Magnetoelastic interaction and domain walls in a magnetic sandwich structure
K. V. Lamonova, A. L. Sukstanskii,* and V. N. Varyukhin
Donetsk Physico-Technical Institute, National Academy of Sciences of Ukraine, 340114, Donetsk, Ukraine
Received 22 February 1999
The domain walls in a sandwichlike structure an ultrathin magnetic layer is situated between two nonmag-
netic plates are investigated theoretically. Taking account of the magnetodipole and magnetoelastic interaction
is shown to lead to the appearance of an effective anisotropy with respect to the direction of the normal to the
wall plane. A type of domain wall, ‘‘angular’’ walls, at which the magnetization vector rotates in the plane
making a certain angle with the wall plane, is proved to exist at a sufficiently small magnetic layer thickness.
The static and dynamic properties of these walls are investigated, in particular, the dependence of the limiting
velocity of the domain wall on the magnetic film thickness is found. S0163-18299900437-3
I. INTRODUCTION
A magnetoelastic interaction is known to play an impor-
tant role in forming properties of magnetically ordered crys-
tals. There are a lot of papers in which various phenomena
associated with this interaction magnetoacoustic resonance,
the appearance of a magnetoelastic gap in the spin-wave
spectrum, etc. are investigated. An effect of the magneto-
elastic interaction on the static and dynamic properties of
large-scale inhomogeneities of the domain-wall type is also
well known, e.g., an existence of 180° domain walls in a
ferromagnetic with cubic magnetic anisotropy is caused just
by the magnetoelastic interaction.
1
The magnetoelastic interaction is especially pronounced
in antiferromagnets and weak ferromagnets, in which this
interaction is enhanced by an exchange interaction. In par-
ticular, some experimentally observed features in the field
dependence of the steady-state velocity of domain walls in
weak ferromagnets
2
are related to Cherenkov radiation of
sound waves, which arises when the wall attains the sound
velocity.
3
An influence of the magnetoelastic interaction on dynam-
ics properties of domain walls in ferromagnets is signifi-
cantly smaller since the impact of the magnetoelastic inter-
action is usually masked by much stronger interaction, for
example, magnetodipole interaction. Moreover, the veloci-
ties of domain walls in ferromagnets are limited by the so-
called Walker limit,
4
the value of which is caused by relativ-
istic interactions rhombic anisotropy, magnetodipole
interaction and is usually much less than the sound velocity,
so that resonance effects such as the Cherenkov radiation of
sound are impossible.
However, there exists a situation in which the role of the
magnetoelastic interaction in the formation of static and dy-
namic properties of domain walls in ferromagnets can be
exceptionally important, namely, in ultrathin magnetic films
with an uniaxial magnetic anisotropy. In a ferromagnet with
such a type of anisotropy the motion of the wall is generally
impossible unless the magnetodipole interaction is taken into
account, in view of the presence of an additional integral of
motion of the system—the total projection of the magnetiza-
tion on the anisotropy axis.
5
Ignoring the magnetodipole in-
teraction leads to a zero limiting velocity
4
and a formally
infinitely large mass of the wall. As shown below, in ultra-
thin ferromagnetic films the magnetodipole interaction in the
main approximation with respect to the small parameter—a
ratio of a film thickness to a characteristic size of a magne-
tization distribution e.g., a domain-wall thickness—results
only in a renormalization of the uniaxial anisotropy constant
and determines neither the mass of the domain wall nor its
limiting velocity.
6
Naturally, in this case, the role of other
interactions affecting these characteristics of the domain wall
becomes more important; this is primarily the magnetoelastic
interaction.
Our paper aims at presenting the theoretical analysis of
the influence of the magnetoelastic interaction on static and
dynamic properties of domain walls in an ultrathin ferromag-
netic film with a uniaxial magnetic anisotropy. Such a film is
usually a part of a multilayer structure consisting of magnetic
and nonmagnetic layers. Below we restrict ourselves to the
simplest sandwichlike structure in which a magnetic film is
situated between two infinitely large nonmagnetic layers.
Let us consider a ferromagnetic film of thickness h with
the normal oriented along the anisotropy axis Z, situated in
the layer -h /2z h /2 between two nonmagnetic plates
z -h /2 and z h /2). Taking into account the magnetodi-
pole and magnetoelastic interactions, the energy of the sys-
tem under consideration can be written in the form
W=W
m
+W
e
+W
me
,
W
m
=
V
0
dV
2
M
2
-
2
M
z
2
-
1
2
MH
d
,
W
me
=b
V
0
dV M
i
M
j
u
ij
1
W
e
=
V
0
dV
2
u
ii
2
+ u
ij
2
+
V
1
dV
˜
2
u
ii
2
+ ˜ u
ij
2
+
V
2
dV
5
2
u
ii
2
+ 5 u
ij
2
,
where M is the magnetization vector, and are the ex-
change interaction and the uniaxial anisotropy constants, re-
PHYSICAL REVIEW B 1 OCTOBER 1999-II VOLUME 60, NUMBER 14
PRB 60 0163-1829/99/6014/102638/$15.00 10 263 ©1999 The American Physical Society