# Single device offset-free magnetic field sensing principle with tunable
sensitivity and linear range based on spin-orbit-torques

Sabri Koraltan<EMAIL_ADDRESS>Physics of Functional Materials,
Faculty of Physics, University of Vienna, Vienna, Austria  Vienna Doctoral
School in Physics, University of Vienna, Vienna, Austria  Christin Schmitt
Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz,
Germany  Florian Bruckner  Physics of Functional Materials, Faculty of
Physics, University of Vienna, Vienna, Austria  Claas Abert  Physics of
Functional Materials, Faculty of Physics, University of Vienna, Vienna,
Austria  Research Platform MMM Mathematics-Magnetism-Materials, University of
Vienna, Vienna, Austria.  Klemens Prügl  Infineon Technologies AG, Regensburg,
Germany  Michael Kirsch  Infineon Technologies AG, Regensburg, Germany  Rahul
Gupta  Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz,
Germany  Sebastian Zeilinger  Physics of Functional Materials, Faculty of
Physics, University of Vienna, Vienna, Austria  Infineon Technologies AG,
Villach, Austria  Joshua M. Salazar-Mejía  Physics of Functional Materials,
Faculty of Physics, University of Vienna, Vienna, Austria  Vienna Doctoral
School in Physics, University of Vienna, Vienna, Austria  Milan Agrawal
Infineon Technologies AG, Munich, Germany  Johannes Güttinger  Infineon
Technologies AG, Villach, Austria  Armin Satz  Infineon Technologies AG,
Villach, Austria  Gerhard Jakob  Institute of Physics, Johannes Gutenberg
University Mainz, 55099 Mainz, Germany  Mathias Kläui  Institute of Physics,
Johannes Gutenberg University Mainz, 55099 Mainz, Germany  Dieter Suess
Physics of Functional Materials, Faculty of Physics, University of Vienna,
Vienna, Austria  Research Platform MMM Mathematics-Magnetism-Materials,
University of Vienna, Vienna, Austria.

###### Abstract

We propose a novel device concept using spin-orbit-torques to realize a
magnetic field sensor, where we eliminate the sensor offset using a
differential measurement concept. We derive a simple analytical formulation
for the sensor signal and demonstrate its validity with numerical
investigations using macrospin simulations. The sensitivity and the measurable
linear sensing range in the proposed concept can be tuned by either varying
the effective magnetic anisotropy or by varying the magnitude of the injected
currents. We show that undesired perturbation fields normal to the sensitive
direction preserve the zero-offset property and only slightly modulate the
sensitivity of the proposed sensor. Higher-harmonics voltage analysis on a
Hall cross experimentally confirms the linearity and tunability via current
strength. Additionally, the sensor exhibits a non-vanishing offset in the
experiment which we attribute to the anomalous Nernst effect.

##  I Introduction

Spintronics is is a very active field of research since the discovery of the
giant magnetoresistive (GMR) effect[1, 2]. With the ability to control the
magnetization of magnetic materials using spin-polarized currents[3, 4, 5],
new possibilities have emerged for spintronic devices[6, 7, 8, 9]. Spin-orbit-
torque (SOT) induced magnetization switching processes have proven to be
ultra-fast and energy efficient[10, 11, 12, 13, 14]. Additionally, spin
currents have been used to propagate chiral objects such as domain walls[15],
or skyrmions[16, 17, 18, 19]. A newly developing topic in this area is the
usage of SOT for the magnetic field sensing[20, 8, 21, 22, 23].

Accurate sensors for magnetic field measurements[24] are very important for
various applications like biosensors and bio-chips for medical usage[25], read
heads for magnetic recording devices[26], or for position measurements with
the vortex sensors used in the automotive industry [27]. Conventional
magnetoresistive field sensors rely on the response of a rather soft magnetic
layer to the external field. An additional hard magnetic layer that is
insensitive to changes of the external field acts as a reference system
introducing a field-dependent magnetoresistivity. The linear regime for these
systems can be tuned to various field ranges by adjusting the material
properties of the soft magnetic layer.

In order to obtain more accurate sensing signals, the conventional
magnetoresistive sensors are operated in Wheatstone bridge configurations,
leading to further problems such as fabrication tolerances of different
elements, thermal drift within and in between different elements. All
magnetoresistive measurements exhibit a positive onset due to Ohm’s law, which
makes it difficult to get accurate measurement results for small external
fields, known as the sensing offset. While the reduction of this offset leads
to an improvement in the sensing process, it can enhance the resolution of the
device to determine vanishing or very small fields. Reducing the offset
present at vanishing fields is a key challenge in magnetic-field measurement
applications. Several designs exist where the offset can be reduced, or nearly
eliminated. These solutions are either based on spinning current techniques
for Hall sensors, or still use Wheatstone bridges, where four distinct
magnetic stacks are required to eliminate the offset [21]. Recently, a 3D
magnetic field sensor was reported using SOTs [22], where the change in domain
wall positions leads to a sensor signal. However, in both cases the linear
range is extremely small.

In this work, we propose a sensor concept for magnetic-field measurements that
exploits the symmetric nature of SOT in order to eliminate the sensing offset.
Using micromagnetic simulations we demonstrate that a differential measurement
method using SOT leads to the elimination of the offset error. We reveal how
increasing the amplitude of the injected current can be used to tune the
sensitivity of the sensor, whereas higher perpendicular magnetic anisotropies
allow for enhancing the linear range. The robustness of the measurement scheme
is tested by investigating the role of perturbation fields normal to the
sensitive direction on the offset, and sensitivity. The proposed concept is
experimentally tested on a Hall-cross stack, where a Ta layer is used as the
heavy-metal layer and spin-polarizer, and a thin film of CoFeB is deposited on
top as the magnetic layer. Second harmonics measurements are performed to
obtain the linear sensor signal via the anomalous Hall effect.

##  II CONCEPT

Figure 1: Schematic illustration of the zero-offset magnetic field sensor. The
ferromagnetic layer (orange, in experiment $\mathrm{Co_{60}Fe_{20}B_{60}}$) is
situated on top of the heavy metal layer (gray, in experiment Ta) which acts
as a spin-polarizer. Upon application of a charge current through the HM-layer
$\bm{J}$ a spin-polarized current flows along $\bm{n}$ into the FM layer with
a polarization $\bm{p}=\left(\bm{J}\times\bm{n}\right)$. The behavior of the
single domained OOP magnetization is illustrated first in a) with a vanishing
external field, where the initial magnetization $\bm{m_{0}}$ is now tilted to
$\bm{m}(I_{+x})$ for a positive charge current, and to $\bm{m}(I_{-x})$ for a
negative charge current, respectively. The deviation angle $\Theta$ is equal
for both current directions. The change induced by an external field applied
in-plane is illustrated in b) where the presence of the external field
disturbs the symmetry of SOT. The resulting angles $\Theta_{+}$ and
$\Theta_{-}$ differ now in magnitude.

The proposed sensor concept is based on a Hall cross with a heavy metal
underlayer and a magnetic top layer with perpendicular magnetic anisotropy
(PMA), see Fig. 1. Subject to an electric current in $\pm x$-direction, the
spin Hall effect [28] in the heavy-metal layer leads to a spin current with a
polarization in $\mp y$-direction flowing into the magnetic layer. This leads
to a spin-torque referred to as spin-orbit torque (SOT) [5].

For a vanishing external field and charge current, the equilibrium
magnetization in the magnetic layer is perfectly aligned out-of-plane (OOP)
due to the PMA. The response of the magnetic system to the spin-orbit torque
is perfectly symmetric [29] when changing the sign of the electric current in
the heavy metal layer, see Fig. 1(a). However, if an external field is
applied, the equilibrium position of the magnetization is tilted and the
response to the SOT becomes asymmetric, see Fig. 1(b).

The main idea of the proposed sensing principle is to perform differential
measurements with alternating currents in order to exploit this symmetry and
eliminate the offset for the zero-field case.

##  III MODELING

In order to verify the feasibility of the proposed concept, we perform a
series of macrospin simulations where the spin-orbit-torque interactions with
the magnetization are described by the Landau-Lifshitz-Gilbert-Equation
(LLG)[30, 31]

$\centering{{\partial}_{t}\bm{m}}={-\gamma\bm{m}\times{\bm{H}}^{\mathrm{eff}}+\alpha{\bm{m}}\times{\partial}_{t}{\bm{m}}+\bm{T}_{\mathrm{damp}}+\bm{T}_{\mathrm{field}}},\@add@centering$
(1)

where $\gamma$ is the gyromagnetic ratio, $\bm{H}^{\mathrm{eff}}$ the
effective field of the FM-layer and $\alpha$ the Gilbert-damping-coefficient.
The LLG is extended by the two additional current induced torques [32, 33, 31]

$\bm{T}_{\mathrm{damp}}={\eta}_{\mathrm{damp}}\frac{j_{e}\gamma\hbar}{2e{\mu}_{0}tM_{s}}\bm{m}\times(\bm{m}\times\bm{p}),$
(2)

and

$\bm{T}_{\mathrm{field}}={\eta}_{\mathrm{field}}\frac{j_{e}\gamma\hbar}{2e{\mu}_{0}tM_{s}}\bm{m}\times\bm{p},$
(3)

with $\hbar$ being the reduced Planck-constant, $e$ being the elementary
charge, and $\mu_{0}$ being the vacuum permeability. The strength of the two
torques is given by the magnitude of the charge current $j_{e}$ applied in the
heavy metal layer, parameters of the magnetic system such as saturation
magnetization $M_{s}$, thickness $t$, as well as the dimensionless SOT
coefficients $\eta_{\mathrm{damp}}$ and $\eta_{\mathrm{field}}$.

Equation (1) can then be written as

$\displaystyle\centering{\partial}_{t}\bm{m}=-\gamma\bm{m}\times\left(\bm{H^{\mathrm{eff}}}-H_{\mathrm{dl}}\bm{m}\times\bm{p}-H_{\mathrm{fl}}\bm{p}\right)+\alpha{\bm{m}}\times{\partial}_{t}{\bm{m}},$
(4)

where

$H_{\mathrm{dl}}=\dfrac{j_{e}\hbar}{2e\mu_{0}tM_{s}}\cdot\eta_{\mathrm{dl}},$
(5)

and

$H_{\mathrm{fl}}=\dfrac{j_{e}\hbar}{2e\mu_{0}tM_{s}}\cdot\eta_{\mathrm{fl}}$
(6)

are the magnitudes of the current induced damping-like and field-like torques.

###  III.1 First-order analytical approximation

The magnetization dynamics of a perpendicularly magnetized system can be
described by macrospin simulations where Eq. (4) is solved numerically, or by
employing an analytical approach like Hayashi et al[34]. Here, we derive an
even simpler first-order approximation based on the Stoner-Wolfarth formalism.

Let us assume now that the ferromagnetic layer is homogeneously magnetized
along the [001] direction, $m_{0}=(0,0,1)^{T}$. When a homogeneous magnetic
field is applied in the $xy$ plane, the effective field $H^{\mathrm{eff}}$ is
given solely by the contribution from the anisotropy and Zeeman energies [31]
as

$H^{\mathrm{eff}}=\begin{pmatrix}H_{x}\\\ H_{y}\\\ H_{k}\end{pmatrix},$ (7)

where $H_{x}$, and $H_{y}$ are the components of the applied in-plane field,
and $H_{k}$ is the effective anisotropy field.

In order to account for the spin-orbit torque, the effective field has to be
complemented according to Eq. (4). In the case of a spin-polarization in
$y$-direction, the field-like torque only contributes via the $y-$component of
the effective field

$H^{\mathrm{eff}}_{y}=H_{y}-p_{y}\cdot H_{\mathrm{fl}},$ (8)

whereas the damping-like torque contributes to the $x$-component as

$H^{\mathrm{eff}}_{x}=H_{x}-(\bm{m}\times\bm{p})_{x}\cdot H_{\mathrm{dl}}.$
(9)

The magnitude of the external field applied along the sensitive direction and
SOT fields is then given by

$H_{\parallel}=\sqrt{(H^{\mathrm{eff}}_{x})^{2}+(H^{\mathrm{eff}}_{y})^{2}}.$
(10)

According to the hard-axis approximation of the Stoner-Wohlfarth model, the
parallel component of the magnetization is given as

$m_{\parallel}=\dfrac{H_{\parallel}}{H_{k}},$ (11)

where $H_{k}$ is the anisotropy field.

The $z$ component of the magnetization can then be simply obtained from the
unit magnetization constraint where

$m_{z}=\pm\sqrt{1-m_{\parallel}^{2}},$ (12)

with the sign depending on the initial sign of $m_{z}$.

Consider now, that we apply a DC current $I_{0}$ along $+x$-direction. For the
sake of simplicity we refer to this current as $I_{+x}$. If the current is
applied along $-x$-direction, $I_{-x}=-I_{0}$. As explained in Sec. II this
leads to a change in magnetization, with the new magnetization state being
$m_{z}(I_{+x})$, which is proportional to the anomalous Hall resistance

$R_{xy}(I_{+x})=\Delta R_{AHE}m_{z}(I_{+x}),$ (13)

where $\Delta R_{\mathrm{AHE}}$ is the anomalous hall coefficient. Thus, one
obtains the hall voltage $V_{xy}(I_{+x})=\Delta R_{AHE}m_{z}(I_{+x})I_{+x}$.

The sensor signal can be obtained either as the difference of measured
resistances

$S=R_{xy}(I_{+x})-R_{xy}(I_{-x}),$ (14)

or as a sum of measured AHE voltages

$S_{xy}=V_{xy}(I_{+x})+V_{xy}(I_{-x}).$ (15)

The linearity of the sensor sensor signal $S$ can be derived from Eq. (12)
directly. Let us now assume the most general case, where both SOT coefficients
are included, and thus both $x,$ and $y$ external fields are applied to be
aligned with the sensitive direction. We can then write

$V_{xy}(I_{+x})=\pm\Delta R_{AHE}I_{+x}\sqrt{1-\dfrac{(H_{x}\mp
H_{\mathrm{dl}})^{2}+(H_{y}+H_{\mathrm{fl}})^{2}}{H_{k}^{2}}}.$ (16)

After expanding Eq.(16) in a Taylor series at $H_{x}=0$ and $H_{y}=0$ and
summing up the resulting voltages given in Appendix Eq. 37 according to Eq.
(15), one obtains the simplified analytical approximation of the sensor signal
where

$S_{xy}=-2\Delta
R_{AHE}I_{0}\left(H_{\mathrm{dl}}\dfrac{H_{x}}{{H_{k}}^{2}}\mp
H_{\mathrm{fl}}\dfrac{H_{y}}{{H_{k}}^{2}}\right)+\mathcal{O}[H_{x}]^{2}+\mathcal{O}[H_{y}]^{2}.$
(17)

The field components $H_{x}$ and $H_{y}$ can be written as
$H_{x}=H\cos{\phi_{H}}$ and $H_{y}=H\sin{\phi_{H}}$, respectively. Hence, the
total sensor signal becomes

$S_{xy}=-2\Delta R_{AHE}I_{0}\dfrac{H(H_{\mathrm{dl}}\cos{\phi_{H}}\mp
H_{\mathrm{fl}}\sin{\phi_{H}})}{H_{k}^{2}}+\mathcal{O}[H_{x}]^{2}+\mathcal{O}[H_{y}]^{2}.$
(18)

###  III.2 Equivalency to higher harmonics voltage measurements

For systems where very low signals are expected, or highly accurate
measurements are desired, a higher harmonics voltage analysis can be
performed[29, 34, 14, 35]. In this case, one applies an AC current instead of
performing two subsequent measurements with opposite currents. The applied
frequencies are in the Hz range.

Consider again the perpendicularly magnetized system, where the magnetization
is modulated due to SOTs. Upon application of the current
$I(t)=I_{0}\sin(\omega t)$, the magnetization component $m_{z}$, and thus,
$R_{xy}$ oscillates with the frequency $2\omega$. The hall voltage can be
expressed as

$V_{xy}(t)=R_{xy}I(t).$ (19)

Substituting $R_{xy}$ with the temporal evolution of Eq. (13) one obtains

$V_{xy}(t)=\Delta R_{\mathrm{AHE}}I(t)m_{z}(t),$ (20)

which can be further completed with the first order approximation for the
$z-$component of the magnetization to

$V_{xy}(t)=\pm\Delta R_{\mathrm{AHE}}I_{0}\sin{(\omega
t)}\sqrt{1-\dfrac{(H_{x}\mp
H_{\mathrm{dl}}(t))^{2}+(H_{y}+H_{\mathrm{fl}}(t))^{2}}{H_{k}^{2}}}.$ (21)

As described in Appendix B one can then perform a second order Fourier series
expansion of Eq. (21) with respect to $\omega t$, and sum up the terms
accordingly to

$\displaystyle V_{xy}(t)=$ $\displaystyle\dfrac{I_{0}\Delta
R_{AHE}\left(H_{\mathrm{dl}}H_{x}\mp
H_{\mathrm{fl}}H_{y}\right)}{2{H_{k}}^{2}}$ (22)
$\displaystyle\mp\dfrac{I_{0}\Delta
R_{AHE}\left(3(H_{\mathrm{dl}}^{2}+H_{\mathrm{fl}}^{2})-16H_{k}^{2}+4(H_{x}^{2}+H_{y}^{2})\right)}{8H_{k}^{2}}\sin(\omega
t)$ $\displaystyle\pm\dfrac{I_{0}\Delta R_{AHE}\left(H_{\mathrm{dl}}H_{x}\mp
H_{\mathrm{fl}}H_{y}\right)}{2{H_{k}}^{2}}\cos(2\omega t).$

Based on the common higher-harmonics voltage analysis where the time dependent
AHE voltage is given as

$V_{xy}(t)=V_{0}+V_{\omega}\sin{(\omega t)}+V_{2\omega}\cos{(2\omega t)},$
(23)

one can express the second-harmonic voltage as

$V_{2\omega}=\pm\dfrac{I_{0}\Delta R_{AHE}\left(H_{\mathrm{dl}}H_{x}\mp
H_{\mathrm{fl}}H_{y}\right)}{2{H_{k}}^{2}},$ (24)

which differs from our sensor signal approximation from Eq. (18) only by a
constant factor.

Thus, one can measure the sensor signal either by the sum of AHE voltages for
positive and negative currents, by applying an AC current and measuring the DC
contribution, or by measuring the field dependence of the second harmonics
voltage using higher-harmonics voltage analysis (HHVA), e.g. with a lock-in
amplifier.

###  III.3 Micromagnetic Simulations

In order to check the accuracy of the analytical model, we additionally
perform fully dynamical LLG simulations. For the sake of simplicity, we use a
single-cell approach for the simulation of the magnetic layer using the
micromagnetic simulation software magnum.np[36]. For the effective field, we
consider the effective anisotropy field and the external field only. The
effect of the demagnetization field is considered in terms of a contribution
to the anisotropy field. As material parameters, we have chosen the saturation
magnetization $\mu_{0}M_{s}=$1.2\text{\,}\mathrm{T}$$ and Gilbert damping
constant $\alpha=1$, as suitable material parameters for a Ta-CoFeB system
[5]. Since the thickness of the magnetic layer is crucial for the strength of
current induced torques, we assume everywhere a thickness of
$t=$1\text{\,}\mathrm{n}\mathrm{m}$$. To obtain the voltage from the
magnetization component $m_{z}$ we consider a cross-section $A=L\times
t=$10\text{\,}\mu\mathrm{m}$\times$1\text{\,}\mathrm{n}\mathrm{m}$$ to
calculate the applied current from the magnitude of the applied current
density $j_{e}$ and we assume $\Delta R_{\mathrm{AHE}}=$1\text{\,}\Omega$$.

##  IV EXPERIMENTAL METHODS

Our numerical investigations are accompanied by experimental validations,
where a SOT stack consisting of an HM layer, Ta, and an FM Layer, CoFeB is
used. Due to the equivalence showed in Sec. III.2, we are measuring the second
harmonics voltage with a Lock-in technique, and investigate the field-
dependency of the sensor signal.

###  IV.1 Sample Fabrication

The SOT structure was grown on an 8” silicon wafer using a Singulus Rotaris
tool. Before depositing the film stack, the wafers got an aluminum
metallization layer to contact the SOT structure from the bottom. The
connection of the SOT structure with the aluminum metal layer through an
insulation $\mathrm{SiO_{2}}$ layer is realized with tungsten vias. In order
to have a smooth surface before depositing the SOT film stack, a chemical
mechanical polish was carried out. The film stack consists of
$\mathrm{Ta}[$6\text{\,}\mathrm{n}\mathrm{m}$]/\mathrm{C_{60}Fe_{20}B_{20}}[$1\text{\,}\mathrm{n}\mathrm{m}$]/\mathrm{MgO}[$1.5\text{\,}\mathrm{n}\mathrm{m}$]/\mathrm{Ta}[$5\text{\,}\mathrm{n}\mathrm{m}$]$.
It was grown without vacuum brake by physical vapor deposition (PVD) with a
base pressure $<5\times 10^{-9}$ torr. The used sputter gas was Argon for all
layers. Metal deposition was done in DC sputtering mode, and MgO deposition
was performed in RF mode. After deposition, the wafers were annealed for
$2\text{\,}\mathrm{h}$ at a temperature of $280\text{\,}{}^{\circ}\mathrm{C}$
in a vacuum. The patterning of the SOT structure can be done by reactive ion
etching (RIE) with chlorine or by ion beam etching (IBE) whereas the latter
needs an additional hard mask. The Hall bars used in this study were patterned
from the films using conventional optical lithography and Ar ion etching. To
prevent the SOT structure from corrosion a passivation layer was deposited.
Note that a further encapsulation by aluminum-filled vias surrounding and
metallization layers on top of the SOT structure leads to better heat
dissipation. Finally, the pads were released by opening the passivation layer.
Standard separation techniques were used (mechanical dicing).

###  IV.2 Second Harmonics Measurements

The harmonic voltages are measured using a standard 2nd harmonic lock-in
amplifier detection-based technique, where we utilize a vector magnet to apply
the magnetic field in different directions. We use standard wire bonding to
connect the Hall bar device with the sample holder. The Hall bar is connected
to 50 $\Omega$ resistance in series to measure the input sinusoidal current
during the measurement. The schematic is shown in Fig.1a). Prior to the
harmonic measurements, the Hall bar device is pre-saturated along the $\pm
z$-direction. We apply a sinusoidal voltage with constant amplitude ($V_{in}$)
using a lock-in amplifier (Model number: HF2LI by Zurich instrument) to the
Hall bar device with a reference frequency
$13.7\text{\,}\mathrm{H}\mathrm{z}$. Note that we will be giving current
densities as input currents to obtain a better analogy to the numerical
studies. Other two lock-in amplifiers (model numbers: 7265 and 7225) were used
to measure the in-phase first and out-of-phase second harmonic voltage
simultaneously.

##  V Results

###  V.1 Role of SOT coefficients

We begin our numerical investigation with a thorough investigation of the role
of the SOT coefficients on the equilibrium states, as this will define the
sensitive direction of our magnetic field sensor. By using a single-cell
approach as explained in Sec. III.3, we vary the field-like and damping-like
coefficients $\eta_{\mathrm{damp}}$ and $\eta_{\mathrm{field}}$ which will
alter the strength of the torques exerted on the magnetization. For this
purpose we are using an effective anisotropy field
$\mu_{0}H_{k}=$200\text{\,}\mathrm{m}\mathrm{T}$$, and apply an SOT current of
magnitude $j_{e}=$10\text{\,}\mathrm{M}\mathrm{A}\mathrm{c}\mathrm{m}^{-2}$$.

In Fig. 2 one can see the influence of the SOT coefficients on the equilibrium
magnetization state. As expected, and briefly mentioned during the derivation
of the first order analytical approximation, the variation of
$\eta_{\mathrm{dl}}$ only changes the $x-$component of the magnetization, see
Fig. 2a), whereas the variation of $\eta_{\mathrm{fl}}$ modulates $m_{y}$, as
it is illustrated in Fig. 2b). Both torques contribute to the inclination of
the magnetization in a symmetrical way, as shown in Fig. 2c). In principle,
the choice of the HM layer will lead to a specific pair of
$\eta_{\mathrm{damp}}$ and $\eta_{\mathrm{field}}$[5], which in return will
then modulate the sensitive direction of the sensor. The strength of the SOT
coefficients can then be determined by measuring the first, and second
harmonic voltages, and applying the Hayashi method for perpendicularly
magnetized systems [34]. If the planar hall-effect is neglectable, than the
expression for $V_{\omega}$ and $V_{2\omega}$ given in Eq. (22) can also be
used. Alternative techniques such as Spin-Torque Ferromagnetic Resonance (ST-
FMR)[5] exist that lead to the precise determination of the damping-like
coefficient.

Figure 2: Dependence of the SOT coefficients on the equilibrium magnetization
when an SOT current is applied to macrospin is shown where from a) we see that
the damping like torque modulates the $x$-component. From b) is obvious that
the field-like coefficient modulates the y-component of the magnetization,
whereas from c) we learn that both torques contribute symmetrically to the
modulation of the z-component of the magnetization.

We have chosen $\eta_{\mathrm{damp}}=-0.4$ and $\eta_{\mathrm{field}}=0.5$ for
all the future investigations, such that the sensitive direction is slightly
away from the [110] axis. The sensitive axis can be calculated also simply
from the ratio of the two current induced torques given in Eqs. (5) and (6)
where $\tan{\phi_{H}}=H_{\mathrm{dl}}/H_{\mathrm{fl}}$. Note that the signs of
the SOT coefficients depend on their definition in the LLG. In our case a
positive field-like coefficient means, that the torque acts parallel to the
Oersted field that is generated by the current flow in the SOT stack[5].

###  V.2 Operation as a linear magnetic field sensor

Using our micromagnetic single cell simulations we apply the charge current
subsequently in opposite directions, where
$j_{e}=$10\text{\,}\mathrm{M}\mathrm{A}\mathrm{c}\mathrm{m}^{-2}$$, and
$\mu_{0}H_{k}=$199\text{\,}\mathrm{m}\mathrm{T}$$. Now, we apply a magnetic
field along the obtained sensitive axis $\bm{H}_{\parallel}$. Figure 3a) shows
the evolution of $V_{xy}$ as the response of the magnetization to the applied
field. As in the simplified analytical model (Eq. (16), lines), the macrospin
simulations (filled symbols) yield a parabolic dependence of $V_{xy}$ from the
measured field $H_{\parallel}$. For opposite charge currents, the
magnetization is modulated symmetrically around the vanishing magnetic field.
If the applied magnetic field exceeds a specific threshold, then the
magnetization switches sign.

Making use of the proposed differential measurement, we obtain the sensor
signal shown in Fig. 3b). The sensor signal is obtained as the sum of Hall
voltages. These signals are obtained from macrospin simulations (red), from
Eq. (15) (blue), and from Eq. (18) (green line). A high linear range is
observed, which is slightly smaller than the anisotropy field. Figure 3c)
illustrates the zoom of gray rectangle from b), which shows a good agreement
between all three curves in the limit of small fields. The sensitivity of the
sensing device, given by the slope of the sensor signal, starts to deviate for
the analytical solutions significantly for higher fields, where we are
approaching the anisotropy field. It is worth reminding that the small field
limit was an approximation we have done in Sec. III.1. Investigation of the
transfer curves clearly yields offset-free sensor signals for all theoretical
models.

Figure 3: The simulated (pink, green) and calculated (red, blue) AHE voltages
using simple approximation from Eq.(12) are depicted in a) for both current
directions starting from $\bm{m}_{0}=(0,0,1)^{\mathrm{T}}$. The calculated
sensor signal is then plotted in b). A very good agreement between the
analytical solutions, as well as the simulations can be observed in the limit
of lower fields, which is highlighted in c), where we zoom into relevant
range.

To better understand the sensing performance, we are investigating the role of
the effective anisotropy $\mu_{0}H_{k}$, as well as of the applied SOT current
$J_{x}$. First, we vary $\mu_{0}H_{k}$, while keeping the applied current
constant at $je=$10\text{\,}\mathrm{M}\mathrm{A}\mathrm{c}\mathrm{m}^{-2}$$.
The numerically obtained sensor signals are illustrated in Fig. 4. We observe
that it is possible to modulate both sensitivity and linear range of the
sensor by the variation of the anisotropy field. The strength of the effective
anisotropy field can be tuned via the thickness of the FM Layer (if the PMA
originates as an interface effect). In materials like Ta-CoFeB or Pt-Co, which
are the materials of choice for investigation of SOT, the anisotropy field at
room temperature can vary between $50\text{\,}\mathrm{m}\mathrm{T}$ to
$300\text{\,}\mathrm{m}\mathrm{T}$ [5]. Based on desired application it is
possible in principle to tune the linear range by either a change of
materials, or adjusting the thickness of the FM layer.

Figure 4: Dependence of the sensor signal on the strength of the perpendicular
magnetic anisotropy is shown in a), where the linear range increases with
$\mu_{0}H_{k}$, but to the cost of the sensitivity. The applied charge current
density is depicted in b), where one can see that a higher current increases
the sensitivity of the sensor signal, but in exchange it reduces the
measurable linear range.

A second measure to tune the sensitivity and linear range of the sensor is the
magnitude of the electric current applied to the heavy metal layer, see Fig.
4b). At higher currents the magnetization switches much faster into the in-
plane state due to the stronger current induced torques, limiting the linear
range to be used for magnetic field detection. Higher modulations can be
obtained using currents below the switching threshold, leading to the increase
of the sensitivity. Note that a similar effect could be obtained by changing
the material acting as HM, in order to scale the exerted torques via the SOT
coefficients, or alternative spin polarizing materials can be chosen that use
different physics, like the orbital orbit torque effect[37]. However, the
dependence of the sensitivity on the applied current allows us to obtain a
magnetic field sensor with tunable sensitivity, and linear range. Independent
of the chosen parameters the sensing signal remains always offset free.

###  V.3 Sensor performance

Magnetic field sensors can be best compared on their properties like linear
range, offset, or sensitivity. Hence, we are interested in the dependence of
the linear range, and sensitivity on the applied current density $J_{x}$ and
anisotropy field $\mu_{0}H_{k}$. The latter is a measure for the used
material. Figure 5a) illustrates the dependence of the chosen effective
anisotropy and applied current on the obtained linear measurement range of
magnetic sensor. From Fig. 5 it can be seen that the linear range is very well
tunable with the change of the $\mu_{0}H_{k}$, and $J_{x}$. Considering the
sensitivity of the sensor signal depicted in Fig. 5b), we distinguish between
two phases. Above a certain given threshold of anisotropy and current the
sensitivity drops significantly, and it cannot be tuned with the current
anymore.

Figure 5: Magnetic field sensing properties as a function of applied SOT
current $J_{x}$ and effective magnetic anisotropy$\mu_{0}H_{k}$, where the
measurable linear range is shown in a), and the sensitivity is shown in b).

Another important measure for the performance of the sensor is its robustness
against field components perpendicular to the measured field direction. In
general, an arbitrary magnetic field will have an in-plane component that can
be projected along the sensitive direction $H_{\parallel}$, and a normal
component $H_{\perp}$, which might change the equilibrium state at vanishing
fields as well, leading to poorer sensor performance.

In order to investigate the influence of perpendicular field components, we
vary the strength of the normal component $H_{\perp}$, as well as the applied
angle $\phi$. We first calculate the sensitive direction, normalize it, and
rotate the normalized vector $90^{\circ}$ around the $z$-axis. Thus, when
$\phi=0$, and $m_{\parallel}=(1,1,0)$, $H_{\perp}$ is applied along
$(-1,1,0)$.

Figure 5a) reveals that a perturbation field does not influence the sensor
offset, independent of amplitude and angle. Note that if $H_{\perp}>H_{k}$
then the magnetization will point along $H_{\perp}$ compromising the general
sensing concept. A weak dependence of the sensitivity on the amplitude of
$H_{\perp}$, and $\phi$ can be seen in Fig. 5. When the normal field is
applied, the sensitivity decreases as the equilibrium state will be slightly
modulated if positive or negative currents are applied. However, there exists
again a perturbation field threshold,
$\mu_{0}H_{\perp}<$10\text{\,}\mathrm{m}\mathrm{T}$$, where the sensitivity of
the sensor signal remains rather constant and independent of the angle of the
magnetization.

Figure 6: Influence of the external bias fields on the offset of the sensor
signal (a) and on the sensor sensitivity (b). While the sensor offset remains
in principle always offset free, the sensitivity changes slightly, yet
insignificantly, with the amplitude of the perpendicular field $H_{\perp}$.

###  V.4 Higher Harmonics Voltage Measurements

To demonstrate the developed sensor concept, we have fabricated a Hall-cross
structure, as explained in Sec. IV. Using the anomalous Hall effect one can
measure the change in $m_{z}$. Since our sensor is based on a differential
measurement in order to eliminate the sensing offset, we employ second
harmonics measurements, as discussed in Sec. IVB. Figure 1a) illustrates the
considered Hall structure. The measured $V_{2\omega}$ curves are plotted as a
function of the applied magnetic field in Fig.7. While the dependence of the
$V_{2\omega}$ on $H_{x}$ is given in Fig. 7a), the dependence of $H_{y}$ is
illustrated in Fig. 7b). Red (blue) curves show the signal if the sample was
pre-saturated along $+z$ ($-z$) directions using an OOP magnetic field. The
charge current density for SOT sensing was
$J_{x}=$28.6\text{\,}\mathrm{M}\mathrm{A}\mathrm{c}\mathrm{m}^{-2}$$.

As expected from our simple analytical derivation, as well as from the
derivations of Hayashi[34], the $2\omega$ signal should change signs when the
magnetic field is applied along the current direction if the initial
magnetization state changes signs. If the magnetic field is applied
perpendicular to the current direction, i.e. $H_{y}$, then the slope of the
measured signal changes, as depicted in Fig. 7.

If one decreases the applied SOT current then the sensitivity of the sensor
signal decreases as theoretically described above, and experimentally
demonstrated and depicted in Fig. 7c).

Overall, all signals are linearly depending on the applied magnetic field,
thus allowing us the experimental validation of sensor principle using SOT.
However, a significant offset is observed if the magnetization was pre-
saturated along $-z$ direction. While in experimental works where the HHVA is
performed one usually disregards this offset attributing them to electronics,
it limits the sensing performance of our single-device magnetic field sensor.
Part of the offset can originate from the electronics in the measurement
setup. However, also the sample can contribute to the offset due to the
anomalous Nernst effect, which gives a contribution to the AHE and can be
quantified. With this correction (that we are not plotting here to show the
real measurement data) only a very minor offset remains, which we attribute to
the electronics of the system.

Figure 7: Experimental results obtained from the second harmonics
measurements, where first the field parallel to current direction ($H_{x}$) is
swept starting from both magnetization states, are shown in a). The
transversal field is swept in b). The tunable sensitivity can be observed in
c) where we sweep the longitudinal field for the charge current densities
$J_{x}=$28.6\text{\,}\mathrm{M}\mathrm{A}\mathrm{c}\mathrm{m}^{-2}$$ (green,
triangles) and
$J_{x}=$42.9\text{\,}\mathrm{M}\mathrm{A}\mathrm{c}\mathrm{m}^{-2}$$ (blue,
squares).

Consider now that current flow in $x-$direction causes an increase in
temperature due to Joule heating. Assuming that the change in temperature is
small enough to not alter any material parameters, it is expected that we will
have temperature gradients ($\nabla T$).

When the magnetization $\bm{m}$ and $\nabla T$ are normal to each other, then
the anomalous Nernst effect (ANE) gives an additional contribution due to the
generated electrical field[38]

$\bm{E}_{\mathrm{ANE}}=Q_{s}\mu_{0}M_{s}\left(\bm{m}\times\nabla T\right),$
(25)

where $Q_{s}$ is the ANE coefficient.

Let us now consider that the gradient of temperature has only a $z$ component,
and we have an arbitrary magnetization state. Thus, the ANE generates an
electric field given by

$\bm{E}_{\mathrm{ANE}}(I_{+x})=Q_{s}\mu_{0}M_{s}\begin{pmatrix}m_{x}\\\
m_{y}\\\ m_{z}\end{pmatrix}\times\begin{pmatrix}0\\\ 0\\\
\partial_{z}T\end{pmatrix}=\begin{pmatrix}m_{y}\partial_{z}T\\\
-m_{x}\partial_{z}T\\\ 0\end{pmatrix}.$ (26)

The measured hall voltage then becomes

$V_{xy}(I_{+x})=V_{\mathrm{AHE}}(I_{+x})+V_{\mathrm{ANE}}(I_{+x}),$ (27)

where

$V_{\mathrm{ANE}}(I_{+x})=wE_{y}(I_{+x})=-wQ_{s}\mu_{0}M_{s}m_{x}\partial_{z}T$
(28)

For the opposite current the magnetization changes signs for the $x,$ and $y$
components leading to

$V_{\mathrm{ANE}}(I_{-x})=wE_{y}(I_{-x})=wQ_{s}\mu_{0}M_{s}m_{x}\partial_{z}T.$
(29)

Summing up the measured hall voltages for the sensing signal according to
Eq.15 yields that contributions from the ANE due to $\partial_{z}T$ are
vanishing. Thus, we conclude that the z-component of the temperature gradient,
that originates in the dissipation of the Joule heating, does not lead to any
offset.

Let us now consider that the one obtains a gradient in temperature where

$\nabla T=\begin{pmatrix}\partial_{x}T\\\ 0\\\ 0\end{pmatrix},$ (30)

which can originate in the imperfections in current flow, and small symmetry
distortion, e.g. slightly non-symmetrical arrangements of vias due to
lithography tolerances.

The ANE contribution to the hall voltage then becomes

$V_{ANE}(I_{+x})=wE_{y}(I_{+x})=wQ_{s}\mu_{0}M_{s}m_{z}\partial_{x}T.$ (31)

Changing the sign of the magnetization leads to the same term

$V_{ANE}(I_{-x})=wE_{y}(I_{-x})=wQ_{s}\mu_{0}M_{s}m_{z}\partial_{x}T,$ (32)

which leads to the offset $2wQ_{s}\mu_{0}M_{s}m_{z}\partial_{x}T$ when the
measured voltages are added. These considerations indicate, that the
nonvanishing offset obtained in the experiments can probably be attributed to
ANE. However not due to the z component of the gradient caused by the cooling
of the sample towards the heat sink, as we have initially expected, but by the
broken symmetry of the hall-bar due to fabrication defects, tolerances, or due
to misalignments.

##  VI CONCLUSION

In this work, we propose a novel concept for magnetic field sensing based on
spin-orbit-torques. In essence, the proposed concept is the equivalent to the
experiments performed to extract the SOT induced torques, as in the well
established Hayashi method[34].

In this sensing concept, the SOT leads to a symmetric inclination of the
initial out-of-plane magnetization for opposite current directions. We
investigate the roles of the SOT-coefficients $\eta_{\mathrm{damp}}$ and
$\eta_{\mathrm{field}}$, which are a measure of the damping-like and field-
like torques respectively.

Our numerical investigations reveal that the sensor signal evolves linearly
with the applied magnetic field in the working regime. This sensing principle
shows a large linear range (hundreds of mT), whereas the offset vanishes in
the absence of an external field. The sensitivity, as well as the linear
range, can be tuned by the variation of the amplitude of the charge current,
and anisotropy fields, respectively.

Perturbation fields applied in the normal plane to the sensitive direction do
not affect the elimination of the offset, however, they slightly change the
sensitivity of the sensor, and thus, lead to a poorer sensor performance.

The performed second harmonics measurements of the AHE allowed us to deliver a
proof-of concept that the proposed sensing method is applicable, and second
harmonics analysis can be performed to obtain a linear sensing signal.
Furthermore, the increase in the applied current leads to a higher sensitivity
allowing for the realization of a tunable modular sensor. However, the
experimental validations show that the AHE is not perfectly suitable to
measure the change in the magnetization, as the anomalous Nernst effect
contribution leads to an offset. This additional offset can be eliminated if
the magnetization is read through a TMR stack[23].

##  Appendix A Taylor Expansion of Sensor Signal

One can expand Eq. (16) in a Taylor series first at $H_{x}=0$, which yields

$\displaystyle V_{xy}(I_{+x})\lvert{{}_{H_{x}=0}}=\Delta
R_{AHE}I_{+x}\sqrt{-\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}-2H_{\mathrm{fl}}H_{y}+H_{k}^{2}-H_{y}^{2}}{H_{k}^{2}}}$
(33) $\displaystyle+\Delta
R_{AHE}I_{+x}H_{\mathrm{dl}}H_{x}\dfrac{\sqrt{\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}-2H_{\mathrm{fl}}H_{y}+H_{k}^{2}-H_{y}^{2}}{H_{k}^{2}}}}{{H_{k}}^{2}\sqrt{1+\dfrac{-H_{\mathrm{dl}}^{2}-(H_{\mathrm{fl}}+H_{y})^{2}}{H_{k}^{2}}}}+\mathcal{O}[H_{x}]^{2}.$

For the opposite current $I_{-x}$, we obtain

$\displaystyle V_{xy}(I_{-x})\lvert{{}_{H_{x}=0}}=\Delta
R_{AHE}I_{-x}\sqrt{-\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{fl}}H_{y}+H_{k}^{2}-H_{y}^{2}}{H_{k}^{2}}}$
(34) $\displaystyle-\Delta
R_{AHE}I_{-x}H_{\mathrm{dl}}H_{x}\dfrac{\sqrt{\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{fl}}H_{y}+H_{k}^{2}-H_{y}^{2}}{H_{k}^{2}}}}{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{fl}}H_{y}+H_{k}^{2}-H_{y}^{2}}+\mathcal{O}[H_{x}]^{2}.$

If one evaluates Eq. 16 at $H_{y}=0$, this yields

$\displaystyle V_{xy}(I_{+x})\lvert{{}_{H_{y}=0}}=\Delta
R_{AHE}I_{+x}\sqrt{-\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{dl}}H_{x}+H_{k}^{2}-H_{x}^{2}}{H_{k}^{2}}}$
(35) $\displaystyle-\Delta
R_{AHE}I_{+x}H_{\mathrm{fl}}H_{y}\dfrac{\sqrt{\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{dl}}H_{x}+H_{k}^{2}-H_{x}^{2}}{H_{k}^{2}}}}{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{dl}}H_{x}+H_{k}^{2}-H_{x}^{2}}+\mathcal{O}[H_{y}]^{2}.$

for positive current $I_{+x}$ and

$\displaystyle V_{xy}(I_{-x})\lvert{{}_{H_{y}=0}}=\Delta
R_{AHE}I_{-x}\sqrt{-\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}-2H_{\mathrm{fl}}H_{y}+H_{k}^{2}-H_{y}^{2}}{H_{k}^{2}}}$
(36) $\displaystyle+\Delta
R_{AHE}I_{-x}H_{\mathrm{fl}}H_{y}\dfrac{\sqrt{\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}-2H_{\mathrm{fl}}H_{y}+H_{k}^{2}-H_{y}^{2}}{H_{k}^{2}}}}{{H_{k}}^{2}\sqrt{1+\dfrac{-H_{\mathrm{fl}}^{2}-(H_{\mathrm{dl}}+H_{x})^{2}}{H_{k}^{2}}}}+\mathcal{O}[H_{y}]^{2},$

for negative current $I_{-x}$.

We can then simplify
$\sqrt{-\dfrac{-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{dl}}H_{x}+H_{k}^{2}-H_{x}^{2}}{H_{k}^{2}}}\approx
1$, since in the first order approximation one has to consider
$-H_{\mathrm{dl}}^{2}-H_{\mathrm{fl}}^{2}+2H_{\mathrm{dl}}H_{x}-H_{x}^{2}\ll
H_{k}^{2}$, which leads to even further simplifications, where

$\displaystyle V_{xy}(I_{+x})\lvert{{}_{H_{x}=0}}=\Delta R_{AHE}I_{0}+\Delta
R_{AHE}I_{0}\dfrac{H_{\mathrm{dl}}Hx}{H_{k}^{2}}+\mathcal{O}[H_{x}]^{2},$ (37)
$\displaystyle V_{xy}(I_{-x})\lvert{{}_{H_{x}=0}}=-\Delta R_{AHE}I_{0}+\Delta
R_{AHE}I_{0}\dfrac{H_{\mathrm{dl}}Hx}{H_{k}^{2}}+\mathcal{O}[H_{x}]^{2}$
$\displaystyle V_{xy}(I_{+x})\lvert{{}_{H_{y}=0}}=\Delta R_{AHE}I_{0}-\Delta
R_{AHE}I_{0}\dfrac{H_{\mathrm{fl}}Hy}{H_{k}^{2}}+\mathcal{O}[H_{y}]^{2},$
$\displaystyle V_{xy}(I_{-x})\lvert{{}_{H_{y}=0}}=-\Delta R_{AHE}I_{0}-\Delta
R_{AHE}I_{0}\dfrac{H_{\mathrm{fl}}Hy}{H_{k}^{2}}+\mathcal{O}[H_{y}]^{2}.$

Summing up all the Taylor expansions one can then obtain the the sensor signal
according to Eq. (15).

##  Appendix B Fourier Expansion of AC Sensor Signal

The measured time-dependent voltage output $V_{xy}(t)$ upon application of an
AC SOT current can be divided into four cases. If one starts with an initial
magnetization pointing along $[001]$, i.e. $m_{z}>0$, and the the magnetic
field being applied along $x-$direction, the AHE voltage is given by

$V_{xy}(t)=\Delta R_{AHE}I_{0}\sin{(\omega
t)}\cdot\sqrt{1-\dfrac{\left(H_{x}-H_{\mathrm{dl}}(t)\right)^{2}}{H_{k}^{2}}},$
(38)

where the current induced damping-like torque is now time-dependent and
becomes

$H_{\mathrm{dl}}(t)=H_{\mathrm{dl}}\sin{(\omega t)},$ (39)

with $\omega$ being the frequency of the applied AC current. Thus one can
write the Eq. (38) as

$V_{xy}(t)=\Delta R_{AHE}I_{0}\sin{(\omega
t)}\cdot\sqrt{1-\dfrac{\left(H_{x}-H_{\mathrm{dl}}\sin{(\omega
t)}\right)^{2}}{H_{k}^{2}}}.$ (40)

Using the approximation
$\sqrt{1-\frac{a}{b^{2}}}\approx\left(1-\frac{a}{2b^{2}}\right)$, for $(a\ll
b^{2})$, one can further simplify Eq. (40) to

$V_{xy}(t)=\Delta R_{AHE}I_{0}\sin{(\omega
t)}\cdot{\left(1-\dfrac{\left(H_{x}-H_{\mathrm{dl}}\sin{(\omega
t)}\right)^{2}}{2H_{k}^{2}}\right)}.$ (41)

The second order Fourier expansion of Eq. (41) at $\omega t$ can then be
written as

$\displaystyle V_{xy}(t)=$ $\displaystyle\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{dl}}H_{x}}{2{H_{k}}^{2}}$ (42)
$\displaystyle-\dfrac{I_{0}\Delta
R_{AHE}\left(3H_{\mathrm{dl}}^{2}-8H_{k}^{2}+4H_{x}^{2}\right)}{8H_{k}^{2}}\sin(\omega
t)$ $\displaystyle-\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{dl}}H_{x}}{2{H_{k}}^{2}}\cos(2\omega t).$

Considering that one starts with the opposite magnetization state
($[00\bar{1}]$, $m_{z}$<0), the the Fourier expansion becomes

$\displaystyle V_{xy}(t)=$ $\displaystyle\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{dl}}H_{x}}{2{H_{k}}^{2}}$ (43)
$\displaystyle+\dfrac{I_{0}\Delta
R_{AHE}\left(3H_{\mathrm{dl}}^{2}-8H_{k}^{2}+4H_{x}^{2}\right)}{8H_{k}^{2}}\sin(\omega
t)$ $\displaystyle-\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{dl}}H_{x}}{2{H_{k}}^{2}}\cos(2\omega t).$

Thus, as a general case on can then write

$\displaystyle V_{xy}(t)=$ $\displaystyle\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{dl}}H_{x}}{2{H_{k}}^{2}}$ (44)
$\displaystyle\mp\dfrac{I_{0}\Delta
R_{AHE}\left(3H_{\mathrm{dl}}^{2}-8H_{k}^{2}+4H_{x}^{2}\right)}{8H_{k}^{2}}\sin(\omega
t)$ $\displaystyle-\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{dl}}H_{x}}{2{H_{k}}^{2}}\cos(2\omega t).$

Analogously one can derive the Fourier expansion if the magnetic field is
applied along the $y-$direction. In this case, the general form can then be
written as

$\displaystyle V_{xy}(t)=$ $\displaystyle\mp\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{fl}}H_{y}}{2{H_{k}}^{2}}$ (45)
$\displaystyle\mp\dfrac{I_{0}\Delta
R_{AHE}\left(3H_{\mathrm{fl}}^{2}-8H_{k}^{2}+4H_{y}^{2}\right)}{8H_{k}^{2}}\sin(\omega
t)$ $\displaystyle\pm\dfrac{I_{0}\Delta
R_{AHE}H_{\mathrm{fl}}H_{y}}{2{H_{k}}^{2}}\cos(2\omega t).$

Considering both initial magnetization states, as well as the different
applied magnetic fields a general equation of $V_{xy}(t)$ can be given:

$\displaystyle V_{xy}(t)=$ $\displaystyle\dfrac{I_{0}\Delta
R_{AHE}\left(H_{\mathrm{dl}}H_{x}\mp
H_{\mathrm{fl}}H_{y}\right)}{2{H_{k}}^{2}}$ (46)
$\displaystyle\mp\dfrac{I_{0}\Delta
R_{AHE}\left(3(H_{\mathrm{dl}}^{2}+H_{\mathrm{fl}}^{2})-16H_{k}^{2}+4(H_{x}^{2}+H_{y}^{2})\right)}{8H_{k}^{2}}\sin(\omega
t)$ $\displaystyle\pm\dfrac{I_{0}\Delta R_{AHE}\left(H_{\mathrm{dl}}H_{x}\mp
H_{\mathrm{fl}}H_{y}\right)}{2{H_{k}}^{2}}\cos(2\omega t).$

## Acknowlegements

The authors acknowledge funding from Senstronic Project. S.K.,S.Z.,F.B., C.A.,
J.S. and D.S. acknowledge funding from Österreichische
Forschungsförderungsgesellschaft (FFG) under the Project Senstronic. C.A
acknowledges funding from Austrian Science Fund (FWF) under Project No. P
34671. Computational results have been achieved, in part, by using the Vienna
Scientific Cluster. C.S., R.G. G.J., M.K. acknowledge funding from Deutsche
Forschung Gemeinschaft (CRC TRR 173 Spin+X, projects A01 and B02).

## AUTHOR DECLARATIONS

### Conflict of interest

The authors have no conflict of interest to disclose.

### Intellectual property

A.S., and D.S. have a licensed Patent under the Paten number uS 17/220,129
[23].

### Author Contributions

S.K., F.B., C.A., D.S. wrote and improved micromagnetic code. S.K.,J.S. and
D.S. derived analytical solutions, S.K. performed all computations and
macrospin simulations. S.Z. assisted in all SOT related evaluations. C.S.,
R.G., G.J., M.K. performed the higher harmonics measurements, Mi.K., K.P.
fabricated the samples. J.G., A.S., M.A., G.J., M.K., and D.S. supervised the
project. S.K. prepared and wrote the initial manuscript. All co-authors have
contributed to the final manuscript.

### Data availability

The data that support the findings of this study are available upon reasonable
request from the corresponding author.

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