# Thermal contribution to current-driven antiferromagnetic-order switching

Myoung-Woo Yoo<EMAIL_ADDRESS>Materials Research Laboratory, University
of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  Department of
Materials Science and Engineering, University of Illinois Urbana-Champaign,
Urbana, Illinois 61801, USA  Virginia O. Lorenz  Materials Research
Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois
61801, USA  Department of Physics, University of Illinois Urbana-Champaign,
Urbana, Illinois 61801, USA  Axel Hoffmann  Materials Research Laboratory,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
Department of Materials Science and Engineering, University of Illinois
Urbana-Champaign, Urbana, Illinois 61801, USA  Department of Physics,
University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA  David G.
Cahill  Materials Research Laboratory, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA  Department of Materials Science and
Engineering, University of Illinois Urbana-Champaign, Urbana, Illinois 61801,
USA  Department of Physics, University of Illinois Urbana-Champaign, Urbana,
Illinois 61801, USA

(August 28, 2024)

###### Abstract

In information technology devices, current-driven state switching is crucial
in various disciplines including spintronics, where the contribution of
heating to the switching mechanism plays an inevitable role. Recently,
current-driven antiferromagnetic order switching has attracted considerable
attention due to its implications for next-generation spintronic devices.
Although the switching mechanisms can be explained by spin dynamics induced by
spin torques, some reports have claimed that demagnetization above the Néel
temperature due to Joule heating is critical for switching. Here we present a
systematic method and an analytical model to quantify the thermal contribution
due to Joule heating in micro-electronic devices, focusing on current-driven
octupole switching in the non-collinear antiferromagnet, Mn3Sn. The results
consistently show that the critical temperature for switching remains
relatively constant above the Néel temperature, while the threshold current
density depends on the choice of substrate and the base temperature. In
addition, we provide an analytical model to calculate the Joule-heating
temperature which quantitatively explains our experimental results. From
numerical calculations, we illustrate the reconfiguration of magnetic orders
during cooling from a demagnetized state of polycrystalline Mn3Sn. This work
not only provides deeper insights into magnetization switching in
antiferromagnets, but also a general guideline for evaluating the Joule-
heating temperature excursions in micro-electronic devices.

##  I Introduction

Novel non-volatile data storage technologies, including magnetic random-access
memory, resistive random-access memory, and phase-change memory, are
fundamentally based on current-induced resistance changes [1, 2, 3]. Despite
the different switching mechanisms inherent in each technology, the
contribution of heating to the mechanisms is inevitable and has been a central
focus of research in various disciplines [4, 5, 6, 7]. With the growing
interest in current-induced state transitions in electronic devices, a
systematic investigation of Joule heating in electronic devices and the
development of comprehensive guidelines for the calculation of temperature
changes are imperative and timely.

In the field of spintronics, current-driven magnetization switching in
antiferromagnets has been studied extensively for the past decade.
Antiferromagnets are a class of magnetic materials in which individual
magnetic moments order in such a way that there is no net magnetization. Due
to their insensitivity to external perturbations and the potential to achieve
high storage densities and fast operating speeds, antiferromagnetic materials
are considered as promising candidates for next-generation spintronic devices
[8, 9, 10, 11].

The use of antiferromagnets for information devices has been challenging due
to the difficulty of detecting antiferromagnetic order on a macroscopic scale
and the local compensation of spin-related electrical and optical responses by
different magnetic sublattices. Recent advances, however, have demonstrated
electrical manipulation and detection of magnetic order in antiferromagnets
due to their characteristic electronic band structures [12, 13, 14, 15, 16,
17, 18, 19, 20]. In these studies, Joule heating must be carefully considered
as it can play a key role in the switching or create a thermal artifacts [18,
21, 22].

Current-driven magnetization switching has been demonstrated in
$D0_{\mathrm{19}}$-type non-collinear antiferromagnets, Mn3$X$ ($X$ = Sn, Ge,
etc.). These non-collinear antiferromagnets, characterized by a kagome atomic
structure and a chiral spin configuration with three sub-lattices, give rise
to a non-vanishing Berry curvature in momentum space, characterized by Weyl
nodes. Consequently, such antiferromagnets exhibit significant macroscopic
electrical and optical responses dependent on their magnetic structure, such
as anomalous Hall effect, anomalous Nernst effect, and magneto-optical Kerr
effect at room temperature, despite their negligible net magnetization [12,
23, 24, 25, 26].

The positions of the Weyl nodes are influenced by the orientation of a cluster
magnetic-multipole, i.e., an octupole moment, which is aligned with a weak net
magnetic moment in the case of Mn3Sn or Mn3Ge [27, 28]. Therefore, the
direction of the octupole moment can be controlled by magnetic fields and
electric currents due to the interplay between Zeeman energy and spin torques
[12, 29, 14].

Since these non-collinear antiferromagnets, such as Mn3Sn and Mn3Ge, have
relatively low Néel temperatures, $\lesssim$ 430 K, Joule heating can play a
critical role. Although it was initially proposed that the octupole switching
is primarily driven by the collective spin rotation induced by the spin-orbit
torque and an in-plane magnetic field [14, 30, 31], other studies have
emphasized the pivotal role of heat, and found that the switching occurs
around the Néel temperature [32, 33]. This issue is a germane problem in
spintronics, where current densities are often close to device breakdown [34,
35, 7, 36, 37]. It is therefore imperative to have a general method for
evaluating the Joule-heating effect.

In this article, we present a systematic methodology and an analytical model
to investigate the thermal contribution to current-driven octupole switching
in the non-collinear antiferromagnet, Mn3Sn. To this end, we varied the
effective thermal resistance independently from the electrical resistance of
the device, allowing us to unambiguously identify the role of thermal versus
electrical effects. Using W/Mn3Sn films on Si/SiO2 substrates with different
SiO2 layer thicknesses, $h_{\mathrm{SiO_{2}}}$, we obtained the threshold
current density for octupole switching, $j_{\mathrm{th}}$, which depends on
both the SiO2 thickness and the base temperature, $T_{0}$. We identified the
heating temperature at the threshold current density, $T_{\mathrm{th}}$, and
consistently showed that $T_{\mathrm{th}}$ remains in all cases above the Néel
temperature, $T_{\mathrm{N}}$. We also developed an analytical model for
calculating the Joule-heating temperature, and showed that it quantitatively
describes our experimental results. From numerical simulations, we illustrate
the reconfiguration of the magnetic octupoles from the demagnetization state.
Our results elucidate the significant role of Joule-heating in current-driven
octupole switching in Mn3Sn, and provide a general method for measuring and
calculating Joule-heating temperatures that is applicable to broader research
areas using micro-electronic devices.

##  II Results and Discussion

### Field-driven octupole switching.

Thermally oxidized Si/SiO2 substrates were prepared with different SiO2
thicknesses, $h_{\mathrm{SiO_{2}}}$, ranging from 100 nm to 1000 nm. A W(7.1
nm) / Mn3Sn(34.4 nm) / MgO(2 nm) film was deposited on these substrates at
room temperature by dc magnetron sputtering. The samples were annealed at
$500\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 30 minutes. The films are
polycrystalline and the atomic ratio of Mn3Sn is Mn:Sn $\approx$ 77:23 [see
Supplementary Material]. The excess Mn atoms stabilize the hexagonal structure
of Mn3Sn [38]. Note that the thickness of the SiO2 layer,
$h_{\mathrm{SiO_{2}}}$, changes the thermal property of the substrates because
the thermal conductivity of SiO2, $\Lambda_{\mathrm{SiO_{2}}}\approx$
$1.3\text{\,}\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}$, is about two orders of
magnitude smaller than that of Si, $\Lambda_{\mathrm{Si}}\approx$
$140\text{\,}\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}$.

Figure 1:  Octupole-moment switching in polycrystalline Mn3Sn. (a) Schematics
of the magnetic dipole moment configurations in Mn3Sn when the octupole
moment, $\hat{\mathbf{g}}$, is oriented in the up (left) and down (right)
directions. The red and green spheres indicate Mn atoms on the first and
second kagome layers, respectively. The insets shows schematics of the
locations of the Weyl nodes in momentum space near the Fermi level [28]. The
red and blue circles in the insets represent the positive (+) and negative (-)
chirality of the Weyl nodes, respectively. (b) $\hat{\mathbf{g}}$ distribution
in a polycrystalline Mn3Sn layer when the average of the octupole moment,
$\langle\hat{\mathbf{g}}\rangle$, is up and down, in which the anomalous Hall
conductivity becomes, $\sigma_{xy}>0$ and $\sigma_{xy}<0$, respectively. (c)
Hall conductivity, $\sigma_{xy}$, as a function of out-of-plane magnetic
field, $H_{z}$, at 300 K for different thicknesses of SiO2 layer,
$h_{\mathrm{SiO_{2}}}$. (d) $\sigma_{xy}$ as a function of $H_{z}$ measured at
different base temperatures, $T_{0}$, for $h_{\mathrm{SiO_{2}}}$ = 300 nm.

We first measured the anomalous Hall conductivity of W/Mn3Sn, which depends on
the positions of the Weyl nodes in momentum space [28]. Since the Weyl points
correlate with the spin ordering in Mn3Sn, the Hall signal can be manipulated
by controlling the octupole-moment direction, $\hat{\mathbf{g}}$ [Fig. 1(a)].
The anomalous Hall effect is maximized when $\hat{\mathbf{g}}$ is
perpendicular to the film plane, and in the case of polycrystalline Mn3Sn, the
signal amplitude depends on the average of the out-of-plane component of the
octupole moment over the sample, $\langle\hat{\mathbf{g}}\rangle$ [Fig. 1(b)].

In Fig. 1(c), we plot the Hall conductivity, $\sigma_{xy}$, as a function of
the perpendicular magnetic field, $H_{z}$, for different
$h_{\mathrm{SiO_{2}}}$ at $T$ = 300 K. As can be seen in Fig. 1(c), all
samples show similar responses to $H_{z}$ regardless of
$h_{\mathrm{SiO_{2}}}$. The residual Hall conductivity is
$\sigma_{xy,0}\approx$
$31\text{\,}\mathrm{\SIUnitSymbolOhm}^{-1}\mathrm{c}\mathrm{m}^{-1}$, which is
comparable to the previously reported value,
$\sim$$20\text{\,}\mathrm{\SIUnitSymbolOhm}^{-1}\mathrm{c}\mathrm{m}^{-1}$, in
a single-layer polycrystalline Mn3Sn film [14]. $\sigma_{xy,0}$ of
polycrystalline Mn3Sn is lower than that of epitaxially grown W/Mn3Sn,
$\sim$$40\text{\,}\mathrm{\SIUnitSymbolOhm}^{-1}\mathrm{c}\mathrm{m}^{-1}$
[30]. The coercive field strength, $\mu_{0}H_{\mathrm{c}}$ =
$0.6\text{\,}\mathrm{T}$, is about three-times larger than that of epitaxially
grown Mn3Sn films, $\mu_{0}H_{\mathrm{c}}\approx$ $0.2\text{\,}\mathrm{T}$
[39, 30].

We also obtained the hysteresis curves at different base temperatures,
$T_{0}$, ranging from 200 K to 400 K. Note that Mn3Sn changes the phase from
the non-collinear state to the incommensurate spin-spiral state below 275 K
[40, 41]. As shown in Fig. 1(d), the shape of the hysteresis loops depends on
$T_{0}$. $\sigma_{xy,0}$ increases with increasing $T_{0}$ from 200 K to 260 K
then decreases with increasing $T_{0}$. $\sigma_{xy,0}$ becomes zero when
$T_{0}$ reaches the Néel temperature, $T_{\mathrm{N}}\approx$ 410 K. The
coercivity decreases monotonically with increasing $T_{0}$ due to thermal
fluctuations, which leads to a decrease of the effective magnetic anisotropy.

### Current-driven octupole switching.

Subsequently, we measured the current-driven octupole-moment switching.
Similar to the case of heavy-metal/ferromagnet structures, the octupole
moments can be controlled deterministically by the spin-orbit torque from the
adjacent heavy-metal layer [14]. Note that the contribution of the intergrain
spin-transfer torque to the switching is negligible because it is much smaller
compared to the spin-orbit torque [29, 42]. To generate the spin-orbit torque,
we applied a dc electric pulse to W/Mn3Sn with a current density $j$ [red
arrow in Fig. 2(a)], and simultaneously applied a small in-plane dc magnetic
field, $H_{x}$, in the current direction [green arrow in Fig. 2(a)]. The pulse
duration time and magnetic field strength were 100 ms and 100 mT,
respectively. The fall time of the pulse is sufficiently long,
$\sim$$1\text{\,}\mathrm{ms}$, so that we can avoid multi-stable octupole
switching [32]. $\mu_{0}H_{x}=$ $100\text{\,}\mathrm{mT}$ is below the
coercivity, $\mu_{0}H_{\mathrm{c}}$ = $0.6\text{\,}\mathrm{T}$, and the value
is optimized to maximized the switching efficiency [33]. The final effective
octupole direction is determined by the directions of the spin-orbit torque
and the magnetic field. After injecting the write current pulse, we measured
$\sigma_{xy}$ with a sufficiently small read current density, $\sim 1.2\times
10^{9}$ $\text{\,}\mathrm{A}\mathrm{m}^{-2}$, to detect the octupole state in
Mn3Sn.

Figure 2:  Current-driven octupole switching in Mn3Sn. (a) An optical
microscopy image of the Hall bar. $j$ and $H_{x}$ indicate a write current
pulse and an in-plane dc magnetic field, respectively. (b) The Kerr rotation
angle, $\theta_{\mathrm{K}}$, as a function of a perpendicular magnetic field,
$H_{z}$. (c)-(d) Magneto-optical Kerr effect microscopy images after applying
a positive ($j>0$) and a negative ($j<0$) pulse. Yellow rectangles indicate
the current flow regions. The brightness reflects the magneto-optical Kerr
effect amplitude. To enhance the contrast, the background image before the
current application was subtracted and a contrast enhancement technique was
used. (e) Hysteresis curves obtained using a current pulse, $j$, for different
thicknesses of the SiO2 layer, $h_{\mathrm{SiO_{2}}}$. (f) Threshold current
density, $j_{\mathrm{th}}$ as a function of $h_{\mathrm{SiO_{2}}}$. The dots
are experimental data and the red line is the calculated $j_{\mathrm{N}}$ from
Eq. 2.

In addition, we observed the octupole switching using the magneto-optical Kerr
effect. First, we measured the Kerr rotation angle, $\theta_{\mathrm{K}}$, as
a function of $H_{z}$ [Fig. 2(b)]. The coercivity in this hysteresis loop is
about 0.6 T which corresponds to the coercivity obtained from the anomalous
Hall effect in Fig. 1(c). The images in Figs. 2(c) and 2(d) show the
differential magneto-optical Kerr effect microscopy images after applying a
positive ($j>0$) and a negative ($j<0$) pulses with positive $H_{x}$,
respectively. The brightness in the current path changes when the pulse
direction is reversed, while the outside brightness is almost conserved. These
Kerr images clearly show the current-driven octupole switching.

We measured the hysteresis loops of the current-driven octupole switching for
different $h_{\mathrm{SiO_{2}}}$ and obtained the threshold current density,
$j_{\mathrm{th}}$, for the switching [see Figs. 2(e) and 2(f)]. Note that we
used the total thickness of the film, $h_{\mathrm{f}}=41.5$ nm, including both
W and Mn3Sn layers to calculate the current density, $j$. In contrast to the
field-driven switching, $j_{\mathrm{th}}$ largely decreases with increasing
$h_{\mathrm{SiO_{2}}}$. $j_{\mathrm{th}}$ decreases by $\sim$40% as the
thickness of $h_{\mathrm{SiO_{2}}}$ increases from 100 nm to 1000 nm. This
result shows that the substrate choice plays a crucial role in spin-orbit-
torque-driven octupole switching.

### Temperature excursion due to Joule heating.

Figure 3:  Effective thermal resistance of Si/SiO2 substrates. (a)-(c)
Longitudinal resistivities, $\rho_{xx}$, of W/Mn3Sn, W, Mn3Sn films as a
function of base temperature, $T_{0}$. (d) $\rho_{xx}$ as a function of
current density, $j$, at $T_{0}$ = 300 K. In (a)-(d), $h_{\mathrm{SiO_{2}}}$ =
$500\text{\,}\mathrm{nm}$. (e) Temperature excursion, $\Delta T$, as a
function of input power, $P$, for different $h_{\mathrm{SiO_{2}}}$. The
symbols and lines indicate the experimental data and the linear fit,
respectively. (f) Effective thermal resistance, $R_{\theta}$, as a function of
$h_{\mathrm{SiO_{2}}}$. The red line is calculated from Eq. 1 with
$\Lambda_{\mathrm{Si}}$ =
$140\text{\,}\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}$ and
$\Lambda_{\mathrm{SiO_{2}}}$ =
$1.3\text{\,}\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}$.

We investigated the temperature excursion due to Joule heating for different
$h_{\mathrm{SiO_{2}}}$ to identify the switching temperature. At steady state,
where the generated heat balances the heat loss to the substrate, the
temperature increase, $\Delta T$, in the W/Mn3Sn microstrip is proportional to
the input power, $P$, i.e., $\Delta T=R_{\mathrm{\theta}}P$, where
$R_{\mathrm{\theta}}$ is the effective thermal resistance of a substrate. It
is important to note that the temperature approaches a saturated state at 100
ms because the heat diffusion distance in silicon, $\sim$2 mm, is much larger
than any length scale in our device,
$\lesssim$$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [see Supplementary
Material].

To measure $R_{\mathrm{\theta}}$ for different $h_{\mathrm{SiO_{2}}}$, we
first determined the longitudinal resistivity, $\rho_{xx}$, of W/Mn3Sn as a
function of the base temperature, $T_{0}$. A small electrical measurement
current density, $\sim 1.2\times 10^{9}$ $\text{\,}\mathrm{A}\mathrm{m}^{-2}$,
was applied to minimize the temperature increase due to Joule heating, $<$1 K.
Figure 3(a) shows a representative $\rho_{xx}$-$T$ relationship for
$h_{\mathrm{SiO_{2}}}$ = 500 nm. $\rho_{xx}$ of the W/Mn3Sn film increases
monotonically, but the slope gradually decreases because $\rho_{xx}$ of W and
Mn3Sn have different temperature dependencies. For W, $\rho_{xx}$ is mostly
proportional to $T_{0}$, as is typical for conventional metals, whereas Mn3Sn
exhibits a non-linear response. In particular, $\rho_{xx}$ of Mn3Sn increases
with increasing temperature below 300 K and then decreases moderately [Figs.
3(b) and 3(c)].

We also measured $\rho_{xx}$ as a function of $j$ at room temperature [Fig.
3(d)]. For this measurement, we applied a 100 ms current pulse, similar to
those used in current-driven switching, and observed the maximum resistivity
before the end of the pulse. As shown in Fig. 3(d), $\rho_{xx}$ increases with
increasing $j$ due to Joule heating. Note that $\rho_{xx}$ largely follows a
quadratic function of $j$, but there are deviations because $\rho_{xx}$ is not
strictly proportional to $T_{0}$.

Using the relationships of $\rho_{xx}$ \- $T_{0}$ and $\rho_{xx}$ \- $j$ in
Figs. 3(a) and 3(d), we determined the temperature increase in W/Mn3Sn due to
Joule heating, $\Delta T$. The results are plotted as a function of $P$ for
different $h_{\mathrm{SiO_{2}}}$ in Fig. 3(e). The plots show that $\Delta T$
is proportional to $P$ and the heating temperature depends on
$h_{\mathrm{SiO_{2}}}$. We obtained the effective thermal resistance,
$R_{\mathrm{\theta}}$, from the slope of the linear fits and plotted
$R_{\mathrm{\theta}}$ as a function of $h_{\mathrm{SiO_{2}}}$ in Fig. 3(f)
(dot symbols). We observed that $R_{\mathrm{\theta}}$ increases monotonically
with increasing $h_{\mathrm{SiO_{2}}}$.

Figure 4:  Temperature, $T_{\mathrm{th}}$, at threshold current density as a
function of $h_{\mathrm{SiO_{2}}}$. The red line shows the Néel temperature,
$T_{\mathrm{N}}\approx$ 410 K.

From $R_{\mathrm{\theta}}$ and $P_{\mathrm{th}}$, we calculated the threshold
temperature, $T_{\mathrm{th}}=T_{0}+R_{\mathrm{\theta}}P_{\mathrm{th}}$, for
various $h_{\mathrm{SiO_{2}}}$ [Fig. 4]. Here, $P_{\mathrm{th}}$ represents
the threshold input power obtained from $j_{\mathrm{th}}$ and $\rho_{xx}(j)$.
$T_{\mathrm{th}}$ is mostly independent of $h_{\mathrm{SiO_{2}}}$, and the
average $T_{\mathrm{th}}$ is about $460\text{\,}\mathrm{K}$, which is higher
than the Néel temperature, $T_{\mathrm{N}}\approx$ $410\text{\,}\mathrm{K}$.
This result shows that temperature plays an important role in the octupole
switching of Mn3Sn, and the switching requires temperatures above the Néel
temperature. Note that the obtained $T_{\mathrm{th}}$ in Fig. 4 is the
threshold temperature in the electrode, and the actual temperature at the
center of the Hall bar cross is lower than $T_{\mathrm{th}}$ due to the heat
and current distribution at the cross structure.

Figure 5:  Current-driven octupole switching below room temperature. (a) Hall
conductivity, $\sigma_{xy}$, as a function of write current density, $j$,
under different base temperatures, $T_{0}$. The SiO2 thickness is 300 nm. (b)
Threshold current density, $j_{\mathrm{th}}$, as a function of $T_{0}$. The
dots are experimental data and the solid line is the calculated
$j_{\mathrm{N}}$ from Eq. 2 (c). Temperature excursions due to Joule heating,
$\Delta T$, as a function of input power, $P$, under different $T_{0}$. The
symbols and lines indicate the experimental data and the linear fit,
respectively. The slope corresponds to the effective thermal resistance,
$R_{\mathrm{\theta}}$ which is about $164.5\text{\,}\mathrm{K}\mathrm{W}^{-1}$
for all $T_{0}$ (d) Threshold temperature, $T_{\mathrm{th}}$, as a function of
$T_{0}$. The red line shows the Néel temperature, $T_{\mathrm{N}}\approx$ 410
K.

We also measured $T_{\mathrm{th}}$ below room temperature. As shown in Fig.
1(d), $\sigma_{xy,0}$ peaks around $T_{0}\approx$260 K, then decreases with
decreasing temperature due to a phase transition [40]. Therefore, here we
measured $T_{\mathrm{th}}$ between 220 K and 300 K. As shown in Fig. 5(a), the
hysteresis behavior varies with $T_{0}$. In particular, $j_{\mathrm{th}}$
decreases monotonically with increasing $T_{0}$ [Fig. 5(b)]. We measured
$\Delta T$ as a function of $P$ at different $T_{0}$, and obtained
$R_{\mathrm{\theta}}\approx$ $164.5\text{\,}\mathrm{K}\mathrm{W}^{-1}$ [Fig.
5(c)], which remains constant in this temperature range. By combining
$j_{\mathrm{th}}$ and $R_{\mathrm{\theta}}$, we calculated $T_{\mathrm{th}}$
[Fig. 5(d)]. $T_{\mathrm{th}}$ is independent of $T_{0}$, averaging
$T_{\mathrm{th}}$ is about 460 K, consistent with the average
$T_{\mathrm{th}}$ in Fig. 4. These consistent results confirm the reliability
of our temperature measurement method.

### Analytical model to calculate the temperature excursion due to Joule
heating.

In Figs. 4 and 5(d), we show that the switching temperature,
$T_{\mathrm{th}}$, remains always above the Néel temperature over all
substrate choices and base temperatures. This result implies that temperature
plays an important role in current-driven octupole switching in Mn3Sn films.
Assuming that switching occurs when the temperature reaches a certain
threshold, $\sim T_{\mathrm{N}}$, the dependence of $j_{\mathrm{th}}$ on
$h_{\mathrm{SiO_{2}}}$ and $T_{0}$ can be explained by the current density
required to reach the Néel temperature, $j_{\mathrm{N}}$.

We calculated the current density required to reach $T_{\mathrm{N}}$. Let us
consider a conductive strip placed on a sufficiently wide Si/SiO2 substrate.
The film thickness, $h_{\mathrm{f}}$, is significantly thinner than the width
of the strip, $w$, i.e., $h_{\mathrm{f}}\ll w$. As soon as an electric current
is applied, the temperature increases due to Joule heating and stabilizes
after a while within 100 ms, because 100 ms is about $10^{3}$ times longer
than the characteristic thermal time scale in the device, $\sim$0.1 ms.
[Supplementary Material]. In this study, we define this condition as steady
state, where the temperature rise can be calculated by $\Delta
T=R_{\mathrm{\theta}}P$ which is independent of time. The effective thermal
resistance of a Si/SiO2 substrate, $R_{\mathrm{\theta}}$, can be obtained from
the thermal resistance of the SiO2 layer, $R_{\theta,\mathrm{SiO_{2}}}$, and
Si substrate, $R_{\mathrm{\theta,\mathrm{Si}}}$.

First, we calculate the effective thermal resistance of the thin SiO2 layer,
$R_{\theta,\mathrm{SiO_{2}}}$. When the SiO2 layer is very thin,
$h_{\mathrm{SiO_{2}}}\ll w$, the generated heat mostly flows perpendicular to
the film plane, and the lateral heat flow can be neglected. In this case,
$R_{\theta,\mathrm{SiO_{2}}}$ can be calculated by
$R_{\theta,\mathrm{SiO_{2}}}=h_{\mathrm{SiO_{2}}}/(wl\Lambda_{\mathrm{SiO_{2}}})$
at steady state, where $\Lambda_{\mathrm{SiO_{2}}}$ is the thermal
conductivity of SiO2 and $l$ is the length of the conductor. Note that the
effect of contact pads on the electrode is excluded here because Joule heating
in these regions is drastically reduced due to the low current density.

Next, we consider the effective thermal resistance of the bulk Si substrate,
$R_{\mathrm{\theta,\mathrm{Si}}}$. In a two-dimensional model with a long
electrode on a large substrate, $R_{\mathrm{\theta,\mathrm{Si}}}$ is given by
$R_{\mathrm{\theta,\mathrm{Si}}}=\ln(\eta
l_{\mathrm{d}}/w)/(\pi\Lambda_{\mathrm{Si}}l)$, where
$l_{\mathrm{d}}=\sqrt{Dt_{\mathrm{p}}}$ is the thermal diffusion length,
$\eta$ is a constant, $D=\Lambda/(\rho C)$ is the thermal diffusivity, $\rho$
is the density, and $C$ is the specific heat [43]. However, in our system,
$R_{\mathrm{\theta,\mathrm{Si}}}$ is almost saturated within $\sim$
$1\text{\,}\mathrm{ms}$, which is much shorter than the pulse length,
$100\text{\,}\mathrm{ms}$ [Supplementary Material]. Therefore,
$R_{\mathrm{\theta,\mathrm{Si}}}$ is considered as a time-independent constant
in this study.

Since the two-dimensional model is not suitable for our system, we numerically
calculated the effective thermal resistance of a Si substrate at steady state,
$R_{\mathrm{\theta,\mathrm{Si}}}$, considering different values of $w$, $D$,
and $l$ [Supplementary Material]. From the simulations, we extrinsically
obtained an analytical form,
$R_{\mathrm{\theta,\mathrm{Si}}}=\ln{(\eta^{\prime}l/w)}/(\pi\Lambda_{\mathrm{Si}}l)$,
where $\eta^{\prime}$ is about 5 when $l\ll h_{\mathrm{Si}}$, where
$h_{\mathrm{Si}}$ is the thickness of the substrate. In our experiments, $l$ =
$120\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $w$ =
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and $h_{\mathrm{Si}}$ =
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

By combining $R_{\mathrm{\theta,SiO_{2}}}$ and $R_{\mathrm{\theta,Si}}$, we
can calculate the effective thermal resistance of Si/SiO2,
$R_{\mathrm{\theta}}$,

$\begin{split}R_{\mathrm{\theta}}=\frac{h_{\mathrm{SiO_{2}}}}{\Lambda_{\mathrm{SiO_{2}}}lw}+\frac{\ln(\eta^{\prime}l/w)}{\pi\Lambda_{\mathrm{Si}}l}.\end{split}$
(1)

We calculate Eq. 1 with $\Lambda_{\mathrm{Si}}$ =
$140\text{\,}\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}$,
$\Lambda_{\mathrm{SiO_{2}}}$ =
$1.3\text{\,}\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}$, and plot the results
in Fig. 3(f) (red line), which shows quantitatively good agreement with our
experimental data. From $\Delta T\leavevmode\nobreak\ =\leavevmode\nobreak\
R_{\theta}P$ and Eq. 1, the current density required to reach the Néel
temperature, $j_{\mathrm{N}}$, can be calculated as follows

$\begin{split}j_{\mathrm{N}}^{2}&=\frac{T_{\mathrm{N}}-T_{0}}{\rho_{xx}h_{\mathrm{f}}wlR_{\theta}}.\end{split}$
(2)

$j_{\mathrm{N}}$ in Eq. 2 is plotted in Figs. 2(f) and 5(b). The calculated
$j_{\mathrm{N}}$ is about 20% lower than $j_{\mathrm{th}}$, but it describes
well the dependence of $j_{\mathrm{th}}$ on both $h_{\mathrm{SiO_{2}}}$ and
$T_{0}$. In all cases, there is a quantitatively and qualitatively reasonable
agreement without any free fitting parameters. Note that the discrepancy
between the calculated $j_{\mathrm{N}}$ and the measured $j_{\mathrm{th}}$ can
be attributed to the differences between the measured threshold temperature,
$T_{\mathrm{th}}$, and the Néel temperature, $T_{\mathrm{N}}$, due to the
reduction of the current density at the center of the Hall bar cross, as shown
in Figs. 4 and 5(d).

To confirm the universality of our model, we calculated the threshold current
density using Eq. 2 in the case of 100% octupole configuration switching in
epitaxial W/Mn3Sn heterostructures [30]. In this study, the threshold current
density is approximately $7.5\times 10^{10}$
$\text{\,}\mathrm{A}\mathrm{m}^{-2}$. Using appropriate parameters,
$\Lambda_{\mathrm{MgO}}$ =
$40\text{\,}\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}$, $T_{\mathrm{th}}$ = 460
K, $\rho_{xx}$ =
$81\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}^{-1}$,
$h_{\mathrm{f}}$ = 37 nm, $w$ = $32\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
and $l$ = $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$, we obtain
$j_{\mathrm{N}}\approx 7.9\times 10^{10}$
$\text{\,}\mathrm{A}\mathrm{m}^{-2}$, which is in good agreement with their
experimental results. This calculation result implies that temperature also
plays a crucial role in octupole switching in epitaxially grown W/Mn3Sn films.

### Numerical calculations of the octupole reconfiguration from the
demagnetization state.

So far, we have demonstrated the importance of temperature in current-driven
octupole switching in Mn3Sn through systematic experiments and an analytical
model to evaluate the Joule heating effects [32, 33]. When an electric pulse
with $j>j_{\mathrm{N}}$ is applied, the temperature rises above
$T_{\mathrm{N}}$, leading to the disappearance of the octupole, while the
individual magnetic moments exhibit random dynamics in the paramagnetic state
[Fig. 6(a)]. As the current decreases after the pulse, the temperature falls
below $T_{\mathrm{N}}$, and the magnetic moments begin to rearrange from a
random orientation. Temperature alone does not provide a preference for the
octupole orientation. Instead, the averaged octupole orientation is influenced
by the residual spin torques below $T_{\mathrm{N}}$. Octupole re-configuration
is initiated near the interface between W and Mn3Sn due to the short spin-
diffusion length in Mn3Sn, $\sim$ $1\text{\,}\mathrm{nm}$, which seeds the
spin texture for the entire layer [33].

Figure 6:  Octupole reconfiguration from demagnetization state in
polycrystalline Mn3Sn. (a) Schematics of the thermal octupole switching
process. The blue and red lines represent the time evolution of a current
pulse, $j$, and temperature, $T$, respectively. The dashed line indicates the
time when the temperature decreases below the Néel temperature, where the
octupole formation starts. The orange arrows show the averaged octupole
polarization, $\langle\hat{\mathbf{g}}\rangle$. (b) Calculated final-octupole
orientations from a thousand random models under different current density,
$j$. Both the initial magnetic moments and the crystal orientations are
randomly distributed [Supplementary Material]. The positive directions of a
magnetic field and the spin torque are parallel to the $+x$ and
$-y$-directions, respectively. (c) The averaged $z$-component of the octupole
moment, $\langle\mathbf{g}_{z}\rangle$, as a function of $j$. The dashed line
is a linear fit. (d) The switching efficiency (red dots and left $y$-axis),
the currenct density at the Néel temperature, $j_{\mathrm{N}}$, (blue line and
right $y$-axis), and threshold current density, $j_{\mathrm{th}}$ (blue
squares and right $y$-axis) as a function of the thickness of SiO2 layers,
$h_{\mathrm{SiO_{2}}}$.

To investigate the control of the octupole state from the demagnetization
state, we performed numerical calculations [Methods and Supplementary
Material]. We computed the spin dynamics of a thousand unit cells with random
crystal orientations and random initial spin configurations to consider the
polycrystalline structure and the demagnetization state just below
$T_{\mathrm{N}}$, respectively. For simplicity, we have not considered the
self-induced spin-transfer torque in the polycrystalline structure because its
effect is much smaller than that of the spin-orbit torque. [29, 42].

Once the simulation starts, the octupole is rapidly formed from a demagnetized
state in $\sim$$0.1\text{\,}\mathrm{ns}$, followed by the rotational dynamics
of the octupole moment [Supplementary Material]. The octupole motion mostly
stops at the energy minimum state in a few ns. The final octupole orientation
is determined by the spin-orbit torque, the magnetic field, and the crystal
orientation. In these calculations, we assumed that the octupole dynamics
occur under nearly constant current and temperature just below
$T_{\mathrm{N}}$, because the fall time of the current,
$\sim$$1\text{\,}\mathrm{ms}$, is much longer than the time for the octupole
dynamics, $\lesssim$ 10 ns.

The distributions of the final octupole orientations are shown in Fig. 6(b)
for different current densities, $j_{\mathrm{hm}}$ = $-6\times 10^{11}$, 0,
and $6\times 10^{11}$ $\text{\,}\mathrm{A}\mathrm{m}^{-2}$ where
$j_{\mathrm{hm}}$ represents the current density in the heavy-metal layer.
These orientations are obtained in the ground state after both current and
field are turned off. When no current is applied, i.e., $j_{\mathrm{hm}}$ = 0,
the octupole directions are mainly oriented in the $+x$ direction due to the
magnetic field, and are uniformly distributed in the $z$-direction, resulting
in an almost zero average out-of-plane octupole moment,
$\langle\mathbf{g}_{z}\rangle$. In the presence of an electric current,
however, the octupoles show a preference for certain orientations depending on
the current direction. Positive and negative currents induce more negative and
more positive $\langle\mathbf{g}_{z}\rangle$, respectively. Figure 6(b) shows
that octupole orientations can be determined stochastically by the current
direction in the polycrystalline Mn3Sn.

Figure 6(c) shows that $\langle\mathbf{g}_{z}\rangle$ is proportional to $j$,
and this implies that the switching efficiency is proportional to
$j_{\mathrm{th}}$ and $j_{\mathrm{N}}$, because the octupole formation occurs
just below $T_{\mathrm{N}}$. The threshold current density, $j_{\mathrm{N}}$,
can be obtained from Eq. 2 by assuming a quasi-static state. To confirm the
relation between $j_{\mathrm{th}}$ and switching efficiency, we plotted the
switching efficiency, $j_{\mathrm{N}}$, and $j_{\mathrm{th}}$ as a function of
$h_{\mathrm{SiO_{2}}}$ in Fig. 6(d). The switching efficiency is obtained from
$\sigma_{xy,0}^{\mathrm{sot}}/\sigma_{xy,0}^{\mathrm{field}}$, where
$\sigma_{xy,0}^{\mathrm{field}}$ and $\sigma_{xy,0}^{\mathrm{sot}}$ are the
residual anomalous Hall conductivity obtained from field- and current-driven
octupole switching, respectively [Figs. 1(c) and 2(e)]. These results confirm
that the switching efficiency is proportional to $j_{\mathrm{N}}$ and
$j_{\mathrm{th}}$, which is consistent with the numerical simulation results
in Fig. 6(c).

##  III Summary and outlook

In this study, we studied a systematic methodology and an analytical model to
evaluate the temperature excursion due to Joule heating in spintronic devices
based on current-driven octupole switching in the non-collinear
antiferromagnet, Mn3Sn. The results show that the threshold current density
for switching depends on both the substrate choice and the base temperature,
while the switching temperature remains essentially constant above the Néel
temperature. The calculated current density for reaching the Néel temperature
from an analytical model quantitatively describes the dependence of the
switching current density. From numerical calculations, we showed the octupole
reconfiguration during cooling in polycrystalline Mn3Sn from the
demagnetization state. This work elucidates the role of temperature in
octupole switching in Mn3Sn and offers fundamental insights into the
demagnetization-mediated octupole reconfiguration. In addition, it provides a
general guideline for characterizing the Joule-heating temperature in micro-
electronic devices, thus allowing a careful assessment of thermal effects,
which is often a concern in spintronic devices.

It is important to note that, in this study, we focused only on spin-orbit-
torque-assisted octupole switching with an adjacent heavy-metal layer.
However, it has recently been reported that switching can also be achieved by
other spin torques, such as the self-induced spin-transfer torque and the
orbital Hall effect, where the role of temperature in octupole switching may
be different [29, 42]. Therefore, further studies with different spin-torque
sources are needed.

##  IV Methods

### Sample preparation and characterization

Thin films of W (7.1 nm)/Mn3Sn (34.4 nm)/MgO (2 nm) were deposited on Si/SiO2
substrates by dc magnetron sputtering. SiO2 layers with eight different
thicknesses ($h_{\mathrm{SiO_{2}}}$ = 100, 200, 300, 400, 500, 700, 900, and
1000 nm) were used in this study . W, Mn3Sn, and MgO films were deposited at
room temperature with a chamber base pressure of $\sim 6\times 10^{-9}$ Torr.
The Mn3Sn films were co-sputtered from Mn and Sn targets. The deposition rates
of W and Mn3Sn were approximately 0.09 nm/s and 0.05 nm/s, respectively. After
deposition, the films were annealed in situ at
$500\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 30 minutes.

Rutherford backscattering was used to determine the composition of the Mn3Sn
film. The stoichiometry of Mn3Sn was determined to be close to Mn:Sn = 77:23
[Supplementary Material].

The film thicknesses were measured by X-ray reflection with a Cu-K$\alpha_{1}$
X-ray at a wavelength of $0.154\text{\,}\mathrm{nm}$. The surface roughness of
W and Mn3Sn is about $\sim$ $0.5\text{\,}\mathrm{nm}$ and $\sim$
$1\text{\,}\mathrm{nm}$, respectively [Supplementary Material].

The magnetization as a function of perpendicular field was measured using a
commercial superconducting quantum interference device (SQUID) magnetometer
(MPMS3, Quantum Design). The saturation magnetization of Mn3Sn is
approximately 0.03 $\mu_{\mathrm{B}}$/f.u. [Supplementary Material].

### Transport measurements

For transport measurements, the films were patterned into Hall bar geometries
with $\sim$ $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ width using
conventional photo-lithography techniques (375-nm maskless laser writer; MLA
150, Heidelberg). The width of the electrodes is about
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and the distance between two Hall
bars is about $120\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The length of the
electrode for the current-driven switching is $\sim$
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. For the electrical
characterization of the devices, Ti (5 nm)/Au (90 nm) contacts were used.

In the current-driven switching experiments, we applied a write current pulse
with a duration of 100 ms. Subsequently, we applied a small dc current, $\sim
1.2\times 10^{9}$ $\text{\,}\mathrm{A}\mathrm{m}^{-2}$, and measured the Hall
voltage. A dc in-plane magnetic field of 0.1 T was applied consistently in the
current direction, $+x$-direction, during both the write and read processes.

To measure the longitudinal resistivity of the Mn3Sn devices, a four-probe
method was used. We applied a 100 ms-long current pulse of $\sim 1.2\times
10^{9}$ $\text{\,}\mathrm{A}\mathrm{m}^{-2}$ and measured the maximum voltage
during the pulse application [Supplementary Material].

The current density, resistivity, and conductivity in the main text are
calculated based on the total thickness of W (7.1 nm)/Mn3Sn (34.4 nm),
$h_{\mathrm{f}}$ = $41.5\text{\,}\mathrm{nm}$. The low and high temperature
measurements were performed using a commercial physical property measurement
system (PPMS, Quantum Design).

### Magneto-optical Kerr effect measurements

The Kerr rotation angle was measured using the polar magneto-optical Kerr
effect (MOKE) system with a 636 nm laser under an out-of-plane external
magnetic field [44]. Both incident and reflected angles were nearly
perpendicular to the film plane. The incident beam was linearly polarized and
the reflected beam was periodically modulated between left and right
circularly polarized light by the photoelastic modulator (PEM-100, Hinds
Instruments). The modulation is performed at a frequency of 50 kHz and a
retardation amplitude of 2.405 radians. The analyzer is set at about
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$. The beam transmits through an
analyzer and is finally detected by a photosensitive diode which provides the
input signal to the lock-in amplifier (Stanford Research Systems, SR830) and a
dc voltmeter.

Based on the Kerr rotation angle measurement, MOKE imaging was performed using
a polar MOKE microscope to observe the current-driven octupole switching. MOKE
provides direct visualization of the spatial distribution of the out-of-plane
component of the magnetic octupole polarization. To increase the image
contrast, we obtained differential images by subtracting the initial image,
and a contrast enhancement technique is used by saturating 3% of the pixels of
the images. All MOKE microscopy measurements were performed at room
temperature.

### Numerical calculations for spin dynamics

We studied the dynamics of the sub-lattice moments based on the Landau-
Lifshitz equation with the spin-orbit torque,

$\begin{split}\frac{d\mathbf{m}_{i}}{dt}=&-\frac{|\gamma|}{1+\lambda^{2}}\left[\mathbf{m}_{i}\times\mathbf{B}^{i}_{\mathrm{eff}}+\lambda\mathbf{m}_{i}\times\left(\mathbf{m}_{i}\times\mathbf{B}^{i}_{\mathrm{eff}}\right)\right]\\\
&+\frac{\theta_{\mathrm{sh}}\hbar|\gamma|j_{\mathrm{hm}}}{2eM_{\mathrm{s}}h}\mathbf{m}_{i}\times\left(\mathbf{m}_{i}\times\mathbf{\sigma}\right)\end{split}$
(3)

where $\mathbf{m}_{i}$ is a unit magnetic moment of site $i$, $\gamma$ is the
gyromagnetic ratio, $\lambda$ is a damping constant, $\theta_{\mathrm{sh}}$ is
a spin-Hall angle, $e$ is the charge of an electron, $M_{\mathbf{s}}$ is the
saturation magnetic moment of the Mn atom, $h$ is a film thickness,
$\mathbf{\sigma}$ is a spin polarization vector, and $j_{\mathrm{hm}}$ is a
current density in the heavy-metal layer. $\mathbf{B}^{i}_{\mathrm{eff}}$ is
the effective magnetic field at each magnetic moment.
$\mathbf{B}^{i}_{\mathrm{eff}}$ can be calculated as

$\begin{split}\mathbf{B}^{i}_{\mathrm{eff}}=-\frac{1}{\mu_{0}M_{\mathrm{s}}}\frac{\partial\cal{H}}{\partial\mathbf{m}_{i}},\end{split}$
(4)

where $\cal{H}$ is the total magnetic energy in a unit cell,

$\begin{split}H=&-J\sum_{\langle
ia,jb\rangle}\mathbf{m}_{ia}\cdot\mathbf{m}_{jb}\\\ &-D\sum_{\langle
i,j\rangle}\hat{\mathbf{z}}\cdot(\mathbf{m}_{i1}\times\mathbf{m}_{j2}+\mathbf{m}_{i2}\times\mathbf{m}_{j3}+\mathbf{m}_{i3}\times\mathbf{m}_{j1})\\\
&-K\sum_{ia}(\mathbf{k}_{a}\cdot\mathbf{m}_{ia})^{2}-\mu_{0}M_{\mathrm{s}}\sum_{ia}\mathbf{m}_{ia}\cdot\mathbf{H},\end{split}$
(5)

where the suffixes $i$ and $j$ denote a unit cell, and $a$ and $b$ denote the
sub-lattices. Our model includes six Mn atoms in a unit cell, taking into
account inter-layer couplings [Fig. 1(a)]. For this calculation, we used the
periodic boundary condition in the $x$-, $y$\- and $z$-directions. $J$, $D$,
and $K$ are the nearest neighbor exchange interaction, the Dzyaloshinskii-
Moriya interaction, and the in-plane magnetic anisotropy, respectively.
$\mathbf{H}$ is an external magnetic field.
$\mathbf{k}_{a}=\left(0,\cos\varphi_{a},\sin\varphi_{a}\right)$ with
$(\varphi_{1},\varphi_{2},\varphi_{3})=(\varphi_{4},\varphi_{5},\varphi_{6})=(\pi,9\pi,5\pi)/6$,
where $\varphi_{a}$ is the angle of the sublattice moments with respect to the
$x$ axis in the kagome plane, i.e., $x$-$y$ plane. In the numerical
calculations, we used $J=-2.8$ meV, $D=-0.22$ meV, and $K$ = 0.187 meV,
$\theta_{\mathrm{sh}}=0.1$, $M_{\mathrm{s}}$ = 3 $\mu_{0}$, and $h$ =
$40\text{\,}\mathrm{nm}$. We did not considered the strain effect which causes
the uniaxial anisotropy. [45, 30].

##  V Acknowledgements

This research is supported by the NSF through the University of Illinois
Urbana-Champaign Materials Research Science and Engineering Center Grant No.
DMR-1720633 and is carried out in part in the Materials Research Laboratory
Central Research Facilities, University of Illinois.

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