text:
\section{Introduction}
\label{sec:introduction}

Domain wall (DW) statics and dynamics in thin-film ferromagnetic
systems have been a subject of intense
experimental\cite{Atkinson03,Yamaguchi04, Parkin:racetrack08,
  Hoffmann2015_review} and theoretical \cite{Tatara04, Thiaville05,
  Tretiakov08,Fert2013, Shibata2011_review} studies over the last
decades due to their direct relevance to spintronic
memory\cite{Parkin08} and logic devices. \cite{Allwood02} Recently, it
has been realized that ferromagnets with Dzyaloshinskii-Moriya
interaction\cite{Dzyaloshinskii58,Moriya60} (DMI) may offer more
benefits in this direction, \cite{Tretiakov_DMI, Thiaville2012,
  Boulle13} which led to an enormous experimental progress for these
systems. \cite{Emori2014,Franken2014,
  Marrows_SAF2015,Boulle2016,Jiawei2016}

Due to their better technological suitability as smaller and more
robust carriers of information in spintronic nanodevices, the DWs in
ultrathin ferromagnetic films with out-of-plane anisotropy and
interfacial DMI have now become the primary objects of experimental
interest. \cite{Emori2014,Franken2014,Marrows_SAF2015,Boulle2016,Jiawei2016}
Moreover, it was discovered that the DWs move much more efficiently in
these systems due to spin-orbit torques.\cite{Garello2013, Fert2013,
  Ado2017} It was also demonstrated that in these systems the DW
equilibrium structure changes from Bloch to N\'eel type in the
presence of strong DMI. \cite{chen13} Following a theoretical
study,\cite{Thiaville2012} we refer below to this new type of magnetic
DWs as {\em Dzyaloshinskii} domain walls.

Boulle \textit{et al.}\cite{Boulle13} were the first to discover
numerically that these DWs develop a tilt under in-plane magnetic
fields and applied currents. This DW tilt was shown to depend on the
DMI and field strengths. It was followed by several more attempts to
investigate this phenomena
theoretically\cite{Martinez2014,Vandermeulen2016} and multiple
experimental studies.\cite{Emori2014,Franken2014,Jiawei2016} However,
up to now there is still lack of a unifying theory of the DW tilt and
its dependence on the in-plane magnetic field.

In this paper, we explain these important findings on more solid
theoretical grounds, using variational analysis of the magnetic energy
functional. We study a DW in a thin nanostrip with perpendicular
magnetic anisotropy (PMA) and interfacial DMI in the presence of a
magnetic field applied in the plane of the strip and perpendicular to
its axis, see Fig.~\ref{fig:tilt2dnum} for an example. For this system
we analytically develop a proper reduced geometric variational model
built on exact one-dimensional (1D) DW solutions to describe the
equilibrium tilted two-dimensional (2D) DW configurations. These exact
1D magnetization profiles are in general neither N\'eel nor Bloch
type, but can be easily computed numerically for all relevant values
of the parameters. We also derive explicit analytical expressions by
expanding the DW energy in the applied field or DMI strengths.

\begin{figure}
  \centering
  \includegraphics[width=3.25in]{tilt2dnum.pdf}
  \caption{An example of a tilted domain wall in a transverse in-plane
    magnetic field computed using micromagnetic simulations.  The
    material parameters are: $A = 10^{-11}$J/m, $K = 1.25 \times 10^6$
    J/m$^3$, $M_s = 1.09 \times 10^6$ A/m, $D = 2.4$ mJ/m$^2$ and
    $B_y = \mu_0 H = 80$ mT (see Sec.~\ref{sec:model} for precise
    definitions).  }
  \label{fig:tilt2dnum}
\end{figure}

In the reduced 2D variational problem, we treat the DW as a curve
whose shape is determined by minimizing an appropriate geometric
energy functional.  As a result we show that the DW in equilibrium
remains straight despite the fact that the wall energy is a function
of its local orientation. In particular, for small fields the tilt
angle is found to be proportional to the transverse magnetic field
strength.  One of the features of our 2D analysis is the necessity to
include the edge DWs found earlier in the context of
skyrmions.\cite{Rohart2013} These edge DWs can be seen in
Fig.~\ref{fig:tilt2dnum} along the upper and lower strip edges. We
show that the contribution of the edge DWs is also essential for
determining the proper tilt angle.  This is because the total DW
energy contains contributions from both the internal and the edge DWs,
and it is the competition among them that determines the tilt
angle. We find that the effect of the edge DWs becomes weaker when the
DMI strength is reduced, whereas the internal DW energy has a
nontrivial dependence on the DMI, magnetic field, and wall orientation
that have to be properly accounted to determine the equilibrium tilt
angle.

The main advantage of our reduced geometric variational model for
tilted DWs is its considerable simplicity compared to the full
micromagnetic description. Specifically, it allows for a detailed
analytical treatment, which highlights the key physical features of
tilted DWs mediated by interfacial DMI in PMA nanostructures. In
particular, it yields explicit closed-form expressions for the
dependence of the equilibrium tilt angle for a wide range of the
material parameters and applied fields. The obtained analytical
predictions are found to be in excellent agreement with the results of
micromagnetic simulations, indicating that the reduced model captures
all the essential physical aspects of the considered system.

The paper is organized as follows. In Sec.~\ref{sec:model} we
introduce the full micromagnetic model and its 2D reduction
appropriate for infinite ultrathin ferromagnetic nanostrips. In
Sec.~\ref{sec:monodomain-states} the theory of edge domain walls is
presented, and in Sec.~\ref{sec:interior-wall} a detailed analysis of
1D interior wall profiles is carried out. Next, in
Sec.~\ref{sec:two-dimens-probl} we demonstrate how the theory of 1D
domain walls developed in the preceding sections is applied to a
Dzyaloshinskii DW in an infinite 2D nanostrip. In
Sec.~\ref{sec:comp-with-numer} we compare our analytical theory with
micromagnetic simulations and show a good agreement between them. Here
the additional effect of dipolar interactions is also discussed.
Finally, a summary and some concluding remarks are presented in
Sec.~\ref{sec:conclusions}.

\section{Model}
\label{sec:model}

We consider a thin ferromagnetic nanostrip exhibiting PMA and
interfacial DMI under the influence of an in-plane magnetic field. We
start with a three-dimensional micromagnetic energy
\cite{Hubert:book,Bogdanov89,Fert90,Bogdanov94} (in the SI units):
\begin{align}
  \label{Ephys}
  E(\mathbf M)
  & =\int_\Omega \left(  {A \over M_s^2}
    |\nabla \mathbf M|^2 +  {K \over M_s^2} |\mathbf M_\perp|^2
    - \mu_0
    \mathbf M \cdot \mathbf H \right) d^3 r \notag \\
  & + \mu_0 \int_{\mathbb R^3} \int_{\mathbb R^3}
    {\nabla \cdot \mathbf M(\mathbf r) \, \nabla \cdot \mathbf
    M(\mathbf r') \over 8 \pi | \mathbf r - \mathbf r'|} \, d^3 r
    \, d^3 r' \notag \\
  & + {D d \over M_s^2} \int_{\partial \Omega_0} \Big(
    \overline{M}_\parallel \nabla \cdot \overline{\mathbf
    M}_\perp - \overline{\mathbf M}_\perp \cdot \nabla
    \overline{M}_\parallel \Big) d^2 r.
\end{align}
Here $\mathbf M = \mathbf M(\mathbf r)$ is the magnetization vector at
point $\mathbf r = (x,y,z) \in \Omega \subset \mathbb R^3$, where
$\Omega = (-L/2, L/2) \times (-W/2, W/2) \times (0, d)$ is the
nanostrip of length $L$, width $W$ and thickness $d$, and
$\mathbf M_\perp$ and $M_\parallel$ are the in-plane and out-of-plane
components of $\mathbf M$, respectively. The terms in
Eq.~\eqref{Ephys} are, respectively: the exchange, uniaxial
perpendicular anisotropy, Zeeman, magnetostatic interactions and the
interfacial DMI terms, and $M_s = |\mathbf M|$, $A$, $K$, $\mathbf H$
and $D$ are the saturation magnetization, exchange stiffness,
anisotropy constant, applied magnetic field and the DMI strength. As
usual, $\mu_0$ is the permeability of vacuum. In the magnetostatic
energy term, the vector field $\mathbf M(\mathbf r)$ is extended by
zero outside $\Omega$, and $\nabla \cdot \mathbf M$ is understood
distributionally (i.e., it includes the contributions of boundary
charges). Since the considered DMI is due to interfacial effects, its
contribution to the energy is via a surface integral over the bottom
film surface $\partial \Omega_0$ corresponding to an interface between
the ferromagnet and a heavy metal, and
$\overline{\mathbf M} = (\overline{\mathbf M}_\perp, \overline
M_\parallel)$
is the value of $\mathbf M$ on $\partial \Omega_0$. However, using the
standard convention, we normalize the DMI strength parameter $D$ to a
unit volume of the ferromagnet.

We assume that the applied magnetic field is in the plane of the film
and is normal to the strip axis, i.e.,
$\mathbf H = H \hat{\mathbf y}$, where $\hat{\mathbf y}$ is the unit
vector in the direction of the $y$-axis. We also consider films which
are much thinner than the exchange length
$\ell_{ex} = \sqrt{2 A / (\mu_0 M_s^2)}$, so that the magnetization in
$\Omega$ is constant along the film thickness. Measuring lengths in
the units of $\ell_{ex}$ and setting
$\mathbf M(x, y, z) = M_s \mathbf{m}(x, y)$ with $|\mathbf{m}| = 1$ in $\Omega$, we
can rewrite the energy, to the leading order\cite{Gioia97} in
$d / \ell_{ex}$, in the units of $A d$ as
\begin{align}
  \label{Eh}
  E(\mathbf{m}) & \simeq \int_{-l/2}^{l/2} \int_{-w/2}^{w/2} \Big[ |\nabla \mathbf{m}|^2 +
          (Q - 1) |\mathbf{m}_\perp|^2 - 2 h \hat{\mathbf y} \cdot \mathbf{m}_\perp
          \notag \\
        & \quad + \kappa \left( m_\parallel \nabla \cdot \mathbf{m}_\perp -
          \mathbf{m}_\perp \cdot \nabla m_\parallel \right) \Big] \, dy \, dx.
\end{align}
Here we defined $\mathbf{m}_\perp \in \mathbb R^2$ and $m_\parallel \in \mathbb R$ to be the respective
in-plane and out-of-plane components of the unit magnetization vector
$\mathbf{m}$, introduced the dimensionless parameters
\begin{align}
  \label{Qkappa}
  Q = {2 K \over \mu_0 M_s^2}, \quad \kappa = D \sqrt{2 \over \mu_0
  M_s^2 A}, \quad h = {H \over M_s},
\end{align}
and defined the rescaled nanostrip dimensions $l = L / \ell_{ex}$ and
$w = W / \ell_{ex}$. In Eq.~\eqref{Qkappa}, $Q > 1$ is the material's
quality factor yielding PMA, $\kappa$ is the dimensionless DMI
strength, which without loss of generality, may be assumed positive,
and $h$ is the dimensionless applied field strength.

We are interested in the case of long nanostrips corresponding to
$l \gg w$. Note that when $l \to \infty$, the energy in Eq.~\eqref{Eh}
diverges even if $h = 0$ because of the presence of edge domain walls
giving $O(l)$ contribution to the energy.\cite{Rohart2013,Muratov2016}
Therefore, in order to pass to the limit $l \to \infty$ we need to
subtract from $E$ the contribution of the one-dimensional ground state
energy $e_0(h, w) = \min E_0(\mathbf{m})$, where
\begin{align}
  \label{E0}
  E_0(\mathbf{m}) & = \int_{-w/2}^{w/2} \Big[ |\mathbf{m}'|^2 + (Q
            - 1) |\mathbf{m}_\perp|^2 - 2 h \hat{\mathbf y} \cdot \mathbf{m}_\perp \notag \\
          & \quad + \kappa \left( (\hat{\mathbf y} \cdot \mathbf{m}_\perp') m_\parallel -
            (\hat{\mathbf y} \cdot \mathbf{m}_\perp) m_\parallel'  \right) \Big] \, dy.
\end{align}
The precise functional form of $e_0(h, w)$ is the subject of
Sec.~\ref{sec:monodomain-states}.

Putting everything together, we now write the expression for the
energy that describes a Dzyaloshinskii domain wall running across the
nanostrip as
\begin{align}
  \label{E}
  E(\mathbf{m}) & = \int_{-\infty}^{\infty} \int_{-w/2}^{w/2} \Big[
          |\nabla \mathbf{m}|^2 + (Q - 1) |\mathbf{m}_\perp|^2  \notag \\
        & \quad - 2 h \hat{\mathbf y}
          \cdot \mathbf{m}_\perp  - w^{-1} e_0(h,w) \notag \\
        & \quad + \kappa \left( m_\parallel \nabla \cdot \mathbf{m}_\perp -
          \mathbf{m}_\perp \cdot \nabla m_\parallel \right) \Big] \, dy \,
          dx.
\end{align}
This formula forms the basis for all of the analysis throughout the
rest of the paper.

\section{Edge domain walls}
\label{sec:monodomain-states}

We next focus on the minimizers of $E_0$ from Eq.~\eqref{E0} in the
case of $w \gg 1$ and $\kappa$ below the threshold of the onset of
helicoidal structures corresponding to $x$-independent ground state
magnetization configurations. \cite{Vedmedenko2004,
  Pfleiderer04,Uchida06,Tretiakov_DMI} From the physical
considerations (for a rigorous mathematical justification in the case
$h = 0$, see Ref.~\onlinecite{Muratov2016}), it is clear that in these
states the magnetization vector will rotate in the $yz$-plane. Hence,
introducing the ansatz:
\begin{align}
  \label{mmono}
  \mathbf{m}(y) = (0, \sin \theta(y), \cos \theta(y)),
\end{align}
into Eq.~\eqref{E0}, we rewrite $E_0(\mathbf{m})$ as
\begin{align}
  \label{E0mono}
  E_0(\mathbf{m}) = \int_{-w/2}^{w/2} \Big[ |\theta'|^2 + (Q - 1) \sin^2
  \theta \notag \\
  - 2 h \sin \theta + \kappa \theta' \Big] \, dy.
\end{align}
The corresponding Euler-Lagrange equation associated with $E_0$ is
\begin{align}
  \label{E0monoEL}
  \theta'' - (Q - 1) \sin \theta \cos \theta + h \cos \theta = 0,
\end{align}
with boundary conditions
\begin{align}
  \label{E0monobc}
  \theta' \Big( \pm \frac{w}{2} \Big) =- {\kappa \over 2}.
\end{align}
Note that Eqs.~\eqref{E0monoEL} and \eqref{E0monobc} obey the
following symmetry relation, which leaves the energy $E_0$ unchanged:
\begin{align}
  \label{monosym}
  \theta \to \pi - \theta, \qquad y \to -y.
\end{align}
Introducing
\begin{align}
  \label{thetah}
  \theta_h = \arcsin \left( {h \over Q - 1} \right),
\end{align}
we first notice that when $w \to \infty$, we should have either
$\theta \to \theta_h$ or $\theta \to \pi - \theta_h$, corresponding to
the two monodomain ground states in the extended film for
$0 \leq h < Q - 1$. In view of the symmetry in
Eq.~\eqref{monosym}, it is enough to consider only the former
case.

In computing the minimal value $e_0(h, w)$ of $E_0$ for $w \gg 1$ one
needs to take into account the contributions of the boundary layers
next to $y = \pm \tfrac12 w$, the so-called edge domain
walls\cite{Muratov2016}. Below we show that in the presence of an
applied field the minimal energy admits an expansion of the
following form as $w \gg 1$:
\begin{align}
  e_0(h, w) & = -(Q - 1)^{-1} h^2 w \notag \\
            &\quad + \sigma_{edge}^+(h) +
              \sigma_{edge}^-(h) + g_0(h, w),
  \label{e0}
\end{align}
where the first term is the contribution of $\theta = \theta_h$ in the
bulk and, the second two terms are the edge domain wall contributions
from the upper and lower edge, respectively, whose explicit form will
be determined shortly, and the last term is an exponentially small
correction that is negligible for $w \gg 1$.

\begin{figure}[t]
  \centering
  \includegraphics[width=3.1in]{edgeexactvslinab}
  \caption{Comparison of the exact edge domain wall energies in
    Eq.~\eqref{sedge} with the approximate ones given by
    Eq.~\eqref{sedgepm} for $Q = 1.674$ and $\kappa = 0.366$ (see
    Sec.~\ref{sec:comp-with-numer} for the corresponding material
    parameters. For those parameters, the value of $h = 0.1$
    corresponds to $\mu_0 H = 132$ mT). (a) Blue and red curves
    correspond to $\sigma_{edge}^+$ and $\sigma_{edge}^-$,
    respectively. Solid lines correspond to the exact values from
    Eq.~\eqref{sedge} and dashed lines represent the approximation of
    Eq.~\eqref{sedgepm}. (b) The relative error in approximating
    $\Delta \sigma_{edge} = \sigma_{edge}^+ - \sigma_{edge}^-$
    using Eq.~\eqref{sedgepm} is shown.}
  \label{fig:edgeexactvslin}
\end{figure}

We now derive Eq.~\eqref{e0}. Close to $y = \pm \tfrac12 w$ the
solutions of Eqs.~\eqref{E0monoEL} and \eqref{E0monobc} approaching
$\theta_h$ in the sample interior are expected to be well approximated
by those on half-line approaching $\theta_h$ far from the edge. After
a straightforward integration, we obtain
$\theta(y) \simeq \theta^\pm(y \mp \tfrac12 w)$,
where\cite{Goussev2013prb}
\begin{align}
  \label{thh}
  \theta^\pm(y)
  & = 2 \tan ^{-1}\Bigg(\tan
    \left(\frac{\theta_h}{2}\right) \notag \\
  &  \quad +\frac{\cos (\theta_h) \sec
    ^2\left(\frac{\theta_h}{2}\right)}{\tan
    \left(\frac{\theta_h}{2}\right) \mp e^{\mp\sqrt{Q-1} (y - y_0^\pm)
    \cos (\theta_h)}}\Bigg).
\end{align}
The unknown values of $y_0^\pm$ are obtained by substituting the above
expression into Eq.~\eqref{E0monobc}, yielding
\begin{align}
  \label{ypm}
  y_0^\pm = \pm \frac{\cosh
  ^{-1}\left(\frac{2 \sqrt{Q-1} \cos ^2 \theta_h}{\kappa } \pm \sin
  \theta_h\right)}{\cos \theta_h
  \sqrt{Q-1}}.
\end{align}
Introducing $\theta_0^\pm = \theta^\pm(0)$, where, after simplifying
the obtained expressions, one gets explicitly
\begin{align}
  \label{th0pm}
  \theta_0^\pm = \arcsin \left( \sin \theta_h \mp {\kappa \over 2
  \sqrt{Q - 1}} \right).
\end{align}
We can then compute the contributions of the profiles in
Eq.~\eqref{thh} by plugging them into the energy in
Eq.~\eqref{E0mono}. After a rather tedious calculation, up to an
exponentially small error $g_0(h, w)$ we obtain Eq.~\eqref{e0} with
$\sigma_{edge}^\pm$ given explicitly by
\begin{align}
  \label{sedge}
  \sigma_{edge}^\pm
  & = 2 \sqrt{Q - 1} \big( \theta_h \sin \theta_h +
    \cos \theta_h \notag \\
  & \quad - \cos \theta_0^\pm - \theta_0^\pm \sin \theta_h
    \big) \pm \kappa (\theta_0^\pm - \theta_h).
\end{align}

Focusing on the regime of moderate values of $h \lesssim 1$, which is
the main regime of practical interest, linearizing Eq.~\eqref{sedge}
in $h$ we get
\begin{align}
  \label{sedgepm}
  \sigma_{edge}^\pm \simeq \sigma_{edge}^0 \pm \sigma_{edge}^1 h,
  \qquad 0 < h \lesssim 1,
\end{align}
where
\begin{align}
  \sigma_{edge}^0 & = 2 \sqrt{Q - 1} \left( 1 - \sqrt{1 - {\kappa^2
                    \over 4 (Q - 1)} } \, \right) \notag \\
                  & \quad \qquad \quad \qquad - \kappa \arcsin \left(
                    {\kappa  \over 2 \sqrt{Q - 1} }
                    \right), \label{sedgepm0} \\
  \sigma_{edge}^1 & = {2 \over \sqrt{Q - 1}} \arcsin \left( {\kappa
                    \over 2 \sqrt{Q - 1}} \right) - {\kappa \over Q -
                    1}. \label{sedgepm1}
\end{align}
We note that Eqs.~\eqref{sedgepm} gives a very good approximation to
the exact expression in Eq.~\eqref{sedge}, see
Fig. \ref{fig:edgeexactvslin}(a). In fact, the difference
$\sigma_{edge}^+ - \sigma_{edge}^-$, which is the relevant quantity
for the domain wall tilt, is captured by Eq.~\eqref{sedge} within a
few percent for practically all values of $h$ and $\kappa$, see
Fig. \ref{fig:edgeexactvslin}(b).

Before concluding this section, we make several observations regarding
Eq.~\eqref{sedgepm}. First, as expected, $\sigma_{edge}^{0,1} \to 0$
as $\kappa \to 0$, indicating that the edge domain walls disappear
without DMI irrespectively of the magnitude of $h$. Of course, the
same conclusion holds for Eq.~\eqref{sedge} as well. Second, for
$h \lesssim 1$ the applied field affects the contribution of the edge
domain walls to the energy only through $\sigma_{edge}^1$. At the same
time, it easy to see that as a function of $\kappa$ we have
$\sigma_{edge}^1 = O(\kappa^3)$, indicating that the effect of the
edge walls is negligible when the surface DMI is sufficiently weak.

\section{One-dimensional interior wall profile}
\label{sec:interior-wall}

We now turn to interior walls and obtain the leading order expressions
for the one-dimensional wall profiles and their energies as functions
of the wall orientation for $h \lesssim 1$. We focus mostly on the two
relevant cases that are amenable to an analytical treatment:
$h \ll \kappa \sim 1$ and $h \sim \kappa \ll 1$, even though our
method is applicable to all values of the parameters $\kappa$ and $h$
for which Dzyaloshinskii walls are expected to exist. For notational
convenience, we introduce the constant two-dimensional vector
\begin{align}
  \label{mpe0}
  \overline{\mathbf{m}}_\perp = (Q - 1)^{-1} h \hat{\mathbf y},
\end{align}
equal to the in-plane component of the equilibrium magnetization in
the film bulk.

We consider a one-dimensional profile in the direction
$\mathbf{n}_\alpha = (\cos \alpha, \sin \alpha)$, namely, a magnetization
configuration $\mathbf{m}(\xi) = (\mathbf{m}_\perp(\xi), m_\parallel(\xi))$, where
$\xi = \mathbf r \cdot \mathbf{n}_\alpha$. Then from Eq.~\eqref{E} with
$w = \infty$ the energy per unit length of the wall profile $\mathbf{m}(\xi)$
is
\begin{eqnarray}
  \label{Eal}
 \!\!\!\! \!\! E_\alpha(\mathbf{m}) \!\!&=& \!\!\int_{-\infty}^\infty \Big[ |\mathbf{m}'|^2 + (Q
  - 1) |\mathbf{m}_\perp - \overline{\mathbf{m}}_\perp|^2  \nonumber \\
  &&\! + \kappa \left( m_\parallel (\mathbf{n}_\alpha \cdot \mathbf{m}_\perp') - m_\parallel' (\mathbf{m}_\perp \cdot
  \mathbf{n}_\alpha) \right) \Big] d\xi.
\end{eqnarray}
The profile is to satisfy the following conditions at infinity:
\begin{align}
  \label{malinf}
  \mathbf{m}_\perp (\pm \infty) = \overline{\mathbf{m}}_\perp, \qquad m_\parallel(\pm
  \infty) = \pm \sqrt{1 - |\overline{\mathbf{m}}_\perp|^2},
\end{align}
and the associated Euler-Lagrange equation is
\begin{align}
  \mathbf{m}_\perp'' - (Q - 1) (\mathbf{m}_\perp - \overline{\mathbf{m}}_\perp) + \kappa m_\parallel' \mathbf{n}_\alpha
  & =
    \lambda(\xi) \mathbf{m}_\perp ,
  \label{ELinperp}
  \\
  m_\parallel'' - \kappa (\mathbf{n}_\alpha \cdot \mathbf{m}_\perp')
  &  = \lambda(\xi) m_\parallel, \label{ELinpar}
\end{align}
where $\lambda(\xi)$ is a scalar Lagrange multiplier due to the
pointwise unit length constraint on $\mathbf{m}$. The wall energy
$\sigma_{wall}$ associated with a solution $\mathbf{m} = \mathbf{m}^\alpha(\xi)$ of
Eqs.~\eqref{ELinperp} and~\eqref{ELinpar} satisfying \eqref{malinf} is
defined as
\begin{align}
  \label{sigwgen}
  \sigma_{wall}(\alpha) = E_\alpha(\mathbf{m}^\alpha).
\end{align}
A distinctive feature of the wall energy in Eq.~\eqref{sigwgen} is
that for $\kappa \not= 0$ it depends on the wall orientation $\mathbf
n_\alpha$.

It is not possible to find an analytical solution to the system of
Eqs.~\eqref{malinf}--\eqref{ELinpar} for general values of $Q$,
$\kappa$, $h$, and $\alpha$. Although it is not difficult to construct
such solutions numerically for any given set of the parameters (see
Sec.~\ref{sec:comp-with-numer}).

\subsection{$h \ll \kappa \sim 1$ regime}
\label{sec:regime1}

We now wish to obtain the leading order expansion of
$\sigma_{wall}(\alpha)$ for $h \ll 1$ and $\kappa \sim 1$.  Setting
$h = 0$ in Eqs.~\eqref{ELinperp} and \eqref{ELinpar} yields the
equation for the profile $\mathbf{m}_0 = \mathbf{m}_0(\xi)$:
\begin{align}
  \mathbf{m}_{0,\perp}'' - (Q - 1) \mathbf{m}_{0,\perp}  + \kappa m_{0,\parallel}' \mathbf{n}_\alpha
  & =
    \lambda_0(\xi) \mathbf{m}_{0,\perp},
  \label{EELinperp}
  \\
  m_{0,\parallel}'' - \kappa (\mathbf{n}_\alpha \cdot \mathbf{m}_{0,\perp}')
  &  = \lambda_0(\xi) m_{0,\parallel}. \label{EELinpar}
\end{align}
The solution of Eqs.~\eqref{EELinperp} and \eqref{EELinpar} that
satisfies \eqref{malinf} is explicitly given by
\begin{align}
  \label{m0}
  \begin{split}
    \mathbf{m}_0(\xi) & = (\mathbf{n}_\alpha \sin \theta_0(\xi), \cos \theta_0(\xi)), \\
    \theta_0(\xi) & = 2 \arctan e^{-\xi \sqrt{Q - 1}},
  \end{split}
\end{align}
and for $h = 0$ we have $E_\alpha(\mathbf{m}_0) = \sigma_{wall}^0$, where
\begin{align}
  \label{sigw0}
    \sigma_{wall}^0 & = 4 \sqrt{Q - 1} - \pi \kappa.
\end{align}
Notice that $\sigma_{wall}^0$ does not depend on $\alpha$.

To obtain the leading order correction to $\sigma_{wall}^0$, we write
$\mathbf{m}^\alpha = \mathbf{m}_0 + \mathbf{m}_1$, where $|\mathbf{m}_1| \ll |\mathbf{m}_0| = 1$, and note
that due to the pointwise unit length constraint, we have
$\mathbf{m}_0 \cdot \mathbf{m}_1 \simeq 0$ to the leading order. Next, we substitute
this expansion into Eq.~\eqref{Eal} to obtain, keeping only the terms
that are linear in $\mathbf{m}_1$ and $\overline{\mathbf{m}}_\perp$:
\begin{align}
  \label{EEal}
  \sigma_{wall}
  & (\alpha)
    \simeq \sigma_{wall}^0 + \int_{-\infty}^\infty
    \Big[ 2 \mathbf{m}_0' \cdot \mathbf{m}_1' \notag \\
  & \,\,\qquad + 2 (Q-1) \mathbf{m}_{0,\perp} \cdot (\mathbf{m}_{1,\perp} - \overline{\mathbf{m}}_\perp) \notag \\
  & \,\,\qquad+ \kappa \Big(m_{0,\parallel} (\mathbf{n}_\alpha \cdot \mathbf{m}_{1,\perp}') + m_{1,\parallel}
    (\mathbf{n}_\alpha  \cdot \mathbf{m}_{0,\perp}') \notag \\
  & \,\,\qquad- (\mathbf{n}_\alpha \cdot \mathbf{m}_{0,\perp}) m_{1,\parallel}' - (\mathbf{n}_\alpha \cdot \mathbf{m}_{1,\perp})
    m_{0,\parallel}' \Big)  \Big] \, d\xi.
\end{align}
Integrating by parts and using Eq.~\eqref{malinf}, this expression may
be rewritten equivalently as
\begin{align}
  \label{EEEal}
  & \sigma_{wall} (\alpha)
    \simeq \sigma_{wall}^0 + 2 \kappa \mathbf{n}_\alpha \cdot \overline{\mathbf{m}}_\perp \notag \\
  & \quad - 2 (Q -
    1) \int_{-\infty}^\infty \mathbf{m}_{0,\perp}  \cdot \overline{\mathbf{m}}_\perp d\xi \nonumber \\
  & \quad - 2 \int_{-\infty}^\infty
    \Big[ \mathbf{m}_0'' \cdot \mathbf{m}_1 - (Q-1) \mathbf{m}_{0,\perp} \cdot \mathbf{m}_{1,\perp} \notag \\
  & \quad - \kappa \Big( m_{1,\parallel} (\mathbf{n}_\alpha
    \cdot \mathbf{m}_{0,\perp}') -  (\mathbf{n}_\alpha \cdot \mathbf{m}_{1,\perp})
    m_{0,\parallel}' \Big)  \Big] \, d\xi.
\end{align}
In fact, in the above formula the integrand in the last integral is
zero to the leading order, which can be seen by multiplying both sides
of the Euler-Lagrange equation in Eqs.~\eqref{EELinperp} and
\eqref{EELinpar} by $\mathbf{m}_1$ and using the condition
$\mathbf{m}_0 \cdot \mathbf{m}_1 = 0$ to the leading order in $h$. Thus, substituting
the profile $\mathbf{m}_0$ into the above expression, after some more algebra
we get that the wall energy up to $O(h^2)$ is
\begin{align}
\label{Sigwall0}
  \sigma_{wall}(\alpha) \simeq \sigma_{wall}^0 - \sigma_{wall}^1 h \sin
  \alpha,
\end{align}
where
\begin{align}
\label{Sigwall1}
  \sigma_{wall}^1 & = \frac{2\pi}{\sqrt{Q-1}} - \frac{2 \kappa}{Q-1}.
\end{align}

We point out that the obtained expression for $\sigma_{wall}(\alpha)$
appears to be meaningless when $\kappa =0$, since Eq.~\eqref{Sigwall1}
suggests that for $h > 0$ the wall energy depends on the angle
$\alpha$ even in the absence of DMI. Yet the energy in Eq.~\eqref{Eal}
is manifestly independent of $\alpha$. The reason for this discrepancy
is the fact that our approximations are justified only when
$\kappa \sim 1 \gg h$, while the limit of $\kappa \to 0$ with $h > 0$
fixed violates this assumption. In fact, when $\kappa \sim 1$ the
magnetization in a domain wall rotates mostly in the plane spanned by
$\mathbf{n}_\alpha$ and $\hat{\mathbf z}$, while when $\kappa = 0$ the
magnetization would prefer to rotate in the plane spanned by
$\hat{\mathbf y}$ and $\hat{\mathbf z}$, even if
$\mathbf{n}_\alpha \not= \hat{\mathbf y}$. To resolve this discrepancy, we
need to consider the case of $\kappa \lesssim h \ll 1$ separately.

\subsection{$h \sim \kappa \ll 1$ regime}
\label{sec:regime2}

When both $h$ and $\kappa$ are small and comparable, we can further
simplify the argument above to obtain the following equation for
$\mathbf{m}_0$ in place of Eqs.~\eqref{EELinperp} and \eqref{EELinpar} to the
leading order:
\begin{align}
  \mathbf{m}_{0,\perp}'' - (Q - 1) \mathbf{m}_{0,\perp}
  & =
    \lambda_0(\xi) \mathbf{m}_{0,\perp},
  \label{EEELinperp}
  \\
  m_{0,\parallel}''
  &  = \lambda_0(\xi) m_{0,\parallel}. \label{EEELinpar}
\end{align}
The solution of Eqs.~\eqref{EEELinperp} and \eqref{EEELinpar} that
satisfies \eqref{malinf} is explicitly given by $\mathbf{m}_0 =
\mathbf{m}_0^\mathbf{n}$, where
\begin{align}
  \label{m0u}
  \begin{split}
    \mathbf{m}_0^\mathbf{n}(\xi)  & = (\mathbf n \sin \theta_0, \cos \theta_0), \\
    \theta_0(\xi) & = 2 \arctan e^{-\xi \sqrt{Q - 1}},
  \end{split}
\end{align}
and $\mathbf n \in \mathbb R^2$ is an arbitrary constant unit vector. For
$h = \kappa = 0$ we have
$E_\alpha(\mathbf{m}_0^\mathbf{n}) = \sigma_{wall}^0$, where
\begin{align}
  \label{sigw0bis}
  \sigma_{wall}^0 & = 4 \sqrt{Q - 1}.
\end{align}
Notice that $\sigma_{wall}^0$ does not depend on $\alpha$ or
$\mathbf n$ and coincides with the energy of the N\'eel wall in the
absence of nonlocal effects.

Now, writing again $\mathbf{m}^\alpha = \mathbf{m}_0^\mathbf{n} + \mathbf{m}_1$ and expanding
the energy to the next order in $h$ and $\kappa$, we obtain
\begin{eqnarray}
  \label{Ealcbis}
\!\!  E_\alpha (\mathbf{m})
            \!\!&\simeq \!\! & \sigma_{wall}^0 + \int_{-\infty}^\infty
            \Big[ 2 \mathbf{m}_0' \cdot \mathbf{m}_1' \nonumber \\
            &&+ \kappa  \left( m_{0,\parallel} (\mathbf{n}_\alpha
            \cdot \mathbf{m}_{0,\perp}') - m_{0,\parallel}' (\mathbf{m}_{0,\perp} \cdot
            \mathbf{n}_\alpha) \right)   \nonumber \\
          && + 2 (Q-1) \mathbf{m}_{0,\perp} \cdot (\mathbf{m}_{1,\perp} - \overline{\mathbf{m}}_\perp)  \Big] d\xi,
\end{eqnarray}
and following the same arguments as in Sec.~\ref{sec:regime1} we
arrive at
\begin{align}
  \label{Ealcubis}
  E_\alpha(\mathbf{m}) = \sigma_{wall}^0 - \frac{2\pi h}{\sqrt{Q-1}} ({\mathbf n}
  \cdot  \hat{\mathbf y}) - \pi \kappa ( {\mathbf n} \cdot \mathbf{n}_\alpha).
\end{align}

Finally, in order to find the direction of vector $\mathbf n$ we need
to minimize the above energy with respect to $\mathbf n$. It is easy
to see that
\begin{align}
  {\mathbf n} = \frac{2\pi h
  \hat{\mathbf y} + \pi \kappa \mathbf{n}_\alpha \sqrt{Q-1}}{\left| 2\pi
  h \hat{\mathbf y} + \pi \kappa \mathbf{n}_\alpha \sqrt{Q - 1} \right|
  }
\end{align}
minimizes the right-hand side of Eq.~\eqref{Ealcubis}, and the minimum
of the energy is given by
\begin{eqnarray}
  \label{swall2}
 \!\!\!\!\!\!\!\! \sigma_{wall}(\alpha) \!\!& \simeq \!\!& 4\sqrt{Q-1} \nonumber \\
  && - \pi \sqrt{ \kappa^2 \cos^2\alpha +
    \left(\kappa\sin\alpha + \frac{2 h}{\sqrt{Q-1}}\right)^2}.
\end{eqnarray}

Thus, the obtained magnetization profile rotates mostly in the plane
spanned by $\mathbf n$ and $\hat{\mathbf z}$, with $\mathbf n$
depending sensitively on both $h$ and $\kappa$. Furthermore, the
obtained result is consistent with the one of
Sec. \ref{sec:regime1}. Indeed, expanding the expression in
Eq.~\eqref{swall2} in the powers of $h$ with $\kappa \ll 1$ fixed
yields Eq.~\eqref{Sigwall0} to linear order in $h$ and the leading
order in $\kappa$. At the same time, setting $\kappa = 0$ with
$0 < h \ll 1$ fixed in Eq.~\eqref{Ealcubis}, we recover the wall
energy
$\sigma_{wall} \simeq 4 \sqrt{Q - 1} - {2 \pi h \over \sqrt{Q - 1}}$,
which is easily seen to be the wall energy for a profile rotating in
the plane spanned by $\hat{\mathbf y}$ and $\hat{\mathbf z}$,
consistent with the discussion at the end of Sec.~\ref{sec:regime1}.

\section{Two-dimensional problem}
\label{sec:two-dimens-probl}

\begin{figure}[t]
  \centering
  \includegraphics[width=3.15in]{twodim}
  \caption{Schematics of a general domain wall geometry in an infinite
    strip. The up/down symbols indicate the direction of the
    magnetization far from the wall and the strip edges.}
  \label{fig:twodim}
\end{figure}

We now demonstrate how the information about one-dimensional domain
walls obtained in the preceding sections may be applied to a single
Dzyaloshinskii domain wall running across an infinite ferromagnetic
nanostrip. For an illustration of the geometry, see
Fig.~\ref{fig:twodim}, where the domain wall is represented by a thick
solid curve. Here we wish to treat the wall as a one-dimensional
object, whose shape is determined by minimizing an appropriate
geometric energy functional. This energy functional is obtained via a
suitable asymptotic reduction of the two-dimensional micromagnetic
energy in Eq.~\eqref{E}. For a rigorous justification of such an
approach in a closely related context, see
Ref.~\onlinecite{Muratov2016}.

Using Eq.~\eqref{e0}, we can rewrite Eq.~\eqref{E} in the following
way:
\begin{align}
  \label{EE}
  E(\mathbf{m}) & = \int_{-\infty}^\infty \int_{-w/2}^{w/2} \Big[ |\nabla \mathbf{m}|^2 + (Q
          - 1) |\mathbf{m}_\perp - \overline{\mathbf{m}}_\perp|^2 \notag \\
        & - w^{-1} (\sigma_{edge}^+ +
          \sigma_{edge}^-) - w^{-1} g_0(h, w) \notag \\
        & +  \kappa \left( m_\parallel \nabla \cdot \mathbf{m}_\perp -
          \mathbf{m}_\perp \cdot \nabla m_\parallel \right) \Big] \, d^2 r.
\end{align}
Recall that $\overline{\mathbf{m}}_\perp$ was defined in Eq.~\eqref{mpe0}.  We
next consider a domain wall whose shape is described by a smooth curve
$\gamma$ which is the graph of a function
$u: (-w/2, w/2) \to \mathbb R$, i.e., for every $\mathbf r \in \gamma$ we
have $\mathbf r = (u(y), y)$ for some $y \in (-w/2, w/2)$. The
associated magnetization profile ${\mathfrak m}_\gamma$ in the
vicinity of this curve will then be close to the optimal
one-dimensional interior wall profile analyzed in
Sec. \ref{sec:interior-wall}. Let $\mathbf r$ be a point in the
vicinity of $\gamma$ and let $\mathbf r_\gamma$ be the orthogonal
projection of $\mathbf r$ on $\gamma$. Denote by
$\mathbf n_\alpha(\mathbf r_\gamma) = (\cos \alpha(\mathbf r_\gamma),
\sin \alpha(\mathbf r_\gamma))$
the unit normal vector to $\gamma$ at point $\mathbf r_\gamma$
pointing towards the region where $m_\parallel > 0$, with
$\alpha(\mathbf r_\gamma)$ the angle that the normal vector
$\mathbf n_\alpha(\mathbf r_\gamma)$ makes with the $x$-axis at
point $\mathbf r_\gamma$.  Then the magnetization profile
$\mathbf{m} = {\mathfrak m}_\gamma$ associated with the curve $\gamma$ is
expected to satisfy
\begin{align}
  \label{ansint}
  {\mathfrak m}_\gamma(\mathbf r) \simeq \mathbf{m}^\alpha ((\mathbf r -
  \mathbf r_\gamma) \cdot \mathbf n_\alpha(\mathbf r_\gamma)),
\end{align}
where $\mathbf{m}^\alpha$ is the optimal profile that minimizes the
one-dimensional interior wall energy $E_\alpha$ in
Eq.~\eqref{Eal}. Assuming that the curvature of $\gamma$ does not
exceed $O(w^{-1})$, the contribution to the energy by the neighborhood
of $\gamma$ for $w \gg 1$ is then dominated by the one-dimensional
wall energy $E_\alpha(\mathbf{m}^\alpha)$ integrated over $\gamma$, which by
Eq.~\eqref{sigwgen} is
\begin{align}
  E_{int}({\mathfrak m}_\gamma) = \int_\gamma
  \sigma_{wall}(\alpha(\mathbf r)) ds(\mathbf r),
\end{align}
where $ds$ is the arclength differential along $\gamma$. Thus, the
energy of the interior wall is characterized by an \emph{anisotropic}
line tension.

Away from the interior wall the magnetization obeys
\begin{align}
  \mathfrak m_\gamma \simeq \left( \overline{\mathbf{m}}_\perp, \pm \sqrt{1 -
  |\overline{\mathbf{m}}_\perp|^2} \right),
\end{align}
consistently with Eq.~\eqref{ansint}. However, this relation is
violated close to the strip edges, where edge domain walls
appear. Therefore, we also need to take into account the edge domain
wall profiles analyzed in Sec.~\ref{sec:monodomain-states} in those
regions. Accordingly, one expects
\begin{align}
  \mathfrak m_\gamma \simeq (0, \sin (\theta^+(y - \tfrac12 w)),
  \cos (\theta^+(y - \tfrac12 w) )),
\end{align}
for $x > u(w/2)$ and $y \sim w/2$, whereas
\begin{align}
  \mathfrak m_\gamma \simeq (0, \sin (\theta^-(y + \tfrac12 w)),
  \cos (\theta^-(y + \tfrac12 w) )),
\end{align}
for $x > u(-w/2)$ and $y \sim -w/2$. Similarly, in view of the
symmetry relation given by Eq.~\eqref{monosym}, we also find
\begin{align}
  \mathfrak m_\gamma \simeq (0, \sin (\theta^-(\tfrac12 w - y)),
  \cos (\theta^-(\tfrac12 w - y) )),
\end{align}
for $x < u(w/2)$ and $y \sim w/2$, whereas
\begin{align}
  \mathfrak m_\gamma \simeq (0, \sin (\theta^+(-\tfrac12 w - y)),
  \cos (\theta^+(-\tfrac12 w - y) )),
\end{align}
for $x < u(-w/2)$ and $y \sim -w/2$. The corresponding edge wall
energy is then
\begin{align}
  E_{edge} (\mathfrak m_\gamma) = (\sigma_{edge}^+ - \sigma_{edge}^-)
  (u(-w/2) - u(w/2)),
\end{align}
recalling that we subtracted the contribution of
$\sigma_{edge}^+ + \sigma_{edge}^-$ in Eq.~\eqref{EE}. Finally, to
match the interior and the edge wall profiles near points
$x = u(\pm \tfrac12 w)$ and $y = \pm \tfrac12 w$, one uses the
construction from Ref.~\onlinecite{Muratov2016}, which can be seen
not to contribute to the energy to the leading order.

Putting all the leading order contributions to the energy in
Eq.~\eqref{EE} together, we obtain
\begin{align}
  E(\mathfrak m_\gamma) \simeq
  E_{int} (\mathfrak m_\gamma) + E_{edge}(\mathfrak m_\gamma).
\end{align}
Then, using the parametrization $x = u(y)$ of the curve $\gamma$,
we find explicitly
\begin{align}
  \label{Egeom0}
  E(\mathfrak m_\gamma)
  & \simeq \int_{-w/2}^{w/2} \sigma_{wall}\big( -
    \arctan u'(y) \big)  \sqrt{1 + |u'(y)|^2} \, dy \notag \\
  & \quad - ( \sigma_{edge}^+ - \sigma_{edge}^-) (u(w/2) - u(-w/2)),
\end{align}
where we recall that $\alpha(\mathbf r) = -\arctan u'(y)$.

As is well known\cite{Herring52} and can be easily seen directly from
Eq.~\eqref{Egeom0}, every critical point $\gamma$ of
$\gamma \mapsto E(\mathfrak m_\gamma)$ is a straight line. In
particular, minimizers of $E(\mathfrak m_\gamma)$ are straight domain
walls running across the strip. Thus, the only free parameter in the
problem is the difference between the $x$-positions $u(w/2) - u(-w/2)$
of the wall at the top and bottom edges. In fact, from the dimensional
considerations this difference is proportional to $w$, i.e., $w$ can
be scaled out of the energy. Thus, the only
free parameter of the minimization problem for $E(\mathfrak m_\gamma)$
is the tilt angle $\beta \in (-{\pi \over 2}, {\pi \over 2})$ that the
line $\gamma = \gamma_\beta$ makes with the $y$-axis. Note that this
angle coincides with the angle $\alpha$ defining the normal vector
$\mathbf n_\alpha$ of $\gamma_\beta$. To compute the tilt angle, we
substitute the straight line ansatz $\gamma_\beta$ into
Eq.~\eqref{Egeom0}, and the angle is then obtained by minimizing the
expression
\begin{align}
  \label{Emb}
  {E(\mathfrak m_{\gamma_\beta}) \over w} \simeq {\sigma_{wall}(\beta)
  \over \cos \beta} + (\sigma_{edge}^+ - \sigma_{edge}^-) \tan \beta.
\end{align}
over $\beta$, which completely characterizes existence and
multiplicity of tilted domain walls in the presence of DMI. It is
clear from Eq.~\eqref{Emb} that the equilibrium tilt angle is
independent of the strip width and depends only on the dimensionless
material parameters $\kappa$ and $Q$ and the dimensionless applied
field strength $h$. In fact, dimensional analysis shows that the
equilibrium tilt angle depends on these parameters only via two
combinations, $\kappa / \sqrt{Q - 1}$ and $h/(Q - 1)$.

To conclude this section, we note that, as expected, the tilt angle
becomes zero when the effect of the DMI vanishes. This can be readily
seen from Eq.~\eqref{Emb}, taking into account that for $\kappa = 0$
we have $\sigma_{edge}^+ = \sigma_{edge}^-$ and $\sigma_{wall}$
becomes independent of $\beta$, see Eqs.~\eqref{sedge} and
\eqref{Eal}.

In the rest of this section, we consider two parameter regimes based
on analytical results presented in Sec. \ref{sec:regime1} and
\ref{sec:regime2} for which explicit expressions for the tilt can be
obtained.

\subsection{$h \ll \kappa \sim 1$ regime}
\label{sec2:regime1}

In this regime, an approximate expression for $\sigma_{wall}(\alpha)$
is given by Eqs.~\eqref{sigw0}, \eqref{Sigwall0} and \eqref{Sigwall1},
and $\sigma_{edge}^\pm$ are given by
Eqs.~\eqref{sedgepm}--\eqref{sedgepm1}. Substituting these expressions
into Eq.~\eqref{Emb}, we obtain
\begin{align}
  {E(\mathfrak m_{\gamma_\beta}) \over w} \simeq {\sigma_{wall}^0
  \over \cos \beta} + h (2 \sigma_{edge}^1 - \sigma_{wall}^1) \tan
  \beta.
\end{align}
Minimizing this expression yields the unique equilibrium tilt angle
\begin{align}
  \label{titlta}
  \beta = \arcsin \left( { \sigma_{wall}^1  - 2 \sigma_{edge}^1 \over
  \sigma_{wall}^0} \, h \right).
\end{align}
In particular, since we are in the regime of small applied fields the
equilibrium tilt angle is linear in $h$:
\begin{align}
  \label{titltalin}
  \beta \simeq  { 4 h \arccos \left( {\kappa \over 2 \sqrt{Q - 1}}
  \right) \over 4 (Q - 1) - \pi \kappa \sqrt{Q - 1}}.
\end{align}
This formula is one of the main findings of our paper.

We note that the expression in Eq.~\eqref{titlta} formally coincides
with the formula for the contact angle of a triple junction between
three distinct phases\cite{Landau5}. Nevertheless, in addition to the
contribution of the difference of line tensions
$\sigma_{edge}^0 \pm \sigma_{edge}^1 h$ associated with the two edges,
the formula also contains a contribution $\sigma_{wall}^1$ due to
anisotropy of the line tension of Dzyaloshinskii wall.

\subsection{$h \sim \kappa \ll 1$ regime}
\label{sec2:regime2}

In this regime, the explicit expressions for $\sigma_{wall}(\alpha)$
is given by Eq.~\eqref{swall2}. At the same time, recalling that
the expression for $\sigma_{edge}^\pm$ in Eq.~\eqref{sedgepm} remains
valid also for $\kappa \ll 1$ and that $\sigma_{edge}^1 =
O(\kappa^3)$, one can see that the contribution of $\sigma_{edge}^+ -
\sigma_{edge}^-$ in Eq.~\eqref{Emb} is negligible. Thus, to the
leading order we arrive at
\begin{align}
  {E(\mathfrak m_{\gamma_\beta}) \over w}
  & \simeq
    \frac{4\sqrt{Q-1}}{\cos\beta} \notag \\
  & \quad - \pi \sqrt{ \kappa^2 +
    \left(\kappa\tan\beta + \frac{2 h}{\sqrt{Q-1} \,
    \cos\beta}\right)^2}.  \label{Embbis}
\end{align}
Note that the second term in Eq.~\eqref{Embbis} is a small
perturbation for the first term, which is a convex even function of
$\beta$ approaching infinity as $\beta \to \pm {\pi \over
  2}$. Therefore, the minimum in Eq.~\eqref{Embbis} is attained for
$|\beta| \ll 1$.

To proceed further, we expand the right-hand side of
Eq.~\eqref{Embbis} in a Taylor series in $\beta$ up to second order
and keep only the leading terms in $h$ and $\kappa$. The result is
\begin{align}
  {E(\mathfrak m_{\gamma_\beta}) \over w}
  & \simeq
    4\sqrt{Q-1} - {2 \pi h \kappa \beta \over \sqrt{4 h^2 + \kappa^2
    (Q - 1)}} \notag \\
  & \quad + \beta^2 \left( 2 \sqrt{Q - 1} - \frac{\pi \kappa}{2} \right).
  \label{Embbisbis}
\end{align}
Minimizing this expression in $\beta$ yields the equilibrium tilt
angle
\begin{align}
  \label{betabis}
  \beta \simeq \frac{\pi  h \kappa }{\left(2 \sqrt{Q-1}-\frac{\pi
  \kappa }{2}\right) \sqrt{4
  h^2+\kappa ^2 (Q-1)}}.
\end{align}
This formula is another main finding of our paper. As expected, the
title angle in Eq.~\eqref{betabis} goes to zero as $h \to
0$.
Moreover, for $h \ll \kappa \ll 1$ we obtain an interesting result:
\begin{align}
  \label{betabis1}
  \beta \simeq {\pi h \over 2 (Q - 1)}, \qquad h \ll \kappa,
\end{align}
i.e., the equilibrium tilt angle becomes {\em independent} of the DMI
strength. In fact, this is in agreement with the prediction of
Eq.~\eqref{titltalin} for vanishingly small $\kappa$.

Similarly, when $\kappa \ll h \ll 1$, we find another surprising
result:
\begin{align}
  \label{betabis2}
  \beta \simeq {\pi \kappa \over 4 \sqrt{Q - 1}}, \qquad \kappa \ll h,
\end{align}
i.e., the equilibrium tilt angle becomes {\em independent} of the
applied field. This indicates that for moderate values of the DMI
strength the measured tilt angle may be used to directly assess the
value of the interfacial DMI constant experimentally.

\section{Comparison with micromagnetic simulations}
\label{sec:comp-with-numer}

To validate the conclusions of our analysis, we performed three types
of numerical tests. For the material parameters, we chose those of
a 0.6 nm-thick film corresponding roughly to two monolayers
of Co, with parameters $A = 10^{-11}$J/m, $K = 1.25 \times 10^6$
J/m$^3$, $M_s = 1.09 \times 10^6$ A/m. The representative values of
the DMI strength and applied field are $D = 1$ mJ/m$^2$ and
$\mu_0 H = 100$ mT, respectively\cite{Boulle13}.

We begin by comparing the tilted Dzyaloshinskii domain wall profiles
from the two-dimensional numerical simulations
obtained using {\sc Mumax3} simulation package within the local
approximation of the magnetostatic energy \footnote{This amounts to
  combining the anisotropy and magnetostatic energy contributions into
  a single effective anisotropy term with constant
  $K_{\rm{eff}}= K -\tfrac{1}{2} \mu_0 M_{s}^{2} =0.504\times 10^6$
  J/m$^3$ and neglecting the rest of dipolar effects.}  [as in
Eq.~\eqref{E}], with the 1D domain wall profiles $\mathbf{m}^\alpha$
minimizing $E_\alpha$ in Eq.~\eqref{Eal}. In the micromagnetic
simulations, we used a conservative discretization step of $1$ nm in
the $xy$-plane. To obtain the one-dimensional profiles $\mathbf{m}^\alpha$
minimizing $E_\alpha$, we solved Eqs.~\eqref{malinf}--\eqref{ELinpar}
by writing $\mathbf{m}^\alpha$ in polar coordinates for $\theta$ and $\phi$:
\begin{align}
  \label{mal}
  \mathbf{m}^\alpha = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos
  \theta),
\end{align}
and solving the following evolution problem:
\begin{align}
  \theta_t & = \theta_{\xi\xi} - \left(\phi_\xi^2+Q-1\right) \sin
             \theta \cos \theta + h \cos \theta \sin \phi \notag \\
           & \quad -\kappa
             \phi_\xi \sin(\phi - \alpha) \sin ^2 \theta , \label{th} \\
  \phi_t & = \phi_{\xi\xi} + 2 \theta_\xi \phi_\xi \cot \theta + h
           \csc \theta \cos \phi \notag \\
           & \quad +\kappa \theta_\xi \sin (\phi - \alpha
             ), \label{phi}
\end{align}
until a steady state was reached. Here the subscripts stand for
 the respective partial derivatives. The equations above correspond to an
overdamped Landau-Lifshitz-Gilbert equation, and their steady states
solve Eqs.~\eqref{ELinperp} and \eqref{ELinperp} upon substitution
into Eq.~\eqref{mal}. Also, in terms of $\theta$ and $\phi$ the wall
energy is
\begin{align}
  E_\alpha(\mathbf{m}^\alpha) & = \int_{-\infty}^\infty \bigg( \theta_\xi^2 +
                        \phi_\xi^2 \sin^2 \theta  + (Q - 1) \sin^2 \theta \notag \\
                      & \quad - 2 h \sin \theta
                        \sin \phi + {h^2 \over Q - 1} + \kappa
                        \theta_\xi \cos (\phi - \alpha) \notag \\
                      & \quad - \kappa \phi_\xi \sin (\phi - \alpha) \cos
                        \theta \sin \theta \bigg) d \xi.
\end{align}

The parameters at the beginning of this section correspond to the
dimensionless parameters $Q = 1.674$, $\kappa = 0.366$ and
$h = 0.073$. For these parameters, we carried out {\sc
    Mumax3}\cite{Mumax} simulations in an 800 nm $\times$ 400 nm
strip, which corresponds to $w = 109 \gg 1$, and obtained the
magnetization profile with the tilt angle $\beta \simeq 11.2^\circ$.
We then solved Eqs.~\eqref{th} and \eqref{phi} with
$\alpha = 11.2^\circ$ and obtained the optimal one-dimensional
wall profile $\mathbf{m}^\alpha$. The result of the two-dimensional
computation is compared with the one-dimensional profile in
Fig.~\ref{fig:numprofiles}, which plots the $z$-component of the
two-dimensional profile $\mathbf{m}$ along the $x$-axis alongside with the
corresponding section of the optimal profile
$\mathfrak m_{\gamma_\alpha}$ obtained from $\mathbf{m}^\alpha$. One can see
an almost perfect agreement between the full two-dimensional
simulation result and the theoretical prediction of
Sec.~\ref{sec:interior-wall}. The same agreement is also observed in
the other two components of the magnetization (not shown). This
justifies the main premise of our theory about the one-dimensional
character of the interior wall profiles.

\begin{figure}[t]
  \centering
  \includegraphics[width=3.15in]{numprofiles}
  \caption{A one-dimensional $y = 0$ cut through the computed
    two-dimensional profile $\mathbf{m}$ (blue dots) vs. a
    one-dimensional cut through the optimal profile
    $\mathfrak m_{\gamma_\alpha}$ (red line). See text for
    details. }
  \label{fig:numprofiles}
\end{figure}

To further test the conclusions of our theory, we computed the energy
$\sigma_{wall}(\alpha)$ of the interior walls as a function of their
orientation angle $\alpha$ from the solutions of Eqs.~\eqref{th} and
\eqref{phi} for the considered values of the parameters. The result is
plotted in Fig.~\ref{fig:numsigmawall}, along with the analytical
approximations given by Eqs.~\eqref{Sigwall0} and \eqref{swall2}. One
can see that both analytical formulas give a fairly good approximation
to the exact interior wall energy $\sigma_{wall}(\alpha)$ for these
parameters. The agreement becomes much better for smaller values of
$h$.

\begin{figure}[t]
  \centering
  \includegraphics[width=3.15in]{numsigmawall}
  \caption{The dependence $\sigma_{wall}(\alpha)$ obtained from the
    numerical minimization of $E_\alpha$ (blue solid curve), the
    analytical expressions given by Eq.~\eqref{Sigwall0} (red dashed
    curve) and Eq.~\eqref{swall2} (green dotted curve),
    corresponding to the dimensionless parameters $Q = 1.674$,
    $\kappa = 0.366$ and $h = 0.073$.}
  \label{fig:numsigmawall}
\end{figure}

We used the interior wall energy $\sigma_{wall}(\alpha)$ obtained
numerically to calculate the equilibrium tilt angle by minimizing the
energy in Eq.~\eqref{Emb} numerically. This resulted in a unique
minimizing angle $\beta = 11.4^\circ$, in excellent agreement with the
result of the full two-dimensional simulation. For comparison, the
formulas in Eqs.~\eqref{titltalin} and \eqref{betabis} yield $\beta =
12.8^\circ$ and $\beta = 13.5^\circ$, respectively, still in a
good agreement with the two-dimensional result, which is reasonable
since both these formulas are at the limits of their applicability for
the considered parameters.

For lower fields $h$, the agreement with the predictions of the
analytical theory becomes much better. We illustrate this by
presenting the results of the full two-dimensional numerical
simulations against the analytical predictions by
Eqs.~\eqref{titltalin} and \eqref{betabis} for smaller fields in the
whole range of values of $\kappa$. Figure~\ref{fig:tiltvsfield} shows
the dependence of the equilibrium tilt angle $\beta$ on the
applied field for several values of the DMI strength. As can be seen
from the figure, the agreement between the theory and the
numerics rapidly increases as the applied magnetic field or the DMI
strength are decreased. This trend can also be seen from the plot of
the equilibrium tilt angle as a function of the DMI strength for
several value of the applied field shown in Fig.~\ref{fig:tiltvsdmi}.

\begin{figure}[t]
  \centering
  \includegraphics[width=3.15in]{tilt_vs_field.pdf}
  \caption{Equilibrium tilt angle, $\beta$, vs. applied field
    $B_y = \mu_0 H$ for several values of the DMI strength $D$,
    alongside with the predictions of Eq.~\eqref{titltalin} (solid
    lines) and Eq.~\eqref{betabis} (dashed lines).}
  \label{fig:tiltvsfield}
\end{figure}

\begin{figure}[t]
  \centering
  \includegraphics[width=3.15in]{tilt_vs_DMI.pdf}
  \caption{Equilibrium tilt angle, $\beta$, vs. DMI strength $D$ for several
    values of the applied field $B_y = \mu_0 H$, alongside with the
    predictions of Eq.~\eqref{titltalin} (solid lines) and
    Eq.~\eqref{betabis} (dashed lines).}
  \label{fig:tiltvsdmi}
\end{figure}

So far we presented the results of micromagnetic simulations for a
nanostrip, using a common approximation that neglects the dipolar
interaction. \cite{Winter1961,Gioia97} More precisely, in the preceding
simulations, the effect of dipolar interactions was accounted only by
introducing a local shape anisotropy term. Let us conclude this
section by discussing the results of micromagnetic simulations for
thin nanostrips with the full account of magnetostatic
interaction. The material parameters and the corresponding
dimensionless parameters were the same as in the simulations without
the dipolar interactions. The length of the nanostrip was extended
$1.5$ times to reduce the effect of magnetic charges at the ends of
the nanostrip. The full accounting of the magnetostatic energy in the
simulations leads to an increase in the DW tilt angle, see
Fig.~\ref{fig:tilt_dipolar_By}. One can see that for the considered
parameters the presence of the dipolar interaction affects the tilt
angle relatively weakly for moderate applied fields and DMI
strengths. Notably, the impact of the dipolar interaction is
practically negligible for sufficiently small DMI strengths, but
increases with increasing DMI strength. We attribute this phenomenon
to a decrease in the DW stiffness as the value of $\kappa$ is
increased, making the domain wall more susceptible to the presence of
the dipolar interactions.

\begin{figure}[t]
  \centering
  \includegraphics[width=3.15in]{with_without_dipolar_simulation_tilt_By.pdf}
  \caption{Equilibrium tilt angle, $\beta$, vs. magnetic field $B_y$
    from simulations with and without the dipolar interactions.}
  \label{fig:tilt_dipolar_By}
\end{figure}

\section{Conclusions}
\label{sec:conclusions}

We have developed an analytical theory of the
Dzyaloshinskii domain wall tilt in a ferromagnetic nanostrip in the
perpendicular in-plane magnetic fields. This type of DW tilt is a
vivid manifestation of the presence of interfacial DMI in ultrathin
ferromagnet/heavy-metal layered structures. Our theory focuses on the
geometric aspect of the problem and treats the DW as a curve, whose
equilibrium shape is determined by minimizing an appropriate geometric
energy functional.

The main ingredients in our theory are the energy densities of edge
and interior domain walls. The former are computed explicitly, and the
latter can be obtained for any given set of parameters, using a
straightforward numerical procedure. We have explicitly considered two
regimes: the regime when the dimensionless magnetic field $h$ is much
smaller than the dimensionless DMI strength $\kappa$ and the regime
when they are both small and comparable. In both regimes, we have
found very good agreement with the micromagnetic simulations for the
tilt angle.

Our theory has three main findings: First, we derived an exact 1D
domain wall profile for any strength of perpendicular in-plane
magnetic field. Second, we proved that the DW is always a tilted
straight line. Third, this allowed us to obtain an explicit expression
for the DW tilt angle. Moreover, in the wide range of DMI strength (as
long as DW does not develop yet helicoidal structure), we find that
the DW configurations are in general neither N{\'e}el nor Bloch type,
and that the DW energy is anisotropic (depends on the tilt angle).

In the regime of small fields $h \ll \kappa \lesssim 1$, we have found
that the equilibrium angle is proportional to the field strength
[Eq.~\eqref{titltalin}]. On the other hand, for small DMI strengths
the tilt angle exhibits a strongly nonlinear dependence on the field
strength, even for relatively small fields
[Eq.~\eqref{betabis}]. Surprisingly, we found that when
$h \ll \kappa \ll 1$ the equilibrium tilt angle becomes independent of
the DMI strength [Eq.~(\ref{betabis1})], which can be a good
experimental test for our theory.  Equally surprisingly, in the
opposite regime $\kappa \ll h \ll 1$, we have shown that the
equilibrium tilt angle becomes independent of the applied magnetic
field [Eq.~(\ref{betabis2})].

Our results indicate that for moderate DMI strengths the tilt angle
may be used to directly assess the value of the interfacial DMI
constant experimentally.  In other words, our theory gives a method to
infer the DMI constant from the tilt angle measurements of the
Dzyaloshinskii domain wall.  We, therefore, propose an experimental
method that requires only a technique for observing the magnetic
structure under external field (e.g. Kerr microscopy). To improve the
accuracy of the DMI determination, one should measure the tilt angle
as a function of magnetic field $B_y$, as shown in
Fig.~\ref{fig:tiltvsfield}, and fit this experimental curve to our
theory [Eqs.~\eqref{titltalin} or~\eqref{betabis}, depending on
smallness of the DMI strength relative to the magnetic field].

\begin{acknowledgments}

  We thank O. Tchernyshyov for helpful discussions. C.\,B.\,M. was
  supported, in part, by NSF via grant DMS-1614948. V.\,S. would like
  to acknowledge support from EPSRC grant EP/K02390X/1 and Leverhulme
  grant RPG-2014-226. O.\,A.\,T. acknowledges support by the
  Grants-in-Aid for Scientific Research (No.\,25247056, No.\,15H01009,
  No.\,17K05511, and No.\,17H05173) from the Ministry of Education,
  Culture, Sports, Science and Technology (MEXT) of Japan; JSPS-RFBR
  grant; and MaHoJeRo grant. V.S. is grateful to Basque Center for
  Applied Mathematics (BCAM) for its hospitality and support.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
NASA's next big space telescope set for blastoff from French Guiana
Steve Gorman | Posted: Dec. 24, 2021, 6 p.m. | Updated: Dec. 24, 2021, 8 p.m. | 3 Min Read
By Steve Gorman
(Reuters) - The James Webb Space Telescope, a NASA instrument designed to allow humankind's first glimpse of the infant universe as it existed when the earliest galaxies are believed to have formed, was set for launch on Saturday from the northeastern coast of South America.
The revolutionary $9 billion infrared telescope, hailed by NASA as the premiere space-science observatory of the next decade, was bundled inside the cargo bay of an Ariane 5 rocket poised for blastoff at 7:20 a.m. EST (1220 GMT) from the European Space Agency's (ESA) launch base in French Guiana.
If all goes as planned, the 14,000-pound instrument, which will unfurl to nearly the size of a tennis court, will be released from the French-built rocket after a 26-minute ride into space.
Over the following month, the Webb telescope will coast to its destination in solar orbit roughly 1 million miles from Earth - about four times farther away than the moon. Webb's special orbital path will keep it in constant alignment with Earth as the planet and telescope circle the sun in tandem.
By comparison, Webb's 30-year-old predecessor, the Hubble Space Telescope, orbits Earth itself from 340 miles away, passing in and out of the planet's shadow every 90 minutes.
Named for NASA's chief during most of the agency's formative decade of the 1960s, Webb is about 100 times more sensitive than Hubble and is expected to profoundly transform scientists' understanding of the universe and our place in it.
Webb mainly will view the cosmos in the infrared spectrum, allowing it to peer through clouds of gas and dust where stars are being born, while Hubble has operated primarily at optical and ultraviolet wavelengths.
COSMOLOGICAL HISTORY LESSON
The new telescope's primary mirror - consisting of 18 hexagonal segments of gold-coated beryllium metal - also has a much bigger light-collecting area, enabling it to observe objects at greater distances, thus farther back into time, than Hubble or any other telescope.
That, astronomers say, will bring into view a glimpse of the cosmos never previously seen - dating to just 100 million years after the Big Bang, the theoretical flashpoint that set in motion the expansion of the observable universe an estimated 13.8 billion years ago.
Hubble's view reaches back to roughly 400 million years following the Big Bang, a period just after the very first galaxies - sprawling clusters and stars, gases and other interstellar matter - are believed to have taken shape.
Aside from examining the formation of the earliest stars and galaxies in the universe with greater clarity, astronomers are eager to study super-massive black holes believed to occupy the centers of distant galaxies.
Webb's instruments also make it ideal to search for evidence of potentially life-supporting atmospheres around scores of newly documented exoplanets - celestial bodies orbiting distant stars - and to observe worlds much closer to home, such as Mars and Saturn's icy moon Titan.
The telescope is an international collaboration led by NASA in partnership with the European and Canadian space agencies. Northrop Grumman Corp was the primary contractor. The Arianespace launch vehicle is part of the European contribution.
Webb was developed at a cost of $8.8 billion, with operational expenses projected to bring its total price tag https://www.reuters.com/article/us-usa-nasa-telescope/northrop-ceo-grilled-by-u-s-lawmakers-over-space-telescope-idUSKBN1KG2US to about $9.66 billion, far higher than planned when NASA was previously aiming for a 2011 launch.
Astronomical operation of the telescope, to be managed from the Space Telescope Science Institute in Baltimore, is expected to begin in the summer of 2022, following about six months of alignment and calibration of Webb's mirrors and instruments. The space observatory is designed to last up to 10 years.
(Reporting by Steve Gorman; Editing by Leslie Adler)

redpajama_set_name: RedPajamaCommonCrawl


text:
Pakistan says it will not allow countries to shelter militants
Digital Desk - Jan 3, 2023
Pakistan's political and military leadership have vowed that no nation will be allowed to shelter militants who stage attacks against the country — an...
UN human rights chief urges Taliban to drop restrictions on women
The United Nations' human rights chief has decried increasing restrictions on women's rights in Afghanistan, urging the country's Taliban rulers to reverse them immediately. Taliban...
Taliban release two Americans in 'goodwill gesture'
The Taliban have released two Americans held in Afghanistan in what appeared to be a "goodwill gesture," State Department spokesman Ned Price said on...
Female Afghan TV presenters ordered to cover their faces by Taliban
Digital Desk - May 19, 2022
Afghanistan's Taliban rulers have ordered all female presenters on TV channels to cover their faces on air, the country's biggest media outlet said. The order...
Afghanistan's Taliban leaders order women to wear burka in public
Afghanistan's Taliban leadership has ordered all Afghan women to wear the all-covering burka in public, officials said. The move evokes similar restrictions imposed on women...
Photographer's family take Taliban chiefs to International Criminal Court
Digital Desk - Mar 22, 2022
The family of a Pulitzer Prize-winning Indian photographer who was killed in Afghanistan last year have filed a formal complaint with the International Criminal...
Taliban dissolves 'unnecessary' election commissions
The Taliban has dissolved Afghanistan's two election commissions as well as the state ministries for peace and parliamentarian affairs. Bilal Karimi, deputy spokesman for Afghanistan's...
Seven killed and 16 wounded in IS attack on Kabul hospital
Digital Desk - Nov 2, 2021
Militants from so-called Islamic State (IS) set off an explosion at the entrance to a military hospital in the Afghan capital, killing at least...
IS claims responsibility after 47 killed in Afghan mosque attack
Digital Desk - Oct 16, 2021
The so-called Islamic State group has claimed responsibility for a deadly suicide bombing on a Shiite mosque in southern Afghanistan that killed 47 people...
Taliban appoint deputy ministers in all-male government
The Taliban have failed to name any women in a list of deputy ministers. The announcement came despite an international outcry when they presented their...


Representing CuraMedix and the United States, Gregory Mercure and Elise Hamann recently attended the STORZ Medical Annual Distributor Meeting in Berlin, Germany.
Mercure and Hamann joined up with 90 others who were there to represent 160 STORZ distributors across 32 countries including UK, Russia, Germany, Sweden, Peru, Mexico, Austria and Turkey — just to name a few.
Meeting sessions were held at the KARL STORZ Visitor and Training Center, formerly the Empress Augusta Hospital which was founded in1869 to treat and care for those wounded in war. KARL STORZ took great care in the extensive conversion and renovation to preserve and maintain the charm of the historical building, while adding modern elements including a state-of-the-art visitor center plus a training and conference center.
Over the course of two days, educational sessions included didactic as well as the clinical application of shock wave, plus the unveiling of new applicators that can be added to the already comprehensive range of STORZ Medical applicators. Conference attendees also got a sneak peak at some new shock wave products to come.
The ninth volume of the successful "Shock Wave Therapy in Practice" series by Level 10 Publishing was also highlighted at the event, and a copy was presented to attendees. This latest publication, "ESWT in Aesthetic Medicine, Burns & Dermatology," is edited by Dr. Karsten Knobloch, an esteemed physician specializing in general, plastic and aesthetic surgery, hand surgery, sports medicine, and emergency medicine. It provides an up-to-date overview of the use of Extracorporeal Shock Wave Therapy (ESWT) in reconstructive surgery, burns surgery, and aesthetic surgery.
For three decades, STORZ Medical's ongoing commitment to innovation and strong partnerships has helped the company remain the leader in the shock wave industry. And for the CuraMedix team, the leading U.S. distributor of STORZ Medical devices, it was particularly exciting to attend the conference and celebrate the company's 30th Anniversary. Having the opportunity to learn about all of the efforts that STORZ is continuing to put forth in order to enhance shock wave, further explore its range, and broaden its indications, leaves little doubt for distributors across the globe that an inspiring future lies ahead. And that's something to celebrate!
Timely tips and valuable insights for better clinical and economic outcomes for your practice.


Your Intranet Search is Over! Noodle is the All-In-One, Easy-To-Use Intranet for the Enterprise. Noodle provides a wide array of functionality. Local or Cloud Deployment, Role Based Permissions, 14 Integrated Applications, Forms & Workflows all with with Single-Sign-On user management. Also, you work with a support team that has been helping companies build intranets for over a decade, Its time to use your Noodle! Visit the website for a FREE Demo.
Noodle helped to bring back life to an old outdated intranet site.
Pros: Noodle has a cloud based system that will allow you to access your website outside of your company's network. They have also integrated a Single Sign On (SSO) option for the cloud based users as wel. This makes it easier to login your intranet site if your company already is using a SSO software especially Windows Active Directory.
The interface allows you to easily build pages and has a drag and drop feature that most people are familiar with. It also has features that will allow you to easily setup a photo gallery, create a poll, make and merge calendars, and create interactive forms. Using Noodle allowed our company to retain existing functionality from our old intranet site while incorporating new ones.
Cons: It would be nice if Noodle would allow you to connect to an external database or use an API widget in order to use existing custom built applications/import data. There are some features that are incorporated in the site (while useful) that are not intuitive to the end users that cannot be disabled.
Pros: Easy to use, low cost, high ROI, outstanding support, flexible, robust, secure and safe.
Cons: Less flexible in design to some of the high end product.
Overall: It is a pleasure recommend Noodle from Vialect. The product is incredibly easy to learn to use from a design perspective and practically intuitive for the standard user.
With help from the Noodle help center we have managed to build and extensive, interactive and informative intranet that is relied upon within our organisation of 500 staff.
Maintenance is easy as long as you have a good map in place for your content.
We have delegated responsibility for content to section contributors thereby avoiding the need for extra resource to maintain the intranet. Further, content is written and maintained by the subject matter experts.
At a fraction of the price of many other products but yet feature rich and with the best support I have ever experienced from a vendor, we highly recommend Noodle from Vialect.
Recommendations to other buyers: At this low cost, go for it. No great scare about the cost of the investment.
Comments: My client was seeking an Intranet portal for a network of over 65 different offices across North America and Noodle was by far the most economical and most flexible platform on the market. One of the greatest strengths of Noodle is the permission levels available per user/group. Intranet applications in general are still relatively behind in terms of their usability and design, but Noodle does a great job of simplifying the Intranet but still includes many valuable tools and applications. In addition, the team at Noodle does an outstanding job of providing quick support and listening to opinions on future improvements. If you're looking for an easy-to-use, cost-effective platform for your intranet, Noodle is one of the most economical and flexible platforms out there. When Noodle works out of the box on mobile, i'll gladly give it a 5 star rating!
Pros: Secure and centralized our intraoffice communication. Because we use Noodle as the hub for policies, human resources, and training documents and videos, it took a lot of time and effort to get this running. The time invested wasn't due to the Noodle programming, but due to the large amount of information we needed to publish.
Logging in and updating is simple. The layout is user friendly and customizable.
Cons: Internal messaging is only delivered if the employee is looking at the Noodle program. There is no desktop function that sends the message or an indicator to look at messages received unless you are in Noodle.
Comments: Our organization has now evaluated many document management, corporate social networking, internal portal and content management systems. Keeping an eye out for ease of use and rich but simple feature set that met our requirements. Noodle was the perfect product for this and also at a price that easily fit into our budget. Additionally, their customer support, ability to adapt to our requirements and energetic attitude was very refreshing. We sent a short list of similar products to a test group of 15 users. The users unanimously voted Noodle as their preferred choice as a document management and internal social networking site.
Cons: There is no out of the box MS Office Integration. However, after speaking with sales and their technical teams we were able to determine a solution that would be included in a future release for integration with MS Office 2013. Does not appear to support NTLM so users are required to login, this makes it difficult to make company announcements that would appear on the users web browsers home page (forcing the user to login to see them).
Pros: I managed to create a useful site for our staff in less than 48 hours. It went up so fast and is incredibly easy to populate with content.
Cons: Sometimes certain things have a mind of their own... the home screen skews percentages of columns depending on what section I put in it... but I can't seem to figure out why... I also don't like that if I lock down editing for an area, users can't 'like' anything there... makes the like button sort of useless.
Overall: I think our site helps highlight staff involvement. It's also an extra level of security & visibility on documents such as policies and procedures that need to be kept off of our public site but still easy to access.
Pros: Easy to use and customize. It is a source of invaluable information for our team.
Cons: Some formatting lost when copy/pasting from other applications.
Overall: Love the software and especially happy with the Customer Support team who helped to customize a solution for our team.
Comments: Our organization has now evaluated many document management, corporate social networking, internal portal and content management systems. Keeping an eye out for ease of use and rich but simple feature set that met our requirements. Noodle was the perfect product for this at a price that easily fit into our budget. Additionally, their customer support, ability to adapt to our requirements and energetic attitude was very refreshing. We sent a short list of similar products to a test group of 15 users. The users unanimously voted Noodle as their preferred choice as a document management and internal social networking site.
- There is no out of the box MS Office Integration. However, after speaking with sales and their technical teams we were able to determine a solution that would be included in a future release for integration with MS Office 2013.
- Does not appear to support NTLM so users are required to login, this makes it difficult to make company announcements that would appear on the users web browsers home page (forcing the user to login to see them).
Pros: Customer and tech support. The features of the platform.
Cons: Can be a bit tricky getting some of the functionality working properly, but that is due to user skill level. Any issues, we contact the support team and they help us get it sorted.
Overall: We used vialect/Noodle as our intranet platform for our UK physiotherapy clinic franchise - Sports Physio UK. The customer support with setting up the platform was excellent, and their tech team are always prompt to rectify any glitches or issues we come across. This platform has given our Team a place to collaborate (formally and informally), share information, disseminate news/updates within the team, and share documents/files. The great thing is, it is future proof for us and can grow with us and adapt to our evolving needs.
Pros: The desktop version is user-friendly and easy to navigate.
Cons: Doesn't have an app. The mobile versions isnt the most user-friendly.
Overall: The customer support was so helpful during our set up process. Anytime I needed anything, he answered emails within minutes. He's able to answer questions and resolve any issues right-of-way.
Pros: Noodle is very well made. Its easy to apply forums, interact with others inside the organization, and you're able to customize your own page just how you want. You also don't have to host it anywhere on the web and you can apply restrictions so that others can't log in unless they're in your network.
Cons: The issues I found were similar to SharePoint and Wordpress in that you are almost limited with how you create your work, but nothing is impossible. Another issue I found was the pricing. If you have Office 365 license you could just use SharePoint. But Noddle is a good alternative if you're wanting to work around all that.
Pros: It's easy for a person who doesn't have IT experience to setup and run. The layout is simple and most users have no issues finding things. The email notifications are handy too.
Cons: It looks very generic and unprofessional. You have to delete your cookies for profile updates to take place. The chat box doesn't have emojis, spell check, or allow you to have the chat up while navigating Noodle. It's just ok, overall.
Overall: It's a more fun way for employees to interact than just via email.
Pros: Noodle is easy to navigate. We use noodle for updates and to find information to help our customers. I never have any problems using Noodle to do this.
Cons: Currently, I do not have any dislikes about Noodle.
Overall: As a customer service representative, it is always helpful to have different information sources when assisting customers. Noodle is our main additional source.
Pros: It looks nice on the surface and if you have the time to dedicate hours to polishing it administratively it could be a great intranet.
Cons: It's very frustrating at times to administrate. The admin gui is not user friendly or intuitive at all. We have been unable to get users to interact with either working on it or using it.
Comments: Very good app, customer service is very responsive. Noodle is a all-in-one intranet package for small and large companies alike. Our account manager/trainer walked us through every aspect of the demo process and go-live. When we have questions, we get quick responses and or he fixes our problems. So far this has been the most economical and best fit solution we have come across. Our company has really benefited from our Noodle site.
Pros: Very easy to use. Has all of the basic needs for a intranet site. Quick to deploy. Great customer support and training.
Cons: Does not integrate with outlook/exchange calendars very well. Messaging needs a little improvement, but I believe will come with time.
Pros: Having a centrally located intranet where all your internal operations important information is at the ready for your employees in a secure environment.
Noodle gave us all the tools and applications we were looking for. Price & support is excellent!
-The tools and applications are very easy to learn and use.
Excellent! I highly recommend this for a smaller corporation on a tight budget.
Comments: We are using the software for our five employees. With this intranet software we are able to publish news and useful links in one central location. It easy to use but also easy manage for the admin.
Pros: If you are not a tech guru, it is okay because the simplicity is nice and capable for those who struggle. Tech support was great as well.
Cons: I would love for this to have Office available right from the get-go. The mobile version can be hit or miss.
Overall: For smaller structure, this is good but I am not sure how it would work out on a big playing field.
Pros: Its cheap, easy to use and no contract!! Its hard to find a program that does everything Noodle does in one place.
Comments: Fun product idea and good functionality. Product is not too intense with a very helpful customer support team.
Pros: Noodle is very user-friendly and the cost is very inexpensive for the value you receive. Customer support is great.
Cons: The design isn't very high-end. The mobile aspect of this software needs improvement and messaging could use improvements as well.
Good value for money with Noodle. Does what it is meant to do without any unnecessary functionality.
Cons: As an intranet tool it does what it needs to do, only drawback is the outlook integration is not very good at this stage.


text:
Q: Normalized or Denormalized Data in Microservices and Service Composition So our development team has been working towards Microservices for the past 6-8 months and have picked up a lot of steam.
We have experienced several gotcha moments in that time, and are humble enough to know that we are in for many more as we move closer to moving our platform to production.
One area that I can't quite put my finger on is how we are to treat our data between our service boundaries. I hear a lot of statements form large companies that have successfully implemented Microservices, but I can never seem to get straight advice and reasoning.
Specifically, given two service domains  User and Contacts,and assuming that a User has a Contact object associated with it, what are the options for each of these two service domains in regards to managing their own data?
Should the User have a ContactID stored with it, or should it store the entire Contact object?
I have seen many reliable service oriented development teams (Netflix,Amazon,Nike,etc) make statements such as the following:
"Normalization is the root of all evil..."
"Break all that is shared..."
"Share nothing..."

A: Well, those statements are wrong. Modularisation is about finding the right combination of low coupling and high cohesion for your context. The user can store a copy of the contact object as long as it is aware that it is only a snapshot of it at a certain moment in time. For many use cases that is fine, and sometimes you just need to ensure that you work with the real contact as known to the responsible domain.

A: Keep master data normalized. But - if you have things like "ContactID" you normalized your data wrongly!
Normalized entity must be not a table but business entity - a document that have business meaning. Why do you need ContactID? If you have list of companies with contacts, keep contacts within companies. If you want contacts to be shared information between, say, companies and contracts - have Contact service that returns contact information by email or company name or any other field that makes sense for business.
Fully denormalized data is for derived data that is built from master data. It is used mainly to build indexes upon it for searching and sorting.

A: It's perfectly fine to correlate data across service boundaries by sharing identities (IDs).
When your UI needs to display correlated data from two services, instead of replicating data between services (which goes against the loosely-coupled microservices grain as you've heard), you can use client-side composition to assemble the data on the client at runtime. Fetch the data from both services, build a ViewModel composing it, and then display it on the screen. The logic that composes it together need not be coupled to either service. It can take whatever data the services return and compose it dynamically. The service that does the client-side composition is sometimes referred to as the "IT/Ops Service."
Here's an example of how that can be accomplished technically:
https://particular.net/blog/secret-of-better-ui-composition

redpajama_set_name: RedPajamaStackExchange


Mill-Duty Rotary Splash Guard - System Seals Inc.
The 421 Series Mill-Duty Rotary Splash Guard is a robust rotary seal that is designed to protect the internal bearing and primary seals from water splash and heavy concentrations of debris. This is commonly used on work rolls and roll neck bearings in steel and aluminum mills. The design of the seal lip is much more robust than traditional rotary face seals providing maximum performance in harsh conditions.
The 421 Mill-Duty Rotary Splash Guard is commonly used in mill duty applications for maximum exclusions resistance where water splash and heavy concentrations of debris are present.
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lobal Infant Formula Market (2019 Edition) – World Market Review By Product Type, By Distribution Channel, By Region, By Country: Opportunities and Forecast (2014-2024)
Date: 1 November 2019
According to a research report published by Azoth Analytics in June 2019, the Global Infant Formula Market was valued at USD 42.4 billion the year 2018. Global Infant Formula market is expected to grow due to a number of factors including rapid urbanization in emerging economies, growing health concern among the consumers in developing region and rising demand by working women population due to lack of time to breastfeed the baby. In addition, ever growing urban and working population and the associated rising income is fuelling the growth of infant formula market.
The Azoth Analytics research report titled 'Global Infant Formula Market' has analysed and segmented the Infant Formula market By Product Type (Infant Milk, Growing-up Milk, Follow on Milk & Specialty Milk), By Distribution Channel (Online Stores, Hypermarket/Supermarket, Pharmacies, Others), By Region (North America, Europe, Asia Pacific, ROW-Latin America, Middle East & Africa) and By Country (U.S, Canada, Mexico, UK, Germany, France, China, India, Japan, Saudi Arabia) for the historical period of 2014-2018 and the forecast period of 2019-2024.
The market condition of infant formula has been improving gradually with ease in availability of infant formula milk in vast number of hypermarkets, supermarkets and online platforms. The Asia Pacific region holds the largest market of Infant Formula and is estimated that it will continue to dominate the market share in the forecast period. Moreover, APAC region will also grow with highest growth rate among the regions.
Growth in female contribution in labour force has encouraged the acceptance of infant formula globally with increased disposable income, busy lifestyle, and high standards of living in countries such as China, India, US, UK, and Japan that boosts the infant formula market growth. Moreover, increasing e-commerce sales, disposable income and accelerating economic growth are expected to drive the market. Market players are focusing on introducing organic infant formula in order to cater to the needs of patrons. Companies equipped with the new edge technologies for innovation, research & development, production, packaging and preserving likely to gain market share in the forecast period.