text:
\section{Introduction}
In noncentrosymmetric ferromagnets the application of
an electric current generates torques on the magnetization
that result from the spin-orbit interaction (SOI) and that
can be used to switch the
magnet (see Ref.~\cite{sot_review} for  a recent review).
In magnetic bilayers
such as Co/Pt, CoFeB/Ta and
CoFeB/W
with structural inversion asymmetry
spin currents mediate an important contribution to these
so-called
spin-orbit torques (SOTs)~\cite{spin_torque_giant_she_tantalum,stt_devices_giant_she_tungsten,ibcsoit,symmetry_spin_orbit_torques}.
At first, these spin currents were generated by the
spin Hall effect (SHE) of nonmagnetic heavy metals such as Pt, W, Ta and Pd.
Later, also the antiferromagnets MnPt, MnPd and MnIr were found to be efficient sources of
spin currents
due to their large SHE
angles~\cite{she_in_afms,switching_sot_AFM_FM_bilayer,current_induced_torques_ultrathin_IrMn}.
Additionally, it has been pointed out
that also the anomalous Hall effect (AHE) and the anisotropic
magnetoresistance (AMR)
of ferromagnets can be used to generate transverse spin currents, because the
transverse charge currents from the AHE and from the planar Hall effect are
spin-polarized~\cite{stt_from_ahe_and_amr}.
Another contribution to the SOT in magnetic bilayers
stems from the interfacial
SOI~\cite{quantum_kinetic_rashba_macdonald,quantum_kinetic_rashba_manchon,CoPt_Haney_Stiles}.

Recently, switching of the magnetization by SOT has been demonstrated
in CoFeB/Ti/CoFeB magnetic trilayers~\cite{sots_interface_generated_spin_currents_baek}.
In the trilayer two CoFeB
ferromagnets (one at the top and one at the bottom)
are separated by a Ti normal metal
spacer (see Fig.~\ref{cocucotrilayer_setup} for illustration of
a Co/Cu/Co trilayer). The magnetization of the
bottom ferromagnet (FM) is fixed to the in-plane $x$ direction.
In order to switch the top FM
an electric current is applied parallel to the magnetization
of the bottom FM.
Experiments show that the spin current that switches the top magnet
is generated at the interface between the bottom magnet and the
normal metal (NM)
spacer and that the spin polarization of the spin current has
components along both the $y$ and the $z$
direction~\cite{sots_interface_generated_spin_currents_baek,
observation_spin_orbit_effects_spin_rotation_symmetry}.

Such interface-generated spin currents can be explained in
terms of spin-orbit filtering and
spin-orbit
precession~\cite{spin_transport_interfaces_soc_formalism,spin_transport_interfaces_soc_phenomenology,interface_generated_spin_currents}
and provide a third mechanism for SOTs, which adds to the
two mechanisms found in magnetic bilayers, i.e., the
contribution from the SHE and the contribution from the
interfacial SOI at the interface between the top magnet and the
normal metal.
Since the spin polarization of the spin current
has a component along the $z$ direction,
the magnetization can be switched deterministically
by the applied electric current without the need
for additional external magnetic fields.
While it has been
observed in \textit{ab-initio}
calculations~\cite{invsot,giant_interface_spin_hall} on
magnetic bilayers that
SHE angles are position-dependent and can
be strongly enhanced close to interfaces,
the interface-generated spin currents from the
bottom FM interface exhibit a distinct spin polarization,
which can be controlled by the bottom FM magnetization
direction.

\begin{figure}
\includegraphics[width=\linewidth,trim=4cm 3cm 4cm 0cm,clip]{cocucotrilayer_setup.eps}
\caption{\label{cocucotrilayer_setup}
Illustration of a Co/Cu/Co trilayer structure.
The magnetization
of the top layer, $\vn{M}^{\rm top}$,
can be switched, while the magnetization of the
bottom layer, $\vn{M}^{\rm bottom}$, is fixed to point into the $x$ direction.
The two Co-ferromagnets are separated by a Cu normal metal spacer.
The applied electric field $\vn{E}$ points in the in-plane direction,
e.g.\ in $x$ direction as shown in the figure. The interface normal direction, i.e., the
out-of-plane direction, points in $z$ direction.
}
\end{figure}

Magnetic bilayer structures such as Co/Pt
are often rotationally symmetric around
the interface normal direction
when the magnetization
points out-of-plane~\cite{symmetry_spin_orbit_torques}. In such cases
the magnitude of the SOT
does not depend on the in-plane direction of the applied electric
current.
However, when the bottom FM magnetization
in a trilayer lies in-plane and the top FM magnetization
points out-of-plane, this symmetry is broken, i.e.,
the SOTs exerted on the top FM are expected to depend on the
direction of the applied electric current.
Since it has been found that SOTs and the
Dzyaloshinskii-Moriya
interaction (DMI) are correlated in various
ways~\cite{sot_dmi_stiles,common_origin_sot_dmi,mothedmisot},
one may expect that also the DMI is anisotropic in this case
such that the wavelength of spin spirals that are stabilized by DMI depends
on the in-plane direction of the spin-spiral wave vector.
Equivalently, when a spin spiral in
the top FM is oriented in a fixed direction, the spin-spiral
wavelength can be tuned by the
magnetization direction in the bottom FM.
Compared to other methods of tuning the DMI,
which require to generate a strong
non-equilibrium situation by applying
femtosecond laser
pulses~\cite{ultrafast_modification_exchange_interaction,spicudmi},
the option to tune DMI
via the magnetization direction of the bottom FM
is attractive by its simplicity.

In this work we investigate SOTs and DMI in Co/Cu/Co trilayers
and in Co/Cu bilayers.
These systems are ideal to investigate SOTs that arise from
interface-generated spin currents, because the SHE angle of
bulk Cu is negligibly small and therefore the possibility that there
might be contributions to the SOT originating from the SHE
of bulk Cu can be ruled out.
So far, current-induced torques in Co/Cu/Co spin-valves have been
investigated experimentally~\cite{stt-cocuco_Katine_Albert_Buhrman_Myers_Ralph}
and theoretically~\cite{stt-cocuco_haney&waldron&duine&nunez&guo&macdonald,stt-cocuco_Heiliger_Czerner_Yavorsky_Mertig_Stiles}
only in the current perpendicular to the plane (CPP) geometry.
Only in Co/Cu bilayers the field-like component of the
SOT has been investigated already
theoretically for the case of electric current applied
parallel to the interfaces~\cite{PhysRevB.93.224420}.

This paper is organized as follows.
In Sec.~\ref{sec_symmetry} we discuss the symmetry of
the SOTs in Co/Cu/Co trilayers.
In Sec.~\ref{sec_formalism} we explain the formalism that
we use in order to calculate SOTs and DMI in these systems.
In Sec.~\ref{sec_sot_in_cocuco} we discuss the
\textit{ab-initio} results for the SOTs in
Co/Cu/Cu trilayers and in Co/Cu bilayers.
In Sec.~\ref{sec_dmi_in_cocuco} we present our result on the
DMI in these systems.
This paper ends with a summary in Sec.~\ref{sec_summary}.

\section{Symmetry properties}
\label{sec_symmetry}
We consider a Co/Cu/Co trilayer as illustrated in
Fig.~\ref{cocucotrilayer_setup}. We assume that the Co and Cu layers are
stacked along their (001) directions.
In this case,
the crystal lattice of the trilayer
has c4 rotational symmetry around the $z$ axis and
the planes $xz$ and $yz$ are mirror planes of the crystal lattice.
We assume that the magnetization of the bottom
FM, $\hat{\vn{M}}^{\rm bottom}$,
points
into the $x$ direction, while the magnetization direction of the top
FM is given
by $\hat{\vn{M}}^{\rm top}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\rm T}$.
In order to describe the torque $\vn{T}^{\rm top}$ on the top FM we define the
torkance tensor $\vn{t}^{\rm top}$ such that
\begin{equation}
\vn{T}^{\rm top}=\vn{t}^{\rm top}\vn{E}.
\end{equation}

We first consider the case with $\theta=0$ and $\phi=0$
when the electric field is applied in $x$ direction.
Since the torque is perpendicular to the magnetization
we only need to consider the two
components $T_{x}^{\rm top}$ and $T_{y}^{\rm top}$.
The $xz$ mirror plane flips $\hat{\vn{M}}^{\rm top}$, $\hat{\vn{M}}^{\rm bottom}$,
and $T_{x}^{\rm top}$ (because they are axial vectors),
but it
leaves $E_x$ (polar vector) and $T^{\rm top}_y$ unchanged.
The c2 rotation around the $z$ axis flips all
in-plane vector components.
Thus, combination of the $zx$ mirror plane and the c2 rotation around the $z$ axis
leaves $T^{\rm top}_x$ and $\hat{\vn{M}}^{\rm bottom}$ unchanged, but flips
$E_x$, $T^{\rm top}_y$, and $\hat{\vn{M}}^{\rm top}$. Consequently, $T^{\rm top}_x$ is odd
in $\hat{\vn{M}}^{\rm top}$, i.e., $T^{\rm top}_x$ changes sign
when $\hat{\vn{M}}^{\rm top}$ is flipped.
Additionally, $T^{\rm top}_y$ is even in the
magnetization $\hat{\vn{M}}^{\rm top}$, i.e., it does not change sign
when $\hat{\vn{M}}^{\rm top}$ is flipped.
When the electric field is applied in $y$ direction, the combination
of the $zx$ mirror plane and of the c2 rotation around the $z$ axis preserves $E_y$.
Therefore, $T^{\rm top}_x$ is even in $\hat{\vn{M}}^{\rm top}$,
and $T^{\rm top}_y$ is odd in $\hat{\vn{M}}^{\rm top}$ when the electric field is applied
in $y$ direction.
Since the c2 rotation around the $z$ axis flips the applied electric
field, $\hat{\vn{M}}^{\rm bottom}$, $T^{\rm top}_x$, and $T^{\rm top}_y$,
it follows that both $T^{\rm top}_x$ and $T^{\rm top}_y$ are even in $\hat{\vn{M}}^{\rm bottom}$.
These properties are summarized in the first row of Table~\ref{tab_symmetry}
and in the first row of Table~\ref{tab_symmetry2}.

Next, we consider the case with $\theta=90^{\circ}$
and $\phi=0^{\circ}$.
The magnetizations of both the top FM
and the bottom FM point in $x$ direction
in this case.
Since the torque is perpendicular to the magnetization
we only consider $T^{\rm top}_y$ and $T^{\rm top}_z$.
We first assume that the electric field is applied in $x$ direction.
The $xz$ mirror plane preserves $E_x$ and $T_y^{\rm top}$,
but it flips $T_z^{\rm top}$, $\hat{\vn{M}}^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$.
Consequently, $T_y^{\rm top}$ does not change
when both $\hat{\vn{M}}^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$ are flipped,
while $T_z^{\rm top}$ changes sign in this case.
When the electric field is applied in $y$ direction,
the $yz$ mirror plane preserves $E_y$, $\hat{\vn{M}}^{\rm top}$,
and $\hat{\vn{M}}^{\rm bottom}$, but it flips
both $T^{\rm top}_y$ and $T^{\rm top}_z$. Thus,
symmetry requires $t^{\rm top}_{yy}=0$ and $t^{\rm top}_{zy}=0$.
These properties are summarized in the second row of Table~\ref{tab_symmetry2}.

\begin{threeparttable}
\caption{Symmetry properties of the torkance tensor $t_{ij}^{\rm top}$
for various directions of the
magnetization $\hat{\vn{M}}^{\rm top}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\rm T}$.
$+$ means that the torkance is even in $\hat{\vn{M}}^{\rm top}$
(i.e., it does not change sign when $\hat{\vn{M}}^{\rm top}$ is flipped
while $\hat{\vn{M}}^{\rm bottom}$ is not flipped) and
$-$ means that the torkance is odd in $\hat{\vn{M}}^{\rm top}$
(i.e., it changes sign when $\hat{\vn{M}}^{\rm top}$ is flipped
while $\hat{\vn{M}}^{\rm bottom}$ is not flipped).
}
\label{tab_symmetry}
\begin{ruledtabular}
\begin{tabular}{c|c|c|c|c|c|c|}
&$t_{xx}^{\rm top}$
&$t_{xy}^{\rm top}$
&$t_{yx}^{\rm top}$
&$t_{yy}^{\rm top}$
&$t_{zx}^{\rm top}$
&$t_{zy}^{\rm top}$
\\
\hline
$\theta=0,\phi=0$
&-&+&+&-&0&0\\
\hline
$\theta=90^{\circ},\phi=90^{\circ}$
&-&+&0&0&+&-\\
\end{tabular}
\end{ruledtabular}
\end{threeparttable}

\begin{threeparttable}
\caption{Symmetry properties of the torkance tensor $t_{ij}^{\rm top}$
for various directions of the
magnetization $\hat{\vn{M}}^{\rm top}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\rm T}$.
$+$ means that the torkance does not change sign when
both $\hat{\vn{M}}^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$ are flipped.
$-$ means that the torkance changes sign when
both $\hat{\vn{M}}^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$ are flipped.
}
\label{tab_symmetry2}
\begin{ruledtabular}
\begin{tabular}{c|c|c|c|c|c|c|}
&$t_{xx}^{\rm top}$
&$t_{xy}^{\rm top}$
&$t_{yx}^{\rm top}$
&$t_{yy}^{\rm top}$
&$t_{zx}^{\rm top}$
&$t_{zy}^{\rm top}$
\\
\hline
$\theta=0,\phi=0$
&-&+&+&-&0&0\\
\hline
$\theta=90^{\circ},\phi=0$
&0&0&+&0&-&0\\
\hline
$\theta=90^{\circ},\phi=90^{\circ}$
&+&+&0&0&-&-\\
\end{tabular}
\end{ruledtabular}
\end{threeparttable}

Finally, we consider the case where $\hat{\vn{M}}^{\rm top}$
points in $y$ direction, i.e., $\theta=90^{\circ}$ and $\phi=90^{\circ}$.
Since the torque is perpendicular to the magnetization we only need to
consider $T^{\rm top}_x$ and $T^{\rm top}_z$.
First, we assume that the electric field is applied in $x$ direction.
The $xz$ mirror plane flips $T_x^{\rm top}$, $T_z^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$,
but it preserves $E_x$ and $\hat{\vn{M}}^{\rm top}$.
Thus, the combination of the c2 rotation around the $z$ axis with
the $xz$ mirror plane
preserves $T_x^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$
but flips $E_x$, $\hat{\vn{M}}^{\rm top}$ and $T_z^{\rm top}$.
Consequently, $T_z^{\rm top}$ is even in $\hat{\vn{M}}^{\rm top}$
and $T_x^{\rm top}$ is odd in $\hat{\vn{M}}^{\rm top}$.
When the electric field points in $y$ direction,
the combination of the c2 rotation around the $z$ axis with
the $xz$ mirror plane preserves $E_y$.
In this case $T_z^{\rm top}$ is odd in $\hat{\vn{M}}^{\rm top}$
and $T_x^{\rm top}$ is even in $\hat{\vn{M}}^{\rm top}$.
A c2 rotation around the $z$ axis
flips $\hat{\vn{M}}^{\rm top}$, $\hat{\vn{M}}^{\rm bottom}$, $\vn{E}$,
and $T_x^{\rm top}$ but preserves $T_z^{\rm top}$.
Consequently, $T_x^{\rm top}$ does not change sign when
both $\hat{\vn{M}}^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$ are flipped,
while $T_z^{\rm top}$ changes sign when both $\hat{\vn{M}}^{\rm top}$
and $\hat{\vn{M}}^{\rm bottom}$ are flipped.
These properties are summarized
in the second row of Table~\ref{tab_symmetry}
and
in the third row of
Table~\ref{tab_symmetry2}.

The
case $\theta=90^{\circ}$ and $\phi=90^{\circ}$
is of special interest,
because it is well-known for magnetic bilayers
that $t_{zx}=0$ and $t_{xx}=0$ when $\theta=90^{\circ}$
and $\phi=90^{\circ}$~\cite{symmetry_spin_orbit_torques}.
Indeed, from Table~\ref{tab_symmetry}
and
Table~\ref{tab_symmetry2}
it is clear that $t_{zx}^{\rm top}$ and $t_{xx}^{\rm top}$
cannot exist without
the bottom magnetic layer:
For $\theta=90^{\circ}$ and $\phi=90^{\circ}$
$t_{zx}^{\rm top}$ is even in
the magnetization of the top layer (see Table~\ref{tab_symmetry}),
but odd in the total
magnetization (see Table~\ref{tab_symmetry2}).
Similarly, $t_{xx}^{\rm top}$ is odd in
the magnetization of the top layer (see Table~\ref{tab_symmetry}),
but even in the total
magnetization (see Table~\ref{tab_symmetry2}).
Clearly, these conditions cannot be satisfied when the
bottom magnetic layer is missing.
Since $t_{zx}^{\rm top}=t_{xx}^{\rm top}=0$ in bilayers, the underlying mechanism
in trilayers has to involve the bottom FM, i.e.,
it has to be a nonlocal mechanism.
Since nonlocal transfer of angular momentum is mediated
by spin currents, we attribute the nonzero $t_{zx}^{\rm top}$ in trilayers
to a spin current with
spin-polarization along $z$ direction flowing from the
bottom FM to the top FM. Such a spin current can be
generated at the bottom FM interface through
spin-orbit precession~\cite{interface_generated_spin_currents,spin_transport_interfaces_soc_formalism,sots_interface_generated_spin_currents_baek}.
$t_{zx}^{\rm top}$ is odd in the magnetization of the bottom magnet,
consistent with the mechanism of spin-orbit precession.
As the $z$-polarized spin current interacts with the top FM it
precesses around the top FM magnetization. This leads to an
additional torque
component in the direction of $\hat{\vn{M}}^{\rm top}\times \hat{\vn{e}}_{z}$,
which is odd in $\hat{\vn{M}}^{\rm top}$.
For  $\hat{\vn{M}}^{\rm top}=\hat{\vn{e}}_{y}$ this torque points
into the $x$ direction, which explains the component $t_{xx}^{\rm top}$.
Therefore, both $t_{xx}^{\rm top}$ and $t_{zx}^{\rm top}$ arise from
the $z$-polarized spin current. Since $t_{xx}^{\rm top}$ is odd in
the top FM magnetization,
it cannot be explained by a spin current with spin-polarization along $x$
that interacts with the top FM through spin transfer, because the
resulting torque would be even in $\hat{\vn{M}}^{\rm top}$.
The $z$-polarized spin current in trilayers is of particular
interest, because it allows field-free
switching and provides an antidamping-torque for the
perpendicular top
FM~\cite{sots_interface_generated_spin_currents_baek,observation_spin_orbit_effects_spin_rotation_symmetry}.
Calculating the SOT at $\theta=\phi=90^{\circ}$ is
an easy way to investigate this $z$-polarized
spin current. In comparison, the SOT at $\phi=0^{\circ}$
and $0<\theta<90^{\circ}$ contains also contributions from the
interfacial SOI at the top FM interface and therefore it does not
give direct access to the $z$-polarized spin current.

\section{Formalism}
\label{sec_formalism}
\subsection{Spin-orbit torque}
We define the torkance $t_{ij\alpha}$ of atom $\alpha$ by
the
equation
\begin{equation}\label{eq_torque_and_torkance}
T_{i\alpha}=\sum_{j}
t_{ij\alpha}E_j,
\end{equation}
where $E_j$ is the $j$-th component of the applied electric field
and $T_{i\alpha}$ is the $i$-th component of the torque exerted
on the magnetic moment of atom $\alpha$.
In the Co/Cu/Co trilayers considered in this
work we will be particularly interested in the sum
of $t_{ij\alpha}$
over all atoms in the
top FM,
\begin{equation}\label{eq_sot_toplayer}
t_{ij}^{\rm top}=
\sum_{\alpha\in {\rm top}}
t_{ij\alpha},
\end{equation}
where $\alpha\in {\rm top}$ denotes the atoms in the top FM.
Since the bottom and top FM are separated by a Cu spacer,
the top FM can switch while the bottom FM can be
kept fixed. For the magnetization dynamics of the top FM
the sum as defined in Eq.~\eqref{eq_sot_toplayer} is the relevant torkance.
In magnetic bilayers with only one ferromagnetic layer,
such as a Co/Cu magnetic bilayer, the relevant
torkance is obtained by summing over all atoms, i.e.,
\begin{equation}\label{eq_sot_total}
t_{ij}=
\sum_{\alpha}
t_{ij\alpha}.
\end{equation}

The torkance is given by the sum
of three terms, i.e.,
$t^{\phantom{II}}_{ij\alpha}=t^{\rm I(a)}_{ij\alpha}+
t^{\rm I(b)}_{ij\alpha}+
t^{\rm II}_{ij\alpha}
$,
where~\cite{ibcsoit,invsot}
\begin{gather}\label{eq_torkance_kubo}
\begin{aligned}
t^{\rm I(a)\phantom{I}}_{ij\alpha}\!\!\!\!&=\phantom{-}\frac{eA}{h}
\int\!\!\frac{{\rm d}^2 k}{(2\pi)^2}
\,{\rm Tr}
\left\langle
\mathcal{T}_{i\alpha}
G^{\rm R}_{\vn{k}}(\mathcal{E}_{\rm F})
v_{j}
G^{\rm A}_{\vn{k}}(\mathcal{E}_{\rm F})
\right\rangle
\\
t^{\rm I(b)\phantom{I}}_{ij\alpha}\!\!\!\!&=-\frac{eA}{h}
\int\!\!\frac{{\rm d}^2 k}{(2\pi)^2}
\,{\rm Re}
\,{\rm Tr}
\left\langle
\mathcal{T}_{i\alpha}
G^{\rm R}_{\vn{k}}(\mathcal{E}_{\rm F})
v_{j}
G^{\rm R}_{\vn{k}}(\mathcal{E}_{\rm F})
\right\rangle
\\
t^{\rm II\phantom{(a)}}_{ij\alpha}\!\!\!\!&=
\phantom{-}\frac{eA}{h}\int_{-\infty}^{\mathcal{E}_{\rm F}}
d\mathcal{E}
\int\!\!\frac{{\rm d}^2 k}{(2\pi)^2}
\,{\rm Re}
\,{\rm Tr}
\left\langle
\mathcal{T}_{i\alpha}G_{\vn{k}}^{\rm R}(\mathcal{E})v_{j}
\frac{dG_{\vn{k}}^{\rm R}(\mathcal{E})}{d\mathcal{E}}\right.\\
 &\quad\quad\quad\quad\quad\quad\quad\quad\,-\left.
\mathcal{T}_{i\alpha}\frac{dG_{\vn{k}}^{\rm R}(\mathcal{E})}{d\mathcal{E}}v_{j}G_{\vn{k}}^{\rm R}(\mathcal{E})
\right\rangle.
\end{aligned
\end{gather}
Here, $A$ is the area of
the in-plane unit cell, such that $T_{i\alpha}$
in Eq.~\eqref{eq_torque_and_torkance} is the $i$-th component
of the torque exerted on the magnetic moments of atom type $\alpha$
in one unit cell.
$e>0$ is the elementary positive charge, $\vn{k}$ denotes a $k$-point in
the two-dimensional Brillouin zone of the bilayers and trilayers,
 $G_{\vn{k}}^{\rm R}(\mathcal{E})$ is the
retarded Green's function,
$G^{\rm A}_{\vn{k}}(\mathcal{E})$ is the advanced Green's
function, $\mathcal{E}_{\rm F}$ is the Fermi energy,
$v_{j}$ is the $j$-th component of the velocity operator, and
$\mathcal{T}_{i\alpha}$ is the $i$-th component of the
torque operator $\vn{\mathcal{T}}_{\!\!\alpha}$ of
atom $\alpha$
with matrix elements
\begin{equation}\label{eq_atom_resolved_torque_operator}
\langle\psi^{\phantom{\dagger}}_{\vn{k}n}
|
\vht{\mathcal{T}}^{\phantom{\dagger}}_{\!\!\alpha}
|
\psi^{\phantom{\dagger}}_{\vn{k}m}\rangle
=-\mu^{\phantom{\dagger}}_{\rm B}\!\!\!\!\int\limits_{\rm MT_{\alpha}}\!\!\!\!d^3r\,
\psi^{\dagger}_{\vn{k}n}(\vn{r})
\vht{\sigma}\!\times\! \vn{\Omega}^{\rm xc}(\vn{r})
\psi^{\phantom{\dagger}}_{\vn{k}m}(\vn{r}),
\end{equation}
where $\mu_{\rm B}$ is the Bohr magneton, $\vn{\sigma}$
is the vector of Pauli spin matrices,
$\vn{\Omega}^{\rm xc}(\vn{r})$ is the exchange field, which
is obtained from the difference between the effective potentials
of minority and majority electrons
as $\vn{\Omega}^{\rm xc}(\vn{r})=\frac{1}{2\mu_{\rm B}}\left[
V^{\rm eff}_{\rm minority}(\vn{r})-V^{\rm eff}_{\rm majority}(\vn{r})
\right]\hat{\vn{M}}(\vn{r})$, and $\hat{\vn{M}}(\vn{r})$ is the magnetization direction.
The volume
integration in Eq.~\eqref{eq_atom_resolved_torque_operator}
is restricted to the MT-sphere of atom $\alpha$.

The torkances depend on the magnetization direction $\hat{\vn{M}}$.
We define the
even torkance $t^{\rm e}_{ij\alpha}(\hat{\vn{M}})=[t_{ij\alpha}(\hat{\vn{M}})+t_{ij\alpha}(-\hat{\vn{M}})]/2$
and the odd torkance $t^{\rm o}_{ij\alpha}(\hat{\vn{M}})=[t_{ij\alpha}(\hat{\vn{M}})-t_{ij\alpha}(-\hat{\vn{M}})]/2$.
Inversion of
magnetization, i.e., $\hat{\vn{M}}\rightarrow -\hat{\vn{M}}$,
does not modify the even torkance,
but it flips the sign of the odd torkance.
The torkance is the sum of its even and odd parts,
i.e., $t^{\phantom{e}}_{ij\alpha}(\hat{\vn{M}})=t^{\rm e}_{ij\alpha}(\hat{\vn{M}})+t^{\rm o}_{ij\alpha}(\hat{\vn{M}})$.
$t^{\rm II\phantom{(a)}}_{ij\alpha}$ in Eq.~\eqref{eq_torkance_kubo}
contributes only to $t^{\rm e}_{ij\alpha}$
and $t^{\rm I(b)\phantom{I}}_{ij\alpha}$ contributes only to $t^{\rm o}_{ij\alpha}$,
while $t^{\rm I(a)\phantom{I}}_{ij\alpha}$ contributes to
both $t^{\rm e}_{ij\alpha}$
and $t^{\rm o}_{ij\alpha}$.
Since the magnetizations in the top and bottom FMs may point into different
directions in trilayers (see Fig.~\ref{cocucotrilayer_setup}), it is important to point out
that $t^{\rm o}_{ij\alpha}(\hat{\vn{M}})$ changes sign when both $\hat{\vn{M}}^{\rm top}$
and $\hat{\vn{M}}^{\rm bottom}$ are flipped,
while $t^{\rm e}_{ij\alpha}(\hat{\vn{M}})$ does not change sign when both $\hat{\vn{M}}^{\rm top}$
and $\hat{\vn{M}}^{\rm bottom}$ are flipped, because the odd character of $t^{\rm o}_{ij\alpha}(\hat{\vn{M}})$
and the even character of $t^{\rm e}_{ij\alpha}(\hat{\vn{M}})$ result from their transformation
properties under time reversal~\cite{ibcsoit}.
Therefore,
the even torkance in Co/Cu/Co trilayers corresponds to the
plus entries in Table~\ref{tab_symmetry2} and the
odd torkance corresponds to the minus entries in Table~\ref{tab_symmetry2}.
As discussed in Table~\ref{tab_symmetry} one can also consider how the torkance
changes when only $\hat{\vn{M}}^{\rm top}$ is flipped while $\hat{\vn{M}}^{\rm bottom}$
is kept fixed and in some cases the torkance for $-\hat{\vn{M}}^{\rm top}$ and $+\hat{\vn{M}}^{\rm bottom}$
is simply equal or opposite to the torkance for $+\hat{\vn{M}}^{\rm top}$ and $+\hat{\vn{M}}^{\rm bottom}$.
Thus, one can also define even and odd torques with respect to $\hat{\vn{M}}^{\rm top}$ when
$\hat{\vn{M}}^{\rm bottom}$ is kept fixed. The fact that the third row in
Table~\ref{tab_symmetry2} differs from the second row in Table~\ref{tab_symmetry} shows
that flipping the bottom magnetization $\hat{\vn{M}}^{\rm bottom}$ may impact the torque on the top magnet.
Therefore, in the following we mean by even torque always that the torque does not change when
both $\hat{\vn{M}}^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$ are flipped, while by odd torque we mean
that the torque changes sign when both $\hat{\vn{M}}^{\rm top}$ and $\hat{\vn{M}}^{\rm bottom}$ are flipped.

\subsection{Dzyaloshinskii-Moriya interaction}
DMI is a chiral interaction, which modifies the free energy of
noncentrosymmetric magnets proportional to
the spatial derivatives of the magnetization.
The contribution of DMI to the free energy density
at position $\vn{r}$ can be
written as~\cite{mothedmisot}
\begin{equation}\label{eq_free_energy_dmi}
F^{\rm DMI}(\vn{r})=\sum_{ij}
D_{ij}(\hat{\vn{M}}(\vn{r}))
\hat{\vn{e}}_{i}\cdot
\left[
\hat{\vn{M}}(\vn{r})
\times\frac{\partial
\hat{\vn{M}}(\vn{r})
}{\partial r_j}
\right],
\end{equation}
where $D_{ij}(\hat{\vn{M}})$ are the DMI coefficients and $\hat{\vn{e}}_{i}$ is a
unit vector pointing in the $i$th spatial direction.
These DMI coefficients can be decomposed into the contributions
associated with the magnetic moments of atom type $\alpha$,
which we denote by $D_{ij\alpha}$,
such that
\begin{equation}
D_{ij}=\sum_{\alpha}D_{ij\alpha}.
\end{equation}
In magnetic trilayers such as Co/Cu/Co we will be
particularly interested in summing $D_{ij\alpha}$ over the layers
of the top magnet in order to quantify the free energy gained
due to the DMI
when the magnetization in the top layer forms a cycloidal spin spiral
while the magnetization in the bottom layer remains unchanged,
i.e., we will be interested in the quantity
\begin{equation}\label{eq_dmi_toplayer}
D^{\rm top}_{ij}=\sum_{\alpha\in{\rm top}}D_{ij\alpha}.
\end{equation}
We evaluate the DMI coefficients associated with the magnetic moments
of atom type $\alpha$ from~\cite{mothedmisot,phase_space_berry,itsot}
\begin{equation}\label{eq_dmi}
D_{ij\alpha}=\int\!\!\frac{{\rm d}^2 k}{(2\pi)^2}
\!\!\sum_{n}
f_{\vn{k}n}
\left[
A_{\vn{k}nij\alpha}-(\mathcal{E}_{\vn{k}n}-\mu)B_{\vn{k}nij\alpha}
\right],
\end{equation}
where
\begin{equation}\label{eq_akn_kubo}
A_{\vn{k}nij\alpha}=\hbar\sum_{m\neq n}\text{Im}
\left[
\frac{
\langle \psi_{\vn{k}n}  |\mathcal{T}_{i\alpha}| \psi_{\vn{k}m}  \rangle
\langle \psi_{\vn{k}m}  |v_{j}| \psi_{\vn{k}n}  \rangle
}
{
\mathcal{E}_{\vn{k}m}-\mathcal{E}_{\vn{k}n}
}
\right]
\end{equation}
and
\begin{equation}\label{eq_bkn_kubo}
B_{\vn{k}nij\alpha}
\!=\!-2\hbar\!\!
\sum_{m\neq n}\!\!\text{Im}
\!\!\left[\!
\frac{
\langle \psi_{\vn{k}n}  |\mathcal{T}_{i\alpha}| \psi_{\vn{k}m}  \rangle
\langle \psi_{\vn{k}m}  |v_{j}| \psi_{\vn{k}n}  \rangle
}
{
(\mathcal{E}_{\vn{k}m}-\mathcal{E}_{\vn{k}n})^2
}
\!\right].
\end{equation}
Here, $\mathcal{E}_{\vn{k}n}$ is the band energy of state $| \psi_{\vn{k}n}  \rangle$.
Eq.~\eqref{eq_dmi} is valid at zero temperature, which is sufficient for this study.
The generalization to finite temperatures has been described
elsewhere~\cite{mothedmisot,phase_space_berry,itsot}.
Eq.~\eqref{eq_dmi}, Eq.~\eqref{eq_akn_kubo},
and Eq.~\eqref{eq_bkn_kubo} differ from previous
works~\cite{mothedmisot,phase_space_berry,itsot}
by the introduction of the atomic index $\alpha$ and by the use of the
atom-resolved torque operator $\mathcal{T}_{i\alpha}$ defined in
Eq.~\eqref{eq_atom_resolved_torque_operator}.
This generalization is necessary in order to study the DMI associated
with one magnetic layer, e.g.\ for the top magnet according to Eq.~\eqref{eq_dmi_toplayer},
in a magnetic multilayer structure.
\section{SOT in C\lowercase{o}/C\lowercase{u}/C\lowercase{o} trilayers}
\label{sec_sot_in_cocuco}
\subsection{Computational details}
We calculate SOTs in magnetic trilayers composed of 3 atomic layers
of Co, 9 atomic layers of Cu, and 3 atomic layers of Co (abbreviated as
Co(3)/Cu(9)/Co(3) in the following). Additionally we compute trilayers
with only 3 or 6 instead of 9 Cu layers, which we abbreviate in the following as
Co(3)/Cu(3)/Co(3) or Co(3)/Cu(6)/Co(3), respectively.
In order to be able to interpret the results on the trilayers
we also calculate a Co/Cu bilayer for
comparison, which is
composed of 3 layers of
Co on 9 layers of Cu (abbreviated as Co(3)/Cu(9)).
In order to calculate the SOTs we first perform
an electronic structure calculation with the FLEUR~\cite{fleurcode} code
using the generalized gradient approximation~\cite{PerdewBurkeEnzerhof}.
We choose the computational unit cell similar to the one in
Ref.~\cite{stt-cocuco_haney&waldron&duine&nunez&guo&macdonald},
where the spin-transfer torque in the current-perpendicular-to-plane-geometry
was investigated.
However,
we do not embed the scattering region between semi-infinite
leads as in Ref.~\cite{stt-cocuco_haney&waldron&duine&nunez&guo&macdonald},
but instead we set up the system in the film geometry, i.e., we calculate
Vacuum/Co(3)/Cu(n)/Co(3)/Vacuum (n=3,6,9),
because we study the SOTs driven by an electric
current that is applied parallel to the interfaces.
For this purpose we use the
film mode of FLEUR, which explicitly takes the vacuum regions into
account~\cite{Krakauer_film_mode}.
We choose the coordinate system such that the $z$ axis is perpendicular to
the interfaces, while the $x$ and $y$ axes are parallel to the
interfaces (see also Fig.~\ref{cocucotrilayer_setup}).
Since the magnetization
in the top FM needs to point into a different direction than
the magnetization in the bottom FM of the Co/Cu/Co trilayer,
we use the noncollinear mode of FLEUR~\cite{noco_flapw}.
The magnetization of the bottom FM points in $x$ direction
in our calculations. We specify the magnetization direction
of the top FM in terms of angles $\theta$ and $\phi$,
i.e., $\vn{m}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\rm T}$.
We include SOI in the calculations.
The computational unit cell of the
Co(3)/Cu(9)/Co(3) trilayer contains 15 atoms in total.
Both Co and Cu are treated as fcc with lattice
constant 3.54~\AA. Consequently, the
distance between adjacent planes is 1.77~\AA.
The MT-radii of Co and Cu are set to 1.22~\AA.

We label the three Co-layers in the top FM as
Co-1, Co-2, and Co-3, where Co-1 is adjacent to vacuum and
Co-3 faces the Cu-layer.
We denote the three Co-layers in the bottom FM
by Co-4, Co-5, and Co-6, where Co-4 is adjacent to the Cu-layer
and Co-6 faces vacuum.
In the Co(3)/Cu(9)/Co(3) trilayer
the Cu-layers are labelled Cu-1 through Cu-9, where Cu-1 is adjacent
to Co-3 of the top FM.

As explained in Ref.~\cite{ibcsoit}
we use Wannier interpolation~\cite{rmp_wannier90}
in order to
evaluate Eq.~\eqref{eq_torkance_kubo} for the SOT
and Eq.~\eqref{eq_dmi} for the DMI computationally efficiently.
For this purpose, we generate
18 maximally localized Wannier functions (MLWFs) per atom
using our interface~\cite{WannierPaper}
between the FLEUR code and
the Wannier90 code~\cite{wannier90}.
For the generation of MLWFs we used an $8\times 8$ $\vn{k}$ mesh and
as trial orbitals we chose 6 $sp^3d^2$, $d_{xy}$, $d_{yz}$, and $d_{zx}$ for
spin-up and for spin-down, i.e., 18 trial orbitals per atom.
In the case of the Co(3)/Cu(9)/Co(3) trilayer we generated 270 MLWFs,
which we disentangled from 378 bands.
In the case of the Co(3)/Cu(9) bilayer we generated 216 MLWFs, which we disentangled
from 304 bands.

In order to investigate the role of spin-orbit coupling in the normal metal spacer
layer we also compute the SOTs in Co(3)/Pt(13)/Co(3) trilayers, where Pt
provides strong SOI. The computational details of the Co/Pt/Co trilayer are
given in Ref.~\cite{ibcsoit}.

\subsection{Even SOT}
In Fig.~\ref{cocuco_eventorque_vs_fermi} we show the
even torkance of the top FM (see Eq.~\eqref{eq_sot_toplayer})
in the Co(3)/Cu(9)/Co(3) trilayer as a function of
the Fermi energy $\mathcal{E}_{\rm F}$ for a lifetime broadening
of $\Gamma=$25meV when the magnetization of the top FM
points in $z$ direction, i.e., $\theta=\phi=0$.
By plotting the results as a function of $\mathcal{E}_{\rm F}$ we
can roughly anticipate chemical trends: Negative $\mathcal{E}_{\rm F}$
mimics the system Fe$_{x}$Co$_{1-x}$/Ni$_{x}$Cu$_{1-x}$/Fe$_{x}$Co$_{1-x}$
while positive $\mathcal{E}_{\rm F}$ mimics the system
Ni$_{x}$Co$_{1-x}$/Zn$_{x}$Cu$_{1-x}$/Ni$_{x}$Co$_{1-x}$.
The pure Co(3)/Cu(9)/Co(3) system corresponds to $\mathcal{E}_{\rm F}=0$.
In agreement with the symmetry analysis in Table~\ref{tab_symmetry2}
the two nonzero components of the even torkance are $t_{xy}^{\rm top,e}$
and $t_{yx}^{\rm top,e}$.
Due to the presence of the bottom FM with magnetization
in $x$ direction the $x$ and $y$ directions are not equivalent and therefore
the torkance components $t^{\rm top,e}_{xy}$ and $t^{\rm top,e}_{yx}$ differ
substantially (in case of equivalent $x$
and $y$ directions $t_{xy}=-t_{yx}$ would hold).
For the range of $\mathcal{E}_{\rm F}$ shown in the
Figure, $t^{\rm top,e}_{yx}$ is much larger than $t^{\rm top,e}_{xy}$.
This means that when the electric field is applied in $x$ direction, i.e., parallel to
the magnetization of the bottom FM,
the SOT is much larger than when the electric field is applied perpendicular to
the magnetization of the bottom FM.

\begin{figure}
\includegraphics[width=\linewidth]{cocuco_eventorque_vs_fermi_compo2and4_fermirel.eps}
\caption{\label{cocuco_eventorque_vs_fermi}
Even torkance $t^{\rm top,e}_{ij}$ on
the top magnetic layer in the
Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ Fermi energy $\mathcal{E}_{\rm F}$
when $\theta=\phi=0$ and $\Gamma=25$meV.
}
\end{figure}

For comparison we show
in Fig.~\ref{eventorque_cocubilayer_vs_fermi} the
total even torkance (see Eq.~\eqref{eq_sot_total})
of the Co(3)/Cu(9) bilayer as a function of
Fermi energy $\mathcal{E}_{\rm F}$ for a lifetime broadening of $\Gamma=$25meV
when the magnetization points in $z$ direction.
In this case, the SOT in the Co(3)/Cu(9) bilayer is
isotropic, i.e., $t^{\rm e}_{xy}=-t^{\rm e}_{yx}$.
The component $t^{\rm e}_{xy}$ in Co(3)/Cu(9) is very similar in size
to $t^{\rm top,e}_{xy}$ in Co(3)/Cu(9)/Co(3),
but $t^{\rm top,e}_{yx}$ in Co(3)/Cu(9)/Co(3) is
much larger than $t^{\rm e}_{yx}$ in Co(3)/Cu(9).
Consequently, the bottom magnet
with magnetization along $x$
in the Co(3)/Cu(9)/Co(3) trilayer strongly enhances $t^{\rm top,e}_{yx}$, while its
effect on $t^{\rm top,e}_{xy}$ is not so strong.

\begin{figure}
\includegraphics[width=\linewidth]{eventorque_cocubilayer_vs_fermi_fermirel.eps}
\caption{\label{eventorque_cocubilayer_vs_fermi}
Even torkance $t^{\rm e}_{ij}$
in the Co(3)/Cu(9) magnetic bilayer
vs.\ Fermi energy $\mathcal{E}_{\rm F}$
when $\theta=\phi=0$ and $\Gamma=25$meV.
}
\end{figure}

In Fig.~\ref{eventorque_cocucotrilayer_vs_sigma} we show the
even torkance of the top FM (see Eq.~\eqref{eq_sot_toplayer})
in the Co(3)/Cu(9)/Co(3) trilayer as a function of
the lifetime broadening $\Gamma$ when the magnetization
points in $z$ direction ($\theta=0$).
At small broadening $\Gamma=1$~meV the component $t^{\rm top,e}_{yx}$
is larger than $t^{\rm top,e}_{xy}$ by a factor of 1.7 and at
large broadening $\Gamma=100$~meV the component $t^{\rm top,e}_{yx}$
is larger than $t^{\rm top,e}_{xy}$ by a factor of 1.5. Thus, with increasing $\Gamma$
the influence of the bottom FM on $t^{\rm top,e}_{ij}$ becomes smaller
such that the anisotropy of the SOT reduces.
This can be explained by assuming that the effect of the bottom FM
on $t^{\rm top,e}_{ij}$ is mediated by spin currents that flow between the
bottom FM and the top FM. Such a spin-current mediated
effect is not possible when the spin-diffusion length is much smaller than the
thickness of the Cu spacer layer. Since the spin-diffusion length decreases with
increasing disorder, which we model by the broadening $\Gamma$, it is
expected that an increase of $\Gamma$ reduces the spin-current-mediated
coupling between the bottom and top FMs.
The enhancement of $t^{\rm top,e}_{yx}$ in comparison
to $t^{\rm top,e}_{xy}$ can be explained by a spin current with spin polarization
in $y$ direction, which flows from the bottom FM into the top FM.
Such a spin current can be generated through spin-orbit filtering
at the bottom FM interface~\cite{interface_generated_spin_currents}.
According to Table~\ref{tab_symmetry} and
Table~\ref{tab_symmetry2}
$t^{\rm top,e}_{yx}$ is even in $\hat{\vn{M}}^{\rm top}$ and
even in $\hat{\vn{M}}^{\rm bottom}$, consistent with this
mechanism. In order to describe the spin-orbit
filtering quantitatively, a model has been proposed that predicts this
contribution to be proportional to the relaxation
time $\tau$~\cite{interface_generated_spin_currents}.
Since $\tau\propto \hbar/(2\Gamma)$, one expects the contribution
from spin-orbit filtering to decrease with increasing $\Gamma$.
Therefore, there are two reasons why the enhancement
of $t^{\rm top,e}_{yx}$ in comparison to $t^{\rm top,e}_{xy}$ decreases with
increasing $\Gamma$: The spin-current is attenuated stronger on the
way from the bottom FM to the top FM and the amount of spin current
generated at the bottom FM interface through spin-orbit filtering decreases.

\begin{figure}
\includegraphics[width=\linewidth]{eventorque_cocucotrilayer_vs_sigma.eps}
\caption{\label{eventorque_cocucotrilayer_vs_sigma}
Even torkance $t^{\rm top,e}_{ij}$ on
the top magnetic layer in the
Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ lifetime broadening $\Gamma$
when $\theta=\phi=0$.
}
\end{figure}

For comparison, we show in Fig.~\ref{eventorque_cocubilayer_vs_sigma}
the total even torkance (see Eq.~\eqref{eq_sot_total})
in the Co(3)/Cu(9) bilayer as a function of
lifetime broadening $\Gamma$.
At large broadening $\Gamma=$100meV
$t^{\rm top,e}_{xy}$ in the Co(3)/Cu(9)/Co(3) trilayer
is larger than
$t^{\rm e}_{xy}$ in the Co(3)/Cu(9) bilayer
by 10\% while at small
broadening $\Gamma=$1meV
$t^{\rm top,e}_{xy}$ in the Co(3)/Cu(9)/Co(3) trilayer
is larger than
$t^{\rm e}_{xy}$ in the Co(3)/Cu(9) bilayer
by 24\%. This trend can be explained again
by assuming that the effect of the bottom FM
on the top FM is mediated by spin-currents and
that this contribution reduces when the disorder increases.
Similarly,
$t^{\rm top,e}_{yx}$ in the Co(3)/Cu(9)/Co(3) trilayer
is larger than
$t^{\rm e}_{yx}$ in the Co(3)/Cu(9) bilayer
by 66\% at large broadening $\Gamma=$100meV,
while at small broadening $\Gamma=$1meV
$t^{\rm top,e}_{yx}$ in the Co(3)/Cu(9)/Co(3) trilayer
is larger than
$t^{\rm e}_{yx}$ in the Co(3)/Cu(9) bilayer
by 113\%.

\begin{figure}
\includegraphics[width=\linewidth]{cocu_eventorque_vs_sigma_compos2and4.eps}
\caption{\label{eventorque_cocubilayer_vs_sigma}
Even torkance $t^{\rm e}_{ij}$
in the Co(3)/Cu(9) magnetic bilayer
vs.\ lifetime broadening $\Gamma$
when $\theta=\phi=0$.
}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{eventorque_vs_sigma_nosoicu.eps}
\caption{\label{eventorque_vs_sigma_nosoicu}
Even torkance $t^{\rm top,e}_{ij}$
on the top magnetic layer
in the Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ lifetime broadening $\Gamma$
when $\theta=\phi=0$ and when SOI on the
Cu atoms is switched off.
}
\end{figure}

Since the SOI in Cu is small, one may expect
that it does not contribute to the even torque. In order
to verify this by calculations we switch off SOI on all Cu atoms.
The resulting even torkance on the top magnetic layer
is shown in Fig.~\ref{eventorque_vs_sigma_nosoicu}
as a function of broadening $\Gamma$ when magnetization
points in $z$ direction ($\theta=0$).
In comparison with Fig.~\ref{eventorque_cocucotrilayer_vs_sigma}
the even torkances are quite similar. Therefore, while switching off SOI on Cu
obviously has an effect on the torkances it is certainly not the dominant
source of $t_{ij}^{\rm top,e}$.
This is a major difference to Co/Pt bilayers, where SOI in Pt provides a strong SHE.

In order to investigate the dependence on the thickness of the
Cu layer, we show in Fig.~\ref{eventorque_3Co3Cu3Co} the
even torkance $t^{\rm top,e}_{ij}$ of the Co(3)/Cu(3)/Co(3) trilayer,
in which the spacer consists of only
3 atomic layers of Cu, and in Fig.~\ref{eventorque_3Co6Cu3Co}
we show
the even torkance $t^{\rm top,e}_{ij}$
of Co(3)/Cu(6)/Co(3), which contains
6 layers of Cu in the spacer layer.
At large broadening of $\Gamma=100$~meV the three systems
Co(3)/Cu(3)/Co(3), Co(3)/Cu(6)/Co(3), and Co(3)/Cu(9)/Co(3)
exhibit very similar even torkances, but at smaller broadenings
the torkances vary differently with $\Gamma$. That the torkance
becomes independent of the layer thicknesses when $\Gamma$
is sufficiently large has been found also for Co/Pt bilayers~\cite{ibcsoit}.
The component $t^{\rm top,e}_{yx}$ is
enhanced in comparison to $t^{\rm top,e}_{xy}$ in both
Co(3)/Cu(3)/Co(3) and Co(3)/Cu(6)/Co(3) at small broadening $\Gamma$,
similar to Co(3)/Cu(9)/Co(3) discussed above.
\begin{figure}
\includegraphics[width=\linewidth]{eventorque_3Co3Cu3Co_vs_sigma.eps}
\caption{\label{eventorque_3Co3Cu3Co}
Even torkance $t^{\rm top,e}_{ij}$
in the Co(3)/Cu(3)/Co(3) magnetic trilayer
vs.\ lifetime broadening $\Gamma$
when $\theta=\phi=0$.
}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{eventorque_vs_sigma_3Co6Cu3Co_layers1234.eps}
\caption{\label{eventorque_3Co6Cu3Co}
Even torkance $t^{\rm top,e}_{ij}$
in the Co(3)/Cu(6)/Co(3) magnetic trilayer
vs.\ lifetime broadening $\Gamma$
when $\theta=\phi=0$.
}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{eventorque_vs_angle.eps}
\caption{\label{eventorque_vs_angle}
Even torkance $t^{\rm top,e}_{\theta y}$ (circles)
and  even torkance $t^{\rm top,e}_{\phi x}$ (squares)
in the Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ angle $\theta$
when $\phi=0$ and $\Gamma=25$meV.
Lines serve as guide to the eye.
}
\end{figure}

Next, we discuss how the even SOT depends on the direction of $\hat{\vn{M}}^{\rm top}$.
Since the torque is always perpendicular
to the magnetization, it is convenient to discuss it using the
unit vectors of the spherical
coordinate system
\begin{equation}
\hat{\vn{e}}_{\theta}
=(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta)^{\rm T}
\end{equation}
and
\begin{equation}
\hat{\vn{e}}_{\phi}=
(-\sin\phi,\cos\phi,0)^{\rm T}.
\end{equation}
We define the torkances $t^{\rm top}_{\theta i}$
and $t^{\rm top}_{\phi i}$ through
\begin{equation}\label{eq_theta_component}
\vn{T}^{\rm top}\cdot \hat{\vn{e}}_{\theta}
=\sum_{i}t^{\rm top}_{\theta i}E_{i}
\end{equation}
and
\begin{equation}\label{eq_phi_component}
\vn{T}^{\rm top}\cdot \hat{\vn{e}}_{\phi}
=\sum_{i}t^{\rm top}_{\phi i}E_{i}.
\end{equation}
In Fig.~\ref{eventorque_vs_angle} we show the even torkance
in the Co(3)/Cu(9)/Co(3) trilayer as
a function of the angle $\theta$ when $\phi=0$
and $\Gamma=25$meV.
In agreement with the symmetry
analysis in Table~\ref{tab_symmetry2} the even SOT is
zero for $\theta=90^{\circ}$ when the electric current is applied
in $y$ direction. However, when the electric field is applied in $x$ direction,
the even torkance increases strongly
from $t^{\rm e}_{\phi x}=-0.36 e a_{0}$ (at $\theta=0^{\circ}$)
to $t^{\rm e}_{\phi x}=-0.88 e a_{0}$ (at $\theta=90^{\circ}$).
A similarly strong increase of the even torkance with the magnetization
angle has been found in experiments on MgO/CoFeB/Ta~\cite{symmetry_spin_orbit_torques}.

\begin{figure}
\includegraphics[width=\linewidth]{cocucotheta90phi90_eventorque_vs_fermi_compo4and5_fermirel.eps}
\caption{\label{eventorque_theta90_phi90}
Even torkance $t^{\rm top,e}_{ij}$
in the Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ Fermi energy $\mathcal{E}_{\rm F}$ when
$\theta=\phi=90^{\circ}$ and $\Gamma=25$meV.
}
\end{figure}

In Fig.~\ref{eventorque_theta90_phi90} we show the even torkance as
a function of Fermi energy $\mathcal{E}_{\rm F}$
when $\theta=\phi=90^{\circ}$.
In agreement with the symmetry
analysis in Table~\ref{tab_symmetry2} the even SOT
points in $x$ direction in this case and it can be generated
by an electric field in $x$ direction, but also by an electric field in $y$ direction.
As discussed at the end of section Sec.~\ref{sec_symmetry}, $t^{\rm top,e}_{xx}$
can be explained by a spin current with spin-polarization in $z$ direction,
which flows from the bottom magnet into the top magnet, and which precesses
around the top FM magnetization, thereby generating a torque
in the direction $\hat{\vn{M}}^{\rm top}\times\hat{\vn{e}}_{z}=\hat{\vn{e}}_{y}\times\hat{\vn{e}}_{z}=\hat{\vn{e}}_{x}$.

In order to compare the SOTs in Co/Cu/Co trilayers, which
contain only 3$d$ transition metals, to Co/Pt/Co trilayers,
in which Pt provides a large SHE,
we show in Fig.~\ref{eventorque_co3pt13co3} the
even torkance in Co/Pt/Co
as a function of broadening $\Gamma$
when $\theta=\phi=0$. Similar to the Co/Cu/Co trilayers, $t^{\rm top,e}_{yx}$
is enhanced when compared to $t^{\rm top,e}_{xy}$ at
small broadening $\Gamma$ due to the presence of the bottom magnet.
The suppression of the SOT with increasing $\Gamma$ is considerably
smaller in the  Co/Pt/Co trilayers than in the Co/Cu/Co trilayers, consistent with
our interpretation that the SOT in  Co/Cu/Co trilayers arises from different
mechanisms.
For small broadening $\Gamma$ the even torkances in Co/Cu/Co and Co/Pt/Co
are of the same order of magnitude.

\begin{figure}
\includegraphics[width=\linewidth]{eventorque_vs_sigma_3Co13Pt3Co_layers1234.eps}
\caption{\label{eventorque_co3pt13co3}
Even torkance $t^{\rm top,e}_{ij}$
in the Co(3)/Pt(13)/Co(3) magnetic trilayer
vs.\ broadening $\Gamma$ when
$\theta=\phi=0^{\circ}$.
}
\end{figure}

At first glance it is suprising that the even torkances in Co/Cu/Co trilayers are comparable
in magnitude to those in Co/Pt bilayers~\cite{ibcsoit,invsot}
and to those in the Co/Pt/Co trilayers considered here,
despite the absence of any 5$d$ or 4$d$ transition metals with
strong SOI in the former.
However, it is well-known that several SOI-generated effects
can be large even in systems that do not contain any 5$d$ or 4$d$ transition metals.
For example, the intrinsic AHE
of bcc Fe~\cite{fipri_ahe_fe} is not much smaller than
the one of $L$1$_{0}$ FePt~\cite{ahe_FePd_FePt}, despite the presence
of a 5$d$ element in the latter.
Also in the Heusler compound Co$_2$MnAl the AHE is large even though
it is composed only of 3$d$ transition metals~\cite{ahe_heuslers_felser}.
Similarly, the MAE in Fe/MgO~\cite{fipri_mae_FeMgO}
is found to be large, despite the absence of 4$d$ and 5$d$ elements
with strong SOI.

According to Eq.~\eqref{eq_torque_and_torkance}
the torkance specifies the torque per electric
field.
However, in order to judge whether the SOT
in a given system is sufficiently large to make it
attractive for applications it is important
to know the torque per electric current.
The torque per electric current is
obtained from the torkance by multiplication with
the electrical resistivity, i.e., $T_{y}/J_{x}=t_{yx}\rho_{xx}$ for example.
Since the resistivity of bulk Cu is much lower than the one of bulk Pt we estimate
that the even torque per current in Co(3)/Cu(9) bilayers
and in Co(3)/Cu(9)/Co(3) trilayers
is smaller by roughly one order of magnitude when compared to
Co/Pt or Co/Pt/Co.

This amplification by the longitudinal resistivity $\rho_{xx}$ is
also known from the anomalous Hall effect and from the spin
Hall effect, where materials with large Hall angles are most
attractive for applications. The Hall angle is the product of the
Hall conductivity and the longitudinal resistivity $\rho_{xx}$.
The antiferromagnets MnIr, MnPt and MnPd have spin Hall angles comparable
to the one of Pt despite smaller spin Hall conductivities,
because they have larger resistivities than Pt~\cite{she_in_afms}.
Also the large spin Hall angles
of $\beta$-W and $\beta$-Ta profit from the high longitudinal
resistivity $\rho_{xx}$ of
the $\beta$-phase~\cite{stt_devices_giant_she_tungsten,spin_torque_giant_she_tantalum}.
In topological insulators one can even achieve infinitely large spin Hall angles,
at least theoretically, and thereby generate large
SOTs~\cite{spin_transfer_torque_generated_by_topological_insulator}.
Therefore, our finding that the even torkances
in Co(3)/Cu(9)/Co(3) and Co(3)/Cu(9) are comparable in size to
those in Co/Pt and Co/Pt/Co signifies that one may expect sufficiently large
torque per current ratios even when only 3$d$ transition metals are
involved, provided that the longitudinal resistivity $\rho_{xx}$
is large enough. While we chose
for the spacer layer in this study Cu, which is known to be a good conductor in bulk, it
can easily be replaced for example by Ti.
Alternatively, one may increase
the resistivity of Cu by alloying and thereby increase the torque per current
ratio.
In this sense Co(3)/Cu(9)/Co(3) and Co(3)/Cu(9) can be considered
as models of trilayers and bilayers that are composed of 3$d$ transition
metals only.
In fact, SOT-switching has recently been demonstated
in FM/Ti/CoFeB trilayers~\cite{sots_interface_generated_spin_currents_baek},
which shows that sufficiently large torque per current ratios can indeed be realized even
when only 3$d$ transition metals are used.

\subsection{Odd SOT}
In Fig.~\ref{oddtorque_toplayer_vs_fermi_cocuco} we show the
odd torkance of the top FM (see Eq.~\eqref{eq_sot_toplayer})
in the Co(3)/Cu(9)/Co(3) trilayer
as a function of the Fermi energy $\mathcal{E}_{\rm F}$
for a lifetime broadening of $\Gamma=25$meV when the magnetization
of the top FM points in $z$ direction.
In agreement with the symmetry analysis in Table~\ref{tab_symmetry2}
the nonzero components of the odd torkance are $t_{xx}^{\rm top,o}$ and $t_{yy}^{\rm top,o}$.
There is a small difference between $t_{xx}^{\rm top,o}$ and $t_{yy}^{\rm top,o}$,
i.e., the odd torkance is slightly anisotropic due to the presence of the
bottom magnetic layer with magnetization in $x$ direction, which lowers the
symmetry. However, this anisotropy is much smaller than in the case of the
even torque discussed above in Fig.~\ref{cocuco_eventorque_vs_fermi}.
\begin{figure}
\includegraphics[width=\linewidth]{oddtorque_vs_fermi_compo1and5_fermirel.eps}
\caption{\label{oddtorque_toplayer_vs_fermi_cocuco}
Odd torkance $t^{\rm top,o}_{ij}$ on
the top magnet in the Co(3)/Cu(9)/Co(3) trilayer
vs.\ Fermi energy $\mathcal{E}_{\rm F}$ when
$\theta=\phi=0$ and $\Gamma=25$meV.
}
\end{figure}

For comparison, we show in
Fig.~\ref{oddtorque_vs_fermi_cocu} the
total odd torkance (see Eq.~\eqref{eq_sot_total}) in the Co(3)/Cu(9) bilayer
as a function of the Fermi energy $\mathcal{E}_{\rm F}$
for a lifetime broadening of $\Gamma=25$meV when the magnetization
points in $z$ direction.
In this case, the odd torkance in the Co(3)/Cu(9) bilayer is
isotropic, i.e., $t_{xx}^{\rm o}=t_{yy}^{\rm o}$.
The odd torkance in Co(3)/Cu(9) is
very similar to the one
in the Co(3)/Cu(9)/Co(3) trilayer.
Thus, the influence of the bottom magnetic layer on the odd torkance
is much smaller than its influence on the even torkance.
This shows that the
odd SOT is dominantly local, i.e., it arises from the SOI in the top FM
and from the interfacial SOI at the top Co/Cu interface (below we will
see that for $\theta=\phi=90^{\circ}$ an additional nonlocal contribution
to the odd torque can be observed).
In contrast, the even SOT
in Co(3)/Cu(9)/Co(3) contains an important
nonlocal contribution mediated by spin-currents that couple the
bottom FM and the top FM.
In the pure Co(3)/Cu(9) bilayer ($\mathcal{E}_{\rm F}=0$) the odd torkance
amounts to $t_{xx}^{\rm o}=0.33ea_{0}$. Interestingly, this value is larger
by roughly a factor of four than the odd torkance for a monolayer of Co on Cu(111)
obtained from a similar approach
with $\Gamma=$25meV (see Fig.~8(b) in Ref.~\cite{PhysRevB.93.224420}).
We attribute this difference to the
different interfaces (in the present work
the interface normal direction corresponds to the (001) direction of Co and Cu,
while in Ref.~\cite{PhysRevB.93.224420} the interface normal direction
corresponds to the (111) direction of Co and Cu)
and to the
different thicknesses of the Co layer (1 layer of Co in
Ref.~\cite{PhysRevB.93.224420} compared
to three layers of Co in the present study).
\begin{figure}
\includegraphics[width=\linewidth]{cocu_oddtorque_vs_fermi_compo1and5_fermirel.eps}
\caption{\label{oddtorque_vs_fermi_cocu}
Odd torkance $t^{\rm o}_{ij}$ in
the Co(3)/Cu(9) bilayer
vs.\ Fermi energy $\mathcal{E}_{\rm F}$
when $\theta=\phi=0$ and $\Gamma=25$meV.
}
\end{figure}

In Fig.~\ref{oddtorque_toplayer_vs_sigma_cocuco} we show the
odd torkance of the top FM (see Eq.~\eqref{eq_sot_toplayer})
in the Co(3)/Cu(9)/Co(3) trilayer
as a function of the lifetime broadening $\Gamma$.
At large broadening of $\Gamma=$100meV
the component $t_{xx}^{\rm top,o}$ is smaller
than $t_{yy}^{\rm top,o}$ by 3\%.
At medium broadening of $\Gamma=$10meV
the component $t_{xx}^{\rm top,o}$ is larger
than $t_{yy}^{\rm top,o}$ by 2.7\%.
At small broadening of $\Gamma=$1meV
the component $t_{xx}^{\rm top,o}$ is larger
than $t_{yy}^{\rm o}$ by a factor of 3.5.
Thus, like in the case of the even SOT,
we find that the anisotropy of the torkance
with respect to the in-plane rotation of the
applied electric field reduces with
increasing lifetime broadening $\Gamma$.
However, the reduction of this anisotropy
with increasing $\Gamma$ is much more abrupt
than in the case of the even SOT and cannot
be explained by the reduction of the
spin-diffusion length. We attribute the strong
anisotropy of the odd SOT at small $\Gamma$
to the anisotropy of the Fermi surface,
which is induced by the bottom FM.
For medium and large values of $\Gamma$
the odd SOT is less sensitive to this Fermi surface
anisotropy than for small $\Gamma$.

\begin{figure}
\includegraphics[width=\linewidth]{oddtorque_cocucotrilayer_vs_sigma.eps}
\caption{\label{oddtorque_toplayer_vs_sigma_cocuco}
Odd torkance $t^{\rm top,o}_{ij}$ on
the top magnet in the Co(3)/Cu(9)/Co(3) trilayer
vs.\ lifetime broadening $\Gamma$
when $\theta=\phi=0$.
}
\end{figure}

For comparison, we show in
Fig.~\ref{oddtorque_vs_sigma_cocu} the
total odd torkance (see Eq.~\eqref{eq_sot_total}) in the Co(3)/Cu(9) bilayer
as a function of the lifetime broadening $\Gamma$.
At large broadening $\Gamma=$100meV
$t^{\rm o}_{xx}$ in the Co(3)/Cu(9) bilayer
is larger than
$t^{\rm top,o}_{xx}$ in the Co(3)/Cu(9)/Co(3) trilayer
by 1.7\% while at small
broadening $\Gamma=$1meV
$t^{\rm top,o}_{xx}$ in the Co(3)/Cu(9)/Co(3) trilayer
is larger than
$t^{\rm o}_{xx}$ in the Co(3)/Cu(9) bilayer
by 24\%.
In contrast,
$t^{\rm top,o}_{yy}$ in the Co(3)/Cu(9)/Co(3) trilayer
is larger than
$t^{\rm o}_{yy}$ in the Co(3)/Cu(9) bilayer
by 1.5\% at large broadening $\Gamma=$100meV,
while at small broadening $\Gamma=$1meV
$t^{\rm o}_{yy}$ in the Co(3)/Cu(9) bilayer
is larger than
$t^{\rm top,o}_{yy}$ in the Co(3)/Cu(9)/Co(3) trilayer
by a factor of 2.8. Thus, the odd torques in the
Co(3)/Cu(9)/Co(3) trilayer are almost identical to those
in the Co(3)/Cu(9) bilayer when the broadening $\Gamma$ is large,
while for small broadening in particular the $yy$ component of
the torkance tensor differs significantly between the bilayer
and the trilayer.

\begin{figure}
\includegraphics[width=\linewidth]{cocu_oddtorque_vs_sigma_compos1and5.eps}
\caption{\label{oddtorque_vs_sigma_cocu}
Odd torkance $t^{\rm o}_{ij}$ in
the Co(3)/Cu(9) bilayer
vs.\ lifetime broadening $\Gamma$
when $\theta=\phi=0$.
}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{oddtorque_vs_gamma_nosoicu.eps}
\caption{\label{oddtorque_vs_gamma_nosoicu}
Odd torkance $t^{\rm top, o}_{ij}$
in the Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ lifetime broadening $\Gamma$ when
$\theta=\phi=0$ and when SOI is switched off on the
Cu atoms.
}
\end{figure}

In order to discuss the contribution of SOI in Cu to the
odd torque we show in Fig.~\ref{oddtorque_vs_gamma_nosoicu}
the torkance $t^{\rm top, o}_{ij}$ as a function of lifetime
broadening $\Gamma$ when SOI is switched off on the Cu atoms.
Compared to Fig.~\ref{oddtorque_toplayer_vs_sigma_cocuco}
the torkance is strongly decreased when SOI is switched off on the Cu atoms.
This is a remarkable difference to the even torques, for which we discussed
above that switching off SOI on Cu has a small influence.

In order to investigate the dependence of the odd torque on
the thickness of the Cu spacer layer, we show the odd torkances
of Co(3)/Cu(3)/Co(3) and Co(3)/Cu(6)/Co(3) in
Fig.~\ref{fig_oddtorque_3Co3Cu3Co_vs_sigma}
and in Fig.~\ref{fig_oddtorque_3Co6Cu3Co_vs_sigma},
respectively.
Like in the case of the even torkance discussed in the
previous subsection, the
odd torkances of Co(3)/Cu(3)/Co(3), Co(3)/Cu(6)/Co(3), and
Co(3)/Cu(9)/Co(3) are similar for sufficiently large $\Gamma$
and differ substantially for small $\Gamma$.

\begin{figure}
\includegraphics[width=\linewidth]{oddtorque_3Co3Cu3Co_vs_sigma.eps}
\caption{\label{fig_oddtorque_3Co3Cu3Co_vs_sigma}
Odd torkance $t^{\rm top, o}_{ij}$
in the Co(3)/Cu(3)/Co(3) magnetic trilayer
vs.\ lifetime broadening $\Gamma$ when
$\theta=\phi=0$.
}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{oddtorque_vs_sigma_3Co6Cu3Co_layers1234.eps}
\caption{\label{fig_oddtorque_3Co6Cu3Co_vs_sigma}
Odd torkance $t^{\rm top, o}_{ij}$
in the Co(3)/Cu(6)/Co(3) magnetic trilayer
vs.\ lifetime broadening $\Gamma$ when
$\theta=\phi=0$.
}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{oddtorque_vs_angle.eps}
\caption{\label{oddtorque_vs_angle}
Odd torkance $t^{\rm top, o}_{\theta x}$ (circles)
and odd torkance $t^{\rm top, o}_{\phi y}$ (squares)
in the Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ angle $\theta$ when $\phi=0$ and $\Gamma=25$meV.
Lines serve as guide to the eye.
}
\end{figure}

In Fig.~\ref{oddtorque_vs_angle} we show the odd torkance
in the Co(3)/Cu(9)/Co(3) trilayer as
a function of the angle $\theta$ when $\phi=0$ and $\Gamma=25$meV.
We plot the $\theta$ and $\phi$ components of the torkance defined in
Eq.~\eqref{eq_theta_component} and
in Eq.~\eqref{eq_phi_component},
respectively.
The torkance $t^{\rm top,o}_{\phi y}$ vanishes at $\theta=90^{\circ}$ in
agreement with the symmetry analysis in Table~\ref{tab_symmetry2}.
However, $t^{\rm top,o}_{\theta x}$ increases strongly
from $t^{\rm top,o}_{\theta x}=-0.44 e a_{0}$ ($\theta=0^{\circ}$)
to $t^{\rm top,o}_{\theta x}=-0.67 e a_{0}$ ($\theta=90^{\circ}$).
As discussed above, a similar increase occurs for the even
torkance $t^{\rm top,e}_{\phi x}$ shown in Fig.~\ref{eventorque_vs_angle}.
One contribution to the increase of $t^{\rm top,o}_{\theta x}$ with
the angle $\theta$ may come from the $z$-polarized spin current present
in trilayers, because it is expected to provide a term proportional to $\sin(\theta)$.
However, experimentally a strong angular dependence of the odd torque is
also found in bilayers such as AlO$_{x}$/Co/Pt
and MgO/CoFeB/Ta~\cite{symmetry_spin_orbit_torques}.
\begin{figure}
\includegraphics[width=\linewidth]{cocucotheta90phi90_oddtorque_vs_fermi_compo1and2_fermirel.eps}
\caption{\label{oddtorque_theta90_phi90}
Odd torkance $t^{\rm top,o}_{ij}$
in the Co(3)/Cu(9)/Co(3) magnetic trilayer
vs.\ Fermi energy $\mathcal{E}_{\rm F}$ when
$\theta=\phi=90^{\circ}$ and $\Gamma=25$meV.
}
\end{figure}

In Fig.~\ref{oddtorque_theta90_phi90} we show the odd torkance as
a function of Fermi energy $\mathcal{E}_{\rm F}$
when $\theta=\phi=90^{\circ}$.
In agreement with the symmetry
analysis in Table~\ref{tab_symmetry2} the odd SOT
points in $z$ direction in this case, and it can be
generated by an electric field in $x$ direction, but also by an
electric field in $y$ direction.
As discussed at the end of Sec.~\ref{sec_symmetry},
we attribute $t^{\rm top,o}_{zx}$ to a spin current
with spin-polarization along $z$ direction flowing from the
bottom Co layer to the top Co layer and assume that this
spin current is generated at the bottom FM interface through
spin-orbit precession.
$t^{\rm top,o}_{zx}$ is small at $\mathcal{E}_{\rm F}=0$ and
therefore also the $z$-polarized spin current is small. The $z$-polarized
spin current therefore provides only a small contribution to the
$\theta$-dependence of  $t^{\rm top,o}_{\theta x}$ shown in
Fig.~\ref{oddtorque_vs_angle}.

Finally, we discuss the odd torque in the Co(3)/Pt(13)/Co(3) trilayer.
In Fig.~\ref{coptco_oddtorque_vs_sigma} we show the odd torkance
as a function of the lifetime broadening $\Gamma$ when $\theta=\phi=0^{\circ}$.
Similar to the Co/Cu/Co trilayers, the difference between $t^{\rm top,o}_{xx}$
and $t^{\rm top,o}_{yy}$ is pronounced only at small $\Gamma$.
Remarkably, for small and medium values of $\Gamma$
the odd torkance in the Co(3)/Pt(13)/Co(3) trilayer is smaller
than the one in the Co/Cu/Co trilayers, showing again that sizable SOTs can
be observed even in trilayers that are composed only of 3$d$ transition metals.
\begin{figure}
\includegraphics[width=\linewidth]{oddtorque_vs_sigma_3Co13Pt3Co_layers1234.eps}
\caption{\label{coptco_oddtorque_vs_sigma}
Odd torkance $t^{\rm top,o}_{ij}$
in the Co(3)/Pt(13)/Co(3) magnetic trilayer
vs.\ broadening $\Gamma$ when
$\theta=\phi=0^{\circ}$.
}
\end{figure}

\section{DMI in C\lowercase{o}/C\lowercase{u}/C\lowercase{o} trilayers}
\label{sec_dmi_in_cocuco}
Since DMI and SOT are related in various
ways~\cite{mothedmisot,sot_dmi_stiles,common_origin_sot_dmi},
we expect the DMI in Co/Cu/Co trilayers to be anisotropic
due to the bottom FM
like
the SOT, the anisotropy of which we discussed above.
We calculate the DMI according to Eq.~\eqref{eq_dmi} for magnetization along $z$ in the
top magnet.
In Table~\ref{table_dmi} we show the layer-resolved DMI coefficients
for the Co/Cu/Co trilayer
and for the Co/Cu bilayer.
The largest contribution to the DMI arises from Co-1, which is adjacent to vacuum.
The contributions from Co-3, which is adjacent to the copper layer, are significantly
smaller and opposite in sign. The contributions from Co-3 are small, because the SOI
in Cu is small.
As expected, the DMI is anisotropic in Co/Cu/Co, because the magnetization
in the bottom FM points in $x$ direction.
The DMI free energy density Eq.~\eqref{eq_free_energy_dmi} of a flat right-handed cycloidal
spin spiral parallel to the $xz$ plane
with wave vector $q$ in $x$ direction is $D_{yx}q$ while the
DMI free energy density of a flat right-handed cycloidal spin spiral
parallel to the $yz$ plane
with wave vector $q$
in $y$ direction is $-D_{xy}q$. These two energy densities differ by 7\%.

\begin{threeparttable}[t]
\caption{\label{table_dmi}
Layer-resolved DMI-coefficients for the
Co/Cu/Co trilayer and for the Co/Cu bilayer.
The sums of the contributions from layers
Co-1, Co-2, Co-3 and Cu-1 are listed in
the line $D_{ij}^{\rm top}$ (see Eq.~\eqref{eq_dmi_toplayer}).
For Co/Cu only $D_{xy}$
is shown, because $D_{yx}=-D_{xy}$ due to
symmetry in this case. $D_{yx}\ne -D_{xy}$ for Co/Cu/Co,
because the bottom magnet reduces the symmetry.
The DMI coefficients are specified in units of
meV\AA\, per unit cell (uc).
}
\begin{ruledtabular}
\begin{tabular}{cccc}
&\multicolumn{2}{c}{Co/Cu/Co} &\multicolumn{1}{c}{Co/Cu}\\
\cline{2-3}\cline{4-4}
&$D_{xy}$ [meV\AA/uc]  &$D_{yx}$ [meV\AA/uc]
&$D_{xy}$ [meV\AA/uc]\\
\hline
Co-1 &-3.07 &2.80 &-3.78 \\
Co-2 &-0.878 &1.07  &-0.58\\
Co-3 &1.24 &-0.94  &0.48\\
Cu-1 &-0.017 &0.014  &-0.01\\
$D_{ij}^{\rm top}$ &-2.73 &2.94  &-3.89
\end{tabular}
\end{ruledtabular}
\end{threeparttable}

This anisotropy provides a simple way to tune the DMI. For a
cycloidal spin spiral that propagates in a given fixed direction in the top magnet
one can change the
angle between the spin-spiral wave vector and the bottom FM magnetization
by rotating the latter.
Thereby one can tune the DMI and as a consequence the wavelength of the spin spiral.
While the anisotropy of DMI is only 7\% in Co/Cu/Co according to our calculations,
we expect that larger anisotropies can be realized by optimizing the composition of
the trilayer with the goal of maximizing this effect.
In systems with anisotropic DMI skyrmions
are of elliptic shape instead of circular shape.
By rotating the magnetization in the bottom layer one may
rotate these ellipsoids and thereby excite the skyrmions.

\section{Summary}
\label{sec_summary}
We find that SOT and DMI in Co/Cu bilayers and Co/Cu/Co trilayers are larger
than what one might expect from the small SOI of Cu.
A small torque per current ratio in Co/Cu bilayers is therefore the result of
the high electrical conductivity of Cu and not the consequence of a
small SOT torkance.
This is consistent with recent experiments showing SOT-switching in trilayers
that contain only 3$d$ transition-metal elements.
Co/Cu/Co can serve as a model system to study SOTs in
magnetic trilayers that are composed of 3$d$ transition metals only.
The SOT is anisotropic in Co/Cu/Co when the
magnetization of the bottom magnet is in-plane and the magnetization of
the top magnet is out-of-plane, i.e., the SOT depends on the in-plane
direction of the applied electric current.
We find the anisotropy of the even torque
to be particularly large and attribute it to
spin currents that are generated at the
bottom magnet.
The even torque is significantly enhanced
when the applied electric field is aligned with the in-plane magnetization of the
bottom magnet. In contrast, when the electric field is applied perpendicularly to the
in-plane magnetization of the bottom layer, the even torque in Co/Cu/Co trilayers
is found to be similar to the even torque in Co/Cu bilayers.
When the magnetizations of both the bottom and the top magnet are in-plane, but
perpendicular to each other, we observe a nonlocal SOT that is mediated by
a spin current with spin polarization along the out-of-plane direction.
Additionally, we find that DMI is anisotropic in Co/Cu/Co trilayers, i.e.,
we predict that the width of domain walls and the spin-spiral wave number of spin-spirals
depend on their in-plane orientation.
Therefore, we expect that the rotation of the magnetization direction of the bottom magnet
in magnetic trilayers similar to Co/Cu/Co provides an easy way to tune the DMI.

\section*{Acknowledgments}
We gratefully acknowledge computing time on the supercomputers
of J\"ulich Supercomputing Center and RWTH Aachen University
as well as funding by
Deutsche Forschungsgemeinschaft (MO 1731/5-1).

redpajama_set_name: RedPajamaArXiv


text:
Clavulariidae es una familia de corales marinos que pertenecen al suborden Stolonifera, dentro de la clase Anthozoa.

Enmarcados comúnmente entre los corales blandos, ya que carecen de esqueleto, como los corales duros del orden Scleractinia, por lo que no son corales hermatípicos.

Son octocorales que comparten la característica de tener pólipos cortos individuales y retráctiles, conectados por estolones. La conexión de los estolones de los corales forma una red de pólipos, a través de una materia sólida, que, usualmente es incrustante, y ocasionalmente, forma elementos verticales o en pliegues.

La mayoría de especies incluidas en esta familia tienen pobremente desarrollado el mesenterio y, por tanto, no están adaptadas a capturar presas. Su alimentación es fotosintética y mediante absorción de materia orgánica disuelta en el agua.

Géneros
Clavulariidae comprende los siguientes géneros:
 Azoriella. (Lopez Gonzalez & Gili, 2001)
 Bathytelesto. Bayer, 1981
 Carijoa. Mueller, 1867
 Clavularia. Blainville, 1830
 Cryptophyton. Williams, 2000
 Cyathopodium. Verrill, 1868
 Denhartogia. Ocaña & van Ofwegen, 2003
 Inconstantia. McFadden & van Ofwegen, 2012
 Incrustatus. van Ofwegen, Häussermann & Försterra, 2007
 Knopia. Alderslade & McFadden, 2007
 Moolabalia. Alderslade, 2001
 Paratelesto. Utinomi, 1958
 Phenganax. Alderslade & McFadden, 2011
 Pseudocladochonus. Versluys, 1907
 Rhodelinda. Bayer, 1981
 Rolandia. de Lacaze-Duthiers, 1900
 Sarcodictyon. Forbes (in Johnston), 1847
 Schizophytum. Studer, 1891
 Scleranthelia. Studer, 1878
 Scyphopodium. Bayer, 1981
 Stereosoma. Hickson, 1930
 Stereotelesto. Bayer, 1981
 Telesto. Lamouroux, 1812
 Telestula. Madsen, 1944
 Tesseranthelia. Bayer, 1981

Referencias

Enlaces externos

Clavulariidae
Stolonifera

redpajama_set_name: RedPajamaWikipedia


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Spring 2019 Johnson String Instrument will visit Tone Academy on Thursday, May 2nd to deliver pre-paid rental instruments and accessories and pick up returns and exchanges upon request. If you would like to have items delivered, please place your order by noon on Wednesday, May 1st, either through our website or by calling 800-359-9351. Orders will be delivered during the day on May 2nd, and will be available for pickup from your teacher in the afternoon. Please note: Orders will be charged as they are received. Exchanges and returns: If you are sending an instrument back to Johnson String for any reason, please be sure to attach your name, phone number, and pickup instructions to the case to ensure proper account crediting. Please leave your instrument with Tone Academy by delivery on the May 2nd in the early afternoon hours. Upgrades: The Johnson String rental program offers a range of rental outfits to accommodate players of all levels. If your child would benefit from a higher-quality rental instrument or bow, consider ordering an Advanced- or Master-level instrument for delivery. Please consult your teacher or call our office at 800-359-9351 for recommendations. Reopening accounts: If you have rented from Johnson String in the past, please let us know when you complete your rental contract, whether by phone or online. Accounts reopened within two years of the date of close will retain and continue to build upon the equity you have already accrued. Equity: All Johnson String customers accrue equity while renting. 100% of the first year's rental fees and 20% of each subsequent years' may be applied as credit toward the purchase of an instrument of like kind from our rental or sales department. Equity remains valid for up to two years from the close of a rental account. This rental service is for students enrolled at Tone Academy of Music. If you are not enrolled, please select the shipping option online or call 800-359-9351 with questions.


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Amazing AccOMplishents
By APRIL WANG, sophomore
Eight months ago, students from JP Stevens embarked on a journey full of hard work, creativity, and competition. It was only after success at multiple qualifying competitions and a grueling twenty-four hour bus ride that they were able to arrive at Iowa State University and represent New Jersey in the 2016 Odyssey of the Mind World Finals from May 25 to May 28.
As a worldwide educational program, Odyssey of the Mind allows students from the age of 5 to 22 to solve a variety of problems, ranging from performance to engineering. Students develop creative problem-solving and team building skills through working together and competing at regional, state, and world levels.
The JP Stevens Odyssey of the Mind has continuously improved throughout the years, with students excelling in different problems. As Mr. Parekkadan, coordinator of JP Stevens Odyssey of the Mind states, "When I first started JP OM, it was really focused on Problem Four, the structure problem. At this stage in the game, I feel as though I have teams in respective to whether it is a structure problem or not that are very diverse in terms of their skills and talents, their dramatic quality of performance elements. I am dealing with very sophisticated students who bring a lot to the table no matter which problem."
The opening ceremony began on May 25 with thousands of flashing glow sticks waving around, laser lights beaming from one end of the stadium to the other, and fire sparklers lighting up at the center of the arena. Introductions of each problem were given, short motivational speeches from the head judges and co-founders of Odyssey of the Mind were presented, and the competition and fun had officially begun.
The seven teams from JP Stevens competed in four different events or "long-term problems," each having different tasks required to be completed. Problem one, "No-Cycle Recycle" required teams to build a vehicle and transport recycled items to different areas throughout their performance. Problem two, "Something Fishy" required teams to simulate different styles of fishing through a technical solution. Problem four, "Stack Attack" challenged teams to build a 15-gram balsa structure and test it to see the maximum amount it could hold. Problem five, "Furs, Fins, Feathers & Friends" called for teams to present a humorous performance involving three animal characters singing and dancing. The scores obtained from these long-term problems were then combined with scores earned from the spontaneous portion of the competition, where teams were required to solve a verbal or hands-on problem presented to them on the spot. The total score contributed to the team's ranking among others in each competition.
On May 28, all teams gathered together once again for the closing ceremony to hear the highly anticipated results. All JP Stevens teams placed in the top twenty. Team B of Problem One received eighteenth place, team A of Problem Two twentieth place, team B of Problem Two tenth place, team A of Problem Four second place, team B of Problem Four fifth place, team C of Problem Four eleventh place, and team A of Problem Five twelfth place. With World Finals almost over, teams headed to the after parties, making their last memories before returning. As senior and executive board member Yixin Jiao explains, "My last World Finals was exceptionally memorable. OM has really changed my time in high school; it has brought me closer to other people and has helped me to develop problem-solving skills while having fun."
Image Source: http://www.coloradoodyssey.org/preparing-for-tournament/world-finals/
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redpajama_set_name: RedPajamaCommonCrawl


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by Vance Martin | Nov 21, 2010
It's been a busy year here at WILD….a lot to recap. AND, we're moving into 20111 with gusto! Below, I've copied our annual letter that went out to WILD donors this past week. I encourage you to also read our 2010 annual report, which covers our FY 2009 audit. Another good read is our recently printed brochure on Nature Needs Half™.
This year, we have a really special year-end giving opportunity. Every donation will be matched, dollar for dollar, up to $25,000 by a generous donor. WILD is a smart choice for giving and is rated by Charity Navigator as a 4-star charity – the highest rating – for our organizational efficiency and their organizational capacity. Give the gift of wilderness today >
Dear WILD friends,
Thank you for making wild nature a top priority. We all love wildlife and the solitude and beauty of wild landscapes, but protecting wild nature is about so much more than these experiences. We evolved within wild nature, formed our bodies, developed our minds and felt our spirit. She provides us – and all life – with irreplaceable "life support" services such as clear air and water, medicines and foods, climate control and more. Quite simply, life on Earth is only possible because of wild nature.
The holiday season helps us re-charge two essential human feelings – thankfulness and hope. Here at WILD we are extremely thankful for your continued support, and hope that together we can generate the positive and necessary actions to protect and sustain wild nature and its many gifts.
As I worked with our partners and projects around the world this year, I realized that it is time to "crank up the energy" – to fully acknowledge the current state of our world, create a collective vision and a personal attitude that befits a healthier planet, and to take action to protect more of our planet so that all plants, animals and people on earth can thrive.
At WILD, we've fearlessly embraced a holistic vision that addresses both the symptoms we see around us and the underlying causes. Nature Needs Half™ is a global call to action to protect and interconnect at least half of the planet's land and water. This vision is supported by the best ecosystem science and is achievable when it is applied from the grass roots, region by region.
Over the past year, we've worked tirelessly to expand and apply this vision in our own community and around the world by:
Working in our neighborhood – Boulder County, where WILD's headquarter office is located, is over half protected already – but this living legacy is at risk of being "loved to death". With local partners, we're highlighting Boulder as a Nature Needs Half™ multi-media case study, while working to address the management challenges and protect this natural treasure for all people, for all time.
Reaching out to partners across the globe – We work daily to expand the network of individuals and organizations who adopt the Nature Needs Half™ vision. Partners from Latin America to Asia are developing Nature Needs Half™ programs within their own organizations and in their local areas, building the global movement and working toward the global goal.
Facilitating collaboration – The North American Wilderness Agreement, signed at WILD9 (the 9th World Wilderness Congress, Mexico 2009), encourages and enhances practical collaboration between land management agencies in Mexico, Canada and US. As the facilitator of this international group, WILD works with the three governments on a continental-scale vision for wilderness, encouraging trans-boundary conservation initiatives, collaborative science and training, and creating new types of protected areas (such as marine and private land wilderness designations) and more.
Implementing on-the-ground field projects – Our ongoing field projects such as the Mali Elephant Project (West Africa) and the Zulu Village project (South Africa) foster the relationship between people and nature. These hands-on programs work with community members and decision makers to ensure that both people and nature benefit.
In 2011, we'll continue these ongoing efforts, and expand our work through:
New Programs & Projects – In 2011, we will expand our work in Canada, teaming up with government and conservation partners to further the Nature Needs Half™ vision. Working with The Wilderness Network, In the Tracks of Giants, a new program, will follow traditional elephant migration routes through southern Africa – from Angola to Swaziland – emphasizing interconnected, wild landscapes.
World Wilderness Congress – Following on the huge success of WILD9, we are working with our European colleagues to determine the possibility of WILD10, with Nature Needs Half™ as a key organizing theme.
Wild-Nature and the Arts – WILD was founded on the principle importance of the human connection to wild-nature. We continue this work through innovative communications and the arts. In 2011, we will share international stories of Nature Needs Half™ through numerous online media outlets, fine arts and publications.
As we look to the New Year, we need your support in order for the Nature Needs Half™ vision to have real,positive and lasting impacts on our planet. Please join us – let's move forward with this bold, new vision for nature and people. Your continued support is essential in our work for wilderness, wildlife and people.
This year, we have a very special year-end matching gift. Every contribution, up to $25,000, will be matched dollar for dollar by an angel donor. Please help us reach this important goal — make a contribution today in support of nature world-wide. Your HALF can make a WHOLE lot of difference!
Thank you!! And, let's go! Best wishes for a WILD New Year,
Vance Martin


The Special Issue "Zeolite Catalysis" published in the online journal Catalysts was recently successfully completed. A good number of peer-reviewed publications were published reflecting the broadness of the zeolite syntheses, characterizations and various application fields. This issue brought together recent research of well-known research teams from all over the world.
The term "zeolite" is based on the Greek words for "to boil" and "stone" and it has been known since 250 years ago. At that time, the Swedish mineralogist, A.F. Cronstedt (1722–1765), observed the formation of a large amount of steam when heating the material Stilbite which indicated its porous character and adsorption capacity. At present, over 200 different zeolite frameworks have been identified. In general, zeolites are crystalline aluminosilicates with a defined micropore structure. Within zeolites, a good number of elements can be isomorphously incorporated and much more elements or their oxides can be hosted by zeolites. In addition, zeolites comprise a large variety in size of pore mouths, channels, crossings, etc. leading also to their designation as molecular sieves and use in membrane applications. Nowadays, various hierarchical and composite materials are designed offering further interesting properties, e.g., by the introduction of mesopores or generation of fibers. Zeolites reveal Brønsted and Lewis acidic properties that can be varied in wide limits as well. Thus, they deserve the name "solid acids". Zeolites have an immense importance in diverse industrial applications such as catalysts and adsorbents, for example in the refinery industry, chemical industry, detergent sector or for solar thermal collectors and adsorption refrigeration.
The aim of the Special Issue was directed to new developments and recent progress with respect to zeolite-catalyzed chemical reactions, adsorption applications and membrane uses, as well as improved syntheses strategies and characterization techniques. Xuan Hoan Vu and colleagues  reported on the synthesis of novel ZSM-5 containing hierarchical composites and their use in catalytic cracking of triglyceride-rich biomass to lower olefins. It could be proven that the yield for propene and butenes can be increased using such composites. Aixia Song et al.  studied the adsorption and diffusion properties of zeolite Beta using three xylene isomers. Adsorption isotherms from microporous and mesoporous zeolite Beta were recorded showing the impact of mesopores on adsorption properties. Houbing Zou and co-workers  reported on facile synthesis of yolk/core-shell structured [email protected] composites, catalytic properties were checked in the challenging hydroxylation of phenol. The catalyst characterizations showed a high surface area of 560–700 m2/g and a hierarchical pore structure with mesochannels and micropores. In comparison to well-known TS-1, the synthesized solids reveal enhanced activity at comparable selectivity. The research group of Fabrizio Cavani  contributed to the Special Issue with an article on the use of zeolite catalysts for phenol benzoylation with benzoic acid. The aim of this work was the synthesis of hydroxybenzophenons which are important intermediates in the chemical industry. H-Beta zeolites offer superior performance compared to H-Y solid. The studies were supported by various mechanistic insights. Radostina Dragomirova and Sebastian Wohlrab  extensively summarized the application of zeolite membranes in catalysis. The detailed review is backed by ca. 300 references on zeolite membrane preparation, separation principles as well as basic considerations on membrane reactors. The given classification according to membrane location considers: (i) membranes spatially decoupled from the reaction zone; (ii) packed bed membrane reactors; (iii) catalytic membrane reactors; and (iv) zeolite capsuled catalyst particles. Ceri Hammond and Giulia Tarantino  reported on post-synthesis modifications of TS-1 to suppress undesirable H2O2 decomposition in hydroxylations. Ti site speciation changes were observed by in-situ spectroscopic techniques. Takashi Tatsumi and colleagues  in their contribution described effects of dealumination and desilication of Beta zeolite and the consequences for their catalytic performance in n-hexane cracking to propene. Dealumination was carried out by HNO3 treatment; desilication was obtained by alkali treatment. The propene selectivity at high n-hexane conversions was increased after alkali treatment followed by acid treatment. This is due to: (i) the decrease in number of acidic sites; and (ii) by an increase in number of mesopores which are beneficial to the diffusion of coke precursor compounds. Jing Han et al.  reported on the manufacture of Ga2O3/ZSM-5 hollow fibers for use as efficient dehydrogenation catalysts for n-butane conversion. Light olefin yields could be increased significantly compared to Ga2O3, ZSM-5 fibers and GaO3 supported on ZSM-5. Guozhu Liu and coworkers  in their article showed the catalytic properties of Pt/H-ZSM-5 in the conversion of lignin-based phenols into xylene isomers. The addition of methanol to the reaction mixtures leads to increased xylene yields. The impact of MeOH addition is attributed to the combined action in both the reaction pathways: methylation of m-cresol into xylenols followed by hydrodeoxygenation to form p-/m-xylene, and hydrodeoxygenation of m-cresol into toluene followed by methylation into p-/m-xylene. Alessandra Silva et al.  reported on the synthesis of ZSM-5 zeolites using biomass such as sugar cane bagasse as structure directing agent. MFI crystals with different morphologies were obtained that were different from the pristine zeolite formed in the absence of biomass. The research team of Roger Gläser  contributed to the Special Issue with a report on photocatalytic oxidation of NO over TiO2/ZSM-5 composites. Various composites were synthesized using different TiO2 sources. The highest NO conversion of ca. 40% was obtained with a catalyst from sol–gel synthesis with equal amounts of the two components after calcination at 250 °C.
This short survey proves the potential of zeolites and zeolite-based materials in modern catalysis and related research areas. I have no doubt that further articles on the above mentioned topics will be published in Catalysts soon.


3292 – What's Love Got to Do with It?
Can a museum be animated by the idea of love? This case study will explore how and why one museum–the San Diego Museum of Man (SDMoM)–passionately believes that it can, and with astonishing results. Fasten your seatbelts and enjoy the ride as you hear about SDMoM's remarkable transformation from a musty, dusty and tired museum to an institution that is thriving inside and out. Drawing its inspiration from core anthropological values, Gandhi, and even the 70s cult classic "Harold and Maud," SDMoM strategically embraces the idea of love as a springboard for creating and holding a space for meaningful discourse about topics that really matter. SDMoM's CEO will share how taking the path less traveled has made all the difference.


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Q: Signal "representation" terminology A paper I'm reading now defines invariant signal "representations" as those functions $\Phi$ of signals $x$ in a Hilbert space such that $\Phi(g\cdot x) = \Phi(x)$ where $g\cdot x$ is the action of some group element.
I'm not quite sure what is meant by "representation" in this case. The paper goes on to cite the Fourier transform modulus and the function $\Phi(x) = x(u-a(x))$ for some function $a$ as examples of these representations. What is meant by this term in general? A google search hasn't produced anything concrete.
Thank you.

A: After some more looking, I found that a represenation is any operator $\Phi : L^2(\mathbb{R}^d) \to H$ for some Hilbert space $H$.

redpajama_set_name: RedPajamaStackExchange


Elkins: architects that build | Architect & Builder | Chippenham: Wiltshire We pride ourselves on being 'architects that build'. This gives us a highly attuned knowledge of the practical use and manipulation of materials, using both traditional and cutting-edge building techniques we have built up a back catalogue of over 500 completed projects. The main emphasis of our past work is in bespoke residential property including both new builds and conversions/renovations, however, we have previously worked on projects ranging from office blocks to tree houses, and schools to factories. We pride ourselves on being able to identify the hidden potential of any space. We're small RIBA chartered practice, but with a team of 5 it means you'll work directly with one of our primary architects throughout the process.


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Ethical Editor
News worth knowing.
Christina Anstead Unveils Dramatic Back Tattoo Amid Divorce From Ant Anstead – E! Online
Diane F. Wilson
A month after their split, the English television presenter told his Instagram followers he was participating in a five-week "Breakup Recovery" program.
"I am now on day 30 of this and I have to say it's been a lifeline for me," he raved in October 2020. "If anyone out there needs this, DO IT!"
Christina has also been healing from heartache in her own way. That same month, she revealed just how she was moving on and focusing on the positives in her life.
"When we have a choice to get caught up in alllllll the nonsense or just choose to find peace…I'm choosing the anomaly," she shared. "It's interesting how you can go from being worried / getting anxious about what people are saying about you to just deciding not to read / read into any of it. When you are officially done 'playing the game.'"
"We all have the choice," she went on, "choose the one that makes you feel good. I've always loved the ocean and find it relaxing—To me—this is peace and perfection."
E! Online (US) – Top Stories
Source – www.eonline.com
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Brazil trial finds efficacy of Sinovac vaccine at 50.4 percent
Covid: Many NHS staff 'traumatised' by first wave of virus, study shows
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Donald, a young fisherman, is overwhelmed when he comes across a group of beautiful women dancing on the shore beneath the moonlight. They are selkies — seals that shed their skin once a year to become human for a few hours. The decision he makes that night, to capture one of these selkies and make her his bride, alters the course of his future.
Back home in his close-knit village in Scotland, he has burgeoning feelings for his selkie bride. With her help, he begins to first question, then change, the town that has been entrenched in the past for generations. He must somehow reconcile his small community's hostility towards outsiders with the responsibilities of love, and find the inner strength required to atone for terrible wrongs.
Based on an entrancing and timeless Scottish legend, Sealskin nevertheless speaks to the modern reader in its rendering of man's need for love and his ability to grow.


The establishment of a school that is aimed at raising a new generation of youths that are sound academically and morally who will serve as fulcrum of the nation development and growth. Even when there are a quite number of schools in the country, the quality and standard of those of schools have remained a huge concerns, a motivating factor for the proposition of this school.
Choice Academy International is an idea born out of the zeal to produce a new generation of young and vibrant minds that are driven with both academic excellence and standard etiquette and moral.
Schools are meant to liberate the minds of students, expand their thinkings and teach them how to be organised and useful to the society. This, we are very much aware of and willing to explore to beat majority of competitions in the education business.
To gain effective communication skills and proper construction of English words, Choice Academy International will organise weekly quiz and debate sessions for her students. The session will provide factual assessment of the students in full glare of academic staffs who will coordinate it, and ensure they make needed corrections or recommendations.
Choice Academy International as an outlined services to be offered students which quite varies from competitors, top of which is its unique quality teaching methods. We have taken note of the ignored reality by other competitors, that different students learn in different ways, and as such, varying teaching applications must be applied to achieve results. So, we will be exploring different teaching techniques, such as: board displays and other visual aids, to make our teachings better effective.
Teaching has been around as long as students. And as long as that fact remains, the need for schools to help trains students to excel will always be demanded by parents/guardians.
However, while the need for tutoring exists, the bulk of tutoring taking place in the market is very limited and unorganized. What will set Choice Academy International apart is its experience and the attention paid to individual needs of students.
Choice Academy International has identified several target market segments that will be pursued. Since the school will only be involved with Elementary, Middle and High School students, parents/guardians of the targeted market segments will be sorted and convinced on the need to enrol them in the school. This, will be achieved by the school's academic, leadership, spiritual and moral training settings, which will be vigorously pursued by the specifics of subjects to be offered, quality academic staffs and extracurricular activities.
The market segments will be two subgroups: Middle School Students (i.e. JSS1-3) and High School Students (i.e. SSS1-2). Choice Academy International will also provide afternoon lesson sessions and holidays/summer tutoring services to enrich students/clients academic standings.