# Measuring interfacial Dzyaloshinskii-Moriya interaction in ultra-thin
magnetic films

M. Kuepferling  A. Casiraghi  G. Soares  G. Durin  Istituto Nazionale di
Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy  F. Garcia-
Sanchez  Dep. of Applied Physics, University of Salamanca, Plaza de los Caidos
s/n 37008, Salamanca, Spain  L. Chen  C. H. Back  Technical University Munich,
James-Frank-Str. 1, 85748 Garching, Germany  C. H. Marrows  School of Physics
and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom  S. Tacchi
CNR, Istituto Officina dei Materiali - Perugia, c/o Dipartimento di Fisica e
Geologia
Università Perugia, Via A. Pascoli, 06123 Perugia, Italy  G. Carlotti
Dipartimento di Fisica e Geologia, Università di Perugia, Via Pascoli, 06123
Perugia, Italy

(August 27, 2024)

###### Abstract

The Dzyaloshinskii-Moriya interaction (DMI), being one of the origins of
chiral magnetism, is currently attracting considerable attention in the
research community focusing on applied magnetism and spintronics. For future
applications, an accurate measurement of its strength is indispensable. Here
we present a review of the state-of-the-art of measuring the coefficient of
the Dzyaloshinskii-Moriya interaction, the DMI constant $D$, focusing on
systems where the interaction arises from the interface between two materials
(i.e. interfacial DMI). We give an overview of the experimental techniques as
well as their theoretical background and models for the quantification of the
DMI constant. The measurement techniques are divided into three categories: a)
domain wall-based measurements, b) spin wave-based measurements and c) spin-
orbit torque-based measurements. We analyze the advantages and disadvantages
of each method and compare $D$ values at different interfaces. The review aims
to obtain a better understanding of the applicability of the different
techniques to various stacks and of the origin of apparent disagreements among
literature values.

###### Contents

  1. I Introduction
  2. II Domain wall methods
    1. II.1 Method overview
    2. II.2 Theory and models
      1. II.2.1 Current-driven domain wall motion
      2. II.2.2 Field-driven domain wall motion
      3. II.2.3 Equilibrium stripe domain pattern
      4. II.2.4 Magnetic stripe annihilation
      5. II.2.5 Nucleation field
      6. II.2.6 Domain wall stray fields
      7. II.2.7 Domain wall internal structure imaging
    3. II.3 Experimental results
      1. II.3.1 Current-driven domain wall motion
      2. II.3.2 Field-driven domain wall motion
      3. II.3.3 Equilibrium stripe domain pattern
      4. II.3.4 Magnetic stripe annihilation
      5. II.3.5 Nucleation field
      6. II.3.6 Domain wall stray fields
      7. II.3.7 Domain wall internal structure imaging
    4. II.4 Advantages and limitations
  3. III Spin wave methods
    1. III.1 Method overview
    2. III.2 Theory and models
      1. III.2.1 Quantum spin wave theory
      2. III.2.2 Classical spin wave theory
    3. III.3 Experimental results
      1. III.3.1 Brillouin light scattering
      2. III.3.2 Time resolved magneto optical imaging
      3. III.3.3 Propagating spin wave spectroscopy
      4. III.3.4 Spin-polarized electron energy loss spectroscopy
    4. III.4 Advantages and Limitations
  4. IV Spin Orbit Torque method
    1. IV.1 Method overview
    2. IV.2 Theory and models
    3. IV.3 Experimental Results
    4. IV.4 Advantages and Limitations
  5. V Comparison of the methods and discussion of problems in the determination of the DMI constant
    1. V.1 Comparing the DMI constants obtained by the different methods
    2. V.2 Comparing the DMI constants in different materials
    3. V.3 Influence of growth conditions on the DMI constant
    4. V.4 Outlook and conclusion

##  I Introduction

Topological spin structures such as chiral domain walls (DWs) and skyrmions
are emerging as promising information carriers for future spintronics
technologies [75]. Electric currents can drive such spin structures with an
unprecedented level of efficiency, which makes them particularly attractive
for innovative storage devices, including the racetrack memory [225]. The
crucial ingredient needed for stabilizing these chiral spin textures is the
Dzyaloshinskii-Moriya Interaction (DMI).

The DMI is an anisotropic exchange interaction favoring a canted spin
arrangement and has a net contribution only in systems without a center of
inversion. It originates from the spin-orbit coupling (SOC) which acts as a
perturbation on localized spin states. Given two neighboring spins
$\vec{S}_{i}$ and $\vec{S}_{j}$, the DMI contributes to the Hamiltonian
locally with a first bilinear energy term described by the following
expression:

$\mathcal{H}_{DMI}=\vec{D}_{ij}\cdot(\vec{S}_{i}\times\vec{S}_{j})$ (1)

where $\vec{D}_{ij}=D_{ij}(\hat{r}_{ij}\times\hat{n})$ is the local DMI vector
which can be described by its magnitude $D_{ij}$, with $\hat{r}_{ij}$ being
the unit vector linking the two neighboring spins and $\hat{n}$ being along
the direction of symmetry breaking. This expression is part of a generalized
exchange interaction, which relates $\vec{D}_{ij}$ to the isotropic exchange
$-J_{ij}\vec{S}_{i}\cdot\vec{S}_{j}$ with exchange constant $J_{ij}$, i.e.
Heisenberg exchange in the case of localized electrons with a direct orbital
overlap, or any other direct or indirect exchange described by the above
Hamiltonian. Contrary to the Heisenberg type exchange, which favors collinear
alignment, the DMI promotes an orthogonal arrangement between $\vec{S}_{i}$
and $\vec{S}_{j}$, with a chirality imposed by the direction of
$\vec{D}_{ij}$.

The DMI was first proposed in the 1950s for antiferromagnets such as
$\alpha$-Fe2O3 to account for the existence of a weak ferromagnetism [85,
217]. In the following decades, several magnetic materials, such as spin
glasses, orthoferrites, manganites or superconducting cuprates [91, 179, 189,
32, 64, 92], were investigated for their non-collinear or helical magnetism
and for the influence of DMI on the magnetic state. An important step towards
multilayers, on which this review focuses, was done by Fert [92], where
anisotropic pair interactions on surfaces or interfaces were considered.

From an experimental point of view, the development of sophisticated imaging
techniques to visualize the magnetic state, such as spin-polarized scanning
tunnel microscopy (SPSTM) or spin-polarized low energy electron microscopy
(SPLEEM), was crucial for triggering a renewed interest in the topic at the
beginning of the century, highlighted in the important works Bode  _et al._
[29], Heinze [113], Heinze  _et al._ [115]. In non-centrosymmetric single
crystals with FeSi structure111Often called B20 according to classification of
the journal “Strukturbericht”, such as the materials MnSi, FeGe, etc., it was
shown that DMI gives rise to skyrmion lattices and other exotic spin textures
at low temperature, which could be directly visualized [218, 237, 302].

More recently, the presence of DMI was also demonstrated in thin magnetic
multilayers with perpendicular magnetic anisotropy and shown to be localised
at the interface between the layers [304, 125, 241, 212]. This interfacial DMI
has since received a broad attention from the magnetic community (see e.g.
Hellman  _et al._ [116], Gallardo  _et al._ [98]). It is of particular
importance in the context of systems consisting of an ultrathin ferromagnetic
(FM) film with perpendicular magnetic anisotropy (PMA), in contact with a
heavy metal (HM) under- or over-layer. In such a system the DMI arises due to
the broken inversion symmetry at the interface between the two materials and
the large SOC of the heavy metal atoms, which mediate the interaction between
neighboring spins $\vec{S}_{i}$ and $\vec{S}_{j}$ of the ferromagnet. It
follows that the DMI has an interfacial nature with a strength decreasing with
the magnetic film thickness. If the symmetry breaking is only due to the
interface and symmetry planes or axes are always normal to the film plane,
according to Moriya’s rules, the DMI vector $\vec{D}$, defined at the
interface, has to be necessarily perpendicular to the film normal, favoring
canting of the out-of-plane magnetisation (see Fig.1).

Figure 1: Diagram of a system possessing interfacial DMI. The exchange
coupling between the spins $\vec{S}_{i}$ and $\vec{S}_{j}$ in a ferromagnetic
(FM) layer is mediated by a heavy metal (HM) atom with large spin-orbit
coupling. In this case, the local DMI vector with magnitude $D_{ij}$ is
perpendicular to the plane formed by the two atoms of the ferromagnetic layer
and the one of the heavy metal layer.

Bilayers composed of a thin magnetic film with PMA and a heavy metal are
particularly interesting from the application point of view, due to the fast
domain wall motion driven by an electric current in presence of DMI. Early
experiments on current-induced domain wall motion in such systems were not
able to fully unravel the driving mechanisms behind such motion. The
conventional spin-transfer torque (STT) theory [257, 26] predicts motion in
the direction of electron flow, opposite to many observations [214, 215]. On
the other hand, the newly proposed spin-orbit torques (SOTs) – caused by the
inverse spin galvanic (or Rashba) effect (iSGE) [209, 210] and the spin Hall
effect (SHE) [184, 107] – did not have the correct symmetry to drive Bloch
walls [140], which were believed to be present in these ultrathin PMA films
from purely magnetostatic considerations. This issue was resolved when it was
suggested222Here direct imaging of the domain wall chirality [29] played a
crucial role. that in presence of DMI, Néel walls, which have a fixed
chirality [112, 275], occur instead of the expected Bloch walls (Fig. 2). A
series of experimental works later confirmed these findings [86, 240],
substantiating the view that DMI-stabilized Néel walls are mostly moved by the
SHE torque. Ever since, DMI-based phenomena have created an extremely active
research field, often referred to as chiral magnetism.

Figure 2: Bloch and Néel configurations for a domain wall in a perpendicularly
magnetized material. The Néel wall has left-handed chirality
($\uparrow\leftarrow\downarrow$) considering the positive Z direction parallel
and fixed to the normal, which is depicted by the arrow $\hat{n}$ and defined
pointing from the substrate to the magnetic film.

Considering the ferromagnetic film itself isotropic (i.e. without special
crystal symmetries, leading to an additional symmetry breaking), the DMI
vector simplifies to $\vec{D}=D(\hat{r}_{ij}\times\hat{n})$ [217, 70, 275],
with a constant $D$ describing the strength of the DMI in the film plane
($\hat{r}_{ij}$ the vector between the two neighboring spins and $\hat{n}$ is
the normal to the film interface). The interfacial DMI can then be thought of
as equivalent to an effective in-plane magnetic field $H_{\mathrm{DMI}}$
which, acting across the domain wall, causes a reorientation of spins from the
Bloch configuration into the Néel configuration with a fixed chirality [275,
54]. The strength of the interfacial DMI is then measured as the DMI field
[275], to which it is proportional. The sign of $D$ dictates the chirality of
the Néel domain wall, and, consequently, together with the sign of the spin
Hall angle, the direction of the domain wall motion under SHE torque 333The
sign of $D$ depends on the convention chosen in relation with the Hamiltonian
1 which can be either positive or negative. Throughout this review we use the
plus sign in the Hamiltonian, so that positive $D$ corresponds to Néel domain
walls with the right-handed (clockwise) chirality
($\uparrow\rightarrow\downarrow$ or $\downarrow\leftarrow\uparrow$), while
negative $D$ corresponds to Néel domain walls with the left-handed
(anticlockwise) chirality ($\uparrow\leftarrow\downarrow$ or
$\downarrow\rightarrow\uparrow$), where up direction is defined by the normal
in Fig.2.

Because of the crucial role that the interfacial DMI plays in stabilizing
chiral spin structures as well as in attaining extremely efficient domain wall
motion, it is of utmost importance for the magnetic community to be able to
accurately measure and predict the magnitude and sign of $D$ in heavy metal
(HM)/ferromagnet (FM) combinations in order to be able to optimize these
relevant effects. However, despite a rapidly increasing number of works
quantifying the DMI through different experimental techniques, a remarkable
spread of data is currently present in the literature and systematic studies
or reviews are still rare [62]. Not only are different measurement techniques
found to provide contradictory values of $D$ for nominally the same system,
but controversies are also present when utilizing the same method on nominally
identical stacks, or different methods on the same sample [261, 254].

The most commonly used experimental techniques that in recent years have been
employed to measure $D$ in HM/FM structures can be broadly divided into three
categories:

  1. 1.

Domain wall methods, where $D$ is extracted by either measuring the domain
wall velocity or energy as a function of an in-plane magnetic field, or by
measuring domain wall spacing in stripe domain phases, or by directly
measuring the domain wall internal structure;

  2. 2.

Spin wave methods, where $D$ is extracted by measuring the non-reciprocity of
propagating spin waves, owing to the presence of DMI, in in-plane magnetised
films;

  3. 3.

Spin-orbit torque methods, where $D$ is extracted by measuring the field shift
of the out-of-plane hysteresis loop under an in-plane magnetic field.

With this literature review we aim at providing an accurate analysis of the
current state-of-the-art regarding measurements of the interfacial DMI
constant $D$ based on the different techniques summarized above. Therefore,
Sections II, III, and IV are devoted to analyze each of the three main classes
of techniques mentioned above. In particular, in each section we start with a
theoretical background, followed by an overview of the relevant experimental
methodologies and then a number of detailed tables with a collection of the
relevant results published in the literature. Finally, for each of the three
above sections, we will provide a critical discussion of advantages and
limitations of each experimental technique. The last section of this review
article, Section V, is devoted to a comparison and discussion of the methods
and results obtained by the three classes of methods analyzed in the previous
sections, as well as to final considerations about open problems and
challenges. In particular, we address crucial aspects of the current research,
such as the reliability of the different techniques, the choice of material
combinations to achieve large/small, negative/positive DMI, and the reasons
for the spread of the results (e.g. sample/interface quality, growth
conditions and reliability of the measuring techniques).

We anticipate that, with regard to the available literature, it is out of the
scope of the present work to give definitive recipes to prepare specific
interfaces with a controlled value of DMI, evaluate the level of uncertainty
of the different measurement methods, or classify the methods clearly
according to their applicability. Therefore, one should consider our synoptic
Figs. 28-29 as general guides to explore measurement methods and ranges of
values of common heterostructures. We emphasize instead the need for more
systematic studies and hope that this review will stimulate the community to
tackle metrological aspects of the measurement of the DMI at interfaces.

##  II Domain wall methods

###  II.1 Method overview

In ultra-thin films with PMA, Bloch domain walls are magnetostatically favored
over Néel DWs444However, Néel DWs become energetically more favorable when the
film is patterned into very narrow wires [161].. However, the interfacial DMI
manifests itself as a local effective in-plane magnetic field
$H_{\mathrm{DMI}}$ that acts on the domain wall which, when large enough to
overcome the magnetostatic energy coming from the shape of the wall, converts
Bloch DWs into Néel DWs with a chirality determined by the sign of $D$ [275,
54]. By tuning the spin texture of the DWs towards a chiral Néel
configuration, the interfacial DMI changes the static and dynamic properties
of the DWs [275].

The direct observation of chiral magnetization states in presence of a DMI
[113, 29] was the key for understanding the modified DW dynamics. Indeed,
first estimates of the DMI strength were obtained by fitting DW energies and
numerical simulations of DW imaging in FM mono-/double-layers on tungsten
heavy metal substrates [304, 29, 112, 90]. A further experimental technique
used to extract a quantitative value of $D$ was the measurement of DW
velocities driven by an electric current as a function of in-plane magnetic
fields [276]. This is not surprising, given the importance of DMI in these
systems in the context of current-induced domain wall motion experiments [86,
240]. At the same time, it was proven that $D$ could also be inferred by
measuring magnetic field driven DW velocities [125, 121]. $H_{\mathrm{DMI}}$,
and thus $D$, is extracted by analyzing the dependence of the DW velocity on
the in-plane magnetic fields, which - added or subtracted to the
$H_{\mathrm{DMI}}$ \- can enhance or reduce the DW speed.

Following these early investigations [240, 276, 125, 121], several further
works have used DW motion, induced by either current or field, to estimate the
strength and sign of the interfacial DMI in different material systems.
Nucleation of reverse domains, which is a necessary step to study the motion
of DWs, was shown to be dependent on $D$ [231], which established another path
to access the magnitude and sign of $D$. Finally, also the static properties
of domains were shown to be altered by the presence of DMI. This was observed
from two points of view: firstly, the domain width is altered due the
modification of the DW energy originated from DMI [216]. Secondly, the DMI
affects the stability of reversed domains and their field of annihilation
[119]. Both effects have been used to estimate the magnitude of $D$.

DW dynamics in the presence of DMI can be understood making use of analytical
1D models. One-dimensional DW models have been established as useful tools to
support computationally costly micromagnetic simulations as well as to
interpret experimental results. As the name suggests, the 1D models for DW
dynamics are based on the approximation that the magnetization $\mathbf{M}$
varies along one direction only, namely the axis of a narrow wire, usually
identified with $x$, as depicted in Fig. 3.

Figure 3: Sketch of the collective coordinates used to describe DW dynamics in
different types of 1D models: $q$ is the domain wall position, $\Phi$ the
angle of the projection of the magnetization $\mathbf{M}$ in the $x$–$y$
plane, $\Delta$ the width, and $\chi$ the tilting angle of the DW.

The origins of the 1D models can be traced back to the original description of
Bloch walls by F. Bloch [28]. The simplest form of the 1D model was introduced
by Walker and Slonczewski [250, 194] in the 1970s555 The 1D model was reported
earlier by L. R. Walker, Bell Telephone Laboratories Memorandum, 1956
(unpublished). A description of this work can be found in J.F. Dillon Jr.,
Domains and Domain Walls, in: Magnetism, Vol. III, edited by G. T. Rado and H.
Suhl (Academic, New York, 1963)., to study DW dynamics in PMA materials under
the influence of an applied perpendicular field. Here DW dynamics was
described in terms of two time-dependent variables (see Fig. 3): the DW
position $q(t)$ along the wire axis, and the DW angle $\Phi(t)$, defined as
the in-plane ($x$–$y$) angle of the internal DW magnetization $\mathbf{M}$
with respect to the positive $x$ axis666According to these conventions,
$\Phi=0$ ($\pi$) corresponds to a right (left) handed chiral Néel DW, while
$\Phi=\pm\pi/2$ corresponds to a Bloch DW.. The $q-\Phi$ model was later
extended by Sobolev  _et al._ [259, 258] to describe DW dynamics in PMA
materials under in-plane fields. Thiaville  _et al._ [273] next adapted the 1D
model to systems with in-plane magnetic anisotropy, using the domain wall
width $\Delta(t)$ as an additional time-dependent variable in a revised
$q-\Phi-\Delta$ model.

More recently, several other contributions have been included in the 1D model
to take into account newly discovered effects, such as the STTs [306, 274],
the SOTs [111, 251, 35, 198, 200] (see Eq. 3), and the DMI [275, 86, 198]. In
this context, $D$ is then derived according to 86:

$D=\mu_{0}H_{\mathrm{DMI}}M_{\mathrm{S}}\Delta,$ (2)

where $M_{\mathrm{S}}$ is the saturation magnetization, and
$\Delta=\sqrt{A/K_{\mathrm{eff}}}$ is the DW width parameter, with $A$ being
the exchange stiffness and
$K_{\mathrm{eff}}=K_{\mathrm{U}}-\frac{1}{2}\mu_{0}M_{\mathrm{S}}^{2}$ the
effective perpendicular anisotropy constant, corresponding to the intrinsic
perpendicular anisotropy constant $K_{\mathrm{U}}$ decreased by the
demagnetizing energy $\frac{1}{2}\mu_{0}M_{\mathrm{S}}^{2}$.

Additions of thermal fluctuations [196] and of spatially dependent pinning
[65] helped to make the 1D model more realistic. Furthermore, considering
experiments of fast current-driven DW motion [241], Boulle  _et al._ [36]
proposed to include DW tilting as an additional time-dependent variable
$\chi(t)$ (defined as the angle of the DW normal plane with respect to the
positive $x$-axis), which led to the development of the $q-\Phi-\chi$ model.
Very recently the 1D model was extended to implement all four collective
coordinates, namely $q-\Phi-\Delta-\chi$, with the aim of improving the
agreement with experimental observations and micromagnetic simulations when
large in-plane fields are applied [220]. The same authors later showed that
the simple two coordinate $q-\Phi$ model can grant higher accuracy when
combined with an ansatz (which links collective coordinates to magnetization
components) that takes into account magnetization canting within the domains
under an in-plane field [219].

Here, we provide a summary for the different DW-based techniques, explaining
the underlying models (Sec. II.2), reviewing the main results achieved
regarding the determination of $D$ (Sec. II.3), and highlighting the
respective advantages and limitations (Sec. II.4).

###  II.2 Theory and models

####  II.2.1 Current-driven domain wall motion

The driving mechanism behind the current-induced motion of DWs in heavy
metal/ferromagnet bilayers with PMA is now widely believed to be due to a
combination of SHE and DMI, while the iSGE torque is considered to be
negligible [86, 240, 199], as the latter acts to stabilize Bloch walls [210],
and does not have the correct symmetry to drive DWs directly [86, 140]. The
SHE in the heavy metal converts an in-plane charge current into a transverse
spin current that gives rise to spin accumulation at the interface between the
two layers, with consequent spin-current diffusion into the ferromagnet. This
spin-current can interact with the local magnetization by exerting a torque on
it, known as the SHE spin-orbit torque (SHE-SOT)777To be precise, field-like
and damping-like torques can occur as a result of iSGE or SHE effects..
According to the 1D model, the amplitude of the effective field associated
with the SHE-SOT is expressed as [275, 140]:

$H_{\mathrm{SHE}}=\frac{\hbar\theta_{\mathrm{SH}}|J_{e}|}{2\mu_{0}|e|M_{\mathrm{S}}d}\cos(\Phi),$
(3)

where $\theta_{\mathrm{SH}}$ is the spin Hall angle, $J_{e}$ is the electron
current density, $d$ is the thickness of the ferromagnetic film, and $\Phi$ is
the internal in-plane DW angle as defined in Fig. 3. As mentioned earlier, the
SHE-SOT can move DWs only if they possess a Néel component (i.e.
$\Phi\neq\pm\pi/2$) in their spin structure, due to the interfacial DMI. The
direction in which DWs move with current depends both on the sign of the spin
Hall angle $\theta_{\mathrm{SH}}$, determined by the spin-orbit coupling
constant of the heavy metal, and on the sign of $D$ (i.e. on the chirality of
the DW). Micromagnetic simulations and 1D model predict that the maximum
velocity of DWs driven by the SHE-SOT increases with the magnitude of $D$, and
saturates for larger currents showing a clear plateau [275, 234, 176].

The current-driven DW dynamics is dramatically affected by the application of
an in-plane magnetic field $H_{x}$ along the current direction. It is indeed
due to this that DW velocity measurements as a function of $H_{x}$ provide a
means to quantify $D$. Given a fixed $J$, it is observed that Néel DWs with
the same chirality $\uparrow\rightarrow\downarrow$ and
$\downarrow\leftarrow\uparrow$ have the same velocity when $H_{x}=0$, while
they move with different velocities under a non-vanishing $H_{x}$. In
particular, for a given sign of $H_{x}$, the DW type for which $H_{x}$ is
parallel to $H_{\mathrm{DMI}}$ moves faster while the other kind, where
$H_{x}$ partially compensates $H_{\mathrm{DMI}}$, slows down with respect to
its velocity at $H_{x}=0$. The situation is reversed by changing the sign of
the applied $H_{x}$. This remarkable behaviour has an important consequence:
both DWs stop moving for a certain $|H_{x}|$, equal in strength but opposite
in sign for $\uparrow\rightarrow\downarrow$ and $\downarrow\leftarrow\uparrow$
DWs respectively, as schematically shown in Fig. 4.

Figure 4: Sketch of the dynamics of DWs driven by current in the absence (a)
or presence (b) of an applied in-plane field $H_{x}$. The white arrows
indicate DW velocity. The strength of $H_{x}$ applied in (b) matches that of
the DMI field $H_{\mathrm{DMI}}$, thus stopping the motion of DW1. A sketch of
the velocity for the two DWs as a function of applied $H_{x}$ is illustrated
in (c).

The applied $H_{x}$ under which these Néel DWs stop moving – typically
referred to as the “compensating” or “stopping” field – is the field that,
opposing the DMI, restores a Bloch DW configuration, for which indeed no
motion is expected via SHE-SOT. As such, this “compensating” field $H_{x}^{*}$
can be considered equivalent in strength (but opposite in sign) to the
$H_{\mathrm{DMI}}$ acting locally across the DWs. In other words, because of
the change in the DW internal magnetization angle [241], Néel DWs move faster
or slower depending on whether the effective in-plane field experienced by the
DW is enhanced ($H_{x}+H_{\mathrm{DMI}}$) or decreased
($H_{x}-H_{\mathrm{DMI}}$), respectively. In the latter case, DWs stop moving
for $H_{x}=H_{x}^{*}=-H_{\mathrm{DMI}}$, when they become Bloch walls, and
they reverse direction of motion for $H_{x}>-H_{\mathrm{DMI}}$, when the Néel
configuration is re-established but with an opposite chirality. However,
identifying $H_{x}^{*}$ with $H_{\mathrm{DMI}}$ is valid only if conventional
STT, due to the spin-polarized current in the ferromagnet, [257, 26] can be
neglected. When a significant STT is present, the relationship between the
$H_{x}^{*}$ and $H_{DMI}$ in the 1D model becomes [86, 240]:

$H_{x}^{*}=H_{\mathrm{DMI}}+\textrm{sgn}(\theta_{\mathrm{SH}})\frac{2}{\pi}\frac{\mu_{\mathrm{B}}P}{\gamma
eM_{\mathrm{S}}\Delta}|J_{\mathrm{e}}|,$ (4)

where $\mu_{\mathrm{B}}$ is the Bohr magneton, $P$ is the spin current
polarization, and $\gamma$ is the gyromagnetic ratio. Eq. 4 implies that
$H_{x}^{*}$ can depend on the amplitude of the current used to drive DWs. When
only a modest dependence of $H_{x}^{*}$ on $J_{e}$ is observed, as for
instance in Ryu  _et al._ [242], Karnad  _et al._ [134], it is possible to
conclude that the contribution from STT is small. In any case, once $H_{DMI}$
is determined, the magnitude of $D$ is derived through Eq. 2, while its sign
is inferred from the direction of the DW motion and the sign of the spin Hall
angle $\theta_{\mathrm{SH}}$.

The dependence of the DW velocity on an in-plane field for a fixed current
density can be analytically described in the context of the 1D model, taking
into account STT, SHE-SOT and DMI [275, 86, 240]. As seen in Fig. 4, the DW
velocity is expected to be approximately linear with $H_{x}$ around the
“compensating” field $H_{x}^{*}$, and some experimental works derive
$H_{\mathrm{DMI}}$ by linearly fitting the data [240, 276]. In some cases, it
has been observed that the DW velocity remains small or null in a quite large
range of $H_{x}$ around $H_{x}^{*}$ [242, 186, 185], consistent with thermally
activated creep regime, and strong pinning effects. To account for it, the 1D
model has been extended to include an effective pinning potential both without
[242, 185], and with thermal fields [186] to describe the influence of thermal
fluctuations. In both cases, the modified 1D model has provided good agreement
with the experimental data and has been used to extract $H_{\mathrm{DMI}}$.
The range of $H_{x}$ values for which the DW velocity is negligible has been
observed to decrease upon increasing the current density, due to a reduced
influence of pinning [242].

A few papers have shown that $D$ can also be quantified through current-driven
DW dynamics by measuring the dependence of the DW depinning efficiency, rather
than the DW velocity, on the in-plane field $H_{x}$ [95, 144]. The efficiency
of DW depinning is defined as:

$\epsilon=\mu_{0}\frac{dH_{\mathrm{SHE}}}{dJ_{\mathrm{e}}},$ (5)

and is measured as the slope of the out-of-plane depinning field as a function
of $J_{e}$. The DW depinning efficiency changes as a function of $H_{x}$, due
to the corresponding variation of the DW internal structure. In particular,
$\epsilon$ is found to vanish at a certain $H_{x}^{*}$, equal in strength but
opposite in sign for $\uparrow\rightarrow\downarrow$ and
$\downarrow\leftarrow\uparrow$ DWs. Similarly to what was already discussed
for the DW velocity dependence on $H_{x}$, this in-plane field $H_{x}^{*}$ for
which $\epsilon=0$ represents the field at which a Bloch DW configuration is
restored, and can thus be identified with $H_{\mathrm{DMI}}$.

####  II.2.2 Field-driven domain wall motion

The simplest and most common way to move DWs in PMA materials is by applying a
perpendicular field $H_{z}$. To minimize the Zeeman energy associated with
$H_{z}$, domains with magnetization along the field direction expand at the
expense of the others, leading to DW motion. Field-driven DW dynamics are
typically classified into three distinctive regimes – creep, depinning, and
flow – which occur in succession upon increasing $H_{z}$, as shown in Fig. 5a.

Figure 5: Dependence of the DW velocity $v_{\mathrm{DW}}$ as a function of
increasing applied field. In (a) the three regimes - creep, depinning and flow
- are visible. The depinning transition, which is abrupt at zero temperature,
shows some rounding due to thermal effects. At higher applied fields (b), two
distinct regimes can be distinguished within the flow regime, namely steady
and precessional flow. The drop in the velocity is called Walker breakdown.

For sufficiently low driving fields DWs move in the thermally activated creep
regime where they interact strongly with disorder, and their velocity grows
exponentially as $v\sim\exp(-H_{z})^{-1/4}$ [174, 50]. Upon increasing $H_{z}$
above a critical value, known as the depinning field $H_{\mathrm{dep}}$,
disorder starts to become irrelevant and the DW velocity grows as
$v\sim(H_{z}-H_{\mathrm{dep}})^{\beta}$ [50], with $\beta$ being the depinning
exponent at $T=0$ as in Fig. 5a. Finally, for $H_{z}\gg H_{\mathrm{dep}}$ the
DW enters the flow regime where the velocity increases linearly with $H_{z}$
up to the so-called Walker field $H_{\mathrm{W}}$, which marks a significant
decrease in DW velocity, due to a change in its internal dynamics [207]. For
$H_{z}\gg H_{\mathrm{W}}$ the DW recovers a linear flow with $H_{z}$, albeit
with a reduced mobility $dv/dH$.

With the exceptions of a few works [280, 127, 229, 3, 263, 163], the DMI has
been mostly quantified through experiments of field-driven DW motion in the
creep regime, which is addressed below. Methods to extract the DMI from DW
dynamics in the flow regime will be discussed later.

##### Creep regime

In the creep regime, DWs are driven by modest fields (typically down to a few
percent of $H_{\mathrm{dep}}$) and move slowly by thermal activation,
interacting strongly with disorder of various origin (pinning defects, film
thickness variations, $M_{\mathrm{S}}$ inhomogeneities, etc). The DW creep
dynamics is understood in terms of the motion of a one-dimensional elastic
line in a two-dimensional disordered potential. The dependence of the DW
velocity $v$ on the applied field $H_{z}$ is described by the so-called creep
law [174, 50]:

$v=v_{0}\exp{[-\zeta(\mu_{0}H_{z})^{-\mu}]},$ (6)

where $v_{0}$ is the characteristic speed proportional to the attempt
frequency for DW propagation, $\zeta$ is a scaling constant, and $\mu=1/4$ is
the scaling exponent for the 1D elastic line.

Creep motion of DWs driven by a perpendicular field $H_{z}$ is significantly
altered by the simultaneous presence of an in-plane field $H_{x}$, similarly
to what already discussed for the current-driven case (see Sec. II.2.1).
Indeed, it is observed experimentally that when a circular magnetic bubble
expands under the application of $H_{z}$ only, the radial symmetry is
maintained and the bubble grows isotropically – thus retaining its original
shape. However, the symmetry is broken when the bubble is expanded under the
application of both $H_{z}$ and $H_{x}$, as Néel
$\uparrow\rightarrow\downarrow$ and $\downarrow\leftarrow\uparrow$ DWs acquire
different velocities along the direction of the applied $H_{x}$. This
circumstance was first observed in Kabanov  _et al._ [129] for continuous
films of Pt/Co/Pt (see Fig. 6), mentioning the interfacial DMI as a possible
origin. Only later it was fully understood and modelled in the context of DW
creep.

Figure 6: First observation by magneto-optical Kerr effect (MOKE) of the
asymmetric expansion of a bubble under in-plane magnetic field in a continuous
film of Pt/Co/Pt. Arrows show the in-plane field directions. From 129.

The first model proposed to explain the asymmetric motion of Néel DWs with the
in-plane field [125], suggested to modify the creep law (Eq. 6) by changing
the scaling parameter $\zeta$ to take into account the dependence of DW energy
on the in-plane field [125]:

$\zeta(H_{x})=\zeta_{0}[\sigma(H_{x})/\sigma(0)]^{1/4},$ (7)

where $\zeta_{0}$ is a scaling constant and $\sigma$ is the DW energy density.
The dependence of DW energy density on the in-plane field has been calculated
as [275, 125]:

$\sigma(H_{x})=\sigma_{0}-\frac{\pi^{2}\Delta\mu_{0}^{2}M_{\mathrm{S}}^{2}}{8K_{\mathrm{D}}}(H_{x}+H_{\mathrm{DMI}})^{2},$
(8)

when the condition
$|H_{x}+H_{\mathrm{DMI}}|<4K_{\mathrm{D}}/\pi\mu_{0}M_{\mathrm{S}}$ is
satisfied. In this case the effective field acting on the DW is not strong
enough to fully convert it into a Néel DW. Otherwise, for higher fields and
fully Néel DWs the expression becomes:

$\sigma(H_{x})=\sigma_{0}+2K_{\mathrm{D}}\Delta-\pi\Delta\mu_{0}M_{\mathrm{S}}|H_{x}+H_{\mathrm{DMI}}|.$
(9)

In these equations, $\sigma_{0}=4\sqrt{AK_{\mathrm{eff}}}$ is the Bloch DW
energy density, $\Delta=\sqrt{A/K_{\mathrm{eff}}}$ is the domain wall width
parameter, and $K_{\mathrm{D}}=N_{x}\mu_{0}M_{\mathrm{S}}^{2}/2$ is the DW
shape anisotropy energy density888$K_{D}$ represents the fact that Bloch DWs
are magnetostatically more stable in the absence of DMI due to the high PMA of
the films., with the demagnetising factor of the DW given by
$N_{x}=\textrm{ln}(2)d/(\pi\Delta)$, with $d$ the magnetic film thickness
[270]. In other words, this model predicts that the effective in-plane field
acting locally on the Néel DWs on either side of the bubble can be increased
(decreased) if $H_{x}$ and $H_{\mathrm{DMI}}$ have the same (opposite) sign,
resulting in smaller (larger) DW energy $\sigma$ and thus a faster (slower)
DW, just as for the current-driven case (see Fig. 7).

Figure 7: (a-b) Dynamics of bubble DWs driven by perpendicular field
$\mu_{0}H_{z}$=3 mT in the absence (a) or presence (b) of an in-plane field
$\mu_{0}H_{x}$=50 mT in a Ta(5 nm)/Pt(2.5 nm)/Co(0.3 nm) thin film with PMA.
Each image is obtained by adding four sequential images with a fixed time step
(0.4 s), which are captured using a magneto-optical Kerr effect microscope.
The blue box in (b) designates the field of measurement. From 125. (c) A
typical symmetric profile of the DW moving left and right as a function of the
in-plane field $H_{x}$ in Ta(5 nm )/Co20Fe60B20(0.8 nm)/MgO(2 nm) thin film.
Here, the velocity minima occur at $|H_{x}|=H_{\mathrm{DMI}}$. From 138.

The field $H_{x}^{*}=-H_{\mathrm{DMI}}$ at which Néel DWs are converted into
Bloch DWs is then equal in magnitude but opposite in sign for
$\uparrow\downarrow$ and $\downarrow\uparrow$ DWs, respectively. It is
important to notice that, differently from the current-driven case, field-
driven Bloch DWs do not stop moving under the in-plane field $H_{x}^{*}$,
since the perpendicular field $H_{z}$ keeps expanding the magnetic bubble to
minimize Zeeman energy. Rather, in field-driven experiments,
$|H_{x}^{*}|=H_{\mathrm{DMI}}$ corresponds to a minimum in DW velocity. In
this simple model, the velocity for each type of DW should be symmetric around
its own minimum, as in Fig. 7c, although in many experimental cases it is not.
It follows that in the absence of DMI, i.e. when the bubble DW is in the Bloch
configuration, both sides of the bubble have the same velocity dependence with
$H_{x}$, show a minimum at $H_{x}=0$ and are symmetric around $H_{x}=0$ [142].
This model can be used to fit the dependence of bubble DW velocities on in-
plane field using three fitting parameters: $v_{0}$, $\zeta_{0}$, and
$H_{DMI}$ itself. Alternatively, the scaling parameters $v_{0}$ and
$\zeta_{0}$ can be extracted separately as the intercept and gradient of a
linear fit to the curve $\textrm{ln}(v)$ vs. $H_{z}^{-1/4}$ for $H_{x}=0$,
thus leaving $H_{\mathrm{DMI}}$ as the only fitting parameter of the $v$
dependence on $H_{x}$. Once $H_{\mathrm{DMI}}$ has been determined, the
magnitude of $D$ is derived through Eq. 2, while its sign is inferred from the
orientation of the bubble asymmetry with respect to the $H_{x}$ direction.

This modified creep model, in which the in-plane field affects DW dynamics
only through a variation of domain wall energy, has been successfully applied
to fit several experimental data and estimate the interfacial DMI constant $D$
[125, 121, 228, 301, 138, 287, 147, 166, 253]. However, for a growing number
of experiments, as for instance in Fig. 8, the model fails to provide an
adequate description of the data, which often show an asymmetric behavior
around the minimum velocity [169, 128, 168, 261, 226, 44, 144, 256, 254], a
local peak in velocity [169, 261, 12], or even a maximum in velocity in the
flow regime [280].

Figure 8: Example of asymmetry of the domain wall velocity profile as a
function of the in-plane field (red dots) for a set of multilayers with
nominal structure Si/SiO2/X($t$)/Co20Fe60B20(1 nm)/MgO(2 nm)/Ta(1 nm) with
four underlayers X($t$): W (2 nm, (a)), W (3 nm, (b)), TaN (1 nm, (c)), and Hf
(1 nm, (d)). All cases are highly asymmetric in the velocity profile, even
showing a local maximum as for the latter case (d). Note that the velocity in
the direction orthogonal to the in-plane field (black squares) remains
symmetric. From 261.

To explain these non-symmetric results, several different mechanisms have been
suggested. Jué  _et al._ [128] proposed to neglect any DW energy contribution,
and consider a DW chirality-dependent damping, which acts as a dissipative SOT
on the DW: this damping would modulate the attempt frequency for DW motion,
and would thus modify the characteristic speed $v_{0}$ of the creep law (see
Eq. 6). On the contrary, Lavrijsen  _et al._ [169] suggested that chiral
damping could not be an explanation of the asymmetry of their own results, as
$v_{0}$ was found to be symmetric with respect to $H_{x}$. The importance to
consider possible $H_{x}$ dependences on both creep parameters $v_{0}$ and
$\zeta$ (not necessarily related to chiral damping) has been highlighted in
Balk  _et al._ [12], Shepley  _et al._ [256]. While Shepley et al. showed that
the dependence on $H_{x}$ is asymmetric for both $v_{0}$ and $\zeta$, Balk et
al. modelled their results through a modified creep law that takes for both
parameters a dependence on $H_{x}$ into account. By combining Eqs. 6, and 7,
with their expression for $v_{0}$, they were able to fit velocity vs $H_{x}$
curves that showed a local peak around $H_{x}=0$, introducing a local
anisotropy due to pinning $K_{\mathrm{pin}}$, and having $\zeta_{0}$ and
$H_{\mathrm{DMI}}$ as fitting parameters.

Several analytical and numerical studies were devoted to understanding these
features. Kim  _et al._ [141] attributed the asymmetry in the DW energy
density $\sigma$, to the asymmetric variation of the DW width with $H_{x}$,
later confirmed by micromagnetic simulations [246]. Lau  _et al._ [168]
described the velocity asymmetry in terms of the Wulff construction, which
yields a methodology to determine the shape of a magnetic bubble, although it
does not explicitly provide a model for the velocity as a function of in-plane
field. They also speculated that nucleation and annihilation of Bloch lines
may be responsible for a peculiar flattening of magnetic bubbles, which could
in turn cause the observed velocity asymmetry. This again was confirmed
through micromagnetic simulations by Sarma  _et al._ [246]. Pellegren  _et
al._ [226] argued that under an applied magnetic field $H_{x}$, where the DW
energy density $\sigma$ becomes anisotropic with respect to the DW orientation
in the film plane, the correct elastic energy that should be considered to
describe the creep regime does not simply identify with $\sigma$, as typically
assumed in the phenomenological model of creep [174]. Rather, to reproduce the
asymmetry in their velocity curves, they propose that $\sigma$ should be
replaced by the DW stiffness
$\widetilde{\sigma}=\sigma+\sigma^{\prime\prime}$, with
$\sigma^{\prime\prime}=\frac{\partial^{2}\sigma}{\partial\Theta^{2}}$, where
$\Theta$ is the azimuthal angle of the DW normal. Interestingly, the
$H_{\mathrm{DMI}}$ value extracted using this stiffness model was found to be
higher than the field at which the minimum in velocity is observed. In a later
work [167], the same authors proposed to upgrade the stiffness model, by
taking into account a possible variation of the characteristic speed $v_{0}$
with $H_{x}$, which they also speculate could be due to a chiral damping
mechanism. Through this improved model, they could fit velocity curves as a
function of $H_{x}$ that are not only asymmetric about the minimum, but also
show a crossover between DW velocities at opposite sides of the bubble. More
recently, another model was advanced by Shahbazi  _et al._ [254] to explain
the presence of both asymmetry and DW crossover in the velocity curves. Here,
the DW depinning field $H_{\mathrm{dep}}$ is allowed to vary with $H_{x}$ in a
way determined from micromagnetic simulations. Notably, this work shows that
the velocity minimum underestimates $H_{\mathrm{DMI}}$, as was also found for
the stiffness model [226]. This is also confirmed by Hartmann  _et al._ [110]
and Géhanne  _et al._ [100]: the former reconsidered the change of the DW
stiffness due to deformation as an angular shape, and minimizing the energy of
the system with a semi-analytical approach they could calculate the velocity
profile and show that the minimum of the velocity does not occur at
$H_{\mathrm{DMI}}$. The latter showed that $H_{x}$ can also modify the
characteristic length scale of pinning, in strong correlation with the DW
width, again implying that the minimum of the velocity could not correspond to
the $H_{\mathrm{DMI}}$ field.

##### Flow regime

According to the 1D model [275], the presence of DMI significantly increases
the Walker field $H_{\mathrm{W}}$. Indeed, in samples with a strong enough DMI
to convert the wall to a fully Néel form, $H_{\mathrm{W}}$ is proportional to
$H_{\mathrm{DMI}}$ :

$H_{\mathrm{W}}\propto\alpha H_{\mathrm{DMI}},$ (10)

where $\alpha$ is the Gilbert damping constant [275]. The DW velocity at
Walker breakdown can thus be expressed as:

$v_{\mathrm{W}}=\gamma_{0}\frac{\Delta}{\alpha}H_{\mathrm{W}}\sim\frac{\pi}{2}\gamma\frac{D}{M_{\mathrm{S}}},$
(11)

where $\gamma_{0}$ is the gyromagnetic ratio. In contrast to the prediction of
the 1D model, it is found experimentally and confirmed by 2D micromagnetic
simulations that in samples with large DMI, the DW velocity does not decrease
at fields larger than $H_{\mathrm{W}}$, but instead reaches a plateau [229,
298, 3]. This fact originates from the complex meander-like structure that the
DW adopts at velocities above $H_{\mathrm{W}}$, with continuously nucleation
and annihilation of pairs of vertical Bloch lines [298, 229]. Measurements of
this roughly constant velocity, which corresponds to $v_{\mathrm{W}}$, provide
a simple way to determine $D$ for samples with large DMI. A combination of
experiments and modelling has been used to show that $D$ is also associated
with the value of $H_{z}$ at which the end of the plateau is reached [163].

In other works [280, 229, 3, 263] measurements of the minimum DW velocity as a
function of $H_{x}$ in the flow regime have also been used to quantify $D$: in
the flow regime the DW velocity depends only on DW width (not DW energy) and
this assumes the minimum value when the DW is Bloch. Indeed, it has been shown
that under certain conditions the 1D model provides an expression for the DW
velocity that exhibits a parabolic dependence on $(H_{x}+H_{\mathrm{DMI}})$
[146].

####  II.2.3 Equilibrium stripe domain pattern

The demagnetized state of PMA materials consists of a complex domain pattern,
usually in the form of labyrinth structures of domains pointing either up or
down. If the thickness of the sample is very small, the equilibrium domain
width can exceed the sample dimensions and such structures are not expected.
Examples of the typical patterns are shown in Fig. 9 for a multilayer of
Pt/Co/Al2O3, taken from Legrand  _et al._ [172]. The exact demagnetized
pattern depends on the direction and the history of the applied field: when an
out of plane field is applied, a maze domain structure is created (Fig. 9a),
while for an in plane field, domain walls remain almost parallel and typically
form a stripe structure (Fig. 9b). In the latter, the width and the density of
domains can be used to estimate the domain wall energy. As suggested by the
theory developed for infinite parallel stripe domains [193, 158], the domain
width is a function of the domain wall energy, which includes magnetostatic,
anisotropy, Zeeman, and exchange terms. Néel walls, favored by DMI, have a
reduced energy and, consequently the DMI markedly affects the equilibrium
domain width.

For domain widths much larger than the domain wall width, as is generally the
case for PMA materials, the magnetostatic contribution comes only from the
surfaces charges at the top and bottom surfaces, so that DW energy is
calculated as:

$\sigma_{DW}=\frac{2A}{\Delta}+2K_{\mathrm{eff}}\Delta-\pi\cos(\Phi)|D|$ (12)

where $\Phi$ is the same angle of the internal DW magnetization defined in
Fig. 3. The minimization of the energy with respect to the domain width
$\Delta$ and core angle $\Phi$ gives the value of the equilibrium domain
width: a critical value $D_{\mathrm{c}}$ is obtained. For $D>D_{\mathrm{c}}$
the preferred configuration is a Néel wall and for $D\leq D_{\mathrm{c}}$ it
gradually transforms into a Bloch wall as $D$ approaches zero.

Initially, this method was limited to Néel walls only, with
$D>D_{\mathrm{c}}$, and hence $\cos(\Phi)=1$, so that the DW energy density
simplifies to $\sigma_{\mathrm{DW}}=\sigma_{0}-\pi|D|$. Later, the inclusion
of the angle of the wall $\Phi$ in Eq.12 removed this limitation [203, 177]. A
further improvement took into account the dipolar terms coming from the
internal structure of the DW or/and from the DW interaction [177]. This is
important in thicker samples, as the dipolar energy causes the internal
magnetization to vary along the thickness, even in the presence of DMI. To
account for all these situations, the model was extended allowing a different
angle $\Phi$ for each layer of the sample [175].

The analytical expression for the DW energy $\sigma_{\mathrm{DW}}$ as a
function of the periodicity $w$ of the domains, which is twice the domain
width, is

$\frac{\sigma_{DW}}{\mu_{0}M_{\mathrm{S}}^{2}d}=\frac{w^{2}}{d^{2}\pi^{3}}\displaystyle\sum_{\mathrm{odd}\;n=1}^{\infty}\frac{1-\left(1+\frac{2\pi
nd}{w}\right)e^{-\frac{2\pi nd}{w}}}{n^{3}}$ (13)

where $d$ is the thickness. From the experimentally obtained domain width one
can then calculate the domain wall energy, and estimate the DMI constant using
one of the previously derived approximations.

This general method has been applied in three different ways: comparisons to
micromagnetic simulations [216, 172], analytical estimations [289, 300, 290,
288], and scaling of the energy of an experimental image [8].

Figure 9: MFM domain patterns for an (a) out-of-plane and (b) in-plane
demagnetized multilayer composed of [Pt(1 nm)/Co(0.8 nm)/Al2O3(1 nm)]20. From
Supplementary Materials of 172.

####  II.2.4 Magnetic stripe annihilation

Two parallel domain walls in a thin film with strong enough DMI are homochiral
Néel walls. Since the chirality of the domain walls is the same, either left-
handed or right-handed, the core magnetization of those parallel domain walls
point in opposite directions. Such a pair of domains constitutes a topological
structure. This situation is different from the case of absence of DMI, where
the core magnetization of the Bloch walls point in the same direction but not
in a fixed orientation. The annihilation of these two parallel Néel walls
depends on the strength of the DMI due to the topological configuration. This
fact was confirmed by simulations [197, 119]. To annihilate the walls an out
of plane field is applied that reduces the size of the domain disfavored by
the field. This domain achieves a minimum size until the walls meet as a so-
called “winding pair” to form a $360^{\circ}$ wall, which annihilates when a
given value of the out-of-plane field is exceeded. From the determination of
the field of annihilation and the minimum domain width the DMI can be
extracted when compared with the corresponding simulations [119]. An example
from [25] of the annihilation field dependence on DMI calculated by means of
micromagnetic simulations can be seen in Fig. 10. More recently, a formula for
the minimum width of stripe domains was derived [177], using the analytical
formulation discussed in the previous section. This allowed the extraction of
the DMI value without the performance of systematic micromagnetic simulations.

The analytical formula for the domain wall energy $\sigma_{\mathrm{DW}}$
obtained from the minimum domain width $w_{\mathrm{min}}$ is given by:

$\frac{\sigma_{\mathrm{DW}}}{\mu_{0}M_{\mathrm{S}}^{2}d}=\frac{1}{2\pi}(\ln[1+(w_{\mathrm{min}}/d)^{2}]+(w_{\mathrm{min}}/d)^{2}\ln[1+(w_{\mathrm{min}}/d)^{-2}]).$
(14)

As in the previous method this allows extracting the value of the domain wall
energy density which is also a function of the DMI constant. Using the same
expressions for the energy density as given in the previous section, one can
estimate its value.

Figure 10: Annihilation field of two homochiral walls as a function of the DMI
value calculated from micromagnetic simulations for two different anisotropy
values. Inset: Magnetic configuration of the two Néel walls squeezed together
by the perpendicular field. From 25. Figure 11: (a) Theoretical nucleation
field as a function of the magnetic field applied perpendicular to the edge of
the sample for edge and bubble (film) nucleation as a function of the field.
(b) Experimental nucleation field as a function of the magnetic field applied
perpendicular to the edge for edge and bubble (film) nucleation. (c)
Schematics of the nucleation of bubble domain (red color) showing the
orientation of the wall core parallel and anti-parallel to the applied field
for in-plane fields below the DMI field. (d) Calculated domain wall energy as
a function of the in-plane field for different DMI fields. (a) and (b) are
from 231, (c) and (d) from 153.

####  II.2.5 Nucleation field

As stated above, the energy of a domain wall is reduced by the presence of
DMI. This modification also affects the energy barrier relevant for nucleation
of reversed domains. Pizzini  _et al._ [231] showed that the nucleation
process is affected by the presence of DMI. They distinguished between
nucleation at the edge of the patterned sample and in the center of the
magnetic film. They showed that the nucleation of reversed domains at the edge
depends on the value of the in-plane applied field and it is asymmetric with
respect to the combination of the DMI sign and the in-plane field direction.
The nucleation is favored at the edge of the sample having a nucleated wall
with the core magnetization in the direction of the in-plane field. The half-
droplet model [284], applied in the paper, describes the nucleation of a
magnetic domain at the side edge under the application of an in-plane magnetic
field. The out-of-plane nucleation field for a reversed domain is:

$H_{\mathrm{n}}=\frac{\pi\sigma^{2}d}{2\mu_{0}M_{\mathrm{s}}pk_{\mathrm{B}}T}$
(15)

where $\sigma$ is the domain wall energy associated with the bubble, $T$ is
the temperature and $p$ is the factor related to the waiting time according to
$\tau=\tau_{0}\exp(p)$ where $\tau_{0}$ is the attempt frequency. The DW
energy is a function of the in-plane field and the DMI constant. From the best
adjustment of their numerical model to the experimental results, they were
able to estimate the DMI constant in Pt/Co/AlOx.

In the case of a bubble domain away from the edges, Pizzini  _et al._ [231]
concluded that the Zeeman energy gained within the half droplet having a DW
magnetization component parallel to the in-plane field is compensated by the
loss of energy within the half droplet with opposite magnetization (see the
orientation of the magnetization in Fig. 11(b)), while this is not true for an
incomplete bubble nucleated at the edge. Therefore, the nucleation field for a
complete bubble is independent of the in-plane field value. This fact agrees
well with their model shown in Fig. 11(a) and their measurements shown in Fig.
11(b).

More recently, Kim  _et al._ [153] realized that such an argument does not
hold for bubbles in films above a critical in-plane field value, corresponding
to the DMI field, where the energy of the bubble is altered due to the fact
that the magnetization at its boundary aligns with the in-plane field. Hence,
the picture in Fig. 11(c) is no longer valid. This fact leads to a reduced
value of the bubble magnetic energy and an associated reduction of the
nucleation field when the in-plane field value is increased beyond the DMI
field. This was in agreement with their numerical calculations as shown in
Fig. 11(d). This fact was not observed in Pizzini  _et al._ [231] due to the
large DMI field in the samples studied in their work.

Therefore, the critical field above which a change from a constant nucleation
field to a field dependent nucleation is observed is a measure of the DMI
constant, since it coincides with the DMI field $H_{DMI}$. Experimentally,
this can be determined from the dependence of the out-of-plane nucleation
field on the in-plane field.

Figure 12: Schematic of the stray field measurement using a NV magnetometer.
The internal DW magnetization angle $\psi$ ($\Phi$, in our notation) depends
on the DMI constant, resulting in different profiles of the stray field
component $B_{x}^{\psi}(x)$. From 104.

####  II.2.6 Domain wall stray fields

Another method taking into account the static DW structure is the direct
measurement of the stray field in the DW. The stray field has a typical
profile along the axis perpendicular to the DW including information on the
angle $\Phi$ and the strength and sign of $D$ (see Fig. 12). This method is
limited to low values of $D\leq D_{\mathrm{c}}$, i.e. when a significant angle
is present. $D_{\mathrm{c}}=2\mu_{0}M_{\mathrm{S}}^{2}d\ln 2/\pi^{2}$ is the
critical $D$ value above which formation of fully oriented Néel walls occurs
($d$ is the film thickness). Therefore only samples with relatively small $D$
can be measured since $D_{\mathrm{c}}$ is typically on the scale of 0.2 mJ/m2
(see Tab. 12). For samples with larger DMI, where fully oriented Néel walls
are present, only a lower limit of $D$ can be given together with its sign
[272]. Since DW widths are of the order of 5-10 nm, the stray field profile
determination requires an excellent spatial resolution, which was
experimentally tackled by nitrogen vacancy (NV) magnetometry [104].

####  II.2.7 Domain wall internal structure imaging

With the progress in the development of advanced microscopy techniques with
nanometric resolution and magnetic contrast, it has become possible to access
not only the size and distribution of magnetic domains, but also to study the
detailed structure of domain walls. As discussed in the previous sections, the
domain wall structure is influenced by the presence of DMI, so that from a
detailed identification of the wall structure one can in principle quantify
the DMI strength. Basically four methods for spin sensitive imaging were used
in literature: pioneering works were performed by spin-polarized scanning
tunneling microscopy/spectroscopy (SP-STM/STS), the most quantitative works
were done by spin-polarized low-energy electron microscopy (SPLEEM) and in a
few more recent works Scanning electron microscopy with polarization analysis
(SEMPA) and Lorentz-transmission electron microscopy (L-TEM) were employed.

Some of the earliest evidences that the DMI inﬂuences domain wall energies and
alters DW conﬁgurations was obtained from SP-STM of ultrathin magnetic films
deposited on heavy metal substrates such as single atomic layers of Mn or Fe
on W(110) substrates [29, 112, 115]. These methods provide information on the
magnetic order at the atomic scale. In order to extract the parameters
entering the total energy of the observed non-collinear spin structures, such
as the DMI constant, simulations of the images (see Fig.13) based on models
describing the tunneling process in the tip [271] and density functional
theory were performed [291]. Since the computational cost of ab initio models
is large, simpliﬁed approaches were devised, combining them with Monte Carlo
or micromagnetic calculations [114, 112]. Successively, using SP-STM, Pietzsch
and Wiesendanger [230], Meckler  _et al._ [202, 201] concentrated on spin
spiral domains in these bilayers and showed that the magnetic structure of the
Fe double layer grown on W(110) is an inhomogeneous right-rotating cycloidal
spin spiral. Meckler  _et al._ [202] extracted the magnitude of the
Dzyaloshinskii-Moriya vector from the experimental data using analytical
micromagnetic calculations. The result was then conﬁrmed by comparison of the
measured saturation ﬁeld along the easy axis to the respective value as
obtained from Monte Carlo simulations.

Figure 13: a) Simulations of SP-STM images based on Monte Carlo calculations
[114] of a monolayer of Mn on W(001) at 13 K (the inset shows the Fourier
transform). b) Detailed spin structure for the white square indicated in a)
where the helical spinstructure can be directly seen in the illustration. From
90.

More recently, spin-polarized low-energy electron microscopy (SPLEEM) has been
successfully exploited to achieve a quantification of the DMI strength in
different systems [54, 52, 55, 53, 295]. While first works give only an
estimate of the DMI constant, a remarkable development towards quantitative
values is observed in literature. Here, the HM thickness dependent transition
from chiral Néel walls to achiral Bloch walls is investigated in wedge
samples. From the critical thickness a quantitative estimate of the DMI
strength is obtained. In addition to SP-STM and SPLEEM, also L-TEM [99] and
SEMPA [66, 156, 205] have been used to quantify the DMI strength by comparing
the measured and the calculated profiles of the domain walls. In most cases
micromagnetic analytical calculations or simulations of the DW energy or
strayfield are employed.

###  II.3 Experimental results

####  II.3.1 Current-driven domain wall motion

Table 1: Overview of DMI measurements for Co/Ni/Co thin films via current-induced domain wall motion experiments. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{\mathrm{DMI}}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Ref
---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (pJ/m) |
Co(0.3)/Ni(0.7)/Co(0.15) | Pt(1) | TaN(5) | 60 | n.a. | n.a. | 240
Pt(1.5) | 190 | n.a. | n.a.
Pt(3) | 240 | n.a. | n.a.
Pt(5) | 220 | n.a. | n.a.
Co(0.3)/Ni(0.7)/Co(0.15) | Pd(5) | TaN(5) | 12 | n.a. | n.a. |
Ir(3) | 18 | n.a. | n.a. | 242
Pt(5) | 140 | n.a. | n.a. |
Table 2: Overview of DMI measurements for CoFeB thin films via current-induced domain wall motion experiments. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{DMI}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (pJ/m) |  |
Co20Fe60B20(1) | Hf(2.6–6) | MgO(2) | n.a. | 0.38 – 0.05111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | 0.38 – 0.05111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | - | 276
Ta(0.5–1.3) | n.a. | 0.08 – 0.07111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | 0.08 – 0.07111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | -
TaN(0.4–6.6) | n.a. | 0.04 – 0.20111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | 0.04 – 0.20111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | +
W(2.1–3.6) | n.a. | 0.24 – 0.37111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | 0.24 – 0.37111The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMI value could occur at mid thickness range). | +
Co20Fe60B20(1) | Ta(5) | MgO(2) | 7.8 | 0.06 | 0.06 | + | 186
Co20Fe60B20(0.8) | Ta(5) | MgO(2) | n.a. | 0.03 | 0.02 | - | 134

Measurements of current-driven DW motion are performed in wires with typical
widths ranging between 1 $\mu$m and 5 $\mu$m and lengths of a few tens of
$\mu$m. Usually, the magnetization in the wire is initially saturated by
applying an out-of-plane field $H_{z}$. $H_{z}$ is then removed and a current
pulse (with duration of a few tens of ns) is applied either directly to the
wire or to an Oersted line patterned on top of the wire. In either case the
current pulse nucleates a reversed magnetic domain and a DW is thus injected
into the wire. Alternatively, one can use a magnetic field pulse to generate
domains in injection pads, and inject them in the wires.

For measurements of DW velocities, current pulses are applied to the wire to
move a DW, while the DW position along the wire is imaged by magneto-optical
Kerr effect (MOKE) microscopy in polar configuration. The number and duration
of the applied current pulses is chosen to obtain a significant displacement
of the DW, at least over a few $\mu$m. Typical values of current densities
fall in the range between 1010 A/m2 and 1012 A/m2. The velocity of the DW is
calculated as the ratio between the current-induced DW displacement and the
total duration of the pulses. Finally, velocities are measured for both
$\uparrow\downarrow$ and $\downarrow\uparrow$ DWs under different strength of
$H_{x}$, keeping the current density fixed, in order to determine the
“compensating” field $H_{x}^{*}$ of Fig. 4.

Table 3: Overview of DMI measurements for Co thin films via current-induced domain wall motion experiments. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{\mathrm{DMI}}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (pJ/m) |  |
Co(0.36) | Pt(4) | Pt(1) | 37 | 0.24 | 0.09 | - | 95
Co(0.36) | Pt(4) | Pt(2) | 12.5 | 0.10 | 0.04 | -
Co(0.5) | Pt(4) | Pt(2) | 11 | 0.09 | 0.04 | -
Co(0.5) | Pt(2) | Pt(4) | 3 | 0.03 | 0.02 | -
Co(0.8) | Pt(4) | AlOx(1.9) |  $\gg$40 | n.a. | n.a. |
Co(1) | Pt(5) | Gd(2) | 280 | 1.00 | 1.00 | - | 280
Co(0.93) | Pt(4) | AlOx(2) | 99 | 0.54 | 0.50 | - |
Co(1.31) | 54 | 0.48 | 0.63 | - | 185
Co(1.37) | 48 | 0.47 | 0.64 | - |
Co(0.9) | Pt(2.5) | Al(2.5) | 107 | 0.87 | 0.78 | - | 144
Co(0.9) | Ti(2.5) | 197 | 1.42 | 1.28 | -
Co(0.9) | W(2.5) | 183 | 1.35 | 1.22 | -
Co(0.5) | Pt(1.5) | 0 | 0 | 0 |

For measurements of DW depinning efficiencies, a constant current density
$J_{e}$ is applied continuously to the wire for a given in-plane field
$H_{x}$, while the out-of-plane field $H_{z}$ is ramped up until DW motion is
detected by MOKE. The DW depinning efficiency $\epsilon$ for a given $H_{x}$
is then determined from the slope of the depinning field $H_{z\mathrm{,dep}}$
as a function of $J_{e}$. Modest current values are used, up to about 1010
A/m2, in order to exclude Joule heating effects. Finally, $\epsilon$ is
measured for both $\uparrow\downarrow$ and $\downarrow\uparrow$ DWs under
different strength of $H_{x}$, and $H_{\mathrm{DMI}}$ is determined as the
“compensating” field $H_{x}^{*}$ for which $\epsilon=0$.

Measurements of current-driven DW motion as a function of the in-plane field
$H_{x}$ have been used to extract $H_{\mathrm{DMI}}$ in several material
stacks, with Co/Ni/Co, Co20Fe60B20 or Co as ferromagnetic films, different
heavy metals as underlayers and either a heavy metal or an oxide as
overlayers. Tables 1, 2, and 3 present a summary of the DMI values measured
for Co/Ni/Co, CoFeB, and Co, respectively.

For Co/Ni/Co films (see Table 1), the DMI seems to increase upon increasing
the thickness of the Pt underlayer [240], while it is very low when using Pd
or Ir as underlayers [242].

For CoFeB films with Ta and MgO as underlayer and overlayer (see Table 2), the
DMI values, generally small, are spread not only in magnitude, but also in
sign [276, 186, 134]. This can be explained by the fact that the two
interfaces contribute with small and opposite DMI values, and that the DMI at
the CoFeB/oxide interface strongly depends on its oxidation state.
Furthermore, Karnad  _et al._ [134] find different signs of DMI for
Ta/CoFeB/MgO depending on whether they measure DW motion driven by current or
field (see table 5 for the field-driven case). Among the underlayers used with
CoFeB, W provides the highest (positive) DMI [276].

Finally, for Co films, Pt is the only underlayer investigated, while
overlayers are either oxides or metals (see Table 3). In the symmetric
structure Pt/Co/Pt, where no DMI is expected if the two interfaces are
identical as experienced with epitaxial layers, either negative [95] or
vanishing [144] DMI values are measured. This lack of perfect compensation
depends on growth conditions, oxidation of the interfaces, and many other
experimental conditions. For all the other stacks the DMI is always negative
and largest in magnitude when using Ti as overlayer [144].

####  II.3.2 Field-driven domain wall motion

Figure 14: Evolution of bubble domain shapes in an applied in-plane magnetic
field in Co/Ni multilayers measured by perpendicular Kerr effect. From 168.

Experiments of field-driven DW dynamics are mostly performed in continuous
films where magnetization reversal proceeds by nucleation and growth of
magnetic bubble domains. Only few works report studies of planar DW dynamics
(i.e. straight walls) in continuous films [226, 147] or in $\upmu$m-wide wires
[127, 144]. The magnetization in the continuous film/wire is initially
saturated by applying a perpendicular field $H_{z}$. A bubble domain (or a
domain separated by a planar DW) is then nucleated by applying a short $H_{z}$
pulse in the opposite direction, through either a coil or an electromagnet.
The bubble domain (or the planar domain) is expanded under simultaneous
application of continuous $H_{x}$ (from an electromagnet) and pulsed or
continuous $H_{z}$ (from a coil or an electromagnet). The initial and final
positions of the DW are imaged through MOKE microscopy in polar configuration.
Typical bubble growths are in the range of at least a few tens of $\upmu$m, to
avoid coalescence with other bubbles. The velocity of the DW is measured along
the direction of the applied $H_{x}$ and is calculated as the ratio between
the DW displacement and the total time during which $H_{z}$ is applied –
whether in pulses or continuously. Finally, velocities are measured for both
$\uparrow\downarrow$ and $\downarrow\uparrow$ DWs (i.e. DWs on opposite sides
of the bubble) under different strength of $H_{x}$, upon keeping $H_{z}$
constant. In this way velocity vs $H_{x}$ curves are constructed for both DWs
and $H_{\mathrm{DMI}}$ is determined either through fitting with one of the
modified creep formula previously discussed (II.2.2), or simply by finding the
field for which the velocity is minimum.

Regarding the strength of the applied fields, values differ greatly depending
on the material and DW motion regime investigated. For measurements in the
creep regime only modest $\mu_{0}H_{z}$ are needed – usually of a few mT
depending on the depinning field – while $\mu_{0}H_{z}$ of hundreds of mT are
used to drive DWs in the flow regime. The maximum values of applied
$\mu_{0}H_{x}$ depend instead on the strength of the DMI. In samples with
large DMI in-plane fields up to 350 mT have been used [121, 169, 44]. However,
since the use of in-plane fields can be accompanied by artefacts (e.g. the
nucleation of bubbles can be influenced, misalignments of the fields and
crosstalk between perpendicular and in-plane electromagnets might be present),
measuring large DMI values can be problematic. A simple scheme has been
proposed to overcome the need for high $H_{x}$ values [147], whereby the DW
velocity is measured at an angle $\theta>0$ with respect to the in-plane field
direction. In this way, the minimum of the DW velocity scales by a factor
$\textrm{cos}\theta$, and becomes measurable also for samples with large DMI
by applying moderate in-plane fields.

Field-driven DW dynamics in the creep regimes have been investigated to
extract $H_{\mathrm{DMI}}$ in material systems with Co/Ni multilayers, CoFeB
or Co as ferromagnetic layers, and different combinations of heavy metals and
oxides as underlayers and overlayers. Regarding the flow regime, instead,
$H_{\mathrm{DMI}}$ has been measured only for Co films as shown in Table 7.
Tables 4, 5 and 6 present a summary of DMI measurements in the creep regime
for Co/Ni, CoFeB and Co, respectively. In the following we highlight some of
the most interesting findings and observations of these experimental studies.

For Co/Ni multilayers (see Tab. 4), Yu  _et al._ [301] studied the effect of
different capping layers with a Pt underlayer and found values for $D$ ranging
from $D=$ 0.05 mJ/m2 ($D_{s}$= 0.05 pJ/m) for MgO to $D=$ 0.39 mJ/m2 ($D_{s}$=
0.35 pJ/m) to Ta. Separately, three works by the same group [168, 226, 167]
have reported on the asymmetry of DW velocity curves about their minima and on
the crossover between DW velocities for $\uparrow\downarrow$ and
$\downarrow\uparrow$ sides of magnetic bubbles, corresponding to a
morphological change from flattened to teardrop bubble shapes (Fig. 14). As
previously mentioned, these velocity curves have been fitted through a model
that takes into account both DW stiffness and a field-dependent prefactor
$v_{0}$, showing that measurements of the minimum in velocity can be
misleading to quantify the DMI [167]. The maximum value corresponds to an
underlayer of Pt and a top layer of Ta/TaN with $D=$ 0.52 mJ/m2 ($D_{s}$= 0.94
pJ/m). The range of values are larger than in Yu  _et al._ [301] but further
analysis is needed to identify its origin.

Regarding CoFeB films (see Tab. 5), both the respective Co and Fe compositions
[124] and the post-growth annealing temperature [138] have been shown to play
a crucial role in the magnitude of DMI. Furthermore, for low DMI values
(namely, $D\lesssim$ 0.2 mJ/m2), the DW motion technique disagrees with the
values obtained by Brillouin Light Scattering (BLS) [261]. Also worth
mentioning is a recent study by Diez  _et al._ [77] which showed that the DMI
can be tuned in a Ta/CoFeB/MgO system through light He+ irradiation, due to an
increasing interface intermixing mostly between Ta and CoFeB layers.

Co is the most widely studied material, specifically in combination with Pt as
underlayer, which can provide high DMI magnitudes depending on the overlayer
choice. DMI measurements performed in the creep regime (see Tab. 6) indicate
that the nominally symmetric stack Pt/Co/Pt can have positive values of
$D_{\mathrm{s}}$ as high as 0.58 pJ/m [121], although the same work shows that
upon ensuring epitaxial growth, $D_{\mathrm{s}}$ reduces to almost 0. An even
larger value of 0.71 pJ/m was calculated from the experimental data by
Hartmann  _et al._ [110] 999In this paper the minimum velocity is found at
$\mu_{0}H_{x}\sim$ 50 mT, a value compatible with other publications which
report much smaller $D/D_{s}$ values as e.g. in Kim  _et al._ [142]. The
difference is the model used to evaluate $D$ from the velocity minimum.
Hartmann  _et al._ [110] calculate $\mu_{0}H_{DMI}\approx$ 170 mT, resulting
in the high value of $D_{s}=$ 0.71 pJ/m.. Two more studies report a vanishing
DMI [229, 3] for Pt/Co/Pt layers measured in the flow regime (see Tab. 7),
while a small negative $D_{\mathrm{s}}$ is found in Shahbazi  _et al._ [253].

Another system widely investigated in the literature is the Pt/Co/Ir stack,
which also gives rise to some controversial results. First principle
calculations predict opposite DMI signs for the Pt/Co and Ir/Co interfaces,
which would result in an additive effect for the Pt/Co/Ir stack [296, 292].
However, Hrabec  _et al._ [121] show that $D$ decreases when introducing a
thin Ir top layer in a Pt/Co/Pt stack, and even changes sign upon increasing
the Ir thickness. Similarly, Shahbazi  _et al._ [254] find that the magnitude
of $D$ is smaller in Pt/Co/Ir/Ta than in Pt/Co/Ta, suggesting that Pt/Co and
Co/Ir interfaces contribute to the net DMI with opposite signs. On the other
hand, a huge increase of DMI has been observed in Pt/Co/MgO layers, upon
insertion of a very thin Mg film between Co and MgO [44], providing among the
highest values of $D_{s}=\leavevmode\nobreak\ 2.32\leavevmode\nobreak\ pJ/m$
($D=$ 2.32 mJ/m2) reported in the literature.

All these results show how important the quality of the interfaces is for the
determination the DMI. Perfect compensation is rarely obtained for nominally
symmetric interfaces, and large variations are possible. The “controlled
damage” of the interfaces by Ar+ ion irradiation can even be an effective way
to tune the sign of DMI in Pt/Co/Pt films [12]. The key aspect is the
influence of the growth conditions on DMI, as extensively investigated in
Lavrijsen  _et al._ [169], Wells  _et al._ [287]. Both studies report a
dramatic variation of the DW velocity dependence with in-plane field, observed
upon changing sputter-deposition conditions, namely Ar gas pressure, substrate
temperature or chamber base pressure. Indeed, it is speculated that such
different growth conditions may lead to different degrees of interfacial
intermixing and/or quality, resulting in a huge range of measured DMI values,
which can even change sign [287].

Table 4: Overview of DMI measurements for Co/Ni multilayers via field-induced domain wall motion experiments in the creep regime. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{\mathrm{DMI}}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (pJ/m) |  |
Co(0.1)/[Ni(0.1)/Co(0.1)]4 | Pt(4) | MgO(2) | 15.59 | 0.05 | 0.05 | - | 301
Cu(2) | 19.23 | 0.12 | 0.11 | -
Pt(2) | 34.88 | 0.20 | 0.18 | +
Ta(2) | 103.86 | 0.39 | 0.35 | -
[Co(0.2)/Ni(0.6)]2/Co(0.2) | Pt(2.5) | Ta(0.5)/TaN(3) | 60 | 0.21 | 0.38 | - | 168
[Co(0.2)/Ni(0.6)]2/Co(0.2) | Pt(2.5) | Ta(0.5)/TaN(6) | 106 | 0.37 | 0.67 | - | 226
[Co(0.2)/Ni(0.6)]2/Co(0.2) | Pt(1.2) | Ta(0.8)/TaN(6) | n.a. | 0.52 | 0.94 | - | 167
Ir(1.2) | Ta(0.8)/TaN(6) | n.a. | 0.07 | 0.13 | -
Pt(2.5) | Ir(2.5) | n.a. | 0.31 | 0.56 | -
Ir(2.5) | Pt(2.5) | n.a. | 0.21 | 0.38 | +
Table 5: Overview of DMI measurements for CoFeB thin films via field-induced domain wall motion experiments in the creep regime. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{\mathrm{DMI}}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (pJ/m) |  |
Co20Fe60B20(0.8) | Ta(5) | MgO(2) | 6.2 – 16.3111Different DMI values correspond to different annealing temperatures. | 0.02 – 0.06111Different DMI values correspond to different annealing temperatures. | 0.02 – 0.05111Different DMI values correspond to different annealing temperatures. | + | 138
CoFeB(1) | W(2) | MgO(2) | 35 | 0.23 | 0.23 | + | 261
W(3) | 15 | 0.12 | 0.12 | +
TaN(1) | 5 | 0.05 | 0.05 | +
Hf(1) | 2 | 0.01 | 0.01 | +
Co20Fe60B20(0.6) | W(5) | MgO(2) | 93 | 0.68 | 0.41 | + | 124
Co40Fe40B20(0.6) | MgO(2)/Ta(5) | 4 | 0.03 | 0.02 | +
Co20Fe60B20(0.8) | Ta(5) | MgO(2) | 8.8 | 0.03 | 0.02 | + | 134
Co20Fe60B20(1) | Ta(5) | MgO(2) | 2.6 – 16222Different DMI values correspond to different doses of He+ ion irradiation. | 0.02 – 0.08222Different DMI values correspond to different doses of He+ ion irradiation. |  0.02 – 0.08222Different DMI values correspond to different doses of He+ ion irradiation., | + | 77
Table 6: Overview of DMI measurements for Co thin films via field-induced domain wall motion experiments in the creep regime. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{\mathrm{DMI}}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (pJ/m) |  |
Co(0.3) | Pt(2.5) | Pt(1.5) | 26.5 | 0.11 | 0.03 | + | 125
Co(0.7) | Pt(5) | Pt(3) | 104 | 0.83 | 0.58 | + | 121
Ir(0.23)/Pt(3) | 10 | 0.08 | 0.06 | +
Ir(0.69)/Pt(3) | 155 | 1.23 | 0.86 | -
Co(0.8) | Pt(10) | Pt(3) | 50 | n.a. | n.a. | n.a. | 228
Co(0.4) | Pt(2) | Pt(2) | 83 | 0.33 | 0.13 | + | 142
Pt(2) | Pd(2) |  $>$200 | n.a. | n.a. | n.a.
Co(0.6) | Pt(3) | AlOx(1.6) | 138 | n.a. | n.a. | n.a. | 147
MgO(2) | 483 | n.a. | n.a. | n.a.
Co(0.54) | Au(4) | NiO(10) | n.a. | 2.04 | 1.11 | + | 166
Co(0.56) | Pt(4) | Ir(5) | 53 | 0.31 | 0.17 | - | 256
Co(1.05) | 8.5 | 0.12 | 0.13 | -
Co(1) | Pt(3) | MgO(0.65)/Pt(5) | 65.8 | 0.77 | 0.77 | n.a. | 44
Mg(0.2)/MgO(1.5-2)/Pt(5) | 311 | 2.32 | 2.32 | n.a.
Co(1.8) | Pt(5) | W(1)/Pt(1) | 25 | 0.19 | 0.34 | n.a. | 180
Co(0.6) | Pt(3) | Pt(3) | 7.8 | 0.07 | 0.04 | - | 253
Pt50Au50(3) | 48 | 0.35 | 0.21 | -
Au(3) | 175 | 1 | 0.6 | -
Co(0.8) | Pt(2.2) | Ta(4) | 140111For the Pt/Co/Ta system, the value provided in the table is obtained considering $H_{\mathrm{DMI}}$ to be the minimum in the velocity curves. However the work estimates the DMI also using a model that assumes $H_{\mathrm{dep}}$ = $H_{\mathrm{dep}}(H_{x})$. In this case $D=$ -2 mJ/m2 and $D_{\mathrm{s}}=$ -1.6 pJ/m. | 1.12111For the Pt/Co/Ta system, the value provided in the table is obtained considering $H_{\mathrm{DMI}}$ to be the minimum in the velocity curves. However the work estimates the DMI also using a model that assumes $H_{\mathrm{dep}}$ = $H_{\mathrm{dep}}(H_{x})$. In this case $D=$ -2 mJ/m2 and $D_{\mathrm{s}}=$ -1.6 pJ/m. | 0.9111For the Pt/Co/Ta system, the value provided in the table is obtained considering $H_{\mathrm{DMI}}$ to be the minimum in the velocity curves. However the work estimates the DMI also using a model that assumes $H_{\mathrm{dep}}$ = $H_{\mathrm{dep}}(H_{x})$. In this case $D=$ -2 mJ/m2 and $D_{\mathrm{s}}=$ -1.6 pJ/m. | - | 254
Ir(0.2 – 2)/Ta(4) | 64.2 – 104.2222The highest and lowest DMI values reported do not necessarily correspond to the extremes of the Ir thickness range. | 0.49 – 0.93222The highest and lowest DMI values reported do not necessarily correspond to the extremes of the Ir thickness range. | 0.39 – 0.74222The highest and lowest DMI values reported do not necessarily correspond to the extremes of the Ir thickness range. | -
Co(0.6) | Ta(4)/Pt(4) | Pt(4) | 170 | 1.20 | 0.71 | -333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | 110
Co(0.8–1.2) | Gd(3)/Pt(2) | 217–77 | 0.43–0.23 | 0.34 | -333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range.
Co(0.8–1) | Ir(4) | 156–92 | 0.39–0.23 | 0.23 | -333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range.
Co(1) | Ta(4)/Pt(4) | Gd(3)/Pt(4) | 280 | 0.37 | 0.37 | n.a. | 43
Co(1) | Gd(3)/Ta(4) | 255 | 0.24 | 0.24 | n.a.
Co(1)/Gd(3)/Co(1) | Ta(4) | 138 | 0.90 | 0.45 | n.a.
Co(0.9) | Pt(5) | Pt(5) | 0 | 0 | 0 | n.a | 100
Pt(5) | Au(5) | 105 | 0.87 | 0.78 | -
Au(5) | Pt(5) | 78 | 0.59 | 0.53 | +

Table 7: Overview of DMI measurements for Co thin films via field-induced domain wall motion experiments in the flow regime. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{\mathrm{DMI}}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. The subscripts 1 and 2 indicate values of $D$ and $D_{s}$ extracted from $H_{DMI}$ and $v_{sat}$, respectively. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D_{1}}$ | $\mathbf{D_{2}}$ | $\mathbf{D_{s1}}$ | $\mathbf{D_{s2}}$ | Sign | Ref
---|---|---|---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (mJ/m2) | (pJ/m) | (pJ/m) |  |
Co(1) | Pt(5) | Gd(5) | 180 | 1.48 | n.a. | 1.48 | n.a. | - | 280
Co(0.8) | Pt(4) | AlOx(3) | 220 | 1.63 / 1.91111Sign convention used here is opposite to the used in the article. | n.a. | 1.3 / 1.53111The two values of DMI reported derive from two different values for the exchange constant $A$. | n.a. | - | 229
Co(1) | GdOx(4) | 200 | 1.48 / 1.73111The two values of DMI reported derive from two different values for the exchange constant $A$. | n.a. | 1.48 / 1.73111The two values of DMI reported derive from two different values for the exchange constant $A$. | n.a. | -
Co(1) | Gd(3) | 300 | 1.52 / 1.78111The two values of DMI reported derive from two different values for the exchange constant $A$. | n.a. | 1.52 / 1.78111The two values of DMI reported derive from two different values for the exchange constant $A$. | n.a. | -
Co(1) | Pt(4) | 0 | 0 | n.a. | 0 | n.a. |
Co(0.6) | Pt(2) | Al(2) | 200 / 212 | 1.52 / 2.02222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | 1.38 / 2.2222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | 0.91 / 1.21222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | 0.83 / 1.32222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | - | 3
Pt(2) | Ir(2) | 96 / 125 | 0.5 / 0.92 222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | 0.37 / 0.67222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | 0.3 / 0.55222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | 0.22 / 0.4222The two values of DMI reported are due to different growth temperatures for the Co layer, either room temperature or $100^{\circ}$C. | -
Pt(2) | Cu(2) | 200 | 0.93 | 1.03 | 0.56 | 0.62 | -
Pt(2) | Pt(2) | 0 | 0 | 0 | 0 | 0 |
Ir(2) | Pt(2) | 106 | 1.08 | 1.08 | 0.65 | 0.65 | +
Co(0.8) | Pt(30) | AlOx(1-3) | n.a. | n.a. | 1 – 1.79333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | n.a. | 0.8 – 1.43333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | - | 263
Co(1) | Pt(4) | GdOx(1-3) | 150 – $>$260333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | 0.6 – 1.42333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | 0.62 – 1.34333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | 0.6 – 1.42333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | 0.62 – 1.34 333The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range. | -
Co(1) | Pt(4) | Gd(4) | n.a. | n.a. | 1.45 | n.a. | 1.45 | - | 163
Co(1) | GdOx(4) | n.a. | n.a. | 1.5 | n.a. | 1.5 | -
GdCo(4) | Ta | n.a. | n.a. | 0.2 | n.a. | 0.8 | -
GdCo(4.8) | Ta | n.a. | n.a. | 0.2 | n.a. | 0.96 | -

####  II.3.3 Equilibrium stripe domain pattern

In the case of the equilibrium stripe domain pattern method, the DMI constant
is estimated by measuring magnetic domain widths from nanometric magnetic
microscopy images. The values and systems analyzed using this method are
presented in Table 8 for CoFeB and Ni/Fe and in Table 9 for Co and FeCo thin
films. As the uncertainty of the measurement depends heavily on the precision
of the imaging techniques, submicrometer range resolution is indispensable.
The most commonly used experimental techniques are: Scanning Transmission
X-ray Microscopy (STXM) [216, 178], Magnetic Transmission X-ray Microscopy
(MTXM) [289, 290], Magnetic Force Microscopy (MFM) [40, 262, 248, 172, 8, 47,
71, 82, 1, 162], and in a few cases MOKE [300, 288, 249]. Most of the
materials analyzed with this method have large DMI values [216, 289, 262]. For
example, Soumyanarayanan  _et al._ [262] reported that the replacement of the
Co layer by a layer of FeCo increases substantially the value of $D$, up to a
maximum of $D_{s}=$ 2.18 pJ/m for Fe(0.6)/Co(0.6). The maximum of $D$
corresponds to $D=$ 2.1 mJ/m2 for a thinner magnetic layer Fe(0.4)/Co(0.4).

Many of the analysed films are multilayers because the additional dipolar
contribution between layers stabilizes skyrmions and increases the sizes of
stable skyrmions [216]. Legrand  _et al._ [172] studied the variation of the
DW internal structure along the multilayer thickness, due to the competition
of dipolar field and DMI in Co multilayers. The domain walls in such
structures have hybrid character between Bloch and Néel types with different
Néel chiralities at bottom and top surfaces. Albeit being hybrid domain walls,
the DMI can be extracted comparing the measured domain width with the one
obtained using micromagnetic simulations. With the same technique symmetric
Pd/Co/Pd samples were also analyzed [71, 82, 162]. The origin of the DMI in
such symmetric trilayers is a different residual stress in the top and bottom
of the Co/Pd interfaces due to different lattice matching [71]. Dugato  _et
al._ [82] were able to optimize the stack with the insertion of W between Co
and Pd, resulting in an increase of $D$, with an optimal size for a W
thickness of 0.2 nm and $D_{s}=$ 0.65 pJ/m ($D=$ 1.3 mJ/m2). Kozlov  _et al._
[162] found a strong variation of the estimated $D$ value, when values of the
exchange constant in the range $A=23$-30 pJ/m were used in the numerical
model. The range of DMI for different exchange constants increases with the
increase of the Co thickness.

Wedges of Ni on top a Fe layer with a Cu(001) substrate were analyzed with
Threshold Photoemission Magnetic Circular Dichroism with PhotoEmission
Electron Microscopy (TP-MCD-PEEM) by Meier  _et al._ [203]. In these samples,
the DMI originates at the Ni/Fe interface due to the lack of inversion
symmetry, but the obtained $D$ values are small because of the lack of a
source of strong SOC. $D$ increases when those bilayers are capped with a Pt
layer. Using this method, the dependence of DMI on temperature was also
measured for Pt/Co/Cu multilayers up to a temperature of 500 K [248]. It was
found that DMI has a stronger dependence on temperature than other magnetic
properties, like magnetocrystalline anisotropy. Polar MOKE was used to measure
the variation of $D$ induced by electric fields in Pt(3 nm)/Co(0.49 nm)/AlOx(6
nm) thin films [249]. The electric field also produces variations of the
magnetization and anisotropy constant, which need to be quantified to obtain
the DMI energy. $D$ was measured only under electric field and the variation
of $D$ measured for an electric field of 133 MV/m was 0.14 mJ/m2 and 0.26
mJ/m2 assuming an exchange constant of 7.5 pJ/m and 16 pJ/m, respectively.

Table 8: Overview of DMI measurements for CoFeB and Ni/Fe thin films via equilibrium stripe domain pattern. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mathbf{D}$$\mid$ |  $\mid$$\mathbf{D_{s}}$$\mid$ | Ref
---|---|---|---|---|---
(nm) | (nm) | (nm) | (mJ/m2) | (pJ/m) |
[Pt(3)/Co40Fe40B20(0.8)/MgO(1.5)]×20 | Ta(3) | Ta(2) | 1.66 | 1.33 | 290
[Pt(2.7)/Co60Fe20B20(0.8)/MgO(1.5)]×15 | Ta(2.3)/Pt(3.7) |  | 1.5 | 1.2 | 40
Co40Fe40B20(1.2) | Ta(5) | TaOx(5) | 0.17 | 0.2 | 300
Ni(6–12 ML)/Fe(1–3 ML) | (001)Cu |  | n.a. | 0.28 $\pm$ 0.14 | 203
Ni(9 ML)/Fe(1-–3 ML)/Ni(4-–14 ML) |  | n.a. | 0
Fe(1 ML)/Ni(6-–12 ML) |  | n.a. | 0.38 $\pm$ 0.14
Fe(1 ML)/Ni(6-–12 ML) | Pt(0.4) | n.a. | 0.6 $\pm$ 0.2
Co20Fe60B20(1.2) | MgO(1) | Ta(5) | 0.65 $\pm$ 0.08 | 0.78 | 288
[Pt(2.7)/CoFeB(0.86)MgO(1.5)]x15 | Ta(3.6)/Pt(1) | Pt(2.7) | 1.76 | 1.51 | 178
$[$MgO(2)/Co20Fe60B20(1)/Ta(5)$]_{\times 15}$ |  | Ta(5) | 0.08 $\pm$ 0.03 | 0.08$\pm$0.03 | 47
$[$MgO(2)/Co20Fe60B20(0.6)/W(5)$]_{\times 15}$ |  | Ta(5) | 0.61$\pm$0.03 | 0.37$\pm$0.02
$[$MgO(1.4)/Co60Fe20B20(0.8)/Pt(3.4)$]_{\times 15}$ | Ta(5.7) | Ta(5) | 1.0$\pm$0.1 | 0.80$\pm$0.08
[Pt(2.5-–7.5)/Co60Fe20B20(0.8)/MgO(1.5)]×13 | Ta(3) | Ta(2) | 1.6 $\pm$ 0.2 | 1.28 | 1

####  II.3.4 Magnetic stripe annihilation

With this method, the first step is to nucleate two domain walls in a
perpendicularly magnetized sample. The two walls are then manipulated with an
out-of-plane field to minimize their distance until the two domain walls
collapse. This annihilation field and the minimum width of the domain
separating the walls before its collapse depend on $D$. The values of $D$
extracted using this method are shown in Table 10. As for the equilibrium
stripe domain pattern technique (Sec. II.3.3), this method is based on
magnetic imaging. For this purpose, MTXM [124, 182], STXM [289, 290] and MOKE
[299] have been used. All the samples are magnetic thin films or wide
patterned tracks, to allow high number of domains and the manipulation of the
walls. In the original study [25], the nucleated domain walls were parallel
and analysed using a combination of Lorentz transmission electron microscopy
(L-TEM) and polar Kerr. In later studies [124, 289, 299, 290, 182], the
annihilation field was studied independently of the shape and boundary of
domains. Benitez  _et al._ [25] measured a value for $D$ of 0.33 mJ/m2
($D_{s}$= 0.27 pJ/m) in a film Pt/Co/AlOx, but they considered it a lower
limit because of a possible underestimation of the annihilation field due to
thermally activated processes. A DMI value of $D_{s}$= 0.44 pJ/m ($D$=0.73
mJ/m2) was measured in W/Co20Fe60B20/MgO [124], while a smaller value of
$D_{s}$= 0.25 pJ/m ($D$=0.25 mJ/m2) was obtained when the same composition was
sandwiched between Ta layers [299].

Table 9: Overview of DMI measurements for Co and FeCo thin films via equilibrium stripe domain pattern. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. The method does not provide the sign. All the $D$ values are absolute values unless noted. FM | Bottom NM | Top NM | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mJ/m2) | (pJ/m) |  |
Co(0.6)/Pt(1)/$[$Ir(1)/Co(0.6)/Pt(1)$]_{\times 10}$ | Pt(10) | Pt(3) | 1.6$\pm$0.2 | 0.96 | n.a. | 216
Co(0.6)/Pt(1)/$[$Co(0.6)/Pt(1)$]_{\times 10}$ | Pt(10) | Pt(3) | 0.2$\pm$0.2 | 0.12 | n.a.
$[$Pt(3)/Co(0.9)/Ta(4)$]_{\times 15}$ | Ta(3) |  | 1.5$\pm$0.2 | 1.35 | n.a. | 289
$[$Ir(1)/Co(0.6)/Pt(1)$]_{\times 20}$ | Ta(3)/Pt(10) | Pt(2) | 1.67 | 1 | n.a. | 262
$[$Ir(1)/Fe(0.2)/Co(0.6)/Pt(1)$]_{\times 20}$ | 1.8 | 1.44 | n.a.
$[$Ir(1)/Fe(0.3)/Co(0.6)/Pt(1)$]_{\times 20}$ | 1.88 | 1.69 | n.a.
$[$Ir(1)/Fe(0.2)/Co(0.5)/Pt(1)$]_{\times 20}$ | 1.98 | 1.39 | n.a.
$[$Ir(1)/Fe(0.4)/Co(0.4)/Pt(1)$]_{\times 20}$ | 2.1 | 1.68 | n.a.
$[$Ir(1)/Fe(0.4)/Co(0.6)/Pt(1)$]_{\times 20}$ | 1.99 | 1.99 | n.a.
$[$Ir(1)/Fe(0.5)/Co(0.5)/Pt(1)$]_{\times 20}$ | 1.96 | 1.96 | n.a.
$[$Ir(1)/Fe(0.6)/Co(0.6))/Pt(1)$]_{\times 20}$ | 1.82 | 2.18 | n.a.
$[$Ir(1)/Co(0.6)Pt(1)$]_{\times 5}$ | Pt(10) | Pt(3) | 2.30 | 1.38 | +111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article. | 172
$[$Ir(1)/Co(0.8)/Pt(1)$]_{\times 5}$ | Pt(10) | Pt(3) | 2.00 | 1.6 | +111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Co(0.8)/Ir(1)/Pt(1)$]_{\times 5}$ | Pt(11) | Pt(3) | 1.37 | 1.1 | -111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Co(0.8)/Ir(1)/Pt(1)$]_{\times 5}$ | Ta(5)/Pt(10) | Pt(3) | 1.63 | 1.3 | -111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Co(0.8)/Ir(1)/Pt(1)$]_{\times 10}$ | Pt(11) | Pt(3) | 1.52 | 1.22 | -111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Co(0.8)/Ir(1)/Pt(1)$]_{\times 10}$ | Ta(5)/Pt(10) | Pt(3) | 2.06 | 1.65 | -111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
Co(0.8)/$[$Pt(1)/Ir(1)/Co(0.8)$]_{\times 10}$ | Ta(15) | Pt(3) | 1.14 | 0.91 | +111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Pt(1)/Co(0.6)/Al2O3(1)$]_{\times 20}$ | Ta(10)/Pt(7) | Pt(3) | 1.29 | 0.77 | -111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Pt(1)/Co(0.8)/Al2O3(1)$]_{\times 20}$ | Ta(10)/Pt(7) | Pt(3) | 1.01 | 0.81 | -111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Al2O3(1)/Co(0.6)/Pt(1)$]_{\times 20}$ | Ta(10) | Pt(7) | 1.94 | 1.16 | +111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Al2O3(1)/Co(0.8)/Pt(1)$]_{\times 20}$ | Ta(10) | Pt(7) | 1.69 | 1.35 | +111The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in the article.
$[$Pt(2)/Co(1.1)/Cu(1)$]_{x15}$ | Ta(3) | Pt(2) | 1.55 (298 K) | 1.7 | n.a. | 248
0.47 (423 K) | 0.52 | n.a.
Co(0.6)/Pt(1)/$[$Ir(1)/Co(0.6)/Pt(1)$]_{\times 5}$ | Pt(10) | Pt(3) | 1.97$\pm$0.02 | 1.18 | n.a. | 8
$[$Pt(2.5-–7.5)/Co(0.8)/Pt(1.5)$]_{\times 13}$ | Ta(3) | Ta(2) | 0 $\pm$ 0.1 | 0 | n.a. | 1
$[$Co(0.8)/Pd(2)$]_{\times 5}$ | Cu(2)/Pd(3) | Pd(3) | 1.6 $\pm$ 0.35222Values assuming $A=20$ pJ/m. Other values are also assumed in the article. | 1.28222Values assuming $A=20$ pJ/m. Other values are also assumed in the article. | -333The negative sign of DMI is inferred in combination with other experiments. | 71
$[$Co(0.8)/Pd(2)$]_{\times 10}$ | 1.85 $\pm$ 0.45222Values assuming $A=20$ pJ/m. Other values are also assumed in the article. | 1.48222Values assuming $A=20$ pJ/m. Other values are also assumed in the article. | -333The negative sign of DMI is inferred in combination with other experiments.
$[$Co(0.8)/Pd(2)$]_{\times 20}$ | 2.3 $\pm$ 0.5222Values assuming $A=20$ pJ/m. Other values are also assumed in the article. | 1.84222Values assuming $A=20$ pJ/m. Other values are also assumed in the article. | -333The negative sign of DMI is inferred in combination with other experiments.
$[$Pd(1)/Co(0.5)/Pd(1)$]_{\times 15}$ |  0.3 $\pm$ 0.1 | 0.15 | n.a. | 82
$[$Pd(1)/Co(0.5)/W(0.1)/Pd(1)$]_{\times 15}$ |  1.1 $\pm$ 0.2 | 0.55 | n.a.
$[$Pd(1)/Co(0.5)/W(0.2)/Pd(1)$]_{\times 15}$ |  1.3 $\pm$ 0.2 | 0.65 | n.a.
$[$Pd(1)/Co(0.5)/W(0.3)/Pd(1)$]_{\times 15}$ |  0.4 $\pm$ 0.1 | 0.2 | n.a.
$[$Pd(1)/Co(0.5)/W(1)/Pd(1)$]_{\times 15}$ |  0.4 $\pm$ 0.1 | 0.2 | n.a.
Table 10: Overview of DMI measurements of Co and CoFeB thin films via magnetic stripe annihilation. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $D$ is the interfacial DMI constant and $D_{s}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM |  $\mid$$\mathbf{D}$$\mid$ |  $\mid$$\mathbf{D_{s}}$$\mid$ | Ref
---|---|---|---|---|---
(nm) | (nm) | (nm) | (mJ/m2) | (pJ/m) |
Co(0.8) | Pt(3) | AlOx(3) | 0.33 $\pm$ 0.05 111Lower limit |  0.27 111Lower limit | 25
[Pt(3)/Co(0.9)/Ta(4)]×15 | Ta(3) |  | 1.1 $\pm$ 0.2 | 1 | 289
Co20Fe60B20(1) | Ta(5) | Ta(0.74-0.9) | 0.25 | 0.25 | 299
Co20Fe60B20(0.6) | W(5) | MgO(2) | 0.73 $\pm$ 0.5 | 0.44 | 124
[Pt(3)/Co40Fe40B20(0.8)/MgO(1.5)]×20 | Ta(3) | Ta(2) | 1.35 | 1.08 | 290
[Pt(3.2)/CoFeB(0.7)/MgO(1.4)]x15 | Ta(3) |  | 1.35 $\pm$ 0.05 | 0.95 | 182

####  II.3.5 Nucleation field

A reversed domain is nucleated in a perpendicular magnetized material and the
out-of-plane nucleation field as a function of the in-plane applied field is
analyzed. The latter can be particularly large, up to 1 T as in Kim  _et al._
[153]. This type of measurement can be divided in two groups: edge nucleation
in patterned wires [231] or asymmetric microstructures (e.g. triangles) [108]
and bubble nucleation in extended films [153, 154]. In all the experiments,
the magnetic images were obtained using MOKE. In Han  _et al._ [108], the
hysteresis loop of asymmetric microstructures measured by wide-field polar
Kerr shows an asymmetry due to DMI similar to exchange biasing. This asymmetry
is attributed to the asymmetric nucleation when an in-plane field is present
and is independent of the structure size. The values obtained for $D$ are
presented in Table 11, mostly single layers of Co [231, 108, 154] or trilayers
of Co/Ni/Co [153]. Pizzini  _et al._ [231] obtained a value of $D_{s}$=-1.32
pJ/m ($D$=-2.2 mJ/m2) for Pt/Co/AlOx. Han  _et al._ [108] measured the
opposite sign for the reversed structure AlOx/Co/Pt with a value of
$D_{s}$=1.62 pJ/m ($D=$1.43 mJ/m2). When the structure was reversed and the
AlOx was replaced by Ir a value of $D_{s}$=-2.03 pJ/m ($D=$-1.69 mJ/m2) was
measured, indicating opposite contributions for Pt and Ir. This method was
used to obtain the value of $D$ as a function of temperature in a Co layer
[154], obtaining a decrease from 1.18 mJ/m2 at 100 K to 0.48 mJ/m2 at 300 K.

Table 11: Overview of DMI measurements of Co and Co/Ni thin films via nucleation field dependence. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. $H_{\mathrm{DMI}}$ is the DMI field, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. Edge nucleation provides $D$ sign, while bubble nucleation provides only the magnitude. Values with ∗ are according to the convention used in this paper and opposite to that in the original manuscript. FM | Bottom NM | Top NM |  $\mid$$\mu_{0}$HDMI$\mid$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Nucleation type | Ref
---|---|---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mT) | (mJ/m2) | (pJ/m) |  |  |
Co(0.6) | Pt(3) | AlOx(2) |  | 2.2∗ | 1.32∗ | - | Edge | 231
Co(1.15) | AlOx(2.5) | Pt(4) |  | 1.43∗ $\pm$ 0.06 | 1.62${}^{*}\pm$0.07 | + | Edge | 108
Co(1.2) | Pt(4) | Ir(4) |  | 1.69∗ $\pm$ 0.03 | 2.03∗ $\pm$ 0.04 | -
Co(0.3)/Ni(0.6)/Co(0.3) | Pt(2) | MgO(1) | 228$\pm$60 | 0.45 $\pm$ 0.15 | 0.54 | n.a. | Bubble | 153
Co(0.5) | Pt(2) | MgO(2) | 372$\pm$30 | 1.18 | 0.59 | n.a. | Bubble (100K) | 154
324$\pm$15 | 1.03 | 0.52 | n.a. | Bubble (150K)
245$\pm$45 | 0.75 | 0.37 | n.a. | Bubble (200K)
166$\pm$50 | 0.48 | 0.24 | n.a. | Bubble (300K)

####  II.3.6 Domain wall stray fields

In NV magnetometry, the value of $D$ is obtained by measuring the magnetic
stray field generated by a 180∘ Bloch wall in PMA materials [272, 104]. The NV
magnetometer measures the Zeeman shift in the electronic spin sublevels of a
NV defect in a diamond crystal in presence of a small magnetic field. The
diamond nanocrystal is placed on the tip of an AFM (atomic force microscope)
and scanned across the DW at a distance of about 100 nm from the surface (see
Fig. 12). The Zeeman shift is proportional to the projection of the external
magnetic stray field (arising from the domains adjacent to the wall, in the
case of a Bloch wall, and of the domains adjacent to the wall and the wall
itself, in the case of a Neél wall) on the quantization axis of the NV center
($B_{\mathrm{NV}}$). Therefore, Bloch and Neél walls can be easily
distinguished by the Zeeman shift profile perpendicular to the wall. Since the
stray field depends on several parameters such as the distance from the
surface, $M_{\mathrm{S}}$ and the DW width, any error in these values reflects
in an uncertainty of $D$. The main source for uncertainty is usually
considered to arise from $A$, the exchange stiffness, but also from
inhomogeneities in $M_{\mathrm{S}}$ or thickness variations.

Table 12: Overview of DMI measurements of CoFeB thin films by stray field using NV magnetometry. FM and NM stand for ferromagnetic and non-magnetic layer, respectively, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. FM | Bottom NM | Top NM | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---
(nm) | (nm) | (nm) | (mJ/m2) | (pJ/m) |  |
Co40Fe40B20(1) | Ta(5) | MgO(2) | 0$\pm$0.01 | _0 $\pm$0.01_ |  | 104
Co20Fe60B20(1) | TaN0.7%(4) | 0.03$\pm$0.01 | _0.03 $\pm$0.01_ | +
TaN0.7%(1) | 0.06$\pm$0.02 | _0.06 $\pm$0.02_ | +

####  II.3.7 Domain wall internal structure imaging

In Bode  _et al._ [29] the spin-polarized scanning tunneling microscopy (SP-
STM) technique was used to investigate the role of the DMI in systems with
chiral spin structure. The specific chirality of the moments in single atomic
layers of Mn on a W(110) substrate was observed directly and the different
energy contributions of the cycloid structure were calculated by employing
density functional theory with a generalized gradient approximation and full-
potential linearized augmented plane waves. The long-range homogeneous spiral
structure was accounted for by the generalized Bloch theorem adding spin-orbit
coupling as a perturbation. The SP-STM experiments (see scheme in Fig.15) were
performed in ultrahigh vacuum at $T=13\pm 1$ K in constant current mode. For
spin resolved measurements Cr- and Fe-covered W tips were used. The tunneling
current $I$ is sensitive to the relative orientation of the magnetization of
the tip ($\mathbf{M}_{\mathrm{T}}$) and the sample
($\mathbf{M}_{\mathrm{S}}$), according to
$I=I_{0}+I_{\mathrm{sp}}\mathbf{M}_{\mathrm{T}}\cdot\mathbf{M}_{\mathrm{S}}$,
where $I_{\mathrm{sp}}$ is the spin polarized part of $I$. This allows for an
atomic scale imaging of magnetic nanostructures [113]. This work was continued
in Ferriani  _et al._ [90], yielding a nearest-neighbor DMI of 4.6 meV, and
$D_{\mathrm{s}}=2.33$ pJ/m with an approximate lattice constant of that of W
(3.165 $\AA$). In Heinze  _et al._ [115] Fe monolayers on Ir(111) were
studied, showing similar values. Double layers of Mn were studied in Yoshida
_et al._ [297] and Fe islands on Cu(111) in Fischer  _et al._ [94]. Similar
studies exploiting SP-STM were also presented by Meckler  _et al._ [202] where
they found that the Dzyaloshinskii-Moriya interaction is too weak to
destabilize the single domain state, but it can deﬁne the sense of rotation
and the type of the cycloidal spiral once the single domain state is
destabilized by dipolar interaction. More recent SP-STM studies are found in
Hsu  _et al._ [122] where the effect of loading double layers of Fe is
investigated, in Perini  _et al._ [227] epitaxial films of Co/Ir(111) and
Pt/Co/Ir(111) are studied making use of density functional theory (DFT) and
shows that in the latter DMI comes almost only from the interface with Pt, in
Romming  _et al._ [236] Rh/Fe atomic bilayers on Ir(111) were studied showing
that higher-order exchange interactions may compete with interfacial DMI.

Figure 15: Schematics of spin-polarized scanning tunneling microscope (SP-STM)
using a magnetically coated scanning tunneling microscope tip: (a) geometry of
the experimental set-up including the magnetization axes of tip
$\mathbf{M}_{\mathrm{T}}$ and sample $\mathbf{M}_{\mathrm{S}}$. (b) electronic
structure of tip and sample. In the case of a perfect parallel alignment of
$\mathbf{M}_{\mathrm{T}}$ and $\mathbf{M}_{\mathrm{S}}$, only majority
electrons can tunnel into unoccupied majority states and only minority
electrons into unoccupied minority states. From 114.

In addition to the above results obtained by SP-STM, also the SPLEEM technique
has yielded a number of quantitative studies of DMI in layered systems. In
this experimental approach, spin-polarized, low-energy electrons are projected
towards the sample surface through an illumination column and reach the sample
at normal incidence [238]. A magnetic beam splitter is exploited to separate
the incoming electrons from the backscattered ones and a magniﬁed image of the
(magnetic) surface is obtained by passing the backscattered beam through an
imaging column, similar to that found in electron microscopes (a schematic is
shown in Fig.16). This technique is extremely surface-sensitive, since it
employs electrons of a few eV energy that do not penetrate more than a few
atomic layers, while the lateral resolution is of the order of several
nanometers. Therefore, SPLEEM requires in-situ analysis of samples prepared in
ultra-high vacuum and it is usually not suitable for samples prepared
elsewhere. By a careful comparison of SPLEEM images on films of different
thicknesses, the film thickness dependent transition from chiral Néel walls
(dominated by the DMI) and non-chiral Bloch walls (dominated by the stray
field) can be directly analyzed, so that one can quantify the DMI relying on
the fact that the DMI energy and the stray field energy are comparable near
the transition thickness. This is because the stray field energy will increase
with increasing film thickness whereas the DMI energy is usually independent
of the film thickness. Thus, the DMI strength can be quantified by calculating
the stray field energy for a wall structure with a given film thickness. This
approach has been followed in a number of quantitative studies that are
collected in Table13.

Figure 16: Schematics of a SPLEEM microscope. Spin-polarized electrons,
photoemitted from a GaAs photocathode, are injected into a spin manipulator
where azimuthal and polar orientation of the polarization is adjusted. Then,
the electron beam passes through an illumination column, before being
decelerated in the objective lens. Electrons finally hit the surface with
normal incidence. Electrons that are backscattered elastically are collected
in an imaging column and focused on a phosphorous screen, where a magnified
image of the surface is obtained. The incoming and reflected electron beams
are separated in a magnetic beam splitter using the Lorentz force. From
Rougemaille and Schmid [238].

As for Lorentz-TEM microscopy, it was exploited to derive the value of the DMI
[99] looking at the mixed Bloch-Néel chiral spin textures in Co/Pd
multilayers. An analysis of the observed intensities under varied imaging
conditions coupled to corroborative micromagnetic simulations permitted to
quantify different magnetic parameters, including the domain wall width, the
exchange stiffness and the DMI. Finally, also SEMPA was used to quantify the
DMI in thin film systems. For instance, in Corredor  _et al._ [66], using the
experimentally determined wall angle and width parameter, a span for the DMI
strength of 0.8 meV/Co$<d<$4.3 meV/Co has been derived, which was in agreement
with previous ab initio calculations for this system. Also, in a recent SEMPA
study of Fe3GeTe2 films [205] a lower bound of $D>$0.09-0.2 mJ/m2 for the DMI
term, films was achieved. A more refined and quantitative result was achieved
by [156], studying the domain wall profiles of Co/Ir(111) films as a function
of Co thickness via SEMPA. They found that below a cobalt thickness of 8.8
MLs, the magnetic domain walls are purely Néel oriented and show a clockwise
sense of rotation. For larger thicknesses the plane of rotation changes and
the domain walls show a signiﬁcant Bloch-like contribution, allowing one to
calculate the strength of the Dzyaloshinskii-Moriya interaction (DMI) from
energy minimization. In particular, from the angle between the plane of
rotation and the domain-wall normal an interfacial DMI parameter $D_{s}=-$1.07
$\pm$ 0.05 pJ/m was determined.

The pioneering approach of these methods was among the first allowing an
experimental determination of the DMI and observing directly its influence on
the magnetic structure, and therefore the area remained very active and
substantial progress towards a quantitative determination of the DMI was
achieved (see Table13).

Table 13: Overview of DMI measurements by Domain wall internal structure imaging. The measurements in Chen  _et al._ [54, 52, 53, 55], Yang  _et al._ [295] were performed by SPLEEM, Meckler  _et al._ [202] by SP-STM, Garlow  _et al._ [99] by L-TEM and Kloodt-Twesten  _et al._ [156] by SEMPA. FM and NM stand for ferromagnetic and non-magnetic layer, respectively, $D^{*}$ is the DMI value given in meV/atom, $D$ is the interfacial DMI constant and $D_{\mathrm{s}}=D\cdot d$, with $d$ being the thickness of the ferromagnetic film. Thicknesses are given either in nm or atomic monolayers (ML). Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. Signs with ∗ are according to the convention used in this paper and opposite to that in the original manuscript. _n.a._ means that the value was not indicated in the original paper nor were parameters available to calculate it. FM | Bottom NM | Top NM | $\mathbf{D^{*}}$ | $\mathbf{D}$ | $\mathbf{D_{s}}$ | Sign | Ref
---|---|---|---|---|---|---|---
|  |  | (meV/atom) | (mJ/m2) | (pJ/m) |  |
Fe(1.7ML) | (110)W |  |  |  | 1.2 | +∗ | 202
Ni(2ML)/Fe(2.5ML) | (001)Cu |  | 0.12-0.17 | 0.08-0.12 |  | - |  52111The values for $D$ given in mJ/m2 were kindly provided by the authors of the original paper. In some cases the sign or thickness values were corrected with respect to the original work.
Ni(2ML)/[Co(1ML)/Ni(2ML)]2 | (111)Pt |  | 1.05 | 0.44 |  | - | 5411footnotemark: 1
(111)Ir | 0.12 | 0.14 |  | +
(111)Pt/Ir(0.6-3ML) | 0.47-0.12 | 0.28-0.14 |  | from - to +
Ni(15ML)/Fe(3.3ML) | (110)W |  | 0.53 | 0.32 |  | + |  5311footnotemark: 1
Co(2.4ML) | (0001)Ru | - | 0.05 | 0.18 |  | + | 29511footnotemark: 1
Co(3.9ML-8.4ML) | Graphene | 0.11 | 0.25 |  | -
Co(0.7nm)[Pd(0.5nm)/Co(0.7nm)]10 | MgO(2nm)/Pt(4nm) | Pt(4nm) |  | 1.05 |  | n.a. | 99
Co(3.5ML-9.7ML) | (111)Ir |  |  |  | 1.07 | +∗ | 156
Co(3ML)/Ni(2ML)/[Co(1ML)/Ni(2ML)]8 | (111)Pd | - | 1.44 | 0.54 |  | - | 5511footnotemark: 1
Co(3ML)/Ni(1ML) | (110)W/Pd(2.6ML) | O(0.2ML) | 0.63 | 0.4 |  | +
Co(3ML)/Ni(1ML) | (110)W/Pd(2.1ML) | H(0.6ML) | 0.01 | 0.01 |  | + |  5611footnotemark: 1

###  II.4 Advantages and limitations

In general, methods based on domain walls require an accurate estimation of
several magnetic parameters in order to evaluate the interfacial DMI (see. Eq.
2). Among these, the exchange stiffness $A$, which enters both in the
expression of the DMI field and in the DW energy, is notoriously difficult to
measure for ultrathin films leading to an uncertainty in the quantification of
$D$. Han  _et al._ [108] pointed out that in the nucleation method a variation
of $A$ from 5 to 15 pJ/m for Co results in a 25% difference for $D$ obtained
using the intermediate value 10 pJ/m. This is not the case for methods based
on spin waves (see Sec. III), where $A$ is not needed to extract the
interfacial DMI. Furthermore, it is worth mentioning that methods that rely on
domain walls can be applied to measure $D$ only in perpendicularly magnetized
materials, while methods based on spin waves (see Sec. III) are able to
quantify $D$ also for systems with in-plane magnetization. Below we review the
main strengths and weaknesses of all methods discussed in this Section.

Both, measurements of current-driven and field-driven DW dynamics in presence
of an in-plane magnetic field allow quantifying not only the magnitude of $D$
but also its sign, which gives information on the DW chirality. On the other
hand, in the creep regime, the dependence of the DW velocity on the in-plane
field does not always follow the simple linear dependence when driven by
current (see Fig. 4), or the symmetric shape with a well defined minimum in
the field-driven case (see Fig. 7). Deviations from these simple theoretical
predictions are in fact quite common (see Fig. 8 as an example) and there is
no consensus on a general model able to predict and fit all the observed
cases. Several models predict that the minimum velocity does not correspond to
the DMI field but agreement on its calculation is still missing, and can lead
to quite different determination of the $D$ value. The main reason is that a
full and general microscopic description of the domain wall dynamics is still
under investigation. While some systems seem to follow the quenched Edward-
Wilkinson equation [4, 103]), there are also experimental evidences that other
system are better described by a quenched-KPZ equation [84], having different
exponents and critical dynamics. The consequences on the DMI estimation of
these two universality classes is still unknown.

Current-driven DW motion is overall a more complex technique to determine the
interfacial DMI, both for experimental realization and for the interpretation
of the results. A standard lithography process is required to pattern films
into wires of appropriate geometry and high in-plane fields, especially for
samples with large DMI, need to be applied and properly aligned. Furthermore,
the torque generated by large applied currents can also result in a tilt of
the DW [87]. Concerning the interpretation of the results, a clear definition
of $H_{DMI}$ might be hard due to the interplay present in complex systems
between different current-induced spin torques (STT, SHE-SOT, iSGE torque),
whose collective effect on DW dynamics is still not fully understood.

Field-driven DW motion has the advantage of being a relatively simple method
to implement experimentally, as it allows the measurement of interfacial DMI
in continuous films, without the need for any lithographic patterning.
However, the application of high in-plane fields, particularly for samples
with significant DMI, requires a perfect alignment of the in-plane field, with
no out-of-plane components. Indeed, in the creep regime any unwanted component
of the field perpendicular to the sample plane would influence the DW velocity
in an exponential manner (see Eq. 6). Several precautions should be taken into
account to achieve an optimal alignment of the in-plane field [125, 169, 261,
44]. A perpendicular stray field component may still arise in MOKE set-ups,
due to a cross-talk between in-plane and perpendicular electromagnets or due
to not completely non-magnetic objectives, which would alter the field
distribution in a way difficult to take into account.

Regarding measurements of DW velocities in the flow regime, here the main
difficulty lies in the generation of large and short magnetic field pulses and
the measurement of very large velocities. The determination of the DMI
constant from the Walker DW velocity (or the saturation velocity after the
Walker field) does not need the knowledge of the exchange constant (see Eq.
11), but on the other hand, may not be accessible when hidden in the creep
regime.

Among the methods that are based on the DW energy to evaluate the interfacial
DMI, imaging the equilibrium stripe domain pattern is particularly
straightforward due to its experimental simplicity. Indeed, this method is
compatible with any magnetic imaging technique like Kerr microscopy or MFM –
which has to be chosen according to the expected domain width – and does not
require the application of perpendicular nor in-plane fields. On the other
hand, imaging the equilibrium domain configuration does not provide
information on the sign of $D$. For multilayer samples with large DMI, small
$A$ or any situation that yields small values of the left hand side of Eq. 13,
high resolution imaging is needed in order to resolve precisely the small
domain widths. Experimentally, several problems can emerge that may render it
unusable (see Supplementary material of 187): for instance, the
demagnetization procedure is unavoidable, since the as-deposited samples may
not be in the ground state. In patterned systems, the domain wall width, the
domain size and configuration may depend on the geometry, due to confinement
and dipolar effects, adding discrepancies with respect to analytical
formulations. Other disadvantages include the fact that the simple analytical
model to extract $D$ may not work for too thick samples which yield neither
Néel nor Bloch wall types (hybrid wall with Néel caps and Bloch core),
although the DMI is expected to be low in these systems.

From the experimental point of view, the domain widths measured may be
different for samples demagnetized with in-plane or out-of-plane fields, up to
20% [172, 71, 162]. For this reason, the in-plane demagnetized configuration
is preferred, as it is closer to the parallel stripes described by the
analytical theory. In principle, these problems may be solved by using the
proper analytical model. The analytical estimations are derived for perfect
parallel stripe domains, while the real domain configuration consists of
rounded meandering structures. Moreover, refined expressions for the DW energy
may yield slightly different values of DMI [177]. This disagreement in the
modeling complicates the extraction of DMI and the interpretation of the
experimental results.

Similar considerations hold for the method based on magnetic stripe
annihilation. Here too, it is only possible to extract information on the
absolute value of $D$, and not its sign. From an experimental point of view,
the original application of this technique requires the presence of parallel
DWs, which may not apply to the case of materials with irregular wall
profiles. Those irregular walls can appear for materials that can host
skyrmion phases, due to the intrinsic faceting of the DWs in such films. This
fact can lead to a deviation from the analytical theory, which was derived for
parallel stripe domains. Another drawback is that as the domain walls are
closer together the corresponding energy barrier for thermally induced domain
wall annihilation is reduced and is also affected by the presence of defects,
which can yield an error in the determination of the DMI value.

Since it relies on the nucleation of a reversed domain, the process of bubble
nucleation to extract the interfacial DMI is intrinsically a statistical
process and several repetitions of the same experiment need to be carried out
to achieve a reliable measurement of the nucleation field. In contrast, for
instance, imaging the equilibrium domain configuration in a film can
automatically provide averaged information of the domain width, from which the
DMI is quantified, if a large enough area is imaged. Bubble nucleation
experiments can be performed both at the edge of patterned structures, such as
nucleation pads or wires, as well as in continuous films. In the former case,
the method can be used to determine both the absolute value of $D$ as well as
its sign. Specifically, the sign of $D$ is related to the nucleated bubble
position with respect to the direction of the applied in-plane field. On the
other hand, bubble nucleation measurements in continuous films provide only
the absolute value of $D$.

Concerning the methods based on domain and domain wall imaging, it has been
shown on different magnetic systems (see Table13) that SP-STM and SPLEEM (and,
to a lesser extent, SEMPA and L-TEM) can be used to achieve a quantitative
estimation of the DMI strength, looking at the domain structure or at the
transition from Néel to Bloch wall as a function of thickness. One should
consider that SP-STM and SPLEEM require a complex apparatus with ultra-high
vacuum and in-situ operation, since they are sensitive to the first atomic
layers, so that they cannot be employed for samples grown elsewhere and the
sample has to be perfectly clean. Use of DFT or Monte Carlo simulations may
help in refining the extraction of DMI, but requires a remarkable
computational effort. However, the main advantage of these methods is the
direct determination of the structure and chirality of the DW.

##  III Spin wave methods

###  III.1 Method overview