Transcript 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
E.C. Stoner, c. 1934 Courtesy of AIP Emilio Segre Visual Archives June 2010 E. C. Stoner, F.R.S.
and E. P. Wohlfarth (no photo) (Note: F.R.S. = “Fellow of the Royal Society”) TU-Chemnitz 2
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 June 2010 TU-Chemnitz 3
Hysteresis loop TU-Chemnitz
M r
= Remanence
M s
= Saturation Magnetization
H c
= Coercivity 4 June 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
TU-Chemnitz 5 June 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 June 2010 TU-Chemnitz 6
Not Stoner Theory of Band Ferromagnetism 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.
June 2010 TU-Chemnitz 7
Small magnetic particles June 2010 TU-Chemnitz 8
Why are we interested? (since 1948!)
Magnetic nanostructures!
Can be single domain, uniform/constant magnetization, no long-range order between particles, anisotropic.
June 2010 TU-Chemnitz 9
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 June 2010 TU-Chemnitz 10
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 June 2010 TU-Chemnitz 11
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 June 2010 TU-Chemnitz 12
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 June 2010 TU-Chemnitz 13
Units, Terminology, Notation E.g.
Gaussian e-m units 1 Oe = 1000/4 π × A/m Older terminology “interchange interaction energy” = “exchange interaction energy” Older notation
I
0 = magnetization vector June 2010 TU-Chemnitz 14
Mathematical Starting Point Applied field energy
E H
HI
0 cos Anisotropy energy
E A
Total energy
E
E H
E A
(what should we use?) (later, drop constants) June 2010 TU-Chemnitz 15
MAGNETIC ANISOTROPY Shape anisotropy (dipole interaction) Strain anisotropy Magnetocrystalline anisotropy Surface anisotropy Interface anisotropy Chemical ordering anisotropy Spin-orbit interaction Local structural anisotropy June 2010 TU-Chemnitz 16
Ellipsoidal particles This gives shape anisotropy – from demagnetizing fields (to be discussed later if there is time).
TU-Chemnitz Spherical particles would not have shape anisotropy, but would have magnetocrystalline and strain anisotropy – leading to the same physics with redefined parameters.
17 June 2010
Ellipsoidal particles We will look at one ellipsoidal particle, then average over a random orientation of particles.
TU-Chemnitz The transverse components of mag netization will cancel, and the net magnetiza tion can be calculated as the component along the applied field direction.
18 June 2010
Demagnetizing fields → anisotropy June 2010
B
H
4
M
,
B
0 , TU-Chemnitz
H
from Bertotti
M
19
Prolate and Oblate Spheroids These show all the essential physics of the more general ellipsoid.
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How do we get hysteresis?
I
0 Easy Axis
H
June 2010 TU-Chemnitz 21
SW Fig. 1 – important notation One can prove (SW outline the proof in Sec. 5(ii)) that for ellipsoids of revolution H, I a plane.
0 , and the easy axis all lie in June 2010 TU-Chemnitz 22
No hysteresis for oblate case
I
0
H
360 o Easy Axis degenerate June 2010 TU-Chemnitz 23
Mathematical Starting Point - again Applied field energy
E H
HI
0 cos Anisotropy energy
E A
1 2
I
0 2
N a
cos 2 Total energy
E
E H
E A
N b
sin 2 (later, drop constants) June 2010 TU-Chemnitz 24
Dimensionless variables Total energy: normalize to drop constant term.
N b
N a
I
0 2 and Dimensionless energy is then 1 4 cos 2
h
cos
h
N b
H N a
I
0 TU-Chemnitz June 2010 25
Energy surface for fixed θ θ = 10 o June 2010 TU-Chemnitz 26
Stationary points (max & min) θ = 10 o June 2010 TU-Chemnitz 27
SW Fig. 2 June 2010 TU-Chemnitz 28
SW Fig. 3 June 2010 TU-Chemnitz 29
Examples in Maple June 2010 (This would be easy to do with Mathematica, also.) [SW_Lectures_energy_surfaces.mw] TU-Chemnitz 30
Calculating the Hysteresis Loop June 2010 TU-Chemnitz 31
June 2010 TU-Chemnitz from Blundell 32
SW Fig. 6 June 2010 TU-Chemnitz 33
Examples in Maple June 2010 [SW_Lectures_hysteresis.mw] TU-Chemnitz 34
H sw and H c June 2010 TU-Chemnitz 35
Hysteresis Loops: 0-45 o – symmetries and 45-90 o June 2010 TU-Chemnitz from Blundell 36
Hysteresis loop for θ = 90 o June 2010 TU-Chemnitz from Jiles 37
Hysteresis loop for θ = 0 o June 2010 TU-Chemnitz from Jiles 38
Hysteresis loop for θ = 45 o June 2010 TU-Chemnitz from Jiles 39
Average over Orientations cos
I H I
0 2 2 0 cos 0 2 2 sin sin
d
d
0 2 cos sin
d
June 2010 TU-Chemnitz 40
SW Fig. 7 June 2010 TU-Chemnitz 41
Part 2
1.
Conditions for large coercivity 2.
Applied field 3.
Various forms of magnetic anisotropy 4.
Conditions for single-domain ellipsoidal particles June 2010 TU-Chemnitz 42
Demagnetization Coefficients: large H
c
possible
m=a/b I
0 ~10 3
h
N b
H N a
I
0 SW Fig. 8 June 2010 TU-Chemnitz 43
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!
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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 June 2010 TU-Chemnitz 45
Demagnetizing Field Energy Energetics of magnetic media are very subtle.
H d
M
H d
is the “demagnetizing field” from Blundell June 2010 TU-Chemnitz 46
Demagnetizing fields → anisotropy June 2010
B
H
4
M
,
B
0 , TU-Chemnitz
H
from Bertotti
M
47
How does depend on shape?
d
H d
is extremely complicated for arbitrarily shaped ferromagnets, but relatively simple for ellipsoidal ones.
H di
j N ij M j
And in principal axis coordinate system for the ellipsoid, June 2010 TU-Chemnitz 48
Ellipsoids
N
Tr Tr
N
N a
0 0
N a N
1 0
N
0
b N b
(SI units) 0 0
N c N c
4 (Gaussian units) June 2010 TU-Chemnitz 49
Examples Sphere
N a
N b
N c
4 3 , Long cylindrical rod
N a
N b
2 ,
N c
0
H d
4 3
M
Flat plate
N a
N b
0 ,
N c
4 June 2010 TU-Chemnitz 50
Ferromagnet of Arbitrary Shape
E d E tot
1 2
V
M E Zeeman
H d
E d d
June 2010 TU-Chemnitz 51
Ellipsoids (again) General
E d
1 2
I
0 2
N a
cos 2
x
N b
cos 2
y
N c
cos 2
z
E d E d
Prolate spheroid
E d
1 2 1 2 1 4
I
0 2
I
0 2
N N I
0 2
N
2 1
a a
cos 2 ( 90 )
c I
sin 0 2 2
N N a a
N
N
cos
c
c
2 cos 2
N
b
cos 2 TU-Chemnitz 90
N c
cos 2 June 2010 52
Magnetocrystalline Anisotropy Uniaxial case is approximately the same mathematics as prolate spheroid. E.g. hexagonal cobalt:
E A h mc
K
sin 2
HI
0 2
K
1
K
1
K
cos 2 2 2 For spherical, single domain particles of Co with easy axes oriented at random, coercivities ~2900 Oe. are possible.
June 2010 TU-Chemnitz 53
Strain Anisotropy Uniaxial strain – again, approximately the same mathematics as prolate spheroid. E.g. magnetostriction coefficient λ, uniform tension σ:
E A
3 2 sin 2 3 4 3 4 cos 2
h
HI
3 0 June 2010 TU-Chemnitz 54
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 I 0 ) Prolate spheroids of Co shape > mc for m > 3 shape > σ for m > 1.08
June 2010 TU-Chemnitz 55
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 June 2010 TU-Chemnitz 56
Domain Walls (Bloch walls) Energies Exchange energy: costs energy to rotate so 2
J
Rotation of N spins through total angle π ,
S
1
S
/ 2
N
2
JS
2 cos , requires energy per unit area
ex
JS
2
Na
2 2 .
Anisotropy energy June 2010 TU-Chemnitz 57
Domain Walls (2) Anisotropy energy: magnetocrystalline easy axis vs. hard axis (from spin-orbit interaction and partial quenching of angular momentum) shape demagnetizing energy It costs energy to rotate out of the easy direction: say,
E
K
sin 2 .
June 2010 TU-Chemnitz 58
Domain Walls (3) Anisotropy energy Taking
E
K
sin 2 ,
an
NKa
, 2 so
BW
for example,
JS
2
Na
2 2
NKa
.
2 June 2010 Then we minimize energy to find
N
S
2
J Ka
3 ,
Na
S
2
J Ka
,
BW
S
2
JK a
.
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Conditions for Single Domain Ellipsoidal Particles (2) Demagnetizing field energy
E D
1 2
N a I
0 2 Uniform magnetization if E
D < E wall
Fe: 10 5 – 10 6 atoms Ni: 10 7 – 10 11 atoms June 2010 TU-Chemnitz 60
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 June 2010 TU-Chemnitz 61