# Dipolar coupled core-shell perpendicular shape anisotropy MTJ with enhanced
write speed and reduced cross-talk

N. Caçoilo  0000-0001-8847-2034 L. D. Buda-Prejbeanu  0000-0002-6105-151X B.
Dieny  0000-0002-0575-5301 O. Fruchart  0000-0001-7717-5229 I. L. Prejbeanu
0000-0001-6577-032X Univ. Grenoble Alpes, CEA, CNRS, Grenoble-INP, SPINTEC,
38000 Grenoble, France

###### Abstract

The concept of Perpendicular Shape-Anisotropy Spin-Transfer-Torque Magnetic
Random-Access Memory tackles the downsize scalability limit of conventional
ultrathin magnetic tunnel junctions (MTJ) below
sub-$20\text{\,}\mathrm{n}\mathrm{m}$ technological nodes. This concept uses a
thicker storage layer with a vertical aspect ratio, enhancing the thermal
stability factor $\Delta$ thanks to the favorable contribution of the shape
anisotropy. However, the increased aspect ratio comes with an increase in
switching time under applied voltage and the cross-over to non-uniform
reversal mechanism at higher aspect ratio, limiting the gain in scalability.
Additionally, the larger volume of the magnetic cell significantly increases
the stray field acting on the neighboring devices compared to thin MTJs. In
this work, we propose the use of a dipolar-coupled core-shell system as a
storage layer. This improves both bottlenecks, as predicted by micromagnetic
simulations for magnetisation reversal, and a macrospin model to estimate the
stray field in a dense array.

Micromagnetism, Magnetic Tunnel Junction, Shape Anisotropy, Stray Field, Spin-
Transfer Torque, Core-Shell, Dipolar Coupling

##  I Introduction

Spin-Transfer-Torque Magnetic Random-Access Memory (STT-MRAM) is one of the
most promising emerging nonvolatile memory technologies. It offers
nonvolatility with very large write endurance, high speed, moderate write
power, and zero off consumption [1]. The retention time is related to the
stability factor $\Delta$ of the device, defined as the ratio between the
energy barrier ($E_{\mathrm{B}}$) of the storage layer and the thermal energy
($\textrm{k}_{\mathrm{B}}\mathrm{T}$, where $\textrm{k}_{\mathrm{B}}$ is
Boltzmann constant and $T$ the operating temperature). The energy barrier for
usual perpendicular magnetic tunnel junctions, made of a stack of ultra-thin
layers with perpendicular magnetisation (p-MTJ), can be expressed as, assuming
coherent reversal:

$E_{\mathrm{B}}=AL\left[\frac{k_{\mathrm{s}}}{L}+\frac{1}{2}\mu_{0}M_{\mathrm{s}}^{2}\left(N_{xx}-N_{zz}\right)+K_{\mathrm{u}}\right]\;,$
(1)

where $A$ is the device area of the magnetic storage layer, $L$ its height,
$k_{s}$ its surface anisotropy per unit area, $\mu_{0}$ the vacuum
permeability, $M_{\mathrm{s}}$ the spontaneous magnetisation, $N_{ii}$ its
demagnetising factors along the plane ($xx$) and out-of-plane ($zz$), and
$K_{\mathrm{u}}$ the bulk uniaxial anisotropy per unit volume. The latter is
usually small in conventional p-MTJs, leaving a competition between surface
and shape anisotropy. For conventional p-MTJ, the lateral dimension of the
storage layer is much larger than its height, resulting in a negative and
large contribution of the shape anisotropy. This lowers the stability of the
out-of-plane state, which is only promoted by surface anisotropy. When the
p-MTJ lateral dimension is reduced, the leading variation in Eq. 1 comes
through the decrease of $A$, which decreases the stability. In practice, this
limits the scalability of the p-MTJ to diameters of around
$20\text{\,}\mathrm{n}\mathrm{m}$ [2]. A proposal to pushback this limit is to
provide an additional source of total anisotropy energy and thus energy
barrier, via the use of a thicker storage layer. Inspecting Eq.(1), we see
that this works out via two aspects: (1) Decrease, in absolute value, the
negative and thus destabilising impact of the in-plane shape effect per unit
volume (the right part, in brackets), or even make it positive for $L/D>0.89$
(for a disk shape, with $D$ the cell diameter and thus $A=\pi D^{2}/4$) (2)
Increase the volume prefactor $AL$, which has a beneficial impact to enhance
the energy barrier in the case of perpendicular shape anisotropy, i.e. for
$L/D>0.89$. This concept has been proven to be effective to extend the range
of stability down to sub-$10\text{\,}\mathrm{n}\mathrm{m}$ diameter [3, 4].
However, this comes at the expense of the need for a larger writing voltage
for STT switching, and longer switching time, both being a handicap for
applicability.

In this manuscript, we propose an alternative approach to address the writing
drawbacks associated with the thick PSA pillars required at small diameter,
introducing a tubular magnetic shell around the storage layer. Making use of
the strong dipolar coupling existing in this system thanks to its high
vertical aspect ratio, it is possible to promote a core-shell synthetic
ferrimagnetic state, with antiparallel storage layer (from now on described as
magnetic core) and shell. The gain introduced by this design is twofold.
First, antiparallel coupling enhances stability. This allows a decrease of the
total height of the core-shell system to reach a given stability, reducing the
volume to switch and delaying the onset of the domain-wall-based reversal
mechanism, thus leading to a lower voltage and faster reversal. Second, the
stray field emanating from the composite cell is sharply reduced, enabling a
denser array than would be possible for single-pillar PSA MTJs. In the
following we first review and refine the evaluation of cell stability (section
II), we then present the core-shell concept and the resulting gain in writing
(section III), and finally describe the additional gain via the reduciton of
stray field and thus cross-talk (section IV).

##  II Capping in scalability of a single PSA storage layer

To understand the increased stability quantitatively, it is necessary to
calculate the energy barrier of the system with a focus on the magnetostatic
energy (related to the shape anisotropy). To calculate this term, we follow
the approach used in [5, 6], where a Fourier-space formalism based on the
shape amplitude $\mathcal{D}(\mathbf{k})$ (in the k-space) of the object is
used to compute the demagnetising factors. For a cylinder, the shape amplitude
is given by (in cylindrical coordinates):

$\mathcal{D}(k_{\perp},k_{z})=\frac{4\pi
R}{k_{\perp}k_{z}}\mathcal{J}_{1}(k_{\perp}R)\sin\left(\frac{L}{2}k_{z}\right)\;,$
(2)

with $k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$ related to direction perpendicular
to the cylinder axis, $R$ the radius of the magnetic cylinder and
$\mathcal{J}_{1}(x)$ the Bessel function of the first kind. It is then
possible to calculate the demagnetising factor along the direction along the
axis of the cylinder:

$\mathcal{N}_{zz}=\frac{1}{8\pi^{3}V}\int\frac{\mathrm{d}^{3}\textbf{k}}{\textbf{k}^{2}}|\mathcal{D}(\textbf{k})|^{2}k_{z}^{2}.$
(3)

After a series of integrations and transformations, it is possible to achieve
a simpler notation like in [5]

$\mathcal{N}_{zz}^{\mathrm{cylinder}}=1+\frac{4}{3\pi\tau}-F_{0}\left(-\frac{1}{\tau^{2}}\right),$
(4)

where $F_{0}$ is the hyper-geometric function, shown here for convenience:

$F_{0}(x)={{}_{2}F^{1}}\left(-\frac{1}{2},\frac{1}{2},2,x\right).$

Figure 1: Stability factor for different dimensions of the storage layer with
a spontaneous magnetisation of
$1\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$ for a single MgO
interface with a surface anisotropy of
$1.4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$. The dashed line
(diagonal) shows the limit between regions with positive and negative values
of the magnetostatic energy.

The two identical transverse demagnetising coefficients can be calculated as
$(1-\mathcal{N}_{zz}^{\mathrm{cylinder}})/2$, and the magnetostatic anisotropy
energy coefficient scales with $(1-3\mathcal{N}_{zz}^{\mathrm{cylinder}})/2$.
Figure 1 shows a stability diagram depending on the geometry of the storage
layer (height and diameter) for a storage layer with the usual FeCoB material
with spontaneous magnetisation of
$1\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$ and a surface anisotropy
of $1.4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$, a typical value
in MgO$|$FeCoB systems. From it, relevant information can be extracted. This
diagram illustrates the limitation of the usual p-MTJ at small nodes as the
stability is reduced with the diameter, going below the required levels at
around $15\text{\,}\mathrm{n}\mathrm{m}$ diameter (typical for a 10 year
retention time values around 60-80 $\textrm{k}_{\textrm{B}}\textrm{T}$ are
used). As the height is increased there is first a reduction in perpendicular
stability, as the magnetostatic energy is still favouring the in-plane
orientation, but now its influence is enhanced by the larger volume of the
storage layer. This can induce an in-plane alignment of the magnetisation, as
seen for $D>$18\text{\,}\mathrm{n}\mathrm{m}$$. However, if the thickness is
further increased there is a stability increase, coming from the progressive
reduction of the difference in axis versus transverse demegnetising factors or
even its cheage in sign, so that perpendicular magnetisation is restored as
the easy axis. The height necessary to restore perpendicular orientation
reduces with reducing diameter (this orientation is aspect-ratio dependent).
Below $D=$18\text{\,}\mathrm{n}\mathrm{m}$$ the easy axis always remains
perpendicular. For a cylindrical pillar with a reasonable high surface
anisotropy value (as in the example of
$1.4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$) it is possible to
maintain a perpendicular orientation with a large stability down to
sub-$5\text{\,}\mathrm{n}\mathrm{m}$ diameter. Although at first glance an
effective path for high density, altogether with an exceptional tolerance for
high temperature applications (since the thermal dependence of the
magnetisation is less steep than that of the surface anisotropy [7, 8, 9]),
there are still several obstacles that need to be addressed to push this
technology forward.

One of the drawbacks of a high aspect ratio PSA pillar is to achieve fast and
efficient switching with spin transfer torque, as it is largely an interfacial
effect in MTJs [10]. Moreover, there is a capping in stability, due to the
formation of a transient domain-wall along the height of the storage layer
during the magnetisation reversal. As already discussed [3], this capping of
stability can be determined considering a tail-to-tail domain wall formed
along the thickness of the magnetic body, which is dependent on the exchange
stiffness $A_{\mathrm{ex}}$ and the spontaneous magnetisation $M_{\mathrm{s}}$
of the magnetic cylinder:

$\Delta^{\mathrm{DW}}=\frac{\mu_{0}M_{\textrm{s}}^{2}}{2k_{\textrm{B}}T}\frac{\pi
D^{2}}{4}\left(\frac{D}{2}+L_{\mathrm{DW}}+\frac{2L_{\mathrm{DW}}^{2}}{D+2L_{\mathrm{DW}}}\right)\;,$
(5)

where $L_{\mathrm{DW}}$ is the width of the domain wall, scaling with
$\sqrt{\frac{4A_{\textrm{ex}}}{\mu_{0}M_{\textrm{s}}^{2}}}$. As the height at
which the capping sets in is related to the domain wall width, a workaround
would be to reduce the total height by increasing the surface anisotropy, for
example, through an additional MgO interface at the top of the storage layer
(as in common double interface p-MTJ). Experimental efforts have been reported
in this direction [11, 12]. However, this approach provides moderate gain at
lower diameters, as the contribution of the surface anisotropy is less
significant. This is observed in Fig. 2, where the stability capping is shown
for different thick layers (Co, FeCoB and Py), with double FeCoB interfaces,
with different surface anisotropies of
$1.4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$ and
$1.0\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$. On this graph, we
also see that the lower the magnetisation, the larger the height needs to be
to achieve a given stability, as for these diameters the main anisotropy
source is magnetostatic.

Figure 2: Stability factor for different PSA storage layers for a
$5\text{\,}\mathrm{n}\mathrm{m}$ diameter storage layer with FeCoB double
interface ($1.4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$ and
$1.0\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$). Predictions based
on macrospin are shown with dashed-dotted lines, while the capping of
stability related to domain wall nucleation-propagation is shown with the
horizontal dotted lines. The open disks stand for micromagnetic simulations of
minimum energy path.

##  III Core-Shell concept for storage layer

To understand the increase in scalability provided by the magnetic shell, it
is helpfull to first calculate its stability as an isolated element. Thus, we
need to calculate its demagnetising factors, since the shape anisotropy is the
only source of magnetic anisotropy. To do so, we use the shape amplitude of
equation 2, with outer radius $R_{2}$ and inner radius $R_{1}$ [13, 14]:

$\mathcal{D}(k_{\perp},k_{z})=\frac{4\pi}{k_{\perp}k_{z}}\big{[}(R_{2}\mathcal{J}_{1}(k_{\perp}R_{2})-\\\
R_{1}\mathcal{J}_{1}(k_{\perp}R_{1})\big{]}\sin{(\frac{L}{2}k_{z})}\;.$ (6)

Following the same reasoning as before, and performing the transformations
$q=k_{\perp}R_{2}$, $\sigma=R_{1}/R_{2}$, and $\tau=L/(2R_{2})$ we obtain the
integral:

$\mathcal{N}_{zz}^{\textrm{shell}}(\sigma,\tau)=\frac{1}{\tau(1-\sigma^{2})}\int^{+\infty}_{0}\frac{\mathrm{d}q}{q^{2}}(1-e^{-2q\tau})\\\
\big{[}(\mathcal{J}_{1}(q)-\sigma\mathcal{J}_{1}(\sigma q)\big{]}^{2}\;.$ (7)

The latter can be expanded to be written with the demagnetising coefficients
of full cylinders ($\sigma=0$):

$\mathcal{N}_{zz}^{\mathrm{shell}}(\sigma,\tau)=\frac{1}{1-\sigma^{2}}\Bigg{[}\mathcal{N}_{zz}^{\textrm{cylinder}}\left(\tau\right)+\sigma^{2}\mathcal{N}_{zz}^{\textrm{cylinder}}\left(\frac{\tau}{\sigma}\right)-\\\
\frac{\sigma^{2}}{\tau}F_{0}(\sigma^{2})+\frac{2\sigma}{\tau}A_{1,1}^{-1}(2\tau,1,\sigma)\Bigg{]},$
(8)

with:

$A_{1,1}^{-1}(2\tau,1,\sigma)=\int^{+\infty}_{0}q^{-2}e^{2q\tau}\mathcal{J}_{1}(q)\mathcal{J}_{1}(\sigma
q)\mathrm{d}q\;,$ (9)

which can be solved making use of a combination of elliptical integrals (see,
for reference, equation 10 and table 1 of [14]).

Figure 3: $a)$ Demagnetising factors $\mathcal{N}_{xx}$ (filled disks) and
$\mathcal{N}_{zz}$ (open disks) for the magnetic shell as a function of the
aspect-ratio $\tau=L/(2R_{2})$ for different for different values of $\sigma$
and $b)$ Stability factor of the shell as a function of the shell thickness
ratio $\sigma$ and aspect-ratio $\tau^{\textrm{shell}}$ for three different
inner radius $R_{1}$ values for the situation of a Ni shell
$0.489\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$, NiFe shell
$0.756\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$ shell and Co shell
$1.446\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$. Patterned region
showns an in-plane orientation of the magnetisation.

Using equation 8 it is possible to calculate the preferred orientation of the
magnetisation in the magnetic shell, as a function of the aspect ratio $\tau$
and $\sigma$. Figure 3 illustrates this versus aspect ratio, for several
values of $\sigma$. There are two important features to be extracted: The
crossing point between the $\mathcal{N}_{xx}^{\textrm{shell}}$ and
$\mathcal{N}_{zz}^{\textrm{shell}}$ curves highlights the crossover between
in-plane versus perpendicular easy directions. This occurs at smaller aspect-
ratios as the value of $\sigma$ gets larger, as the shell resembles a thin
rolled sheet. In the opposite situation of small $\sigma$, the aspect-ratio
for crossover is similar to that of a full cylinder. Above the crossover, the
difference between the two demagnetising factors keeps increasing as the
aspect ratio increases, liable to provide a larger stability to the magnetic
shell.

This stability, product of the volume of the shell and the difference of its
demagnetising coefficients shown in Fig. 3 ($a$), is displayed in figure 3 b)
versus the height of the shell and $\sigma$ ($R_{1}/R_{2}$), for different
radius of the inner shell (columns) and different materials (rows). Consistent
with Fig. 3 ($a$), perpendicular anisotropy is promoted by a high aspect ratio
and small shell thickness ($R_{2}-R_{1}$). In these diagrams, we clearly see
that in the case of perpendicular anisotropy, the regions with highest
stability are those for rather thick shells (low $\sigma$), which comes from
its large volume and therefore a large energy barrier. Also, as is the case
with standard PSA pillars, the stability scales with $M_{\mathrm{s}}^{2}$.

We now turn to considering both the core and the shell in dipolar interaction
to evaluate how the several geometrical parameters can be best tuned to
enhanced stability. In addition to $R_{1}$, $R_{2}$ and $L$ already described,
we introduce $R_{0}$ as the radius of the core cylinder, with
$R_{0}<R_{1}<R_{2}$.

As described in [14], the magnetostatic coupling energy of the core and the
shell ($E_{\textrm{cs}}$) is given by the interaction between the shape
amplitude of the two objects:

$E_{\textrm{cs}}=\frac{\mu_{0}M_{\textrm{core}}M_{\textrm{shell}}}{8\pi^{3}}\\\
\int\frac{\textrm{d}\mathrm{\textbf{k}}}{k^{2}}D_{\textrm{core}}(\textbf{k})D_{\textrm{shell}}(\textbf{k})(\mathrm{\textbf{m}}_{1}\cdot\textbf{{k}})(\textrm{{m}}_{2}\cdot\textbf{{k}}),$
(10)

where $D_{\textrm{core}}$ and $D_{\textrm{shell}}$ are defined by equations 2
and 6, respectively. Equation 10 can be formally simplified to:

$E_{\textrm{cs}}=\pi\mu_{0}M_{\textrm{core}}M_{\textrm{shell}}R_{0}\\\
\big{[}2\cos\theta_{1}\cos\theta_{2}-\sin\theta_{1}\sin\theta_{2}\cos(\phi_{1}-\phi_{2})\big{]}\\\
\times\int^{+\infty}_{0}\frac{\mathrm{d}k_{\perp}}{{k_{\perp}^{2}}}\bigg{[}R_{2}\mathcal{J}_{1}(k_{\perp}R_{2})\mathcal{J}_{1}(k_{\perp}R_{0})-\\\
R_{1}\mathcal{J}_{1}(k_{\perp}R_{1})\mathcal{J}_{1}(k_{\perp}R_{0})\bigg{]}\left(1-e^{-k_{\perp}L}\right)\;.$
(11)

Compared with the initial approach [14, 15], we expand the modeling to the
case in which the shell is not in contact with the magnetic shell. Making use
of $q_{i}=k_{\perp}R_{i}$, $\sigma_{i}=R_{0}/R_{i}$ and $\tau_{i}=L/2R_{i}$
(where $i$ is related to the inner radius $R_{1}$ and outer radius $R_{2}$)
the integral of Eq. 11 it can be further expanded to

$\mathcal{J}_{cs}=\frac{1}{R_{0}^{2}}\int^{+\infty}_{0}\frac{\mathrm{d}q_{2}}{\sigma_{2}^{2}q_{2}^{2}}\mathcal{J}(q_{2})\mathcal{J}(\sigma_{2}q_{2})\left(1-e^{-2q_{2}\tau_{2}}\right)-\\\
\int^{+\infty}_{0}\frac{\mathrm{d}q_{1}}{\sigma_{1}^{2}q_{1}^{2}}\mathcal{J}(q_{1})\mathcal{J}(\sigma_{1}q_{1})\left(1-e^{-2q_{1}\tau_{1}}\right).$
(12)

Making use of the last results from equations 8 and 9, it is possible to get
the results for the dipolar coupling of core and shell $\mathcal{J}_{cs}$ as

$\mathcal{J}_{cs}R_{0}^{2}=\frac{1}{\sigma_{2}^{2}}\left[\sigma_{2}F_{0}(\sigma_{2}^{2})+A_{1,1}^{-1}(2\tau_{2},1,\sigma_{2})\right]-\\\
\frac{1}{\sigma_{1}^{2}}\left[\sigma_{1}F_{0}(\sigma_{1}^{2})+A_{1,1}^{-1}(2\tau_{1},1,\sigma_{1})\right].$
(13)

In addition, these results can be linked to the demagnetising factors of the
shell and core through equation 8.

Figure 4: (left side) Schematics of the core-shell system where $R_{0}$ is the
inner radius of the magnetic core (red colour) with magnetisation orientation
$\overrightarrow{M}_{\mathrm{core}}$ with angles $\theta_{1},\phi_{1}$;
magnetic shell (blue color) with magnetisation orientation
$\overrightarrow{M}_{\mathrm{shell}}$ with angles $\theta_{2},\phi_{2}$. Both
parts have a common thickness of L. Right side: dipolar coupling magnitude
$\mathcal{J}_{\mathrm{cs}}$ for a fixed value of $\sigma_{2}=0.5$ for
different $R_{0}$, as a function of its thickness L and inner radius $R_{1}$.

In the end, the magnetostatic coupling energy depends on the geometry, and it
is proportional to the magnetisation of both the shell and the core:

$E_{\textrm{cs}}=\mu_{0}\pi
R_{0}^{3}M_{\mathrm{core}}M_{\textrm{shell}}\times\underbrace{\mathcal{J}_{cs}(\tau_{0},\tau_{1},\tau_{2},\sigma_{1},\sigma_{2})}_{\textrm{Geometrical
scaling factor}}\\\
\times(2\cos\theta_{1}\cos\theta_{2}-\sin\theta_{1}\sin\theta_{2}).$ (14)

The dependency of the scaling factor ($\mathcal{J}_{cs}$) versus the system’s
geometry is illustrated in Fig. 4 for different values of core radius $R_{0}$
and keeping constant the ratio $\sigma_{2}=0.5$. Thus, the three plots differ
in core radius, separation between the inner shell and core, as well as the
ratio between the inner and outer radius of the shell
$\sigma_{s}=R_{1}/R_{2}$. It is observed that a stronger geometrical scaling
is achieved at thinner separations and longer aspect-ratios. Besides tuning
$\mathcal{J}_{cs}$, The total magnetostatic coupling energy can be further
increased through an increase in $R_{0}$ , related to the magnetic volume, or
through the increase in the magnetisation of both shell and magnetic core
$M_{\textrm{core}}M_{\textrm{shell}}$.

As a final step, we now compute the total energy of the dipolar-coupled core-
shell magnetic system $E_{\mathrm{tot}}$ as [14]

$E_{\mathrm{tot}}(\theta_{1},\theta_{2},\phi_{1},\phi_{2})=E_{\mathrm{c}}\sin^{2}{\theta_{1}}+E_{\mathrm{s}}\sin^{2}{\theta_{2}}+\\\
E_{\textrm{cs}}(\theta_{1},\theta_{2},\phi_{1},\phi_{2})\;,$ (15)

where $E_{\mathrm{c}}$ is the energy barrier of the isolated magnetic core,
$E_{\mathrm{s}}$ is that of the shell alone, and $E_{\textrm{cs}}$ the
coupling energy. For the sake of simplicity and generality, we write the
energy of the system using the coefficients $\mathcal{A}$, $\mathcal{B}$,
$\mathcal{C}$ and $\mathcal{D}$ for the different angular dependencies:

$E_{\mathrm{tot}}=\mathcal{A}\sin^{2}{\theta_{1}}+\mathcal{B}\sin^{2}{\theta_{2}}-\\\
\mathcal{C}\sin\theta_{1}\sin\theta_{2}\cos(\phi_{1}-\phi_{2})+\mathcal{D}\cos\theta_{1}\cos\theta_{2}\;.$
(16)

$\mathcal{A}$ is the self energy barrier of the magnetic core, $\mathcal{B}$
the self energy barrier of the magnetic shell, $\mathcal{C}$ and $\mathcal{D}$
are the core-shell coupling energies in the parallel state for the transverse
and axial directions, respectively. With the present geometry all four
coefficients are positive and $\mathcal{D}=2C$, see Eq. 11. Thus, the lowest-
energy state is expected when both shell and core are out-of-plane-magnetised
and antiparallel on to one another, and the most energetic state when both
shell and core are in-plane-magnetised and antiparallel to one another. More
generally, for arbitrary $\theta_{1}$ and $\theta_{2}$ the inspection of Eq.
15 shows that the minimum energy path for switching is expected to satisfy
$\phi_{1}=\phi_{2}+\pi$ at all time, which we consider satisfied for now on.

The angular dependence of the energy barrier described by equation 16 is shown
in Fig. 5 $(a)$, for the case of a fixed core radius of
$7\text{\,}\mathrm{n}\mathrm{m}$, a shell inner radius of
$8\text{\,}\mathrm{n}\mathrm{m}$ and an outer radius of
$10\text{\,}\mathrm{n}\mathrm{m}$ for several lengths (6, 8, 10 and
$12\text{\,}\mathrm{n}\mathrm{m}$). The core magnetisation is
$1\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$, the surface anisotropy
is $1.4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$ and the shell
magnetisation is $1.446\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$. The
corresponding coefficients $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$
involved in the dipolar energy are listed in Tab.1. For each vertical aspect
ratio the minimum energy path is computed [16] and displayed in Fig. 5 (b).

Figure 5: (a) Angular dependency of the energy barrier of the magnetic system for different core-shell length with a fixed core radius of $7\text{\,}\mathrm{n}\mathrm{m}$, inner shell radius of $8\text{\,}\mathrm{n}\mathrm{m}$ and outer radius of $10\text{\,}\mathrm{n}\mathrm{m}$, for the saturation magnetisation of $1\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$ for the magnetic core and $1.446\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$ for the magnetic shell. Negative angles can be viewed as standing for the case $\phi_{1}=\phi_{2}+\pi$, while formally only positive angles are defined in spherical coordinates. The trace of the minimum energy path for each geometry is shown with filled white circles. (b) Relative orientation of the minimum energy path for increasing thickness of the core-shell system and (c) 3D snapshots for the height of 8 and $12\text{\,}\mathrm{n}\mathrm{m}$, along the minimum energy path. Table 1: Coefficients of the energy of the coupled core-shell systems shown on Fig. 5 in units of k${}_{\textrm{B}}$T, i.e., for different heights and fixed core-shell radius of $R_{0}=$7\text{\,}\mathrm{nm}$$, $R_{1}=$8\text{\,}\mathrm{nm}$$ and $R_{2}=$10\text{\,}\mathrm{nm}$$. Height (nm) |  $\mathcal{A}$ ($k_{\textrm{B}}T$) |  $\mathcal{B}$ ($k_{\textrm{B}}T$) |  $\mathcal{C}$ ($k_{\textrm{B}}T$)
---|---|---|---
6 | 15 | 17 | 19
8 | 22 | 43 | 25
10 | 30 | 72 | 32
12 | 42 | 100 | 35

In all four cases, we confirm that the ground state is perpendicular
anisotropy in an antiparallel state. The energy barrier increases with the
vertical aspect ratio, as expected from our previous discussion on individual
core, individual shell, and magnetostatic coupling. For the system with height
$6\text{\,}\mathrm{n}\mathrm{m}$ the minimum energy path is a line with slope
$-1$, which means that the two parts rotate simultaneously. We indeed expect
this if the coupling between shell and core is comparable or even larger than
the anisotropy energy of each element taken separately, which is the case
here. To the contrary, in the system with height
$12\text{\,}\mathrm{n}\mathrm{m}$ the coupling is smaller than any of the
energy barriers of the two elements taken independently. This implies that a
sequential reversal allows to minimise the total energy barrier, which indeed
is seen in the energy maps and the energy plot in Fig.5 (b). The heights
$8\text{\,}\mathrm{n}\mathrm{m}$ and $10\text{\,}\mathrm{n}\mathrm{m}$ appear
as cross-over situations, with a combination of sequential and coupled paths.
From a device point of view, a strong coupling situation may imply a
prohibitively-large energy barrier, while the low-coupling one is not suited
either if only the core is connected through the leads for STT switching, as
the coupling is not sufficient to switch the shell once the core has been
switched. Thus, we expect that the optimum situation is to be found in the
cross-over regime, to be identified using simulation.

The discussion of the energetics of a coupled set of macrospins and its
minimum-energy path is valuable to draw the general trends and identify the
suitable range of operating parameters. Now, we turn to micromagnetic
simulations to refine the conclusions and determine the optimal geometrical
and magnetic parameters in the framework of the PSA-MTJ, taking into account
precessional effects, the excitation means (current injected via an MTJ in the
core only) and possible non-macrospin effects. For this purpose, we use a
finite-difference 3D micromagnetic solver with a cubic computational cell of
$1\text{\,}\mathrm{n}\mathrm{m}^{3}$ [10, 17], in which the magnetisation
dynamics is described by the Landau-Lifshitz-Gilbert-Slonzcewski (LLGS)
equation. Here we consider that an injected spin-polarised current interacts
with the magnetic body of the core via a damping-like torque
($\Gamma_{\textrm{STT}}$), while we neglect the effect of the field-like
torque:

$\partial_{t}\textbf{m}=-|\gamma|\mu_{0}(\textbf{m}\times\textrm{H}_{\textrm{eff}})+\alpha(\textbf{m}\times\partial_{t}\textbf{m})+\Gamma_{\textrm{STT}}\;,$
(17)

with $\gamma$ the gyromagnetic ratio, and:

$\Gamma_{\textrm{STT}}=-|\gamma|a_{\parallel}\textrm{V}\textbf{m}\times(\textbf{m}\times\textbf{m}_{\textbf{RL}}),$
(18)

where V is the applied bias voltage, m is the unit vector for magnetisation of
the storage layer at the location considered, m${}_{\textrm{RL}}$ is the unit
vector for magnetisation of the reference layer and $a_{\parallel}$ is the STT
pre-factor. The latter is expected to physically decay along the thickness of
the magnetic body through the interaction with magnetisation [18, 19].
Contrary to the case of ultrathin MTJs, the torque is expected to be largely
nonhomogeneous and predominantly exerted close to the interface with the
tunnel barrier. To reflect this, we apply the full STT torque in the first
layer of cells, whose strength matches the expected torque integrated when
moving away from the interface, with an exponential decay

$\langle
a_{\parallel}\rangle=a_{\parallel}(0)\times\sum_{n=0}^{N-1}\exp{\frac{-n\lambda_{STT}}{\delta_{z}}}\;,$
(19)

in which $\lambda_{\textrm{STT}}$ is the characteristic STT decay length
(taken as $1\text{\,}\mathrm{n}\mathrm{m}$) and $\delta_{z}$ is the cell size
($1\text{\,}\mathrm{n}\mathrm{m}$). Making use of the definition

$a_{\parallel}(0)=\frac{\hbar}{2e}\frac{\eta_{\textrm{STT}}}{\textrm{R}\times\textrm{A}}\frac{1}{M_{\mathrm{s}}\delta_{z}}$
(20)

where $\eta_{\textrm{STT}}$ is the STT efficiency [20], it is possible to
calculate the strength of the equivalent interfacial STT pre-factor. In this
study, we consider that the core and thus the interface with Mg0 is made of a
FeCoB base alloy, with spontaneous magnetisation of
$M_{\mathrm{s}}=$1\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$$,
exchange stiffness of
$A_{\mathrm{ex}}=$15\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$, and a
damping value of $\alpha=0.01$,
$a_{\parallel}=$140\text{\,}\mathrm{m}\mathrm{T}\mathrm{/}\mathrm{V}$$
considering a typical TMR value of 100$\%$ and a R$\times$A product of
$1.5\text{\,}\Omega\cdot\mu\mathrm{m}^{2}$. The voltage is only applied at the
interface of the the magnetic core, so that the dynamics of the magnetic shell
may only result from its dipolar interaction with the core. Surface anisotropy
is considered to only be present at the interface between the core and the
tunnel barrier, with a value of
$1.4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$.

Figure 6: ($a$) Time trace of the mean magnetisation components $\langle
M_{z}\rangle$ for a core-shell system, considering two different damping
parameters for a Co magnetic shell. The top plots shows magnetisation averaged
over the entire composite system, while the bottom plots show the components
of the core and shell independently. ($b$) 3D snapshots of the time evolution
of a core-shell system under an applied voltage of $-1\text{\,}\mathrm{V}$ for
the situation of low and high damping. Colorbar shows the magnitude of the z
component of the average magnetisation in each computational cell.

We considered the switching dynamics for two different material parameters for
the magnetic shell. The first situation makes use of the system presented in
Fig. 5 with a length of $8\text{\,}\mathrm{n}\mathrm{m}$ (chosen such that the
stability determined from the minimum energy path is around
$80\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}$),
$R_{0}=$7\text{\,}\mathrm{n}\mathrm{m}$$,
$R_{1}=$8\text{\,}\mathrm{n}\mathrm{m}$$ and
$R_{2}=$10\text{\,}\mathrm{n}\mathrm{m}$$, where a magnetic shell of Co is
used, with $M_{\mathrm{s}}$ of
$1.446\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$, exchange stiffness
$A_{\mathrm{ex}}$ of $15\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$ and
damping either $\alpha=0.01$ or $\alpha=0.1$. The applied voltage is
$-1\text{\,}\mathrm{V}$. The time trace of the mean perpendicular
magnetisation components of the core, shell, and full system are shown in Fig.
6. The magnetisation switching of the isolated magnetic core (without magnetic
shell) happens at an earlier stage compared with the core coupled with a
magnetic shell. This is expected, since the core alone has a lower stability,
around $22\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}$, and it is also not
stabilised along the initial direction by the stray field of the shell. When
considering the shell, we can see that its reversal is delayed with that of
the core, which makes sense, as current is injected only into the latter.
Besides, the switching of the core is delayed in the case of a shell with
higher damping, which we attribute to an incubation effect, with a lower
efficiency to set in precessional dynamics: part of the injected energy may be
absorbed in the shell, via dipolar coupling. A striking effect is that for
lower damping of the shell, even though both the core and the shell start
their reversal faster, the shell takes longer to relax to the stable final
state. Indeed, if the damping of the shell is large, then the incubation time
for precession is longer, but then there is a faster relaxation to the stable
state. This is consistent with the large oscillations of perpendicular
magnetisation seen in the case of lower damping.

Figure 7: ($a$) Time trace of the mean magnetisation components $\langle
M_{z}\rangle$ for a core-shell system, considering two different damping
parameters for a Ni magnetic shell. The top plots shows magnetisation averaged
over the entire composite system, while the bottom plots show the components
of the core and shell independently. ($b$) 3D snapshots of the time evolution
of a core-shell system under an applied voltage of $-1\text{\,}\mathrm{V}$ for
the situation of low and high damping. Colorbar shows the magnitude of the z
component of the average magnetisation in each computational cell.

We now consider a shell made of a low-magnetisation material, considering Ni
with spontaneous magnetisation of
$489\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$ and exchange stiffness
of $8\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$, again for both low
and high damping situations (Fig. 7). As concerns the switching time of the
core, it is seen that it is similar to the one without a magnetic shell, both
in incubation time and switching dynamics, and it is also barely dependent on
the damping parameter of the shell. This is understandable as the lower stray
field off the shell has a lower stabilising effect than that shown on Fig. 6;
and also it will be less effective in pumping energy out of the initial
oscillations during the incubation time. As concerns the shell, a close
examination of the time scale shows that its reversal starts when the core has
not fully reversed, but rather has an in-plane magnetisation. Besides, its
reversal takes longer than for a high-magnetisation shell (figure 6),
especially for low damping. This may result from the small energy barrier of
the magnetic shell (around $5\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}$), so
that it is mostly and directly affected by the dipolar coupling with the
magnetic core. Figure 7 (b) shows 3D snapshots of the reversal for the high
and low damping situations, illustrating the ringing effect in the latter.

In conclusion, all geometrical and magnetic parameters influence the switching
process in a core-shell process. Unlike the case of simple MRAM cells, either
flat or PSA, higher damping may be beneficial in the shell. In practice, the
parameter space shall probably be explored in more detail to meet the
desirable criteria of any given situation, in terms of stability, switching
time and power consumption.

##  IV Reduced cross-talk making use of the core-shell concept

We examine here another advantage of the core-shell storage layer, which is
the possibility to mitigate the mutual coupling via stray field from
neighboring bits in a high dense array via the optimisation of both geometry
and magnetic properties. To evaluate the gain quantitatively, we compute the
magnetic field emanating from a pillar with the formalism implemented in [21].
Starting from the magnetic potential, it is possible to derive the radial
($H_{r}$) and perpendicular components ($H_{z}$) of the magnetic field,
respectively:

$H_{r}(r,z)=2\pi
R_{0}M_{\mathrm{s}}\int_{0}^{\infty}\mathrm{d}k\big{[}e^{-k|Z-L|}-e^{-k|Z|}\big{]}\\\
\mathcal{J}_{1}(kr)\mathcal{J}_{1}(kR_{0}).$ (21)

$H_{z}(r,z)=2\pi
R_{0}M_{\mathrm{s}}\int_{0}^{\infty}\mathrm{d}k\big{[}\mathrm{sign}(Z-L)e^{-k|Z-L|}\\\
-\mathrm{sign}(Z)e^{-k|Z|}\big{]}\mathcal{J}_{0}(kr)\mathcal{J}_{1}(kR_{0})\;,$
(22)

where $\mathcal{J}_{x}$ are the Bessel functions of the order $x$, $R_{0}$ is
the radius of the magnetic cylinder, $L$ its height and $M_{\mathrm{s}}$ its
spontaneous magnetisation. These equations can be used to calculate both the
demagnetising field (inside the magnetic body $0<Z<L$ and $r<R_{0}$, since
there is a discontinuity for Z = L and Z = 0 and $r<|R_{0}|$) and the stray
field outside (for $Z>L$ and $r>|R_{0}|$).

Figure 8: (a) 2D stray field lines arising from a pillar with high aspect
ratio (thickness of $16.5\text{\,}\mathrm{n}\mathrm{m}$ and
$20\text{\,}\mathrm{n}\mathrm{m}$ diameter, orange-outlined) with up
magnetisation along $\hat{z}$. (b) magnetic field strength at mid thickness of
the magnetic pillar, for 4 different aspect ratios.

hese are displayed in Fig. 8 (a) for a magnetic pillar with
$2R_{0}=$20\text{\,}\mathrm{n}\mathrm{m}$$ diameter and height
$L=$16.5\text{\,}\mathrm{n}\mathrm{m}$$. We consider perpendicular
magnetisation, which in practice would arise from the combination of shape and
surface anisotropy. The magnetic field strength at midheight of the pillar is
shown in Fig. 8 (b) for pillars with different aspect ratios, all assumed to
have perpendicular magnetisation. In the situation of a low aspect ratio (such
as the usual p-MTJ with $20\text{\,}\mathrm{n}\mathrm{m}$ diameter and
$1.4\text{\,}\mathrm{n}\mathrm{m}$ height), we can observe a thin-film-like
demagnetising field for $r<|R_{0}|$, with a sharp decay outside the pillar.
For PSA-like pillars the demagnetising field is reduced, but the stray field
is greatly enhanced due to the larger volume involved and hence the magnetic
moment. As in two dimensions, the extension of stray field is short-ranged,
typically of the order of the system thickness or so [22], one may evaluate
its impact based on a finite and small array, which we do in the following.

Figure 9: Volume-averaged stray field at the central pillar in a $5\times 5$
array for (top) PSA-MTJs with a diameter of $20\text{\,}\mathrm{n}\mathrm{m}$
and height $16.5\text{\,}\mathrm{n}\mathrm{m}$ and (bottom) p-MTJs with a
diameter of $20\text{\,}\mathrm{n}\mathrm{m}$ and thickness
$1.4\text{\,}\mathrm{n}\mathrm{m}$. The contribution from the first shell of
neighbours is shown with black open disks. The contribution of the second
shell of neighbours is shown with red open disk. The sum of both contributions
is shown with open blue disks. The dependency of the thermal stability as a
function of the pitch for both PSA-MTJ and p-MTJ is shown as an inset.

To evaluate the strength of stray field in a dense array of pillars with
perpendicular magnetisation, we consider a 5x5 pillar array, as depicted in
the inset of the Fig. 9. We define the pitch as the distance between the
center of two adjacent pillars, and thus, when in contact, it equals the
pillar diameter. We do not consider the stray field arising from a possible
reference layer below the pillar, as this can be engineered to be negligible
[23].

Figure 10: 2D stray field lines of (a) two magnetic cores with a pitch of
$30\text{\,}\mathrm{n}\mathrm{m}$. Each magnetic core has a diameter of
$14\text{\,}\mathrm{n}\mathrm{m}$ and a height of
$8\text{\,}\mathrm{n}\mathrm{m}$ and (b) two core-shell systems with a pitch
of $30\text{\,}\mathrm{n}\mathrm{m}$. Each system is composed of a magnetic
core with a radius of $7\text{\,}\mathrm{n}\mathrm{m}$, inner radius
$8\text{\,}\mathrm{n}\mathrm{m}$, an outer radius
$10\text{\,}\mathrm{n}\mathrm{m}$, and a height of
$8\text{\,}\mathrm{n}\mathrm{m}$. In both situations, the system is shown with
a filled gray color, and the magnetisation direction of the core is shown in
black, pointing up. For the core-shell situation, the magnetisation of the
shell is white, pointing down. In both cases, a central element is shown with
an orange colour.

Figure 9 shows the stray field in an array as a function of its pitch, for
pillars of high vertical aspect-ratio ($20\text{\,}\mathrm{n}\mathrm{m}$
diameter and $16.5\text{\,}\mathrm{n}\mathrm{m}$ height, upper panel), and for
cells with a flat aspect ratio ($20\text{\,}\mathrm{n}\mathrm{m}$ diameter and
$1.4\text{\,}\mathrm{n}\mathrm{m}$ height, lower panel), both with spontaneous
magnetisation of $1\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$. These
situations correspond, respectively, to PSA-MTJs and p-MTJs. Both panels show
the contribution of the first shell, the second shell, and the total. For
p-MTJs, the stray field is significant when pillars are in contact, but
becomes negligible for pitch $50\text{\,}\mathrm{n}\mathrm{m}$ or greater, as
is standard for usual integration processes [24]. The impact of the second
shell of neighbors is insignificant in all cases, so considering the impact of
the first nearest neighbors is enough [25]. For PSA MTJs, the stray field from
the first shell is stronger, and that of the second is non-negligible. The
next shells may be safely ignored thanks to the $1/r^{2}$ decay of the stray
field emanating from a complete shell. The resulting total stray field is very
significant and is expected to affect the stability factor of the device. It
could be increased (reduced) if the average stray field points in the same
(opposite) direction as the pillar magnetisation [26]:

$\Delta_{\mathrm{k}_{\mathrm{B}}\mathrm{T}}^{\textrm{stray}}=\Delta_{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\left(1\pm\frac{H_{z}^{\textrm{stray}}}{H_{\textrm{K}}^{\textrm{eff}}}\right)\;,$
(23)

where $H_{\textrm{K}}$ is the effective anisotropy field of the magnetic
element and $H_{z}^{\textrm{stray}}$ the average stray field felt by it. Both
increase and decrease of stability are detrimental for device operation, the
former related to the write operation, the latter related to the thermal
stability [25]. The dependence of the stability with the pitch of the array is
shown in the inset of Fig. 9 (bottom panel). In the case of an array of p-MTJs
it is seen that a pitch around $30\text{\,}\mathrm{n}\mathrm{m}$ is sufficient
to avoid a significant variation in stability. This is related to the fact
that in 2D systems the range of dipolar interactions scales with the height of
the system. To the contrary, for PSA-MTJ, it would be necessary to drastically
increase the pitch to $100\text{\,}\mathrm{n}\mathrm{m}$ or more to reach the
same level of crosstalk reduction. The stray field would even be enough to
flip magnetisation of the central pillar smaller than
$50\text{\,}\mathrm{n}\mathrm{m}$.

Figure 11: Volume-averaged stray field at the central pillar in an array of
PSA-MTJs with a diameter of $20\text{\,}\mathrm{n}\mathrm{m}$ and height
$16.5\text{\,}\mathrm{n}\mathrm{m}$ as a function of the pitch for situation
of unshelled pillar (open black circles) and shelled pillars (colored open
circles) for an inner radius of $8\text{\,}\mathrm{n}\mathrm{m}$ and different
outer radius, from compensated magnetic volume (red open circles) to a
$0.5\text{\,}\mathrm{n}\mathrm{m}$ smaller outer radius. The system is shown
in the inset where the neighbouring core-shell have the opposite orientation
of that of the central core.

We expect that the core-shell design can also contributs to a mitigation of
the stray field from the neighboring dots, thanks to the antiparallel and
mostly compensated core and shell magnetic moments. This is illustrated in
Fig. 10 (b), considering the effect of two neighboring cells with center-to-
center distance of $30\text{\,}\mathrm{n}\mathrm{m}$, each with a CoFeB core
with $14\text{\,}\mathrm{n}\mathrm{m}$ diameter and
$8\text{\,}\mathrm{n}\mathrm{m}$ height. The top panel shows the case of the
simple pillar, while in the bottom panel these are surrounded by a shell of Co
with an outer radius of $10\text{\,}\mathrm{n}\mathrm{m}$ The field lines are
indeed drastically different in the core-shell system, characteristic of a
quadrupolar arrangement, as the core and shell moments are nearly balanced, in
contrast with the situation with no magnetic shell, characteristic of a
dipolar system.

Figure 11 quantitatively shows the volume-averaged dipolar field felt by the
core of the central cell in a $5\times 5$ array, for the case of a core radius
of $7\text{\,}\mathrm{n}\mathrm{m}$ with spontaneous magnetisation of
$1\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$, an inner shell radius of
$8\text{\,}\mathrm{n}\mathrm{m}$ and a varying outer shell radius from the
situation where there is compensation of magnetic volume
($M_{\mathrm{c}}V_{\mathrm{c}}=M_{\mathrm{s}}V_{\mathrm{s}}$, where
$V_{\mathrm{c},\mathrm{s}}$ is the volume from the core or shell) down to
smaller outer radius, with a spontaneous magnetisation of
$1.446\text{\,}\mathrm{M}\mathrm{A}\mathrm{/}\mathrm{m}$. Small variations in
the outer radius lead to qualitative variations of the stray field, which,
however, always decays much faster with a magnetic shell than without it (open
black disks). The difference is related to the $\sim 1/r^{4}$ decay of the
stray field for quadrupoles.

##  V Conclusion

We quantitatively evaluated the advantages of using composite storage cells
for perpendicular-shape anisotropy MTJs, made of a core pillar surrounded by a
magnetic shell decoupled from exchange but coupled by dipolar interaction, so
that it behaves as a synthetic ferrimagnet with out-of-plane magnetisation.
The practical realisation of such cells could make use of conformal coating
techniques, such as atomic layer deposition for the formation of the magnetic
shell, around a patterned magnetic tunnel junction. Using both analytical
models and micromagnetic simulations, we outline the benefits in terms of
enhanced stability and stray-field mitigation in dense arrays. Due to the
antiparallel state further promoting perpendicular magnetisation besides the
shape effect of the individual core, it is possible to consider reducing the
core-shell height for a given stability factor, thus reducing the switching
voltage necessary to trigger reversal and lowering its switching time. Making
use of an appropriate geometry for stability of around
$80\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}$, the use of magnetic shells of
Ni versus Co are compared. For both materials, a larger damping allows for a
faster reversal, a criteria opposite to the one to follow for classical MTJs .
The use of a magnetic shell also dramatic reduces the stray field arising from
neighbouring cells, allowing to extend the technology to very dense arrays.
These assets are general to core-shell structures, so they remain valid for
other geometries, such as prismatic.

###### Acknowledgements.

This work was supported by Samsung Electronics Co., LTD. (IO190709-06540-02).

## References

  * Dieny  _et al._ [2020] B. Dieny, I. L. Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. Ebels, S. O. Demokritov, J. Akerman, A. Deac, P. Pirro, C. Adelmann, A. Anane, A. V. Chumak, A. Hirohata, S. Mangin, S. O. Valenzuela, M. C. Onbaşlı, M. d’Aquino, G. Prenat, G. Finocchio, L. Lopez-Diaz, R. Chantrell, O. Chubykalo-Fesenko, and P. Bortolotti, Nat Electron 3, 446 (2020).
  * Sato  _et al._ [2017] H. Sato, S. Ikeda, and H. Ohno, Jpn. J. Appl. Phys. 56, 0802A6 (2017).
  * Perrissin  _et al._ [2018] N. Perrissin, S. Lequeux, N. Strelkov, A. Chavent, L. Vila, L. D. Buda-Prejbeanu, S. Auffret, R. C. Sousa, I. L. Prejbeanu, and B. Dieny, Nanoscale 10, 12187 (2018).
  * Watanabe  _et al._ [2018] K. Watanabe, B. Jinnai, S. Fukami, H. Sato, and H. Ohno, Nature Communications 9, 663 (2018).
  * Beleggia and De Graef [2005] M. Beleggia and M. De Graef, Journal of Magnetism and Magnetic Materials 285, L1 (2005).
  * Tandon  _et al._ [2004] S. Tandon, M. Beleggia, Y. Zhu, and M. De Graef, Journal of Magnetism and Magnetic Materials 271, 9 (2004).
  * Almeida  _et al._ [2022] T. P. Almeida, S. Lequeux, A. Palomino, R. C. Sousa, O. Fruchart, I.-L. Prejbeanu, B. Dieny, A. Masseboeuf, and D. Cooper, Nano Lett. 22, 4000 (2022).
  * Lequeux  _et al._ [2021] S. Lequeux, T. Almeida, N. Caçoilo, A. Palomino, I. L. Prejbeanu, R. C. Sousa, D. Cooper, and B. Dieny, in  _2021 IEEE International Memory Workshop (IMW)_ (2021) pp. 1–4.
  * Lequeux  _et al._ [2020] S. Lequeux, N. Perrissin, G. Grégoire, L. Tillie, A. Chavent, N. Strelkov, L. Vila, L. D. Buda-Prejbeanu, S. Auffret, R. C. Sousa, I. L. Prejbeanu, E. D. Russo, E. Gautier, A. P. Conlan, D. Cooper, and B. Dieny, Nanoscale 12, 6378 (2020).
  * Caçoilo  _et al._ [2021] N. Caçoilo, S. Lequeux, B. Teixeira, B. Dieny, R. Sousa, N. Sobolev, O. Fruchart, I. Prejbeanu, and L. Buda-Prejbeanu, Phys. Rev. Appl. 16, 024020 (2021).
  * Jinnai  _et al._ [2020] B. Jinnai, J. Igarashi, K. Watanabe, T. Funatsu, H. Sato, S. Fukami, and H. Ohno, in  _2020 IEEE International Electron Devices Meeting (IEDM)_ (2020) pp. 24.6.1–24.6.4.
  * Jinnai  _et al._ [2021] B. Jinnai, J. Igarashi, T. Shinoda, K. Watanabe, S. Fukami, and H. Ohno, in  _2021 IEEE International Electron Devices Meeting (IEDM)_ (2021) pp. 1–4.
  * Beleggia  _et al._ [2006] M. Beleggia, J. Lau, M. Schofield, Y. Zhu, S. Tandon, and M. De Graef, Journal of Magnetism and Magnetic Materials 301, 131 (2006).
  * Beleggia  _et al._ [2009] M. Beleggia, D. Vokoun, and M. De Graef, Journal of Magnetism and Magnetic Materials 321, 1306 (2009).
  * Richter and Dobin [2006] H. J. Richter and A. Y. Dobin, Journal of Applied Physics 99, 08Q905 (2006).
  * Marcos-Alcalde  _et al._ [2015] I. Marcos-Alcalde, J. Setoain, J. I. Mendieta-Moreno, J. Mendieta, and P. Gómez-Puertas, Bioinformatics 31, 3853 (2015).
  * Buda  _et al._ [2002] L. D. Buda, I. L. Prejbeanu, U. Ebels, and K. Ounadjela, Computational Materials Science 24, 181 (2002).
  * Slonczewski and Sun [2007] J. C. Slonczewski and J. Z. Sun, Journal of Magnetism and Magnetic Materials 310, 169 (2007).
  * Sun and Ralph [2008] J. Z. Sun and D. C. Ralph, Journal of Magnetism and Magnetic Materials 320, 1227 (2008).
  * Sun  _et al._ [2013] J. Z. Sun, S. L. Brown, W. Chen, E. A. Delenia, M. C. Gaidis, J. Harms, G. Hu, X. Jiang, R. Kilaru, W. Kula, G. Lauer, L. Q. Liu, S. Murthy, J. Nowak, E. J. O’Sullivan, S. S. P. Parkin, R. P. Robertazzi, P. M. Rice, G. Sandhu, T. Topuria, and D. C. Worledge, Phys. Rev. B 88, 104426 (2013).
  * Taniguchi [2018] T. Taniguchi, Journal of Magnetism and Magnetic Materials 452, 464 (2018).
  * [22] O. Fruchart, J.-P. Nozières, B. Kevorkian, J.-C. Toussaint, D. Givord, F. Rousseaux, D. Decanini, and F. Carcenac, Phys. Rev. B 57, 2596.
  * Veiga  _et al._ [2023] P. B. Veiga, A. Mora-Hernandez, M. Dammak, S. Auffret, I. Joumard, L. Vila, L. D. Buda-Prejbeanu, I. L. Prejbeanu, B. Dieny, and R. C. Sousa, AIP Advances 13, 025253 (2023).
  * Wan  _et al._ [2022] L. Wan, T.-W. Wu, N. Smith, T. Santos, G. Mihajlovic, J.-L. Li, K. Patel, N. D. Melendez, B. Terris, and J. A. Katine, IEEE Transactions on Magnetics 58, 1 (2022), conference Name: IEEE Transactions on Magnetics.
  * Wu  _et al._ [2020] L. Wu, S. Rao, M. Taouil, E. J. Marinissen, G. Sankar Kar, and S. Hamdioui, in  _2020 Design, Automation & Test in Europe Conference & Exhibition (DATE)_ (2020) pp. 1211–1216.
  * Khvalkovskiy  _et al._ [2013] A. V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R. S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W. H. Butler, P. B. Visscher, D. Lottis, E. Chen, V. Nikitin, and M. Krounbi, J. Phys. D: Appl. Phys. 46, 074001 (2013).