14
1 Introduction
(a)
(b)
Fig. 1.4 Depiction of a magnon, as viewed a in side profile and b from a top-down perspective,
where the red curve represents the relative phase of the spins
A charged particle exposed to an electric and magnetic field is subject to the
Lorentz force F = q(E + v × B). When exposed to the electric field of light atoms or
ions in a material are displaced from their equilibrium positions, and the interactions
between adjacent atoms or ions in the crystalline lattice result in collective vibrational
modes, known as phonons. The magnetic field of light can have an analagous effect
on the magnetic moments of the ions in the crystal lattice. In a magnetically ordered
material, the magnetic torque applied by the optical magnetic field causes displacements of the spins from their equilibrium positions; due to interactions between
adjacent spins in the crystal structure, these displacements propagate through the
material as a spin wave, the quasiparticle of which is known as a magnon, shown
schematically in Fig 1.4.
In order to describe the behaviour of magnons in a material, an equation of motion
for the spin angular momentum of the i
th spin S i can be expressed in terms of an
effective magnetic field H eff , used to represent all the interactions felt by a single
spin. This is given by the Landau-Lifshitz-Gilbert (LLG) equation
dS i
dt
= γH eff × S i −
λ
S i × (H eff × S i )
S 2
,
(1.33)
where γ = g|e|/2m is the gyromagnetic ratio and λ is the Gilbert damping parameter.
The effective field drives the spins to align in its direction, causing the spins to precess
about their equilibrium position. The effective field can be expressed as the partial
derivative of the free energy F of the system with respect to S i . In the condition
where the temperature of the system is much lower than the ordering temperature,
the free energy can be approximated by the semi-classical spin Hamiltonian H of
the system, allowing us to define the effective field as
H eff =
1
γ
∂H
∂S i
.
(1.34)
Using this definition, the spin dynamics in a particular system can be investigated
by: defining a specific spin Hamiltonian accounting for all the interactions between
spins in the system, calculating the resultant effective field acting on the spins, then
solving the LLG equation to find properties of interest, such as the magnon dispersion
relation and the nature of the magnetic dynamics.
1 Introduction
(a)
(b)
Fig. 1.4 Depiction of a magnon, as viewed a in side profile and b from a top-down perspective,
where the red curve represents the relative phase of the spins
A charged particle exposed to an electric and magnetic field is subject to the
Lorentz force F = q(E + v × B). When exposed to the electric field of light atoms or
ions in a material are displaced from their equilibrium positions, and the interactions
between adjacent atoms or ions in the crystalline lattice result in collective vibrational
modes, known as phonons. The magnetic field of light can have an analagous effect
on the magnetic moments of the ions in the crystal lattice. In a magnetically ordered
material, the magnetic torque applied by the optical magnetic field causes displacements of the spins from their equilibrium positions; due to interactions between
adjacent spins in the crystal structure, these displacements propagate through the
material as a spin wave, the quasiparticle of which is known as a magnon, shown
schematically in Fig 1.4.
In order to describe the behaviour of magnons in a material, an equation of motion
for the spin angular momentum of the i
th spin S i can be expressed in terms of an
effective magnetic field H eff , used to represent all the interactions felt by a single
spin. This is given by the Landau-Lifshitz-Gilbert (LLG) equation
dS i
dt
= γH eff × S i −
λ
S i × (H eff × S i )
S 2
,
(1.33)
where γ = g|e|/2m is the gyromagnetic ratio and λ is the Gilbert damping parameter.
The effective field drives the spins to align in its direction, causing the spins to precess
about their equilibrium position. The effective field can be expressed as the partial
derivative of the free energy F of the system with respect to S i . In the condition
where the temperature of the system is much lower than the ordering temperature,
the free energy can be approximated by the semi-classical spin Hamiltonian H of
the system, allowing us to define the effective field as
H eff =
1
γ
∂H
∂S i
.
(1.34)
Using this definition, the spin dynamics in a particular system can be investigated
by: defining a specific spin Hamiltonian accounting for all the interactions between
spins in the system, calculating the resultant effective field acting on the spins, then
solving the LLG equation to find properties of interest, such as the magnon dispersion
relation and the nature of the magnetic dynamics.