# Switching between Magnetic Bloch and Néel Domain Walls with Anisotropy
Modulations

Kévin J. A. Franke  School of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, United Kingdom  Colin Ophus  National Center for Electron
Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Andreas K. Schmid  National Center for
Electron Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Christopher H. Marrows  School of Physics and
Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom

###### Abstract

It has been shown previously that the presence of a Dzyaloshinskii-Moriya
interaction in perpendicularly magnetized thin films stabilizes Néel type
domain walls. We demonstrate, using micromagnetic simulations and analytical
modeling, that the presence of a uniaxial in-plane magnetic anisotropy can
also lead to the formation of Néel walls in the absence of a Dzyaloshinskii-
Moriya interaction. It is possible to abruptly switch between Bloch and Néel
walls via a small modulation of both the in-plane, but also the perpendicular
magnetic anisotropy. This opens up a route towards electric field control of
the domain wall type with small applied voltages through electric field
controlled anisotropies.

The presence of an interfacial Dzyaloshinskii-Moriya interaction (DMI) in
perpendicular (denoted “PP”) magnetized thin films stabilizes Néel type domain
walls (DWs) of fixed chiralityThiaville  _et al._ (2012); Chen  _et al._
(2013) as opposed to the Bloch DWs favored by magnetostatics that are formed
in the absence of a DMI. Hubert and Schäfer (1998) In nanowires, Nééel DWs of
fixed chirality have been shown to be driven efficiently in the same direction
as the conventional electric current by interfacial spin-orbit torques,Ryu
_et al._ (2013); Emori  _et al._ (2013) making them appealing for potential DW
devices. Parkin  _et al._ (2008)
At the same time, electric field control of magnetism holds the promise of
low-power spintronic devices. Particularly the modulation of both in-plane
(IP) and PP magnetic anisotropies is well-established. Control is achieved
either via interfacial strain transfer from a ferroelectric or piezoelectric
substrate and inverse magnetostriction, Lahtinen  _et al._ (2011); Streubel
_et al._ (2013); Yu  _et al._ (2015); Shepley  _et al._ (2015); Finizio  _et
al._ (2014) or via direct charge modulation at the interface with an
insulator. Niranjan  _et al._ (2010); Maruyama  _et al._ (2009); Shiota  _et
al._ (2011); Wang  _et al._ (2012); Bauer  _et al._ (2015) The latter
modulates the interface anisotropy, which arises from the broken translational
symmetry at the interface and spin-orbit coupling (SOC), and can give rise to
PP magnetic anisotropy (PMA). Johnson  _et al._ (1996); Aharoni (2001)
Broken spatial inversion symmetry and SOC are also the ingredients that give
rise to the DMI. It emerges at the interface of a ferromagnet with a heavy
metal, Fert and Levy (1980) or more generally at the interface with a
different material due to Rashba SOC, as a result of the electrostatic
potential difference between the materials.Kundu and Zhang (2015) The latter
induces a DMI at the interface between a ferromagnetic film and an insulator,
and can thus be sensitive to a gate voltage. Nawaoka  _et al._ (2015);
Srivastava  _et al._ (2018); Zhang  _et al._ (2018); Suwardy  _et al._ (2019);
Yang  _et al._ (2020); Schott  _et al._ (2021) This has been used for electric
field control of magnetic DW motion via the modulation of the DMI.Koyama  _et
al._ (2018, 2020) The DMI has also been shown to be sensitive to the
application of strain,Gusev  _et al._ (2020); Yang  _et al._ (2020) which
opens up the route towards electric field control of DMI via coupling to a
piezoelectric or ferroelectric substrate. Still, both mechanisms for tuning
the DMI will also affect the magnetic anisotropy, making it difficult to
disentangle their effect on magnetic DWs. Furthermore, a switch between DW
types (Bloch and Néel), or a reversal of chirality with voltage remains
elusive. Similarly, the voltage control of skyrmions is currently being
investigated, and the electric field induced creation, annihilation and even
motion have been demonstrated. Hsu  _et al._ (2017); Schott  _et al._ (2017);
Srivastava  _et al._ (2018); Ma  _et al._ (2019) As for the case of DWs,
electric fields generally affect several material parameters, making it
difficult to determine the mechanism that allows for this voltage control.
Recently, Chen  _et al._ (2015) reported that in a magnetic multilayer
exhibiting PMA and DMI, the type of DW depends on the relative angle between
the DW and a uniaxial IP magnetic anisotropy (IMA) of constant magnitude.
Given the strong dependence of spin-orbit torques on DW type and the fact that
magnetic anisotropies can be induced and modulated in various ways, Hubert and
Schäfer (1998) this observation raises the question about control – and
possibly switching – of DW type with anisotropy modulations.
In this letter, we therefore demonstrate an alternative mechanism for the
control of DW type: using micromagnetic simulations and analytical modeling,
we show that the presence of a uniaxial IMA of fixed orientation can also lead
to the formation of Néel DWs in the absence of a DMI. It is possible to
abruptly switch between Bloch and Néel DWs via a small modulation of the
strength of both the IMA and PMA. This opens up a route towards efficient
electric field control of the DW type with small applied voltages, as the
magnetic anisotropy strength can be modulated via the direct voltage
controlled magnetic anisotropy mechanism,Maruyama  _et al._ (2009); Shiota
_et al._ (2011) or via magnetoelastic anisotropy induced through coupling to a
piezoelectric element.Li  _et al._ (2015); Yu  _et al._ (2015)
We investigated this control of DW type through micromagnetic simulations
using the OOMMF software package.Donahue and Porter (1999) The simulated
geometry is sketched in Fig. 1: it consists of a thin film of thickness $t=1$
nm.

Figure 1:  (left) Sketch of the simulation geometry with definition of
directions. (right) Definition of in-plane color wheel and domain wall
magnetization angle $\phi$ between the magnetization $\overrightarrow{M}$ at
the centre of the domain wall and the normal $\hat{n}$ to the domain wall.

The in-plane dimensions are $400\times 200$ nm2, and two-dimensional periodic
boundary conditions Wang  _et al._ (2010) are used to simulate an infinite
film. Simulations are initialized such that two DWs are stabilized. We choose
reasonable values for the saturation magnetization $M_{\mathrm{s}}=1\times
10^{6}$ A/m and exchange stiffness $A=3\times 10^{-11}$ J/m.Stöhr and Siegmann
(2006); Eyrich  _et al._ (2012); Devolder  _et al._ (2016) We consider the
effects of a PMA with anisotropy constant $K_{\mathrm{pp}}$, a uniaxial IMA
along the $x$-direction (perpendicular to the DWs) with anisotropy constant
$K_{\mathrm{ip}}$, and an interfacial DMI with constant $D$.
To simulate a nanowire geometry, the two-dimensional periodic boundary
conditions are omitted and the width of the simulations altered in the
$y$-direction. The extent of simulations in the $x$-direction is chosen such
that DWs are not affected by finite size effects along this dimension. The DW
magnetization angle $\phi$ is defined relative to the DW normal $\hat{n}$
(Fig. 1). For Bloch DWs $\phi=\pm 90^{\circ}$, while for Néel DWs
$\phi=0^{\circ}$ or $180^{\circ}$. The DW width
$\delta=\int_{-\infty}^{+\infty}\cos^{2}(\theta)dx$ is defined as an integral
over the magnetization profile of the DW, where
$\theta=\sin^{-1}\left(\nicefrac{{M_{\mathrm{z}}}}{{M_{\mathrm{}s}}}\right)$
is the polar angle between the magnetization direction and the film plane.
Jakubovics (1978) For an ideal Bloch DW this definition yields
$\delta=2\sqrt{\nicefrac{{A}}{{K}}}$, where $K$ is the effective anisotropy.
Hubert and Schäfer (1998)
We start by reproducing the well-known effect the DMI has on the chirality of
magnetic DWs in PP magnetized thin films. Images of a DW as a function of
increasing DMI constant $D$ for $K_{\mathrm{pp}}=1\\!\times\\!10^{6}$ J/m3 are
shown in Fig. 2(a). The corresponding $\phi$ is plotted in Fig. 2(d). As
reported previously,Thiaville  _et al._ (2012) the DW magnetization angle
rotates continuously from a Bloch towards a Néel configuration as soon as a
DMI is present. Above a certain value of $D$, $\phi$ saturates at $0$, i.e. a
Néel DW.
In the absence of a DMI, an IMA with easy axis perpendicular to the DW also
allows for a tuning between Bloch and Néel DWs. The effect of an increasing
IMA is shown in Fig. 2(b). Unlike the DMI, the anisotropy does not immediately
affect the DW magnetization angle. As a function of increasing
$K_{\mathrm{ip}}$, the DW first remains of Bloch type until it switches
abruptly to a Néel DW. This behaviour is highlighted in Fig. 2(e), where
$\phi$ is shown as a function of $K_{\mathrm{ip}}$. Note, that the magnitude
of the IMA required to switch between DW types is about two orders of
magnitude smaller than the PMA strength and does thus not significantly affect
the magnetization in the domains.
For $K_{\mathrm{pp}}=1\\!\times\\!10^{6}$ J/m3 and
$K_{\mathrm{ip}}=3\\!\times\\!10^{4}$ J/m3, a Néel DW is stabilized. As shown
in the images of Fig. 2(c), and the graph in panel (f), an increase in the PMA
strength eventually leads to an abrupt switch to a Bloch DW. It is thus
possible to switch between DW types by either tuning the IMA or PMA strength.

Figure 2:  Domain wall images as a function of (a) DMI constant $D$ and (b)
in-plane anisotropy constant $K_{\mathrm{ip}}$ for a perpendicular anisotropy
constant $K_{\mathrm{pp}}=1\\!\times\\!10^{6}$ J/m3. (c) Images as a function
of $K_{\mathrm{pp}}$ for $K_{\mathrm{ip}}=3\\!\times\\!10^{4}$ J/m3.
Corresponding domain wall magnetization angles $\phi$ as a function of (d)
$D$, (e) $K_{\mathrm{ip}}$, and (f) $K_{\mathrm{pp}}$.

We further investigate this in a phase diagram (Fig. 3(a)), establishing
regions where Néel or Bloch DWs are stabilized as a function of
$K_{\mathrm{pp}}$ and $K_{\mathrm{ip}}$. We find that for higher values of
$K_{\mathrm{ip}}$ and lower values of $K_{\mathrm{pp}}$, Néel DWs form.
Conversely, for smaller values of $K_{\mathrm{ip}}$ and larger values of
$K_{\mathrm{pp}}$, Bloch DWs are observed. The transition between Bloch and
Néel DWs appears sharp, which is in stark contrast to the continuous
transition observed when the DMI constant is changed. A $K_{\mathrm{pp}}$ – vs
– $D$ phase diagram in the Supplemental InformationSup  (SI) furthermore
reveals that in the presence of a DMI, but absence of IMA, a tuning of the
magnitude of $K_{\mathrm{pp}}$ has no effect on the DW magnetization angle.

Figure 3: (a) Phase diagram of the domain wall magnetization angle $\phi$ as a
function of perpendicular ($K_{\mathrm{pp}}$) and in-plane ($K_{\mathrm{ip}}$)
anisotropy constants. (b) Location of the transition between Bloch and Néel
walls for a thin film (blue line), for a nanowire (orange dashed line), and
according to the analytical model (black dotted line). (c) Widths $\delta$ of
Bloch (green line) and Néel (red dashed line) domain walls just below and
above the transition. The analytical expression
$\delta=2\,\sqrt{\nicefrac{{A}}{{K_{\mathrm{pp,eff}}}}}$ is shown as a black
dotted line.

It is thus only this new mechanism, involving an IMA, that allows for
switching between DW types via a modulation of the PMA strength, at least for
the experimentaly achievable parameters considered here.
We investigate this surprising result further by plotting the IMA strength at
which the transition occurs as a function of the effective PMA (blue line in
Fig. 3(b). We find a linear dependence of $\log\left(K_{\mathrm{ip}}\right)$
on $\log\left(K_{\mathrm{pp,eff}}\right)$ with slope
$s\\!=\\!\nicefrac{{1}}{{2}}$. The IMA strength at which the switch between DW
types occurs thus shows a square root dependence on the effective PMA
strength.
To understand this dependence, we construct a simple analytical model. The
full derivation can be found in the SI.Sup  The model compares the total
energies of Bloch and Néel DWs for a given magnitude of $K_{\mathrm{pp}}$ and
$K_{\mathrm{ip}}$. The widths of both Néel and Bloch DWs are shown as a
function of $K_{\mathrm{pp,eff}}$ in Fig. 3(c), along with the theoretical
value $\delta=2\,\sqrt{\nicefrac{{A}}{{K_{\mathrm{pp,eff}}}}}$. We find
excellent agreement between them, and therefore make the simplification that
both types of DW types exhibit the same width.
We find that in first approximation the difference in DW surface energy
$\sigma$ between Néel ($\sigma_{\mathrm{N}}$) and Bloch
($\sigma_{\mathrm{B}}$) DWs is given by:

$\Delta\sigma=\sigma_{\mathrm{N}}-\sigma_{\mathrm{B}}=K_{\mathrm{ip}}\,\delta-\frac{\mathrm{ln}2}{\pi}\,\mu_{0}\,M_{\mathrm{s}}^{2}\,t.$
(1)

The first term results from the IMA, while the second term is a consequence of
magnetostatics. The magnetostatic contribution arises from magnetic volume
charges only.Sup ; Skaugen  _et al._ (2019); García-Cervera (2004); Kohn and
Slastikov (2005) For low values of $K_{\mathrm{ip}}$, the magnetostatic energy
favoring Bloch DWs dominates. At large values of $K_{\mathrm{ip}}$, the
anisotropy energy favoring Néel DWs overcomes the magetostatic energy. The
transition between DW types is expected to occur when the difference in energy
is zero. As a result, the DW is expected to switch between Bloch and Néel type
when:

$K_{\mathrm{ip}}=\frac{\mathrm{ln}2\,\mu_{0}\,}{2\,\pi}\frac{M_{\mathrm{s}}^{2}\,t}{\sqrt{A}}\sqrt{K_{\mathrm{pp,eff}}}.$
(2)

This dependence is plotted in Fig. 3(b), where excellent agreement between
results from micromagnetic simulations (blue line) and the analytical model
(black dotted line) is observed. The analytical model thus explains the square
root dependence of the IMA strength at which the switch between DW types
occurs on the effective PMA strength. The fact that the type of DW that is
stabilized also depends on $K_{\mathrm{pp}}$ is due to the fact that the IMA
contribution in $\Delta\sigma$ (Eq. 1) depends on the DW width, which in turn
depends on the PMA.
Our simulations correspond to an experimental system where IP and PP magnetic
anisotropies can be tuned independently. Electric field control of DW type
could be achieved by tuning the strength of one of these anisotropies with a
voltage. One way would be to deposit a magnetic multilayer exhibiting PMA onto
a piezoelectric substrate to induce a voltage tuneable uniaxial IMA via
interfacial strain transfer and inverse magnetostriction. Finizio  _et al._
(2014); Li  _et al._ (2015) Another approach, that would also allow for local
control, would be to tune the PMA strength via charge modulation at an
interface. Niranjan  _et al._ (2010); Maruyama  _et al._ (2009); Shiota  _et
al._ (2011); Wang  _et al._ (2012); Bauer  _et al._ (2015) This would of
course require the presence of an uniaxial IMA, which could be induced in
various ways. One way to achieve this would be to simply utilize the shape
anisotropy in a magnetic nanowire to induce a uniaxial IMA. DWs tend to form
perpendicular to the nanowire length, while the shape anisotropy induces a
uniaxial anisotropy along it, which corresponds to the geometry investigated
here. Nanowires are used in most DW applications, and this approach would
eliminate the need for a separate mechanism to induce the uniaxial IMA.
Fig. 4(a) displays the phase diagram for the DW type as a function of
$K_{\mathrm{pp}}$ and nanowire width $w$. It confirms previous observations of
a transition from a Bloch to a Néel DW when the nanowire width is reduced.
Martinez  _et al._ (2011); DeJong and Livesey (2015)

Figure 4: (a) Phase diagram of the domain wall magnetization angle $\phi$ as a
function of perpendicular magnetic anisotropy constant $K_{\mathrm{pp}}$ and
nanowire width $w$. (b) Domain wall images as a function of $K_{\mathrm{pp}}$
for $w=32$ nm.

It does also show that this transition depends on the strength of the PMA.
Therefore, it is possible to switch between Bloch and Néel DWs for a given
nanowire width when $K_{\mathrm{pp}}$ is modulated. We extract the location of
the transition and express it in terms of effective anisotropies.Sup  The
resulting curve is plotted as an orange dashed line in Fig. 3(b). It matches
the results for thin films and the analytical model well, except for low
values of the effective IMA and PMA strengths. We ascribe this to the fact
that for wide nanowires (corresponding to a low effective $K_{\mathrm{ip}}$),
expressing magnetostatic effects as a simple uniaxial anisotropy is too crude
an approximation.
Unlike the case of thin films, the transition between DW types in nanowires
does not result from a competition between a magnetic anisotropy and
magnetostatics. it is purely the results of magnetostatics: magnetic volume
charges favor Bloch DWs, while magnetic surface charges on the edges of the
nanowire are minimized for Néel DWs. For a given width of the nanowire,
increasing $K_{\mathrm{pp}}$ decreases the width of the DW which leads to a
reduction of magnetic surface charges. As a result, Bloch DWs become
energetically favorable. Conversely, decreasing $K_{\mathrm{pp}}$ increases
the build up of magnetic surface charges, thus favoring Néel DWs.
The nanowire geometry also allows for the stabilization of intermediate DW
magnetization angles $\phi$. As highlighted by the inset in Fig. 4(a), the
transition between Bloch and Néel DWs is not as sharp as in the thin film
case. This has already been observed as a function of $w$.Boehm  _et al._
(2017) As shown in Fig. 4(b), tuning between Néel and Bloch DWs with a PMA in
nanowires also involves DWs with intermediate $\phi$.
Stabilizing Néel DWs with an IMA does not favor one chirality, unlike the DMI.
Left- and right-handed DWs are energetically degenerate. For applications, it
might be necessary to obtain Néel DWs with a fixed chirality. We now show that
it is still possible to tune DW type with an anisotropy in the presence of a
small DMI that yields DWs of fixed chirality. Phase diagrams as a function of
$K_{\mathrm{pp}}$ and $K_{\mathrm{ip}}$ for fixed values of $D$ are shown in
the SI.Sup  Here, we focus on the phase diagram as a function of $D$ and
$K_{\mathrm{ip}}$ for a fixed value of $K_{\mathrm{pp}}=1\times 10^{6}$ J/m3
shown in Fig. 5.

Figure 5:  Phase diagram of the domain wall magnetization angle $\phi$ as a
function of DMI constant $D$ and in-plane anisotropy constant
$K_{\mathrm{ip}}$ for a fixed magnitude of the perpendicular anisotropy
constant $K_{\mathrm{pp}}=1\\!\times\\!10^{6}$ J/m3. Solid blue lines are
contour lines of the simulation data for given domain wall magnetization
angles of $15^{\circ}$, $45^{\circ}$, and $75^{\circ}$. Dotted black lines are
the contour lines expected from the analytical model.

We observe that while a positive value of $K_{\mathrm{ip}}$ can be used to
switch from a Bloch to a Néel DW when $D=0$, a negative value of
$K_{\mathrm{ip}}$ tunes the Néel DW obtained for large values of $D$ towards a
Bloch DW. For values of $D$, where an intermediate $\phi$ is obtained,
negative and positive $K_{\mathrm{ip}}$ values tune the DW towards the Bloch
and Néel type, respectively. A negative $K_{\mathrm{ip}}$ corresponds to an
easy axis along the DW. We observe furthermore that while the transition
between Bloch and Néel DWs is abrupt for $D=0$, it becomes increasingly wider
as $D$ increases. This is highlighted by the contour lines (blue) for DW
magnetization angles of $15^{\circ}$, $45^{\circ}$, and $75^{\circ}$.
The contour lines can be obtained from our analytical model by including the
DMI energy in the DW surface energy. It yields the black dotted lines in Fig.
5, showing excellent agreement between micromagnetic simulations and the
model, and demonstrating that the contour lines are linear in $-D$.Sup
We have therefore shown, using micromagnetic simulations, that the presence of
a uniaxial in-plane magnetic anisotropy can lead to the formation of Néel
domain walls in the absence of a DMI. It is possible to abruptly switch
between Bloch and Néel walls via a small modulation of not only the in-plane,
but also the perpendicular magnetic anisotropy. In nanowires, the shape
anisotropy can be used to induce the in-plane anisotropy. In this case, tuning
between domain wall types with a perpendicular magnetic anisotropy proceeds
via intermediate domain wall magnetization angles. The presence of a DMI
widens the transition between domain wall types. A simple analytical model
accounts for the dependence of domain wall type on both the in-plane and
perpendicular magnetic anisotropies, and the DMI. Our results open up the
route towards voltage control of domain wall type with small applied voltages
through electric field controlled anisotropies. As only Néel domain walls are
driven by interfacial spin orbit torques in nanowires, while Bloch domain
walls are not, this could allow for efficient control of doman wall motion
with electric fields. We expect that our results obtained for DWs can be
extended to other chiral spin textures such as skyrmions.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 750147. K.J.A.F. acknowledges support from the Jane and Aatos
Erkko Foundation. Work at the Molecular Foundry was supported by the Office of
Science, Office of Basic Energy Sciences, of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231. This research used the Lawrencium
computational cluster resource provided by the IT Division at the Lawrence
Berkeley National Laboratory (Supported by the Director, Office of Science,
Office of Basic Energy Sciences, of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231.

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