1.3 Electromagnons in Improper Ferroelectrics
15
1.3.2 Electromagnons
While the dynamics of magnons may be understood solely by the interaction of the
magnetic field of light with the spins in a material, the electric dipole-active nature of
electromagnons requires that coupling between the spins and phonons in the material
are taken into account in the Hamiltonian. Two distinct mechanisms that give rise to
electromagnons have been discussed in the literature: Dzyaloshinskii-Moriya electromagnons, and exchange-striction (ES) electromagnons. The different origins for
these mechanisms are shown schematically by Fig. 1.5a for the DM electromagnon
and Fig. 1.5b for the ES electromagnon.
The static polarisation in an improper ferroelectric phase with spin-cycloidal
ordering can be understood as arising from the spin-current or inverse DzyaloshinskiiMoriya (DM) interaction [33, 36]. The DM interaction can be expressed as
H DM =
a
d a (r i ) · (S i × S i+1 ) ,
(1.35)
where S i and S i+1 are adjacent spins in the cycloid. The DM vectors d a are parameterised by
d a = γ (δ i × r i ) ,
(1.36)
where γ is the interaction strength, δ is the vector connecting two adjacent spins in the
cycloid, and r is the displacement of the superexchange-mediating oxygen ions from
the line connecting the spins, shown schematically in Fig. 1.5a. The energy of the DM
interaction can therefore be lowered by inducing a polar lattice distortion, resulting
in a net polarisation P ∝ (S i × S i+1 ). DM electromagnons result from the dynamic
coupling of this polarisation to oscillating electric fields, causing an alteration in the
oxygen displacements r which in turn alters the magnetic interactions and drives a
magnon, and are eigenmodes of the spin-cycloid [43].
Exchange-striction electromagnons, on the other hand, arise from a modulation of
the isotropic Heisenberg exchange interaction by lattice vibrations, proportional to
a (S i · S j ) term in the Hamiltonian [47, 48], accounting for interactions between the
i
th and j
th spins. In RMnO 3 (where R = rare earth), distortions of the orthorhombic
crystal structure cause displacements in the positions of the superexchange-mediating
oxygen ions, resulting in a change in the Mn-O-Mn bond angle. According to the
Goodenough-Kanamori rules, the strength of the antiferromagnetic exchange interaction decreases as the Mn-O-Mn bond angle decreases from 180
◦ . The application of an electric field along the a-direction displaces all the oxygen ions by an
equal distance along this direction; this produces an alternating rotation of the MnO 6
octahedra, changing the Mn-O-Mn bond angles and altering the nearest-neighbour
exchange interaction along the spin-spiral propagation vector [48].
The different mechanisms giving rise to electromagnons exhibit seperate optical
selection rules, which may be used to differentiate between the two experimentally:
the DM electromagnon selection rules are directly linked to the plane of the spin-
15
1.3.2 Electromagnons
While the dynamics of magnons may be understood solely by the interaction of the
magnetic field of light with the spins in a material, the electric dipole-active nature of
electromagnons requires that coupling between the spins and phonons in the material
are taken into account in the Hamiltonian. Two distinct mechanisms that give rise to
electromagnons have been discussed in the literature: Dzyaloshinskii-Moriya electromagnons, and exchange-striction (ES) electromagnons. The different origins for
these mechanisms are shown schematically by Fig. 1.5a for the DM electromagnon
and Fig. 1.5b for the ES electromagnon.
The static polarisation in an improper ferroelectric phase with spin-cycloidal
ordering can be understood as arising from the spin-current or inverse DzyaloshinskiiMoriya (DM) interaction [33, 36]. The DM interaction can be expressed as
H DM =
a
d a (r i ) · (S i × S i+1 ) ,
(1.35)
where S i and S i+1 are adjacent spins in the cycloid. The DM vectors d a are parameterised by
d a = γ (δ i × r i ) ,
(1.36)
where γ is the interaction strength, δ is the vector connecting two adjacent spins in the
cycloid, and r is the displacement of the superexchange-mediating oxygen ions from
the line connecting the spins, shown schematically in Fig. 1.5a. The energy of the DM
interaction can therefore be lowered by inducing a polar lattice distortion, resulting
in a net polarisation P ∝ (S i × S i+1 ). DM electromagnons result from the dynamic
coupling of this polarisation to oscillating electric fields, causing an alteration in the
oxygen displacements r which in turn alters the magnetic interactions and drives a
magnon, and are eigenmodes of the spin-cycloid [43].
Exchange-striction electromagnons, on the other hand, arise from a modulation of
the isotropic Heisenberg exchange interaction by lattice vibrations, proportional to
a (S i · S j ) term in the Hamiltonian [47, 48], accounting for interactions between the
i
th and j
th spins. In RMnO 3 (where R = rare earth), distortions of the orthorhombic
crystal structure cause displacements in the positions of the superexchange-mediating
oxygen ions, resulting in a change in the Mn-O-Mn bond angle. According to the
Goodenough-Kanamori rules, the strength of the antiferromagnetic exchange interaction decreases as the Mn-O-Mn bond angle decreases from 180
◦ . The application of an electric field along the a-direction displaces all the oxygen ions by an
equal distance along this direction; this produces an alternating rotation of the MnO 6
octahedra, changing the Mn-O-Mn bond angles and altering the nearest-neighbour
exchange interaction along the spin-spiral propagation vector [48].
The different mechanisms giving rise to electromagnons exhibit seperate optical
selection rules, which may be used to differentiate between the two experimentally:
the DM electromagnon selection rules are directly linked to the plane of the spin-