#  Tunable spin-flop transition in artificial ferrimagnets

N. O. Antropov  Institute of Metal Physics, 620180 Ekaterinburg, Russia  Ural
Federal University, 620002 Ekaterinburg, Russia  E. A. Kravtsov  Institute of
Metal Physics, 620180 Ekaterinburg, Russia  Ural Federal University, 620002
Ekaterinburg, Russia  M. V. Makarova  Institute of Metal Physics, 620180
Ekaterinburg, Russia  Ural Federal University, 620002 Ekaterinburg, Russia  V.
V. Proglyado  Institute of Metal Physics, 620180 Ekaterinburg, Russia  T.
Keller  Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1,
D-70569 Stuttgart, Germany  Max Planck Society Outstation at the Heinz Maier-
Leibnitz Zentrum (MLZ), D-85748 Garching, Germany  I. A. Subbotin  National
Research Center ”Kurchatov Institute”, 123182 Moscow, Russia  E. M. Pashaev
National Research Center ”Kurchatov Institute”, 123182 Moscow, Russia  G. V.
Prutskov  National Research Center ”Kurchatov Institute”, 123182 Moscow,
Russia  A. L. Vasiliev  National Research Center ”Kurchatov Institute”, 123182
Moscow, Russia  Yu. M. Chesnokov  National Research Center ”Kurchatov
Institute”, 123182 Moscow, Russia  N. G. Bebenin  Institute of Metal Physics,
620180 Ekaterinburg, Russia  V. V. Ustinov  Institute of Metal Physics, 620180
Ekaterinburg, Russia  B. Keimer  Max-Planck-Institut für Festkörperforschung,
Heisenbergstraße 1, D-70569 Stuttgart, Germany  Yu. N. Khaydukov  Max-Planck-
Institut für Festkörperforschung, Heisenbergstraße 1, D-70569 Stuttgart,
Germany  Max Planck Society Outstation at the Heinz Maier-Leibnitz Zentrum
(MLZ), D-85748 Garching, Germany  Skobeltsyn Institute of Nuclear Physics,
Moscow State University, Moscow 119991, Russia

###### Abstract

Spin-flop transition (SFT) consists in a jump-like reversal of
antiferromagnetic magnetic moments into a non-collinear state when the
magnetic field increases above the critical value. Potentially the SFT can be
utilized in many applications of a rapidly developing antiferromagnetic
spintronics. However, the difficulty of using them in conventional
antiferromagnets lies in (a) too large switching magnetic fields (b) the need
for presence of a magnetic anisotropy, and (c) requirement to apply magnetic
field along the correspondent anisotropy axis. In this work we propose to use
artificial ferrimagnets in which the spin-flop transition occurs without
anisotropy and the transition field can be lowered by adjusting exchange
coupling in the structure. This is proved by experiment on artificial Fe-Gd
ferrimagnets where usage of Pd spacers allowed us to suppress the transition
field by two orders of magnitude.

Antiferromagnetic (AF) spintronic is nowadays a rapidly developing area [1, 2,
3, 4, 5]. In addition to non-volatility of conventional ferromagnetic
spintronics the AF devices can offer immunity to external magnetic
disturbances, absence of cross-talks between small-area devices and much
faster dynamics (THz vs MHz). The antiferromagnetic systems are featured by
spin-flop transition (SFT) when there is the transition from antiferromagnetic
ordering to noncollinear (NC) state at magnetic field exceeding certain value
$H_{SP}$. Creation of noncollinear magnetic state and possibility to switch
between AF and NC states may have useful applications by utilizing anomalous
Hall or Nernst effects [6, 7, 8, 9, 10, 11]. In addition, proximity of
noncollinear magnetic texture to superconducting layer generates long-range
triplet superconductivity which may also find diverse applications in
superconducting spintronics [12, 13, 14, 15, 16].

The utilization of the spin-flop effect in AF systems is overly complicated
due to at least two reasons. The first thing is the existence of SFT in AF
requires uniaxial anisotropy and an external field applied along the
corresponding axis. Secondly, typical transition fields $H_{SP}$ in bulk
antiferromagnets are tens of Tesla [17, 18, 19, 20] thus they are too high for
real applications. The need to have anisotropy inside the system can be
circumvented by replacing antiferromagnets with ferrimagnets (FEMs). In the
FEMs one does not require presence of anisotropy and the SFT takes place at
$H_{SP}=\lambda|m_{1}-m_{2}|$ [21], where $m_{1,2}$ are the magnetic moment of
first and second sublattices and $\lambda$ is the exchange parameter. In bulk
systems the $H_{SP}$ are still too high for applications and can hardly be
tuned.

In contrast, artificial ferrimagnets based on magnetic heterostructures give a
possibility to tune the SFT field by varying parameters of ferromagnetic
layers and by introducing non-magnetic spacers. Heterostructures based on 3d
transition metals (TM) and heavy 4f rare-earth (RE) metals, like Fe/Gd, are
model ferrimagnetic systems demonstrating a rich magnetic phase diagram with
complex types of magnetic ordering [22, 23, 24, 25, 26, 27]. Coupling between
4f electrons of Gd and 3d electrons of Fe leads to the antiferromagnetic
alignment of TM and RE magnetic moments which due to the difference in
magnetic moments of Fe($\sim 2\mu_{B}$) and Gd ($\sim 7\mu_{B}$) leads to the
emergence of a one-dimensional ferrimagnetic lattice. The spin-flop transition
was found in Gd/Fe systems at typical value $H_{SP}\sim$3kOe [28], which is
much smaller than that for bulk FEMs but still quite high for applications.
Further tuning of $H_{SP}$ can be gained by suppression of interlayer exchange
coupling which can be performed by spacing of Fe and Gd with a non-magnetic
material like Cr [29, 30], Pt [31] or Si [32].

The SFT can be detected by integral magnetic techniques as a kink on a
magnetic hysteresis loop at $H_{SP}$. In case of artificial FEMs magnetic
signal from thin films is heavily polluted by dia- or paramagnetic signal of
thick substrates.This makes it difficult, if not impossible at all, to use
integral magnetometric methods to study the SFTs. Neutron scattering, being a
depth-selective magnetometric method is a widely used method for studying AFs
and FEMs [33, 34, 35]. Similar to X-ray and light, neutrons diffract at
periodic lattice with period $D$ according to the well-known Bragg law
$n\lambda=2D\sin\theta$. Here $\lambda$ and $\theta$ are the neutron
wavelength and incident angle, and $n$ is integer number corresponding to
order of Bragg peak. Presence of spin one-half makes neutron scattering
sensitive to the magnetic lattice. In case of antiferromagnetic lattice
magnetic peak is doubled comparing to the structural one, so that the magnetic
Bragg peak appears on the positions of $n/2$ of the structural Bragg peaks.
Applying spin analysis, that is detecting neutron spin-states before and after
scattering, allows one to get additional information about magnetic
configuration. The non-spin-flip (NSF) channels (++) and (- -) are sensitive
to the sum and difference of nuclear potential and collinear to the neutron
polarization part of magnetization. Here first and second sign codes neutron
polarization along the external magnetic field $H$ before and after the
scattering process. Presence of non-collinear magnetization causes spin-flip
(SF) scattering (+-) and (-+). In Born approximation the amplitude of the SF
scattering is proportional to the spatial profile of the noncollinear
magnetization in reciprocal space. Thus the SF scattering is very sensitive
channel to detect the SFTs.

In our prior work [36] we studied superlattice
[Fe(3.5nm)/Pd(1.2nm)/Gd(5nm)/Pd(1.2nm)]12. In the neutron experiment we
measured intensity of SF scattering at the position of the first Bragg peak
$R^{SF}_{1}$ as a function of external magnetic field at a temperature of 10K.
Above magnetic field of $H_{SP}$=1.5kOe we detected a 20-fold increase of SF
scattering which is the direct evidence for the presence of SFT in our system.
We note that the $H_{SP}$ field is much smaller than in spacer free Fe/Gd
systems. Subsequent structural studies by transmission electron microscopy and
synchrotron radiation [37] indicated presence of mutual diffusion at Gd/Pd
interface. For thin ($\sim$1nm) Pd spacers this interdiffusion leads to almost
complete dissolution of Pd in Gd. As a result the Curie temperature (and hence
exchange energy) of the (nominal) Gd layer decreases from 294K for bulk Gd to
$\lesssim$ 100K. Thus ability of Pd and Gd to form an alloy with controllable
suppression of exchange energy paves the way for tuning of SFT by varying
thickness of Pd spacer. To do this we prepared series of samples of nominal
composition [Fe(3.5nm)/Pd(t)/Gd(5.0nm)/Pd(t)]12 varying $t$ from 1.0 to 1.6 nm
(details can be found in our prior works [36, 37]). Further we will code
samples as PdYY, where YY is thickness of Pd layer in Angstroms.

Fig. 1a shows the X-ray low-angle diffraction patterns (reflectivities)
measured at a wavelength of $\lambda$=1.54Å from the samples under study. More
than 10 orders of Bragg reflection are seen on the reflectivities, which
indicates good repeatability of the Fe/Gd unit cell. Fig. 1b shows the energy
dispersive X-ray (EDX) microanalysis of scanning transmission electron
microscopy (STEM) of Pd12 sample. The EDX analysis shows well-defined Fe
layers depicted by blue color and yellow layers of GdPd alloy instead of
separate red Gd layers and green Pd spacers. For the sake of simplicity, we
will keep naming Gd layer, remembering however that in reality the layer is a
GdxPd1-x alloy.

Figure 1:  (a) X-ray low-angle diffraction (reflectivity) of samples under
study. Vertical arrows show the position of several Bragg peaks for sample
Pd10. (b) The energy dispersive X-ray (EDX) microanalysis of Pd12 sample.

Polarized neutron reflectometry (PNR) experiment was conducted on the
monochromatic ($\lambda$=4.3Å) reflectometer NREX of the research reactor
FRM-2 (Garching, Germany). Fig.2 shows the PNR data measured on sample Pd10 at
$T$=10 K in magnetic field $H$=1kOe and additional SF curve at $T$=10 K in
magnetic field $H$=3kOe (solid line). In the neutron experiment 4 Bragg peaks
were confidently measured. A large splitting of (++) and (- -) NSF Bragg peaks
indicates the presence of a collinear magnetic moment in the system. At the
same time we observed a much weaker (1-2 orders below NSF signal) SF
scattering at Bragg peaks. The origin of this small, though not negligible SF
signal can be associated with noncollinear inhomogeneities at the Fe/Gd
interfaces. The data at $H$=1kOe can be quantitatively described by a
predominantly collinear AF state with magnetic moments of Gd $M_{Gd}\approx
5\mu_{B}$ and Fe $M_{Fe}\approx 2\mu_{B}$ aligned parallel and antiparallel to
$H$. By increasing the magnetic field above $H_{SP}$=2.3kOe (inset in Fig.2)
we observed a 20-fold increase of SF scattering at the first Bragg peak
$R^{SF}_{1}$. This SFT is similar to observed previously spin-flop in Pd12
sample though taking place at 1kOe higher magnetic field.

Figure 2:  Polarized neutron reflectivities of sample Pd10 measured at $T=10$
K at magnetic field $H=1$ kOe (symbols) and SF curve at $T$=10 K, $H$=3kOe
(solid line) Inset shows the field dependence of intensity of SF scattering at
the first Bragg peak $R^{SF}_{1}(H)$. Vertical arrow denotes the magnetic
field at which spin-flop transition takes place.

By measuring family of $R^{SF}_{1}$(H) scans at different temperatures we were
able to construct the noncollinear magnetic phase diagram for the sample Pd10
in $H$-$T$ coordinates (Fig. 3a). For this sample we observe a collinear AF
state in the temperature range up to 30 K in magnetic fields not exceeding 2
kOe. Above this field, the collinear AF state is replaced by a NC spin-flop
state. Increasing the temperature to 60K leads to a gradual shift of the SFT
field towards lower values. Finally, above 60K, the spin-flip signal
disappears due to the absence of magnetic ordering in Gd layer. Fig.3b and
Fig.3c shows similar phase diagrams for Pd12 and Pd14 samples. One can see
that the transition field $H_{SP}$ decreases with increase of $t$. For the
samples with $t$=1.6nm (not shown) we did not observe any detectable SF signal
evidencing absence of coupling of Fe and Gd layers.

Figure 3: (a)-(c) Experimental ($H$,$T$) maps of $R^{SF}_{1}$ for samples with
different Pd spacer. (d) Simulated map for Pd10 sample (e) Fit-resulted
$J_{1}$ and $J_{2}$ terms vs temperature for Pd10 sample. (f) Thickness
dependence of bilinear and biquadratic energies $J_{1}$ and $J_{2}$ obtained
for $T$=10K.

To describe magnetic state of our systems we applied extended Stoner-Wohlfarth
model widely used for description of magnetic multilayers [38, 8]. Density of
magnetic energy of one Fe/Gd unit cell can be written as

$\begin{split}E(\alpha_{Gd},\alpha_{Fe})=-H[m_{Gd}cos(\alpha_{Gd})+m_{Fe}cos(\alpha_{Fe})]+\\\
J_{1}cos(\alpha_{Gd}-\alpha_{Fe})+J_{2}cos^{2}(\alpha_{Gd}-\alpha_{Fe}).\\\
\end{split}$ (1)

In Eq.1 $m_{X}=M_{X}d_{X}$ is a product of magnetization and thickness
(magnetic moment), $\alpha_{X}$ is the angle between magnetization and $H$ of
a layer $X$ ($X$=Fe,Gd). The first term in (1) is Zeeman coupling which tends
to align magnetic moments of the layers along the external field. The second
term is bilinear antiferromagnetic exchange coupling of Fe and Gd layers with
strength parameter $J_{1}$. The third term describes biquadratic coupling
tending to align the magnetic moments non-collinearly. As seen from (1) in
case $J_{2}$=0 the transition field can be estimated as $H_{SP}\approx
J_{1}|m_{Gd}-m_{Fe}|/m_{Gd}\cdot m_{Fe}$.

For every magnetic field $H$ the magnetic configuration of the system as a
function of $J_{1,2}$ can be obtained by minimizing energy (1) varying angles
$\alpha_{Gd}$ and $\alpha_{Fe}$. The magnetization amplitudes $M_{Gd,Fe}$ and
thicknesses $d_{Gd,Fe}$ were taken from PNR and SQUID data and fixed during
calculations. The angles $\alpha^{{}^{\prime}}_{Gd}$ and
$\alpha^{{}^{\prime}}_{Fe}$ corresponding to the minimum of energy for a given
set of $H$ and $J_{1,2}$ is used to construct a theoretical SF reflectivity at
the first Bragg peak in Born approximation:

$\begin{split}R^{SF}_{1,th}=c[m_{Gd,\bot}^{2}+m_{Fe,\bot}^{2}+\\\
2m_{Gd,\bot}m_{Fe,\bot}\cos\frac{d_{Fe}}{d_{Fe}+d_{Gd}}]+R_{bg},\end{split}$
(2)

where $m_{Gd(Fe),\bot}=m_{Gd(Fe)}\sin\alpha^{{}^{\prime}}_{Gd(Fe)}$ is the
non-collinear component of magnetic moment of Gd(Fe) layer, $c$ is scaling
constant and $R_{bg}$ is background intensity. The latter two values were
adjusted manually before the fit. We fitted then theoretical $R^{SF}_{1,th}$
to the experimental $H$-dependencies $R^{SF}_{1}$ by varying $J_{1}$ and
$J_{2}$. The procedure was repeated for every $T$ so that for every sample we
obtained temperature dependencies of $J_{1,2}$. Fig.3d shows results of such a
fit for sample Pd10. It is rather noticeable that despite of the simplicity of
the Stoner-Wohlfarth approach it allows to reproduce experimental features
quite well. Fig.3e shows the fit-resulted $T$-dependence of the exchange
energies $J_{1}$ and $J_{2}$ for Pd10 sample. It can be seen that the bilinear
term has a predominant contribution, which gradually decreases with decreasing
temperature. Thus our analysis showed that for a qualitative description of
the SFT, a bilinear term is sufficient, but quantitatively the data are
described better by including an additional biquadratic term.

The data for the other samples were fitted in a similar way. Fig.3f shows the
dependency of coupling energies on thickness of Pd spacer. As follows from the
figure, the bilinear energy decreases almost linearly from 1.5 erg/cm2 at
$t$=1nm to 0 at $t$=1.6nm. Biquadratic energy in turn increases with $t$. The
obtained values are of the same orders as $J_{1}\sim$ 0.8 erg/cm2 and
$J_{2}\sim$ 0.2 erg/cm2 obtained in Ref.[39] for Gd/Pt/Co multilayers at
$T$=10K.

The decrease in the bilinear component with the increase in $t$ can obviously
be correlated with a decrease in the effective concentration of Gd in the GdPd
layer. At the same time, structural studies carried out earlier [37] indicate
an increase in structural inhomogeneities with increasing of $t$ . It seems
prudent to correlate this growth with an increase in the biquadratic
component.

In conclusion, using PNR we performed a systematic study of magnetic
configuration of [Fe(3.5nm)/Pd(t)/Gd(5.0nm)/Pd(t)]12 heterostructures with
t=1.0-1.6nm. By measuring neutron spin-flip scattering we have detected
presence of magnetically non-collinear state at temperatures $T\lesssim$ 50 K
in magnetic fields of above $H>$500 Oe for the samples with 1nm$<t<$1.4nm. By
using of an extended Stoner-Wohlfarth model we were able to describe the
observed transition as a competition of Zeeman energy, bilinear interaction of
order of 1 erg/cm2 and biquadratic addition of order of 0.5 erg/cm2. The
coupling energies can be tuned by varying thickness of spacer between 1nm and
1.4nm leading to the shift of the transition field below kilo-Oersted range.
Our study opens perspectives for a purposeful design of artificial FEMs with
adjustable field of spin-flop transition. Thus, the FEMs systems with low
Curie temperature components studied in this work can be used in
superconducting spintronics for generation of triplet superconductivitiy. An
additional advantage here is the good compatibility of gadolinium with
superconducting niobium [40, 41]. For the room temperature applications one
can use well-studied synthetic AFs such as Fe/Cr [34, 33, 35], Fe/V [42, 43]
or Co/Cu [44, 45] where subsequent adjustment can be carried out by tuning of
the coupling energy and the imbalance of the magnetic moments of the sub-
lattices.

We would like to thank M.A. Milyaev for assistance in preparation of the
samples, A.B. Drovosekov and D.I. Kholin for fruitful discussion of the
results. This work is partially based on experiments performed at the NREX
instrument operated by the Max Planck Society at the MLZ), Garching, Germany,
and supported by the Deutsche Forschungsgemeinschaft (Project No.
107745057-TRR80). Research in Ekaterinburg was performed within the state
assignment of Minobrnauki of Russia (theme ”Spin” No. AAAA-A18-118020290104-2)
and was partly supported by Russian Foundation for Basic Research (Project No.
19-02-00674).

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