Ritwik Mondal presents a framework for describing relativistic magnetization dynamics using the Dirac equation. The document outlines several key points: 1) A Dirac-Kohn-Sham Hamiltonian is used to describe electrons in magnetic materials excited by a laser pulse. A Foldy-Wouthuysen transformation is applied to obtain a two-component Hamiltonian. 2) Spin-orbit coupling terms are included which describe intrinsic coupling to ionic potentials and extrinsic coupling to external laser fields. 3) Expressions for the Landau-Lifshitz-Gilbert equation are derived from the Hamiltonian, with the Gilbert damping parameter dependent on spin-orbit coupling and magnetic susceptibility. 4) For

1. Ritwik Mondal Department of Physics and Astronomy
Relativistic theory in
magnetisation dynamics
Ritwik Mondal, Marco Berritta,
Peter M. Oppeneer

2. Ritwik Mondal Department of Physics and Astronomy
Ultrafast demagnetisation
E. Beaurepaire et al. PRL 76, 4250 (1996) C. Stamm et al. Nat. Mater. 6, 740 (2007)

3. Ritwik Mondal Department of Physics and Astronomy
Relativistic mechanism?

4. Ritwik Mondal Department of Physics and Astronomy
What is needed?
• Proper relativistic formalism
• Inclusion of exchange effect
• Different relativistic light-spin interactions
• ab initio expressions for the parameters which
describe the system dynamics
• ab initio calculations

5. Ritwik Mondal Department of Physics and Astronomy
Outline
•Relativistic Hamiltonian formulation
•Foldy-Wouthuysen transformation
•Landau-Lifshitz-Gilbert equation
•Gilbert damping parameter
•ac harmonic field
•time-dependent general magnetic field
•LLG to LL equation of motion
•Conclusions

6. Ritwik Mondal Department of Physics and Astronomy
Dirac-Kohn-Sham Hamiltonian
• Dirac Hamiltonian for electrons in a (ferro)magnetic
materials, excited by laser pulse:
• four component Hamiltonian
• fully relativistic
• non-relativistic limit is important for description of low-
energy systems
• two-component system: large components and small
components
• Foldy-Wouthyusen transformation

7. Ritwik Mondal Department of Physics and Astronomy
Foldy-Wouthuysen transformation
• unitary transformation
• find an unitary matrix:
• time-dependent FW transformation
• only keeping even terms
• The large component upto :
Mondal et al. Phys. Rev. B 91, 174415 (2015)

8. Ritwik Mondal Department of Physics and Astronomy
Spin-orbit coupling Hamiltonian
• gauge invariant
• hermitian
• two types of spin-orbit coupling: (1) with the ionic potential -
intrinsic SOC, (2) with the field from laser - extrinsic SOC
• intrinsic SOC time-independent electric field
• extrinsic SOC time-dependent electric field

9. Ritwik Mondal Department of Physics and Astronomy
Fast magnetisation dynamics
Landau-Lifshitz-Gilbert equation

10. Ritwik Mondal Department of Physics and Astronomy
LLG equation of motion
• Is it possible to derive from a
fundamental Dirac equation?
• What is the expression for Gilbert damping
parameter?
• How the relativistic effects play a role?
• Can we have a Gilbert damping parameter which is
ab initio calculable?

11. Ritwik Mondal Department of Physics and Astronomy
Definitions
• wavelength of the incident light, = 800 nm
• thickness of the sample, = 20 nm (multilayered system)
•
• within Coulomb gauge ( ), uniform/slowly varying
magnetic field:
• spherically symmetric potential:
• spin angular momentum:

12. Ritwik Mondal Department of Physics and Astronomy
Extrinsic SOC
• Is it hermitian?
• in the component form:
• but the full Hamiltonian is hermitian
• Magnetisation dynamics definition:
Mondal, Berritta, Oppeneer Phys. Rev. B 94, 144419 (2016)
Hickey and Moodera Phys. Rev. Lett. 102, 137601 (2009)

13. Ritwik Mondal Department of Physics and Astronomy
Magnetisation dynamics
• Zeeman-like non relativistic field-spin interactions are
responsible for precession.
• intrinsic SOC contributes to the relativistic counterpart in
precession.
• extrinsic SOC dynamics:
• relation between magnetisation and magnetic flux:
• for a time-dependent magnetic field:
• for an ac harmonic field: differential magnetic susceptibility

14. Ritwik Mondal Department of Physics and Astronomy
ac harmonic field
• for an ac harmonic field:
• damping tensor :
• electronic part (interband):
• damping tensor can be expressed as: a isotropic Gilbert
(scalar) + Anisotropic Ising-like (tensor) + anti-symmetric
Dzyaloshinskii-Moriya-like (vector) contributions.
Mondal, Berritta, Oppeneer Phys. Rev. B 94, 144419 (2016)

15. Ritwik Mondal Department of Physics and Astronomy
General time-dependent field
• for a general time-dependent field
• damping parameters:
• New torque: field derivative torque in LLG equation!
time (t)
f(t)
Derivative
time (t)

16. Ritwik Mondal Department of Physics and Astronomy
LLG to LL equations
• for harmonic field:
• DM-like contributions act as renormalisation factor to LL
equation.
• Take and : we are back to the original LL
equation.
• for a general time-dependent magnetic field

17. Ritwik Mondal Department of Physics and Astronomy
Conclusions
• New expression for Gilbert damping parameter - very suitable
for ab initio calculations!
• Gilbert damping parameter for an ac harmonic field is a
tensor that depends on susceptibility and .
• Gilbert damping parameter for a general time-dependent
field depends only on .
• Gilbert damping tensor contains an isotropic (scalar), an
anisotropic Ising-like (tensor) and a chiral Dzyaloshinskii-
Moriya like contribution.
• For a Gilbert damping tensor the transformation of LLG to LL
equation is non-trivial.
• Dzyaloshinskii-Moriya like contribution serves as a
renormalisation factor in LL equation.

18. Ritwik Mondal Department of Physics and Astronomy
Acknowledgements
• Pablo Maldonado
• Alex Aperis
• Karel Carva
• Boris A. Ivanov

19. Ritwik Mondal Department of Physics and Astronomy
Different Hamiltonians