# Entropy-assisted nanosecond stochastic operation in perpendicular superparamagnetic tunnel junctions

Lucile Soumah, Louise Desplat, Nhat-Tan Phan, Ahmed Sidi El Valli, Advait Madhavan, Florian Disdier, Stéphane Auffret, Ricardo C Sousa, Ursula Ebels, Mark D Stiles, et al.

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Lucile Soumah, Louise Desplat, Nhat-Tan Phan, Ahmed Sidi El Valli, Advait Madhavan, et al. "Entropy-assisted nanosecond stochastic operation in perpendicular superparamagnetic tunnel junctions." Physical Review Applied, 2025, 24 (1), pp.L011002. 10.1103/v53b-mz5b. hal-05149179

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Distributed under a Creative Commons Attribution 4.0 International License
Entropy-Assisted Nanosecond Stochastic Operation
in Perpendicular Superparamagnetic Tunnel Junctions

Lucile Soumah<sup>1, *, †</sup> Louise Desplat<sup>1, 2, †</sup> Nhat-Tan Phan<sup>1</sup> Ahmed Sidi El Valli<sup>1</sup> Advait Madhavan<sup>1, 3, 4</sup> Florian Disdier<sup>1</sup> Stéphane Auffret<sup>1</sup> Ricardo C. Sousa<sup>1</sup> Ursula Ebels<sup>1</sup> Mark D. Stiles<sup>1, 3</sup> and Philippe Talatchian<sup>1, ‡</sup>

<sup>1</sup>Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP, SPINTEC, 38000 Grenoble, France

<sup>2</sup>Nanomat/Q-mat/CESAM, Université de Liège, B-4000 Sart Tilman, Belgium

<sup>3</sup>Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

<sup>4</sup>Institute for Research in Electronics and Applied Physics,
University of Maryland, College Park, Maryland 20742, USA

We demonstrate good agreement between mean dwell times measured in 50 nm diameter, perpendicularly magnetized superparamagnetic tunnel junctions (SMTJ), and theoretical calculations based on Langer's theory. Due to a large entropic contribution, the theory yields Arrhenius prefactors in the femtosecond range for the measured junctions, in stark contrast to the typically assumed value of 1 ns. Thanks to the low prefactors, and fine-tuning of the perpendicular magnetic anisotropy, we report measured mean dwell times as low as 2.7 ns under an in-plane applied field at negligible bias voltage. Under a perpendicular applied field, we predict a Meyer-Neldel compensation phenomenon, whereby the prefactor scales like an exponential of the activation energy, in line with the exponential dependence of the measured dwell time on the field. We further predict the occurrence of (sub)nanosecond dwell times as a function of effective anisotropy and junction diameter at zero bias voltage. These findings pave the way towards the development of ultrafast, low-power, unconventional computing schemes operating by leveraging thermal noise in perpendicular SMTJs, which can be scaled down below 20 nm.

In magnetism, mastering thermal activation is essen-
tial for understanding the complex interplay of temper-
ature and dynamics that governs the behavior of spin-
tronic nanostructures. While enhancing thermal stability
is crucial for preventing information loss in nonvolatile
memory [1–3], reduced stability facilitates rapid state
switchings, enabling energy-efficient cognitive computing
schemes [4, 5].

Magnetic tunnel junctions (MTJs) are a prime exam-
ple of devices in which the study of thermal activation
is important. MTJs consist of two ferromagnetic lay-
ers separated by an insulating oxide, and exhibit two
metastable states, corresponding to the relative orien-
tations of the magnetization in the two layers, namely,
parallel (P) and antiparallel (AP). These states, readable
through the tunneling magnetoresistance (TMR), have
distinct resistance levels, and are separated by an energy
barrier. In particular, we refer to junctions in which ther-
mal fluctuations induce random switchings between the
two states at a scale of a few seconds and below [6–8]
as superparamagnetic tunnel junctions (SMTJs). De-
spite their inherently stochastic resistance fluctuations,
the corresponding state probabilities can be controlled
deterministically, either through current-induced spin-
transfer torques (STT) [9–11], or via an external mag-
netic field. This tunability, coupled with their energy
efficiency, has made SMTJs highly appealing for cogni-
tive applications, including stochastic implementations
of artificial neural networks [12], brain-inspired [13], and
probabilistic schemes [14, 15].

A strategy to further reduce energy consumption in
these schemes is to lower the mean dwell times between

magnetization reversals by reducing the energy barrier
between states to a few $kT_{RT}$ [16], where $k$ is Boltzmann's
constant, and $T_{RT}$ is room temperature. Macrospin mod-
els suggest that the mean dwell times cannot be lower
than a characteristic attempt time, $\tau_0 \approx 1$ ns, even
with negligible energy barriers, implying a significant
speed limitation [17]. Such models imply [6, 17] that
in-plane MTJs have larger limiting speeds than perpen-
dicular ones, focusing research on in-plane MTJs, which
are reported to reach dwell times of a few nanosec-
onds [6, 18, 19].

There are several reasons to study perpendicular
MTJs. The importance of dipolar interactions in in-
plane MTJs makes them sensitive to the junction's shape.
When scaled down to nanometer sizes, they become hard
to control due to patterning-induced defects, resulting in
high device-to-device variability. The magnetic proper-
ties of perpendicular MTJs are less susceptible to the
same shape variability, allowing for downscaling below
20 nm [20, 21] and integration into industrial foundry
processes [22, 23].

Experiments including Refs. [24, 25] show that systems
that exhibit domain wall reversal mechanisms have much
lower Arrhenius prefactors than a typical macrospin.
Theoretical studies [26–28] show that entropic contribu-
tions are responsible for this effect, which arises through
the importance of the Helmholtz free energy rather than
the internal energy in the reversal process. These ideas
are validated by the good agreement between calcula-
tions [26] and experiment [24]. As such, the Arrhenius
prefactor of a typical perpendicular MTJ used for data
storage was predicted to vary by many orders of mag-
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Scanning electron microscopy image of a nominal 50 nm diameter perpendicular SMTJ pillar obtained after ion beam etching. (b) Random telegraph noise signal relative to the 85 mV mean voltage of a 50 nm diameter perpendicular SMTJ under an in-plane applied field of $\mu_0H_\| = 60$ mT, and applied current of 15 µA. A 40 ns portion of the recorded voltage-time trace is shown in blue, together with its numerically digitized replica, shown in red. (c) Histogram of the overall voltage values measured over 10 ms and containing more than $10^6$ transitions. (d) Autocorrelation function of the recorded voltage-time trace, partially shown in (b) in blue, with an exponential fit shown in red, with a 3.6 ns time constant.</figcaption></figure>

nitude with material parameters and reach extreme values down to $10^{-21}$ s [27]. The latter stems from the domain-wall-mediated reversal pathway, which not only increases the switching rate through the reduced barrier to reversal, but also favors the transition state against the collinear state due to the increased number of available degrees of freedom at energies close to the extremum. In some cases, this effect yields a case of the Meyer-Neldel compensation rule [27], whereby the Arrhenius prefactor scales like an exponential of the activation energy [29–31]. While compensation was predicted to reduce the stability of memory elements [27, 28], it should be beneficial to achieve faster switching rates for computing.

Kaneko et al. [32] measured the temperature dependence of a set of perpendicular MTJs with diameters below 30 nm, and dwell times just under 1 s at room temperature. Based on an Arrhenius fit to the dwell times, they found prefactors in the ($10^{-18}$ to $10^{-14}$) s range, but dismissed these values as they did not match macrospin simulations. They analyzed their results using Brown's model [33] for a uniaxial macrospin, and concluded that magnon modes play an essential role in the reversal process. Assuming coherent magnetization reversal, Elyasi et al. [34] developed a model, based on the theory of the Suhl instability [35] to account for nonlinear magnon interactions in the Kittel macrospin, and concluded that their approach can explain the disagreement found in the measurements of Ref. [32].

In this work, we experimentally validate the prediction of extremely low Arrhenius prefactors in magnetic tunnel junctions [27]. By measuring mean dwell times, $\tau_{dwell}$, in perpendicular, 50 nm nominal diameter SMTJs under applied in- and out-of-plane magnetic field at low bias voltage, we obtain an excellent agreement with predictions based on Langer's theory [36, 37]. These reveal prefactors at the femtosecond scale. We show that the reduction in prefactor values is instrumental for achieving mean dwell times as short as a few nanoseconds in our experiments. Finally, we theoretically predict the

occurrence of (sub)nanosecond dwell times at zero voltage under a low in-plane field, by reducing the effective perpendicular anisotropy and/or junction diameter.

The stack structure of the fabricated SMTJs consists of Si substrate / SiO$_2$ / Ta(15) / Pt(5) / [Co(0.5)/Pt(0.25)]$_6$ / Ru(0.9) / [Co(0.5)/Pt(0.25)]$_2$ / Co(0.5) / W(0.25) / CoFeB($t_{fixed}$) / MgO(1.25) / CoFeB($t_{free}$) / W(2) / Pt(5). Numbers in parentheses denote nominal thickness in nanometers, and subscripts on brackets show bilayer repeat counts. The two uniform, perpendicularly magnetized ferromagnetic layers of CoFeB, fixed and free, with thicknesses $t_{fixed}$ and $t_{free}$ respectively, are separated by an insulating MgO layer. In order to reach the superparamagnetic regime while keeping the perpendicular easy axis, we tune the effective magnetic anisotropy (see the Supplemental Material (SM) [38] Sec. III with Refs. [39–42]) through the interfacial anisotropy term $K_{int} \propto 1/t_{free}$, by adjusting the nominal free layer thickness from 1.3 nm to 1.8 nm. Measurements are performed on two different stacks labeled 1 and 2, with $t_{free}^{stack1} = (1.21 \pm 0.20)$ nm and $t_{free}^{stack2} = (1.16 \pm 0.20)$ nm, and nominal diameter $d = (50 \pm 20)$ nm (see Fig. 1(a)). The above uncertainties correspond to one standard deviation, with the diameter uncertainty estimated from scanning electron microscopy observations of nanopillars fabricated on the studied wafers, and the thickness uncertainty determined from deposition speed calibration measurements performed by the sputtering tool used during nanofabrication. The free layer's perpendicular anisotropy can be evaluated for both nanopatterned stacks (see SM [38] Sec. IV with Ref. [43]).

We measure voltage fluctuations across the fabricated SMTJs using a high-bandwidth circuit designed to capture submicrosecond events (see SM [38] Sec. I with Ref. [44]). Figure 1(b) depicts a voltage-time trace obtained from measurements performed on stack 1 under a 60 mT in-plane magnetic field and 15 µA mean current (85 mV mean voltage). The signal exhibits nanosecond timescale random telegraph noise (RTN). To ensure
reliable statistics, we collected more than 10⁶ stochastic transitions, allowing the identification of two distinct voltage levels, as shown in the corresponding voltage histogram in Fig. 1(c). These levels are associated with low and high resistance states, respectively, consistent with P and AP magnetic configurations. The nanosecond scale of the observed mean dwell times is further confirmed in Fig. 1(d) by the autocorrelation function calculated over the full duration of the recorded signal, which exhibits approximately exponential decay with a time constant of 3.6 ns.

We demonstrate two different experimental realizations of nanosecond mean dwell times in perpendicular SMTJs, using separately in- and out-of-plane applied magnetic fields. In what follows, experimental results are shown with a red colormap, while theoretical ones use a blue colormap (see Fig. 2(b)).

Figure 2(c) shows the effect of an in-plane magnetic field, $H_\parallel$, on the mean dwell times for stack 1. For electrical readout, we apply a minimal voltage $V_{AP}$ ($V_P$) of 61 mV (55 mV) across the SMTJ and vary $\mu_0H_\parallel$ from 25 mT to 70 mT. The field reduces the energy barriers between states, $\Delta E_{P(AP)}$, (Fig. 2(a)), by reducing the energy of the transition state (see SM [38] Sec. X). These fields are small compared to the anisotropy fields, so that the perpendicular component of the dimensionless magnetization at the minimum energy state satisfies $|m_z| > 0.8$ at zero temperature (see SM [38] Sec. VIII). We observe a large reduction of the mean dwell times with field, from about 10 ms to just above 1 ns, without the need for a large bias voltage.

Figure 2(d) shows the effect of a perpendicular magnetic field, $H_\perp$, for stack 2. The field decreases (increases) $\Delta E_{P(AP)}$ by increasing (decreasing) the energy of state P (AP) (see SM [38] Sec. X). We vary $\mu_0H_\perp$ from 75 mT to 90 mT under a minimal voltage $V_{AP}$ ($V_P$) of -12 mV (-9 mV) across the MTJ.

Figure 2(e) shows that combining the perpendicular field with larger voltages gives smaller timescales yielding a range of mean dwell times, from about 0.2 ms, to 20 ns at the edge of the magnetic field range above 120 mT, for voltages $V_{AP}$ of 360 mV (270 mV) (see SM [38] Sec. II). In the simplest macrospin approximation, STT plays a role similar to the magnetic field in modifying the switching rates [7, 19]. A possible interpretation of the voltage dependence of the dwell times is Joule heating. However, we estimate that the corresponding temperature increase is only 25 K, which would not explain the changes we observe (see SM [38] Sec. VI with Ref. [45]). The phenomenology that has been used to describe the voltage and field dependence [46] of mean dwell times suggests that the two affect reversal differently. Another possibility is that, even though Joule heating is small, the STT excites the magnons [47], thereby increasing the transition rates.

The average energy per switching event, given by

$I^2R\tau_{dwell}$, where I is the applied current, and R is the mean device resistance, can be as low as 1.89 fJ (see SM [38] Sec. V with Ref. [48]) for the conditions used to obtain the results in Fig. 1(b), where $I = 15$ µA, $R = 2.8$ kΩ, and $\tau_{dwell} \approx 3$ ns. This represents a significant energy reduction compared to the smallest values reported in in-plane MTJs [6].

The mean dwell times presented in Fig. 2(d) reveal a seemingly exponential dependence on $H_\perp$ (see SM [38] Sec. VII.b). The rate of thermally activated transitions can be described by the Arrhenius law,

$$\tau_{\text{dwell}}^{\text{P(AP)}} = \tau_{0}^{\text{P(AP)}} e^{\beta \Delta E_{\text{P(AP)}}}, \quad (1)$$

where $\beta = (kT_{RT})^{-1}$, and $\tau_0^{P(AP)}$ is the Arrhenius prefactor. For $H_K \gg H_\perp$, with $H_K$ the anisotropy field, $\Delta E_{P(AP)}$ is approximately linear in $H_\perp$ [49], so the Boltzmann factor in Eq. (1) scales exponentially with $H_\perp$. In what follows, we show that the Arrhenius prefactor also scales like an exponential of $H_\perp$. We find that this explains the overall exponential dependence of the dwell times on the field. Note that the barrier dependence on $H_\parallel$ is not trivial [50], and neither is that of the prefactor, so we cannot infer the dwell time dependence on $H_\parallel$.

To shed light on the mechanism behind the observed dwell times, we perform micromagnetic simulations [27, 28]. For both stacks, we use the exchange stiffness $A = 10$ pJ/m [51], and the Dzyaloshinkii-Moriya interaction (DMI) $D = 0.14$ mJ/m² [52]. For stack 1, we use a saturation magnetization $M_S = 1.16$ MA/m, a mesh size equal to the free layer thickness $a = t_{free} = 1.4$ nm, a diameter $d = 50$ nm, and an effective perpendicular anisotropy $K = 75$ kJ/m³. For stack 2, we use $M_S = 1.03$ MA/m, $a = t_{free} = 1.1$ nm, $d = 40$ nm, and $K = 85$ kJ/m³ (see SM [38] Sec. VII). We also show simulations for $K = 45$ kJ/m³.

Following previous work [27], we use the geodesic nudged elastic band method (GNEB) [53] to compute reversal pathways (see SM [38] Sec. X) and corresponding energy barriers. Figures 3(a, c) show the values of $\Delta E_P$ in units of $kT_{RT}$ with $T_{RT} = 294$ K, for stacks 1 and 2 as a function of in- and out-of-plane fields. In agreement with previous studies, the reversal occurs through the nucleation and propagation of a domain wall through the disk [27, 28, 51, 54, 55]. The saddle point configurations with the domain wall in the center are shown as insets for two values of $H_\parallel$ (Fig. 3(a)), and $K$ (Fig. 3(c)). As shown, the barriers in Fig. 3(c) scale like $\Delta E_0 - \mu_0H_\perp M_S V$, where $\Delta E_0 \propto t_{free}d\sqrt{AK}$ is the barrier at zero field [49], and $V$ is the volume of the free layer.

The Arrhenius prefactor for the P → AP transition is computed with Langer's theory [36, 37] (see SM [38] Sec. IX with Ref. [56]) and is expressed as

$$\tau_0^P = \left( \frac{\lambda_+^P}{2\pi} \Omega_0^P \right)^{-1}, \quad (2)$$
<figure><img src="image_3.png" /><figcaption>FIG. 2. (a) Sketch of the energy profile along the reaction coordinate for P and AP states, where the associated energy barriers are indicated as orange arrows. (b) General legend for (c, d, e). (c, d, e) Experimental (red squares) and theoretical (blue circles) mean dwell times as a function of applied (c) in-plane field for stack 1 (d, e) out-of-plane field for stack 2. The insets in (c, d) show a comparison of the measured dwell times with dwell times computed with the barriers in Fig. 3(a, c) and $\tau_0 = 1$ ns in gray. The error bars on the experimental results indicate single standard deviation uncertainties due to the counting statistics of the dwell times.</figcaption></figure>

in which $\Omega_0^P$ is the dimensionless entropic contribution that contains the curvatures of the energy landscape at the initial state and the saddle point, and $\lambda_+^P$ is the characteristic rate of the unstable translation of the wall at the saddle point.

Figures 3(b, d) show the computed values of $\tau_0^P$. They are divided by two to account for the two equivalent saddle points [27]. We find $\tau_0^P$ in the 0.1 fs to 10 fs range, up to 7 orders of magnitude lower than the value of 1 ns typically considered in the literature [17, 49, 51, 57–59]. Such values stem from entropic contributions from the larger number of states available to thermal fluctuations at the saddle point in the presence of a domain wall, compared to the uniform state. In that sense, the Arrhenius prefactor captures more about the system than

<figure><img src="image_4.png" /><figcaption>FIG. 3. Details of the dwell time computations. (a, c) Energy barriers and (b, d) prefactor as a function of (a, b) in-plane field for stack 1 (c, d) out-of-plane field for stack 2. The insets in (a, c) show the spin configuration at the saddle point for different (a) fields and (c) anisotropies, where the dashed isolines correspond to $m_z = \pm 0.1$ to indicate the position of the wall. The insets in (b, d) show the prefactor as a function of the barrier on a logarithmic scale, highlighting in (d) the compensation effect. The colorscale is the one in Fig. 2(b).</figcaption></figure>

a physical attempt time. Moreover, for families of transitions obtained by varying the anisotropy, and the DMI, this system was shown to exhibit Meyer-Neldel compensation [27, 28], whereby $\tau_0 \propto e^{-\Delta E/E_0}$, where $E_0$ is the characteristic energy of the family of transitions [29–31]. In the insets of Figs. 3(b, d), $\tau_0^P$ is shown in logarithmic scale as a function of the energy barriers obtained as a function of $H_\parallel$, and $H_\perp$. For $H_\perp$, the graphs are approximately straight lines, implying that compensation also tends to occur under a perpendicular magnetic field. We find $E_0 = 6.6 kT_{RT}$ (see SM [38] Sec. VII. b). Along with the linear barrier dependence on $H_\perp$, we obtain

$$ \tau_{\text{dwell}}^{\text{P(AP)}} \propto e^{\mp \mu_0 H_{\perp} M_S V (\beta - E_0^{-1})}. \quad (3) $$

This agrees with the observation that the measured dwell times in Fig. 2(d) exhibit an exponential dependence on the field. Under $H_\parallel$, the system does not seem to exhibit compensation, although the prefactor assumes similarly low values.

We compute the mean dwell times with Eqs. (1)-(2), and compare them in Figs. 2(c, d) with the experimental values, as a function of in- and out-of-plane field. Under $H_\parallel$, the barrier remains symmetric, so $\tau_P = \tau_{AP}$. For $H_\perp$, we compute $\tau_P$, and note that $\tau_{AP}(-H_\perp) = \tau_P(H_\perp)$. In both cases, the computed dwell times agree well with the
experimental values obtained at negligible bias voltage, with barriers of about 5 kT<sub>RT</sub> and 25 kT<sub>RT</sub> respectively yielding nanoscond, and microsecond dwell times, due to the sub-femtosecond prefactors. Additionally, we predict that with a lower anisotropy of 45 kJ/m<sup>3</sup> (pale blue points in Fig. 2(d)), nanosecond dwell times can be obtained even at zero field and current. In the insets of Figs. 2(c, d), we show that dwell times computed from the GNEB barriers and τ<sub>0</sub><sup>P(AP)</sup> = 1 ns (in gray) fail to reproduce the measured dwell times (in pale red) by over 6 orders of magnitude. This discrepancy is even more drastic when using macrospin energy barriers.

Last, given the scalability of perpendicular MTJs, and
the drive for low energy operation, we compute dwell
times for junction diameters from 60 nm to 10 nm to
demonstrate that (sub)nanosecond dwell times can also
be obtained at low field and zero current. We use the
parameters of stack 1 and vary the effective anisotropy
from 100 kJ/m³ to 25 kJ/m³. The in-plane field is set
to 25 mT so that the equilibrium magnetization satisfies
|m<sub>z</sub>| ≥ 0.8 for all anisotropy values. While the in-plane
field reduces the energy barrier, it is in principle not a re-
quirement for nanosecond dwell times, but was included
in the calculations as it allows us to circumvent the sep-
arate treatment of the rotational Goldstone mode that
arises at the saddle point for smaller diameters in axi-
ally symmetric systems [60], as it is not present in the
experimental system.

The energy barriers and the prefactors are given in
Figs. 4(a) and (b). The macrospin model is also shown
in dotted lines (see SM [38] Sec. X), where the prefac-
tor was derived analytically from Langer’s theory [61].
Although the barriers coincide with the macrospin pic-
ture for d ≤ 40 nm, the prefactor still differs from it
by orders of magnitude down to 20 nm diameter, due to
the presence of nonuniform magnon modes. The mean
dwell times thus reach the (sub)nanosecond regime for
K ≤ 35 kJ/m³ at d = 60 nm, for K ≤ 60 kJ/m³ at
d = 30 nm, and for K ≤ 100 kJ/m³ at d ≤ 20 nm. We
note that as Langer’s theory neglects barrier recrossings,
the computed dwell times are typically underestimated
by about one order of magnitude [27], but the underly-
ing physics remains the same.

In this work, we demonstrate good agreement between
mean dwell times measured in perpendicular SMTJs with
50 nm nominal diameter at negligible bias voltage, and
theoretical predictions based on Langer’s theory. The lat-
ter yields Arrhenius prefactors in the femtosecond range
for the measured junctions due to large entropic contri-
butions. Under a perpendicular field, we find Meyer-
Neldel compensation, where the prefactor scales like an
exponential of the energy barrier. This is experimentally
verified by the exponential dependence of the dwell times
on the field.

Fine-tuning of the effective anisotropy in our junc-
tions gives measured nanosecond mean dwell times as

<figure><img src="image_5.png" /><figcaption>FIG. 4. Theoretical predictions of (sub)nanosecond dwell times at zero current and 25 mT in-plane field, in a 1.4 nm thick MTJ using the parameters of stack 1. (a) Energy barriers, (b) Arrhenius prefactors, and (c) mean dwell times as a function of effective anisotropy, for junction diameters between 10 nm and 60 nm. The dotted lines in (a, b) correspond to the macrospin model. The inset in (c) shows a zoom of the dwell times around 1 ns.</figcaption></figure>

low as 2.7 ns under a 70 mT in-plane field at negligible
bias voltage–approximately 4 orders of magnitude shorter
than previously reported in perpendicular SMTJs [62–
65], and similar to values so far reported in in-plane junc-
tions [6, 18, 19], with even lower energy per fluctuation
down to 1.89 fJ. We reach comparable values under an
effective perpendicular field of ±10 mT at larger voltage,
through the effect of STT.

Comparing our dwell time calculations to those of
macrospin models provides evidence that the common
assumption of a constant Arrhenius prefactor in the
nanosecond range is not justified. That assumption gives
dwell times over six orders of magnitude above our mea-
sured values.

We further predict (sub)nanosecond dwell times at zero
current i. for the measured stack 2 parameters under zero
applied field and low anisotropy, and ii. under a 25 mT
in-plane field, as a function of junction diameter and ef-
fective anisotropy.

With minor stack modifications, the perpendicular
SMTJs studied here are well-suited for operating at zero
applied magnetic field [66], which is an important require-
ment for industrial applications. The perpendicular mag-
netic field used in our experiments compensates for the
residual stray field from the magnetic layers surrounding
the free layer; this stray field can be entirely eliminated
through well-known optimizations that are manageable
with current nanofabrication practices. Similarly, if in-
plane fields are needed, they can be included by intro-
ducing a ferromagnetic in-plane layer that generates a
fixed stray field [67], requiring only the optimization of
its thickness and distance from the free layer.

By advancing the fundamental understanding of
stochastic magnetization switchings in magnetic tunnel
junctions, our work hereby paves the way towards the
implementation of (sub)nanosecond-operating stochastic
units for cognitive computing, with perpendicular mag-
netic stacks that can be scaled to tens of nanometers.

We graciously thank L. Buda-Prejbeanu, L. Hutin, E. Yildiz, J.-V. Kim, O. Fruchart, M. Chshiev, H. Béa, and O. Boulle for fruitful discussions. This work was partially supported by the ANR (ANR-21-CE94-0002-01, StochNet project), the NSF (CCF-CISE-ANR-FET-2121957), the Carnot project PRIME SPOT ANR P-22-03813, the French RENATECH network, and the University of Liège under Special Funds for Research, IPD-STEMA Programme.

* L. S. and L. D. contributed equally.

† L. S. and L. D. contributed equally.; louise.desplat@cea.fr

‡ philippe.talatchian@cea.fr

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