Presentation mini-simphosium
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Transcript of Presentation mini-simphosium
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Ritwik Mondal Department of Physics and Astronomy
Relativistic theory in magnetisation dynamics
Ritwik Mondal, Marco Berritta, Peter M. Oppeneer
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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)
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Ritwik Mondal Department of Physics and Astronomy
Relativistic mechanism?
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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
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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
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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
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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)
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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
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Ritwik Mondal Department of Physics and Astronomy
Fast magnetisation dynamics
Landau-Lifshitz-Gilbert equation
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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?
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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:
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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)
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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
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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)
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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)
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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
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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.
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Ritwik Mondal Department of Physics and Astronomy
Acknowledgements• Pablo Maldonado • Alex Aperis • Karel Carva • Boris A. Ivanov
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Ritwik Mondal Department of Physics and Astronomy
Different Hamiltonians