††thanks: Contributed equally††thanks: Contributed equally

# Non-equilibrium spin polarisation via interfacial exchange-field spin
filtering

Prasanta Muduli ‡<EMAIL_ADDRESS>Institute for Solid State Physics,
University of Tokyo, Kashiwa 277-8581, Japan  Department of Physics, Indian
Institute of Technology Madras, Chennai 600036, India  Naëmi Leo ‡
<EMAIL_ADDRESS>Institute for Solid State Physics, University of Tokyo,
Kashiwa 277-8581, Japan  CIC nanoGUNE BRTA, 20018 Donostia-San Sebastián,
Spain  Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC-
Universidad de Zaragoza, 50009 Zaragoza, Spain  Mingran Xu  Institute for
Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan  Center for
Emergent Matter Science, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan  École
polytechnique fédérale de Lausanne (EPFL), School of Engineering, Institute of
Materials, Laboratory of Nanoscale Magnetic Materials and Magnonics, 1015
Lausanne, Switzerland  Zheng Zhu  Institute for Solid State Physics,
University of Tokyo, Kashiwa 277-8581, Japan  Jorge Puebla  Center for
Emergent Matter Science, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan  Christian
Ortiz  Fraunhofer ITWM, 67663 Kaiserslautern, Germany  Hironari Isshiki
Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581,
Japan  YoshiChika Otani  Institute for Solid State Physics, University of
Tokyo, Kashiwa 277-8581, Japan  Center for Emergent Matter Science, RIKEN, 2-1
Hirosawa, Wako 351-0198, Japan

###### Abstract

A key phenomenon that enables nanoscale spintronic devices is the efficient
inter-conversion between spin and charge degrees of freedom. Here, we
experimentally demonstrate a pathway to generate current-induced spin
polarization at the interface between an insulating ferromagnet and a non-
magnetic metal using interfacial exchange-field spin filtering. Measuring
current-in-plane giant magnetoresistance in Py$|$Cu$|$EuS trilayer Hall cross
devices, we induce a non-equilibrium spin polarization of
$P_{\text{neq}}^{\text{Cu|EuS}}=$0.6\text{\,}\mathrm{\char 37\relax}$$ at a
low charge current density of
$1.88\text{\times}{10}^{3}\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$.
This efficiency is comparable to that of conventional charge-to-spin
conversion of spin-Hall or Rashba-Edelstein effects enabled by relativistic
spin-orbit coupling. Interfacial exchange field filtering allows operation
with largely reduced power consumption and magnetic reconfigurability, opening
new pathways to nanoscale low-power insulator spintronics.

spintronics, spin polarised transport, spin transport effects

Future green information technologies with fast, non-volatile, and scalable
low-power operation rely on the creation and detection of spin currents that
carry information in the form of spin angular momentum instead of charge
carriers. These desirable properties are crucial to spintronics applications
ranging from magnetic memories, sensors, logic gates, and unconventional
computing architectures [1, 2, 3, 4] To achieve further miniaturization of
nanoscale spintronic devices and for the implementation with complementary
metal-oxide semiconductor (CMOS) technologies, a key challenge is the
efficient generation of spin polarization at low charge current densities [2,
5, 4].

Table 1 summarizes contemporary current-driven phenomena to generate a non-
equilibrium spin polarization. These mechanisms can be classified by two
characteristics, namely whether they are mediated by relativistic spin-orbit
interaction or magnetic exchange, and whether the conversion occurs in bulk or
is confined to interfaces.

In the case of the spin Hall effect (SHE), spin-orbit coupling in heavy metals
such as Pt, Ta, and W leads to spin-dependent deflection of charge carriers in
bulk and at interfaces, leading to a non-equilibrium spin accumulation [6].
With SHE, spin polarization values in the order of 3% to 12% have been
demonstrated [7]. The Rashba-Edelstein effect allows for even larger current-
induced spin polarization, with values up to 60% observed [8, 9, 10]. The
effect originates from the spin-momentum locking and momentum-dependent spin
splitting at interfaces with broken inversion symmetry and large spin-orbit
coupling, and occurs in topological insulators and two-dimensional electron
gases [11, 12, 13, 14, 15]. While interfacial effects in topological
insulators are, in principle, great to scale devices to the nanoscale,
spurious bulk conduction produces significant Joule heating and thus makes
miniaturization unfeasible.

The generation of spin currents and spin-polarised currents based on generally
stronger exchange interaction therefore might pose an attractive alternative.
Indeed, bulk ferromagnetic metals are a commonly-used source for spin-
polarized currents generated from unpolarised charge currents, with typically
a few percent conversion efficiency [16]. Almost perfectly spin-polarised
currents can theoretically be achieved in half-metals, however, these
materials are not yet viable for technological applications [17].

A pathway to interfacial exchange-mediated generation of spin currents is
explored within the field of insulator spintronics [18, 19, 20], and spin
polarised currents at (heavy metal)|(ferromagnetic insulator) interfaces have
been demonstrated via the spin-Hall magnetoresistance (SMR) [21] and the
unidirectional spin Hall magnetoresistance (USMR) [22]. Nevertheless, in
addition to the exchange interaction at the interface, both SMR and USMR rely
on the presence of spin-orbit coupling. As such, these effects require
expensive heavy metals, which furthermore have high resistances that are
undesirable for energy-efficient device operation.

Table 1: Overview of current-induced mechanisms that generate a non-equilibrium spin polarization $P_{\text{neq}}$, with prototypical materials. These effects can be mediated by either the spin-orbit or the exchange interaction, and occur eitehr in bulk or are confined to interfaces.  Spin-orbit field |  Exchange field
---|---
Spin current generation in bulk
Spin Hall effect
Heavy metals (e.g. Pt, Ta, W) |  Spin-polarised currents
Ferromagnetic metals
(e.g. Fe, Co, Ni) and
half-metals (e.g. Co2FeSi)
Spin current generation at interfaces
Rashba-Edelstein effect
Topological insulators (e.g. Bi)
or 2D electron gases
(e.g. LAO$|$STO) |  Exchange-field spin filtering
Ferromagnetic insulators
here: EuS

In this work, we experimentally demonstrate the generation of a current-
induced spin polarisation without the need for heavy metals via interfacial
exchange-field spin filtering (IESF) between a normal metal and an insulating
ferromagnet, here highly-conducting copper (Cu) and prototypical europium
sulphate (EuS). We quantify the non-equilibrium spin polarization via magneto-
resistance measurements on trilayer Py$|$Cu$|$EuS Hall-bar devices, obtaining
an interfacial spin polarization of
$P^{\text{Cu|EuS}}_{\text{neq}}=$0.6\text{\,}\mathrm{\char 37\relax}$$ at a
low charge current density of
$1.88\text{\times}{10}^{3}\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$.
We show that this efficiency is comparable to that of conventional methods
based on spin-orbit coupling, such as the spin-Hall or the Rashba-Edelstein
effect. Interfacial exchange-field spin filtering thus offers itself for the
implementation of low-power spintronic devices.

##  I Interfacial exchange-field
spin filtering (IESF)

The interfacial exchange field (IEF) arises from short-ranged direct spin-spin
interactions between adjacent layers of different – not necessarily magnetic –
materials and thus is highly localized to the common interface, with a decay
depth of a couple of nanometres only [23]. The contemporary technological
importance of the interfacial exchange field is given by static equilibrium
phenomena, including the exchange bias between ferromagnets and
antiferromagnets [24], proximity-induced magnetization [25, 26] and field-free
spin splitting of up to several hundred Tesla in superconductors [27, 28].

Figure 1:  Current-induced spin polarization by interfacial exchange-field
spin filtering. Schematic illustration of the mechanism of non-equilibrium
spin polarization via spin-dependent reflection from an interfacial exchange
field (IEF). An unpolarised in-plane charge current $\mathbf{j}_{c}$ in the
conductor Cu layer (brown) leads to the scattering of the charge carriers at
the IEF localized at the Cu$|$EuS interface (green). No charge current flows
in EuS (blue), due to it being highly insulating. If the charge carrier spin
(black and silver) is not aligned with the EuS magnetization (large black
arrow), the exerted torque at the interfacial exchange field leads to
different spin-dependent resistivities $\rho_{\uparrow}\neq\rho_{\downarrow}$,
and thus to net non-equilibrium spin polarization in non-magnetic Cu.

In Fig. 1 we illustrate how the interfacial exchange field can also generate a
non-equilibrium current-induced spin polarization at the interface between a
ferromagnetic insulator and a non-magnetic conductor such as Cu. An in-plane
unpolarised charge current $\mathbf{j}_{c}\neq 0$ (with equal spin-dependent
resistivities $\rho_{\uparrow}=\rho_{\downarrow}$, resulting in
$\mathbf{j}_{s}=0$) flowing in the metallic layer (brown) scatters off the
interface to the ferromagnetic insulator (blue). Depending on the relative
orientation between charge carrier spins and the local magnetization of the
ferromagnetic insulator (black arrows), spin-dependent reflection from the
interfacial exchange field (IEF, green) modifies the spin-dependent
resistivities, $\rho_{\uparrow}\neq\rho_{\downarrow}$, which leads to a net
spin current $\mathbf{j}_{s}\neq 0$ (this effect is also called spin
filtering). As consequence, a non-equilibrium interfacial spin polarization
$P^{i}_{\text{neq}}$ emerges with

$P^{i}_{\text{neq}}=\frac{\rho_{\uparrow}^{i}-\rho_{\downarrow}^{i}}{\rho_{\uparrow}^{i}+\rho_{\downarrow}^{i}}\,.$
(1)

A suitable material to experimentally prove the non-equilibrium current-
induced spin polarization mediated by the IESF is europium sulphite (EuS), a
highly insulating ferromagnetic semiconductor with a Curie temperature of
$T_{C}^{\text{EuS}}=$16\text{\,}\mathrm{K}$$ [29]. EuS is a popular material
for superconducting tunnel junctions [30, 31, 23], and the strength of the IEF
from spin-splitting in superconductors in proximity with EuS has been
estimated to be about $14\text{\,}\mathrm{T}$ [32]. The influence of the IEF
for EuS has also been demonstrated as exchange bias [33, 34], and in contact
with various 2D materials [32, 35] as well as topological insulators [36].
Recent results of in EuS$|$Pt bilayers furthermore demonstrated the creation
of an interfacial non-equilibrium spin accumulation via the spin Hall
magnetoresistance [37].

Based on these properties, we chose EuS in combination with highly conducting
non-magnetic Cu (a metal without sizable spin-orbit coupling, which avoids
spurious additional spin-Hall or Rashba-Edelstein effects) for our proof-of-
principle experiments for IESF-mediated current-induced spin polarization.

Experimentally, the current-induced spin polarization can be detected in
various ways, including using optical techniques [38, 39], spin-polarized
positron beams [40, 7], and electrical methods [41]. One of the conceptually
simplest techniques to quantitatively determine the value $P$ of the
interfacial spin polarization is via the giant magnetoresistance (GMR) [42],
which is largely equivalent to spin potentiometric measurements used to
characterize interfacial spin-orbit effects in heavy metals [43] and
topological insulators [44, 45, 8].

Here, our GMR device is based on a (conducting ferromagnet)|(non-magnetic
conductor)|(insulating ferromagnet) trilayer, where the conducting ferromagnet
acts as a source for a spin-polarized current arising from partial shunting of
a pure charge current – here, we chose widely used permalloy (Py, Fe0.2Ni0.8)
(Supplemental Material Sec. S1). If the intermediate non-magnetic conducting
layer is much thinner than its spin diffusion length (for Cu this value is
about $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$), the spin of the
charge carrier is preserved during the transport between the two ferromagnetic
interfaces. Spin-dependent charge carrier scattering at the insulating
ferromagnet (i.e., the detector) then leads to a magnetoresistance

$\text{MR}=\frac{R_{\uparrow\downarrow}-R_{\uparrow\uparrow}}{\min\left(R_{\uparrow\downarrow},\,R_{\uparrow\uparrow}\right)}\,,$
(2)

where $R_{\uparrow\uparrow}$ and $R_{\uparrow\downarrow}$ denote the
resistance in the state with parallel and antiparallel alignment of the two
ferromagnets, respectively. For a current-in-plane geometry (CIP GMR) with a
Py|Cu|EuS trilayer, the magnetoresistance MR then also can be written as
(Supplementary Information Sec. S2) [2]:

$\text{MR}=\frac{{P^{\text{Py|Cu}}_{\text{eq}}\,P^{\text{EuS|Cu}}_{\text{neq}}}}{{1-P^{\text{Py|Cu}}_{\text{eq}}\,P^{\text{EuS|Cu}}_{\text{neq}}}}\,.$
(3)

Knowledge of the spin polarization between the two metals, which is a
material-dependent equilibrium property of the interface (here
$P^{\text{Py|Cu}}_{\text{eq}}=48\%$ [1]), allows determination of the unknown
non-equilibrium spin polarisation $P^{\text{EuS|Cu}}_{\text{neq}}$ at the
metal|insulator interface.

Figure 2: Hall cross devices. Schematic of the multi-layer Hall-cross device
with arm width of $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ used to measure
the effect of interfacial exchange field on the magnetotransport. The non-
equilibrium spin polarization $P^{\text{Cu|EuS}}_{\text{neq}}$ induced by the
IESF at the Cu|EuS interface (green) is estimated from a current-in-plane GMR
experiment, which compares it to the known value
$P^{\text{Cu|Py}}_{\text{eq}}$ at the reference (conductor)|(conducting
ferromagnet) interface.

##  II Methods

Hall bar devices for magneto-transport measurements were fabricated using
mask-less photo-lithography. To ensure good electrical contact to the bottom
Py electrode, and as EuS is insulating and therefore does not allow for post-
growth wire-bonded contacts, a two-step lithography procedure was used: In the
first step, gold contact pads were patterned on Si/SiO2 substrates, then
$2\text{\,}\mathrm{n}\mathrm{m}$ Ti to promote adhesion and
$60\text{\,}\mathrm{n}\mathrm{m}$ Au were deposited using electron-beam
evaporation, followed by a lift-off process. In a second lithography step,
Hall devices as shown in Fig. 2 with
$5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ leg width and crosses
separated by $200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$,
$320\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, and
$440\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ were patterned.

Trilayer Py($10\text{\,}\mathrm{n}\mathrm{m}$)|Cu(10 nm or 20
nm)|EuS($10\text{\,}\mathrm{n}\mathrm{m}$) thin films for both the Hall-bar
devices as well as macroscopic reference measurements were co-deposited by
electron-beam evaporation in an ultra-high vacuum chamber (Kitano Seiki Co.,
Ltd.) at a base pressure of about
$2\text{\times}{10}^{-8}\text{\,}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}$,
without breaking the vacuum between deposition of the different layers. During
deposition, the substrate stage was cooled to
$5\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to improve the crystalline quality
and ensure the magnetically-soft behaviour of the EuS thin films [31, 1]. All
films and Hall-cross devices were capped with $5\text{\,}\mathrm{n}\mathrm{m}$
AlOx to prevent degradation, deposited by RF sputtering at room temperature in
a separate UHV chamber.

The crystal structure of the EuS thin films was characterized using x-ray
diffraction, indicating polycrystalline growth [1]. Surface morphology
measurements via atomic force microscopy (AFM) verified smooth films with a
root-mean-square (rms) roughness of $0.28\text{\,}\mathrm{n}\mathrm{m}$.

The EuS thin films are highly insulating, with resistances measured in a two-
point configuration much larger than $50\text{\,}\mathrm{M\SIUnitSymbolOhm}$,
exceeding the limit of the digital multimeter (Fluke). Previous publications
estimate the resistance of EuS thin films from four-point measurements to
$68\text{\,}\mathrm{M\SIUnitSymbolOhm}$ at $300\text{\,}\mathrm{K}$,
increasing to $10\text{\,}\mathrm{T}\mathrm{\SIUnitSymbolOhm}$ at
$200\text{\,}\mathrm{K}$ [37]. As all our measurements have been performed
below $50\text{\,}\mathrm{K}$, we therefore, can safely assume that any
spurious current through the EuS layer can be neglected.

Temperature- and field-dependent magnetization profiles of multilayer films
were measured using DC magnetometry in a Quantum Design MPMS magnetometer.
Both longitudinal and transverse magneto-resistances were measured using the
four-probe method in a custom-built flow cryostat, using a phase-sensitive
lock-in technique with an applied AC current of
$I=$$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{A}$-$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{A}$
modulated at a frequency of $f=$173\text{\,}\mathrm{H}\mathrm{z}$$.

Figure 3:  Magnetometry of Py$|$Cu($10\text{\,}\mathrm{n}\mathrm{m}$)$|$EuS
sample. (a) Temperature-dependent magnetization, measured upon cooling with an
in-plane magnetic field
$\mu_{0}H_{\text{FC}}=$100\text{\,}\mathrm{m}\mathrm{T}$$. Below
$T_{C}^{\mathrm{EuS}}\approx$16\text{\,}\mathrm{K}$$ EuS is ferromagnetic. (b)
Normalized hysteresis loops $M(H)$ measured at $30\text{\,}\mathrm{K}$ (gray
circles) and $5\text{\,}\mathrm{K}$ (red squares) with an in-plane field. The
high-temperature easy-axis loop has a coercive field of
$H_{c}($30\text{\,}\mathrm{K}$)\approx$0.5\text{\,}\mathrm{m}\mathrm{T}$$. As
EuS becomes ferromagnetic at low temperatures, the coercive field increases
$H_{c}($5\text{\,}\mathrm{K}$)\approx$1\text{\,}\mathrm{m}\mathrm{T}$$ and the
loop broadens, indicating interactions between the two magnetic layers and
formation of compensated and non-collinear spin configurations at low fields.
Figure 4: Longitudinal magnetoresistance, measured with different in-plane
fields at $5\text{\,}\mathrm{K}$, where both Py and EuS are ferromagnetic
(blue and red lines for field up and down sweep, respectively), and at
$30\text{\,}\mathrm{K}$ (only Py ferromagnetic, gray lines). (a)
$R_{xx}(H_{x})$ and $R_{xx}(H_{y})$ for the
Py|Cu($10\text{\,}\mathrm{n}\mathrm{m}$)|EuS sample. $R_{xx}(H_{x})$ exhibits
a drop in resistance at low temperatures due to the GMR-like effect arising
from spin scattering between the rotated Py and EuS magnetization at low
fields. For $R_{xx}(H_{y})$ an additional AMR arises from non-collinear spin
textures in Py, and therefore persists in the $30\text{\,}\mathrm{K}$ control
experiment. (b) $R_{xx}(H_{x})$ and $R_{xx}(H_{y})$ measured for the
Py|Cu($20\text{\,}\mathrm{n}\mathrm{m}$)|EuS sample. Its behaviour is
equivalent to that of the other sample, albeit with a lower resistance due to
the thicker Cu layer, and a lower value of MR.

##  III Results

Table 2: Comparison of interfacial current-induced spin polarization, with experimental results for typical materials, with measured current-induced non-equilibrium spin polarization $P_{\text{neq}}$ at a given in-plane current density $j_{c}$. The last column gives the conversion efficiency, which we define as the normalized spin polarization $P^{\text{norm}}_{\text{neq}}$ at a reference current density $j_{c}^{\text{ref}}=${10}^{4}\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$. As demonstrated here, the interface exchange spin filtering using insulating ferromagnetic EuS has comparable efficiency to more conventional charge-to-spin conversion effects.  [Reference] Material |  $|P_{\text{neq}}|$ (%) |  $j_{\text{c}}$ ($\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$) |  $P^{\text{norm}}_{\text{neq}}$ (%)
---|---|---|---
Direct spin Hall effect in heavy metals
[7] |  $\beta$-Ta | $7$ | $5.0\text{\times}{10}^{4}$ | $1.4$
[7] |  $\beta$-W | $9$ | $1.0\text{\times}{10}^{5}$ | $0.9$
[7] |  $\alpha$-W | $6$ | $1.0\text{\times}{10}^{5}$ | $0.6$
[7] | Pt(111) | $11$ | $2.0\text{\times}{10}^{5}$ | $0.6$
[7] |  $\alpha$-Ta | $12$ | $2.5\text{\times}{10}^{5}$ | $0.5$
[7] | Pd(111) | $8$ | $2.4\text{\times}{10}^{5}$ | $0.3$
Rashba-Edelstein effect in topological insulators
[10] | Bi1.5Sb0.5Te1.7Se1.3 | $60$ | $7.14\text{\times}{10}^{4}$ | $8.4$
[44] | Bi2Se3 | $36$ | $2.5\text{\times}{10}^{5}$ | $1.4$
[48] | Bi2Se3 | $15$ | $1.25\text{\times}{10}^{5}$ | $1.2$
[49] | InAlAs|InGaAs | $6.5$ | $5.4\text{\times}{10}^{4}$ | $1.2$
[50] | Bi|Ag | $5.0$ | $1.5\text{\times}{10}^{5}$ | $0.3$
Interfacial exchange-field spin filtering
[this work] EuS|Cu($10\text{\,}\mathrm{n}\mathrm{m}$) | $0.6$ | $1.88\text{\times}{10}^{3}$ | $3.2$
[this work] EuS|Cu($20\text{\,}\mathrm{n}\mathrm{m}$) | $0.3$ | $9.7\text{\times}{10}^{3}$ | $0.3$

As shown in Fig. 3(a), the field-cooled magnetization in the Py$|$Cu$|$EuS
trilayer shows an increase below $16\text{\,}\mathrm{K}$, which coincides with
the ferromagnetic ordering temperature $T_{C}^{\text{EuS}}$ of high-quality
EuS films [23, 1]. The non-zero baseline magnetization of about
$230\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{m}\mathrm{u}$ is due
to Py, which has a much larger Curie temperature
($T_{C}^{\text{Py}}\approx$800\text{\,}\mathrm{K}$$). Like Py, EuS films
exhibit an in-plane anisotropy (Supplemental Materials Fig. S1 and Sec. S3).

Figure 3(b) shows normalized hysteresis loops $M(H)/M_{\text{sat}}$ of the
Py|Cu($10\text{\,}\mathrm{n}\mathrm{m}$)|EuS trilayer for in-plane field $H$,
measured at $30\text{\,}\mathrm{K}$ and $5\text{\,}\mathrm{K}$ (gray circles
and red squares, respectively). Compared to the soft-magnetic hysteresis loop
at $30\text{\,}\mathrm{K}$, where only Py is ferromagnetic, the low-
temperature loop is broader and shows a more complex behaviour, indicative of
non-collinear spin states arising from interactions between Py and EuS.
Regardless of the interaction, a low field of $4\text{\,}\mathrm{m}\mathrm{T}$
is sufficient to achieve complete switching.

With the observation of collinear and non-collinear magnetic configurations in
Py|Cu|EuS trilayers, we fulfill a key requirement for the observation of a GMR
signal related to the interfacial exchange-field spin filtering process. As
shown in Fig. 2, we now discuss the longitudinal magnetoresistance
$R_{xx}=V_{x}/{I_{x}}$ only, with a voltage $V_{x}$ measured parallel to the
charge current $\mathbf{j}_{c}\parallel x$ and with in-plane magnetic fields
applied along $x$ and $y$ directions.

Temperature-dependent magnetoresistance measurements show an upturn below the
EuS transition temperature $T_{C}^{\text{EuS}}$, indicative of the onset of
IESF process at the Cu|EuS interface as EuS turns ferromagnetic (see
Supplemental Material Fig. S2 and Sec. S4).

In order to reliably quantify the GMR related to IESF and to avoid
contamination by magnetoresistive effects in Py, we compare measurements above
and below $T_{C}^{\text{EuS}}$, which allows us to separate anisotropic
magnetoresistance (AMR) and GMR contributions occurring in the device. Figure
4 shows the longitudinal magnetoresistance $R_{xx}(H_{i})$ ($i=x,y$) for the
two Py|Cu($t_{\text{Cu}}$)|EuS samples with
$t_{\text{Cu}}=$10\text{\,}\mathrm{n}\mathrm{m}$$ and
$20\text{\,}\mathrm{n}\mathrm{m}$ shown in Fig. 4(a) and (b), respectively. In
particular, for $R_{xx}(H_{x})$ the $5\text{\,}\mathrm{K}$ data shows dips
around $\pm 3\text{\,}\mathrm{m}\mathrm{T}$ (taking into account a field
offset of about $-3\text{\,}\mathrm{m}\mathrm{T}$) for either sample (red and
blue lines), which is absent in the high-temperature $30\text{\,}\mathrm{K}$
data (gray lines), giving clear evidence of GMR signals occurring below
$T_{C}^{\text{EuS}}$.

From the low-temperature $R_{xx}(H_{x})$ curves shown in Fig. 4 the
magnetoresistance MR is determined to be
$\text{MR}_{$10\text{\,}\mathrm{n}\mathrm{m}$}=$-0.28\text{\,}\mathrm{\char
37\relax}$$ for the $t_{\text{Cu}}=$10\text{\,}\mathrm{n}\mathrm{m}$$ device
($\text{MR}_{$20\text{\,}\mathrm{n}\mathrm{m}$}=$-0.14\text{\,}\mathrm{\char
37\relax}$$). The negative GMR signal is related to the opposite sign of
$P^{\text{Py|Cu}}_{\text{eq}}$ and $P^{\text{Cu|EuS}}_{\text{neq}}$
originating from the electronic structure at each interface, as known from
other material combinations for current-in-plane GMR devices [51, 52, 53].

Comparable GMR values are obtained for $R_{xx}(H_{y})$, however, with a larger
uncertainty due to the additional AMR which originates from the complex non-
collinear spin textures in Py close to zero fields [and thus also is
observable at $30\text{\,}\mathrm{K}$ (Supplemental Material Sec. S5)].

As the spin polarization of the Py|Cu interface is known from previous works
to be $P^{\text{Py|Cu}}_{\text{eq}}=$48\text{\,}\mathrm{\char 37\relax}$$ at
$10\text{\,}\mathrm{K}$ [1], we can now employ Eq. (3) to calculate the
conductor-insulator interfacial spin polarization
$P_{\text{neq}}^{\text{Cu|EuS}}$. We obtain
$P_{\text{neq}}^{\text{Cu|EuS}}=$0.6\text{\,}\mathrm{\char 37\relax}$$ for
$t_{\text{Cu}}=$10\text{\,}\mathrm{n}\mathrm{m}$$ at a current density of
$j_{c}=$1.88\text{\times}{10}^{3}\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$,
and $P_{\text{neq}}^{\text{Cu|EuS}}=$0.3\text{\,}\mathrm{\char 37\relax}$$ for
$t_{\text{Cu}}=$20\text{\,}\mathrm{n}\mathrm{m}$$ at
$9.7\text{\times}{10}^{3}\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$.

In general, the obtained values for $P_{\text{neq}}^{\text{Cu|EuS}}$ are
conservative estimates, and the actual spin polarization generated by the IESF
process might be even higher, if spurious AMR contributions, the influence of
sweep rates, or incomplete antiparallel coupling between EuS and Py could be
taken into account.

##  IV Discussion

A spin polarization of half a percent might look small compared to the large
values of $P_{\text{neq}}^{\text{SHE}}\approx$11\text{\,}\mathrm{\char
37\relax}$$ obtained from the spin-Hall effect at Pt interfaces [7], or even
$P_{\text{neq}}^{\text{REE}}\approx$60\text{\,}\mathrm{\char 37\relax}$$ for
the Rashba-Edelstein effect in topological insulators [8, 9, 10]. However,
this comparison ignores that our experiments were performed at current
densities which are lower than in many reported experiments.

To quantitatively compare the efficiency of the spin polarization generated
via different effects, we make use of the fact that for all these mechanisms,
including the IESF, the current-induced non-equilibrium spin polarization
scales linearly with the charge current density $P_{\text{neq}}\propto j_{c}$
[6, 8]. This linear relationship allows us to calculate a normalized
polarization
$P^{\text{norm}}_{\text{neq}}=|P_{\text{neq}}/j_{c}|j_{c}^{\text{ref}}$ for
literature values, evaluated at the reference charge current density
$j_{c}^{\text{ref}}$. Here, we choose
$j_{c}^{\text{ref}}=${10}^{4}\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$,
a value compatible with CMOS operation and thus a desirable target for low-
power spintronic devices [54].

Table 2 compares the normalized polarization $P_{\text{neq}}^{\text{norm}}$
for different materials from literature and our measurements. Taking the
normalization into account, the efficiency of interfacial exchange-field spin
filtering is on par with more conventional non-equilibrium charge-to-spin
conversion effects, like the spin-Hall and Rashba-Edelstein effect. The
reduction of $P_{\text{neq}}^{\text{norm}}$ for
$t_{\text{Cu}}=$20\text{\,}\mathrm{n}\mathrm{m}$$ compared to the
$10\text{\,}\mathrm{n}\mathrm{m}$ device from 3.2% to 0.3% most likely is a
direct consequence of the highly localized interfacial exchange field confined
to the Cu|EuS interface, which thus reduces the efficiency in thicker films.

Interfacial exchange-field spin filtering in the bilayer (normal non-heavy
metal)|(ferromagnetic insulator) devices, therefore, offers an attractive
method for future nanoscale generators of current-induced spin polarization.
As the converting material (here EuS) is highly insulating, parasitical
resistivities are greatly reduced, and the dissipated power $P_{\text{loss}}$
in the metallic part with resistivity $\rho$ and wire cross section $A=wt$
(with wire width $w$ and thickness $t$) scales as

$P_{\text{loss}}=RI^{2}\propto\frac{\rho}{A}\left(j_{c}A\right)^{2}\propto\rho
j_{c}^{2}t\,.$ (4)

This equation promises a profound reduction of waste heat generation by
several orders of magnitude: First, via the lower device resistance $R$ by
using highly-conductive CMOS-compatible metals like Cu or Al. Second, via the
effective reduction of required current densities $j_{c}$ via the IESF, as
listed in Tab. 2. And third, the favourable scaling $P_{\text{loss}}\propto t$
with the conductor thickness is further enhanced by the confinement of the
interfacial exchange field to within nanometres at the metal|insulator
interface, which leads to an increase of $P_{\text{neq}}^{\text{norm}}$
(respectively a reduction of the required current density $j_{c}$ to achieve
the same non-equilibrium spin polarization).

Finally, the polarization $\mathbf{s}$ of the spin-polarized current generated
via interfacial exchange field filtering can both be tuned externally by
fields, as well as allows for field-free operation due to the non-volatile
magnetization in the ferromagnetic insulator. As Cu has a long spin diffusion
length, the resulting spin current furthermore can be transported over long
distances. These properties are useful for applications, e.g., for the
implementation of next-generation low-power magnetic random access memory
devices or for coupling spintronic oscillators [55, 3, 56].

##  V Outlook

In this work, we demonstrate a new pathway for current-induced spin
polarization via interfacial exchange field filtering (IESF), based on the
spin-dependent scattering at a ferromagnetic insulator that is highly confined
to the interface between a nonmagnetic metal and an insulating ferromagnet.
From magnetoresistance measurements in Py|Cu|EuS trilayer Hall bar devices, we
quantified the spin polarization arising from this effect to be in the order
of $P_{\text{neq}}^{\text{Cu|EuS}}\approx$3\text{\,}\mathrm{\char 37\relax}$$
at low current densities of
${10}^{4}\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$, which is
comparable to the charge-to-spin conversion efficiency based on spin-orbit
coupling mechanisms, such as the spin-Hall and Rashba-Edelstein effect.

Interfacial exchange field filtering therefore is an attractive pathway
towards insulating spintronics devices operating at low current densities and
with additional magnetic reconfigurability and non-volatility. Future
developments, therefore, should aim at identifying highly-insulating
ferromagnets, such as garnets [57, 58] or other semiconductors and insulators,
in particular those whose spin-mixing conductance exhibit a large imaginary
contribution [59, 37, 60]. Such materials would allow harnessing interchange
exchange-field spin filtering for implementation in functional spintronic
devices, ideally at room temperature and with CMOS compatibility.

###### Acknowledgements.

We thank Touru Yamauchi for giving access to the MPMS facility at ISSP, and
Jonathan Leliaert for helpful suggestions regarding the implementation of
micromagnetic simulations. This work was supported by a Grant-in-Aid for
Scientific Research on Innovative Area, Nano Spin Conversion Science, MEXT,
Japan (Grant No. 26103002). N.L. received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program under Marie Skłodowska Curie Grant Agreement No. 844304 (LICONAMCO),
as well as support from the European Community under the Horizon 2020 program,
Contract No. 101001290 (3DNANOMAG). P.K.M. acknowledges funding from IIT
Madras (Grant Numbers IP21221798PHNFSC008989 and RF21221392PHNFIG008989) and
Science and Engineering Research Board (SERB), India Grant No.
SRG/2022/000438.

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Non-equilibrium spin polarisation via interfacial exchange-field spin
filteringSupplemental Material

##  S1 Current shunting in trilayers

Typical resistances of EuS films at low temperatures are very large (see
Methods), and we can safely disregard any charge conduction through the EuS
layer. In the Py|Cu|EuS trilayer Hall bars therefore the net charge current
$I_{T}$ flows parallel through both the Cu and Py layer, so that
$I_{T}=I_{\mathrm{Py}}+I_{\mathrm{Cu}}$ (current shunting). Due to the partial
shunting in the magnetic Py layer, the charge current $I_{\text{Cu}}$ in the
Cu layer is also spin-polarised with a spatially varying polarization
direction determined by the local moment at the Py$|$Cu interface.

The ratio $\eta$ between currents $I_{\mathrm{Py}}$ and $I_{\mathrm{Cu}}$ is
inversely proportional to the resistances of the respective layers,

$\eta=\frac{I_{\mathrm{Cu}}}{I_{\mathrm{Py}}}=\frac{R_{\mathrm{Py}}}{R_{\mathrm{Cu}}}=\frac{\rho_{\mathrm{Py}}}{\rho_{\mathrm{Cu}}}\,\frac{t_{\mathrm{Cu}}}{t_{\mathrm{Py}}}\,,$
(S1)

and the fraction of the respective current flowing through both the Cu and Py
layer then can be written as

$\frac{I_{\mathrm{Cu}}}{I_{T}}=\frac{\eta}{1+\eta}\qquad\mathrm{and}\qquad\frac{I_{\mathrm{Py}}}{I_{T}}=\frac{1}{1+\eta}\,.$
(S2)

The resistances in the wires is given by
$R_{i}=\rho_{i}\frac{l_{i}}{w_{i}t_{i}}$ with material-dependent resistivity
$\rho_{i}$ ($i$ = Py, Cu) and $l_{i}$, $w_{i}$, and $t_{i}$ as lengths,
widths, and thicknesses of the respective layer of the wire. As the length and
width of the wires are equal, ${l_{\mathrm{Cu}}=l_{\mathrm{Py}}}$ and
${w_{\mathrm{Cu}}=w_{\mathrm{Py}}}$, only the layer thicknesses appear in Eq.
(S1).

At $10\text{\,}\mathrm{K}$ the resistivities are
$\rho_{\mathrm{Cu}}\approx$2\text{\,}\mathrm{\SIUnitSymbolMicro}\Omega\,\mathrm{c}\mathrm{m}$$
and
$\rho_{\mathrm{Py}}\approx$32\text{\,}\mathrm{\SIUnitSymbolMicro}\Omega\,\mathrm{c}\mathrm{m}$$
[1]. With these values we obtain current ratios
${\eta(t_{\mathrm{Cu}}=$10\text{\,}\mathrm{n}\mathrm{m}$)\approx 16}$ and
${\eta(t_{\mathrm{Cu}}=$20\text{\,}\mathrm{n}\mathrm{m}$)\approx 32}$ for the
Py$|$Cu$|$EuS devices used in this work. We thus estimate that
$94\text{\,}\mathrm{\char 37\relax}$ of the net charge current flows through
the trilayer device with Cu layer thickness
$t_{\mathrm{Cu}}=$10\text{\,}\mathrm{n}\mathrm{m}$$ (respectively
$97\text{\,}\mathrm{\char 37\relax}$ for
$t_{\mathrm{Cu}}=$20\text{\,}\mathrm{n}\mathrm{m}$$).

Figure S1: Magnetic anisotropy. Hysteresis loops of a
$70\text{\,}\mathrm{n}\mathrm{m}$-thick EuS film measured at
$5\text{\,}\mathrm{K}$ with in-plane (closed red circles) and out-of-plane
(closed blue circles) fields, showing easy-plane anisotropy. The negative
slope at high fields originates from the diamagnetic contribution of the
Si/SiO2 substrate.  Figure S2: Temperature-dependent (a-c) longitudinal and
(d-f) transverse magnetoresistance of multi-layer structures consisting of
permalloy (gray), copper (orange), and EuS (blue), as indicated by schematics
on top. For EuS devices, the magnetoresistances (a,b) $R_{xx}(T)$ and (d,e)
$R_{xy}(T)$ shows a clear upturn below
$T_{C}^{\text{EuS}}\approx$16\text{\,}\mathrm{K}$$ (gray dashed line), with
$R_{xx}(T)$ also showing a thermal hysteresis at zero fields (arrows indicate
the sweep direction). (c,f) In contrast, the control Py|Cu device shows a low-
temperature saturation of the magnetoresistance only.

##  S2 CIP-GMR for Py|Cu|EuS trilayers

According to the two-carrier conduction model, the charge conduction can be
considered in up-spin and down-spin channels, with spin-dependent
resistivities $\rho^{i}_{\uparrow}$ and $\rho^{i}_{\downarrow}$ at the
magnet|(non-magnetic metal) interfaces $i$, where $i$ can be either Py$|$Cu or
Cu$|$EuS. With this, the magnetoresistance $\mathrm{MR}$ of a CIP GMR trilayer
is [2]

$\mathrm{MR}=\frac{(\alpha_{\text{Py|Cu}}-1)(\alpha_{\text{Cu|EuS}}-1)}{\alpha_{\text{Py|Cu}}(1+q)+\alpha_{\text{Cu|EuS}}(1+q^{-1})}\,.$
(S3)

Here, ${\alpha}^{i}=\rho_{\downarrow}^{i}/\rho^{i}_{\uparrow}$ is the spin
asymmetry at the interface, and
${q=\rho^{\text{Cu$|$EuS}}_{\uparrow}/\rho^{\text{Py$|$Cu}}_{\uparrow}}$
denotes the ratio of the up-spin resistivities of the two ferromagnets.

Due to its high resistance, no charge current flows through the EuS layer,
and, as described above, only a small fraction of the net current passes
through Py. Thus we can assume that the spin-dependent resistances
($\rho^{i}_{\uparrow}$) at each interface are dominated by the Cu resistance,
and $q\approx 1$:

$q=\frac{\rho^{\text{Cu|EuS}}_{\uparrow}}{\rho^{\text{Py|Cu}}_{\uparrow}}\approx\frac{\rho^{\text{Cu|EuS}}}{\rho^{\text{Py|Cu}}}\approx
1\,.$ (S4)

The spin polarization $P_{i}$ then can be calculated from the interface spin
asymmetry parameter $\alpha^{i}$ as

$P_{i}=\frac{\rho_{\downarrow}^{i}-\rho^{i}_{\uparrow}}{\rho^{i}_{\downarrow}+\rho^{i}_{\uparrow}}=\frac{1-\alpha^{i}}{1+\alpha^{i}}.$
(S5)

From Eqs. (S3) and (S5) we obtain Eq. (3) in the manuscript:

$\text{MR}=\frac{{P_{\text{Py|Cu}}P_{\text{EuS|Cu}}}}{{1-P_{\text{Py|Cu}}P_{\text{EuS|Cu}}}}\,.$
(S6)

##  S3 Magnetic anisotropy of EuS

Below the Curie temperature of
$T_{C}^{\mathrm{EuS}}\approx$16\text{\,}\mathrm{K}$$, EuS is a soft
ferromagnet with an in-plane magnetic anisotropy, as demonstrated by
hysteresis loops shown in Fig. S1. Here, a small field of
$4\text{\,}\mathrm{m}\mathrm{T}$ is sufficient to switch the in-plane
magnetization, whereas a much higher field of about $1.5\text{\,}\mathrm{T}$
is necessary to saturate the magnetization out-of-plane.

##  S4 Temperature-dependent magnetoresistance

Fig. S2 summarizes the temperature dependence of the longitudinal $R_{xx}(T)$
and transverse $R_{xy}(T)$ magnetoresistance. For the Py|Cu|EuS devices, Fig.
S2(a,b,d) the zero-field curves show a clear upturn at
$T_{C}^{\text{EuS}}\approx$16\text{\,}\mathrm{K}$$, below which EuS becomes
ferromagnetic and interfacial exchange field spin filtering plays a role (red
squares). The upturn can be suppressed by applied magnetic fields (blue dots),
which stabilise the collinear magnetization alignment between Py and EuS. The
longitudinal magnetoresistance $R_{xx}(T)$ was also found to be hysteretic
below $T_{C}^{\text{EuS}}$ (sweep directions are indicated by arrows), likely
related to the emergence of complex multi-domain states

The downturn (instead of an upturn) in the
Py|Cu($10\text{\,}\mathrm{n}\mathrm{m}$)|EuS device for the transverse
magnetoresistance $R_{xy}(T)$ [Fig. S2(e)] probably is due to the high
shunting of the Hall voltage via the thicker Cu layer.

As shown in Fig. S2(c,f), the clear change of the magnetoresistance at
$T_{C}^{\text{EuS}}$ is notably absent in a Py|Cu reference device, where the
temperature dependence of $R_{xy}(T)$ and $R_{xx}(T)$ follows regular metallic
behaviour, saturating to a constant resistance value below
$10\text{\,}\mathrm{K}$.

To summarize, we observe a clear variation of $R_{xx}(T)$ or $R_{xy}(T)$ in
our Py|Cu|EuS devices emerging at $T_{C}^{\text{EuS}}$. This provides direct
evidence for the spin-dependent scattering of conduction electrons inside Cu
from the IESF at the Cu|EuS interface.

##  S5 Micromagnetic Simulations

Figure S3: Micromagnetic behaviour of Py Hall cross, from mumax3 simulations.
(a) Average magnetization $M_{x}$ and $M_{y}$ in the central region of the
Hall cross [white outline in (c)]. (b) Expected shape of possible
magnetoresistance contributions due to field-dependent non-collinear spin
textures (for perpendicular $\mathbf{j}_{c}\parallel x$ and
$\mathbf{B}\parallel y$). (c) Micromagnetic configurations at field values
marked with blue circles in (a).

Finite-difference micromagnetic simulations were implemented using the GPU-
accelerated software package MuMax3 [3]. The geometry of a permalloy Hall
cross of $5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ leg width,
$17.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ leg length, and a
thickness of $t=$10\text{\,}\mathrm{n}\mathrm{m}$$ was discretised with cuboid
cells of $5\text{\,}\mathrm{n}\mathrm{m}$ x $5\text{\,}\mathrm{n}\mathrm{m}$ x
$10\text{\,}\mathrm{n}\mathrm{m}$ size. Material constants of bulk permalloy
were used, with saturation magnetization
${M_{\text{sat}}=$860\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$}$,
exchange constant
${A_{\text{ex}}=$13\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$}$, and
vanishing magneto-crystalline anisotropy $K=0$. The equilibrium magnetization
configuration was simulated for static magnetic fields applied along the
vertical $y$ direction, with a small field of
$0.1\text{\,}\mathrm{m}\mathrm{T}$ applied along the horizontal $x$ direction
to break the symmetry. To emulate the response of an infinite wire in either
direction, the magnetic surface charges at the bounding box boundary were
compensated.

Figure S3(a) shows the magnetization components in the central region of the
Hall cross under application vertical fields $B_{y}$: While $M_{y}$ shows a
conventional hysteresis loop, due to the soft magnetic behaviour sizable in-
plane contributions $M_{x}$ emerge [see snapshots in (c)].

The complex non-uniform spin textures within the Py Hall cross allow for
additional magnetoresistance effects to emerge, proportional to specific
products of the magnetization components, as shown in Fig. S3(b). These
contributions make quantification of the GMR due to the interfacial exchange
field spin filtering more difficult.

In the case of the longitudinal magnetoresistance $R_{xx}$, the response is
dominated by the anisotropic magnetoresistance (AMR) [4] with

$R_{\text{AMR}}=R_{\bot}+(R_{\parallel}-R_{\bot})M^{2}_{x}\,,$ (S7)

where $R_{\bot}$ and $R_{\parallel}$ are resistance in the configuration
$\mathbf{j}\bot\mathbf{M}$ and $\mathbf{j}\parallel\mathbf{M}$, respectively.
In the results shown in Fig. 4 of the main manuscript, the AMR leads to the
peaks when the current is perpendicular to the field $(\mathbf{j}_{c}\parallel
x)\perp H_{y}$, and the field approaches zero $H_{y}\rightarrow 0$, which are
especially noticeable at $30\text{\,}\mathrm{K}$, where EuS is paramagnetic.

The transverse magnetoresistance $R_{xy}$ is [4, 5]

$R_{xy}=\frac{\mu_{0}R_{0}}{t}H_{z}+\frac{\mu_{0}R_{S}}{t}M_{z}+\frac{\Delta
R}{t}M_{x}M_{y}\,.$ (S8)

Here, $R_{0}$ and $R_{S}$ are the coefficients for the ordinary Hall effect
(OHE) and anomalous Hall effect (AHE), respectively. The third term
corresponds to the planar Hall effect (PHE), which is related to the AMR of
Eq. (S7) via the coefficient ${\Delta R=(R_{\parallel}-R_{\bot})}$. As shown
in Fig. S3(b), due to the dependence of the PHE on the product $M_{x}M_{y}$ we
expect an asymmetric $R_{xy}(\mu_{0}H_{x,y})$ profile upon sweeping the in-
plane magnetic field.

## References

  * Muduli  _et al._ [2018] P. K. Muduli, M. Kimata, Y. Omori, T. Wakamura, S. P. Dash, and Y. Otani, Detection of the interfacial exchange field at a ferromagnetic insulator-nonmagnetic metal interface with pure spin currents, Phys. Rev. B 98, 024416 (2018).
  * Tsymbal  _et al._ [2001] E. Y. Tsymbal, D. G. Pettifor, H. Ehrenreich, and F. Spaepen, Perspectives of giant magnetoresistance, in  _Solid State Physics_, Vol. 56 (Academic Press, 2001) pp. 113–237.
  * Vansteenkiste  _et al._ [2014] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The design and verification of MuMax3, AIP Advances 4, 107133 (2014).
  * McGuire and Potter [1975] T. McGuire and R. Potter, Anisotropic magnetoresistance in ferromagnetic 3d alloys, IEEE Transactions on Magnetics 11, 1018 (1975).
  * Goldberg and Davis [1954] C. Goldberg and R. E. Davis, New galvanomagnetic effect, Phys. Rev. 94, 1121 (1954).


# Magnetoelastic standing waves induced in UO2 by microsecond magnetic field
pulses

Rico Schönemann<EMAIL_ADDRESS>MPA-MAGLAB, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA.  George Rodriguez  Q-6, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA.  Dwight Rickel  MPA-MAGLAB,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA.  Fedor Balakirev
MPA-MAGLAB, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.  Ross
D. McDonald  MPA-MAGLAB, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA.  Jordan Evans  MPA-MAGLAB, Los Alamos National Laboratory, Los Alamos, NM
87545, USA.  Boris Maiorov  MPA-MAGLAB, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA.  Charles Paillard  Université Paris-Saclay, CNRS,
UMR8580, Lab SPMS, Cent Supelec, 8-10 Rue Joliot Curie, F-91190 Gif Sur
Yvette, France  Physics Department and Institute for Nanoscience and
Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA
Laurent Bellaiche  Physics Department and Institute for Nanoscience and
Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA  Myron
B. Salamon  MPA-MAGLAB, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA.  Krzysztof Gofryk  Idaho National Laboratory, Idaho Falls, Idaho 83415,
USA  Marcelo Jaime<EMAIL_ADDRESS>MPA-MAGLAB, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA.  Physikalisch-Technische Bundesanstalt,
38116 Brauschweig, Germany

###### Abstract

Magnetoelastic measurements in the piezomagnetic antiferromagnet UO2 were
performed via the fiber Bragg grating method in magnetic fields up to
$150\,\mathrm{T}$ generated by a single-turn coil setup. We show that in short
timescales, order of a few micro seconds, pulsed-magnetic fields excite
mechanical resonances at temperatures ranging from $10\,\mathrm{K}$ to
$300\,\mathrm{K}$, in the paramagnetic as well as within the robust
antiferromagnetic state of the material. These resonances, which are barely
attenuated within the 100 ms observations, are attributed to the strong
magnetoelastic coupling in UO2 combined with the high crystallographic quality
of the single crystal samples. They compare well with mechanical resonances
obtained by a resonant ultrasound technique and superimpose on the known non-
monotonic magnetostriction background. A clear phase-shift of $\pi$ in the
lattice oscillations is, unexpectedly, observed in the antiferromagnetic state
when the magnetic field overcomes the piezomagnetic switch-field
$H_{c}\simeq-18\,\mathrm{T}$. We further present simulations and a theoretical
argument to explain the observed phenomena.

##  I Introduction

The antiferromagnetic (AFM) insulator Uranium dioxide UO2 has been the subject
of extensive research during the last decades predominantly due to its
widespread use as nuclear fuel in pressurized heavy water reactors. Besides
efforts to understand the unusually poor thermal conductivity of UO2 which
impacts its performance as nuclear fuel Gofryk  _et al._ (2014), a recent
magnetostriction study in pulsed magnetic fields to $92\,\mathrm{T}$ uncovered
linear magnetostriction in UO2 Jaime  _et al._ (2017a) \- a hallmark of
piezomagnetism.

Piezomagnetism is characterized by the induction of a magnetic polarization by
application of mechanical strain, which, in the case of UO2, is enabled by
broken time-reversal symmetry in the 3-$k$ antiferromagnetic structure which
emerges below $T_{\mathrm{N}}=30.8\,\mathrm{K}$ Burlet  _et al._ (1986);
Caciuffo  _et al._ (1999); Blackburn  _et al._ (2005); Caciuffo  _et al._
(2007) and is accompanied by a Jahn-Teller distortion of the oxygen cage
Giannozzi and Erdös (1987); Ikushima  _et al._ (2001); Santini  _et al._
(2009); Caciuffo  _et al._ (2011). This also leads to a complex hysteretic
magnetoelastic memory behavior where magnetic domain switching occurs at
fields around $\pm 18\,\mathrm{T}$ at $T=2.5\,\mathrm{K}$. Interestingly, the
very large applied magnetic fields proved unable to suppress the AFM state
that sets in at $T_{\mathrm{N}}$ Jaime  _et al._ (2017a). These earlier
results provide direct evidence for the unusually high energy scale of spin-
lattice interactions, and call for further studies in higher magnetic fields.

Figure 1: (a) Uniaxial magnetostriction $\Delta l/l_{0}$ of a UO2 single
crystal along the [111] axis for $T=10\,\mathrm{K}$ (red line) and
$40\,\mathrm{K}$ (black line). The dashed blue line is a 2nd order polynomial
fit of the $40\,\mathrm{K}$ data representing the expected magnetostriction
behavior. The arrows marking up and down sweep of the magnetic field. The
inset shows a picture of the UO2 single crystal attached to the FBG-furnished
fiber. Note that the sample is not physically attached to the sample holder,
other than through the $125\,\mu m$ optical fiber. A copper coil, located next
to the sample was used to measure the magnetic field. (b) $\Delta l/l_{0}$ of
UO2 as a function of time for pulses at different temperatures between 10 and
$300\,\mathrm{K}$. The spike in $\Delta l/l_{0}$ during the first few
microseconds is caused by the magnetostriction of the sample (shown in (a))
due to the changing magnetic field during the field pulse (d), whereas the
oscillatory part of the signal continues after the field decayed to zero. A
detailed view of the oscillations at $40\,\mathrm{K}$ is shown in the inset
(c) as well as the magnetic field vs. time during the first $20\,\mathrm{\mu
s}$ (d).

Here we present axial magnetostriction data obtained in a UO2 single crystal
in magnetic fields to $150\,\mathrm{T}$. These ultra-high fields were produced
by single-turn coil pulsed resistive magnets and applied along the [111]
crystallographic axis at various temperatures between $10\,\mathrm{K}$ and
room temperature. We see, at all temperatures, a dominant negative
magnetostriction proportional to the square of the applied field accompanied
by unexpectedly strong oscillations that establish a mechanical resonance in
the sample virtually instantly upon delivery of the ultra-fast,
$10^{2}\,\mathrm{T/\mu\mathrm{s}}$, magnetic field rate-of-change. The
oscillations observed quickly set, well within a single oscillation period,
are long-lasting due to very low losses, with frequencies in the hundreds of
kilohertz that match mechanical resonances obtained with a resonant ultrasound
spectroscopy (RUS) technique Balakirev  _et al._ (2019). When the sample
temperature is reduced, the frequencies soften, consistent with observations
in studies of the UO2 elastic constant $c_{44}$ as a function of temperature
Brandt and Walker (1967). When the magnetic field is applied at temperatures
$T<30.8\,\mathrm{K}$ in the AFM state, the magnetic field changes sign to a
negative field magnitude (a characteristic of destructive magnets) in excess
of the UO2 AFM domain-switch-field of $\simeq$ -18 T. This negative field that
follows a positive field pulse exposes yet another unexpected result, namely a
$\pi$ (180o) phase-shift in the magnetoelastic oscillations. We use a driven
harmonic oscillator and a analytical model to shed light on the origin of our
findings.

##  II Methods

In this work, the magnetostriction signal of UO2 was measured with a
$100\,\mathrm{MHz}$ coherent pulse fiber Bragg interrogation method. The setup
is driven by a modelocked pulsed $90\,\mathrm{fs}$ Er laser with a
$100\,\mathrm{MHz}$ repetition rate and allows interrogation speed on the
$10\,\mathrm{ns}$ scale. This method offers a faster readout rate than
traditional fiber Bragg grating (FBG) interrogation systems which operate in
the range of several kHz Jaime  _et al._ (2017b). The UO2 single crystal was
attached to the optical fiber using an epoxy encapsulant with the
crystallographic [111] axis aligned parallel to the fiber and the magnetic
field. A picture of the sample is shown as an inset in figure 1(a). Details
about the FBG setup can be found in Ref. Rodriguez  _et al._ (2015). During
the field pulse, the induced voltage in a small copper coil, located in close
proximity to the sample (see picture in figure 1(a)), was used to measure the
magnetic field.

Figure 2: Magnetic field vs. time of a $146\,\mathrm{T}$ pulse in a single-
turn coil (a) and a $60\,\mathrm{T}$ pulse in a non-destructive short pulse
magnet (b). (c) Fourier transformations of the field vs. time curves shown in
(a, b). FFT’s of the short pulse magnet/single-turn pulses are depicted by
red/black lines respectively.

The magnetic field was generated with a semi destructive capacitor-driven
single-turn coil magnet system at the National High Magnetic Field
Laboratory’s pulse field facility at Los Alamos National Laboratory. The
system is designed for fields up to $300\,\mathrm{T}$ with a rise time of
approximately $2.5\,\mathrm{\mu s}$. Note that for magnetic fields in the
region of $200\,\mathrm{T}$ and above, damage to the cryostat and sample
becomes increasingly likely. Therefore magnetic fields were limited to
$\approx 150\,\mathrm{T}$ in this study, with a peak rate of change in the
order of $10^{2}\,\mathrm{T/\mu\mathrm{s}}$. Further details about the single-
turn coil setup are presented in Refs. Mielke and Novac (2010); Mielke and
McDonald (2006). Optical measurement techniques, like the FBG method used
here, are in general advantageous in single-turn experiments when compared to,
$e.g.$, electrical capacitance-based dilatometry measurements predominantly
due to optical fibers being impervious to the large induced voltages generated
inside even small metallic loops caused by the large
$\mathrm{d}B/\mathrm{d}t$, as well as the associated electromagnetic noise.

The measurement of the natural mechanical resonances of elastic vibration
where several normal modes of the sample are determined, is obtained with a
set of piezoelectric transducers using a technique known as resonant
ultrasound spectroscopy. Here, one transducer serves as source of the tunable
sinusoidal wave of frequency $f$ and the other serves as detector at the
synchronous frequency of the sample’s response. The electronics and room
temperature apparatus was described in detail by Balakirev et al. Balakirev
_et al._ (2019). In our case, the transducer had an Al2O3 hemisphere that
allows precise and reproducible point contact on desired positions of the
crystal Evans  _et al._ (2021). As a frequency scan is performed, a resonance
peak is observed at each of the normal modes. We performed resonant mode
measurements on UO2 single crystals alone as well with the $125\,\mathrm{\mu
m}$ optical fiber attached.

Figure 3: (a) Fourier transforms of $\Delta l/l_{0}$ shown in figure 1(b)
displaying three dominant frequencies labeled as $f_{1}$, $f_{2}$ and $f_{3}$.
The low frequency peaks originate from the background magnetostriction. (b)
Resonant ultrasound spectroscopy (RUS) spectra at $T=300\,\mathrm{K}$. The RUS
spectra of the bare sample (black line) and the sample with the attached
optical fiber (red line) were recorded along the [111] axis. (c) Normalized
temperature dependence of $f_{1,2,3}$.  Figure 4: (a) $\Delta l/l_{0}$ vs.
time at $10\,\mathrm{K}$ with a peak field of $126\,\mathrm{T}$ before (black
line) and after (red line) the 700-$1200\,\mathrm{kHz}$ band pass filter was
applied. The experimental data is compared to a driven harmonic oscillator
model with a characteristic frequency of $800\,\mathrm{kHz}$ matching the
dominant oscillations seen during the field pulse in the magnetostriction
signal (grey curve). The experimental data and the model show a $\pi$ phase
shift around $7.3\,\mathrm{\mu s}$ marked by the grey vertical line. (b)
Magnetoelastic oscillations at 10, 40 and $70\,\mathrm{K}$ after application
of a 700-$1200\,\mathrm{kHz}$ band pass filter to the experimental data.
Curves are shifted for clarity in figures (a, b). (c) Fast Fourier Transform
of $\Delta l/l_{0}$ at $T=10\,\mathrm{K}$. The windows of the band pass filter
(W1, W2, W3) are highlighted in different colors. The filtered curves (black
lines) for each window are shown in (d) and are compared with sinusoidal
functions indicated as red dashed lines.

##  III Results

Magnetostriction $\Delta l/l_{0}$ vs. magnetic field curves at $T$ =
$10\,\mathrm{K}$ and $40\,\mathrm{K}$ are displayed in Fig. 1. We observe an
overall negative magnetostriction signal at high fields as expected from
previous results in pulsed magnetic fields to $92\,\mathrm{T}$ Jaime  _et al._
(2017a) with no indication of suppression of the robust AFM order. However,
the signal is highly hysteretic due to the large mechanical resonances that
are superimposed on the magnetostriction signal. The magnetostriction signal
itself roughly follows a second order polynomial field dependence (blue dashed
line).

The field-induced mechanical resonances become clearer when $\Delta l/l_{0}$
is plotted as a function of time (Fig. 1(b)). Oscillations start with the
onset of the field pulse and persist during the entire data acquisition period
($100\,\mathrm{\mu s}$ at $300\,\mathrm{K}$, $50\,\mathrm{\mu s}$ for lower
temperatures). The high quality of our UO2 crystal is probably a key factor
behind the low attenuation of the mechanical resonances observed in the
experiment. This is validated by a large quality factor $Q=f/w$ found by RUS
and ranging from 2500 to 5000. Here, $f$ is the frequency and $w$ the width of
the resonance (fitted by a Lorenzian). The onset of the mechanical resonances
is approximately instantaneous, which indicates that they arise as a response
to the magnetic field change and the strong magnetoelastic coupling. Hence, it
does not appear that this mechanism is triggered by the shock wave generated
by the disintegration of the single turn coil which would need a few
microseconds to reach the sample. A similar experiment run with identical
interrogation parameters and a bare FBG sensor, i.e. with no sample attached
to the fiber, yielded no detectable mechanical resonances.

In order to understand the origin of the observed mechanical resonances, we
compare the typical field vs. time profile of a $146\,\mathrm{T}$ pulse
performed in a single turn coil (Fig. 2(a)) with a $60\,\mathrm{T}$ shot in a
short-pulse magnet (Fig. 2(b)). The field generated by the short pulse magnet
has a total duration of about $100\,\mathrm{ms}$ with a rise time of
$10\,\mathrm{ms}$. The single turn coil on the other hand has a pulse duration
in the order of $25\,\mathrm{\mu s}$ and a rise time of $2.5\,\mathrm{\mu s}$
($\times$4000 faster compared to the non-destructive magnet). Furthermore, the
field switches sign several times, referred to as magnetic field recoil,
displaying a significantly less attenuated behavior than the short-pulse
magnet. The extremely short timescales in the single turn pulse results in a
shift of the Fast Fourier Transform (FFT) of the field pulse towards higher
frequencies up to several MHz (i.e., high frequencies are $50\,\mathrm{dB}$
more intense) which is displayed in Fig. 2(c). Thus, if the system under study
is magnetic - the field-pulse itself can, by virtue of the strong
magnetoelastic coupling present in UO2 Dolling  _et al._ (1965); Dolling and
Cowley (1966); Brandt and Walker (1967), excite mechanical resonances in the
range of several $100\,\mathrm{kHz}$.

The FFT of $\Delta l/l_{0}(t)$ in UO2 reveals three distinct frequencies
labeled $f_{1}$, $f_{2}$ and $f_{3}$ for temperatures between 40 and
$300\,\mathrm{K}$, as shown in Fig. 3(a). All modes display a softening as the
temperature is lowered and a stiffening below $T_{\mathrm{N}}$ (Fig. 3(c)), in
agreement with previous measurements of the elastic constants of UO2 which
show a similar behavior Brandt and Walker (1967). Three independent elastic
constants $c_{11}$, $c_{12}$ and $c_{44}$ exist for a lattice with cubic
symmetry. The distinct temperature dependence of all three mechanical
resonance frequencies $f_{1-3}$ indicates that we probe predominantly
resonances that are associated with $c_{44}$ Brandt and Walker (1967). The
observed frequencies are in agreement with RUS data obtained at
$300\,\mathrm{K}$, shown in Fig. 3(b). The RUS measurements reveal a rich
spectrum of sharp resonances in the frequency range up to $1\,\mathrm{MHz}$.
When the crystal is attached to the fiber, a broadening of resonance peaks is
observed, as well as a decrease in amplitude, indicating a larger damping,
product of a larger system consisting of the crystal, glue and fiber.
Nevertheless, the overall range of mechanical resonance frequencies observed
in the magnetostriction measurements is comparable with RUS spectra. In the
FBG measurements we observe three distinct frequencies. The additional
mechanical resonance frequencies present in the RUS spectra are potential
resonances that are either completely damped by the attached fiber and
encapsulant or they only have a small compressive component parallel to the
fiber since the FBG method is less sensitive to sheer strain. Note that the
RUS measurements on the bare UO2 crystal were performed on a $\approx
2\,\mathrm{mm}$ longer sample than the FBG and RUS measurements with the
optical fiber attached. The change of the sample geometry also affects the
mechanical resonance frequencies.

Upon cooling the sample below the antiferromagnetic transition we observe a
substantial change in the mechanical resonances when compared to temperatures
above $T_{\mathrm{N}}$:

(i) In the data recorded at $10\,\mathrm{K}$, within the AFM ordered phase,
the resonances appear to be significantly damped after $t\approx
7\mu\mathrm{s}$, particularly the FFT amplitude of $f_{3}=800\,\mathrm{kHz}$
is suppressed when compared to $f_{1}$ and $f_{2}$, in contrast to the Fourier
transforms of the data sets above $T_{\mathrm{N}}$ where $f_{3}$ is clearly
the dominant resonance (Fig. 3(a)). The attenuation effect becomes more
apparent when band pass filters are applied to the experimental data,
isolating the individual resonances and removing the magnetostriction
background (Fig. 4(a, b, d)). We show that, when compared to $f_{3}$, the
amplitude of lower resonances $f_{2}$ and $f_{3}$ does not display a drastic
change for $t>7\,\mathrm{\mu s}$. The beating pattern observed in $f_{3}$
indicates the presence of two resonances very close to each other, which we
cannot easily resolve in the FFT. These details are also impacted by the lower
and upper limits chosen for the band pass filter.

(ii) A $\pi$ phase shift can be observed in the mechanical resonances at
$t\approx 7\,\mathrm{\mu s}$ which is accompanied by the observed attenuation
effect. One can clearly identify the $\pi$ phase shift after a
700-$1200\,\mathrm{kHz}$ high pass filter was applied to the experimental data
(Fig. 4(d)). Interestingly only $f_{3}$ shows the phase shift and both lower
resonances $f_{1}$ and $f_{2}$ seem not to be affected and follow a single
sinusoidal function as shown in Fig. 4(d)).

The phase shift around $t\approx 7\,\mathrm{\mu s}$ coincides with a magnetic
field value of approximately $-18\,\mathrm{T}$ close to the field value where
a abrupt sign change of the AFM ordering vector $L_{0}$ (as defined in Ref.
Bar’yakhtar  _et al._ (1985)) leads to a jump in the lattice distortion and
the characteristic piezomagnetic butterfly that was reported in Ref. Jaime
_et al._ (2017a). This effect is illustrated in Fig. 5 with the piezomagnetic
butterfly shown in the inset. The origin of the phase shift can be found in
the sudden reversal of $L_{0}$ (red) to the $-L_{0}$ (blue) as we demonstrate
in the following section.

Figure 5: Magnetic field (upper panel) and magnetostriction of UO2 (lower
panel) as a function of time. The magnetostriction data was mapped and
extrapolated onto the single turn field profile using previously published
pulse field data (Ref. Jaime  _et al._ (2017a)) shown in the inset. The
vertical line at $t\approx 7\,\mathrm{\mu s}$ separates the two states with
positive (red) and negative (blue) AFM ordering vector $L_{0}$ by reaching the
switching field of $\approx-18\,\mathrm{T}$ in this case.

##  IV Hamiltonian

Following Bar’yakhtar et al. Bar’yakhtar  _et al._ (1985), we denote as
$\bm{L}_{1}$, $\bm{L}_{2}$ and $\bm{L}_{3}$ the different AFM vectors that
describe the 3-$k$ order in UO2, and $\bm{M}$ the average magnetization. The
magnetic unit cell below the AFM transition is made of 4 formula units, each
formula unit carrying a magnetic moment $\bm{S}_{1}$, $\bm{S}_{2}$,
$\bm{S}_{3}$, and $\bm{S}_{4}$. The above AFM vectors and magnetization can be
expressed, in terms of the individual magnetic moments as:

$\displaystyle\bm{M}$ $\displaystyle=$
$\displaystyle\bm{S}_{1}+\bm{S}_{2}+\bm{S}_{3}+\bm{S}_{4},$
$\displaystyle\bm{L}_{1}$ $\displaystyle=$
$\displaystyle\bm{S}_{1}+\bm{S}_{2}-\bm{S}_{3}-\bm{S}_{4},$
$\displaystyle\bm{L}_{2}$ $\displaystyle=$
$\displaystyle\bm{S}_{1}-\bm{S}_{2}+\bm{S}_{3}-\bm{S}_{4},$
$\displaystyle\bm{L}_{3}$ $\displaystyle=$
$\displaystyle\bm{S}_{1}-\bm{S}_{2}-\bm{S}_{3}+\bm{S}_{4},$

In the Landau approach, the thermodynamic potential must be able to describe
both the paramagnetic and AFM phases. This is achieved by performing a
polynomial expansion of the free energy whose terms respect the symmetry of
the highest symmetry phase Tolédano and Tolédano (1987). Such terms have
already been worked out in Ref. Bar’yakhtar  _et al._ (1985). The Hamiltonian
of the system can thus be written as:

$\hat{H}=\int d^{3}\bm{r}\hat{\mathcal{H}}(\bm{r}),$ (1)

where $\hat{\mathcal{H}}$ can be found in the supplemental material. Since we
are primarily interested in explaining the properties of the mechanical
resonances and the $\pi$ phase shift in response to an applied pulsed magnetic
fields, we make the simplifying following assumption: the magnetic response of
the sample is primarily determined by the external magnetic field, and elastic
vibrations do not affect it significantly. As a result, we write each magnetic
quantity under the form $\bm{\mu}=\bm{\mu}_{eq}+\delta\bm{\mu}$ (with
$\mu=L_{1},L_{2},L_{3},M$). $\bm{\mu}_{eq}$ represents an equilibrium value,
and $\delta\bm{\mu}$ the deviation away from that value, i.e. the response to
the magnetic pulse.

We then write the mechanical equations of motion, from which we retain
$\rho\ddot{u}_{x}$

$\rho\ddot{u}_{x}=\frac{\partial}{\partial
x}\left(\frac{\partial\hat{\mathcal{H}}}{\partial\eta_{1}}\right)+\frac{\partial}{\partial
y}\left(\frac{\partial\hat{\mathcal{H}}}{\partial\eta_{6}}\right)+\frac{\partial}{\partial
z}\left(\frac{\partial\hat{\mathcal{H}}}{\partial\eta_{5}}\right),$ (2)

In the mechanical equations, $\rho$ is the volumic mass and $u_{\alpha}$ is
the displacement field in direction $\alpha$. Since the derivations are
lengthy, we will focus on the component $u_{x}$ of the displacement field
which, after replacing with the Hamiltonian expression (see SI) and assuming,
as demonstrated in Ref. Bar’yakhtar  _et al._ (1985), that at equilibrium,
$L_{1x}^{eq}=L_{0}$, $L_{2y}^{eq}=L_{0}$ and $L_{3z}^{eq}=L_{0}$, we write
$L_{1x}=L_{0}+\delta L_{1x}$, $L_{1y}=\delta L_{1y}$, $L_{1z}=\delta L_{1z}$,
etc. Similarly, Ref. Jaime  _et al._ (2017a) shows that no magnetization seems
to exist in the antiferromagnetic phase, so we write $M_{x}=\delta M_{x}$,
etc. We also recall that $\eta_{1}=\frac{\partial u_{x}}{\partial x}$,
$\eta_{5}=\frac{\partial u_{x}}{\partial z}+\frac{\partial u_{z}}{\partial
x}$, the equation can now be written, for linear terms, as

$\displaystyle\rho\ddot{u}_{x}$ $\displaystyle\approx$ $\displaystyle
c_{11}\frac{\partial^{2}u_{x}}{\partial
x^{2}}+c_{12}\left(\frac{\partial^{2}u_{y}}{\partial x\partial
y}+\frac{\partial^{2}u_{z}}{\partial x\partial
z}\right)+c_{44}\left(\frac{\partial^{2}u_{x}}{\partial
y^{2}}+\frac{\partial^{2}u_{y}}{\partial y\partial
x}+\frac{\partial^{2}u_{z}}{\partial z\partial
x}+\frac{\partial^{2}u_{x}}{\partial z^{2}}\right)$ (3) $\displaystyle+$
$\displaystyle L_{0}(\lambda_{1}+\lambda_{1}^{\prime})\frac{\partial\delta
L_{1x}}{\partial x}+\lambda_{1}^{\prime}L_{0}\left(\frac{\partial\delta
L_{2y}}{\partial x}+\frac{\partial\delta L_{3z}}{\partial
x}\right)+\frac{\lambda}{2}L_{0}\left(\frac{\partial\delta L_{1y}}{\partial
y}+\frac{\partial\delta L_{2x}}{\partial y}+\frac{\partial\delta
L_{1z}}{\partial z}+\frac{\partial\delta L_{3x}}{\partial z}\right).$

In Equation 3, the terms proportional to $\lambda_{1}$, $\lambda_{1}^{\prime}$
and $\lambda$ relate the linear change of shape of an antiferromagnetic
uniform domain with respect to antiferromagnetic excitation.

If we perform a Fourier transform
$u_{x}=\int\frac{d\bm{q}}{2\pi}\int\frac{d\omega}{2\pi}u_{x}(\bm{q},\omega)e^{i(\bm{q}.\bm{r}-\omega
t)}$, etc., yielding:

$\displaystyle\left(c_{11}q_{x}^{2}+c_{44}\left[q_{y}^{2}+q_{z}^{2}\right]-\rho\omega^{2}\right)u_{x}(\bm{q},\omega)$
$\displaystyle=$
$\displaystyle-(c_{44}+c_{12})q_{x}\left[q_{y}u_{y}(\bm{q},\omega)+q_{z}u_{z}(\bm{q},\omega)\right]$
(4) $\displaystyle+$ $\displaystyle
iq_{x}{\color[rgb]{1,0,0}L_{0}}\left[(\lambda_{1}+\lambda_{1}^{\prime})\delta
L_{1x}(\bm{q},\omega)+\lambda_{1}^{\prime}\left(\delta
L_{2y}(\bm{q},\omega)+\delta L_{3z}(\bm{q},\omega)\right)\right]$
$\displaystyle+$
$\displaystyle\frac{\lambda}{2}{\color[rgb]{1,0,0}L_{0}}\left(q_{y}\left[\delta
L_{1y}(\bm{q},\omega)+\delta L_{2x}(\bm{q},\omega)\right]+q_{z}\left[\delta
L_{1z}(\bm{q},\omega)+\delta L_{3x}(\bm{q},\omega)\right]\right)$

We now have sets of coupled harmonic oscillators which are driven by a force
which is proportional to $L_{0}$ (a cyclic permutation $x\rightarrow
y\rightarrow z$ allows to get the equations for the other components). In
other words, given a proper change of basis, we can diagonalize this set of
equations and write the displacement fields dynamical equations under the form

$(\omega_{0}(\bm{q})^{2}-\omega^{2})u_{1}(\bm{q},\omega)=F_{1}(\bm{q},\omega),$

with $F_{1}(\bm{q},\omega)$ being the force driving the oscillation of $u_{1}$
at pulsation $\omega$ with wavevector $\bm{q}$. This has an obvious solution,
which is

$u_{1}(\bm{q},\omega)=\frac{F_{1}(\bm{q},\omega)}{\omega_{0}(\bm{q})^{2}-\omega^{2}}.$

We note that $F_{1}$ is proportional to $L_{0}$, the antiferromagnetic order
parameter. It is then clear that upon reversal of the AFM order,
$L_{0}\longrightarrow-L_{0}$, the force applied on the set of harmonic
oscillators reverses sign, i.e. $F_{1}\rightarrow-F_{1}=e^{i\pi}F_{1}$ and
thus a $\pi$ phase shift must be experienced in the elastic oscillatory
response of the sample long enough after the pulse. Hence, sufficient
switching of the AFM order is likely the cause of the phase shift observed in
Fig. 4. It is to be noted that the main energy couplings responsible for such
an effect are quadratic in the AFM vectors and linear in strain; in other
words, they are typical (antiferro)magnetostriction terms. We note that some
of those energy couplings are the same ones from which piezomagnetism arises
(see Equation 31-32 from Ref. Bar’yakhtar  _et al._ (1985)).

Therefore, by assuming that the force driving the oscillations is proportional
to the systems strain (shown in Fig. 5) and fixing the frequency at
$800\,\mathrm{kHz}$, we are able to model the experimental data with a simple
driven harmonic oscillator. As depicted in Fig. 4(a) this harmonic oscillator
model reproduces the magnetostriction background as well as the $\pi$ phase
shift in the $800\,\mathrm{kHz}$ oscillations. The attenuation observed in the
oscillations after the switching of $L_{0}$ is not captured by the model and
will be discussed in detail below.

##  V Discussion

A recent X-ray study on UO2 single crystals evidences the presence of AFM
domains and subsequently the coexistence of AFM phases $L_{0}$ and $-L_{0}$
connected by time-reversal even in magnetic fields beyond the piezomagnetic
switching field Antonio  _et al._ (2021). This is also supported by
magnetostriction measurements Jaime  _et al._ (2017a), which show that the
first pulse taken below $T_{\mathrm{N}}$ always has a smaller magnetostriction
slope for fields below the switching field. This could be caused by the
coexistence of all possible domains, with some contracting and some expanding
as the field increases. In our measurements we observe a large attenuation
effect around the switching field but oscillations seem not to be further
damped afterwards. Therefore, the attenuation of $f_{3}$ appears to be caused
by the coupling of the mechanical resonances to critical spin fluctuations
and/or domain movement close to the piezomagnetic switching field which can
lead to a significant attenuation of the mechanical resonances similar to the
dramatically increased ultrasonic attenuation that was observed in UO2 in the
vicinity of the AFM phase transition Brandt and Walker (1967). The mechanical
resonances $f_{1}$ and $f_{2}$ might have a predominantly transversal
character which would explain the smaller amplitude and the absence of
attenuation below $T_{N}$ since the longitudinal or compressive modes are
expected to be more affected by spin fluctuations Itoh (1975). For future
experiments we plan to perform magnetocaloric measurements to detect possible
heating effects at the switching field caused by dissipative processes like
domain movement.

Another interesting point is that the $\pi$ phase shift only occurs in
$f_{3}$. The effect is completely absent in $f_{2}$ and much less clear in
$f_{1}$ which is only slightly out of phase when compared to the single
sinusoidal function in the time interval between 0 and $20\,\mathrm{\mu s}$
(Fig. 4(d)). As of now we do not have a conclusive argument on why the phase
shift is only visible in $f_{3}$. Depending on the involved antiferromagnetic
excitations and the anisotropy of the magneto-elastic couplings, longitudinal
and transversal mechanical resonances can display different phase shifts. A
possible way to test this in future experiments is to use a birefringent FBG
which can yield an orthogonal biaxial strain response along two directions
with polarization based probing techniques.

We demonstrate that mechanical resonances can be a useful tool to detect
otherwise-elusive AFM domain flips, and possibly also other types of
crystallographic domain dynamics (e.g., in liquid crystals). On the other
hand, our results indicate that mechanical resonances can also cause issues in
experiments where they are unwanted. A mitigation strategy in experiments
where excessive noise is prevalent could consist of clamping the sample as
well as to conduct runs with different sample and/or sample-holders geometries
and dimensions to minimize mechanical resonances triggered by the magnetic
field. This phenomena is reminiscent of wire-motion resonances in electrical
transport experiments performed in short-pulse magnets, which can be quite
detrimental to the data quality and which effects are minimized by fixing the
wires and in this way effectively shifting their resonances to frequencies
outside of the experimental range of interest.

##  VI Summary

We measured the lattice dilation along [111] for the first time up to 150T in
UO2, in the AFM as well as in the paramagnetic states. Surprisingly, the AFM
state is robust against a field $\geq$ 120T at $10\,\mathrm{K}$, energy-wise
$\sim 4\times$ stronger than TN = 30K (if $g=2$). This result confirms the
large energy scale for correlations in UO2. We show that mechanical resonances
can be induced virtually instantaneously via the magnetoelastic coupling in
UO2 by $\mu s$ field pulses generated with the single turn coil technique,
making this material an interesting candidate for magneto-elastic transducers.
We demonstrate the impact of the piezomagnetic switching in UO2 on the
standing wave indicated by a $\pi$ phase shift and a distinct mode dependent
attenuation of the mechanical resonances. Our findings present a novel way to
study magnetic dynamics in high magnetic fields and could have an impact on
the interpretation of past and future data collected in experiments involving
semi destructive pulsed magnetic fields as well practical implications, e.g.,
as a way to trigger resonators at faster speeds.

## Acknowledgements

A portion of this work was performed at the National High Magnetic Field
Laboratory, which is supported by the National Science Foundation Cooperative
Agreement No. DMR-1644779 and the state of Florida. R.S. acknowledges funding
through the Seaborg Institute and the NHMFL UCGP program. C.P. and L.B thank
the DARPA Grant No. HR0011-15-2-0038 (MATRIX program). K.G. acknowledges
support from the US DOE BES Energy Frontier Research Centre ”Thermal Energy
Transport under Irradiation” (TETI). M.J. acknowledges support from the US DOE
Basic Energy Science program through the project ”Science at 100T” at LANL.
MJ. and G.R. acknowledge support from the LANL Institute for Materials
Science.

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