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Institute of Solid State PhysicsTechnische Universität Graz
15. Magnetism/ Superconductivity
Nov 25, 2019
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
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
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.
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.
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
Motors, generators, speakers, microphone
Applications of hard magnets
Giant magnetoresistance
D(E)
Espin up
spin down
spin value
Peter Gruenberg Nobel Lecture 2007:From Spinwaves to Giant Magnetoresistance (GMR) and Beyond
Institute of Solid State PhysicsTechnische Universität Graz
Superconductivity
Primary characteristic: zero resistance at dc
There is a critical temperature Tc above which superconductivity disappears
About 1/3 of all metals are superconductors
Metals are usually superconductors OR magnetic, not both
Good conductors are bad superconductors
Kittel chapter 10
Superconductivity
Heike Kammerling-Onnes
Superconductivity was discovered in 1911
Nobel Lecture 1913
Superconductivity
Heike Kammerling-Onnes
Superconductivity was discovered in 1911
Nobel Lecture 1913
Critical temperature
Superconductivity
No measurable decay in current after 2.5 years. < 10-25 m.
Antiaromatic molecules are unstable and highly reactive
Meissner effect
Superconductors are perfect diamagnets at low fields. B = 0 inside a bulk superconductor.
Superconductors are used for magnetic shielding.
T >Tc T <Tc
SuperconductivityCritical temperature Tc
Critical current density Jc
Critical field Hc
2 2 210 2 22B c c c
mn nk T H nmv Jne
YBa2Cu3Ox
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(Sn5In)Ba4Ca2Cu10Oy Tc = 212 K
SuperconductivityCritical temperature Tc
Critical current density Jc
Critical field Hc
2 2 210 2 22B c c c
mn nk T H nmv Jne
Superconductivity
Perfect diamagnetism
Jump in the specific heat like a 2nd order phase transition, not a structural transition
Superconductors are good electrical conductors but poor thermal conductors, electrons no longer conduct heat
There is a dramatic decrease of acoustic attenuation at the phase transition, no electron-phonon scattering
Dissipationless currents - quantum effect
Electrons condense into a single quantum state - low entropy.
Electron decrease their energy by but loose their entropy.
Probability current
21 ( )2
i i qA Vt m
Schrödinger equation for a charged particle in an electric and magnetic field is
write out the term 2( )i qA
ie i ie i e
22 2 22i i i ie i e i e e
2 2 2 21 -2
i i qA ihq A q A Vt m
write the wave function in polar form
Probability current
22 2 2
2 2
1 - 22
i i it t m
i qA i ihq A q A V
Schrödinger equation becomes:
Real part:
222-
2q A V
t m
Imaginary part:
22 21 - 2 22
i qA q At m
Probability current
Imaginary part:
22 21 - 2 22
i qA q At m
Multiply by || and rearrange
2 2 0q At m
This is a continuity equation for probability
0P St
2 qS Am
The probability current:
Probability current / supercurrent
2 qS Am
The probability current:
All superconducting electrons are in the same state so
2cp
e
e n ej Am
2 cpj en S
In superconductivity the particles are Cooper pairs q = -2e, m = 2me, ||2 = ncp.
London gauge 0 2 22 cp s
e e
n e n ej A Am m
ns = 2ncp
This result holds for all charged particles in a magnetic field.
1st London equation
First London equation:
2s
e
n ej Am
2 2s s
e e
n e n edj dA Edt m dt m
dA Edt
2s
e
n edj Edt m
s
dv m djeE mdt n e dt
Classical derivation:
2s
e
n edj Edt m
Heinz & Fritz
2nd London equation
Second London equation:
2s
e
n ej Am
2s
e
n ej Am
2s
e
n ej Bm