# Double-Free-Layer Stochastic Magnetic Tunnel Junctions
with Synthetic Antiferromagnets

Kemal Selcuk  Department of Electrical and Computer Engineering, University of
California, Santa Barbara, CA, 93106, USA Shun Kanai  Research Institute of
Electrical Communication, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai
980-8577, Japan Graduate School of Engineering, Tohoku University, 6-6 Aramaki
Aza Aoba, Aoba-ku, Sendai 980-8579, Japan WPI Advanced Institute for Materials
Research (WPI-AIMR), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai
980-8577, Japan  Center for Science and Innovation in Spintronics (CSIS),
Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan  PRESTO,
Japan Science and Technology Agency (JST), Kawaguchi 332-0012, Japan Division
for the Establishment of Frontier Sciences of Organization for Advanced
Studies at Tohoku University, Tohoku University, Sendai 980-8577, Japan
National Institutes for Quantum Science and Technology, Takasaki 370-1207,
Japan Rikuto Ota  Research Institute of Electrical Communication, Tohoku
University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan Graduate School of
Engineering, Tohoku University, 6-6 Aramaki Aza Aoba, Aoba-ku, Sendai
980-0845, Japan Hideo Ohno  Research Institute of Electrical Communication,
Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan WPI
Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, 2-1-1
Katahira, Aoba-ku, Sendai 980-8577, Japan  Center for Science and Innovation
in Spintronics (CSIS), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai
980-8577, Japan  Center for Innovative Integrated Electronic Systems (CIES),
Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku, Sendai 980-0845, Japan
Shunsuke Fukami  Research Institute of Electrical Communication, Tohoku
University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan Graduate School of
Engineering, Tohoku University, 6-6 Aramaki Aza Aoba, Aoba-ku, Sendai
980-0845, Japan WPI Advanced Institute for Materials Research (WPI-AIMR),
Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan  Center for
Science and Innovation in Spintronics (CSIS), Tohoku University, 2-1-1
Katahira, Aoba-ku, Sendai 980-8577, Japan  Center for Innovative Integrated
Electronic Systems (CIES), Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku,
Sendai 980-0845, Japan  Inamori Research Institute of Science (InaRIS), Kyoto
600-8411, Japan Kerem Y. Camsari  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, CA, 93106, USA

###### Abstract

Stochastic magnetic tunnel junctions (sMTJ) using low-barrier nanomagnets have
shown promise as fast, energy-efficient, and scalable building blocks for
probabilistic computing. Despite recent experimental and theoretical progress,
sMTJs exhibiting the ideal characteristics necessary for probabilistic bits
(p-bit) are still lacking. Ideally, the sMTJs should have (a) voltage bias
independence preventing read disturbance (b) uniform randomness in the
magnetization angle between the free layers, and (c) fast fluctuations without
requiring external magnetic fields while being robust to magnetic field
perturbations. Here, we propose a new design satisfying all of these
requirements, using double-free-layer sMTJs with synthetic antiferromagnets
(SAF). We evaluate the proposed sMTJ design with experimentally benchmarked
spin-circuit models accounting for transport physics, coupled with the
stochastic Landau-Lifshitz-Gilbert equation for magnetization dynamics. We
find that the use of low-barrier SAF layers reduces dipolar coupling,
achieving uncorrelated fluctuations at zero-magnetic field surviving up to
diameters exceeding ($D\approx 100$ nm) if the nanomagnets can be made thin
enough ($\approx 1$-$2$ nm). The double-free-layer structure retains bias-
independence and the circular nature of the nanomagnets provides near-uniform
randomness with fast fluctuations. Combining our full sMTJ model with advanced
transistor models, we estimate the energy to generate a random bit as
$\approx$ 3.6 fJ, with fluctuation rates of $\approx$ 3.3 GHz per p-bit. Our
results will guide the experimental development of superior stochastic
magnetic tunnel junctions for large-scale and energy-efficient probabilistic
computation for problems relevant to machine learning and artificial
intelligence.

##  I Introduction

The slowing down of Moore’s law has been driving the development of domain-
specific computers in different fields. One approach is to build physics-
inspired computers exploiting the natural physics of materials and devices for
efficiency. A prominent emerging example of “let physics do the computing”
Feynman (1982); Porod (1997) is based on probabilistic or $p$-bits Camsari et
al. (2017a); Borders et al. (2019); Chowdhury et al. (2023a). The central idea
in this field is to build _programmable_ networks of $p$-bits, or
$p$-circuits, whose natural dynamical evolution leads to the solution of a
problem of interest. The mathematical formulation of $p$-bits is related to
the widely used Monte Carlo or Markov Chain Monte Carlo algorithms Koller and
Friedman (2009), casting a wide net for their application domains from random
number generation Vodenicarevic et al. (2017); Talatchian et al. (2021),
machine learning Kaiser et al. (2022); Daniels et al. (2020), combinatorial
optimization Borders et al. (2019); Aadit et al. (2022) to a subset of quantum
simulation problems Camsari et al. (2019); Chowdhury et al. (2023b); Grimaldi
et al. (2022); Chowdhury et al. (2023c).

Since their appearance as building blocks for probabilistic computation, there
have been many proposals and implementations of $p$-bits in different material
systems using a variety of stochastic phenomena Woo et al. (2022); Liu et al.
(2022); Park et al. (2022); Deng et al. (2023); Gutiérrez-Finol et al. (2023).
A particularly promising possibility is the use of magnetic devices in the
superparamagnetic regime which have recently been shown to produce $\approx$
GHz fluctuations at room temperature Hayakawa et al. (2021); Safranski et al.
(2021); Sun et al. (2023); Schnitzspan et al. (2023) using in-plane magnetic
tunnel junctions (MTJ). Combined with the proven manufacturability of magnetic
memory that has integrated billions of _deterministic_ MTJs with standard
complementary metal-oxide semiconductor (CMOS) technology Aggarwal et al.
(2019); Lee et al. (2019), the development of highly integrated and fast
probabilistic computers is highly probable.

Despite successful recent demonstrations of fast superparamagnetism in in-
plane MTJs, following earlier work on perpendicular MTJs Parks et al. (2018);
Funatsu et al. (2022), ideal stochastic MTJs for _circuit-level_ p-bits have
been difficult to realize. In particular, a three transistor/1 MTJ design
(3T/1MTJ) Camsari et al. (2017b) has been explored as a compact and energy-
efficient implementation of a p-bit in discrete Borders et al. (2019); Kaiser
et al. (2022); Grimaldi et al. (2022); Kobayashi et al. (2023) and integrated
implementations Daniel et al. (2023). Various other possibilities beyond the
3T/1MTJ p-bit design have been considered, including a spin-orbit-torque
design (proposed in Ref. Camsari et al. (2017a)) and later experimentally
demonstrated in Ref. Yin et al. (2022) and a comparator based design Grimaldi
et al. (2022); Kobayashi et al. (2023), with possibly different sMTJ
requirements. Moreover, entirely different approaches to create p-bits
exploiting the stochasticity observed in the switching characteristics of
_deterministic_ MTJs have been proposed Fukushima et al. (2014); Rehm et al.
(2023); Shim et al. (2017); Zink et al. (2022); McCray et al. (2020). In this
paper, we focus on the necessary design requirements for the 3T/1MTJ p-bit.
This design makes minimal modifications to the currently integrated 1T/1MTJ
circuit in the commercialized spin-transfer torque magnetoresistive random
access memory (STT-MRAM). The 3T/1MTJ p-bit does not explicitly require spin-
torque control on magnetization characteristics, and its variation tolerance
is greatly improved since it uses low-barrier nanomagnets.

There are several requirements for scalable sMTJ-based p-bits in this 3T/1MTJ
p-bit circuit. First, the sMTJ design should be magnetically soft, fluctuating
without requiring external magnetic fields but electronically hard so that
even at large bias currents the free layer magnetization is not pinned.
Second, the sMTJs should exhibit reasonable tunneling magnetoresistance (TMR),
with _uniform_ randomness in resistance values Hassan et al. (2021). Third,
the sMTJ should exhibit _fast_ fluctuations. A previous design Camsari et al.
(2021) using double-free-layers nearly satisfies all these requirements by
providing fast zero-field fluctuations and bias-independence (FIG. 1). The
simplicity of the double-free-layer structure is appealing as opposed to
asymmetrically designed typical MTJs with fixed and free layers. However,
achieving zero-bias and zero-field fluctuations comes at a price: for the
thermal noise to overcome the dipolar coupling between free layers the
magnetic layers need to be scaled down to $\approx 20$ nm or below, imposing
fabrication challenges.

In this paper, we propose an improvement to this earlier design by considering
double-free-layer sMTJs using _synthetic antiferromagnetic_ (SAF) free layers
(FIG. 1). The magnetic compensation of SAFs ensures minimized dipolar coupling
between their constituent magnets, greatly easing the scalability requirements
of double-free sMTJs. In the rest of this paper, we evaluate this proposed
design by the language of spin-circuits, pioneered by Brataas, Bauer and Kelly
Brataas et al. (2006). Further, we employ coupled stochastic Landau-Lifshitz-
Gilbert (sLLG) equations to take magnetization dynamics into account. We apply
a microscopic approach to calculate the dipolar tensors between magnets and
provide an approximate autocorrelation theory with analytical results. We then
simulate the proposed sMTJ in conjunction with advanced transistor nodes in
SPICE, estimating energy and delays to produce random bits.

##  II Spin-circuit and stochastic LLG analysis of the proposed sMTJ

Our approach in this paper is to start from the powerful spin-circuit
approach, pioneered by Brataas, Bauer and Kelly Brataas et al. (2006), that
has later been converted into explicit circuit models that can be modularly
simulated in SPICE-like simulators Srinivasan et al. (2013); Manipatruni et
al. (2012); Camsari (2015). These two approaches have been extensively
compared and found to be exactly equivalent Camsari (2015); Camsari et al.
(2015). The advantage of writing down explicit SPICE-compatible $4\times 4$
conductances is the ability to combine spin the transport physics obtained
from the spin-circuit approach with time-dependent stochastic LLG solvers and
advanced CMOS transistor models, all in standard circuit simulators. The sLLG
solvers are rigorously benchmarked against probabilistic models such as the
Fokker-Planck Equation applied to nanomagnet dynamics Torunbalci et al.
(2018); Kanai et al. (2021). This approach allows us to consider each
interface separately and it can be used to provide a detailed understanding of
spin currents as they travel through the 4-magnet system. Technically, the use
of spin-circuit theory is strictly applicable to _metallic_ interfaces Brataas
et al. (2006) and modeling _tunneling_ junctions requires a different
treatment such as multiplying the conductance matrices Camsari et al. (2014)
to get a tunneling magnetoresistance ratio of $2P^{2}/(1-P^{2})$ rather than
the giant magnetoresistance ratio of $P^{2}/(1-P^{2})$ where $P$ is the
interface polarization. Nevertheless, to keep our model fine-grained and
microscopic at each interface, we use the metallic spin-circuit models (as in
Ref. Manipatruni et al. (2014)) to study the proposed double-SAF-free MTJs as
a series of ferromagnet-normal metal interfaces. To match experiments, we
choose the interface polarizations ($P$) appropriately, corresponding to
experimental TMR values (shown in Table 1).

Figure 1: Stochastic MTJ (sMTJ) designs (a) This work: Double-SAF-free sMTJ
where the low barrier free layers are replaced by magnetically inert SAF
layers. (b) Double-free-layer sMTJ which no fixed layer but two low-barrier
free layers. (c) Standard sMTJ with a fixed layer and a low-barrier free
layer: commonly used in literature since it minimally modifies existing stable
MTJs with a fixed layer to have a low-barrier free layer. Figure 2: Modular
spin-circuit model for the double-SAF-free layer structure (a) Close-up look
on the spin-circuit transport model for a single SAF stack-layer: represented
with two ferromagnet-normal metal (F$|$N) interfaces and a normal metal (NM)
block. (b) The full model coupling magnetizaton dynamics (sLLG) with spin-
transport modules. The transport module produces spin-polarized currents input
to sLLGs. For the two interfaces of the same magnet, we add the spin-currents
incident to the magnets vectorially (e.g.,
$\vec{I}_{s2}=\vec{I}_{s21}+\vec{I}_{s22}$) as an input to the sLLGs, since we
are in the monodomain approximation for the magnets. LLGs in turn produce
instantaneous magnetization vectors back to the interface modules. In
addition, the two interfaces of the same magnet receive the same magnetization
vector from their corresponding sLLG’s as shown. Dipolar, exchange and thermal
noise are considered by the sLLG model. Details of the 4$\times$4 conductances
and the sLLG module can be found in Ref.’s Camsari (2015); Camsari et al.
(2015); Torunbalci et al. (2018); nanohub.org . Numerical parameters are
summarized in Table 1.

FIG. 2 shows how we decompose the full double-SAF-free layers into
corresponding interfaces. F$|$N denotes a ferromagnet-normal metal interface,
NM denotes a normal, non-magnetic metal. All spin-circuits are described by
4-component currents (3 for spin and 1 for charge) related to 4$\times$4
conductance matrices (FIG. 2a). For the spacer in the SAF layers we explicitly
consider the NM layers (modeling the Ruthenium) but we do not include an
explicit NM layer in between the free layers for the MgO. In our full model
(FIG. 2) coupling transport and magnetism, we parameterize the F$|$N
interfaces with instantaneous magnetization vectors obtained from the sLLG
equations. We assume that the transport timescales are much faster than
magnetization dynamics, hence we assume a lumped circuit model for each
magnetization vector Torunbalci et al. (2018). The details of the 4$\times$4
conductance models and the sLLG solver we used in this paper have been
explained in detail in Ref.’s Camsari (2015); Camsari et al. (2015);
Torunbalci et al. (2018).

Figure 3: Dipolar coupling in the double-SAF-free layer stucture (a) Dipolar
coupling coefficients $\mathrm{D_{ij}}$ between magnets $(i,j)$ are shown at
varying diameters (1 nm thickness), numerically calculated based on the
approach described in Ref. Camsari et al. (2021). The effective dipolar
($\text{D}_{\text{eff}}$) coupling (Eq. 3) is also shown. $\mathrm{D_{12}}$
and $\mathrm{D_{34}}$ are combined with exchange tensor ($J_{ij}$) for SAF
couples in the numerical model but not shown in the plot. (b) Average angle
between layers (2,3), $\langle\cos\ \theta\rangle$, are calculated numerically
(markers) and analytically (solid and dashed lines), based on (Eq. 4), as a
function of varying magnet diameters and thicknesses. Numerical calculations
for the sLLG model uses a 1 ps time-step and averages are taken over 5 $\mu
s$.

##  III Equilibrium (Zero-bias) behavior

In this section, we study the zero-bias behavior of the double-SAF free layer
sMTJs under the influence of thermal noise, dipolar and interlayer exchange
interaction fields. We first describe our magnetostatics model we use in this
paper. We consider 4 identical magnets with the same volume, interacting with
dipolar and exchange interactions. The magnets are assumed to be perfectly
circular and zero-barrier ($k_{B}T\ll 1$) with easy-plane anisotropy in the
$(x-y)$ plane. These assumptions may not exactly hold in practical systems
however they should provide qualitative guidance for experiments. The
equilibrium energy description for this system is:

$\displaystyle E$ $\displaystyle=-2\pi
M_{s}^{2}\text{Vol.}\Bigg{(}\sum_{i=1}^{n=4}\hat{m}_{i}^{T}{N}_{ii}\hat{m}_{i}$
$\displaystyle\quad+\sum_{{i,j\atop i\neq
j}}^{n=4}\hat{m}_{i}^{T}({J}_{ij}+\mathrm{{D}_{ij}})\hat{m}_{j}\Bigg{)}$ (1)

where $M_{s}$ represents the saturation magnetization, Vol. is the volume of
each magnet, $\hat{m}_{i}$ are the 3-dimensional magnetization vectors on the
unit sphere, and $N_{ii}$, and $\mathrm{D_{ij}}$, $J_{ij}$, are the
demagnetizations, dipolar and exchange tensors, respectively. We consider
exchange interactions only between SAF-pairs (1,2) and (3,4), while dipolar
interactions are considered between all $\binom{4}{2}=6$ pairs of magnets.
Throughout this paper, we assume $N_{zz}=-1$ for all magnets with all other
components of the demagnetization tensor being zero.

In Eq. 1, we assume an isotropic and constant exchange model where the
exchange tensor $J_{ij}$ is parameterized by a single number, i.e.,
$J_{ij}=J_{0}\delta_{ij}$. We renormalize the exchange coupling $J_{ex}$,
typically measured in units of J/$\mathrm{m^{2}}$ Yakushiji et al. (2015),
such that $J_{0}=J_{\mathrm{ex}}/(2\pi M_{s}^{2}\sf{t})$, where t is the
thickness of the magnets (assumed equal). In this form, $J_{ij}$ has the same
units with dipolar tensors and can be directly compared. Throughout, we use a
$|J_{ex}|$ value of 2.5 mJ/$\mathrm{m}^{2}$, a reasonable choice considering
experimental measurements Yakushiji et al. (2015); Parkin and Mauri (1991). To
calculate dipolar tensors, $\mathrm{D_{ij}}$, we follow the approach described
in Ref. Camsari et al. (2021). The basic idea is to obtain position-dependent
dipolar tensors $\rm D_{ij}(x,y,z)$ starting from the Poisson equation for the
magnetic potential, $\Psi$, $\nabla^{2}\Psi=\nabla\cdot\hat{m}$ to calculate
dipolar tensors, and then to _average_ these tensors over the volume of the
target magnet to obtain single $\rm D_{ij}$ values. Until the averaging step,
this approach is exact. Due to the cylindrical symmetries present, the only
non-zero components are $\rm D_{xx}=D_{yy}=-D_{zz}/2$ where $(x,y)$ are the
easy-plane and z is the out-of-plane components. In the rest of the paper, we
drop the coordinate subscripts and use $\rm D$ for $\rm D_{xx}=D_{yy}$.

To study the equilibrium behavior of this system, we perform numerical and
theoretical analysis. For our numerical results, after calculating all 6 D’s
for each pair of magnets and assuming a fixed interlayer exchange coupling
between the SAF layers, we employ our full model in FIG. 2 with 4 coupled
sLLGs to obtain the average cos$\ \theta$ between layers 2 and 3, which
determines the output signal. The dipolar coefficients and the numerical
average cos$\ \theta$ are shown in FIG. 3a and FIG. 3b, respectively. We study
the average angle as a function of changing magnet thickness, t. Using a
revised model without the use of SAF free layers but two FM$|$NM layers in
series, we also closely reproduce the results in Ref. Camsari et al. (2021).
Ideally, $\langle\mathrm{cos}\ \theta\rangle$ should be zero corresponding to
freely fluctuating and truly independent stochastic free layers. As shown by
the triangle gray plots in FIG. 3b, for the double-free sMTJ (without the use
of SAFs), reaching such an independent regime requires extremely thin magnets
at small diameters ($<20$ nm). The proposed MTJ with double SAF layers on the
other hand shows near independence even at large diameters for a range of
ferromagnet thicknesses, indicating the magnetic inertness of the SAF layers.

Figure 4: Autocorrelation of magnetizations (a) shows numerical and
theoretical autocorrelation for the in-plane component of layers (2,3). (b)
shows the numerical and theoretical autocorrelation of $\cos\theta$. Numerical
results are obtained for a diameter of 50 nm and a thickness, t = 1 nm.
Numerical results are obtained over 5 $\mu$s using a timestep of 1 ps in sLLG
simulations. (c) Time-dependent magnetizations of all layers from the full
numerical model. Histogram for the cosine of the angle between layers (2,3)
$\cos\ \theta$ demonstrating the necessary near uniform randomness between -1
and +1 for p-bit operation.

Next, we provide an approximate theoretical analysis for the results shown in
FIG. 3. To proceed, we make simplifying assumptions. First, we assume an
infinite exchange coupling between layers (1,2) and (3,4). This allows us to
simplify the dipolar part of the energy equation by substituting
$m_{1}^{x,y,z}$ with $-m_{2}^{x,y,z}$ and $m_{4}^{x,y,z}$ with
$-m_{3}^{x,y,z}$, yielding:

$\displaystyle E_{\mathrm{dip.}}$ $\displaystyle=-2\pi
M_{s}^{2}\text{Vol.}\Big{(}\text{{D}}_{\text{23}}\hat{m}_{2}^{T}\hat{m}_{3}-\text{{D}}_{\text{13}}\hat{m}_{2}^{T}\hat{m}_{3}$
$\displaystyle\qquad-\text{{D}}_{\text{24}}\hat{m}_{2}^{T}\hat{m}_{3}+\text{{D}}_{\text{14}}\hat{m}_{2}^{T}\hat{m}_{3}\Big{)}$
(2)

allowing us to obtain an “effective” dipolar tensor between layers 2 and 3:

$\displaystyle E_{\mathrm{dip.}}=-2\pi
M_{s}^{2}\text{Vol.}\left[(\mathrm{{D}}_{\text{eff}})\hat{m}_{2}^{T}\hat{m}_{3}\right]$
(3)

where
$\mathrm{D_{eff}}$=$\mathrm{D_{23}}$$-$$\mathrm{D_{13}}$$-$$\mathrm{D_{24}}$+$\mathrm{D_{14}}$.
FIG. 3a shows $\mathrm{D_{eff}}$ for magnets with t=$1$ nm at varying
diameters, $\Phi$. We observe an approximate $\rm
D_{\mathrm{eff}}\propto\Phi^{-2}$ dependence, with a decreasing effective
dipolar coupling at larger diameters. These results indicate how the double
SAF layer reduces the effective coupling between layers 2 and 3. Next, we use
the theoretical calculation presented in Ref. Camsari et al. (2021) where the
equilibrium $\langle\cos\ \theta\rangle$ is related to the dipolar coupling
coefficient D between two magnets. The idea behind this calculation is to
assume a large demagnetization field and expand the Boltzmann probability,
$\rho=1/Z\exp(-E/k_{B}T)$ around the out-of-plane component since
$m_{z}\approx 0$ for analytical approximation. With this approach, one obtains
(in mks units):

$\left\langle\cos\theta_{2,3}\right\rangle\approx\frac{I_{1}\left(\text{d}_{\text{eff}}\right)}{I_{0}\left(\text{d}_{\text{eff}}\right)}$
(4)

where $I_{n}$ represents the modified Bessel function of the first kind and
$\text{d}_{\text{eff}}=h_{d}{\text{D}_{\text{eff}}}$ where
$h_{d}=\mu_{0}M_{s}^{2}\mathrm{Vol.}/k_{B}T$, $\mu_{0}$ being permeability.
Using the $\mathrm{D_{eff}}$ values obtained at different diameters with Eq.
4, we obtain excellent agreement between the theory and our numerical model
that does not assume an infinite exchange coupling between layers (1,2) and
(3,4) as shown in FIG. 3b. We note that the reason for the roughly constant
and small $\mathrm{cos(\theta)}\approx 0-0.1$ at small thickness (t=1 nm) can
be explained by the cancellation of the approximate inverse square law
dependence of the dipolar coefficient on the diameter and the square law
dependence of the dipolar energy (in $h_{d}$) on the diameter. The results
shown in FIG. 3 indicate that to overcome the dipolar energy between layers
(2,3), using small volume and small thickness magnets are still favorable as
in double-free layer sMTJs Camsari et al. (2021). Making CoFeB-type magnets
too thin (below $<$ 2 nm) may induce perpendicular anisotropy in circular
magnets Kobayashi et al. (2022), possibly necessitating further experimental
optimization or different material systems.

##  IV Autocorrelation Theory

Next, we provide an autocorrelation theory for the relative angle between
magnetic layers (2,3) and compare it against our full numerical model (FIG.
2). We are interested in computing the quantity:

$C[\cos\ \theta(\tau)]=\frac{1}{T}\int_{0}^{T}\cos\ \theta(t+\tau)\cos\
\theta(t)dt$ (5)

where $\cos\ \theta$ represents the angle between layers (2,3). To simplify
our analysis, to first order, we will assume that the magnets always remain in
the $(x,y)$ plane and assume $m_{z}(t)\approx 0$, making $\cos\
\theta(t)=m^{x}_{2}(t)m^{x}_{3}(t)+m^{y}_{2}(t)m^{3}_{y}(t)$. Technically the
$(x,y)$ components of magnets (2,3) are always correlated as established in
FIG. 3, however, at low thicknesses, they are only _weakly_ correlated. As
such, for our theoretical analysis we make another simplifying assumption and
assume that the magnets are independent of each other. This essentially means
that we only need an analytical approximation for the autocorrelation of the
$m_{x}$/$m_{y}$ components. Following the analysis shown in Refs. Hassan et
al. (2019); Kaiser et al. (2019), we proceed as follows. In circular
nanomagnets, the mechanism of random fluctuations arise from the random
precession of the in-plane magnetization vector around the stochastically
fluctuating demagnetization field which is a function of the out-of-plane
component. The autocorrelation of the in-plane magnetization vector $m_{x}(t)$
can be written as $C[m_{x}(\tau)]=\langle m_{x}(\tau)m_{x}(0)\rangle$ which
reads:

${\displaystyle\int\limits_{-\infty}^{\infty}\cos(\alpha m_{z})\exp(-\beta
m_{z}^{2})\,dm_{z}}\bigg{/}{\displaystyle\int\limits_{-\infty}^{\infty}\exp(-\beta
m_{z}^{2})\,dm_{z}}$ (6)

where $\alpha$, $\beta$ are defined to be $\gamma H_{d}\tau$ and
${H_{d}M_{s}\text{Vol.}}/{2k_{B}T}$, respectively, while $H_{d}$ is the out-
of-plane demagnetizaton field and the $\gamma$ is the gyromagnetic ratio,. The
integral limits are extended to infinity to obtain a closed-form expression
with minimal errors, due to the exponentially low probability of large $m_{z}$
components. This expression evaluates to,

$C[m_{x}(\tau)]=\exp\left(-\frac{\alpha^{2}}{2\beta}\right)$ (7)

Figure 5: Bias dependence of sMTJs (a) Shows the time-averaged $\cos\ \theta$
as a function of the bias voltage across the MTJs for the double-SAF-free
sMTJ. Device exhibits independence between the free layers, $\langle\cos\
\theta\rangle\approx 0$, with bias-independence. (b) Shows the time-averaged
$\cos\ \theta$ as a function of the bias voltage for a double-free sMTJ,
modeled as two F$|$N interfaces in series. These results are in quantitative
agreement with the phenomenological model introduced in Ref. Camsari et al.
(2021). Unlike that model however we do not legislate anti-symmetric currents
to the magnets at the interfaces. This behavior comes out of our fine-grained
interface model. Moreover, due to the uncompensated dipolar coupling the
double-free sMTJ does not exhibit nearly independent fluctuations between its
layers, $\langle\cos\ \theta\rangle\neq 0$. All magnets assumed to have a
thickness of of 1 nm. Each data point is averaged over 1 $\mu$s with a time
step of 1 ps using the .trannoise function in HSPICE.

In the present context, however, we are interested in the cosine of the angle
between layers 2 and 3. As such, the autocorrelation expression (assuming both
magnets remain in their $(x,y)$ plane) becomes:

$\frac{\displaystyle\iint\limits_{-\infty}^{+\infty}\cos[\alpha(m_{z1}-m_{z2})]\exp[{-\beta(m_{z1}^{2}+m_{z2}^{2})}]d\mathbf{m_{z}}}{\displaystyle\iint\limits_{-\infty}^{+\infty}\exp[{-\beta(m_{z1}^{2}+m_{z2}^{2})}]d\mathbf{m_{z}}}$
(8)

evaluating to,

$C[\cos(\theta(\tau))]=\exp\left(-\frac{\alpha^{2}}{\beta}\right)$ (9)

We observe an interesting difference between the previously obtained Eq. 7
Kaiser et al. (2019) and the new Eq. 9. It turns out that the autocorrelation
of the angle between the two independent magnets decays to $\exp(-1)$ of its
value $\sqrt{2}$ times as fast as the autocorrelation of a single magnet. We
corroborate this theoretical calculation with our full numerical model without
any independence or in-plane rotation assumptions. We observe that at moderate
diameters of 50 nm’s with thicknesses of 1 nm’s, the SAF layers behave as if
they are nearly independent of one another. The theoretically obtained
$\sqrt{2}$ factor is also observed from our numerical simulations in FIG. 4.
Faster decays of the autocorrelation on $\cos\theta$ bode well for
probabilistic computing applications since they directly influence the speed
with which new and independent random numbers can be obtained from sMTJ-based
p-bits.

Table 1: Simulation parameters Parameter | Value
---|---
CoFeB/MgO/CoFeB RA-product | 9 $\Omega\ \mu\text{m}^{2}$ Lin et al. (2009)
CoFeB/Ru/CoFeB RA-product |  $5.2\times 10^{-3}$ $\Omega\ \mu\text{m}^{2}$ Pal et al. (1976) Zhang et al. (2016)
corresponding $G_{0}$ | (1/RA-product)/Area $\mho$
Temperature | 300 K
$M_{s}$ |  $8\times 10^{5}$ A/m Camsari et al. (2021)
Damping Coefficient ($\alpha$) | 0.01 Sankey et al. (2008)
Real part of $g_{mix}$, ($G_{0}a$) | a=1 Camsari et al. (2015); Tokaç et al. (2015)
Imaginary part of $g_{mix}$, ($G_{0}b$) | b=0 Camsari et al. (2015)
$\lambda_{\text{sf}}$ of Ru Layer | 14 nm Eid et al. (2002)
$P_{0}$ of CoFeB/MgO/CoFeB, TMR | 0.73, TMR=115$\%$ Camsari et al. (2021)
$P_{0}$ of CoFeB/Ru/CoFeB, GMR | 0.077, GMR=0.6$\%$ Gubbiotti et al. (2005)
Diameters of sMTJs, used in p-bit |  $\\{10,\ldots,200\\}$ nm, 25 nm
Aspect ratio of Free layer | 1
Thickness of Free layer, Ru Layer | 1-2 nm, 0.8 nm Kobayashi et al. (2022)
Exchange Field ($J_{ex}$) |  $\rm-2.5\times 10^{-3}\ J/m^{2}$ Yakushiji et al. (2015)
CMOS Models | 14 nm HP-FinFET pre

##  V Non-equilibrium (bias-dependent) behavior

We now investigate the bias-dependence of our proposed device, using the full
numerical model of FIG. 2. Considering self-consistent finite-temperature
magnetization, dipolar, exchange fields and spin-circuits for transport, we
numerically measure $\cos\ \theta$ at different bias voltages as shown in FIG.
5a-b. We choose experimentally-informed conductance and polarization
parameters for the MTJ and Ruthenium interfaces, shown in Table 1. Even though
we treat the SAFs layer as spin-valves composed of F$|$N interfaces, due to
their relatively small interface polarizations, the main resistance modulation
arises between layers 2 and 3 that are separated by the MgO layer.

To test our fine-grained spin-circuit model that splits the system into
individual F$|$N interfaces, we first design a spin-valve structure with two
F$|$N interfaces to study the double-free-layer sMTJ proposed in Ref. Camsari
et al. (2021). Without legislating any assumptions on the spin-currents, we
observe the same bias independence using our spin-circuit model with
quantitatively similar results to those in Ref. Camsari et al. (2021) which
used a phenomenological model with explicitly defined anti-symmetric spin-
currents (FIG. 5b). This indicates that our fine-grained interface model
captures detailed physics accurately. We observe that the double-free layer
sMTJ shows significant correlations, measured by $\cos\ \theta$ for diameters
above 50 nm’s, where the magnets are completely pinned in anti-parallel
directions.

Next, we use our full numerical model to investigate the bias-dependence of
$\cos\ \theta$ for the double SAF-layer structure (FIG. 5a). We observe that
for a large range of diameters $\cos\ \theta$ shows weak bias dependence and
is close to zero. In addition, the double-SAF-free sMTJ should be robust to
external magnetic fields, due to the magnetic insensitivity of the SAF layers
Kobayashi et al. (2022). We believe that our new proposed design is the most
promising stochastic MTJ to date since it satisfies the three important
conditions of (a) bias-independence (FIG. 5a), (b) uniform randomness in the
relative magnetization angle (FIG. 4c) and (c) fast fluctuations without
requiring external magnetic fields (FIG. 4b). Our design improves upon earlier
ones where the dipolar interaction between two closely separated nanomagnets
was either ignored entirely Camsari et al. (2017b) or required single-digit
magnet diameters Camsari et al. (2021) with challenging fabrication
requirements. Given the recent experimental success in observing nanosecond
fluctuations in nanomagnets Hayakawa et al. (2021); Safranski et al. (2021);
Schnitzspan et al. (2023), the prospects for experimentally demonstrating the
key features of double SAF sMTJs seem promising.

##  VI cmos circuit behavior and performance

One of the main advantages of the self-consistent spin-circuit approach
coupling microscopic transport physics with magnet dynamics in standard
circuit simulators is in the ability to integrate them with existing CMOS
models. Here, we combine our full device model presented in FIG. 2 with
advanced predictive technology models (PTM) for 14 nm FinFET transistors. FIG.
6 shows the 3T/1MTJ topology Camsari et al. (2017b) to design a p-bit with a
binary output whose probability is controlled by an input voltage $\rm
V_{IN}$. The behavioral equation for the ideal p-bit is given by:

$m_{i}=\operatorname{sgn}\left[\tanh\left(I_{i}\right)-\mathrm{rand_{U}}\right]$
(10)

For this equation to produce a smoothly varying output probability and a mean
of $\langle m\rangle=\mathrm{tanh}(I_{i})$, an essential requirement is that
the random noise denoted by $\mathrm{rand_{U}}$ is _uniform_ and continuous
between -1 and +1 Hassan et al. (2021). To see how the 3T/1MTJ circuit maps to
Eq. 10, we consider the time-varying drain voltage in FIG. 6a given by:

$\frac{V_{\text{DRAIN}}}{2V_{\text{DD}}}=\frac{1}{2}-\frac{R_{\text{MTJ}}(t)}{R_{\text{MTJ}}(t)+R_{\text{DS}}\left(V_{\text{IN}})\right.}$
(11)

where $\mathrm{R_{DS}(V_{IN})}$ is the transistor resistance which varies as a
function of the input voltage. Eq. 11 indicates another important design
criterion for the 3T/1MTJ circuit to function as a p-bit, namely, for $\rm
V_{DRAIN}$ to produce equal 1 and -1 fluctuations at the output of the
inverter, the _average_ MTJ resistance needs to be matched to the transistor
resistance when $\rm V_{IN}$ is at the midpoint of the sigmoid (in our circuit
this is when $\rm V_{IN}=0$). When biased in its midpoint ($\rm V_{DRAIN}=0)$,
the main function of the inverter is to amplify $\rm V_{DRAIN}$, which can be
approximated by a sign function, making the output:

$\begin{gathered}\frac{V_{\text{OUT}}}{V_{\text{DD}}}\sim\operatorname{sgn}\left[R_{\text{DS}}\left(V_{\text{IN}}\right)-R_{\text{MTJ}}(t)\right]\end{gathered}$
(12)

Figure 6: 3T/1MTJ p-bit circuit with double-SAF-free MTJ (a) We use the full circuit model described in FIG. 2in conjunction with a 14 nm HP-FinFET predictive model Zhao and Cao (2007), simulated in HSPICE. Magnet diameters are 25 nm and thicknesses are 1 nm. The diameter is chosen to match transistor resistance. (b) The blue trace is observed by observing $\rm V_{OUT}$ as a function of $\rm V_{IN}$ while $\rm V_{IN}$ is linearly swept from $-$0.2 V to 0.2 V in a 250 ns time-window. The red dots are obtained as separate measurements where $\rm V_{IN}$ is fixed to a given value and a time-average of the output over 5 $\mu$s is taken with a 1 ps time step. The black line is a tanh fit to this average. Table 2: Performance measurements Power dissipation on sMTJ branch |  6.6 $\mu$W
---|---
Power dissipation on Inverter branch |  5.3 $\mu$W
Flips per second |  3.3 flips/ns
Energy per RNG |  3.6 fJ

FIG. 6 shows numerical results for this full circuit indicating how the
proposed double-SAF-free MTJ approximately produces the tunable randomness
necessary for the p-bit. Next, using our SPICE model we estimate several key
metrics for p-bits. We delineate the power dissipation of the circuit into two
branches, the sMTJ branch and the inverter branch. We obtain the power
dissipation on the MTJ branch by measuring the instantaneous current flow over
a 5 $\mu$s time window and multiplying this with the supply voltage, where the
product is around 6.6 $\mu$W. Similarly, we measure the power dissipation on
the inverter branch to be around 5.3 $\mu$W. We define a probabilistic flip
based on the autocorrelation values we reported in FIG. 4 where a new flip is
defined as the time when the autocorrelation roughly saturates. We estimate
such an independent flip time to be around $\tau_{f}=0.3$ ns for the double-
SAF-free MTJ. Using these numbers the energy per random number generation is
estimated to be $P_{total}\times\tau_{f}\approx 3.6$ fJ. These values are
reported in Table 2 and they are in rough agreement with those estimated in
Ref. Hassan et al. (2021). The important point to note however is that the
proposed sMTJ does not make any simplifying assumptions about dipolar fields,
the necessary uniform randomness in the MTJ or the speed of fluctuations. The
total flips per second in a system of integrated p-bits, say with a million
p-bits fluctuating in sparsely connected networks could lead to a system flips
per second of $N/\tau_{f}=3.3\times 10^{15}$ which is four to five orders of
magnitude larger than what can be achieved with present-day CMOS technology
Aadit et al. (2022); romero2020high .

##  VII Conclusion

We proposed and analyzed a new stochastic magnetic tunnel junction with double
synthetic-antiferromagnetic free layers with a modular spin-circuit model. By
theoretical analysis and numerical calculations, we have shown that the
proposed stochastic MTJ can meet the important requirements of bias
independence, uniform randomness and fast fluctuations without any external
magnetic fields. Similar to earlier double-free sMTJ designs, the structure
should be relatively easy to build. The low-barrier nature of the constituent
magnets should provide robustness against variations. Circuit analysis shows
that the proposed device can function as a fast and energy-efficient p-bit
design which can aid the development of large-scale probabilistic computers,
useful for a wide variety of applications relevant to machine learning and
Artificial Intelligence algorithms.

## Acknowledgment

KS and KYC acknowledge the U.S. National Science Foundation (NSF) support
through CCF 2106260 and SAMSUNG Global Research Outreach (GRO) grant. Use was
made of computational facilities purchased with funds from the National
Science Foundation (CNS-1725797) and administered by the Center for Scientific
Computing (CSC). The CSC is supported by the California NanoSystems Institute
and the Materials Research Science and Engineering Center (MRSEC; NSF DMR
2308708) at UC Santa Barbara. SF and SK acknowledge JST-CREST JPMJCR19K3, JST-
PRESTO JPMJPR21B2, and JST-AdCORP JPMJKB2305.

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