# Steerable current-driven emission of spin waves in magnetic vortex pairs

Sabri Koraltan  Faculty of Physics, University of Vienna, Kolingasse 14-16,
A-1090, Vienna, Austria  Vienna Doctoral School in Physics, University of
Vienna, A-1090, Vienna, Austria  Research Platform MMM Mathematics - Magnetism
- Materials, University of Vienna, Vienna 1090, Austria
<EMAIL_ADDRESS>Katrin Schultheiss  Helmholtz-Zentrum Dresden-
Rossendorf, Institute of Ion Beam Physics and Materials Research, 01328
Dresden, Germany  Florian Bruckner  Faculty of Physics, University of Vienna,
Kolingasse 14-16, A-1090, Vienna, Austria  Markus Weigand  Institut für
Nanospektroskopie,Helmholtz-Zentrum Berlin für Materialien und Energie GmbH,
12489 Berlin, Germany  Claas Abert  Faculty of Physics, University of Vienna,
Kolingasse 14-16, A-1090, Vienna, Austria  Research Platform MMM Mathematics -
Magnetism - Materials, University of Vienna, Vienna 1090, Austria  Dieter
Suess  Faculty of Physics, University of Vienna, Kolingasse 14-16, A-1090,
Vienna, Austria  Research Platform MMM Mathematics - Magnetism - Materials,
University of Vienna, Vienna 1090, Austria  Sebastian Wintz  Institut für
Nanospektroskopie,Helmholtz-Zentrum Berlin für Materialien und Energie GmbH,
12489 Berlin, Germany  Max Planck Institute for Intelligent Systems, 70569
Stuttgart, Germany<EMAIL_ADDRESS>
###### Abstract

The efficient excitation of spin waves is a key challenge in the realization
of magnonic devices. We demonstrate the current-driven generation of spin
waves in antiferromagnetically coupled magnetic vortices. We employ time-
resolved scanning transmission X-ray microscopy (TR-STXM) to directly image
the emission of spin waves upon the application of an alternating current
flowing directly through the magnetic stack. Micromagnetic simulations allow
us to identify the origin of the excitation to be the current-driven Oersted
field, which in the present system proves to be orders of magnitude more
efficient than the commonly used excitation via stripline antennas. Our
numerical studies also reveal that the spin-transfer torque can lead to the
emission of spin waves as well, yet only at much higher current amplitudes. By
using magnetostrictive materials, we futhermore demonstrate that the direction
of the magnon propagation can be steered by increasing the excitation
amplitude, which modifies the underlying magnetization profile through an
additional anisotropy in the magnetic layers. The demonstrated methods allow
for the efficient and tunable excitation of spin waves, marking a significant
advance in the generation and control of spin waves in magnonic devices.

## Introduction

State-of-the-art data processing is nowadays based on complementary metal
oxide semiconductor (CMOS) technology, where the control of current flow in
transistors is used for logic operations and routing of information [1, 2].
Accompanied by memory units such as random access memory (RAM) for temporary
purposes or flash memory for persistent data storage, they build the backbone
of today’s computer designs. However, the relatively high power consumption of
CMOS devices, due to ohmic losses, volatile refresh memories, and limits for
further miniaturization, represents severe challenges for developing a
sustainable information and communication technology for the future [3].

Magnonic devices have emerged as a candidate solution to the challenges that
CMOS is facing today [4, 5]. Within such devices, one makes use of magnons
(the quanta of spin waves) [6] to transfer low-loss information. As a local
deflection of the magnetic orientation, spin waves typically exhibit
wavelengths from subnanometers to centimeters at frequencies ranging from the
megahertz to the terahertz range [5, 7]. Based on their nonlinear propagation
and coupling mechanisms, they can be used to operate various kinds of devices
such as reprogrammable magnonic crystals [8], magnonic directional couplers
[9], boolean computing devices [10], magnonic nano-ring resonators [11],
magnon memory devices [12], magnonic logic circuits [13], offset-free magnetic
field sensor [14], spin-wave majority gates [15] or even multi-purpose devices
[16, 17, 18] and many more [19, 20]. For real-world implementation of most
proposed magnonic applications, it will be crucial to utilize spin waves of
nanoscale wavelengths. This is mainly for two reasons: (i) miniaturization,
where the wavelength imposes a constraint on the device footprint, and (ii)
high group velocities in the short-wavelength (exchange-dominated) regime for
sufficient processing speeds and propagation lengths.

Typically, the excitation of coherent spin waves is achieved by using
lithographically patterned microwave antennas, for example metallic coplanar
waveguides [21]. However, this method has limited efficiency, in particular
for nanoscale wavelengths. Here, the wavelengths are restricted on the lower
end to the minimum patterning sizes involved and at the same time such
antennas still need to allow for sufficient impedance matching of the electric
circuitry. One way to overcome this limitation, among others, is to couple a
global alternating external magnetic field, to local internal demagnetization
fields [22, 23, 24] or to the dynamics of confined spin textures such as
vortex cores [25, 26, 27, 28, 29, 30], skyrmions [31, 32, 33, 34], or domain
walls [35, 36, 37, 38, 39, 40, 41, 42]. The latter coupling to spin textures
was found to be particularly resourceful in synthetic ferrimagnets (SFi),
where two ferromagnetic layers are coupled antiparallelly through a non-
ferromagnetic interlayer [43, 44, 25, 45]. On the other hand, it was
previously reported that oscillatory dynamics of spin textures can also be
excited using the effects from electric currents flowing through these
textures themselves, as a result of Oersted fields, spin-transfer torques, or
spin-orbit torques [46, 47, 48]. This can be the case for both direct and
alternating currents as well as for lateral and vertical flow directions.

Figure 1: Schematics of experimental setup and SFi Sample #1. (A) Scanning
transmission X-ray microscopy setup, where the monochromatic X-rays are
diffracted by the zone plate to focus in a single spot. The combination of the
zone plate center stop and the order sorting aperture blocks the undiffracted
beam and higher diffraction orders (latter not shown) from reaching the
sample. The sample itself is grown on a thin membrane and mounted on a piezo
stage that is raster scanned (x/y) through the focal point. The transmitted
X-rays are collected by a point detector. (B) Schematic of the
$\mathrm{Co/Ru/Ni_{81}Fe_{19}}$ microdisk with adjacent, partially overlapping
copper leads (aspect ratios not to scale). White arrows indicate the general
antiparallel orientation of the two ferromagnetic layers of the SFi.
Alternating current injected from the leads flow laterally through the disk as
indicated by the orange arrow. (C) Static STXM image of the sample recorded at
the Fe $\mathrm{L_{3}}$ edge. The SFi microdisk (9 $\mu$m diameter) and the
copper leads appear as gray contrast areas, with darker contrast in the
overlapping regions. Within the disk, there is already faint magnetic contrast
visible as well.

Thus, it has been a key question, both in fundamental and practical respects,
whether electric currents flowing through spin textures directly can be
employed to generate short-wavelength spin waves and how efficient and
versatile such a process would be as compared to the global field approach.
Here, we demonstrate the realization of such current-driven spin-wave emission
in SFi vortex pairs in a lateral alternating-current geometry and without the
need for any magnetic bias field. We use high-resolution time-resolved X-ray
microscopy to directly image the resulting spin-wave dynamics. Large-scale
micromagnetic simulations allow us to understand the origin of the spin-wave
emission, distinguish between different excitation mechanisms, and compare
their efficiency. When the SFi consists of magnetostrictive materials, we
demonstrate that the emission direction of the generated spin waves can be
steered by the magnitude of the applied current.

## Results

Current-driven spin-wave generation. For our study, we require a time-resolved
magnetic probe beyond the spatial resolution limit of visible-light
techniques. Therefore, we image our samples using scanning transmission X-ray
microscopy (STXM) with approximately 25 nm lateral resolution and by
exploiting the X-ray magnetic circular dichroism [49] (XMCD) effect as
magnetic contrast mechanism (see Methods). The typical STXM measurement setup
is illustrated schematically in Fig. 1A. Measurements were carried out at the
Maxymus end station [50] at the BESSYII electron storage ring operated by the
Helmholtz-Zentrum Berlin für Materialien und Energie. The first sample we
examine is a circular thin film disk of 9 $\mu$m diameter made out of a
Co($47.8\text{\,}\mathrm{n}\mathrm{m}$)/Ru($0.8\text{\,}\mathrm{n}\mathrm{m}$)/$\mathrm{Ni_{81}Fe_{19}}$($42.8\text{\,}\mathrm{n}\mathrm{m}$)
SFi stack (Sample #1), as shown schematically in Fig. 1B. Here, we also
highlight the electric contacting to the sample which is provided via copper
thin film leads, overlapping at about 2 $\mu$m on either side with the
microdisk. A static STXM image of Sample #1 recorded at the Fe
$\mathrm{L_{3}}$ absorption edge is given in Fig. 1C. While the substrate
background appears as bright contrast (high photon transmission), both the SFi
microdisk and the copper leads appear as gray (medium transmission) or even
dark gray where they overlap (low transmission). There is also already a hint
for additional magnetic contrast within the disk, which we will turn to in the
following.

Figure 2: STXM imaging of current-driven spin-wave emission in SFi Sample #1.
(A,B) Static STXM images with partial in-plane magnetic sensitivity ($\sim
M_{x}$), recorded at the Fe $\mathrm{L_{3}}$ edge (A) and the Co
$\mathrm{L_{3}}$ edge (B), respectively. Zoom-in panels on the right
correspond to the solid squares in (A) and (B), with orange arrows indicating
the general magnetic vortex pair orientations as given by the black and white
contrast. (C) Normalized TR-STXM snapshot of spin-wave dynamics in response to
an alternating current of
$f_{AC}=$1.1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ flowing laterally
through the disk, recorded with perpendicular magnetic sensitivity
($\sim\Delta M_{z}$) at the Co $\mathrm{L_{3}}$ edge. (D) Spin-wave amplitude
profile extracted along the green arrow in (C) for a relative time delay of
$\Delta t=235$ ps.

To identify the magnetic state of Sample #1, we performed static STXM imaging
with partial in-plane magnetic sensitivity, separately for the two
ferromagnetic layers. In addition to topographic information, Figures 2A and B
reveal the horizontal projection ($M_{x}$) of the magnetic orientation for the
$\mathrm{Ni_{81}Fe_{19}}$ layer (recorded at Fe $\mathrm{L_{3}}$) and the Co
layer (recorded at Co $\mathrm{L_{3}}$), respectively. By means of these
images, it becomes obvious that the in-plane magnetic components of the two
layers indeed are coupled antiparallelly (opposite relative contrast) and that
there is a multivortex-domain state within the microdisk (multiple vortex
cores in each layer). Despite this multidomain state, a large portion of the
disk area can be identified as being part of a single interlayer vortex pair
with antiparallel vortex circulations that have a slightly discrete and
distorted rotation character (wall formation with rotation angles above and
below $90^{\circ}$). The corresponding vortex cores are being somewhat
displaced congruently to the upper right from the center of the disk. Zoom-in
panels of the center regions (solid red and blue frames) allow for a closer
view of the core area, with orange arrows illustrating the general vortex
pattern. Although the core polarity cannot be directly obtained from these
particular two images, it was found that typically the cores are aligned
parallel in this kind of system [25] (cf. Supplemental Fig. LABEL:fig:figs01).
Note that, in general, the STXM images of the Co layer have a higher signal-
to-noise ratio than those of the $\mathrm{Ni_{81}Fe_{19}}$ layer, as there are
many more absorbing Co atoms than Fe atoms in the sample, at a similar level
of XMCD.

To evaluate whether current-driven spin-wave emission is possible in this kind
of system, we send an alternating current laterally through Sample #1, which
is injected via the adjacent copper leads. The microwave current had a
frequency of $f_{AC}=$1.11\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, a
relevant current density amplitude of
$j=2.9\times${10}^{10}\text{\,}\mathrm{Am^{-2}}$$ and is flowing mainly along
the $x$-axis, connecting the leads. Note that for the calculation of the
current density we assume a uniform current flow in the SFi stack, where the
cross-section area is defined as $A=l\times d$, where $l$ is the diameter of
the disks ($l=$9\text{\,}\mu\mathrm{m}$$) and $d$ is the total thickness.
Using TR-STXM (see Methods), we directly observe the dynamic response of the
system. Figure 2C shows a TR-STXM normalized snapshot in time $\Delta
M_{z}(t_{0})$, being sensitive to dynamic changes of the perpendicular
component $M_{z}$ of the magnetization of the Co layer. The area represented
by the dashed blue box in Fig. 2B is the corresponding image area. Indeed, we
observe a clear spin-wave pattern of significant amplitude with an average
wavelength of $307\pm 30$ nm. By looking at consecutive snapshots in an
animated way (Supplemental Movie M1), it becomes clear that these waves are
mainly emitted from the somewhat bend, higher-angle domain walls (highlighted
by orange dashed lines in Fig. 2C) and the vortex core. Note that the
background magnetic state in Fig. 2C is different from that in Fig. 2 A/B with
respect to the state of the domain wall (sample was remounted in between). The
emission pattern can be seen in further detail in Supplemental Fig.
LABEL:fig:figs01, which displays the dynamic snapshots for the two layers
separately in both absolute and normalized visualization, also revealing an
in-phase dynamics of the $M_{z}$ component of the two layers. Thereby we can
conclude, in line with earlier reports [44, 25], that the observed spin waves
are an instance of the acoustic, layer-collective SFi mode in the Damon-
Eshbach geometry. Figure 2D further highlights the propagating nature of the
observed spin waves by plotting the amplitude profile along the green arrow in
Fig. 2C for a relative time delay of 235 ps and with additional lines for
guiding the eye. The phase propagation is clearly visible. Note that the
observed excitation process is not restricted to a certain frequency or
resonance, but rather covers a broadband frequency range from hundreds of MHz
to multiple GHz [42] as shown exemplary in Supplemental Fig. LABEL:fig:figs02
and Supplemental Movie M2. Our experiments evidently demonstrate that short-
wavelength spin-wave emission in spin textures can be driven by alternating
currents flowing through these textures.

Origin of the current-driven emission. In the following, we investigate the
origin of the spin-wave emission from spin textures by employing micromagnetic
simulations with magnum.np[51], see Methods for more details. When a charge
current flows directly through a ferromagnetic material, several effects may
occur to back-act on its magnetic state, two out of which are most prominent
for our case. On the one hand, the charge current flowing along a given
direction will generate an Oersted field within the conductor that is often
axially symmetric. On the other hand, electrons that have acquired a spin
polarization may be exerting a torque on the magnetization, termed the spin-
transfer torque (STT) [52, 53]. Both effects can be described by means of a
micromagnetic continuum model [54]. Although, in reality, they are occurring
at the same time, we can use micromagnetic simulations to apply them together,
or individually, to investigate their role on the emission of spin waves.
Thus, we perform such micromagnetic simulations, numerically solving the
Landau-Lifshitz-Gilbert equation (LLG) in order to understand the origin of
the spin-wave emission that we observed experimentally. Note that we chose a
system analogous to Sample #1, but limited the lateral size of the simulated
disk to $4~{}\mu$m for computational practicability reasons. Nevertheless, we
confirmed that very similar results arise in even smaller disks, which
validates our approach. Noteworthy, it is crucial to use double precision
floating point operations to perform the numerical calculations in order to
achieve the necessary accuracy.

Figure 3 summarizes the different excitation methods simulated for an ideal
vortex pair state with opposite circulations and parallel cores. It allows us
to identify, in general, the main contributor to the current-driven emission
of spin waves in our case. Figure 3A depicts the steady-state magnetization
profile in the Co layer during the application of an AC current density of
$j=$1\text{\times}{10}^{9}\text{\,}\mathrm{Am^{-2}}$$ and
$f_{AC}=$1.11\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, where contributions
from both the STT and the Oersted field are included in the modeling. For the
sake of simplicity, we assume a uniform current flow along the $x$ axis
throughout the stack. In reality, position dependent areal cross-sections,
different resistivities of the materials and misalignments of the copper leads
might induce inhomogeneities. One can see the emission of coherent, high-
amplitude spin waves with wavelengths similar to those in the experiment. In
contrast, there is no emission of spin waves anymore if one takes into
consideration only the STT as shown in Fig. 3B. Even though the STT acts
locally wherever the magnetization profile has a spatial gradient, the
amplitude of the applied current is not sufficient to excite the magnetic
vortex cores to gyrate in order to emit spin waves. If one considers only the
generated Oersted field, then the resulting spin-wave pattern can be again
observed in Fig. 3C, which is almost identical to the one in Fig. 3A.
Therefore, we can conclude that it is solely the Oersted field that causes the
efficient excitation of spin waves in our experiment. The magnetization
dynamics only driven by Oersted fields is shown in the Supplemental Movie M3,
and the Supplemental Movie M4 contains animations of spin-wave emissions for
different excitation methods. In addition, we provide the results from
simulations with excitation current densities of the same order as in the
experiment in Supplementary Fig. LABEL:fig:figs03, where the magnetization and
the normalized perpendicular dynamic magnetization are displayed separately
for the Co and the $\mathrm{Ni_{81}Fe_{19}}$ layer, respectively. These
simulations demonstrate that the spin-wave amplitude increases with the
applied current density, while we also observe a noticeable double-frequency
($2f$) excitation of spin waves at the same time.

Figure 3: Origin of current-driven spin-wave emission in SFi vortex pairs
(simulations). Snapshots of the normalized perpendicular dynamic magnetization
($\Delta m_{z}$) in the Co layer (averaged over the layer thickness) in a
system analogous to Sample #1 obtained from micromagnetic simulations with the
contrast according to the blue-white-red color bar. The current density was
modeled to $j=$1\text{\times}{10}^{9}\text{\,}\mathrm{Am^{-2}}$$ at
$f_{AC}=$1.11\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$. In (A), both the STT
and the current-driven Oersted field were included in the effective field,
while in (B) only the STT and in (C) only the Oersted field were considered,
respectively.

Excitation efficiency. As we identified the Oersted field as the origin of the
current-driven spin-wave emission, we further evaluate the excitation
efficiency of this method. To this end, we compare the current-driven Oersted
field-dominated spin-wave emission, to the previously used stripline antenna
driven spin-wave emission from vortex cores[25, 42], and also to the STT
excitation above, yet considering two orders of magnitude higher current
densities.

Figure 4: Comparison of the spin-wave excitation efficiency in SFi vortex
pairs (simulations). Averaged spin-wave amplitude maps of the
$\mathrm{Ni_{81}Fe_{19}}$ layer, given in terms of $\Delta m_{z}$ snapshots
during excitation with (A) current-driven Oersted field, (B) uniform field of
equal magnitude
$max(H^{\mathrm{Oersted}}_{y})=H^{\mathrm{uniform}}_{y}=$400\text{\,}\mathrm{A}\mathrm{m}^{-1}$$
(as if the magnetic field was generated by a stripline antenna), and (C)
enhanced STT excitation with a 100-fold increased current density. The
excitation frequency is $f=$2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$. The
spin-wave amplitude along the black dotted line depicted in (A) is plotted in
(D) for the three different cases according to the color code below (A).

Figure 4 gives an overview of the spin-wave amplitude profiles obtained from
our micromagnetic simulations; this time at a frequency of $f_{\mathrm{AC}}=2$
GHz. The current amplitude is again
$j_{AC}=$1\text{\times}{10}^{9}\text{\,}\mathrm{A/m^{2}}$$. In contrast to
Fig. 3, we now depict the magnetization of the $\mathrm{Ni_{81}Fe_{19}}$
layer, yet, as shown above, the Co layer behaves equivalently in terms of the
perpendicular spin-wave component. The color-coded change in the normalized
dynamic $\Delta m_{z}-$component of the magnetization, this time averaged over
the $\mathrm{Ni_{81}Fe_{19}}$ layer, is strongest if the spin waves are
excited via the current-driven Oersted field method, as can be seen in Fig.
4A. As referred to above, two other cases of spin-wave excitation were modeled
numerically. The first involves a stripline antenna situated underneath the
vortex pair, which produces an Oersted field that acts on the SFi (Fig. 4B).
The second case involves only the STT from the current flowing through the
sample, yet increasing the injected current density by a factor of $100$ (Fig.
4C). Both of these methods can excite spin waves, but the current-driven
Oersted field generates spin waves with much higher amplitudes, making it the
most effective excitation method of the three cases discussed, see also
Supplemental Movie M4. The excitation efficiency can be further highlighted by
means of the amplitude profiles in Fig. 4D. Note that the stripline antenna is
assumed to generate a uniform field of equal magnitude compared to the
current-driven Oersted field
$H^{\mathrm{uniform}}_{y}=\mathrm{max}(H^{\mathrm{Oersted}}_{y})=$20.9\text{\,}\mathrm{A}\mathrm{m}^{-1}$$.
Comparing the spin-wave excitation efficiency of the three methods, we see
that the amplitude of the spin waves that were excited by current-driven
excitation is a factor of $\sim 10$ higher compared to the enhanced STT
excitation, and a factor of $\sim 30$ higher compared to the excitation with
the stripline antenna. In terms of energy coupling efficiency (considering the
square of the amplitude), it means that current-driven Oersted field
excitation has an impressive benefit of three orders of magnitude for the case
of the given system.

Figure 5: SFi spin-wave profile and excitation symmetries (schematics). (A)
Cross-sectional profile of the dynamic magnetization components (white arrows)
of an acoustic Damon-Eshbach spin wave in the SFi system with colors (blue and
red). Note that here acoustic means that the perpendicular dynamic components
are in-phase (in contrast to the dynamic in-plane components which are anti-
phased). (B,C) Equilibrium magnetization in the antiparallel SFi vortex pairs
(blue and red arrows) in the respective layers (blue and red disks), Oersted
fields (gray arrows) at a snapshot in time for the Oersted-driven case (B) and
stripline-antenna case (C). The locally resulting torque
$(-\mathbf{M}\times\mathbf{H})$ is depicted by green (up) and purple (down)
arrows. From symmetry perspective, only the Oersted driven excitation can
directly excite the acoustic Damon-Eshbach SFi mode, as here the $z$-torques
are in-phase between the two layers (B), in contrast to the stripline antenna
excitation, where they are antiphased (C).

Mechanisms of excitation.The reason why the spin-wave generation efficiency is
much higher for the current-driven case than for the uniform (stripline) case
presumably lies in the symmetry of the excited spin-wave mode. This mode is
the acoustic collective Damon-Eshbach mode of the SFi, which has in-phase
perpendicular dynamic components and antiphase in-plane dynamic components for
the two ferromagnetic magnetic layers [25], Fig. 5A. Since the two layers have
antiparallel in-plane equilibrium magnetic orientations, this mode cannot be
directly excited by means of uniform in-plane fields as a consequence of
symmetry considerations, see Fig. 5C, (yet it could be excited for
perpendicular uniform fields). For the Oersted field, however, the actual
field orientations acting on the two ferromagnetic layers are opposite, see
Fig. 5B, and, therefore, a direct excitation of the acoustic mode is possible
from symmetry arguments. The above considerations are valid for the quasi-
uniform precession case, but not necessarily for the excitation of short-
wavelength spin waves. In the uniform in-plane field case (and the enhanced
STT case), wave excitation requires the presence and gyration dynamics of the
vortex cores, that can be driven by in-plane fields and then act as kind of a
perpendicular nano-stirrer, leading to a spiraling wave pattern[25, 42]. For
the Oersted field case, the situation is different. Here, the vortex cores are
not strictly required for the excitation of waves. In fact, simulations with
artificially removed cores show comparable results to those conducted with the
cores present, as shown in Supplementary Fig. LABEL:fig:figs09.
Consequentially, there is no spiraling signature in the resulting emission
pattern, typically originating from vortex core gyration. The reason why
nevertheless a finite wavelength is excited, lies in the very strong
dispersion nonreciprocity of the acoustic Damon-Eshbach mode in the SFi [25,
45]. Here, nonreciprocity refers to the fact that counterpropagating waves of
the same frequency have significantly different wavelengths (by a factor of
the order of 10). Therefore, the global Oersted field can excite a quasi-
homogeneous precession of the order of half the structure size (radially), and
upon lateral radial reflection, this dynamics appears as a short-wavelength
branch propagating away from the cores (or the rim for the other antiparallel
relative vortex circulation combination possible). We refer to this effect as
nonreciprocal excitation. Note that in the simulations there is also a
dependence of the spin-wave amplitude on the azimuthal angle, exhibiting a
two-fold symmetry. In particular, there appears a quasi-nodal line, crossing
the disk center from left to right. This effect stems from the azimuth-
dependent resulting torque of the form
$\propto(-\mathbf{M}\times\mathbf{H}^{\mathrm{Oersted}})$, when considering
the Oersted field geometry and the vortex magnetization. Figure 5B illustrates
the torque generated by the Oersted field, and Fig. 5C depicts the torque
resulting from the uniform external field from a stripline antenna. The
possible extent to which the spin-texture dynamics (vortex core gyration,
domain wall oscillation) still contributes to the excitation of waves in
parallel to the nonreciprocal oersted-field excitation effect remains a
challenge to be quantified.

Direction-steerable spin-wave emission. In the above experiments and
simulations, the spin-wave emission pattern was fixed by the specific
geometries of the spin texture and the current flow. In the following, we will
demonstrate a proof-of-principle system where the emission pattern and, in
particular, its directional properties can be tuned by the drive current. To
this end, we used another SFi disk of the same diameter (9 $\mu$m) made of a
$\mathrm{Ni_{81}Fe_{19}}$($44.9\text{\,}\mathrm{n}\mathrm{m}$)/Ru($0.8\text{\,}\mathrm{n}\mathrm{m}$)/$\mathrm{Co_{40}Fe_{40}B_{20}}$($46.6\text{\,}\mathrm{n}\mathrm{m}$)
stack (Sample #2), which is illustrated in Supplemental Fig. LABEL:fig:figs04.
$\mathrm{Co_{40}Fe_{40}B_{20}}$ is well-known to be a magnetostrictive
material [55, 56], i.e., it changes its anisotropy with mechanical strain
through the magneto-elastic effect.

Figure 6: TR-STXM imaging of direction-steerable spin-wave emission in SFi
Sample #2. Response to alternating currents of
$f_{AC}=$571\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ with current densities
as provided below the individual columns. Images recorded at the Co
$\mathrm{L_{3}}$ edge. Top row (A,B,C), direct absorption snapshots showing
both topographic and magnetic contrast, the latter with mixed sensitivity
($\sim M_{x,z}$). Bottom row (D,E,F), normalized snapshots that highlight the
magnetic dynamics ($\sim\Delta M_{x,z}$). The orange arrows indicate the
magnetic background orientation.

Figure 6 A shows a STXM image with partial in-plane sensitivity of the
$\mathrm{Co_{40}Fe_{40}B_{20}}$ layer of Sample #2. Although excited by an
alternating current, the overall magnetic state is analogous to the
equilibrium state of the sample. As a consequence of an intrinsic uniaxial
magnetic anisotropy along the $x$-axis (horizontal), the vortex circulation is
broken up into two oppositely oriented domains as indicated by the orange
arrows. These domains are connected by 180∘ domain walls that have a vortex
core remaining in the center [42]. The origin of the intrinsic uniaxial
anisotropy presumably stems either directly from the film growth or from a
potential tensile strain exerted by the copper leads. As for Sample #1, the Ru
mediated interlayer exchange coupling enforces an antiparallel alignment of
the magnetic in-plane components between the two ferromagnetic layers of
Sample #2, as it can be seen in Supplemental Fig. LABEL:fig:figs04.

In fact, Fig. 6A shows the response of the sample as TR-STXM snapshots with
mixed in-plane and out-of-plane sensitivity ($\sim M_{x,z}$) towards an
alternating current of $f_{AC}=$571\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$
flowing through the sample at
$j=$0.36\text{\times}{10}^{10}\text{\,}\mathrm{Am^{-2}}$$ amplitude. It shows
the direct absorption view, where magnetic and nonmagnetic features are
visible (the disk itself, the copper leads, the magnetic domains, and some
signature of spin waves). Figure 6D shows the corresponding snapshot as
normalized view, with sensitivity to magnetic changes over time compared to
the time-average state ($\sim\Delta M_{x,z}$). In this view, we see a clear
emission pattern of rather plane spin waves from the horizontal domain walls
into both directions (up and down). There are also oscillations of the domain
wall itself visible with some additional complexity in the regions outside the
center area, below the leads (possible wall branching). For the animated
version of Fig. 6, we refer to Supplemental Movie M5. When the current density
is increased to
$j=$1.24\text{\times}{10}^{10}\text{\,}\mathrm{A}\mathrm{m}^{-2}$$, we see in
Figs. 6B,E that the time average state has changed compared to the low-
current/static state. The earlier well-defined horizontal domain walls have
changed into a more continuous vortex state, with some diagonal wall features
remaining. As one can see in the normalized snapshot in Fig. 6E, the spin-wave
emission pattern is now more isotropic than before. When the current density
increases further to
$j=$2.02\text{\times}{10}^{10}\text{\,}\mathrm{A}\mathrm{m}^{-2}$$, the time
averaged state changes again, as shown in Figs 6C,F. There is a reappearance
of well defined $180^{\circ}$ domain walls, however, this time they are
oriented along the $y$-axis (vertical). The resulting spin-wave emission
pattern in Fig. 6F mainly consists of plane waves again emitted from the
domain walls, now propagating along the horizontal axis. By tuning the
excitation current density, we can control the orientation of the domain walls
in Sample #2 to be horizontal or vertical, which in turn gives us control of
the axis along which plane spin waves are emitted. For intermediate current
densities, the system assumes a less anisotropic state, with almost isotropic
spin-wave emission. At the same time, the spin-wave wavelength appears to be
almost unaffected by this way of steering the emission direction.

Figure 7: Direction-steerable spin-wave emission in anisotropic SFi vortex
pairs (simulation) (A) magnetization of the bottom $\mathrm{Ni_{81}Fe_{19}}$
layer as color-coded snapshot, where the horizontal in-plane domains with the
vortex core are visible, as well as the modulation in the magnetization due to
the spin-wave amplitude, which is depicted more clearly in (D) as $\Delta
m_{z}$ snapshots. In A and D, the strain-induced uniaxial anisotropy in the
$\mathrm{{Co_{40}Fe_{40}B_{20}}}$ layer was set to be parallel to the applied
current ($\varphi=0$) with an anisotropy constant of
$K_{u}=$3\text{\,}\mathrm{k}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$$. This
anisotropy was set to zero in B and E, to simulate the current/strain-induced
compensation of the anisotropy. To simulate the rotation of the anisotropy,
leading to the emission of spin waves in the vertical direction, the
anisotropy axis was rotated to align along the $y$ axis ($\varphi=90^{\circ}$)
in C and F.

The most likely explanation for the reorientation of the domain walls lies in
a combination of joule heating and magneto-elastic effects (inverse
magnetostriction). With increasing current density, the disk starts to warm
up; urging it to expand. Here, the copper leads provide a certain constraint
for a horizontal expansion, which exerts essentially a compressive strain on
the disk. In the course of this action, the initially horizontal anisotropy is
first reduced, and eventually realigned to a vertical orientation. This
explanation is in line with the positive magnetostrictive constant of
$\mathrm{Co_{40}Fe_{40}B_{20}}$ [57, 58], while the magnetosrictive effects in
$\mathrm{Ni_{81}Fe_{19}}$ are negligible [59]. It is also consistent with the
fact that the anisotropy control is rather independent of the driving
frequency, as can be seen in the Supplemental Fig. LABEL:fig:figs05 and
Supplemental Movie M6, showing similar state responses at
$f_{AC}=$71\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ and
$f_{AC}=$1071\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$. We underpin this
hypothesis by further micromagnetic simulations, where we model Sample #2 with
dimensions equal to those used for Sample #1. To this end, at first we
introduce an in-plane anisotropy to the effective field along the $x$-axis. We
demonstrate in Supplemental Fig. LABEL:fig:figs08 that by increasing the
anisotropy constant $K_{u}$, a bidomain state is induced, similar to that
observed in our experimental findings. Then, we simulate the current-driven
Oersted field excitation to model the corresponding spin-wave generation. In
Fig. 7, we depict three cases. First, the anisotropy axis is chosen to be
along the $x$-axis and parallel to the applied current (Figs. 7A and D), where
$K_{u}=$3\text{\,}\mathrm{k}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$$. Figure 7D
shows a clear anisotropic emission of plane spin waves, originating at the
domain wall. By that we reproduce the low current density regime in Fig. 6 D.
If we assume that upon the intermediate increase of the current density, the
resulting magneto-elastic effect compensates for the original anisotropy, then
a regular vortex pair is obtained again as shown in Fig. 7B, where the spin-
wave emission is now rather isotropic, as can be seen in Fig. 7E. Finally,
when the magneto-elastic effect induces the domain wall to rotate by
$90^{\circ}$, the emission of spin waves occurs collinear to the current
direction (Fig. 7C,F). We provide the animations analogous to Fig. 7 in the
Supplemental Movie M7.

Note that while there are certainly clear benefits in being able to control
the spin-wave emission direction by the drive current, there are also other,
potentially more practical ways to control strain/anisotropy directly, e.g. by
piezoelectrics or voltage controlled magnetic aniostropy (VCMA). In addition
to direction-steerable emission, Sample #2 can also be used to demonstrate
single-pulse driven excitation of individual spin-wave wave packages (see
Supplemental Fig. LABEL:fig:figs07 and Supplemental Movie M8). This effect
further highlights the agility of current-induced spin-wave excitation in SFi,
neither requiring particular resonance frequencies nor multiple onset
excitation cycles.

## Discussion

We demonstrated efficient current-driven excitation of spin waves in an
antiferromagnetically coupled magnetic vortex pair. By employing TR-STXM, we
showed that the spin waves are generated when the SFi is subjected to an
alternating current flowing directly through it. Micromagnetic simulations
revealed that the Oersted field generated in the SFi by the current is the
main contributor to the emission of spin waves in the magnetic vortex pair. By
further comparing this to the contributions from both STT and the uniform
magnetic field of a stripline antenna, we determined that the excitation
efficiency of the current-driven Oersted field in terms of energy is at least
three orders of magnitude higher than the other two methods investigated. To
demonstrate an additional advancement of current-driven spin-wave generation,
we prepared the SFi from a magnetostrictive material, where initial strain
induces an additional anisotropy. This additional anisotropy distortes the
vortex state, leading to the formation of a bidomain state with a pronounced
domain wall in between the domains in both magnetic layers. We demonstrated
that the propagation direction of plane spin waves excited in this system can
be redirected if the amplitude of the applied current is varied, allowing even
for a $90^{\circ}$ change in the emission direction. Our findings represent a
significant advance in the current-driven, energy-efficient excitation and
control of spin waves for potential magnonic nanodevices.

## Materials and Methods

Sample Preparation

Analogous to Ref. [42] and [25], the magnetic multilayers were deposited by
means of magnetron sputtering onto X-ray transparent silicon-nitride window
membranes. For oxidation protection, an aluminium capping layer of 3 nm
thickness was used. The microdisks were fabricated by means of electron beam
lithography (EBL) and consecutive ion beam etching. After an initial oxygen
plasma treatment for adhesive purposes, a negative resist (MA-N 2910) was spun
onto the films and baked out. In a second step, the microdisks were exposed by
EBL and the samples were then developed for 300 s in MA-D 525 and subsequently
rinsed in deionized water (60 s). Finally, the samples were milled using an
argon ion beam at two different angles (85° and 5°) with endpoint detection
for approximately one hour to physically etch the magnetic microdisks out of
the continuous films. An acetone bath (12 hours) and a second oxygen plasma
step (20 minutes) were applied to remove the remaining resist. For being able
to apply electric currents flowing laterally through the samples,
copper/aluminium leads of 200/5 nm thickness were fabricated to overlap at
about 2 $\mu$m with a microdisk at two opposing rim positions (see Fig. 1).
For this purpose EBL, electron beam evaporation and lift-off processing was
used. The resulting resistances of the samples were typically between 10
$\Omega$ and 100 $\Omega$.

Scanning-Transmission X-ray Microscopy

To image the magnetic orientation in the multilayer disks, synchrotron-based
scanning-transmission X-ray microscopy (STXM) was employed [50], as
schematically depicted in Fig. 1A. For STXM, a Fresnel zone plate is used to
focus monochromatic X-rays onto the sample, while undiffracted X-rays and
those of higher diffraction order are blocked by a combination of the zone
plate center stop and a circular order selecting aperture. The X-ray intensity
transmitted through the sample is collected by a point detector. To get an
image, the sample is raster scanned through the focused beam, yielding a
lateral resolution of approximately 25 nm. Magnetic contrast is achieved by
exploiting the X-ray magnetic circular dichroism effect (XMCD) [49], occurring
at the element specific resonant absorption edges. For the given magnetic
multilayer of Sample #1, therefore, the two layers can be separately imaged in
terms of magnetic orientation by measuring at photon energies of Co L3
$\sim$778 eV, and Fe L3 $\sim$708 eV or Ni L3 $\sim$853 eV, respectively. For
Sample #2, on the other hand, Co and Ni photon energies lead to layer-
selective images, while the Fe energy provides an integrated signal of both
layers. In general, the logarithmic transmission contrast detected is
proportional to the projection of the magnetization on the X-ray propagation
direction, which means that in normal incidence XMCD is sensitive to the
perpendicular magnetization component, while inclination of the sample
provides additional information on in-plane magnetization components.

The spin-wave dynamics in the multilayer disks was imaged using pump-and-probe
time-resolved STXM (TR-STXM) [50]. For TR-STXM, the inherent time structure of
the incident X-ray pulses (500 MHz repetition rate and $\sim$100 ps effective
pulse width) is used. The accessible sinusoidal excitation frequencies are of
the form $f_{M}=M/N*500\mathrm{MHz}$ where $M$ is an integer multiplier and
$N$ corresponds to the integer number of phases acquired simultaneously. The
level of the excitation signal was measured both before and after the sample
(via a pick-off tee) by means of an oscilloscope. The mean of the input and
transmitted signal was used to estimate the resulting current density in the
sample by assuming the diameter and height of the disk as relevant areas for
the current cross section.

Micromagnetic Simulations.

We use magnum.np[51], an open-source GPU-accelerated micromagnetic simulation
library based on PyTorch, to conduct our numerical investigations. We modeled
the SFi stack in a total simulation box of $1000\times 1000\times 20$ cells
with discretization lengths
$($4\text{\,}\mathrm{n}\mathrm{m}$,$4\text{\,}\mathrm{n}\mathrm{m}$,$4.5\text{\,}\mathrm{n}\mathrm{m}$)$,
where the first ten cells along the dimension $z$ are chosen to be the bottom
layer and the other ten cells represent the top layer. For Sample #1, we use
the following material parameters
$M_{s}^{Co}=$1240\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$,
$A^{Co}=$16\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$, and
$K_{u}^{Co}=0$ for the bottom Co layer, as well as
$M_{s}^{\mathrm{Ni_{81}Fe_{19}}}=$750\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$,
$A^{\mathrm{Ni_{81}Fe_{19}}}=$7.5\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$,
and $K_{u}^{\mathrm{Ni_{81}Fe_{19}}}=0$, for the top $\mathrm{Ni_{81}Fe_{19}}$
layer, in agreement with previous work and literature[25, 42]. Each layer is
modeled as a disk with a diameter of $4\text{\,}\mu\mathrm{m}$, where cells
outside the disk are initialized with $M_{s},A,K_{u}=0$, as usual in finite-
difference-based micromagnetic simulations. We use second-order Ruderman-
Kittel-Kasuya-Yosida (RKKY) coupling to ensure the correct antiferromagnetic
coupling between cells[60]. The coupling strength is chosen to be
$J_{\mathrm{RKKY}}=$-0.1\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$$.
For Sample #2, the material parameters were chosen as
$M_{s}^{\mathrm{Co_{40}Fe_{40}B_{20}}}=$1240\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$,
$A^{\mathrm{Co_{40}Fe_{40}B_{20}}}=$12\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$,
and $K_{u}^{\mathrm{Co_{40}Fe_{40}B_{20}}}$ was varied for the top
$\mathrm{Co_{40}Fe_{40}B_{20}}$ layer. We used
$M_{s}^{\mathrm{Ni_{81}Fe_{19}}}=$750\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$,
$A^{\mathrm{Ni_{81}Fe_{19}}}=$8\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$,
and $K_{u}^{\mathrm{Ni_{81}Fe_{19}}}=0$, for the bottom
$\mathrm{Ni_{81}Fe_{19}}$ layer. The Gilbert damping is $\alpha=0.01$. In all
cases, we assume an artificially increased damping region of $\alpha=0.1$ for
the annulus of the outermost 50 nm. This allows us to avoid reflections of the
spin waves at the rim of the sample and thus to achieve the steady state much
faster.

We begin our simulations by parametrizing a SFi with two magnetic vortices of
opposite circulations but parallel polarities of the vortex cores. We relax
the magnetization state by numerically integrating the Landau-Lifshitz-Gilbert
(LLG) equation at a high Gilbert damping parameter of $\alpha=1$ for
$20\text{\,}\mathrm{n}\mathrm{s}$, where

$\centering{\dfrac{\partial\mathbf{m}}{{\partial}t}}={-\gamma\mathbf{m}\times{\mathbf{H}}^{\mathrm{eff}}+\alpha{\mathbf{m}}\times{\dfrac{\partial\mathbf{m}}{{\partial}t}}},\@add@centering$
(1)

describes the temporal evolution of the magnetization vector field. Here,
$\gamma=$2.2128\text{\times}{10}^{5}\text{\,}\mathrm{m}\mathrm{/}\mathrm{A}\mathrm{s}$$
is the gyromagnetic ratio, and $\mathbf{H}^{\mathrm{eff}}$ is the effective
field that can be derived as the variational derivative of the different
relevant energetic contributions. For the statical relaxation process of the
total energy, the effective field includes only contributions from the
demagnetization, exchange, uniaxial anisotropy, and interlayer exchange (RKKY)
energies. Note that we do not use a physical spacing layer, but a second-order
finite-difference implementation of the RKKY [60]. This will slightly change
the stray fields in between the two magnetic layers in comparison to the
experiments, but it suffices to reproduce the most relevant effects. Once the
magnetization is relaxed, we include additional dynamic contributions such as
the current-induced Oersted field, the alternating uniform Zeeman field, or
the spin-transfer torque (STT). Note that all terms are already implemented in
magnum.np and can be used together or individually as desired.

Oersted Field. The charge current that passes through a conducting material,
such as we consider in our experiments and modeling, will generate an Oersted
field that can be calculated by means of the Biot-Savart law, where

$\mathbf{H}^{\mathrm{Oersted}}(\mathbf{x},t)=\frac{1}{4\pi}\int\mathbf{j}(\mathbf{x^{\prime}},t)\times\frac{(\mathbf{x}-\mathbf{x^{\prime}})}{\lvert\mathbf{x}-\mathbf{x^{\prime}}\rvert^{3}}\,\mathrm{d}\mathbf{x^{\prime}},$
(2)

where $\mathbf{j}(\mathbf{x^{\prime}},t)$ is the space and time dependent
current density flowing through the conductor. In our modeling we assume that
the current density is homogenous, and thus, a spatially uniform and constant
vector that is only time-dependent. This equation is solved in the Fourier
space, similar to the calculation of the demagnetization field, using a Fast
Fourier-Transform algorithm, which provides a substantial speed-up. For
simulations where we consider the current-driven excitation of spin waves, the
Oersted field is solved for each time step according to the adaptive time-
stepped Runge-Kutta-Fehlberg time integrator provided as default in magnum.np.
For the stripline antenna simulations, we precompute the Oersted field at a
constant current density
$j=$1\text{\times}{10}^{9}\text{\,}\mathrm{A}\mathrm{m}^{-2}$$, and take the
maximum of the $y$ component of the calculated Oersted field as the value for
the spatially homogeneous, stripline generated field.

Spin-Transfer Torque. The spin of the elecrons flowing through our magnetic
stack can be polarized due to the magnetization. Such spin-polarized electrons
will exert a torque on the magnetization, called the spin-transfer torque.
This additional, current-induced torque can be modeled in a continuum theory
by using the Zhang and Li approach[54], where the STT is modeled as an
additional torque in the Landau-Lifshitz-Gilbert equation mentioned above, and
hence becoming the Landau-Lifshitz-Gilbert-Slonczewski equation, where

$\centering{\dfrac{\partial\mathbf{m}}{{\partial}t}}={-\gamma\mathbf{m}\times{\mathbf{H}}^{\mathrm{eff}}+\alpha{\mathbf{m}}\times{\dfrac{\partial\mathbf{m}}{{\partial}t}}+\mathbf{T}_{\mathrm{STT}}}.\@add@centering$
(3)

The additional torque

$T_{\mathrm{STT}}=-b\times\mathbf{m}\times\left[\mathbf{m}\times\left(\mathbf{j}_{e}\cdot\bm{\nabla}\right)\mathbf{m}\right]-b\xi\mathbf{m}\times\left(\mathbf{j}_{e}\cdot\bm{\nabla}\right)\mathbf{m},$
(4)

describes how the STT acts due to the local gradient of the magnetization,
where $\xi$ is the degree of nonadiabacity, and $b$ is described as

$b=\dfrac{\beta\mu_{B}}{eM_{s}(1+\xi^{2})}.$ (5)

The polarization rate of the conducting electrons, which possess an elementary
charge of $e$, is symbolized by $\beta$. The Bohr magneton is denoted by
$\mu_{B}$. For all simulations involving STT, we use $\xi=0.05$ and
$b=$72.17\text{\times}{10}^{-12}\text{\,}$$.

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Acknowledgements: The computational results presented have been achieved, in
part, using the Vienna Scientific Cluster (VSC). F.B. and D.S. gratefully
acknowledge the Austrian Science Fund (FWF) for support through Grant No. I
4917 (MagFunc). S.K. and C.A. gratefully acknowledge the Austrian Science Fund
(FWF) for support through Grant No. P34671 (Vladimir). S.K. and D.S.
acknowledge the Austrian Science Fund (FWF) for support through Grant No. I
6267 (CHIRALSPIN). We thank Helmholtz-Zentrum Berlin for the allocation of
synchrotron radiation beamtime. We thank Roland Mattheis (IPHT Jena) for the
sputter deposition of the magnetic multilayers, Sina Mayr for fruitful
discussions. Support by the Nanofabrication Facilities Rossendorf at IBC is
gratefully acknowledged.

Author Contributions: S.K., C.A., D.S., and S.W. conceived the project. S.K.,
F.B., C.A., and D.S wrote and improved the micromagnetic code. S.K. performed
the micromagnetic simulations. S.K. and S.W. analyzed the simulation data.
K.S. and S.W. patterned the samples. K.S., M.W., and S.W. performed the
measurements. K.S., and S.W. analyzed the experimental data. S.K. and S.W.
wrote the initial manuscript with input from all authors. All authors have
reviewed and commented on the final manuscript.

Competing Interests: The authors declare that they have no competing
interests.

Date and materials availability: All data needed to evaluate the conclusions
in the paper are present in the paper and/or the Supplementary Materials.
Additional data can be requested from the corresponding author upon request.


[a]Sabarnya Mitra

# A new way to resum Lattice QCD equation of state at finite chemical
potential

Prasad Hegde  Christian Schmidt

###### Abstract

The Taylor expansion of thermodynamic observables at a finite baryon chemical
potential $\mu_{B}$ is an oft-used method to circumvent the well-known sign
problem of Lattice QCD. A reliable Taylor estimate demands sufficiently high-
ordered calculations in chemical potential $\mu$ for a proper estimate of its
radius of convergence. Owing to the associated difficulty and limitations of
precision in calculating these high-order Taylor coefficients, it becomes
essential to look for various alternative resummation schemes which can work
around this computational hurdle. Recently, a way to resum exponentially, the
contributions of the first $N$ baryon charge density correlation functions
$D_{1},\dots,D_{N}$ to the Taylor series to all orders in $\mu_{B}$ was
proposed in Phys. Rev. Lett. 128, 2, 022001 (2022). Since the correlation
functions $D_{n}$ are calculated stochastically using estimates from different
random volume sources, the resummation formulation gets affected by biased
estimates, which can become very drastic and can radically misdirect the
calculations for large values of $N$, $\mu$ and also higher order $\mu$
derivatives of free energy.In this work, we present a cumulant expansion
procedure that allows to investigate and regulate these biased estimates at
different orders in $\mu$. We find that the unbiased estimates in the cumulant
expansion can truly capture the genuine higher-order stochastic fluctuations
of the higher order correlation functions, which got suppressed by the
exponential resummation formulation. Finally, we introduce an unbiased
formalism of exponential resummation, which when expanded in a series, can
exactly reproduce the Taylor series upto a desired order in $\mu$. This allows
to regain the knowledge of reweighting factor and many other important
properties of the partition function, which got entirely lost while
implementing the cumulant expansion scheme.

##  1 Introduction

The QCD Equation of State (EoS), illustrating the QCD Phase diagram is of
significant importance in the parlance of QCD phase transitions and also in
the study of heavy-ion collisions [1, 2, 3, 4]. In principle, the entire phase
diagram can be completely explained from a comprehensive study of the gauge
theory of QCD. But, in reality, it is still a conjecture from a practical
standpoint, with many salient and robust features remaining to be established.
Hence, for a proper unfazed conclusion, an unambiguous thermodynamic approach
is adopted, which revolves around the important calculations of the estimates
of various thermodynamic observables.

The system considered, resembles a grand canonical ensemble of quarks
interacting via gluons, described by a grand canonical partition function
$\mathcal{Z}$($\mu,V,T$), which, in principle, is given as a path integral
over all the constituent particle (quark) and gauge field (gluon)
configurations. Unfortunately, this path integral formulation yields an
intractable, infinite-dimensional integral. Although lattice QCD averts this
problem by rendering this integral to a finite-dimensional one, the complex
integral measure at a finite $\mu$ inhibits the implementation of Monte-Carlo
importance sampling (MCIS). By virtue of the reweighting procedure [5, 6, 7,
8], although the measure being weighted at zero $\mu$ becomes real, the
complex measure problem assumes the form of the sign problem [9, 10, 11],
which manifest in the observable part of the integral. On a positive note,
reweighting enables the application of MCIS for calculating $\mathcal{Z}$ by
making the integral measure semi-positive definite.

The Taylor expansion of thermodynamic observables upto the first $N$
coefficients [12, 13] as a function of $\mu$ is one of the numerous methods
[14, 15, 16, 17, 18, 19, 20] adopted to evade the sign problem in Lattice QCD.
The slow rate of convergence and non-monotonic behaviour of the Taylor series
for a wide range of temperatures necessitate computations upto sufficiently
high orders in $\mu$, invoking calculation of higher-order Taylor
coefficients. This directs one towards resummation of Taylor series [21, 22,
23, 24, 25, 26], which allows to conduct an all-ordered calculation with the
knowledge of a few Taylor coefficients. The exponential resummation [27] is
one such resummation method, instrumental in our work.

In this work, we present the mathematical form of Taylor expansion and
exponential resummation. We then comprehensively discuss about the emergence
of biased estimates in exponential resummation, which has the potential to
become highly problematic in the regime of large values and higher orders of
$\mu$. We then come across the formulation of cumulant expansion [28, 29, 30,
31, 32], which allows an order-by-order analysis of biased estimates, but
unfortunately at the cost of the reweighting factor and hence, the invaluable
partition function itself. Finally, we present an unbiased formulation of
exponential resummation, which reproduces the Taylor (QNS) expansion upto a
given order of $\mu$ apart from a newly defined reweighting factor and
partition function altogether.

##  2 Setup of the simulation

In our work, we have used Highly Improved Staggered Quark (HISQ) action [33,
34, 35] for the fermions and tree-level improved Symanzik gauge action [36,
37] for the gauge fields. The work has been done on a $32^{3}\times 8$
lattice, using 2+1 flavor QCD with the quark masses chosen to satisfy
$m_{u}=m_{d}=m_{s}/27$. With a fixed lattice spacing and coupling parameter
$\beta$, these masses are tuned appropriately to their physical values, so
that they produce physical pion and kaon masses, as directed by chiral
perturbation theory. This therefore fixes the line of constant physics for our
work [12, 38, 39]. We have collected gauge configurations for two temperatures
at T $=135$ and $157$ MeV, which in $\beta$ scale, corresponds to
$\beta=6.245$ and $6.390$ respectively. We have worked with 20K configurations
for both baryon ($\mu_{B}$) and isospin ($\mu_{I}$) chemical potentials.
Recent work for $176$ MeV is in progress and also the number of gauge field
configurations is increased for $\mu_{B}$ for more statistics. Although we
worked mostly upto $D_{4}$, all the eight derivatives till $D_{8}$ are
calculated stochastically using $\mathcal{O}(500)$ random volume sources (RVS)
per configuration. All these correlation functions $D_{n}$ for $n\leq N$ can
be expressed as different linear combinations of traces [25, 40], involving
products of fermion propagator $\mathcal{M}^{-1}$ and different ordered $\mu$
derivatives of fermion matrix $\mathcal{M}$. The stochastic calculation of
these traces arises due to the inexact computation of $\mathcal{M}^{-1}$. A
detailed description of the gauge ensembles and scale setting can be found in
Ref. [13].

##  3 Taylor series and Exponential Resummation

The Taylor expansion of excess pressure $\Delta
P/T^{4}=P(\mu,T)/T^{4}-P(0,T)/T^{4}$ and number density $\mathcal{N}/T^{3}$,
in terms of $\mu_{B}$ upto the first $N$ derivatives [12, 13] are given by

$\frac{\Delta
P_{N}^{Q}(T,\mu_{B})}{T^{4}}=\frac{1}{VT^{3}}\ln\bigg{[}\frac{\mathcal{Z(\mu)}}{\mathcal{Z}(0)}\bigg{]}=\sum_{n=1}^{N}\frac{X_{2n}}{(2n)!}\hskip
2.84526pt\hat{\mu}_{B}^{2n}$ (1)

where $X_{n}$ are the $n$th order quark number susceptibilities (QNS) and
$\hat{\mu}_{B}=\mu_{B}/T$. The CP symmetry of QCD [25] ensures that pressure
and number density constitutes even and odd series in $\mu_{B}$ respectively.
The exponentially resummed estimate of excess pressure is given by

$\begin{split}\frac{\Delta
P^{R}_{N}(T,\mu_{B})}{T^{4}}&=\frac{1}{VT^{3}}\ln\Bigg{\langle}\text{Re}\Bigg{[}\exp\left(\frac{1}{4}\hskip
1.42262pt\sum_{n=1}^{N}\overline{D}_{n}(T)\hskip
2.84526pt\hat{\mu}_{B}^{n}\right)\Bigg{]}\Bigg{\rangle},\hskip
5.69054pt\overline{D}_{n}(T)=\frac{1}{N_{R}}\sum_{r=1}^{N_{R}}\tilde{D}_{n}^{(r)}(T)\\\
\end{split}$ (2)

$\displaystyle\tilde{D}_{n}^{(r)}(T)=\frac{D_{n}^{(r)}(T)}{n!}$
$\displaystyle\textnormal{where}\ \hskip
4.2679ptD_{n}^{(r)}(T)=\frac{\partial^{n}}{\partial\hat{\mu}_{B}^{n}}\ln\big{[}\det
M^{(r)}\left(T,\mu_{B}\right)\big{]}\Big{|}_{\mu_{B}=0}$

The corresponding resummed and Taylor estimate of number density is given by

$\frac{\mathcal{N}_{N}^{R,Q}}{T^{3}}=\frac{\partial}{\partial\hat{\mu}_{B}}\bigg{[}\frac{\Delta
P_{N}^{R,Q}(T,\mu_{B})}{T^{4}}\bigg{]}$ (3)

The factor of $1/4$, as mentioned in Eqn. (2), represents staggered signature
of fermion action. Also in this equation, the angular brackets
$\langle.\rangle$ represent the gauge ensemble average and $D_{n}^{(r)}$ is
the $r$th estimate of the derivative $D_{n}$. In continuum limit, these
derivatives are the integrated $n$-point correlation functions of the product
of the zeroth component of the four baryon current density $J_{\alpha}$ =
$(J_{0},\vec{\mathbf{J}}\hskip 0.85358pt)$ at different spacetime points $x$,
which are given by $D_{n}=\int d^{4}x_{1}\hskip 2.27621ptd^{4}x_{2}\hskip
2.27621pt...\hskip 2.27621ptd^{4}x_{n}\hskip 2.27621ptJ_{0}(x_{1})\hskip
2.27621ptJ_{0}(x_{2})\hskip 2.27621pt...\hskip 2.27621ptJ_{0}(x_{n})$ [27].
The CP symmetry ensures real-valued $\mathcal{Z}$, thereby dictating that all
$D_{n}$ are real for even $n$ and imaginary for odd $n$. causing only the real
part of the exponential to be considered as shown in eqn. (2). It is therefore
evident that

$\Delta P_{N}^{R}/T^{4}=\Delta
P_{N}^{Q}/T^{4}+\sum_{n>N}^{\infty}\Big{\langle}(\overline{D}_{1})^{A_{1}}(\overline{D}_{2})^{A_{2}}\dots(\overline{D}_{N})^{A_{N}}\Big{\rangle}\;\hat{\mu}_{B}^{n}$
(4)

where every $k$-point correlation function $D_{k}$ satisfies
$\sum_{k=1}^{N}k\cdot A_{k}=n$. The number density $\mathcal{N}_{N}/T^{3}$
also exhibits similar comparative behaviour resembling eqn. (4).

##  4 Problem of Biased Estimates and Cumulant Expansion

###  4.1 Biased estimates

Figure 1:  Isospin pressure (left) and number density (right) plots for T =
135 MeV.

As shown in Fig. 1 and also in [27], the resummed results differ appreciably
from the QNS counterparts in the regime of higher values of $\mu_{I}$. This
stark difference arises from the higher order contribution terms, as indicated
in eqn. (4). More significantly, as given in eqn. (2), the different powers of
these different derivative estimates, from quadratic power onwards give rise
to biased estimates. This is because, some given random vector estimates are
raised to higher powers than the others, thereby treating different estimates
on different footing in the sample of estimates.

$\begin{split}\big{(}\overline{D}_{n}\big{)}^{m}=\Bigg{[}\frac{1}{N_{R}}\sum_{r=1}^{N_{R}}D_{n}^{(r)}\Bigg{]}^{m}=\Bigg{[}\bigg{(}\frac{1}{N_{R}}\bigg{)}^{m}\sum_{r_{1}=1}^{N_{R}}...\sum_{r_{m}=1}^{N_{R}}D_{n}^{(r_{1})}...D_{n}^{(r_{m})}\Bigg{]}\\\
\approx\text{Biased
estimate}+\bigg{(}\frac{1}{N_{R}}\bigg{)}^{m}\mathop{\sum^{N_{R}}...\sum^{N_{R}}}_{r_{1}\neq...\neq
r_{m}}D_{n}^{(r_{1})}...D_{n}^{(r_{m})}\end{split}$ (5)

The effects of these problematic biased estimates can become very pronounced
and drastic, specially in the regime of large values and higher orders of
$\mu_{I}$ and estimating observables which are higher order $\mu_{I}$
derivatives of free energy. This therefore motivates one to truncate the
resummed series in terms of different powers of $\mu_{I}$ and analyse the
biased estimates for different orders of $\mu_{I}$.

###  4.2 Cumulant Expansion

The cumulant expansion of eqn. (2) upto $M$ cumulants in $\mu_{I}$ yield
(barring the $1/VT^{3}$ factor)

$\ln\left\langle
e^{X_{N}}\right\rangle=\sum_{n=1}^{M}\frac{\kappa_{n}^{N}}{n!}+\mathcal{O}(\kappa_{M+1}^{N}),\hskip
5.69054ptX_{N}=\sum_{n=1}^{N}\overline{D}_{2n}(T)\hskip
2.84526pt\hat{\mu}_{I}^{2n}$ (6)

We exploited the efficacy of cumulant expansion for $\mu_{I}$, where there is
no sign problem, because of the vanishing odd-ordered derivatives and also
because of which, $X_{N}$ in eqn. (6) is manifestly real, ensuring
$\mathcal{O}(10K)$ gauge configurations is good enough for an appreciable
signal. We worked with only the first $M=4$ cumulants and computed biased and
unbiased cumulants, where the biased cumulants $\kappa_{b,n}^{N}$ are given by

$\displaystyle\kappa_{b,1}^{N}$ $\displaystyle=\left\langle
X_{N}\right\rangle$ (7) $\displaystyle\kappa_{b,2}^{N}$
$\displaystyle=\left\langle X_{N}^{2}\right\rangle-\left\langle
X_{N}\right\rangle^{2}$ $\displaystyle\kappa_{b,3}^{N}$
$\displaystyle=\left\langle X_{N}^{3}\right\rangle-3\left\langle
X_{N}^{2}\right\rangle\left\langle X_{N}\right\rangle+2\left\langle
X_{N}\right\rangle^{3}$ $\displaystyle\kappa_{b,4}^{N}$
$\displaystyle=\left\langle X_{N}^{4}\right\rangle-4\left\langle
X_{N}^{3}\right\rangle\left\langle X_{N}\right\rangle+12\left\langle
X_{N}^{2}\right\rangle\left\langle X_{N}\right\rangle^{2}-6\left\langle
X_{N}\right\rangle^{4}-3\left\langle X_{N}^{2}\right\rangle^{2}$

For unbiased cumulants $\kappa_{u,n}^{N}$, we replace $X_{N}^{n}$ with
$U_{n}[X_{N}]$ for each $n$, in the cumulants of eqn. (7). Here $U_{n}[X_{N}]$
is the unbiased $n$th power of $X_{N}$, where
$X_{N}=\sum_{n=1}^{N}\overline{D}_{n}\hskip 2.84526pt\hat{\mu}_{B}^{n}$ and
unbiased $n$th power of $D_{m}$ is given by

$U_{n}[D_{m}]=\frac{n!}{\prod_{k=0}^{n-1}(N_{R}-k)!}\mathop{\sum^{N_{R}}...\sum^{N_{R}}}_{r_{1}\neq...\neq
r_{n}}D_{m}^{(r_{1})}...D_{m}^{(r_{n})}$ (8)

As shown in the plots in Fig. 1, the biased and unbiased results of pressure
and number density are in good agreement with the resummed and QNS results of
similar orders respectively. The unbiased cumulants managed to capture more
higher-order fluctuations, which got suppressed by the exponential behaviour
of the resummed series [28]. The unbiased cumulant expansion results hence,
demonstrated that the difference between the resummed and QNS results is
attributable to the difference between biased and unbiased estimates.

But, while incorporating unbiasedness at different orders, the truncation of
the resummed series led to the loss of the reweighting factor and partition
function altogether. This inspired the idea of a newly defined exponential
resummation scheme which would, in principle reproduce QNS upto the desired
order in $\mu$. In addition, a numerically different partition function with
an associated new reweighting factor is obtained, thereby re-enabling the
essential calculations of phasefactor and roots of partition function.

##  5 Unbiased Exponential Resummation

Motivated by the isospin results, we have implemented this new formalism of an
unbiased exponential resummation using $\mu_{B}$. Unlike $\mu_{I}$, the odd-
ordered derivatives are non-vanishing and imaginary for $\mu_{B}$ and hence,
it is necessary to extract the real part following eqn. (2) to obtain the
expression of the partition function $\mathcal{Z}$. In this formalism, all
mathematical manipulations are done at the level of individual RVS present
within every gauge configuration constituting the gauge ensemble. We have
worked in two bases, which are stated as follows:

###  5.1 Chemical potential basis

In $\mu$ basis, with this new formalism, we define the unbiased pressure from
a newly defined partition function following the usual prescription of the
exponential resummation as follows:

$\displaystyle\Delta P_{ub}^{N}(\mu)=\frac{1}{VT^{3}}\hskip 2.84526pt\ln\hskip
2.84526pt\mathcal{Z}_{ub}^{N}(\mu),\hskip
11.38109pt\mathcal{Z}_{ub}^{N}(\mu)=\left\langle\text{Re}\bigg{[}\exp\Big{(}A_{N}(\mu)\Big{)}\bigg{]}\right\rangle,\hskip
11.38109ptA_{N}(\mu)$ $\displaystyle=\sum_{n=1}^{N}\mu^{n}\hskip
1.42262pt\frac{\mathcal{C}_{n}}{n!}$ (9)

where the $\mathcal{C}_{n}$ for $1\leq n\leq 4$ are given as follows:

$\displaystyle\mathcal{C}_{1}$ $\displaystyle=\overline{D_{1}},$
$\displaystyle\mathcal{C}_{2}$
$\displaystyle=\overline{D_{2}}+\left(\overline{D_{1}^{2}}-\overline{D_{1}}^{2}\right),$
$\displaystyle\mathcal{C}_{3}$
$\displaystyle=\overline{D_{3}}+3\left(\overline{D_{2}D_{1}}-\overline{D_{2}}\;\overline{D_{1}}\right)+\left(\overline{D_{1}^{3}}-3\,\overline{D_{1}^{2}}\;\overline{D_{1}}+2\,\overline{D_{1}}^{3}\right),$
$\displaystyle\mathcal{C}_{4}$
$\displaystyle=\overline{D_{4}}+3\left(\overline{D_{2}^{2}}-\overline{D_{2}}^{2}\right)+4\left(\overline{D_{3}D_{1}}-\overline{D_{3}}\;\overline{D_{1}}\right)+6\left(\overline{D_{2}D_{1}^{2}}-\overline{D_{2}}\;\overline{D_{1}^{2}}\right)$
$\displaystyle-12\left(\overline{D_{2}D_{1}}\;\overline{D_{1}}-\overline{D_{2}}\;\overline{D_{1}}^{2}\right)+\left(\overline{D_{1}^{4}}-4\,\overline{D_{1}^{3}}\;\overline{D_{1}}+12\,\overline{D_{1}^{2}}\;\overline{D_{1}}^{2}-6\,\overline{D_{1}}^{4}-3\,(\overline{D_{1}^{2}})^{2}\right)$
(10)

Here, the powers of different $D_{n}$ are the unbiased powers of the
respective different ordered derivatives, calculated as per eqn. (8). The
analysis from this basis is important in the sense, that the degree of the
unbiasedness in $\mu$ is exactly identical with the degree of the polynomial
$A(\mu)$ as given in eqn. (9). This therefore ascertains the exact order of
Taylor or QNS expansion in $\mu$, it will achieve, apart from the prescence of
important beyond the QNS-order contributions, still comprising biased
estimates.

###  5.2 Cumulant basis

In cumulant basis, a new variable $W$ is defined, where
$W_{N}=\sum_{n=1}^{N}\frac{\mu^{n}}{n!}D_{n}$, we have

$\displaystyle\Delta P_{ub}^{M}(W_{N})=\frac{1}{VT^{3}}\hskip
2.84526pt\ln\hskip 2.84526pt\mathcal{Z}_{ub}^{M}(W_{N}),\hskip
11.38109pt\mathcal{Z}_{ub}^{M}(W_{N})=\left\langle\text{Re}\bigg{[}\exp\Big{(}Y_{M}(W_{N})\Big{)}\bigg{]}\right\rangle,\hskip
11.38109ptY_{M}(W_{N})=\sum_{n=1}^{M}\frac{\mathcal{L}_{n}(W_{N})}{n!}$ (11)

which would reproduce exactly the first M cumulants in unbiased cumulant
expansion of excess pressure. The different unbiased powers of derivatives are
calculated as before, as given in eqn. (8). The $\mathcal{L}_{n}(W)$ of eqn.
(11) upto $M=4$ for $1\leq n\leq 4$ are as follows:

$\displaystyle\mathcal{L}_{1}(W)$ $\displaystyle=(\overline{W})$ (12)
$\displaystyle\mathcal{L}_{2}(W)$
$\displaystyle=\bigg{[}\Big{(}\overline{W^{2}}\Big{)}-\Big{(}\overline{W}\Big{)}^{2}\bigg{]}$
$\displaystyle\mathcal{L}_{3}(W)$
$\displaystyle=\bigg{[}\Big{(}\overline{W^{3}}\Big{)}-3\hskip
2.84526pt\Big{(}\overline{W^{2}}\Big{)}\hskip
2.84526pt\Big{(}\overline{W}\Big{)}+2\hskip
2.84526pt\Big{(}\overline{W}\Big{)}^{3}\bigg{]}$
$\displaystyle\mathcal{L}_{4}(W)$
$\displaystyle=\bigg{[}\Big{(}\overline{W^{4}}\Big{)}-4\hskip
2.84526pt\Big{(}\overline{W^{3}}\Big{)}\hskip
2.84526pt\Big{(}\overline{W}\Big{)}+12\hskip
2.84526pt\Big{(}\overline{W^{2}}\Big{)}\hskip
2.84526pt\Big{(}\overline{W}\Big{)}^{2}-6\hskip
2.84526pt\Big{(}\overline{W}\Big{)}^{4}-3\hskip
2.84526pt\Big{(}\overline{W^{2}}\Big{)}^{2}\bigg{]}$

##  6 Results: Comparison between Biased and Unbiased formalism

The cumulant basis provides much more number of terms in addition to those of
$\mu$ basis. Although $\mu$ basis is important for simplicity and first-hand
understanding, the cumulant basis ensures a faster rate of convergence and
agrees well with the $\mu$ basis results, as the additional terms get almost
cancelled out among themselves. This also vindicates a genuine series
expansion.The results in unbiased resummation are carried out therefore
primarily, in cumulant basis.

Figure 2: Baryon pressure (left) and phasefactor (right) plots for T = 135 and
157 MeV

The $2^{nd}$ and $4^{th}$ ordered unbiased pressure results in Fig. 2 are in
better agreement with the $4^{th}$ ordered QNS results than the old biased
counterparts and the difference is stark and highly pronounced for 135 MeV.
Surely, one can argue for higher statistics reducing the gauge noise, allowing
the comparison for higher values of $\mu_{B}/T$. Also, one can even vouch to
increase the number of RVS from 500 to even more, per gauge configuration,
specially for the noisiest $D_{1}$.

However, the solutions to these arguments come at the cost of huge
computational time and storage space for data extraction of every $D_{n}$. The
pressure plots demonstrate that even with a meagre $20K$ configurations with
$\mathcal{O}(500)$ random vectors per configuration, the new formalism attains
excellent agreement with QNS over the old one, thereby saving profound
computational time and storage space. The imaginary part of the argument in
the exponential function constitutes the phase-angle, the cosine of which
forms the phasefactor in the biased and unbiased cases. The biased and
unbiased phasefactor results in Fig. 2 vary slightly, besides showing
appreciable order-by-order agreement, indicating that the difference between
biased and unbiased pressure at 135 MeV is predominantly arising from phase
quenched reweighting factor.

##  7 Conclusions

We have introduced a cumulant expansion which allows us to introspect the
biased estimates and substitute them with unbiased counterparts order-by-
order, in terms of $\mu_{I}$. The unbiased cumulant expansion, although
truncated, managed to capture the higher-order fluctuations which the old
exponential resummation could not efficiently serve to perform. Eventually,
this results in the loss of reweighting factor and partition function
$\mathcal{Z}$ itself. We then, therefore introduce a new exponential
resummation formalism, which unlike the old resummation, exudes an excellent
agreement with the QNS results, even using $20K$ configurations for $\mu_{B}$,
with $\mathcal{O}(500)$ RVS per configuration. This enables to retrieve the
partition function and hence, preserve the thermodynamics altogether. More
significantly, this partially unbiased exponential resummation gives an all-
ordered unbiased exponential resummation reproducing the exact all-ordered QNS
in the limit of an infinite cumulant expansion series, apart from providing a
much faster convergence with the QNS results.

The unbiased exponential resummed approach, outlined here is a new way of
extending the QCD EoS. Nevertheless, the possible connections between the
approach presented here and various other proposals in the literature [22, 41,
42, 43] still remain to be explored and therefore serve to be the promising
ingredients for numerous future works.

## Acknowledgments

We sincerely thank all the members of the HotQCD collaboration for their
inputs and for valuable discussions, as well as for allowing us to use their
data from the Taylor expansion calculations. The computations in this work
were performed on the GPU cluster at Bielefeld University, Germany. We thank
the Bielefeld HPC.NRW team for their support.

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