Page 3: Stoner-Wohlfarth Theory

TU-Chemnitz 3June 2010

Stoner-Wohlfarth Motivation How to account for very high

coercivities Domain wall motion cannot explain

How to deal with small magnetic particles (e.g. grains or imbedded magnetic clusters in an alloy or mixture) Sufficiently small particles can only have

a single domain

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TU-Chemnitz 4June 2010

Hysteresis loop

Mr = Remanence

Ms = Saturation Magnetization

Hc = Coercivity

Page 5: Stoner-Wohlfarth Theory

TU-Chemnitz 5June 2010

Domain Walls Weiss proposed

the existence of magnetic domains in 1906-1907 What elementary

evidence suggests these structures?

www.cms.tuwien.ac.at/Nanoscience/Magnetism/magnetic-domains/magnetic_domains.htm

Page 6: Stoner-Wohlfarth Theory

TU-Chemnitz 6June 2010

Stoner-Wohlfarth Problem Single domain particles (too small for

domain walls) Magnetization of a particle is uniform

and of constant magnitude Magnetization of a particle responds

to external magnetic field and anisotropy energy

Page 7: Stoner-Wohlfarth Theory

TU-Chemnitz 7June 2010

Not Stoner Theory of BandFerromagnetism

The Stoner-Wohlfarth theory of hysteresis does not refer to the Stoner (or Stoner-Slater) theory of band ferromagnetism or to such terms as “Stoner criterion”, “Stoner excitations”, etc.

Page 8: Stoner-Wohlfarth Theory

TU-Chemnitz 8June 2010

Small magnetic particles

Page 9: Stoner-Wohlfarth Theory

TU-Chemnitz 9June 2010

Why are we interested? (since 1948!)

Magneticnanostructures!

Can be single domain, uniform/constant magnetization, no long-range order between particles, anisotropic.

Page 10: Stoner-Wohlfarth Theory

TU-Chemnitz 10June 2010

Physics in SW Theory

Classical e & m (demagnetization fields, dipole)

Weiss molecular field (exchange) Ellipsoidal particles for shape anisotropy Phenomenological magnetocrystalline

and strain anisotropies Energy minimization

Page 11: Stoner-Wohlfarth Theory

TU-Chemnitz 11June 2010

Outline of SW 1948 (1) 1. Introduction

review of existing theories of domain wall motion (energy, process, effect of internal stress variations, effect of changing domain wall area – especially due to nonmagnetic inclusions)

critique of boundary movement theory Alternative process: rotation of single

domains (small magnetic particles – superparamagnetism) – roles of magneto-crystalline, strain, and shape anisotropies

Page 12: Stoner-Wohlfarth Theory

TU-Chemnitz 12June 2010

Outline of SW 1948 (2)

2. Field Dependence of Magnetization Direction of a Uniformly

Magnetized Ellipsoid – shape anisotropy

3. Computational Details 4. Prolate Spheroid Case 5. Oblate Spheroid and General

Ellipsoid

Page 13: Stoner-Wohlfarth Theory

TU-Chemnitz 13June 2010

Outline of SW 1948 (3)

6. Conditions for Single Domain Ellipsoidal Particles

7. Physical Implications types of magnetic anisotropy

magnetocrystalline, strain, shape ferromagnetic materials

metals & alloys containing FM impurities powder magnets high coercivity alloys

Page 14: Stoner-Wohlfarth Theory

TU-Chemnitz 14June 2010

Units, Terminology, NotationE.g. Gaussian e-m units

1 Oe = 1000/4π × A/m Older terminology

“interchange interaction energy” = “exchange interaction energy”

Older notation I0 = magnetization vector

Page 15: Stoner-Wohlfarth Theory

TU-Chemnitz 15June 2010

Mathematical Starting Point Applied field energy

Anisotropy energy

Total energyAE

AH EEE

cos0HIEH

(later, drop constants)

(what should we use?)

Page 16: Stoner-Wohlfarth Theory

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MAGNETIC ANISOTROPY Shape anisotropy (dipole interaction) Strain anisotropy Magnetocrystalline anisotropy Surface anisotropy Interface anisotropy Chemical ordering anisotropy Spin-orbit interaction Local structural anisotropy

Page 17: Stoner-Wohlfarth Theory

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Ellipsoidal particlesThis gives shape anisotropy – from demagnetizing fields (to be discussed later if there is time).

Spherical particles would not have shape anisotropy, but would have magnetocrystalline and strain anisotropy – leading to the same physics with redefined parameters.

Page 18: Stoner-Wohlfarth Theory

TU-Chemnitz 18June 2010

Ellipsoidal particlesWe will look at one ellipsoidal particle, then average over a random orientation of particles.

The transverse components of mag-netization will cancel, and the net magnetiza-tion can be calculated as the component along the applied field direction.

Page 19: Stoner-Wohlfarth Theory

TU-Chemnitz 19June 2010

Demagnetizing fields → anisotropy

MHBMHB

,0,4from Bertotti

Page 20: Stoner-Wohlfarth Theory

TU-Chemnitz 20June 2010

Prolate and Oblate Spheroids

These show all the essential physics of the more general ellipsoid.

Page 21: Stoner-Wohlfarth Theory

TU-Chemnitz 21June 2010

How do we get hysteresis?

H

I0Easy Axis

Page 22: Stoner-Wohlfarth Theory

TU-Chemnitz 22June 2010

SW Fig. 1 – important notation

One can prove (SW outline the proof in Sec. 5(ii)) that for ellipsoids of revolution H, I0, and the easy axis all lie ina plane.

Page 23: Stoner-Wohlfarth Theory

TU-Chemnitz 23June 2010

No hysteresis for oblate case

Easy Axis360o degenerate

H

I0

Page 24: Stoner-Wohlfarth Theory

TU-Chemnitz 24June 2010

Mathematical Starting Point - again Applied field energy

Anisotropy energy

Total energy 222

021 sincos baA NNIE

AH EEE

cos0HIEH

(later, drop constants)

Page 25: Stoner-Wohlfarth Theory

TU-Chemnitz 25June 2010

Dimensionless variables

Total energy: normalize to and drop constant term.

Dimensionless energy is then

20INN ab

cos2cos41 h

0INNHh

ab

Page 26: Stoner-Wohlfarth Theory

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Energy surface for fixed θ

θ = 10o

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TU-Chemnitz 27June 2010

Stationary points (max & min)

θ = 10o

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SW Fig. 2

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SW Fig. 3

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Examples in Maple

(This would be easy to do with Mathematica, also.)

[SW_Lectures_energy_surfaces.mw]

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Calculating the Hysteresis Loop

Page 32: Stoner-Wohlfarth Theory

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from Blundell

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SW Fig. 6

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Examples in Maple

[SW_Lectures_hysteresis.mw]

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Hsw and Hc

Page 36: Stoner-Wohlfarth Theory

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fromBlundell

Hysteresis Loops: 0-45o and 45-90o

– symmetries

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TU-Chemnitz 37June 2010

Hysteresis loop for θ = 90o

fromJiles

Page 38: Stoner-Wohlfarth Theory

TU-Chemnitz 38June 2010

Hysteresis loop for θ = 0o

fromJiles

Page 39: Stoner-Wohlfarth Theory

TU-Chemnitz 39June 2010

Hysteresis loop for θ = 45o

fromJiles

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TU-Chemnitz 40June 2010

Average over Orientations

2

0

2

0

2

0

0

sincos

sin2

sincos2cos

d

d

d

IIH

Page 41: Stoner-Wohlfarth Theory

TU-Chemnitz 41June 2010

SW Fig. 7

Page 42: Stoner-Wohlfarth Theory

TU-Chemnitz 42

Part 21. Conditions for large coercivity2. Applied field3. Various forms of magnetic

anisotropy4. Conditions for single-domain

ellipsoidal particlesJune 2010

Page 43: Stoner-Wohlfarth Theory

TU-Chemnitz 43June 2010

DemagnetizationCoefficients: large Hc possible

SW Fig. 8m=a/bI0~103 0INN

Hhab

Page 44: Stoner-Wohlfarth Theory

TU-Chemnitz 44June 2010

Applied Field, H Important! This is the total field

experienced by an individual particle.It must include the field due to the magnetizations of all the other particles around the one we calculate!

Page 45: Stoner-Wohlfarth Theory

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Magnetic Anisotropy Regardless of the origin of the

anisotropy energy, the basic physics is approximately the same as we have calculated for prolate spheroids.

This is explicitly true for Shape anisotropy Magnetocrystalline anisotropy (uniaxial) Strain anisotropy

Page 46: Stoner-Wohlfarth Theory

TU-Chemnitz 46June 2010

Demagnetizing Field Energy Energetics of magnetic media are very

subtle.

is the “demagnetizing field”

MH d

dH

from Blundell

Page 47: Stoner-Wohlfarth Theory

TU-Chemnitz 47June 2010

Demagnetizing fields → anisotropy

MHBMHB

,0,4from Bertotti

Page 48: Stoner-Wohlfarth Theory

TU-Chemnitz 48June 2010

How does depend on shape?

dH

dH

is extremely complicated for arbitrarily shaped ferromagnets, but relatively simple for ellipsoidal ones.

j

jijdi MNH

And in principal axis coordinate system for the ellipsoid,

Page 49: Stoner-Wohlfarth Theory

TU-Chemnitz 49June 2010

1Tr4Tr

000000

NNNNN

NN

NN

cba

c

b

a

Ellipsoids

(SI units)(Gaussian units)

Page 50: Stoner-Wohlfarth Theory

TU-Chemnitz 50June 2010

Examples Sphere

Long cylindrical rod

Flat plate0,2 cba NNN

4,0 cba NNN

MHNNN dcba

3

43

4 ,

Page 51: Stoner-Wohlfarth Theory

TU-Chemnitz 51June 2010

Ferromagnet of Arbitrary Shape

dZeemantot

Vdd

EEE

dHME

21

Page 52: Stoner-Wohlfarth Theory

TU-Chemnitz 52June 2010

Ellipsoids (again) General

Prolate spheroid

zcybxad NNNIE 222202

1 coscoscos

222

021

222202

1

cossin

cos90cos)90(cos

cad

cbad

NNIE

NNNIE

2cos2

041

202

1

ac

cad

NNI

NNIE

Page 53: Stoner-Wohlfarth Theory

TU-Chemnitz 53June 2010

Magnetocrystalline Anisotropy Uniaxial case is approximately the same

mathematics as prolate spheroid. E.g. hexagonal cobalt:

2cossin 21

212 KKKEA

KHIhmc 2

0For spherical, single domain particles of Co with easy axes oriented at random, coercivities ~2900 Oe. are possible.

Page 54: Stoner-Wohlfarth Theory

TU-Chemnitz 54June 2010

Strain Anisotropy Uniaxial strain – again, approximately the

same mathematics as prolate spheroid. E.g. magnetostriction coefficient λ, uniform tension σ:

2cossin 43

432

23 AE

30HIh

Page 55: Stoner-Wohlfarth Theory

TU-Chemnitz 55June 2010

Magnitudes of Anisotropies Prolate spheroids of Fe (m = a/b)

shape > mc for m > 1.05 shape > σ for m > 1.08

Prolate spheroids of Ni shape > mc for m > 1.09 σ > shape for all m (large λ, small I0)

Prolate spheroids of Co shape > mc for m > 3 shape > σ for m > 1.08

Page 56: Stoner-Wohlfarth Theory

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Conditions for Single Domain Ellipsoidal Particles

Number of atoms must be large enough for ferromagnetic order

within the particle small enough so that domain boundary

formation is not energetically possible

Page 57: Stoner-Wohlfarth Theory

TU-Chemnitz 57June 2010

Domain Walls (Bloch walls) Energies

Exchange energy: costs energy to rotate neighboring spins

Rotation of N spins through total angle π, so , requires energy per unit area

Anisotropy energy

cos22 221 JSSSJ

N/

.22

2

NaJSex

Page 58: Stoner-Wohlfarth Theory

TU-Chemnitz 58June 2010

Domain Walls (2) Anisotropy energy:

magnetocrystallineeasy axis vs. hard axis(from spin-orbit interaction and partial quenching of angular momentum)shapedemagnetizing energy

It costs energy to rotate out of the easydirection: say, .sin2 KE

Page 59: Stoner-Wohlfarth Theory

TU-Chemnitz 59June 2010

Domain Walls (3) Anisotropy energy

Taking for example,

Then we minimize energy to find

,sin2 KE

so,2NKa

an .22

22 NKaNa

JSBW

,2 3KaJSN ,2 KaJSNa

.2 aJKSBW

Page 60: Stoner-Wohlfarth Theory

TU-Chemnitz 60June 2010

Conditions for Single Domain Ellipsoidal Particles (2) Demagnetizing field energy

Uniform magnetization if ED < Ewall

Fe: 105 – 106 atoms

Ni: 107 – 1011 atoms

202

1 INE aD

Page 61: Stoner-Wohlfarth Theory

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Thanks

Friends at Uni-Konstanz, where this work was first carried out – some of this group are now at TU-Chemnitz

Prof. Manfred Albrecht for invitation, hospitality and support