Basics of magnetic materials We will briefly review magnetism and magnetic materials as necessary background for understanding magnetoelectronics, particularly the data storage industry. • Definitions in SI • Types of magnetic materials • Origins of magnetic properties • Ferromagnetism • Domains • Hysteresis loops

Definitions in SI Units are the worst part of dealing with magnetic systems! B: • magnetic induction (usually what physicists mean by magnetic field). • most physically significant quantity - what shows up in Lorentz force law, in determining NMR frequencies, etc. • Unit is the Tesla. Earth’s magnetic field = 6 x 10-5 T. • More convenient cgs unit is the Gauss = 10-4 T. • Important boundary condition:

∇ ⋅B = 0

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Definitions in SI H: • the magnetic field. • Caused by currents of free charge. • Unit is the Amp/m. • Important relation:

∇×H = J

• With no magnetic materials around,

“permeability of free space” = 4π x 10-7 Tm/A

B = µ0H

M: • the magnetization. • magnetic moment per unit volume of a material. • Unit is the Amp/m. • For a material with a permanent magnetization M0,

M = χH + M 0 magnetic susceptibility

Susceptibility and permeability Combining effects of external currents and material response,

B = µ 0 (H + M) We define the permeability by:

B = µH

So, for a material with a magnetic susceptibility χ,

B = µH = µ 0 (1 + χ )H → µ = µ 0 (1 + χ ) Note that for real materials χ and µ are tensorial. Relative permeability is defined as µr = µ/µ0.

→ B = µ0µr H

Susceptibility is most useful when discussing diamagnetic (χ < 0) and paramagnetic (χ > 0) materials, rather than systems with nonzero M0.

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Magnetic dipoles and magnetization m

A magnetic dipole can be modeled as a loop of current: m = IA n Units: Am2 = J/T

eh Convenient = 9.27 × 10 − 24 J/T µB ≡ number: 2m A group of identical dipoles in a plane can be replaced by one big dipole with the same current circulating around the perimeter.

A 3d stack of dipoles can be replaced by thinking about a sheet current running around the perimeter.

Forces, torques, and fields B

Torque on a magnetic dipole: τ = m×B

m

Force on a magnetic dipole:

F = m ⋅ (∇B ) Energy of magnetic dipole:

U = −m ⋅ B

What is B at the surface of a long uniformly magnetized body with magnetization M in the absence of other fields? Answer: just µ0M, since normal component of B must be continuous.

M

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Field from a dipole B (r ) = µ 0

3(m ⋅ r )r − r 2m 4πr 5

Note that B(r) ~ r -3

Current loop only approximates a dipole field in far field limit (r >> loop radius).

Image from Purcell.

Diamagnetism Some materials develop a magnetization that is antialigned with the applied external field H. Such materials are diamagnetic, and have χ < 0. Simple classical picture for diamagnetism: Lenz’s Law Try ramping up B = µ0H. B

m

Result is a circumferential electric field that opposes the direction of the current in the loop. This would act to reduce the dipole moment along H, and would be diamagnetic.

Correct quantum treatment involves 2nd order perturbation theory - can end up with either sign, depending on particulars of atoms. Larmor diamagnetism or Van Vleck paramagnetism.

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Paramagnetism Also common is paramagnetism, when χ > 0. Two common origins of paramagnetism: • Curie paramagnetism - localized moments free to flip. • Pauli paramagnetism - requires “free” electrons in a metal. Start w/ Curie case, spin J and gyromagnetic ratio g, in magnetic flux density B.

1 k BT Do statistical physics here. Alignment of spin with external field lowers spin energy. g ≡ (m / J ) / µ B

β≡

Can write down partition function and solve to find equilibrium magnetization.

Curie paramagnetism Result:

M = nµ B gJB J ( βµ B gJB)

2J +1 1  x   2J +1  x − coth  coth   2J  2J  2J  2J  Bj is called the Brillouin function. B J ( x) ≡

As long as µBB