Ritwik Mondal Department of Physics and Astronomy

Relativistic theory in magnetisation dynamics

Ritwik Mondal, Marco Berritta, Peter M. Oppeneer

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)

Ritwik Mondal Department of Physics and Astronomy

Relativistic mechanism?

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

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

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

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)

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

Ritwik Mondal Department of Physics and Astronomy

Fast magnetisation dynamics

Landau-Lifshitz-Gilbert equation

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?

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:

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)

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

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)

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)

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

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.

Ritwik Mondal Department of Physics and Astronomy

Acknowledgements• Pablo Maldonado • Alex Aperis • Karel Carva • Boris A. Ivanov

Ritwik Mondal Department of Physics and Astronomy

Different Hamiltonians