# Antidamping torque-induced switching in biaxial antiferromagnetic insulators

X. Z. Chen,1 R. Zarzuela,2 J. Zhang,3 C. Song,1,∗ X. F. Zhou,1 G. Y. Shi,1 F.
Li,1 H. A. Zhou,4 W. J. Jiang,4 F. Pan,1 and Y. Tserkovnyak2 1Key Laboratory
of Advanced Materials (MOE), School of Materials Science and Engineering,
Tsinghua University, Beijing 100084, China
2Department of Physics and Astronomy, University of California, Los Angeles,
California 90095, USA
3School of Physics and Wuhan National High Magnetic Field Center, Huazhong
University of Science and Technology, Wuhan 430074, China
4State Key Laboratory of Low-Dimensional Quantum Physics and Department of
Physics, Tsinghua University, Beijing 100084, China<EMAIL_ADDRESS>
###### Abstract

We investigate the current-induced switching of the Néel order in NiO(001)/Pt
heterostructures, which is manifested electrically via the spin Hall
magnetoresistance. Significant reversible changes in the longitudinal and
transverse resistances are found at room temperature for a current threshold
lying in the range of $10^{7}$ A/cm2. The order-parameter switching is
ascribed to the antiferromagnetic dynamics triggered by the (current-induced)
antidamping torque, which orients the Néel order towards the direction of the
writing current. This is in stark contrast to the case of antiferromagnets
such as Mn2Au and CuMnAs, where fieldlike torques induced by the Edelstein
effect drive the Néel switching, therefore resulting in an orthogonal
alignment between the Néel order and the writing current. Our findings can be
readily generalized to other biaxial antiferromagnets, providing broad
opportunities for all-electrical writing and readout in antiferromagnetic
spintronics.

Introduction.—Antiferromagnets exhibit ultrafast spin dynamics with
characteristic frequencies in the THz range, produce negligible stray fields
and are robust against magnetic perturbations, offering prospects for the
design of reliable high-density memories with fast operation speeds Jungwirth-
NN2016 ; Wadley-Science2016 ; Olejnik-NatComm2017 ; Chen-NatComm2017 . Thus,
it is crucial to develop new avenues for manipulating the antiferromagnetic
(AFM) order, of particular interest being electrical methods based on the
spin-transfer effect. Spin-orbit torques (SOT) can be split into two generic
classes, namely fieldlike and antidamping torques:
$\partial_{t}\bm{m}|_{\textrm{FL}}\sim\bm{m}\times\bm{p}$ Wadley-Science2016 ;
Olejnik-NatComm2017  and
$\partial_{t}\bm{m}|_{\textrm{AD}}\sim\bm{m}\times(\bm{m}\times\bm{p})$ Liu-
Science2012 ; Sinova-RMP2015 , respectively, where $\bm{m}$ denotes the
orientation of a magnetic sublattice and $\bm{p}$ represents the
nonequilibrium spin polarization. For bipartite antiferromagnets, if $\bm{p}$
is the same on both sublattices, the fieldlike torque is equivalent to that
exerted by a uniform magnetic field, forcing the Néel order to orient
perpendicular $\propto\bm{p}$. While this could mimic the antiferromagnetic
spin-flop transition, the corresponding critical field is in the range of tens
of Tesla for conventional antiferromagnets, which would translate into
prohibitively large charge currents.

Recently, Wadley et al. Wadley-Science2016  reported a current-induced
switching of the AFM order in CuMnAs, setting a milestone for the manipulation
of the staggered order parameter. In such material, where the breaking of
inversion symmetry occurs at the sublattice level, opposite spin polarizations
are induced in the two (inversion-partner) sublattices,
$\bm{p}_{1}=-\bm{p}_{2}$, via the Edelstein effect. The ensuing fieldlike
torque can reorient the Néel order towards the (current-induced) effective
field $\propto\bm{p}_{1}-\bm{p}_{2}$, whenever this field overcomes the
threshold given by the in-plane anisotropy (several tens of Oersted), as
dictated by the effective energetics of the order parameter. Even though this
SOT seems attractive for the low-current switching of the Néel order, the
underlying inverse spin-galvanic mechanism imposes stringent requirements on
the crystallographic structure and quality of the antiferromagnet, namely
global centrosymmetry plus broken sublattice inversion symmetry. At present,
only a few antiferromagnets such as CuMnAs and Mn2Au can meet this demand
Wadley-Science2016 ; Olejnik-NatComm2017 ; Zelezny-PRL2014 .

Antidamping torques have also been proposed to trigger oscillations and even
the switching of AFM moments regardless of the crystal symmetries Zelezny-
PRL2014 ; Gomonay-PRB2010 ; Cheng-PRL2016 ; Hals-PRL2011 ; Zarzuela-PRBRC2017
. This antidamping switching mechanism could be extended to a wide range of
antiferromagnets with biaxial anisotropy, allowing for the extensive materials
usage. However, no conclusive experimental observation of the antidamping
torque-induced switching has been reported yet in AFM/HM (heavy metal)
heterostructures. Moriyama et al. Moriyama-2017  reported very recently the
control by SOT of AFM moments in Pt/NiO(111)/Pt trilayers. While the
antidamping torques were invoked as the driving force behind this
magnetization switching, the symmetries of the structure instead seem to point
to the fieldlike scenario. Here, we investigate the switching of AFM moments
mediated by antidamping torques in the biaxial NiO(001)/Pt heterostructure. We
utilize the spin Hall magnetoresistance (SMR) as a probe for the (dynamics of
the) magnetic moments of this AFM insulator Nakayama-PRL2013 ; SMR ; Yokoyama-
PRB2014 ; Manchon-PSS2017 ; Han-PRB2014 , which is well known to display a
negative SMR signal Hoogeboom-APL2017 ; Hou-PRL2017 ; Lin-PRL2017 . We observe
significant changes in the longitudinal and transverse resistances when
current pulses are applied along the orthogonal easy axes of NiO,
corresponding to the Néel switching towards the direction of the current.

Figure 1: (a) X-ray diffraction pattern, (b) high resolution TEM images and
(c) room-temperature hysteresis loop of the SrTiO3(001)/NiO/Pt(5 nm)
heterostructure. The inset of subfigure (b) shows the image of the NiO
lattice, which has been magnified twice. (d) Spatial (angular) dependence of
the magnetocrystalline anisotropy energy for the tetragonal NiO. The easy axis
points towards $[1\bar{1}0]$, while the local easy axis is $[110]$. (e)
Optical image of a NiO/Pt 8-terminal device and schematic of current-driven
switching measurements. (f) Application of writing currents along $[110]$ and
$[1\bar{1}0]$ yields reversible changes in the Hall resistance that can be
detected at $T=300$ K. Hall resistances are measured $10$ s after applying the
current pulses.

Experimental results.—The in-plane biaxial NiO films were obtained by
deposition under compressive strain on SrTiO3 (STO) substrates
($a_{\textrm{NiO}}\sim 0.4177$ nm and $a_{\textrm{STO}}\sim 0.3905$ nm) by
using magnetron sputtering at 473 K, and capped with 5 nm-thick Pt after
cooling down to room temperature. Afterwards, the NiO(5 nm)/Pt(5 nm) bilayers
are patterned to the desired geometry by using electron beam lithography and
Ar ion etching.

Figure 1(a) shows the X-ray diffraction pattern of the NiO(100 nm)/Pt(5 nm)
stack on STO(001), where only (002) and (004) NiO peaks and the substrate
peaks are visible in the range from 20∘ to 120∘, therefore reflecting the
quasi-epitaxial growth of the NiO. The parameter $c\sim 0.4205$ nm, deduced
from the (004) peak at 94.2∘, confirms that NiO on STO is elongated out-of-
plane or, equivalently, constrained within the plane. As shown in Fig. 1(b),
the cross-sectional transmission electron microscopy (TEM) images of the
STO/NiO(5 nm)/Pt(5 nm) heterostructure illustrate the high sample quality of
NiO, with the stack relationship of NiO(001)[100]$\parallel$STO(001)[100] and
a sharp interface between the NiO and Pt. The NiO lattice is highlighted in
the inset panel, magnified twice as compared to the whole TEM image.
Magnetization measurements were carried out at $300$ K in a Superconducting
Quantum Interference Device to reveal the AFM order of NiO. Magnetic fields
were swept from $-5$ T to $5$ T, and only the diamagnetic background of the
STO substrate was observed [Fig. 1(c)].

Based on the lattice parameters deduced from XRD, first-principles
calculations of the magnetocrystalline anisotropy (MCA) of the tetragonal NiO
were performed. Figure 1(d) depicts the spatial dependence of the MCA energy
in terms of the angles $\theta,\varphi$ (spherical representation of the order
parameter). Note that $[1\bar{1}0]$ and $[001]$ are the easy and hard axes of
the NiO sample, respectively, while a second (local) easy axis aligns along
$[110]$. The easy plane rotates from $(111)$ to $(001)$ due to the fourfold
in-plane constraint imposed by the substrate. Also visible is that the NiO MCA
is greatly enhanced as compared to the bulk SM . An optical image of a NiO/Pt
8-terminal device and the schematic of the current-driven switching
measurement are depicted in Fig. 1(e). As the SMR measurements in Fig. 1(f)
show, two orthogonal orientations (easy axes) of the Néel order can be
reversibly written by applying current pulses at room temperature. For this
experiment, the writing current channels along the two orthogonal directions
$[110]$ and $[1\bar{1}0]$ of the NiO film, Write 1 and 2, respectively, allow
for the control of the spin polarization of the injected spin current. The
$[100]$ and $[010]$ channels are used for probing AFM moments via the
longitudinal and transverse SMR, where current and voltage ($I$ and $V$) stand
for the detection. The change in the Hall resistance is $\sim 30$ m$\Omega$,
corresponding to a ratio of $\sim$0.05% when divided by the longitudinal
resistance. An inspection of the figure shows the gradual change in the Hall
resistance with consecutive current pulses, indicating the multidomain
switching behavior.

Figure 2 illustrates the current-induced Néel switching in the tetragonal
(biaxial) NiO. In Figs. 2(a)-(d) both the magnitude and direction of the
writing current are identical, but the configurations of the detected $I$ and
$V$ are different, as highlighted in the left column. Five 1-ms current pulses
with the delay time of 10 s were applied along the writing channels. The
current amplitude used was $4\times 10^{7}$ A/cm2, comparable to the
ferrimagnetic system Avci-NatMat2017 ; Peng-NatCommun2016 , and a small probe
dc current of $4\times 10^{5}$ A/cm2 was applied along $[100]$. Figures 2(a)
and (b) show the detection of the AFM moments in the transverse geometry. Two
distinct values for the transverse resistance were observed reversibly, see
Fig. 2(a), the total change in resistance being about 30 m$\Omega$. Since
$\rho_{xy}^{\textrm{sH}}=\Delta\rho\sin 2\theta$ SMR , the change in the Hall
resistance is maximal (in absolute value) at the relative angles of
$\theta=45,135^{\circ}$ between the probe current and the Néel order;
therefore, writing protocols along $[110]$ (high resistance) and $[1\bar{1}0]$
(low resistance) correspond to $\theta=45^{\circ}$ and $\theta=135^{\circ}$,
respectively, i.e. the AFM order switches towards the direction of the writing
current. In contrast, the inset panel shows the current-induced Néel switching
in our Mn2Au control samples for the identical experimental configuration,
where the anisotropic magnetoresistance of Mn2Au and the SMR of NiO/Pt
possesses the same in-plane symmetry Nakayama-PRL2013 ; SMR . The concomitant
sign of the changes in the Hall resistance is opposite to that of the NiO/Pt
scenario, because the (fieldlike) Edelstein SOT would switch the Néel order
towards the spin polarization direction, which is transverse to the writing
current, similar to the CuMnAs case Wadley-Science2016 ; Zelezny-PRL2014 .

For the sake of completeness we considered a different Hall detection scheme,
as shown in Fig. 2(b). Insignificant changes in the transverse resistance were
measured after running the same current pulse protocol, which correspond to
the angles $\theta=0^{\circ}$ (red circles) and $\theta=90^{\circ}$ (blue
circles). This is in accordance with the aforementioned expression for
$\rho_{xy}^{\textrm{sH}}$. Furthermore, we studied the symmetry of SMR for the
longitudinal geometry, see Figs. 2(c) and (d). The longitudinal resistance
increases when the writing current is parallel to the probe current. Theory of
SMR predicts $\rho_{xx}^{\textrm{sH}}=\rho_{0}+\Delta\rho\cos 2\theta$ SMR ,
so that larger values of the longitudinal resistance indicate the parallel
alignment of the AFM moments with the probe current. Again, the control
experiment depicted in Fig. 2(d) shows negligible changes in the longitudinal
resistance, which agrees with the theoretical predictions. Therefore, we
conclude that the Néel order switches towards the direction of the writing
current. Note that both transverse and longitudinal resistances were measured
one minute after the current pulses to minimize the Joule heating.

Figure 2: Detection of the Néel order switching for transverse [(a)-(b)] and
longitudinal [(c)-(d)] geometries. Blue (red) arrows indicate the current
being applied along the writing channel $[110]$ $([1\bar{1}0])$. $I$ and $V$
represent the readout scheme. The writing current density used is $4\times
10^{7}$ A/cm2. Each circle in the right column results from the application of
five 1-ms pulses with delay time of 5 s, where the color corresponds to the
writing channel considered.

Model.—Dynamics of the Néel order, defined by
$\bm{l}=(\bm{m}_{1}-\bm{m}_{2})/2$, with $\bm{m}_{1},\bm{m}_{2}$ being the
sublattice spin fields, are described by the equation of motion

$s\chi\partial_{t}^{2}\bm{l}\times\bm{l}=\bm{H}_{\textrm{an}}\times\bm{l}+\alpha\bm{l}\times\partial_{t}\bm{l}+(\vartheta
J/s)\bm{l}\times(\bm{l}\times\bm{p}),$ (1)

along with the normalization condition $\bm{l}^{2}\equiv 1$. Here, $s$ is the
saturated spin density, $\chi$ is the (transverse) spin susceptibility,
$\bm{H}_{\textrm{an}}=-\delta_{\bm{l}}\mathcal{E}_{\textrm{MCA}}/s$ is the
anisotropy field, $\alpha=0.007$ is the Gilbert damping constant due to spin
pumping in our 5-nm film Tserkovnyak-PRL2002 ; Khymyn-SRep2017 , which is also
the value used in Ref. Cheng-PRL2016, , and $\vartheta$ is a phenomenological
parameter characterizing the charge-to-spin current conversion at the NiO/Pt
interface, which can be approximated by $\vartheta\sim 0.1\hbar/2e$. The last
term of the right-hand side represents the current-induced antidamping torque,
which vanishes for $\bm{l}\parallel\bm{p}$. Contrary to the ferromagnetic
case, this configuration of the Néel order is unstable against perturbations,
as predicted by the stability analysis of Eq. (1) Zarzuela-PRBRC2017 ; Hirch-
Book2004 : Thermal fluctuations will induce slight deviations of the Néel
order from the ground state $\bm{l}\parallel\bm{p}$, which, above some current
threshold, will evolve in time towards the plane perpendicular to the spin
polarization, $\bm{l}\perp\bm{p}$. Note, however, that for the case of
uniaxial anisotropy (NiO on MgO substrate) the Néel order will switch back to
the (only) easy axis when the applied current is switched off, irrespective of
its intensity SM . This is not the case for the (in-plane) biaxial anisotropy,
for which writing currents along orthogonal (easy-axis) directions allow for
the reversible manipulation of the AFM moments between the easy axes. It is
worth remarking that reactive fieldlike torques $\propto\bm{l}\times\bm{p}$
are forbidden if the sublattice symmetry $\\{\bm{l}\rightarrow-\bm{l}\\}$ is
preserved.

Further insight into this switching process can be gained through monodomain
simulations of the Néel dynamics driven by antidamping torques. Zero-
temperature values of the anisotropy constants are obtained by fitting the MCA
energy calculated by the DFT+U method SM  to the expression
$\mathcal{E}_{\textrm{MCA}}[\bm{l}]=K_{0}+(K_{2\perp}/2)l_{z}^{2}+(K_{4\parallel}/4)l_{x}^{2}l_{y}^{2}+K_{2\parallel}(l_{x}^{2}-l_{y}^{2})$,
which contains all (dominant) terms allowed by the tetragonal symmetry up to
4-th order in the Néel field. Current pulses of amplitude $4\times 10^{7}$
A/cm2 and 1-ms width would heat the sample to $T=350$ K SM . Thermal effects
can be incorporated into our model by invoking Bloch’s $T^{2}$ law for the
sublattice spin density $\mathfrak{s}$ in antiferromagnets Kubo-PR52 ; Oguchi-
PR60  and the power law
$K_{l\parallel},K_{l\perp}\propto\mathfrak{s}^{l(l+1)/2}$ for the MCA
coefficients Callen-1966 . The resultant estimates for the anisotropy
constants are $K_{2\perp}=3.8\times 10^{-8}$ J/cm2, $K_{2\parallel}=4.3\times
10^{-9}$ J/cm2 and $K_{4\parallel}=1.4\times 10^{-9}$ J/cm2. The ground Néel
state thus lies along the $y$ direction (absolute minimum of the MCA energy)
within the easy basal ($xy$) plane. Note that in the above expression the
frame of reference has been rotated so that $\hat{y}\,(\hat{x})$ is the
(local) easy axis. We take the writing current along the $x$ axis in what
follows. Stability analysis of the fixed points $\bm{l}_{0}=\pm\hat{y}$ of Eq.
(1) indicates that the Néel order along the easy axis is stable up to the
critical current threshold

$J_{x,c}^{\textrm{th}}=\frac{1}{2\vartheta}\sqrt{\mathcal{K}_{\perp}^{2}+\alpha^{2}\mathcal{K}_{\parallel}/\chi}\simeq
5.8\times 10^{8}\textrm{ A/cm}^{2},$ (2)

where $\mathcal{K}_{\perp}=K_{2\perp}-K_{2\parallel}-K_{4\parallel}/2$ and
$\mathcal{K}_{\parallel}=6K_{2\parallel}+2K_{2\perp}+K_{4\parallel}$ are
effective anisotropy constants. Numerical integration of Eq. (1) leads to the
Néel dynamics depicted in Fig. 3: Transverse deviations from the easy-axis
ground state die out below $J_{x,c}^{\textrm{th}}$, and therefore the Néel
order eventually evolves back to the $y$ axis, as depicted in Fig. 3(a). Above
the current threshold, however, transverse excitations of the Néel order blow
up until the staggered order parameter becomes confined to the $xz$ plane, see
Fig. 3(b), where its (limiting) dynamics take the form of a limit cycle. When
the writing current is turned off (end of the pulse), no Néel dynamics occur
below $J_{x,c}^{\textrm{th}}$, i.e., the order parameter remains aligned with
the easy axis. Differently, above the critical threshold, the AFM order
evolves in time to the minimum of MCA energy within the $xz$ plane, which
corresponds to the local easy axis $\hat{x}$ in Fig. 3(b).

For writing currents along the $y$ axis, stability analysis of the fixed
points $\pm\hat{x}$ of the dynamical system (1) yields their instability no
matter what the injected current is. Irrespective of their steady state
(either a stable fixed point or a limit cycle), the Néel order always evolves
to the absolute minima of the MCA energy, $\pm\hat{y}$, when the current is
turned off . Antidamping torque-induced switching of the order parameter in
biaxial antiferromagnets is then summarized in Fig. 4, which is described by a
two-step process, namely (1) deviation from the easy axis and confinement to
the plane perpendicular to the spin polarization, and (2) switching towards
the local easy axis contained in the same plane.

Figure 3: Monodomain simulation of the Néel switching mediated by antidamping
torques for a charge current injected along the $x$ axis. Panels (a) and (b)
show the fixed points and limit cycles of the dynamical system in the Néel-
order sphere (up) and the time evolution of the Néel order (bottom) for
$J_{x}=4.9\times 10^{8}$ A/cm2 and $J_{x}=2.2\times 10^{9}$ A/cm2,
respectively. Blue dots represent the initial configuration
$\bm{l}=(-0.1,0.99,0.1)$ for the Néel order and blue trajectories describe the
complete time evolution of the order parameter towards (a) the $y$ axis and
(b) the limit cycle. Red arrows denote the fixed points of the dynamical
system, along with the easy axis (green arrows). The limit cycle in (b) lies
within the $xz$ plane. After turning off the current, the Néel order switches
towards the $x$ axis (local minimum of energy) above the critical threshold
$J_{x,c}=5.8\times 10^{8}$ A/cm2. Figure 4: Schematic of the two-step process
involved in the antidamping torque-induced switching. Green (red) arrows
depict initial (final) configurations for the Néel order. Blue circle depicts
the limit cycle in the $xz$ plane.

Discussion and conclusion.—The critical threshold (2) is one order of
magnitude larger than the measured threshold current. For ferromagnetic
switching, similar overestimates of the threshold current within the
monodomain treatment are well known Lee-PRB2K14 ; Zhang-APL2K15 . A switching
process based on domain-wall nucleation and propagation can significantly
lower the critical current. Indeed, the multistage character of switching in
our pulse-sequence experiment suggests such multidomain mechanism for the
order-parameter reorientation. While extending our theoretical picture towards
this regime is left as an open problem, we remark that the monodomain
treatment does appear to capture the physics qualitatively. More refined
calculations and direct measurements of the magnetic anisotropies for our
structures can further reconcile quantitative aspects. The reversible
switching and detection reported here make biaxial AFM insulator/HM
heterostructures promising candidates for all-electrical writing and readout
of the Néel order.

Acknowledgments.—We thank Dr. R. Cheng, Dr. D. Z. Hou and Y. Z. Tan for
helpful discussions. The authors in Tsinghua thank the support of Beijing
Innovation Center for Future Chip (ICFC) and Young Chang Jiang Scholars
Program. The work was partially supported by the National Key R&D Program of
China under Grant No. 2017YFB0405704, and the NSFC under Grant Nos. 51671110,
51571128 and 11704135. The work at UCLA was supported by NSF-funded MRSEC
under Grant No. DMR-1420451 and NSF under Grant No. DMR-1742928.

Note added: During the preparation of this manuscript, we became aware of a
report of the switching of AFM moments in Pt-sandwiched NiO films, with the
critical current of $\sim 5\times 10^{7}$ A/cm2 Moriyama-2017 . Measured
changes in the transverse Hall resistance were ascribed there to the time
evolution of the Néel order towards $\bm{l}\parallel\bm{p}$, i.e., the order
parameter being transverse to the writing current. This interpretation
contrasts with the antidamping scenario, which could be explained that the
trilayer Pt/NiO/Pt geometry might reduce the inversion asymmetry and thus
diminish the relative importance of the antidamping torques. These issues need
to be clarified in future studies on multilayers.

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