# Equilibrium properties of the lattice system with SALR interaction potential
on a square lattice: quasi-chemical approximation versus Monte Carlo
simulation

Ya.G. Groda${}^{\textsf{\footnotesize{\mbox{?}}}}$, V.S.
Vikhrenko${}^{\textsf{\footnotesize{\mbox{?}}}}$, D. di
Caprio${}^{\textsf{\footnotesize{\mbox{?}}}}$

(Received November 2, 2018, in final form November 12, 2018)

###### Abstract

The lattice system with competing interactions that models biological objects
(colloids, ensembles of protein molecules, etc.) is considered. This system is
the lattice fluid on a square lattice with attractive interaction between
nearest neighbours and repulsive interaction between next-next-nearest
neighbours. The geometric order parameter is introduced for describing the
ordered phases in this system. The critical value of the order parameter is
estimated and the phase diagram of the system is constructed. The simple
quasi-chemical approximation (QChA) is proposed for the system under
consideration. The data of Monte Carlo simulation of equilibrium properties of
the model are compared with the results of QChA. It is shown that QChA
provides reasonable semiquantitative results for the systems studied and can
be used as the basis for next order approximations.

Key words: lattice fluid model, competing interaction, order-disorder phase
transition, Monte Carlo simulation, quasi-chemical approximation, phase
diagram

PACS: 05.10.Ln, 64.60.De, 64.60.Cn

###### Abstract

Ðîçãëÿäàòüñÿ ðàòêîâà ñèñòåìà ç êîíêóðåíòíèìè âçàìîäÿìè, ùî ìîäåëþ áîëîãí
îá’êòè (êîëî¿äè, àíñàìáë ïðîòå¿íîâèõ ìîëåêóë ò.ä.). Öÿ ñèñòåìà ïðåäñòàâëÿòüñÿ
ðàòêîâèì ïëèíîì íà êâàäðàòíé ðàòö ç ïðèòÿãàëüíîþ âçàìîäþ ìæ íàéáëèæèìè ñóñäàìè
âäøòîâõóâàëüíîþ âçàìîäþ ìæ íàñòóïíèìè çà íàñòóïíèìè äî íàéáëèæèõ ñóñäâ. Äëÿ
îïèñó âïîðÿäêîâàíî¿ ôàçè â òàêé ñèñòåì ââîäèòüñÿ ãåîìåòðèíèé ïàðàìåòð ïîðÿäêó.
Çðîáëåíà îöíêà êðèòèíîãî çíàåííÿ ïàðàìåòðà ïîðÿäêó ïîáóäîâàíà ôàçîâà äàãðàìà
ñèñòåìè. Äëÿ ñèñòåìè, ùî ðîçãëÿäàòüñÿ, çàïðîïîíîâàíî ïðîñòå êâàçõìíå
íàáëèæåííÿ (ÊÕÍ). Äàí ñèìóëÿöé Ìîíòå Êàðëî ðâíîâàæíèõ âëàñòèâîñòåé ìîäåë
ïîðâíþþòüñÿ ç ðåçóëüòàòàìè ÊÕÍ. Ïîêàçàíî, ùî ÊÕÍ çàáåçïåó ïðèéíÿòí íàïâêëüêñí
ðåçóëüòàòè äëÿ ñèñòåìè, ùî âèâàòüñÿ, ìîæå âèêîðèñòîâóâàòèñÿ ÿê áàçèñ äëÿ
íàñòóïíèõ íàáëèæåíü.

Ключов слова: ðàòêîâà ìîäåëü ïëèíó, êîíêóðóþà âçàìîäÿ, ôàçîâèé ïåðåõä ïîðÿäîê-
áåçëàä, ñèìóëÿö¿ Ìîíòå Êàðëî, êâàçõìíå íàáëèæåííÿ, ôàçîâà äàãðàìà

##  1 Introduction

At present, there is a great interest in studying the processes of self-
organization and self-assembly in systems of a nanoscale range. As the
elements of such systems are supramolecular formations with a sufficiently
large molecular mass, this leads to low velocities of their thermal motion and
to large, on the molecular scale, characteristic times of the processes within
the system. At the same time, the interaction between these elements is very
complex. Despite their rather large dimensions, the interactions remain of the
same order as the thermal energy. This leads to a large variety of
possibilities for various phase transitions in such systems at room
temperature. Examples are solutions of protein molecules [1], clays and soil
suspensions [2], ecosystems [3], etc.

In general, structure elements of such systems attract each other at small
distances due, for example, to the Van der Waals attraction, and repulse on
longer separations because of the electrostatic interactions (SALR systems,
short-range attractive and long-range repulsive interaction) [4, 5]. In the
case of biological molecules, repulsion can also be caused by elastic
deformations of the lipid membranes. In any case, the attraction between the
structural elements of the system ensures the phase separation, and repulsion
— the formation of clusters.

One of the simplest methods for investigating the general properties of SALR
systems is to consider their lattice models. These models are simple enough
and allow one to make a detailed analysis both by analytical methods and
computer simulation, using the Monte Carlo method. A large number of common
properties of such systems can be obtained within these frameworks.

For example, in papers [6, 7], a lattice fluid was studied with attraction of
nearest neighbours and repulsion of the third neighbours on a plane triangular
lattice. Possible configurations of the ensemble of particles were
investigated. A phase diagram of the system in the mean field approximation
and by Monte Carlo simulation was constructed. It revealed the existence of
several phase transitions in the system.

In [8] the generalized quasi-chemical approximation (QChA) was proposed for
lattice systems with SALR interaction potential on a triangular lattice. This
approximation has demonstrated its applicability for estimating the
equilibrium properties of the model in the disordered phase.

In this paper, we present the results for a similar model of the lattice fluid
on a square lattice and propose a geometric order parameter, which makes it
possible to investigate the ordered phases in the system.

##  2 The model and its order parameter

We consider the lattice fluid consisting of $n$ particles on a square lattice
of $N$ lattice sites. Multiple occupation of sites is forbidden. Particles
that occupy nearest lattice sites and sites that are neighbours of the third
order interact with each other. The interaction energies are equal to $J_{1}$
and $J_{3}$, respectively. It is assumed that $J_{1}=-J<0$, and $J_{3}=3J>0$,
which corresponds to the attraction of the nearest neighbours and the
repulsion of the third ones. The second, forth and more distant neighbours are
considered as noninteracting ones.

The simulation of the equilibrium characteristics of the system under
consideration in the grand canonical ensemble using the Monte Carlo method is
performed within the framework of the Metropolis algorithm [9]. For
simulation, we used a lattice containing $2^{14}$ lattice sites with periodic
boundary conditions. The total length of the simulation procedure consisted of
$70000$ steps of the Monte Carlo algorithm (MCS). The first $20000$ MCSs were
used to equilibrate the system and were not taken into account at subsequent
averaging.

A preliminary simulation on the lattice containing $2^{10}$ lattice sites has
shown that two different types of ordered phases are formed in the system at
low temperatures (below the critical temperature $T_{\text{c}}$). The both
types of ordered phases are shown in figure 1.

Figure 1: The final screenshot of the system at $\mu=4J$ and
$J/k_{\text{B}}T=1.0$. The left-hand and right-hand parts of the figure
correspond to different launches of the MC-simulation program.

On smaller lattices, such as those shown in figure 1, the phases appear
sporadically on different trajectories. On larger lattices (e.g., of $2^{16}$
lattice sites) the system breaks up into domains with different types of the
ordered structures. The ground state of the model is degenerated with the
energy $-J$ per particle for both configurations. Subsequently, with an
increase in temperature, both states are realized with approximately equal
probabilities.

To describe the ordered phases, the initial square lattice was divided into a
system of 8 identical sublattices rotated by $45^{\circ}$ with the spacing
$2a\sqrt{2}$, where $a$ is the lattice spacing of the initial lattice. In the
case of complete ordering of the system at the lattice concentration $c=0.5$
and at low temperatures, four sublattices are completely filled
($p$-sublattices) and four sublattices are completely vacant
($v$-sublattices). This makes it possible to determine the system order
parameter $\delta c$ as the difference between the particle concentrations on
the sublattices

$\displaystyle\delta{c}=\frac{c_{p}-c_{v}}{2}.$ (2.1)

If the sublattices are numbered, we can consider the order parameter matrix

$\displaystyle\delta{c}_{ij}=\frac{|c_{i}-c_{j}|}{2}\,,$ (2.2)

where $c_{i(j)}$ is the concentration of particles on the sublattice $i(j)$.

This is a symmetric matrix with the diagonal elements equal to zero and 28
independent non-diagonal elements equal to one or zero for the ordered
structures. It is possible to distinguish the ordered phases by the order of
filled sublattices or by the structure of the matrix; e.g., even numbers (two
or four) of 1 or 0 appear in the sequences in rows and columns of the matrix
for squares, while odd numbers (three) are characteristic of stripes.
Importantly, for large lattices when both structures are present
simultaneously, the scalar order parameter equation (2.1) distinguishes
ordered states from disordered ones.

Figure 2: The order parameter versus the inverse temperature at $\mu=4J$
($c=0.5$).

The order parameter characterizes the strength of the ordering of the ordered
state and is equal to zero in a disordered state. The total lattice
concentration $c$ and the sublattices concentrations $c_{p}$ and $c_{v}$ are
connected by the expressions (the subscripts 1 and 0 are related to particles
and vacancies, correspondently)

$\displaystyle c_{p}=c_{1}^{p}=c+\delta{c},\qquad
c_{0}^{p}=1-c_{1}^{p}=1-c-\delta{c},$ (2.3) $\displaystyle
c_{v}=c_{1}^{v}=c-\delta{c},\qquad c_{0}^{v}=1-c_{1}^{v}=1-c+\delta{c},$ (2.4)
$\displaystyle c=c_{1}=\frac{n}{N}=\frac{c_{1}^{p}+c_{1}^{v}}{2}\,,\qquad
c_{0}=1-c.$ (2.5)

The MC simulation shows (see figure 2) that the order parameter increases
sharply at $J/k_{\text{B}}T_{\text{c}}=0.655$ for the chemical potential
$\mu=4J$ which corresponds to the system with the average concentration
$c=0.5$.

Such an increase of the order parameter corresponds to the order-disorder
phase transition, which is of the second order like in the system with the
nearest neighbour repulsive interaction [10]. In this case, the value $0.655$
can be interpreted as a critical parameter of the system.

Thus, the value of the order parameter can be used to localize points of the
structural phase transitions and to construct the phase diagram of the model.

##  3 The quasi-chemical approximation

The free energy of the system can be represented as a sum of the free energy
of the reference system $F_{\text{r}}$ and the diagrammatic part
$F_{\text{d}}$ [11, 12]:

$\displaystyle F=F_{\text{r}}+F_{\text{d}}.$ (3.1)

The reference system is characterized by the mean potentials
$\phi_{j}^{\,\beta}(n_{i}^{\alpha})$ describing the interaction of a particle
($n_{i}^{\alpha}=1$) or vacancy ($n_{i}^{\alpha}=0$) on the site $i$ of
$\alpha$-sublattice with site $j$ of $\beta$-sublattice. Equation (3.1) is an
identity and the free energy should not be dependent on the choice of the mean
potentials. Therefore, the mean potentials can be found from the minimal
susceptibility principle [13]

$\displaystyle\frac{\partial{F}}{\partial{\phi_{i}^{\alpha}}}=0.$ (3.2)

The free energy is a function of the sublattice concentrations. It is useful
to represent it as a function of the lattice concentration $c$ and the order
parameter $\delta{c}$. The latter can be determined from the extremity
condition

$\displaystyle\frac{\partial{F}}{\partial{\delta{c}}}=0,$ (3.3)

which is equivalent to the requirement that the chemical potentials on all the
sublattices are equal.

As a first step, one can consider a quasi-chemical approximation when the
diagrammatic part of free energy contains the two-vertex graph contribution
only

$\displaystyle
F=\frac{k_{\text{B}}T}{2}\sum_{\alpha=p}^{v}\sum_{i=0}^{1}c_{i}^{\alpha}\left[\ln{c_{i}^{\alpha}}-\sum_{k}z_{k}\ln
X_{i}^{{\alpha}(k)}\right]-\frac{k_{\text{B}}T}{2}\sum_{k}\frac{z_{k}}{2}\sum_{\alpha,\beta=p}^{v}\sum_{i,j=0}^{1}c_{i}^{\alpha(k)}c_{j}^{\,\beta(k)}\left[\frac{W_{ij}^{(k)}}{X_{i}^{{\alpha}(k)}X_{j}^{{\beta}(k)}}-1\right],$
(3.4)

where

$\displaystyle
X_{i}^{{\alpha}(k)}=\exp\left[-\beta\phi^{k}(n_{i}^{\alpha})\right],$ (3.5)

$z_{k}$ is the coordination number for neighbours of $k$-order
($z_{1}=z_{3}=4$ for a square lattice).

In this approximation, the diagrammatic part of free energy is equal to zero
and the mean potentials for nearest-neighbours do not depend on the sublattice
structure. The free energy in the QChA reads

$\displaystyle F^{\text{(QChA)}}(c,\delta c)$
$\displaystyle=\frac{k_{\text{B}}T}{2}\sum_{i}c_{i}\left[\ln(c_{i}^{2}-\delta
c^{2})-2z_{1}\ln
X_{i}\right]-\frac{k_{\text{B}}T}{2}z_{3}\left(\ln{Z_{0}^{p}Z_{0}^{v}}+c\ln{\xi_{v}\xi_{p}}\right)$
$\displaystyle+\frac{k_{\text{B}}T}{2}\delta
c\left(\sum_{k}\ln{\frac{c_{i}+\delta c}{c_{i}-\delta
c}}-z_{3}\ln\frac{\xi_{p}}{\xi_{v}}\right),$ (3.6)

where

$\displaystyle
W=\exp\left(-\frac{J_{1}}{k_{\text{B}}T}\right);\qquad\Omega=\exp\left(-\frac{J_{3}}{k_{\text{B}}T}\right);$
(3.7)
$\displaystyle\eta=-\frac{c_{1}-c_{0}}{2c_{0}}+\sqrt{\left(\frac{c_{1}-c_{0}}{2c_{0}}\right)^{2}+\frac{c_{1}}{c_{0}}W};$
(3.8) $\displaystyle X_{0}=\sqrt{c_{0}+\frac{c_{1}}{\eta}}\,,\qquad X_{1}=\eta
X_{0};$ (3.9) $\displaystyle\xi_{p(v)}=-\frac{c_{1}-c_{0}\pm 2\Omega\delta
c}{2(c_{0}\mp\delta c)}+\sqrt{\left[\frac{c_{1}-c_{0}\pm 2\Omega\delta
c}{2(c_{0}\mp\delta c)}\right]^{2}+\frac{c_{1}\pm\delta c}{c_{0}\mp\delta
c}\Omega};$ (3.10) $\displaystyle
Z_{0}^{v}Z_{0}^{p}=c_{0}^{v}+\frac{c_{1}^{v}}{\xi_{v}}=c_{0}^{p}+\frac{c_{1}^{p}}{\xi_{p}}.$
(3.11)

All the thermodynamic characteristics can be investigated with equation (3.6)
for the free energy. Thus, the chemical potential $\mu$, the thermodynamic
factor $\chi_{T}$ and the correlation function $g_{k}(1;1)$ for two nearest-
and next-next-nearest neighbours (at $k=1$ and $3$, respectively) are
determined by the expressions

$\displaystyle\beta\mu=\left(\frac{\partial(\beta F)}{\partial c}\right)_{T},$
(3.12) $\displaystyle\chi_{T}=\frac{\partial(\beta\mu)}{\partial\ln c}\,,$
(3.13) $\displaystyle g_{k}(1;1)=\frac{2}{z_{k}c^{2}}\left(\frac{\partial
F}{\partial J_{k}}\right)_{T}.$ (3.14)

##  4 Calculation and simulation results

The most important structural feature of an ordered state is the order
parameter. The comparisons of the calculation and simulation results for the
order parameter are shown in figure 3.

Figure 3: The order parameter versus concentration at $\beta J=0.8$. The solid
line 1 represents the QChA results, the full circles 2 are the MC simulation
data.

The order parameter is used to determine the phase transition curve. The
corresponding phase diagram is represented in figure 4. One can note that QChA
leads to a wider area of the ordered phase in the system as compared to the
Monte Carlo simulation results. In addition, the critical temperature is
overestimated by approximately 30% in this approximation. These deviations are
known to be typical of QChA [14, 15].

Figure 4: The order-disorder phase transition curves. The solid line 1
represents the QChA results, the full circles 2 are the MC simulation data.

It should be noted that the phase diagram of the system with the competing
interaction on a square lattice is much simpler than that of the system on a
triangular lattice [7] which is probably the consequence of a close-packed
structure of the latter.

The chemical potential isotherms are shown in figure 5. The ordered phase
exists at temperatures below critical ($\beta J=0.6$, $0.7$, $0.8$ and $1.0$)
where a steep increase of the chemical potential with an increase of the
particle concentration is observed. The concentration derivative of the
chemical potential or the thermodynamic factor (3.13) indicates the second-
order phase transition by discontinuities at the concentrations that
correspond to the phase transition points. The strong peaks at $c=0.5$
correspond to the most ordered states of the system at corresponding
temperatures (see figure 6).

Figure 5: (Colour online) The chemical potential (in units of the nearest
neighbour interaction energy $J$) versus concentration at $\beta J=0.3$ (1);
0.5 (2); 0.6 (3); 0.7 (4); 0.8 (5) and 1.0 (6). The solid lines represent the
QChA results, the full circles are the MC simulation data. Each group of
curves is shifted down by 10 units along the $\mu$ axis with respect to the
previous one for better visibility. The unshifted curve (3) is characterized
by $\mu/J=4$ at $c=0.5$, and this point is the same for all the temperatures.
Thus, the groups of curves (1) and (2) are shifted up from their true
position, while (4), (5) and (6) are shifted down. The same curves numbers are
kept in figures 6–8.

Figure 6: (Colour online) The thermodynamic factor versus concentration at
$\beta J=0.3$ (1, square); 0.5 (2, circle); 0.7 (4, down triangle) and 0.8 (5,
diamond). The solid lines represent the QChA results, the symbols are the MC
simulation data.

The thermodynamic factor is inverse to the concentration fluctuations. The
latter grow immediately after the transition from a disordered to an ordered
state and then decrease systematically until concentration $0.5$ is reached
when the most ordered state is possible. Physically, this means that
concentration fluctuations are suppressed in the ordered states.

In QChA, the chemical potential and the thermodynamic factor were calculated
by numerical differentiating of the free energy expression (3.6). However,
such a differentiation of the chemical potential extracted from MC simulations
may not be used. In this case, the thermodynamic factor can be calculated as
the value inversely proportional to the mean square concentration fluctuations

$\displaystyle\chi_{T}=\frac{\langle n\rangle}{\langle(n-\langle
n\rangle)^{2}\rangle}.$ (4.1)

The parameter $\chi_{T}$ plays an important role in describing the diffusion
process in lattice fluids [16].

Figure 7: (Colour online) The correlation functions for nearest neighbours
versus concentration at $\beta J=0.3$ (1, square); 0.6 (3, up triangle); 0.8
(5, diamond) and 1.0 (6, left-hand triangle). The solid lines represent the
QChA results, the symbols are the MC simulation data.

The calculation and simulation results for the chemical potential and
thermodynamic factor are in a good agreement except for those at close
vicinity of the second order phase transition curve where the QChA curves show
abrupt jumps.

A short range ordering is characterized by the correlation functions (3.14).
These functions are more informative objects than the distribution functions
because they represent the deviation of the short range correlations in
interacting systems from the case of noninteracting (Langmuir) lattice gases
where they are equal to one.

The correlation functions for nearest and next-next-nearest neighbours are
show in figures 7 and 8, respectively.

Figure 8: (Colour online) The correlation functions for the next-next-nearest
neighbours versus concentration at $\beta J=0.3$ (1, square); 0.6 (3, up
triangle); 0.8 (5, diamond) and 1.0 (6, left-hand triangle). The solid lines
represent the QChA results, the symbols are the MC simulation data.

The long range ordering manifests itself in a short range structure as well.
At temperatures $T<T_{\text{c}}$ (see curves 3, 5 and 6) the probability to
find two next-next-nearest neighbour sites occupied by particles becomes very
low. Of course, the higher is the temperature, the less pronounced difference
from the Langmuir gas behaviour is observed.

The simulation and analytical results satisfactorily match each other in the
disordered phase. They significantly differ at intermediate concentrations and
at low temperatures due to the problems concerning the determination of the
order parameter and the critical temperature in QChA. The most significant
differences appear for the correlations on the nearest neighbour lattice
sites.

##  5 Conclusion

The lattice system with an attractive interaction between nearest neighbours
and repulsive interaction between next-next-nearest neighbours has been
studied.

It is shown that the competing interactions lead to the order-disorder phase
transitions. The order parameter $\delta c$ is used as the indicator of the
second order phase transitions. With this parameter, it was established that
the critical value of the interaction parameter is equal to
$|J_{1}|/k_{\text{B}}T_{\text{c}}=0.655\pm 0.005$, and the phase diagram of
the system was constructed.

The quasi-chemical approximation is found to be self-consistently constructed
through the mean potentials that describe the interaction of a particle or a
vacancy with its nearest and next-next-nearest neighbour lattice sites. The
chemical potential, thermodynamic factor and correlation functions are
determined both in the QChA and in the Monte Carlo simulations. The chemical
potential demonstrates irregular behaviour in the phase transition region,
while the thermodynamic factor indicates strong suppressing of fluctuations
that are inherent to the ordered states of the system. The complicated
behaviour of the correlation functions that reflects structural peculiarities
of the system demonstrate a great importance of competing interactions.

The order parameter of the system $\delta c$ is determined in the QChA with
significant errors. This leads to errors in determining the critical
temperature of the system, which is overestimated by approximately 30%. As a
result, the quasichemical approximation fails to reproduce the structural
characteristics of the system with competing interactions. At the same time,
the thermodynamic characteristics such as the chemical potential and the
thermodynamic factor are determined in the quasi-chemical approximation with a
sufficiently high accuracy.

Thus, the developed approach allows us to correctly describe the qualitative
features of the structural properties of the systems with competing
interactions, and can be used to quantify the thermodynamic characteristics of
these systems.

## Acknowledgements

The project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 73427, Institute for Nuclear Problems of Belarusian State
University (agreement No 209/103) and the Ministry of Education of Belarus.

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Ukrainian 0

Ðâíîâàæí âëàñòèâîñò ðàòêîâî¿ ñèñòåìè ç ïîòåíöàëîì âçàìîä¿ òèïó SALR íà
êâàäðàòíé ðàòö: êâàçiõiìiíå íàáëèæåííÿ ó ïîðâíÿíí ç ñèìóëÿöiÿìè Ìîíòå Êàðëî
ß.Ã. Ãðîäà${}^{\textsf{\footnotesize{\mbox{?}}}}$, Â.Ñ.
Âõðåíêî${}^{\textsf{\footnotesize{\mbox{?}}}}$, Ä. ä
Êàïðî${}^{\textsf{\footnotesize{\mbox{?}}}}$

${}^{\textsf{\footnotesize 1}}$ Áëîðóñüêèé äåðæàâíèé òåõíîëîãíèé óíâåðñèòåò,
âóë. Ñâðäëîâà, 13a, 220006 Ìíñüê, Áëîðóñü ${}^{\textsf{\footnotesize 2}}$
Äîñëiäíèöüêèé óíiâåðñèòåò íàóêè òà ëiòåðàòóðè Ïàðèæó, ChimieParisTech CNRS,
Iíñòèòóò õiìiíèõ äîñëiäæåíü Ïàðèæó, Ïàðèæ, Ôðàíöiÿ


# Peculiarities of spin- orbit interaction systematically measured in FeCoB
nanomagnets

Vadym Zayets  National Institute of Advanced Industrial Science and Technology
(AIST), Umezono 1-1-1, Tsukuba, Ibaraki, Japan

###### Abstract

This study introduces a method to measure strength of spin-orbit interaction
(SO) in a nanomagnet, investigating fundamental phenomena governing magnetic
anisotropy. The method explores the fundamental property of SO in its linear
proportionality to the external magnetic field, a relationship validated
through experimental observation. Systematic study of SO in FeCoB nanomagnets
reveals distinct SO behaviors in bulk and at an interface, its substantial
disparities in single- and multi-layer nanomagnets, intriguing periodic
oscillations in SO strength, and the systematic relationship between SO
strength, demagnetization field, and magnetic anisotropy based on surface
imperfections. These findings provide crucial insights into diverse spin-orbit
interaction behaviors, crucial for understanding and optimizing magnetic
anisotropy and nanomagnet properties.

Spin-orbit interaction (SO) refers to a magnetic field $H_{SO}$ of
relativistic origin [1, 2] experienced by an electron while moving within an
electric field $E$:

$H_{so}=\frac{v}{c^{2}}E$ (1)

where $v$ is a component of the electron velocity perpendicular to $E$.

Within a nanomagnet, the spin-orbit interaction plays a crucial role in
establishing the magnetic anisotropy, which effectively maintains the
nanomagnet’s magnetization along its easy magnetic axis [3, 4]. This magnetic
anisotropy leads to two stable magnetization directions within the nanomagnet,
allowing data to be stored using these stable states. As a result, a
nanomagnet becomes valuable for non-volatile memory applications [5, 6, 7].

As the density of magnetic memory increases, the size of nanomagnets decreases
accordingly. Consequently, the magnetic energy diminishes and reaches a point
where it becomes comparable to thermal energy [5, 6, 7, 8]. In such a
scenario, thermal fluctuations can lead to magnetization reversal, resulting
in data loss within the magnetic memory. To prevent this undesirable
occurrence, it is essential to enhance the strength of the magnetic anisotropy
in nanomagnets, which can be achieved through the optimization and
reinforcement of the spin-orbit interaction.

In amorphous ferromagnetic metals such as FeB and FeCoB, which are mainly used
as a material for MRAM, the magnetic anisotropy originates primarily at the
interface [5, 6, 7]. The orbits of localized electrons at the interface
undergo significant deformation towards the neighboring atomic layer. This
substantial deformation leads to a remarkable enhancement in the strength of
the spin-orbit interaction and results in perpendicular magnetic anisotropy
(PMA). Due to this remarkable interface enhancement of magnetic anisotropy,
the magnetic energy of a nanomagnet significantly increases, making the
nanomagnet thermally stable even at smaller sizes. Consequently, a nanomagnet
with PMA becomes the preferred choice for magnetic memory applications [5, 6,
7].

As seen in Eq. 1, $H_{so}$ is proportional to $1/c^{2}$ and $E$. Therefore,
the magnetic field of the spin-orbit interaction $H_{so}$ is only significant
when the electron moves through a large electric field at a speed close to a
substantial fraction of the speed of light $c$. This occurs during the orbital
movement around the atomic nucleus. However, certain conditions must be met
for the spin-orbit interaction to be large in the case of orbital movement.
The key requirement is the breaking of time-inversion symmetry (T-sym), as
without such breaking, there would be no spin-orbit interaction (SO). For
instance, the orbital movement of the electron can be separated into clockwise
(CW) and counterclockwise (CCW) components, each contributing to $H_{so}$. In
the absence of T-sym breaking, $H_{so}$ generated by the CW and CCW components
are of equal strength but in opposite directions, resulting in a cancellation,
leading to no overall $H_{so}$.

The time-inversion symmetry is broken for an electron having a non-zero
orbital moment. For instance, conduction electrons possess an orbital moment,
leading to a substantial contribution to its energy from the spin-orbit
interaction [9]. Another method of breaking T-sym is by applying an external
magnetic field $H_{ext}$. This magnetic field induces a Lorentz force acting
in opposite directions for the CW and CCW components of the wavefunction.
Consequently, the Lorentz force leads to a modification of the wavefunction,
causing one component to, on average, move closer to the nucleus while the
other component moves away from it. This imbalance in the contributions from
these components to the Spin-Orbit Interaction results in the creation of
magnetic field $H_{so}$. Since in the absence of $H_{ext}$, T-sym remains
unbroken and $H_{so}$ equals zero, a linear relationship between $H_{so}$ and
$H_{ext}$ can be assumed:

$H_{so}=k_{so}\cdot H_{ext}$ (2)

where $k_{so}$ is the coefficient of spin- orbit interaction.

In situations where the electron’s orbital lacks symmetry, the strength of the
spin-orbit interaction (SO) becomes dependent on the orientation of the
external magnetic field $H_{ext}$. The SO magnetic field $H_{so}$ arises due
to the imbalance created by the Lorentz force between two components of the
orbital, and, consequently, the initial shape and symmetry of the electron
orbital substantially influence SO strength. This implies that the value of
$k_{so}$ is orientation-dependent, particularly concerning the direction in
which the magnetic field is applied relative to the electron orbital’s
asymmetry.

The amplification of SO occurs when the electron orbital is in closer
proximity to the atomic nucleus, where the nucleus’s electrical field is most
potent (Eq. 1). This proximity is achieved when the center of the orbital
deviates from the nucleus’s position. Such a deviation can occur in the
vicinity of an interface where the electron interacts with two distinct types
of atoms on either side, prompting the necessary shift. Consequently, the
enhancement of SO for an electron at the interface predominantly occurs when
the magnetic field is applied perpendicular to the interface. Even when the
magnetic field is applied parallel to the interface, SO remains present but
with a smaller $k_{so}$.

The creation of the magnetic anisotropy in a ferromagnetic material is the key
feature of the spin- orbit interaction [3, 4]. Inside a ferromagnet, there
exists an intrinsic magnetic field aligned with the direction of ferromagnet
magnetization. The intrinsic magnetic field creates $H_{so}$ as described by
Eq.2. Since $k_{so}$ is larger in the direction perpendicular- to -plane
($\perp$-plane), $H_{so}$ larger and magnetic energy higher in the
$\perp$-plane direction compared to the in-plane direction. This implies that
the spin-orbit interaction makes the $\perp$-plane magnetization direction
energetically favorable.

Another factor influencing the equilibrium magnetization direction in a
ferromagnetic nanomagnet is demagnetization. The demagnetization field is
oriented in the $\perp$-plane direction and opposes the direction of
magnetization. The demagnetization field effectively lowers the magnetic
energy in the $\perp$-plane direction. Consequently, in the absence of spin-
orbit interaction, the equilibrium magnetization direction is the in-plane
direction [3].

The presence of the Spin-Orbit Interaction counteracts the demagnetization
effect and promotes the $\perp$-plane direction as the equilibrium
magnetization orientation. For instance, this phenomenon is observed in thin
amorphous nanomagnets, where the substantial spin-orbit interaction exists
only for electrons at the interface. In the case of a thicker nanomagnet, the
bulk contribution dominates, the demagnetization effect overcomes SO
interaction and the equilibrium magnetization direction is in-plane.
Conversely, for a thinner nanomagnet, the interface contribution dominates,
the SO interaction counteracts the demagnetization effect and the equilibrium
magnetization direction becomes $\perp$-plane [10].

Since the strength of the magnetic anisotropy is inherently linked to the
strength of the spin-orbit interaction, the most intuitive and direct approach
for a measurement of the SO strength is through measurements of magnetic
anisotropy. This approach further benefits from the linear dependence of SO
strength on the external magnetic field (Eq.2). Consequently, by measuring the
dependence of the strength of magnetic anisotropy on the external magnetic
field, it becomes possible to discern and quantify the contribution of the
spin-orbit interaction. In this study, we have developed and employed this
measurement method to systematically evaluate and study the strength of SO in
FeCoB nanomagnets.

To measure the strength of magnetic anisotropy, an external magnetic field
$H_{x}$ is applied perpendicular to the easy magnetic axis, causing the
magnetization to tilt. The strength of the magnetic anisotropy is evaluated
from the magnetization tilt. There is an intrinsic magnetic field in a
nanomagnet, which holds magnetization along its easy axis. The intrinsic
magnetic field is larger in a nanomagnet with a stronger magnetic anisotropy.
Consequently, as the intrinsic magnetic field strengthens for a stronger
magnetic anisotropy, a stronger perpendicular field $H_{x}$ becomes necessary
to induce an equivalent tilt in the magnetization. The external magnetic field
$H_{x}$, which aligns the magnetization fully along the hard magnetic axis, is
referred to as the anisotropy field $H_{ani}$. $H_{ani}$ serves as a
descriptor of the magnetic anisotropy strength. In this study, the tilt of the
nanomagnet magnetization was measured by a measurement of the Hall angle. For
this purpose, the FeCoB nanomagnets were fabricated on top of a Ta- or W- made
Hall bar (See Appendix B).

Figure 1:  Anisotropy field $H_{ani}$ vs. external magnetic field $H_{z}$
measured (a) in $Ta(2):FeB(1.1)$ and (b)
$W(3):[FeB(0.55):W(0.5)]_{5}:FeB(0.55)$ nanomagnets. The slope of the
dependence is proportional to SO strength and gives the coefficient of SO
interaction.

There exists a linear relationship between the in-plane component of
magnetization $M_{x}$ and $H_{x}$ (See Appendix A):

$\frac{M_{x}}{M}=\frac{H_{x}}{H_{ani}}$ (3)

By performing a linear fit of the measured $M_{x}/M$ against $H_{x}$, the
value of $H_{ani}$ was evaluated. Additionally, an external magnetic field
$H_{z}$ was applied perpendicular to the plane, aligning with the easy
magnetic axis, serving as a parameter for the measurement. For each $H_{z}$
value, an in-plane field $H_{x}$ was systematically scanned, enabling the
evaluation of $H_{ani}$ vs. $H_{z}$. Equilibrium magnetization direction of
all measured nanomagnet is $\perp$-plane.

The equilibrium magnetization direction under an applied tilted magnetic field
was calculated in Appendix A by minimizing the magnetic energy. Subsequently,
it gives the dependence of the anisotropy field $H_{ani}$ on the external
magnetic field $H_{z}$ as follows:

$H_{ani}=H^{0}_{ani}+H_{z}+k_{so}H_{z}$ (4)

where $H^{0}_{ani}$ is the anisotropy field in absence of $H_{z}$.

As anticipated, $H_{ani}$ exhibits a linear proportionality to $H_{z}$. The
second term in the equation corresponds to the bulk contribution, representing
the alignment of magnetization along $H_{z}$, which doesn’t provide
informative insights. Consequently, when conducting data analysis, it is more
advantageous to utilize the relationship between $H_{ani}-H_{z}$, and $H_{z}$.
This approach ensures that the significant bulk contribution does not obscure
a potentially weaker dependence on $k_{so}$.

Figure 1 displays the measured relationship between $H_{ani}-H_{z}$ and the
external magnetic field $H_{z}$ in a single-layer nanomagnet and a multi-layer
nanomagnet. In both cases, the relationship exhibits an approximately linear
trend, but the slopes differ significantly. For the single-layer nanomagnet,
the slope is negative, while for the multi-layer nanomagnet, it is positive.
It means that the strength of the spin-orbit interaction is opposite for those
nanomagnets. $k_{so}$ is negative for the single-layer nanomagnet but positive
for the multi-layer nanomagnet. This divergence is a consistent pattern. In
Figure 2, the measured values of $H^{0}_{ani}$ are plotted against the
measured values of $k_{so}$ for nanomagnets of various sizes and structures.
In all multi-layer nanomagnets, $k_{so}$ consistently exhibits a large and
positive value. In contrast, for single-layer nanomagnets, $k_{so}$ is
generally smaller and can be either positive or negative. A negative value of
$k_{so}$ indicates that, when considering an average over bulk and interface
contributions, the strength of the spin-orbit interaction in the in-plane
direction is greater than in the $\perp$-plane direction (See Eq. 11).

The significant disparity in strength of spin-orbit interaction between
single- and multi-layer nanomagnets underscores the substantial influence of
interfaces on the overall strength. A greater number of interfaces correlates
with a larger $k_{so}$. Moreover, each interface incrementally adds to the
overall strength, irrespective of the interface’s polarity. For instance, both
Pt/FeB and FeB/Pt interfaces enlarge the overall strength. That is the reason
why $k_{so}$ in a multi- layer nanomagnet is substantially larger than in a
single- layer nanomagnet.

Comparative analysis of the measured $k_{so}$ in single- and multi-layer
nanomagnets reveals that the interface contribution to $k_{so}$ is positive
and large. Conversely, the contribution from the bulk of the nanomagnet is
small and negative. In single-layer nanomagnets, there’s a balancing between
the bulk and interface contributions, resulting in a small $k_{so}$ that can
be either positive or negative.

Figure 2:  Measured anisotropy field $H^{0}_{ani}$ vs. the measured
coefficient of spin-orbit interaction $k_{so}$ in nanomagnets of varying sizes
and structures. Each dot represents an individual nanomagnet measurement. Dots
of the same color and shape correspond to nanomagnets fabricated at different
locations on the same wafer. Stars indicate multilayer nanomagnets, while
circles represent single-layer nanomagnets.

It indicates a notable distinction in the deformation of Fe and Co orbitals
between the interface and the bulk regions. In the bulk, orbital deformation
remains minimal, resulting in an almost isotropic tensor $\hat{k}_{so}$.
Elements $k_{so,x}$ and $k_{so,y}$ are slightly larger than $k_{so,z}$, which
contributes to an overall small and negative effective $k_{so}$ within the
bulk.

Conversely, at the interface, Fe and Co orbitals undergo significant
deformation along the interface’s normal. This deformation arises from
differing orbital interactions with orbitals situated below and above the
interface. Consequently, the tensor $\hat{k}_{so}$ assumes a uniaxial
orientation, where $k_{so,z}$ becomes notably larger than the in-plane
elements $k_{so,x}$ and $k_{so,y}$. This configuration gives rise to a
substantial and positive effective $k_{so}$ for the orbitals positioned at the
interface (See Eq. 11).

Additional observation in Fig. 2 shows scattering among the data points of
nanomagnets fabricated at different locations on the same wafer. Some wafers
display data convergence within a small area, while others exhibit data
spreading over a larger range. A more homogeneous and smoother wafer
corresponds to reduced data spreading. This spread in data serves as an
indicator of fabrication technology perfection. Both variations in thickness
and surface roughness contribute to data dispersion on a wafer. Given that
nanomagnet thickness is around 1 nm, even a variance within a single atomic
layer range markedly impacts $k_{so}$ and $H^{0}_{ani}$.

Moreover, the surface’s perfection plays an even more substantial role,
exerting a notable influence on $k_{so}$ and $H^{0}_{ani}$. As previously
discussed, robust spin-orbit interaction at an interface arises from
substantial disparities in electron interactions with neighboring orbitals
positioned above and below the interface. This leads to substantial and
asymmetric orbital deformation along the surface normal. Deterioration in
interface smoothness and sharpness weakens this deformation, causing its
direction to randomly incline from the interface normal, resulting in a
reduced $k_{so}$. Consequently, the value of $k_{so}$ acts as an indicator of
interface smoothness and sharpness, providing another insight into the quality
of fabrication technology. Keeping other nanomagnet properties constant, a
higher $k_{so}$ value signifies a smoother and sharper interface.

The data from nanomagnets from a single wafer in Fig. 2 don’t spread randomly;
instead, they follow a linear trend. In the case of single-layer nanomagnets,
the slope of this trend line is consistently positive. However, for multi-
layer nanomagnets, the slope is primarily negative, turning positive only for
nanomagnets with a minimal number of layers. The correlation between
variations in $H^{0}_{ani}$ and $k_{so}$ is expected because $H^{0}_{ani}$ is
linearly proportional to $k_{so}$ (see Eq.25). Thus, the variation in data for
$H^{0}_{ani}$ vs. $k_{so}$ should follow a line, and the slope of this line
should be positive. The existence of a negative slope indicates that, in
addition to $k_{so}$, there is another parameter affecting $H^{0}_{ani}$, and
its influence is opposite to that of $k_{so}$. This is why the slope polarity
may change from positive to negative.

The parameter that satisfies these properties is demagnetization.
Demagnetization is the magnetic field created by the magnetic dipole formed at
opposite surfaces of a magnetic film. The direction of the demagnetization
field $H_{demag}$ is opposite to the direction of the internal magnetic field,
thus reducing $H^{0}_{ani}$ (see Eq. 25). Surface roughness and diminished
interface sharpness reduce the demagnetization field. Interface roughness
introduces random deviations in the demagnetization direction from the
interface normal at each local point, resulting in the effective reduction of
the average demagnetization field. A variation in interface roughness and
sharpness leads to variations in the $H_{demag}$ for nanomagnets fabricated at
different locations on the same wafer. This contributes to the data spread
observed in Fig. 2.

In total, a smoother interface makes $k_{so}$ larger, leading to a larger
$H^{0}_{ani}$. However, a smoother interface also enhances $H_{demag}$, which
conversely reduces $H^{0}_{ani}$. Therefore, there exists a delicate balance
between these two effects. For a single-layer nanomagnet, the first effect
prevails, resulting in the positive slope seen in Fig.2. Conversely, for
multi-layer nanomagnets, the impact of the demagnetization field prevails,
yielding the negative slope. The reason for the negative slope and, therefore,
the dominance of $H_{demag}$ can be explained as follows: Both $k_{so}$ and
$H_{demag}$ are large in a multi- layer nanomagnet. As the demagnetization
field becomes nearly equal to the magnetization’s field, the intrinsic
magnetic field within the nanomagnet becomes small. Consequently, even a
slight variation in interface roughness and subsequently the demagnetization
field induces a notable relative change in the already small intrinsic
magnetic field, having a large effect on the anisotropy field.

Figure 3:  Oscillation amplitude of the strength of the spin-orbit interaction
under an increasing external magnetic field vs. the coefficient of spin-orbit
interaction. Each dot represents an individual nanomagnet measurement.

Another intriguing feature observed in the measurement of SO strength is the
presence of oscillations atop the linear dependence of $H_{ani}$ versus
$H_{z}$ (See Fig.1), which are consistent across all measured nanomagnets. The
oscillations indicate periodic variations in the strength of the spin-orbit
interaction under the influence of the magnetic field $H_{z}$. The precise
reason for this periodic modulation of SO is not fully understood. One
possible explanation could be the interplay between the various influences
exerted by different neighboring orbitals on the modification of the electron
orbital under the $H_{z}$\- induced Lorentz force.

The amplitude of the oscillations decreases for thicker nanomagnets, where the
bulk contribution is more prominent, clearly indicating the interface origin
of the oscillations. Figure 3 shows the measured amplitude of the oscillations
vs. $k_{so}$. The spread of data along a straight line suggests a proportional
relationship between oscillation amplitude and $k_{so}$. Although the slope
varies for each wafer, it consistently remains positive. The absence of a
negative slope implies that the demagnetization field has no influence on the
oscillations.

In conclusion, a measurement method of the strength of spin-orbit interaction
has been introduced, yielding insights into this complex fundamental
phenomenon. Its systematic study in FeCoB nanomagnets has unveiled distinct
behaviors characterizing the spin-orbit interaction in the bulk and at
interface levels, the substantial disparity in the coefficients of spin-orbit
interaction between single- and multi-layer nanomagnets, intriguing periodic
oscillations in spin-orbit interaction strength under an external magnetic
field, the impact of interface imperfections on both spin-orbit interaction
and the demagnetization field and their competitive influence on the magnetic
anisotropy.

The measurement method can provide deep experimental insights into the
physical processes governing magnetic anisotropy. For instance, it reveals
intriguing particularities in the relationships between the strengths of spin-
orbit interaction and magnetic anisotropy affected by a gate voltage [11, 12]
or electrical current [4, 13] or spin accumulation[14, 15, 16].

## References

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##  Appendix A Calculation of anisotropy field when an external magnetic field
is applied along the easy axis

The equilibrium magnetization direction of a nanomagnet is the direction at
which magnetic energy is the smallest. The calculation of magnetic energy is
as follows:

$-E=\vec{H}_{total}\cdot\vec{M}=(\vec{H}+\vec{M}+\vec{H}_{demag}+\vec{H}_{so})\cdot\vec{M}$
(5)

where $\vec{M}$ is the magnetization, $\vec{H}_{total}$ is the total magnetic
field inside the nanomagnet, $\vec{H}_{demag}$ is the demagnetization field
and $\vec{H}_{so}$ is the magnetic field of spin- orbit interaction (SO).

The magnetic field $\vec{H}_{so}$ is directly related to the total magnetic
field experienced by an electron. In the presence of an anisotropy,
$\vec{H}_{so}$ can be described using the tensor $\hat{k}_{so}$ In this
context, Equation 2 is modified as follows:

$H_{so}=\hat{k}_{so}(\vec{H}+\vec{M}+\vec{H}_{demag})$ (6)

The z-axis is established as perpendicular to the plane, while the x-axis is
set within the plane. In the case of an amorphous nanomagnet, anisotropy in
spin-orbit interaction can occur only between the z- and x-axes. Consequently,
the tensor $\hat{k}_{so}$ can be written as follows:

$\hat{k}_{so}=\begin{pmatrix}k_{so,x}&0&0\\\ 0&k_{so,x}&0\\\
0&0&k_{so,z}\end{pmatrix}$ (7)

The demagnetization field $H_{d}emag$ is always directed along the z-axis and
is proportional to z- component of magnetization $M_{z}$

$H_{demag,z}=-k_{demag}\cdot M_{z}$ (8)

Rewriting Eq. 5 in components gives

$\begin{array}[]{l}-E=M_{x}\cdot(H_{x}+M_{x}+H_{so,x})+\\\
+M_{z}\cdot(H_{z}+M_{z}+H_{demag}+H_{so,z})\end{array}$ (9)

Substitution of Eqs. 6,7,8 into Eq. 9 gives

$\begin{array}[]{l}-E=M_{x}\cdot(1+k_{so,x})(H_{x}+M_{x})+\\\
+M_{z}\cdot(1+k_{so,z})[H_{z}+(1-k_{demag})\cdot M_{z}]\end{array}$ (10)

The effective coefficient $k_{so}$ of SO is defined as:

$1+k_{so}=\frac{1+k_{so,z}}{1+k_{so,x}}$ (11)

Under an applied external magnetic field the magnetization $M$ is tilted, but
does not change its value. The ratio between two components of $M$ is:

$M_{z}=\sqrt{M^{2}-M^{2}_{x}}$ (12)

Substitution of Eqs. 11, 12 into Eq. 10 gives

$\begin{array}[]{l}\frac{-E}{1+k_{so,x}}=M^{2}+M_{x}H_{x}+H_{z}(1+k_{so})\sqrt{M^{2}-M^{2}_{x}}+\\\
+(M^{2}-M^{2}_{x})[(1+k_{so})(1-k_{demag})-1]\end{array}$ (13)

The equilibrium magnetization angle corresponds to the orientation where the
magnetic energy is at its minimum. The minimum can be found from the
condition:

$\frac{\partial E}{\partial M_{x}}=0$ (14)

When there is no perpendicular external magnetic field ($H_{z}=0$), the
condition specified in Eq.14 yields:

$H_{x}-2M_{x}[(1+k_{so})(1-k_{demag})-1]=0$ (15)

Solution of Eq. 15 gives the linear relation between $M_{x}$ and $H_{x}$ as:

$\frac{M_{x}}{M}=\frac{H_{x}}{H^{0}_{ani}}$ (16)

where the anisotropy field $H^{0}_{ani}$ in absence of the external field is
calculated as:

$H^{0}_{ani}=2M[(1+k_{so})(1-k_{demag})-1]$ (17)

Substitution of Eq. 17 into Eq. 13 gives:

$\begin{array}[]{l}\frac{-E}{1+k_{so,x}}=M^{2}+M_{x}H_{x}+H_{z}(1+k_{so})\sqrt{M^{2}-M^{2}_{x}}+\\\
+(M^{2}-M^{2}_{x})H^{0}_{ani}/(2M)\end{array}$ (18)

Substitution of Eq. 18 into condition the condition 14 gives:

$\begin{array}[]{l}H_{x}\cdot(1-\frac{M_{x}}{M}H^{0}_{ani})-H_{z}(1+k_{so})\frac{M_{x}}{\sqrt{M^{2}-M^{2}_{x}}}=0\end{array}$
(19)

The solution of Eq. 18 is similar to the solution 17 and can be express as:

$\frac{M_{x}}{M}=\frac{H_{x}}{H_{ani}}$ (20)

where the anisotropy field $H_{ani}$ is calculated as:

$H_{ani}=H^{0}_{ani}+H_{z}\frac{1+k_{so}}{\sqrt{1-\frac{M^{2}_{x}}{M^{2}}}}$
(21)

The following condition holds true for small to moderate tilting angles:

$\frac{M^{2}_{x}}{M^{2}}\ll 1$ (22)

Thus, even in the presence of an external magnetic field, there is a linear
relation between $M_{x}$ and $H_{x}$. The anisotropy field $H_{ani}$ is
calculated from Eq. 21 as:

$H_{ani}=H^{0}_{ani}+H_{z}+k_{so}H_{z}$ (23)

Within a nanomagnet, an internal magnetic field $H_{int}$ maintains the
magnetization along its easy axis even without the external magnetic field
$H_{z}$. Given that the internal and external magnetic fields share the same
nature, their impact on the anisotropy field should be analogous. This implies
that $H_{ani}$ should solely depend on $H_{int}+H_{z}$. Noting that, Eq. 23
can be reformulated as:

$H_{ani}=(H_{z}+H_{int})+k_{so}(H_{z}+H_{int})$ (24)

where

$H^{0}_{ani}=H_{int}+k_{so}H_{int}$ (25)

The internal magnetic field $H_{int}$ can be calculated from Eq. 25 as:

$H_{int}=\frac{H^{0}_{ani}}{1+k_{so}}$ (26)

##  Appendix B  Details of fabrication and measurement

The equilibrium magnetization for all examined nanomagnets was oriented
perpendicular to the plane. The perpendicular magnetic anisotropy (PMA) in
these nanomagnets was induced at the interface. Sufficient PMA is only
observed in nanomagnets featuring a smooth interface, emphasizing the pivotal
role of fabrication technology in achieving this property.

Figure 4:  Experimental setup for measuring strength of spin- orbit
interaction. An external magnetic field $H_{z}$ is applied along the easy axis
of the FeCoB nanomagnet and used as a parameter. An additional magnetic field
$H_{x}$ is scanned perpendicular to the easy axis. $M$ denotes the
magnetization.

The sample was fabricated on a $Si$/$SiO_{2}$ substrate through sputtering at
room temperature, followed by annealing at $T=250^{0}C$. To attain a smooth
surface for the nanomagnet, a buffer made of a non-magnetic metal such as
tantalum or tungsten, thicker than 2 nm, was utilized. It’s noteworthy that
the desired perpendicular magnetic anisotropy (PMA) wasn’t attained with a
thinner buffer.

The thickness of FeCoB or FeB single-layer nanomagnets ranged between 0.8 nm
and 1.4 nm. Thinner nanomagnets showed either non-ferromagnetic behavior or
in-plane magnetization, attributed to greater surface roughness or even film
discontinuity. Thicker nanomagnets exhibited in-plane magnetization due to a
significant bulk contribution, resulting in a reduced overall strength of the
spin-orbit (SO) interaction (See A). Multilayer nanomagnets can have a greater
thickness, but increasing the number of ferromagnetic layers inevitably
results in an increase of surface roughness and , as a consequence, a decrease
of the SO strength and a decrease of the anisotropy field (See Fig. 3).

The nanomagnet was covered by a 10 nm-thick MgO layer to induce a sufficient
PMA. To shield the MgO from air exposure, a Ta/Ru layer was used. The
fabrication procedure involved multiple nanofabrication stages, employing
electron-beam (EB) or optical lithography, alongside Ar milling and lift-off
techniques. Each fabrication step was aligned within at least a 5 nm
precision. Monitoring the etching material in real-time was carried out using
a Secondary-ion-mass-spectroscopy (SIMS) detector to maintain the required
etching depth accuracy.

Nanomagnets of varying sizes, ranging from 50 nm x 50 nm to 2000 nm x 2000 nm,
were fabricated at different locations on a single wafer. Etching of the
nanomagnet stopped at the top of the W or Ta buffer layer. Subsequently, a W
or Ta nanowire with a $SiO_{2}$ (100 nm) isolation layer was fabricated. A
pair of Hall probes was precisely aligned with the position of the nanomagnet,
and the width of the nanowire matched the width of the corresponding
nanomagnet.The nanowire’s etching stopped at the top of the $SiO_{2}$
substrate. Lastly, Cr (2 nm)/Au (200 nm) contacts were fabricated within the
etched $SiO_{2}$, with the contact etching process ending in the middle of the
Ta or W buffer layer.

Figure 5:  Measurement principle. Schematic diagram. The in-plane component of
magnetization $M_{x}$ vs. in-plane magnetic field $H_{x}$. $M_{x}$ is linear
proportional to $H_{x}$ until the magnetization M aligns in-plane. The
dependence is asymmetric versus a reversal of $H_{x}$ due to existence of the
intrinsic magnetic field $H_{||}$.

The experiments were carried out at room temperature, well below the Curie
temperature of FeCoB and FeB. A nanovoltmeter measured the Hall voltage, while
a current source both supplied and gauged the electrical current. An
electromagnet produced the magnetic field, offering the flexibility to align
it in any direction. This allowed separate control over the in-plane and
$\perp$-plane components of the applied magnetic field. To ensure accuracy,
the electromagnet underwent calibration via Hall measurements conducted on
non-magnetic nanowires made of Ta, W, and Ru.

The measurement procedure involved recording a measurement of the Hall angle
$\alpha_{Hall}$ while sweeping an in-plane external magnetic field $H_{x}$ in
two opposing directions. The $\perp$-plane magnetic field was used as a
parameter in this measurement and kept unchanged during the $H_{x}$ sweep (See
Fig. 4).

The anisotropy field $H_{ani}$ was evaluated from the linear relation (Eq.3).
The Hal angle $\alpha_{Hall}$ is linearly proportional to the perpendicular-
to- plane component of the magnetization $M_{z}$. The in- plane magnetization
component $M_{x}$ was calculated as:

$\frac{M_{x}}{M}=\sqrt{1-\left(\frac{M_{z}}{M}\right)^{2}}=\sqrt{1-\left(\frac{\alpha_{Hall}}{\alpha_{Hall,0}}\right)^{2}}$
(27)

where $\alpha_{Hall,0}$ is the maximum of the Hall angle when the
magnetization is perpendicular to the plane.

Figure 5 shows the dependence of $M_{x}$ on $H_{x}$. $M_{x}$ is linear
proportional to $H_{x}$ until the magnetization M is aligned in-plane. The
shift in dependence [16, 15] is a result of the presence of the in-plane
magnetic field $H_{||}$. The magnetic field generated by spin accumulation and
the Oersted magnetic field resulting from the current in the non-magnetic
metal both contribute to this magnetic field. Disregarding the influence of
$H_{||}$ could lead to a significant systematic error. To eliminate the
contribution of $H_{||}$, the positive and negative segments of the measured
spectrum were simultaneously fitted, allowing the determination of $H_{||}$.
Subsequently, the measured spectrum was shifted based on $H_{||}$. $M_{x}$ was
obtained using Eq. 27, following the evaluation of $H_{ani}$ derived from the
linear fitting of $M_{x}$ versus $H_{x}$. For each value of $H_{z}$, $H_{x}$
was systematically scanned and $H_{ani}$ was evaluated. $H_{ani}$ consistently
increases as $H_{z}$ increases.