# Fast non-volatile electric control of antiferromagnetic states

S. Ghara111email<EMAIL_ADDRESS>Experimental Physics V,
Center for Electronic Correlations and Magnetism, University of Augsburg,
86159 Augsburg, Germany  M. Winkler  Experimental Physics V, Center for
Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg,
Germany  K. Geirhos  Experimental Physics V, Center for Electronic
Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany
L. Prodan  Experimental Physics V, Center for Electronic Correlations and
Magnetism, University of Augsburg, 86159 Augsburg, Germany  V. Tsurkan
Experimental Physics V, Center for Electronic Correlations and Magnetism,
University of Augsburg, 86159 Augsburg, Germany  Institute of Applied Physics,
Moldova  S. Krohns  Experimental Physics V, Center for Electronic Correlations
and Magnetism, University of Augsburg, 86159 Augsburg, Germany  I. Kézsmárki
Experimental Physics V, Center for Electronic Correlations and Magnetism,
University of Augsburg, 86159 Augsburg, Germany

###### Abstract

Electrical manipulation of antiferromagnetic states, a cornerstone of
antiferromagnetic spintronics, is a great challenge, requiring novel material
platforms. Here we report the full control over antiferromagnetic states by
voltage pulses in the insulating Co3O4 spinel. We show that the strong linear
magnetoelectric effect emerging in its antiferromagnetic state is fully
governed by the orientation of the Néel vector. As a unique feature of Co3O4,
the magnetoelectric energy can easily overcome the weak magnetocrystalline
anisotropy, thus, the Néel vector can be manipulated on demand, either rotated
smoothly or reversed suddenly, by combined electric and magnetic fields. We
succeed with switching between antiferromagnetic states of opposite Néel
vectors by voltage pulses within a few microsecond in macroscopic volumes.
These observations render quasi-cubic antiferromagnets, like Co3O4, an ideal
platform for the ultrafast (pico- to nanosecond) manipulation of microscopic
antiferromagnetic domains and may pave the way for the realization of
antiferromagnetic spintronic devices.

Ferromagnets still represent an important material platform for magnetic
storage and spintronic devices 1; 2. However, they set restrictions on further
miniaturization and speed-up of these devices due to several drawbacks, such
as the cross-talking between the adjacent units via their stray fields, high
sensitivity to external magnetic fields and relatively slow spin dynamics 3;
4.

Though antiferromagnets are free from these limitations, their applicability
has not been put into consideration for long. Even Louis Néel, who discovered
them, stated in his Nobel lecture that antiferromagnets had no practical
importance 5. Contradicting to his prediction, a plethora of recent studies
imply that antiferromagnets can be viable replacements for ferromagnetic
components in spintronic devices 3; 4; 6; 7. In fact, emergent phenomena
associated with antiferromagnets, such as anisotropic magnetoresistance 8; 9,
spin Hall effect 10; 11; 12, spin Seebeck effects 13; 14; 15, Néel spin-orbit
torque 16; 17, ultrafast THz manipulation of antiferromagnetic (AFM) spin
waves 18 and dynamics of AFM skyrmions 19; 20; 21; 22, open up new
opportunities in spintronics. In fact, AFM spintronics, which is an
impressively growing field at the frontiers of magnetism and electronics, is
foreseen to trigger a change of paradigm in information technology 3; 4; 6;
23.

In the last years, the electrical manipulation of AFM domains and the
orientation of the AFM Néel vector ($\mathbf{L}$) have attracted much
attention, as a possible route to write-in and read-out information in AFM
devices 3; 4; 7; 24; 25; 26; 27; 28. The spin-orbit coupling plays a key role
in controlling the orientation of $\mathbf{L}$ in itinerant and insulating
antiferromagnets via electric currents and voltages, respectively 29. In non-
centrosymmetric AFM metals, like CuMnAs 17, Mn2Au 30 and Fe1/3NbS2 31, the
current induced rotation of $\mathbf{L}$ is governed by the spin-orbit torque
and has led to the realization of the first prototype of AFM random access
memories. In non-centrosymmetric AFM insulators, such as Cr2O3 32; 33, LiCoPO4
34; 35; 36 and MnTiO3 37; 38, the linear magnetoelectric (ME) effect, that is
another manifestation of the spin-orbit coupling, can be exploited for the
electric field-driven switching of AFM domain states, as a working principle
for AFM-ME random access memories 39. While the voltage-driven switching of
$\mathbf{L}$ has an advantage over current-driven switching in terms of heat
dissipation, especially for devices with high density of information and/or
active elements, such in situ voltage-driven switching has been observed only
in a few compounds so far, and in all cases only in the vicinity of $T_{N}$
40; 33; 39; 37.

In this proof-of-concept study, we demonstrate the in situ isothermal
switching of $\mathbf{L}$ driven by an electric field deep in the AFM state of
insulating Co3O4. This compound belongs to the family of only $A$-site
magnetic spinels, where Co2+ ions, occupying the diamond lattice formed by the
$A$ sites, have spins S=3/2 in tetrahedral oxygen coordination 41; 42; 43. The
pyrochlore lattice of the $B$ sites is occupied by non-magnetic Co3+ ions.
While Co3O4 has a centrosymmetric crystal structure (space group $Fd\bar{3}m$)
in the paramagnetic state, the Co2+ ions have no inversion symmetry (site
symmetry $\bar{4}3m$) 44 and the onset of Néel-type AFM order at $T_{N}$=30 K
breaks the global inversion symmetry of the lattice, making Co3O4 a non-
centrosymmetric insulating antiferromagnet. Earlier reports showed that Co3O4
undergoes a transition to a simple textbook-like collinear AFM state at
$T_{N}$ 41; 44, in strong contrast to the other highly frustrated $A$-site
spinels, such as $M$Sc2S4 ($M$=Mn, Fe) 45; 46. However, the direction of the
magnetic easy-axis, which can be either along the $\langle$111$\rangle$-type
body-diagonals or along the $\langle$001$\rangle$-type main cubic axes, has
not been resolved yet due to the scarce of single crystals.

This compound, along with a few other $A$-site magnetic spinels, such as
CoAl2O4 and Mn$B_{2}$O4 ($B$=Al, Ga), has recently been reported to exhibit a
linear ME effect 47; 44; 48; 49. Due to the non-centrosymmetric tetrahedral
environment ($\bar{4}3m$) of the magnetic Co2+ ions, the ME free energy has
the following form 47:

$F_{ME}=L_{x}(M_{y}P_{z}+M_{z}P_{y})+L_{y}(M_{z}P_{x}+M_{x}P_{z})\\\
+L_{z}(M_{x}P_{y}+M_{y}P_{x}).$ (1)

Here, the $x$, $y$ and $z$ axes are the main cubic axes, whereas $M$, $L$ and
$P$ stands for the magnetization, the AFM Néel vector and the electric
polarization, respectively. For weak magnetic fields, when $M\ll L$ and the
magnitude of $\mathbf{L}$ is nearly field independent, Eq. (1) yields a linear
ME susceptibility ($\hat{\alpha}$), which solely depends on $\mathbf{L}$
according to,

$\hat{\alpha}\sim\begin{pmatrix}0&L_{z}&L_{y}\\\ L_{z}&0&L_{x}\\\
L_{y}&L_{x}&0\\\ \end{pmatrix}.\quad$ (2)

This specific form of its ME coupling together with its weak magnetic
anisotropy make Co3O4 an ideal benchmarking material to investigate the
electrical control of $\mathbf{L}$. For example, when applying an electric
field along one of the main cubic axes together with a magnetic field
arbitrarily oriented in the plane perpendicular to it, the formation of an AFM
mono-domain state with $\mathbf{L}$ perpendicular to both fields is expected,
if the ME free energy can overcome the magnetocrystalline anisotropy energy.

Here, we demonstrate the full control over the orientation of $\mathbf{L}$,
achieved by simultaneous electric and magnetic fields in the AFM state of
Co3O4. The electrical switching of $\mathbf{L}$ in macroscopic volumes is
accomplished only within a few microseconds, pointing to an ultrafast
intrinsic switching mechanism for microscopic AFM domains.

Figure 1: $\mid$ Realization of two distinct AFM mono-domain states via ME
poling. a, Temperature-dependent electric polarization $P_{z}$ measured after
ME poling with $E_{p}$=+4.2 kV/cm ($\parallel z$ axis) and various
$\mu_{0}H_{p}$ ($\parallel x$ axis). The inset shows the temperature-dependent
ME coefficient $\alpha$. b, Magnetic field-dependent electric polarization at
4 K measured after ME poling with all four sign combinations of the poling
fields, $E_{p}$ = $\pm$4.2 kV/cm and $\mu_{0}H_{p}$ = $\pm$14 T. In all these
cases, the electric polarization was measured with $E_{p}$ being switched off
after ME poling. c, Schematics of $+L_{y}$ and $-L_{y}$ AFM mono-domain states
in Co3O4. The red dots between the Co2+ ions of opposite magnetization
indicate the centers of inversion in the paramagnetic cubic state, which are
broken by the AFM order.  Figure 2: $\mid$ In situ switching and rotation of
$\mathbf{L}$. a, Magnetic field ($\mathbf{H}$ $\parallel x$ axis) dependence
of the electric polarization at 4 K measured in the presence of electric
fields ($\mathbf{E}$ $\parallel z$ axis). b, Sketch showing the orthogonality
of the AFM Néel vector $\mathbf{L}$ (red arrow) to the magnetic field (blue
arrow) and introducing the angles $\phi$ and $\theta$. c, Schematic diagram
showing the favoured direction of $\mathbf{L}$, either $+L_{z}$ or $-L_{z}$,
depending on the orientation of the magnetic field rotated within the $xy$
plane, when the electric field is applied along the $y$ axis. d, Schematics of
the in-plane rotation of $\mathbf{L}$ with either preceding (yellow shaded
angular range) or following (green shaded angular range) the magnetic field
rotating in the $xy$ plane, when the electric field is applied along the $z$
axis. e/f, Dependence of the electric polarization component $P_{y}$/$P_{z}$
on the orientation of the magnetic field (14 T) rotating within the $xy$ plane
at 4 K in the presence of an electric field (+17.4 kV/cm) applied along
$y$/$z$ axis. The purple curves in the two panels are the experimental data.
The solid/dashed black curve in panel e indicates the calculated behaviour of
$P_{y}$, according to Eq. (3), for the $+L_{z}$/$-L_{z}$ mono-domain state,
and those in panel f show the expected behaviour of $P_{z}$, according to Eq.
(4), when $\mathbf{L}$ rotates with preceding/following the magnetic field
within the $xy$ plane. In all panels, the yellow and green background colours
indicate field or angular regions, where $\mathbf{L}$ is preserved (panels a,
c & e) or continuously rotated (panels d & f). At the borders of these regions
$\mathbf{L}$ is switched suddenly.

AFM domain selection via ME effect

Figure 1 summarizes our results on the linear ME effect obtained after poling
with orthogonal electric and magnetic fields, namely $E_{p}\parallel z$ axis
and $H_{p}\parallel x$ axis. As clear from Fig. 1a, the electric polarization
emerges below $T_{N}$ in proportion to the magnetic field, characteristic to
the linear ME effect. The value of $\alpha$ is found to be $\sim$11.3 ps/m at
4 K (inset of Fig. 1a), which is four times larger compared to that of the
prototypical ME material Cr2O3 33. Note that $\alpha$ stands for the $xz$
component of $\hat{\alpha}$ throughout this work. The linear behaviour of the
ME effect is directly evidenced in the magnetic field-dependent electric
polarization (Fig. 1b). The sign of $\alpha$, that is the sign of the slope of
the $P$-$H$ curve, depends on the relative signs of the poling fields:
$\alpha$ is positive/negative when the product $H_{p}E_{p}$ is
positive/negative. These results, together with the presence/absence of
electric polarization for perpendicular/parallel orientation of the electric
and magnetic fields (shown in Fig. 2e and discussed below), confirm the
specific form of the linear ME effect, as given in Eqs. (1) & (2).

For electric and magnetic fields, respectively, applied along the $z$ and the
$x$ axes, the ME free energy term has the simple form $L_{y}M_{x}P_{z}$ and,
thus, the $+L_{y}$ AFM mono-domain state is expected to emerge for
$H_{p}E_{p}>0$ and the $-L_{y}$ mono-domain state for $H_{p}E_{p}<0$ (see Fig.
1c). Indeed, such selection of the $\pm L_{y}$ mono-domain states for opposite
signs of $H_{p}E_{p}$ is implied by $P$-$H$ curves in Fig. 1b. Please note
that in the present case, $\pm L_{y}$ AFM states are interchanged not only by
time reversal but also by spatial inversion, that is why the sign of
$\mathbf{L}$ is linked to the sign of $H_{p}E_{p}$. Before measuring the
electric polarization at $T$ = 4 K, shown in Fig. 1b, the poling electric
field was switched off. Since the $P$-$H$ curve is linear in the whole
magnetic field range, we can conclude that the AFM mono-domain state,
established by poling with crossed $E_{p}$ and $H_{p}$, was preserved upon the
entire magnetic field sweep. We note that magnetic anisotropy with easy axes
along the cubic $\langle$111$\rangle$-type directions could prevent the
formation of the $\pm L_{y}$ AFM mono-domain states. However, the
magnetization of Co3O4 is found to be independent of the orientation of the
magnetic field (see the supplementary figure 1), which indicates the weakness
of magnetic anisotropy and the dominance of the external fields in determining
the orientation of $\mathbf{L}$ via the ME coupling.

Figure 3: $\mid$ Voltage pulse-driven switching of $\mathbf{L}$. a, The $+$,
$+$, $-$, $-$, … electric pulse sequence (dashed line/left scale) for
$E\parallel z$ axis and the measured switching current (solid line/right
scale) at 4 K in the presence of +14 T ($\parallel x$ axis). b, ME hysteresis
loop at 4 K in +14 T. The difference between the $+$ and $-$ remanent value of
the ME coefficient, $\Delta\alpha$, is a measure of the switching efficiency.
The sequence of pulses is indicated by colours according to panel a. c,
Dependence of the ME coefficient $\alpha$ on the strength of the pulsed
electric field (purple), when +14 T is constantly applied, and on the strength
of the static magnetic field (green), while exposing the sample to $\pm$17.4
kV/cm electric pulses. The $+$/$-$ signs of $\alpha$ corresponds to the
$+L_{y}$/$-L_{y}$ AFM mono-domain state. d, Time traces of the driving
electric pulses (light blue line) for various rise times ($t_{r}$) and the
time traces of the corresponding switching currents (dark blue line). The
latter are plotted on a common vertical scale after multiplication with
numerical factors, as indicated. The estimated switching time of the AFM
domains is indicated by $t_{s}$. e, Dependence of the switching time $t_{s}$
(left scale) and the switching efficiency $\Delta\alpha$ (right scale) on the
rise time $t_{r}$ of the driving electric pulse. The data indicated by solid
and open symbols were obtained on a thick and a thin crystal, respectively
(see Methods). f, Dependence of the switching efficiency on the number of $+$,
$-$, … pulses sequences at 4 K in +14 T. The solid and dashed lines in panel
c, e and f are guides to the eyes.

Magnetic control and kinetics of $\mathbf{L}$

In the following, we demonstrate that the AFM mono-domain state can be
achieved not only by ME poling but can also be created in situ from a multi-
domain state by orthogonal electric and magnetic fields, even at temperatures
well below $T_{N}$. In addition to the mono-domain creation, we control the
orientation of $\mathbf{L}$ in two ways: We reverse it via the reversal of
either the electric or the magnetic field, and rotate it by rotating the
magnetic field, while the electric field is kept constant.

For these studies, the sample was cooled to 4 K without any poling fields. In
the absence of electric field, no magnetically induced polarization was
observed, indicating that the net $\mathbf{L}$ averages to zero in the multi-
domain state. When starting from the virgin multi-domain state and applying a
constant electric field of $E$=$+$17.4 kV/cm along the $z$ axis, the sample is
set to the $+L_{y}$ mono-domain state by magnetic fields exceeding 2 - 3 T,
applied along the $x$ axis. This is shown in Fig. 2a, where the initial
parabolic increase of the $P$-$H$ curve turns to linear above this field
range, implying no further changes in domain population. Indeed, in $+$14 T
the electric polarization reaches the same value as recorded after ME poling
(cf. Figs. 1a and 1b), also evidencing that the $+L_{y}$ mono-domain state has
been achieved. Intriguingly, when sweeping the magnetic field back with
keeping the constant electric field on, the slope of the $P$-$H$ curve changes
sign for negative magnetic fields, indicating the switching to the $-L_{y}$
mono-domain state. This is opposite to the case with no electric field
applied, when the poled $+L_{y}$ mono-domain state is preserved upon the sign
change of the magnetic field (Fig. 1b). Electric fields as low as $+$1.7 kV/cm
are still efficient in increasing the population of the $+L_{y}$ state and
switching the sign of the excess $+L_{y}$ component, but the mono-domain state
is only reached for $E\geq$ $+$8.7 kV/cm, as seen in Fig. 2a. The same domain
selection and switching processes are also demonstrated in Fig. 2a for the
opposite sign of the electric field.

After demonstrating the in situ ME domain control, we investigate the kinetics
of $\pm\mathbf{L}$ switching and the rotation of $\mathbf{L}$. This is done by
analysing the dependence of electric polarization on the orientation of the
magnetic field using Eqs. (1)-(2). Due to the negligibly small anisotropy
energy, $\mathbf{M}$ coaligns with magnetic fields larger than 2-3 T,
irrespective of the field direction, and $\mathbf{L}$ can point along any
direction perpendicular to $\mathbf{M}$, as depicted in Fig. 2b. Via the ME
free energy, the application of an additional electric field can select an
unique $\mathbf{L}$, which determines the electric polarization, probed
experimentally, according to Eq. (2).

Figure 2e and f respectively show the dependence of $P_{y}$ and $P_{z}$ on the
orientation of the magnetic field, rotated in the $xy$ plane. In the two
cases, the electric field was applied along the measured electric polarization
component, i.e., along the $y$ or $z$ axis. Based on Eq. (1) and the
orthogonality of $\mathbf{L}$ to the magnetic field, the angular dependences
of the two electric polarization components have the following forms:

$\displaystyle P_{y}\propto M_{z}L_{x}+M_{x}L_{z}=\eta\cos(\phi)\sin(\theta),$
(3) $\displaystyle P_{z}\propto
M_{x}L_{y}+M_{y}L_{x}=\eta\cos(2\phi)\cos(\theta).$ (4)

where $\eta$ is a constant factor, $\phi$ is the angle between the magnetic
field and the $x$ axis, and $\theta$ is the angle of $\mathbf{L}$ with respect
to the $xy$ plane, as shown in Fig. 2b. Since the magnetic anisotropy is
negligible, the angle $\theta$ is determined by the ME free energy. When the
electric field is applied along the $y$ axis, the ME free energy is minimized
by the $+L_{z}$ state ($\theta=+90\degree$) and the $-L_{z}$ state
($\theta=-90\degree$) for $0\degree<\phi<90\degree$ and
$90\degree<\phi<180\degree$, respectively, as depicted in Fig. 2c (For the
details of the calculation see the supplementary note 2 and supplementary
figure 2). Thus, upon the rotation of the magnetic field, the
$+L_{z}\rightarrow-L_{z}$ switching is expected when the magnetic and electric
fields become parallel, while the back-switching should take place when they
are antiparallel. The solid/dashed black curve in Fig. 2e indicates the
calculated behaviour of $P_{y}$, according to Eq. (3), for the
$+L_{z}$/$-L_{z}$ mono-domain state. The angular dependence of the measured
$P_{y}$ (purple curve in Fig. 2e) is well captured by this model, confirming
the sudden switching between the $\pm L_{z}$ states.

When the electric field was applied along the $z$ axis, the ME energy is
minimized by $\theta=0\degree$ for $-45\degree<\phi<45\degree$ and
$135\degree<\phi<225\degree$, and by $\theta=180\degree$ for
$45\degree<\phi<135\degree$ and $225\degree<\phi<315\degree$, as depicted in
Fig. 2d. This means $\mathbf{L}$ lies within in the $xy$ plane during the
entire rotation of $\mathbf{H}$. More specifically, the mutual orthogonality
among $\mathbf{H}$, $\mathbf{E}$ and $\mathbf{L}$, according to Eq. (1),
results in the smooth rotation of $\mathbf{L}$, within each of the four
$90\degree$ segments mentioned above. However, at the edges of these segments,
i.e. at $\phi$ = $45\degree$, $135\degree$, $225\degree$ and $315\degree$,
$\mathbf{L}$ is suddenly reversed. Therefore, if $\mathbf{L}$ precedes the
magnetic field in one segment, then it follows the field in the next segment,
as sketched in Fig. 2d. (For more details see supplementary note 2 and
supplementary figure 3). The experimentally observed behaviour of $P_{z}$, as
shown by purple curve in Fig. 2f, is also confirmed by our model. These
results speak for the full controllability of $\mathbf{L}$ via the ME effect,
and no significant influence of magnetic anisotropy on the kinetics of
$\mathbf{L}$. Such an unconstrained in situ ME control of $\mathbf{L}$ even at
$T\ll T_{N}$ makes Co3O4 unique among antiferromagnets.

Pulsed voltage control of AFM domains

Next, we demonstrate the in situ switching of $\mathbf{L}$ by pulsed electric
fields. Fig. 3a displays the effect of $+$, $+$, $-$, $-$, … voltage pulse
sequences on an initially multi-domain crystal at 4 K. The pulsed electric
field and the static magnetic field were oriented along the $z$ and $x$ axis,
respectively. The current measured with the applied electric field exhibits
peaks only when the sign of the voltage pulse is changed, i.e. at the first
$+$ and first $-$ pulses. This current peak originates from the electric field
induced switching of the ME polarization. The lack of current peaks associated
with the second $+$ and the second $-$ pulses indicate that the first $+$ and
$-$ pulses established the $+L_{y}$ and $-L_{y}$ mono-domain states,
respectively. During the switching process, the time dependence of $\alpha$ is
obtained by dividing the electric polarization (the time-integral of the
current) by the magnetic field value. The switching process is more easily
accessible in Fig. 3b, where we directly plot the magnetoelectric coefficient
versus the electric field, which resembles a typical ferroelectric hysteresis
loop, and demonstrates that $\alpha$ is reversed when the electric field is
inverted. Note that the AFM domain switching is non-volatile, as indicated by
the high-remanent $\alpha$ observed after all electric pulses in Fig. 3b. This
magnitude of remanent $\alpha$ agrees with the value in the inset of Fig. 1a,
measured on a mono-domain sample after switching off the electric poling
field. Though a similar in situ ME control of AFM domains has been reported in
Cr2O3 and MnTiO3, the switching in those cases are limited to the vicinity of
$T_{N}$ 40; 33; 39; 37.

In Fig. 3c, we analyse the dependence of switching efficiency on the strength
of the pulsed electric field in +14 T at 4 K. While electric fields $\gtrsim$
9 kV/cm are required to create $\pm L_{y}$ mono-domain states, weaker electric
pulses still lead to considerably high $\pm\alpha$ values in a reproducible
fashion. This implies that a full control over $\mathbf{L}$ can be achieved by
lower electric fields when microscopic AFM domains are subject to manipulation
and not macroscopic volumes, like in the present case. A similar behaviour of
$\alpha$ on the strength of the static magnetic field indicates that magnetic
fields $\gtrsim$ 6 T are necessary to create in situ $\pm L_{y}$ mono-domains
by electric pulses of $\pm$17.4 kV/cm, as seen in Fig. 3c.

Next, we investigate the dynamics of the switching with particular interest on
the switching speed, a parameter highly relevant to device applications. The
dependence of the switching time ($t_{s}$) on the rise time of the driving
electric pulse ($t_{r}$) is shown in Fig. 3e. Time traces of the switching
current are displayed in Fig. 3d for three different values of $t_{r}$, where
$t_{s}$ is obtained as the full width of the current peak at half maximum. The
switching is found to be as fast as $t_{s}\approx$ 5.5 $\mu$s for $t_{r}=$ 10
$\mu$s. We have to stress that for all $t_{r}$ values studied here, the
switching was completed during the rise of the driving voltage, with the
current peak located typically at $t\lesssim\frac{t_{r}}{2}$. This
observation, together with the linear dependence of $t_{s}$ on $t_{r}$ without
any sign of a saturation, suggests that the values of the switching time,
indicated in Figs. 3d & 3e, are limited by the rise time of the drive pulse of
our experimental setup ($t_{r}$ $\geq$10 $\mu$s), and the shortest intrinsic
switching time is less than 5.5 $\mu$s.

The sound velocity associated with the AFM magnon band in Co3O4, $v_{m}$ = 2.3
km/s 50, imposes a fundamental limit for the speed of the AFM domain walls.
This maximum domain wall speed leads to a switching time of $t_{s}$ = 100 ps
for 200 nm sized AFM domain in Co3O4, which is considerably faster than
$t_{s}$ = 100 ns observed for heterostructure based on 200 nm thick Cr2O3
films 51. Whether this fundamental speed limit can be reached or closely
approached by AFM domain walls in Co3O4 requires additional studies,
investigating the switching dynamics on the nano- to picosecond timescale.

Fig. 3e also demonstrates that the change in $\alpha$ ($\Delta\alpha$)
associated with the switching between $\pm L_{y}$, i.e. the switching
efficiency, remains unchanged while the rise time of the driving pulse is
reduced to $t_{r}$ = 10 $\mu$s. Finally, we show that $\mathbf{L}$ can be
switched reversibly and repetitively in Co3O4. The switching efficiency is not
reduced upon subsequent switchings of $\mathbf{L}$ up to million switching
events, as seen in Fig. 3f. Instead a weak increase is observed, which is
attributed to the training effect, often observed in ferroelectrics 52; 53.

In summary, we demonstrated the fast manipulation of AFM states in the
isotropic collinear antiferromagnet Co3O4 by utilizing its strong linear ME
effect. We showed that the AFM Néel vector can be either rotated smoothly or
reversed isothermally by electric fields. Furthermore, our results revealed
the microsecond dynamics of the non-volatile and reversible switching process
of macroscopic AFM domains and indicate that the switching process can be even
faster for microscopic domains. The present study represents an important step
towards the on-demand electric control of AFM states, that may eventually lead
to the realization of fast-operation AFM spintronic device.

Data availability. The experimental data that support the findings of this
manuscript are available from the corresponding author upon reasonable
request.

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## Methods

Sample synthesis. The single crystals of Co3O4 were grown by chemical
transport reaction method using the preliminary treated high purity powder of
Co3O4 together with anhydrous TeCl4 transport agent. The growth was performed
in a horizontal two-zone furnace with the source temperature of 830 $\degree$C
and a temperature gradient of 40 to 25 $\degree$C. The crystallographic
orientation of the crystals were determined by X-ray Laue diffraction.

ME measurements. Crystals with flat (001) surfaces were prepared for the
measurements. Contacts on two parallel (001) surfaces were made by silver
paint. All measurements were carried out in an Oxford helium-flow cryostat,
where the electrical measurements were performed up to the lowest temperature
of 4 K and the highest magnetic field of 14 T. Temperature- and magnetic
field-dependent electric polarization were obtained via the pyroelectric and
ME current measurements, respectively, with a Keysight electrometer (B2987A).
For angular-dependent electric polarization study, the ME current was recorded
under a rotating magnetic field of 14 T at a constant electric field of +17.4
kV/cm by using a home-designed probe, where the sample platform can be rotated
by a stepper motor. For the data shown in Fig. 1, the sample was cooled with
$E_{p}$ and $H_{p}$ from 45 K to 4 K and the pyroelectric/ME current was
measured after switching off the electric field. For the data shown in Fig. 2
& 3, the measurements were performed on a virgin multi-domain sample without
any ME cooling. In all these cases, the electric polarization were obtained by
performing integration of the measured pyroelectric/ME current over time.

Pulsed electric field measurements. Electric polarization measurements in
pulsed electric fields were carried out by a AixACCT TF Analyzer 2000 equipped
with high voltage booster HVB 1000 and Krohn-Hite model 7500 amplifier. In
order to obtain the purely ME contribution, we subtracted the current arising
from the dielectric response in zero magnetic field from the current measured
in applied magnetic fields. The data indicated by the closed and opened
symbols in Fig. 3e were obtained by the pulse field measurements on crystal
with thickness of 250 $\mu$m and 40 $\mu$m, respectively.

Acknowledgements We thank Anton Jesche for assistance in X-ray Laue
diffraction experiment and Peter Lunkenheimer, Sándor Bordács, Maxim Mostovoy,
N. V. Ter-Oganessian and Sundaresan Athinarayanan for fruitful discussions.
This work was supported by the DFG via the Transregional Research
Collaboration TRR 80 From Electronic Correlations to Functionality
(Augsburg/Munich/Stuttgart).

Author Contributions: S.G. initiated this project; L.P. and V.T. synthesized
the single crystals; M.W., S.G., K.G. and S.K. performed the experiments and
analyzed the data; S.G. and I.K. performed the theoretical modeling and wrote
the manuscript; I.K. supervised this project. All authors discussed the
content of the manuscript.

Competing interests: The authors declare no competing interests.