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