Micromagnetic Description of Symmetry-Breaking Effects in Curvilinear Ferromagnetic Shells

The behaviour of any physical system is governed by the order parameter,
determined by the geometry of the physical space of the object, namely their
dimensionality and curvature. Usually, the effects of curvature are described
using local interactions only, e.g. local spin-orbit- or curvature-induced
Rashba and Dzyaloshinskii-Moriya interactions (DMI). In the specific case
of ferromagnetism, until recently, there was no analytical framework, which
was treating curvature effects stemming from local [1] and non-local [2]
interactions on the same footing. The lack of a proper theoretical foundation
impedes the description of essential micromagnetic textures like magnetic
domains, skyrmion-bubbles and vortices. Here, we present a micromag-
netic theory of curvilinear ferromagnetic shells, which allows to describe the
geometry-driven effects stemming from exchange and magnetostatics within
the same framework [3]. A general description of magnetic curvilinear shells
can be done using tangential derivatives of the unit magnetization vector.
Tangential derivatives are represented by the covariant derivatives of in-
surface components and the regular derivative of the normal magnetiza-
tion component, normalized by the square root of the corresponding metric
tensor coefficient. This allows to separate the explicit effects of curvature
and spurious effects of the reference frame. The shape of a given thin shell
can be determined by two principal curvatures k1 and k2, which are functions
of coordinate. The respective classification of curvilinear surfaces operates
with (i) developable surfaces, where one of the principal curvatures equals
to zero; (ii) minimal ones, where the mean curvature k1 + k2 = 0; and (iii)
the general case. The local geometry-driven energy contributions are repre-
sented by the DMI and anisotropy, whose coefficients are determined by
powers of the principal curvatures. This allows to cancel the influence of one
of the DMI terms for the developable surfaces for any magnetic texture. The
magnetostatic interaction is a source of new chiral effects, which are essen-
tially non-local in contrast to the conventional DMI. The physical origin is
the non-zero mean curvature of a shell and the non-equivalence between the
top and bottom surfaces of the shell. We demonstrate that the analysis of
non-local effects in curvilinear thin shells can become more straightforward
when introducing three magnetostatic charges. In this respect, in contrast
to the classical approach by Brown [4], we split a conventional volume
magnetostatic charge into two terms: (i) magnetostatic charge, governed by
the tangent to the sample’s surface, and (ii) geometrical charge, given by the
normal component of magnetization and the mean curvature. In addition to
the shape anisotropy (local effect), there appear four additional non-local
terms, determined by the surface curvature. Three of them are zero for any
magnetic texture in shells with the geometry of minimal surfaces. The fourth
term is determined by the non-equivalence of the top and bottom surfaces
of the shell and becomes zero only for the special symmetries of magnetic
textures. The discovered non-local magnetochiral effects introduce hand-
edness in an intrinsically achiral material and enables the design of magne-
to-electric and ferro-toroidic responses. This will stimulate to rethink the
origin of chiral effects in different systems, e.g. in fundamentally appealing
and technologically relevant skyrmionic systems, and further theoretical
investigations in the field of curvilinear magnetism as well as experimental
validation of these theoretical predictions. These developments will pave the
way towards new device ideas relying on curvature effects in magnetic nano-
structures. The impact of effects predicted in this work goes well beyond
the magnetism community. Our description of the vector field behaviour
can be applied to different emergent field of studies of curvature effects.
The prospective applications include curved superconductors [5], twisted
graphene bilayers [6], flexible ferroelectrics [7], curved liquid crystals [8].

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