# Skyrmionic device for three dimensional magnetic field sensing enabled by
spin-orbit torques

Sabri Koraltan<EMAIL_ADDRESS>Physics of Functional Materials,
Faculty of Physics, University of Vienna, A-1090 Vienna, Austria  Vienna
Doctoral School in Physics, University of Vienna, Vienna, Austria  Research
Platform MMM Mathematics-Magnetism-Materials, University of Vienna, A-1090
Vienna, Austria.  Rahul Gupta  Institute of Physics, Johannes Gutenberg
University Mainz, 55099 Mainz, Germany  Reshma Peremadathil Pradeep  Empa,
Swiss Federal Laboratories for Materials Science and Technology, Dübendorf
CH-8600, Switzerland  Fabian Kammerbauer  Institute of Physics, Johannes
Gutenberg University Mainz, 55099 Mainz, Germany  Iryna Kononenko  Institute
of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
National Academy of Sciences of Ukraine, Institute of Applied Physics, 58
Petropavlivska St., Sumy, 40000, Ukraine  Klemens Prügl  Infineon Technologies
AG, 93049 Regensburg, Germany  Michael Kirsch  Infineon Technologies AG, 93049
Regensburg, Germany  Bernd Aichner  Faculty of Physics, University of Vienna,
A-1090 Vienna, Austria  Santiago Helbig  Faculty of Physics, University of
Vienna, A-1090 Vienna, Austria  Florian Bruckner  Physics of Functional
Materials, Faculty of Physics, University of Vienna, A-1090 Vienna, Austria
Claas Abert  Physics of Functional Materials, Faculty of Physics, University
of Vienna, A-1090 Vienna, Austria  Research Platform MMM Mathematics-
Magnetism-Materials, University of Vienna, A-1090 Vienna, Austria.  Andrada
Oana Mandru  Empa, Swiss Federal Laboratories for Materials Science and
Technology, Dübendorf CH-8600, Switzerland  Armin Satz  Infineon Technologies
AG, 9500 Villach, Austria  Gerhard Jakob  Institute of Physics, Johannes
Gutenberg University Mainz, 55099 Mainz, Germany  Hans Josef Hug  Empa, Swiss
Federal Laboratories for Materials Science and Technology, Dübendorf CH-8600,
Switzerland  Department of Physics, University of Basel, 4056 Basel,
Switzerland  Mathias Kläui  Institute of Physics, Johannes Gutenberg
University Mainz, 55099 Mainz, Germany  Dieter Suess  Physics of Functional
Materials, Faculty of Physics, University of Vienna, A-1090 Vienna, Austria
Research Platform MMM Mathematics-Magnetism-Materials, University of Vienna,
A-1090 Vienna, Austria.

###### Abstract

Magnetic skyrmions are topologically protected local magnetic solitons that
are promising for storage, logic or general computing applications. In this
work, we demonstrate that we can use a skyrmion device based on
$\mathrm{[W/CoFeB/MgO]}_{10}$ multilayers for three-dimensional magnetic field
sensing enabled by spin-orbit torques (SOT). We stabilize isolated chiral
skyrmions and stripe domains in the multilayers, as shown by magnetic force
microscopy images and micromagnetic simulations. We perform magnetic transport
measurements to show that we can sense both in-plane and out-of-plane magnetic
fields by means of a differential measurement scheme in which the symmetry of
the SOT leads to cancelation of the DC offset. With the magnetic parameters
obtained by vibrating sample magnetometry and ferromagnetic resonance
measurements, we perform finite-temperature micromagnetic simulations, where
we investigate the fundamental origin of the sensing signal. We identify the
topological transformation between skyrmions, stripes and type-II bubbles that
leads to a change in the resistance that is read-out by the anomalous Hall
effect. Our study presents a novel application for skyrmions, where a
differential measurement sensing concept is applied to quantify external
magnetic fields paving the way towards more energy efficient applications in
skyrmionics based spintronics.

### Introduction

Magnetic skyrmions are localized magnetic solitons that are topologically
protected [1, 2, 3]. Their topological protection means that they cannot be
annihilated into the ferromagnetic state by a purely uniform and continuous
transformation without the appearance of magnetic singularites [4]. Skyrmions
can have different stabilization mechanisms. Most commonly, they are found in
systems where the inversion symmetry is broken due to the presence of the
Dzyaloshinskii-Moriya interaction(DMI) . Multiple systems exist where bulk [5]
or interfacial DMI [6] is responsible for the formation of skyrmions. The
latter relies on the DMI that arises at the interface between a ferromagnetic
layer and heavy metal layers [7]. Thus, metallic multilayers can be used to
enhance the stability region of skyrmions, allowing them to also appear at
room temperature [8]. These chiral skyrmions are called Néel skyrmions as
their cross-section reveals a Néel domain wall [4]. Note that skyrmions can
also be stabilized in materials without DMI, where the competition between
demagnetizing energies and low perpendicular magnetic anisotropy leads to the
formation of Bloch skyrmions[9, 10] and even higher-order skyrmions and
antiskyrmions [11].

Because the skyrmions can be found over a wide range of temperatures where
their sizes depend on the applied external fields and because of their energy
efficient propagation by electric currents [12, 13], they have been proposed
as information carriers in skyrmion race track memories [14, 15], logic
devices [16], spin-wave emitters [17, 18] and even in unconventional computing
tasks [19, 20] such as reservoir computers [21, 22, 23]. The latter is based
on the change in resistance that is obtained when skyrmions are breathing or
propagating. Generally, a transverse voltage results from the electrons
flowing in a skyrmionic circuit [24]. The most common contributions are the
anomalous Hall effect (AHE) [25] and topological Hall effect (THE) [26]. While
the former arises due to the change in the magnetization (for instance,
breathing of skyrmions), the latter emerges due to the presence of skyrmions
as topological defects that deflect the electrons. Thus, AHE and THE can be
used to characterize the change in the magnetization of a skyrmionic system.

Recently, AHE-based magnetic field sensing devices have triggered increased
attention [27, 28, 29]. The concept of magnetic field sensing can be
summarized quite simply: The aim is to measure the change in resistance when
the externally applied magnetic field changes [30]. Most designs rely on Hall
effect or magnetoresistive effects: anisotropic magnetoresistance (AMR), giant
magnetoresistance (GMR), or tunnel magnetoresistance (TMR) [31]. TMR sensors
are indeed gaining popularity in various applications due to their high
sensitivity compared to traditional Hall effect sensors. The drawback of
state-of-the-art TMR sensors is that they rely only on the shape anistropy of
an ellipsoidal element, where only small linear ranges can be achieved, strong
hysteretic effects appear, while high phase noise can be measured. Novel
concepts have been proposed and even commercially adapted, where magnetic
vortices were used to overcome these drawbacks [32]. Recently, AHE-based
sensors have been shown to be highly sensitive [27, 28], still lower than the
sensitivities achieved by TMR sensors. In most AHE-based sensors, the
measurable linear ranges do not exceed $4\text{\,}\mathrm{m}\mathrm{T}$, which
limits the application range of such sensors [33].

Further research work focussed on the use of spin-orbit torques [34, 35, 36]
(SOT) for an energy-efficient switching of the magnetization in prototypical
devices. Additional work used SOT for the controlled propagation of a domain
wall for three-dimensional field sensing [37], whereas our own work
demonstrated the suppression of inevitable sensor offsets by SOT [29].
However, while the linear field range remained limited for the 3D field
sensing devices [37], the sensor concepts developed in our own work suffered
from a relatively small field sensitivity [29].

In this work, we design and fabricate a skyrmionic hall bar type device to
measure in-plane (IP) and out-of-plane (OOP) components of the applied
magnetic field. The electrical read out makes use of the anomalous Hall
effect. Magnetic force microscopy performed under vacuum conditions is
employed to reveal the evolution of stripe domains to skyrmions and to test
the fidelity of micromagnetic simulations. We then use the latter to
demonstrate that the formation or annihilation of skyrmions, trivial bubbles
and stripe domains becomes symmetric with respect to the applied current
direction due to the SOT, which enables offset cancelation for IP fields.
Micromagnetic simulations accompany our experimental studies and reveal the
origin of the offset-free sensing principle and how the SOT introduced a
symmetry in the orientation of stripe domains and formation of spin textures.
We achieve a measurable linear range of $\pm 17\text{\,}\mathrm{m}\mathrm{T}$
for IP fields and $\pm 30\text{\,}\mathrm{m}\mathrm{T}$ for OOP fields,
respectively, while keeping the sensitivity moderately high. Our sensing
performance exceeds both previous academic and commercial devices, opening a
new paradigm for research in skyrmions-based spintronics.

### Néel Skyrmions in W/CoFeB/MgO Multilayers

Figure 1: Formation of Néel Skyrmions. The investigated thin-film
[W(5)/CoFeB(0.7)/MgO(1.2)]N multilayer stack is illustrated in (a) where the
layers are grown on a Ta(5) seed layer and capped with $\mathrm{HfO_{2}}$ (3)
layer to avoid oxidation. The numbers in parantheses represent the thickness
of each layer in nanometers. The hysteresis curve experimentally acquired by
VSM (red) and micromagnetically simulated curve (blue) are compared in (b).
Isolated Néel skyrmions as obtained from micromagnetic simulations for
negative (c) and positive (d) skyrmion cores, and as obtained from MFM images
in (e) and (f), respectively. The evolution of the magnetization states for
different magnetic fields is illustrated in (g)-(j) as snapshots of
micromagnetic simulations and as experimentally imaged using MFM, showing the
measured frequency shift patterns arising from the magnetization patterns in
(k)-(n). From the MFM images at vanishing fields, an average domain width
$\sigma_{DW}=$166.34\text{\,}\mathrm{n}\mathrm{m}$\pm$22.56\text{\,}\mathrm{n}\mathrm{m}$$
was calculated. The micromagnetic simulations use the input of the MFM state
as an initial state before applying the OOP field to observe the formation of
skyrmions.

The interfacial DMI originating from the spin-orbit coupling arising at the
interface between a heavy metal and ferromagnetic layers is a well-studied
mechanism for the nucleation of skyrmions under an external field [8, 38]. In
this article, we study a multilayer
Si/SiO2/Ta(5)/[W(5)/CoFeB(0.7)/MgO(1.2)]x10 sample (all thicknesses in
nanometers), which was deposited by DC and RF magnetron sputtering (as
depicted in Fig. 1a); see Methods for more details.

In our study, the relevant system parameters of our magnetic thin films were
analyzed through the application of vibrating sample magnetometry (VSM),
ferromagnetic resonance (FMR), and magnetic force microscopy (MFM). A
saturation magnetization
$M_{s}=$1197.5\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$ was
determined by VSM and then used to determine the anisotropy constant
$K_{u}=$1042\text{\,}\mathrm{k}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$$ from
frequency of the Kittel resonance mode [39, 40] obtained from FMR. For the FMR
measurements, the samples were capped by a Hf2O to mitigate electromagnetic
interference with the coplanar waveguide. Detailed methodologies of these
measurements are elaborated in the Methods section. Note that the anisotropy
originates from the W/CoFeB/MgO interfaces [41, 42]. The obtained anisotropy
and saturation magnetization agree well with previously reported values [38,
43] for similar systems. We successfully derived temperature-dependent
material parameters via VSM and FMR, which we subsequently used for our
micromagnetic simulations conducted on magnum.np [44]. These measurements are
depicted in Extended Data Figure ED1 and Extended Data Figure ED2 for VSM and
FMR outcomes, respectively. For futher simulations, the exchange constant
$A_{\mathrm{ex}}$ was taken from literature [43]. The value for DMI was
numerically optimized to obtain the same zero field domain morhopology as
measured by MFM; see Methods and Extended Data Fig. ED3 for more details.

In Fig. 1b we depict the response of magnetization to an out-of-plane (OOP)
field (MH-Loops) as measured with VSM (red) and as simulated (blue) at room
temperature. We observe the two hysteresis pockets, where the system collapses
into a multi-domain/striped domain phase from which skyrmions can form. To
directly observe the existence of skyrmions in our samples, we acquired
magnetic force microscopy (MFM) [45, 46] images for different magnetic fields
applied in the OOP direction. Additionally, we accompany our experimental
findings with snapshots of the magnetization states from micromagnetic
simulations, where we used the zero-field MFM state as the initial state.
Snapshots of the magnetization for the simulation results of the blue curve in
Fig. 1b are given in Extended Data Fig. ED3, where a high density of skyrmions
is observed if one starts from a random magnetization state for each field.
However, this represents rather a local energy minimum, and in reality, a
sparse skyrmion nucleation is obtained. Isolated Néel skyrmions are depicted
for both core polarities from simulations (Fig. 1c,d) and MFM experiments
(Fig. 1e,f). The micromagnetic simulations (Fig. 1g-j) and MFM images (Fig.
1k-n) reveal that a rich multidomain (stripe) state is achieved at zero field.
By performing a domain width analysis, we find the average domain width
$\sigma_{DW}=$166.34\text{\,}\mathrm{n}\mathrm{m}$\pm$22.56\text{\,}\mathrm{n}\mathrm{m}$$.
Note that we used the magnetization state obtained from MFM as the initial
state in our micromagnetic simulations, which allows for an exceptionally good
agreement. If the magnetic field is increased, the stripes start to shrink,
and eventually Néel skyrmions form for sufficiently large fields
($>$40\text{\,}\mathrm{m}\mathrm{T}$$). In general, we confirmed the presence
of skyrmions in our sample by employing MFM imaging and micromagnetic
simulations, where we found skyrmions with an average radius in the range of
$r_{Sk}\approx$60\text{\,}\mathrm{n}\mathrm{m}$$. Further MFM images at
different magnetic fields are provided in the Extended Data Fig. ED4

.

### Three Dimensional Magnetic Field Sensor

In the following, we discuss how our skyrmionic device can be used as a
magnetic-field sensor to sense all three directions. For this purpose, we use
Si/SiO2/Ta(5)/[W(5)/CoFeB(0.7)/MgO(1.2)] $\times$ 10/Ta(3) multilayers that
have also been grown by DC magnetron sputtering; see Fig. 2a. We structured
6-armed Hall bars using convential photolithography and $\mathrm{Ar^{+}}$ ion
etching using a hard mask. More details are given in the Methods section. The
Hall bar has a width of $w=$2\text{\,}\mu\mathrm{m}$$. Throughout this paper,
we follow a differential DC measurement protocol. That is, the current is
first injected along the positive direction and the longitudinal ($R_{xx}$)
and transverse ($R_{xy}$) resistances are recorded. Then, the current polarity
is reversed and the resistances are recorded again. Only after the resistances
for both current directions have been recorded, the externally applied
magnetic field is changed.

Figure 2: Three dimensional sensing using a skyrmionic device. The multilayer
stack (a) used to pattern the Hall bars shown artistically in the upper
pannels, where the contacting for a positive flowing current and the direction
of the recorded transverse voltage are highlighted. In contrast to thin films,
we use $\mathrm{Ta}$(3) as a capping layer. The transverse resistance $R_{xy}$
as a function of the applied OOP magnetic field for both current polarities
$I=$\pm 1\text{\,}\mathrm{m}\mathrm{A}$$ (b) reproducing the VSM curve from
Fig. 1b. The change in $R_{xy}$ when an OOP field is applied under a strong
injection of SOT current ($I=$30\text{\,}\mathrm{m}\mathrm{A}$$) in (c), where
the hysteresis from (b) vanishes and a very linear signal
$\sigma^{\mathrm{OOP}}_{xy}=R_{xy}(+I)+R_{xy}(-I)$ can be obtained by summing
the two curves for the two current polarities (d), demonstrating that the
skyrmionic device can be used to sense OOP magnetic fields. The evolution of
$R_{xy}$ when an IP field is applied for positive and negative currents of
magnitude $|I|=$25\text{\,}\mathrm{m}\mathrm{A}$$ is depicted in (e), where
the symmetry of the SOT reverses the curves, allowing us to employ the offset-
free sensing principle to obtain the sensor transfer curve (f) where
$\sigma^{\mathrm{IP}}_{xy}=R_{xy}(+I)-R_{xy}(-I)$ is quantified. All
measurements are performed at an environmental temperature
$T=$300\text{\,}\mathrm{K}$$.

We start the transport measurements by injecting a low-amplitude current
$I=$1\text{\,}\mathrm{m}\mathrm{A}$$ into both positive and negative $x$
directions and recording the transverse resistance ($R_{xy}$) as a function of
OOP field, as depicted in Fig. 2b. More details of the measurements are given
in the Methods section. When the injected currents are small, the current
density is insufficient to induce the motion of the skyrmions or change the
magnetization state. Furthermore, the impact of SOT is negligible. This allows
us to qualitatively analyze conventional MH loops electrically. The
experimental data shown in Fig. 2b closely match the MH loops obtained
previously by VSM. It is important to note that both AHE and THE could
potentially contribute to $R_{xy}$. Although THE could be quantified from this
signal, it is beyond the scope of this study. The electrical signal obtained
can already function as a sensor signal, as changes in resistance are observed
as a result of the shrinking of stripes and subsequent formation of skyrmions.
The linear measurement range of the sensor is $LR=$\pm
10\text{\,}\mathrm{m}\mathrm{T}$$. A slight hysteresis is noticeable, possibly
indicating variations in the population of the multi-domain state with stripes
when returning from a saturated state.

The overall scenario undergoes a substantial transformation when the amplitude
of the injected current is greatly increased. As the current density
approaches orders of magnitude close to electromigration, the systems
temperature increases significantly due to Joule heating. To understand how
the system heats up with respect to the injected in-plane current, we perform
a series of transport measurements in which (I) we quantify the longitudinal
resistance $R_{xx}$ as a function of environmental temperature, and (II) we
measure $R_{xx}$ for different injected currents at room temperature while the
system was magnetically saturated. We then correlate the two measurements to
obtain the sample temperature as a function of the applied current; see
Extended Data Fig. ED5. For $I=$\pm 30\text{\,}\mathrm{m}\mathrm{A}$$ we
deduced the sample temperature
$T_{\mathrm{sample}}=$562.9\text{\,}\mathrm{K}$$. For the calculation of the
current density, we assume that the current flow is homogeneous in all layers.
Taking into account the cross section of current flow with thickness
$t=$74\text{\,}\mathrm{n}\mathrm{m}$$ and width $w=$2\text{\,}\mu\mathrm{m}$$,
the resulting current density is
$j_{e}\approx$2\text{\times}{10}^{11}\text{\,}\mathrm{A}\mathrm{/}\mathrm{m}^{2}$$.
This current density is sufficient to harness very strong effective SOT
fields, which can be summarized as field-like ($\mathbf{T}_{\mathrm{fl}}$) and
damping-like ($\mathbf{T}_{\mathrm{dl}}$) torques that extend the regular LLG
equation [47, 48]. Both the damping-like torque (DLT) and the field-like
torque (FLT) depend on the local direction of the spin polarization
$\mathbf{p}$ and of the magnetization $\mathbf{m}$, and can be expressed via

$\mathbf{T}_{\mathrm{DL}}=\mathbf{m}\times\mathbf{H}_{\mathrm{DL}}=\eta_{\mathrm{DL}}\dfrac{j_{e}\gamma\hbar}{2e\mu_{0}tM_{s}}{\mathbf{m}\times\left(\mathbf{m}\times\mathbf{p}\right)},$
(1)

and

$\mathbf{T}_{\mathrm{FL}}=\mathbf{m}\times\mathbf{H}_{\mathrm{FL}}=\eta_{\mathrm{FL}}\dfrac{j_{e}\gamma\hbar}{2e\mu_{0}tM_{s}}{\mathbf{m}\times\mathbf{p}},$
(2)

where
$\gamma=$2.21\text{\times}{10}^{5}\text{\,}\mathrm{(}\mathrm{m}\mathrm{/}\mathrm{A}\mathrm{s}\mathrm{)}$$
represents the gyromagnetic ratio, $\hbar$ denotes the reduced Planck
constant, $e$ stands for the elementary charge, and $t$ signifies the
thickness of the ferromagnetic layer. The effective SOT fields
$\mathbf{H}_{\mathrm{DL}}$ and $\mathbf{H}_{\mathrm{FL}}$ directly enter the
LLG effective field term, where their magnitude is given by the dimensionless
SOT coefficients $\eta_{\mathrm{DL}}$ and $\eta_{\mathrm{FL}}$, respectively.
The electrons injected into the system acquire spin polarization as a result
of the spin Hall [49] and Rashba-Edelstein [34] effects. When the current is
injected along the $x$ axis, and the spin-polarized current flows into the
skyrmionic device following the $+z$ direction, the electrons acquire spin
polarization along $-y$ [47, 29].

Now, we increase the applied current to $I=$\pm
30\text{\,}\mathrm{m}\mathrm{A}$$. The experimentally recorded $R_{xy}$ as a
function of the applied OOP field $\mu_{0}H_{z}$ is depicted in Fig. 2c for
both positive and negative current directions. Compared to the VSM hysteresis
loops, or low-current measurements, we observe that the hysteresis and the
characteristic pockets disappear, allowing us to harness a highly linear and
hysteresis-free magnetic signal. By adding the resistances for positive and
negative currents, a highly linear sensing signal
$\sigma^{\mathrm{OOP}}_{x}y(H_{z})=R_{xy}(+I)+R_{xy}(-I)$ is obtained, as
shown in Fig. 2d. The measurable linear range increases to
$LR\approx\pm$30\text{\,}\mathrm{m}\mathrm{T}$$, while the sensitivity is
quantified as
$S_{z}=\dfrac{\mathrm{d}^{2}V_{xy}}{\mu_{0}\mathrm{d}H_{z}\mathrm{d}I}=\dfrac{\mathrm{d}\sigma^{\mathrm{OOP}}_{xy}}{\mu_{0}\mathrm{d}H_{z}}\approx$0.8\text{\,}\mathrm{(}\mathrm{V}\mathrm{/}\mathrm{A}\mathrm{)}\mathrm{/}\mathrm{T}$$.
Note that we do not average the two curves, thus, the magnitude of the sensing
signal is twice as high as individual transfer curves. In Extended Data Fig.
ED6 we demonstrate how the hysteresis decreases with increasing injected
current and thus leads to an improvement in the measurable linear range of the
sensing signal. As for many applications, it is important that the sensors
also work at lower temperatures; we repeated our experiments at lower
temperatures ($T=$200\text{\,}\mathrm{K}$$; see again Extended Data Fig. ED6).
When the environmental temperature is decreased, the magnetic parameters, e.g.
$M_{s}$, and $K_{u}$, increase, while the effective SOT fields induced by
current are not significantly affected, as the change in the magnetic
parameters can be compensated by the temperature dependence of the SOT
coefficients. Overall, our measurements indicate that our skyrmionic device
can be utilized to sense OOP magnetic fields with a high linear working range
and moderate sensitivities.

So far we have considered only OOP fields, while the current was applied along
the $x$-axis. We now change our focus to in-plane (IP) magnetic fields, which
are always applied parallel to the injected current. In Fig. 2e we compare the
measured resistances for a positive current (red) and a negative current
(blue). The two curves cross each other at vanishing external fields. When
comparing these curves to those under OOP fields, we understand following
behavior: while a positive current is injected, a positive (negative) IP field
strives for achieving negative (positive) magnetic saturation. By reversing
the current polarity, now a positive fields tries to reach positive
saturation. For sufficiently high IP magnetic fields, it is possible to
saturate the magnetization OOP, before turning the entire system IP, when
$H\approx H_{k}$. As in our previous work [29], we exploit this symmetry and
our differential measurement protocol allows us to obtain a sensing signal
($\sigma^{\mathrm{IP}}_{xy}(H_{x})=R_{xy}(+I)-R_{xy}(-I)$) for IP magnetic
fields. Due to the symmetry of the SOT, the sensing offset is nearly
eliminated. Hence, our skyrmionic device can also be used to sense IP magnetic
fields with a linear range of $LR\approx\pm$17\text{\,}\mathrm{m}\mathrm{T}$$,
negligible DC offset, and a sensitivity
($S_{x}=\dfrac{\mathrm{d}^{2}V_{xy}}{\mu_{0}\mathrm{d}H_{x}\mathrm{d}I}=\dfrac{\mathrm{d}\sigma^{\mathrm{IP}}_{xy}}{\mu_{0}\mathrm{d}H_{x}}\approx$1\text{\,}\mathrm{(}\mathrm{(}\mathrm{V}\mathrm{/}\mathrm{A}\mathrm{)}\mathrm{/}\mathrm{T}\mathrm{)}$$,
which is greatly improved compare to our previous work
$(S_{x}=$0.02\text{\,}\mathrm{(}\mathrm{(}\mathrm{V}\mathrm{/}\mathrm{A}\mathrm{)}\mathrm{/}\mathrm{T}\mathrm{)}$)$
[29]. Similar to OOP sensing, we investigated the sensing signals as a
function of injected in-plane currents at room temperature and at
$T=$200\text{\,}\mathrm{K}$$ (Extended Data Fig. ED7), demonstrating once
again the robustness of using the proposed skyrmionic device to generate a
sensing signal.

Generally, the IP sensitive direction is that direction that is parallel to
the applied current. Thus, it is to also sense $H_{y}$ fields simply by
injecting the current along this direction. As this is equivalent to the case
we discussed above due to simple symmetry arguments, we will not be providing
an experimental demonstration.

### Sensing Performance

Magnetic field sensors can be classified based on basic properties such as
measurable linear range, sensitivity, noise, offset and detectability. To
better highlight the performance of our skyrmionic device as a magnetic field
sensor, consider now the sensor signals from Extended Data Figs. ED6 and ED7,
from which one can quantify the reachable sensitivities and DC offsets, as
summarized in Fig. 3. Figure 3a depicts the sensitivity of our sensor for an
IP field ($H_{x}$) at $T=$300\text{\,}\mathrm{K}$$ and
$T=$200\text{\,}\mathrm{K}$$. The sensitivity can be increased by injecting
more current into the skyrmionic device, until we reach a peak sensitivity of
$S_{x}=$1.85\text{\,}\mathrm{(}\mathrm{V}\mathrm{/}\mathrm{A}\mathrm{)}\mathrm{/}\mathrm{T}$$
for $I=$20\text{\,}\mathrm{m}\mathrm{A}$$. For an OOP field, the sensitivity
is not affected as strongly as for IP fields; as we show in Fig. 3c. Here, the
sensitivity is in the range of
$S_{z}=$0.95\text{\,}\mathrm{(}\mathrm{V}\mathrm{/}\mathrm{A}\mathrm{)}\mathrm{/}\mathrm{T}$$,
and it deviates only from $\pm 10\%$ for higher and lower currents,
respectively. Compared to commercially available TMR sensors, the
sensitivities we reach are orders of magnitude lower, but we need to keep in
mind that we read out the signal via the AHE, which is generally a much
smaller effect. If one reads out the change in magnetization via magnetic
tunnel junctions (MTJ) [24], one could achieve resistivities as high as those
of commercially available products. We demonstrate in Extended Data Fig. ED8
that we can enhance our sensitivity to IP fields by a factor of 4 simply by
reducing the thickness of MgO to $1.1\text{\,}\mathrm{n}\mathrm{m}$.

In a previous work [29], in which we demonstrated the first offset-free
sensing principle enabled by SOT with a linear range higher than
$10\text{\,}\mathrm{m}\mathrm{T}$, the sensitivities reached were as low as
$0.017\text{\,}\mathrm{(}\mathrm{V}\mathrm{/}\mathrm{A}\mathrm{)}\mathrm{/}\mathrm{T}$.
Thus, the skyrmionic device that we realized in this work increases the
sensitivity by two orders of magnitude. In the work of Li et al. [37], they
reach sensitivities of the order of
$280\text{\,}\mathrm{(}\mathrm{V}\mathrm{/}\mathrm{A}\mathrm{)}\mathrm{/}\mathrm{T}$
for IP fields and one order of magnitude higher for OOP fields. However, their
linear range is limited to only $\pm 1\text{\,}\mathrm{m}\mathrm{T}$ (IP) and
$\pm 0.4\text{\,}\mathrm{m}\mathrm{T}$ (OOP), respectively. Similarly, AHE
based sensors report very high sensitivities [27, 28] up to three orders of
magnitude higher than our sensor. While their linear ranges are again very
limited ($<$4\text{\,}\mathrm{m}\mathrm{T}$$), the thickness of the total
magnetic active component is lower. As the AHE decreases with increasing
thickness of the magnetic layer [50], lower resistances are recorded in our
skyrmionic sensor. Fortunately, this opens also new possibilities for
skyrmion-based sensors of the future, where thin-film skyrmionic systems can
be used to sense magnetic fields with much higher sensitivities and
potentially even higher linear ranges.

Figure 3: Sensor performance at different environmental temperatures. The
calculated sensitivities for IP fields (a) and OOP fields (c) as a function of
injected SOT current for both $T=$300\text{\,}\mathrm{K}$$ (red) and
$T=$200\text{\,}\mathrm{K}$$ (blue) as environmental temperatures set in the
cryostat. The slopes of the sensor transfer curves were fitted by a linear
function to extract the sensitivities. The DC offset was then extracted from
the transfer curves as the magnetic IP (b) and OOP (d) fields where the
measured signal vanishes.

As explained in more detail in the Methods section, we use the Quantum Design
Physical Property Measurement System (PPMS), where the magnetic field is
applied by a superconducting magnet. For a superconducting magnet, the applied
field is linear to the sent current; thus, a very precise calculation of the
magnetic field can be reached internally. However, we performed a linearity
test by measuring the electron-paramagnetic resonance of
2,2-diphenyl-1-picrylhydrazyl (DPPH) to calibrate the applied field versus the
expected resonance field. We find that the PPMS field has an offset of
$-0.59\text{\,}\mathrm{m}\mathrm{T}$ and a linearity error of $1.4\%$. This
error becomes significant for very high magnetic fields, which we are not
considering. However, the zero-field offset will, of course, limit the DC
offset of our sensors. Without correcting this error, we provide in Fig. 3b
($O_{x}$) and in Fig. 3d ($O_{z}$) the DC offsets of our magnetic field
sensor, which are the magnetic field for which $\sigma_{xy}=0$. In the IP
field, where we applied the differential measurement scheme and the offset-
free sensing principle, very low offsets are recorded,
$<$1.5\text{\,}\mathrm{m}\mathrm{T}$$ at $T=$300\text{\,}\mathrm{K}$$. When
the field of the PPMS is corrected, this error is reduced to
$<$1.0\text{\,}\mathrm{m}\mathrm{T}$$ and disappears almost completely for
$I=$10\text{\,}\mathrm{m}\mathrm{A}$$ and
$I=$15\text{\,}\mathrm{m}\mathrm{A}$$. For the OOP field, the situation
changes significantly. Although the error is rather small at lower currents,
as we simply measure the hysteresis curve of a conventional skyrmionic device,
the DC offset error increases with the applied SOT current. Taking into
account also the fact that we do not apply any type of magnetic field
correction here, the measured offsets are rather high for the high current
regime ($5\text{\,}\mathrm{m}\mathrm{T}$ at
$I=$30\text{\,}\mathrm{m}\mathrm{A}$$). However, this error can also be
eliminated by applying a spinning current concept [51]. If the current is
applied along $y$ instead of $x$, basically the same transverse voltage can be
obtained, but mirrored with respect to the field if the contacts are chosen
accordingly. Then, the two signals can be subtracted again from each other,
leaving again an offset free signal behind for an OOP field as well.

### Origin of the sensing signal

In the following we will take an in-depth look at the working mechanism of our
3D magnetic field sensor, where we make use of micromagnetic simulations at
finite temperature using magnum.np [44] to analyze the field-dependent
evolution of the magnetic states.

Figure 4: Influence of effective spin-orbit torques on the nonuniform
magnetization states. Finite-temperature micromagnetic simulations
demonstrating different scenarios for injected SOT current along $+x$ with a
magnitude $I=$30\text{\,}\mathrm{m}\mathrm{A}$$ and applied magnetic fields
where each row represents a combination: (up) vanishing field, (center) IP
field with $\mu_{0}H_{x}=$-36\text{\,}\mathrm{m}\mathrm{T}$$, and (down) OOP
field with $\mu_{0}H_{z}=$-48\text{\,}\mathrm{m}\mathrm{T}$$

. The $x$, $y$ and $z$ components of the magnetizations are given in (a)-(c),
(d)-(f), and (g)-(i), respectively. The effective normalized SOT field
$\mathbf{H}^{\mathrm{SOT}}$ color coded in (j)-(l) by the color wheel at the
bottom, as well as the normalized effective $\mathbf{H}^{\mathrm{DL}}$ to
highlight the dominating effect of SOT in (m)-(o).

To understand the main mechanisms of SOT in combination of high Joule heating
and magnetic fields, we studied three different cases with high SOT current of
$I=$30\text{\,}\mathrm{m}\mathrm{A}$$ in the $x$ direction. We start from a
random magnetization state, turn on the current, and solve the LLG equation
extended with the DLT and FLT from Eqs. (1) and (2). We include thermal
fluctuations [52] and temperature-dependent material parameters and
numerically integrate the LLG until we reach an equilibrium state. No external
magnetic field is applied. For the SOT coefficients, we assume
$\eta_{\mathrm{DL}}=-0.3$ and $\eta_{\mathrm{FL}}=0.05$ as appropriate values
for the SOT efficiencies in W/CoFeB/MgO trilayers [47]. It is worth mentioning
that these are then reduced by the ratio of thickness of magnetic and
nonmagnetic layers as a part of our effective media modeling; see Methods for
more details.

The magnetization components are shown in Fig. 4a, b, c, where a well-ordered
stripe pattern is visible, where the domain walls are all oriented along $-y$.
Figure 4d illustrates the normalized direction of the total effective SOT
field that acts on the magnetic layer, while Fig. 4e depicts only the
normalized $\mathbf{H}^{\mathrm{DL}}$. We do not illustrate
$\mathbf{H}^{\mathrm{FL}}$ as this is a constant field along the polarization
$\mathbf{p}$. Let us consider the expressions for the effective SOT fields in
Eqs. 1 and Eq.2. $\mathbf{H}_{\mathrm{FL}}$ tries to align the magnetization
along the direction of the polarization, as an external field, while the
direction of $\mathbf{H}_{\mathrm{DL}}$ depends on the local magnetization
vector. Our numerical investigations reveal that the DLT is mainly responsible
for the periodic alignment of the stripes that exist intrinsically in the
sample at vanishing fields and SOT currents. The FLT then forces all the
domain walls and stripes to align towards $-y$. Note that we do not include
the Oersted fields directly in our calculations but assume that the FLT
already is an effective FLT where the contributions of Oersted fields are
considered, as this separation is usually not done in the quantification of
SOT parameters. In vanishing fields, DLT similarly favors an alignment along
$-y$ if the magnetization is initially oriented OOP. To better understand the
role of each torque individually, we varied the strength of both SOT
coefficients. The higher $|\eta_{\mathrm{DL}}|$, the lower the average width
of the stripes.

After the stripe domain is formed, we now apply an IP field
$\mu_{0}H_{x}=$-36\text{\,}\mathrm{m}\mathrm{T}$$. We observe that the
external field affects the chirality of the system induced by FLT and
reorients the walls approximately $40\text{\,}{}^{\circ}$ from the $y$ axis
towards $x$; as depicted in Fig. 4b,e,h. The strength of the applied field
changes the angle of the stripe domains as a consequence of an effective field
from the external field and FLT. Normalized $\mathbf{H}_{\mathrm{SOT}}$ is
illustrated in Fig. 4k, while $\mathbf{H}_{\mathrm{DL}}$ is shown in Fig. 4n.
Since the external field introduces a stronger $x$ component of magnetization,
the DLT then is completed by an additional term along the $z$ component due to
the cross-product $\mathbf{m}\times\mathbf{p}$. Furthermore, the DLT acting on
the previously negative domains is enhanced, transforming them more IP, while
the torque acting on the positive domains is minimized. Thus, the negative
(down) domains shrink, while the positive (up) ones grow. Note that this
effect is fully reversible with respect to the injected current, which means
that, for a negative current and negative IP fields, the negative OOP domains
grow, while the positive ones shrink, as $H_{\mathrm{DL},z}$ is now negative.
The magnetization state behaves differently for an OOP field
$\mu_{0}H_{z}=$-48\text{\,}\mathrm{m}\mathrm{T}$$, increasing the effect of
the DL torque acting on the regions originally magnetized along $-x$ that
favors a reorientation of the stripes along $+x$. Thus, the induced $x$
component of the magnetization will generate an effective OOP field, where
effective field of FLT and the Zeeman term then additionally shrink the
domains, as the new direction is tilted increasingly towards $-z$. Reversing
the current polarity will reorient the stripes rather along $-x$, but the
applied negative field will still lead to the shrinking of the positive
domains due to the induced OOP SOT field.

Figure 5: Origin of sensing signal for IP fields. (a) Micromagnetically
simulated transfer curves for IP magnetic fields equivalent to Fig. 2e, and
the calculated linear sensor signal (b) as the difference of the two curves
from (a). Snapshots of the magnetizations (c) for positive current (top row)
and negative current (bottom row) for magnetic fields in the range $\pm
30\text{\,}\mathrm{m}\mathrm{T}$. Selected enlarged view of a stripe (left) at
zero field, type-II bubbles with a negative core (center) and a positive core
(right) in (d) at $\mu_{0}H_{x}=$26\text{\,}\mathrm{m}\mathrm{T}$$ and
$\mu_{0}H_{x}=$-28\text{\,}\mathrm{m}\mathrm{T}$$, respectively. The arrows in
(d) show the in-plane component of the magnetization.

As the magnetization changes significantly for these fields, we obtain a
transverse voltage dominated by the AHE, which we use to record a sensor
signal in our experiments. Note that the AHE is proportional to $M_{z}$ in the
micromagnetic simulations. Figure 5 summarizes the results of our
micromagnetic simulations for the IP sensing mechanism. The numerically
obtained magnetization responses to the applied IP magnetic field are given in
Fig. 5a, while the calculated sensor transfer curve can be seen in Fig. 5b.
The applied current is reduced to $I=$\pm 25\text{\,}\mathrm{m}\mathrm{A}$$.
We have exceptionally good qualitative agreement with our experiments; thus,
we are confident that we can make use of the underlying microstates to reveal
the origin of the sensing signal. In Fig. 5c,d we provide snapshots of the
magnetization states at different IP fields for both current directions. At
vanishing fields, both currents lead to a similar stripe pattern. As the
current is reduced compared to Fig. 4, the stripe domains are not as well
structured as in the previous case. The domain walls around a stripe domain
favor a parallel alignment with the FLT ($-y$ axis), as can be seen in Fig.
5d-left. In the upper row we provide the magnetization states for a positive
current, where we observe that a positive field slightly reorients the
stripes. The DLT favors negative domains with increasing $+x$ component of the
magnetization due to cross-product $\mathbf{m}\times\mathbf{p}$. Therefore,
the positive stripes break down into what looks like skyrmionic textures
(type-II trivial bubbles). The vector field depicted in Fig. 5d shows that the
FLT and DLT destroy the topological protection of the skyrmions, and rather
topologically trivial type-II bubbles are observed [53]. Note that skyrmionic
states with an integer topological number would be moved out of the sample
because of strong SOT currents [15]. The type-II bubbles are stable because
their boundaries are aligned parallel to the polarization of the spin current;
thus, the resulting torque vanishes. Due to the symmetry of the SOT, reversing
the current polarity allows the DLT to favor positive domains, thus allowing
the negative stripe to shrink down and ultimately collapse into type-II bubble
states. Thus, a positive ferromagnetic state can be achieved by injecting a
negative current and positive field, or a positive current and negative field,
and a negatively polarized state by a negative current and negative field, or
a positive current and positive field. Overall, the IP field enhances or
diminishes the effect of DLT on positive or negative magnetized domains, based
on the applied current direction. This in exchange leads to a transverse
voltage due to the collapse of stripe domains in to skyrmionic states before
fully saturating.

Figure 6: Domain morphology during OOP field sensing. Analogously to Fig. 5,
we provide the transfer curves for positive and negative injected SOT currents
in (a), and the sensor signal in (b) as the average of two curves from (a) as
obtained from finite-temperature micromagnetics. The domain morphology is
illustrated as snapshots of magnetizations for positive currents (c) and
negative currents (d). The zero-field magnetization profiles are similar to
those from Fig. 5, thus, we do not repeat them here. With increasing
(decreasing) magnetic field, stripes with negative (positive) domains start to
shrink, approaching a positive (negative) polarized ferromagnetic state.

The case of an OOP magnetic field is much simpler. At vanishing fields, we
find the same periodic stripe domain state with the orientation of the domain
walls being dictated by the SOT. The simulated magnetization responses to the
OOP field are depicted in Fig. 6a, and the sensor signal calculated from these
two curves is shown in Fig. 6b. As described above, an OOP field slightly
reorients the stripes along the $\pm x$ axis, which is also visible in the
snapshots provided in Fig. 6c (positive current) and in Fig. 6d (negative
current). Our finite-temperature micromagnetics reveal that with increasing
(decreasing) OOP field positive (negative) domains are favored, while the
system thrives to reach a saturated state. As the SOT induced a well-defined
directionality and chirality of the stripe domains, the transformation of
stripe domains into the polarized states occurs hysteresis-free, leading to a
highly linear sensor signal.

## Discussion and Conclusions

In this work, we demonstrated that we can use a skyrmionic device to sense
both IP and OOP magnetic fields with linear ranges up to $\pm
17\text{\,}\mathrm{m}\mathrm{T}$ and $\pm 30\text{\,}\mathrm{m}\mathrm{T}$,
respectively. The applied fields will either act with or against the damping-
like torque and lead to the collapse of periodically arranged stripes into
skyrmionic states (type-II bubbles) of given core polarity, as we reveal
employing finite-temperature micromagnetic simulations. The change in the
magnetization can be measured as a change in the transverse voltage, where the
main contributor is the anomalous Hall effect. Repetition of our experiments
also at lower temperatures ($T=$200\text{\,}\mathrm{K}$$) shows that our
skyrmionic sensor could potentially also be used at cryogenic conditions. As
the main objective, we demonstrated offset cancelation, where the DC offset
can be nearly eliminated for IP magnetic fields by employing an offset-free
sensing principle, where the symmetry of the SOTs is exploited. The offset
present for the OOP fields can eventually be eliminated by applying a spining-
current technique based on the symmetry of the interactions. The proposed
skyrmionic device works very well for larger systems, where we have enough
stripes to transform to type-II bubbles and to the polarized state.
Furthermore, the domain wall width as well as the pinning fields of the
samples will significantly impact the sensing performance.

The sensitivity is orders of magnitude lower when compared to commercially
available TMR sensors, as well as AHE-based sensors with ultrathin
ferromagnetic layers. Several improvements can be made to improve the
sensitivity. We have demonstrated here that one can reduce the thickness of
the MgO layer to further increase the resistivity of our rather thick
multilayer system and hence ameliorate the sensitivity. A different option is
to change the material of the skyrmionic device. A multilayer skyrmionic stack
composed of [Ir / Fe / Co / Pt]N can be engineered to host a high density of
skyrmions and stripe domains [8]. As these stacks are much thinner, the
measured resistances are expected to increase accordingly. Because of the Pt
and Ir interfaces, very strong SOTs are expected in such a stack. While the
thickness reduction improves the sensitivity, the high density of
skyrmions/stripes and stronger torques are expected to increase the linear
ranges to the levels of commercially available TMR sensors. Furthermore,
stronger effective SOT fields can be reached within such a stack due to
additive spin current generation, which will further increase the linear
range. Ultimately, one could combine the present approach (or [Ir/Co/Fe/Pt]N
based stack) with the read-out of the OOP component of the magnetization by
using a magnetic tunnel junction and exploiting the TMR effect [24],
specifically enhancing the sensitivity, while keeping the DC offset minimal.
To do so, the MTJ needs to consist of an OOP syntethic antiferromagnet, for
instance, Co/Pt/X/Pt/Co multilayers, where X can be Ir, Ru or a similar
material that ensures a high interlayer exchange interaction [54].

Our results represent an innovative approach to combine both skyrmions and SOT
based spintronic devices for magnetic field sensing applications. Our
experimental and numerical investigations will pave the way towards a new
research direction for magnetic skyrmions and magnets with nonuniform
magnetization states to be explored as potential candidates for magnetic field
sensing applications.

## Methods

Sample Growth and Patterning
Thin films. For the characterization of magnetic parameters the
[W(5)/CoFeB(0.7)/MgO(1.2)]${}_{\mathrm{N}}/\mathrm{HfO_{2}}$ multilayer stack
was sputter deposited on an 8-in silicon wafer using a Singulus Rotaris tool
deposition system with base pressure
$<$3\text{\times}{10}^{-8}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$$
using thermally oxidized silicon substrates. The stack is then capped with
$\mathrm{HfO_{2}}(3)$ to avoid oxidation and prevent electromagnetic coupling
to the coplanar waveguide for FMR measurements. Metals were deposited in the
DC mode, while oxides ($\mathrm{MgO},\,\mathrm{HfO_{2}}$) were deposited in
the RF mode.
Hall Bars. We apply the same strategy as in [29]. That is, before film
deposition, the wafers received an aluminum metallization layer to realize the
contacts for current flow into the stack from the bottom, where W vias are
used through an insulation $\mathrm{SiO_{2}}$ layer. A mechanical chemical
polish is then carried out to ensure a smooth surface for the deposition of
the [W(5)/CoFeB(0.7)/MgO(1.2)]${}_{\mathrm{N}}/\mathrm{Ta}(3)$ stack. We use
the Singulus Rotaris tool to sputter-deposit the multilayer stack on the
prepared substrate, as described above for thin films. The structure is then
capped with Ta(3) to avoid oxidation. Hall bar patterning is achieved using
conventional optical lithography and $\mathrm{Ar^{+}}$ ion etching. The
average milling rate for the SOT stack was found to be around
$10\text{\,}\mathrm{n}\mathrm{m}\mathrm{/}\mathrm{m}\mathrm{i}\mathrm{n}$.

Vibrating Sample Magnetometry. For experimental investigations, we use the
Physical Property Measurement System (PPMS) from Quantum Design where magnetic
fields up to $\pm 9\text{\,}\mathrm{T}$ can be applied, while the temperature
can be varied between $$1.8\text{\,}\mathrm{K}$-$400\text{\,}\mathrm{K}$$. In
order to quantify the magnetic moment as a function of the applied magnetic
field, we use the Vibrating Sample Magnetometer module. For this, we
mechanically cut the thin film specimen to approximately $2.5\times
2.5\,\mathrm{mm^{2}}$, in order to measure the hysteresis curves while
applying the field both parallel (IP) and normal (OOP) to the film plane. The
oscillation amplitude for the VSM is chosen as
$2\text{\,}\mathrm{m}\mathrm{m}$, while the frequency is
$40\text{\,}\mathrm{H}\mathrm{z}$. We measured the hysteresis loops for both
configurations for temperatures between
$$25\text{\,}\mathrm{K}$-$400\text{\,}\mathrm{K}$$ in $25\text{\,}\mathrm{K}$
steps, as provided in Extended Data Fig. ED1. As we measure the total magnetic
moment of the sample in $\mathrm{Am^{2}}$, we then use the magnetic volume to
obtain the saturation magnetization $M_{s}$ as a function of temperature. We
found
$M_{s}=1375.25\times\left(1-\left(T/T_{C}\right)^{1.77}\right)\,\mathrm{(kA/m)}$,
where $T_{C}$ is the Curie temperature of the magnet. The fit yields
$T_{C}=$952.63\text{\,}\mathrm{K}$$. Note that we normalize $M_{s}$ to the
magnetic volume only ($t_{\mathrm{m}}=$7\text{\,}\mathrm{n}\mathrm{m}$$). For
the micromagnetic simulations, we scale the magnetic moment to the total
volume. We do this to employ an effective media model for our micromagnetic
modeling, as discussed later. In our case, we have a magnetic thickness
$t_{\mathrm{m}}=$7\text{\,}\mathrm{n}\mathrm{m}$$ and a nonmangetic thickness
of $t_{\mathrm{nm}}=10t_{W}+10t_{MgO}=$62\text{\,}\mathrm{n}\mathrm{m}$$.
Thus, for the effective media model’s saturation magnetization, we obtain
$M^{\mathrm{emm}}_{s}=139.52\times\left(1-\left(T/T_{C}\right)^{1.77}\right)\,\mathrm{(kA/m)}$
as the best fit.

Ferromagnetic Resonance We perform ferromagnetic resonance (FMR) measurements
to quantify the perpendicular magnetic anisotropy in our samples. For this
purpose, we use a sample-holder with a coplanar waveguide (NanOsc) such that
the magnetic field applied within the cryostat is IP, where we employ the
flip-chip method to investigate the field dependence of the resonance
frequencies. To do so, we use a Rohde&Schwarz Z40A Vector Network Analyzer to
apply and detect microwave signals with frequencies up to
$40\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The signal power is
$5\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$, to obtain high signal-to-noise
ratios and well-defined resonance peaks. We then measure the scattering
parameter $S_{21}$ in a frequency sweep mode in a fixed magnetic field. The
obtained resonance frequencies $f_{\mathrm{res}}$ of the fundamental peak are
evaluated against the resonance fields $H_{\mathrm{res}}$. For an IP field
configuration the resonance condition is given by the Kittel formula
$f_{\mathrm{res}}=\gamma^{\prime}\mu_{0}\sqrt{H_{\mathrm{res}}(M_{\mathrm{eff}}+H_{\mathrm{res}}{})}$,
where $\gamma^{\prime}$ is the reduced gyromagneitc ratio, $\mu_{0}$ is the
vacuum permeability, and $M_{\mathrm{eff}}$ is the effective magnetization,
which is given by
$M_{\mathrm{eff}}=M_{s}\left(1-\dfrac{2K_{u}}{\mu_{0}M_{s}^{2}}\right)$, with
$K_{u}$ being the uniaxial anisotropy constant. In our case, this equals the
strength of the perpendicular magnetic anisotropy originating from the heavy
metal feromagnet interface. We repeat the measurements for the same
temperature range as the VSM measurements. Since $M_{s}$ is known, we use the
Kittel resonance condition to fit both the reduced gyromagnetic ratio and
$M_{\mathrm{eff}}$ using least squares methods. The average reduced
gyromagnetic ratio is found to be
$\gamma^{\prime}=$27.66\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{T}$\pm$0.27\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{T}$$.
In Extended Data Fig. ED2 we display the measurement setup, as well as the
extracted temperature dependent quantities $K_{u}$, $M_{\mathrm{eff}}$,
$K_{\mathrm{eff}}$, which denotes the effective anisotropy constant,
$H_{k,\mathrm{eff}}$ which is the effective anisotropy field. The temperature
dependence of $K_{u}$ is found to be
$K^{\mathrm{emm}}_{u}(T)=1.43\times(M_{s}(T)/M_{s}(0))^{2.28}\,\mathrm{(MJ/m^{3})}$,
if one considers $M_{s}$ normalized to magnetic layers only. For micromagnetic
simulations, we use $M^{\mathrm{emm}}_{s}$ for which we obtain
$K^{\mathrm{emm}}_{u}(T)=36.7\times(M_{s}(T)/M_{s}(0))^{3.21}\,\mathrm{(kJ/m^{3})}$.

Magnetotransport Measurements. Patterned 6-arm Hall bars are used to
electrically investigate the feasibility of the skyrmionic device as a
magnetic field sensor. For this purpose, four contacts are established by
wire-bonding $25\text{\,}\mu\mathrm{m}$ gold wires. This enables to measure
the transverse voltage through a four-point measurement, whereas the
longitudinal resistance $R_{xx}$ is determined in two-point geometry at the
current contacts. We use a self-built box to control the channels along which
the current is applied. The current, always injected along the x axis, is
delivered by a Keithley 6221 current source, which can apply currents up to
$100\text{\,}\mathrm{m}\mathrm{A}$. We use a Keithley nanovoltmeter (model
K2182) for voltage measurements, which has two channels and offers nV
resolution. With the first channel, we record the transverse resistance
$R_{xy}$, with the second channel the longitudinal resistance $R_{xx}$. In
order to apply IP and OOP fields during measurements, we mount the sample in a
horizontal rotator, which allows to control the angle of the magnetic field by
a linear motor. The sample holder and the chamber are continuously grounded
while the sample is inserted into the PPMS. For all experiments, we start from
the highest positive magnetic field and apply a differential measurement
scheme. That is, we consequently measure the resistances for positive and
negative currents (three times each, and then we take the average for each
current direction). Afterwards, the magnetic field is lowered and we repeat
this procedure until the lowest magnetic field has been reached. As we are
using the horizontal rotator, small angle deviations must be expected and
might disturb the sensing signal, leading to poor performance. However, our
results indicate that we have successfully omitted large errors. As mentioned
in the main text, we have tested the calibration of the superconducting
magnet. We found that we have an offset error of
$-0.59\text{\,}\mathrm{m}\mathrm{T}$ and a linearity error of 1.4. To be more
transparent, we have not corrected the magnetic fields in our post-processing,
as we focus rather on the concept we are demonstrating.

Magnetic Force Microscopy The MFM measurements presented in this study were
performed using a home-built high vacuum
$(\approx${10}^{-6}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$)$
scanning probe microscopy system where magnetic fields as high as
$300\text{\,}\mathrm{m}\mathrm{T}$ can be applied [46]. The MFM is operated in
vacuum, drastically improving the mechanical quality factor $Q$ for the
cantiveler to $Q\approx$2\text{\times}{10}^{5}\text{\,}$$. In addition to this
improvement over conventional MFM measurements in air, which increases the
sensitivity by a factor of 40, we also deposit a very thin Co layer on the tip
to minimize stray-field interactions with the sample. We choose SS-ISC
cantilevers from Team Nanotech GmbH, where the tip radius is below
$5\text{\,}\mathrm{n}\mathrm{m}$. To enhance the sensitivity of the cantilever
tip to magnetic fields, we utilized sputter deposition to apply a 3 nm layer
of Co at room temperature on top of a Ta seed layer (2 nm), followed by
capping it with a 4 nm layer of Ta to protect against oxidation. The
oscillation of the cantilever at resonance with a constant amplitude of 7 nm
and the detection of frequency shifts resulting from the derivative of the
tip-sample interaction force were carried out using a Zurich Instruments
phase-locked loop (PLL) system. It is important to note that the frequency
shift indicates an attractive force (derivative) when it is negative. In
Figs.1 and ED4, if a negative OOP field is applied, the MFM tip is magnetized
down, while skyrmions (stripes) will have a positively magnetized core. As the
force generated between the spin textures and the tip is repulsive, the
recorded frequency shift is negative. Hence, we revert the colors of the MFM
images based on the magnetization of the tip to depict the magnetization
states similar to those obtained from micromagnetic simulations and to omit
artificial confusions.

Micromagnetic Simulations
Finite-temperature micromagnetics. We use magnum.np [44] for GPU-accelerated
micromagnetic simulations using a finite-difference algorithm for the
numerical solution of the Landau-Lifshitz-Gilbert equation [48] which
describes the temporal evolution of the magnetization, where

$\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\mathbf{m}\times\mathbf{H}^{\mathrm{eff}}+\alpha\mathbf{m}\times\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t},$
(3)

with $\alpha$ being the Gilbert damping parameter, and
$\mathbf{H}^{\mathrm{eff}}$ the effective field term that is derived from the
total energy of the system via the variational derivative as

$\mathbf{H}^{\mathrm{eff}}=-\dfrac{1}{\mu_{0}M_{s}}\dfrac{\delta
E_{tot}}{\delta\mathbf{m}},$ (4)

where $E_{\mathrm{tot}}$ denotes the total energy of the system. In our
simulations, the energy contributions considered are the demagnetizing energy,
exchange energy, perpendicular magnetic anisotropy energy, and DMI energy. All
simulations are performed at finite temperatures, where a stochastic term is
added to introduce thermal fluctuations into the system, which is directly
added to the effective field via the thermal field
$\mathbf{H}^{\mathrm{th}}=\bm{\epsilon}\sqrt{\dfrac{2\alpha
k_{B}T}{{\mu}_{0}M_{s}\gamma V\Delta t}}$, where $\bm{\epsilon}$ is a random
vector that is normally distributed for each timestep $\Delta t$, $V$ is the
cell volume, $k_{B}$ is the Boltzmann constant and $T$ is the temperature of
the system. The now stochastic LLG is integrated using a Runge-Kutta-Fehlberg
algorithm with an adaptive time step [52]. Note that the results can depend on
the mesh size, and a rescaling of parameters can improve the reproducibility
of the results [55]. To reduce computation times, we used an effective media
model, where it is assumed that the magnetization is homogeneous along the $z$
axis, and that all skyrmions and stripes are complete throughout the
thickness. In this model, the effective media parameters are usually
calculated from the parameters of a single magnetic layer using the relation
$M_{s}=M_{s}^{\prime}\times t_{\mathrm{m}}/t_{\mathrm{t_{n}m}}$, where
$M_{s}^{\prime}$ is the parameter of a single layer, $t_{\mathrm{m}}$ denotes
the total thickness of the magnetic layers and $t_{\mathrm{nm}}$ the total
thickness of the non-mangetic layers, respectively. However, we follow a
different, yet equivalent, route to apply the effective media model. That is,
we assume that the sample is magnetic throughout the thickness and that the
measured magnetic moment from VSM results from the entire volume. Thus, we
simulate the multilayer as a single-layer magnet with reduced saturation
magnetization. The temperature dependence
$M_{s}=139.52\times\left(1-\left(T/T_{C}\right)^{1.78}\right)\,\mathrm{(kA/m)}$
is obtained, which is then used to fit the strength of the perpendicular
magnetic anisotropy and its temperature dependence as
$K_{u}(T)=36.7\times(M_{s}(T)/M_{s}(0))^{3.2}\,\mathrm{(kJ/m^{3})}$. The
exchange constant $A^{\prime}_{\mathrm{ex}}$ for one W/CoFeB/MgO trilayer is
taken from the literature as
$A^{\prime}_{\mathrm{ex}}=$15\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$.
We assume that $A^{\prime}_{\mathrm{ex}}$ follows a dependence of $M_{s}$,
where
$A^{\prime}_{\mathrm{ex}}(T)=A^{\prime}_{\mathrm{ex}}(0)\times\left(\dfrac{M_{s}(T)}{M_{s}(0)}\right)^{1.7}$.
This value is then reduced according to the effective media model as
$A_{\mathrm{ex}}(T)=A^{\prime}_{\mathrm{ex}}(T)\times\dfrac{t_{\mathrm{m}}}{t_{\mathrm{nm}}}$.
Note that all relevant material parameters ($M_{s}$, $K_{u}$,
$A_{\mathrm{ex}}$ and $D$) are assumed as average values in our simulations.
To better reproduce the reality, we assume that $M_{s}$, $A_{\mathrm{ex}}$ and
$D$ are normally distributed around their mean values with a standard
deviation of $20\%$, and $K_{u}$ with a standard deviation of $30\%$.
Furthermore, as it is expected that the perpendicular anisotropy will not be
perfect in the defects, we distribute the anisotropy axis in a cone around
[001], where the maximal deviation angle is $30\text{\,}{}^{\circ}$.
Numerical optimization of the DMI constant. The value of the DMI constant at
$T=$0\text{\,}\mathrm{K}$$ was optimized numerically as demonstrated in Fig.
ED3. For this purpose, we read the initial magnetization pattern from the MFM
image of Fig.1k. The simulation box is discretized in $(1024\times 1024\times
1)$ cells, with total dimensions of
$($4\text{\,}\mu\mathrm{m}$\times$4\text{\,}\mu\mathrm{m}$\times$69\text{\,}\mathrm{n}\mathrm{m}$)$.
The domain walls are then randomly magnetized in each cell, since we do not
have any information about this from the MFM data. We relax the structure with
moderately high damping $\alpha=0.1$ for $10\text{\,}\mathrm{n}\mathrm{s}$.
The average absolute difference between the final relaxed state and the
initial state is then calculated and plotted against the DMI value. We take
the value at the local minimum as the optimal DMI value to reproduce our
experiments, which is
$D(0)=$0.14\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$$. For the
remainder of the simulations, we will use this value, which depends on
$M_{s}(T)$ as $D(T)=D_{0}\times(M_{s}(T)(M_{s}(0))^{2}$.
MH-Loops.For the simulations in Fig. 1 we use the same simulation box and
parameters. For the hysteresis curve in Fig. 1b we always start from a
randomly magnetized state, set an OOP magnetic field, and numerically solve
the LLG for $10\text{\,}\mathrm{n}\mathrm{s}$ for each field, including
thermal fluctuations. Snapshots of the magnetizations in Fig. 1 were obtained
from a set of simulations, where we always start from the magnetization
pattern that was measured by MFM, set an OOP magnetic field, and relax the
system for $10\text{\,}\mathrm{n}\mathrm{s}$.
Spin Orbit Torques simulations. For these simulations, we reduce the total
size of the simulation box to
$($1\text{\,}\mu\mathrm{m}$\times$1\text{\,}\mu\mathrm{m}$\times$69\text{\,}\mathrm{n}\mathrm{m}$)$
which is discretized in $(256\times 256\times 1)$ cells. The influence of the
SOT current in the simulations is considered together with the effect of Joule
heating. In our experiments, we quantified the sample temperature as a
function of the applied lateral current, resulting in
$T(I_{x})=T_{0}\times(a\times I_{x}^{2}+b\times I_{x}+c)-T_{1}$, where
$T_{0}=$27.81\text{\,}\mathrm{K}\mathrm{/}\Omega$$,
$a=$9\text{\times}{10}^{-3}\text{\,}\Omega\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{A}\mathrm{)}^{2}$$,
$b=$4.1\text{\times}{10}^{-3}\text{\,}\Omega\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{A}\mathrm{)}$$,
$c=$68.31\text{\,}\Omega$$ and $T_{1}=$1567.3\text{\,}\mathrm{K}$$. Following
again the effective media model approach, it is assumed that the current
density that contributes to the spin polarization is achieved in individual
layers, thus the current density employed in the effective SOT fields is
calculated with the cross section
$j_{e}=I/A=I/($1\text{\,}\mu\mathrm{m}$\times$6.9\text{\,}\mathrm{n}\mathrm{m}$)$.
Furthermore, the dimensionless SOT coefficients $\eta_{\mathrm{dl}}$ and
$\eta_{\mathrm{fl}}$ are scaled according to the effective medium approach by
$t_{\mathrm{m}}/t_{\mathrm{nm}}$. So in reality, one could keep the original
cross section and assume that the total effective SOT parameters are
unchanged. We tried to quantify the SOT parameters for our multilayer stack,
but the signal obtained from second-harmonic measurements was too low to make
a conclusive statement about the strength of the parameters. For the
simulations presented in Figs. 4-6 we always start from a random magnetization
in each simulation cell. For each magnetic field, we first apply the positive
current and relax the system for $10\text{\,}\mathrm{n}\mathrm{s}$, and then
reverse the current polarity and relax for another
$10\text{\,}\mathrm{n}\mathrm{s}$. By doing so, we can apply the differential
measurement protocol from the experiments.

### Acknowledgements

S.K. thanks Barbora Budinska for support with PPMS measurements, Silvia
Damerio, Alejandro de Souza, Stefano Fedel and Can Onur Avci for trying
experiments of higher harmonic measurements for the quantification of SOT
efficiencies, MOKE imaging, and fruitful discussions, and Wolfgang Lang and
Andrii V. Chumak for the measurement equipment and use of their laboratories.
We thank Maria-Andromachi Syskaki for the optimization of the HfOx growth
process. The computational results presented have been achieved, in part,
using the Vienna Scientific Cluster (VSC). S.K. and C.A. gratefully
acknowledge the Austrian Science Fund (FWF) for support through Grant No.
P34671 (Vladimir). S.K. and D.S. acknowledge the Austrian Science Fund (FWF)
for support through Grant No. I 6267 (CHIRALSPIN). S.K. and D.S. acknowledge
funding from Österreichische Forschungsförderungsgesellschaft (FFG) under the
project Senstronic. B.A. was supported by the Austrian Science Fund (FWF)
through Grant No. I4865-N (FLUXPIN). Thin film deposition used infrastructure
provided by the ForLab MagSens. R.G., F.K., I.K., G.J., and M.Kl. acknowledge
support by the Deutsche Förderung Geselschaft (SFV TAR, 73 SPIN+x, A01, B02)
and Infineon. The infrastructure for thin-film deposition was provided by the
ForLab MagSens.

### Author Contributions

A.S. and D.S. conceived the project for a magnetic field sensor enabled by
SOT. D.S. conceived the idea of using multilayers. S.K. conceived the idea of
skyrmionic textures for sensing. S.K. and B.A. built the experimental setup at
the PPMS, S.K. performed all measurements (VSM, FMR, and transport
measurements) and analyzed the data. S.K. performed and analyzed all
micromagnetic simulations. R.G., F.K., I.K, G.J, M.Kl. sputter-deposited the
thin films. R.P., A.O.M., and H.J.H. performed the MFM measurements, M.Ki. and
K.P. fabricated the Hall bars, S.H. and F.B. implemented the thermal
fluctuations in the micromagnetic code, and S.K., F.B., C.A. and D.S. wrote
and improved the micromagnetic code. D.S. supervised the project. S.K. wrote
the initial manuscript. All authors have discussed the results, commented on,
and improved the initial manuscript.

### Additional information

### Correspondence and requests for materials

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Figure ED1: Vibrating Sample Magnetomery measurements (experiments).
Normalized MH-loops for OOP fields at different temperatures are given in (a),
and for IP fields in (b). The saturation magnetization as a function of
temperature was quantified in (c), where the magnetic moment was normalized to
the magnetic volume only, while in (d) it was magnetized to the total volume
according to the effective media approach. Figure ED2: Ferromagnetic Resonance
Measurements (experiments). (a) Schematic illustration of the experimental
setup, where we employ the flip-chip method, with the sample being placed in
direct contact to a commercially available coplanar waveguide (CPW) on a
printed circuit board (PCB). The magnetic field is applied IP, and the
resonance frequencies are measured for fields where the sample is saturated
IP. From the Kittel modes, we extract the effective magnetization
$M_{\mathrm{eff}}$ as a function of temperature. For known values of the
saturation magnetization, the effective anisotropy field shown in (c) is
$H_{\mathrm{k,eff}}=\dfrac{2K_{\mathrm{eff}}}{\mu_{0}M_{s}}$, where the
effective anisotropy constant is
$K_{\mathrm{eff}}=K_{u}-K_{d}=-\dfrac{K_{d}\times M_{\mathrm{eff}}}{M_{s}}$,
where $K_{d}$ is the shape anisotropy of the sample given by
$\dfrac{1}{2}\mu_{0}M_{s}^{2}$. The reduced gyromagnetic ratio
$\gamma^{\prime}$ shown in (d) is also fitted from the Kittel modes. If the
$M_{s}$ is normalized to the magnetic volume only we obtain $K_{\mathrm{eff}}$
and $K_{u}$ as illustrated in (e) and (f), respectively. If one uses the
effective media model, the dependencies from (g) and (h) are obtained. Figure
ED3: Finite-temperatue micromagnetic and numerical optimization of DMI
constant (simulations). Evolution of all components of the magnetization
averaged spatially plotted as a function of the applied OOP field
$\mu_{0}H_{z}$ in (a), while the zero field domain morhopology is displayed in
(b) as the OOP component of the magnetization and with a Hue-Saturation-Value
(HSV) coding scheme, where the color represents the orientation of IP
components, while the dark ($-z$) and bright ($+z$) contrasts show the OOP
components. Snapshots of the magnetization state for positive fields are given
in (c) for positive OOP fields, and in (d) for negative OOP fields. Note that
these magnetization states have been obtained by always starting from
normalized random magnetization in each cell and then relaxing the system at
each OOP field sequentially. The value of the DMI constant $D$ was numerically
optimized where we provide in (e) the average value of the objective function
$\langle|\mathbf{M}-\mathbf{M_{\mathrm{target}}}|\rangle$ that calculates the
difference between the magnetization state at a given value for $D$ and the
target magnetization configuration corresponding to the MFM data measured at
room temperature. The evolution of the magnetization state and of the cellwise
value of $|\mathbf{M}-\mathbf{M_{\mathrm{target}}}|$ is given in (f). Figure
ED4: Magnetic Force Microscopy Imaging (experiments). Schematic illustration
of the measurement setup (a), where the stack is scanned with a piezoresistive
cantilever that is attracted or repulsed based on the orientation of the
stray-field generated by the sample. The tip of the cantilever received
additional Ta(2)/Co(3)/Ta(4) deposition to enhance the sentivity and reduce
the moment of the tip. The evolution of the magnetization with the applied
field is given in (b), where the shrinking of stripes into skyrmions is
observed for both magnetic field polarities. Figure ED5: Quantification of
Joule heating (experiments). The longitudinal resistance is measured as a
function of the environmental temperature set in the PPMS cryostat while the
sample is saturated by an OOP field
$\mu_{0}H_{z}=$400\text{\,}\mathrm{m}\mathrm{T}$$ and a low amplitude current
$I=$1\text{\,}\mathrm{m}\mathrm{A}$$ is injected, which does not introduce any
Joule heating. Then we set the environmental temperature to
$T=$300\text{\,}\mathrm{K}$$, vary the applied current and record $R_{xx}$,
which is depicted in (b). Ultimately, we correlate the two measurements and
obtain the quadratic dependence of the sample temperature as a function of
applied lateral current in (c). Figure ED6: Temperature and current dependence
of the OOP sensing signal (experiments). The OOP sensing signals
$\sigma^{\mathrm{OOP}}_{xy}$ measured for different current amplitudes in (a)
at the environmental temperature $T=$300\text{\,}\mathrm{K}$$, and in (g) at
$T=$200\text{\,}\mathrm{K}$$, and the individual $R_{xy}$ curves recorded for
both current polarities (b) to (f) at $T=$300\text{\,}\mathrm{K}$$, and (h) to
(l) at $T=$200\text{\,}\mathrm{K}$$. Figure ED7: Temperature and current
dependence of the IP sensing signal (experiments). Same as Extended Data Fig.
ED6 but for IP magnetic fields. Figure ED8: Improving sensing performance in
W(5)/CoFeB(0.7)/MgO(1.1)]10 multilayers (experiments). The sensor signals for
an IP magnetic field are shown in (a) at $T=$200\text{\,}\mathrm{K}$$ and in
(b) at $T=$300\text{\,}\mathrm{K}$$ for different lateral current amplitudes.
The DC offset $O_{x}$ was then quantified as a function of the SOT current in
(c), while the sensitivity $S_{x}$ is shown in (d). We observe that the
sensitivity increases by a factor of 4 for the thinner MgO layer. Individual
curves $R_{xy}$ are given in (e) to (f) at $T=$200\text{\,}\mathrm{K}$$, and
in (g) to (h) at $T=$300\text{\,}\mathrm{K}$$, with the corresponding sensing
signals in (i) to (l).


# On the Bombieri-Lang Conjecture over finitely generated fields

Giulio Bresciani  Freie Universität Berlin, Arnimallee 3, 14195, Berlin,
Germany<EMAIL_ADDRESS>
###### Abstract.

We prove that the strong Bombieri-Lang conjecture for $\mathbb{Q}$ implies it
for fields finitely generated over $\mathbb{Q}$. We prove that the geometric
Lang conjecture plus the weak Bombieri-Lang conjecture over $\mathbb{Q}$ imply
the strong Bombieri-Lang conjecture for fields finitely generated over
$\mathbb{Q}$. Thus we reduce the arithmetic part of the conjectures to
$\mathbb{Q}$ removing the need, for arithmetic purposes, of Lang’s conjecture
for function fields.

The author is supported by the DFG Priority Program "Homotopy Theory and
Algebraic Geometry" SPP 1786

##  1\. Introduction

In [Lan86], Lang famously stated a series of conjectures centered around the
scarcity of rational points of a variety of general type over fields finitely
generated over $\mathbb{Q}$. Independently, Bombieri stated part of these
conjectures. Let us recall some definitions.

######  Definition.

A variety $X$ over $k$ is _mordellic_ if $X(k^{\prime})$ is finite for every
finitely generated extension $k$, and it is _pseudo mordellic_ if it has a
non-empty open subset which is mordellic.

A variety $X$ over $k$ is _geometrically mordellic_ , or _GeM_ , if every
subvariety of $X_{\bar{k}}$ is of general type, and it is _pseudo GeM_ if it
has a non-empty open subset which is GeM.

In order to organize Lang’s program, let us give names to some statements. The
list is taken from [AV96]. Fields are of characteristic $0$, but not
necessarily finitely generated over $\mathbb{Q}$.

LG:

(Geometric Lang Conjecture) If $X$ is a variety of general type over a field
$k$ of characteristic $0$, then it is pseudo GeM.

LF:

(Lang Conjecture for function fields) Let $K/k$ be a finitely generated
extension of fields of characteristic $0$ with $k$ algebraically closed in
$K$, and $X/K$ a variety of general type. If $X(K)$ is Zariski dense in $X$,
then $X$ is birational to a variety $X_{0}$ defined over $k$ and the "non-
constant points" of $X(K)\setminus X_{0}(k)$ are not Zariski-dense in $X$.

LW$(k)$:

If $X/k$ is a positive dimensional variety of general type, the set of
rational points $X(k)$ is not Zariski dense.

LW:

(Weak Bombieri-Lang Conjecture) ${\bf LW}(k)$ holds for all finitely generated
extensions $k$ of $\mathbb{Q}$.

LS$(k)$:

If $X/k$ is a variety of general type, then it is pseudo mordellic.

LS:

(Strong Bombieri-Lang Conjecture) ${\bf LS}(k)$ holds for all finitely
generated extensions $k$ of $\mathbb{Q}$.

Lang conjectured that ${\bf LG}$, ${\bf LF}$, ${\bf LW}$ and ${\bf LS}$ hold.
Faltings proved in [Fal94] that these conjectures hold for subvarieties of
abelian varieties, but the general case remains widely open.

We are interested in the passage from number fields to finitely generated
extensions of $\mathbb{Q}$ both in ${\bf LW}$ and ${\bf LS}$. In the case of
curves, this passage was settled by Grauert [Gra65] and Manin [Man63] who
proved the Mordell conjecture for function fields years before the Mordell
conjecture for number fields was proved by Faltings [Fal83].

Lang’s program suggests following the same route: ${\bf LF}$ should replace
Grauert and Manin’s theorem. In fact, if ${\bf LF}$ holds, then ${\bf LW}(k)$
for all number fields $k$ clearly implies ${\bf LW}$. Moreover, it is easy to
show that ${\bf LG}\wedge{\bf LW}\Rightarrow{\bf LS}$, thus

${\bf LG}\wedge{\bf LF}\wedge(\forall{\rm\leavevmode\nobreak\
n.f.\leavevmode\nobreak\ }k\mathrel{\mathop{\ordinarycolon}}{\bf
LW}(k))\Rightarrow{\bf LS}.$

We take another route, reducing the arithmetic part of the conjectures to
$\mathbb{Q}$ and avoiding ${\bf LF}$.

######  Theorem A.

Let $k$ be a field of characteristic $0$. If ${\bf LS}(k)$ holds, then ${\bf
LS}(K)$ holds for all finitely generated extensions $K$ of $k$.

In particular, ${\bf LS}(\mathbb{Q})$ implies ${\bf LS}$.

######  Theorem B.

Let $k$ be a field of characteristic $0$, and assume that ${\bf LG}$ holds. If
${\bf LW}(k)$ holds, then ${\bf LS}(K)$ holds for all finitely generated
extensions $K$ of $k$.

In particular, ${\bf LG}$ and ${\bf LW}(\mathbb{Q})$ imply ${\bf LS}$.

##  2\. Conventions

Fields are of characteristic $0$. Even though we are mainly interested in
fields finitely generated over $\mathbb{Q}$, our arguments are general,
assuming characteristic $0$ is enough. A non-standard example of a field for
which the conjectures might hold is $\mathbb{Q}(x_{1},x_{2},\dots)$: in fact,
it is easy to show that ${\bf LS}$ implies ${\bf
LS}(\mathbb{Q}(x_{1},x_{2},\dots))$.

A variety over $k$ is a quasi-projective, geometrically integral scheme. A
smooth, projective variety is of general type if its Kodaira dimension is
equal to its dimension: in particular, a point is a variety of general type. A
possibly singular, non-projective variety is of general type if it is
birational to a variety of general type.

##  3\. Proof of Theorem A

######  Lemma 3.1.

Let $h/k$ be a finite extension of fields of characteristic $0$. A variety $X$
over $k$ is pseudo mordellic if and only if $X_{h}$ is pseudo mordellic.

###### Proof.

If $U\subseteq X$ is a mordellic open subset, then $U_{h}\subseteq X_{h}$ is
mordellic too. On the other hand, let $U\subseteq X_{h}$ be a mordellic open
subset, and let $h^{\prime}$ be a Galois closure of $h/k$. Then
$\bigcap_{\sigma\in\operatorname{Gal}(h^{\prime}/k)}\sigma(U_{h^{\prime}})$ is
Galois invariant, thus it descends to a mordellic open subset of $X$. ∎

######  Lemma 3.2.

Let $X_{1},\dots,X_{n}$ be varieties over a field $k$, and assume that
$\prod_{i}X_{i}$ is pseudo mordellic. Then each $X_{i}$ is pseudo mordellic.

###### Proof.

Let $U$ be a non-empty mordellic open subset of $\prod_{i}X_{i}$. For the sake
of simplicity, assume $i=1$. Thanks to 3.1 we may assume that we have a
rational point $(x_{i})_{i}\in U(k)$. Then
$U\cap\left(X_{1}\times(x_{2},\dots,x_{n})\right)$ is a non-empty mordellic
open subset of $X_{1}\times(x_{2},\dots,x_{n})=X_{1}$. ∎

If $h/k$ is a finite extension and $X$ is a variety over $h$, recall that the
Weil restriction $R_{h/k}(X)$ is a variety over $k$ representing the functor
$S\mapsto X(S_{h})$ for schemes $S$ over $k$. The Weil restriction always
exists for quasi-projective varieties, see [BLR90, §7.6].

If $\sigma\mathrel{\mathop{\ordinarycolon}}h\to h^{\prime}$ is a field
extension, denote by $\sigma^{*}S$ the $h$-scheme
$S\to\operatorname{Spec}h\xrightarrow{\sigma^{\\#}}\operatorname{Spec}h$.

######  Lemma 3.3.

Let $h/k$ be a finite, separable extension with a Galois closure $h^{\prime}$,
$\Sigma$ the set of $k$-linear embeddings
$\sigma\mathrel{\mathop{\ordinarycolon}}h\to h^{\prime}$, $X$ a variety over
$h$. There is a natural isomorphism
$R_{h/k}(X)_{h^{\prime}}\simeq\prod_{\sigma\in\Sigma}\sigma^{*}X$.

###### Proof.

The scheme $R_{h/k}(X)_{h^{\prime}}$ over $h^{\prime}$ represents the functor
$S\mapsto\operatorname{Hom}_{h}(S\times_{\operatorname{Spec}k}\operatorname{Spec}h,X)$.
Fix an embedding $h\subseteq h^{\prime}$, and for every $\sigma\in\Sigma$
choose $\varphi_{\sigma}\in\operatorname{Gal}(h^{\prime}/k)$ such that
$\sigma$ is equal to $h\subseteq
h^{\prime}\xrightarrow{\varphi_{\sigma}^{-1}}h^{\prime}$.

If $S$ is a scheme over $h^{\prime}$, the fibered product
$S\times_{\operatorname{Spec}k}\operatorname{Spec}h$ is naturally isomorphic
to $\bigsqcup_{\sigma\in\Sigma}\varphi_{\sigma}^{-1*}S$ as an $h$-scheme.
Moreover,
$\operatorname{Hom}_{h}(\varphi_{\sigma}^{-1*}S,X)=\operatorname{Hom}_{h^{\prime}}(S,\sigma^{*}X)$.
It follows that $R_{h/k}(X)_{h^{\prime}}$ represents

$S\mapsto\prod_{\sigma\in\Sigma}\operatorname{Hom}(S,\sigma^{*}X),$

and the same clearly holds for $\prod_{\Sigma}\sigma^{*}X$. By Yoneda’s lemma,
we get the desired isomorphism. ∎

######  Lemma 3.4.

Let $h/k$ be a finite extension of fields of characteristic $0$. Then ${\bf
LS}(k)$ holds iff ${\bf LS}(h)$ holds.

###### Proof.

If ${\bf LS}(h)$ holds, then ${\bf LS}(k)$ holds thanks to 3.1. Assume now
that ${\bf LS}(k)$ holds, we want to prove that ${\bf LS}(h)$ holds. Thanks to
the preceding case, we may pass to a Galois closure and assume that $h/k$ is
Galois.

Since a product of varieties of general type is of general type and thanks to
3.3, we have that $R_{h/k}(X)$ is of general type. Since we are assuming that
${\bf LS}(k)$ holds, $R_{h/k}(X)$ is pseudo mordellic, and thus
$R_{h/k}(X)_{h}=\prod_{\sigma}\sigma^{*}X$ is pseudo mordellic too. By 3.2, it
follows that $X$ is pseudo mordellic. ∎

Assume that ${\bf LS}(k)$ holds. Let $K/k$ be a finitely generated extension,
we want to prove the ${\bf LS}(K)$ holds. Thanks to 3.4 we may assume that $k$
is algebraically closed in $K$, let $V/k$ be some variety such that $k(V)=K$.
Thanks again to 3.4, we may pass to finite extensions of $K$, and thus we may
assume that $V$ is of general type.

Let $X$ be a variety of general type over $K$, we want to show that it is
pseudo mordellic. Spread out $X$ to a family $\widetilde{X}\to V$, we have
that $\widetilde{X}$ is of general type by subadditivity of the Kodaira
dimension with fibers of general type [Kol87]. Since we are assuming that
${\bf LS}(k)$ holds, there exists a mordellic open subset
$U\subseteq\widetilde{X}$, its generic fiber $U_{k(V)}$ is a mordellic open
subset of $X$. This completes the proof of Theorem A.

##  4\. Proof of Theorem B

In this section, assume that ${\bf LG}$ and ${\bf LW}(k)$ hold. Thanks to
Theorem A, in order to prove Theorem B it is enough to show that ${\bf LS}(k)$
holds. Since we are assuming ${\bf LG}$, it is enough to show that, if $X/k$
is a GeM variety and $K/k$ is finitely generated, then $X(K)$ is finite.

We want to apply a theorem of Abramovich and Voloch [AV96, Theorem 1.7],
[Abr97] which implies that there exists an uniform bound on $|X(h)|$ for
finite extensions $h/k$ of bounded degree, but to do so we need to make a
small remark on their hypotheses. In fact, they assume that the base field $k$
is finitely generated over $\mathbb{Q}$ and that ${\bf LW}$ holds, but a
careful analysis of their proof shows that the only hypothesis on $k$ they
actually use is that ${\bf LW}(k)$ holds.

Since we are assuming that ${\bf LW}(k)$ holds, we may thus apply the uniform
bound theorem by Abramovich and Voloch. Let $V$ be an integral scheme of
finite type over $k$ whose function field is $K$, choose any generically
finite, dominant rational map $V\dashrightarrow\mathbb{P}^{n}$ and let $d$ be
its degree. Since rational points are dense in $\mathbb{P}^{n}$, it follows
that $V_{d}=\\{p\in V(\bar{k})\mid[k(p)\mathrel{\mathop{\ordinarycolon}}k]\leq
d\\}$ is dense in $V$. Thanks to the uniform bound theorem, there exists an
$N$ such that $|X(h)|\leq N$ for every finite extension $h/k$ with
$[h\mathrel{\mathop{\ordinarycolon}}k]\leq d$. Let us prove that $|X(K)|\leq
N$.

If by contradiction we have $N+1$ different sections $\operatorname{Spec}K\to
X$, up to shrinking $V$ we may assume that they extend to $N+1$ morphisms
$f_{1},\dots,f_{N+1}\mathrel{\mathop{\ordinarycolon}}V\to X$. Since
$|X(h)|\leq N$ for every finite extension $h/k$ with
$[h\mathrel{\mathop{\ordinarycolon}}k]\leq d$, we have that for every $p\in
V_{d}$ there exists a pair of different indexes $i\neq j$ with
$f_{i}(p)=f_{j}(p)$. Since $V_{d}$ is dense and the pairs of different indexes
are finite, there exists a pair $i\neq j$ and a subset $S\subseteq V_{d}$
dense in $V$ such that $f_{i}(p)=f_{j}(p)$ for every $p\in S$, thus
$f_{i}=f_{j}$ which gives a contradiction.

##  5\. Acknowledgements

I would like to thank Dan Abramovich for suggesting some simplifications.

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