150
J. Masell and K. Everschor-Sitte
irrelevant on larger scales. Other terms, such as the DMI term with D
α
i j , are only
non-vanishing because of the finite spin-orbit coupling, which is usually also small.
Here we list the most common and relevant examples focusing on magnetic systems
with their dominant lowest order chiral interaction
• For a time-reversal invariant system all terms with an odd power of m vanish.
• For an inversion symmetric system there is no chiral interaction, i.e. the DMI term
vanishes, D
α
i j = 0, and so do all terms with an odd number of derivatives.
• For a bulk chiral magnet with a cubic unit cell and a three-fold screw axis in
the [109] direction, like the prototypical chiral magnets MnSi or FeGe, the DMI
tensor simplifies to what is denoted as Bloch-type DMI in the literature, i.e.,
D
α
i j = D iα j , with i jk being the Levi-Civita symbol. The exchange interaction
becomes A
αβ
i j = Aδ i j δ αβ + A
δ i j δ αβ δ iα with the Kronecker delta δ i j . The last term
proportional to A
reflects an anisotropic exchange coupling which can be present
in cubic systems, but for MnSi and FeGe it turns out to be negligible [16].
• In thin films or monolayers, the inversion symmetry along the film normal (e.g., the
ˆ
z-direction) is explicitly broken by the sandwich structure of the material or the substrate, but is usually preserved in the other directions. In such a setup the DMI tensor simplifies to what is known as Néel-type DMI, i.e. D
α
i j = D(δ iα δ jz − δ iz δ jα ).
The exchange interaction simplifies to A
αβ
i j = Aδ i j δ αβ + A
z
δ i j δ iz δ αβ . Besides
exchange and DMI, the term that is often relevant in such systems is the uniaxial anisotropy K i j = K δ i j δ iz . In combination with the demagnetizing field, it
can lead to the stabilization of magnetic bubbles.
• For systems with lower symmetry, the emerging terms and the corresponding
tensor entries become more and more complex. We still would like to highlight
systems with C 2ν symmetry, where the two-fold rotational symmetry allows not
only to realize magnetic skyrmions but also antiskyrmions [17], see Fig. 7.1. In a
basis where the ˆ
z-axis is the two-fold rotational symmetry and the ˆ
x and ˆ
y-axes
are defined to be along the two reflection planes of the C 2v point group [18], the
exchange parameters are A
αβ
i j ∂ α m i ∂ β m j = A i δ i j δ αβ and there are seven independent DMI tensor components given by D
x
xz , D
x
zx , D
y
yz , D
y
zy , D
z
zz , D
z
xx , and D
z
yy .
For further interesting systems we refer to [19, 20].
To summarize, the specific systems determine which magnetic interaction scales
are relevant and which magnetic structures can be realized as (meta-)stable states.
Over the past century, magnets with strong uniaxial anisotropy have been in the
focus of material research, mostly application-oriented. With the advances made
over the past decades, more detailed engineering of the properties of magnetic materials became possible and experimental techniques were developed that enable the
observation of magnetic structures on the nanometer scale. With these new techniques at hand, more exotic materials can be studied where other interactions are
dominant and stabilize new forms of magnetic textures.
J. Masell and K. Everschor-Sitte
irrelevant on larger scales. Other terms, such as the DMI term with D
α
i j , are only
non-vanishing because of the finite spin-orbit coupling, which is usually also small.
Here we list the most common and relevant examples focusing on magnetic systems
with their dominant lowest order chiral interaction
• For a time-reversal invariant system all terms with an odd power of m vanish.
• For an inversion symmetric system there is no chiral interaction, i.e. the DMI term
vanishes, D
α
i j = 0, and so do all terms with an odd number of derivatives.
• For a bulk chiral magnet with a cubic unit cell and a three-fold screw axis in
the [109] direction, like the prototypical chiral magnets MnSi or FeGe, the DMI
tensor simplifies to what is denoted as Bloch-type DMI in the literature, i.e.,
D
α
i j = D iα j , with i jk being the Levi-Civita symbol. The exchange interaction
becomes A
αβ
i j = Aδ i j δ αβ + A
δ i j δ αβ δ iα with the Kronecker delta δ i j . The last term
proportional to A
reflects an anisotropic exchange coupling which can be present
in cubic systems, but for MnSi and FeGe it turns out to be negligible [16].
• In thin films or monolayers, the inversion symmetry along the film normal (e.g., the
ˆ
z-direction) is explicitly broken by the sandwich structure of the material or the substrate, but is usually preserved in the other directions. In such a setup the DMI tensor simplifies to what is known as Néel-type DMI, i.e. D
α
i j = D(δ iα δ jz − δ iz δ jα ).
The exchange interaction simplifies to A
αβ
i j = Aδ i j δ αβ + A
z
δ i j δ iz δ αβ . Besides
exchange and DMI, the term that is often relevant in such systems is the uniaxial anisotropy K i j = K δ i j δ iz . In combination with the demagnetizing field, it
can lead to the stabilization of magnetic bubbles.
• For systems with lower symmetry, the emerging terms and the corresponding
tensor entries become more and more complex. We still would like to highlight
systems with C 2ν symmetry, where the two-fold rotational symmetry allows not
only to realize magnetic skyrmions but also antiskyrmions [17], see Fig. 7.1. In a
basis where the ˆ
z-axis is the two-fold rotational symmetry and the ˆ
x and ˆ
y-axes
are defined to be along the two reflection planes of the C 2v point group [18], the
exchange parameters are A
αβ
i j ∂ α m i ∂ β m j = A i δ i j δ αβ and there are seven independent DMI tensor components given by D
x
xz , D
x
zx , D
y
yz , D
y
zy , D
z
zz , D
z
xx , and D
z
yy .
For further interesting systems we refer to [19, 20].
To summarize, the specific systems determine which magnetic interaction scales
are relevant and which magnetic structures can be realized as (meta-)stable states.
Over the past century, magnets with strong uniaxial anisotropy have been in the
focus of material research, mostly application-oriented. With the advances made
over the past decades, more detailed engineering of the properties of magnetic materials became possible and experimental techniques were developed that enable the
observation of magnetic structures on the nanometer scale. With these new techniques at hand, more exotic materials can be studied where other interactions are
dominant and stabilize new forms of magnetic textures.