Spin waves Part I Sylvain Petit Laboratoire Lon Brillouin - - PowerPoint PPT Presentation

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Oct 13, 2023 307 likes •775 views

Spin waves Part I Sylvain Petit Laboratoire Lon Brillouin CE-Saclay F-91191 Gif sur Yvette sylvain.petit@cea.fr Why spin waves ? Time-dependent phenomenon precession of the spin Theory developed to describe the excited states of the

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Sylvain Petit Laboratoire Léon Brillouin CE-Saclay F-91191 Gif sur Yvette sylvain.petit@cea.fr

Spin waves Part I

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Why spin waves ?

Time-dependent phenomenon precession of the spin Theory developed to describe the excited states of the Heisenberg Hamiltonian And determine exchange interaction (and anisotropies) via experiments

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Why spin waves ?

Bulk systems ~A few THz, meV, cm-1 in bulk system k ~ 0.1 A-1 Observed by neutron scattering in (k, ) space, but also NMR, optical techniques (Raman, =0) Part I General considerations Ferromagnet Antiferromagnet Failure of the theory Part II Neutron scattering Examples

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Molecular field

A spin experiences a molecular field due to interaction with its neighbours Long range ordering Heisenberg Hamiltonian

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Molecular field

Example : AF ordering Depending on interactions, this molecular field can induce a new periodicity

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Molecular field

m labels the unit cell i labels the ion within the unit cell New periodicity, « magnetic unit cell »

J J J

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Molecular field

m labels the unit cell i labels the ion within the unit cell New periodicity, « magnetic unit cell »

J

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Molecular field

Define Interactions

= 0

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Molecular field

Define Interactions

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Molecular field

Define Interactions

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Molecular field

Define Interactions

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Easy diagonalization

Molecular field

Mean field approximation One site Hamiltonian, broken symmetry T (K)

!

Mermin and Wagner theorem : no spontaneous broken symmetry at finite temperature in 1and 2 dimension TN

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Spin in a field

A spin experiences a molecular field due to the interaction with its neighbors

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Spin in a field

Precession of a spin in a magnetic field

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Spin in a field

Classical mechanics Equation of motion

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Spin in a field

The spin precesses around Sz with a frequency proportional to h 1 degree of freedom Classical mechanics

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Spin in a field

e3 S

S-1

S-2

Quantum mechanics Spin operators in the local basis Eigenvalues

e1 e2

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Spin in a field

e3 S

S-1

S-2

The spin rotates around Sz with a frequency proportional to h Quantum mechanics Equation of motion

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Coupled spins

S S-1 S S-1 S S-1

Back to the problem of coupled spins …

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Transformation to local basis

S S-1 S S-1 Cartesian coordinates Local coordinates

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Equation of motion

HDR

… non linear equations N coupled … Classical mechanics Equation of motion :

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1. Molecular field : small deviations around the direction of the ordered moment 2. Take advantage of the new periodicity (Fourier transform) : reduce the number

f coupled equations

3. Exchange

Equation of motion

!

Magnetic unit cell

Classical mechanics

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1. Fourier transform
2. Ordered moment + small deviations : linearization

Equation of motion

effective magnetic field acting on

Classical mechanics

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Local coordinates (use local transformation) L magnetic ions per magnetic unit cell : L coupled linear equations Approximations :

1. Ordered phase
2. Small deviations around the ordered moment : « linear spin wave theory » (large

S, low T)

Equation of motion

Classical mechanics

Effective magnetic field acting on

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From the general equations of motion back to the simple ferromagnetic case :

Ferromagnet

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Ferromagnet

Phase wavelength Coupled precessions of the spins around the ordered moment; propagate through the lattice The dispersion relation connects the wavevector and the frequency (energy)

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Ferromagnet

Zone center Zone boundary Zone center of the next Brillouin zone Parabolic dispersion

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The thermal fluctuations prevents long range

rdering for

Breakdown of the spin wave theory is consistent with Mermin and Wagner theorem

Ferromagnet

Back to quantum mechanics : spin waves are (quasi) independent Bose modes Check the approximations (correction to the magnetization)

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Antiferromagnet

From the general equations of motion 1 2

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Antiferromagnet

Local coordinates (use local transformation) 4 : There are 2 degrees of freedom 1: Sublattices 1 and 2 are still coupled 2 : Projection on e3 is constant 3 : Exchange the role of sublattices 1 and 2 (degenerate modes)

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Antiferromagnet

Additional transformation to decouple sublattice 1 and 2 Spin wave energies Details of the transformation :

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Antiferromagnet

Two degenerate modes

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1 2 Linear dispersion

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Antiferromagnet

Zone center

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Antiferromagnet

Zone center of the magnetic unit cell, (Zone boundray of the lattice unit cell)

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Antiferromagnet

(Zone boundray of the magnetic unit cell)

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Antiferromagnet

Thermal fluctuations Check the approximations (correction to the magnetization) Breakdown of the spin wave theory is consistent with Mermin and Wagner theorem Quantum fluctuations

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Summary

Spin waves : excited states of the Heisenberg Hamiltonian L ions per magnetic unit cell : L branches Approximations 1) Ordered phase 2) Small deviations around the ordered moment : large S, low T Quasi independent modes (bosons) and important role of quantum fluctuations (low dimension)

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Beyond spin wave theory

Spin ½ : no long range order, no spin waves A spin 1 excitation = 2 spinons : continuum and no dispersion relation

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Beyond spin wave theory

Kagome Lattice Degenerate ground state : no long range order The system keeps fluctuating : liquid and co-planar regimes (order by disorder) configuration

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Beyond spin wave theory : calculate the equation of motion for each spin (~ molecular dynamics) in classical mechanics (no approximation): Propagative modes as well as soft modes

Beyond spin wave theory

Robert et al, PRL 101, 117207 (2008)

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Beyond spin wave theory

Configuration

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Beyond spin wave theory

Configuration

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Thanks for your attention Questions Practical To be continued …Part II : how to observe spin waves ?

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References

[1] P.W. Anderson, Phys. Rev. 83, 1260 (1951) [2] R. Kubo, Phys. Rev. 87, 568 (1952) [3] T. Oguchi, Phys. Rev 117, 117 (1960) [4] D.C. Mattis, Theory of Magnetism I, Springer Verlag, 1988 [5] R.M. White, Quantum Theory of Magnetism, Springer Verlag, 1987 [6] A. Auerbach, Interacting electrons and Quantum Magnetism, Springer Verlag, 1994.

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Holstein-Primakov representation of the spin « seen from the classical picture » : New variable : deviation

Quantum mechanics

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