# Deterministic and stochastic aspects of current-induced magnetization
reversal in perpendicular nanomagnets

G. Sala<EMAIL_ADDRESS>Department of Materials, ETH Zurich, 8093
Zurich, Switzerland  J. Meyer  Department of Materials, ETH Zurich, 8093
Zurich, Switzerland  A. Flechsig  Department of Materials, ETH Zurich, 8093
Zurich, Switzerland  L. Gabriel  Department of Materials, ETH Zurich, 8093
Zurich, Switzerland  P. Gambardella<EMAIL_ADDRESS>Department
of Materials, ETH Zurich, 8093 Zurich, Switzerland

###### Abstract

We study the incubation and transition times that characterize the
magnetization switching induced by spin-orbit torques in nanomagnets with
perpendicular anisotropy. We present a phenomenological model to interpret the
dependence of the incubation time on the amplitude of the voltage pulse and
assisting magnetic field, and estimate the volume of the seed domain that
triggers the switching. Our measurements evidence a correlation between the
incubation and transition times that is mediated by the temperature variation
during the electric pulse. In addition, we discuss the stochastic
distributions of the two times in terms of the energy barriers opposing the
nucleation and expansion of the seed domain. We propose two models based on
the log-normal and gamma functions to account for the different origin of the
variability of the incubation and transition times, which are associated with
a single nucleation barrier and multiple pinning sites, respectively.

††preprint: AIP/123-QED

The original manuscript has been published with DOI
10.1103/PhysRevB.107.214447 by the American Physical Society, which ows the
copyright of the published article. Publication on a free-access e-print
server of files prepared and formatted by the authors is allowed by the
American Physical Society. The authors own the copyright of these files.

##  I Introduction

Current-induced magnetization switching by either spin-transfer torque (STT)
[Myers  _et al._ , 1999; Wegrowe  _et al._ , 1999; Stiles and Zangwill, 2002]
or spin-orbit torque (SOT) [Miron  _et al._ , 2011; Liu  _et al._ , 2012;
Manchon  _et al._ , 2019] provides the most effective means to achieve
electrical control of nanoscale magnetic devices [Khvalkovskiy  _et al._ ,
2013; Krizakova  _et al._ , 2022]. Although the switching can be described
theoretically by macrospin models [Sun, 2000; Lee  _et al._ , 2013], the
magnetization reversal caused by STT and SOT is in general a complex and non-
uniform process that involves the nucleation and expansion of magnetic domains
[Bernstein  _et al._ , 2011; Baumgartner  _et al._ , 2017]. This mechanism
prevails over the pure macrospin dynamics even in devices as small as few tens
of nm because it is more energetically favorable than the coherent rotation of
all the magnetic moments [Sun  _et al._ , 2011; Chaves-O’Flynn  _et al._ ,
2015; Beik Mohammadi and Kent, 2021; Bouquin  _et al._ , 2021]. Time-resolved
measurements in magnetic tunnel junctions [Devolder  _et al._ , 2008; Tomita
_et al._ , 2008; Grimaldi  _et al._ , 2020; Krizakova  _et al._ , 2020;
Bouquin  _et al._ , 2021] and Hall crosses [Sala  _et al._ , 2021] have
revealed that the non-uniform switching comprises two phases at the ns and
sub-ns timescale: an initial waiting time $t_{0}$ during which the
magnetization is at rest and the actual transition of duration $\Delta t$. The
physical mechanisms underlying $t_{0}$ and $\Delta t$ are similar but not
identical in the STT and SOT scenarios.

In the STT-driven dynamics, the magnetization is switched by the torque
exerted by the spin-polarized current originating from a reference
ferromagnetic layer. The initial waiting time $t_{0}$ results from the
combination of two factors. First, if the electronic spin polarization and the
magnetization are collinear, the torque is initially zero and stochastic
thermal fluctuations are necessary to trigger the reversal [Bultynck  _et al._
, 2018]. Second, the formation of the seed domain that initiates the reversal
always requires to overcome the energy barrier created by the effective
magnetic anisotropy. The duration of the nucleation phase is, therefore,
influenced by the temperature increase caused by Joule heating, the magnetic
field $B$, and the amplitude of the electric pulses (current density $j$ or
voltage $V_{\textrm{P}}$) that drive the dynamics. In this case, $t_{0}$ is
modeled by an Arrhenius-type law with an energy barrier $\Delta U$

$t_{0}=\tau e^{\frac{\Delta U}{k_{\textrm{B}}T}},$ (1)

which describes thermally activated processes [Wegrowe  _et al._ , 1999; Koch,
Katine, and Sun, 2004; Li and Zhang, 2004; Bedau  _et al._ , 2010; Bultynck
_et al._ , 2018; Desplat and Kim, 2020]. Here, $\tau^{-1}$ is the attempt
frequency and $k_{\textrm{B}}T$ is the thermal energy. $\Delta U=\Delta
U(V_{\textrm{P}},B)$ generally depends on both $V_{\textrm{P}}$ and $B$ such
that higher pulse amplitudes and/or larger magnetic fields result in a strong
decrease of the nucleation time [Iga  _et al._ , 2012; Hahn  _et al._ , 2016;
Bultynck  _et al._ , 2018; Grimaldi  _et al._ , 2020]. The ensuing domain
expansion determines the transition time $\Delta t$. Yet, the details of this
process depend on the magnetic properties and dimension of the specific sample
under investigation as well as the pulse amplitude [Hahn  _et al._ , 2016;
Visscher, Munira, and Rosati, 2016; Bultynck  _et al._ , 2018]. The nucleation
can take place at the device edge, and the magnetization reversal can be
driven by the motion of a single domain wall across the device. In this case,
$\Delta t$ is inversely proportional to the domain wall speed [Hahn  _et al._
, 2016; Devolder  _et al._ , 2018; Bouquin  _et al._ , 2021]. However, theory
shows that the switching may also be triggered by magnetostatic instabilities
in which different device regions are perturbed differently by the STT [Munira
and Visscher, 2015; Visscher, Munira, and Rosati, 2016; Beik Mohammadi and
Kent, 2021]. Under these circumstances, $t_{0}$ and $\Delta t$ may not be
associated with two distinct processes.

In the SOT scenario, a spin current is injected into the ferromagnet from an
adjacent non-magnetic layer as a consequence of the spin Hall effect and spin-
orbit scattering occurring in the non-magnet [Manchon  _et al._ , 2019;
Krizakova  _et al._ , 2022] (Fig. 1a). Therefore, the spin polarization and
the magnetization are orthogonal in perpendicular magnets, and no initial
latency is expected [Bedau  _et al._ , 2010; Liu  _et al._ , 2014; Garello
_et al._ , 2014]. However, recent experiments have shown that a finite $t_{0}$
exists also in the SOT-induced dynamics near the critical switching threshold,
because the strong energy barrier due to the perpendicular magnetic anisotropy
can be more easily overcome with thermal assistance [Grimaldi  _et al._ ,
2020; Krizakova  _et al._ , 2020; Sala  _et al._ , 2021; Cai  _et al._ ,
2021]. The similarity between the STT and SOT dynamics suggests that the
nucleation time may be described by an expression similar to Eq. 1 even in the
SOT scenario. However, this possibility and the validity of statistical models
that capture the stochastic aspects of SOT switching have not been tested thus
far. Additionally, the dependence of $t_{0}$ on $V_{\textrm{P}}$ and $B$
(applied parallel to the current direction) in the SOT case has never been
investigated quantitatively. In the SOT-driven switching of perpendicular
magnets, the nucleation of the seed domain usually happens at the device edge
because of the combination of SOT, Dzyaloshinkii-Moriya interaction (DMI), and
external magnetic field [Mikuszeit  _et al._ , 2015; Martinez  _et al._ ,
2015; Baumgartner  _et al._ , 2017; Sala  _et al._ , 2022; Pizzini  _et al._ ,
2014] (see Fig. 1c,d). Thus, in the SOT scenario, $\Delta t$ is usually
identified with the duration of the domain wall displacement between opposite
device edges [Mikuszeit  _et al._ , 2015; Martinez  _et al._ , 2015;
Baumgartner  _et al._ , 2017; Sala  _et al._ , 2022].

This interpretation remains valid as long as domains can be clearly
identified. However, the magnetization reversal in small devices assumes a
mixed character that is neither pure macrospin nor domain-driven [Krizakova
_et al._ , 2022]. In this crossover regime, $t_{0}$ and $\Delta t$ are ill-
defined and cannot be associated with distinct physical processes. This
happens in devices with dimensions comparable to the domain wall width,
typically in the order of 10-50 nm in thin films with perpendicular
magnetization [Chaves-O’Flynn  _et al._ , 2015]. Further, the switching
dynamics is expected to change if ps-long current pulses are used to drive the
magnetization out-of-equilibrium by reducing the saturation magnetization and
magnetic anisotropy at a faster rate than the precessional motion [Jhuria  _et
al._ , 2020]. However, the exact magnetic dynamics at these timescales and the
relative importance of torques and Joule heating remain unclear [Yang  _et
al._ , 2017; Wilson  _et al._ , 2017].

These considerations show that temperature plays a two-fold role in both the
STT- and SOT-driven dynamics. On the one hand, the temperature increase causes
a deterministic variation of the magnetic parameters and, hence, of $t_{0}$
and $\Delta t$. On the other hand, thermal fluctuations lead to a distribution
of the nucleation and transition times. Indeed, non-reproducible dynamics have
been observed by time-resolved single-shot measurements of the STT-induced
switching [Tomita  _et al._ , 2008; Bultynck  _et al._ , 2018; Bouquin  _et
al._ , 2021; Cui  _et al._ , 2010; Zhao  _et al._ , 2012; Grimaldi  _et al._ ,
2020; Krizakova  _et al._ , 2020, 2021; Cai  _et al._ , 2021] and, recently,
of the SOT-induced reversal [Grimaldi  _et al._ , 2020; Krizakova  _et al._ ,
2020; Sala  _et al._ , 2021; Krizakova  _et al._ , 2021; Cai  _et al._ ,
2021]. These measurements show that both $t_{0}$ and $\Delta t$ present in
general skewed statistical distributions [Cui  _et al._ , 2010; Iga  _et al._
, 2012; Zhao  _et al._ , 2012; Bouquin  _et al._ , 2021; Grimaldi  _et al._ ,
2020; Krizakova  _et al._ , 2020; Sala  _et al._ , 2021]. The stochasticity of
the STT-driven dynamics has also been analyzed theoretically in a few
publications [Diao  _et al._ , 2007; Vincent  _et al._ , 2015; Moon, Lee, and
You, 2018; Siracusano  _et al._ , 2018; D’Aquino, Perna, and Serpico, 2020;
Shukla, Parthasarathy, and Rakheja, 2020], but there has been little synergy
between the experimental and theoretical approaches. At the experimental
level, the choice of a specific distribution to fit the data has been seldom
guided by physical considerations. At the theoretical level, exact but rather
complex and mathematically non-explicit distribution functions have been
proposed [Diao  _et al._ , 2007; Vincent  _et al._ , 2015; Moon, Lee, and You,
2018; Siracusano  _et al._ , 2018; D’Aquino, Perna, and Serpico, 2020; Shukla,
Parthasarathy, and Rakheja, 2020], which prevent an intuitive understanding of
the underlying physical processes and limit the applicability of the derived
equations. In addition, most theoretical models rest on the assumption of
coherent macrospin dynamics, which is rarely fulfilled.

Figure 1: (a) Schematic side view of the device. A perpendicularly-magnetized
GdFeCo dot (blue) is fabricated on a Pt Hall cross (yellow). Current-induced
SOT cause the magnetization switching by domain wall motion in the presence of
a magnetic field $\mathbf{B}_{x}$ collinear to the current direction
$\mathbf{j}$. (b) Schematic top view of the device. The detection of the
anomalous Hall voltage $V_{\textrm{H}}$ during the injection of ns-long
electric pulses allows us to track the magnetization switching in real time
[Sala  _et al._ , 2021]. (c) Schematic side view of the magnetization profile
before the nucleation of a domain when the current $\mathbf{j}$ and magnetic
field $\mathbf{B}_{x}$ are oriented along $+x$. In Pt/ferromagnet and
Pt/ferrimagnet bilayers, the DMI favors inward canting of the magnetic moment
when the magnetization is on average pointing up. This canting and the global
tilt towards $+x$ caused by $\mathbf{B}$ assist the nucleation of the seed
domain by the SOT field $\mathbf{B}_{\textrm{SOT}}$ on the left side. When the
magnetization is pointing down, the same combination of current and field
leads to the nucleation on the right side (not shown). (d) Schematic of the
SOT-induced switching of perpendicular magnets as a function of time (defined
by the labels I-V). The black-white contrast indicates the sign of the out-of-
plane component of the magnetization ($m_{z}$). The switching is driven by the
nucleation of a domain at the device edge and the ensuing domain wall motion.
$t_{0}$ and $\Delta t$ are the duration of the two phases, respectively. The
sketches in (c) and (d) are adapted from Ref. [Baumgartner  _et al._ , 2017]

.

Here, we provide a simple phenomenological interpretation of the deterministic
and stochastic dynamics induced by SOT in perpendicular magnets. We have
performed systematic time-resolved Hall measurements of the SOT-induced
magnetization switching in ferrimagnetic Pt/GdFeCo dots with perpendicular
magnetic anisotropy [Sala  _et al._ , 2021] and have analyzed the influence of
the pulse amplitude $V_{\textrm{P}}$, assisting in-plane field $B_{x}$, and
device size on the mean $t_{0}$ and $\Delta t$. We propose a phenomenological
model to describe the dependence of $t_{0}$ on the SOT amplitude, magnetic
field, and self-heating. This model validates the qualitative interpretation
of the SOT switching as a thermally-assisted process, as proposed in previous
works [Grimaldi  _et al._ , 2020; Krizakova  _et al._ , 2020, 2021; Cai  _et
al._ , 2021; Sala  _et al._ , 2021], and allows us to estimate the volume of
the seed domain that initiates the reversal. Our data further evidence a clear
correlation between $t_{0}$ and $\Delta t$ that was not identified before and
that we attribute to the temperature variation during the electric pulse.
Finally, we consider the statistical distributions of $t_{0}$ and $\Delta t$
and propose two physically intuitive models based on the log-normal and gamma
functions to interpret the stochasticity of the two times. These models
suggest a different origin of the stochastical variations of $t_{0}$ and
$\Delta t$, which we associate with a single energy barrier and multiple
pinning sites, respectively. Although our analysis focuses on SOT-induced
switching, our considerations are general and apply also to the magnetization
reversal induced by STT. They are also equally valid for ferro- and
ferrimagnets because our models are independent of the ferrimagnetic order of
GdFeCo. The only assumption that we make is that switching occurs by the
nucleation and expansion of domains.

##  II Experiment

Figure 1a,b shows the structure of our devices and experimental scheme. A
15-nm-thick Gd29Fe64Co7 pillar is fabricated above a 5-nm-thick Pt Hall cross
by optical lithography and ion-beam etching. These devices may be considered
equivalent to the lower half of a top-pinned magnetic tunnel junction.
Gd29Fe64Co7 is a ferrimagnet with room-temperature net magnetization
$M=M_{\textrm{FeCo}}-M_{\textrm{Gd}}$ dominated by the FeCo sublattice and
with perpendicular magnetic anisotropy, as indicated by the hysteresis loops
in Fig. 2a. We chose a rare-earth, transition-metal ferrimagnet because of the
large anomalous Hall effect of these materials [Mimura  _et al._ , 1978],
which facilitates time-resolved measurements in Hall crosses [Sala  _et al._ ,
2021; Krishnaswamy  _et al._ , 2022; Sala and Gambardella, 2022]. However, the
analysis and considerations presented below are independent of the
antiferromagnetic order of ferrimagnetic materials and, hence, apply also to
ferromagnets. In addition, in the following, we consider explicitly a pillar
with 5 µm diameter, but we have verified that our considerations are valid in
devices with smaller dimensions (1, 3 and 4 µm) as well as different
composition. In these samples, we have performed time-resolved single-shot
Hall measurements of the magnetization switching induced by SOT, as described
in detail in Ref. [Sala  _et al._ , 2021]. In short, two electric pulses with
the same amplitude but opposite polarity are injected simultaneously into the
Pt Hall cross (Fig. 1b). The electric current exerts SOT on the magnetization
and, at the same time, generates the anomalous Hall voltage $V_{\textrm{H}}$
proportional to the out-of-plane magnetization. Therefore, the real-time
detection of $V_{\textrm{H}}$ allows us to track with 50 ps temporal
resolution the orientation of the magnetization excited by SOT. Exemplary
time-resolved measurements performed with this approach are presented in Fig.
2b. Here, the temporal variation of the normalized voltage $V_{\textrm{H}}$
indicates that the reversal comprises three phases: the initial quiescent
phase of duration $t_{0}$, the transition during the time $\Delta t$, and the
final equilibrium phase. As discussed before [Grimaldi  _et al._ , 2020; Sala
_et al._ , 2021], we assume that $t_{0}$ and $\Delta t$ are the times required
to nucleate a domain and move the domain wall across the device, respectively
(Fig. 1d). Both $t_{0}$ and $\Delta t$ are expected to depend on the pulse
amplitude $V_{\textrm{P}}$ and magnetic field $B_{x}$, which is applied
collinear to the current direction to ensure the deterministic magnetization
reversal [Manchon  _et al._ , 2019]. For each combination of these two
parameters, we have taken a set of 250 measurements that, after fitting a
piece-wise linear function to the normalized voltage traces (see Fig. 2b),
yielded the distributions of the nucleation and transition times. Hereafter,
$t_{0}$ and $\Delta t$ will stand for the statistical means of the respective
distributions.

##  III Dependence of $t_{0}$ and $\Delta t$ on the pulse amplitude and
magnetic field

Figure 3a-d shows the dependence of $t_{0}$ and $\Delta t$ on the pulse
amplitude and in-plane magnetic field. Both times decrease upon increasing
either parameter, and so do their standard deviations, as also observed in
previous works [Grimaldi  _et al._ , 2020; Krizakova  _et al._ , 2020; Sala
_et al._ , 2021]. Therefore, the dynamics becomes faster and more reproducible
when overdriven. Moreover, for a given pair of current density and magnetic
field, both $t_{0}$ and $\Delta t$ decrease with the device diameter $d$, as
shown in Fig. 3e,f. The linear scaling of $\Delta t$ with $d$ corroborates the
assumption that the transition time is inversely proportional to the domain
wall speed. Indeed, the dependence of $\Delta t$ on $B_{x}$ and
$V_{\textrm{P}}$ agrees qualitatively with predictions of the dynamics of
domain walls driven by SOT in the presence of a magnetic field [Haazen  _et
al._ , 2013; Emori  _et al._ , 2013, 2014; Martinez  _et al._ , 2014]. In
these models, a more intense current exerts stronger torques on the
magnetization of the domain wall and speeds up its motion. However, since SOT
also push the magnetization $\mathbf{m}$ away from the current direction, the
domain wall velocity tends to saturate [Martinez  _et al._ , 2014; Yang, Ryu,
and Parkin, 2015]. The saturation occurs because the damping-like SOT is
proportional to the product $m_{x}j_{\textrm{c}}\sim m_{x}V_{\textrm{P}}$,
which becomes constant in the high current limit. This explains why $\Delta t$
approaches an asymptotic value at large pulse amplitudes. As shown in Fig. 3d,
the asymptote is influenced by the magnetic field because the latter reorients
the saturation magnetization along $x$, thereby increasing $m_{x}$. The same
argument explains why $\Delta t$ decreases as the field is increased at
constant pulse amplitude (Fig. 3c).

The decrease of $\Delta t$ with $d$ is beneficial to the scalability and speed
of spintronic devices. For example, we extrapolate from Fig. 3e that the
transition time of a 100-nm-wide device could be of the order of 120 ps for a
current density of $j_{\textrm{c}}=9\cdot 10^{11}$ A/m2 and a magnetic field
$B=60$ mT. Thus, the switching could be accomplished with pulses as short as
$\approx 150$ ps. Decreasing further the diameter would reduce the minimum
pulse length necessary to switch the magnetization, but a simple estimate of
$\Delta t$ for $d\ll 100$ nm cannot be drawn from our data because of the
onset of coherent dynamics in small devices.

Figure 2: (a) Out-of-plane magnetization of Gd29Fe64Co7 measured by the
anomalous Hall effect during a magnetic field sweep in the $xz$ plane at a
polar angle $\theta_{B}=0^{\circ}$ and $\theta_{B}=87^{\circ}$ from the normal
direction. (b) Exemplary single-shot measurements of the up-down magnetization
reversal in a 5-µm-wide device switched by electric pulses with
$V_{\textrm{P}}=30$ V and 15 ns duration and $B=50$ mT applied parallel to the
current direction. The straight lines are piece-wise linear fits of the
quiescent, reversal, and equilibrium phase. The voltage traces are offset for
clarity, and the times $t_{0}$ and $\Delta t$ of the first switching event are
indicated. The labels I-V refer to Fig. 1d.

###  III.1 Phenomenological model of the dependence of $t_{0}$ on the pulse
amplitude and magnetic field

Figure 3: (a), (b) Dependence of the mean nucleation time $t_{0}$ on $B_{x}$
and $V_{\textrm{P}}$, respectively. The shaded areas define the standard
deviation of the distributions of $t_{0}$. The lines are fits with shared
parameters of Eq. 4 to the data. (c), (d) Same as (a), (b) for the transition
time $\Delta t$. (e) Scaling of $\Delta t$ with the device diameter $d$ for
three different combinations of current density and magnetic field. The lines
are fit to the data with zero intercept. (f) Scaling of $t_{0}$ and
$t_{\infty}$ with $d$. The legend of $t_{0}$ is the same as that of $\Delta t$
in (e). The gray shaded area of $t_{\infty}$ represents the uncertainty in the
fit of Eq. 4 to the field dependence of $t_{0}$.

As mentioned before, the dependence of $t_{0}$ on $V_{\textrm{P}}$ and $B_{x}$
has been discussed in the framework of the STT-induced magnetization reversal
[Wegrowe  _et al._ , 1999; Koch, Katine, and Sun, 2004; Li and Zhang, 2004;
Bultynck  _et al._ , 2018; Desplat and Kim, 2020], but these models have not
been extended to SOT yet. To fill the gap, we propose here a simple
phenomenological model that builds upon the interpretation of the SOT-induced
switching as a thermally-assisted process [Lee  _et al._ , 2014; Grimaldi  _et
al._ , 2020; Krizakova  _et al._ , 2022]. We argue that the non-zero
nucleation time is determined by the energy barrier $\Delta U$ that hinders
the formation of a reversed domain and its domain wall. This barrier is
lowered by the synergetic action of SOT, magnetic field, DMI, and Joule
heating [Mikuszeit  _et al._ , 2015; Baumgartner  _et al._ , 2017; Grimaldi
_et al._ , 2020; Sala  _et al._ , 2022]. The relationship between $t_{0}$ and
$\Delta U$ is captured by an Arrhenius-type law similar to Eq. 1, as typical
of thermal processes and similar to the STT-induced dynamics [Li and Zhang,
2004; Lo Conte  _et al._ , 2014]. The spread of $t_{0}$ is then the result of
thermal fluctuations, as discussed later. In this scenario, $\Delta U$ equals
the energy cost of rotating in plane some of the magnetic moments at the edge
of the dot, where the reversal starts [Mikuszeit  _et al._ , 2015; Baumgartner
_et al._ , 2017] (see Figs. 1d and 4a), and may be expressed as follows:

$\displaystyle\Delta U=\Delta U_{B}+\Delta U_{\textrm{ex}}+\Delta
U_{\textrm{k}}+\Delta U_{\textrm{DMI}}=\Delta U_{B}+\Delta U_{0}.$ (2)

Here, we consider the contributions of the Zeeman energy $\Delta U_{B}$,
exchange energy $\Delta U_{\textrm{ex}}$, effective anisotropy energy $\Delta
U_{\textrm{k}}$, and DMI $\Delta U_{\textrm{DMI}}$, and assume that $\Delta
U_{0}=\Delta U_{\textrm{ex}}+\Delta U_{\textrm{k}}+\Delta U_{\textrm{DMI}}$ is
independent of the applied magnetic field. This assumption does not affect the
validity of our model, which could be refined to include, for example, the
field-dependence of the effective anisotropy. The independence of the
exchange, anisotropy, and DMI energy of the magnetic field is consistent with
the so-called "droplet model" [Pizzini  _et al._ , 2014; Kim  _et al._ ,
2017]. The present analysis can be considered as a reformulation of this model
with no explicit reference to the analytical form of $\Delta U_{0}$. In
general, we expect a decrease of the magnetization and anisotropy caused by
Joule heating [Grimaldi  _et al._ , 2020] and, hence, a dependence of $\Delta
U$ on the applied voltage. Further, we postulate that the SOT itself
contributes to the reduction of the energy barrier. In principle, a well-
defined energy associated with the torques does not exist because the
corresponding effective fields cannot be derived from an energy functional [Li
and Zhang, 2004; Manchon  _et al._ , 2019]. However, it is still possible to
conceive an effective torque contribution to the energy barrier, as it is
often assumed in the STT theory [Koch, Katine, and Sun, 2004; Li and Zhang,
2004; Bedau  _et al._ , 2010; Lo Conte  _et al._ , 2014; Munira and Visscher,
2015; Bultynck  _et al._ , 2018]. The direct (via the SOT) and indirect (via
Joule heating) effects of the pulse amplitude on the energy barrier are
unknown and may not have the same functional dependence on $V_{\textrm{P}}$.
Therefore, we assume for simplicity that, to the lowest order, $\Delta U$ can
be corrected by a term proportional to $V_{\textrm{P}}$, i.e., $\Delta
U\rightarrow\Delta U-\gamma V_{\textrm{P}}$. Then, $t_{0}$ may be calculated
as

$\displaystyle t_{0}(V_{\textrm{P}},B_{x})=\tau(V_{\textrm{P}})e^{\frac{\Delta
U(B_{x})-\gamma V_{\textrm{P}}}{k_{\textrm{B}}T+\eta V_{\textrm{P}}^{2}}},$
(3)

where $\eta V_{\textrm{P}}^{2}$ accounts for the temperature increase caused
by the current. $\tau(V_{\textrm{P}})$ is a characteristic time that could
depend on both $V_{\textrm{P}}$ and $B_{x}$ [Wegrowe  _et al._ , 1999; Desplat
and Kim, 2020]. Since the dependence of $t_{0}$ on $B_{x}$ resembles a simple
exponential (see Fig. 3a), we assume that the effect of the field on $\tau$,
if any, is negligible. This assumption is corroborated a posteriori by the
good quality of the fits in Fig. 3a, as discussed later. On the other hand,
the data in Fig. 3b do not exclude that $\tau$ might depend on the voltage, so
that we keep the notation $\tau(V_{\textrm{P}})$.

A final point to consider is the finite asymptotic value of $t_{0}$ when
$B_{x},V_{\textrm{P}}\rightarrow\infty$. We do not have a conclusive
explanation for the existence of a finite intrinsic nucleation time
$t_{\infty}$, which is of the order of 1.5 ns in Fig. 3a-b. However, we have
observed that $t_{\infty}$ decreases with the dimension of the device and
becomes negligible within the temporal resolution of our measurements and the
precision of the fits in 3-µm-wide dots. A likely explanation is the
following. The incubation time $t_{0}$ is obtained from the fits under the
assumption that the nucleation leads to an appreciable variation of the time-
resolved anomalous Hall voltage. If we assume that the volume of the seed
domain is independent of the device size (see below), the relative variation
of this signal becomes smaller and more difficult to detect in larger devices.
In such a case, $t_{\infty}$ represents an artifact of the measurement without
any physical meaning. This explanation is supported by the decrease of
$t_{\infty}$ with the device diameter $d$ in Fig. 3f. We also note that the
rather large value of $t_{\infty}$ cannot be ascribed to any known magnetic
dynamics, which should have much shorter timescales in the limit
$B_{x},V_{\textrm{P}}\rightarrow\infty$. In any case, $t_{\infty}$ does not
impact the validity of the model of $t_{0}$, its stochastic properties, and
the estimate of the seed volume.

Taking into account this temporal offset, we express the dependence of $t_{0}$
on $B_{x}$ and $V_{\textrm{P}}$ as:

$t_{0}(V_{\textrm{P}},B_{x})=t_{\infty}+\tau^{\prime}(V_{\textrm{P}})e^{-B_{x}/\beta},$
(4)

where

$\displaystyle\beta=\frac{(k_{\textrm{B}}T+\eta
V_{\textrm{P}}^{2})B_{x}}{\Delta U_{B}(B_{x})}=\frac{(k_{\textrm{B}}T+\eta
V_{\textrm{P}}^{2})}{cM_{\textrm{s}}},$ (5)
$\displaystyle\tau^{\prime}(V_{\textrm{P}})=\tau(V_{\textrm{P}})e^{\frac{\Delta
U_{0}-\gamma V_{\textrm{P}}}{k_{B}T+\eta V_{\textrm{P}}^{2}}}.$ (6)

The parameter $\beta$ represents an effective magnetic field, and $c$ depends
on the volume of the seed domain and on the magnetization orientation, as
discussed below. Given the dependence of $\beta$ on the temperature and
saturation magnetization $M_{\textrm{s}}$, we interpret it as a fictitious
field assisting the reversal, analogous to the thermal field used in
micromagnetics to account for stochastic thermal effects [Brown, 1963; García-
Palacios and Lázaro, 1998]. As shown in Fig. 3a, Eq. 4 captures well the
dependence of the nucleation time on the magnetic field. The fits of $t_{0}$
to Eq. 4 yield $\beta\approx 23\pm 2$ mT, with a slight increase of $\beta$
with the voltage, in accordance with Eq. 5. Equation 4 reproduces also the
dependence of $t_{0}$ on the voltage, which is not simply exponential because
of Joule heating (Fig. 3b). The good quality of the fits confirms that the
nucleation is a dynamical process associated with a single energy barrier and
validates our model, which also allows for estimating the volume $\mathcal{V}$
of the reversed domain, as shown below.

###  III.2 Estimate of the activation volume

We assume that the smallest activated volume has a length $L$ along $x$,
starting from the device edge, that equals at least half of a domain wall
width $L=\frac{\delta}{2}$ (see Fig. 4a). Here, we estimate $\delta\approx 35$
nm based on the measured magnetic anisotropy and exchange stiffness of about 2
pJ/m reported in the literature for GdFeCo [Xian-Ying  _et al._ , 2003; Raasch
_et al._ , 1994; Radu and Sánchez-Barriga, 2018].
$\mathcal{A}=wt_{\textrm{FM}}$ is the surface of the domain wall in the $yz$
plane, and $t_{\textrm{FM}}=15$ nm is the thickness of GdFeCo. Before the
nucleation, the magnetic moments are oriented at an average angle $\theta$ to
the $z$ axis determined by the external field. This angle can be determined
from the hysteresis loop in Fig. 2a, for example $\theta=32^{\circ}$ at 100
mT. With the field angle $\theta_{B}=90^{\circ}$, we can write the initial
Zeeman energy as $U_{B}=\frac{\delta}{2}\mathcal{A}M_{\textrm{s}}B\sin\theta$,
where $M_{\textrm{s}}=$ 113 kA/m. To calculate the Zeeman energy
$U_{B}^{\prime}$ after the nucleation, we assume that the domain wall is of
Néel type because of the finite DMI of Pt/ferrimagnet bilayers [Finley and
Liu, 2016; Shahbazi  _et al._ , 2019; Kim  _et al._ , 2019] and applied in-
plane magnetic field. In this case,
$\theta^{\prime}(x)=2\arctan(e^{\frac{x}{\delta}})$ [Mougin  _et al._ , 2007]
and
$U^{\prime}_{B}\approx\int_{0}^{\frac{\delta}{2}}\mathcal{A}M_{\textrm{s}}B\sin\theta^{\prime}\,dx=0.48\delta\mathcal{A}M_{\textrm{s}}B$.
Thus, $c=\delta\mathcal{A}(0.48-\frac{\sin\theta}{2})$ and
$\beta=\frac{k_{\textrm{B}}T+\eta
V_{\textrm{P}}^{2}}{\delta\mathcal{A}M_{\textrm{s}}(0.48-\frac{\sin\theta}{2})}$.
If we take $k_{\textrm{B}}T+\eta V_{\textrm{P}}^{2}\approx k_{\textrm{B}}\cdot
350\,\textrm{K}$, we find $\mathcal{V}=\frac{\delta}{2}\mathcal{A}=4300$ nm3,
$\mathcal{A}=$ 240 nm2, and $w=$ 16 nm. The estimated activation volume is in
good agreement with other estimates obtained from measurements of the field-
induced magnetization reversal and domain wall depinning, which yield
$\mathcal{V}\approx 8\cdot 10^{3}$ nm3 [Wernsdorfer  _et al._ , 1997; Sun  _et
al._ , 2011; Wuth, Lendecke, and Meier, 2012]. This calculation indicates that
the activated region is much smaller than the dot size and corresponds to the
smallest reversed volume compatible with micromagnetic constraints. We also
note that this estimation is independent of the dot dimension as long as the
device is much larger than the domain wall width. In future studies, it may be
interesting to verify how the device shape influences the size of the seed
volume. Estimating the dimension of the seed domain can be useful to identify
the critical dimension at which the transition from the domain-wall driven to
the macrospin dynamics occurs, which is relevant to the downscaling of
devices.

Figure 4: (a) Model of the nucleation of the seed domain at the edge of a
magnetic dot starting from an initial state magnetized "down". The
magnetization $\mathbf{m}$ at the device edge rotates toward the plane. The
gray shading represents this rotation from $-z$ (right) to the $xy$ plane
(left), which occurs on a lengthscale determined by micromagnetic constraints,
i.e., half of the domain wall width $L=\frac{\delta}{2}$. $w$ is the width of
the activated area. (b) Variation of the mean transition time with the mean
nucleation time. For a given magnetic field, distinct data points correspond
to different pulse amplitudes. The solid line is a fit to Eq. 7.

##  IV Correlation between $t_{0}$ and $\Delta t$

Because the nucleation always precedes the expansion of the seed domain, it is
interesting to investigate the relationship between $t_{0}$ and $\Delta t$.
Figure 4b shows that $t_{0}$ and $\Delta t$ correlate over several
combinations of the pulse amplitude and magnetic field. In general, we find
that the data points follow the same universal curve, and the longer $t_{0}$,
the shorter $\Delta t$ is. The relation is not simply linear. However, a
simple way to explain the overall trend in Fig. 4b is based on the assumption
that temperature mediates the correlation between the two times. The domain
wall speed $v$ in a ferrimagnet depends on several parameters but, if we
retain only those that are in principle temperature dependent, we may write
$v\sim\frac{\delta\theta_{\textrm{SHE}}}{\alpha(g_{2}M_{1}+g_{1}M_{2})}$
($g_{i}$ and $M_{i}$ are the Landé g-factor and the saturation magnetization
of the $i^{\textrm{th}}$ sublattice) [Martínez, Raposo, and Alejos, 2019]. The
spin Hall angle $\theta_{\textrm{SHE}}$ is not expected to vary significantly
with temperature since the spin Hall effect in Pt is of intrinsic origin, nor
is the effective damping $\alpha$ of GdFeCo [Kim  _et al._ , 2019]. Since the
exchange stiffness and anisotropy can be expressed as powers of
$\sim\left(\frac{M(T)}{M_{\textrm{s}}}\right)^{p}$ of the net magnetization
[Grimaldi  _et al._ , 2020; Moreno  _et al._ , 2016; Hirata  _et al._ , 2018],
which in turn depends on temperature according to a power law of the form
$M(T)\sim\left(1-\frac{T}{T_{\textrm{c}}}\right)^{q}$, we can write that
$\Delta t\sim
1/v\sim\left(g_{2}M_{1}+g_{1}M_{2}\right)\sqrt{\frac{k_{\textrm{eff}}}{A_{\textrm{ex}}}}\sim\left(1-\frac{T}{T_{\textrm{c}}}\right)^{k}$,
with $T_{\textrm{c}}$ the Curie temperature and $k$ an appropriate
exponential. Further, finite-element COMSOL simulations show that the increase
in time of temperature caused by Joule heating is governed by a law of the
type $T(t)=T_{0}+\Delta T\left(1-e^{-\frac{t}{\tau_{T}}}\right)$, where
$T_{0}$ is the ambient temperature, $\Delta T$ the maximum temperature
variation, and $\tau_{T}$ the corresponding time constant. Upon setting
$t=t_{0}$, we obtain

$\displaystyle\Delta t\sim\left(E+Fe^{-\frac{t_{0}}{\tau_{T}}}\right)^{k},$
(7)

where $E^{k}\approx 8$ ns and $(E+F)^{k}\approx 2.8$ ns are the transition
time in the long and short nucleation time regime, respectively. As shown in
Fig. 4b, Eq. 7 can fit well the dependence of $\Delta t$ on $t_{0}$ with
$k\approx-0.6$ and $\tau_{T}\approx 2$ ns despite our simplifications. This
result supports our hypothesis of a correlation between the two times mediated
by temperature.

##  V Statistical distributions of $t_{0}$ and $\Delta t$

Time-resolved Hall measurements provide insight also into the stochastic
nature of the dynamics. Figure 5a,b shows exemplary distributions of the
nucleation and transition times at different in-plane magnetic fields. As the
field increases, both the mean and standard deviation of the distributions
decrease [Grimaldi  _et al._ , 2020; Krizakova  _et al._ , 2020; Hahn  _et
al._ , 2016], in accordance with the trends in Fig. 3a,c. The distributions
show a similar trend when the control parameter is the pulse amplitude because
in the limit of large $V_{\textrm{P}}$ the dynamics is essentially dictated by
SOT [Vincent  _et al._ , 2015].

Figure 5: (a), (b) Distributions of the nucleation and transition time,
respectively, at different magnetic fields and for a pulse amplitude
$V_{\textrm{P}}=30$ V. The datasets include 250 measurements. The bin size is
100 ps, larger than the nominal temporal resolution of 50 ps, for ease of
visualization. (c), (d) Cumulative distributions obtained from the data in
(a)-(b) and fits (solid lines) to the log-normal and Gamma cumulative
distribution function, respectively.

As a first attempt, we may choose the Gaussian function to describe the
variability of $t_{0}$ and $\Delta t$ because thermal fluctuations are usually
assumed to be Gaussian-distributed [Brown, 1963; Martinez  _et al._ , 2007].
We find, indeed, that the Gaussian function fits reasonably well the
distributions of both $t_{0}$ and $\Delta t$ (not shown). However, we argue
that this function does not provide the correct statistics of the two
characteristic times. In fact, both $t_{0}$ and $\Delta t$ are lower bounded,
i.e., they cannot be smaller than 0. Therefore, we expect that in the limit of
large $V_{\textrm{P}}$ and $B_{x}$ the skewness of the distributions increases
because of the accumulation of data points near 0. Although this tendency is
not clearly observable in Fig. 5a,b, skewed distributions were found in former
measurements of similar ferro- and ferrimagnetic samples [Bouquin  _et al._ ,
2021; Grimaldi  _et al._ , 2020; Krizakova  _et al._ , 2020; Sala  _et al._ ,
2021; Tomita  _et al._ , 2008; Zhao  _et al._ , 2012]. This evidence indicates
the need for alternative statistical models, which, in the case of $\Delta t$,
cannot simply be the exponential function that is sometimes used to model the
depinning time of domain walls [Cui  _et al._ , 2010; Bultynck  _et al._ ,
2018; Burrowes  _et al._ , 2010]. Here we adopt a phenomenological approach
and propose distinct distributions for $t_{0}$ and $\Delta t$ with a direct
physical interpretation.

###  V.1 Interpretation of the stochasticity of $t_{0}$

According to Eq. 1, the nucleation time is a function of the energy barrier
$\Delta U$, hence, its variability stems directly from fluctuations of $\Delta
U$. If we assume that such fluctuations are of thermal origin and hence
Gaussian-distributed [Brown, 1963; Martinez  _et al._ , 2007], then the
statistics of $t_{0}$ is log-normal [Taniguchi  _et al._ , 2014] because the
exponential function $Y=e^{X}$ of a normally-distributed random variable $X$
has log-normal distribution. As shown in Fig. 5c, the cumulative distribution
of $t_{0}$ is well fitted by the cumulative log-normal distribution function.
Although the Gaussian and gamma functions fit equally well to the cumulative
distributions in Fig. 5c, we consider the log-normal function a
mathematically- and physically-sound statistics for the nucleation time. It is
associated with the overcome of an energy barrier, independently of whether
SOT or another force drives the nucleation.

Indeed, in a mean-field model, the energy barrier $\Delta U$ is a macroscopic
quantity obtained by summing the microscopic energies $U_{i}$ of all the
magnetic moments involved in the nucleation of a seed domain. The macroscopic
fluctuations of $\Delta U$ are approximately the sum of the microscopic random
variations of $U_{i}$, that is, $\Delta U$ is a random variable with Gaussian
distribution. As a consequence, $t_{0}\sim e^{\frac{\Delta
U}{k_{\textrm{B}}T}}\approx\prod_{i}e^{\frac{U_{i}}{k_{\textrm{B}}T}}$ is a
log-normal variable because the product of many independent random variables
approaches the log-normal distribution, in the same way as the sum of many
independent variables is normally distributed. This reasoning indicates that,
at the macroscopic level, independent thermal fluctuations are multiplicative,
not additive. Moreover, it implies that the standard deviation of $t_{0}$
grows exponentially with the number of involved magnetic moments, that is,
with the dimension of the seed domain. This implication may be examined in
future works. Note that a log-normal distribution only admits positive values,
consistently with $t_{0},\Delta t>0$.

###  V.2 Interpretation of the stochasticity of $\Delta t$

The motion of a domain wall is influenced by both thermal fluctuations and
local variations of the pinning potential. Micromagnetic simulations that only
include the thermal field predict a Gaussian distribution of depinning times
[Martinez  _et al._ , 2007]. However, previous experimental work has found
evidence of more complex statistics [Im  _et al._ , 2009; Vincent  _et al._ ,
2015], and exponential [Burrowes  _et al._ , 2010; Bedau  _et al._ , 2010],
log-normal [Taniguchi  _et al._ , 2014], and gamma [Metaxas  _et al._ , 2013]
functions have been used to describe the distributions of depinning times from
notches or constrictions. Simple physical considerations, however, support the
choice of a specific distribution. The transition time is associated with the
high-speed and nearly-continuous motion of the domain wall in a small and
confined device [Baumgartner  _et al._ , 2017; Sala  _et al._ , 2022]. During
its propagation, the domain wall can encounter multiple independent pinning
sites in the form of local variations of the magnetic properties, surface and
edge roughness, and device geometry. The transition time is therefore the sum
of the times $\Delta t_{i}$ taken to move from one pinning point to the next
and the depinning times at each site $t_{j}$: $\Delta t=\sum_{i}\Delta
t_{i}+\sum_{j}t_{j}$. We assume for simplicity that the time intervals $\Delta
t_{i}$ are nearly deterministic and determined uniquely by the domain wall
speed, which is in turn set by the external magnetic field and SOT. In
contrast, we consider the depinning times $t_{j}$ as the main source of
stochasticity. We show now that the nature of the pinning determines the
overall statistical distribution of $\Delta t$.

If the depinning process is statistically described by an exponential
distribution, as often assumed [Attané  _et al._ , 2006; Burrowes  _et al._ ,
2010; Bultynck  _et al._ , 2018; Cui  _et al._ , 2010], then the sum
$\sum_{j}t_{j}$ and, hence, $\Delta t$ is mathematically a random variable
with gamma distribution [Gupta, 1960; McGill and Gibbon, 1965]. If, instead,
we assume that each depinning time behaves similarly to $t_{0}$ and is thus a
log-normal variable, then $\Delta t$ is a random variable with no explicit
statistical distribution because the sum of log-normal variables resembles but
is not a log-normal distribution. In practice, the overall pinning landscape
may encompass different pinning potentials with distinct distribution
functions, such that the overall statistic of $\Delta t$ depends on the
details of the sample [Im  _et al._ , 2009]. In the present case, we find that
the cumulative gamma function fits well to the cumulative distributions of
$\Delta t$, as shown in Fig. 5d. A more extensive investigation as a function
of the sample material and geometry is needed to ascertain the actual nature
of the stochasticity of $\Delta t$. We note, however, that the gamma
distribution offers an interesting connection with other physical phenomena.
Gamma-distributed events are particular cases of a Lévy process, which is the
simplest process consisting of a continuous motion with jumps of random size
occurring at random times [Applebaum, 2004]. For instance, the Poisson process
and the Brownian motion, which is the continuous-time version of a random
walk, are Lévy processes. The field- or current-driven dynamics of domain
walls share these features because the domain expansion may be interpreted as
the continuous motion of a domain wall interspersed by random occurrences
(pinning) that alter its speed. The accumulation of successive random events
represents a fundamental difference between the variability of $\Delta t$ and
$t_{0}$.

Our considerations on the stochasticity of $t_{0}$ and $\Delta t$ are based on
the physical processes described by the nucleation and transition times and do
not refer to specific experimental conditions. They are general and remain
valid independently of the amplitude and duration of the electric pulses or
the strength of the magnetic field. Moreover, our model is independent of the
magnetic properties of the sample under investigation and is expected to lose
validity only when the magnetic dynamics approach the macrospin regime, i.e.,
in devices smaller than a few tens of nm.

##  VI Conclusions

We have analyzed the dependence of the nucleation time $t_{0}$ and transition
time $\Delta t$ on the pulse amplitude $V_{\textrm{P}}$ and assisting in-plane
magnetic field $B_{x}$ during the SOT-induced switching of perpendicularly-
magnetized GdFeCo dots. We have found that the decrease of $\Delta t$ with
$V_{\textrm{P}}$ and $B_{x}$ can be interpreted according to existing models
of the domain wall motion driven by SOT. On the other hand, we have proposed a
new framework to understand the variation of $t_{0}$ with current and field
and provided an estimate of the dimensions of the nucleated domain. In our
model, $t_{0}$ depends on the activation energy required to nucleate a seed
domain against a single energy barrier determined by the magnetic anisotropy,
the Zeeman energy, and the DMI. We have also revealed the existence of a
correlation between $t_{0}$ and $\Delta t$ established by the temperature
variation during the switching.

Finally, we have discussed the physical origin of the stochasticity of the two
times, which decreases with strong pulses or magnetic fields. The variability
of $t_{0}$ is associated with thermal fluctuations of a single energy barrier
and is captured by a log-normal function. Instead, the distributions of
$\Delta t$ can be interpreted in terms of a cumulative process of random
events associated with distinct pinning sites. The exact analytical function
describing the statistical distribution of $\Delta t$ depends on the details
of the pinning landscape, but can be approximated by a gamma distribution. We
expect these statistical considerations to be valid also for the STT-induced
magnetization switching because the reversal mechanism (nucleation and domain
expansion) is to some extents similar to that triggered by SOT, and we do not
make explicit reference to the driving force of the dynamics. Overall, our
study provides new insight into the deterministic and stochastic aspects of
the magnetization switching induced by electric currents.

##  VII Acknowledgement

We acknowledge the support of the Swiss National Science Foundation (Grant No.
200020-200465).

##  VIII Bibliography

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