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Home , Ferromagnetism

Band Theory of Ian McDonald in Northeastern University Boston, Ma

03/24/15 Contents • Free vs Band Theory • • Ferromagnetism of Magnetism • Slater-Pauling Curve in Metals

Application of band theory to magnetic materials was first performed by Stoner, Mott & Slater (1933-1936) 03/24/15 1 Electronic structure of free atoms vs solids

• Atoms separated by large distances (free atoms) • occupy well-defined energy levels – Pauli exclusion principle = 2 electrons per energy level

• Atoms in close proximity (solids) • clouds overlap • Energy levels split to accommodate 4 electrons that would be at each energy level

B.D. Cullity. “Introduction to Magnetic Materials” Wiley. pg. 135. 2009. 1 Electronic structure of free atoms vs solids

• In transition elements, 3d and 4s levels split the most because they’re farther from the nucleus (i.e. interact first) • Lower levels (2p, 2s, 1s) less split because closer to nucleus • More electrons = more splitting • These levels are so closely spaced that they can be approximated into an energy band. • Electrons become delocalized (itinerant)

B.D. Cullity. “Introduction to Magnetic Materials” Wiley. pg. 135. 2009. 2 Practical Example of Band Theory

• 1 mg of Fe contains ~1019 atoms. • Pauli exclusion principle requires each separate energy level in the free must necessarily be split into 1019 levels in the .

• N(E) – defines the density of energy levels at a given energy • N(E) dE is the number of available energy levels between E and E + dE

B.D. Cullity. “Introduction to Magnetic Materials” Wiley. pg. 135. 2009. 3 Density of States

• 3d band has a much larger density because there are five 3d levels per atom each with a capacity of 10 electrons whereas only one 4s with 2 electrons • Area under the curve is equal to the total available number of energy levels in a band. • Fermi level shows topmost filled energy level

B.D. Cullity. “Introduction to Magnetic Materials” Wiley. pg. 135. 2009. 4 Density of States

• Imagine 3d band split into two sections ( up / spin down) • And these two bands as two connected tanks of water • Fermi energy is the water level • Filled energy bands cannot contribute a because opposing spins cancel out

H.S. Nalwa. “Handbook of Thin Film Materials” Academic Press. pg. 520. 2002. 5 Spin up VS Spin down

• An electron may reverse its spin 0 through (Stoner criterion) • Exchange force acts like a dam allowing for spin imbalance resulting in a magnetic moment +2 • This only happens if the levels are very close together

S. Chikazumi. “ of Magnetism” Wiley & Sons. Pg. 74. 1964. 6 Ferromagnetism in Solids

Non-magnetic Ferromagnetic

N = N N > N

H.S. Nalwa. “Handbook of Thin Film Materials” Academic Press. pg. 520. 2002. 7 Slater-Pauling Curve

μ = (10.6-n3d)μβ

B.D. Cullity. “Introduction to Magnetic Materials” Wiley. pg. 135. 2009. 8 Summary

Requirement for ferromagnetism from 3d band 1) Partially filled bands so there’s available energy levels for unpaired spin to move into. 2) Density of states in the band must be high so that change in spin orientation will produce only small change in energy. 3) Interatomic distance must facilitate exchange forces

• Only Fe, Co, and Ni meet all requirements • RE-elements have spontaneous due to spin unbalance in 4f band