# Probing Magnetars Using Spectral Lines with Future Telescopes

Demet Kırmızıbayrak1 Jeremy Heyl1 1Department of Physics & Astronomy,
University of British Columbia, Vancouver, BC V6T 1Z1, Canada; e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(2021)

###### Abstract

We present our findings on magnetar spectral line analysis in the context of
upcoming high resolution, high effective area, high throughput X-ray
telescopes for two cases: persistent magnetar emission and magnetar bursts.
For magnetars in quiescence, we present our preliminary work on modelling for
phase-resolved emission. Our results reveal the necessity of constraining line
depth and width concurrently with line energy to conclusively determine
hotspot emission and corresponding magnetic field geometry. We then present
the results of our simulations using effective area and response of various
current and upcoming X-ray telescopes for magnetar spectral line detection and
expand on the exciting opportunities upcoming telescopes provide to probe
quiescent and burst emission region geometry and propagation in the extreme
magnetic field of a magnetar.

###### keywords:

stars: neutron – stars: magnetic fields - X-rays: bursts - X-rays: general -
techniques: spectroscopic - telescopes

††volume: 363††journal: Neutron Star Astrophysics at the Crossroads: Magnetars
and the Multimessenger Revolution††editors: E. Troja & M. Baring, eds.

##  1 Introduction

Magnetar spectral lines are thought to result from proton cyclotron scattering
in the magnetosphere, where photons emerging from a hotspot on the surface are
scattered at the cyclotron frequency, resulting in absorption features in the
spectrum. However, relativistic effects due to the magnetar’s gravitational
field, as well as changes in the extreme magnetic field near the emission
region highly broaden line features, making them much harder to detect.

Studying magnetar spectral lines promises insights into many unknown features
of magnetars and physics in strong magnetic fields. First, if interpreted in
the context of the magnetar model, spectral lines yield an estimate of the
magnetar’s magnetic and gravitational fields near the crust and the size of
the emission region. Moreover, spectral lines offer a unique tool to resolve
the geometry of the emission region as well as that of the field lines near
the surface. Understanding such geometry would enable astronomers to identify
exact causes of the quiescent and burst emission.

##  2 Modelling Phase-Resolved Spectroscopy

In the proton cyclotron model, changes in spectral lines throughout a
magnetar’s rotation phase can be explained as resulting from the geometry of
the magnetic field lines surrounding the emitting region. Here, we use the
cyclotron absorption model

$F(E)=F_{0}\
\exp{\Big{[}-D\dfrac{(WE/E_{c})^{2}}{(E-E_{c})^{2}+W^{2}}\Big{]}}$ (1)

from [Tiengo et al. (2013), Tiengo et. al. (2013)] and [Makishima et al.
(1990), Makishima et. al. (1990)] where $D$ and $W$ are line depth and width,
$F_{0}$ is the level of the continuum, $F(E)$ is the flux at the given energy,
and $E_{c}$ is the cyclotron energy. A key observation was discovered by
[Tiengo et al. (2013), Tiengo et. al. (2013)] where absorption lines were
observed to vary in energy, width and depth as the magnetar SGR 0418+5729
rotates during its persistent emission. To model this behaviour, we assume a
solar prominence-like scenario with a hotspot on the surface emitting photons
that travel through a current-loaded loop of plasma along magnetic field lines
as in [Tiengo et al. (2013), Tiengo et. al. (2013)]. As the magnetar rotates,
photons emerging from the hotspot travel through different locations on the
transverse loop to reach our line of sight. The location on the loop where
photons traverse affects the line energy, as photons meeting with the narrower
base of the loop encounter a stronger field, resulting in a higher line
energy. On the left panel of Figure 1, we illustrate this model and several
other parameters expected to affect line shape: hotspot location ($\alpha$),
loop orientation ($\phi$) and loop transverse angle ($\beta$).

Figure 1: Left: Visualization of magnetar emission geometry model with a
hotspot (orange) and loop of magnetic field lines carrying along a current
(pink). $\hat{r},\hat{n},\hat{d}$ and $\hat{m}$ represent the rotation axis,
spot normal, loop diameter and spot meridian respectively. Middle:
Visualization of a ribbon-like loop. Right: Visualization of a tube-like loop.

We plot the resulting model using parameters provided in [Tiengo et al.
(2013), Tiengo et. al. (2013)] on top of the phase-resolved spectrum of SGR
0418+5729 observed with XMM-Newton EPIC-pn on the left panel of Figure 2. In
the phase-resolved spectrum, darker regions indicating an absorption line
trace a V-shape as the magnetar rotates between phases $\sim$0$-$0.4. The
model is able to obtain this with $\alpha=20^{\circ}$, $\phi=90^{\circ}$ and
the line of sight at $70^{\circ}$. To demonstrate the sensitivity for
different geometries, we plot the same model with a hotspot closer to the
magnetar’s equator ($\alpha=50^{\circ}$) keeping all remaining parameters
constant on the middle panel of Figure 2. In this case, the model fails to
trace the observed line energy, exhibiting our ability to identify hotspot
location, loop orientation and line of sight using spectral line energy.

Figure 2: Phase Resolved Spectroscopy of SGR 0418+5729 from [Tiengo et al.
(2013), Tiengo et. al. (2013)] and model discussed in text. Darker regions
indicate a lack of counts and green lines depict our model with a hotspot at
Left: $20^{\circ}$ and Middle: $50^{\circ}$ from the rotation axis. Right:
Model with a second hotspot (purple) with larger width gradient at
$37^{\circ}$ from the rotation axis.

We note that a hint of a second broader line with a lower dip is present
between phases $\sim$0.4$-$0.7. We are not able to explain this feature using
a single hotspot model. However, with a second hotspot at $\alpha=37^{\circ}$
from the rotation axis, and a larger field gradient ($f=7.23\times 10^{15}\
G$), we obtain the purple line on the right panel of Figure 2. Yet, since the
detection is weak, it is not possible to conclusively determine the hotspot
parameters or rule out any other scenarios. More data, especially with future
higher effective area telescopes could allow to probe such features to
understand hotspot origins.

Although line width and depth are also observed to vary as SGR 0418+5729
rotates, this behaviour was not yet studied. To demonstrate what these
features can tell us about magnetic field geometry, we consider two cases: 1)
A ribbon-like transverse loop 2) A tube-like loop with a certain thickness as
well as a width (illustrated in Figure 1). Here, the line equivalent width can
be expressed as

$\Gamma(E)=\int(1-F(E)/F_{0})\,dE$ (2)

where $\Gamma$ is the equivalent width, $F_{0}$ the continuum level, and
$F(E)$ the line flux across the energy of interest. Line equivalent width
depends on the number of field lines photons encounter on the loop, and the
magnetic field scales with the field line density, yielding:

$\Gamma\propto L^{-1}\ \textrm{(ribbon \& tube)}\hskip 28.45274ptB\propto
L^{-1}\ \textrm{(ribbon)}\hskip 28.45274ptB\propto L^{-2}\ \textrm{(tube)}$
(3)

where $B$ is the magnetic field and $L$ is the length scale of the loop.
Considering all spectral lines detected for SGR 0418+5279 from [Tiengo et al.
(2013), Tiengo et. al. 2013], our best power-law fits show that line energy is
linear with line width within error bars ($W\propto A\times E^{\alpha}$ where
$A$ is the amplitude and $\alpha=1.07\pm 0.26$, Figure 3). Assuming that this
is correct, and using Equation 3, and $\Gamma\propto DW$ we get:

$D\propto E^{0}\ \textrm{(ribbon)}\hskip 28.45274ptD\propto E^{-0.5}\
\textrm{(tube)}$ (4)

where $D$ is the line depth, and $E$ is the line energy. For SGR 0418+5279
line depth vs. energy (Figure 3), we obtain a best fit power of 0.5 within
error bars (($D\propto A\times E^{\alpha}$ where $A$ is the amplitude and
$\alpha=0.47\pm 0.16$), in line with the tube-like scenario.

Our preliminary calculations above show that the magnetic flux tube shape can
be determined conclusively if width, depth and energy of spectral lines are
studied concurrently. Indeed, with these calculations we demonstrate that the
loop shape differs from previously discussed and resembles a more complex 3D
structure, while other loop shapes remain to be tested. This emphasizes the
need in modelling for spectral line width and depth to obtain a more
comprehensive picture of magnetar emission and geometry.

Figure 3: Left: Width (keV) vs. Energy (keV), Middle: Depth vs. Energy (keV)
of all lines detected throughout the phase of SGR 0418+5729 in [Tiengo et al.
(2013), Tiengo et. al. 2013]. Orange lines indicate best power-law fits with
$\textrm{Width}\sim\textrm{Energy}$ and
$\textrm{Depth}\sim\textrm{Energy}^{-0.5}$ within error bars, indicating a
scenario closer to a tube-like loop in Figure 1. Right: Average 1$\sigma$
Energy (blue), Depth (orange) and Width (green) errors per fitted value for
all SGR 0418+5729 spectral lines between between phases 0$-$0.3 in [Tiengo et
al. (2013), Tiengo et. al. 2013] vs. simulated detectors.

##  3 Future of Spectral Lines with New Detectors

Motivated by our modelling results, we conducted several simulations to obtain
a general view on magnetar spectral line detections with future X-ray
telescopes. For all simulations, we created fake spectra using the fakeit
function on XSPEC v12.11.1 ([Arnaud (1996), Arnaud 1996])) with the most up-
to-date public effective area and response of several detectors. For
persistent emission, we represented the spectrum as an absorbed blackbody plus
power law with a cylotron absorption feature as in Equation 1 (wabs $\times$
(bb+po) $\times$ cyclabs on XSPEC) setting all parameters to those reported in
[Tiengo et al. (2013), Tiengo et. al. (2013)] (best-fit model of the phase-
averaged spectrum for the continuum model and reported parameters between
phases 0$-$0.3 for the line model). For bursts, we used an absorbed
Bremsstrahlung model with a Gaussian line (wabs $\times$ brems+gauss on XSPEC)
with parameters set to the 6.4 keV line detection by [Strohmayer (2000),
Strohmayer et. al. (2000)] during the 1998 burst precursor of SGR 1900+14, but
scaled continuum normalization to a burst-like count rate of $\sim$ 35k in 0.2
seconds of burst duration. We then fit the simulated spectra with each
detector’s associated response and effective area, setting all parameters free
with the exception of the persistent emission continuum model (which we fixed
to the reported phase-averaged best-fits in [Tiengo et al. (2013), Tiengo et.
al. (2013)]) and SGR 1900+14 Hydrogen Column Density (fixed to the best-fit
value in [Strohmayer (2000), Strohmayer et. al. (2000)]).

For persistent emission of SGR 0418+5729, we plot average 1$\sigma$ line
parameter errors per fitted value on the right panel of Figure 3. We find that
line width, energy and depth are further constrained by over a factor of three
with the advent of eXTP, Athena and Colibrì (see [Heyl2019, Heyl et. al.
(2019)] for a description of Colibrì) compared to current detectors. The
simulated burst lines and model fits for SGR 1900+14 in Figure 4, show that
high effective area is crucial, with eXTP and Colibrì providing most accurate
detections. The advent of large effective area telescopes with high pile-up
limitations therefore place us on the edge of an exciting new era to probe
magnetars with spectral lines.

a b c d

Figure 4: SGR 1900+14 6.4 keV burst line fits and residuals with Colibrì (a)
Athena X-IFU (b), eXTP LAD (c) and NICER (d) response and effective area.

## References

  * [Arnaud (1996)] Arnaud, K. A. 1996, Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * [Beloborodov (2002)] Beloborodov, A. M. 2002, APJL, 566,L85
  * [Heyl (2019)] Heyl, J., Ilaria, C., Hoffman, K., et. al. 2019, BAAS, 51, 175
  * [Makishima et al. (1990)] Makishima, K., Mihara, T., Ishida, M. et al. 1990, Publ. Astron. Soc. Japan, 42, 295–315
  * [Mereghetti et al. (2015)] Mereghetti, S., Pons, J. A., & Melatos, A. 2015, SSRv, 191,315
  * [Strohmayer (2000)] Strohmayer, T. E., & Ibrahim, A. I. 2000, APJL, 537, L111
  * [Tiengo et al. (2013)] Tiengo, A., Esposito, P., Mereghetti, S., et al. 2013, Nature, 500,312


spacing

# Freezing and thawing magnetic droplet solitons

Martina Ahlberg1,∗ Sunjae Chung1,3,∗,+ Sheng Jiang1,4,5 Q. Tuan Le1,5 Roman
Khymyn1 Hamid Mazraati2,5 Markus Weigand6 Iuliia Bykova6 Felix Groß6 Eberhard
Goering6 Gisela Schütz6 Joachim Gräfe6 & Johan Åkerman1,2,5,+

###### Abstract

Magnetic droplets are non-topological magnetodynamical solitons displaying a
wide range of complex dynamic phenomena with potential for microwave signal
generation. Bubbles, on the other hand, are internally static cylindrical
magnetic domains, stabilized by external fields and magnetostatic
interactions. In its original theory, the droplet was described as an
imminently collapsing bubble stabilized by spin transfer torque and, in its
zero-frequency limit, as equivalent to a bubble. Without nanoscale lateral
confinement, pinning, or an external applied field, such a nanobubble is
unstable, and should collapse. Here, we show that we can freeze dynamic
droplets into static nanobubbles by decreasing the magnetic field. While the
bubble has virtually the same resistance as the droplet, all signs of low-
frequency microwave noise disappear. The transition is fully reversible and
the bubble can be thawed back into a droplet if the magnetic field is
increased under current. Whereas the droplet collapses without a sustaining
current, the bubble is highly stable and remains intact for days without
external drive. Electrical measurements are complemented by direct observation
using scanning transmission x-ray microscopy, which corroborates the analysis
and confirms that the bubble is stabilized by pinning.

Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden

NanOsc AB, 164 40 Kista, Sweden

Department of Physics Education, Korea National University of Education,
Cheongju 28173, Korea

School of Microelectronics, Northwestern Polytechnical University, Xi’an
710072, China

Department of Applied Physics, School of Engineering Sciences, KTH Royal
Institute of Technology, 100 44 Stockholm, Sweden

Max Planck Institute for Intelligent Systems, Stuttgart, Germany

∗These authors contributed equally to this work.

Figure 1:  Droplet vs. bubble, device structure and layout, and magnetic
characterization.  (a) Schematic of dynamical magnetic droplet soliton. (b)
Schematic of a static magnetic bubble. (c) Schematic of an all-perpendicular
STNO composed of [Co/Pd] (fixed) and [Co/Ni] (free) multilayers with a Cu
spacer fabricated on a SiN membrane structure. The insets underneath show
optical micrographs of the SiN membrane areas through which the different
metal layers of the device can be seen. (d) Hysteresis loops of single Co/Pd
and Co/Ni layers. (e) Hysteresis loop of a full [Co/Pd]/Cu/[Co/Ni] stack.

Magnetic droplets are intrinsically dynamic, non-topological, magnetodynamical
solitons[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], which can be
nucleated and sustained both in spin torque nano-oscillators (STNOs)[3, 6, 8,
12, 14] and spin Hall nano-oscillators (SHNOs)[16], provided the
magnetodynamically active layer has sufficient perpendicular magnetic
anisotropy (PMA). Magnetic droplets are characterized by a reversed core
separated from the surrounding magnetization via a perimeter of precessing
spins (See Fig.1(a))[2, 3]. While first predicted over 40 years ago in an
ideal zero-damping medium[1], their possible experimental realization was
later suggested theoretically[2] in STNOs with PMA free layers[17, 18]. After
the first experimental demonstration of magnetic droplets, reported in STNOs
with a PMA Co/Ni free layer and a Co fixed layer[3], interest in magnetic
droplets continues to increase due to its interesting characteristics, such as
a highly nonlinear dynamics[2, 19, 11], large power emission[3, 20, 10, 21],
and possible applications in microwave-assisted magnetic recording (MAMR)[22,
23] and neuromorphic chips as nonlinear oscillators[24, 25, 26]. Several
theoretical[19, 27, 5, 28, 11, 29, 30, 15, 31, 32, 33] and experimental[6, 7,
9, 8, 34, 35, 10, 12, 16, 36, 21, 37, 38] studies on magnetic droplets have
since been presented.

As pointed out by Hoefer _et al._ , the droplet is reminiscent of a magnetic
bubble[2] (Fig.1(b)) and they identify a possible zero-frequency droplet with
a topologically trivial magnetic bubble[39, 40, 41, 42, 43]. Despite the large
number of experimental droplet studies, the low-field/low-frequency behavior
of droplets has not yet been explored and the relation between droplets and
bubbles — as well as a possible transition between the two — remain unclear.
In order to explore these phenomena, we here study magnetic droplets
specifically in the low-field regime using both electrical and microwave
spectroscopy measurements as well as the direct microscopical observation
based on Scanning Transmission X-ray Microscopy (STXM). We find clear
experimental evidence for a droplet-to-bubble transition as the field
strength, and hence the droplet frequency, is reduced, and a reversible
bubble-to-droplet transition as the field is again increased in an attempt to
squash the bubble, provided stabilizing spin transfer torque is still present
via the STNO current. Our experimental results hence corroborate the picture,
first expressed by Hoefer _et al._ , that a magnetic droplet can be viewed “
_as an imminently collapsing bubble that is critically stabilized by the
localized injection of spin torque_ ”.

Figure 1 shows a schematic of the studied all-perpendicular STNOs, comprised
of a [Co/Pd]/ Cu/[Co/Ni] GMR stack deposited on a Si3N4 membrane (for
fabrication details, please see Methods). Underneath the schematic we show two
optical microscopy images taken from opposite directions to highlight the
optical transmission of the Si3N4 membrane. In Fig. 1(d) we show the magnetic
properties of the individual free and fixed layers based on calibration
samples, and their combined behavior in full STNO stacks in Fig. 1(e).

Figure 2 presents the resistance and microwave signal as a function of field
for an applied current of –5 mA. The field is first increased from –0.51 T to
0.51 T in Fig. 2(a) and then decreased from positive to negative field in Fig.
2(b). At large negative fields, the STNO is in its lowest resistance state,
consistent with a parallel (P) relative orientation of its free and fixed
layers. At about –0.49 T, the resistance increases about 20 mOhm in a step-
like fashion and there is a slight increase in the microwave noise background,
both strong indications of the nucleation of a droplet. At about –0.38 T,
there is a second step-like increase in the STNO resistance and a marked
further increase in the microwave noise. We interpret this as a transition
into a larger droplet as the opposing applied field is reduced. At yet lower
fields the droplet continues to grow in size (the STNO resistance increases),
while its stability seems to deteriorate as indicated by the growing intensity
of the microwave noise background. At about –0.04 T, the microwave noise
rapidly reaches a maximum and then suddenly disappears altogether, while the
resistance exhibits a small jump of about 5 mOhm. The complete microwave
silence indicates that the magnetic state is now static, and we are lead to
conclude that the droplet precession has stopped entirely and that the droplet
has transitioned into a nanobubble state.

The nanobubble resistance exhibits jumps reminiscent of Barkhausen noise[44,
45, 46], indicating pinning possibly at grain boundaries or defects of the
sputtered film. When the field is further increased, the bubble resistance
increases gradually, indicating a continued growth of its size. At about 0.06
T, the entire free layer switches its magnetization direction and the
antiparallel (AP) state is clearly identifiable in the resistance. When the
fixed layer switches at 0.23 T, a droplet is immediately nucleated. With
further increasing of the opposing field, the droplet again shows a gradual
transition to a smaller size; the droplet finally disappears as the STNO
transitions into a full P state at about 0.47 T. The overall behavior is very
similar for decreasing fields (Fig.2b) where the same P/AP/droplet/nanobubble
states can be clearly identified via the STNO resistance and the microwave
noise.

As mentioned above, the microwave noise power is far from constant for the
whole droplet region and peaks at certain fields. We identify these peaks as
marks of mode hopping between different droplet states[14]. While the details
of the spectrum is highly reproducible (cf. increasing and decreasing fields)
and serves as a fingerprint for each device, the patterns at negative and
positive fields are quite different. The magnetoresistance implies that a
relatively small and stable droplet ($\mu_{0}H<-0.4$ T) is abruptly followed
by a larger but similarly stable mode. In contrast to the symmetric noise
patterns around the droplet-to-droplet transitions, the strong increase in
microwave noise power around the droplet-to-bubble transition is highly
asymmetric. There is first an extended field region of monotonic increase in
the noise, which is then abruptly cut off and replaced by a completely silent
bubble state. This highlights the very different non-dynamical nature of the
nanobubble and suggests that mode hopping out of the nanobubble state and back
into a droplet state is negligible, once the nanobubble has formed.

Figure 2:  Microwave noise and STNO resistance vs. field. (a)-(d) Color plot
of the power spectral density (PSD) of the microwave noise as a function of
decreasing (a,c) and increasing (b,d) field, with the STNO resistance (white
line) overlayed; the applied current is $-5$ mA. (a,b) Wide field sweep
covering full saturation at both positive and negative fields. P/AP indicate
the parallel/antiparallel state of the STNO; red arrow indicates the droplet
region, and green arrow the bubble region. (c,d) Minor field sweeps showing
how the droplet/bubble transition is fully reversible.

Figure 2(c) and (d) demonstrate that it is possible to freeze the dynamic
droplet into a static bubble and then thaw it back into a droplet using only
the magnetic field under constant spin transfer torque. In particular, Fig.
2(d) shows how the nanobubble first is about to collapse at 0.025 T as it is
getting squeezed by the opposing pressure from the increasing applied field.
There is some slight Barkhausen noise in the rapidly dropping resistance, but
otherwise no measurable microwave noise. However, instead of switching to a P
state, the resistance then exhibits a sharp minimum after which it shows a
rapid increase, which is accompanied by a high level of microwave noise. The
collapsing nanobubble is hence rescued by the stabilizing spin transfer
torque, which sets the spins in the bubble perimeter into precessional motion
and restores the full dynamics of a magnetic droplet. Judging from the
resistance, it is noteworthy that the droplet is slightly larger than the
smallest nanobubble. Within the experimental accuracy (a field step of 2 mT),
we do not observe any significant hysteresis in this transition. Hence there
is a negligible energy barrier between the two states and the bubble can
indeed be viewed as a zero-frequency droplet, albeit still likely affected by
pinning.

Figure 3(a) presents a phase diagram based on a two-dimensional map of the
STNO resistance as functions of current and field. All data was acquired in a
decreasing field at a constant current level. The parallel (P) and
antiparallel (AP) configurations are easily identified by the dark blue and
dark red colors, respectively, and for current magnitudes below 1.8 mA, these
are the only two available states, as expected for a GMR device. However, even
at these weak currents, the P$\rightarrow$AP switching field is clearly
affected by the STT from the nanocontact; in contrast, the AP$\rightarrow$P
switching field is entirely unaffected. In an intermediate current region,
from about –1.8 to –3.5 mA, the STT can not yet sustain a droplet but is
sufficient to create a nanobubble directly from the P state. As magnetic
switching typically involves both domain nucleation and domain propagation, we
interpret this current dependent switching in the following way (see
Supplementary Materials for a zoom-in of this particular part of the phase
diagram). For current magnitudes below 1 mA, magnetic switching is limited by
the field required for domain nucleation and, in addition, the location of
initial domain nucleation is far from the nanocontact region as STT from the
current has no discernible impact. However, for current magnitudes above 1 mA,
where we observe a strong current dependence of the switching field, we
conclude that the domain nucleation has moved to underneath the nanocontact.
If we reduce the field magnitude, we need a stronger current to assist in the
domain nucleation, but once formed, it propagates through the entire free
layer. However, at fields weaker than the field needed for domain propagation,
i.e. the pinning field, which we read out as about 60 mT, the nucleated domain
is no longer able to propagate and instead remains as a nanobubble directly
underneath the nanocontact. The nanobubble can hence form either from the P
state or from a droplet.

The droplet shows two discernable states, a high-field/low-current mode that
exhibits a rather small MR (light blue). This mode moves to higher fields with
increasing current and is no longer visible above $\approx-6$ mA. The other
distinguishable droplet mode is characterized by an intermediate resistance
(green-yellow). The bubble is almost indiscernible from the latter droplet
state, even though a subtle line traces out the transition between the two.
Moreover, the bubble resistance is not a smooth function of applied field, but
displays notches and steps, indicative of Barkhausen noise due to pinning.

In contrast to their almost identical resistance, a stark difference between
the droplet and the bubble is uncovered in Fig. 3(b), where we show the
microwave signal integrated over 0–0.5 GHz. The droplet exhibits non-zero
power levels of low frequency microwave noise, while the P, AP, and bubble
states are definitely static and silent. Figure 3(b) also further unveils the
complex relation between the applied field and current, and the particular
droplet characteristics. A strong microwave noise signal denotes mode hopping
and these events exhibit a strong dependence on both field and current. We can
identify three traces of mode hopping for positive fields, while there is only
two weak trails at negative fields. There is also regions where the droplet is
very stable and the noise level is almost zero. These features act like
fingerprints for each measured device and are highly reproducible in
consecutive measurements, but differ between STNOs. We then overlay the
microwave noise data onto the resistance data, now plotted with a gray scale
that highlights intermediate resistance levels (Fig. 3(c)). Parts of the low-
field/low-current droplet regime (light blue in Fig. 3(a)) does not exhibit
any measurable microwave noise. It is possible that its dynamics is on a
slower time scale than the microwave frequencies our set-up is sensitive to.

Figure 3:  Phase diagrams based on the resistance and the microwave noise. (a)
STNO resistance and (b) integrated (0–0.5 GHz) microwave noise level as a
function of field and current. (c) shows the noise level in (b) overlaid on
the resistance in (a) displayed using a gray scale highlighting intermediate
resistance levels indicative of droplets/bubbles. The dash-dotted black line
corresponds to the field-sweep at $I$ = –5 mA given in Fig. 2. The parallel
(P) and antiparallel (AP) states are easily discernible in the MR-map (a) as
dark blue and dark red, while both the droplet and the bubble are
characterized by intermediate resistance in green–yellow. The stark difference
between the droplet and the bubble is revealed in the noise spectrum (b),
where the stability of the bubble is manifested. Note however that the light-
blue flanges in (a) correspond to a different droplet regime not captured in
the microwave signal presented in (b).

We finally turn to the results of the scanning transmission X-ray microscopy
results, illustrated in Fig. 4. Images of the droplet/bubble are shown in Fig.
4(a)-(f), and the corresponding magnetoresistance and microwave signal are
presented in Fig. 4(g) with the matching field of the images marked by their
letter. The STXM and the electrical measurements were performed in separate
setups, hence there is a small uncertainty in comparing the field values of
the two, although both measurements seem highly consistent with each other.
The dashed white or black circles mark the position of the nanocontact. It has
been placed by assuming that the droplet/bubble in Fig. 4(d) is centered under
the NC and by comparing the non-magnetic contrast of the different images. The
method works very well as confirmed by the good overlap of the perimeters in
the inset of Fig. 4(g), but it should be remembered that the absolute position
is still based on this assumption.

Figure 4:  Scanning Transmission X-ray Microscopy (STXM). (a)–(f) STXM images
of the nanocontact region vs. decreasing field for a current of -7 mA. Blue
corresponds to magnetization alligned with the applied field, red corresponds
to magnetization anti-alligned with the applied field, whereas white indicates
in-plane magnetization. The STNO resistance and the microwave noise PSD vs.
decreasing field are shown in (g) where the points corresponding to the STXM
images have been labelled a–f. The inset in (g) highlights the perimeter of
the droplet/bubble as the applied field is decreased from 200 mT (dark blue)
to 60 mT (blue), and further reduced to -40 mT (red) in steps of 20 mT.

Figure 4(a) is measured at 270 mT, and shows a mode associated with a high
noise level in Fig. 4(g). Only a weak and mostly white feature is captured in
the STXM image. STXM measures a time averaged image and the droplet is in this
highly noisy regime expected to experience large drift instabilities and
continuously vanish and renucleate underneath the nanocontact. As a
consequence, only a washed out and poorly reversed feature results. In
contrast, Fig. 4(b)-(d) display more stable and more clearly reversed
droplets. They have approximately the same radius as the nanocontact, although
the size definitely increases slightly with decreasing field, as expected. We
have in earlier STXM work observed a significant effect of the Zhang-Li torque
on the droplet size[21, 47]. The magnitude of this effect depends on the
current density ($j_{\mathrm{dc}}$) and we have performed simulations which
confirm that the difference between the droplet diameter here and in our
former publication is indeed due to a weaker $j_{\mathrm{dc}}$. At zero field,
a bubble is clearly formed and it prevails down to -40 mT (Fig. 4(e)-(f)). It
is no longer centered on the nanocontact, but has mostly expanded in one
direction. It should be noted though, that the field in the microscope is
given by rotating permanent magnets and the sample may have been subjected to
in-plane fields between two set values. Nevertheless, the inset in Fig. 4(g)
presents the perimeter of the droplet/bubble as the field decreases from 200
mT (dark blue) to -40 mT (red), and the initial bubble at 40 mT (light blue)
grows in distinct steps, which implies that the size is controlled by pinning.

Returning to the original droplet theory of Hoefer et al, we note that pinning
was not included.[2] It is clear from our experimental observations that
pinning has a strong and immediate impact on the relation between droplets and
nanobubbles and must be included in the low-field/low-current regime. Instead
of exhibiting a continuous slow-down and frequency decrease to zero with
decreasing field, there is a minimum droplet precession frequency that spin
transfer torque can sustain before pinning overcomes the precession. As this
minimum frequency is approached from above, the broad-band microwave noise
diverges as the competition between the inertia of the precession and the
pinning makes the droplet dynamics increasingly erratic until pinning finally
gets complete control of the precession abruptly stops, leaving complete
microwave silence in its wake.

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##  0.1 Sample Preparation

A sample stack is consisted of a Ta ($4$ nm)/ Cu ($14$ nm) / Ta ($4$ nm) / Pd
($2$ nm) seed layer and an all-perpendicular pseudo-spin valve [Co ($0.35$ nm)
/ Pd ($0.7$ nm)]${\times 5}$ / Co ($0.35$ nm) / Cu ($5$ nm) / [Co ($0.22$ nm)
/ Ni ($0.68$ nm)]${\times 4}$ / Co ($0.22$ nm), capped by a Cu ($2$ nm) / Pd
($2$ nm) layer, which was deposited by magnetron sputtering on Si wafer with
300 nm thick LPCVD silicon nitride layer. Using a conventional photo-
lithography and metal-etching techniques, 8 $\mu$m $\times$ 16 $\mu$m mesas
were fabricated on above stack wafer and all mesas were insulated by a 30-nm-
thick SiO2 layer deposited by using chemical vapor deposition (CVD). To
pattern nanocontacts (NCs) on the top of each mesa having different diameters
from 50 to 150 nm, electron beam lithography was used. SiO2 layer was then
etched through by the reactive ion etching (RIE) technique to open NCs. The
NC-STO device fabrication was completed by the deposition of Cu $200$ nm / Au
$100$ nm top electrode and lift-off processing. For STXM measurements, Si was
removed from backside using highly selective RIE process and leave only SiN
membrane to allow X-ray transmission underneath NC-STOs. (See, Figure 1(c))
For magnetic and electrical charaterization of NC-STOs, same stack was
prepared on Si thermally oxidized Si wafer and then similar fabrication
processing were done except a deep etching for a membrane structure.

##  0.2 Magnetic and Electrical Characterization

the magnetization hysteresis loops was measured using Alternating Gradient
Magnetometry (AGM) with the unpatterned material stacks. dc and microwave
measurements of the fabricated STOs were carried out using our custom-built
setup, where magnetic field strength, polarity, and angle can be controlled. A
magnetic field between -0.5 to +0.5 can be manipulated using electromagnet.
The device is connected using GSG probe to a dc-current source (Keithley
6221), a nanovoltmeter (Keithley 2182A), and a spectrum analyzer (R & S
FSQ26). A 0–40GHz bias-tee is used to separate the bias input and the
generated microwave signal. The microwave sinalg is amplified by a low-noise
amplifier (operational range: 0.1–26.5 GHz) before being sent to the spectrum
analyzer.

##  0.3 Scanning transmission x-ray microscopy

The STXM measurements were performed at the BESSY II synchrotron, using the
MPI IS operated MAXYMUS end station at the UE46-PGM2 beam line.[48] The out-
of-plane component of the magnetization was probed using circularly polarized
light at normal incidence. The applied field, with a maximum value of 300 mT,
was generated by a set of four rotatable permanent magnets[49]. An optimal
XMCD contrast was achieved by setting the photon energy to the Ni $L_{3}$
edge, which resulted in clear images. The size of each pixel is 10$\times$10
$nm^{2}$, while the nominal resolution of the focusing plate is 18 nm.

This work was supported by the Swedish Research Council (VR; 2017-06711 and
2019-04229). Helmholtz Zentrum Berlin is acknowledged for allocating beam time
at the BESSY II synchrotron radiation facility. M. W., E.G., G.S. and J.G.
acknowledge the financial support by the Federal Ministry of Education and
Research of Germany in the frame work of DynaMAX (Project No. 05K18EYA). This
work was supported by the National Research Foundation of Korea(NRF) grant
funded by the Korea government(MSIT) (No. 2020R1F1A1049642)

These authors contributed equally: Martina Ahlberg and Sunjae Chung.

M.A., S.C., and J.Å. conceived the project, S.C. S. J. and T.Q.L. performed
the electrical measurements. S.C., T.Q.L., S.J., and A.H. fabricated the
devices. M.A., S. J., J.G., M.W., F.G. and I.B. carried out the STXM
measurements. J.Å. coordinated the project. All authors analyzed the results
and co-wrote the manuscript.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to S. Chung
(email<EMAIL_ADDRESS>and J. Åkerman (email:
johan.akerman@physics.gu.se).

##  0.4 Supplementary information.

Figure S1: Supplementary Fig.1.  A zoom-in of the phase diagram in Fig. 3(a)
of the main text. The color code represents the different states: droplet
(green), bubble (gray) and antiparallel (red). A droplet is nucleated at high
currents and fields. Below a certain positive field ($H_{\mathrm{balance}}$)
the droplet is stabilized as a static bubble due to magnetostatic effects[2].
The bubble is pinned below the nanocontact until the negative field is high
enough to let the bubble domain expand throughout the film at
$H_{\mathrm{pinning}}$. At low currents the magnetic switching is only
governed by the coercive field ($H_{\mathrm{c}}$) of the free layer.