First-order reversal curves (FORCs) of nano-engineered 3D Co-Fe structures

Abstract

In this paper we present first-order reversal curve (FORC) diagrams of ensembles of three-dimensional Co3Fe nanostructures as 2 × 2 arrays of nano-cubes and nano-trees. The structures are fabricated and investigated by an advanced platform of focused electron beam induced deposition combined with high-resolution detection of magnetic stray fields using a home-built micro-Hall magnetometer based on an AlGaAs/GaAs heterostructure. The experimental FORC diagrams are compared to macrospin simulations for both geometries at different angles of the externally applied magnetic field. The measured FORC diagrams are in good agreement with the simulated ones and reflect non-uniform magnetization reversal dominated by multi-vortex states within, and strong magnetic coupling between, the building blocks of our nanostructures. Thus, a FORC analysis of small arrays of 3D magnetic nanostructures provides more detailed insights into the mechanisms of magnetization reversal beyond standard major hysteresis loop measurements.

Full text

AIP Advances ARTICLE scitation.org/journal/adv
First-order reversal curves (FORCs)
of nano-engineered 3D Co-Fe structures
Cite as: AIP Advances 10, 015319 (2020); doi: 10.1063/1.5129850
Presented: 5 November 2019 •Submitted: 2 October 2019 •
Accepted: 16 December 2019 •Published Online: 10 January 2020
Mohanad Al Mamoori,1,2,a) Christian Schröder,3,4 Lukas Keller,1Michael Huth,1and Jens Müller1
AFFILIATIONS
1Institute of Physics, Goethe-University Frankfurt, 60438 Frankfurt am Main, Germany
2Institute of Materials Science, Technical University of Darmstadt, 64287 Darmstadt, Germany
3Institute for Applied Materials Research, Bielefeld University of Applied Science, 33619 Bielefeld, Germany
4Faculty of Physics, Bielefeld University, 33501 Bielefeld, Germany
Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials.
a)Electronic mail: AlMamoori@Physik.uni-frankfurt.de
ABSTRACT
In this paper we present first-order reversal curve (FORC) diagrams of ensembles of three-dimensional Co3Fe nanostructures as 2 ×2 arrays
of nano-cubes and nano-trees. The structures are fabricated and investigated by an advanced platform of focused electron beam induced
deposition combined with high-resolution detection of magnetic stray fields using a home-built micro-Hall magnetometer based on an
AlGaAs/GaAs heterostructure. The experimental FORC diagrams are compared to macrospin simulations for both geometries at different
angles of the externally applied magnetic field. The measured FORC diagrams are in good agreement with the simulated ones and reflect non-
uniform magnetization reversal dominated by multi-vortex states within, and strong magnetic coupling between, the building blocks of our
nanostructures. Thus, a FORC analysis of small arrays of 3D magnetic nanostructures provides more detailed insights into the mechanisms
of magnetization reversal beyond standard major hysteresis loop measurements.
©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5129850
., s
I. INTRODUCTION
Magnetic nanostructures have become key building blocks in
many areas of fundamental research in magnetism as well as in
nano-technological fabrication of functional applications. Identify-
ing intrinsic magnetic properties as well as the nature of interactions
in larger networks is a challenging task that requires both advanced
fabrication tools and investigation techniques. Advanced methods to
study ferromagnetic hysteresis, like the measurement of first-order
reversal curves (FORCs),1represent an appropriate tool for inves-
tigating processes that take place during the often complex magne-
tization reversal, which may be inaccessible by measuring a global
hysteresis loop. The attractiveness of the FORC method lies in its
straightforward application to large ensembles of nanoscale mag-
nets, such as arrays of interacting one- or two-dimensional magnetic
nanostructures, see, e.g. Ref. 2–4
Therefore, accessing also three-dimensional (3D) nanomag-
netic structures with this method appears as a promising approach
to probing the behaviour of the expected complex magnetization
reversal processes. Recently, we have demonstrated the fabrica-
tion of building blocks for artificial magnetic lattices, aiming to
advance beyond planar arrays toward the creation of 3D nanomag-
netic networks, which were investigated experimentally by ultra-
sensitive micro-Hall magnetometers and simulated by macro- and
micromagnetic approaches.5,6
Here, we discuss FORC diagram signatures of these ensembles
of 3D Co3Fe (CoFe) structures as small (2 ×2) arrays of nano-cubes
and nano-trees measured by micro-Hall magnetometry7to gain fur-
ther insights into the observed magnetic reversal processes. Exper-
imental results are compared to simulations based on an idealized
theoretical model using single-dipole macrospins (SDMS).
II. EXPERIMENTAL METHODS
The samples consist of metallic bcc Co3Fe grown as nano-cubes
and nano-trees that were directly written on the active area of a
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micro-Hall sensor surface using focused electron beam deposition
(FEBID). Details about FEBID nano-fabrication of 3D geometries
can be found elsewhere.6,8–10 The perpendicular stray field ⟨Bz⟩aver-
aged over the active area of the Hall-cross, which is directly linked
to the sample magnetization,11 is measured by detecting the gen-
erated Hall voltage VHin the micro-Hall sensor plane (formed by
FIG. 1. (a) SEM image of two 5 ×5μm2Hall crosses with arrays of 2 ×2 CoFe
nano-cubes and nano-trees grown by FEBID directly on the surface. The Hall
voltage generated in the sensor plane is directly proportional to the sample magne-
tization. (b) Schematic drawing of the FORC data acquisition protocol. Calculation
of the second-order mixed derivative of the magnetic signal with respect to Hrand
Haresults in the FORC distribution.
a high-mobility two-dimensional electron gas at the interface of an
AlGaAs/GaAs heterostructure).12 The large linear Hall-effect back-
ground signal is cancelled in situ by a differential measurement of an
empty reference Hall cross in a so-called gradiometry set-up accord-
ing to ΔVH=1
ne I<Bz>, where nis the carrier density and Ithe
applied current.
The FORC data acquisition routine using the electronic read-
out of the Hall sensor7is extended to perform FORC protocols
for our 3D CoFe nanostructures. The FORC acquisition proto-
col1,13 is depicted in Fig. 1(b). First, the sample is saturated in a
large positive applied magnetic field Hsat. Second, the magnetic
field is ramped down to the reversal field Hr. The first FORC
is obtained when the applied field Hais increased up to satura-
tion. This process is repeated for many values of approximately
evenly spaced Hr, from Hrto Hsat. The FORC distribution is con-
veniently represented by a rotated coordinate system, the hori-
zontal axis (representing the switching fields) given by Hc= (Ha
−Hr)/2 and the vertical axis (representing the interaction fields)
given by Hu= (Ha+Hr)/2. To obtain the FORC distribution,
the raw magnetization data are then further processed with a
suitable code (FORCinel)14 using smoothing factors ranging from
6 to 10.
III. RESULTS AND DISCUSSION
FORC distributions provide additional information to that
revealed by global hysteresis loops, thereby allowing a better under-
standing of the mechanisms of magnetization reversal through the
analysis of coercive and interaction field distributions. Many mag-
netic structures, such as nanoparticles and -dots,15 nanowires16 or
GMR devices,17 have been investigated using this method. In the
following, we discuss FORC diagrams of our 3D CoFe nano-cubes
and nano-trees, that have been investigated by global hysteresis loop
measurements in Refs. 5and 6. Here, FORC data from micro-Hall
magnetometry measurements have been obtained at a maximum
saturation field μ0Hsat = 150 mT at T= 25 K, with a sweep rate of
10 mT/min and reversal field steps Δμ0Hr= 2 mT. FORC distribu-
tions were calculated from the raw data on an equidistant grid using
the FORCinel code.14
Figure 2(a) shows the experimental FORC diagram for an array
of 2 ×2 CoFe nano-cubes at θ= 0, i.e. the applied field perpendicular
to the sensor plane and parallel to the anisotropy axis of the nano-
cube stems. The inset shows corresponding raw FORCs for the mea-
sured structures. As reported before,5,6 the large magnetic volume of
the nano-cubes consisting of many connected bistable rods leads to
step-like switching events seen in the global hysteresis loop, where
the magnetisation reversal likely develops by inhomogeneous multi-
vortex states nucleating, propagating and annihilating through the
individual building blocks as well as the three- and four-leg joints.
It should be mentioned that the detailed account of multi-vortex
FORC signatures was firstly reported in geologic materials.18 In our
FORC diagram, this results in complex signatures in the Hu-Hc
plane accompanied by circular loops of intensities forming around
the central ridge. The first dominant signature is a strong and broad
peak in the FORC diagram with a slight offset to the Hu= 0 axis and
overlapping with peak-like features lined up vertically. This central
peak is caused by the magnetization switching (μ0HFORC
c=22.5 mT
at μ0Hu= 0) of the stems that are aligned parallel to the applied
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FIG. 2. (a) Experimental FORC distribution of 2 ×2 individual CoFe nano-cubes
obtained at a 0○applied field direction (parallel to the stems) at T= 25 K. Upper
inset: family of FORCs from which the corresponding contour plot is calculated.
Lower inset: scanning electron microscope image of the CoFe nano-cubes. (b)
Simulated FORC distribution from macrospin simulations for the corresponding
structures.
magnetic field, which is approximately equivalent to the coercive
field of the global hysteresis loop μ0Hhyst.
c=22.1 mT. The different
orientations of the edge elements lead to a distribution of switch-
ing fields and hence to a “stretched” central peak in the FORC
diagram. It should be noted that the shape of the coercivity distribu-
tion likely reflects magnetic vortex nucleation and annihilation. Fur-
thermore, the predominant vertically elongated features (subsidiary
overlapping vertical peaks) might be due to vortex nucleation and
annihilation field distributions in non-uniformly magnetized edges
and interactions between vortices of the same or neighboring edges.
A quantitative analysis of the interaction fields at the coercivity peak
is best illustrated by a vertical profile (not shown here). From the lat-
ter, the full width at half maximum (FWHM) is determined to about
33.3 mT.
We compare our experimental FORC diagrams with simula-
tions based on an idealized theoretical model using single-dipole
macrospins (SDMS),11 where one SDMS reflects the magnetic
behaviour of an ideal, i.e. single-domain, stem or edge. Each SDMS
is located at the center of the building block positions. Assuming
each building block to represent a prolate spheroid with a homoge-
neous magnetization allows us to relate their magnetic properties to
the Stoner-Wohlfarth model19 of a particle with uniaxial anisotropy.
The SDMS model consists of four SDMS per nano-tree and thirteen
SDMS per nano-cube. The SDMS interact via dipole-dipole interac-
tion only. The FORCs of these systems have been calculated for T= 0
by solving the stochastic Landau-Lifshitz equation11 following essen-
tially the same acquisition protocol as used in the experiment. Since
the macrospin model only contains interacting SDMS, all features
observed in the simulated FORC diagrams merely correspond to
the properties of simple Stoner-Wolfarth19 particles and the dipole-
dipole interactions among them. This means that generally there
are fewer features expected in simulated FORC diagrams, which in
turn can be used to identify additional features in the experimental
FORC diagrams that originate from physical mechanisms beyond
the macrospin model.
Figure 2(b) shows the FORC diagram for the nano-cubes cal-
culated from the simulated FORCs shown in the upper inset. Com-
pared to the experimental FORC, essential features on the central
ridge can be reproduced, however, with less intensity and more
peaked (i.e. with a narrower distribution) on the central ridge. The
less dominant coercivity peaks at negative and positive μ0Huvalues
are shifted further away from the reversal side and coincide at μ0Hu
= 0 mT, unlike the experiment. The coercivity distribution is peaked
at μ0HFORC
c=29.9 mT comparable to the coercivity from global hys-
teresis loop measurements. Broadening of the experimental FORC
features on the central ridge as well as peaks in the positive and
negative μ0Huare likely attributed to switching of non-uniformly
magnetized edges and stems.
Changing the field angle with respect to the stem and edges
to θ= 90○(the applied field parallel to the sensor plane and per-
pendicular to the nano-cube stems) leads to a more complex FORC
diagram as shown in Fig. 3. One observes many prominent features
distributed in the upper and lower half planes and weaker nega-
tive features at different positions. The most pronounced features
consisting of circular positive regions are those located on the nega-
tive diagonal highlighted by the blue arrow. For example, in simple
arrays of Co nanodots,15 such circular peaks have been attributed to
distinct annihilation and nucleation paths. In our case, the occur-
rence of such circular peaks may be attributed to the nucleation and
annihilation of vortex states within the magnetic edges, stems and
the vertices of the nano-cubes since they have different anisotropy
orientations giving rise to multiple separated peaks in the FORC
diagram progressively shifting to higher coercivities. We find the
coercivity at θ= 90○approximately peaked at μ0HFORC
c=48 mT
(at μ0Hu= 0), a value higher than for θ= 0○(Fig. 2). This is expected
from the anisotropy of the structures and may be explained by the
dominating contribution of the stems with inhomogeneous spin
texture involving creation and annihilation of magnetic vortices.
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FIG. 3. (a) Experimental FORC distribution of 2 ×2 individual CoFe nano-cubes
obtained at a 90○applied field direction (perpendicular to the stems) and at
T= 25 K. (b) Simulated FORC distribution from macrospin simulations.
Additionally, shape irregularities play a vital role in vortex annihila-
tion and nucleation processes. The magnetization reversal via differ-
ent intermediate states causes non-deterministic switching events,
i.e. they do not necessarily occur at the same field in ascending and
descending branches of the FORC leading to the observed asymmet-
ric isolated peaks (see the peaks marked C and M). Such behaviour
might indicate strong interaction effects (the connected two upper
and separated lower peaks of the diagonal reveal a reasonable verti-
cal spread filling the FORC space). The simulated FORC distribu-
tion for this selected angle shown in Fig. 3(b) remarkably reveals
some essential features, however, with lower peak intensity. The
diagonal circular peaks are reproduced, although at different posi-
tions. The side circular peaks (C and M) also appear in the simu-
lated FORC diagram. As explained above, the lower intensity and
sharper peak structure reflect the properties of interacting idealized
Stoner-Wolfarth19 particles.
The experimental and simulated FORC diagrams obtained for
the 2 ×2 nano-tree array are shown in Fig. 4. The experimental
FORC diagram is shown in Fig. 4(a) for θ= 0○, where the anisotropy
axis of the stems points in the field direction, with the lower inset
showing a scanning electron microscope image of the structures.
We find that the FORC distribution is dominated by an asymmet-
ric, fish-like positive central area centered along the horizontal axis.
FIG. 4. (a) Experimental FORC distribution of 2 ×2 individual CoFe nano-trees
obtained at a 0○applied field direction (parallel to the stems) and at T= 25 K. Inset:
SEM image of CoFe nano-trees. (b) Simulated FORC distribution from macrospin
simulations for the corresponding structures.
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There are also weak negative features visible in the FORC space.
The central area has a strong peak at μ0HFORC
c=40 mT. The hor-
izontal positive distribution of contours reflects a broad coercivity
distribution (the horizontal line projection, not shown here, has a
Gaussian curve). Switching of the nano-trees develops by inhomo-
geneous vortex states along the edges and the stems.5,6 It commences
by the edges switching before reversal of the major part of the sam-
ple volume, i.e. the stems, which are the last to switch causing the
sharp step in the major hysteresis loop.5,6 The low-end of coerciv-
ity (highlighted by z-zone) is shifted toward negative μ0Hu, while
the p-zone is shifted to the positive direction. Such vertical spread
is a direct measure of a broad interaction distribution. A vertical
profile at the coercivity peak yields a FWHM = 11.5 mT, which is
a measure of interaction effects both between the building blocks
within a nano-tree structure and between neighbouring structures.
The simulated FORC diagram is shown in Fig. 4(b). Strikingly, it
is in good qualitative and reasonable quantitative agreement with
the experiment. The coercivity peak is found at μ0HFORC
c=49 mT.
The tails (p- and z-zones) are less shifted along the Hu-axis, which
is again due to the assumption of idealized single-domain stems and
edges in our simulations. In addition, the high coercivity end (red
arrow) coincides on the horizontal line (at μ0Hu= 0). The vertical
profile (not shown here) reveals a FWHM = 12.5 mT, as a measure
of the interaction field strength among the magnetic entities which
is in good quantitative agreement with the experiment.
IV. CONCLUSIONS
In this paper, we present the first FORC diagrams for FEBID-
fabricated 3D CoFe nanostructures grown as stem-mounted nano-
cubes and nano-trees directly on top of 5 ×5μm2Hall crosses of
an ultra-sensitive micro-Hall sensor. We have applied the external
magnetic field at different angles and complemented the experi-
mental results with simulations based on a theoretical model of
single-dipole macrospins (SDMS), which reflect the magnetic behav-
ior of ideal, i.e. single-domain building blocks of the nanostructures.
The FORC signatures for both geometries have complex coercive
field and interaction field distributions. This reflects non-uniform
magnetization reversal of the stems and edges dominated by multi-
vortex states, and strong magnetic coupling between the magnetic
entities. The FORC diagrams calculated from SDMS simulations
agree well with the experiments and prove to be a useful charac-
terization tool for the switching dynamics also of structures of this
kind. The observations in this study provide an avenue for future
research to unveil complex magnetic switching and interaction sce-
narios of various 3D nanomagnets by adapting the FORC technique
to combined FEBID fabrication/micro-Hall magnetometry.
ACKNOWLEDGMENTS
Mohanad Al Mamoori acknowledges financial support from
the Deutscher Akademischer Austauschdienst (DAAD) within the
doctoral program for research studies in Germany. The high-
mobility wafer material, grown by molecular beam epitaxy, that
was used to build the Hall magnetometer was kindly provided by
Jürgen Weis, Max-Planck-Institute for Solid State Research,
Stuttgart, Germany. Jens Müller and Mohanad Al Mamoori thank
Merlin Pohlit for help with sensor fabrication and Jonathan Pieper
for help with FORC analysis. Michael Huth and Lukas Keller
acknowledge financial support by the Deutsche Forschungsgemein-
schaft (DFG) through the Collaborative Research Centre SFB/TR49.
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