Summary of Lecture 13Interlayer exchange coupling (IEC)

Quantum-well model for IECQuantum well• Transmission and reflection amplitudes, multiple reflection• Two configurations (P and AP)• Spin-polarized quantum well states

Exchange coupling• Bilinear• Energy difference• Critical nesting vectors (Fermi surface; free electrons, Cu(001))• Coupling strength

Measurements• Thickness fluctuations• Magnetization fluctuations (→ biquadratic coupling)

Summary of Lecture 13Magnetocrystalline anisotropy

Magnetic force theorem• DFT+LSDA calculations without SOC• SOC treated as perturbation• Band energy differences

• Bulk: μeV/atom• Surface: meV/atom

Summary: Shape anisotropy• Magnetostatic energy• Dipole-dipole interaction• Depends on the shape of the sample• For thin films and surfaces: favours in-plane anisotropy

Magnetocrystalline anisotropyInterpretation of the thin-film MAE – a superposition principleQuestion: What are the effects of the interfaces?

Dyson equation for the film Green function

Bulk GF Perturbations

Spin

Series expansion

`Left‘ perturbation `Right‘perturbation

Multiple scatteringNeglect!

Magnetocrystalline anisotropyInterpretation of the thin-film MAE – a superposition principle

Neglect!

Magnetic profile

Neglects finite-size effectsImportant for very thin films

Superposition principle for the band energy

Layer-resolved magneticmoments

Magnetocrystalline anisotropyInterpretation of the thin-film MAE – a superposition principleSuperposition principle for the band energySOC as perturbation

Band energy profile

SOC

Magnetocrystalline anisotropySummary: Superposition principle• Dyson equation for the film Green function• Neglect of multiple-scattering contributions

• Finite-size effects neglected• Quantum-well states not accounted for

• Profiles decomposed into• Surface part• Interface part• Bulk part

• Unambigous interpretation• Magnetic profiles• Band energy profiles

Magnetocrystalline anisotropySummary so far...Magnetocrystalline anisotropy• Alignment of the magnetization aling crystallographic axes

Effect of spin-orbit coupling on the band structure

Anisotropy constants

Magnetic force theorem• Compute the anisotropy energy within DFT + LSDA

Shape anisotropy• Dipole-dipole interaction of local magnetic moments

Superposition principle• Interpretation of the magnetization and band-energy profiles

Magnetocrystalline anisotropyNi/Cu(001)Experiment Cu/Ni/Cu(001)XPEEMMagnetic dichroism

Wedge-shaped Ni film

In-plane

Perpendicular

In-plane

Spin reorientation transitions(SRT) @ 10 ML and 19 ML

Origin:Tetragonal distortion in the Ni film

Magnetocrystalline anisotropyNi/Cu(001)Band energy profiles

Ni film

thickness

Solid: Film calculationDashed: Superposition

• Oscillations• Superposition• QW effects (thin films)

Magnetocrystalline anisotropyNi/Cu(001)Spin reorientation transition

In-plane

Out-of-plane

SRT

Magnetocrystalline anisotropyNi/Cu(001)Anisotropy constantsAngular dependence of the free energy

Magnetocrystalline anisotropySummary: Ni/Cu(001)• Band energy profiles• Superposition principle• Effect of the tetragonal distortion in the Ni film• Spin reorientation transition in agreement with experiments• Anisotropy constants

Solid state magnetismOrigin of the magnetism in transition metals?

• Atomic magnetism: Hund‘s rules• Landau diamagnetism• Pauli paramagnetism• Itinerant ferromagnetism

• Magnetism in thin films

Solid state magnetismAtomic magnetism: Hund‘s rules1. Full shells and subshells do not contribute to total S, the total spin angular

momentum and L, the total orbital angular momentum quantum numbers.

2. The term with maximum multiplicity (maximum S) has the lowest energy level.

3. For a given multiplicity, the term with the largest value of L has the lowest energy.

4. For atoms with less than half-filled shells, the level with the lowest value of J lies lowest in energy. Otherwise, if the outermost shell is more than half-filled the term with highest value of J is the one with the lowest energy.

Solid state magnetismHund‘s rules applied to Fe

2. rule: S maximum

1. rule: s-shell does not contribute

3. rule: minimizes SOC

Solid state magnetismHund‘s rules applied to transition metals

Solid state magnetismLandau diamagnetism Nd magnetsGraphite (diamagnet)

Paramagnetism(neglect)

Diamagnetism

Ansatz

Cyclotron frequency

Solid state magnetismLandau diamagnetism

Harmonic oscillator

Landau tubes

Solid state magnetismLandau diamagnetism

Degeneracy

B = 0:

Area between two levels:

Degeneracy

Solid state magnetism

SusceptibilityLandau diamagnetism

Occupation of the highest block:

Energy of the highest block

Solid state magnetismLandau diamagnetismSusceptibility

Energy difference

Including spin degeneracy and number of states in kz

Susceptibility

Diamagnetic elements: Cu, Ag, Bi,

Solid state magnetismPauli paramagnetism

ParamagnetismDiamagnetism(neglect)

Energy shift

Spin-resolved DOS

Solid state magnetismPauli paramagnetism

Magnetization

Susceptibility

Landau vs. Pauli susceptibility

Total susceptibility including effective mass

Solid state magnetismBand ferromagnetismStoner criterion – energetic preference of ferromagnetism

Kinetic energy (cost)

Density of states

Magnetization

Free energy

Potential energy (gain)

with

Electrons move in a mean field of strength λPauli paramagnetism → magnetizationPositive feedback

(Stoner exchange integral)

Solid state magnetismBand ferromagnetismStoner criterion – energetic preference of ferromagnetism

Total energy balance

Spontaneous magnetism if

→ Stoner criterion

Solid state magnetismBand ferromagnetismStoner amplification in paramagnets

Paramagnetism

Increased Pauli susceptibility

Important for Pt and Pd (`close to being ferromagnetic‘)Can one make Pd magnetic?

Solid state magnetismQuantum-well induced ferromagnetism in thin Pd films

ConstantIncrease

Idea: thin Pd films on a Ag(001) substrate

magnetic

nonmagnetic

→ ferromagnetism for N = 3, 4, 8, 9, ...

Pd film thickness

Solid state magnetismQuantum-well induced ferromagnetism in thin Pd films→ ferromagnetism for N = 3, 4, 8, 9, ...

Magnetization profiles

Solid state magnetismQuantum-well induced ferromagnetism in thin Pd filmsBand structures

Not confined

ConfinedQuantum well states

Solid state magnetismQuantum-well induced ferromagnetism in thin Pd filmsQuantum well statesFerromagnetism for N = 3, 4, 8, 9, ...Spin-split QWS

→ paramagnetic QWS closer than 0.1 eV to the Fermi level

Solid state magnetismSummaryAtomic magnetism: Hund‘s rules• Application to Fe, transition metals

Landau diamagnetism• Free-electron Hamiltonian• Landau tubes• Susceptibility

Pauli paramagnetism• Energy shift by an external magnetic field• Susceptibility

Itinerant ferromagnetism (band ferromagnetism)• Energy gain → magnetic ground state• Stoner criterion• Stoner amplification of the Pauli susceptibility

Magnetism in thin films induced by quantum well states (Stoner criterion)