Magnetism - Graz University of Technologyhadley/ss2/lectures17W/nov30.pdf · Magnet Br(T) Hci(kA/m)...
Transcript of Magnetism - Graz University of Technologyhadley/ss2/lectures17W/nov30.pdf · Magnet Br(T) Hci(kA/m)...
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Magnetism
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Mean field theory
Experimental points for Ni.tanh mm
t
tanh cS
s
T MM MT M
Source: Kittel and 2 4S B c
B
N zM g T JV k
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FerromagnetismMaterial Curie temp. (K)
Co 1388Fe 1043FeOFe2O3 858NiOFe2O3 858CuOFe2O3 728MgOFe2O3 713MnBi 630Ni 627MnSb 587MnOFe2O3 573Y3Fe5O12 560CrO2 386MnAs 318Gd 292Dy 88EuO 69 Electrical insulatorNd2Fe14B 353 Ms = 10 Ms(Fe) Sm2Co17 700 rare earth magnets
2
4
S B
cB
NM gV
zT Jk
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Curie - Weiss law
1 tanh2 2
B MF aB
B
g B BNM gV k T
2 22 2
14 B a
B B
N VM g zJM BVk T Ng
Above Tc we can expand the hyperbolic tangent
2 2
4aB
B c
Bg NMVk T T
c
dM CdH T T
2 2MFB
VB zJMNg
Critical fluctuations near Tc
Solve for M
Curie Weiss Law
4cB
zT Jk
tanh( ) for 1x x x
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Ferromagnets are paramagnetic above Tc
Ferromagnetic
Paramagnetic
Critical fluctuations near Tc.
Source: Kittel
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Magnetic ordering
Ferromagnetism
Ferrimagnetism
Antiferromagnetism
Helimagnetism
All ordered magnetic states have excitations called magnons
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Ferrimagnets
Magnetite Fe3O4(Magneteisen)
Ferrites MO.Fe2O3
M = Fe, Zn, Cd, Ni, Cu, Co, Mg
Spinel crystal structure XY2O4
8 tetrahedral sites A (surrounded by 4 O) 5B
16 octahedral sites B (surrounded by 6 O) 9B
per unit cell
Two sublattices A and B.MgAl2O4
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Ferrimagnets
Magnetite Fe3O4
Ferrites MO.Fe2O3
M = Fe, Zn, Cd, Ni, Cu, Co, Mg
Exchange integrals JAA, JAB, and JBB are all negative (antiparallel preferred)
|JAB| > |JAA|,|JBB|
Magnetite
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Ferrimagnetismgauss = 10-4 Toersted = 10-4/4×10-7 A/m
KittelD. Gignoux, magnetic properties of Metallic systems
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At low temperatures, below the Neel temperature TN, the spins are aligned antiparallel and the macroscopic magnetization is zero.
Spin ordering can be observed by neutron scattering.
At high temperature antiferromagnets become paramagnetic. The macroscopic magnetization is zero and the spins are disordered in zero field.
Antiferromagnetism
Negative exchange energy JAB < 0.
0A B
a
M M CTB
Curie-Weiss temperature
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Antiferromagnetism
Average spontaneous magnetization is zero at all temperatures.
Source: Kittel
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from Kittel
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Applications of magnetism
Hard magnets: permanent magnets, motors, generators, microphones
Soft magnets: transformers
Magnetic recording
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Magnetic domains (weisssche Bezirke )
Magnetic energy density2
02B
Costs energy to introduce domain walls where spin up regions are adjacent to spin down regions.
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Irreproducible jump between c and d.
Ferromagnetic domains
Weak fields: favorable domains expandStrong fields: domains rotate to align with field
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Magnetizing a magnet
Weak fields: favorable domains expandStrong fields: domains rotate to align with field
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Hysteresis
M
H
0B H M
Area of the loop is proportional to energy dissipated in traversing the loop.
1r
Remnant magnetization
Coercive field M H
0 0(1 ) rB H H
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Anisotropy energy
easy axis
hard axis
Spin-orbit coupling couples the shape of the wavefunction to the spin. The exchange energy depends on the overlap of the wavefunctions and thus on spin direction.
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Bloch wall
22 2cos 1
2i jw JS S JS JS
N
2 2
2JSNw
N
energy of two spins
energy of a line of spins
22
2w JS
neglecting the constant part
energy is lower if the Bloch wall is wide
energy per unit area 2 2
22JS
Na
a is the lattice constant
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Bloch wall
2 22
2 [J/m ]2JSE KNa
Na
N ~ 300 for iron
2 2
2 2
2 2
3
0 02
2
dE JS KadN N a
JSNKa
Na = thickness of wall
smaller for large N
Anisotropy energy depends on the number of spins pointing in the hard direction
KNa
anisotropy constant J/m3
Total energy per unit area:
smaller for small N
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Soft magnetic materials
Coercive field
Remnant Fieldsoft magnets
transformersmagnetic shieldingferrites have low eddy current losses
0rB H
0B H M
1r
B [T]
M H
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Single domain particles
Small 10 - 100 nm particles have single domains.
Elongated particles have the magnetization along the long axis. M
Single domains are used for magnetic recording. Long crystals can be magnetized in either of the two directions along the long axis.
Shape anisotropy.
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Hard magnets
Grains too small to contain Bloch walls must be flipped entirely by the field.
Alnico: 8-12% Al, 15-26% Ni, 5-24% Co, up to 6% Cu, up to 1% Ti, rest is Fe
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Coercive field
Remnant Field
hard magnetsMagnet Br (T) Hci (kA/m) (BH)max (kJ/m3) Tc (°C)
Nd2Fe14B (sintered) 1.0–1.4 750–2000 200–440 310–400
Nd2Fe14B (bonded) 0.6–0.7 600–1200 60–100 310–400
SmCo5 (sintered) 0.8–1.1 600–2000 120–200 720
Sm(Co,Fe,Cu,Zr)7(sintered) 0.9–1.15 450–1300 150–240 800
Alnico (sintered) 0.6–1.4 275 10–88 700–860
Permanent magnets, magnetron, motors, generatorsferrites can also be hard magnets
Hard magnetic materials
Defects are introduced to pin the Bloch walls in a hard magnet.
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Motors, generators, speakers, microphone
Applications of hard magnets
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Superparamagnetism
Below the Curie temperature the thermal energy changes the direction of magnetization of the entire crystallites.
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Composite magnets
Injection molded magnets are a composite of various types of resin and magnetic powders
Flexible magnets are made by embedding magnetic particles in vinyl.
Powers deposited on tapes for magnetic storage.
Magnetic tapes are much cheaper per GB than hard disks.
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magnetic recording
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Giant magnetoresistance
D(E)
Espin up
spin down
spin value
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Peter Gruenberg Nobel Lecture 2007:From Spinwaves to Giant Magnetoresistance (GMR) and Beyond
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Magnetic force microscope
In-plane magnetization of a hard disk.