Warsaw University of Technology, Faculty of Physics, 00-662 Warsaw, Poland AGH
University of Science and Technology, Faculty of Physics and Applied Computer
Science, 30-059 Kraków, Poland Institute of Molecular Physics, Polish Academy
of Sciences, ul. M.Smoluchowskiego 17, 60-179 Poznań, Poland AGH University of
Science and Technology, Faculty of Physics and Applied Computer Science,
30-059 Kraków, Poland FMR, SOT-FMR, SOT, DW

# Study of Spin-Orbit Interactions and Interlayer Ferromagnetic Coupling in
Co/Pt/Co Trilayers in Wide Range of Heavy Metal Thickness

Piotr Ogrodnik<EMAIL_ADDRESS>AGH University of Science and
Technology, Institute of Electronics, 30-059 Kraków, Poland  Krzysztof Grochot
AGH University of Science and Technology, Institute of Electronics, 30-059
Kraków, Poland  Łukasz Karwacki  Institute for Theoretical Physics, Utrecht
University, Princetonplein 5, 3584 CC Utrecht, Netherlands  Jarosław Kanak
AGH University of Science and Technology, Institute of Electronics, 30-059
Kraków, Poland  Michał Prokop  Catalan Institute of Nanoscience and
Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona,
Spain  Jakub Chęciński  AGH University of Science and Technology, Institute of
Electronics, 30-059 Kraków, Poland  Witold Skowroński  AGH University of
Science and Technology, Institute of Electronics, 30-059 Kraków, Poland
Sławomir Ziętek  AGH University of Science and Technology, Institute of
Electronics, 30-059 Kraków, Poland  Tomasz Stobiecki  AGH University of
Science and Technology, Institute of Electronics, 30-059 Kraków, Poland

###### Abstract

The spin-orbit torque, a torque induced by a charge current flowing through
the heavy-metal conducting layer with strong spin-orbit interactions, provides
an efficient way to control the magnetization direction in heavy-
metal/ferromagnet nanostructures, required for applications in the emergent
magnetic technologies like random access memories, high-frequency nano
oscillators, or bio-inspired neuromorphic computations. We study the interface
properties, magnetization dynamics, magnetostatic features and spin-orbit
interactions within the multilayer system
Ti(2)/Co(1)/Pt(0-4)/Co(1)/MgO(2)/Ti(2) (thicknesses in nanometers) patterned
by optical lithography on micrometer-sized bars. In the investigated devices,
Pt is used as a source of the spin current and as a non-magnetic spacer with
variable thickness, which enables the magnitude of the interlayer
ferromagnetic exchange coupling to be effectively tuned. We also find the Pt
thickness-dependent changes in magnetic anisotropies, magnetoresistance,
effective Hall angle and, eventually, spin-orbit torque fields at interfaces.
The experimental findings are supported by the relevant interface structure-
related simulations, micromagnetic, macrospin, as well as the spin drift-
diffusion models. Finally, the contribution of the spin-orbital Edelstein-
Rashba interfacial fields is also briefly discussed in the analysis.

###### keywords:

Ferromagnetic Resonance, Spin Hall Effect, Magnetoresistance, Spin Orbit
Torques, Spin Accumulation, Edelstein-Rashba Effect, X-Ray Diffraction

##  1 Introduction

The magnetic multilayer structures consisting of thin ferromagnetic (F) layers
and non-magnetic spacers are known to exhibit a plenty of phenomena, among
which one can find those extensively studied for the last decades like
anisotropic, giant and tunneling magnetoresistace or spin-transfer torque
effect (STT)1, 2 and recently current-driven spin orbit torque (SOT)
magnetization switching.3 These effects are widely exploited in the spintronic
devices - magnetic random access memories (MRAM), like STT-MRAM and SOT-
MRAM,4, 5, 6 as well as may be exploited in magnetic sensors (including
magnetic nanoparticles) and nanooscillators.7, 8 Such devices include non-
magnetic layers that are crucial for their features and performance. These
layers may be both insulating (e.g. MgO in magnetic tunnel junctions) or
metallic (e.g. Cu, Au in GMR devices).8 Recently,the non-magnetic layers made
of heavy metallic (HM) elements (W, Ta, Pt and their alloys9, 10, 11) are
extensively studied because of their large spin-orbit coupling (SOC).12 Such
layers combined with ferromagnetic ones (typically Co, CoFeB) are expected to
have new spin transport properties related to the SOC, e.g. spin Hall effect
(SHE) and Rashba-Edelstein effect (REE).13, 14 Although the SHE occurs in
single HM layer,15 it is detectable in heterostructures with ferromagnets
only, such as F/HM bi- 16 and F/HM/F trilayers,17, 18. In these structures the
spin-polarized electrons can accumulate at the HM/F interfaces and then may be
efficiently injected into F layer exerting the spin-orbit torque (SOT) on its
magnetic moment. This effect has been predicted theoretically19, 20, 21 and
reported in many experimental works on SOT-induced magnetic dynamics22 and
magnetic switching.18, 23, 24, 25 Especially for the F layers with a magnetic
perpedicular anisotropy, the SOT enables a promising way to design efficient,
ultra-low power consumption spintronic devices. Apart from the SHE and related
spin accumulation, the other pure interfacial effects, like charge-spin
conversion REE at interfaces, contribute to the SOT.26, 27, 28 Therefore the
interface engineering becomes significant for optimization of SOT-based
devices.29, 30, 31 The spin-currents injected to F layer and SOTs may be
examined by electric measurements through its magnetoresistance32, 33, 34 and
the Anomalous Hall effect (AHE).35 The change of the resistance of the hybride
structure caused by the above effects are refered to as spin magnetoresistace
(SMR).36 Up to date, the specific multilayer structures (bi- and trilayers)
were studied in details. Among them we find CoFeB-based structures like
W/i-CoFeB/Pt,37 p-CoFeB/Ta, as well as the Co-based multilayers like 38
p-Co/Pt/i-Co,18 p-Co/Pt,39, 40, 41 i-Co/Ta,41 Ta/i-Co/Pt,42 where p(i) stands
for perpendicular(in-plane) anisotropy. In this paper we present the detailed
studies of the Co/Pt/Co system with use of the electrically-detected FMR
(ferromagnetic resonance) as well as low-frequency harmonic Hall voltage and
static magnetotransport measurements. We provide the results on the resonance
frequencies and the SOT effective fields depending on the Pt thickness. Also,
we analyse the magnetic parameters of the system like anisotropies, saturation
magnetizations and the interlayer exchange coupling (IEC). We show that
anisotropies and the IEC strongly depend on the Pt thickness, particularly for
Pt layer thicknesses less than 2 nm. For such a thin Pt, the transition from
the effective in-plane Co anisotropy to the perpendicular one may occur. We
account for the features by providing a reliable theoretical macrospin models
of magnetization dynamics, magnetoresistance and effective spin Hall effect
angle.

##  2 Experimental

###  2.1 Sample preparation

Multilayers are deposited on thermally oxidized Si substrates using magnetron
sputtering at room temperature. We study Co/Pt/Co trilayer within
Ti(2)/Co(1)/Pt(0-4)/Co(1)/MgO(2) /Ti(2) structure (the numbers in brackets
indicate the nominal thickness of the individual layers in nanometers).

Figure 1: (a) Experimental multilayer stack with a wedge of Pt. The red thick
(thin) wavy arrows indicate strong (weak) IEC, whereas the grey arrows show
the change in the magnetization alignment with Pt thickness, (b) the patterned
device for a certain thickness of Pt - the arrows indicate the direction of
current flow ($\vec{j_{c}}$) and associated spin current ($\vec{j_{s}}$) due
to SHE. The short arrows depicted in Co layers mean their magnetization
vectors for a given Pt thickness at remanence, (c) the polar and azimuthal
angles describing the magnetization direction within the Co layers.

The Co/Pt/Co trilayer was designed so it allows to study the influence of Pt
thickness on the magnetic anisotropy of bottom and top Co layers, the IEC
between Co layers through Pt spacer, magnetization dynamics, and SHE-driven
SOT acting on the F layers. For this purpose, both bottom and top thin Co
layers should have small anisotropy (differing by interfaces Ti/Co and Co/MgO)
with values close to the transition from in-plane to perpendicular. The Ti
underlayer improves subsequent layers’ adhesion and smooths the substrate
surface. Moreover, as shown in Ref.43, Ti/Co interface is alloyed due to
mixing during magnetron deposition, while Co/MgO interface is sharp.44
Therefore, the top Co layer is characterized by higher interface perpendicular
anisotropy. The Pt layer was deposited in wedge-shaped form with thickness
varied from 0 to 4 nm along 20 mm long sample edge (x coordinate). The
resulting thickness gradient was achieved by the controlled movement of a
shutter. Thicknesses of all layers were determined from the deposition growth
rate of particular materials calibrated using x-ray reflectivity measurements.
Next, before patterning to the form of bar devices, all as-deposited samples
were characterized by x-ray diffraction $\theta-2\theta$ (XRD) and grazing
incidence diffraction (GIXD), and also examined by the polar Kerr magnetometer
(p-MOKE) and time-resolved TR-MOKE in order to determine the static and
dynamic magnetization parameters, which studies have been described in detail
in a separate work.45 After basic characterization of continuous samples,
multilayers were patterned using optical direct imaging lithography and ion
etching to create a matrix of Hall- and resistance-bar devices, with different
thicknesses of Pt for subsequent electrical measurements (Figure 1a). The
sizes of prepared structures were 100$\mu$m x 20$\mu$m for magnetoresistance
and spin diode effect measurements, whereas 100 $\mu m$ x 10 $\mu m$ for the
AHE, and harmonics measurements. Al(20)/Au(30) electrical leads of 100 $\mu$m
x 100 $\mu$m were deposited in a second lithography step followed by the lift-
off process. Specific locations of pads near the Hall-bars were designed for
measurement in a custom-made rotating probe station allowing 2- or 4-points
measurement of electrical transport properties in the presence of the magnetic
field applied at an arbitrary azimuthal and polar angle with respect to the
Hall-bar axis. The scheme of experimental setup for longitudinal ($R_{xx}$)
and Hall ($R_{xy}$) resistance measurements is shown in Figure 2.

Figure 2: The Hall resistance ($R_{xy}$) (a) and 4-points longitudinal
magnetoresistance (b) measurement setups depicted on micrographs of the
devices.

The resistance was measured using a four-point method46 and resistivities of
Pt and Co layers were determined using a parallel resistors model and the
method described by Kawaguchi et al.47. The Pt and Co resistivities analysis
yielded 59$\mu\Omega$ cm and 72.5 $\mu\Omega$ cm, respectively.

###  2.2 Structural characterization

High-resolution X’Pert–MPD diffractometer with a Cu anode was used for x-ray
diffraction (XRD) characterization. Figure 3 shows the XRD $\theta-2\theta$
profiles of the Si/SiO2/Ti/Co(1)/Pt(0-4)/Co(1)/MgO/Ti multilayer measured at
different positions of the Pt wedge. The $\theta-2\theta$ measurements show
the preferred growth of the Pt/Co in [111] direction of fcc structure. The
peak of Co layers is invisible because of their tiny thicknesses
($t_{Co}\approx 1$ nm). The arrows indicate the Co (111) peak position present
in the thick Co layer case (see supplemental material in Refs.18, 48). Peak on
the right side of the Pt(111) is a Laue satellite18 that confirms the
asymmetry in top Pt/Co and bottom Co/Pt interfaces. The intensity of the
profiles depends on the number of Pt atoms. Therefore, in the case of very
thin (0.2 nm) Pt layers, the Laue peaks are out of detection of the
experimental method. However, one can see that the Pt peak slightly shifts to
the right for thin Pt layers. It suggests that most of the Pt layer becomes
mixed with Co atoms making a rather Co-Pt compound than a well-separated
layer.

Figure 3: The XRD $\theta-2\theta$ profiles of a
Si/SiO2/Ti/Co(1)/Pt(0-4)/Co/MgO/Ti measured at different position of the Pt
wedge. The arrow indicates $2\theta$ position of the structural Co (111) peak
visible in the reference sample with 8 nm of Co and 4 nm of Pt layer thickness
(inset).

Figure 4a shows the profile measured for the
Si/SiO2/Ti(2)/Co(1)/Pt(4)/Co(1)/MgO(2) /Ti(2) at thick Pt $\approx$ 4 nm,
together with the profile calculated using the simulated48 structure (Figure
4b). An excellent agreement between the experimental and theoretical values is
achieved. The structure was simulated with the assumption of Pt and Co mixing
at the interfaces. The simulations represent the columnar grains in the Pt and
Co layers and a transition area with the Pt-Co mixed interface. Mixing of the
Pt and Co atoms at interface causes decreasing Pt lattice planes spacing
compared to pure Pt.

The simulation assumes more significant mixing at the bottom side of the Pt
than at the top one. In the former case, the heavy Pt atoms can penetrate the
Co layer more easily than in the latter case. Moreover, for Pt within Co, the
interfacial enthalpy is -33 kJ/(mole of atoms), whereas for Co in Pt is -26
kJ/(mole of atoms). Higher negative enthalpy results in easier mixing at the
bottom Co/Pt interface.

Figure 4: Measured and calculated XRD $\theta-2\theta$ profiles (a). The
assumed thicknesses of the Pt, Co layers and transition area with the Pt-Co
mixed interfaces (inset). A snapshot of the Monte Carlo simulation of the
interface structure in Co/Pt/Co (b).

###  2.3 Spin diode effect measurements

The magnetic dynamics of the patterned samples was electrically-detected with
the FMR measurements through the spin diode effect.49 The scheme of
measurement setup is shown in Figure 5. The effect occurs when the rf current
flows through the magnetoresistive element that in the case of our samples
exhibits the anisotropic magnetoresistance (AMR) and SMR. Then, the current-
related effective magnetic fields (as Oersted field or SOT fields) force the
sample magnetization to oscillate. The magnetization oscillations, in turn,
result in time-dependent resistance of the sample which mixes up with the rf-
current.

Figure 5: The resistance-bar attached to the spin-diode FMR measurement setup
(see main text for details) depicted on the set of four patterned devices
photomicrography. The dashed arrow indicates the direction of the external
magnetic field $\vec{H}_{ext}$.

Therfore the measured output voltage may be expressed as
$V_{out}=I_{0}\cos(\omega t)\cdot R(\omega t+\Phi)$, where the $\Phi$ is the
phase shift between the current and resistance. One notes that $V_{out}$
includes ac and dc contributions, namely: $V_{out}=V_{dc}+V_{ac}=I_{0}\delta
R\cos\Phi+I_{0}\delta R(2\omega t+\Phi)$. The dc output voltage depends on the
angular frequency, external magnetic field and parameters of the sample.

In this paper, the spin-diode FMR measurements are performed with an
amplitude-modulated radio-frequency (rf) current with a corresponding power of
P = 16 dBm and frequencies ranging from 1 to 25 GHz. The mixing voltage
($V_{out}$) is measured using a lock-in amplifier synchronized to the rf
signal. The in-plane magnetic field ($H_{ext}$) is applied at $\phi$ = 45 deg
with respect to the microstrip long axis and was swept from 0 up to 9 kOe.

###  2.4 Harmonic measurements

In order to determine spin-orbit torque fields (damping- and field-like
components), as well as the spin Hall angle, we used the methods based on the
harmonic measurements.50, 51, 10 For these measurements, we apply a low-
frequency constant-amplitude sinusoidal voltage to the Hall bar device with
current density from j= $5.75\times 10^{9}$ $A/m^{2}$ to j= $9.62\times
10^{9}$ $A/m^{2}$ depending on the Pt layer thickness. Using two lock-in
amplifiers, we measure simultaneously the in-phase first harmonic
($V_{\omega}$) and the out-of-phase second harmonic Hall voltages
($V_{2\omega}$) as a function of an external magnetic field $H_{ext}$. The
sample is rotated within the x-y plane, making an azimuthal angle $\phi_{H}$
with the x-axis as depicted in Figure 6 on which the scheme of measurement
setup is shown.

Figure 6: The Hall-bar and the experimental setup for harmonic Hall voltage
($V_{\omega,2\omega}$) measurements.

The measurements were conducted in two configurations: the first one is
referred to as field measurements, and the samples were probed with the
different magnitudes of the external magnetic field applied along both x and y
directions.52, while the second configuration is the angular measurements. The
sample is rotated in x-y plane while the $V_{\omega,2\omega}$ is recorded 52,
50 for fixed magnitudes of external magnetic field. The field measurements are
relevant in the case of samples with out-of-plane effective anisotropies. On
the contrary, the angular measurements allow to detect harmonic signals in
samples with in-plane effective anisotropy.

##  3 Theory

###  3.1 Resonance model

This subsection presents the macrospin model that allow us to calculate
resonance frequencies of the considered Co/Pt/Co structure. Since the Co
layers may be either coupled or uncoupled we employ the approach that has been
already presented in details and successfully applied in our previous work.53
We describe magnetic moments of each layer by spherical angles (polar
$\theta_{i}$ and azimuthal $\phi_{i}$):

$\bm{M_{i}}=M_{S,i}[\sin\theta_{i}\cos\phi_{i},\,\sin\theta_{i}\sin\phi_{i},\,\cos\theta_{i}],$
(1)

where $i$=1(2) is reffered as to bottom(top) cobalt layer. The magnetization
dynamics of the system is described by two coupled Landau–Lifshitz–Gilbert
(LLG) equations,

$\displaystyle\frac{d\bm{M_{i}}}{dt}=-\gamma_{e}\bm{M_{i}}\times\bm{H}_{\rm
eff,i}+\frac{\alpha_{g}}{M_{S,i}}\bm{M_{i}}\times\frac{d\bm{M_{i}}}{dt}+$
$\displaystyle+\gamma_{e}(\bm{\tau}_{DL,i}+\bm{\tau}_{FL,i}),$ (2)

where $\gamma_{e}\approx 1.760859644\times 10^{11}\
\frac{\mathrm{rad}}{\mathrm{sT}}$ is the gyromagnetic ratio, and $\alpha_{g}$
is the Gilbert damping parameter for each layer.

The terms
$\bm{\tau}_{DL}=H_{DL}(\bm{m}_{i}\times\bm{m}_{i}\times\bm{\hat{e}}_{y})$ and
$\bm{\tau}_{FL}=H_{FL}(\bm{m}_{i}\times\bm{\hat{e}}_{y})$ stand for SOT
damping-like (DL) and field-like(FL) components with the unit vector
$\bm{m_{1(2)}}=\frac{\bm{M_{1(2)}}}{M_{S,1(2)}}$ and the amplitudes $H_{DL}$
and $H_{FL}$ respectively.

The effective field ($H_{eff}$) can be expressed as functional derivative of
the follwing total magnetic energy of the system:

$\displaystyle
U=\sum_{i=1}^{2}K_{\perp,i}t_{Co_{i}}\left[\left(\cos\beta_{i}\sin\theta_{i}\sin\phi_{i}-\sin\beta_{i}\cos\theta_{i}\right)^{2}+\right.$
$\displaystyle+\left(\cos\delta_{i}\left(\cos\theta_{i}\cos\beta_{i}+\sin\theta_{i}\sin\phi_{i}\sin\beta_{i}\right)-\right.$
$\displaystyle\left.\left.-\cos\phi_{i}\sin\theta_{i}\sin\delta\right)^{2}\right]+K_{\parallel,i}\left(\cos^{2}\theta_{i}+\sin^{2}\theta_{i}\sin^{2}\phi_{i}\right)-$
$\displaystyle-t_{Co_{i}}\bm{M_{i}}\cdot\bm{H}_{\rm
ext}-t_{Co_{i}}\bm{M_{i}}\cdot\bm{H}_{dem,i}-J\bm{M_{i}}\cdot\bm{M_{j}},$ (3)

The complex expression for the anisotropies originates from the rotation of
the easy-axes around x and y directions with use of the relevant Euler
rotation matrices. The angles $\beta$ and $\delta$ have been introduced in
order to account for a small deviation of the perpendicular
anisotropies($K_{\perp,i}$) from perpendicular(z) direction
($\delta,\beta\ll\pi/2$). The perpendicular anisotropy terms simplify into
well-known form $K_{\perp,i}\sin^{2}\theta_{i}$ when $\delta_{i},\beta_{i}=0$.
Also, we have added a small in-plane contribution $K_{\parallel}\ll K_{\perp}$
along $y$ direction. As long as they are small, they slightly improve the
fitting the macrospin model to the experimental data. In Eq.(3), $t_{Co,i}$,
$\bm{H}_{\rm ext}$, $\bm{H}_{dem,i}$ and $J$ stand for magnetic layer
thickness, external magnetic field, demagnetizing field and IEC respectively.

The LLG equation (3.1) in polar coordinates can be written in the general form

$\dot{\bm{\alpha}}=\bm{v}(\theta_{i},\phi_{i})^{T},$ (4)

where $\dot{\bm{\alpha}}$ and $\bm{v}$ are the vectors containing the
spherical angles ($\theta_{1,2},\phi_{1,2}$) time-derivatives and the right-
hand-side (RHS) of the LLG equation, respectively. After linearization of
$\bm{v}$ with respect to small deviations in $\theta_{i}$ and $\phi_{i}$ from
their stationary values, one can write Eq. (4) in the form

$\dot{\bm{\alpha}}=\hat{X}\bm{\Gamma}(t),$ (5)

where $\hat{X}$ is a $4\times 4$ matrix consisting of the derivatives of the
RHS of Eq. (4) with respect to the angles $\theta_{i},\,\phi_{i}$ (i.e.,
$X_{kj}\equiv\frac{\partial v_{k}}{\partial\alpha_{j}}$), while
$\bm{\Gamma}(t)=\bm{(}\delta\alpha_{1}(t),\ldots,\delta\alpha_{4}(t)\bm{)}^{T}$
is a vector containing time-dependent angle differentials, i.e.
$\delta\alpha_{1}(t)\equiv\delta\theta_{1}(t)$,
$\delta\alpha_{2}(t)\equiv\delta\phi_{1}(t)$, etc. When small oscillations
assumed and in the absence of the external driving force (i.e. SOT or Oersted
field) Eq. (5) can be rewritten as an eigenvalue problem of the matrix
$\hat{X}$:

$\left|\hat{X}-\omega\hat{I}\right|=0.$ (6)

The solution of the problem provides the complex eigenvalues $\omega_{i}$
determine 2 distinct natural resonance angular frequencies of the system,
$\omega_{R,i}=\text{Re}\,\omega_{i}$.

###  3.2 Diffusive model of the magnetoresistance

The average longitudinal resistance of our trilayer stack is in general
dependent on the orientation of magnetizations $\bm{m}_{1(2)}$ in both
ferromagnetic layers and reads

$\displaystyle
R_{xx}(\bm{m_{1(2)}})=\frac{L}{w}\left[\frac{1}{E_{x}}\frac{1}{\sum_{\chi}t_{\chi}}\sum_{\chi}\int_{\chi}dzj_{c,x}^{\chi}(z,\bm{m_{1(2)}})\right]^{-1}\,,$
(7)

where $E_{x}$ is the electric field in $x$ direction, $L$ is the length, $w$
the width, and $t_{\chi}$ the thickness of layer $\chi=HM,F1,F2$ and
$\int_{\chi}$ denotes integral with limits corresponding to the position of
layer $\chi$ in the stack. For Pt, i.e. for $\chi=HM$, the charge current
density reads

$\displaystyle
j_{c,x}^{HM}(z,\bm{m_{1(2)}})=\frac{1}{\rho_{HM}}E_{x}-\frac{\theta_{SH}}{2e\rho_{HM}}\frac{\partial\mu_{s,y}^{HM}(z,\bm{m_{1(2)}})}{\partial
z}\,,$ (8)

where $\rho_{HM}$ is the resistivity of the Pt layer, $\theta_{SH}$ is the
spin Hall angle defined as charge-to-spin current conversion efficiency at a
very thick HM layer limit, and $\mu_{s,y}^{HM}(z,\bm{m_{1(2)}})$ is the spin
accumulation, while for the ferromagnetic layers, i.e. $\chi=F1$ ($\chi=F2$)

$\displaystyle
j_{c,x}^{F1(F2)}(z,\bm{m_{1(2)}})=\frac{\left[1-\theta_{AMR}(\bm{m_{1(2)}}\cdot\hat{\bm{x}})^{2}\right]}{\rho_{F1(F2)}}E_{x}\,,$
(9)

where $\rho_{F1(F2)}$ is the resistivity of the corresponding ferromagnetic
layer, $\theta_{AMR}$ is the AMR in the thick ferromagnet limit (assumed for
simplicity the same in both ferromagnetic layers). For more details see e.g.
32.

To obtain spin accumulation in the Pt layer we consider the spin current
density flowing in Pt,

$\displaystyle\mathbf{j}_{s}^{HM}(z,\bm{m_{1(2)}})=-\theta_{SH}\frac{1}{\rho_{HM}}E_{x}\hat{\bm{y}}+\frac{1}{2e\rho_{HM}}\frac{\partial\bm{\mu}_{s}^{HM}(z,\bm{m_{1(2)}})}{\partial
z}\,,$ (10)

along with the boundary conditions:

$\displaystyle\mathbf{j}_{s}^{HM}(-t_{H},\bm{m_{1}})$
$\displaystyle=\mathbf{q}_{1}(\bm{m_{1}})\,,$ (11a)
$\displaystyle\mathbf{j}_{s}^{HM}(0,\bm{m_{2}})$
$\displaystyle=-\mathbf{q}_{2}(\bm{m_{2}})\,,$ (11b) $\displaystyle
j_{s}^{F1}(-t_{HM}-t_{F1})$ $\displaystyle=0\,,$ (11c) $\displaystyle
j_{s}^{F1}(-t_{HM})$
$\displaystyle=\mathbf{m_{1}}\cdot\mathbf{q}_{1}(\bm{m_{1}})\,,$ (11d)
$\displaystyle j_{s}^{F2}(0)$
$\displaystyle=\mathbf{m_{2}}\cdot\mathbf{q}_{2}(\bm{m_{2}})\,,$ (11e)
$\displaystyle j_{s}^{F2}(t_{F2})$ $\displaystyle=0\,,$ (11f)

where the spin current in ferromagnetic metals assumes the following form

$j_{s}^{F1(F2)}(z)=\frac{(1-\beta_{F1(F2)}^{2})}{2e\rho_{F1(F2)}}\frac{\partial\mu_{s}^{F1(F2)}(z)}{\partial
z}\,,$ (12)

and the interfacial spin currents:

$\displaystyle\bm{q}_{1}(\bm{m_{1}})$
$\displaystyle=G_{s}^{(1)}\left[\mu_{s}^{F1}(-t_{HM})-\bm{m_{1}}\cdot\bm{\mu}_{s}^{HM}(-t_{HM},\bm{m_{1(2)}})\right]\bm{m_{1}}$
$\displaystyle+G_{r}^{(1)}\bm{m_{1}}\times\mathbf{m_{1}}\times\bm{\mu}_{s}^{HM}(-t_{HM},\bm{m_{1(2)}})$
$\displaystyle+G_{i}^{(1)}\bm{m_{1}}\times\bm{\mu}_{s}^{HM}(-t_{HM},\bm{m_{1(2)}})$
(13a) $\displaystyle\bm{q}_{2}(\bm{m_{2}})$
$\displaystyle=G_{s}^{(2)}\left[\mu_{s}^{F2}(0)-\bm{m_{2}}\cdot\bm{\mu}_{s}^{HM}(0,\bm{m_{1(2)}})\right]\bm{m_{2}}$
$\displaystyle+G_{r}^{(2)}\bm{m_{2}}\times\bm{m_{2}}\times\bm{\mu}_{s}^{HM}(0,\bm{m_{1(2)}})$
$\displaystyle+G_{i}^{(2)}\bm{m_{2}}\times\bm{\mu}_{s}^{HM}(0,\bm{m_{1(2)}})\,,$
(13b)

where $G_{s}^{(1)}$ and $G_{s}^{(2)}$ are spin conductances and
$G_{r(i)}^{(1)}$ and $G_{r(i)}^{(2)}$ are real (imaginary) parts of spin-
mixing conductances for interface 1 (F1/HM) and 2 (HM/F2), respectively.
Moreover, the effective fields: $H_{DL}$ and $H_{FL}$ (cf. Sec.3.1) due to SHE
and spin accumulation at the interfaces can be expressed as follows:

$H_{DL}^{1(2)}=-\frac{\hbar}{2e^{2}}\frac{1}{\mu_{0}M_{S,1(2)}t_{F1(F2)}}\mathbf{x}\cdot(\mathbf{m_{1(2)}}\times\mathbf{q_{1(2)}})$
(14)

and

$H_{FL}^{1(2)}=-\frac{\hbar}{2e^{2}}\frac{1}{\mu_{0}M_{S,1(2)}t_{F1(F2)}}\mathbf{y}\cdot(\mathbf{m_{1(2)}}\times\mathbf{q_{1(2)}}).$
(15)

To fit the appropriate magnetoresistance relations obtained with use of Eq.
(7) and shown in Figure 10, we use the following parameters 32:
$\rho_{HM}=59~{}\mu\Omega\text{cm}$,
$\rho_{F1(F2)}=72.5~{}\mu\Omega\text{cm}$, $\lambda_{HM}=1.8~{}\text{nm}$,
$\lambda_{F1}=\lambda_{F2}=7~{}\text{nm}$, $\theta_{SH}=8\%$,
$\theta_{AMR}=0.15\%$, $\beta_{1}=\beta_{2}=0.3$,
$G_{s}^{(1)}=G_{s}^{(2)}=G_{r}^{(1)}=G_{i}^{(1)}=10^{15}~{}\Omega^{-1}m^{-2}$,
$G_{r}^{(2)}=G_{i}^{(2)}=0.4G_{r}^{(1)}$ The parameters were also used to
calculate SOT effective fields that turn out to be pivotal in the
interpretation of the experimental data presented in Sec.4.3.

##  4 Results

###  4.1 FMR and interlayer coupling

First, we measured the magnetization dynamics of the Co/Pt/Co sample in a wide
range of Pt thickness from strong through moderate coupling to completely
decoupled Co layers. The dynamics was investigated using the electrically
detected FMR through the spin diode effect49 as described in Sec.2.3. We
observed the dispersion relations changing once the Pt thickness reaches
boundary values. Thus, we could point out the distinct regions where the
system behaves differently. This feature is illustrated in Figure 7. On the
right panel of Figure 7 one can see the color FMR spectral line-shapes. On the
left panel, points correspond to the experimental resonance frequencies. On
both sides of this figure, we show that for thin Pt spacer (below 1 nm), the
dispersion relations are typical Kittel-like dependencies and move towards
lower frequencies when the $t_{Pt}$ is growing. Next, for $t_{Pt}>1nm$, the
$f(H_{r})$ change their slopes. Also, their branches part from each other,
especially at low frequencies when a sort of resonance mode gaps in
experimental data occur. The increase of the Pt thickness ($t_{Pt}$ > 2 nm)
provides the Kittel-like dependencies again. However, for thick Pt ($t_{Pt}$>3
nm) the experimental $f(H_{r})$ practically does not change anymore with the
Pt thickness.

In order to understand the dependence on Pt thickness, we did the macrospin
simulations. We performed a vast number of fittings for the whole range of the
Pt thickness, i.e. from 0.09 nm to 4.04 nm (Figure 7). We reproduced the
difference in the dynamical behavior of the structure, and we were able to
identify the boundaries between different regions of Pt thickness where these
behaviors occur, namely region I (thin Pt), II (medium Pt), III (intermediate
Pt) and region IV (thick Pt). Despite the similarity of $f(H_{r})$ in regions
III and IV, we refer to the former as the intermediate since the differences
occur in magnetoresistance results discussed in the further part of the paper.
The presence of the additional modes (especially in regions III and IV) that
were not registered in the experiment, we explain as follows. The theoretical
results based on the model presented in Sec.3.1 come from the solution of the
eigenvalue problem, i.e., the model predicts all possible steady-state modes,
regardless of the source of their excitations. On the other hand, the
different way of forcing the excitation (by SOTs or Oersted field) is
inherent. In general, the resonance modes are not always exicted, depending on
the force amplitude and its origin.

Figure 7: Experimental versus theoretical relations of dispersion for samples
from region I ((a) and (b)), II (c-d), III (e-f), IV (g-h). Left column: the
sets of theoretical (lines) and experimental (points) dependencies for each
region. Right column: the experimental $V_{DC}$ spectra shown as color map
(color is the magnitude of SD signal) and the source raw spectra measured at
frequency ranging from 2 to 20 GHz (light color lines) with corresponding
theoretical f(H) dependencies (solid black lines). The macrospin simulation
magnetic parameters are the same as presented in Figure 8(a-d).

The macrospin parameters are summarized in Figure 8. The vertical dashed lines
indicate the boundaries between regions I, II, III, and IV. Figure 8(a-d)
presents: perpendicular anisotropies $K_{\perp,1,2}$, saturation magnetization
$M_{S,1,2}$, as well as the strength of the interlayer coupling. We also
depicted the effective anisotropies $K_{eff,1,2}\equiv
K_{\perp}-\frac{\mu_{0}}{2}M^{2}_{S}$. One can see that the Co layer (indexed
as 1) covered by the MgO layer exhibits a larger perpendicular anisotropy than
the one adjacent to the Ti layer (indexed as 2), similarly as in the system
Si/SiO2/Ti(2)/Co(3)/Pt(tPt)/Co(1)/MgO(2)/Ti(2) examined in our previous
work.18 Moreover, on the basis of magnetization measurements in an external
perpendicular field, using vibrating sample magnetometer (VSM), we showed that
sample Pt(4)/Co(1)/MgO(2)/Ti(2) has smaller effective anisotropy field
($H_{K,eff}$=1.3 kOe) than Ti(2)/Co(1)/Pt(4) ($H_{K,eff}$=1.65 kOe). In Figure
8a,b we show that the effective anisotropy $K_{eff,1}$ changes its sign, while
the $K_{eff,2}$ is negative for all Pt thicknesses. The change of the sign of
the effective anisotropy is related to the boundary between regions I and II.
Furthermore, one can see that $K_{\perp,1}$ increases with the Pt thickness up
to $t_{Pt}$ = 2 nm, whereas the value of $K_{\perp,2}$ is growing just up to 1
nm. Above this thickness, $K_{\perp,2}$ is rather stable, and its values are
more or less 0.2 $MJ/m^{3}$. The $K_{\perp,1}$ reaches its the highest value
at $t_{Pt}\approx 2nm$ when it drops to the level about 0.75 $MJ/m^{3}$. The
different values of $K_{\perp,1}$ than $K_{\perp,2}$ we relate to the much
more efficient perpendicular anisotropy at Co/MgO than Co/Ti interface. On the
other hand, the mean value magnetization saturation averaged over the whole
range of the Pt thickness is about 1.07 T for both Co layers. The actual value
for a given $t_{Pt}$ differs by $\pm$ 10 $\%$. The abrupt decrease in $M_{S}$
for very thin Pt layer (0.09 nm) is caused by the quality of the interfaces
and related inter-mixing of Pt and Co atoms.

The last but not least parameter derived from the macrospin is the interlayer
coupling energy $J$. The polarization of the Pt is the mechanism of the
interaction between two magnetic moments in Co layers. Such an interaction is
ferromagnetic by its very nature,54 whereas the dipolar coupling (neglected
here) is antiferromagnetic. The indirect way to probe the coupling (and the
polarization of the Pt) by electrical detection is to measure FMR by
rectification of radiofrequency current.55, 56, 57 We derived the coupling
($J$) from the macrospin simulations, similarly as magnetizations and
anisotropies. The coupling dependence on Pt thickness is shown in Figure 8d.
However, we could estimate $J$ down to Pt thickness of 1.36 nm, at which $J$ =
5.5 $mJ/m^{2}$. Below this thickness, the coupling has no effect on the
resonance fields at frequencies experimentally accessible (< 25 GHz). The
strong coupling causes both Co layers to rotate in the same manner, and they
can be treated as one layer rather than two separate layers. In addition, two
magnetizations of Co layers oscillate in phase (acoustic mode). It is seen in
the experiment as observed low-frequency mode.53 On the contrary, the
magnetization oscillations with opposite phases correspond to high-frequency
optical mode (> 30 GHz), not achievable in the experimental method due to
large losses in the power of the microwave current injected to the sample.57
For this reason, although the exact value of $J$ is undeterminable for
$t_{Pt}<1.36nm$ we just kept the value $J=5.5mJ/m^{2}$ for simulations. This
$J$ magnitude is marked as a horizontal dashed line in Figure 8d. Its real
value may reach any point from the hatched region, and particularly it may
follow the exponential dependence, as predicted in Ref.54 and shown in Figure
8d too. For the thicker Pt ($t_{Pt}\geq 1.36nm$, the fitting procedure to the
Pt polarization model returned the Pt polarization depth parameter $\xi\approx
0.47nm$, that is 1.5 times greater than $\xi$ reported for Py/Pt/Py
structure.54

Figure 8: Magnetic parameters of Co layers as a function of Pt layer thickness
derived from the macrospin simulations of the spin-diode FMR spectra:
perpendicular and effective anisotropies (a) and (b), magnetization saturation
(c) and interlayer coupling (d).

Summarizing, we emphasize that the coupling strength correlates with the
regions from I to IV. The constant value of $J$ within the region I
corresponds to large and undetectable coupling, whereas in region II, $J$ is
still significant and measurable. The intermediate region III is characterized
by weak coupling, while samples within the region IV are practically
decoupled.

Having the magnetic parameters derived from the macrospin simulations of the
spin-diode FMR dynamics, we calculated longitudinal static magnetoresistance
($R_{xx}$) dependencies on the external magnetic field in $H_{x},H_{y}$, and
$H_{z}$ directions. We also modeled the anomalous Hall resistance ($R_{xy}$)
when the external magnetic field is applied in $z$ direction.

###  4.2 Magnetoresistance and anomalous Hall effect

The Pt-based magnetic multilayers are expected to exhibit a large spin
magentoresistance due to substantial spin-orbit interaction within the HM
layer. These interactions cause the relatively large spin currents to be
generated and injected into ferromagnetic layers. The spin currents and spin
accumulations at the Co/Pt interfaces influence the magnetoresistance of the
sample, as predicted by the theoretical model presented in Sec.3.2. Here we
focus on the spin orbit interactions that are reflected in SMR. The SMR is
defined as difference in longitudinal resistance measured in saturated
magnetization of Co layers under the external magnetic field applied in $y$
and $z$ directions, i.e. $SMR=R_{xx}(H_{y})-R_{xx}(H_{z})$, while the AMR is
defined similarly as in Sec.3.2 as $AMR=R_{xx}(H_{x})-R_{xx}(H_{z})$.36 Also,
we measured the AHE configuration ($R_{xy}$) in the field applied in $z$
direction. All magnetoresistance and AHE measurements were perfomed by DC
current method sweeping the external magnetic field up to 10 kOe.

Then, we modeled the magnetoresistance dependencies with use of the macrospin
model. We used the parameters dervied previously by fitting the model to the
FMR experimental results (shown in Figure 8). For sake of simplicity, we treat
the considered sample as a doubled bilayers: Co/Pt and Pt/Co. It allows us to
calculate the resistance of Co/Pt/Co structure as the equivalent resistance of
layers connected in parallel: $R_{xx,xy}=\frac{R_{1}\cdot R_{2}}{R_{1}+R_{2}}$
where composite layer resistances are described by
$R_{xx1(2)}=R_{0,1(2)}+\Delta R_{AMR}m_{x1(2)}^{2}+\Delta
R_{SMR}m_{y1(2)}^{2}$,36 while the AHE-related resistances are given by
$R_{xy1(2)}=\Delta R_{AHE}m_{z1(2)}$.36 In Figure 9 we show the typical MR
curves for samples from regions I to IV.

Figure 9: The magnetoresistances $R_{xx}$ and $R_{xy}$ (inset) as a function
of the magnetic field in the samples with Pt thickness: (a) 0.52 nm as an
example from region I, (b) 1.74 nm from region II, (c) 2.63 nm from region III
and (d) 3.76 nm from region IV: $R_{xx}$ experimental data measured at
magnetic field applied in $x$ (blue triangles), $y$ (green circles), $z$ (grey
squares) directions. Depicted diagrams for all regions indicate the direction
of magnetizations of magnetic layers at remanence. The macrospin (black solid
and dashed lines) and micromagnetic(dashed red and blue lines) simulations of
$R_{xx}(H_{y})$. Micromagnetic simulations for $t_{Pt}$ = 3.76 nm were
performed using the same parameters(cf. Figure 8) as derived from macrospin
model (red dashed line) as well as for an $K_{1}$ anisotropy increased by 0.17
$MJ/m^{3}$ (blue dashed line).

For $t_{Pt}=0.52$ nm, the macrospin qualitatively reproduces a narrow peak in
MR. It also accounts for a more complicated dependence of $R_{xx}(H_{z})$ (see
Figure 9a). The AHE curve does not exhibit a hysteresis and its shape is
typical for hard-axis rotation of both Co layers magnetized in-plane in the
remanent state. The hysteresis in $R_{xx}(H_{z})$ is due to a competition
between a different anisotropies: in-plane and perpendicular that affect how
the magnetizations rotate. Moreover, as one can see in Figure 10b, the AMR
effect dominates in the region I with the thinnest Pt layers.

On the contrary, in region II, the SMR is the highest as predicted by the
spin-diffusive model and macrospin simlutaions (cf. Figure 10a). The
representative sample from region II ($t_{Pt}$ = 1.74 nm) exhibits parabolic-
like $R(H_{y})$ dependence. It means that two Co layers are magnetized
perpendicularly to the sample plane in the remanent state. Therefore the AHE
reveals the clear switching-like shape (see inset in Figure 9b). The both,
simulation and experimental results, show the negligible contribution of the
AMR.

For the sample from the region III ($t_{Pt}=2.63nm$), the $R(H_{y})$ is rather
convex-shaped than parabolic. On the other hand, the AHE curve still exhibits
a switching-like behavior. It suggests that the magnetization of layer 2 is
tilted away from the perpendicular towards in-plane direction.

For the thickest Pt layer (e.g. $t_{pt}$ = 3.76 nm in Figure 9d) when the Co
layers are weakly coupled (region IV), one can see $R_{xx}(H_{y})$ having
parabolic-like shape in high magnetic fields. This part of the curve is due to
rotation the perpendicularly magnetized Co layer from $z$ to $y$ direction. On
the contrary, at low fields, there is a characteristic sharp peak related to
rotation of the in-plane magnetized Co layer from its remanent state direction
to the y direction. The dependence was well reproduced by macrospin model
(black solid line in Figure 9d). The same macrospin parameters provide the
satisfactory agreement of AHE magnetoresistance with experimental points (see
inset in the same figure). The AHE curve exhibits smooth-edged shape
hysteresis, characteristic for the simulatneous rotation of the bottom Co
layer magnetization ($\vec{M}_{2}$) in its hard-direction and switching of the
top Co layer between two states: $\pm\vec{M}_{1,z}$.

We supported the macrospin model with micromagnetic simulations in the case of
almost decoupled Co layers. The relevant calculations were performed with
MUMAX3,58 where the LLG equation was integrated numerically for each
simulation cell. Due to memory and time usage limits, the simulated area was
nominally restricted to 5 $\times$ 20 $\mu$m2. However, we also utilized
periodic boundary conditions along the $x$ direction in order to produce a
demagnetization tensor matching the actual experimental conditions. To
optimize simulation performance, the cell size was chosen as 4.88 $\times$
4.88 $\times$ 0.87 nm for $t_{Pt}$ = 1.74 nm and as 4.88 $\times$ 4.88
$\times$ 0.94 nm for $t_{Pt}$ = 3.76 nm case. In both cases, the external
magnetic field $H_{y}$ was increased with 500 Oe step, and the magnetization
of Co layers was allowed to relax fully before moving to the next step. Then,
the averaged magnetization vector for each layer was registered and used as an
input for further resistance calculations. The micromagnetics revealed the
same shapes of $R_{xx}$ curves as the macrospin model, for the same parameters
(or very close), as shown in Figure 8 (see caption of Figure 9 for details).
The agreement between macrospin and micromagnetic simulations confirmed that
the macrospin parameters are reliable.

For the sake of completness, in Figure 10 we show the $\Delta R_{AMR}$ and
$\Delta R_{SMR}$ that were derived in whole range of the Pt thickness from
experiment and prediced by the spin-diffusive model described in Sec.3.2. The
obtained amplitudes agree to a satisfactory extent. As one can see from Eqs.
(8) and (9), the SMR and AMR depend on the charge current flowing in HM and F
layers, respectively. However, the currents in HM layer are also influenced by
spin accumulation at interfaces of this layer due to inverse SHE. The spin
accumulation is mainly determined by the mean spin diffusion length
($\lambda_{HM}$) and spin Hall angle ($\theta_{SH}$). The negative value of
the SMR reaches its maximum at $t_{Pt}\approx 1.5$ nm and decreases for
thicker Pt layers, for which the spin decoherence affects the spin current,
which, in turn, reduces the effective spin accumulation at F/HM interface. On
the contrary, the positive value of the AMR rapidly and monotonically
decreases with HM thickness since the average charge current density flowing
into F layer decreases for thicker Pt layer.Discrepancies between the
experimental and theoretical MR dependencies in the region I results from
strongly mixed and alloyed interfaces for small thicknesses of Pt.

Figure 10: SMR and AMR amplitudes derived directly from the measurements(empty
grey triangles) and calculated within the diffusive model (red dashed lines)
as function of the Pt thickness.

###  4.3 Spin Hall angle and spin orbit torques

In order to quantitatively characterize the spin-orbit interactions in the
Co/Pt/Co trilayers, we performed the harmonic measurements briefly described
in Sec.2.4. For the samples for which both or one of the Co layers is
magnetized in-plane (region I and IV) in the remanent state, we applied the
angular harmonic voltage measurement method.50, 10 On the contrary, in the
case of the Co layers that magnetizations are perpendicularly oriented (region
II and III), we measured the field dependence of the relevant harmonic
voltages.52 In the latter method, the damping-like (DL) and field-like (FL)
components of SOT fields are determined using the following formula:

$\Delta H_{DL(FL)}=-2\frac{B_{x(y)}\pm 2\xi B_{y(x)}}{1-4\xi^{2}}$ (16)

where $B_{x(y)}\equiv\frac{\partial V_{2\omega}}{\partial
H_{x(y)}}/\frac{\partial^{2}V_{\omega}}{\partial H_{x(y)}^{2}}$ and $H_{x(y)}$
stands for the in-plane external magnetic field applied in x(y) direction (cf.
Figure 1). The parametetr $\xi=\frac{\Delta R_{PHE}}{\Delta R_{AHE}}$ is the
planar to anomalous Hall effect ratio.

Figure 11: Experimental harmonic voltages $V_{\omega}$ and $V_{2\omega}$ for
the representative samples from regions: (a-b) II ($t_{Pt}=1.74$ nm), (c-d)
III ( $t_{Pt}=2.63$ nm) both measured at the in-plane magnetic fields ($H_{x}$
and $H_{y}$) swept from -1.5 to +1.5 kOe (cf. Eq.16). The fitted linear and
quadric functions corresponding to the field-sweeping method (see Ref.52).

The first and second harmonic voltages ($V_{\omega}$ and $V_{2\omega}$)
measured as a function of the applied magnetic field ($H_{x}$ and $H_{y}$) are
plotted in Figure 11. The results shown in the Figure 11 are representative
for the samples from region II ($t_{Pt}=1.74$ nm) and III ($t_{Pt}=2.63$ nm).
Next, we could use the Eq.(16) and follow the method described in Ref.52 to
calculate SOT effective fields $\Delta H_{DL(FL)}$ in samples from regions II
and III. The results are shown in Figure 14a and marked as squares.
Nevertheless, the above method turns out to be ineffective in the case of
samples with one or both layers magnetized in-plane. In this case, in order to
determine $\Delta H_{DL(FL)}$, we measured the angular dependence of
$V_{2\omega}$ on the magnetic field applied in the sample plane. Such a
dependence may be expressed as follows10, 50 :

$V_{2\omega}=\left(-\frac{\Delta H_{FL}}{H_{ext}}R_{PHE}\cos
2\phi_{H}-\right.$ $\left.-\frac{1}{2}\frac{\Delta
H_{DL}}{H_{eff}}R_{AHE}+\alpha_{0}\right)I\cos\phi_{H}$ (17)

where $\phi_{H}$ stands for the in-plane angle of magnetic field. The term
$\alpha_{0}$ is the anomalous Nernst effect(ANE) coefficient due to thermal
gradients within the samples induced by the Joule heating.50 The experimental
angular dependencies $V_{2\omega}(\phi_{H})$ for the samples with
$t_{Pt}=0.52$ nm (region I) and $t_{Pt}=3.76$ nm (region IV) are shown in
Figure 12a,b. As one can see the damping-like SOT effective field ($\Delta
H_{DL}$) is proportional to the $\cos\phi_{H}$, whereas the field-like one
($\Delta H_{FL}$) is proportional to the $\cos\phi_{H}\cos 2\phi_{H}$.
Moreover, the $H_{eff}=H_{ext}-\frac{2K_{eff}}{M_{S}}$, where
$K_{eff}$(defined as in Sec.4.1) and $M_{S}$ are the parameters of the Co
layers magnetized in-plane. As long as we knew the magnetic parameters
(summarized in Figure 8) of the layers, we could fit Eq.(17) to the
experimental data and consequenlty determine the both, field-like and damping-
like SOT components.

Figure 12: Experimental harmonic voltages $V_{2\omega}$ for the representative
samples from regions: (a) I ($t_{Pt}=0.52$ nm) and (b) IV ($t_{Pt}=3.76$ nm)
both measured using the angular harmonic voltage method (cf. Eq.(17)). The
fitted trigonometric functions follow the Eq.(17).

In addition, by plotting the terms proportional to $\Delta H_{DL}$ as a
function of $1/H_{ext}$ we could estimate the contribution of the ANE. One
should note that the offset of linear fit (at $1/H_{ext}=0$) visible in Figure
13a,b is the ANE contribution $\alpha_{0}I_{0}$. We show the relevant plots
for two samples ($t_{Pt}$ = 0.52 and 3.76 nm). The small offsets of fitted
lines, ($\approx-0.3\mu V$) and ($\approx-0.15\mu V$) corresponding to the
ANE-related electric fields $E_{ANE}\approx-0.03V/m$ and
$E_{ANE}\approx-0.015V/m$ respectively, are much smaller than the values in
the Co/Pt systems present in the literature, e.g. in Ref.50. Therefore, it
suggests that the ANE contribution is negligible in our devices with thin and
thick Pt layers. It is worthy to notice that the dependencies shown in Figure
13 can be used to examine the applicability of Eq.17. At low magnetic fields
the dependencies deviate from the linear and the Eq.17 is not fulfilled. On
the other hand, according to the model, the dependencies are linear at high
fields.

Figure 13: The amplitude of Eq.(17) plotted as a function of magnetic field
1/$H_{ext}$ measured in the samples with Pt thickness 0.52 nm (a) and 3.76 nm
(b) at angle $\phi_{H}=45^{\circ}$. The interceptions with $y$ axis indicate
the ANE contribution.

We summarize the results for the samples from all regions (I to IV) in Figure
14a. For very thin Pt (region I), both field-like and damping-like components
are small, although the former contributes slightly more than the latter one.
In the region II and III, both components increase in their magnitudes;
however, in the region III, the DL component (filled red points), due to
intermixing and alloying Co and Pt, surpasses the FL component’s (blue filled
points) magnitude. The most intriguing is region IV, where the damping-like
component dominates over the field-like, especially for $t_{Pt}$ > 3 nm. For
such a thick Pt layer, the $H_{DL}$ saturates, while the $H_{FL}$ drops again
towards small values. Similarly the effective spin Hall angle defined by:
$\theta_{SH,eff}=\frac{2e}{\hbar}\frac{H_{DL}M_{S}t_{Co}}{j_{Pt}}$ starts to
increase in region II where the Pt thickness is sufficient to generate a
significant spin currents, and consequently the $H_{DL}$ SOT field. The
$\theta_{SH,eff}$ continues increasing in region III, and then it reaches its
maximum value c.a. 14$\%$ at $t_{Pt}=3.24$ nm. Next it slightly decreases with
the Pt thickness in region IV.

Figure 14:  Experimental spin-orbit torque: damping-like and field-like
components obtained with macrospin magnetic parameters from Figure 8 (blue and
red filled points), the effective theoretical sum of SOT fields ($H_{FL}$
(blue empty points), and separately $H_{DL}$(red empty points)) acting on two
Co layers. The amplitude of Oersted field($H_{Oe}$) (orange and grey dashed
line) and Rashba-Edelstein spin-orbital field ($H_{S-O}$) (black dash-dotted
line). The resultant field-like SOT (filled green squares) including $H_{S-O}$
field in region I and, additionaly, the Oersted field in region IV. Orange
triangles correspond to the case when the Oersted field only is added to the
$H_{FL}$. (b) The effective spin Hall angle determined from field (red
circles) and angular (blue squares) harmonics measurements with the $H_{DL}$
amplitudes from (a). The red dashed line indicates the theoretical value of
spin Hall angle fitted to the experimental MR dependencies (cf. Figure 10)

One should note that $\theta_{SH,eff}$ accounts for the spin accumulation
effect at interfaces in the trilayer system. For this reason, the experimental
value of $\theta_{SH,eff}$ may substantially differ from the theoretical one
($\theta_{SH}$ in Sec.3.2), introduced as a material parameter of pure SHE
efficiency at a very thick HM thickness limit). More precisely, the effective
spin Hall angle is smaller than the theoretical one ($\theta_{SH}$) for
$t_{Pt}$ < 2.6 nm. For thicker Pt layers, $\theta_{SH,eff}$ increases and
becomes higher than the theoretical value. It is correlated with the change of
magnetization direction from perpendicular to the in-plane for the thick Pt
layer, particularly in region IV. In order to shed light on $\theta_{SH,eff}$
dependence on Pt thickness, one needs to go back to the Figure 14a based on
which the $\theta_{SH,eff}$ was determined. The two modes of experimental
harmonics method allow the measurement of the effective SOT fields acting on
the Co layers magnetized only in the plane (angular method) or perpendicular
to the plane (field method). This limitation arises from very high fields
required to saturate the layers magnetized in-plane (perpendicularly) at
remanence into the perpendicular (in-plane) direction. Therefore, the data
recorded with the experimental harmonic method accounts for both magnetic
layers when magnetized in the same direction. That is the case of samples from
regions I, II, and III. On the contrary, only one magnetic layer can be sensed
by the measurement setup in region IV, where both magnetizations are
orthogonal. Therefore, the experimental conditions differ in regions I-III and
IV. As a consequence, the experimentally determined $H_{DL}$ and $H_{FL}$
fields in regions I-III are the sums of the related SOT fields acting on
F1(top) and F2(bottom) magnetic layers. Since the spin currents generated by
the SHE have opposite polarizations at F/HM and HM/F interface, the SOT
effective fields also have the opposite signs. The residual fields, defined as
$H_{DL}=H_{DL,1}+(-H_{DL,2})$ and $H_{FL}=-H_{FL1}+H_{FL2}$ come from the
different interface properties, included in real parts of mixing conductances
$G_{R1}$ and $G_{R2}$ of diffusive model(cf. Sec.3.2). In order to support our
analysis, we calculated the SHE-induced SOT effective fields using the
formulas Eqs.(14)-(15) and magnetic parameters from Figure 8. Next, we plotted
the difference of $H_{DL,1}$ and $H_{DL_{2}}$ in regions from I to III at the
experimental thicknesses of Pt. Nevertheless, considering the experimental
conditions discussed above, we plotted $H_{DL1}$ field only in region IV. The
results are shown in Figure 14a as solid empty circles and agree with the
experimental ones (red filled points - triangles and circles) to a
satisfactory extent.

On the contrary, a similar procedure is insufficient to reproduce the
experimental $H_{FL}$ field. The theoretical $H_{FL}$ (empty blue circles)
depends differently on HM thickness than the experimental one (blue-filled
circles and empty blue triangles). However, the diffusive model does not
account for an Oersted field ($H_{Oe}$) coming from the charge current as well
as a spin-orbital field ($H_{S-O}$) at interfaces, arising from the Rashba-
Edelstein effect. Both fields have the same direction as SHE induced $H_{FL}$.
Thus, the additional terms have to be included in the analysis of the
effective field-like field. We assumed that the Oersted field linearly
increases with the Pt thickness (grey and orange dashed line in Figure 14a),
whereas the $H_{S-O}$(black dash-dotted line) is independent of the HM
thickness.18, 37 Since the Oersted field has the same amplitude with opposite
signs in both F layers, its impact on the resultant $H_{FL}$ cancels out, and
it does not affect the field-like SOT field in I-III regions. For the same
reason, as discussed above, the $H_{Oe}$ adds to $H_{FL}$ in region IV (see
full orange triangles in Figure 14a). Conversely, the $H_{S-O}$ fields do not
cancel out due to the difference in F/HM and HM/F interfaces. Therefore their
difference contributes to the total $H_{FL}$ term in regions I-III. In region
IV the amplitude $H_{S-O}$ significantly increases because the perpendicularly
magnetized Co layer is out of the experimental detection. Thus, the estimated
magnitude generated at the Co/Pt interface was 0.86 kA/m. One should note that
such a substantial value of the SOT field due to REE is much higher than its
FL counterpart coming from the SHE. Moreover, it is of the order of the
charge-current induced effective Oersted field. However, all fields SHE-
related $H_{FL}$, $H_{Oe}$ and $H_{S-O}$ must be considered in the total FL
SOT component in order to achieve a satisfactory agreement with experimental
data in F/HM/F trilayer system (see solid blue line in Figure 14a). The
discrepancies at the border between regions III and IV are due to the
unreliability of the experimental methods (field and angular harmonics)
applicable only when magnetic layers are magnetized fully in the plane or
fully perpendicularly to the plane. This requirement is not fulfilled in the
intermediate case, especially at $t_{Pt}=2.87$ nm. It has its consequence in
spin Hall angle value that slightly drops at this Pt thickness. On the other
hand, at $t_{Pt}=3.24$ nm, the SOT fields determined from the field and
angular harmonics methods differ. In this case, the field harmonics method
detects the magnetic layer magnetized perpendicularly. Therefore, as discussed
earlier, the SOT fields at the Pt/Co interface are different from the SOT
fields at the Co/Pt interface. However, the spin Hall angle (Figure 14b)
treated as the HM material parameter is the same for both experimental
methods. For thicker Pt, the detection signal in the field harmonics
measurement was too weak to properly determine the $H_{DL}$ and $H_{FL}$
effective fields.

##  5 Conclusions

The paper presents the detailed results on structural, static, and dynamic
properties of Co/Pt/Co trilayer in a wide range of the Pt layer thickness.
First, we showed the Co/Pt and Pt/Co inherent interface asymmetry that
resulted in different interfacial spin-orbit-related properties of
ferromagnetic layers, magnetic anisotropies, and the effective spin-orbit
torque fields due to SHE and REE. The difference in anisotropies makes the Co
layer with stronger perpendicular anisotropy be a primary layer that
determines the magnetization direction of the secondary Co layer through the
interlayer exchange coupling. Therefore we were able to determine four ranges
of Pt thickness where the trilayer reveals different static and dynamic
behaviors correlated with the strength of coupling: region I (two Co
magnetizations are in-plane), region II (Co magnetizations are both
perpendicular to the plane), region III (one of Co magnetization is tilted
from the perpendicular direction) and, finally, region IV (one Co
magnetization is in-plane, whereas the second one is perpendicular). We showed
that the experimental relation of dispersions and magnetoresistances differ in
each region. This difference is accounted for by the macrospin models that we
used, and therefore, both the experimental magnetoresistance and SOT-FMR
relations of dispersion were reproduced by theoretical calculations to a
satisfactory extent. Moreover, we made a detailed analysis of the SOT
effective fields determined using harmonics measurements. We showed that the
experimental method applied to trilayers with two Co magnetizations aligned
both in-plane or both out-of-plane allows measuring a difference of the
effective SOT fields coming from two F/HM and HM/F interfaces. However, when
two magnetizations are orthogonal, the experimental technique enables
measuring SOT fields from the single F layer. The experimental results
revealed this feature and were successfully parametrized with the magnitudes
of damping-like and field-like SOT fields obtained from the diffusive model.
Finally, both experimental and theoretical data allowed us to determine the
contribution of Oersted ($H_{Oe}$) and spin-orbital ($H_{S-O}$) fields to the
resultant experimental field-like SOT. We showed that the latter contribution
due to REE might be comparable to the effective Oersted field and more
significant than the field-like SOT field caused by the SHE.

## ACKNOWLEDGMENTS

This work was supported by the National Science Centre, Poland, Grant No.
2016/23/B/ST3/ 01430 (SPINORBITRONICS). Numerical calculations were supported
by PL-GRID infrastructure. As part of cooperation, the multilayer systems were
deposited in the Institute of Molecular Physics Polish Academy of Sciences and
nanofabrication was performed at the Academic Centre for Materials and
Nanotechnology of AGH University of Science and Technology. We would like to
express our gratitude to Prof. F. Stobiecki for helpful discussions on data
analysis.

## References

  * Ralph and Stiles 2008 Ralph, D.; Stiles, M. Spin transfer torques. _J. Magn. Magn. Mater._ 2008, _320_ , 1190 – 1216
  * Brataas et al. 2012 Brataas, A.; Kent, A. D.; Ohno, H. Current-induced torques in magnetic materials. _Nat. Mater._ 2012, _11_ , 372–381
  * Manchon et al. 2019 Manchon, A.; Železnỳ, J.; Miron, I. M.; Jungwirth, T.; Sinova, J.; Thiaville, A.; Garello, K.; Gambardella, P. Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems. _Rev. Mod. Phys._ 2019, _91_ , 035004
  * Zhang et al. 2021 Zhang, C.; Takeuchi, Y.; Fukami, S.; Ohno, H. Field-free and sub-ns magnetization switching of magnetic tunnel junctions by combining spin-transfer torque and spin–orbit torque. _Appl. Phys. Lett._ 2021, _118_ , 092406
  * Bhatti et al. 2017 Bhatti, S.; Sbiaa, R.; Hirohata, A.; Ohno, H.; Fukami, S.; Piramanayagam, S. Spintronics based random access memory: a review. _Mater. Today_ 2017, _20_ , 530–548
  * Ikegawa et al. 2020 Ikegawa, S.; Mancoff, F. B.; Janesky, J.; Aggarwal, S. Magnetoresistive random access memory: Present and future. _IEEE T. Electron Dev._ 2020, _67_ , 1407–1419
  * Dieny et al. 2020 Dieny, B.; Prejbeanu, I. L.; Garello, K.; Gambardella, P.; Freitas, P.; Lehndorff, R.; Raberg, W.; Ebels, U.; Demokritov, S. O.; Akerman, J. Opportunities and challenges for spintronics in the microelectronics industry. _Nat. Electron._ 2020, _3_ , 446–459
  * Hirohata et al. 2020 Hirohata, A.; Yamada, K.; Nakatani, Y.; Prejbeanu, I.-L.; Diény, B.; Pirro, P.; Hillebrands, B. Review on spintronics: Principles and device applications. _J. Magn. Magn. Mater._ 2020, _509_ , 166711
  * Obstbaum et al. 2016 Obstbaum, M.; Decker, M.; Greitner, A. K.; Haertinger, M.; Meier, T. N. G.; Kronseder, M.; Chadova, K.; Wimmer, S.; Ködderitzsch, D.; Ebert, H.; Back, C. H. Tuning Spin Hall Angles by Alloying. _Phys. Rev. Lett._ 2016, _117_ , 167204
  * Fritz et al. 2018 Fritz, K.; Wimmer, S.; Ebert, H.; Meinert, M. Large spin Hall effect in an amorphous binary alloy. _Phys. Rev. B_ 2018, _98_ , 094433
  * Wang et al. 2017 Wang, T.; Xiao, J. Q.; Fan, X. Spin–Orbit Torques in Metallic Magnetic Multilayers: Challenges and New Opportunities. 2017; p 1740013
  * Song et al. 2021 Song, C.; Zhang, R.; Liao, L.; Zhou, Y.; Zhou, X.; Chen, R.; You, Y.; Chen, X.; Pan, F. Spin-orbit torques: Materials, mechanisms, performances, and potential applications. _Prog. Mater. Sci._ 2021, _118_ , 100761
  * Sinova et al. 2015 Sinova, J.; Valenzuela, S. O.; Wunderlich, J.; Back, C.; Jungwirth, T. Spin Hall effects. _Rev. Mod. Phys._ 2015, _87_ , 1213
  * Edelstein 1990 Edelstein, V. M. Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems. _Solid State Commun._ 1990, _73_ , 233–235
  * Hirsch 1999 Hirsch, J. E. Spin Hall Effect. _Phys. Rev. Lett._ 1999, _83_ , 1834–1837
  * Mihajlović et al. 2016 Mihajlović, G.; Mosendz, O.; Wan, L.; Smith, N.; Choi, Y.; Wang, Y.; Katine, J. Pt thickness dependence of spin Hall effect switching of in-plane magnetized CoFeB free layers studied by differential planar Hall effect. _Appl. Phys. Lett._ 2016, _109_ , 192404
  * Lim et al. 2020 Lim, G. J.; Gan, W.; Lew, W. S. Effect of seed and interlayer Pt thickness on spin-orbit torque efficiency in Co/Pt multilayer with perpendicular magnetic anisotropy. _J. Phys. D Appl. Phys._ 2020, _53_ , 505002
  * Łazarski et al. 2019 Łazarski, S.; Skowroński, W.; Kanak, J.; Karwacki, L.; Ziętek, S.; Grochot, K.; Stobiecki, T.; Stobiecki, F. Field-Free Spin-Orbit-Torque Switching in $\mathrm{Co}$/$\mathrm{Pt}$/$\mathrm{Co}$ Multilayer with Mixed Magnetic Anisotropies. _Phys. Rev. Applied_ 2019, _12_ , 014006
  * Chen et al. 2013 Chen, Y.-T.; Takahashi, S.; Nakayama, H.; Althammer, M.; Goennenwein, S. T. B.; Saitoh, E.; Bauer, G. E. W. Theory of spin Hall magnetoresistance. _Phys. Rev. B_ 2013, _87_ , 144411
  * Choi et al. 2017 Choi, J.-G.; Lee, J. W.; Park, B.-G. Spin Hall magnetoresistance in heavy-metal/metallic-ferromagnet multilayer structures. _Phys. Rev. B_ 2017, _96_ , 174412
  * Hals and Brataas 2013 Hals, K. M.; Brataas, A. Phenomenology of current-induced spin-orbit torques. _Phys. Rev. B_ 2013, _88_ , 085423
  * Liu et al. 2011 Liu, L.; Moriyama, T.; Ralph, D. C.; Buhrman, R. A. Spin-Torque Ferromagnetic Resonance Induced by the Spin Hall Effect. _Phys. Rev. Lett._ 2011, _106_ , 036601
  * Zhang et al. 2015 Zhang, C.; Fukami, S.; Sato, H.; Matsukura, F.; Ohno, H. Spin-orbit torque induced magnetization switching in nano-scale Ta/CoFeB/MgO. _Appl. Phys. Lett._ 2015, _107_ , 012401
  * Garello et al. 2014 Garello, K.; Avci, C. O.; Miron, I. M.; Baumgartner, M.; Ghosh, A.; Auffret, S.; Boulle, O.; Gaudin, G.; Gambardella, P. Ultrafast magnetization switching by spin-orbit torques. _Appl. Phys. Lett._ 2014, _105_ , 212402
  * Lau et al. 2016 Lau, Y.-C.; Betto, D.; Rode, K.; Coey, J.; Stamenov, P. Spin–orbit torque switching without an external field using interlayer exchange coupling. _Nat. Nanotechnol._ 2016, _11_ , 758–762
  * Du et al. 2020 Du, Y.; Gamou, H.; Takahashi, S.; Karube, S.; Kohda, M.; Nitta, J. Disentanglement of Spin-Orbit Torques in $\mathrm{Pt}$/$\mathrm{Co}$ Bilayers with the Presence of Spin Hall Effect and Rashba-Edelstein Effect. _Phys. Rev. Applied_ 2020, _13_ , 054014
  * Fan et al. 2014 Fan, X.; Celik, H.; Wu, J.; Ni, C.; Lee, K.-J.; Lorenz, V. O.; Xiao, J. Q. Quantifying interface and bulk contributions to spin–orbit torque in magnetic bilayers. _Nat. Commun._ 2014, _5_ , 1–8
  * Cui et al. 2019 Cui, B.; Wu, H.; Li, D.; Razavi, S. A.; Wu, D.; Wong, K. L.; Chang, M.; Gao, M.; Zuo, Y.; Xi, L.; Wang, K. L. Field-Free Spin–Orbit Torque Switching of Perpendicular Magnetization by the Rashba Interface. _ACS Appl. Mater. Interfaces_ 2019, _11_ , 39369–39375, PMID: 31603641
  * Lee et al. 2019 Lee, H.-Y.; Kim, S.; Park, J.-Y.; Oh, Y.-W.; Park, S.-Y.; Ham, W.; Kotani, Y.; Nakamura, T.; Suzuki, M.; Ono, T.; Lee, K.-J.; Park, B.-G. Enhanced spin–orbit torque via interface engineering in Pt/CoFeB/MgO heterostructures. _APL Mater._ 2019, _7_ , 031110
  * Avci et al. 2019 Avci, C. O.; Beach, G. S. D.; Gambardella, P. Effects of transition metal spacers on spin-orbit torques, spin Hall magnetoresistance, and magnetic anisotropy of Pt/Co bilayers. _Phys. Rev. B_ 2019, _100_ , 235454
  * Zhu et al. 2019 Zhu, L.; Zhu, L.; Shi, S.; Sui, M.; Ralph, D.; Buhrman, R. Enhancing Spin-Orbit Torque by Strong Interfacial Scattering From Ultrathin Insertion Layers. _Phys. Rev. Applied_ 2019, _11_ , 061004
  * Karwacki et al. 2020 Karwacki, Ł.; Grochot, K.; Łazarski, S.; Skowroński, W.; Kanak, J.; Powroźnik, W.; Barnaś, J.; Stobiecki, F.; Stobiecki, T. Optimization of spin Hall magnetoresistance in heavy-metal/ferromagnetic-metal bilayers. _Sci. Rep._ 2020, _10_ , 1–8
  * Skowroński et al. 2016 Skowroński, W.; Cecot, M.; Kanak, J.; Ziętek, S.; Stobiecki, T.; Yao, L.; Van Dijken, S.; Nozaki, T.; Yakushiji, K.; Yuasa, S. Temperature dependence of spin-orbit torques in W/CoFeB bilayers. _Appl. Phys. Lett._ 2016, _109_ , 062407
  * Cecot et al. 2017 Cecot, M.; Karwacki, Ł.; Skowroński, W.; Kanak, J.; Wrona, J.; Żywczak, A.; Yao, L.; van Dijken, S.; Barnaś, J.; Stobiecki, T. Influence of intermixing at the Ta/CoFeB interface on spin Hall angle in Ta/CoFeB/MgO heterostructures. _Sci. Rep._ 2017, _7_ , 1–11
  * Yamaguchi et al. 2009 Yamaguchi, A.; Motoi, K.; Hirohata, A.; Miyajima, H. Anomalous Hall voltage rectification and quantized spin-wave excitation induced by simultaneous application of dc and rf currents in a single-layered ${\text{Ni}}_{81}{\text{Fe}}_{19}$ nanoscale wire. _Phys. Rev. B_ 2009, _79_ , 224409
  * Kim et al. 2016 Kim, J.; Sheng, P.; Takahashi, S.; Mitani, S.; Hayashi, M. Spin Hall Magnetoresistance in Metallic Bilayers. _Phys. Rev. Lett._ 2016, _116_ , 097201
  * Skowroński et al. 2019 Skowroński, W.; Karwacki, L.; Ziętek, S.; Kanak, J.; Łazarski, S.; Grochot, K.; Stobiecki, T.; Kuświk, P.; Stobiecki, F.; Barnaś, J. Determination of Spin Hall Angle in Heavy-Metal/$\mathrm{Co}\text{$-$}\mathrm{Fe}\text{$-$}\mathrm{B}$-Based Heterostructures with Interfacial Spin-Orbit Fields. _Phys. Rev. Applied_ 2019, _11_ , 024039
  * Liu et al. 2012 Liu, L.; Pai, C.-F.; Li, Y.; Tseng, H. W.; Ralph, D. C.; Buhrman, R. A. Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. _Science_ 2012, _336_ , 555–558
  * Ryu et al. 2019 Ryu, J.; Avci, C. O.; Karube, S.; Kohda, M.; Beach, G. S. D.; Nitta, J. Crystal orientation dependence of spin-orbit torques in Co/Pt bilayers. _Appl. Phys. Lett._ 2019, _114_ , 142402
  * Hayashi et al. 2021 Hayashi, H.; Musha, A.; Sakimura, H.; Ando, K. Spin-orbit torques originating from the bulk and interface in Pt-based structures. _Phys. Rev. Res._ 2021, _3_ , 013042
  * Avci et al. 2015 Avci, C. O.; Garello, K.; Ghosh, A.; Gabureac, M.; Alvarado, S. F.; Gambardella, P. Unidirectional spin Hall magnetoresistance in ferromagnet/normal metal bilayers. _Nat. Phys._ 2015, _11_ , 570–575
  * Luo et al. 2019 Luo, F.; Wong, Q. Y.; Li, S.; Tan, F.; Lim, G. J.; Wang, X.; Lew, W. S. Dependence of spin-orbit torque effective fields on magnetization uniformity in Ta/Co/Pt structure. _Sci. Rep._ 2019, _9_ , 1–7
  * Süle et al. 2008 Süle, P.; Kotis, L.; Toth, L.; Menyhard, M.; Egelhoff Jr, W. Asymmetric intermixing in Co/Ti bilayer. _Nucl. Instrum. Meth. B_ 2008, _266_ , 904–910
  * Gweon et al. 2018 Gweon, H. K.; Yun, S. J.; Lim, S. H. A very large perpendicular magnetic anisotropy in Pt/Co/MgO trilayers fabricated by controlling the MgO sputtering power and its thickness. _Sci. Rep._ 2018, _8_ , 1–8
  * Bonda et al. 2020 Bonda, A.; Uba, S.; Uba, L.; Skowronski, W.; Stobiecki, T.; Stobiecki, F. Laser-induced magnetization precession parameters dependence on Pt spacer layer thickness in mixed magnetic anisotropies Co/Pt/Co trilayer. _J. Magn. Magn. Mater._ 2020, _505_ , 166702
  * Smits 1958 Smits, F. M. Measurement of Sheet Resistivities with the Four-Point Probe. _Bell Syst. Tech. J._ 1958, _37_ , 711–718
  * Kawaguchi et al. 2018 Kawaguchi, M.; Towa, D.; Lau, Y.-C.; Takahashi, S.; Hayashi, M. Anomalous spin Hall magnetoresistance in Pt/Co bilayers. _Appl. Phys. Lett._ 2018, _112_ , 202405
  * Kanak et al. 2013 Kanak, J.; Wiśniowski, P.; Stobiecki, T.; Zaleski, A.; Powroźnik, W.; Cardoso, S.; Freitas, P. X-ray diffraction analysis and Monte Carlo simulations of CoFeB-MgO based magnetic tunnel junctions. _J. Appl. Phys._ 2013, _113_ , 023915
  * Tulapurkar et al. 2005 Tulapurkar, A.; Suzuki, Y.; Fukushima, A.; Kubota, H.; Maehara, H.; Tsunekawa, K.; Djayaprawira, D.; Watanabe, N.; Yuasa, S. Spin-torque diode effect in magnetic tunnel junctions. _Nature_ 2005, _438_ , 339–342
  * Avci et al. 2014 Avci, C. O.; Garello, K.; Gabureac, M.; Ghosh, A.; Fuhrer, A.; Alvarado, S. F.; Gambardella, P. Interplay of spin-orbit torque and thermoelectric effects in ferromagnet/normal-metal bilayers. _Phys. Rev. B_ 2014, _90_ , 224427
  * Lau and Hayashi 2017 Lau, Y.-C.; Hayashi, M. Spin torque efficiency of Ta, W, and Pt in metallic bilayers evaluated by harmonic Hall and spin Hall magnetoresistance measurements. _Jpn. J. Appl. Phys._ 2017, _56_ , 0802B5
  * Hayashi et al. 2014 Hayashi, M.; Kim, J.; Yamanouchi, M.; Ohno, H. Quantitative characterization of the spin-orbit torque using harmonic Hall voltage measurements. _Phys. Rev. B_ 2014, _89_ , 144425
  * Ogrodnik et al. 2019 Ogrodnik, P.; Kanak, J.; Czapkiewicz, M.; Ziętek, S.; Pietruczik, A.; Morawiec, K.; Dłużewski, P.; Dybko, K.; Wawro, A.; Stobiecki, T. Structural, magnetostatic, and magnetodynamic studies of Co/Mo-based uncompensated synthetic antiferromagnets. _Phys. Rev. Materials_ 2019, _3_ , 124401
  * Omelchenko et al. 2018 Omelchenko, P.; Heinrich, B.; Girt, E. Measurements of interlayer exchange coupling of Pt in Py|Pt|Py system. _Applied Physics Letters_ 2018, _113_ , 142401
  * Harder et al. 2011 Harder, M.; Cao, Z. X.; Gui, Y. S.; Fan, X. L.; Hu, C.-M. Analysis of the line shape of electrically detected ferromagnetic resonance. _Phys. Rev. B_ 2011, _84_ , 054423
  * Ziętek et al. 2015 Ziętek, S.; Ogrodnik, P.; Frankowski, M.; Chęciński, J.; Wiśniowski, P.; Skowroński, W.; Wrona, J.; Stobiecki, T.; Żywczak, A.; Barnaś, J. Rectification of radio-frequency current in a giant-magnetoresistance spin valve. _Phys. Rev. B_ 2015, _91_ , 014430
  * Ziętek et al. 2015 Ziętek, S.; Ogrodnik, P.; Skowroński, W.; Wiśniowski, P.; Czapkiewicz, M.; Stobiecki, T.; Barnaś, J. The influence of interlayer exchange coupling in giant-magnetoresistive devices on spin diode effect in wide frequency range. _Appl. Phys. Lett._ 2015, _107_ , 122410
  * Vansteenkiste et al. 2014 Vansteenkiste, A.; Leliaert, J.; Dvornik, M.; Helsen, M.; Garcia-Sanchez, F.; Van Waeyenberge, B. The design and verification of MuMax3. _AIP Adv._ 2014, _4_ , 107133